Alma Mater Studiorum Universita di Bologna

FACOLTA DI SCIENZE MATEMATICHE, FISICHE E NATURALI

Corso di Laurea Specialistica in Fisica

MAJORANA SPINORPAIR CREATION

IN ACCELERATED FRAMES

Tesi di Laurea in Teoria dei Campi

Relatore:

Chiar.mo Prof.ROBERTO SOLDATI

Presentata da:PIETRO LONGHI

Sessione I

Anno Accademico 2009-2010

Ai miei genitori Nadia e Franco,

con gratitudine.

Contents

Abstract 1

Introduction 3

Acknowledgements 7

1 Quantum Field Theory in non-Minkowskian geometries 9

1.1 Generalized spin-0 field theory 9

1.2 Spin-0 fields in Rindler geometry 12

a. The Rindler frame 12

b. The scalar field 14

1.3 Defining spinors: the generalized theory of spin-12

fields 16

1.4 The Majorana field in Minkowski spacetime 21

1.5 Majorana spinors in Rindler geometry 27

a. Solving the Majorana-Rindler equation 28

b. The basis of helicity eigenstates 38

c. Study of the hermiticity of the Hamiltonian 42

2 The Unruh effect 45

2.1 The Bogolyubov transform 46

2.2 General theory of the Unruh effect 48

I

CONTENTS

2.3 Unruh effect for the spin-0 field 53

2.4 Unruh effect for the Majorana field 57

a. Finding the spinor algebraic RS-to-MS transformation 58

b. Consistency with the general theory of spinors in curved

spacetimes 62

c. Helicity-eigenstate normal modes 63

d. Canonical normal modes 65

e. Choosing the proper representation for helicity eigenstates 66

f. Choosing the proper representation for canonical modes 71

g. Digression: normalization of Rindler modes in RS 75

h. Comparing the helicity-eigenstate scheme with the canon-

ical modes one: advantages of each scheme 76

i. The thermal spectrum for Majorana fermions 78

2.5 A different derivation of the Unruh effect: helicity structure 80

2.6 Criticisms and discussions on the Unruh effect 86

3 Dark Matter 89

3.1 The idea: a connection with Dark Matter 89

3.2 Dark matter models 91

3.3 Heuristic evaluation of the energy density 94

3.4 Majorana-Unruh fermions in strong gravitational fields 98

4 Conclusions 103

Appendix 107

A.1 Orthonormality and completeness of Rindler modes 107

a. A study of the scalar Rindler modes 107

b. Proof: orthonormality of Majorana Rindler modes in MS 111

c. Proof: completeness of Majorana Rindler modes in MS 113

d. Proof: completeness of Unruh modes in MS 114

A.2 Alternative derivation of the Unruh effect 115

Table of constants 118

Bibliography i

II

Abstract

Nel primo capitolo presenteremo gli strumenti necessari alla riformu-

lazione della teoria dei campi in maniera generalmente covariante, studieremo

poi le teorie di campo scalare e di Majorana dal punto di vista di un osserva-

tore uniformemente accelerato. Eseguiremo uno studio esplicito e dettagliato

di entrambe le teorie, dal punto di vista classico dapprima, quantistico poi.

L’obiettivo del capitolo e quello di acquisire tutti gli strumenti necessari ad

un’analisi approfondita dell’effetto Unruh.

Il secondo capitolo e dedicato allo studio dell’effetto Unruh per i campi

scalare e di Majorana. Dopo aver speso qualche cenno sulla teoria delle

trasformazioni di Bogolyubov, tratteremo in maniera del tutto generale la

teoria dell’effetto Unruh: mostreremo che un oggetto del tutto naturale in

relativita, come una trasformazione generale di coordinate, puo indurre effetti

drammatici sullo schema di quantizzazione come portare all’inequivalenza tra

spazi di Fock. Procederemo analizzando questo inscindibile legame tra op-

eratori di seconda quantizzazione e sistemi di coordinate nei casi di campo

scalare e di Majorana. Nel caso di quest’ultimo seguiremo due possibili strade

equivalenti, di cui una ci permettera di formulare la teoria quantistica in

maniera particolarmente agevole, mentre l’altra avra il pregio di preservare

il significato fisico della trattazione. Il taglio della trattazione e prettamente

tecnico e particolare attenzione e posta nello studio dei modi di Unruh ot-

tenuti, si dimostra in particolare che: sono analitici su tutto lo spaziotempo

1

Abstract

di Minkowski, si riducono ai modi di Rindler opportunamente trasformati nei

rispettivi settori, sono un set ortonormale e completo e sono dunque adatti

per costruirvi una teoria quantistica. Lo studio dell’effetto Unruh si conclude

con un’analisi della struttura dei coefficienti di Bogolyubov per lo spinore di

Majorana, in cui stabiliremo le relazioni tra stati fisicamente osservabili dagli

osservatori inerziale e non. Il capitolo chiude con una rassegna sulle recenti

critiche e dispute su problemi di natura matematica legati a certe derivazioni

dell’effetto Unruh.

Nel terzo capitolo sfrutteremo infine i risultati ottenuti per il campo di

Majorana studiando la possibilita di generare materia oscura tramite il mec-

canismo di Unruh. Dapprima introdurremo alcune ipotesi necessarie per

giustificare la possibile presenza di un ipotetico campo di Majorana. Pre-

senteremo poi in sintesi alcuni tra i modelli piu recenti di distribuzioni di

materia oscura e i rispettivi candidati particellari. Svolgeremo dunque un

derivazione euristica della densita di tali fermioni generati tramite meccan-

ismo di Unruh, ipotizzando l’accelerazione cosmica come causa scatenante di

tale effetto. In una seconda parte studieremo lo stesso meccanismo in pre-

senza di accelerazione gravitazionale da buco nero, esplicitando l’analogia con

l’effetto Hawking. Il capitolo conclude con una rassegna sugli attuali risultati

circa la distribuzione di materia oscura ai livelli galattico e di grande scala.

2

Introduction

Physics is just the refinement of everyday thinking

Albert Einstein

The unification of quantum field theories with the theory of general rela-

tivity is being, since the seventies, among the greatest efforts in fundamental

theoretical Physics, probably the most ambitious one. Einstein’s elegantly

simple idea is that of a geometrical universe, wherein spacetime and matter

are both main actors, shaping each other according to the laws of general

relativity. Quantum field theory is instead a conceptually complex theory,

which successfully describes the behavior and the properties of the matter

forming the universe at its most fundamental level. While in general rel-

ativity one completely ignores the fundamental structure of matter, in the

quantum theory of fields it is spacetime that is neglected, being treated as a

rigid stage which ignores the effects of the events taking place on it. Both

theories have been confirmed experimentally to the highest orders of preci-

sion, however all the data in our possess regard contexts wherein the effects

of one theory or the other become negligible. Nonetheless there are situations

in which both theories are important : it could be extreme phenomena like

those happening in presence of a black hole or like the origin of universe, but

it could be much more common situations as the presence of dark matter.

3

Introduction

Of the various attempts to unify these theories, we will deal with that

known as quantum field theory in curved spacetime. This semi-classical ap-

proach consists in generalizing the quantum theory of fields through a gener-

ally covariant formulation which makes is possible to incorporate the equiv-

alence principle within the theory. It is not expected to be an exact theory

of nature, but it should provide a good approximate description of those cir-

cumstances in which the effects of quantum gravity do not play a dominant

role. The most striking application of the theory is Hawking’s prediction that

black holes behave as black bodies, emitting a thermal spectrum of radiation

with temperature T κ2π. There was however a very disturbing aspect of

Hawking’s calculation: it appeared to show a divergent density of ultrahigh

energy particles in proximity of the horizon of the black hole. In order to gain

insight on this issue, Unruh made an operational choice of particle: a particle

is a state of the field which can induce a transition in a certain detector appa-

ratus. What Unruh found out was surprising: whenever in flat spacetime a

certain field is in the ordinary Minkowskian vacuum, an accelerated observer

perceives a thermal spectrum of particles of temperature T a2π. These

apparently paradoxical phenomena have their roots in a fundamental fact

that actually lies at the heart of quantum field theory in curved spacetimes:

the notion of particle is not fundamental in QFT, the quantum theory of fields

is, indeed, a theory of fields not particles. In order to better understand the

meaning of this last claim it is necessary to delve technically in the theory

of QFT in curved spacetimes. There are actually three main approaches

to the Unruh effect: (i) analysis of the response of accelerated detectors in

Minkowski spacetime (ii) Unruh’s original derivation which is based on QFT,

without reference to the details of detectors (iii) the algebraic approach, based

largely on the Bisognano-Wichmann theorem which essentially says that the

ordinary Minkowski vacuum, when restricted to observables localized in the

right Rindler wedge, satisfies the Kubo-Martin-Schwinger condition. In this

work we will deal exclusively with Unruh’s original derivation.

The first chapter begins with an overview of the tools that are necessary

to reformulate QFT in a generally covariant way. After reviewing the gen-

eralized Klein Gordon theory we begin to study the case of a scalar field in

Rindler spacetime. We solve the classical theory by finding normal modes

that are suitable for quantization, we then proceed for the quantum theory.

4

Introduction

Thereafter we turn to the case of spinor fields. After explaining how the

connection with the Lorentz group can be achieved in a curved spacetime,

we begin to study the Majorana field in the frame of a Rindler observer. We

first give a schematic treatment of the generalized theory of spinor fields,

explaining how it is possible to preserve the connection with the Lorentz

transformation properties of these fields. Then we begin to study the par-

ticular case of a Majorana field in Rindler spacetime. We found that, in the

literature regarding the Unruh effect, scalar fields are overwhelmingly much

more studied than spinor ones, so we decided to put particular emphasis on

the development of such theory by analyzing it in detail. In particular, we

develop two quantization schemes: one allowing for a cleaner quantization

while the other providing a physical meaning to one-particle states in terms

of physical observables. The aim of this chapter is to acquire all the necessary

tools that we will need for a detailed study of the Unruh effect.

The second chapter is devoted to the study of the Unruh effect for the

scalar and the Majorana fields. We begin with a review of the theory of

Bogolyubov transformation for both bosonic and fermionic systems. Then

we outline the general theory of the Unruh effect: the aim is to show, on

the most general grounds, how a coordinate transformation, which is just a

natural operation in relativity, can have dramatic effects on the quantization

of fields, such as bringing to inequivalences between Fock spaces. The Unruh

effect is then studied first for the scalar field and then for the Majorana one.

The content of this chapter is to a large extent technical and its aim is to

prove that the Unruh modes that we find for the Majorana field have all the

required properties, such as: they are analytical over the whole Minkowski

spacetime, they reduce to Rindler modes within the corresponding sectors of

Rindler spacetime, they are complete and orthonormal. We eventually derive

the spectrum of Majorana particles which turns out to be an exactly thermal

fermionic distribution. We end our study of the Unruh effects with another

analysis of the structure of Bogolyubov coefficients for the Majorana spinor,

wherein we determine the relation between physically observable states for

the inertial and Rindler observers. The chapter ends with a review of the

recent critics and discussions regarding some mathematical issues involved in

certain derivations of the Unruh effect.

In the third chapter we take advantage of the results obtained for the Ma-

5

Introduction

jorana field by studying the possibility to generate dark matter by means of

the Unruh mechanism. We first make some necessary assumptions concerning

the hypothetical existence of a Majorana field, explaining why the Majorana

field could be plausible dark matter candidate. We then briefly review the

most recent models of galactic and extra-galactic distributions of DM and

the corresponding particle candidates. Thereafter we proceed evaluating the

energy density that the Unruh mechanism would produce, we assume the cos-

mic acceleration as the source of the effect. The derivation is heuristic since

the aim is to obtain an order of magnitude for the energy density, in order to

make a comparison with the observed values of dark matter density. Finally

we repeat the evaluation taking into account the strongest non-inertial field

that occurs in our universe, namely the gravitational field of a black hole.

We first exploit the analogy with the Hawking effect and then proceed to

evaluate the Majorana energy density produced by such gravitational field

by means of the Unruh mechanism.

6

Acknowledgements

Acknowledgements

This work was originally conceived as the investigation of an idea sug-

gested by Prof. Roberto Soldati, supervisor of this thesis. First and foremost

I would like to thank him for the guidance and the support he has provided

during the course of this work. Being both new to this fascinating field of

physics, I had the pleasure to discuss with him on many issues that came up

along the path, learning from him an enormous amount. I am also grateful

to Prof. Soldati for he gave me my first exposure to quantum field theory, in

a superb course which pushed me to undertake the way of theoretical physics

(at the time I was a student in experimental high energy physics). A thank

goes out also to Prof. Fabio Ortolani for valuable discussions on issues of

mathematical nature that arised within this work. I had the fortune to meet

many valuable professors and teachers during my studies, I cannot exempt

myself from spending a few words for two of them. Of Prof. Giovanni Carlo

Bonsignori I cannot forget the endless passion for science and for teaching

in all their forms, together with his intuitive way of understanding physics.

Of Stefano Valli, a dear teacher of mathematics of mine, I treasure the rigor-

ous and clear approach to mathematics and physics; his willingness to make

every student understand the subjects certainly finds satisfaction in his in-

credible teaching skills. Last, but not least, I am grateful to Luca Zambelli

and Aurelio Patelli for several enlightening discussions and for their interest

in this work.

The typesetting of this work has been carried out using the freeware program

Kile based on the LATEX standard. The figures were realized using the asymptote

language. I am grateful to all those people who made it possible for these projects

to begin and to remain free.

7

CHAPTER 1

Quantum Field Theory in non-Minkowskian geometries

I believe that ideas such as absolute certitude, absolute exactness, final

truth, etc., are figments of the imagination which should not be

admissible in any field of science... This loosening of thinking seems to

me to be the greatest blessing which modern science has given us. For

the belief in a single truth and in being the possessor thereof is the root

cause of all evil in the world.

Max Born

First the general treatment of Quantum Field Theory in curved space-

times is presented in this chapter for both the Klein-Gordon and the Dirac

fields. We will then turn to the study of Rindler spacetime (we will refer to

it as RS) and its causal structure. Finally we will present a detailed study of

the real scalar and Majorana spinor fields in RS. We will use natural units

i.e. ~ c 1

1.1 Generalized spin-0 field theory

This section is devoted to the study of the Klein-Gordon field as perceived

from a non-inertial point of view: the generalized theory of spin-0 fields

9

Chapter 1. Quantum Field Theory in non-Minkowskian geometries

is presented in order to give all the necessary tools to deal with general

coordinate transformations. In the next section we will focus on the case of

constant proper acceleration, i.e. the Rindler case.

We start with the generalized Lagrangian density for a real scalar field

φpxq of mass m:

Lpxq 1

2rgpxqs12 gµνpxqφ,µpxqφ,νpxq rm2 ξRpxqsφ2pxq( (1.1)

where gpxq is the metric determinant. The coupling between the scalar field

and the gravitational field is given by the term ξRφ2, where ξ is a numerical

factor and R(x) is the Ricci scalar, the only possible local, scalar coupling of

this sort with the correct dimensions. The resulting action is

S »Lpxq d4x . (1.2)

By varying S with respect to φ and setting δS 0 we arrive at the generalized

Klein Gordon equationlm2 ξRpxqφpxq 0 (1.3)

where the D’Alembert operator reads

lφ gµν∇µ∇νφ pgq12Bµpgq12gµνBνφ (1.4)

There are two particular cases, corresponding to two values of ξ that are

interesting: the minimally coupled case (ξ 0) and the conformally coupled

case (ξ p14qrpn 2qpn 1qs) where the positive integer n is the number

of spacetime dimensions. In the latter case, if m 0 the scalar field equation

turns out to be conformally invariant.

Since we will be dealing with flat spacetimes, let us drop the gravitational

coupling term; then it is easy to check that a conserved current of the La-

grangian is

Jµ iφÐÑB µφ ∇ J pgq12Bµrpgq12Jµs 0 (1.5)

and more generally, also the vector current

Jµ12pxq iφ1pxqÐÑB µ

φ2pxq (1.6)

10

1.1. Generalized spin-0 field theory

satisfies the continuity equation, as long as φ1, φ2 satisfy (1.3). It is then

possible to define an invariant scalar product for the field as

pφ1, φ2q i

»Σ

φ1pxqÐÑBλφ2pxq dΣλ (1.7)

where Σ is a three dimensional spacelike hypersurface and

dΣλ 1

3!ελρστ dxρ ^ dxσ ^ dxτ

agpxq (1.8)

ε0123 1 εµνκλ gµτ ετνκλ (1.9)

is the invariant future-oriented hypersurface element. As for the Minkowskian

case, the solutions of (1.3) can be expanded in normal modes

φpxq ¸i

aiuipxq a:iu

i pxq

(1.10)

with the tuiu being a complete and orthonormal set of mode solutions.

Covariant quantization is achieved by imposing the commutation relations

rai, aj:s δij , etc. (1.11)

It is then straightforward to construct a vacuum state, a Fock space and

proceed as usual for the Minkowskian case. Although from the purely math-

ematical point of view we haven’t encountered any difficulties, these show

up as soon as we try to give a physical interpretation of what we just ob-

tained. Indeed it was clearly pointed out by Fulling in [22] that, while in

the Minkowskian case we could readily make a distinction between positive-

frequency and negative-frequency normal modes, this is no longer obvious in

a nonflat metric.

In MS (Minkowski spacetime), the vector BBt is a Killing vector of the space,

orthogonal to the spacelike hypersurfaces t constant and the well-known

modes

rp2πq32ωks12 exp tiωkt ik xu (1.12)

are eigenfunctions of this Killing vector. These modes are closely associated

with the natural coordinates pt, x, y, xq. In turn these coordinates are asso-

ciated with the Poincare group, which leaves the line element unchanged.

In curved spacetime the Poincare group is no longer a symmetry group. As

11

Chapter 1. Quantum Field Theory in non-Minkowskian geometries

a consequence there will be no Killing vectors that can be used to define

positive-frequency modes, at least in general. Further detail is given in [57]

This could be expected, as the very first and most important consequence

of the principle of General Covariance is just that coordinate systems are

physically irrelevant.

