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Département de Physique
Thèse de Doctorat en Sciences
Spécialité: Physique Théorique
On the Hamiltonian Formalism of the Einstein-Cartan
Gravity with Fermionic Matter
Par
LAGRAA Meriem Hadjer
Soutenue le Jeudi 19 avril 2018 à l'Université d'Oran 1 Ahmed Ben Bella
Membres du Jury
Président
Professeur TAHIRI Mohamed, Université Oran I Ahmed Ben Bella.
Examinateurs
Professeur ABDESSELAM Boucif, Centre Universitaire de Aïn Témouchent.
Professeur BALASKA Smaïn, Université Oran I Ahmed Ben Bella.
Professeur BOUDA Ahmed, Université de Bejaïa.
Professeur NOUICER Khireddine, Université de Jijel.
Directeur de Thèse
Professeur LAGRAA Mohammed, Université Oran I Ahmed Ben Bella.
Laboratoire de Physique Théorique d'Oran (LPTO)
Département de Physique
Thèse de Doctorat en Sciences
Spécialité: Physique Théorique
On the Hamiltonian Formalism of the Einstein-Cartan
Gravity with Fermionic Matter
Par
LAGRAA Meriem Hadjer
Soutenue le Jeudi 19 avril 2018 à l'Université d'Oran 1 Ahmed Ben Bella
Membres du Jury
Président
Professeur TAHIRI Mohamed, Université Oran I Ahmed Ben Bella.
Examinateurs
Professeur ABDESSELAM Boucif, Centre Universitaire de Aïn Témouchent.
Professeur BALASKA Smaïn, Université Oran I Ahmed Ben Bella.
Professeur BOUDA Ahmed, Université de Bejaïa.
Professeur NOUICER Khireddine, Université de Jijel.
Directeur de Thèse
Professeur LAGRAA Mohammed, Université Oran I Ahmed Ben Bella.
Laboratoire de Physique Théorique d'Oran (LPTO)
Remerciements
J'exprime mes remerciements à mon encadreur de thèse,
le Pr. Mohammed LAGRAA, pour son aide et sans
lequel ce travail n'aurait pas pu aboutir. Je remercie
également les membres du jury d'avoir accepté
d'examiner ce travail, je cite le Pr. Abdesselam Boucif,
le Pr. Balaska Smaïn, le Pr. Bouda Ahmed et le Pr.
Nouicer Khireddine et enfin je remercie le Pr. Tahiri
Mohamed pour ses conseils et d'avoir accepté de
présider le jury de cette thèse.
Mes remerciements vont également à mes parents, à ma
sœur et à mes deux frères pour leur soutien.
Enfin, aux membres du Laboratoire de Physique
Théorique d'Oran.
---------------------------------------------------------------
Abstract
This thesis presents a detailed analysis of the formalism Hamiltonian of the tetrad-connection Gravity
at d-dimension without decomposing the tangent space of the space-time manifold by the ADM
decomposition. To avoid the problematic constraints in the Hamiltonian treatment of the theory, we
decompose the connection into dynamical and non-dynamical parts, and fix the non-dynamical part
to zero. The application of the Dirac procedure with this fixing has allowed to obtain a covariant
formalism under the Lorentz transformations and an algebra of first-class constraints closing on
structure constants rather than the usual structure functions resulting from the ADM decomposition.
The Dirac brackets deliver the same algebra of the first-class constraints, while the second-class ones
are eliminated by the strong equality. At the end, we obtain the same physical degrees of freedom of
the theory of General Relativity.
The Hamiltonian analysis is also performed when Gravity is coupled to fermionic matter. The
obtained results establish an algebra of first-class constraints closing on structure constants as the
case of pure Gravity. We also obtain, by canonical transformations, a new reduced Phase space
equipped with canonical Dirac brackets leading to the same algebra of first-class constraints.
Keywords: Tetrad-connection Gravity, Hamiltonian formalism, Dirac brackets, Dirac spinors,
fermions, algebra of first-class constraints.
Résumé
Cette thèse est consacrée à une analyse détaillée du formalisme hamiltonien de la Gravitation tetrad-
connection à d-dimension sans décomposer l’espace tangent de la variété espace-temps par la
procédure ADM. Pour éviter les contraintes qui posent problème dans le traitement hamiltonien de la
théorie, nous décomposons la connexion en la projetons en parties dynamique et non-dynamique et
nous fixons la partie non-dynamique à zéro. L'application de la procédure de Dirac avec cette fixation
à permis d'avoir une étude covariante sous les transformations de Lorentz et une algèbre des
contraintes de première-classe qui se ferme avec des constantes de structure au lieu des fonctions de
structure qui en résultent habituellement de la décomposition ADM. Concernant les contraintes de
seconde-classe, elles sont éliminées par l'égalité forte en utilisant les crochets de Dirac. Nous
montrons aussi que ce formalisme donne le même nombre de degrés de liberté physique de la théorie
de la Relativité Générale.
Le formalisme Hamiltonien est également développé quand la Gravitation est couplée à la matière
fermionique. Les résultats obtenus conduisent à la même algèbre des contraintes de première-classe
avec des constantes de structure comme dans le cas de la Gravitation pure. Nous avons également
obtenu, grâce à des transformations canoniques, un nouvel espace de Phase équipé de crochets de
Dirac conduisant à la même algèbre des contraintes de Première-classe.
Mots clés: Gravité tetrad-connection, formalisme hamiltonien, crochets de Dirac, spineurs de Dirac,
fermions, algèbre des contraintes de première-classe.
Contents
Introduction 3
1 The Hamiltonian Formalism of Pure Gravity 7
1.1 The Metric Formulation of Gravity (The ADM Decomposition) . . . . . . 8
1.1.1 The Hamiltonian analysis of the Einstein-Hilbert action . . . . . . . 9
1.2 The Tetrad-Gravity actions . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.1 The Palatini action . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.2 The Covariant Hamiltonian Analysis . . . . . . . . . . . . . . . . . 25
2 The Hamiltonian Analysis of Gravity with Fermions 28
2.1 Gravity with matter: The general algebra . . . . . . . . . . . . . . . . . . 28
2.2 Gravity coupled with fermions . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.1 Gravity with fermions the minimal coupling . . . . . . . . . . . . . 30
2.2.2 Torsion in presence of fermions: The e¤ective action . . . . . . . . . 32
2.2.3 The constraints a¤ected by the Fermions . . . . . . . . . . . . . . . 33
3 On the Hamiltonian formalism of the Tetrad-Gravity 40
3.1 The ABC notation and its properties . . . . . . . . . . . . . . . . . . . . . 40
3.2 Constructing the Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 The constraints algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 The xing of the non-dynamical connection . . . . . . . . . . . . . . . . . 53
3.5 The constraints algebra in terms of Dirac brackets . . . . . . . . . . . . . . 60
3.6 conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4 The Hamiltonian Formalism of Tetrad-Gravity Coupled to Fermions 68
4.1 The constraints in presence of fermions . . . . . . . . . . . . . . . . . . . . 68
4.2 The consistency of The Hamiltonian Formalism . . . . . . . . . . . . . . . 71
4.3 Dening the total Hamiltonian and the constraints algebra . . . . . . . . . 76
4.4 The algebra of constraints in terms of Diracs bracket . . . . . . . . . . . . 83
Conclusion 90
1
Appendix A 92
Appendix B 96
Appendix C 102
2
Introduction
Since the emergence of the theory of General Relativity during the early decades of the
twentieth century, this theory continues to show its accuracy. As example, we cite the
latest experimental results appearing in [1]; describing the rst direct detection of gravita-
tional waves signal. However, General Relativity also came up with the hard challenge of
conciliating its foundations with those of Quantum mechanics to edify Quantum Gravity.
[2]
Over the years, candidates has emerged to represent Quantum Gravity. Precisely, one
can list two major theories as the canonical quantization of Gravity, covering many di¤er-
ent approaches as Loop Quantum Gravity (LQG) [3], [4], [6], [7] and [5] (and references
therein) or Spinfoam formalism [8] and [9], which are based on the background indepen-
dence and non-perturbative approach, or the other perturbative theories as String theory
[10], with its variants and SuperGravity [11]. About the results presented in this the-
sis, which concern the Hamiltonian analysis of the Tetrad-Gravity at ddimension, theyare to be considered as rst steps necessary to establish theories representing Quantum
Gravity in the non-perturbative approach.
Through the last decades, the Holst action [12] was principally used to obtain LQG,
by applying the ADM decomposition [13] on the space-time manifold and then the Dirac
procedure [14]. This action brings many advantages as solving issues related to reality
conditions [15] and [17] or the possibility to solve second-class constraints [16]. However,
this action contains the dimensionless Barbero-Immirzi parameter [18], which leads to
many issues discussed in [19], [20] and [21]. This parameter also a¤ects the area and
volume operators predicted by LQG [23], and the black hole entropy formula [24]: In
addition, the Holst action uses the real version of the Ashtekar variables [32] leading to
consider SO (3) group rather than the Lorentz group. Furthermore, the algebra of the
rst-class constraints obtained from this action contains structure functions depending on
the phase space variables and its Hamiltonian constraint is non-polynomial. In particular,
the presence of structure functions is problematic, bringing issues related to the anomaly
problem and reveals that the rst-class constraints do not correspond to symmetries based
on true Lie group. Many attempts has been made to avoid these di¢ culties [25] and [26].
3
Among these attempts we cite one way to treat the problem related to the structure
functions given by T. Thiemann in his master constraint Program [27] and [28], where as
basic idea, it was suggested to reformulate the Hamiltonian constraintsM in combination
of all the smeared Hamiltonian constraints taken with all the smearing functions, leading
to vanishing brackets fM;Mg = 0:
However, in this thesis we take di¤erent path from the usual Hamiltonian procedure
and we do not completely use the ADM decomposition that decomposes the metric or the
tetrad used in the tetrad-connection Gravity formalism. Instead, we take the notation
of the tetrad variables used in (Frolov and all work [29]) to derive the phase space of
the tetrad-Gravity considered at d-dimension. Then, by using the Dirac procedure, in
addition to the rst-class constraints that represent the Gauss, spatial di¤eomorphism
and the scalar constraints, we nd a second-class constraint that is di¢ cult to solve. To
avoid this problematic constraint, we proceed by splitting the connection variables into
dynamical and non-dynamical parts and x the second part to zero. Thus, we obtain new
action that leads to a coherent and consistent Hamiltonian, plus an algebra of rst-class
constraints that closes on structure constants rather of the usual structure functions. The
procedure is pushed forward and the second-class constraints are eliminated via strong
equalities by the Dirac procedure. At the end we retrieve the same physical degrees of
freedom of the theory of General Relativity.
We also reinvestigate the Hamiltonian analysis when fermionic matter are supplied
to the theory. The same procedure is maintained where the non-dynamical part of the
connection is set to zero and the Dirac program is applied. The obtained results show
that the rst-class constraints close on algebra with structure constants as it is the case
for pure Gravity. At the end, we also give canonical transformations leading to new phase
space equipped with new canonical Dirac brackets leading to the same algebra of rst-class
constraints.
The presented thesis is subdivided in two major parts, the rst one contains two
chapters gathering the calculations of the Hamiltonian formalism both when Gravity is
free or coupled to fermions. While for the second part, composed of the third and the
fourth chapters, it presents results appearing in [30] and [31].
After this general introduction, the rst chapter starts by introducing the ADM de-
composition applied on the space-time manifold, when Gravity is described by the metric
formalism. This is achieved in order to develop the Hamiltonian formalism of the Einstein-
Hilbert action, then we give brief discussion on the obtained algebra of Gravity described
by the metric eld. After, we develop the tetrad-connection Gravity formalism. We start
by dening the geometrical objects that are involved in the Palatini action, which is of rst
4
order treating the tetrad and the connection as independent variables. The development
of the Dirac procedure applied on the Palatini action reveals that other constraints have
to be introduced to match the physical degrees of freedom of Gravity. However, these
second-class constraints are di¢ cult to solve. This leads us to study the other approach
proposed by Abhay Ashtekar in 1986 [32] which introduces complex self-dual connection
AI as the main variable. The development of the Hamiltonian analysis of the self-dual
action leads to the rst-class constraints Gi, Va and S that represent the SO (3) groupgenerators, the vectorial and the scalar constraints respectively. However, complex vari-
ables will bring the issue of the reality conditions, leading to introduce the Holst action
which discards this issue, making of this action a good candidate to represent the tetrad-
connection Gravity. Finally, we end this rst chapter by a discussion on the main results
that has been obtained these last decades in the treatment of the Hamiltonian formalism
in covariant way.
In the second chapter we consider the Hamiltonian analysis of Gravity when it is
coupled to fermionic matter. We start by searching the algebra of Gravity when it is
coupled to scalar elds. Then, we continue with the case of Gravity coupled to fermions
by adding the spinorial Dirac action. With the use of the variational principle we derive
the equation of motion of the connection which is modied by the presence of fermions.
The solutions of this equation give a non vanishing torsion. Dividing the connection
into torsion free part and co-torsion part will lead to an e¤ective action that contains
an extra term expressed by an axial current-current interaction [33]. Then, we take the
obtained e¤ective action and submit it to the Hamiltonian analysis. This chapter covers
mostly the results obtained in [34]. In addition to the obtained Gauss constraints, the
vectorial and the scalar constraints, these constraints are divided into gravitational part
and matter part. It is noticed that whenever the system is taken with or without fermions,
all the obtained algebras always tend to deliver structure functions when one calculates
the scalar-scalar Poisson bracket.
The third and the fourth chapters are exclusively dedicated to present the new re-
sults appearing in [30] and [31]. We start by the third chapter where we perform the
development of the tetrad-Gravity on the space-time manifold M of ddimension. Wekeep the internal Lorentz group covariant, and we do not completely rely on the ADM
decomposition of the space-time. Particularly for the last chapter, we retest the results
presented in the third one with fermions supplied to Gravity. We start directly by con-
sidering the action with the gauge xing of the non-dynamical part of the connection,
and then applying the Dirac procedure. We take the same process, however, the obtained
results show that the algebra closes only when the second-class constraints are solved.
5
At the end, the reader may nd three appendices:
a) The appendix A: we have decided to derive the classical vacuum Einstein equations
from the di¤erent actions that has been used as the starting point for the development of
the Hamiltonian formalism. As it is well known, a well-dened action to describe Gravity
must at least reproduce the classical Einstein equations in the vacuum, and based on
what has been presented in the rst chapter, we have chosen to focus on the process of
getting these equations from the Einstein-Hilbert action, for the metric formalism, and
from the Palatini and the Holst actions for the tetrad-connection one. In this appendix,
during the development of the calculations we also managed to introduce notations and
symbols that refer to the geometrical objects and their properties.
b) The appendix B: The appendix is based mostly on the Dirac monograph [14]. The
ideas that constitute the Dirac procedure are introduced to the reader in general, where
we start by giving the initials steps of the program, as passing from the Lagrangian
formalism to the Hamiltonian one via the Legendre transformations, from where one can
have the primary constraints of a given system. Later, we give the denition and the
properties of the Poisson brackets, we pass then to the second point which concerns the
classication of the constraints where we distinguish between two di¤erent ones. The rst
classication that labels constraints as primary, secondary, ... and which is related to the
derivation order with respect to time. The second classication lists the constraints in
two categories, known as the rst-class constraints and the second-class constraints. We
then focus on clarifying the di¤erence between these two types of constraints, where the
rst-class ones are used to identify gauge invariances, and the second-class constraints are
linked to gauge xing.
c) The appendix C: Since the thesis also concerns the Hamiltonian formalism of Gravity
when it is coupled to fermions, we dedicate this last appendix to the notations used in
the Dirac spinors, and the Cli¤ord algebra with its properties.
6
Chapter 1
The Hamiltonian Formalism of Pure
Gravity
This chapter mainly concerns the description of some renowned works that have been pro-
duced these last few decades in the subject of the Hamiltonian formalism of gravitational
theories. In the non-perturbative attempts to build a quantum version of General Relativ-
ity, a consistent Hamiltonian formulation is needed. However, Studying the Hamiltonian
formalism of Gravity revealed to be particular compared to the other theories that restrict
them selves to elds evolving in at xed background. The theory of General Relativity
(GR) aims to gather space and time as one and indivisible entity, on the other hand the
Hamiltonian formulation of any theory is strongly related to the concept of time and this
contradicts the foundations of GR . Indeed, in purpose to develop the Hamiltonian of any
theory, it is necessary to calculate the velocities, to pass through the Legendre transforms,
and compute the conjugate momentums. This is possible only by singling out a particular
time, tied to a particular observer, which seems to conict the initial statement about the
theory of General Relativity brought in this paragraph.
As we will see, it is possible to use the Hamiltonian formalism over General Relativity
since it is a parameterized theory [35] and [36]. An important point of these theories
is that the obtained Hamiltonian which generates the translation along time, should be
constrained to vanish and this is precisely the case for General Relativity. In this chapter,
we try to give a picture of the Hamiltonian formulation while Gravity is described either by
the metric eld or by the tetrad-connection variables and then we discuss the main issues
related to those descriptions. Finally, we end the chapter by listing the most renowned
covariant Hamiltonian studies of Gravity.
7
1.1 The Metric Formulation of Gravity (The ADM
Decomposition)
In 1959, Richard Arnowitt, Stanley Deser and Charles W. Misner came up with a major
contribution to the development of the Hamiltonian formalism of General Relativity in a
series of articles from 1959 to 1962, summarized in [13]. The essential of their study is
based on the splitting of the metric eld that carries the dynamical informations and then
the application of Dirac procedure [14] on the Eintein-Hilbert action. To go into further
details, we start by giving some denitions and tools that serve the theory of General
Relativity and its Hamiltonian prospecting.
General Relativity is the theory that describes the dynamic of Space-time, meaning
that Space-time is itself subject to evolution equations and it is no more considered
as statical background 1. The physical Space-time is depicted mathematically by a 4-
dimensional topological smooth manifold M equipped with the (0; 2) non-degenerate
symmetric metric tensor g (with the Lorentzian signature). With the choice of coordinates
basis the components g of the metric appear in
ds2 = gdx dx (1.1)
where the greek letters ; ; ::: refer to the tensorial space-time indices.
The non-degenerate metric g is compatible with the Levi-Civita connectionrg = 0.(More details about the covariant derivative r is given in the Appendix A). Providingthese denitions it occurs that the metric takes the role of the dynamical eld describing
the change happening on the Space-time.
The Space-time manifoldM is sliced into Cauchy hypersurfaces of d 1 dimensionwithout boundaries (with spacelike signature) providing the manifold M the following
topologyM =R , henceM is particularly foliated into a t := t () parametrized
by t 2 R.
: M! R
x ! t (x) := X (t; xa) (1.2)
the latin letters a; b; c:::: label the "spatial" tensors belonging to the hypersurfaces em-
bedded via the foliation t:
By inheritance, the leaves that constitute this set of hypersurfaces will be deformed
when a variation over the manifoldM occurs. Hence, the deformations of the hypersurface
t induce a deviation along the introduced shift vector N tangent to the hypersurface
1The reader may refer to the following books for the theory of general relativity [37], [38] and [39].
8
t and along the vector N normal to t and proportional to the lapse function N .
Consequently, the deformation is given by the following expression
T (X) :=@X (!x ; t)
@t= N (X) +N (X) (X) (1.3)
From this decomposition of the manifold M, it follows the reformulation of the metric
tensor as
ds2 = g (x) dx dx = g (X (
!x ; t)) dX (!x ; t) dX (!x ; t) (1.4)
to not burden the next calculation, we do not use X (!x ; t) and simply write X
ds2 = g
@X
@tdt+
@X
@xadxa@X
@tdt+
@X
@xbdxb
= g
(N +N) dt+
@X
@xadxa(N +N ) dt+
@X
@xbdxb
= gN2 +NN
dt dt+ 2g
(N +N)
@X
@xb
dt dxb
+g
@X
@xa
@X
@xb
dxa dxb (1.5)
We substitute the denition of the time ow (1.3) in the second line of the precedent
equation. Terms containing gN vanish since the vectors and N are orthogonal
due to their denitions. The vector is orthogonal to the hypersurface, hence it is time
like vector and because of the Lorentzian signature, the relation g = 1 is used tocalculate the ds2 expression. We will rewrite (1.5) in non-covariant form to perform the
calculation in the next subsection
ds2 =N2 +NN
dt dt+ 2Naqabdt dxb + qabdx
a dxb: (1.6)
As consequence, every function of the metric g should be written in terms of the new
components gtt = (N2 +NN), gtb = Naqab and qab given by the relation (1.6).
1.1.1 The Hamiltonian analysis of the Einstein-Hilbert action
In order to achieve the Hamiltonian development of the metrical Gravity, we introduce the
Einstein-Hilbert action from which the classical vacuum Einstein equations are derived by
using the variational principle with respect of the fundamental variable dynamical eld:
The metric. (see Appendix A)
The Einstein-Hilbert action is constructed via the Ricci curvature in addition to the
9
square root of the metrics determinant that guarantees the invariance of the action under
the general coordinate transformations.
sEH (g) =1
2k
ZMRp det(g)d4x: (1.7)
Where k equals 8G in the unit of the speed of light in the vacuum c = 1 (With G is the
gravitational constant).
The next task consists in the formulation of the Einstein-Hilbert action as it is allowed
by the chosen foliation of the Riemannian manifold M. Initially, we will perform it in
a covariant way: Therefore, the Riemannian tensor R, function of the metric and its
second derivative, will be decomposed. Performing tensor calculus over the submanifold
t needs to introduce the intrinsic metric on t, q := g (consequently we willhave det (g) = N2 det (q)); and the extrinsic curvature K := qq
rn = 1
2(Lnq) ;
where q is identied as projector on the submanifolds t dened by q = gq and
Ln is the Lie derivative along n: The tensor K is symmetric and it shows how the
hypersurface t curves in its neighborhood, precisely along the normal n:
These new dened elements allows to write
R
u =hr;
riu
=hq
0
q0
q0
q0R
0
000 KK
+KK
iu: (1.8)
r is the covariant derivative acting on spatial tensors compatible with the intrinsic metricrq = 0 and respecting
hr;
rif = 0 for scalars f:
The insertion of (1.8) into the scalarR =
Rq
q yields to
R = K2 KK + qqR (1.9)
Finally, replacing by the denition of the intrinsic metric and r () = 0, we can ndthe nal form of the Gauss-Codazzi equation.
R =RK2 +KK 2r (r r) : (1.10)
The last term is boundary term that vanishes once (1.10) is inserted into (1.7).
In the end, the use of (1.2) and (1.6) gives to the Einstein-Hilbert action its nal form,
ready to be developed in purpose to analyze the obtained Hamiltonian. From now, the
tensors are no more taken as covariant but they will carry time-like or space-like indices.
10
Thus, the action (1.7) reads
S (qab; Na; N) =
1
2k
ZN (det (q))
R(3)
K2 +KabKab
dtd3x: (1.11)
Obviously, the metric g is no more the dynamical variable to be taken into account but
rather its components qab; Na and N emerging from the relation (1.6) when the Space-
time manifold is foliated. Now that we have written the starting action in convenient
way to calculate the conjugate momentums, we can directly notice that the velocities of
both the shift vector and the lapse function are missing from the Lagrangian density Lof (1.11). Consequently, primary constraints C (t;!x ) and Ca (t;!x ) will emerge
C (t;!x ) : = (t;!x ) = @L@N (t;!x ) 0
Ca (t;!x ) : = a (t;
!x ) = @L@Na (t;!x ) 0: (1.12)
This leads us to use the Dirac algorithm for constrained systems [14] (See Appendix
B). Whereas ab; the conjugate momentum of qab, can be calculated with the use of
Kab =12N
qab L!N qab
2, to obtain
ab =@L
@qab (t;
!x )= det (q)
Kab Kqab
: (1.13)
Now that the phase space is dened, it remains to nd the Hamiltonian H proper to
the EinsteinHilbert action. To achieve this, one should use H =R
pq L
d3x without
omitting to take into account the primary constraints (1.12) with their respective Lagrange
multiplier elds (t;!x ) and a (t;!x ).
HT =1
k
Zdt
Zd3x [C + aCa +NaHa +NH] : (1.14)
From the action (1.11) and the relation (1.13), and after tedious and straighforward
computing, we have
Ha = 2qabDccb (1.15)
H =1pdet (q)
qacqbd
1
d 1qabqcdabcd
pdet (q)R: (1.16)
2To show this relation, we simply start by considering the Lie derivative Lnqab = crcqab+qacrbc+qbcrac and by the use of the metric compatibility rcgab = 0, the precedent relation can be rewritten asLnqab = crc (ab)+rba+rab; which in it turns can be reformulated as Lnqab = (ca + ac)rcb+(cb + b
c)rca to nally get Kab =12Lnqab: Now it remains to explicit
= 1N (T
N) to obtain the
nal form Kab =12N
qab L!N qab
:
11
In the canonical formalism, the total Hamiltonian generates the translation through the
time dimension. For now, it is clear that Ha and H are not in function of the conjugate
momentums of the shift and the lapse, hence an innitesimal temporal variation of the
lapse tN = HTt = t and the shift tNa = HT
at = at will correspond to the
arbitrary and completely undetermined lagrange multipliers introduced in (1.14).
The constraints (1.12) require to be derivated over time according to the Dirac proce-
dure. This is performed via the Poisson brackets between (1.12) and the total Hamiltonian
density (1.14) leading to
fCa;HTg = Ha 0 and fC;HTg = H 0; (1.17)
with the use of the fundamental Poisson Brackets
fqab (!x ) ; qcd (!y )g = 0,ab (!x ) ; cd (!y )
= 0
and qab (!x ) ; cd (!y )
= k
1
2
ca
db + da
cb
(!x !y ) : (1.18)
Furthermore, the Hamiltonian equations of the variables and their conjugate momentums,
which means:qab = fqab; HTg,
:cd=cd; HT
and those of the shift and the lapse, remain
insensitive to the constraints Ca and C. Therefore, the Hamiltonian can be reduced to
HT =1
k
Zdt
Zd3x [NaHa +NH] : (1.19)
This latest result shows that General Relativity is a parametrized theory since its Hamil-
tonian is combination of constraints. The relations (1.17) impose on Ha and H to be
constrained in their turns as secondary constraints. We have to continue this process
until it gives back all the constraints that dene the theory. So, we have to derivate once
more over time the obtained secondary constraints. It turns out that this last opera-
tion will conclude the iteration process and allows us to nd the following hypersurface
deformation algebra.
Ha (fa) ;Hb
f 0b
= kHaLfbf 0a
fHa (fa) ;H (f)g = kH (Lfaf)
fH (f) ;H (f 0)g = kHaqab (f@bf
0 f 0@bf)
(1.20)
where f and f 0 with the appropriate structure indices are used as test functions to smear
the constraints on the Riemannian space-time manifold. These test functions are the free
12
shift!N and lapse N .
By the algebra given in (1.20), we can classify the constraints as rst-class constraints
according to the Dirac terminology where Ha and H generate innitesimal deformations
on the hypersurfaces. Precisely, vectorial constraint Ha is responsible of deformationsoccurring tangentially to the hypersurface parametrized by the shift vector. WhereasH,the scalar di¤eomorphism, deforms along the normal to the hypersurface. Apparently,
the Hamiltonian does not generate time-transformation but rather di¤eomorphisms, which
rises the conict between General Relativity and Quantum Gravity, know as "Problem
of time"[40]. We also mention that the vectorial di¤eomorphism constraints form by
themselves a clean subalgebra, but for the case of the scalar constraint H, the situationcan be tricky. The last equation, which concerns the scalar di¤eomorphism, is in function
of the dynamical 3-dimension metric eld qab. This dependence is due to the chosen
foliation when the Hamiltonian formalism is developed. The structure functions can be
potential sources of anomalies and reveals many issues as for example the factor ordering
if one wants to naively promote the constraints to operators. The structure functions
forbid the scalar constraint to form a proper group and many issues related to their
presence is investigated in [41]. We mention that the obtaining of the Poisson brackets can
be achieved by geometrical constructions based on the path independence requirement.
Performing successively two innitesimal transformations and along the orthogonal
to the hypersurface to reach the hypersurface 0 and then making the same operation
but in inverse order, may lead to another hypersurface 00 di¤erent from 0. In order
to respect the path independent requirement one must complete the trajectory by the
variation from 00 to 0, where the innitesimal involves the spatial metric [42].
Therefore, it imposes to consider only the On-shell level to avoid the structure function
obstacles. Besides, we mention that the Hamiltonian equationsab; HT
= HT
qabis non-
linear and includes second-order derivatives, hence solving the Wheeler-DeWitt equation
[43] turns to be complicated. Thus, nding the corresponding operators to the ADM
metrical constraints failed, leading to reconsider the used variables to describe Gravity.
13
1.2 The Tetrad-Gravity actions
1.2.1 The Palatini action
The Tetrad-connection formalism
The Palatini action, instead of the metric eld g , relies of the use of the tetrad and the
connection to describe the dynamic of Gravity [44]. To be a good candidate to describe
Gravity, the Palatini action must also lead to the classical Einstein equations by using
the variational principle (See Appendix A). This action has many advantages as being of
rst order which will drastically ease the study. We can also use the Tetrad-connection
formulation to couple the gravitational eld to fermionic matter. Therefore, the Standard
Model or its possible supersymmetric expansions can be incorporated in the theory.
In the Tetrad-connection Gravity formulation, the co-tetrad eld eI (x) = eI (x) dx
is 1-form valued in Minkowski space and where its components set map from the tan-
gent space of M at x : TxM to the Minkowskis space endowed with the at metric
IJ (1; 1; 1; 1) : The Latin alphabet labels the components of Minkowskian tensors. Geo-metrically, the co-tetrad can be interpreted as a frame sticked to the point x belonging to
the Riemannian manifoldM. The co-tetrad is not singular and its inverse is the tetrad
e1I (x) = eI@ (1.21)
The relation between the metric and the co-tetrad is given by g = eIeJIJ . This
leads to det (g) = e2; with e is the determinant of the tetrad.The 1-form connection !IJ = !IJdx
is the second eld involved in the description
of Gravity, !IJ is valued in the Lie algera of the Lorentz group SO(1; 3) and denes the
covariant derivative.