Summarizing, we have seen that, at least mathematically, a curved space-

time scalar quantum field theory is viable, although it is unclear what physi-

cal meaning could be attributed to it. We will now leave this interesting and

crucial point and return to it later on. Actually there are some very special

classes of spacetimes in which ’natural coordinates’ analogous to pt, x, y, zqfor MS may exist, together with a timelike Killing vector. This is indeed the

case for Rindler spacetime.

1.2 Spin-0 fields in Rindler geometry

We will now examine the behavior of a scalar field from the point of

view of an accelerated observer. This case, in which the metric is flat, is

nonetheless interesting, since it enjoys some important features due to non-

inertial quantum effects.

a. The Rindler frame

Consider an object moving with constant proper acceleration along the

xaxis through Minkowski spacetime. A typical example of this situation

could be a spaceship with an infinite energy supply and a propulsion engine

that exerts a constant force, another could be an electron inside an infinitely-

wide parallel plane condensator. Let us define the laboratory frame as the

usual inertial reference frame with the coordinates pt, x, y, xq, and the proper

frame as the accelerated system of reference that moves together with the

observer. To describe quantum fields as seen by an accelerated observer,

we need to use the proper coordinates pη, ξ, y, zq, which are also known as

12

1.2. Spin-0 fields in Rindler geometry

ξ=0+ ,

η=+∞ξ

=0 −,η= −∞

ξ=0 +,η= −∞

ξ=0− ,

η=+∞ξ=

const.

η = const.

x

ct

R

L

Figure 1.1: The Rindler wedge: the dashed lines are the event horizons where ξ 0 and

η 8, the hyperbolae are the trajectories of particles at rest with respect to the Rindler

observer (ξ constant), while the stright lines are equal-time spacelike hypersurfaces

with respect to the Rindler time variable (η constant). Notice that η increases towards

Minkowski’s past in the left wedge. The origin is a singularity of coordinate system.

Rindler coordinates. They are defined as

pt, x, y, zq Ñ pη, ξ, y, zq (1.13)

t ξ sinh aη x ξ cosh aη (1.14)

where a ¡ 0, these can be inverted to yield

ξ ?x2 t2 η a1arctanhptxq (1.15)

from the inversion formulae it is evident that this system of coordinates covers

only the region t2 x2, 8 y, z 8, which is called ”Rindler wedge”.

The rest of Minkowski’s space can be covered by changing the signs in the

right-hand side of equations (1.14). In the new frame of reference the sign of

time is reversed in the region x 0, i.e. for ξ 0 η increases as t decreases

and vice versa.

Finally, the metric tensor in Rindler space is gµν diagpa2ξ2,1,1,1q;A remarkable property of RS is that its causal structure is deeply different

from the MS one. Indeed the hypersurfaces tpt, x, y, zq;x tu play the role

13

Chapter 1. Quantum Field Theory in non-Minkowskian geometries

of event horizons, dividing MS into causally disconnected parts. This feature

of RS and its effects on the fields will be discussed in more detail at the end

of this chapter.

x

c t

P

Q

x

c t

P

Q

Figure 1.2: On the left: depicted is the world line of a particle moving through MS

as seen by a Minkowskian observer, the two points P,Q are causally connected from this

point of view. On the right: the particle lives in RS and propagates through an accelerated

background, from the point of view of a Rindler observer there is not a world line connecting

P to Q, any signal emitted from P will fall towards the future light-cone branch, which

plays the role of an event horizon. The dashed lines are world lines of particles at rest in

P in MS and RS respectively.

b. The scalar field

Rindler spacetime is flat, which means that equation 1.3 reduces to the

usual Klein Gordon equation, being Rpxq 0 at every point. The field, as

seen by the Rindler observer, will satisfy the KG equation which reads1

a2ξ2 B

2

Bη2 1

ξ BBξ

B2

Bξ2 B2

By2 B2

Bz2m2

φpη, ξ, y, zq 0

(1.16)

To find a functional form of φ satisfying (1.16) it is convenient to make use

of a partial Fourier transform

φpη, ξ, y, zq »R3

dk0 dkKp2πq32 φpk0,kK; ξq eik0ηikKxK (1.17)

xK py, zq kK pky, kzq (1.18)

14

1.2. Spin-0 fields in Rindler geometry

so that equation (1.16) becomesB2

Bξ2 1ξ BBξ β2 k0

2

a2x2

φpk0,kK; ξq 0 (1.19)

β a

kK2 m2 (1.20)

If we perform a Wick rotation by substituting ik0 E, we are left with the

well-known Bessel equation B2

Bξ2 1

ξ BBξ

β2 k0

2

a2x2

φpk0,kK; ξq 0 (1.21)

the solutions of the above differential equation are known to be

φpk0,kK; ξq C1pkKqIiνpβξq C2pkKqKiνpβξq (1.22)

iν E

a ik0

a(1.23)

the functions Iiνpβξq diverge exponentially for large positive ξ and must

consequently be rejected. Finally the full solution to (1.16) reads

φpη, ξ,xKq » 0

8» 8

0

dk0?

2π

»d2kK2π

fpkKqKik0apβξqeik0ηikKxK

p2πq32» 8

0

dk0

»d2kK

rfpkKq exp pikK xK ik0ηq fpkKq exp pik0η ikK xKqs

where we took into account the reality condition for the scalar field that

brings

φpk0,kK; ξq φpk0,kK; ξq Ñ fpkKq fpkKq (1.24)

being Kik0apβξq Kik0apβξq. This solution holds within the right part of

the Rindler wedge; as discussed earlier it suffices to make the appropriate

changes of sign in the coordinates and repeat the above procedure to obtain

solutions of (1.16) in the remaining parts of MS. The normal modes

uk0,kKpη, ξ,xKq Cp2πq32θpξqKik0apβξq exp tik0η ikK xKu(1.25)

15

Chapter 1. Quantum Field Theory in non-Minkowskian geometries

are also known as Fulling modes for the scalar field, and they are orthonormal

with respect to the inner product (1.7), indeed if we choose the initial time

3-hypersurface

dΣ0 θpξq paξq1 dξ d2xK dΣi 0, pi 1, 2, 3q (1.26)

puk0,kK , uh0,hKq |C|2 π asinhpπk0aq δ

2pkK hKq δpk0 h0q (1.27)

where θpξq is the standard Heaviside distribution, it can be seen by di-

rect inspection (see appendix for an explicit calculation) that choosing C asinh pπk0aqpπaq the Fulling modes are normalized according to

puk0,kK , uh0,hKq δ2pkK hKq δpk0 h0q (1.28)

A proof of the above result is given in the Appendix.

The Fulling modes have the standard canonical engeneering dimensions

ruk0,kKs eV12 in natural units. Consequently the normal modes expansion

of the real scalar quantized field on the right Rindler wedge reads

φpη, ξ,xKq » 8

0

dk0

»d2kK rak0,kKuk0,kKpη, ξ,xKq ak0,kK

:uk0,kKpη, ξ,xKqs

where the standard second-quantization operators are those satisfying

rak0,kK , ah0,hK:s δpkK hKq δpk0 h0q rak0,kK , ah0,hKs 0

(1.29)

such that rφs eV and rak0,kKs eV32.

1.3 Defining spinors: the generalized theory

of spin-12 fields

In flat spacetime QFT the definition of the spin for a tensor field is

closely connected with the field’s transformation properties under infinitesi-

mal Lorentz transformations, indeed it is a cornerstone of QFT that all the

irreducible representations of the Lorentz group (in Minkowski spacetime)

can be classified by two integers or half integers pA,Bq; for example the

p0, 0q representation is that carried by the scalar field, while the p0, 12q and

16

1.3. Defining spinors: the generalized theory of spin-12

fields

p12, 0q are the spin 12 irreducible representations. For the details of this

correspondence between the Lorentz algebra and two angular momentum

algebras see for example [41] or [51].

A problem with this method of classifying fields readily arises as we turn

to curved spacetimes: is it possible to generalize the above considerations

without losing the connection with the Lorentz group? The answer is yes,

this can be achieved by employing the vierbein formalism. This approach

consists of erecting normal coordinates yαX at each spacetime point X, in

the sense that in terms of yαX the metric is Minkowskian. The vierbeins are

defined as

V αµpXq

ByαXBxµ

xX

, α 0, 1, 2, 3 (1.30)

so to satisfy:

gµνpxq V αµpxqV β

νpxqηαβ (1.31)

We shall adopt the convention that letters from the beginning of the Greek

alphabet refer to the normal coordinate system, while those from the end

refer to the general coordinate system. For the properties of the vierbeins

see [60] (”The Tetrad Formalism”). Given the metric, the vierbeins are not

uniquely defined, their form can be chosen following one’s convenience (for

more details see e.g. [27]); in our case we shall adopt diagonal vierbeins:

V αµpξq

aξ

1

1

1

V µα pξq

paξq1

1

1

1

Vαµpξq

aξ

1

1

1

(1.32)

they satisfy the important relation

V αµV

βνηαβ gµν (1.33)

where gµν is the Rindler metric.

17

Chapter 1. Quantum Field Theory in non-Minkowskian geometries

Thus, by use of the vierbeins, one can convert general tensors into lo-

cal Lorentz-transforming tensors as the additional spacetime dependence is

shifted into the vierbeins. Hence we can now properly talk about spin 12fields within a general framework.

So far we have all the necessary tools to build a generally covariant gener-

alization of Dirac’s theory, an elegant and concise derivation by principles can

be found in [7]. Nonetheless we shall follow a personal, more pedagogical,

approach.

In order to write down an action behaving as a scalar under general

coordinate transformations it is necessary to introduce a covariant derivative

for the spinor field. A suitable choice is (see [41], [7])

∇α ψpxq V µα pBµ Γµqψpxq (1.34)

Γµpxq 1

2Σαβ V ν

α pxqVβν;µpxq Vβν;µ BµVβν ΓλνµVβλ

whereas Σαβ 14r 9γα, 9γβs being the generators of the Lorentz group associ-

ated with the p0, 12q ` p12, 0q representation, while the symbol Γλµν is the

standard Christoffel symbol

Γλµν λµν

( 1

2gλκ pgκν,µ gµκ,ν gµν,κq (1.35)

the lagrangian density is then

Lpxq pdetV q"

1

2iψγµ∇µψ p∇µψqγµψ

mψψ

*(1.36)

S »Lpxq d4x (1.37)

where γµ V µα 9γα. The gamma matrices satisfy

tγµ, γνu 2gµν (1.38)

the generalized Dirac equation follows readily upon variation of the action

with respect to ψ

iγµ∇µψ mψ 0 (1.39)

on the other hand, varying with respect to ψ we obtain the conjugate equation

i∇µψ

γµ mψ 0 (1.40)

18

1.3. Defining spinors: the generalized theory of spin-12

fields

notice however that a priori we do not necessarily know how ψpxq is defined,

nor how does ∇µ act on it. Let us investigate on these points: if we take the

hermitean conjugate of (1.39) we obtain

0 ψ:pxqi

ÐÝB µ Γ:µpγµq: m

(1.41)

recalling that

p 9γαq: 9γ09γα 9γ0 pγµpxqq: 9γ0 γµpxq 9γ0 (1.42)

and recalling the form of the Fock-Ivanenko coefficients Γµ, it is straightfor-

ward to achieve that

Γµ:pxq 9γ0 Γµpxq 9γ0 (1.43)

inserting in (1.41) we get

0 ψ:pxqiÐÝB µ 9γ0 Γµ 9γ

0

9γ0 γµ 9γ0 m

ψ:pxq 9γ0iÐÝB µ Γµ

γµ m

ψpxq

ÐÝ∇µ γµ m

(1.44)

it follows that, in order to be consistent with (1.40) we must identify

ψpxq ψ:pxq 9γ0 (1.45)

∇µ ψpxq pBµ Γµq ψpxq (1.46)

comparing with (1.34) it is evident how ∇µ acts in two different ways on ψ

and ψ: indeed the Fock-Ivanenko coefficients are nothing but a Christoffel-

like symbol for the spinor field

Γµ Γaµb (1.47)

where the spinor indices a, b are usually left understood. The change of sign

tells us that ψ, ψ transform differently under general coordinate transforma-

tions, just similarly to covariant and contravariant vectors.

Having understood how ψ,∇µ behave, we might check the consistency of

our results by testing them directly on the action:

S »

d4x?g

"i

2

ψγµ∇µψ p∇µψqγµψ

mψψ

*

»d4x

?g"i

2ψÐÑB ψ i

2pΓµ γµ γµ Γµq mψψ

*

»d4x

?g ψ pi B γµ Γµ mq ψ »

d4x?g i

2ψ rΓρ, γρsψ

19

Chapter 1. Quantum Field Theory in non-Minkowskian geometries

recalling the explicit form of Γµ, the last piece reads

i

2

»d4x

?g ψ V ρα V

να Vνβ;ρ 9γβ ψ

i

2

»d4x

?g ψ Vρβ;ρ 9γβ ψ

ψ γρ ψBΩ

»

d4x∇ρ

?g ψ γρ ψ 0

the first term vanishes by virtue of the condition on the asymptotic behavior

of fields; the second term vanishes as well by virtue of Noether’s theorem,

since it is the integral of the divergence of the current associated to the Up1qglobal symmetry of the Lagrangean.

Hence we come to a form of the action

S »

d4x?g ψ pi B γµ Γµ mq ψ (1.48)

in full consistency with equation (1.39).

Actually one may adopt the following representation (for a detailed dis-

cussion see [7] ) for general coordinate transformations on ψ, let the coordi-

nate transformation be X Ñ x with X coordinate of a certain Lorentz frame,

then

ψ1pxq NpXqψLpXq (1.49)

where ψL is the spinor field in a Lorentz frame, requiring general covariance

of (1.39) it turns out that

NpXq exp

» Xpxq

dx1µ Γµpx1q

exp

i

» x

dx1µ V να Vβν;µ

i

2Σαβ

(1.50)

id est the field transforms according to a local Lorentz-like transformation:

something between a Lorentz transformation and a gauge transformation,

the coefficients are given by the indefinite integral

ωαβpxq i

» x

dx1µ V να pxqVβν;µpxq (1.51)

The form of such transformation rule reminds immediately of the principle of

general covariance: associated to each point of the manifold there is a class

20

1.4. The Majorana field in Minkowski spacetime

of Lorentz frames in which the laws of special relativity hold true. Equation

1.50 can be interpreted as follows: upon a general change of coordinates the

spinor field undergoes a local Lorentz transformation in each point of the

(4-dimensional) manifold. We will see an explicit example of such transfor-

mation rule further on, in subsection 2.4 a. where we will study the case of

Rindler coordinates in MS that correspond to a global Lorentz transformation

varying with time but constant over the whole 3 dimensional space.

1.4 The Majorana field in Minkowski space-

time

In the last section we introduced all the basic ingredients necessary for

the study of the Majorana field in a Rindler spacetime. However, let us

first review the theory of Majorana fermions in MS, as this particular case

is rarely treated in depth in the literature; actually an exception is provided

in [51], in this section we will essentially follow that reference, focusing on

some important aspects that we will need.

Let us start from the study of the two classical Weyl fields, i.e. mas-

sive two-component spinors carrying separately the representations p0, 12q and

p12, 0q of the Lorentz group:

ψ1Lpx1q ΛLψLpxq ψ1Rpx1q ΛRψRpxq (1.52)

where

ψLpxq ψL 1pxqψL 2pxq

ψRpxq

ψR 1pxqψR 2pxq

(1.53)

Let us call a generic field of one of these two kinds χapxq pa 1, 2q and let

us consider χa as a classical anticommuting field, i.e. a Grassmann valued

spinor field function, which enjoys the property

tχapxq, χbpyqu 0 px, y PM a, b 1, 2q (1.54)

together with the complex conjugation rule

pχT1 χ2q pχ2qTχ1 pχ1qTχ2 (1.55)

21

Chapter 1. Quantum Field Theory in non-Minkowskian geometries

in such a way that it resembles the hermitean conjugation properties of quan-

tum spinor fields. We can write the Lagrange density as follows

L 12χ: σµ i

ÐÑBµ χ 12m

χT σ2 χ χ: σ2 χ

(1.56)

9 χ: σµ i Bµ χ 12m

χT σ2 χ χ: σ2 χ

(1.57)

where σµ p12,σkq and 9 means that equivalence holds up to a irrelevant

four-divergence additional term. Let us consider χ and χ as independent

lagrangian coordinates, then the Euler-Lagrange field equations yield

i Bµχ: σµ mχT σ2 (1.58)

transposing and taking the complex-conjugate

i σµ Bµχ mσ2 χ 0 (1.59)

this is the Majorana field equation for a Weyl spinor field with a Majorana

mass term; multiplying from the left by σ2 and taking complex conjugation

we obtain

i σµ Bµχ mσ2 χ 0 (1.60)

where σµ p1, σkq and we took into account that σ2σkσ2 σk; this is

the equivalent form of the Majorana equation. Acting from the left with the

operator iσνBν on eq. (1.59) and use eq. (1.60) we end up with

plm2qχpxq 0 (1.61)

that is, the two components of this field separately satisfy the KG equation.

Recall the conjugation rule for Dirac spinor fields: given ψ solution of the

Dirac equation, ψc γ2ψ is the corresponding conjugate spinor. Let us now

introduce the well known Majorana self-conjugated bispinor, its expression

is

χM

χ

σ2χ

χcM (1.62)

notice that, if we were working with a right-handed Weyl field, we would

have buildt

ψM σ2ψ

R

ψR

ψcM (1.63)

22

1.4. The Majorana field in Minkowski spacetime

enjoying autoconjugation as well.