D, the covariant derivative gives zero by acting on the at metric IJ ; as
DIJ = @IJ + ! NI NJ + ! N
J NI
= !IJ + !JI
= 0 (1.22)
Given the vector eld V I belonging to the Minkowski space, one has
DVI = @V
I + !I JVJ (1.23)
The covariant derivative also denes the 2-form Torsion via the rst Cartan structure
equation, as
I = DeI De
I: (1.24)
14
Such manifold equipped with non-vanishing torsion are called Einstein-Cartan manifold.
Contrary to the standard Riemannian manifold where the torsion is set to zero from the
beginning, in the Einstein-Cartan theory the torsion vanishes by the equation of motion
of the connection, to recover the classical vacuum Einstein equations (See Appendix A).
Finally, The 2-form curvature K L =12 K Ldx
^ dx associated to the Lorentzconnection is dened by the second Cartan structure equation K
LVL = (DD DD)V
K
which gives
K L = @!
KL @!KL + !KN!
NL !KN!NL (1.25)
and satises the Bianchi identity D = 0, where in terms of components, reads
DKL
+DKL
+DKL
= 0 (1.26)
The constrained Hamiltonian of the Palatini action
The Palatini action is given in terms of the tetrad and the components of the 2-form
curvature
SP (e; !) = Z
e
2k(eKe
L eKe
L)
KL d4x: (1.27)
(1.27) has to be invariant under both Lorentz and general coordinate transformations. As
performed before, the ManifoldM is once again sliced into parameterized hypersurfaces
t where we have to re-express this action under the ADM decomposition. For this we
rewrite the components of the tetrad by using the time ow T = N + N, where
is normal to the shift N. if we take the tetrad projection over the hypersurfaces
of the foliation, which is eeI := q eI = ( + ) e
I ; it leads to express the tetrad as
eI = eeI I ; where KeeK = 0:SP (e; !) =
Z pdet(q)
kN (eeK K) (eeL L) KL
d4x
= Z p
det(q)
k[NeeKeeL 2NKeeL] KL
d4x
= Z p
det(q)
k
NeeKeeL 2N T N
N
KeeL KL
d4x: (1.28)
e = Nee = Npdet(q) is the decomposed tetrad determinant. At this stage, we use
T KL = T
@!
KL D!
KL
= LT
! KL
D
T ! KL
(1.29)
15
in the precedent expression to obtain
SP (e; !) = Z p
det(q)
k[NeeKeeL + 2NKeeL] KL
d4x
Z p
det(q)
k
2KeeLD
T ! KL
2KeeLLT ! KL
d4x: (1.30)
Next, separating the coordinates x into x (t; xa) gives
SP (e; !) = Z p
det(q)
k
NeeaKeebL + 2NaKeebL KL
ab dtd3x
Z p
det(q)
k
2KeeaLDa
! KLt
2KeeaL@t ! KL
a
dtd3x: (1.31)
Written this way, the action (1.31) will give the Hamiltonian density HP simply by usingH = pi
:qi L as
HP = Da
2pdet(q)
k[KeeaL]
!! KLt
+Na
2pdet(q)
k[KeeaL] KL
ab
!+N
pdet(q)
k
ea
[K
eb
L]KL
ab
!: (1.32)
As expected, the Hamiltonian is completely constrained, which leads to list the following
constraints:
MKL : = Da
aKL
0
Da : = bKLabKL 0
D : =k
4pdet(q)
aKIb LI abKL 0: (1.33)
where aKL is the conjugate momentum of the component !aKL obtained via the Legendre
transformation aKL = @L@@t!aKL
= 2ek[KeeaL]:The relations (1.33) are classied as rst-
class constraints and are well-known as the Gauss, the vector and the scalar constraints
respectively. They appear in the Hamiltonian (1.32) with the Lagrange multipliers: the
temporal part of the connection !tKL; the shift vector Na and the lapse function N:
However, before claiming any statements, one must follow the regular path shown for
the Dirac procedure [14] as it has been done in the section treating the Einstein-Hilbert
action, and proceeds to compute the evolution over time of the listed constraints by using
the fundamental Poisson brackets
!aKL (x) ;
bIJ (y)= kba
1
2
IK
JL ILJK
3 (x y) : (1.34)
16
f!aKL (x) ; !bIJ (y)g = 0 andaKL (x) ; bIJ (y)
= 0
The calculations are straightforward and lead to the following algebranM (!) ;M
!0o
= Mh!; !
0i
nM (!) ;Dsp
!No
= ML!N!
fM (!) ;D (N)g = 0nDsp
!N;Dsp
!Mo
= DaL!NMan
Dsp!N;D (M)
o= D
L!NM
fD (N) ;D (M)g = Da eeaIeeIbdet(q)
(N@bM M@bN)
: (1.35)
Obviously, the constraints MKL are nothing but the generators of the Lorentz group
SO(1; 3), new constraints that were missing from the precedent metric formalism inves-
tigation; where there is no Minkowski locality feature: Whereas the constraint Dsp!N;
precisely Dsp!N= Na
Da +M IJ!aIJ
; represents the spatial di¤eomorphism symme-
try, dragging the free transformation parameters along the shift!N . The third result is
predictable, since D(N) does not carry Minkowski indices, the generators MKL do not
transform the scalar constraint. However, the last equation of (1.35) still remain prob-
lematic. Indeed, the function structures are maintained in the poisson bracket between
the scalar constraints, as it is the case for the Gravity described by the metric.
The Physical Degrees of Freedom
At this stage, one should pay attention to the number of the Degrees of Freedom involved
in this study. Indeed with the use of
2
Number of physical
degrees of freedom
!=
Total number of
canonical variables
!
2 Number of rst-class
constraints
!
Number of original
second-class constraints
!(1.36)
we have to nd the two physical degrees of freedom which characterize Gravity.
We begin with
Total number of
canonical variables
!given by the variables of the phase space.
17
This means, according to the action (1.31), the number of the spatial part of the connec-
tion and its conjugate momentums which equals 36:
The
Number of rst-class
constraints
!equals 10, counted from the constraints M IJ ; Da and
Dt of (1.33).Consequently, by using (1.36) one should nd 36210 = 16: This is still far from the
two degrees expected for the theory of Gravity. So certainly, some constraints are missing
from the theory. To manage this, the study has been supplied with new constraints
ab = IJKLaIJaKL 0: (1.37)
If one considers (1.37), then the Hamiltonian has to be supplied with the constraint
ab; side-by-side with the Lagrange multiplier eld ab; and the Dirac procedure should be
reconducted. Since the Hamiltonian that describes the theory is only made of constraints,
the study of abs time evolution is equivalent to compute its di¤erent Poisson brackets
with all the other ones. Particularly, from the bracket of ab and Dt another constraintemerges, noted by ab and equivalent to
ab = IJKLc MI
(aMJ
Dc
b)KL
0 (1.38)
As (1.37), the constraint ab is of second-class type, leading to no more constraints this
time, but rather xing ab = 0 the Lagrange multiplier eld of ab; and ab. We inform
the reader that further details about this part are to be found in [32] and in the good
review [45].
Now, one could once again count the physical degrees of freedom where the contri-
bution of
Number of second-class
constraints
!of ab and ab is to be taken into consideration.
Hence, the new counting will deliver 36 2 10 (6 + 6) = 4; the expected physical
degrees of freedom of Gravity.
The Ashtekar variables (Self-Dual Action)
In 1986, Abhay Ashtekar presented an alternative approach to cancel the di¢ culties due to
the second-class constraints that are involed in the Hamiltonian development of the tetrad-
connection Gravity presented previously. This approach introduces complexe variables to
describe Gravity, more precisely, it suggests to use the complex Lorentz connection [32],
18
enhancing the role played by the connection3. The Hamiltonian developed by Ashtekar is
considered to be a good stepping-stone to ward the theory of Loop Quantum Gravity by
the non-perturbative process [3], [4] and [7] (and references therein), or to the spinfoam
theory by the Path integral [8]. it has permitted to integer and use tools of Yang-Mills
(for instances the Wilsons loop [47]) to the general theory of relativity4, where the gauge
group corresponds to SU (2).
The Ashtekars formalism starts with the lorentz connection which is decomposed into
self dual and anti-self dual parts ! = A(+) + A()
A(+)IJ =1
2
!IJ
i
2KLNM!
NM
A()IJ =
1
2
!IJ +
i
2KLNM!
NM
(1.39)
Hence, the self dual (or the anti-self dual) action is obtained by the curvature F KL
A(+)
associated to the self-dual connection A(+)IJ
S (A) = 12k
Ze (eKe
L eKe
L)F
KL
A(+)
d4x: (1.40)
The self-dual property is also transmitted to the curvature leading to
iF KL =
1
2KLNMF
NM :
As we did precedently, we use the ADM decomposition of the tetrad we nd
S (A) = 1k
ZNpdet(q) (eeK K) (eeL L)F KL
d4x
= 1k
ZNpdet(q)eeKeeLF KL
d4x
+1
k
ZNpdet(q)2K
T N
N
eeLF KL d4x: (1.41)
Because of the free role that plays K in the construction of the action, it is possible to
x it without changing the dynamic of the other elds, and hence to work under specic
time gauge xing. Precisely, because of its time-like feature, the vector K can simply be
xed as K = (1; 0; 0; 0) ; consequently K = (1; 0; 0; 0) ; leading to ee0a = 0: Thus, a major
consequence of this xing is that the internal group is no longer the Lorentz group but
rather its rotational subgroup. By convention, we have KKLIJ = 0lij = lij
0 = lij
3 [46] R. Capovilla, J. Dell and T. Jacobson have also a type of formulation that relies exclusively onthe connection and which links to the Ashtekar Hamiltonian formulation via Legendre transformations.
4The rst hint to consider General Relativity as gauge theory able to use the Yang-Mills technics hasbeen suggested in [49]. Utiyama upgraded the invariance of the given system to wider group transfor-mations, for instance GR and local Lorentz transformations, introduced new elds and determined thetype of interaction these elds have with the original ones and then constructed the suitable invariantlagrangian.
19
where the letters l; i; j:: label the spatial indices of the local Minkowski space, and lij are
taken to be analogous to the constants structure of the rotational group SO (3) :We have
alsoEa
k =pdet(q)eeak, leading to nally re-express (1.41) as
S(A) = 1k
Z ZN
1pdet(q)
eEak eEbl F klab d3x
!dt
1k
Z ZNa2 eEbl F 0l
ab d3x
dt
+1
k
Z Z2 eEbl F 0l
tb d3x
dt: (1.42)
The dentizied triad eEak is the canonical conjugate momentum and can be considered
similar to the Electric eld in Yang-Mills theory. At this stage one has to explicit the
objects involved in the action (1.42) and transforms the used notations in an easier fashion
to handle. Thus, F 0ltb reads
F 0ltb = @tA
0lb @bA0lt + A0mt A l
b m A0mb A lt m
= @tA0lb @bA0lt A0mb A l
t m A lb mA
0mt ; (1.43)
by using the self-dual property, we can susbtitute A lt m = imliA0it in the precedent
expression (1.43) to nd
F 0ltb = @tA
0lb @bA0lt + 2ilmiA0mb A0it (1.44)
We should also explicit the variable i2A0i = Ai and use it to rewrite (1.43)
2F 0ltb = i
@tA
lb @bA
lt + lmiAmb A
it
= i
@tA
lb D
(A)b Alt
:= iF itb; (1.45)
hence the action reads
S(A) = ik
Z ZN
i
2pdet(q)
eEak eEbl kliF iabd3x!dt
= ik
Z ZNa eEbl F l
ab d3x
dt
+i
k
Z Z eEbl @tAlb D(A)b Alt
d3x
dt: (1.46)
The relation i2A0i = Ai allows to redene the variable Ai as
Ai := i iKi
20
where the equation of motion of i =12kli!kl leads to a spin-connection compatible with
the tetrad, while Ki = !0i is related to the extrinsic curvature: In terms of these new
dened components, we have
F iab = @aAib @aAib + lmiAlaA
mb
= Riab i2D()[a K
ib] lmiK l
aKmb ;
Where Riab is associated to the spin-connection ia, with R
iab = 2@[a
ib] ijkia
jb:
The canonical variables used in the usual Hamiltonian process (K;E) can be changed
to (A;E) Aia (x) ; A
bj (y)
PB= 0;
Ea
i (y) ;eEbj (y)
PB
= 0
and nAia (x) ;
eEbj (y)oPB=k
iba
ij3 (x y) : (1.47)
From the action (1.46), the constraints are identied as
Gi : =i
kD(A)b
eEbi 0 (1.48)
Va : =i
keEbiF i
ab 0 (1.49)
S : =1
2pdet(q)
eEak eEbl kliF iab 0 (1.50)
The constraints (1.48), (1.49) and (1.50) need to be smeared in order to obtain the con-
straints algebra. Thus, the lapse N and shift Na will remain as usual for the vector
Va and scalar S. However, for the Gauss constraint, i = Ait = it iKit is used as
the transformation parameter. The spatial di¤eomorphism constraint can be dened by
Ha = Va AaiGi: Finally, the algebra is given by
Gii;Gj0j
= Gkijki0j
S (N) ;Gi
i
= 0nGii;Ha
!No
= GiL!
Nin
Ha!N;Hb
!Mo
= HaL!NMan
S (N) ;Ha!No
= SL!
NN
fS (N) ;S (M)g = Ha eEak eEbk (N@bM M@bN)
(1.51)
As nal step, it is important to inspect again the degrees of freedom. The canonical
variablesAia;
eEbi give 18; there are 3 given by the generators Gj and 4 from Ha and thescalar constraint S. If we do the calculations according to the formula (1.36), we recover
21
the accurate degrees of freedom of the theory of General Relativity. Despite this, the
algebra still constains the structure functions, but on the other hand this new formulation
will lead to simpler Hamiltonian equations compared to those of the traditional General
Relativity metric [50], even if the constraint S has to be rescaled due to 1=pdet(q)
involved in its expression [48].
Reality conditions and Barberos Hamiltonian constraints
The cost to pay for such simplication is the introduction of complex variables to describe
Gravity. However, at some point, one should manage to get a real space-time. Fullling
this requirement needs to implement reality conditions. This is supposed to be achieved
by "hand", meaning that these reality conditions do not arise from the Hamiltonian as
primary or secondary constraints. The reality conditions are imposed over the dentized
inverse of the 3metric
eQab = eEai eEbi=
pdet(q)qab: (1.52)
This issue is investigated depending whereas the inverse of the metric is degenerate or
not [15] . The condition (1.52) has been submitted to the Dirac procedure like the other
constraints, where in its turn, has imposed "secondary" reality conditions where
eP ab = ijk eEci eE(aj DceEb)k 0 (1.53)
must be real. It has been deduced that when eQab is degenerate, the reality conditionscan no longer be consistently imposed, specially for the vector and the scalar constraints
and the theory of General Relativity remains complex. However if theQab
> 0 (excluding
the degenerate case) the Einsteins theory can be recovered. On the other hand, Barbero
accepted to deal with more complicated Hamiltonian scalar constraint in [18] for both
Euclidean and Lorenztian cases, where he managed to rewrite (1.50) as
S 0 = Ea
k
Eb
l kliF iab +
2 ( 2 1) 2
Ea
[k
Eb
l]
Aka ka
Alb lb
0 (1.54)
to avoid the use of complex formulation to describe Gravity.
The extended constraint S 0 is in function of an unphysical parameter : The Ashtekarvariables can always be recovered once 2 1 = 0 is respected. If the signature = 1,one works in the lorentzian case, and by xing the parameter = 1, the constraint S 0
becomesEa
k
Eb
l kliF iab 4
Ea
[k
Eb
l]
Aka ka
Alb lb
0; (1.55)
22
keeping the Poissont brackets unchanged.
The Holst Action The Holst action [12] is introduced to discard the issues of the
reality conditions that must be taken in consideration when one deals with the Ashtekars
formulations. By using this action, one should obtain the Barberos constraints. This
action is introduced by implementing an extra term in the usual Palatini action. This
new term is function of a real parameter without physical dimension . In the vacuum, at
the classical level, this new action gives the Einstein equations. However, once Gravity is
coupled to the fermionic matter, it happens that this unphysical parameter has e¤ects due
to the non-vanishing torsion [19]. The parameter manifests itself as well at the quantum
level, where it takes place in the area and volume operators [23], and in the Black Hole
entropy formula [24]. A fully covariant treatment of this action is to be found in [16].
We begin by
S = 1k
ZeeI e
JT
IJKL
KL (!) d4x (1.56)
where the tensor T IJKL is dened as TIJKL =
12
IK
JL JKIL
1
2IJKL
, and is a
parameter without physical dimension, which could be taken as complex parameter in
some cases, to have the Ashtekars action. For the case, when ! 1, (1.57) we havethe usual Palatini action.
S (e; !) = 1
k
Zee[Ie
J ]
IJ (!) 1
IJKL
KL (!)
d4x (1.57)
In the nal part of Appendix B, we derive the classical Einstein equations from (1.57),
where we can see that the supplied term can not be considered as topological term but
rather related to the vanishing cyclicity of the Riemannian tensor. We mention that the
Holst term can also be linked to the Nieh-Yan topological invariant [20], this issue has
been investigated in [21]. For now, our task consists to follow the regular ADM procedure
to decompose the action into three distinct parts
S (e; !) = SG (e; !) + Sk (e; !) + S? (e; !) : (1.58)
We start by the rst term appearing in the right hand side of (1.58)
23
ST (e; !) =1
2k
Z Z
0lta 1
20lij ij
ta
eEal d3x dt=
1
2k
Z Z !0l
a @a!
0lt + 2!
0k[t !
la]k
eEal d3x dt+1
2k
Z Z 1 lij!ij
a @a!
ijt + 2!
iI[t !
ja]I
eEal d3x dt: (1.59)This explicit form of SG (e; !) allows to get the conjugate momentum eEal related to thevariable A0la = !0la 1
2 lij!ija and to achieve the following canonical transformations
1 A0la ; eEal !
Ala;1 eEal withAla =:
la + K l
a =1
2lij!ija !0la : (1.60)
Where the fundamental Poisson brackets of the theory arenAia (x) ;
eP bj (y)o = kbaij3 (x y) (1.61)
Aia (x) ; A
Jb (x)
= 0 and
n eP ai (y) ; eP bj (y)o = 0 (1.62)
After straighforward computations, (1.59) is written as
ST (e; !) =1
k
Z ZiD(A)
a
eP ai + 1 + 2 jmi!0jt ! 0maeP ai d3x dt: (1.63)
The rst term appearing in the right hand side of (1.63) refers to the Gauss constraints
where i = 12lij!ijt !0lt is the transformation angles. While from the second term,
another constraint takes place identied as Cj = jmi! 0maeP ai 0; related to the vanishing
antisymmetric part of extrinsic curvature K[ab] = 0, where Kab := Kma eebm: To derive the
nal form of Sk (e; !) and S? (e; !) expressions, one has to introduce the curvature Fi( )ab
Fi( )ab = @aA
ib @bAia + lmiAlaA
mb
= @aib + Ki
b
@b
ia + Ki
a
+lmi
lb + K l
b
(ma + Km
a ) (1.64)
which can be expressed as F i( )ab = Riab 2 D()[a K
ib]+
2lmiK laK
mb ; where R
iab =
12i klR
klab:
After straighforward computations, we nd
Sk (e; !) =1
k
Z ZNa eP bi F i( )ab +
1 + 2
iklKk
aKlb
d3x
dt (1.65)
24
and
S? (e; !) =1
k
ZN 2 eP ai eP bjpdet (q)
Rijab + 2K
i[aK
jb]
2
lijD
()[a K
kb]
d3x
!dt: (1.66)
Once again, we manage to obtain action with constraints related to the Lagrange mul-
tipliers N and Na leading to a total constrained Hamiltonian. We must point out that
there is another constraint emerging from the variation of the spin connection lb which
allows us to determine lb uniquely in function of the tetrad. A detailed analysis will show
that constraint Cj = jmi! 0maeEai 0 is of second-class type, while all the other remaining
are rst-class constraints. Hence Cj can be solved and xed to zero. While from Sk (e; !)
and S? (e; !) we can nd the nal form of
H( )a = eP bi F i( )ab +1 + 2
KiaC
i 0 (1.67)
and
H( ) = 2 eP ai eP bj2pdet (q)
kijFk( )ab +
1 + 2
knmKn
aKmb
1 + 2
eP ai D()a Cip
det (q) 0
(1.68)
H( )a and H( ) correspond to the dened vectorial and scalar constraints. We can dropthe last term of each of these constraints, making possible to construct the algebra given
in (1.51). Finally, it results constraints algebra made by real variables, which avoids
the doubling of the degrees of freedom and where there is no need to incorporate reality
conditions . However, the constructed algebra will still include the structure functions.
[16] and [22]
1.2.2 The Covariant Hamiltonian Analysis
In this last part, we present briey other works related to the Hamiltonian analysis of
Gravity, in particular we focus on the covariant treatment. We begin by the (2+1) Tetrad
Gravity Hamiltonian covariantly investigated in [29]. As it is known (2+1) dimension
makes a good ground for testing technics and tools to understand Gravity [51] and [52].
The Frolovs and all work [29] is special in the sense that it is free from the decomposition
of the metric by the ADM foliation and it does not use canonical transformations on the
dynamical variables. The authors are motivated by answering the main question "What
transformations do correspond to the gauge generators given by the Poisson brackets al-
gebra?".Their obtained results stated that the transformations correspond to the rotation
and translation forming the Poincaré group due to the form of the obtained algebra. Ef-
25
fectively, the secondary constraints X tI := :tI= Hc
etIand X tIJ :=
:
PtIJ= Hc
!tIJwill
combine the following algebra. X tI ;X tJ
= 0 (1.69)
X tIJ ;X tK
=1
2
JKX tI IKX tJ
(1.70)
X tIJ ;X tKL
=1
2
JKX tIL IKX tJL + JLX tKI ILX tKJ
(1.71)
The authors noticed that the constraints lack of the external indices and are only labelled
with the Lorentz ones. Therefore, they claimed that even if the action of the start is gen-
eral coordinates invariant, this can not be reected in the Hamiltonian rst-class algebra.
These results correspond to the ones appearing in [52]. An attempt to generalize these
outcomes to higher dimension is to be found in [53]. Unlike the usual Ashtekar formula-
tion that tends to x the time gauge and use the SO (3) (or SU (2)) subgroup because
of its compactness feature, these listed approaches keep the internal Lorentz group fully
covariant. As we will see, the results presented in this thesis are di¤erents from those
obtained in [29] and [53].
In [54], one can also nd a detailed application of the Dirac program on the Pala-
tini action at higher dimensions where the corresponding internal group is SO(1; D)(or
SO(D+1) for Euclideen signature). It is performed by extending ADM phase space, where
the set of constraints is supplied with the Simplicity constraints S5 to solve the missmatch-
ing between the degrees of freedom of the conjugate momentumP tIJ
= D2 (D + 1) =2
and its denitions determined by the tetradeIa= D (D + 1) : In [54],
:
S led to another
constraint D, these two constraints are of second-class type and the system is submitted
to an unxing gauge process by introducing a corrective term in the Hamiltonian without
a¤ecting the algebra of the rst-class constraint, hence the constraint S is promoted to
be a rst-class and the full algebra closes without D, but yet the scalar Hamiltonians
Poisson brackets still deliver structure functions in the results. They also established a
correspondance with the higher dimension BF-theory described in [55].
During the same period, the spin foam models derived from BF-theory gained much
attention, which conducts to investigate the Hamiltonian of BF-type action [56]. It starts
from the generalization of the action given in [57] that couples the curvature F [A] ; with
the Immirzi parameter ; to the 2form BF-eld BIJ = eI ^ eJ
6; where BIJ is sub-
mitted to conditions because of its other coupling with the internal tensor IJKL7; where
5The autors called the constrains S simplicity constraint analogously to the one of the Plebanskiformulation.
6We have mainly worked with components and projected form but we recall that ^ is the wedgeproduct and designs the hodge product
2 = 1
on the Lorentz indices.
7The tensor IJKL has the following permutation symmetric and antisymmetric properties IJKL =
26
we can recover the tetrad formulation by solving this imposed conditions. The analysis de-
livers second-class constraints, in addition we have the usual Gauss and di¤eomorphism
constraints, which are solved, and with the gauge-time xing one nds the Ashtekar-
Barbero variables and the results given in [16]. Among other attempts to study the
covariant Hamiltonian Gravity, we mention the work presented in [58], where it has been
suggested to give to the connection variable values in so(4; C) algebra, rather than su (2)
imposed by the time gauge xing, to investigate the ambiguities related to the Immirzi
parameter and its e¤ects on the quantum level.
In this last subsection, we have tried to sum up the most relevant studies related to
the topic of the Hamiltonian formalism. Yet, it seems that all these attempts tend to give
an open constraint algebra.
We point out that all along this rst chapter, we have not considered the cosmological
constant while developing these di¤erent formulations of the Hamiltonian Gravity. these
works can be generalized and one can rigorously derive the constraints new expressions
by supplying the appropriate terms to each starting action [45]. Certainly, the cosmo-
logical constant comes along term that has to be invariant under the general coordinate
transformations, so one should expect that the algebra of the di¤eomorphism constraints
(1.20) remains unchanged, but not the constraint expressions.
KLIJ = JIKL = IJLK :
27
Chapter 2
The Hamiltonian Analysis of Gravity
with Fermions
Once that we have set the Hamiltonian formalism of the gravitational theory in the
vacuum case, under various formulations, and enumerate di¤erent issues related to the
source-free Hamiltonian formalism, we continue the analysis by including the matter to
the theory and study the consequences of this insertion on the constraints. The second
chapter details the Hamiltonian procedure when fermions are incorporated to the theory.
For instance, the reader may refer to many accomplished works and studies listed non
exhaustively in [59]. Now and before giving more details, we shall start by giving a simple
example of the scalar eld coupled to the metric gravity to give an insight into how should
the Hamiltonian constraints be a¤ected.
2.1 Gravity with matter: The general algebra
The matter is supplied to the theory by extending the Lagrangian density appearing in
the starting action with the appropriate term L(M) describing the dynamic of the matter
in question. We have then
S = S(G) + S(M) =
Zd4x
1
2k
p det(g)R + L(M)
(2.1)
where, in this case , L(M) depends on the scalar eld ' and the metric eld g ; however
we insist that in most of cases L(M) is independent of the g derivatives. L(M) will dene
the Energy-momentum stress tensor appearing in the Einstein equations with T =2 1p
det(g)S(M)
g: In the metric formalism, S(M) can have the following expression
S(M) =
Z p det(g)
1
2g@'@' V (')
d4x (2.2)
28
From which we can obtain the Klein-Gordon equation for massless scalar eld ' =
@@' = 0, when the potential term in considered zero. To develop the Hamiltonian of
the theory, we explicitly write the derivation along the time coordinate of the scalar ';
which leads to express S(M) as
S(M) =
Z p det(g)
1
2gtt (
:')
2+ gat
:'@a'+
1
2gab@a'@b' V (')
dtd3x: (2.3)
From which we can compute the conjugate momentum (') related to the scalar eld vari-
able '; as (') = @L(M)
@:'=p det(g) [gtt :'+ gat@a'] : In addition, this action is quadratic
in terms of the velocities, so we are able to dene the expression of the velocity:' in terms
of ('). Once the expression of:' is injected, we retrieve the following form of the action
(2.2), where
S(M) =
Z 1
2
(')
2p det(g)gtt
+1
2
p det(g)
gab gatgbt
(gtt)
@a'@b'
!dtd3x
Z p
det(g)V (')dtd3x: (2.4)
From which we can compute the following Hamiltonian H(M) as usual to gain
H(M) =
Z (')
2p det(g)gtt
12
gtb
gtt(')
@b'
p det(g)V (')
!d3x
+
Z 12
p det(g)
gab gatgbt
(gtt)
@a'@b'+ V (')
d3x (2.5)
Once we decompose the metric by the ADM foliation1, we obtain
H(M) =
ZN
(')
22p det(q)
12
p det(q)qab@a'@b'+
p det(q)V (')
!d3x
+
ZNa
1
2(')@a'
d3x (2.6)
From which we can have the following expressions by gathering similar terms, as
H(M)a =
1
2(')@a' (2.7)
1We have used the following metric components and its inverse g , in terms of the lapse N , the shiftvector
!N and the spatial metric qab
g =
N2 +NaNa Na
Na qab
and g =
1N2
Na
N2
Na
N2 qab NaNb
N2
!
and the decomposition of the metric determinantpdet (g) = N
pdet (q):
29
and
H(M) = (')
22p det(q)
12
p det(q)qab@a'@b'+
p det(q)V (') (2.8)
These obtained terms do not form constraints by themselves but they rather supply the
ones that already exist and which have been calculated in the previous chapter. Hence,
each part of the constraints (1.15) and (1.16) will receive contribution from (2.7) and
(2.8), respectively. Therefore, if we consider the general Poisson bracket of the extended
scalar constraint we have the following expression
fH (N) ;H (M)g =H(G) (N) +H(M) (N) ;H(G) (M) +H(M) (M)
=
H(G) (N) ;H(G) (M)
+H(G) (N) ;H(M) (M)
+H(M) (N) ;H(G) (M)
+H(M) (N) ;H(M) (M)
(2.9)
If there are no derivatives of the lapse M and N , functions smearing the full scalar
constraints, then the precedent relation (2.9) reduces to
fH (N) ;H (M)g =H(G) (N) ;H(G) (M)
+H(M) (N) ;H(M) (M)
(2.10)
This happens specially, when the scalar constraint does not contain the momentum deriv-
atives. We choose to consider the scalar eld action for its simplicity in the study, in
addition the action is appropriate to represent the physics of the Higgs boson in purpose
to extend the theory by the Standard Model. The same procedure will be applied to the
Dirac action. We recall that the aim is to establish the formalism that will be partly used
in the fourth chapter. Finally, before we begin the next part of this chapter and thus giv-
ing details and adding spinors to Gravity described by the tetrad-connection formalism,
we end this chapter by pointing out that an attempt to couple General Relativity in pure
metrical formalism to Dirac spinors is to be found in [60]
2.2 Gravity coupled with fermions
2.2.1 Gravity with fermions the minimal coupling
Short after the success of the introduction of the Ashtekar variables, the inclusion of matter
was taken into account in [61]. When the real variables was introduced by Barbero to
describe Gravity, fermions were added in parallel to the theory in [62]. However, in this
subsection, we rely mostly on [34]. We start by introducing the action of Gravity coupled
to fermions. As it has been shown in the previous chapter, the most appealing action to
start from, in purpose to obtain LQG, is the Holst action [12]. To couple Gravity with
30
fermions, the Dirac action will be incorporated along the Holst action
S (e; !;) = S(G) (e; !) + S(F ) (e; !;)
= 1k
ZMeeI e
JT
IJKL
KL (!) d4x
i2
ZMeeI I (D) (D) I
d4x (2.11)
In S(G) (e; !), we retrieve the tensor
T IJKL =
1
2
IK
JL JKIL
1
2IJKL
; (2.12)
which inverse is
T1 KLIJ =
2
2 + 1
1
2
KI
LJ LJ KI
+1
2 KLIJ
: (2.13)
T1 KLIJ is required to compute the motion equation of the connection. In the third line
of the action (2.11) we have the Dirac part to describe the dynamic of the fermions. This
part involves the spinors and its conjugate = ()T 0, where the covariant derivative
acts on the spinors by
D = @+1
2!IJ IJ D = @
1
2!IJ IJ (2.14)
where IJ = 14[ I ; J ], and I are the Dirac matrices obeying the Cli¤ord algebra (See
Appendix C). We insist that, at this stage, the interaction Gravity-Matter is hidden within
D acting on fermions. For now, to compute the equation of motion of the connection,
we should consider the variation of the full action with respect of the connection
!S (e; !;) = !S(G) (e; !) + !S
(F ) (e; !;) = 0: (2.15)
The rst part, which is S(G) (e; !) has already been investigated in the previous chapter,
giving
!S(G) (e; !) =
ZM
2
kD
eeI e
J
T IJKL!