The Lagrange density of χM can be easily obtained to be

LM 1

4χM i

ÐÑB χM 1

2mχM χM (1.64)

it follows, by direct inspection, that Majorana’s action is not invariant under

the phase transformation

χpxq Ñ χ1pxq χpxq eiα (1.65)

Due to the self-conjugation constraint (1.62), it easily verifiable that the pair

of equations 1.59 and 1.60 are equivalent to the single bispinor equation

rαµiBµ βmsχMpxq 0 (1.66)

with

αµ σµ 0

0 σµ

β γ0

0 12

12 0

γµ βαµ (1.67)

The constraint (1.62) relates the upper two components of the Majorana

field to the lower two components, hence instead of four complex fields, the

degrees of fredom get divided by two, therefore it must be possible to find

a representation in which there are only four real fields. To obtain this real

representation, we note that

χM

0 σ2

σ2 0

χM (1.68)

this property lets us proceed to write a real bispinor ψM ψM by the trans-

formation

χM SψM (1.69)

where S must be constructed so to satisfy

S

0 σ2

σ2 0

S (1.70)

if we set0 σ2

σ2 0

iρ2 pρ2q2 14 (1.71)

23

Chapter 1. Quantum Field Theory in non-Minkowskian geometries

we can cast the solution of the above equation as

S exp tπiρ24u 1?2p14 iρ2q (1.72)

which fulfills S: S1. It follows that we can suitably make use of the

Majorana representation of the gamma matrices

γµM S:γµS (1.73)

Explicitly they read

γ0M

σ2 0

0 σ2

γ1M

iσ3 0

0 iσ3

(1.74)

γ2M

0 σ2

σ2 0

γ3M

iσ1 0

0 iσ1

(1.75)

γ5M

0 σ2

σ2 0

(1.76)

These matrices are just another representation of the four Dirac gamma

matrices, indeed they still satisfy Clifford algebra

tγµM , γνMu 2gµν tγµM , γ5Mu 0 (1.77)

γ0M γ0 :

M γkM γk :M γ5M γ5 :

M (1.78)

γµM γµM γ5M γ5

M (1.79)

Then the Majorana lagrangean and its field equations read

LM 14ψTMpxqανM iÐÑBν ψMpxq m

2ψTMpxqβMψMpxq (1.80)

piBM mqψMpxq 0 ψMpxq ψMpxq (1.81)

αM γ0Mγ

νM βM γ0

M (1.82)

The only relic symmetry of the Majorana field is the discrete symmetry Z2:

ψMpxq Ñ ψMpxq In orded to solve the Majorana wave equation we proceed

as usual by passing to momentum space

ψMpxq p2πq32»

dp ψMppqeipx ψMppq ψMppq (1.83)

where the last condition follows naturally from conjugating Majorana’s field

equation and from the fact that Majorana matrices are purely imaginary,

24

1.4. The Majorana field in Minkowski spacetime

and it provides a reality condition for ψM . The wave equation in momentum

space reads

ppM mqψMppq 0

by acting on the left with ppM mq and by employing the anticommuta-

tion relations for Majorana’s gammas, we find that it is possible to express

ψppq through the action of a projector onto a collection of four scalar fields

satisfying the KG equation

ψMppq ppM mqφppq φ

φ1

φ2

φ3

φ4

pp2 m2qφppq 0

φαppq δpp2 m2qfαppq

furthermore, from the reality condition on the Majorana spinor field that we

mentioned just above, we find out that fαppq fαppq. Then we can write

ψMpxq »

d4p rp2πq3ωps12θpp0q ppM mqfppqδpp0 ωpqeipx

»d4p rp2πq3ωps12θpp0q

ppM mqfppqδpp0 ωpqeipx def p2πq32Σp

εMppqapeipx εMppqapeipx

ψMpxq (1.84)

with ap 2mfppq p2ωpq12 and

εMppq pm pMq2m ωp a

p2 m2 (1.85)

εMppq εMppq

εMppq

2 εMppqεMppqεMppq 0 trεMppq 2

εMppq εMppq I

Now, by direct inspection of the spin operators for the Majorana field, it

turns out that the matrix γ0M shares the same eigenspinors as the ΣM,2 spin

25

Chapter 1. Quantum Field Theory in non-Minkowskian geometries

matrix, indeed

Σ2 iγ3Mγ

1M

σ2 0

0 σ2

(1.86)

ξ

0

0

1

i

ξ

i

1

0

0

which satisfy

γ0Mξ ξ ΣM,2ξ ξ ξ:rξs 2δrs (1.87)

Hence, the spin states ξ correspond to the two degenerate eigenstates of the

Majorana hamiltonian in the particle rest frame (in which H mγ0M), with

opposite spin projections along the OY axis, since

σ13Mξ

1

2ΣM,2ξ 1

2ξ (1.88)

Turning back to ψpxq, the spin states now read explicitly#urppq 2mp2ωp 2mq 1

2 εMppqξrur ppq 2mp2ωp 2mq 1

2 εMppqξrr (1.89)

they are the eigenstates of the positive-energy projector εM and they are

normalized according to

u:rppqusppq 2ωpδrs (1.90)

In conclusion, the solution to Majorana’s wave equation reads

ψMpxq ¸p,r

ap,r up,rpxq ap,r u

p,rpxq

with

up,rpxq rp2πq32ωps12urppq exptiωpt ip xu0 tap,r, aq,su tap,r, aq,su tap,r, aq,su

@p, q P R3 @r, s ,

26

1.5. Majorana spinors in Rindler geometry

Quantization is achieved introducing the second-quantization creation-annihilation

operators satisfying

tap,r, aq,su 0 ta:p,r, a:q,sutap,r, a:q,su δrs δpp qq

so that the quantized field reads

ψMpxq ¸p,r

ap,r up,rpxq a:p,r u

p,rpxq

(1.91)

ψ:MpxqLet us stress that eq. (1.63) tells that one could repeat the whole procedure

and build a quantum theory for a Majorana spinor as derived from a right-

handed Weyl field.

Summarizing, so far we have learned that

• The Majorana field carries the p12, 0q ` p0, 1

2q representation of the

Lorentz group, i.e it carries both Left- and Right-handed components,

just as expected for massive Dirac fields.

• However from eq.s (1.62), (1.63) it is evident that a Majorana field

possesses only half the degrees of freedom of a general Dirac field, more

precisely the right-handed part depends on the left-handed one (or

viceversa).

• There exists a representation in which ψM is real and the γ’s are imag-

inary. It is the Majorana representation and within this representation

the Majorana equation enjoys manifest reality (self-conjugated under

complex conjugation).

• The global Up1q symmetry is not preserved, and from the Majorana

representation we may infer that it reduces to a discrete Z2 symmetry,

i.e. SrψM s SrψM s. Hence the field carries no charge, this is the

spinor analogue to the relation between complex and real scalar fields.

1.5 Majorana spinors in Rindler geometry

We are now ready to study the theory of a Majorana field, as it is seen

by a Rindler observer, in full detail.

27

Chapter 1. Quantum Field Theory in non-Minkowskian geometries

a. Solving the Majorana-Rindler equation

Putting ourselves in Rindler coordinates, equation (1.39) is the one that

describes how a Dirac field behaves; in order to obtain a covariant form

of Majorana’s equation it is sufficient to transform the spinors and Dirac’s

matrices as

ψM S1ψ ψ: ψ:MS: ψTMS

: (1.92)

9γαM S19γαS (1.93)

being ψM ψM , the matrix S is unitary (see (1.72)) and reads

S 1?2

1 0 0 i0 1 i 0

0 i 1 0

i 0 0 1

. (1.94)

Substituting in (1.36) we readily obtain

LM pdetV q"i

2ψ:Mα

µM∇M µψM pBµψ:M 9γ0

M

ψ:M 9γ0MΓM µqγµMψM mψ:M 9γ0

MψM

)(1.95)

∇M µ S1∇µS Bµ 1

2V να pVβν;µqS1ΣαβS Bµ ΓMµ (1.96)

ανM 9γ0Mγ

νM (1.97)

the Euler-Lagrange equation for ψM reads

0 riγµM∇M µ msψM (1.98)

riγµMBµ msψM iγµMΓM µψM (1.99)

To lighten the notation, let us drop the subfix M from now on, keeping in

mind that we’ll always be dealing with Majorana’s matrices and spinors.

To proceed it is useful to evaluate the four Γµ’s; it is easily achieved that

Γξ Γy Γz 0 (1.100)

28

1.5. Majorana spinors in Rindler geometry

since

∇ξVβν BξVβν ΓκξνVβκ

1

2Σαβ

δηβδ

ηα δηαδ

ηβ

0 (1.101)

∇yVβν ByVβν ΓκyνVβκ

1

2Vβκg

κλpgyλ,κ gλκ,y gyκ,λq 0 (1.102)

∇zVβν BzVβν ΓκzνVβκ

1

2Vβκg

κλpgzλ,κ gλκ,z gzκ,λq 0 (1.103)

while for Γη we have: λην

( a2ξδλη δ

ξν paξq2 δλξ δ

ην

(1.104)

Γη a

29γ09γ1 (1.105)

We now turn to the study of the second order differential equation, that

reads

0 tiγµ∇µ mu tiγµ∇µ muψ

B2η

paξq2 Bξ 1

2ξ

2

B2y B2

z m2 1

aξ29γ09γ1Bη

ψ

the matrix 9γ0 9γ1 has two doubly degenerate eigenvalues λ 1, and the

corresponding eigenspinors are:

Θ1

1

1

0

0

, Θ2

0

0

1

1

, Θ1

1

1

0

0

, Θ2

0

0

1

1

.We look for solutions of the form

fa pη, ξ, y, zq ga pη, ξ, y, zqΘa (1.106)

where g is a scalar function. It is useful to turn to momentum space for the

three variables η, y, z, we readily get:

ga pη, ξ, y, zq »dk0 dkKp2πq32 eik

0ηikKxK ga pξ; k0,kKq (1.107)

29

Chapter 1. Quantum Field Theory in non-Minkowskian geometries

the equation then reads:B2ξ

1

ξBξ

14 pk0aq2

ξ2 k2

K m2

i

k0

aξ2

ga pξ; k0,kKq 0

(1.108)

with the upper sign correspoding to eigenvalue λ 1 and the lower to

λ 1. This equation is clearly similar to Bessel’s modified equation, indeed

performing a Wick rotation pk0aq Ñ iω we obtain

B2ξ

1

ξBξ

14 ω2 ω

ξ2 k2

K m2

(1.109)

where it is understood that it is to be carefully chosen whether to perform a

clockwise or anti-clockwise Wick rotation, in order to get the above form for

the two different cases λ 1, λ 1.

λ 1 ñ k0 i a ω (1.110)

λ 1 ñ k0 i a ω (1.111)

now let us perform the substitution: ξ pk2K m2q12τ , we then have"

B2τ

1

τBτ

pω 12q2τ 2

1

*gpω,kK, ξq 0 (1.112)

which is the standard form of the modified Bessel equation. Its solutions are

gpω,kK, ξq pRqgpω,kK, ξq pLqgpω,kK, ξq (1.113)pRqgpω,kK, ξq Kω12pβξq (region R) (1.114)pLqgpω,kK, ξq Kω12pβξq (region L) (1.115)

β b

k2K m2

we do not take into account the solutions of the form Iαpµq, as they diverge

for |µ| Ñ 8.

We proceed building up the solutions of (1.99) in the R region, then we will

expand them on the whole Rindler wedge.

30

1.5. Majorana spinors in Rindler geometry

Case λ 1

k0 i a ω η iτ (1.116)

pRqgpη, ξ,xKq a

»i dω dkKp2πq32 eipτaωkKxKq Kω12pβξq (1.117)

»dk0 dkKp2πq32 eipk

0ηkKxKq Kik0a12pβξq (1.118)

to obtain solutions of (1.99) it is sufficient to make use of the projector ppmq

ppq

Bξ 1

2ξωξ ikz 0 iky

ωξ ikz

Bξ 1

2ξ

iky 0

0 iky Bξ 12ξ

ωξ ikz

iky 0 ωξ ikz

Bξ 1

2ξ

ppq m

Θ

1

Bξ 1

2ξ ω

ξ ikz m

Bξ 1

2ξ ω

ξ

ikz m

ikyiky

DΘ

1 ikyΘ2 pm ikzqΘ

1

DΘ1 Θ

1 (1.119)

ppq m

Θ

2

ikyiky

Bξ 12ξ ω

ξ ikz m

Bξ 12ξ ω

ξ ikz m

DΘ

2 ikyΘ1 pm ikzqΘ

2

DΘ2 Θ

2 (1.120)

where we understand

D Bξ 12 ω

ξ

(1.121)

from [25] eq.s 8.486 10-11, we know that

d

dzKνpzq 1

2pKν1pzq Kν1pzqq (1.122)

ν

zKνpzq 1

2pKν1pzq Kν1pzqq (1.123)

31

Chapter 1. Quantum Field Theory in non-Minkowskian geometries

in such a way that the following useful relation holds

DK 12ωpβξq β K 1

2ωpβξq (1.124)

consequently, turning back from the Euclidean formalism to the Minkowskian

one, the Fourier transform of Majorana-Rindler modes in the R region reads

pRqvpqa,k0,kKpξq θpξq cpR,qa rppq ms Θ

a Kω12pβ ξq (1.125)

θpξq cpR,qa

β K12ik0apβ ξqΘa

K12ik0apβ ξqΘa

)where a 1, 2 and c

pR,qa c

pR,qa pk0,kKq is a normalization factor and we

understand

Θ1 ikyΘ

2 pm ikzqΘ1

ikz m

ikz m

ikyiky

(1.126)

Θ2 ikyΘ

1 pm ikzqΘ2

ikyiky

ikz m

ikz m

(1.127)

In the same way, using (1.115) instead of (1.114), and bearing in mind

the change of sign in eqs. (1.122),(1.123), we get the Fourier transform of

Majorana-Rindler modes in the L region

pLqvpqa,k0,kKpξq θpξq cpL,qa rppq ms Θ

a Kω12pβ ξq (1.128)

θpξq cpL,qa

β K12ik0apβ ξqΘa

K12ik0apβ ξqΘa

)Finally, the complete form of these modes over the Rindler wedge is:

vpqa,k0,kK

pξq λRpRqvpqa,k0,kK

pξq λLpLqvpqa,k0,kK

pξq (1.129)

where λR,L are phase factors to be found by imposing analiticity constraints,

we shall treat this further on together with the Unrih effect.

32

1.5. Majorana spinors in Rindler geometry

Case λ 1

k0 i a ω η iτ (1.130)

pRqgpη, ξ,xKq a

» i dω dkKp2πq32 eipτaωkKxKq Kω12pβξq (1.131)

»dk0 dkKp2πq32 eipk

0ηkKxKq Kik0a12pβξq (1.132)

to obtain solutions of (1.99) it is sufficient to make use of the projector ppmq

ppq

Bξ 1

2ξωξ ikz 0 iky

ωξ ikz

Bξ 1

2ξ

iky 0

0 iky Bξ 12ξ

ωξ ikz

iky 0 ωξ ikz

Bξ 1

2ξ

ppq m

Θ

1

Bξ 1

2ξ ω

ξ ikz m

Bξ 12ξ ω

ξ ikz m

iky

iky

DΘ

1 ikyΘ2 pm ikzqΘ

1

DΘ1 Θ

1 (1.133)

ppq m

Θ

2

iky

ikyBξ 1

2ξ ω

ξ ikz m

Bξ 1

2ξ ω

ξ

ikz m

DΘ

2 ikyΘ1 pm ikzqΘ

2

DΘ2 Θ

2 (1.134)

again, using (1.124) and turning back from the Euclidean formalism to the

Minkowskian one, we obtain the Fourier transform of the Majorana-Rindler

modes in the R region

pRqvpqa,k0,kKpξq θpξq cpR,qa rppq ms Θ

a Kω12pβ ξq (1.135)

θpξq cpR,qa

β K12ik0apβ ξqΘa

K12ik0apβ ξqΘa

)(1.136)

33

Chapter 1. Quantum Field Theory in non-Minkowskian geometries

where a 1, 2 and cpR,qa c

pR,qa pk0,kKq is a normalization factor and we

understand

Θ1 ikyΘ

2 pm ikzqΘ

1

ikz m

ikz m

iky

iky

(1.137)

Θ2 ikyΘ

1 pm ikzqΘ

2

iky

ikyikz m

ikz m

(1.138)

In the same way, using (1.115) instead of (1.114) we get the Fourier transform

of Majorana-Rindler modes in the L region

pLqvpqa,k0,kKpξq θpξq cpL,qa rppq ms Θ

a Kω12pβ ξq θpξq cpL,qa

β K12ik0apβ ξqΘa

K12ik0apβ ξqΘa

)(1.139)

Finally, the complete form of these modes over the Rindler wedge is:

vpqa,k0,kK

pξq ρRpRqvpqa,k0,kK

pξq ρLpLqvpqa,k0,kK

pξq (1.140)

where ρR,L are phase factors to be found by imposing analiticity constraints,

we shall treat this further on together with the Unruh effect.

Given the invariant scalar product for spinors, namely

pψ1, ψ2q »

Σ

dΣλpxq ψ1pxqγλpxqψ2pxq (1.141)

»

Σ

dΣ0pxq ψ1pxqγ0pxqψ2pxq

we can evaluate the norm of these normal modes making the suitable choice

of the spacelike hypersurface η 0, on which dΣ0 θpξqpaξqdξd2xKpR,Lqvpqa,k0,kK

pξq eik0ηikKxK , pR,Lqvpq

a,k01,kK1pξq eik

01ηikK1xK

|cpqpa, k0,kKq|2 p2πq2 2β2 δp2qpkK kK1q Ik0,k01

(1.142)

34

1.5. Majorana spinors in Rindler geometry

where

Ik0,k01 » 8

0

dξK ik0

a 1

2

pβξqK ik01

a 1

2

pβξq pk0 Ø k10q

(1.143)

we can take advantage of eq. 6.576 4. in [25]» 8

0

dx xλKiµ 12pβxqKiν 1

2pβxq 22λ βλ1

Γp1 λqΓ

1 λ iµ iν

2

Γ

1 λ iµ iν

2

Γ

2 λ iµ iν

2

Γ

λ iµ iν

2

holding for <λ 0. Since the r.h.s. is analytic for λ P R it is possible to

perform an analytic regularization by taking the limit λ Ñ 0, this limit

yields » 8

0

dxKiµ 12pβxqKiν 1

2pβxq iπ2

4β coshµν

2

sinh

µν

2

(1.144)

which is manifestly antisymmetric under pµ Ø νq, henceforth Ik0,k10 0 at

least for k0 k10. Instead if we study the behavior of such integral when

µ ν, by setting µν ξ iε then we can take the limit εÑ 0 and obtain» 8

0

dx xλKiµ 12pβxqKiν 1

2pβxq ε×0 iπ

2β coshπµ

1

ξ iεCPV

1

ξ

iπδpξq

iπ

2β cosh πµ

ε×0Ñ π2

2β cosh πµδpµ νq (1.145)

in such a way that

cpqpa, k0,kKq cpk0,kKq d

coshpπk0aqaπβ

(1.146)pR,Lqvpqa,k0,kK

pξq eik0ηikKxK , pR,Lqvpq

a,k01,kK1pξq eik

01ηikK1xK

p2πq3 δpk0 k10q δpkK kK1q (1.147)

pR,Lqvpqa,k0,kKpξq eik

0ηikKxK , pR,Lqvpqa,k01,kK

1pξq eik01ηikK1xK

0 (1.148)

As we will see further on, we will have the opportunity to evaluate Ik0,k10

by making use of new tools that will be acquired, involving only very basic

35

Chapter 1. Quantum Field Theory in non-Minkowskian geometries

analytical techniques. The result will be in full accordance with the method

of the regularization trick.

Completeness can also be checked, we will leave this task for the next

chapter for convenience. In particular, we will see how the set of pq modes

and the set of pq modes are complete sets independently from each other,

so that one may discard one of the two sets of solutions.

We may introduce the canonical normal modes

pR,LqUa,k0,kKpη, ξ,xKq p2πq32 pR,Lqvpqa,k0,kKpξq eik

0ηikKxK

1

2π2

dcoshpπk0aq

aβeik

0ηikKxK

$&% θpξq

βΘ

a K ik0

a 1

2

pβξq Θa K ik0

a 1

2

pβξq

pin Rqθpξq

βΘ

a K ik0

a 1

2

pβξq Θa K ik0

a 1

2

pβξq

pin Lq(1.149)

pR,LqVa,k0,kKpη, ξ,xKq p2πq32 pR,Lqvpqa,k0,kKpξq eik

0ηikKxK

1

2π2

dcoshpπk0aq

aβeik

0ηikKxK

$&% θpξq

βΘ

a K ik0

a 1

2

pβξq Θa K ik0

a 1

2

pβξq

pin Rqθpξq

βΘ

a K ik0

a 1

2

pβξq Θa K ik0

a 1

2

pβξq

pin Lq

Then the full solution to the Covariant Majorana equation (1.99) in RS

reads:

ψpxq pRqψpxq pLqψpxq (1.150)

whereas

pR,Lqψpxq ¸

a,k0,kK

pR,Lqfa,k0,kKpR,LqUa,k0,kKpη, ξ,xKq

((1.151)

¸

a,k0,kK

pR,Lqga,k0,kKpR,LqVa,k0,kKpη, ξ,xKq

((1.152)

It is straightforward to check that the following identities hold:

pR,LqUa,k0,kKpη, ξ,xKq pR,LqUa,k0,kKpη, ξ,xKq

pR,LqVa,k0,kKpη, ξ,xKq

pR,LqVa,k0,kKpη, ξ,xKq

(1.153)

36

1.5. Majorana spinors in Rindler geometry

then, by virtue of the purely imaginary form of the Majorana gammas and by

conjugating the Majorana equation, we see that the classical Majorana field

must be real also in RS. It follows immediately that fa,k0,kK rfa,k0,kKsand so for ga,k0,kK . Then the form (1.151) of the field is then manifestly self-

conjugated (complex-conjugation). Note that, unlikely to the Minkowskian

case, positive- and negative-frequency solutions are not well-separated here,

indeed we have an integral over k0 which ranges over R. In order to achieve

field quantization we will need to separate positive frequency modes from the

negative frequency ones, which can be obtained by splitting the domain of

integration, as follows

pR,Lqψpxq ¸a,kK

» 8

0

» 0

8

dk0 pR,Lqfa,k0,kK

pR,LqUa,k0,kKpη, ξ,xKq

¸a,kK

» 8

0

dk0pR,Lqfa,k0,kK

pR,LqUa,k0,kKpη, ξ,xKq pc.c.q

(1.154)

the same holding true for the V modes. Notice that expression (1.154) ex-

hibits clearly PT invariance.

By virtue of eq.s (1.149) and (1.147) we can infer that the standard or-

thonormality relations hold true for the canonical modespAqUa,k0,kK ,

pA1qUa1,k10,kK1 δA,A1 δpk0 k10q δpkK kK

1q (1.155)

where A,A1 R,L. Moreover, independence of the positive-frequency modes

from the negative frequency ones occurspAqUa,k0,kK ,

pA1qUa1,k10,kK

1

δpk0 k10q δpkK kK

1q δA,A1 (1.156)

0

since we are restricting the field expansions to positive values of the variable

k0.

Quantization is achieved introducing the usual creation-annihilation op-

erators satisfying

δaa1 δA,A1δpkK kK1q δpk0 k0

1q tpAqfa,k0,kK ,pA1qf :

a1,k01,kK1u (1.157)

tpAqga,k0,kK ,pA1qg:

a1,k01,kK1u

all the other anticommutators vanishing.

37

Chapter 1. Quantum Field Theory in non-Minkowskian geometries

b. The basis of helicity eigenstates

We have seen how it is possible to solve the Majorana equation and cast

its solution in the form of an expansion on normal modes, in such a way that

the procedure of quantization is straightforward. Actually the expansion we

buildt doesn’t help us understand what physical meaning to give each normal

mode, in order to recover some more information let us stress that our solu-

tions are built on the distinction between the eigenspinors of the matrix 9γ0 9γ1,

whose eigenvalues are doubly degenerate. This means that the eigenspinors

Θpqa , with a 1, 2 are not univocally defined, indeed any combination

Υ AΘpq1 BΘ

pq2 (1.158)

will still satisfy 9γ0 9γ1 Υ Υ.

We can then search for any observable commuting with 9γ0 9γ1, so to pre-

serve our construction of stationary solutions. If we try, by analogy with the

Minkowskian case, to check r 9γ0 9γ1, hs we find out that spinors with helicity

12

along the acceleration axis are the only suitable ones. This could have

been naively expected, since an acceleration along the Ox-axis means that

none of the stationary solutions can have zero-momentum in that direction,

which does not allow for stationary solutions with spin along the Oy or Oz

axes.

Indeed, it is straightforward to check that

Σ1M i

0 σ3

σ3 0

Σ1M , 9γ

09γ1 0 (1.159)

the matrix Σ1M has two doubly degenerate eigenvalues λ 1, we can then

look for combinations of stationary normal modes that satisfy

Σ1M

A pR,LqU1,k0,kKpη, ξ,xKq B pR,LqU2,k01,kK

1pη, ξ,xKq

A pR,LqU1,k0,kKpη, ξ,xKq B pR,LqU2,k01,kK

1pη, ξ,xKq

(1.160)

since we look for stationary solutions, we must immediately impose the con-

dition k0 k01. Before we try to solve the abovementioned problem, let us

point out some formulae that will turn out to be useful, actually one can

verify by direct inspection that

Σ1M Θ

1 iΘ2 Σ1

M Θ1 pkKq i Θ

2 pkKq (1.161)

38

1.5. Majorana spinors in Rindler geometry

and, since pΣ1Mq2 1, (1.161) implies also that

Σ1M Θ

2 iΘ1 Σ1

M Θ2 pkKq i Θ

1 pkKq (1.162)

if, for simplicity, we restrict ourselves to the right-Rindler wedge, it follows

that

Σ1M Ut1

2u,k0,kKpxq i p2πq32 eik

0ηikKxK vt21u,k0,kK

pξq (1.163)

then we can recast equation (1.160) asΣ1M

Σ1M

ψÒpxqψÓpxq

ψÒpxqψÓpxq

(1.164)

together withψÒpxqψÓpxq

A B

C D

U1,k0,kKpxqU2,k0,kK

1pxq

U

v1,k0,kKpxq eikKxK

v2,k0,kK1pxq eikK

1xK

p2πq32eik

0η

(1.165)

where A,B,C,D are complex-valued 4 4 linear operators. The above con-

ditions explicitly read:

i

Av2,k0,kK

1 eikK1xK B v1,k0,kK eikKxK

C v2,k0,kK1 eikK

1xK D v1,k0,kK eikKxK

Av1,k0,kK eikKxK B v2,k0,kK1 eikK

1xK

C v1,k0,kK eikKxK D v2,k0,kK1 eikK

1xK

we see that it is sufficient that we take A,B,C,D P C and we readily get

the conditions

A iB C iD (1.166)

β β1 kK 0 kK1 (1.167)

i.e. only for particles travelling in the direction of the acceleration we get a

helicity eigenstate:

ψÒÓ ψÒÓk0,kKΣ1M ψÒÓk0,0 ψÒÓk0,0 (1.168)

39

Chapter 1. Quantum Field Theory in non-Minkowskian geometries

the corresponding with Ò while the with Ó. Hence by requiring ψÒÓ to

be normalized, and making the convenient choice kK1 kK, we finally get,

up to a phase factor

U 1?2

1 i1 i

U :U 1 (1.169)

Then we obtain the new normal modes

ψÒ,k0,kKpxq α θpξq eik

0ηikKxK

iβΥÒ

K 12 ik0

a

pβξq uÒpkKqK 12 ik0

a

pβξq

(1.170)

ψÓ,k0,kKpxq α θpξq eik

0ηikKxK

iβΥÓ

K 12 ik0

a

pβξq uÓpkKqK 12 ik0

a

pβξq

(1.171)

together with

α cpk0,kKq2

π32 12π2

bcoshpπk0aq

aβ

ΥÒ 1

2

i

i

1

1

ΥÓ 1

2

ii1

1

uÒpkKq 12

m ky ikz

m ky ikz

im iky kz

im iky kz

uÓpkKq 12

m ky ikz

m ky ikz

im iky kz

im iky kz

In the same way one obtains the second set of eigenspinors of Σ1

M :ψÒ,k0,kK

pxqψÓ,k0,kK

pxq

1?

2

i 1

i 1

V1,k0,kKpxqV2,k0,kKpxq

(1.172)

ψÒ,k0,kKpxq

ψÓ,k0,kKpxq

1?

2V

V1,k0,kKpxqV2,k0,kKpxq

ψÒ,k0,kK

pxq α θpξq eik0ηikKxK (1.173)

iβΥÒ

K 12 ik0

a

pβξq uÒpkKqK 12 ik0

a

pβξq

ψÓ,k0,kKpxq α θpξq eik

0ηikKxK

iβΥÓ

K 12 ik0

a

pβξq uÓpkKqK 12 ik0

a

pβξq

40

1.5. Majorana spinors in Rindler geometry

together with

ΥÒ 1

2

1

1

ii

ΥÓ 1

2

1

1

i

i

uÒpkKq 12

im iky kz

im iky kz

m ky ikz

m ky ikz

uÓpkKq 12

im iky kz

im iky kz

m ky ikz

m ky ikz

Notice that, had we used U in place of V , we would have obtained a phase

factor i for ψÒ,k0,kKand a i for ψÓ,k0,kK

. We chose to use V just for a

matter of convenience.

These new normal modes are mutually orthogonal and normalized. Let

us compactify our notation according to

ι pr, k0,kK, σq ι P O (1.174)

where r Ò, Ó, σ ,.

The same reasoning can be applied in order to recover left-Rindler wedge

helicity eigenstates, the procedure is the same. For the sake of brevity let

us just summarize the results since we will need them later on. It turns out

that these modes are the left-Rindler wedge counterparts of those in (1.170)

and (1.171):

LψÒ,k0,kKpxq α θpξq eik

0ηikKxK (1.175)

iβΥÒ

K 12 ik0

a

pβξq uÒpkKqK 12 ik0

a

pβξq

LψÓ,k0,kKpxq α θpξq eik

0ηikKxK (1.176)

iβΥÓ

K 12 ik0

a

pβξq uÓpkKqK 12 ik0

a

pβξq

Now that the explicit form of ψιpxq is completely clear, once again we

stress the fact that one has

Σ1M uÒpkKq uÒpkKq (1.177)

Σ1M uÓpkKq uÓpkKq

i.e. ψιpxq has a definite helicity iff kK 0.

41

Chapter 1. Quantum Field Theory in non-Minkowskian geometries

This could be expected since we have seen that the only component of

the relativistic spin operator that commutes with 9γ0 9γ1, i.e. with the operator

that defines the stationary normal modes, is Σ1M , hence helicity could be well-

defined only for particles moving along the x direction.

The modes ψipxq satisfy Majorana’s equation by their definition, it readily

follows that

HMψιpxq iBηψιpxq k0ψιpxq (1.178)

by virtue of their status of stationary solutions, actually the last identity

could be proven by direct inspection quite easily (the proof involves the use

of recursion relations for the McDonald functions), however we will omit

these passages since they are not interesting.

Hence these normal modes correspond to particles of opposite helicity

(when they move along the direction of the acceleration) and positve energy

k0 in the Rindler frame.

We could write the field expansion by substituting (1.165) and (1.172)

into (1.154) however this would lead to a complication since we’d need to

introduce other constant spinors together with the ΥÒÓ and the uÒÓ pkKq. A

simpler way to write down the field expansion is using equation (1.151), in

which the energy index runs continuously over R, that brings

pRqψpxq ¸

pÒÓq,µ,kK

aÒÓµkK ψ

ÒÓµkKpxq aÒÓµkK ψ

ÒÓµkKpxq c.c.

(1.179)

c. Study of the hermiticity of the Hamiltonian

Let us now check that the Majorana-Rindler Hamiltonian is actually Her-

mitean: rewriting equation (1.99) as

iBη aξ mloomoonHm

i aξ 9γ09γ1

Bξ 1

2ξ

looooooooooomooooooooooon

Hξ

i aξ 9γ09γ2 By 9γ0

9γ3 Bzloooooooooooooomoooooooooooooon

HK

(1.180)

42

1.5. Majorana spinors in Rindler geometry

Hermiticity must be checked w.r.t. the invariant scalar product defined by

pψ1pxq , ψ2pxqq »

Σ

dΣα ψ1pxq γαpxqψ2pxq (1.181)

if we choose Σ tx P RS|η 0udΣα dΣ0

?g ε0βγδ3!

dxβdxγdxδ

aξ dξdydz

ñ pψ1pxq , ψ2pxqq »

Σ

dξdxK ψ:1pxqψ2pxq (1.182)

hermiticity of Hm and of HK can be easily checked by standard procedures,

let us focus on the two terms within Hξ separately: without loss of generality

we can work with right-Rindler-wedge modes only and getpRqψ1 ,i aξ 9γ0

9γ1 Bξ pRqψ2

» 8

0

dξ

»R2

dxK ψ:1

i aξ 9γ0

9γ1 Bξψ2

»R2

dxKi aξ ψ:1 9γ0

9γ1 ψ2

ξ8ξ0

» 8

0

dξ

»R2

dxKi aξ 9γ0

9γ1 Bξ ψ1

:ψ2

» 8

0

dξ

»R2

dxKi a 9γ0

9γ1 ψ1

:ψ2 (1.183)

while the second piece readspRqψ1 ,

i aξ 9γ0

9γ1 1

2ξ

pRqψ2

i aξ 9γ0

9γ1 1

2ξ

pRqψ1 ,

pRqψ2

the same happens in left Rindler wedge. Eventually we come to the hermitic-

ity conditionpRqψ1 , HξpRqψ2

pRqψ1Hξ,pRqψ2

»R2

dxK iaξ ψ:1 9γ

09γ1ψ2

ξ0

ξ ψ:1 9γ09γ1ψ2

ξ0

In conclusion it turns out that the Hamiltonian is hermitean only if we restrict

to fields satisfying

ξψpxq ξÑ0Ñ 0 (1.184)

This is Physically relevant, as it tells us that, in order for the evolution oper-

ator to be unitary, the event horizons must play the role of mirrors. Equiva-

lently, any field that does not satisfy condition (1.184) enjoys an absorption-

creation contribution to the evolution operator, as if it were manipulated

43

Chapter 1. Quantum Field Theory in non-Minkowskian geometries

from a hidden source. Hence a field crossing the light-cone event horizon

would be unphysical. This means that we should work in a restricted Hilbert

space, satisfying the mirror condition.

Eventually we can safely say that, since H is hermitean, its eigenstates

form a complete and orthonormal basis of the one-particle Hilbert space,

henceforth they are suitable for quantization. This ultimately confirms the

validity of our quantization scheme.

44

CHAPTER 2

The Unruh effect

Any intelligent fool can make things bigger, more complex, and more

violent. It takes a touch of genius -and a lot of courage- to move in the

opposite direction.

Albert Einstein

This chapter is devoted to the presentation of the Unruh effect, which

arises as a natural consequence for the inequivalence between the Minkowski-

Fock and the Rindler-Fock quantization schemes. We will see how a Rindler

observer perceives a thermal bath of particles as he moves through the quan-

tum state that coincides with the vacuum in the Minkowski-Fock represen-

tation. At this point, before we begin our treatment of the Unruh effect, we

stress that the Physical interpretation of these results is not completely clear,

thus they should be regarded as a working mathematical scheme, based on

some assumptions which may be too strong; nowadays the debate over the

validity of these predictions is still open between theoretical Physicists as no

experiment has yet shed light on this rather complicated argument.

45

Chapter 2. The Unruh effect

2.1 The Bogolyubov transform

To begin our study of the Unruh effect we will first introduce a well-

known mathematical framework known as the Bogolyubov transform. As we

will show in the next sections, the role of Bogolyubov transforms is absolutely

central in QFT in curved spacetime. This is due to the fact that they de-

scribe the relations among the normal-mode solutions of the field equations

in different frames, which in turn yield a connection between the different

quantization schemes. In other words, Bogolyubov transforms encode how

canonical quantization is affected by the frame of reference.