KL d4x: (2.16)
While for the part containing fermions, the detail of the variation is given below
!S(F ) (e; !;) =
ZM i2eeI I! (D) !(D) I
d4x
=
ZM
i4eeI( IKL + KL I)
!KL d4x
=
ZM
i4eeI [ I ; KL]+
!KL d4x: (2.17)
31
Hence, when fermionic matter is supplied to the theory, the principle of the least action
will deliver the following equation
!S (e; !;) = 0 =)1
kD
ee[Ke
]L
T IJKL
i
4eeI [ I ; KL]+
= 0 (2.18)
Where we can see that contrary to the source free case where its solution gives a connection
compatible with the tetrad, when fermions are present it is no more the case and instead
we have
D
ee[Ne
]M
=ki
4T1 KLNM e
eI [ I ; KL]+
; (2.19)
leading to axial fermionic current noted by JN = 5
N:
2.2.2 Torsion in presence of fermions: The e¤ective action
As consequence when we add fermionic elds to the theory, the equation of motion of the
connection is modied and contains the axial fermionic current. Our aim for now is to
express the total action under the form of
S = S(G) (e; e!) + S(F ) (e; e!;) + S(int) (e; J) (2.20)
where we will explicitly show the interaction part, function of the current JN : To achieve
this we will have to decompose the connection into torsion free part e! IJ [e] compatible
with the tetrad and a co-torsion part, as
! IJ = e! IJ
[e] + C IJ : (2.21)
The cotorsion C IJ is related to the torsion by T I
=C IK eK C IK
eK: Thus, it
follows that the curvature will also split into
IJ = e IJ
[e] + 2 eDIJ[ ] + 2CI[NCNJ] : (2.22)
Where e IJ [e] is the curvature associated to the torsion free spin connection e! IJ
: The
insertion of the equations (2.21) and (2.22) into the action (2.11) delivers the following
expression
S (e; !;) = 12k
ZMeeI e
JT
IJKLe KL d4x
i2
ZMeheI I
eD ( eD) I
eeI e
JT
IJKLCK[NCNL]
id4x
14
ZMeeIC KL
IKLNJNd4x: (2.23)
32
From the equation (2.21), we immediately have DeI = CIJ eJ leading to express (2.19)
as
D
ee[Ne
]M
= C J
N
ee[J e
]M
+ C J
M
ee[Ne
]J
12eC JN eMe
J +
1
2eC JM eNe
J =
k
8T1 KLNM eeIIKLPJ
P ; (2.24)
by contracting with eQ, we get
eNCMQ =k
4T1 KLNM QPKLJ
P
=k
4
2
2 + 1
QPNMJ
P 2 N [PQ]MJ
P
: (2.25)
Now we have to inject (2.25) and after straightforward calculations, particularly per-
formed on the third and the fourth lines of (2.23), we can have the nal form of the
e¤ective action
Seff (e; !;) = 12k
ZMeeI e
JT
IJKLe KL d4x
i2
ZMeheI I
eD ( eD) I
id4x
+3k
16
2
2 + 1
ZMJ IJId
4x: (2.26)
The obtained e¤ective action exhibits an interaction term, weakened by the gravitational
constant k. If the parameter ! 1; we recover the usual Einstein-Cartan theory. In[19], it has been claimed that the e¤ects of this four-fermions interaction term should be
observable even outside the Quantum Gravity sector. However, this statement tends to
break the equivalence between the General Relativity and Einstein-Cartan theory, when
Gravity is minimally coupled to fermions. To restore this equivalence and to discard
the ambiguities that take place in the theory because of this emerging interaction term,
other generalized Einstein-cartan action has been used in [63]. The generalization of the
Palatini, or Palatini-Holst action, was initially introduced in [64] and [65] for the vacuum
case leading to the classical Einstein equations.
2.2.3 The constraints a¤ected by the Fermions
We had previously given a simple example of the scalar matter changing the Gravity
constraints. In its turn the presence of fermions contributes, for instance, to the set of
33
constraints (1.63), (1.65) and (1.66). For this, we begin by dividing the Dirac action as
S(F ) (e; !;) = ST (F ) (e; !;) + Sk(F ) (e; !;) + S?(F ) (e; !;) (2.27)
We start as usual by rewriting the tetrad with its components given in the ADM decom-
position, leading to
S(F ) (e; !;) = i2
ZMNee (eeI I) I (D) (D)
Id4x (2.28)
The next step consists in obtaining the conjugate momentums of the various elds implied
here by simply expliciting the time ow where we insert as usual = TN
N, to have
the term
i2
ZMNeeT N
N
I I (D) (D)
Id4x (2.29)
we remark that we can already have ST (F ) (e; !;) of (2.27). We must keep in mind that
we are coupling fermionic matter to the Holst term, that uses the Ashtekar variables, so
at some point by setting the time gauge as I = (1; 0; 0; 0) ; we have to change fromSL(2;C) gauge group to SU (2) ; hence the spinors will rather be represented explicitlyby and part of = ( )T and its conjugate. Therefore, we nd
ST (F ) (e; !;) =i
2
Z ee 0 (Dt) (Dt) 0dtd3x
=i
2
Z ee y : + y: c:c
dtd3x
14
Z eeijk! ijt Jkdtd
3x (2.30)
where y := ( )T 0; and c:c are the complex conjugate. We remark that contrary to
the scalar eld, the Dirac action is of rst order with respect of the time dimension,
hence other constraints related to the dynamic of the spinors will arise and dene the
momentums of these spinors, but in the following part, we start focusing on the Gauss
constraints. For the nal line in the previous equation we use [ 0; ij]+
= i0ijkJk.
At this stage, we introduce the canonical variables as it was performed in (1.60), where
Al =: l+ K
l =
12lij!ij + !
0l . We just have to notice that for now we can not identify
l as the torsion free spin connection compatible with the tetrad since the torsion is still
dynamically present because of (2.19).
The Gauss constraint
To obtain the Gauss constraint, we can express the fermionic part ST (F ) (e; !;) as it
follows, where we nd the free parameter k; already dened as k := lt + K lt, leading
34
to
ST (F ) (e; !;) =i
2
Z ee y : + y: c:c
dtd3x
12
Z ee k Kkt
Jkdtd
3x: (2.31)
We have mainly paid attention to the fermionic part of the total action, since the part
which concerns Gravity has already been detailed in the previous chapter, now by adding
the expression (2.31) to (1.63) we obtain
ST (e; !;) =1
k
Z ZiD(A)
a
eP ai + 1 + 2 jmiKjtK
maeP ai d3x dt
1
k
Z Z k2ee k Kk
t
Jkd
3x
dt (2.32)
1
k
Z ik
2
Z ee y : + y: c:c
d3x
dt: (2.33)
From which we can derive the following equations. The rst one represents the presence
of the fermionic source that a¤ects the Gauss constraint
iD(A)a
eP ai k
2eeJi = 0; (2.34)
the second equation shows the e¤ects of fermions on the external curvature because of
the non vanishing torsion
Kjt
mij Km
aeP ai 2k
(1 + 2) 2eeJj = 0 (2.35)
These constraints are equivalent, and can be melted into
Gi : = D(A)a
eP ai k
2eeJi (2.36)
= mij Km
aeP ai 2k
(1 + 2) 2eeJj 0: (2.37)
The vectorial di¤eomorphism constraint
We have successfully obtained the Gauss constraint of Gravity in presence of fermions.
From now, we will focus on developing the next term
Sk(F ) (e; !;) = i2
ZNaee 0 (Da) (Da)
0dtd3x:
35
To have the di¤eomorphism constraints we have to explicit Sk(F ) (e; !;) to obtain
Sk(F ) (A;) = i2
ZNaee y@a + y@a c:c
dtd3x
1
2
ZNaeekaJkdtd3x; (2.38)
by developing the expressions of Da and Da in order to get 0; IJ
+!aIJ
2.
The Lorentz indices I and J could only be spatial allowing us to replace by iijkJk !ij
a
2=
ikaJk to nd the second line involved in (2.38). We mention that the current Jk is dened
as Jk := yk + yk in this representation.
By inserting the variable Ai, we can rewrite the Sk(F ) (e; !;) part as
Sk(F ) (A;) = i2
ZNaee yD(A)
a + yD(A)a c:c
dtd3x
+
2
ZNaeeKk
aJkdtd3x; (2.39)
with D(A) = @ + Ai i ; and i :=
i2i:
Once again, by adding the gravitational contribution (1.65) to the precedent detailed
expression, we can have the full vectorial constraint density dened as
Ha = H(G)a +H(F )a = eP bi F i( )ab +1 + 2
iklKk
aKlb
i2ee yD(A)
a + yD(A)a c:c
+
2eeKk
aJk
0; (2.40)
which can be reduced to
Ha = H(G)a +H(F )a = eP bi F i( )ab
i2ee yD(A)
a + yD(A)a c:c
(1 +
2)
GkKk
a
0
by using the complete form of the obtained gauss constraint (2.36).
The scalar constraint
It remains to investigate one last expression which is S?(F ) (e; !;) : As we have done be-
fore for the vectorial constraint case, we only have to add the gained results of S?(G) (e; !;)
36
to compile the full constraint later.
S?(F ) (e; !;) = i2
ZNeeeeai i (Da) (Da)
idtd3x
= i2
ZNeeeeai yi@a + yi@a c:c
dtd3x
i2
ZNpdet (q)eeai iikl!akl2 J0
dtd3x
i2
ZNpdet (q)eeai iikl!0ka Jl dtd3x
By switching to the Aia variables and the expression of eEai , we can express S?(F ) asS?(F ) = i
2
ZN eEai yi@a + yi@a c:c
dtd3x
12
ZN eP ai iaJ0dtd3x+ 12
ZN lik eP ai Kk
i
Jldtd
3x: (2.41)
We can put the precedent action under the form of
S?(F ) = i2
ZN eEai yiD(A)
a + yiD(A)a c:c
dtd3x
+1
2
ZN lik eP ai Kk
a
Jldtd
3x:
After this detailed reformulation, we are nally able to inject the gravitational part given
by the Holst term, to compile the full scalar constraint as
H = H(G) +H(F ) =
2 eP ai eP bj2pdet (q)
kijFk( )ab +
1 + 2
knmKn
aKmb
+
1 + 2
eP al D()a
lik eP biKk
b
pdet (q)
ik2 eP ai yiD(A)
a + yiD(A)a c:c
+ k
2
lik eP ai Kk
a
Jl
0: (2.42)
This not nished yet. As we have said previously, this chapter is based mostly on the
canonical analysis developed in [34], which explores the constraints deeper, in the sense
that the obtained constraints are decomposed in the torsion-free part and second part
that depends explicitly on the torsion arising from the presence of the fermions. In order
to achieve this, the splitting will be performed over the connection Aia = eAia + Ai
a, witheAia = eia+ eKia and A
i
a = Cia+ kia:We retrieve eia, the SO(3) spin connection compatible
with the tetrad:
37
Thus, with the use of (2.25) after its contraction with eN , we can dene
Cia =1
2"iklC kl
a =k
4
2
(1 + 2)
1
ikleekaJ l ejaJ0 ; (2.43)
and
kia := C 0ia =
k
4
2
(1 + 2)
ikleekaJ l + 1 ejaJ0
; (2.44)
and gather these obtained solutions into the expression of Aia to write
Aia = eAia + Ai
a =eia + eKi
a
= eia + eKia +
k
4ikleekaJ l (2.45)
it remains to explicit the constraints (2.36), (2.40) and (2.42) into torsion-free and torsion-
dependant parts. For the Gauss constraints, we get
Gi = D(A)a
eP ai k
2eeJi
= eD(A)a
eP ai + ijkAj
a
eP ai k
2eeJi
= eD(A)a
eP ai = eGi 0: (2.46)
Where the source contribution involved in the gauss constraint is suppressed by the torsion
that takes part in Aj
a. Thus, the constraint eGi keeps its role that underlines the invarianceunder SU (2) transformations. However, for the vectorial di¤eomorphim constraint, the
calculations are harder and the splitting acts as
Ha = eP bi eF i( )ab + 2@[aAi
b] + inmAn
aAm
b + inmAn
aeAmb + imn eAnaAmb
(1 + 2)
GkKk
a i
2ee y eD()
a + y eD()a c:c
12CiaeeJi +
2KiaeeJi; (2.47)
with eF i( )ab is the curvature associated to the torsion-free self-dual connection eAia:Many steps are required to establish the nal form of Ha;where we have essentially
used the explicit formulas of Cia; kia and those of An
a to have
Ha =1
eEbi eF i( )ab +
k
4abc eEci eD(A)
a
eeJ i i2ee y eD(A)
a + y eD(A)a c:c
+
(1 + 2)
Kia
k
4jklJkeeal eGi (2.48)
38
Finally, the same procedure will be applied to the expression of the scalar constraint and
after some tedious manipulations this constraint may be simplied as
H = ee1 eEai eEbj ijk eF i( )ab 2 2 + 1
eKi[aeKjb]
+i
2k eEai yi eD(A)
a + yi eD(A)a c:c
3k16
2
1 + 2ee J02
+3k
16
2
1 + 2eeJ lJl + (1 + 2)
eD(A)a
ee1 eEai eGi+ k
2eGiJ i
0;
which can be resumed to
H = eH(G) + eH(F ) + eHint (1 + 2)
eD(A)a
ee1 eEai eGi+ k
2eGiJ i 0;
by identifying the interaction part eHint as eHint = 3k16
2
1+ 2ee hJ lJl (J0)2i :
These expressions are enough to be the starting point of LQG supplied with fermions.
However it is more suitable to work with the half-densitized Dirac spinor elds as it was
originally pointed in [66]. Thus, it brings us to use these variables := (det (q))1=4
and := (det (q))1=4 (same for y := (ee)1=2 y and y := (ee)1=2 y). in addition, formore complete study, the massive fermions can be considered in the theory by supplying
the initial action withZemd4x, where m is the fermions mass: Finally, we mention
that we have focused on fermionic elds minimally coupled to Gravity, a version which
takes into account the non-minimal coupling is investigated in detail in [67]. From the
expressions that has been developed in the rst subsection of this second chapter (2.10),
the structure functions still appear in the constraints algebra, since there is no possible
way that H(G) and H(F ) cancel each other.
39
Chapter 3
On the Hamiltonian formalism of the
Tetrad-Gravity
In the previous chapters, we summarized the state of the art of the Hamiltonian formalism
dedicated to describe Gravity, both in vacuum and coupled to matter. We listed some
issues related to Quantum Gravity developed specially by the Hamiltonian formalism.
These approaches are mostly based on the ADMdecomposition leading to open constraints
algebra because of the structure functions. Unlike, the precedent works, we short-circuit
the ADM decomposition, but we will respect the Dirac procedure carefully. In this study,
we derive the conjugate momentums and follow the procedure until we calculate the
algebra of the rst-class constraints, and as interesting result we have obtained a closed
algebra, particularly the Poisson bracket of the smeared scalar constraints is under the
form of fDt (N) ; Dt (M)g = 0; meaning that the algebra is free from structure functions
that emerge from the ADM foliation. However, the set of constraints contains a condition
which is problematic. This condition turns to be impossible to solve via the Dirac brackets.
Yet, a possible way to handle this condition is to split the connection into dynamical
and non-dynamical parts; and x the non-dynamical part to zero. As we will see, this
procedure will not change the physical degrees of freedom. We recall that this third
chapter is based on the article appearing in [30]. Presently, we have to introduce some
useful and convenient notations that make the calculations easier.
3.1 The ABC notation and its properties
In this chapter, we often use some functions formed by the tetrad or the co-tetrad un-
der compressed form and dened at any dimension d. These functions are obtained by
the consecutive variation of the tetrad determinant e with respect to the co-tetrad. As
consequence, these obtained functions will obey to some antisymmetric properties.
40
We begin with
eJ(e) = eeJ (3.1)
If we variate this function with respect to the co-tetrad, we obtain A as
eAIJ =
eIeeJ
=
1
(d 2)!I0:::Id3IJe0I0 :::ed3Id3"
0:::d3 (3.2)
or
eAIJ = eeIeJ eJeI
(3.3)
From its denition, the function eAIJ is antisymmetric while we permute the indices and ; or I and J . For example, if and both label time, then we have eAtItJ = 0:We iterate this variation again, and we dene the B function as
eBLIJ =
eLeAIJ
(3.4)
=1
(d 3)!I0:::Id4LIJe0I0 :::ed4Id4"
0:::d4 (3.5)
Both the Lorentz and di¤eomorphism indices obey the following cyclicity
eBLIJ = eLeAIJ + eJeALI + eIeAJL
= eLeAIJ + eLeAIJ + eLeAIJ (3.6)
Remark when the dimension of the manifold d = 3, we have eBaKbIcJ = 0 when all thedi¤eomorphism indices are spatial.
If we continue, we can obtain the function C as the following
eCKLIJ =
eKeBLIJ
(3.7)
=1
(d 4)!I0:::Id5KLIJe0I0 :::ed5Id5"
0:::d5; (3.8)
where
eCKLIJ = eKeBLIJ eKeBLIJ + eKeBLIJ eKeBLIJ
= eKeBLIJ eJeBKLI + eIeBJKL eLeBIJK : (3.9)
41
In the following analysis, we need to use the following function BNKL dened as
BNKL =1
2
eNAKLd 2 + eN
AKLd 2 + eNAKL
(3.10)
=1
2
ANKd 2 eL +
ALNd 2 eK +AKLeN
:
Where ANK = eNeK eNeK :Unlike BNIJ which is antisymmetric with respect of the indices ; and or the
Lorentz ones N; I and J: The function BNKL is only antisymmetric with respect to and ; or K and L:
BNKL denes the inverse of BNIJ , so we have
BNKLBNIJ = IK
JL ILJK
(3.11)
and
BNKLBMKL = MN
(3.12)
Based on the properties detailed earlier, by xing for example = b and = a then
the relation (3.11) reads
BcNtKaLBcNtIbJ +1
2BcNdKaLBcNdIbJ =
1
2baIK
JL ILJK
(3.13)
and the relation (3.12), when = = t; reduces to
BcNtKaLBbMtKaL = MN bc; (3.14)
and for = t and = d; we get
BcNdKaLBbMtKaL = 0: (3.15)
3.2 Constructing the Hamiltonian
After introducing the precedent notations, we focus on the tetrad-connection Gravity
action at ddimension and the construction of the Hamiltonian derived from this action.We start by writing the expression of this action in terms of forms where
S (e; !) =
ZM
1
(d 2)!eI0 ^ :::eId3 ^ Id2Id1I0:::Id1 (3.16)
42
where we have used the expression of 2form curvature given in
IJ =1
2IJdx
^ dx
=1
2
@!IJ @!IJ + ! N
I !NJ ! NI !NJ
; (3.17)
and the expressions of the A-function detailed in (3.2) and (3.3), to have the action as
S (e; !) = ZMeAKLKL
2ddx; (3.18)
The next step is to show explicitly the time derivative in order to perform the Legendre
transformations as it mentioned in Appendix B. The action (3.18) reads
S (e; !) =
ZM
eAaKtL (@t!aKL Da!tKL) eAaKbL
abKL2
ddx; (3.19)
from which we obtain the conjugate momentums N and PKL of the co-tetrad eN andthe so(1; d 1) connection !KL
N (x) =S (e; !)
@teN (x)= 0;PaKL = S (e; !)
@t!aKL (x)= eAaKtL (x)
and
P tKL = S (e; !)
@t!tKL (x)= 0;
which are subject to the following Poisson brackets
eI (!x ) ; N (!y )
=
NI (!x !y )
!IJ (!x ) ;PKL (!y )
=
1
2
KI
LJ LI KJ
(!x !y ) : (3.20)
We have also the primary constraints as
tN (x) = 0;P tKL = 0
bN (x) = 0 and CaKL = PaKL eAaKtL = 0: (3.21)
The constraints appearing in the second line of (3.21) satisfy the following non-zero Pois-
son brackets bN (!x ) ; CaKL (!y )
= eBbNtKaL (!x !y ) ; (3.22)
43
where we have used the denition of eBbNtKaL and then its antisymmetric properties.Now, we can obtain the total Hamiltonian as
HT =Z
tNtN + P tKL
AtKL2
+ bNbN + CaKLAaKL2
+H0; (3.23)
where tN ; AtKL;bN and AaKL are the Lagrange multipliers of the primary constraintslisted in (3.21) and H0 is dened by
H0 =Z
eAaKbLabKL
2+ eAaKtLDa!tKL
: (3.24)
We have to check the consistency of the obtained Hamiltonian formalism. To achieve
this, the constraints must be preserved under the time evolution given in terms of total
Hamiltonian (3.23) leading to
tN ;HT
= eBtNaKbLabKL
2= PN = 0; (3.25)
P tNM ;HT
= DaeAaNtM = eBcKaNtMDaecK =MNM = 0 (3.26)
bN ;HT
= eBbNtKaL
AaKL2Da!tKL
eBbNaKcLacKL
2= 0 (3.27)
and
CaKL;HT
= eBbNtKaL
bN + ! M
tN ebM+Dc
eAcKaL
= eBbNtKaL
bN + ! M
tN ebM DbetN+ eBbNcKaLDcebN
= 0; (3.28)
where we have used (3.22) andN ; eAKL
=
eN
eAKL
= eBNKL:
From these consistency conditions emerge new constraints, referred as the secondary
constraints PN and MNM obtained by the evolution of the Primary ones tN and P tNM ,respectively. Whereas the evolutions of bN and CaKL lead to equations involving the
Lagrange multipliers AaKL and bN :We have to continue the procedure and check again the consistency of the secondary
constraints.
For MKL we get
MKL;HT
= Da
eBbNaKtLbN
+1
2A Ka NeAaNtL +
1
2A La NeAaKtN ;
44
by using (3.28)
Da
eBbNaKtLbN
= Da
eBbNtKaLbN
= Da
eBbNtKaL!tNMeMb
+Da
eBbNtKaLDbeN
and for Da
eBbNtKaLDbeN
we have
Da
eBNbKaLDbeN
= DaDb
eAbKaL
= 1
2(DaDb DbDa) eAaKbL
= 12
Kab NeAaNbL + L
ab NeAaKbN:
From the property of the B-Matrix (3.6), a direct computation leads, for any antisym-metric tensor TNM = TMN ; to the identity
T KNeANL + T L
NeAKN =1
2
eK eBLNM eLeBKNM
TNM
=1
2
eM eBNKL eN eBMKL
TNM ; (3.29)
leading to
Da
eBbNtKaL!tNMeMb
= Da
! Kt NeAtNaL + ! L
t NeAtKaN
= eAaNtLDa!Kt N eAaKtNDa!
Lt N
! Kt NM
NL + ! Lt NM
KN; (3.30)
from which we get
MKL;HT
=
! Kt NM
NL + ! Lt NM
KN
A Ka N
2Da!
Kt N
eAtNaL
A La N
2Da!
Lt N
eAtKaN
12
Kab NeAaNbL + L
ab NeAaKbN: (3.31)
By using (3.29), the precedent equation can be expressed as
MKL;HT
= 1
2
eKb
eBbLtNaM
AaNM2Da!tNM
eBbNaNcM acNM
2
+(K ! L)
+1
2
eKt P
L eLt PK! Kt NM
NL + ! Lt NM
KN
45
Which leads to
MKL;HT
=1
2
eKt P
L eLt PK! Kt NM
NL + ! Lt NM
KN' 0 (3.32)
ensuring the consistency of the constraint MKL; when the constraints (3.27) and (3.28)
are satised.
The next step is to search for the evolution of the constraint PN , which is given by
PN ;HT
= eCaMtNbKcLbcKL
2aM eBtNbKaLDb
AaKL2
(3.33)
which involves the C function (3.7) introduced in the previous subsection. By using theexpressions of (3.27) and (3.28) and with the help of (3.9), we get the evolution of PN
under the form of combination of constraints
PN ;HT
= ! N
t MPM
ecNPM
cM + !tMKe
Kc DcetM
+MKL
AcKL2Dc!tKL
' 0 (3.34)
The term appearing in the second line of (3.34) is orthogonal to etN , therefore the
evolution of etNPN gives
etNP
N ;HT=tN + ! M
tN etMPN ' 0: (3.35)
The part proportional to ecN in (3.34) shows, by using the consistency of the constraint
MKL (3.32), that the evolution of D(sp)a = eaNPN +!aKLM
KL; linear combination of PN
and MKL is given by
eaNP
N + !aKLMKL;HT
= PN@aetN +MKL@a!tKL ' 0 (3.36)
while we have
Dt = PNetN = eAaKbLabKL2
; (3.37)
where we have used etNeBtNaKbL = eAaKbL:By the use of
N cecNPN = N cecNeBtNaKbL
abKL2
= eAaKtLN ccaKL
= eAaKtLL!N(!aKL)Da (N
c!cKL); (3.38)
46
the smeared spatial projection is
D(sp)!N=
Z
NaeaNP
N + !aKLMKL=
Z
eAaKtLL!N(!aKL) ; (3.39)
which exhibits L!N(!aKL) = N b@b (!aKL) + @a
N b!aKL; the Lie derivative along the
vector eld!N : The next equation shows the consistency of the constraints D(sp)
!N
nD(sp)
!N;HT
o=
Z
PNL!
N(etN) +MKLL!
N(!tKL)
(3.40)
where, from L!N(etN) = Na@aetN and L!
N(!tKL) = Na@a!tKL; we can see that the
variables etN and !tKL transform as scalars under the Lie derivative L!N:
Iterating the Procedure will not give other constraints. The total Hamiltonian HT iscoherent if the constraints (3.27) and (3.28) are satised. These constraints have to be
solved. For this, we rely on the properties of of the B-Matrix and its inverse (3.10), andwe use (3.14) and (3.15) to obtain the solution of (3.27) as
1
2AaKL = Da!tKL
1
2BbNtKaLBbNcPdQcdPQ + BbNcKaLbNc (3.41)
with arbitrary bNc:
bN = ! MtN ebM +DbetN BbNtKaLBcMdKaLDdecM
= ! MtN ebM +DbetN + e1BbNtKtLMKL ' ! M
tN ebM +DbetN (3.42)
whereas this last solution which is the solution of bN is obtained from (3.28).
If we multiply the constraint (3.28) by BdNeKaL and use (3.15), we obtain
BdMaKeLBbNcKeLDcebN = 0 =NMbd
ca bacd
BdMaKtLBbNcKtL
DcebN
= DaedM DdeaM e1BdMaKtLMKL = 0 (3.43)
3.3 The constraints algebra
To construct the new Hamiltonian one has to implement the xed Lagrange multipliers
given in (3.41) and (3.42), solutions of the equations (3.27) and (3.28) into the expression
47
(3.23). This leads to
H0T =
Z
P tKLAtKL2
+ tNtN
Z
aKeLa aLeKa
+ 2Da
CaKL + eAaKtL
!tKL2
+
Z
eAaKbLabKL
2+ aNDaetN CaKLBbNtKaLBbNcPdQ
cdPQ2
+
Z
CaKLBbNcKaLbNc (3.44)
We have to apply the procedure again and check the consistency of the constraints with
the new Hamiltonian H0T . We proceed by computing the time evolution of cN which
gives
cN ;H0T
= ! N
t KcK
+1
2CaKL
ecN
BbNtKaLBbNePdQ
edPQ
12CaKL
ecN(BbNdKaL) bNd; (3.45)
showing that the constraint is consistent since its Poisson bracket with H0T results incombination of cK and CaKL:
While for the constraint CaKL we obtain
CaNM ;H0T
= ! N
t KCaKM ! M
t KCaKN 1
2
aNeMt aMeNt
+Db
eAbNaM
+ eBbQtNaMDbetQ
Dc
CdKLBbNtKdLBbNcNaM
: (3.46)
Where its second line vanishes by using
Db
eAbNaM
+ eBbQtNaMDbetQ =
1
2eBcQbNaM (DbecQ DcebQ)
=1
2eBcQbNaMe1BcQbKtLMKL
' 0: (3.47)
This consistency of the constraint CaKL is veried only when the condition (3.43) is
satised. It is mainly because (3.43) is itself a condition to solve the equation (3.28)
which result from the previous time evolution of CaNM : We expect that the consistency
of the constraints MKL;Dt and D(sp) will also depend on (3.43).
48
In fact, a direct computation of the time evolution of these constraints leads to
MKL;H0T
= ! K
t NMNL ! L
t NMKN +
1
2
eKt P
L eLt PK
12Da
eBdNbKaL (DbedN DdebN)
; (3.48)
fDt;H0Tg = N + !tNMe
Mt
PN + BdNcKbLDa
eAaKbL
dNc;
and nD(sp)
!N;H0T
o=
Z
PNL!
N(etN) +MKLL!
N(!tKL)
Z
1
2eBdNbKaL (DbedN DdebN)L!N!aKL; (3.49)
which shows that MKL;Dt and D(sp) are consistent only when the condition (3.43) issatised. For the constraint tN , we obtain
tN ;H0T
= eBtNaKbLabKL
2+Da
aN
+CaKL(e1etNBbMtKaLeBbMdPeQ
+e1
etN(BbMtKaL) eBbMdPeQ
+e1BbMtKaLeCtNbMdPeQ)dePQ2
= P0N ' 0; (3.50)
where we have used etN
e1 = e1etN and the properties of the C-function.By the use of etN
etN(BbMtKaL) = BbMtKaL, the projected constraint P
0N ; where D0t =P
0NetN ; is simplied into combination of constraints as
D0t = Dt aNDaetN + CaKLBbNtKaLBbNcPdQcdPQ2' 0 (3.51)
The projected part of P0N on the spatial tetrad component ecN will be computed by using
the relation ecN ecN
(BbMtKaL) = BbMcKaL rst, and then
ecNeCtNbMdPeQdePQ2
= BbMtPeQcePQ + bcPM (3.52)
49
to get
CaKLBbMtKaLecNCtNbMdPeQdePQ2
= CaKLcaPQ +1
2CaKLBbMdKaLBbMdPeQdePQ
+CaKLBcMtKaLPM ; (3.53)
from which we nally obtain
ecNP0N =
CaKL + eAaKtL
caKL + ecNDa
aN + CaKLe1BcMtKaLPM
+CaKLbMcKaLBbMdPeQdePQ
2+bMdKaL BbMdPeQcePQ
2
; (3.54)
leading to the linear combination of the smeared constraints
Z
N cecNP
0N +!cKL2
M0KL + aN (DaecN DceaN)
Z
N cCaKLbMcKaLBbMdPeQdePQ
2+bMdKaL BbMdPeQcePQ
2
N cCaKLe1BcMtKaLP
M
= Z
aML!
N(eaM) +
CaKL + eAaKtL
L!N(!aKL)
: (3.55)
From the consistency of the constraint P tKL checked with the new Hamiltonain H0T ; wehave
P tKL;H0T
=
Da
CaKL + eAaKtL
+1
2
aKeLa aLeKa
= M 0KL ' 0: (3.56)
The secondary constraints M 0KL is deduced and expressed in terms of constraints combi-
nation.
This leads to the nal form of the Hamiltonian
H0T =
Z
1
2P tKLAtKL + tNtN D0t M 0KL!tKL
2
+
Z
CaKLBbNcKaLbNc: (3.57)
Remark that the constraintD0(sp)!Ncan be completed by adding the term tML!
N(etM),
so
D0(sp)!N=
Z
ML!
N(eM) +
CaKL + eAaKtL
L!N(!aKL)
; (3.58)
50
Which satises nD0(sp)
!N;D0(sp)
!N 0o
= D0(sp)h!N ;!N 0i
(3.59)
whereh!N ;!N 0iis the Lie bracket.
D0(sp)!N; on the variables co-tetrad components eN and on the connection !aKL;
reproduces di¤eomorphisms of the hypersurfaces t:neN ;D0(sp)
!No
= L!N(eN) ;
n!aKL;D0(sp)
!No
= L!N(!aKL) (3.60)
While the primary constraints transform asnN ;D0(sp)
!No
= L!N
N
;nCaNM ;D0(sp)
!No
= L!N
CaNM
; (3.61)
which shows that the Poisson brackets of D0(sp)!Nwith N and CaNM vanish weakly,
and D0(sp)!Ntransforms D0t as scalar of weight one.nD0(sp)
!N;D0to
= L!N(D0t) = @a (NaD0t)
=)nD0(sp)
!N;D0t (M)
o= D0t
L!NM
(3.62)
where D0t (M) =RMD0t is the smeared scalar constraint with M as an arbitrary scalar
function serving to test the Poisson brackets behavior.
The same can be deduced for the constraintM0 () =RM 0KL KL
2; where KL corre-
spond to the dimensionless innitesimal arbitrary parameters KL = t!tKL;nD0(sp)
!N;M 0KL (!x )
o= L!
N
M 0KL (!x )
= @a
NaM 0KL (!x )
=)
nD0(sp)
!N;M0 ()
o=M0 L!
N: (3.63)
We can also calculate the following Poisson brackets, by the use of (3.60) and (3.61)D0(sp)
!N;
Z
CaKLBbNcKaLbNc=
Z
CaKLBbNcKaLL!NbNc ' 0
showing that D0(sp)!Nis preserved under time evolution. For the condition (3.43) we
get D0(sp)
!N;
Z
BdMaKeLBbNcKeLDcebN
= L!
N
BdMaKeLBbNcKeLDcebN
' 0
which classies D0(sp)!Nas rst-class constraint.
51
We focus now on the Lorentz constraints M 0KL that can be completed by adding the
constraint tN
M 0KL
2= Da
CaKL + eAaKtL
+1
2
KeL LeK
The constraints M 0KL acts on the tetrad and the connection as local innitesimal gauge
transformations
feN ;M0 ()g = LN eL; f!aNM ;M0 ()g = DaNM ; (3.64)
while on the primary constraint N and CaNM , M 0KL acts as
N ;M0 ()
= NL
L;CaNM ;M0 ()
= NLC
aLM + MLCaNL: (3.65)
The space-time manifold indices do not transform under the Lorentz transformations,
hence it makes easy to calculate the transformations generated byM0 () : The constraint
D0t is a scalar regarding to the Lorentz transformation, leading to
fD0t (!x ) ;M0 ()g = 0 =) fD0t (M) ;M0 ()g = 0: (3.66)
M 0KL is contravariant tensor, where
M 0KL;M0 ()
= KNM
0NL + L NM0KN
which also leads to
M 0NM (!x ) ;M 0KL (!y )
= (NLM 0MK (!x ) + MKM 0NL (!x )
NKM 0ML (!x ) MLM 0NK (!x )) (!x !y ) ;
giving the so(1; d 1) Lie algebra. We can also achieve the following transformationsM0 () ;
Z
CaKLBbNcKaLbNc=
Z
MN CaKLBbMcKaL
bNc
specially for the condition (3.43), where we have
M0 () ;BdMaKeLBbNcKeLDcebN
= Q
M BdQaKeLBbNcKeLDcebN
which demonstrates that MKL is a rst-class constraint.
For the last Poisson bracket that involves the smeared scalar constraints, after straight-
52
forward computations, we nd that the only surviving term is
fD0t (M) ;D0t (M 0)g = (M@nM0 M 0@nM)
eAnIaJBbP tIaJBbPcKdL
cdKL2
(3.67)
which leads to an exact zero when we explicit the product of A and B functions appearingin the precedent equation
fD0t (M) ;D0t (M 0)g = 0
However if we take the Poisson bracket between D0t (M) and the primary constraintsCaKL; or with
RCaKLBbNcKaLbNc; we can see that that it vanishes only when (3.43)
is satised. Therefore, D0t can be classied as rst-class constraint only if the condition(3.43) holds.
we conclude this subsection by stating that the Hamiltonian formalism of the tetrad
connection Gravity provides a set of rst-class constraints formed by tN ; P tNM , theLorentz constraints M0 () and the spatial di¤eomorphism constraints D0(sp)
!N: On
the other side, D0t (M),M0 (), D0(sp)!Nand the scalar constraint D0t (M) form a closed
algebra. However, the Poisson Brackets of D0t (M) with CaKL will vanish only if we solve(3.43).
3.4 The xing of the non-dynamical connection
Even if the obtained algebra closes in terms of structure constants, the resulted condition
(3.43) is problematic making di¢ cult to check its consistency. In order to discard this
condition from the Hamiltonian analysis, we decompose the connection into
!aKL = !1aKL + !2aKL (3.68)
where
!1aKL = P PdQ1KaL !dPQ = BbNtKaLBbNtPdQ!dPQ (3.69)
and
!2aKL = P PdQ2KaL !dPQ =
1
2BbNcKaLBbNcPdQ!dPQ (3.70)
as consequence of (3.12), P PdQ1KaL and P PdQ
2KaL are projectors that verify
P PdQ1KaL + P PdQ
2KaL =1
2da
PK
QL PL
QK
; (3.71)
P NbM1KaL P PdQ
1NbM = P PdQ1KaL ; (3.72)
53
P NbM2KaL P PdQ
2NbM = P PdQ2KaL ; (3.73)
and
P NbM1KaL P PdQ
2NbM = 0 (3.74)
As we will see, this decomposition is motivated by the fact that the time derivative of
!2aKL does not contribute to the kinematical part of the action (3.19). We can use the
relations
BbNcKaLeaL = BbNcKaLeaK = BbNcKaLetL = BbNcKaLetK = 0
to get
eAaKtLP PdQ2KaL = 0 and eAaKtLP PdQ
1KaL = eAdPtQ (3.75)
leading to
@teAaKtL
!2aKL = @t
eAaKtL!2aKL
eAaKtL@t (!2aKL)
= BbNtKaL (@tebN)!2aKL = 0 (3.76)
which clearly shows that in addition to the temporal components of tetrad and the con-
nection, we also have the projected spatial connection !2aKL as non-dynamical variable.
We can also compute
eAaKtLD2a!tKL = eAaKtLP PdQ2KaL Dd!tPQ = 0 (3.77)
which implies
eAaKtLDa!tKL = eAaKtLP PdQ1KaL Dd!tPQ = eAaKtLD!1
1a!tKL: (3.78)
Hence, only the projected part P PdQ1KaL Dd!tPQ = D!1
1a!tKL of the covariant derivative
of !tKL; given in terms of the projected part of the connection !1aKL; contributes to the
action (3.19), leading to the expression
S (e; !) =
ZM
eAaKtL (@t!1aKL D!1
1a!tKL) eAaKbLabKL2
(3.79)
The second part of the spatial connection !2aKL contributes only to its third term. We
have shown that the temporal components of the connection are Lagrange multipliers, and
KL = t!tKL plays the role of the innitesimal dimensionless parameters of the Lorentz
group local transformations, which transform the spatial connection as !aKL = DaKL:
Since the projected part D2a!tKL does not contribute to the action, it is possible to x it
to zero without a¤ecting the action (3.19), we will see that this gauge xing is consequence
54
from xing !2aKL:
But rst, we determine the rank of the propagators P PdQ1KaL and P PdQ
2KaL respec-
tively1
2ad
PK
QL PL
QK
P PdQ1KaL = d (d 1) (3.80)
and1
2ad
PK
QL PL
QK
P PdQ2KaL =
1
2d (d 1) (d 3) (3.81)
which seems to equal the numbers of !1aKL and !2aKL: On the other side, if we subtract
the d(d 1) identities given in
eBbNcPtQDcedN DdecN e1BdNcKtLMKL
= 2MPQ 2MPQ = 0 (3.82)
from 12d (d 1) (d 2) relations in (3.43), we nd the number of independent relations in
(3.43) which is 12d (d 1) (d 3) : This corresponds exactly to the number of components
!2aKL: Also, if this action does not explicitly contain !2aKL then the problematic condition
(3.43) will not appear.
We proceed now by xing the non-dynamical spatial connection. If we apply inni-
tesimal Lorentz transformations on eK = NK eN and !aKL = DaKL; we deduce
!1aKL = NK !1aNL + N
L !1aKN P PdQ1KaL @dPQ (3.83)
and
!2aKL = NK !2aNL + N
L !2aKN P PdQ2KaL @dPQ (3.84)
showing that each projected part of the connection dened in (3.69) and (3.70) transforms
independently from the other. By xing the non-dynamical part of the connection to zero,
we get
!02aKL = !2aKL + NK !2aNL + N
L !2aKN P PdQ2KaL @dPQ = 0 (3.85)
if we apply another transformation on !02aKL we get
!002aKL = !02aKL + 0 NK !02aNL + 0 N
L !02aKN PPdQ
2KaL @d0PQ
= P PdQ2KaL @d
0PQ (3.86)
which shows that the xing of the non-dynamical part of the connection remains invariant
if
P PdQ2KaL @d
0PQ = @2a
0KL = 0 =) @aKL = P PdQ
1KaL @dPQ (3.87)
55
on the other hand we have to impose
2!1aKL = P PdQ2KaL !1dPQ = P PdQ
2KaL
NP !1dNQ + N
Q !1dPN= 0 (3.88)
in order to preserve the same number of the degrees of freedom during the gauge trans-
formation of !1aKL: Both equations (3.87) and (3.88) show that this xing does not a¤ect
the gauge parameters, but only the gauge transformations of the spatial connection as it
is expressed below
eK = NK eN ; !tNM = DtNM and !1aKL = D!1
1a KL (3.89)
subject to conditions
D!12a KL = 0 and @2aKL = 0.
At this stage we focus on the third term appearing on (3.79) and check its invariance,
when we x !2aKL, hence we get
eAaKbL (!1)abKL
2
=
NK eAaNbL + N
L eAaKbN (!1)abKL
2
eAaKbLD!1a D
!11b KL
= NK eAaNbL + N
L eAaKbN (!1)abKL
2eAaKbLD!1
a D!1b KL
= NK eAaNbL + N
L eAaKbN (!1)abKL
2
eAaKbL (!1)abKL
2NL +
(!1)abLN2
NK
= 0; (3.90)
leading to rewrite the action (3.79) as
Sf (e; !1) =
ZM
eAaKtL (@t!1aKL D!1
1a!tKL) eAaKbL (!1)abKL
2
; (3.91)
where !2aKL has been xed to zero. The xed phase space is then equipped with the
following non-zero Poisson brackets
eI (!x ) ; N (!y )
=
NI (!x !y ) ;
!tIJ (!x ) ;P tKL (!y )
=
1
2
KP
LQ KQLP
(!x !y ) ;
!1aIJ (!x ) ;PbKL1 (!y )
= P KbL
1IaJ (!x !y )
56
The primary constraints are
tN = 0;P tKL = 0; bN = 0; CaKL1 = PaKL1 eAaKtL = 0 (3.92)
Remark that we can retrieve, in [54], the almost same procedure, where the simplicity
constraint, which corresponds to the primary constraint CaKL has been split into boost
and non-boost parts. However, in our case the decomposition is done by projectors as
CaKL = CaKL1 +CaKL2 with CaKL2 = PaKL2 = PcNMP KaL2NcM is the conjugate momentum
of the non-dynamical part of the connection.
Using the gauge xing leads to the total xed Hamiltonian
HfT =Z
M
tNtN + P tKL
AtKL2
+ bNbN + CaKL1
A1aKL2
+Hf0 ; (3.93)
where
Hf0 =Z
MeAaKbL (!1)abKL
2+ eAaKtLD!1
1a!tKL: (3.94)
Before performing the Hamiltonian analysis, we list some properties of the Poisson Brack-
ets involving the xed phase space. For instance, we project the Poisson brackets acting
on the projected element of the phase space as
f:; !1aKLg = P PdQ1KaL f:; !1dPQg
leading to
N ; !1aKL
1= P PdQ
1KaL
N ; !1dPQ
= P PdQ
1KaL
N ; P RbS
1PdQ !bRS
= P PdQ1KaL
N ; P RbS
1PdQ
!bRS
= P PdQ1KaL
N ; P RbS
1PdQ
!1bRS = 0 (3.95)
where we have used !2bRS and P (P )P = 0, true for any projectors. The same compu-
tation gives N (!x ) ;PaKL1 (!y )
1= 0; (3.96)
from which we deduce
aN (!x ) ; CbKL1 (!y )
1= eBaNtKbL (!x !y ) :
With these projected Poisson brackets the Jacobi identity are satised.
For example
N ;
!1aIJ ;PbKL1
1
1+PbKL1 ;
N ; !1aIJ
1
1+!1aIJ ;
PbKL1 ; N
1
1= 0
57
by using (3.95), (3.96) and
N ;
!1aIJ ;PbKL1
1
1= P PdQ
1IaJ
N ;
!1dPQ;PcNM1
P KbL1NcM
= P PdQ1IaJ
N ; P NcM
1PdQ
P KbL1NcM = 0:
After these given details on the consistency of the projected Poisson brackets, the Hamil-
tonian analysis is ready to be performed on (3.93) by using the projected Poisson Brackets.
The time evolution of the constraint bN is given by (3.27), where AaKL is replaced byA1aKL; and which is exclusively expressed in terms of !1aKL: We add that the solutionof A1aKL is free from the parameter bNc: For the consistency of CaKL1 , we nd the same
equation as (3.28) where DcAcKaL is replaced by D!11cAcKaL and which solution does not
give the condition (3.43).
With the new expressions of bN and AtKL; the Hamiltonian takes the form of
HfT =Z
tNtN + P tKL
AtKL2Df
t M fKL!tKL2
;
where
Dft = eAaKbL
(!1)abKL2
aND!1a etN + CaKL1 BbNtKaLBbNcPdQ
(!1)cdPQ2
and1
2M fNM =
D!1a
CaNM1 + eAaNtM
+1
2
NeM MeN
:
The xed di¤eomorphism constraint is
Df(sp)!N=
Z
ML!
N(eM) +
CaKL1 + eAaKtL
L!N(!1aKL)
:
A direct computation shows that Df(sp)!Nsatises the algebra
nDf(sp)
!N;Df(sp)
!N 0o
= Df(sp)h!N ;!N 0i
The transformation induced by Df(sp)!Non the primary constraints are given by
nN ;Df(sp)
!No
= L!N
N
;nCaNM1 ;Df(sp)
!No
= L1!N
CaNM1
and on the co-tetrad and the spatial components of the dynamical connection, we have
neN ;Df(sp)
!No
= L!N(eN) ;
n!1aNM ;Df(sp)
!No
= L1!N(!1aNM) :
58
We can also list the transformations induced byMf () on the primary constraints as
N ;Mf ()
= NL
L;CaNM1 ;Mf ()
= NLC
aLM1 + MLC
aLN1 ;
and for the co-tetrad and the spatial components of the dynamical connection, we have
eN ;Mf ()
= L
N eL;!1aNM ;Mf ()
= D1aNM
where NM satises the condition D!12a NM = 0 and @2aNM = 0.
The above transformations show that the Poisson brackets of Df(sp)!NandMf () ;
taken with Primary constraints vanish weakly. Like in the previous section, Dft andM
fKL
are transformed as scalar densities of weight onenDf(sp)
!N; Df
t (M)o= Df
t
L!NM;nDf(sp)
!N;Mf ()
o=Mf ()
L!N()
Dft is transformed underMf () as a scalarn
Mf () ; Dft (M)
o= 0
and M fKL as tensors
M fKL;Mf ()
= KNM
fNL + L NMfKN
from which we can deduce the so(1; d 1) Lie algebra for the generators M fNL.
The Poisson bracket Dft and
aN (!x ) gives
naN (!x ) ; Df
t (!y )o= 1
2CeKL1
eaN (!x )
BbMtKeLBbNcPdQ (!y )
(!1)cdPQ ' 0
and with CaNM1 (!x ) ; it givesnCaNM1 (!x ) ; Df
t (!y )o
= D!11b
eAbNaM
+ eBbQtNaMD!1
b etQ
+D!11c
CdKL1 BbNtKdLBbNcNaM
= P PdQ
1KaL
eBcQbKdL
D!1b ecQ
+D!11c
CdKL1 BbNtKdLBbNcNaM
= BeP tKtLBeP tNaMMKL
+D!11c
CdKL1 BbNtKdLBbNcNaM
' 0
which shows that the constraint Dft commutes weakly.
59
Finally, from direct calculation we getnDft (M) ; D
ft (M
0)o= 0:
Although this gauge xing, the Hamiltonian formalism is still coherent and the algebra
of the rst-class constraints Mf () ;Df(sp)!Nand Df
t (M) form a closed algebra with
structure constants. The smearing functions M and!N taken as testing functions for the
Poisson Brackets, behave as the lapse function and the shift vector, although they do not
result from the decomposition of the tetrad by the ADM formalism.
The physical degrees of freedom per point of the xed phase space are obtained by
subtracting, from the d (5d 3), the number 2d (d 1) of the second-class constraintsand twice the number of d (d+ 1) of the rst-class constraints to get d (d 3) ; whichcorresponds to the number of the physical phase space of the d-dimensional Gravity.
Next, we investigate the case when instead of xing the non-dynamical part of the
spatial connection to zero, we search for solutions of its equation of motion and then
replace these solutions in the action as in [68], Since the decomposition of the spatial
connection into the dynamical and the non-dynamical parts is unique, the part of the
solution of the zero torsion, which corresponds to !2aKL must be of the form of
!2aKL = P PdQ2KaL ePrbeQ = P PdQ
2KaL eP@be
Q +
be
Q
where reM = @e
M + e
M is the covariant derivative with respect to Christo¤els
symbol where its expression contains linear derivatives of the components of the
co-tetrad. As consequence, tN remains rst-class constraint but for aN we get
aN =D!1b eAbKaL + ! K
2b MeAbMaL + ! L2b MeAbKaM
!2aKL@teaN
where !2aKL@teaN
= P PbQ2KaL eP
@teaN
(b) eQ is not null. This will leads to second-order
formalism.
3.5 The constraints algebra in terms of Dirac brack-
ets
Even after discarding the problematic condition (3.43), other second-class constraints
remain in the formalism which are aN and CaKL1 : In this subsection, we have to consider
these constraints as strong equalities to eliminate them. In this case, the algebra must be
computed with the projected Dirac brackets dened in terms of the Poisson Brackets as
60
(see Appendix B)
[A;B] = fA;Bg1 fA;Cig1 fCi; Cjg11 fCj; Bg
where Ci =aN ; CaKL1
:
The inverse of this Poisson bracket
bN (!x ) ; CaKL1 (!y )
1' eBbNaKtL (!x !y ) = eBbNtKaL (!x !y )
is given by bN ; CaKL1
11= e1BbNtKaL
satisfying bN ; CaKL1
11
CaKL1 ; cM
1= cb
MN
and CaKL1 ; bN
11
nbN ; CdPQ1
o1= P PdQ
1KaL :
Now, it is possible to consider the constraints of the second-class bN and CaKL1 as strong
equalities by eliminating them from the total Hamiltonian H0T to get the reduced Hamil-tonian
HrT =Z
tNtN + P tKL
AtKL2Dr
t M rKL!tKL2
(3.97)
where
M rKL =tKeLt tLeKt
+ 2D!1
a
eAaKtL
(3.98)
and
Drt = eAaKbL
(!1)abKL2
(3.99)
The di¤eomorphism constraint reduces to
Dr(sp)!N=
Z
tNL!
N(etN) + eAaKtLL!
N(!1aKL)
: (3.100)
The Hamiltonian given in (3.97) is dened on the reduced phase space eN ; !1aKL; etN ; tN ; !tKLand P tKL equipped with the following non-zero Dirac brackets
etN (!x ) ; tM (!y )
= MN (
!x !y )
!tKL (
!x ) ;P tKL (!y )=1
2
KI
LJ KJ LI
(!x !y )
and
[eaN (!x ) ; !1bKL (!y )] = e1BbNtKaL (!x !y ) : (3.101)
61
This is di¤erent from the Dirac brackets obtained in [58], the dynamical connection is
Dirac self commutating because of (3.95).
These projected Dirac brackets verify the Jacobi identity. For the non trivial example
[ecN ; [!1aKL; !1bPQ]]+ [!1bPQ; [ecN ; !1aKL]
]+ [!1aKL; [!1bPQ; ecN ]
]= 0; (3.102)
The rst term vanishes, while the second and the third term give
P RdS1KaL
ehI
e1BcNtRdS
e1BhItPdQ + P RdS
1PbQ
ehI
e1BcNtRdS
e1BhItKaL
= ehIe1BcNtKaLe1BhItPdQ ehIe1BcNtPdQe1BhItKaL
P RdS1KaL e1
ehI(BcNtRdS) e1BhItPdQ
+e1P RdS1PbQ
ehI(BcNtRdS) e1BhItKaL
where we have used
!1aKL; e
1BcNtPdQD= e1P RdS
1PbQ
ehI
e1BcNtRdS
f!1aKL; ehIgD
= P RdS1PbQ
ehI
e1BcNtRdS
e1BhItKaL
and ehI
e1 = e1ehI :A direct computation leads to
+e1P RdS1PbQ
ehI
e1BcNtRdS
e1BhItKaL P RdS
1KaL
ehI
e1BcNtRdS
e1BhItPdQ
= ehIe1BcNtKaLe1BhItPdQ + ehIe1BcNtPdQe1BhItKaL = 0
which shows that the Jacobi identity is satised.
With these new elements we can continue the Hamiltonian analysis, where we get the
smeared Lorentz Constraint
Mr () =
Z
tK1 eLt tK1 eLt
+ 2D!1
a eAaKtL KL2=
Z
M rKL KL2
(3.103)
which acts on eN and !1aKL like local innitesimal transformations of gauge
[eN ;Mr ()] = MN eM ; [!1aKL;Mr ()] = D1aKL; (3.104)
leading to
[Mr () ; Drt (M)]
= 0 (3.105)
62
and to M rKL;Mr ()
= KNM
rNL + L NMrKN ; (3.106)
from which we get again the so(1; d 1) Lie algebra of the generators M rKL:
From a direct computation we have the following expression for the spatial di¤eomor-
phism hDr(sp)
!N;Dr(sp)
!N 0i
= Dr(sp)h!N ;!N 0i
(3.107)
The transformations induced by Dr(sp)!Non eN and !1aKL are given by
heN ;Dr(sp)
!Ni
= L!N(eN) ;
h!1aKL;Dr(sp)
!Ni
= L!N(!1aKL) : (3.108)
In view of these transformations we can deduce that Drt and M
rKL transform as scalar
densities of weight one, by the spatial di¤eomorphism, leading tohDr(sp)
!N; Dr
t (M)i= Dr
t
L!N(M)
;
hDr(sp)
!N;Mr ()
i=Mr
L!N():
(3.109)
As nal step, a direct calculation gives for the smeared scalar constraint
[Drt (M) ; D
rt (M
0)]=
Z
MeAaKbL (!1)abKL2
;
Z
M 0eAcNdM (!1)cdNM2
=
Z
Da
MeAaKbL
M 0eBhQcNdMe1BhQtKbL
(!1)cdNM2
Z
Dc
M 0eAcNdM
MeBhQaKbLe1BhQtNdM
(!1)abKL2
=
Z
MM 0Da
eAaKbL
eBhQcNdMe1BhQtKbL
(!1)cdNM2
_Z
MM 0Dc
eAcNdM
eBhQaKbLe1BhQtNdM
(!1)abKL2
= 0; (3.110)
where we have usedZ
@a (M) eAaKbLBhQcNdMBhQtKbL (!1)cdNM
2
=
Z
@a (M) eAaKbL1
2 (d 2)etQAbKhLBhQcNdM (!1)cdNM
2
= 0; (3.111)
consequence of AaKbLAtKcL = 0 and etQBhQcNdM = 0:
The previous Dirac brackets betweenMr () ; Dr(sp)!Nand Dr
t (M) show that the
algebra of the rst-class constraints is closed in terms of structure constants. If we search
63
for the Physical degrees of freedom, as we did previously, we then have to subtract from
d (5d 3), twice the number d (d+ 1) that corresponds to the number of rst-class con-straints and the number of second-class ones which is 2d (d 1) leading to
d (5d 3) 2d (d+ 1) 2d (d 1) = d (d 3) ; (3.112)
where we nd that the number of the phase space is d (d 3), corresponding exactly tothose of the Gravity at d-dimension.
The obtained Hamiltonian (3.97) propagates the phase space from the hypersurface
t0 to the hypersurface t0+t: This is done along the innitesimal time parameter t or
Mt depending on whether we use the scalar density Drt or its smeared version D
rt (M) in
the Hamiltonian equation. While the Lorentz constraint induces a gauge transformation
of the innitesimal parameter KL = t!tKL:
Finally, we reach the level of solving Hamilton equations in terms of Dirac brackets.
For the co-tetrad components ecN ;we get
@tecN (x) = [ecN ;HrT ] = D!1
a
eAaKbL
e1BcNtKbL
+eAaKtLe1BcNtKaM! M
t L + e1BcNtLaM! MtK
:
The rst term of the right hand side gives
D!1a
eAaKbL
e1BcNtKbL = e1BcNtKbLeBMaKbLD!1
a eM
= D!1a etM e1BcNtKbLeBdMaKbLD!1
a edM
= D!1a etM + e1BcNtKtLMKL; (3.113)
while a direct computation gives for the second term of the right hand side
eAaKtLe1BcNtKaM! M
t L + e1BcNtLaM! MtK
= ! M
tN ecM ; (3.114)
leading to
DteaN D!1a etN = BcNtKtLMKL: (3.115)
Since !tKL can be considered as Lagrange multiplier, in addition to the above equation
we have
BdMaKtLDaedM = D!1a
eAaKtL
=MKL = 0; (3.116)
leading to
DteaN D!1a etN = 0:
64
The solution of D!1a
eAaKtL
=MKL = 0 are
!saKL = eKraeL = eK (@aeL +
e
L) :
Once the solution !saKL is injected in the second equation, we got
DteaN D!s
a etN = @teaN + !tNMeMa @aetN !saNMeMt
= @teaN + !tNMeMa @aetN + (raeN) e
Me
Mt
= @teaN + !tNMeMa
ateM (3.117)
leading to !tNM = eNrteM = eN (@teM +
te
M) ; which exhibits solutions of the zero
torsion, condition to have an equivalence between the tetrad-connection Gravity and the
General Relativity.
3.6 conclusion
Through this work we have shown the possibility to construct a consistent Hamiltonian
formulation, without relying on the decomposition of the tetrad by the ADM foliation of
the space-time, or introducing extra term [12] to the original Palatini action, that contains
the Barbero-Immirzi parameter [18]. Indeed, this has been achieved by considering a
modied action where the non-dynamical connection !2aKL is xed to zero.
Unlike the other studies that lead to algebras of rst-class constraints with structure
functions, this presented algebra underlines the fact that the action is invariant under a
true Lie group where its generators are rst-class constraints.
As we have explained in the initial chapter, in the Hamiltonian analysis of the Einstein-
Hilbert action part (1.20), the main reason behind the presence of structure functions in
the algebra is the decomposition of the metric, or the co-tetrad in the case of the tetrad-
connection Gravity, following the ADM foliation of the space-time. This is not the case
here, where the scalar function M and the spatial vector!N are introduced as functions
testing the algebra, independently from the co-tetrad components and not as objects
resulting from the ADM decomposition.