Since we will be dealing with canonical quantization, we are mainly inter-

ested in Bogolyubov transformations applied to continuous sets of harmonic

oscillators. Let us begin with the simplest case of just one bosonic har-

monic oscillator, let a, a: be the annihilation-creation operators satisfying

ra, a:s 1, then one may define a hermitean operator N a:a which eigen-

states form a complete-orthonormal discrete set of the one-particle Hilbert

space (for a rigorous and exhaustive treatment see [6], chap. V)

t|nyun N |ny n |ny n 0, 1, 2 a|0y 0 |ny pa:qn?

n!|0y

a:|ny ?n 1|n 1y a|ny ?

n |n 1y¸|nyxn| 1 xm|ny δm,n

then one can introduce another set of operators, b, b: satisfying the same

algebra and use them to build another complete-orthonormal basis of the

one-particle Hilbert space

t|nyun N |ny n |ny n 0, 1, 2 b|0y 0 |ny pa:qn?

n!|0y

b:|ny ?n 1|n 1y b|ny ?

n |n 1y¸|nyxn| 1 xm|ny δm,n

since both bases are complete, it must be possible to express each state of

the first in terms the second basis’ states, and vice versa

|ly ¸n

cln |ny (2.1)

46

2.1. The Bogolyubov transform

in order to find the expansion coefficients, it is necessary to introduce a

general transformation between the two couples of operators

b α a β a: b: α a: β a (2.2)

requiring that b, b: actually satisfy the usual algebra one finds the fundamen-

tal relation

1 rb, b:s rα a β a: , α a: β as |α|2 |β|2 (2.3)

that is characteristic of the bosonic Bogolyubov transform. Henceforth the

α, β coefficient are not wholly independent and the whole Bogolyubov trans-

formation may be described by three parameters θ1, θ2, γ such that α eiθ1 cosh γ, β eiθ2 sinh γ. It is important to notice that if β 0 then the

two ground states are inequivalent, indeed

b|0y pα a β a:q |0y β|1y βÑ0Ñ 0 (2.4)

hence |0y |0y. Indeed

0 b|0y pα a β a:q

¸j

c0j |jy

by virtue of the orthonormality, it follows immediately that

c0p2kq β

α

k p2k 1q!!p2kq!!

12

c00 (2.5)

c01 0 c0p2k1q β

α

k p2kq!!p2k 1q!!

12

c01 0

the coefficient c00 can be evaluated by requiring that x0|0y 1. Indeed

convergence of the series is guaranteed by

|c00|2 x0|0y 8k0

β

α

k p2k 1q!!p2kq!!

12

¤ 8

k0

βαk 1

1 |β||α|

since |β| |α| by virtue of 2.3.

All the other coefficients follow by application of (2.5). Moreover also the

cjk with j ¥ 1 are found by recursive action of b: on |0y.

47

Chapter 2. The Unruh effect

It is interesting to analyze the Bogolyubov transform when involving

fermionic harmonic oscillators. To cut a long story short, the Hilbert space

only consists of the states |0y, |1y and all one has to do is replace eq. (2.3)

with

1 tb, b:u tα a β a: , α a: β au |α|2 |β|2 (2.6)

however when one tries to repeat the above machinery to find the connection

between |0y, |1y and |0y, |1y it turns out that they are independent.

One can as well generalize to multiple sets of harmonic oscillators, defining

a Bogolyubov transform that mixes operators with different frequencies, this

is just what occurs in QFT in curved spacetime. We will investigate this

interesting case further on, while beginning to dig into the Unruh effect.

2.2 General theory of the Unruh effect

We saw in Chapter 1 that it is generally possible, both for scalars and

spinors, to solve the field equations and to quantize the field once we find

a suitable (i.e. complete and orthonormal) basis; what is still missing is a

physical interpretation of the theory: the major weakness of QFT in curved

spacetime with respect to standard QFT is that the latter gives a notion of

particle which is consistent with what is observed, while the former doesn’t

generally give the notion of particle! What makes a particle a particle in

the framework of standard QFT is the clear distinction between positive and

negative-frequency normal modes. This distinction is made according to the

action of the Killing vector Bt; the existence of such a vector is not trivial in

general, indeed most classes of spacetimes do not admit the existence of such a

Killing vector, this clearly precludes the possibility of distinguishing positive-

frequency modes from the negative-frequency ones, in such a situation the

concept of particle remains obscure.

However there are also situations with a high degree of symmetry which

admit a Killing vector playing the role of Bt. In such cases it is possible

to use a quantization scheme analogous to the standard one: associating

annihilation operators with the positive-frequency modes in the field decom-

position and the creation operator with the negative-frequency ones. Rindler

spacetime is obviously one of such cases.

48

2.2. General theory of the Unruh effect

The next question to ask is whether this quantization scheme is equivalent

to the common one, i.e. to that of standard QFT, or more appropriately if

there is any kind of relation between the two. Indeed, what can one expect

as equivalence between quantization schemes? The form of normal modes

depends on the coordinate system, more precisely on the metric. To clarify

the concept of equivalence between two different quantization schemes, let

us introduce the related Bogolyubov transformation.

For the sake of simplicity, let us introduce a free autoconjugated field

φpxq, over a certain Riemaniann manifold D, its Lagrangian density Lrφ, gswill be a functional of the metric and of the field. As usual it will be possible

to derive the field equations and to express the solutions in form of normal

modes decomposition. If we also assume that a time-like Killing vector is

admitted, we can divide the normal modes with respect to the sign of their

frequency and come to the standard form for the quantized field

φpxq ¸ι

aι uιpxq aι

: uιpxq

(2.7)

where the index ι is a label for a set of eigenvalues belonging to a complete set

of commuting observables which we shall call O, while uιpxq are positive fre-

quency modes and uιpxq are the negative-frequency counterparts. The sum

is understood to be extended over the whole set of complete and orthonormal

modes. The construction of the Fock space proceeds from the definition of

the vacuum state |0y

aι|0y 0 @ι P O (2.8)

and by recurrent action of the creation operator on |0y, paying attention

to the operator algebra, depending on the field’s spin. Next we turn to a

different coordinate frame, in which the metric takes a different form from

the previous case, and solve again the equations of motion, generally they

will differ from the ones above, due to the fact that gµν is different in the

lagrangean and so will the differential operators; hence the corresponding

normal modes will be different from the first set. Let us express the field in

the new coordinate system as

ψpx1q ¸κ

bκ vκpx1q bκ

: vκpx1q, x1 T x (2.9)

49

Chapter 2. The Unruh effect

Again the set of all the vκpx1q will be complete and orthonormal, they are di-

vided into positive-frequency (vκpx1q) and negative frequency (vκpx1q) modes,

and the construction of a Fock space proceeds in the usual way. The new

vacuum state will be called |0q. Let us assume that the new normal modes

tvκpx1qu are complete and orthonormal, then it must be possible to express

it in relation to the former set of modes, through a set of coefficients:

vκpx1q ¸ι

rακιuιpxq βκιuιpxqs

vκpx1q

¸ι

rακιuι pxq βκιuιpxqs (2.10)

The field lagrangean is covariant, and so are the field equations, hence the

following must hold true

φpxq ψpT xq @x P D (2.11)

ψpx1q ¸κ

#bκ

¸ι

rακιuιpxq βκιuιpxqs

b:κ¸ι

rακιuι pxq βκιuιpxqs+

¸ι

¸κ

ακιaκ βκιa

:κ

vκpx1q

ακιa

:κ βκιaκ

vκpx1q

(by comparison with (2.7) we obtain the Bogolyubov transformation between

creation/annihilation operators belonging to the two different Fock represen-

tations.

bκ ¸ι

ακιaι βκιa

:ι

b:κ

¸ι

ακιa

:ι βκιaι

(2.12)

We can now give a clear definition for the equivalence of two Fock represen-

tations, precisely the two representations will be said equivalent iff

βικ 0 @ι, κ (2.13)

As it is, the above condition looks just as a mathematical condition, which

tells us that if two representations are equivalent their creation/annihilation

50

2.2. General theory of the Unruh effect

operator do not mix in the Bogolyubov transformation which relates them,

and vice versa. To catch a glimpse of how profound this condition actually is,

let us suppose that the representations were inequivalent, then it is instructive

to inspect the state |0y by the point of view of the second observer, to this

end we shall evaluate the number of particles seen by the two observers, it is

straightforward from (2.12) that

x0|N1|0y x0|¸ι

aι:aι

|0y 0 (2.14)

x0|N2|0y x0|¸κ

bκ:bκ

|0y ¸κ

¸ι

|βκι|2

(2.15)

hence, while for the first observer the |0y state contains no particles (as by

definition), the second perceives the presence of particles belonging to any

state, depending on the nature of the Bogolyubov transformation; precisely

the number of particles in state κ is given by°ι |βκι|2.

Let us go back a little: where does it all come from? The key feature of

this inequivalence is the nature of the Bogolyubov transformation, which is

given by eq. (2.10), indeed it is easy to check that

βκι puι , vκq (2.16)

in turn this only depends on the analytical form of the two sets of normal

modes, which in turn depend on the coordinate transformation. Finally, we

obtained that a coordinate transformation, which is just a natural operation

in general relativity, can have dramatic effects on the quantum treatment of

fields, leading to discrepant particle interpretations.

The Unruh effect is just a particular case of such a situation, precisely it

predicts that a thermal bath of particles is detected by an observer moving

with constant acceleration through Minkowski’s vacuum. In particular the

spectrum of particles perceived is Planckian for a scalar field and has a similar

form for higher spin fields.

The unfamiliar reader would (reasonably) be skeptical about any theory

in disagreement with the field algebra of the Minkowski-Fock representation.

To cite Fulling in [22], ’...if any other proposed theory disagrees with this

one, so much the worse for that theory...’. Indeed the Unruh effect has been

both sustained and pitched by theoretical Physicists during the years, as it

51

Chapter 2. The Unruh effect

presents some peculiarities, one for all, Rindler coordinates do not cover the

whole Minkowski spacetime, not to mention the fact that its causal structure

is deeply inequivalent to the MS one. A complete understanding of this

problem doesn’t seem to be achieved yet, nor from the mathematical point

of view (what effects does incompleteness of space have on quantization?),

nor from the Physical one. On the other hand, experimental data do not tell

us much more.

The Unruh effect is particularly interesting, since in presence of a gravi-

tational (or cosmological!) field every point of space time has a gravitational

acceleration associated with it, hence the study of the Unruh effect is con-

nected with the local behavior of quantum fields in presence of gravity. Let

us give a well-known example of this fact: let us consider the Schwarzschild

metric

ds2

1 2GM

r

dt2

1 2GM

r

dr2 r2 dΩ2 (2.17)

if we perform the coordinate change ξ r1 2GMrs12 the metric turns

into

ds2 ξ2 dt2

4GM

r1 ξ2s22

dξ2

2GM

1 ξ2

2

dΩ2 (2.18)

in such a way that, in the vicinity of the Schwarzschild radius one has ξ 0,

bringing

ds2 ξ2 dt2 p4GMq2 dξ2 p2GMq2 dΩ2 (2.19)

id est, upon a rescaling of the radial coordinate we get a Rindler-like metric

with corresponding acceleration along the radial direction.

Nonetheless the main reason of interest in the Unruh effect is that it

provides the simplest case for understanding how to properly quantize a field

in a non-Minkowskian background: Rindler spacetime enjoys many features

(such as event horizons and a singularity of the coordinate system) in common

with more complicated metrics, and these features are the fundamental ones

that give rise to many of the issues encountered in curved-spacetime QFT.

In the following we give a treatment of this interesting effect both for

scalar and for Majorana fields. At the end of the chapter we will instead

discuss some notable opinions about it.

52

2.3. Unruh effect for the spin-0 field

2.3 Unruh effect for the spin-0 field

In order to study the Unruh effect for a scalar field, we need two inequiva-

lent quantization schemes to be compared. We derived the so-called Rindler-

Fulling quantized field in the previous chapter, so we proceed with calculat-

ing its Bogolyubov coefficients with respect to the Minkowsian quantization

scheme. To begin we evaluate these for modes within the right-Rindler wedge,

a generalization to other regions of spacetime are straightforward.

The well-known normal modes expansion for a scalar field in an inertial

frame reads

φpxq »

dkckhkpxq c:kh

kpxq

(2.20)

where ck, ck: are the standard second quantization annihilation-creation op-

erators, which obey the canonical commutation relationsck, ck1

: δ pk k1q (2.21)

rck, ck1s ck:, ck1:

0 (2.22)

while hkpxq are the Minkowski modes and they can be cast in the form

hkpxq r2ωkp2πq3s12 exp tiωkt ik xu (2.23)

ωk pk2 m2q12. (2.24)

In the Rindler-Fulling scheme, we have instead

φpxq »

dkK

» 8

0

dk0ak0,kKuk0,kKpxq a:k0,kK

uk0,kKpxq

(2.25)

where the normal modes read

uk0,kKpxq d

sinhπ k0

a

8π4 a

Kipk0aqpβ ξq eik0ηikKxK (2.26)

as we discussed previously. We start by evaluating the α coefficients, let us

introduce the shorthand notation k pk0,kKq.

αkk1 phk, uk0,kKq i

» 8

0

dξ

»dxK

?g hkgηηÐÑB ηuk0,kK

η0

53

Chapter 2. The Unruh effect

in order to evaluate this amplitude, let us employ a trick due to Takagi,

which we already used for the normalization of Rindler-Fulling modes (see

appendix). Since the scalar product does not depend on the hypersurface Σ

that we choose, as long as it is spacelike, we push it up close to the horizon

H (see figure 1 in the appendix). To make it more explicit, we introduce

the null coordinates defined as

u aη log pξλq v aη log pξλq λ P R (2.27)

then the condition of pushing the hypersurface towards the H horizon con-

sists in taking the limit uÑ 8 inside the integral; it is easy to see that in

these coordinates the hypersurface oriented element reads

dΣv dv dxK (2.28)

so that integration over the transverse coordinates keeps unchanged and we

are left with:

αkk1 i C

»dv lim

uÑ8

eiλ2peveuqpω1k11qÐÑBv (2.29)

Kipk0aqβ λ e

vu2

ei

k0

auv

2

with

C δpkK k1Kqc

sinhpπk0aq8 π3 aω1

(2.30)

in the limit uÑ 8 we can take advantage of an expansion for small values

of the argument of Bessel functions (see formula (16))

Kipk0aqβ λ e

vu2

α

ei

k0

avu

2 Reik0

avu

2

α αpk0,kKq iπ

2 sinhπk0

a

Γ1 ik

0

a

β λ

2

i k0

a

, R α

α

the amplitude then reads

C α

»dv lim

uÑ8

eiλ2peveuqpω1k11q

i k0 aR

eik0

av

eik0

au R ei

k0

aviλ

2pω1 k11qev ei

λ2pω1k11qpeveuq

C α

»dv

eiλ2

evpω1k11qi k0

av R

k0

a λ

2pω1 k11q ev

eik0

au ei

λ2pω1k11qevv λ

2pω1 k11q

54

2.3. Unruh effect for the spin-0 field

where we dropped the terms eu in taking the limit; besides this, let us note

that the amplitude we are evaluating is between normal modes, in particular

the Mikowskian ones (the plane waves), do have an infinite norm as they

actually lie out of the proper one-particle Hilbert space. Under this light, we

might expect to get an infinite value of this amplitude, indeed we’ll see that

this is the case, reasonably. Actually when one deals with Physical particle

states, one works with wave packets, i.e. one smears the field operator with

a test function fpkq. Indeed what we are evaluating is precisely an improper

Bogolyubov coefficient, which is to be intended in a distributional sense. In

this spirit, and taking into account that we are still working in the limit

u Ñ 8, we may drop the last line by virtue of the Riemann Lebesgue

lemma, as intended for integration over k0 when smearing over a Rindler

wave packet

C αR

»dv ei

λ2

evpω1k11qi k0

av

k0

a λ

2pω1 k11q ev

(2.31)

by changing variable according to z ev, we get

C αR

» 8

0

dz eiλ2z pω1k11q zi

k0

a

k0

a z λ

2pω1 k11q

(2.32)

and, by employing another substitution y z λ2pω1 k11q, where ω1 ¡ k11 by

definition, we finally come to

C αR

» 8

0

dy

k0

a y

yi

k0

a1 eiy

λ

2pω1 k11q

i k0

a

the last integral can be easily evaluated by contour integration, with a rota-

tion of π2, and it reads

i k0

aeπk0

2a Γpik0aq eπk0

2a Γp1 ik0aq

2 eπk0

2a Γp1 ik0aqfinally, we found that the value of αk,k1 reads

αk,k1 i δpkK kK1qω1 k11ω1 k11

ik0

2a

eπk0

2a

8 π aω1 sinh

πk0

a

12

(2.33)

55

Chapter 2. The Unruh effect

The β coefficient can be obtained with a similar procedure, by substituting

the normal mode uk0,kK by its complex-conjugate, as in equation (2.16), this

corresponds to switching

k1 Ñ k1 ωk1 Ñ ωk1 (2.34)

finally one finds

βk,k1 i δpkK kK1qω1 k11ω1 k11

ik0

2a

eπk0

2a

8 π aω1 sinh

πk0

a

12

(2.35)

it is now easy to compute the number of Rindler-Fulling particles seen by

an accelerated observer moving through the state |0y, by virtue of formula

(2.15)

Npk0,kKq »

dk1 |βk,k1 |2 (2.36)

1

4 π a

1

e2πk0

a 1

»dk11

1apk11q2 β2

just as we expected, this quantity is logarithmically divergent; in order to

recover a mathematical sense, let us go back a little and employ the so-called

proper states for the Miknowski-normal modes:

Hk1pxq »

dk1 gpk1qhk1pxq

where g is the Fourier transform of some wave-packet within the one-particle

Hilbert space.

Then the number of detected particles is given by suitably modifying eqs

(2.7) and (2.15) as follows

φP pxq ¸k1

ak1Hk1pxq a:k1H

k1pxq

(2.37)

x0|N2,P |0y ¸k

¸k1

|Bk,k1 |2

(2.38)

(2.39)

where

Bk,k1 pHk1 , ukq phk1 , ukq gpk1q βk,k1 g

pk1q (2.40)

56

2.4. Unruh effect for the Majorana field

in full analogy with the reasoning of the previous section, whence the number

of detected particles reads

x0|N2,P |0y »

dk

»dk1 |βk,k1 |2 |gpk1q|2 (2.41)

1

4π a

» 8

0

dk0 1

e2πk0

a 1

»dkK

»dk11

|gpk11,kKq|2bkK

2 pk11q2 m2

hence, we obtained the spectrum of particles, up to a multiplicative factor,

which is given by the last two integrals, the convergence being ensured by our

assumptions on g. The spectrum of particles is Planckian, with temperature

T a

2π ~a

2πckB(2.42)

this represents a canonical ensemble with temperature T, called Davies-

Unruh temperature.