We observe that all the reduced rst-class constraints are polynomial, but because
of the presence of e1 in (3.101) the phase space variables will obey a non-polynomial
Dirac bracket. Since the rank of the projector P KaL1NcM is d(d 1); it delivers an equal
number between the independent components of the dynamical connection and those of
the co-tetrad.
65
This allows to perform an invertible transformation
!1aKL ! PbN = eBbNtKaL!1aKL () !1aKL = PbNe1BbNtKaL;
to obtain the reduced phase space obeying the following commutation relations in terms
of Dirac brackets eaN (
!x ) ;PbM (!y )= ba
MN (!x !y ) ;
[eaN (!x ) ; ebM (!y )] = 0 and
PaN (!x ) ;PbM (!y )
= 0: (3.118)
By the substitution of !1aKL with PbNe1BbNtKaL; we get the following polynomial con-straints
M rKL = @aeAaKtL +1
2
PaKeLa PaLeKa
and
Dr(sp)!N=
Z
tNL!
N(etN) + PaNL!N (eaN)
:
However the scalar constraint becomes non-polynomial because of the quadratic term
of the dynamical connection !1aKL (e;P) = PbNe1BbNtKaL in the curvature (!)abKL :From the same calculation performed in (3.102), we deduce
[!1aKL (!x ) ; !1dPQ (!y )] =
PbNe1BbNtKaL;PcMe1BCMtPdQ
= 0;
where with the use of the new Dirac brackets introduced in (3.118), it leads to the same
rst-class constraints given previously.
We end this conclusion by giving a few words on the three dimension case where
eBbNcKaL vanishes, as a result of the antisymmetry of the spacial indices a; b and c;
MNM (!x ) ;MKL (!y )
= (NLM 0MK (!x ) + MKM 0NL (!x )
NKM 0ML (!x ) MLM 0NK (!x )) (!x !y )
PN (!x ) ;MKL (!y )
=NLPK (!x ) NKPL (!x )
(!x !y )
and PN (!x ) ; PM (!y )
= 0;
which exhibits the Lie algebra of the Poincaré group. Here tetN takes the role of the
innitesimal local translation parameter. Hence, when d = 3; we can either consider the
Poincaré group or the Lorentz and the di¤eomorphism ones. In this case the physical
degrees of freedom vanish leading to topological theory of Gravity. [52]
It is has been shown that the equivalence between the Poincaré group and Lorentz and
66
di¤eomorphism groups can be established O¤-shell and the symmetries of the Hamiltonian
formalism of tetrad Gravity at three dimension result from the rst class-constraints, these
results are di¤erent from the claiming found in [29] where it is stated that the di¤eomor-
phism invariant can not be a gauge symmetry derived from the rst-class constraints.
These results are also di¤erent from those of [52], where the equivalence between the
Poincaré group and Lorentz and di¤eomorphism groups is only established at the On-
shell level.
for d > 4, the smeared rst-class constraint PN can be decomposed into
P (N) =
Z
NNeBtNaKbLabKL2
= Dt (M) +D(sp)!N+Mr ()
whereD(sp)!N=
Z
eAaKtLL!N(!1aKL) ;Mr () =
Z
D!1a eAaKtLKL; NN is Lorentzian
vector, N = NNetN ; KL = NNe
aN!1aKL and NNeaN are the (d 1) components of thevector
!N tangent to the hypersurfaces t:
If we take the Dirac brackets between the smeared translation generators, we can
see that it does not vanish, but it rather gives a sum which is linear in rst-class con-
straints with structure functions and other quadratic. This is mainly due to the fact
that the smearing function N;!N and KL are dened in way that they depend on the
co-tetrad and the connection. This shows that P (N) does not satisfy the Dirac bracketPN (!x ) ; PM (!y )
= 0 and hence P (N) can not correspond to translation transforma-
tions in the Poincaré group. Therefore, only the Lorentz group and the di¤eomorphisms
are the symmetries that occur on the system, not the translation part of the Poincaré
group, which is not the case in [53].
67
Chapter 4
The Hamiltonian Formalism of
Tetrad-Gravity Coupled to Fermions
In this nal chapter, we complete the analysis by considering the precedent results detailed
in the third chapter, which were published in [30], and add fermions. We consider the
action given in (3.79) where the connection is divided into two distinct parts, where the
non-dynamical part is xed to zero, and couple the tetrad-Gravity action to the Dirac
one. We apply then the Dirac procedure on this new completed action, where all the steps
of the Dirac program are respected. The results of this chapter appear [31].
4.1 The constraints in presence of fermions
Our main task is to study the new constraints once that the tetrad-Gravity is coupled
to fermions. We proceed by adding the fermionic part to the functional action of the
tetrad-Gravity, where the Space-time manifoldM is of ddimension
S(e; !;) = S(e; !) + Sf (e; !;): (4.1)
The fermionic part is expressed by
Sf (e; !;) =
ZM
eeIi
2 ID
i
2(D) I
em (4.2)
It contains the Dirac spinors, and = y 0. This time the study includes massive
fermions. We retrieve the Dirac matrices I ; of dimension 22d22
for d even and 2
2d32
for d odd, satisfying the relation
I J + J I = 2IJ (4.3)
68
more informations on the Cli¤ord algebra is to be found in the Appendix C, the relations
given there are still valid in d dimension. We also mention that we keep using the A; Band C functions that have been introduced in the precedent chapter.We start by the full action, that combines the Gravity and the fermionic parts, which
is rewritten in terms of temporal and spatial components
S(e; !) =
ZM
eAaKtL (@t!1aKL D1a!tKL) eAaKbL
1abKL2
ddx
+
ZM
eetIi
2 I@t
i
2(@t) I
+
ZM
eeaIi
2
ID1a (D1a) I
em
+
ZM
eetIi
2
( I
KL
2+KL
2 I)
!tKL (4.4)
where the non-dynamical part of the connection is xed to zero.
The Lagrange density (4.4) is invariant under the innitesimal gauge transformations
eK = NK eN , !tNM = DtNM , !1aKL = D!1
1a KL,
=KL
2KL and =
KL
2KL (4.5)
subject to the condition
D!12a KL = 0, and @2aKL = 0. (4.6)
We can have directly from action (4.4) the momentums N , P tKL and PaKL1 conjugate
to the co-tetrad eN and to the so (1; d 1) connection !tKL and !1aKL respectively,
N(x) =S(e; !)
@teN(x)= 0, PaKL1 (x) =
S(e; !)
@t!1aKL(x)= eAaKtL(x) (4.7)
and
P tKL(x) = S(e; !)
@t!tKL(x)= 0. (4.8)
By the left functional derivatives we compute the fermionic momentums and
=i
2eetN N and =
i
2eetN N. (4.9)
conjugate to and respectively.
The bosonic phase space elements obey the following non-zero fundamental Poisson
brackets at the xed time
69
eI(!x ); N(!y )
=
NI (!x !y );
!1aIJ(!x );PbKL1 (!y ))
= P KbL
1IaJ (!x !y ): (4.10)
Whereas the fermionic elds, they are subject to the following non-zero fundamental
anticommutating Poisson brackets.
fA(!x );B(!y )g+ = AB(!x !y ) (4.11)
and A(!x );B(!y )
+= AB(
!x !y ). (4.12)
Where the Dirac indices take the values A;B 2n1; 2; :::; 2
2d22
ofor d even and A;B 2n
1; 2; :::; 22d32
ofor d odd.
We can calculate the following primary bosonic constraints, as we have done in the
chapter 3
tN ' 0; P tKL ' 0 ;
bN ' 0, CaKL1 = PaKL1 eAaKtL ' 0 (4.13)
In addition, the fermionic ones that emerge from (4.9)
C = i
2eetN N , C =
i
2eetN N; (4.14)
which are subject to the following non-zero Poisson brackets when they are taken with
the other constraints, as it is written below
aN(!x ); CbKL1 (!y )
= eBaNtKbL(!x !y ), (4.15)
aN(!x ); C(!y )
=i
2eAaNtM(!y ) M(!x !y ), (4.16)
aN(!x ); C(!y )
=i
2eAaNtM M(
!y )(!x !y ) (4.17)
and CA(!x ); CB(!y )
+= ieetI IBA. (4.18)
70
Finally, the Total Hamiltonian is given by
HT =Z
(tNtN + P tKLAtKL2
+ bNbN + CaKL1
A1aKL2
+ C C) +H0 (4.19)
where
H0 =
Z
(eAaKbL1abKL2
+ eAaKtLD1a!tKL)
Z
eeaKi
2 KD1a
i
2(D1a) K
+
Z
em
i8
Z
eetI( I K L + K L I)!tKL:
In addition to the bosonic Lagrange multipliers tN , AtKL; bN and A1aKL, we retrievethe fermionic ones and related the constraints (4.14).
4.2 The consistency of The Hamiltonian Formalism
Now that we have obtained the primary constraints and have constructed the rst version
of the total Hamiltonian of the Tetrad-Gravity coupled to fermions, we proceed by check-
ing the consistency of these constraints. We write the time evolution of each constraints
that has been listed previously by using the fundamental Poisson brackets given in (4.10),
(4.11) and (4.12).
A direct computation for each constraints will deliver
tN ;HT
= 1
2eBtNaKbL1abKL +
i
2eAtNaK
KD1a (D1a) K
eetNm = PN ' 0; (4.20)
P tKL;HT
= DaeA
aKtL +i
8eetI( I
K L + K L I)
= MKL ' 0; (4.21)
71
bN ;HT
= eBbNtKaL
A1aKL2D1a!tKL
eBbNaKcL1acKL
2
+i
2eAbNtK( K K)
+i
2eAbNaK
KD1a (D1a) K
eebNm
+i
2eAbNtI
( I
KL
2+KL
2 I)
!tKL = RbN = 0, (4.22)
and
CaKL1 ;HT
= eBbNtKaL
bN + ! M
tN ebM+D1ceA
cKaL
+i
2(eeaI( I
KL
2+KL
2 I))1 = RaKL = 0. (4.23)
These four relations give the modication of the constraints used in the chapter 3 by the
presence of Dirac elds. However, we have to check the consistency of (4.14), where we
nd
fC;HTg = Cfer ' 0 = ieetN N i
2eAbNtMbN M em
Da(i
2eeaM) M
i
2eeaM(Da) M
+i
2eetM( M
KL
2+KL
2 M)!tKL (4.24)
and
C;HT
= Cfer ' 0 = ieetN N
i
2eAbNtMbN M+ em
Da(i
2eeaI I)
i
2eeaI I(Da)
i2eetI( I
KL
2+KL
2 I)!tKL (4.25)
The evolutions of (4.20) and (4.21) lead to secondary constraints, while the other expres-
sions will lead to equations imposing conditions on the Lagrange multipliers tN , AtKL;bN , A1aKL and also on the fermionic Lagrange multiplier elds.Now we have to check the consistency of the secondary constraints. We use the same
useful relation where for any antisymmetric tensor T KL = T LK ; we have
T KN eANL + T LNeAKN =1
2
eK eB
LNMTNM eLeBKNMTNM. (4.26)
72
Hence, we get the evolution of the constraints MKL
MKL;HT
= 1
2
eKb RbL eLbRbK
+1
2(KL
2Cfer + Cfer
KL
2)
+1
2
eKt P
L eLt PK! Kt NM
NL + ! Lt NM
KN
(4.27)
which is consistent.
When RbL; Cfer and Cfer are satised; MKL is reduced to
MKL;HT
=1
2
eKt P
L eLt PK! Kt NM
NL + ! Lt NM
KN' 0: (4.28)
As it has been done in the chapter 3, we project the constraint (4.20) over the temporal
component of the co-tetrad etN to dene
Dt = etNPN = eAaKbLabKL
2+e
2eaK(i KDa iDa K) em: (4.29)
While its smeared spatial projection is given by
Dsp(!N ) =
Z
Na(eaNPN + !aKLM
KL)
=
Z
eAaKtLL!N(!aKL) +
i
2eetK( KL!N L!N K): (4.30)
L!N(!aKL) = N b@b (!aKL) + @a
N b!aKL is the Lie derivative along the vector eld
!N of (d 1) dimension. Its e¤ects on the spinors is given by L!
N = N b@b and
L!N = N b@b; which show that the Dirac elds are treated like scalars under spatial
di¤eomorphism.
Now if we compute the evolution of Dt; we obtain
fDt;HTg = PNN
eAaKbLDaA1aKL2
+i
2eeaI( I
KL
2+KL
2 I)
A1aKL2
eBcNaKbLabKL2
cN +i
2eAcNaI( IDaDa I)cN
eecNmcN+i
2eeaI( IDa + IDa)
i
2eeaI(Da I +Da I)
em em: (4.31)
73
with the use of
etNRcN = eAcKaL(A1aKL2Da!tKL)
i
2eecK( K K)
i2eecI
( I
KL
2+KL
2 I)
!tKL (4.32)
it can be reduced to
fDt;HTg =N + ! M
tN etMPN Dc
etNRcN
+RaKL
A1aKL2Da!tKL
+RcN
cN + ! M
tN ecM
+Cfer +
KL
2!tKL
KL
2!tKL
Cfer:
Then if the constraints RcN ;RaKL; Cfer and Cfer are satised, this bracket reduces to itsnal form, where
fDt;HTg =N + ! M
tN etMPN ' 0: (4.33)
For the consistency of Dsp(!N ), we get
nDsp(!N );HT
o=
Z
(PNL!N(etN) +MKLL!
N(!tKL))RaKLL!N (!aKL)
RcNL!N(ecN) CferL!N+ L!N ()Cfer
leading to nDsp(!N );HT
o=
Z
(PNL!N(etN) +MKLL!
N(!tKL)):
Once N is xed by
N = ! MtN etM =) tNN =
1
2(NeMt MeNt )!tNM ; (4.34)
the Poisson bracket (4.33) becomes strong equality.
From the time evolution of the Primary and the emerging secondary constraints
MKL;Dsp(!N ) and Dt; we have shown that the total Hamiltonian HT is consistent if
the constraints RcN ;RaKL; Cfer and Cfer are satised.Now we have to solve the other consistency equations that involve the Lagrange mul-
tipliers, we start by the equation (4.23), where we have to multiply it by BbNtKaL and use
BbNtKaLBcMtKaL = cbMN (4.35)
74
to get
bN = ! MtN ebM +DbetN e1BbNtKtLMKL
i4AbKtL( N
KL
2+KL
2 N)
' ! MtN ebM +DbetN
i
4AbKtL( N
KL
2+KL
2 N).
Whereas the solution of (4.22) is obtained as
1
2A1aKL = D1a!tKL BbNtKaLBbNcPdQcdPQ
2
+i
2BbNtKaLA
bNtM( M M)BbNtKaLebNm
+i
2BbNtKaLA
bNcM( MDcDc M)
+i
2BbNtKaLA
bNtM( MPQ
2+PQ
2 M)!tPQ: (4.36)
For the fermionic Lagrange multipliers and , with the use of
i2eAbNtMbN =
i
2!Mt Ne
tN +Db(i
2ebM) +
i
2MNMetN
' i
2!Mt Ne
tN +Db(i
2ebM);
and by multiplying on the left the equation (4.24) with i e1
gttetM M ; we get
= 14 K L!tKL
i
gttmetM M
1gttebKetM M KDb: (4.37)
On the other side, for we have to multiply (4.25) with i e1
gttetM M to obtain
=1
4 K L!tKL +
i
gttmetM M
1gttebKetMDb K M (4.38)
where gtt = etNetN :
75
If we substitute the expressions of and into (4.36), we get
1
2A1aKL = D1a!tKL BbNtKaL(BbNcPdQcdPQ
2
i2AbNcM
MDcDc M
+i
2AbNtP ecMetI( P I MDcDc M I P)
+m
gttebP etP e
tN): (4.39)
4.3 Dening the total Hamiltonian and the constraints
algebra
Now that we have dened all the constraints and solve the Lagrange multiplier equations,
it remains to construct the total Hamiltonian by replacing these solutions into (4.19) to
obtain
H0T =
Z
P tKLAtKL2
Z
(1
2(KeL LeK ) +Da(C
aKL1 + eAaKtL)
+(C +i
2eetI I)
KL
2 +
KL
2(C +
i
2eetI I))!tKL
+
Z
(eAaKbL1abKL2
i
2eeaK( KDaDa K) + em
+
Z
aNDaetN
i
4AaKtL( N
KL
2+KL
2 N)
Z
CaKL1 BbNtKaL(BbNcPdQcdPQ
2 i
2AbNcM( MDcDc M)
+i
2gttetP ecIAbNtM( M P IDcDc I P M)
+1
gttetMe
bMetNm)Z
mietM
gtt(C M+ MC)
Z
1
gttetMeaI(C M IDaDa I MC): (4.40)
76
We have to check again the consistency to prove that we are using a coherent Hamiltonian.
For this, we start by computing the Poisson bracket of P tKL with H0T as
P tKL;H0T
= Da(C
aKL1 + eAaKtL) +
1
2(KeL LeK ))
+(C +i
2etI I)
KL
2
+KL
2(C +
i
2etI I)
=1
2M 0KL ' 0 (4.41)
WhereM 0KL takes the place ofMKL in the analysis. By using the relation etN etN
(BbMtKaL) =
BbMtKaL; and
etN
tN ;
eMtgtt
= e
tM
gttand etN
tN ; eMb
= 0
we can rewrite the Hamiltonian (4.40) as
H0T =Z
(1
2P tKLAtKL D0t M 0KL!tKL
2) (4.42)
Where D0t is the projection of the evolution of tN , meaning that
etNtN ;H0T
= etN!NtMtM + P 0NetN
= etN!NtMtM +D0t ' 0 =) D0t ' 0: (4.43)
by using ecN etN
(e1BbMtKaL) = e1BbMcKaL; CaKL1 BbMcKaL = 0 andZ
CaKL1 BbMtKaLNcecNCtNbMePdQedPQ
2
= Z
CaKL1 L!
N(!aKL) +Da
eAaKtL
N c!cKL
Z
N cCaKL1 e1BbMtKaL BtMePdQedPQ2
;
77
we get for the projection ecNP 0N ; the following smeared constraintZ
N c(ecNP0N +
!cKL2
M 0KL + aN(DaecN DceaN))
i4
Z
N caNAaKcL( NKL
2+KL
2 N)
Z
N cCaKLe1BcMtKaLPM
= Z
(aML!N(eaM) + (C
aKL + eAaKtL)L!N(!aKL))
:Z
(C +
i
2eetN N)L!N () + L!N ()(C +
i
2eetN N)
: (4.44)
This can be completed by adding tNL!N(etN) to get
D0sp(!N ) =
Z
(ML!N(eM) + (C
aKL + eAaKtL)L!N(!aKL))
+
Z
((C +i
2eetN N)L!N () + L!N ()(C +
i
2eetN N)): (4.45)
As we have seen, D0(sp)(!N ) represents the generator of the spatial di¤eomorphisms. The
transformations on the co-tetrad components eN and the connection !aNM areneN ;D0sp(
!N )o= L!
N(eN),
n!aNM ;D0sp(
!N )o= L!
N(!aNM) (4.46)
and also for the spinors and , we haven;D0sp(
!N )o= L!
N(),
n;D0sp(
!N )o= L!
N(): (4.47)
It also drags the primary constraints N ; CaNM1 ; C and C along the vector eld!N , asn
N ;D0sp(!N )o= L!
N(N),
nCaNM ;D0sp(
!N )o= L!
N(CaNM); (4.48)
nC;D0sp(
!N )o= L!
N(C) and
nC;D0sp(
!N )o= L!
N(C); (4.49)
which shows that the Poisson bracketsD0sp(!N ) with the primary constraints vanish weakly.
We also deduce the transformations of D0t induced by D0sp(!N ), showing that they are
78
treated as scalar densities of weight onenD0t;D0sp(
!N )o
= L!N(D0t) = @c(N
cD0t)
=)nD0sp(!N );D0t(M)
o= D0t(L!N (M))
and for MKL we havenMKL;D0sp(
!N )o
= L!N(MKL) = @c(N
cMKL)
=)nD0sp(!N );M()
o=M(L!
N());
Where D0t(M) =RMD0t and M() =
RM
0KL KL2are the smeared constraints. The
calculations performed earlier show that the constraint D0sp(!N ) is of rst-class type.
M() acts in terms of Poisson brackets on the co-tetrad components eN ; the connec-
tion !1aNM ; the spinors and , as the generator of the innitesimal Lorentz transfor-
mation group, where
feN ;M()g = LN eL, f!1aNM ;M()g = D1a(NM), (4.50)
f;M()g = KL
2KL,
;M()
=
KL
2KL: (4.51)
M() acts on the primary constraints aN and CaNM1 as
N ;M()
= NL
L,CaNM1 ;M()
= NLC
aLM1 + MLC
aNL1 ; (4.52)
whereas on the fermionic primary constraints, it acts as
fC;M()g = CKL
2KL,
C;M()
=KL
2CKL: (4.53)
Showing thatM() Poisson brackets, with the primary constraints aN ; CaKL1 ; C and C,
vanish weakly.
In addition M 0NM transforms as a covariant tensor
M 0NM ;M()
= NLM
0LM + MLM0NL;
leading to the so(1; d 1) Lie algebra
M 0NM(!x );M 0KL(!y )
D= (NLM 0MK(!x ) + MKM 0NL(!x )
NKM 0ML(!x ) MLM 0NK(!x ))(!x !y ):(4.54)
79
We point out thatM() does not act on the space-time indices and treats them as scalars,
which makes easy to compute the transformations that occur on D0t(!x ); leading to
fD0t(!x );M()g = 0 =) fD0t(M);M()g = 0, (4.55)
which makes M 0NM a rst-class constraint.
The next step is to investigate the consistency of the constraint D0t(M): We havealready shown that the Poisson brackets of D0t(M) with the constraints D0sp(
!N ) and
M() vanish weakly: Let us now focus on the computing of the brackets that involve the
remaining primary constraints. We start by the constraints cN ; where straightforward
computation givesncN(!x );D0
t(!y )o' +(BcNaKbLabKL
2
i2AcNaK( KDaDa K) + ee
cN)(!x !y )
eBcNtKaLBbMtKaL(BbMcPdQcdPQ
2 i
2AbMdP ( PDdDd P)
+i
2gttetP ecIAbMtQ( Q P IDcDc I P Q)
+m1
gttetIe
bIetM)(!x !y )m 1
gttetKAcNtK(!x !y )
+1
gttetMeaI
i
2eAcNtK( K M IDaDa I M K)(
!x !y ): (4.56)
We have only kept terms that are not proportional to the primary constraints. At the
end,cN(!x );D0
t(!y )vanishes weakly by using BbMtKaLBcNtKaL = cb
NM .
For the Poisson brackets of the constraint CaKL1 with D0t, we getn
CaKL1 (!x );D0
t(!y )o' (DbeAbKaL)1(!x !y )
i2(eaN( N
KL
2+KL
2 N))1(
!x !y )
eBcNtKaLDcetN(!x !y )
+i
4eBcNtKaLAcP tQ( N
PQ
2+PQ
2 N))(
!x !y ). (4.57)
The rst term of the right hand side of the precedent relation gives
(DbeAbKaL)1= BcMtPdQBcMtKalDbeAbPdQ
= BcMtPdQBcMtKal(eBtIbPdQDbetI + eBeIbPdQDbeeI)
= eBcMtKalDcetM BcMtPtQBcMtKalDbeAbP tQ (4.58)
80
The relation
BcMtPdQedN = (BcMtPtQe
tN +1
2(AcMtP
d 2 NQ +AcQtMd 2
NP +AcP tQNM);
deduced from the denition of the B matrix and
1
2(AcMtP
d 2 NQ +
AcQtMd 2
NP )( N
PQ
2+PQ
2 N) = 0
implies for the second term
i2(eaN( N
K L + K L N))1
= i2BcMtPdQB
cMtKaledN( NPQ
2+PQ
2 N)
= i2BcMtPtQB
cMtKaletN( NPQ
2+PQ
2 N)
i4eBcMtKaLAcP tQ( M
PQ
2+PQ
2 M)); (4.59)
leading to nCaKL1 (!x );D0
t(!y )o
' BcMtPtQBcMtKal(DbeA
bP tQ +i
2(etN( N
PQ
2+PQ
2 N))(
!x !y )
' BcMtPtQBcMtKalMKL(!x !y ) ' 0. (4.60)
For the other fermionic constraint, we get
nC(!x );D0
t(!y )o'
Da(
i
2eaK) +
i
2eaKDa) K + em
(!x !y )
i
2eAcNtK K(DcetN
i4AcP tQ( N
PQ
2+PQ
2 N))(
!x !y ):
i
gttetM(!y ) M
C (!x ) ; C (!y )
+i
gttetMeaIDa(
!y ) I MC (!x ) ; C (!y )
(4.61)
The relations listed in the following
eAcNtKDcetN = Dc(eAcNtKetN)Dc(eAcNtK)etN= Dc(ee
cK)Dc(eAcNtK)etN ,
81
AcNtKAcP tQ = NP etKetQ NQetKetP KP etNetQ + KQe
tNetP ;
(NP etKetQ NQetKetP )( N
PQ
2+PQ
2 N) = 0;
with etMetN M N = gtt andCA(!x ); CB(!y )
+= ieetI IBA, lead ton
C(!x );Dft (!y )o' i
2(Dc(eA
cNtK)etN
+i
2etP( P
NK
2+NK
2 P ))etN K(
!x !y )
' i2MNKetN K(
!x !y ) ' O. (4.62)
The same computation showsnC(!x );D0
t(!y )o' i
2MNK KetN(
!x !y ) ' O: (4.63)
A straightforward computations givesnD0
t(M);D0
t(M0)o=
ZM
(M@aM
0 M 0@aM)F a (C1) +Ga
C;C
' 0 (4.64)
where
F a (C1) = 1
2 (d 2) gttCaKL1 AbKtLebQ(metQ
edIetP i2( Q P IDdDd I P Q))
+1
2 (d 2) (gtt)2CbKL1 ebKe
aJetP etM(m( L P J M + M J P L)
edIi L P J M IDdDd I M J P L
) (4.65)
and
GaC;C
=
1
(gtt)2etNetMeaK(im
C N K M+ N K MC
+edI
C N K M IDdDd I N K MC
: (4.66)
Compared to the results obtained in the vacuum case, where the Poisson brackets
between the smeared scalar constraints strongly vanish, (4.64) vanishes only weakly. The
above Poisson brackets have shown that the total Hamiltonian (4.40), even in the pres-
ence of fermionic matter, is consistent. The constraints of the system are composed of
rst-class constraints which are tN ;P tNM ; the Lorentz constraintM () ; the spatial dif-
feomorphism constraint D(sp)!Nand nally the scalar constraint Dt (M) : In addition,
we have the second-class constraints composed by the bosonic ones aN and CaNM and
82
the fermionic constraints C and C:
For the physical degrees of freedom per point in space-time, as usual, we subtract
the number of the second-class constraints and twice the number of the rst-class con-
straints from the degrees of freedom of the xed phase-space which is d (5d 3) : Hencethe computation leads to
d(5d 3) 2d(d 1) 2d(d+ 1) = d(d 3) (4.67)
which corresponds to the physical phase space degrees of freedom of the theory of General
Relativity at d-dimension. Whereas for the fermionic phase space, we have either 42d2
2
minus the number of the second-class fermionic constraints 2
2d2
2
for d even, or 4
2d3
2
minus 2
2d3
2
for d odd, divided by 2 it leads to the physical degrees of freedom of the
Dirac spinors.
4.4 The algebra of constraints in terms of Diracs
bracket
As the Dirac procedure requires, we have to use the Dirac brackets to reduce the phase
space and then eliminate the second-class constraints aN ; CaKL1 ; C and C and use the
strong equality. Whereas the algebra of the rst-class must be computed in terms of the
projected Dirac brackets dened in terms of the projected Poisson brackets via
[A;B] = fA;Bg fA;Cig fCi; Cjg1 fCj; Bg (4.68)
where Ci =bN ; CaKL1 ; C; C
: The non-zero elements of the inverse matrix fCi; Cjg1
are given by
bN(!x ); CaKL1 (!y )
1= e1BbNtKal(!x !y ) =
CaKL1 (!x ); bN(!y )
1, (4.69)
bN(!x ); C(!y )
1= 0 =
bN(!x ); C(!y )
1, (4.70)
CA(!x ); CB(!y )
1+=ie1
gttetI IAB(
!x !y ) =CB(!y ); CA(!x )
1+, (4.71)
83
CA(!x ); CaKL1 (!y )
1= 1
2
egtt1
etIBbNtKaLAbNtM( I M)A(!x !y )
=CaKL1 (!y ); CA(!x )
1(4.72)
CA(!x ); CaKL1 (!y )
1=
1
2
egtt1
etIBbNtKalAbNtM( M I)A(!x !y )
=CaKL1 (!y ); CA(!x )
1(4.73)
and hCaKL1 (!x ); CbPQ1 (!y )
i= i
4
egtt1
etIBdNtKaLAdNtMBcJtPbQAcJtR
( M I R R I M)(!x !y ). (4.74)
When we insert bN = 0; CaKL1 = 0; C = 0 and C = 0; the total Hamiltonian is reduced
to
HrT = Z
(Drt +M rKL!tKL2) (4.75)
where
M rKL = (tKeLt tLeKt ) + 2D!1a (eAaKtL) + ieetI( I
KL
2+KL
2 I) (4.76)
are the reduced Lorentz constraints and
Drt = eAaKbLabKL2
+i
2eeaK( KDaDa K) em. (4.77)
is the reduced scalar constraint.
The di¤eomorphism constraint is given by
Drsp(!N ) =
Z
tNL!N(etN) + eAaKtLL!
N(!1aKL) +
i
2eetK( KL!N L!N K).