2.4 Unruh effect for the Majorana field

As we pointed out, in order to study the Unruh effect for the Majorana

field one needs to compare the Rindler quantization scheme with the usual

Minkowskian one. More precisely it is necessary to evaluate the coefficients

of the Bogolyubov transformation that occurs between the two sets of normal

modes, the modes on which the quantization schemes are built on. Once these

coefficients are found one can use them to determine how the Rindler cre-

ation/annihilation operators are related to the original Minkowskian ones.

Eventually this machinery allows one to evaluate the spectrum of Rindler

quanta that are present in Minkowski’s vacuum state, which is just the oc-

currence of the Unruh effect.

Let us begin; first of all we shall compactify our notation for the Majorana

field: we’ll drop the indices kK as they do not play an important role in our

discussion, so the Rindler modes will read

ψÒÓ,µpxq (2.43)

where it is understood that µ k0a and that it carries with itself the

quantum numbers kK. Unless otherwise specified, we understand x as the

generic Rindler-space coordinate pη, ξ,xKq.

57

Chapter 2. The Unruh effect

a. Finding the spinor algebraic RS-to-MS transfor-

mation

The first important difference between the scalar and the spinor cases is

the following: when we compared the Rindler modes to the Minkowskian

modes for the scalar field all we had to do was to make a change of vari-

able, instead in the case of a spinor field one needs to take into account the

algebraic transformation under which the spinor field undergoes when a gen-

eral coordinate transformation is performed. Our first step will then be to

find such transformation operator: let us consider the particular coordinate

transformation between Minkowski and Rindler observers

t ξ sinh aη x ξ cosh aη (2.44)

our aim is to find the spinor algebraic transformation that accounts for this

coordinate transformation. One could naively argue that the case of an

observer with constant proper acceleration is nothing but that of a time-

varying Lorentz boost with velocity β apη η0q, where η is the Rindler

time. This naive point of view is intuitive and actually appropriate, as we

will see.

By differentiating (2.44) one obtains

dt dξ sinh aη aξ dη cosh aη (2.45)

dx dξ cosh aη aξ dη sinh aη

id est the variation of Minkowskian coordinates subject to variation of the

Rindler ones, from these formulæ one can extrapolate the local linearized

version of the coordinate transformation, which in the neighbourhood of a

certain pη0, ξ0,xK0q reads: t

x

xK

Lµν

τ

ξ

xK

cosh aη0 sinh aη0

sinh aη0 cosh aη0

I2

τ

ξ

xK

(2.46)

58

2.4. Unruh effect for the Majorana field

where we introduced the Rindler proper time τ , that is the line element of an

observer at rest w.r.t the Rindler frame, indeed the metric turns Minkowskian

upon the substitution ξ0η Ñ τ :

ds2 paξq2 dη2 dξ2 dxK2

Ñ dτ 2 dξ2 dxK2 (2.47)

actually Lµν can be expressed in term of its generator:

Lµν exp paη0 J1q (2.48)

J1

0 1 0 0

1 0 0 0

0 0 0 0

0 0 0 0

(2.49)

that is just the boost generator along the x-axis for flat-spacetime QFT, in-

deed the whole structure of this transformation is that of a boost with the

identification β Ø aη0, hence this is actually a boost of time-varying velocity

which increases linearly with time, the action of a constant proper accelera-

tion is manifest. Moreover, the fact that this transformation holds locally is

a consequence of the fact that one can find a Lorentz boost transformation

only between MS and an instantaneous rest frame of a Rindler observer i.e.

an inertial frame which at time η η0 has the same velocity as the Rindler

observer but that can be related to MS by a simple Lorentz boost by virtue

of its inertiality.

Since the metric is given by (2.47), the gamma matrices are just the

Minkowskian gamma matrices since the Vierbeins read V µα δµα and they

transform as contravariant vectors under a coordinate transformation

pxµq1 Lµν xν Ñ pγµpxqq1 Lµνpxq γνpxq (2.50)

Lµνpxq 9γν

by direct inspection one has

gµν Lµα L

νβ gαβ (2.51)

hence confirming that L is a Lorentz-like transformation in the sense that

it leaves the metric invariant (one must keep in mind that L is just the

59

Chapter 2. The Unruh effect

local, linearized version of (2.44)); henceforth the new set of gammas will be

equivalent to the former in the sense that

tpγµq1, pγνq1u 2gµν tγµ, γνu (2.52)

this means that one can find a certain real matrix S satisfying

pγµq1 S1γµS (2.53)

the matrix S can be rendered unique, up to a sign, by imposing the normal-

ization constraint

detS 1 (2.54)

indeed, if we assume that there exist another matrix T such that S1γαS T1γαT then it must also hold that γα ST1 ST1 γα which entails ST1 λI i.e. S λT , this proves that the normalization condition fixes the arbi-

trary multiplicative factor up to a sign.

From (2.50) and (2.53) it follows that

Lµν γν S1 γµ S (2.55)

since we have required S to be normalized we can, without loss of gener-

ality, set S exp

12εαβ J αβ

and expand (2.55) up to first order in the

transformation parameters

γµ εµνγν

I εαβ

2J αβ

γµ

I εαβ

2J αβ

(2.56)

εµνγν 1

2εαβ

γµ,J αβ

(2.57)

now, let us recall that εαβ εβα since L leaves the metric unchanged,

henceforth if we antisymmetrize the l.h.s. of the last equation according to

1

2εαβ pgµαγβ gµβγαq 1

2εαβ

γµ,J αβ

(2.58)

we can finally get rid of the transformation parameters ε without the indeter-

minacy of a possible additive symmetric factor. The solution to this equation

is well known and reads

J αβ 1

4

γα, γβ

(2.59)

60

2.4. Unruh effect for the Majorana field

just as one could naively expect, it turns out that the generators of the spin

algebraic transformation for this particular Lorentz-like transformation are

actually the Lorentz group generators.

Hence we have found the general form of a Lorentz-like spinor algebraic

transformation, whereas in our case

ε01 ε10 aη0 all other components vanishing (2.60)

that leads to a Minkowski-to-Rindler algebraic spinor transformation which

infinitesimal form reads

S I 1

29γ09γ1 aη0 I i

2εµνΣ

µν (2.61)

Σµν i

4rγµ, γνs

hence the finite transformation reads

ψ1px1q SpLqψpxq exp

aη0

29γ09γ1ψpxq (2.62)

finally, if we require a transformation that follows the Rindler observer through-

out its whole motion, we can just rewrite (2.62) according to

SpLq expaη

29γ09γ1

(2.63)

Notice that one could obtain the above transformation rule also by means

of eq (1.50), indeed recalling that the only non-vanishing Fock-Ivanenko co-

efficient is Γη, that was obtained in (1.105) it is straightforward that

ψ1RSpxq NpXqψMSpXqNpXpxqq exp

!» x

dx1µ Γµpx1q) exp

!» x

dη1a

29γ09γ1)

exp! aη

29γ09γ1) S1pLq (2.64)

in full accordance with our previous derivation.

The fact that the spinor transformation rule resembles a time-varying

boost along the acceleration axis is clearly intuitive in terms of classical me-

chanics. Nonetheless it also tells us that Rindler-Fulling modes are actually

eigenstates of the generator of boosts along the acceleration axis, indeed in

literature they are sometimes referred to as boost modes.

61

Chapter 2. The Unruh effect

b. Consistency with the general theory of spinors in

curved spacetimes

In order to enforce our derivation of the spinor representation of the RS-

to-MS coordinate transformation, let us show that it is consistent with the

well-known generally covariant form of the (Dirac) Majorana equation. In

what follows we shall call ψMpXq, ψRpxq respectively the Minkowskian and

Rindler spinors linked by

X L x ψMpXq SpLqψRpxq (2.65)

X P MS x P RS

then it is convenient to start from the flat Majorana equation:

0 i 9γ0Bt i 9γ1Bx i~9γK ~BK m

ψMpXq

i 9γ0Bt i 9γ1Bx i~9γK ~BK m

eaη2

9γ09γ1

ψRpxqinverting (2.44) one gets

ξ ?x2 t2 aη arctanh

t

x(2.66)

Bt t?x2 t2

Bξ 1

ax

1

1 tx

2 Bη

sinhpaηq Bξ 1

aξcoshpaηq Bη (2.67)

Bx x?x2 t2

Bξ 1

a

1

1 tx

2

t

x2

Bη

coshpaηq Bξ 1

aξsinhpaηq Bη (2.68)

substituting into our wave equation bringsi 9γ0

sinhpaηq Bξ 1

aξcoshpaηq Bη

i 9γ1

coshpaηq Bξ 1

aξsinhpaηq Bη

i~9γK ~BK m

eaη2

9γ09γ1

ψRpxq

(2.69)

62

2.4. Unruh effect for the Majorana field

#

eaη2

9γ09γ1

i 9γ0

sinhpaηq Bξ 1

aξcoshpaηq Bη 9γ0 9γ1

2ξcoshpaηq

i 9γ1

coshpaηq Bξ 1

aξsinhpaηq Bη 9γ0 9γ1

2ξsinhpaηq

eaη2

9γ09γ1 i~9γK ~BK m

+ψRpxq

#

eaη2

9γ09γ1

i9γ0

aξ

coshpaηq 9γ0

9γ1 sinhpaηq Bη a

29γ09γ1

i 9γ1coshpaηq 9γ0

9γ1 sinhpaηq Bξ

eaη2

9γ09γ1 i~9γK ~BK m

+ψRpxq

eaη2

9γ09γ1 pi∇ mq ψRpxq (2.70)

which completes the proof.

c. Helicity-eigenstate normal modes

So far we have obtained the explicit algebraic spinor operator that carries

out the transformation of spinors from Rindler coordinates to Minkowskian

ones, and confirmed its validity. What we still lack is a suitable expression for

the McDonald functions in Minkowski coordinates, however literature offers a

variety of integral representations and we may employ the following suitable

one (see [59] §6.22)

Kνpzq 1

2e

iπν2

» 8

8dθ eiz sinh θνθ <pzq ¡ 0 (2.71)

indeed it is straightforward that

Kνpβξq eaην 1

2e

iπν2

» 8

8dθ eiβξ sinh θνpθaηq

1

2e

iπν2

» 8

8dθ eiβξpcosh aη sinh θ sinh aη cosh θqνθ

1

2e

iπν2

» 8

8dθ eipωtkxxqνθ

63

Chapter 2. The Unruh effect

where we set ω β cosh θ and kx β sinh θ and used relations (2.44) to

emphasize that these functions can be regarded as a superposition of positive-

frequency or negative-frequency 2-dimensional plane waves, since the funda-

mental relation β2 k2x ω2 of flat-spacetime QFT is satisfied. Actually, in

what follows, we will use the representation

Kνpβξq eaην 1

2e

iπν2

» 8

8dθ eikxxiωtνθ (2.72)

If we recall that, by construction:

9γ09γ1 ΥÒÓ

ΥÒÓ 9γ0

9γ1 uÒÓ uÒÓ (2.73)

SpLqΥÒÓ e

aη2 ΥÒÓ

SpLquÒÓ eaη2 uÒÓ (2.74)

eventually the Rindler modes in Minkowskian coordinates read, within the

right Rindler wedge

RpMqψ

ÒÓ,µpxq α θpξq eiµaηikKxK

σÒÓ iβ SpLqΥÒÓ

Kiµ12pβξq SpLquÒÓKiµ12pβξq

α θp?x2 t2q eikKxK

σÒÓ iβ eaηpiµ 1

2qΥÒÓKiµ12pβξq eaηpiµ 1

2quÒÓKiµ12pβξq

α

2θp?x2 t2q eikKxK

σÒÓ iβΥÒÓ

eiπ2 piµ 1

2q» 8

8dθ eiωtikxxpiµ 1

2qθ

uÒÓ eiπ2 piµ 1

2q» 8

8dθ eiωtikxxpiµ 1

2qθ R

pMqψÒÓ,µpXq

σÒ , σÓ

together with their counterparts

RpMqψ

ÒÓ,µpxq α θpξq eiµaηikKxK

σÒÓ iβ SpLqΥÒÓ

Kiµ12pβξq SpLquÒÓKiµ12pβξq

(2.75)

64

2.4. Unruh effect for the Majorana field

α θp?x2 t2q eikKxK

σÒÓ iβ eaηpiµ 1

2qΥÒÓKiµ12pβξq eaηpiµ 1

2quÒÓKiµ12pβξq

α

2θp?x2 t2q eikKxK

σÒÓ iβΥÒÓ

eiπ2 piµ 1

2q» 8

8dθ eiωtikxxpiµ 1

2qθ

uÒÓ eiπ2 piµ 1

2q» 8

8dθ eiωtikxxpiµ 1

2qθ R

pMqψÒÓ,µpXq

σÒ , σÓ

where X P MS; clearly one can choose among all the four combinations of

integral representations for each set of modes, all being equivalent within the

right Rindler wedge.

d. Canonical normal modes

For reasons that will become clear later, we want to find the explicit

expression of modes (1.149) for a Minkowskian observer.

By making use of (2.72) and noticing that

9γ09γ1 Θ

a Θa 9γ0

9γ1 Θa pkKq Θ

a pkKq (2.76)

SpLqΘa e

aη2 Θ

a SpLq Θa pkKq e

aη2 Θ

a pkKq (2.77)

if, just for convenience, we call canonical modes those given in (1.149), then

we can write down their Minkowskian version within the right Rindler wedge

as:

RpMqUa,µ,kKpxq

cpk0,kKqp2πq 3

2

θpξq eiµaηikKxK

β SpLqΘ

aKiµ12pβξq SpLqΘaKiµ12pβξq

cpk0,kKq

p2πq 32

θp?x2 t2q eikKxK

β eaηpiµ 1

2qΘaKiµ12pβξq eaηpiµ 1

2qΘaKiµ12pβξq

(2.78)

65

Chapter 2. The Unruh effect

cpk0,kKq2

52π

32

θp?x2 t2q eikKxK

βΘ

a eiπ2 piµ 1

2q» 8

8dθ eiωtikxxpiµ 1

2qθ

Θa ei

π2 piµ 1

2q» 8

8dθ eiωtikxxpiµ 1

2qθ R

pMqUa,µ,kKpXq(2.79)

together with their counterparts

RpMqVa,µ,kKpxq

cpk0,kKqp2πq 3

2

θpξq eiµaηikKxK

β SpLqΘ

aKiµ12pβξq SpLqΘaKiµ12pβξq

(2.80)

cpk0,kKqp2πq 3

2

θp?x2 t2q eikKxK

β eaηpiµ 1

2qΘaKiµ12pβξq eaηpiµ 1

2qΘaKiµ12pβξq

cpk0,kKq

252π

32

θp?x2 t2q eikKxK

βΘ

a eiπ2 piµ 1

2q» 8

8dθ eiωtikxxpiµ 1

2qθ

Θa ei

π2 piµ 1

2q» 8

8dθ eiωtikxxpiµ 1

2qθ R

pMqVa,µ,kKpXq(2.81)

where X P MS; again one can choose among all the four combinations of

integral representations for each set of modes, all being equivalent within the

right Rindler wedge.

e. Choosing the proper representation for helicity eigen-

states

We now come to a crucial point: we need to compare two quantization

schemes, but in order to build a Fock space related to Minkowski space-

time one needs a complete-orthonormal basis that covers the whole MS. Un-

66

2.4. Unruh effect for the Majorana field

ruh found that it is possile to extend the above expressions of the Rindler-

Majorana modes, by combining different integral representations: in the fol-

lowing we present Unruh’s original procedure.

We will first discuss the procedure for the Majorana-Rindler helicity-

eigenstate modes, thereafter we will repeat the whole procedure for the canon-

ical modes, finally the benefits of each of the two representations will be clear.

Recall that the two representation are completely equivalent and linked to

each other by a unitary transformation, as we have already shown.

First we must drop the Heaviside θ-terms, we shall then make a choice

for the integral representations to use: we’ll define

pMqψÒÓ,µpX|aq def Aa

2eikKxK (2.82)

σÒÓ iβΥÒÓ

eiπ2 piµ 1

2q» 8

8dθ eipωtkxxqpiµ 1

2qθ

uÒÓ eiπ2 piµ 1

2q» 8

8dθ eipωtkxxqpiµ 1

2qθ

pMqψÒÓ,µpX|`q def A`

2eikKxK (2.83)

σÒÓ iβΥÒÓ

eiπ2 piµ 1

2q» 8

8dθ eipωtkxxqpiµ 1

2qθ

uÒÓ eiπ2 piµ 1

2q» 8

8dθ eipωtkxxqpiµ 1

2qθ

where we introduced the two different normalization coefficientsAa, A` which

are due since normalization in MS is generally different from the one in RS;

actually it is not difficult (see appendix) to obtain thatpMqψ

ÒÓ,µpX|aq,pMq ψ

òó,µ1pX|aq

MS

|Aa|2eπµ8aπ4β δÒÓ,òóδpk0 k01qδpkK kK

1qpMqψ

ÒÓ,µpX|`q,pMq ψ

òó,µ1pX|`q

MS

|A`|2eπµ8aπ4β δÒÓ,òóδpk0 k01qδpkK kK

1qpMqψ

ÒÓ,µpX|`q,pMq ψ

òó,µ1pX|aq

MS

0 @pµ,kK, ÒÓq; pµ1,kK1,òóq(2.84)

so that we may suitably choose, up to a phase factor

Aa eπµ2

π2?

8aβA` e

πµ2

π2?

8aβ(2.85)

67

Chapter 2. The Unruh effect

Actually the same can be done for the pMqψÒÓ,µpXq, namely we define two

different representations as:

pMqψÒÓ,µpX|aq def Ba

2eikKxK (2.86)

σÒÓ iβΥÒÓ

eiπ2 piµ 1

2q» 8

8dθ eipωtkxxqpiµ 1

2qθ

uÒÓ eiπ2 piµ 1

2q» 8

8dθ eipωtkxxqpiµ 1

2qθ

pMqψÒÓ,µpX|`q def B`

2eikKxK (2.87)

σÒÓ iβΥÒÓ

eiπ2 piµ 1

2q» 8

8dθ eipωtkxxqpiµ 1

2qθ

uÒÓ eiπ2 piµ 1

2q» 8

8dθ eipωtkxxqpiµ 1

2qθ

where we introduced the two different normalization coefficients Ba, B`; ac-

tually it can be obtained in the way as before thatpMqψ

ÒÓ,µpX|aq,pMq ψ

òó,µ1pX|aq

MS

|Ba|2eπµ8aπ4β δÒÓ,òóδpk0 k01qδpkK kK

1qpMqψ

ÒÓ,µpX|`q,pMq ψ

òó,µ1pX|`q

MS

|B`|2eπµ8aπ4β δÒÓ,òóδpk0 k01qδpkK kK

1qpMqψ

ÒÓ,µpX|`q,pMq ψ

òó,µ1pX|aq

MS

0 @pµ,kK, ÒÓq; pµ1,kK1,òóq(2.88)

so that we may suitably choose, up to a phase factor

Ba Aa B` A` (2.89)

Actually, one can check (see appendix) that these two sets of modes enjoy

completeness separately w.r.t. MS, i.e.¸ÒÓ,µ,kK

pMqψ

ÒÓ,µpX|`q b pMqψ

ÒÓ :,µpX 1|`q (2.90)

pMqψÒÓ,µpX|aq b pMqψ

ÒÓ :,µpX 1|aq

X0X01

δpXX1q¸ÒÓ,µ,kK

pMqψ

ÒÓ,µpX|`q b pMqψ

ÒÓ :,µpX 1|`q (2.91)

pMqψÒÓ,µpX|aq b pMqψ

ÒÓ :,µpX 1|aq

X0X01

δpXX1q

68

2.4. Unruh effect for the Majorana field

By virtue of (2.72) one can easily infer that the following identity holds

true within the left Rindler wedge

Kνpβξq eaην 1

2e

iπν2

» 8

8dθ eipωtkxxqνθ

1

2e

iπν2

» 8

8dθ eiωtikxxνθ (2.92)

as we will see shortly, this clarifies the reason of our choice on integral rep-

resentations (2.86) and (2.87). From this crucial observation follows the

celebrated Unruh trick: let’s consider the combinations

RÒÓ,µpXq e

πµ2 pMqψ

ÒÓ,µpX|aq e

πµ2 pMqψ

ÒÓ,µpX|`q?

2 coshπµ(2.93)

LÒÓ,µpXq eπµ2 pMqψ

ÒÓ,µpX|aq e

πµ2 pMqψ

ÒÓ,µpX|`q?

2 coshπµ(2.94)

these modes enjoy some crucial features:

• They all satisfy the MS version of the Majorana equation, by construc-

tion.

• They are analytical over the whole MS, by construction.

• They are orthonormal in MS, by virtue of equations (2.84).

• As shown in A.1 d. they form a complete set in MS, by virtue of

equation (2.90).

• They enjoy:$'''&'''%RÒÓ,µpXq R

pMqψÒÓ,µpxq in the right Rindler wedge

RÒÓ,µpXq 0 in the left Rindler wedge

LÒÓ,µpXq LpMqψ

ÒÓ,µpxq in the left Rindler wedge

LÒÓ,µpXq 0 in the right Rindler wedge

the last point needs to be verified, let us do so.

69

Chapter 2. The Unruh effect

If we confine ourselves to the right part of the wedge

RÒÓ,µpXq e

πµ2 Aa e

πµ2 A`a

2 coshpπµq eikKxK (2.95)

σÒÓ iβ eaηpiµ 1

2qΥÒÓKiµ12pβξq eaηpiµ 1

2quÒÓKiµ12pβξq

1

α 1

2π2

dcoshpπµq

aβR

pMqψÒÓ,µpXq (2.96)

SpLq RpRq ψ

ÒÓ,µpxq x L1 X

LÒÓ,µpXq eπµ

2 Aa eπµ2 A`a

2 coshpπµq eikKxK

σÒÓ iβ eaηpiµ 1

2qΥÒÓKiµ12pβξq eaηpiµ 1

2quÒÓKiµ12pβξq

0 (2.97)

upon recalling that µ k0a. While, turning to the left Rindler wedge

RÒÓ,µpXq 1

4π2aaβ coshpπµq eikKxK (2.98)

#σÒÓ iβΥÒÓ

eπµi

π2 piµ 1

2q» 8

8dθ eiωtikxxpiµ 1

2qθ

eπµiπ2 piµ 1

2q» 8

8dθ eiωtikxxpiµ 1

2qθ

uÒÓ

eπµi

π2 piµ 1

2q» 8

8dθ eiωtikxxpiµ 1

2qθ

eπµiπ2 piµ 1

2q» 8

8dθ eiωtikxxpiµ 1

2qθ+

it is now sufficient to use the following formulæ

eπµiπ2 piµ 1

2q eiπ2 piµ 1

2q piq (2.99)

eπµiπ2 piµ 1

2q eiπ2 piµ 1

2q piq (2.100)

together with equation (2.92) to achieve that

RÒÓ,µpXq 0 (2.101)

by a similar, straightforward reasoning one can see that within the left part

of the wedge also the following holds true

LÒÓ,µpXq LpMqψ

ÒÓ,µpxq (2.102)

70

2.4. Unruh effect for the Majorana field

which completes the proof of the properties of Unruh modes.

One can define as well the combinations

RÒÓ,µpXq e

πµ2 pMqψ

ÒÓ,µpX|aq e

πµ2 pMqψ

ÒÓ,µpX|`q?

2 coshπµ(2.103)

LÒÓ,µpXq eπµ2 pMqψ

ÒÓ,µpX|aq e

πµ2 pMqψ

ÒÓ,µpX|`q?

2 coshπµ(2.104)

and verify quite easily that, also for these modes, all the above properties are

satisfied.

Summarizing, we found two sets of normal modes:

• The!pMqψ

ÒÓ,µpX|aq , pMqψ

ÒÓ,µpX|`q , pMqψ

ÒÓ,µpX|aq , pMqψ

ÒÓ,µpX|`q

), which

are orthonormal and complete (orthonormality and completeness hold

for the set of ψ separately from the set of ψ) and analytical on MS.

Another important feature of these modes is that they can be regarded

as the superposition of purely positive-frequency plane waves or purely

negative frequency ones with respect to Minskowskian time. Henceforth

a Fock space built one them must be equivalent to the usual standard

flat QFT Fock space.

• The!LÒÓ,µpXq , RÒÓ

,µpXq , LÒÓ,µpXq , RÒÓ,µpXq

), these are the so-called

Unruh modes; they are complete, orthonormal (again the set enjoys

orthonormality and completneness separately from the set) and an-

alytical on MS and reduce to Rindler modes within the corresponding

sectors of the wedge.

f. Choosing the proper representation for canonical

modes

Within this subsection we use the so-called canonical modes and repeat

the machinery of the previous subsection, quite quickly one might definepMqU1,µ,kKpX|aqpMqU2,µ,kKpX|aq

U :

pMqψ

Ò,µpX|aq

pMqψÓ,µpX|aq

(2.105)

pMqU1,µ,kKpX|`qpMqU2,µ,kKpX|`q

U :

pMqψ

Ò,µpX|`q

pMqψÓ,µpX|`q

(2.106)

71

Chapter 2. The Unruh effect

where U is given by (1.165), in this manner we just get

pMqUa,µ,kKpX|aq def Aa?8

eikKxK (2.107)

βΘ

a eiπ2 piµ 1

2q» 8

8dθ eipωtkxxqpiµ 1

2qθ

Θa ei

π2 piµ 1

2q» 8

8dθ eipωtkxxqpiµ 1

2qθ

pMqUa,µ,kKpX|`q def A`?8

eikKxK (2.108)

βΘ

a eiπ2 piµ 1

2q» 8

8dθ eipωtkxxqpiµ 1

2qθ

Θa ei

π2 piµ 1

2q» 8

8dθ eipωtkxxqpiµ 1

2qθ

notice that the normalization factor differs by a multiplicative factor?

2

from eqs. (2.82), (2.83), this accounts for the fact that the tΘa , Θ

a u aren’t

normalized while the tΥÒÓ , u

ÒÓ u are, indeed:

U

Θ1

Θ2

i

?2

ΥÒ

ΥÓ

(2.109)

V

Θ1

Θ2

i

?2

ΥÒ

ΥÓ

(2.110)

U

Θ1 pkKq

Θ2 pkKq

?2

uÒpkKquÓpkKq

(2.111)

V

Θ1 pkKq

Θ2 pkKq

?2

uÒpkKquÓpkKq

(2.112)

By virtue of (2.84) it is obvious thatpMqUa,µ,kKpX|aq,pMq Ua1,µ1,kK1pX|aq

MS

δa,a1 δpk0 k01q δpkK kK

1qpMqUa,µ,kKpX|`q,pMq Ua,µ,kKpX|`q

MS

δa,a1 δpk0 k01q δpkK kK

1qpMqUa,µ,kKpX|`q,pMq Ua,µ,kKpX|aq

MS

0 @pµ,kK, ÒÓq; pµ1,kK1,òóq(2.113)

72

2.4. Unruh effect for the Majorana field

Actually the same can be done for the pMqVa,µ,kKpXq, namely

pMqV1,µ,kKpX|aqpMqV2,µ,kKpX|aq

V :

pMqψ

Ò,µpX|aq

pMqψÓ,µpX|aq

(2.114)

pMqV1,µ,kKpX|`qpMqV2,µ,kKpX|`q

V :

pMqψ

Ò,µpX|`q

pMqψÓ,µpX|`q

(2.115)

where V is given by (1.172), in this manner we just get

pMqVa,µ,kKpX|aq def Aa?8

eikKxK (2.116)

βΘ

a eiπ2 piµ 1

2q» 8

8dθ eipωtkxxqpiµ 1

2qθ

Θa ei

π2 piµ 1

2q» 8

8dθ eipωtkxxqpiµ 1

2qθ

pMqVa,µ,kKpX|`q def A`?8

eikKxK (2.117)

βΘ

a eiπ2 piµ 1

2q» 8

8dθ eipωtkxxqpiµ 1

2qθ

Θa ei

π2 piµ 1

2q» 8

8dθ eipωtkxxqpiµ 1

2qθ

By virtue of (2.88) it is obvious that

pMqVa,µ,kKpX|aq,pMq Va1,µ1,kK1pX|aq

MS

δa,a1 δpk0 k01q δpkK kK

1qpMqVa,µ,kKpX|`q,pMq Va,µ,kKpX|`q

MS

δa,a1 δpk0 k01q δpkK kK

1qpMqVa,µ,kKpX|`q,pMq Va,µ,kKpX|aq

MS

0 @pµ,kK, ÒÓq; pµ1,kK1,òóq(2.118)

Also for canonical modes, it can be shown (see appendix) that they enjoy

73

Chapter 2. The Unruh effect

completeness w.r.t. MS separately, in the sense that¸a,µ,kK

pMqUa,µ,kKpX|`q b pMqU :a,µ,kKpX 1|`q (2.119)

pMqUa,µ,kKpX|aq b pMqU :a,µ,kKpX 1|aqX0X01

δpXX1q¸a,µ,kK

pMqVa,µ,kKpX|`q b pMqV:a,µ,kKpX 1|`q (2.120)

pMqVa,µ,kKpX|aq b pMqV:a,µ,kKpX 1|aqX0X01

δpXX1qOnce again, if we notice that the following identity holds true within the

left Rindler wedge

Kνpβξq eaην 1

2e

iπν2

» 8

8dθ eiωtikxxνθ (2.121)

From this crucial observation follows the celebrated Unruh trick: let’s con-

sider the combinations

URa,µ,kKpXq eπµ2 pMqUa,µ,kKpX|aq e

πµ2 pMqUa,µ,kKpX|`q?

2 coshπµ(2.122)

ULa,µ,kKpXq eπµ2 pMqUa,µ,kKpX|aq e

πµ2 pMqUa,µ,kKpX|`q?

2 coshπµ(2.123)

As for helicity-eigenstate Unruh modes, also the canonical Unruh modes

enjoy some crucial features:

• They all satisfy the MS version of the Majorana equation, by construc-

tion.

• They are analytical over the whole MS, by construction.

• They enjoy orthonormality in MS, by virtue of equations (2.113).

• They enjoy completeness in MS, by virtue of equations (2.119).

• They enjoy:$'''&'''%URa,µ,kKpXq R

pMqUa,µ,kKpxq in the right Rindler wedge

URa,µ,kKpXq 0 in the left Rindler wedge

ULa,µ,kKpXq LpMqUa,µ,kKpxq in the left Rindler wedge

ULa,µ,kKpXq 0 in the right Rindler wedge

74

2.4. Unruh effect for the Majorana field

proof of the last point follows readily from the results on helicity-eigenstate

Unruh modes, we shall not repeat the procedure here.

One can define as well the combinations

VRa,µ,kKpXq eπµ2 pMqVa,µ,kKpX|aq e

πµ2 pMqVa,µ,kKpX|`q?

2 coshπµ(2.124)

VLa,µ,kKpXq eπµ2 pMqVa,µ,kKpX|aq e

πµ2 pMqVa,µ,kKpX|`q?

2 coshπµ(2.125)

and verify quite easily that, also for these modes, all the above properties are

satisfied.

Summarizing, we found again two sets of normal modes:

• pMqUa,µ,kKpX|aq , pMqUa,µ,kKpX|`q , pMqVa,µ,kKpX|aq , pMqVa,µ,kKpX|`q

(,

which are orthonormal and complete (orthonormality and completeness

hold for the set of U separately from the set of V) and analytical on

MS. Another important feature of these modes is that they can be re-

garded as the superposition of purely positive-frequency plane waves

or purely negative frequency ones with respect to Minskowskian time.

Henceforth a Fock space built one them must be equivalent to the usual

standard flat QFT Fock space.

• tURa,µ,kKpXq , ULa,µ,kKpXq , VRa,µ,kKpXq , VLa,µ,kKpXqu, these are the

corresponding canonical Unruh modes ; they are complete and orthonor-

mal (again, orthonormality and completeness hold separately for the Uand for the V) and analytical on MS and reduce to Rindler modes

within the corresponding sectors of the wedge.

g. Digression: normalization of Rindler modes in RS

In the last two subsections we obtained classes of normal modes that

are orthonormal, complete and reduce exactly to the MS version of the RS

Rindler-Majorana modes, both for canonical and helicity-eigenstate modes.

Let us now take advantage of the results we just obtained and employ

them to check the consistency with our previous result on the difficult integral

of equation (1.143): the normalization of Rinlder modes w.r.t. RS spinor

inner product. We have seen thatRÒÓ,µ, R

òó,µ1

MS

δÒÓ,òó δpk0 k01q δpkK kK1q (2.126)

75

Chapter 2. The Unruh effect

however this inner product is just a particular case of the more general form»dΣ0pxq RÒÓ

,µ γ0pxqRòó

,µ1 (2.127)

corresponding to the choice of Minkowskian coordinates and of the t 0

spacelike hypersurface. Indeed expression (2.127) is, by definition, invariant

under general coordinate transformations, and we may express it in Rindler

coordinates using (2.95) and (2.101) as» 8

0

dξ p2πq2 δpkK kK1q δÒÓ,òó β2

π2?

8aβ2

a4 coshpπµq coshpπµ1q

Kiµ 1

2pβξqKiµ1 1

2pβξq Kiµ 1

2pβξqKiµ1 1

2pβξq

(2.128)

δpkK kK1q δÒÓ,òó β

aπ2

acoshpπµq coshpπµ1q Ipµ, µ1q

Ipµ, µ1q » 8

0

dξKiµ 1

2pβξqKiµ1 1

2pβξq Kiµ 1

2pβξqKiµ1 1

2pβξq

(2.129)

which brings to the solution of this difficult integral:

Ipµ, µ1q π2

β coshpπµq δpµ µ1q (2.130)

This is another example reminding how powerful the formalism of general

covariance is, the result being in full accordance with (1.145).

h. Comparing the helicity-eigenstate scheme with the

canonical modes one: advantages of each scheme

Let us explain why we decided to develop two parallel Unruh schemes.

The first one is based on helicity-eigenstate normal modes, the main ad-

vantage of this scheme is its physical interpretation: each mode corresponds

to the C.S.C.O.1 labelled by pk0,kK, ÒÓq. Let us study second quantization on

these modes within the right Rindler wedge: first of all we need to separate

the positive-frequency modes from the negative-frequency ones, for this pur-

pose we will choose to restrict ourselves to the ψÒÓ,µ,kK modes, since we have

1Complete Set of Commuting Observables

76

2.4. Unruh effect for the Majorana field

shown that they enjoy completeness without the need to introduce ψÒÓ,µ,kK

ψpxq ¸ÒÓ

»R3

dk0 dkK aÒÓ,µ,kK ψ

ÒÓ,µ,kKpxq (2.131)

¸ÒÓ

» 8

0

dk0

»R2

dkK aÒÓ,µ,kK ψ

ÒÓ,µ,kKpxq aÒÓ,pµq,kK ψ

ÒÓ,pµq,kKpxq

hence quantization proceeds as usual, by performing the formal replacements

aÒÓ,µ,kK Ñ aÒÓ,µ,kK aÒÓ,pµq,kK Ñ bÒÓ :,pµq,kK (2.132)

and imposing canonical anticommutation relations. The disadvantage of this

scheme is that the nature of self-conjugateness is not manifest, this leads us

to introduce a new set of operators which, nonetheless, are not independent

from the first ones.

On the other hand, if we study quantization on the canonical modes, we

readily obtain

ψpxq ¸a

»R3

dk0 dkK fa,µ,kK Ua,µ,kKpxq (2.133)

¸a

» 8

0

dk0

»R2

dkK fa,µ,kK Ua,µ,kKpxq fa,pµq,kK Ua,pµq,kKpxq

by virtue of (1.149) and (1.153) we can recast the above expression as

ψpxq ¸a

» 8

0

dk0

»R2

dkK fa,µ,kK Ua,µ,kKpxq fa,pµq,pkKq pUa,µ,kKpxqq

where we performed a change of variable on the second part of the integrand,

according to kK Ñ kK; quantization can be achieved by performing the

formal replacements

fa,µ,kK Ñ fa,µ,kK fa,pµq,pkKq Ñ f :a,µ,kK (2.134)

within this scheme of normal modes the self-conjugation of Majorana’s field

can be expressed in a manifest way, which considerably simplifies the math-

ematical treatment of the Unruh effect.