The Hamiltonian (4.75) is dened on the reduced phase space eaN ; !1aKL;;; etKand tK endowed with the following non-zero Dirac brackets given in terms of the above
inverse elements fCi; Cjg1 as
etN(!x ); tM(!y )
= MN (
!x !y ) (4.78)
[eaN(!x ); !1bKL(!y )] = e1BaNtKbL(
!x !y ), (4.79)
84
[!aKL(!x ); !1bPQ(!y )] =
nCaKL1 (!x ); CbPQ1 (!y )
o1, (4.80)
[A(!x ); !1bPQ(!y )] =
CA(!x ); CaKL1 (!y )
1= [!1bPQ(
!y );A(!x )] , (4.81)
A(!x ); !1bPQ(!y )
=CA(!x ); CaKL1 (!y )
1=!1bPQ(
!y );A(!x ); (4.82)
and
A(!x );B(!y )
==
CA(!x ); CB(!y )
1+=B(!y );A(!x )
: (4.83)
Using these Dirac brackets between the elements of the reduced phase space gives, after
a direct computation, the Dirac bracket between the spatial di¤eomorphism constraints
as hDrsp(!N );Drsp(
!N 0)i= Drsp(
h!N ;!N 0i). (4.84)
The reduced space phase elements transform under spatial di¤eomorphism as
eN =heN ;Drsp(
!N )i= L!
NeN , (4.85)
!aKL =h!aKL;Drsp(
!N )i= L
1!N!aKL, (4.86)
A =hA;Drsp(
!N )i= L!
NA, (4.87)
and
A =hA;Dr(sp)(
!N )i= L!
NA; (4.88)
from which we deduce that the scalar and the Lorentz constraints are treated as densities
scalars of weight onehDr(sp)(
!N );Drt
i= L!
N(Drt ) = @a(NaDrt )
=)hDr(sp)(
!N );Drt (M)
i= Drt (L!N (M)) (4.89)
and hDr(sp)(
!N );M rKL
i= L!
NM rKL = @a(NaM rKL)
=)hDr(sp)(
!N );Mr()
i=Mr(L!
N()). (4.90)
85
The reduced phase space elements transform underMr() as innitesimal gauge trans-
formations, where we have
eN = [eN ;Mr()] = MN eM , (4.91)
!1aKL = [!1aKL;Mr()] = D1aKL, (4.92)
= [;Mr()] =KL
2KL, (4.93)
and
=;Mr()
=
KL
2KL: (4.94)
While M rKL transforms as contravariant tensor
M rKL;Mr()
= KPM
PL + LPMKP
leading for arbitrary KL (x) ; to the Lie algebra of the Lorentz group
M rKL(!x );M rNM(!y )
=
KMM rLN(!x ) + LNM rKM(!x ) (N !M)
(!x !y ). (4.95)
From the transformations of the reduced phase space the smeared scalar constraint Drt (M)transforms under the condition imposed on KL as scalar
fMr();Drt (M)gD = 0.
Now we have to detail some computations to obtain the Dirac bracket between the
smeared scalar constraint.We start byZMeAaKbL1abKL
2;
ZM 0eAcNdM 1cdNM
2
=
Z@aM@cM
0 i
4gtteeaKecNetI( K I N N I K)Z
((M@aM0 M 0@aM)
i
4gtt(etIeaKBdRtNdMAdRtS
Dc(eAcNdM)( K I S S I K)); (4.96)
86
ZMeAaKbL1abKL
2;
ZM 0 i
2eecN( NDcDc N) +M
0em
+
ZM i
2eeaK( KDaDa K) +Mem;
ZM 0eAcNdM 1cdNM
2
=
ZMeAaKbL
1abKL2
;
ZM 0 i
2eecN( NDcDc N) +M
0em
;
(M , M 0) =
Zi
2gtt@aM@cM
0eaKecNetI( K I N N I K)
= (M@aM0 M 0@aM)(
em
gttetMe
aM
1
8gtteeaKebLetI( L
PQ
2+PQ
2 L)BdRtPbQAdRtM( K I M M I K)
1
4gtteeaKebLetI( K I LDbDb L I K)
1
4gtteaKetI( K I LDb(ee
bL)Db(eebL) L I K)
Z((M@aM
0 M 0@aM))i
4gttetIeaKBdRtPbQAdRtM
Dc
eAcNdM
( K I S S I K));
and ZM i
2eeaK( KDaDa K);
ZM 0 i
2eecN(i NDc iDc N)
+
ZM i
2eeaK( KDaDa K);
ZM 0em
+
ZMem;
ZM 0 i
2eecN(i NDc iDc N)
+
ZMem;
ZM 0em
= (M@aM
0 M 0@aM)(i
4gtteeaKebLetI( K I LDbDb L I K)
+i
4gtteaKetI( K I LDb(ee
bL)Db(eebL) L I K)
em
gttetMe
aM
+1
8gtt(eaKebLetI( L
P Q
4+ P Q
4 L)
BdRtPbQAdRtM( K I M M I K)))
Z
i
4gtt@aM@cM
0eeaKecNetI( K I N N I K)
87
leading to
[Drt (M);Drt (M 0)]= 0
Hence, the above relations show that the algebra of the reduced rst-class constraints
closes with structure functions calculated via the Dirac brackets. This is the opposite of
the results obtained in the chapter 3, and the Physical degrees of freedom correspond to
those of General Relativity.
We notice that all the reduced rst-class constraints are polynomial, but not the Dirac
brackets of the reduced phase space elements. Furthermore, the Dirac brackets between
the spatial connection with itself, or taken with the fermionic eld do not vanish.
We can achieve the following canonical transformations
eaN ! eaN
!1aKL ! PcNe; !1;;
= eBcNtKaL!1aKL +
i
2eAcNtK K
!
! e;
= ieetN N (4.97)
We recall that this is possible because the number of independent components of the
spatial connection equals the number of the co-tetrad ones. This is due to the rank of the
projector P RdS1PbQ which is d (d 1) :
These listed transformations are consequences of the new symplectic form of the action
containing time derivativesZM
eAaNtK@t!1aKL +
i
2eetI
I@t (@t) I
=
ZM
BcNtKaL!1aKL +
i
2eAcNtK K
@tecN + ieetN N@t:
The primary constraints become
cN ! CaN = PaN eBcNtKaL!1aKL +i
2eAcNtK K;
CaKL1 ! CaKL1 = PaKL1 ' 0;
C ! C = ieetI ' 0;
and
C ! C = ' 0: (4.98)
However the super matrix fCi; Cjg elements are the same of the ones given earlier, hence
88
the inverse matrix fCi; Cjg1 keeps the same form.This new reduced phase space is canonical, and the non-zero Dirac brackets are
eaN (
!x ) ;PbM (!y )
= baMN (!x !y )
[A (!x ) ;B (!y )] = AB (
!x !y ) (4.99)
If we inverse the transformations (4.97), we can have !1aKL and as functions of the
canonical phase space
!1aKL (e;P ;;) = PbMe1BbMtKaL BbMtKaLi
2AbMtN N
and
(e;) = iegtt1
etI I
By inserting these denitions of !1aKL and back in the formalism, we can have poly-
nomial constraints of rst-class of the form
Dr(sp)(!N ) =
Z
tKL!
N(etK) + PaKL!N (eaK) + L!N ()
(4.100)
and
M () =
Z
@aeAaKtL
+1
2
PaKeLa PaLeKa
+
KL
2
KL
=
Z
M rKL KL2; (4.101)
contrary to the scalar constraint that become rather non-polynomial complicated one.
89
Conclusion
This part summarizes the main results that has been detailed in the third and the fourth
chapters and published in [30] and [31]. Through these chapters, we have given in details
the procedure that leads to the nal results establishing a closed algebra of the tetrad-
Gravity constraints on structure constants rather than structure functions. This study
has been conducted in both vacuum and when Gravity is coupled to fermionic matter. In
the vacuum case, the results are obtained without introducing the dimensionless Barbero-
Immirzi parameter [18], or the use of the ADM decomposition on the tetrad, so the
shift vector!N and the lapse function M are simply introduced to test the smeared
constraints and their Poisson brackets. Hence M and!N are not considered part of the
co-tetrad. In the new procedure presented here, we split the Lorentz spin connection
!aKL into dynamical and non-dynamical parts, where the second part is set to zero. Such
proceeding has allowed to cancel the problematic condition (3.43), that emerges while
one is computing and detailing the Dirac program, and consequently the new obtained
action has been taken as the starting point to perform again the Hamiltonian analysis at
d-dimension. At the end of the procedure and the application of this program, the reduced
rst-class constraints that we have obtained are polynomials and more importantly they
obey an algebra that closes on structure constants, however the obtained phase space Dirac
brackets are not polynomial and they rather contain the inverse of the tetrad determinant
e1: Nevertheless, these brackets can be transformed by an invertible transformation by
substituting the dynamical connection but which also leads to non-polynomial scalar
constraint.
We recall that the study has also been performed for the particular three dimensional
case, d = 3; where the object eBbNaKcL vanishes and as consequence the projected part ofthe connection !2aKL completely disappears from the action. Thus, all the previous results
are maintained. In addition, the rst-class constraints PN can be taken into account
instead of the projected ones Dt (M) and Dsp!N: The remaining rst-class constraints
which are PN and MKL will form an algebra that corresponds to the Lie algebra of the
Poincaré group. In this case, tetN is taken as the innitesimal local translation. However,
this isnt the same for d 4;
90
where PN will not satisfy the Poisson bracket fP (N) ; P (M)g = 0:We end this part,by recalling that the three dimension case has been investigated in [52], and also in [29],
where in the last cited article, the obtained results are di¤erent of ours where it has been
claimed that the di¤eomorphism transformations do not form a gauge symmetry derived
from the rst-class constraints of the tetrad Gravity, neither in 3-dimension case or in
higher dimension.
The other study achieved in this thesis is to consider the previous Hamiltonian analysis
at ddimension and take into account the Dirac action. This has shown that tetrad-Gravity coupled with fermions, when the non-dynamical part of the connection is set to
zero, will also lead to consistent Hamiltonian formalism as in the vacuum case. Therefore,
we were also been able to reproduce an algebra of reduced rst-class constraints closing
over structure constants, once the second-class constraints are solved. At the end, we
achieved canonical transformations leading to rst-class constraints of polynomial form
except for the scalar constraint.
The presence of structure functions, which are consequence of the ADM decompo-
sition, has been criticized in [41], where it has been said that with their presence the
di¤eomorphism transformations cannot be considered as a true group. In our approach,
we have avoid the ADM decomposition of the metric and importantly we have kept the
internal group, which is the Lorentz group, fully covariant. However keeping the internal
group covariant may complicate the following steps since the group is no compact and
hence it rises the di¢ culty of constructing an appropriate Hilbert space as it was achieved
in LQG [69], [3] (and references therein), and even before reaching these steps, one has to
deal with the new form of the obtained constraints, particularly the scalar constraint that
isnt polynomial and obviously isnt evident to solve. However, these addressed issues has
to be considered as the next challenges to tackle.
91
Appendix A
The Classical Vacuum Einstein Equations
From the Einstein-Hilbert action
The rst action used in the initial chapter is the Einstein-Hilbert action constructed via
the Ricci scalar R = gR and the square root of the determinant of the metric.
sEH (g) =1
2k
ZMRp det(g)d4x (4.102)
The dynamical variable involved in this action is the non-degenerate metric g compat-
ible with the Levi-Civita connection free from the torsion T , where:
rg = 0; (4.103)
The operator r is the covariant derivative, acting over the contravariant component v
as rv = @v + v
, where the vector v = v@ belongs to the tangent space of the
manifold T (M) ; with @ a natural basis of T (M) :
Over the covariant component u of the 1-form u = udx, r acts by ru =
@u u: where u = udx belongs to cotangent space T (M) dual of T (M) ;
whose basis is dx: Where
dx (@) = (4.104)
These denitions allow to express (4.103)
rg = @g g g = 0; (4.105)
and the components of the Torsion T ; T =12
= 0.
The expression of the Riemannian curvature tensor is given by the Levi-Civita con-
nection. From the denition [r;r]u = R u, we get its formulation
R = @ @ +
(4.106)
92
The fully covariant form R = gR of the Riamannian curvature tensor satises
the following relations
(A) Symmetry
R = R (4.107)
(B) Antisymmetry
R = R = R = R (4.108)
(C) Cyclicity
R +R +R = 0 (4.109)
Now that we have dened all the necessary mathematical objects that construct the
action (4.102), we proceed by applying the principal of the least action in order to de-
rive the classical vacuum Einstein equations. The variation, with respect of the metric
gsEH (g) = 0; gives
1
2k
ZM
hg (R) g
p det(g) +Rg
gp det(g)
id4x = 0 (4.110)
The variation of the Ricci tensor is gR = gR = rg rg; hence
the rst part of (4.110) can be set as a total derivative, which vanishes.
we need to variate the metrics inverse gg = ggg; and the determinantg
p det(g)
= 1
2
p det(g)gg; allowing to rewrite the second part of (4.110) as
1
2k
ZM
hRg
g
p det(g)
+p det(g)
Rgg
id4x
1
2k
ZM
p det(g)
1
2Rg Rgg
gd
4x
= 0 (4.111)
leading to the classical vacuum Einstein equations
R 12Rg = 0:
From the Palatini action
Now we will briey derive the Einstein Field equations in the vacuum from the
Palatini action. For this, we have to treat the tetrad and the connection as di¤erent and
unrelated variables.
As rst step, we consider the variational principal of the Palatini action with respect
93
to the Lorentz connection.
!SP (e; !) = Z
e
2k(eKe
L eKe
L) !
KL d4x = 0 (4.112)
The explicit expression of the innitesimal variation of the curvature K L with
respect to the connection is detailed by
!K
L = @!KL @!KL !KN!NL !KN!NL
!KN!NL !KN!NL= D!
KL D!
KL: (4.113)
Inserting (4.113) into the variation of the action !SP (e; !) yields to a vanishing covariant
derivative over the tetrad DeL = reL + ! K
L eK = 0; or in other words to a vanishing
torsion. Once contracted with the inverse of the tetrad eK , it gives
!K
L [e] = eK reL = eLreK
; (4.114)
which is the unique torsion free connection compatible with the tetrad eK :
This equation will restore the number of the degrees of freedom, reducing the 40
degrees of the connection !KL and the tetrad eK to 16: In the remained 16; 6 degrees
will be absorbed by the Lorentz invariance while 10 degrees which left correspond to those
of the fundamental eld g used in the Einstein metric formalism.
The second part is to variate the Palatini action with respect to the tetrad this time
eSP (e; !) = 0 =)2
keBIKLKL (!)
eI = 0
=)2
keBIKLKL (!)
= 0 (4.115)
where we have used the properties of the functionsA;B; C to have eeAKL = eBIKLeI :Inserting the solution (4.114) in the expression of the curvature gives
KL
! [e]
= eK e
L@
@ +
(4.116)
and then it gives KL in function of the Riemannian curvature tensor
KL = eK e
LR : (4.117)
Finally inserting (4.117) in (4.115), with some simplications, will lead to this vanishing
94
equation2e
k
eIR 2geIR
= 0; (4.118)
which multiplied by eI ; leads to the Classical Einstein eld equations
R 12gR = 0; (4.119)
which shows the equivalence between General Relativity and the Einstein-Cartan theory.
From the Holst action It remains to derive the classical vacuum Einstein equation
from the action introduced in [12]
S (e; !) = 1
2k
ZeeKeL eLeK
KL (!)
1
2 IJKL IJ (!)
d4x (4.120)
The variation of this action with respect of the connection will give the same result
obtained in (4.114). But it is not the case for the variation with respect to the tetrad,
where we obtain
eS (e; !) = 0 =) eBNKL
KL (!)
1
2 IJKL IJ (!)
= 0: (4.121)
By inserting (4.114), the precedent equation reads
eBIKLeK e
LR 1
2 IJKL eIe
JR
= 0: (4.122)
Let us focus on the second term of (4.122), which is
1 2eBIKL IJ
KL eIeJR
: (4.123)
By using the relation
eBIKL = eeIAKL + eIAKL + eIAKL
; (4.124)
and eKeLeI eJ
IJKL = e1", it is possible to rewrite (4.123) as
1
eI" + eI" + eI"
R = 0 (4.125)
which vanishes because of the cyclicity property of the Riemannian tensor. Hence, the
only remaining term in (4.122) is the rst one, which delivers the usual Einstein equation
as it was already performed in the Palatini action part (4.118) and (4.119).
95
Appendix B
The Dirac Quantization Program
In this part, we summarize some steps and tools that are part of the Dirac quantization
program. We rely mostly on this program, as rst step to achieve the quantum version of
the studied theory in non-perturbative methods. The idea of such program was developed
by Dirac [14] and Bergmann independently. While Bergmann focused on the Hamiltonian
development of GR specially [70], Dirac devoted himself to build the general program,
such that it could be applied to any theory. Besides Diracs monograph, the reader may
nd good textbooks in [36] and [71].
From the Lagrangian formalism to the Hamiltonian formalism
Given an action
S =
ZL (qi;
:qi) dt;
the Lagrangian L is function of the dynamical coordinates qi and theirs velocities:qi =
dqidt
from the tangent space of the conguration manifold C to R; L : TC ! R; (qi;:qi) ! L
(qi;:qi), where i = 1; :::; N with N is the number of degrees of freedom. From variation of
the action integral, one can obtain the Lagrange equations
d
dt
@L
@:qi
=@L
@qi:
To pass from the Lagrangian formalism to the Hamiltonian one, we need to perform the
Legendre transformations which introduce the momentums of the coordinates variables
via
TC ! T C; (qi) !qi; Pi =
@L (q;:q)
@:qi
: (4.126)
There is particular case when it is impossible to determine the velocities in function of the
variables and the momentums involved in the description of the physical system, attention
will be focused to such systems because they possess Gauge Degrees of Freedom.
Now, the search for the momentums via Pi =@L(q;
:q)
@:qi
may lead to number of vanishing
equations m (p; q) = 0; where 1 6 m 6M; referred as primary constraints which embed-
96
ded the primary constraint surface, submanifold of the phase space. These constraints
are relations between the variables q0s and the momentums p0s:
The variation of the Hamiltonian quantity H = pi:qi L is free from the variation of
the velocities, hence the Hamiltonian H is only in function of the q0s and the p0s: However
this denition is not entirely accurate if one does not add the constraints m (p; q) along
arbitrary m known as the Lagrange multipliers.
HT = H + mm (p; q) : (4.127)
At this stage, dening the total Hamiltonian will allow us to study the evolution over
time of arbitrary functions by the use of:
dF (q; p)
dt= fF (q; p) ; HTg fF (q; p) ; Hg+ m fF (q; p) ; mg ; (4.128)
where " " refers to the weak equality that veries the non-zero Poisson brackets withthe canonical variables. The denition of the bilinear operator Poisson bracket f:; :g isgiven by
fF;Gg = @F
@qi
@G
@pi @F
@pi
@G
@qi
from which one can list the following properties: a) Antisymmetry fA;Bg = fB;Ag ; b)Distributivity fA;B + Cg = fA;Bg+fA;Cg ; c) the Leibniz rule fA;BCg = fA;BgC+B fA;Cg and Finally d) the Jacobi identity
fA; fB;Cgg+ fC; fA;Bgg+ fB; fC;Agg = 0 (4.129)
The classication of the constraints
Primary, secondary, tertiary .... constraints The constraints are constant of mo-
tion thus m (p; q) should be preserved over time, consequently:
m (q; p) should weakly
vanishes too. This is known by consistency conditions
dmdt
= fm; HTg
= fm; Hg+ m0 fm; m0g 0; (4.130)
which may conduct to four di¤erent cases
1) The rst case may lead to absurdity, such non zero number 0; showing thatthe system is incoherent, or the Lagrangian of the start taken to describe the physical
system is ill-dened.
2) The second case may be irrelevant, as 0 0:
97
3) From the third case, one can obtain equations from which the Lagrange multipliers
m0 can be dened.
4) Finally, the equation (4.128) could bring other possible relations between the q0s
and the p0s, m (q; p) 0, known as secondary constraints.The secondary constraints should be submitted to this process in their turns and then
one must check if these constraints potentially lead to what will be labelled as tertiary
constraints and so on. This iteration should be maintained until the system is exhausted
and can deliver back no more new constraints.
Classes of constraints Now, that all the constraints has been determinated by the
operations explained earlier. We have to classify the constraints in two distinct subsets,
known as rst-class and second class. This new classication is di¤erent from the pre-
vious distinction between the primary, secondary, tertiary... constraints introduced by
Bergmann. The rst-second class classication is more relevant to describe the physics
of the system through the Hamiltonian procedure. A constrained function a is classied
as rst class, as it is given in [14], if its Poisson brackets taken with all the constraints
dened previously, vanish weakly
fa; mg 0 m = 1; :::;M (4.131)
All the other constraints that fail this test are identied as second-class constraints. From
this given denition, one can deduce that the Poisson brackets of two rst-class constraints
yield to rst class constraints, this can be proven by using the Jacobi identity. Now, let us
submit the rst class constraint m to the consistency conditions (4.130), as consequence
from the denition of the rst-class constraint and (4.127), the Hamiltonian H can only
be classied as rst class in its turn.
In the second enumerated case, we have mentioned that the consistency condition may
leads to equations involving restriction over the Lagrange multipliers .
fn; Hg+ m fn; mg 0 (4.132)
The general solutions of this set of equations is of the form m = Um+Vm; where Um (q; p)
is a particular solution of the inhomogenious equations (4.132), while Vm represents the
most general solution of the homogenious part in (4.132), where
Vm fn; mg = 0
which can be expressed as Vm = vaVam (p; q) and va are completely arbitrary undetermined
98
functions, with a = 1; :::; A is the number of independent solutions V am . If one replaces
this solution in the expression of the total Hamiltonian, one gets
HT = H + Um (q; p)m + vaVam (p; q)m (4.133)
by setting H 0 = H + Um (q; p)m and a = Vam (p; q)m; where a labels number of
independent rst-class constraints, we get
HT = H 0 + vaa (4.134)
Generators of Gauge transformations
The fact that the total Hamiltonian contains arbitrary functions va, will be reected in
the Hamiltonian equations of motion. Considering innitesimal evolution of the system
through time, the canonical variables (qi; pi) change as
qi (t+ t) = qi (t) + t:q (t)
= qi (t) + t (fqi; H 0g+ va fqi; ag) (4.135)
and
pi (t+ t) = pi (t) + t (fpi; H 0g+ va fpi; ag) ; (4.136)
Because of the dependence of the equations of motion to va, given di¤erent values of the
arbitrary functions va, the equations of motion will give di¤erent trajectories. Hence one
can consider various initial (qi; pi) yielding to the same nal physical state and search
for transformations linking these (qi; pi) that left the physical state unchanged. Such
transformations are referred as gauge transformations. In practical, considering an inni-
tesimal time evolution t from the initial xed time t = t0, the evolution of any dynamical
function F reads
F (t0 + t) = F (t0) + t:
F (t)
= F (t0) + t (fF;H 0g+ va fF; ag) ; (4.137)
if we choose to consider v0a instead of va, this will obviously a¤ect the evolution of F .
Hence, this change will be measured by
Fv (t0 + t) Fv0 ((t0 + t)) = tva v0a
fF; ag (4.138)
99
which can be simplied as,
F = a fF; ag (4.139)
once we put a = tva v0a
. Actually, it is the transformations conserving F
0 that are relevant and are the transformations that we were looking for, i.e. gauge
transformations. It has been claimed by Dirac that such transformations are strongly
tied to rst-class constraints, where these constraints in the Hamiltonian theory generate
the transformations that preserve the Physical states. E¤ectively, the number a indexing
the arbitrary functions or parameters is the same of the independent rst-class constraints
forming the Hamiltonian (4.134). One can also show that the Poisson bracket of two rst-
class constraints fa; a0g 1, or in other words the Poisson bracket between two generatorsof gauge tranformations remains a generator of gauge tranformations. It follows that
the Poisson bracketa; H
0generates gauge transformations in its turn too. We also
mention that if a are of rst-class, where fa; a0g = Caa0a00a00 ;then this reveals symmetry
where its generators are those of the Lie group. At nal, this show the relevance of rst-
class second-class classication over the primary, secondary classication and turns to be
appreciable tool if the symmetries underlying the Lagrangian of the system are not so
obvious.
Second-class constraints and Dirac brackets
As opposite to the rst class constraints, Second-class ones do not generate gauge trans-
formations. Thus, they deserve special treatment in the Hamiltonian procedure. Such
constraints are dened by the non-vanishing matrix 0 over the constraints surface,
where 0 =; 0
and det
0
6= 0: Hence, this matrix possesses an inverse such
00 = ; leading to extend the usual Poisson brackets to more general brackets to
include the second-class constraints. These are known as the Dirac brackets and dened
by
[F;G] = fF;Gg F;
; 0
1 0 ; G
= fF;Gg
F;
0 0 ; G
(4.140)
The Dirac brackets should inherit the Poisson brackets properties as 1) the antisymmetry
[F;G] = [G;F ]b) distributivity [F;G+ J ] = [F;G] + [G; J ] ; c) the Leibniz rule
1However, we point that the denition of the rst class constraints literally given in [14] is tricky,infact (4.131) hides the ambiguity when the brackets fn; mg = fcnmc delivers structure functions interms of the phase space variables, hence it corresponds to gauge transformations only on the constraintsurface.
100
[F;GJ ] = [F;G] J +G [F; J ] and Finally d) the Jacobi identity
[F [G; J ]]+ [J [F;G]]
+ [G [J; F ]]
= 0: (4.141)
For the second-class constraints, we will then have
; F
=
; F
; 0
; F
=
; F
; F
= 0; (4.142)
which enables us to discard the second-class constraints by setting them to zero. On the
other hand, the Dirac brackets of the rst-class constraints a with any function F will
deliver simply the Poisson brackets
[a; F ] = fa; Fg ; (4.143)
After deducing that the second-class constraints can be dismissed from the theory
via (4.142) and referring to the rst-class constraints as gauge generators, the phase
space can be released from its superuous canonical variables and then address one-to-one
correspondence from the remaining independent canonical variables and a given physical
state, and hence nding the exact degrees of freedom of the evolving system.
101
Appendix C
The Spinors formalism
In this part, we succinctly develop the spinor formalism in order to incorporate fermionic
matter to Gravity in the second chapter. We start by the introduction of the Dirac
matrices that verify
I J + J I = 2IJI44
Where I44 =
I22 0
0 I22
!is the matrix identity, which is itself constituted by
blocks of matrix identity I22 in Weyl representation. The I matrices correspond to
0 =
0 I22
I22 0
!and i =
0 i
i 0
!: (4.144)
Where i are the Pauli matrices, satisfying [i; j] = 2ij. Giving these denitions, we can
link the fermions to Gravity by the covariant derivative that involves the spinor connection
!IJ by
D = @+1
2!IJ IJ D = @
1
2!IJ IJ ; (4.145)
where IJ = 14[ I ; J ] are the generators of the so (1; 3) Lie algebra in the Dirac repre-
sentation. The matrices I and KL verify the following expressions of the commutator
and the anti-commutator
[ I ; KL] = IK L IL K (4.146)
and
[ I ; KL]+ = iIKLJ 5 J (4.147)
with 5 =i4IJKL I J K L = i 0 1 2 3 where the matrix 5 has these properties
( 5)2 = I44 and +5 = 5 and [ 5; I ] = 0: In addition, the matrices KL will provide a
102
representation of the lorentz group, where one can demonstrate
[IJ ; KL] = (JKIL IKJL + ILJK JLIK) (4.148)
From the Dirac spinors, one can build many Lorentz covariant items as the scalar ;
the Lorentz vector ; pseudo-scalar 5; or axial vector as 5 ; where this
nal is dened as the current J in the second chapter.
One can generalize the denition of the Dirac matrices at higher dimensions, hence
these matrices will be of 2m 2m dimension, where the values of m depend wherever the
dimension d is odd or even. Thus, d = 2m if d is even, or d = 2m+ 1 if d odd. We have
also the generalization of the matrix 5 as
d+1 = (i)m+1 0 1::: d1
where, d+1
is hermitian and
d+1
2= Idd in every d even.
103
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1
Classical and Quantum Gravity
On the Hamiltonian formalism of the tetrad-connection gravity
M H Lagraa1,2, M Lagraa2 and N Touhami2
1 Ecole Préparatoire en Sciences et Techniques d’Oran (EPSTO), PO Box 64 CH2 Achaba Hanifi, Oran, Algeria2 Laboratoire de physique théorique d’Oran (LPTO), Université d’Oran I, Ahmed Benbella, PO Box 1524, El M’Naouer, 31000 Es-Sénia, Oran, Algeria
E-mail: [email protected], [email protected] and [email protected]
Received 7 October 2016, revised 29 March 2017Accepted for publication 11 April 2017Published 19 May 2017
AbstractWe present a detailed analysis of the Hamiltonian constraints of the d-dimensional tetrad-connection gravity where the non-dynamic part of the spatial connection is fixed to zero by an adequate gauge transformation. This new action leads to a coherent Hamiltonian formalism where the Lorentz, scalar and vectorial first-class constraints obey a closed algebra in terms of Poisson brackets. This algebra closes with structure constants instead of structure functions resulting from the Hamiltonian formalisms based on the A.D.M. decomposition. The same algebra of the reduced first-class constraints, where the second-class constraints are eliminated as strong equalities, is obtained in terms of Dirac brackets. These first-class constraints lead to the same physical degrees of freedom of the general relativity.
Keywords: tetrad-connection gravity, Hamiltonian formalism, Dirac brackets
1. Introduction
Any coherent canonical quantification of a theory requires a correct treatment of its classi-cal Hamiltonian formalism. During the last half-century, canonical quantization of general relativity has attracted much attention, especially these last few decades with the development of the loop quantum gravity [1, 2] and [3] (and references therein). Despite a lot of progress made in the different approaches of the canonical quantization of the gravity, these approaches are not complete in the sense that the algebra of the first-class constraints closes with struc-ture functions both in the metrical [4] and the tetrad formulation of gravity [1]. The presence of structure functions can be a potential source of anomalies and reveals that the first-class constraints do not correspond to symmetries based on true Lie groups. This shows the special attention that must be paid to the construction of the Hamiltonian formalism of gravity.
M H Lagraa et al
On the Hamiltonian formalism of the tetrad-connection gravity
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Class. Quantum Grav.
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1361-6382
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2
Among these different approaches, we cite the one where the Hamiltonian formalism is derived from a generalized action [5] where the time gauge is fixed in order to reduce Lorentz’s manifest invariance of the action to SO(3) = SU(2)/Z2. The interest in SO(3) came from the fact that SO(3) is compact allowing the construction of the Hilbert space in loop quantum gravity. The Holst action does not modify the classical Einstein equation in the vac-uum but contains a new dimensionless parameter known as the Barbero–Immirzi parameter [6] which appears in the spectra of area and volume operators at the quantum level [7]. It also appears in the black hole entropy formula [8]. Note that even at the classical level, once grav-ity is coupled to fermionic matter, the Barbero–Immirzi parameter appears on-shell via the non zero torsion [9].
The local Lorentz invariance of the canonical vierbein form of general relativity has been done in the second order formalism in [10]. The main complication of the covariant canoni-cal formulation of the first order tetrad-connection gravity is the presence of the second-class constraints which require the Dirac brackets. The covariant Hamiltonian treatments of the generalized 4 − dimensional action [5] have been developed either in terms of Dirac brackets in [13], where two copies of su(2) Barbero tedrad-connection gravity are combined to get a SO(4,C) covariant Hamiltonian, or in [14] where the resolution of second-class constraints leads to a reduced symplectic form where the phase space elements obey canonical Poisson’s brackets.
Almost at the same period, the enthusiasm aroused by the spin foam model of the BF-theory [11] and [12] has encouraged the investigation of the covariant Hamiltonian of the tetrad grav-ity formulated as a BF-theory with extra constraints on the 2 − form B. In [15] the extra con-straints on the 2 − form B are solved leading to similar results as the ones of [14].
For higher dimensions, the analysis of the covariant Hamiltonian formalism of the tetrad-connection action was performed in [16] by considering an extension of the A.D.M. phase space where the Lagrange multipliers, the lapse and the shift, are considered as part of the phase space. After solving the second-class constraints, a canonical reduced symplectic form is obtained leading to an algebra of constraints which closes with structure functions.
Until now all the Hamiltonian formulations of gravity where one starts from the very begin-ning by the A.D.M. Decomposition of the tetrad components in terms of lapse and shift lead to an algebra of first-class constraints involving structure functions.
Rather than proceeding as in [16], we begin from the phase space resulting directly from the tetrad-connection action without using the A.D.M. formalism. We show that the connection splits in dynamic and non dynamic parts. By fixing the non dynamic part of the connection to zero we obtain a coherent Hamiltonian formalism where all the stages of the Dirac procedure [17] for constrained systems are scrupulously respected. In particular, the second-class con-straints are eliminated as strong equalities only after the Dirac brackets are established.
The paper is organized as follows:In section 2, we investigate and classify the different constraints according to Dirac ter-
minology. The resolution of the equations involving the Lagrange multiplier leads to a prob-lematic constraint whose consistency is difficult to check. In section 3, we show that this problematic constraint is tied to the non-dynamic part of the spatial connection which can be fixed to zero by an adequate gauge fixing. The formalism derived from the new action where the non-dynamic part of the connection is fixed to zero leads to a consistent Hamiltonian treat-ment where the Lorentz, scalar and vectorial first-class constraints form a Poisson algebra that closes with structure constants. In section 4, we establish the Dirac brackets of the reduced phase space elements where the second-class constraints are eliminated as strong equalities. The reduced first-class constraints becomes polynomial and obey, in terms of Dirac brack-ets, the same algebra as that of the previous section and lead to the same physical degrees
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
3
of freedom of the general relativity. We end this section by showing that the solutions of Hamiltonian’s equations of the reduced phase space elements lead to a zero torsion which is required to establish the equivalence between the tetrad-connection gravity and the general relativity.
2. Hamiltonian formalism of tetrad-gravity
In d − dimension space-time manifold M, the functional action of the tetrad-connection gravity is:
S(e,ω) =∫
M
1(d − 2)!
eI0 ∧ ...eId−3 ∧ ΩId−2Id−1εI0...Id−1 (1)
where the capital Latin letters I0, ..., Id−1 ∈ [0, .., d − 1] denote internal indices of the tensor representation spaces of the Lorentz group, εI0...Id−1
are the components of the totally antisymmetric Levi-Cevita symbol, ε0...d−1 = −ε0...d−1 = 1, satisfying
εI0...InIn+1...Id−1εJ0...JnIn+1...Id−1 = −(d − n)!δ[J0...Jn]
I0...In . eI = eµIdxµ is the co-tetrad one-form valued
in the vectorial representation space endowed with the flat metric ηIJ = diag(−1, 1, ..., 1) and xµ are local coordinates of the manifold M where the Greek letters µ, ν ∈ [0, 1, .., d − 1] denote space-time indices (t represents the time, t = x0 = xt). Time-like indices will be labelled ‘t’ in the tangent space and space-like indices will be labelled with small Latin let-ters a, b, c ∈ [1, .., d − 1]. The metric ηIJ and its inverse ηIJ are used to lower and to lift the Lorentz indices and to determine the metric gµν = eI
µeJνηIJ of the tangent space of the mani-
fold M. ∧ is the wedge product and ΩIJ = −ΩJI = dωIJ + ωNI ∧ ωNJ is the curvature two-
form associated to the connection one-form ωIJ = −ωJI = ωµIJdxµ valued in the so(1,d − 1) Lie-algebra. The co-tetrad and the connection are supposed to be independent variables.
Before starting the Hamiltonian analysis of the action (1), let us recall that in the Lagrangian formalism of fields theories the basic variables are fields φi(x) and their time derivative ∂tφi(x) which, in our case, are the co-tetrad components eµI, the connection components ωµIJ and their time derivative ∂tφi(x) = (∂teµI , ∂tωµIJ). These variables and their time derivatives are considered as independent variables and constitute the configuration space. In this frame-work the dynamic is presupposed to be determined by evolution equations of second order with respect to time and the configuration space is nothing but a space isomorph to the set of initial conditions of the solutions of the evolution equations. It is not the case here where we are dealing with a first order theory where the equations of motion are of the first order, so they are only constraints in the configuration space. In addition, since the action (1) is given in terms of differ ential forms, it follows that the time derivative of the temporal components of the co-tetrad and the connection are absent from the evolution equations that govern the tetrad-connection gravity theory and therefore the evolution in time of these variables is unde-termined. In the following, these points will be investigated in the Hamiltonian formalism which is more suitable for constrained systems.
To pass from the Lagrangian formalism to the Hamiltonian formalism, we will suppose that the manifold M has topology R × Σ, where R represents the time which is an evolution parameter of d − 1 dimensional space-like hypersurfaces Σt into the d − dimensional mani-fold M. In order to get the momenta conjugate to the configuration fields eµI and ωµIJ we must develop the action (1) in terms of components of the co-tetrad and the connection
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
4
S(e,ω) =∫
M
1(d − 2)!
eI0 ∧ ...eId−3 ∧ ΩId−2Id−1εI0...Id−1
=
∫
M
12(d − 2)!
εI0...Id−1 eµ0I0 ...eµd−3Id−3Ωµd−2µd−1Id−2Id−1εµ0...µd−1 ddx
= −∫
MeAµKνL ΩµνKL
2ddx
(2)
where
ΩIJ =12ΩµνIJdxµ ∧ dxν =
12(∂µωνIJ − ∂νωµIJ + ωN
µIωνNJ − ωNνIωµNJ
)dxµ ∧ dxν ,
eAµKνL =1
(d − 2)!εI0...Id−3KLeµ0I0 ...eµd−3Id−3ε
µ0...µd−3µν
= e(eµKeνL − eνKeµL) = −AνKµL = −AµLνK ,
(3)
e = det(eµI), and eµK is the inverse of eµL , eµKeµL = δKL , eµKeνK = δµν .
To carry out the Legendre transformations, the time derivatives must appear explicitly in the action (2)
S(e,ω) = −∫
M
(eAaKtL (∂tωaKL − DaωtKL)− eAaKbL ΩabKL
2
)ddx (4)
from which we deduce the conjugate momenta πβN and PβKL of the co-tetrad eβN and the so(1, d − 1) connection ωβKL
πβN(x) =δS(e,ω)δ∂teβN(x)
= 0,PaKL(x) =δS(e,ω)
δ∂tωaKL(x)= eAaKtL(x)
and
P tKL(x) =δS(e,ω)δ∂tωtKL(x)
= 0
obeying the following non-zero Poisson brackets
eαI(−→x ),πβN(−→y )
= δβαδ
NI δ(
−→x −−→y ),ωαIJ(
−→x ),PβKL (−→y )= δβα
12(δK
I δLJ − δL
I δKJ )δ(
−→x −−→y ).
(5)
where −→x denotes the local coordinates xa of Σt .The expressions of the conjugate momenta lead to the following primary constraints
πtN = 0,P tKL = 0,
πbN = 0 and CaKL = PaKL − eAaKtL = 0 (6)
which satisfy the following non-zero Poisson bracketsπaN(−→x ), CbKL(−→y )
= −eBaNtKbLδ(−→x −−→y ) (7)
where BβNµKνL is defined as
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
5
eBβNµKνL =1
(d − 3)!εI0...Id−4NKLeµ0I0 ...eµd−4Id−4ε
µ0...µd−4βµν
= e(eβNAµKνL + eβKAµLνN + eβLAµNνK)
= e(eβNAµKνL + eµNAνKβL + eνNAβKµL) =δ
δeβNeAµKνL.
(8)
The total Hamiltonian is defined by
HT =
∫
Σ
(πtNΛtN + P tKL AtKL
2+ πbNΛbN + CaKL AaKL
2) + H0 (9)
where ΛtN, AtKL, ΛbN and AbKL are the Lagrange multipliers for primary constraints (6) and
H0=
∫
Σ
(eAaKbL ΩabKL
2+ eAaKtLDaωtKL).
The consistency of the Hamiltonian formalism requires that these constraints must be pre-served under the time evolution given in term of total Hamiltonian (9) in the standard form:
πtN ,HT
= −eBtNaKbL ΩabKL
2= PN = 0, (10)
P tNM ,HT
= DaeAaNtM = eBcKaNtMDaecK = MNM = 0, (11)
πbN ,HT
= −eBbNtKaL
(AaKL
2− DaωtKL
)− eBbNaKcL ΩacKL
2= 0, (12)
and
CaKL,HT= eBbNtKaL (ΛbN + ωM
tNebM)+ Dc(eAcKaL)
= eBbNtKaL (ΛbN + ωMtNebM − DbetN
)+ eBbNcKaLDcebN = 0
(13)where we have used (7) and
παN , eAµKνL
= − δ
δeαNeAµKνL = −eBαNµKνL .
These consistency conditions show that the evolution of the constraint πtN and P tNM leads to the secondary constraints PN and MNM respectively, while the evolution of the constraints πbN and CaKL leads to the equations for the Lagrange multipliers AaKLand ΛaN .
Now we have to check the consistency of the secondary constraints. For the constraint (11) we get
MKL,HT
= Da(eBbNaKtLΛbN) +
12AK
aNeAaNtL +12AL
aNeAaKtN . (14)
Using (13) we obtain
Da(eBbNaKtLΛbN
)= −Da
(eBbNtKaLΛbN
)
= Da(eBbNtKaLωtNMeM
b
)+ Da
(eBµNbKaLDbeµN
)
where the second term of the right hand side of the equality above is written as
Da(eBµNbKaLDbeµN
)= DaDbeAbKal = −1
2(DaDb − DbDa) eAaKbL
= −12(ΩK
abNeAaNbL +ΩLabNeAaKbN) .
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
6
From the properties (8) of the B-matrix, an explicit computation leads, for any antisym-metric tensor DNM = −DMN , to the identity
DKN eAνNρL +DL
NeAνKρN =12(eK
βeBβLνNρM − eLβeBβKνNρM)DNM
=12(eM
β eBβNνKρL − eNβeBβMνKρL)DNM
(15)
leading to
Da(eBbNtKaLωtNMeM
b
)= Da(ω
KtNeAtNaL + ωL
tNeAtKaN)
= −eAaNtLDaωKtN − eAaKtNDaω
LtN
−(ωK
tNMNL + ωLtNMKN)
from which we get
MKL,HT= −
(ωK
tNMNL + ωLtNMKN)
− (AK
aN
2− Daω
KtN)eAtNaL − (
ALaN
2− Daω
LtN)eAtKaN
− 12(ΩK
abNeAaNbL +ΩLabNeAaKbN) .
(16)
As a consequence of (15), (16) is written in the form
MKL,HT
= −12
(eK
b
(eBbLtNaM
(12AaNM − DaωtNM
)+
12
eBbLaNcMΩacNM
))
+ (K ↔ L)
+12(eK
t PL − eLt PK)− (
ωKtNMNL + ωL
tNMKN)
which, when (12) and (13) are satisfied, reduces to
MKL,HT
=
12(eK
t PL − eLt PK)− (
ωKtNMNL + ωL
tNMKN) 0 (17)
ensuring the consistency of the constraint MKL. Here ‘’ denotes weak equality which means equality modulo the constraints.
The evolution of the constraint PN is given by
PN ,HT
= −eCaMtNbKcL ΩbcKL
2ΛaM − eBtNbKalDb
AaKL
2 (18)
where
eCµMνNαKβL =1
(d − 4)!εI0...Id−5MNKLeµ0I0 ...eµd−5Id−5ε
µ0...µd−5µναβ
= eµMeBνNαKβL − eβMeBµNνKαL + eαMeBβNµKνL − eνMeBαNβKµL
= eµMeBνNαKβL − eµLeBνMαNβK + eµKeBνLαMβN − eµNeBνKαLβM
=δeBνNαKβL
δeµM.
(19)
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
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By using (12), (13) and the properties of the C-matrix (19) we get, after lengthy computa-tion, the evolution of the constraint PN under the form of combination of constraints
PN ,HT
= −ωN
tMPM
−ecN(
PM (ΛcM + ωtMKeK
c − DcetM)+ MKL
(AcKL
2− DcωtKL
)) 0.
(20)Note that the second term of the right hand side of (20) is orthogonal to etN, therefore the
evolution of etNPN gives
etNPN ,HT=
(ΛtN + ωM
tNetM)
PN 0 (21)
while the part of (20) proportional to ecN shows, by using the consistency of the constraint MKL (17), that the evolution of the linear combination of the constraints PN and MKL, Dspa = eaNPN + ωaKLMKL is given by
eaNPN + ωaKLMKL,HT
= PN∂aetN + MKL∂aωtKL 0. (22)
The consistency conditions (21) and (22) show that contrary to the constraint PN the evo-lution of its projections PNetN and Dspa = eaNPN + ωaKLMKL are simple and independent of the Lagrange multipliers ΛaN and AaKL .
In what follows, instead of the constraint PN we consider its temporal projection
Dt = PNetN = −eAaKbL ΩabKL
2 (23)
and its smeared spatial projection
Dsp(−→N ) = −
∫
Σ
Na(eaNPN + ωaKLMKL) =
∫
Σ
eAaKtLL−→N (ωaKL) (24)
where we have used eBtNaKbLetN = eAaKbL to obtain (23) and
−NcecNPN = NcecNeBtNaKbL ΩabKL
2= eAaKtLNcΩcaKL
= eAaKtL(L−→N (ωaKL)− Da(NcωaKL))
(25)
to obtain (24). L−→N (ωaKL) = Nb∂bωaKL + ∂a(Nb)ωbKL is the Lie derivative along the arbitrary
vector field −→N tangent to Σt . This Lie derivative does not affect the Lorentz indices.
The relation (22) exhibits the consistency of the constraint Dsp(−→N ) as
Dsp(
−→N ),HT
=
∫
Σ
(PNL−→N (etN) + MKLL−→
N (ωtKL)) 0
which shows, by comparing to (22), that L−→N (etN) = Na∂aetN and L−→
N (ωtKL) = Na∂aωtKL from which we see that the Lie derivative L−→
N treats the temporal components etN and ωtKL as scalars.
This analysis of the primary constraints πtN , P tKL , πaN and CaKLand the secondary con-straints Dt, Dsp and MKL shows that the set of constrains is complete meaning that the total Hamiltonian HT is coherent provided that (12 ) and (13) are satisfied.
To complete this analysis, we have to solve the equations (12) and (13). We can check that
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
8
BµNνKαL =12
(eµN
AνKαL
d − 2+ eνN
AαKµL
d − 2+ eαNAµKνL
)
=12
(AµNνK
d − 2eαL +
AµLνN
d − 2eαK + AµKνLeαN
)
(26)
is the inverse of BβNµKνL in the sense that
BµNνKαLBµNνPβQ = δβα(δPKδ
QL − δP
LδQK ) (27)
and
BµNνKαLBρMσKαL = δMN (δρµδ
σν − δρνδ
σµ). (28)
We see from (26) that, contrary to BµNνKαL which is antisymmetric with respect of the indices µ, ν and α and of the indices N, K and L, BµNνKαL is antisymmetric with respect of the indices µ and ν and of the indices Kand L only.
For β = b and α = a, (27) gives
BcNtKaLBcNtPbQ +12
BcNdKaLBcNdPbQ =12δb
a(δPKδ
QL − δP
LδQK ) . (29)
As a consequence of the antisymmetric of the indices µ, ν and α of BµMνKαL, (28) gives for σ = ν = t
BcNtKaLBbMtKaL = δMN δb
c . (30)
and for σ = t and ν = d
BcNdKaLBbMtKaL = 0. (31)
Using (30) and (31) we get the solution of (12) as
12AaKL = DaωtKL −
12
BbNtKaLBbNcPdQΩcdPQ + BbNcKaLΛbNc (32)
with arbitrary ΛbNc . The third term of the right hand side of (32) is the part of solution of the homogeneous equation associated with (12).
The determination of the lagrange multipliers ΛbN is obtained by multiplying (13) by BbNtKaLand using (30) to get
ΛbN = −ωMtNebM + DbetN − BbNtKaLBcMdKαLDdecM
= −ωMtNebM + DbetN + e−1BbNtKtLMKL −ωM
tNebM + DbetN . (33)
By multiplying (13) by BdNeKaLand using (31) we get
BdMaKeLBbNcKeLDcebN = 0 = (δNM(δ
bdδ
ca − δb
aδcd)− BdMaKtLBbNcKtL)DcebN
= DaedM − DdeaM − e−1BdMaKtLMKL = 0.
(34)
The above condition is not a solution of the homogeneous equation associated with (13). It result neither from the Legendre transform nor from the consistency of the constraints. This condition shows that the spatial components of the torsion ΘMed = 1
2 (DeedM − DdeeM) are a combination of constraints MKL. It is a condition to have the general solution of (13). In fact if we multiply (33) by BbNtPeQ and use (29) we get
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
9
BbNtPeQ (ΛbN + ωM
tNebM − DbetN)+ BbNtPeQ (
BbNtKaLBcMdKaLDdecM)
=BbNtPeQ (ΛbN + ωM
tNebM − DbetN)+ BcMdPeQDdecM
− 12
BbNfPeQBbNfKaLBcMdKaLDdecM
implying, by virtue of (13),
12
BbNfPeQ (BbNfKaLBcMdKaLDdecM
)= 0
which is equivalent to the condition (34).By substituting (32) and (33) in HT, we get
H′T =
∫
Σ
P tKL AtKL
2+ πtNΛtN
−∫
Σ
((πaKeLa − πaLeK
a ) + 2Da(CaKL + eAaKtL))ωtKL
2
+
∫
Σ
(eAaKbL ΩabKL
2+ πaNDaetN − CaKLBbNtKaLBbNcPdQ ΩcdPQ
2)
+
∫
Σ
CaKLBbNcKaLΛbNc.
(35)
Now we check the consistency of constraints with the Hamiltonian H′T. The evolution
of the primary constraint πaN is consistent in the sense that its Poisson brackets with the Hamiltonian H′
T give combinations of πaN and CaNM asπcN ,H′
T
=− ωN
tKπcK
+12
CaKL δ
δecN(BbMtKaLBbMcPdQ)
ΩcdPQ
2
− 12
CaKL δ
δecN(BbMdKaL)Λ
bMd 0.
(36)
For the constraint CaNM, we obtain
CaNM ,H′
T
=− ωN
tKCaKM − ωMtKCaNK − 1
2(πaNeM
t − πaMeNt )
+ Db(eAbNaM) + eBbQtNaMDbetQ − Dc(CdKLBbNtKdLBbNcNaM)
(37)
which vanishes weakly if we use (34) to get
Db(eAbNaM) + eBbQtNaMDbetQ =12
eBcQbNaM(DbecQ − DcebQ)
=12
eBcQbNaMe−1BcQbKtLMKL 0
showing that the evolution of the constraint CaNM is consistent only when (34) is satisfied. This is due to the fact that (34) is a condition to solve the equation (13) which result from the con-sistency of the constraint CaNM. From that, we expect that the consistency of the constraints
MKL, Dt and Dsp(−→N ) depends on (34) also. In fact a direct computation gives
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
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MKL,H′
T
=− ωK
tNMNL − ωLtNMKN +
12(eK
t PL − eLt PK)
− 12
Da(eBdNbKaL(DbedN − DdebN)),
Dt,H′T = −(ΛN + ωtNMeM
t )PN + BdNcKbLDa(eAaKbL)ΛdNc
andDsp(
−→N ),H′
T
=−
∫
Σ
(PNL−→N etN + MKLL−→
N ωtKL)
−∫
Σ
12
eBdNbKaL(DbedN − DdebN)L−→N ωaKL
which show that the secondary constraints MKL, Dt, and Dsp(−→N ) are consistent only when (34)
is satisfied.The consistency of the constraint πtN gives
πtN ,H′
T
=− eBtNaKbL ΩabKL
2+ Daπ
aN
+ CaKL(−e−1etNBbMtKaLeBbMdPeQ
+ e−1 δ
δetN(BbMtKaL)eBbMdPeQ
+ e−1BbMtKaLeCtNbMdPeQ)ΩdePQ
2= P
′N 0
(38)
where we have used δδetN
e−1 = −e−1etN and (19).From the relation etN
δδetN
(BbMtKaL) = BbMtKaL , the projection of the constraint P′N , D′
t = P′NetN , is written in a combination of constraints
D′t = Dt − πaNDaetN + CaKLBbNtKaLBbNcPdQ ΩcdPQ
2 0 (39)
where we have used eCtNbMdPeQetN = eBbMdPeQ obtained from the properties of the C-matrix (19).
For the projection on the spatial component of the co-tedrad we use the relation
ecNδ
δetN(BbMtKaL) = BbMcKaL and
ecNCtNbMdPeQ ΩdePQ
2= −BbMtPeQΩcePQ + δb
c PM ,
obtained from the properties of the C-matrix (19), to get
CaKLBbMtKaLecNCtNbMdPeQ ΩdePQ
2
= −CaKLΩcaKL +12
CaKLBbMdKaLBbMdPeQΩcePQ + CaKLBcMtKaLPM
from which we obtain
ecNP′N =− (CaKL + eAaKtL)ΩcaKL + ecNDaπaN + CaKLe−1BcMtKaLPM
+ CaKL(BbMcKaLBbMdPeQ ΩdePQ
2+ BbMdKaLBbMdPeQ ΩcePQ
2)
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
11
leading to a linear combination of smeared constraints as∫
Σ
Nc(ecNP′N +ωcKL
2M′KL + πaN(DaecN − DceaN))
−∫
Σ
NcCaKL(BbNcKaLBbNdPeQ ΩdePQ
2+ BbNdKaLBbNdPeQ ΩcePQ
2)
−∫
Σ
NcCaKLBcMtKaLPM
= −∫
Σ
(πaML−→N (eaM) + (CaKL + eAaKtL)L−→
N (ωaKL)).
(40)
Here the new secondary constraint M′KLis deduced from the consistency of the primary constraint P tNM
P tKL,H′
T
= (Da(CaKL + eAaKtL) +
12(πaKeL
a − πaLeKa ))
=12
M′KL 0.
(41)
From the expressions (39) and (41), the total Hamiltonian takes the compact form
H′T =
∫
Σ
(12P tKLAtKL + πtNΛtN −D′
t − M′KL ωtKL
2)
+
∫
Σ
CaKLBbNcKaLΛbNc.
The constraint (40) can be completed by adding the constraint πtNL−→N (etN), where
L−→N (etN) = Na∂a(etN), to get
D′sp(
−→N ) =
∫
Σ
(πµML−→N (eµM) + (CaKL + eAaKtL)L−→
N (ωaKL)).
which satisfyD′
sp(−→N ),D′
sp(−→N′)
= D′
sp([−→
N ,−→N′
])
where[−→
N ,−→N′
] is the Lie bracket. D′
sp(−→N ) acts on the co-tetrad components eµN and on the
connection ωaNM as diffeomorphisms of the hypersurface Σt
eµN ,D′sp(
−→N )
= L−→
N (eµN),ωaNM ,D′
sp(−→N )
= L−→
N (ωaNM). (42)
The primary constraints πµN and CaNM transform asπµN ,D′
sp(−→N )
= L−→
N (πµN),
CaNM ,D′sp(
−→N )
= L−→
N (CaNM) (43)
which show that, contrary to the constraint Dsp(−→N ), the Poisson brackets of D′
sp(N) with the
primary constraints πµN and CaNM vanish weakly. On the other hand the above transformations imply that the constraints D′
t and M′KLare treated by the spatial diffeomorphism constraint
D′sp(
−→N ) as scalar densities of weight one
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
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D′
sp(−→N ),D′
t
= −L−→
N (D′t) = −∂a(NaD′
t) =⇒D′
sp(−→N ),D′
t(M)= D′
t(L−→N M)
andD′
sp(−→N ), M
′KL(−→x )= −L−→
N (M′KL(−→x )) = −∂a(NaM
′KL(−→x ))
=⇒D′
sp(−→N ),M′(θ)
= M′(L−→
N θ)
which can be verified by a direct computation. Here D′t(M) =
∫Σ
MD′t is the smeared scalar
constraint where M is an arbitrary function and M′(θ) =∫Σ
M′KL θKL2 where θKL may be iden-
tified to the dimensionless infinitesimal arbitrary parameters θKL = δtωtKL.From (42) and (43), We deduce
D′
sp(−→N ),
∫
Σ
CaKLBbNcKaLΛbNc
=
∫
Σ
CaKLBbNcKaLL−→N (ΛbNc) 0
showing that the constraint D′sp(
−→N ) is preserved in the time evolution. In addition, the Poisson
bracket of D′sp(
−→N ) with the condition (34) givesD′
sp(−→N ), BdMaKeLBbNcKeLDcebN
= L−→
N (BdMaKeLBbNcKeLDcebN) 0
which shows that D′sp(
−→N ) is a first-class constraint.
We may also complete the Lorentz constraint M′KL by adding the constraint πtN as
M′KL
2= (Da(CaKL + eAaKtL) +
12(πµKeL
µ − πµLeKµ))
which acts on the co-tetrad components eµN and on the connection ωaNM like local infinitesi-mal transformations of gauge
eµN ,M′(θ) = θLNeµL, ωaNM ,M′(θ) = −DaθNM . (44)
The primary constraints πµN and CaNM transform like the contravariant tensorsπµN ,M′(θ)
= θN
L πµL,
CaNM ,M′(θ)
= θN
L CaLM + θML CaNL (45)
from which we deduce that the Poisson brackets of M′(θ) with the primary constraints πaN and CaNM vanish weakly. The fact that the space-time indices do not transform facilitate the calculation of transformations that M′(θ) generates.
The constraint Dt is a scalar under Lorentz transformations
Dt(−→x ),M′(θ) = 0 =⇒ Dt(
−→x ),M′(θ) = 0
and M′KL is a contravariant tensor
M′KL,M′(θ)= θK
N M′NL + θLNM′KN
leading to the so(1, d − 1) Lie algebra
M′NM(−→x ), M′KL(−→y )=(ηNLM′MK(−→x ) + ηMKM′NL(−→x )
− ηNKM′ML(−→x )− ηMLM′NK(−→x ))δ(−→x −−→y ).
The transformations rules (44) and (45) lead to
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
13
M′(θ),
∫
Σ
CaKLBbNcKaLΛbNc
=
∫
Σ
θMN CaKLBbMcKaLΛ
bNc
andM′(θ), BdMaKeLBbNcKeLDcebN
= θQ
MBdQaKeLBbNcKeLDcebN
showing that M′KL is first-class constraint.Finally, a straightforward computation gives
D′t(M),D′
t(M′) = 0.
But (37) shows that the Poisson brackets of D′t(M) with the primary constraints CaNM and
with ∫Σ
CaKLBbNcKaLΛbNc vanish modulo the constraint (34). Therefore, the scalar constraint
D′t is preserved under the time evolution and can be considered as a first-class constraint only
if the condition (34) is satisfied.In conclusion, we are in presence of a Hamiltonian formalism of the tetrad-connection
gravity composed of first-class constraints, πtN ,P tNM , the Lorentz constraintM′(θ) and the
spatial diffeomorphism constraint D′sp(
−→N ). Although the scalar constraint D′
t(M) forms with
M′(θ) and D′sp(
−→N ) a closed algebra with structure constants, its Poisson bracket with the
primary constraint CaNM vanishes weakly only if the condition (34) is resolved.
3. The fixing of the non-dynamical connection
In spite of the fact that we have obtained a closed algebra in terms of structure constants, the constraint (34) is problematic because of the difficulties to check its consistency. To avoid this problem, we decompose the spatial connection as
ωaKL = ω1aKL + ω2aKL
where
ω1aKL = PPdQ1KaLωdPQ = BbNtKaLBbNtPdQωdPQ
and
ω2aKL = PPdQ2KaLωdPQ =
12
BbNcKaLBbNcPdQωdPQ.
It is easy to check from (29)–(31) that
PPdQ1KaL + PPdQ
2KaL =12δd
a
(δP
KδQL − δP
LδQK
),
PNbM1KaLPPdQ
1NbM = PPdQ1KaL, PNbM
2KaLPPdQ2NbM = PPdQ
2Kal
and
PNbM1KaLPPdQ
2NbM = 0
which show that PPdQ1KaL and PPdQ
2KaL are projectors.These projections of the connection is motivated by the fact that the time derivative of
ω2aKL does not contribute to the kinematic part of the action (4). In fact, from the identities
BbNcKaLeaL = BbNcKaLeaK = BbNcKaLetL = BbNcKaLetK = 0
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
14
we deduce the relations
eAaKtLPPdQ2KaL = 0 and eAaKtLPPdQ
1KaL = eAdPtQ
from which we get
eAaKtL∂tω2aKL = ∂t(eAaKtLω2aKL)− ∂t(eAaKtL)ω2aKL
= −eBbNtKaL(∂tebN)ω2aKL = 0
as a consequence of (31). Therefore, like for the temporal component of the co-tetrad and of the connection, the projected spatial connection ω2aKL is non-dynamic.
We also have
eAaKtLD2aωtKL = eAaKtLPPdQ2KaLDdωtPQ = 0
implying
eAaKtLDaωtKL = eAaKtLPPdQ1KaLDdωtPQ = eAaKtLDω1
1aωtKL
which shows that only the projected part PPdQ1KaLDdωtPQ = Dω1
1aωtKL of the covariant derivative of ωtKL given in terms of the connection ω1aKL contributes to the action. So, the action (4) can be rewritten under the form
S(e,ω) =∫
M
(eAaKtL (∂tω1aKL − Dω1
1aωtKL)− eAaKbL ΩabKL
2
) (46)
showing that the two parts of the spacial connection do not play the same role. The non-dynamic spatial connection ω2aKL contributes only to the third term. The above section showed us that the temporal components of the connection are Lagrange multipliers and δtωtKL = θKL play the role of infinitesimal dimensionless parameters of local transformations of the Lorentz group under which the spatial connection transforms as δωaKL = −DaθKL = DaδtωtKL. Since the projected part D2aωtKL does not contribute to the action, we can fix it to zero without modifying the action (4). We will show that this gauge fixing results from the fixing of the non-dynamic connection ω2aKL = 0.
Before showing how the gauge transformations of the connection allow us to fix the non-dynamic connection ω2aKL to zero, let us note that the ranks of the propagators PPdQ
1KaL and PPdQ
2KaL, given by their trace,
12δa
d
(δK
P δLQ − δK
P δLP
)PPdQ
1KaL = d(d − 1)
and
12δa
d
(δK
P δLQ − δK
P δLP
)PPdQ
2KaL =12
d(d − 1)(d − 3)
are equal exactly to the number of independent components ω1aKL and ω2aKL respectively. On the other side the number 1
2 d(d − 1)(d − 3) of independent relations (34) which is equal to 12 d(d − 1)(d − 2) relations in (34) minus 12 d(d − 1) identities
eBdNcPtQ (DcedN − DdecN − e−1BdNcKtLMKL) = 2MPQ − 2MPQ = 0
corresponds exactly to the number of components ω2aKL. Since the equation which results from the functional derivative of the action (46) with respect to ω2aKL is (34), an action which does not contain explicitly this projected spacial connection does not lead to (34).
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
15
Now, let us see how to fix the non-dynamic spatial connection. From the Lorentz infinitesi-mal transformations δeµK = θN
K eµN and δωaKL = −DaθKL we deduce
δω1aKL = θNKω1aNL + θN
L ω1aKN − PPdQ1KaL∂dθPQ
and
δω2aKL = θNKω2aNL + θN
L ω2aKN − PPdQ2KaL∂dθPQ
which show that each part of the projected spatial connection transforms independently of the other. This allows us to fix the non-dynamic part of the connection to zero
ω′2aKl = ω2aKL + θN
Kω2aNL + θNL ω2aKN − PPdQ
2KaL∂dθPQ = 0.
A transformation of ω′2aKl gives
ω′′
2aKl = ω′2aKL + θ
′NK ω′
2aNL + θ′NL ω′
2aKN − PPdQ2KaL∂dθ
′PQ
= −PPdQ2KaL∂dθ
′PQ
showing that this fixing of the non-dynamic part of the connection remains invariant if
PPdQ2KaL∂dθ
′PQ = ∂2aθ
′KL = 0 =⇒ ∂aθKL = PPdQ
1KaL∂dθPQ. (47)
On the other hand, in order to keep the same degrees of freedom during gauge transforma-tions of ω1aKL, we impose
δ2ω1aKL = PPdQ2KaLδω1dPQ = −PPdQ
2KaL(θNPω1aNQ + θN
Qω1aPN) = 0 (48)
implying
(θNKω1aNL + θN
L ω1aKN) = PPdQ1KaL(θ
NPω1aNQ + θN
Qω1aPN) = 0.
The relations (47) and (48) show that the fixing of the non-dynamic connection to zero does not restrict the gauge parameters but only the gauge transformations of the dynamic part of the spatial connection to
δeµK = θNK eµN , δωtNM = −DtθNM and δω1aKL = −Dω1
1a θKL (49)
subject to the condition
Dω12a θKL = 0 and ∂2aθKL = 0. (50)
Since the first and the second term of (46) do not depend of ω2aKL, to verify that the Lagrangian density (46), where ω2aKL = 0, is invariant under the infinitesimal gauge transfor-mations (49) subject to the conditions (50), it suffices to check the invariance of the third term
δ
(eAaKbL Ω(ω1)abKL
2
)= (θK
N eAaNbL + θLNeAaKbN)
Ω(ω1)abKL
2
− eAaKbLDω1a Dω1
1b θKL
= (θKN eAaNbL + θL
NeAaKbN)Ω(ω1)abKL
2− eAaKbLDω1
a Dω1b θKL
= (θKN eAaNbL + θL
NeAaKbN)Ω(ω1)abKL
2
− eAaKbL(Ω(ω1)abKN
2θN
L +Ω(ω1)abLN
2θN
K
)= 0.
(51)
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
16
In what follows, we consider the action (4) where we fix ω2aKL to zero:
S f (e,ω1) =
∫
M
(eAaKtL (∂tω1aKL − Dω1
1aωtKL)− eAaKbL Ω(ω1)abKL
2
). (52)
The fixed phase space, etN, eaN, ωtKL, ω1aKL and their conjugate momenta πtN , πaN , P tKL and PaKL
1 = PKaLPbQPbPQ , is equipped by the following non zero Poisson brackets
eαI(
−→x ),πβN(−→y )= δβαδ
NI δ(
−→x −−→y ),ωtIJ(
−→x ),P tKL (−→y )=
12(δK
I δLJ − δL
I δKJ )δ(
−→x −−→y ),ω1aIJ(
−→x ),PbKL1 (−→y )
= PKbL
1IaJδ(−→x −−→y ).
The primary constraints are
πtN = 0,P tKL = 0,πbN = 0, CaKL1 = PaKL
1 − eAaKtL = 0
and the total fixed Hamiltonian is
H fT =
∫
Σ
(πtNΛN + P tKL AtKL
2+ πbNΛbN + CaKL
1A1aKL
2) + H f
0 (53)
where
H f0 =
∫
Σ
(eAaKbL Ω(ω1)abKL
2+ eAaKtLDω1
1aωtKL).
Before we start the analysis of this Hamiltonian, which will be performed step by step in complete analogy with the treatment of the previous section, let us precise some remarks con-cerning the Poisson brackets between the elements of the fixed phase space. Like for the gauge transformations of ω1aKL (49), to keep the same degrees of freedom, we project the Poisson brackets acting on the projected elements of the phase space as
.,ω1aKL1 = PPdQ1KaL .,ω1dPQ
leading toπαN ,ω1aKL
1 = PPbQ
1KaL
παN ,ω1dbQ
= PPdQ
1KaL
παN , PRdS
1PbQωdRS
= PPdQ1KaL
παN , PRdS
1PbQ
ωdRS
= PPdQ1KaL
παN , PRdS
1PbQ
ω1dRS = 0
(54)
where we have used ω2dRS = 0 and P(δP)P = 0 true for any projector P. The same computa-tion gives
παN(−→x ),PaKL
1 (−→y )
1 = 0 (55)
from which we deduceπaN(−→x ), CbKL
1 (−→y )
1 = −eBaNtKbLδ(−→x −−→y ).
With these projected Poisson brackets the Jacobi identities are satisfied. For exampleπαN ,
ω1aIJ ,PbKL
1
1
1+PbKL
1 ,παN ,ω1aIJ
1
1+ω1aIJ ,
PbKL
1 ,παN1
1= 0
as a consequence of (54), (55) and
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
17
παN ,
ω1aIJ ,PbKL
1
1
1= PPdQ
1KaL
παN ,
ω1dPQ,PcNM
1
PKbL
1NcM
= PPdQ1KaL
παN , PNcM
1PdQ
PKbL
1NcM = 0.
Now we are ready to perform the treatment of the Hamiltonian (53) by using the pro-jected Poisson brackets. The consistency of the constraint πbN is given by the equation (12) expressed in term of ω1aKL where AaKL is replaced by A1aKL and whose solution is (32) with-out the term containing ΛbNc . The consistency of the constraint CaKL
1 is given by the same equation (13) where DceAcKaL is replaced by its projection Dω1
1c eAcKaL and whose solution is (33) independently of the condition (34). The substitution of Dω1
1c eAcKaL in (13) results from the Poisson bracket of the fixed phase space.
With the new expressions of ΛbN and A1aKL the Hamiltonian takes the compact form
H fT =
∫
Σ
(πtNΛtN + P tKL AtKL
2−D f
t − M fKL ωtKL
2)
where
D ft = −eAaKbL Ω(ω1)abKL
2− πaNDω1
a etN + CaKL1 BbNtKaLBbNcPdQ Ω(ω1)cdPQ
2 0
and
12
M fNM = (Dω1a (CaNM
1 + eAaNtM) +12(πµNeM
µ − πµMeNµ)) 0.
The fixed diffeomorphism constraint is
D fsp(
−→N ) =
∫
Σ
(πµML−→N (eµM) + (CaKL
1 + eAaKtL)L−→N (ω1aKL)).
A direct computation shows that D fsp(
−→N ) satisfies the algebra
D f
sp(−→N ),D f
sp(−→N′)
= D f
sp([−→
N ,−→N′
]). (56)
The transformations induced by D fsp(
−→N ) on the primary constraints are given by
πµN ,D f
sp(−→N )
= L−→
N (πµN),
CaNM1 ,D f
sp(−→N )
= L1
−→N (CaNM
1 )
and on the co-tetrad and the spatial components of the dynamic connection by
eµN ,D fsp(
−→N )
= L−→
N (eµN),ω1aNM ,D f
sp(−→N )
= L1
−→N (ω1aNM).
The transformations induced by M f (θ) on the primary constraints are given byπµN ,M f (θ)
= θN
L πµL,
CaNM
1 ,M f (θ)= θN
L CaLM1 + θM
L CaNL1
and on the co-tetrad and the spatial components of the dynamic connection by
eµN ,M f (θ)= θL
NeµL,ω1aNM ,M f (θ)
= −D1aθNM
where θKL are subject to the condition (50).
The above transformations show that the Poisson brackets of D fsp(
−→N ) and M f (θ) with the
primary constraints vanish weakly. Like in the previous section, D ft and MfNM are transformed
as scalar densities of weight one by the diffeomorphisms
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
18
D f
sp(−→N ),D f
t (M)= D f
t (L−→N (M)),
D f
sp(−→N ),M(θ)
= M f (L−→
N (θ)).
(57)
D ft is transformed under M f (θ) as a scalare
M f (θ),D f
t (M)= 0 (58)
and MfNM as a tensor
M fKL,M f (θ)= θK
N M fNL + θLNM fKN
from which we deduce the so(1, d − 1) Lie algebra for the generators MfKL.The Poisson bracket of D f
t with πaN(−→x ) givesπaN(−→x ),D f
t (−→y )
=
−12
CeKL1 (−→y )
δ
δeaN(−→x )
(BbMtKeLBbMcPdQ(−→y ))ΩcdPQ(
−→y )
2 0
and with CaNM1 (−→x ) gives
CaNM
1 (−→x ),D ft (−→y )
=Dω1
1b (eAbNaM) + eBbQtNaMDω1b etQ
+ Dω11c (C
dKL1 BbNtKdLBbNcNaM)
=PNaM1KdL(eBcQbKdL)Dω1
b ecQ
+ Dω11c (C
dKL1 BbNtKdLBbNcNaM)
=− BePtKtLBePtNaMMKL
+ Dω11c (C
dKL1 BbNtKdLBbNcNaM) 0
which show that the constraint D ft commutes weakly, in terms of Poisson brackets, with the
primary constraints πaK and CaKL1 . Finally, from a direct calculation we get
D f
t (M),D ft (M
′)= 0 (59)
which shows that the Hamiltonian treatment is coherent and the algebra of the first-class con-
straints M f (θ), D fsp(
−→N ) and D f
t (M) closes with structure constants. The function M and the
vector field −→N may be identified with the usual lapse and shift respectively although they do
not result from the A.D.M. formalism.
In addition of the first-class constraints M f (θ), D fsp(
−→N ) and D f
t (M), we have the first-class
constraints πtN and P tKL and the second-class constraints πaN and CaNM1 . The physical degrees
of freedom per point in space-time are obtained by subtracting from the d(5d − 3) degrees of freedom of the fixed phase space the number 2d(d − 1) of the second-class constraints and twice the number d(d + 1) of the first-class constraints to get d(d − 3) which is exactly the number of the degrees of freedom of the physical phase space of the d-dimensional general relativity.
Remark: in order to avoid the constraint (34), one might wonder what happen if instead of fixing the non dynamic part of the connection to zero one solve its equation of motion and then put the solution back into the action as in [18]. Since the decomposition of the connection in dynamic and non-dynamic part is unique, the part of solution of the zero-torsion [19] which corresponds to ω2aKL must be of the form
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
19
ω2aKL = PbPQ2aKLeµP∇beµQ = PbPQ
2aKLeµP(∂beµQ + ΓµbνeνQ)
where ∇aeµM = ∂aeµM + ΓµaνeνM is the covariant derivative with respect to Christoffel’s symbol
Γµaν whose expression contains linearly the derivatives of the components of the co-tetrad. an
explicit computation shows that ω2aKL contains linearly only the time derivatives of the spatial components of the co-tetrad. This leaves πtN as a primary constraint but not πaN which takes the form
πaN =(Dω1
b eAbKaL + ωK2bMeAbMaL + ωL
2bMeAbKaM) δω2aKL
δ∂teaN
where δω2aKLδ∂teaN
= PbPQ2aKLeµP
δδ∂teaN
(Γµaν)e
νQ is not null. The linear dependence of πaN with respect
to the time derivative of the spatial components of the co-tetrad leads to a second order formalism.
4. The algebra of constraints in terms of Dirac brackets
In the previous section we have showed that the set of constraints is complete and closed meaning that the total Hamiltonian H f
T (53) is consistent. In this section we consider the second-class constraints πaN and CaKL
1 as strong equalities by eliminating them. In this case the algebra of the first-class constraints must be computed with the projected Dirac brackets defined in terms of the Poisson brackets of the previous section as
A, BD = A, B1 − A, Ci1 Ci, Cj−11 Cj, B1
where Ci =(πaN , CaKL
1
). The inverse of the Poisson bracket
πbN(−→x ), CaKL
1 (−→y )
1 eBbNaKtLδ(−→x −−→y ) = −eBbNtKaLδ(−→x −−→y )
is given by πbN , CaKL
1
−11 = e−1BbNtKaL satisfying
πbN , CaKL
1
−11
CaKL
1 ,πcM1 = δc
bδMN
and
CaKL1 ,πbN−1
1
πbN , CdPQ
1
1= PPdQ
1KaL.
Now we can consider the constraints of second-class πbN and CaKL1 as strong equalities by
eliminating them from the total Hamiltonian H′T to get the reduced Hamiltonian
HrT =
∫
Σ
(πtNΛtN + P tKL AtKL
2−Dr
t − MrKL ωtKL
2) (60)
where
MrKL = (πtKeLt − πtLeK
t ) + 2Dω1a (eAaKtL)
and
Drt = −eAaKbL Ω(ω1)abKL
2. (61)
The diffeomorphism constraint reduces to
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
20
Drsp(
−→N ) =
∫
Σ
(πtNL−→
N (etN) + eAaKtLL−→N (ω1aKL)
).
The Hamiltonian (60) is defined on the reduced phase space eaN, ω1aKL, etN, πtK , ωtKL, and P tKL equipped with the following non zero Dirac brackets:
etN(
−→x ),πtM(−→y )
D = δMN δ(−→x −−→y ),
ωtIJ(
−→x ),P tKL (−→y )
D =12(δK
I δLJ − δL
I δKJ )δ(
−→x −−→y )
and
eaN(−→x ),ω1bKL(
−→y )D = e−1BaNtKbLδ(−→x −−→y ). (62)
Note that as opposite to the results obtained in [13], the dynamic connection is Dirac self commuting as a consequence of (54).
These projected Dirac brackets guarantee the Jacobi identities. In fact for the non trivial example
ecN , ω1aKL,ω1bPQD
D + ω1bPQ, ecN ,ω1aKLDD +
ω1aKL, ω1bPQ, ecND
D
the first term vanishes and the second and third terms give
− PRdS1Kal
δ
δehI(e−1BcNtRdS)e−1BhItPbQ + PRdS
1PbQδ
δehI(e−1BcNtRdS)e−1BhItKaL
= ehIe−1BcNtKaLe−1BhItPbQ − ehIe−1BcNtPbQe−1BhItKaL
− PRdS1Kale
−1 δ
δehI(BcNtRdS)e−1BhItPbQ + e−1PRdS
1PbQδ
δehI(BcNtRdS)e−1BhItKaL
where we have used
ω1aKL, e−1BcNtPbQ
D = PRdS
1PbQδ
δehI(e−1BcNtRdS) ω1aKL, ehID
= −PRdS1PbQ
δ
δehI(e−1BcNtRdS)e−1BhItKaL
and δδehI
e−1 = −e−1ehI . A direct computation leads to
+ e−1PRdS1PbQ
δ
δehI(BcNtRdS)e−1BhItKaL − PRdS
1Kale−1 δ
δehI(BcNtRdS)e−1BhItPbQ
= −ehIe−1BcNtKaLe−1BhItPbQ + ehIe−1BcNtPbQe−1BhItKaL
showing that the Jacobi identities are satisfied.Now we are ready to calculate the algebra of constraints in terms of the projected Dirac
brackets. The smeared Lorentz Constraint
Mr(θ) =
∫
Σ
((πtK1 eL
t − πtL1 eK
t ) + 2Dω1a eAaKtL)
θKL
2=
∫
Σ
MrKL θKL
2
acts on eµN and ω1aKL like local infinitesimal transformations of gauge
eµN ,Mr(θ)D = θMN eµM , ω1aKL,Mr(θ)D = −D1aθKL
leading to
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
21
Mr(θ),Drt (M)D = 0
as a consequence of (51) and to
MrKL,Mr(θ)
D = θKN MrNL + θL
NMrKN
from which we get the so(1, d − 1) Lie algebra for the generators MrKL.From a direct computation we get for the spatial diffeomorphism constraint
Dr
sp(−→N ),Dr
sp(−→N′)
D= Dr
sp([−→
N ,−→N′
]).
The transformations induced by Drsp(
−→N ) on eµN and ω1aKL are given by
eµN ,Dr
sp(−→N )
D= L−→
N (eµN),ω1aKL,Dr
sp(−→N )
D= L1
−→N (ω1aKL).
In view of these transformations we then deduce that Drt and MrKL transform under the
spatial diffeomorphisms as scalar densities of weight one leading toDr
sp(−→N ),Dr
t (M)
D= Dr
t (L−→N (M)),
Dr
sp(−→N ),Mr(θ)
= Mr(L−→
N (θ)).
Finally, a direct calculation gives for the smeared scalar constraint
Drt (M),Dr
t (M′)D =
∫
Σ
MeAaKbL Ω(ω1)abKL
2,∫
Σ
M′eAcNdM Ω(ω1)cdNM
2
D
=
∫
Σ
Da(MeAaKbL)M′eBhQcNdMe−1BhQtKbLΩ(ω1)cdNM
2
−∫
Σ
Dc(M′eAcNdM)MeBhQaKbLe−1BhQtNdMΩ(ω1)abKL
2
=
∫
Σ
MM′Da(eAaKbL)eBhQcNdMe−1BhQtKbLΩ(ω1)cdNM
2
−∫
Σ
MM′Dc(eAcNdM)eBhQaKbLe−1BhQtNdMΩ(ω1)abKL
2= 0
where we have used∫
Σ
∂a(M)eAaKbLBhQcNdMBhQtKbLΩ(ω1)cdNM
2
=
∫
Σ
∂a(M)eAaKbL 12(d − 2)
etQAbKhLBhQcNdM Ω(ω1)cdNM
2= 0
due to AaKbLAtKcL = 0 and etQBhQcNdM = 0. The above Dirac brackets between Drsp(
−→N ),
Mr(θ) and Drt (M) show that the algebra of the reduced first-class constraints closes with
structure constants.As in the previous section, the degrees of freedom of the physical phase space d(d − 3) are
exactly the number of the degrees of freedom of the physical phase space of the d − dimen-sional general relativity.
In terms of Dirac brackets, the Hamiltonian (60) propagates the phase space variable from an initial hypersurface Σt0 to the hypersurface Σt0+δt . In this sense, we can interpret geo-metrically that the Hamiltonian (60) propagates the hypersurface Σt0 in the space-time. This
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
22
propagation is done along the infinitesimal time δt or Mδt depending on whether we consider the scalar density Dr
t or Drt (M) in the Hamilton equation while the Lorentz constraint induces,
during this propagation, a gauge transformation of infinitesimal parameter θKL = δtωtKL. The constraint (61) acts independently of (60) and induces diffeomorphisms on each hypersurfaces.
We end this section by solving Hamilton’s equations in terms of Dirac brackets. For the co-tetrad components ecN, we get
∂tecN(x) = ecN(x),HrTD = −Dω1
a (eAaKbL)e−1BcNtKbL
+ eAaKtL(e−1BcNtKaMωMtL + e−1BcNtLaMω
MtK).
The first term of the right hand side gives
−Dω1a (eAaKbL)e−1BcNtKbL = −e−1BcNtKbLeBµMaKbLDω1
a eµM
= Dω1c etM − eBcNtKbLe−1BdMaKbLDω1
a edM
= Dω1c etM + e−1BcNtKtLMKL
and a direct computation gives for the second term of the right hand side
eAaKtL(e−1BcNtKaMωMtL + e−1BcNtLaMω
MtK) = −ωM
tNecM
leading to
DteaN − Dω1a etN = BcNtKtLMKL.
Since ωtKLcan be considered as Lagrange multiplier, in addition to the above equation we have
BdMaKtLDaedM = Dω1a
(eAaKtL) = MKL = 0
leading
DteaN − Dω1a etN = 0.
The solutions of Dω1a
(eAaKtL
)= MKL = 0 are ωs
aKL = eµN∇aeµM = eµN(∂aeµM + ΓµaνeνM).
The solution ωsaKL is injected in the second equation to get
DteaN − Dωs
a etN = ∂teaN + ωtNMeMa − ∂aetN − ωs
aNMeMt
= ∂teaN + ωtNMeMa − ∂aetN +∇aeµNeµMeM
t
= ∂teaN + ωtNMeMa − Γµ
ateµM
leading to ωtNM = eµN∇teµM = eµN(∂te
µM + Γµ
tνeνM) [19] which exhibits solutions of the zero torsion, condition to have an equivalence between the tetrad-connection gravity and the gen-eral relativity.
5. Conclusion
The results presented in this paper show that a modified action of the tetrad-gravity where the non-dynamic connection ω2aKL is fixed to zero makes possible the construction of a consistent Hamiltonian formulation for any dimension d 3 without Barbero–Immirzi’s parameter nei-ther the A.D.M. decomposition of the action.
Unlike the works where the ADM decomposition of action is taken as starting point leading to a Hamiltonian system where the algebra of the first-class constraints closes with structure functions we have showed that, by starting from a phase space without using the A.D.M.
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
23
decomposition, we get a coherent Hamiltonian formalism with an algebra of the first-class constraints which closes with structure constants. This algebra expresses the invariance of the action under a true Lie group whose generators are the first class constraints.
The absence of structure functions is due to the fact that the scalar function M and the spatial vector field
−→N are introduced as test functions independently of the co-tetrad not as
objects which result from the A.D.M. decomposition of the tangent space where the pure deformation of the space-like hypersurface, expressed in term of the lapse, requires for its definition the metric of space-time which appears in the structure functions [20]. It was shown in [21] that it is possible to obtain an algebra of the diffeomorphism constraints which closes with structure constants by considering the general transformations of the coordinates which depend on the metric.
In [16] the simplicity constraint, which corresponds to the primary constraint CaKL of the section 2, is split into boost and non-boost part while in our case the decomposed is done by the projectors like CaKL = CaKL
1 + CaKL2 where CaKL
1 is the primary constraint of the section 3 and CaKL
2 = PaKL2 = PcNMPKaL
2NcM is the conjugate momenta of the non-dynamic part of the connection which is a primary constraint whose the consistency condition is (34). The fixing in the action of the non-dynamic part of connection to zero allowed us to eliminate the con-straint (34) to get a coherent Hamiltonian formalism.
Note that all reduced first-class constraints are polynomial but the phase space variables obey a non polynomial Dirac bracket because of the presence of e−1 (62). Since the rank of the projector PKaL
1NcM is d(d − 1), the number of the independents components of the dynamic connection is equal to the ones of the co-tetrad eaK. This allows us to perform an invertible transformation
ω1aKL −→ PbN = eBbNtKaLω1aKL ⇐⇒ ω1aKL = PbNe−1BbNtKaL
to get a reduced phase space obeying the following canonical commutation relations in terms of Dirac brackets
eaN(
−→x ),PbM(−→y )
D = δbaδ
MN δ(−→x −−→y ),
eaN(−→x ), ebM(
−→y )D = 0 andPaN(−→x ),PbM(−→y )
D = 0.
(63)
Substituting ω1aKL by PbNe−1BbNtKaL, we get polynomial constraints for
MrKL = ∂aeAaKtL +12(PaKeL
a − PaLeKa )
and
Drsp(
−→N ) =
∫
Σ
(πtNL−→
N (etN) + PaNL−→N (eaN)
)
whereas the scalar constraint becomes non-polynomial because of the quadratic term of the dynamic connection ω1aKL(e,P) = PbNe−1BbNtKaL in the curvature Ω(ω1)abKL. From the same calculation done in section 4 to check the Jacobi identities we deduce
ω1aKL,ω1dPQD =PbNe−1BbNtKaL,PcMe−1BcMtPdQ
D = 0
conducting, with the use of (63), to the same algebra of first class constraints of the previous section. A similar approach with the same phase space was adopted in [18] where the con-nection is decomposed in dynamic and non-dynamic part and where the latter is replaced by the solution of its equation of motion. If the algebra of the lorentz constraints is explicitly computed, because they depend only on the dynamic part of the connection, the computation
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
24
of the bracket between the translation constraints raises problems because of the reasons given in the remark of the section 3.
In three dimension eBbNaKcL vanishes, as a result of the antisymmetry of the spacial indices b,a and c, implying that the projected connection ω2aKL disappears in the same way. All the previous results are valid in the three dimensional case. In addition, the first class constraint PN
can be retained instead of Dt(M) and Dsp(−→N ). PN and MKL obey the following Dirac bracket
MNM(−→x ), MKL(−→y )
D =(ηNLMMK(−→x ) + ηMKMNL(−→x )
− ηNKMML(−→x )− ηMLMNK(−→x ))δ(−→x −−→y ),
(64)
PN(−→x ), MKL(−→y )
D =(ηNLPK(−→x )− ηNKPL(−→y )
)δ(−→x −−→y ) (65)
and
PN(−→x ), PM(−→y )
D = 0 (66)
which exhibit the Lie algebra of the Poincaré group, where δtetN plays the role of the infini-tesimal local translation. This shows that, in the three dimensional case, we can either consider the Poincaré group or the Lorentz and diffeomorphism group. The physical degrees of free-dom vanish, expressing the topological character of the d = 3 gravity.
We end this paper by noticing that, as opposed to [22] where it is claimed that diffeomor-phism invariance is not a gauge symmetry derived from the first-class constraints of the tetrad-gravity or in [23] where the equivalence between the translation transformations of Poincaré group and the diffeomorphism can only be established on-shell. In this paper, we have shown that the symmetries of the Hamiltonian formalism of three dimensional tetrad-connection gravity are obtained either through the first-class constraints of Poincaré group or the ones of Lorentz and diffeomorphisms which show that this equivalence is established off-shell.
For d 4, the smeared first-class constraint PNcan be written as a sum of first-class constraints
P(N) =
∫NNeBtNaKbL ΩabKL
2= −Dt(N) +Dsp(
−→N ) +Mr(θ)
where Dsp(−→N ) =
∫Σ
eAaKtLL−→N (ω1aKL), Mr(θ) =
∫Σ
Dω1a eAaKtLθKL, NN is a Lorentzian vec-
tor, N = NNetN , θKL = NNeaNωaKL and NNeaN are the d − 1 components of the vector −→N
tangent to Σt . The Dirac bracket between the smeared translation constraints does not vanish but gives a sum of which a part is linear in the first-class constraints with structure functions and the other quadratic. The quadratic part results from the fact that the functions N,
−→N and
θKL depend on the components of the tetrad and thus contribute to the results of Dirac’s bracket. This shows that PN does not satisfy (66) and therefore does not correspond to the translational part of the Lie algebra of the Poincaré group. Only the Lorentz group and the diffeomorphisms are symmetries, not the translation part of the Poincaré group, contrary to what is claimed in [24].
Acknowledgments
M Lagraa would like to thank Michel Dubois-Violette for helpful discussions.
M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
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M H Lagraa et alClass. Quantum Grav. 34 (2017) 115010
Abstract
This thesis presents a detailed analysis of the formalism Hamiltonian of the tetrad-connection Gravity at
d-dimension without decomposing the tangent space of the space-time manifold by the ADM
decomposition. To avoid the problematic constraints in the Hamiltonian treatment of the theory, we
decompose the connection into dynamical and non-dynamical parts, and fix the non-dynamical part to
zero. The application of the Dirac procedure with this fixing has allowed to obtain a covariant formalism
under the Lorentz transformations and an algebra of first-class constraints closing on structure constants
rather than the usual structure functions resulting from the ADM decomposition. The Dirac brackets
deliver the same algebra of the first-class constraints, while the second-class ones are eliminated by the
strong equality. At the end, we obtain the same physical degrees of freedom of the theory of General
Relativity. The Hamiltonian analysis is also performed when Gravity is coupled to fermionic matter. The
obtained results establish an algebra of first-class constraints closing on structure constants as the case of
pure Gravity. We also obtain, by canonical transformations, a new reduced Phase space equipped with
canonical Dirac brackets leading to the same algebra of first-class constraints.
Keywords:
Tetrad-Connection Gravity; Hamiltonian Formalism; Einstein-Cartan Theory; Spin-connection Projection ;
Algebra of First-class Constraints; Gravity Constraints ; Dirac Brackets ; Dirac Quantification Program;
Dirac Spinors; Fermions and Gravity.