Summarizing: the canonical-modes quantization scheme allows for a cleaner

quantization procedure, in the sense that we do not need to introduce a new

77

Chapter 2. The Unruh effect

set of operators, as it was the case for helicity-eigenstate normal modes2. On

the other hand, the physical interpretation of these modes is not as clear as

that of the former set, indeed canonical modes are a linear combination of

modes with opposite helicity.

The advantages of canonical normal modes will be clear within the next

section.

i. The thermal spectrum for Majorana fermions

We are now ready to find the particle spectrum: first we build the Fock

space on MS, employing the canonical modes, so to take advantage of their

self-conjugation properties. Recalling that the tMUa,µ,kKpX|`q , MUa,µ,kKpX|`quform a complete set, the field expansion reads:

pMqψpXq ¸a

»R3

aa,µ,kK pMqUa,µ,kKpX|aq ca,µ,kK pMqUa,µ,kKpX|`q

dk0 dkK

if we notice, by direct inspection, that

pMqUa,µ,kKpX|`q pMqUa,µ,kKpX|aq

(2.135)

(2.136)

then, by virtue of the self conjugation property of the Majorana field (that

follows directly from the field equation), we can infer the necessary relation-

ships

ca,µ,kK paa,µ,kKq (2.137)

so that quantization is achieved by performing the formal replacements

aa,µ,kK Ñ aa,µ,kK ca,µ,kK Ñ a:a,µ,kK (2.138)

together with the usual anticommutation rules. The quantized Minkowskian

field reads

pMqψpXq ¸a

»R3

dk0 dkKaa,µ,kK pMqUa,µ,kKpX|aq h.c.

(2.139)

¸a

»R3

dk0 dkKaa,µ,kK pMqUa,µ,kKpX|aq a:a,µ,kK pMqUa,µ,kKpX|`q

2The number of degrees of freedom however is the same, since the operators bÒÓ :

,pµq,kK

and aÒÓ,µ,kK are not independent, a relation is found requiring the self-cojugation of the

field

78

2.4. Unruh effect for the Majorana field

The self-conjugation property of the field is manifest in (2.139), moreover no-

tice that, had we used the helicity-eigenstate normal modes, we would need

to introduce two more sets of creation/annihilation operators. Remind that

modes Ua,µ,kKpX|aq are a superposition of positive-frequency plane waves,

whereas modes Ua,µ,kKpX|`q are a superposition of negative-frequency plane

waves; henceforth we shall regard this quantization scheme as the one equiv-

alent to the standard flat-QFT one, i.e. it is the Fock space corresponding

to an inertial observer.

By inverting relations (2.122), (2.123) as

pMqUa,µ,kKpX|aq eπµ2 URa,µ,kKpXq e

πµ2 ULa,µ,kKpXq?

2 coshπµ(2.140)

pMqUa,µ,kKpX|`q eπµ2 URa,µ,kKpXq e

πµ2 ULa,µ,kKpXq?

2 coshπµ(2.141)

we come to the second, inequivalent quantization scheme:

pMqψpXq ¸a

»R3

dk0 dkKUra,µ,kK URa,µ,kKpXq U la,µ,kK ULa,µ,kKpXq

such expansion univocally defines the operators

Ura,µ,kK eπµ2 aa,µ,kK e

πµ2 a:a,µ,kKa

2 coshpπµq (2.142)

U la,µ,kK eπµ

2 aa,µ,kK eπµ2 a:a,µ,kKa

2 coshpπµq (2.143)

that satisfy, by direct inspection, the anticommutation relations!Ura,µ,kK , Ur

:a1,µ1,kK

1

) δa,a1 δpkK kK

1q δpk0 k01q (2.144)!U la,µ,kK , U l

:a1,µ1,kK

1

) δa,a1 δpkK kK

1q δpk0 k01q (2.145)

all other anticommutators vanishing.

With the aid of (2.135) we easily notice that

URa,µ,kK URa,µ,kK

(2.146)

ULa,µ,kK ULa,µ,kK (2.147)

and also that, by construction:

Ura,µ,kK Ur:a,µ,kK

(2.148)

U la,µ,kK U l:a,µ,kK

(2.149)

79

Chapter 2. The Unruh effect

which lead us to write the field expansion in a manifestly self-conjugated

form:

pMqψpXq ¸a

» 8

0

dµ

»R2

dkKUra,µ,kK URa,µ,kKpXq

U la,µ,kK ULa,µ,kKpXq h.c.

Unruh suggested that this latter quantization scheme is just the equivalent

to the Rindler one. Indeed this claim is supported by the fact that the normal

modes URa,µ,kKpXq , ULa,µ,kKpXq reduce (up to a multiplicative factor) to

Rindler-Majorana modes in the respective sectors, and by the fact that the

creation-annihilation operators can be obtained as

Ura,µ,kK pURa,µ,kKpXq , ψpXqqMS (2.150)

U la,µ,kK pULa,µ,kKpXq , ψpXqqMS (2.151)

whereas these scalar products are taken just on the hypersurface t 0

where URa,µ,kKpXq , ULa,µ,kKpXq coincide exactly with the left/right Rindler-

Majorana modes, up to multiplicative factors analogous to those exploited

in (2.96) and (2.102).

Finally, the spectrum of quanta detected within the right Rindler wedge

by an accelerated observer as he moves through Minkowski’s vacuum is

x0M| Ur:a,µ,kK Ura,µ,kK |0My eπµ

2 cosh pπµq 1

e2πµ 1(2.152)

notice that, had we done the calculation for the V modes, we would have

obtained the same spectrum. Recalling that µ k0a where k0 is the Rindler

energy of the Majorana quantum, one indeed finds a Fermi-Dirac distribution

of corresponding temperature

TMajorana ~a2πckB

(2.153)

2.5 A different derivation of the Unruh effect:

helicity structure

We now want to study the helicity structure of the Bogolyubov coeffi-

cients. In order to do so, we shall need to find spinor normal modes that

80

2.5. A different derivation of the Unruh effect: helicity structure

are physically meaningful. Let us consider the matrix 9γ0: its two doubly

degenerate eigenvalues 1 are associated to the eigenspinors ξr, ηr

9γ0 ξr ξr 9γ0 ηr ηrit follows that ηr ξr by virtue of the purely imaginary nature of the gam-

mas.

If we consider the projector εppq that was introduced in (1.85) then

εppqξr is a solution of the Majorana equation when associated to the plane

wave eipx.From Noether’s theorem it is easily achieved that the intrinsic spin density

tensor reads

Sαβγpxq ψT pxq 9γ0

1

2

9γα,Σβγ

(ψpxq (2.154)

hence the third component of the spin operator reads

S12 »

dxψT pxq 9γ0

1

2

9γα,Σβγ

(ψpxq

»

dxψT pxq 9γ0

sz ψpxq

if p p0, 0, pq it is easy to achieve that

d

dtS12 0 (2.155)

If we notice that r 9γ0, szs 0, then we may choose a combination of ξ1, ξ2

such that

τÒ a11ξ1 a12ξ2 szτÒ 1

2τÒ

τÓ a21ξ1 a22ξ2 szτÓ 1

2τÓ (2.156)

in such a way that, when p 0

ετÒ τÒ ετÓ τÓ (2.157)

are both solutions to the Majorana equation and have respectively a posi-

tive/negative projection along the z direction.

If we define

uÒÓppq 2mpωp mq12 εppq τÒÓ (2.158)

81

Chapter 2. The Unruh effect

which satisfy

puÒ, uÒq 2ωp puÓ, uÓq puÓ, uÒq 0

eventually the normal modes

uÒÓ,ppxq rp2πq32ωps12 exptip xuuÒÓppq (2.159)

are normalized and have a definite spin projection along the z direction within

the rest frame.

Now that we have a physically meaningful expression for the spinor nor-

mal modes in MS, we are ready to evaluate the Bogolyubov coefficients be-

tween these and the Rindler helicity-eigenstate modes.

R,LαÒÓ,òópp; k0,kKq uÒÓ,p,

R,LpMqψ

òó,k0,kK

R,LβÒÓ,òópp; k0,kKq

uÒÓ,p,

R,LpMqψ

òó,k0,kK

Where the helicity eigenstates are understood as

R,LpMqψ

òó,k0,kK

pXq SpLq R,Lψòó,k0,kKpxq (2.160)

X L x X P MS x P RS

The Rψòó,k0,kKpxq being explicit in (1.170) and (1.171) while the Lψòó,k0,kK

pxqare given in (1.175) and (1.176).

For convenience we shall evaluate the above products in Minkowskian

coordinates, since the final result does not depend on the frame of reference.

If we take

ξ1 1?2

0

0

i1

ξ2 1?2

i

1

0

0

(2.161)

together with

τÒ 1

2

i

1

i

1

1?2pξ1 ξ2q τÓ 1

2

i

1

i1

1?2pξ1 ξ2q

(2.162)

82

2.5. A different derivation of the Unruh effect: helicity structure

then all the above requirements are satisfied. Consequently the spinor normal

modes that we need for our purposes read explicitly

uÒ,ppxq eipx

rp2πq32ωps 12

pm ωpq 12 θÒppq

uÓ,ppxq eipx

rp2πq32ωps 12

pm ωpq 12 θÓppq

where

θÒppq 1

2

i pm ω ipx py pzqm ω ipx py pz

i pm ω ipx py pzqm ω ipx py pz

θÓppq 1

2

i pm ω ipx py pzqm ω ipx py pz

i pm ω ipx py pzqm ω ipx py pz

We can now begin to evaluate the coefficients:

RαÒ,òpp; k0,kKq 1

2π2

dcoshpπk0aq

aβ p2πq3 2ωppωp mq p2πq2δpkK pKq

iβ pθ :

Ò Υòq Iνppx; k0q pθ :

Ò uòpkKqq Iνppx; k0q

where we understand ν ik0a 12, and making use of eq. 6.611 3. of

[25] the Iν read

Iνppx; k0q » 8

0

dx eipxxKνpβxq

iπ2ωp coshpπk0aq

piβqν ppx ωpqν piβqν ppx ωpqν

iπ2ωp coshpπk0aq

px ωp

px ωp

ν2

px ωp

px ωp

ν2

iπ

2ωp coshpπk0aq Γνppq (2.163)

83

Chapter 2. The Unruh effect

if we define

Rpp; k0q 2

dcoshpπk0aq

aβ p2πq3 2ωppωp mqiπ

2ωp coshpπk0aq

i πr2 a β p2 π ωpq3 coshpπk0aq pωp mqs 1

2

(2.164)

we can eventually write down the Bogolyubov coefficients in a conveniently

compact form:

RαÒ,òpp; k0,kKq Rpp; k0q2

δpkK pKq pm ωp px ipy pzq

iβ Γνppq ppx ωpqΓνppq

RαÒ,ópp; k0,kKq iRpp; k0q

2δpkK pKq pm ωp px ipy pzq

iβ Γνppq ppx ωpqΓνppq

RαÓ,òpp; k0,kKq i

Rpp; k0q2

δpkK pKq pm ωp px ipy pzq

iβ Γνppq ppx ωpqΓνppq

RαÓ,ópp; k0,kKq Rpp; k0q2

δpkK pKq pm ωp px ipy pzq

iβ Γνppq ppx ωpqΓνppq

in the same way we obtain the β’s, that read

RβÒ,òpp; k0,kKq iRpp; k0q

2δpkK pKq pm ωp px ipy pzq

iβ Γνppq ppx ωpqΓνppq

RβÒ,ópp; k0,kKq Rpp; k0q2

δpkK pKq pm ωp px ipy pzq

iβ Γνppq ppx ωpqΓνppqRβÓ,òpp; k0,kKq Rpp; k0q

2δpkK pKq pm ωp px ipy pzq

iβ Γνppq ppx ωpqΓνppq

RβÓ,ópp; k0,kKq iRpp; k0q

2δpkK pKq pm ωp px ipy pzq

iβ Γνppq ppx ωpqΓνppq

84

2.5. A different derivation of the Unruh effect: helicity structure

Similarly we can evaluate the left-Rindler wedge coefficients as

LαÒ,òpp; k0,kKq 1

2π2

dcoshpπk0aq

aβ p2πq3 2ωppωp mq p2πq2δpkK pKq

iβ pθ :

Ò ΥòqJνppx; k0q pθ :

Ò uòpkKqqJνppx; k0q

where we understand ν ik0a 12, and making use of eq. 6.611 3. of

[25] the Iν read

Jνppx; k0q » 0

8dx eipxxKνpβxq

» 8

0

dx eipxxKνpβxq

Iνppx; k0q iπ2ωp coshpπk0aq Γνppq

then we can cast the left-Rindler wedge Bogolyubov coefficients in a conve-

niently compact form:

LαÒ,òpp; k0,kKq Rpp; k0q2

δpkK pKq pm ωp px ipy pzq

iβ Γνppq ppx ωpqΓνppqLαÒ,ópp; k0,kKq i Rpp; k0q

2δpkK pKq pm ωp px ipy pzq

iβ Γνppq ppx ωpqΓνppq

LαÓ,òpp; k0,kKq i Rpp; k0q2

δpkK pKq pm ωp px ipy pzq

iβ Γνppq ppx ωpqΓνppqLαÓ,ópp; k0,kKq Rpp; k0q

2δpkK pKq pm ωp px ipy pzq

iβ Γνppq ppx ωpqΓνppq

85

Chapter 2. The Unruh effect

together with

LβÒ,òpp; k0,kKq i Rpp; k0q2

δpkK pKq pm ωp px ipy pzq

iβ Γνppq ppx ωpqΓνppq

LβÒ,ópp; k0,kKq Rpp; k0q2

δpkK pKq pm ωp px ipy pzq

iβ Γνppq ppx ωpqΓνppq

LβÓ,òpp; k0,kKq Rpp; k0q2

δpkK pKq pm ωp px ipy pzq

iβ Γνppq ppx ωpqΓνppq

LβÓ,ópp; k0,kKq i Rpp; k0q2

δpkK pKq pm ωp px ipy pzq

iβ Γνppq ppx ωpqΓνppq

2.6 Criticisms and discussions on the Unruh

effect

It wasn’t until about 10 years after the discovery of the Unruh effect that

the question of whether the system actually radiates or not was questioned.

Grove [26] was the first to go against the prevailing opinion and argue that

radiation does not occur. Supported later by Raine et al [46], the full-fledged

controversy began.

Those in the opposition such as Barut et al [2] claim the thermal photons have

no independent existence outside the accelerating detector. Hu et al [32] at

Maryland, very confidently joined in the debate, claiming ‘there is absolutely

no emitted radiation from a uniformly accelerated oscillator in equilibrium

conditions’. Other authors remain quite skeptical about some of the math-

ematical and physical details in the derivations of the Unruh effect. Some

of these issues are discussed in [18], [38], [3], [43]. Perhaps the mostly crit-

icized point of the Unruh effect is the universality of the detector response.

For example, it was shown by Nikishov and Ritus in [42] that elementary

particles placed in a constant electric field do not demonstrate a universal

thermal response, while other authors (see [23]) asserted for the same for-

mulation of the problem that the Unruh effect existed. Also Belinski et al.

have illustrated in [19] a case in which the response of the detectors turns

86

2.6. Criticisms and discussions on the Unruh effect

out to be non thermal. In particular in [18], [38] and [3] the authors claim

that such universality does not hold due to some serious gaps in the foun-

dations of the Unruh effect. The first objection comes from the observation

that a microscopic detector is a quantum object that is supposed to move

along a classical trajectory, this clearly is in contrast with the uncertainty

principle. A more technical observation comes from a condition that we al-

ready exploited when we studied the hermiticity of the Majorana-Rindler

hamiltonian: the field is required to satisfy the null condition at the spatial

boundary of the manifold on which quantization is performed. This spe-

cial condition is, according to these authors, enough for exclusion of the MS

vacuum from the set of possible quantum states. This conclusion substan-

tially follows from the property of translational invariance of the Minkowski

vacuum, which cannot be maintained in the space with the null boundary

condition. Translational invariance would indeed mean that the field should

be vanishing over the whole MS. The authors also criticize the interpretation

of the Bose/Fermi distributions in terms of a temperature, pointing out that

there are several other cases in which such distributions arise without the

necessity of a thermal viewpoint. In conclusion these authors claim that the

Unruh-Fulling quantization scheme lacks completeness of the set of modes in

MS, the final opinion is that the principles of QFT do not give convincing

arguments in favor of a universal thermal response. Most recently Ford et

al [20], in an effort to subdue the controversy, have produced a pedagogical

paper with solid, easy-to-understand arguments that support the notion of

no radiation.

Still, these treatments seem to be the minority as numerous others assume

that Unruh radiation exists [1], [31] [56] [58] [54] [45] [47] [35] [5] Others

also believe it may soon be detectable using lasers such as Chen [8] and

Schutzhold [48], as well as through mircrowave cavity conditions [49]. Per-

haps the most vocal believer is Unruh, as he is reported to have said “It is

real enough to roast a steak.”. [50]

87

CHAPTER 3

Dark Matter

Behind it all is surely an idea so simple, so beautiful that when we

grasp it - in a decade, a century, or a millennium- we will all say to

each other, how could it have been otherwise? How could we have been

so stupid for so long?

John Archibald Wheeler

So far we have reviewed the Unruh effect for the Majorana field. The

main result of the previous chapter can be summarized as follows: given

a field in a certain state, e.g. the vacuum w.r.t. an inertial observer, a

uniformly accelerated observer will find that the same field is in a thermal

state of temperature T a2π.

In this chapter we will evaluate the energy density perceived by the accel-

erated observer. Let us first discuss why we are interested in such quantity.

3.1 The idea: a connection with Dark Matter

The idea we will investigate was suggested by R.Soldati (supervisor of

this thesis) and relies on two well known facts:

89

Chapter 3. Dark Matter

• While examining the Coma galaxy cluster in 1933, Fritz Zwicky was

the first to use the virial theorem to infer that galaxies do not spin

correctly according to Newtonian dynamics. He was able to infer the

average mass of galaxies within the cluster, and obtained a value about

160 times greater than expected from their luminosity, and proposed

that most of the matter was dark. The same calculation today shows

a smaller factor, based on greater values for the mass of luminous ma-

terial; but it is still clear that the great majority of matter is dark.

Several hypotheses have been made in the attempt to explain this be-

havior, one of them is the existence of Dark Matter. This hypothesis

is supported also by the fact that Dark Matter correctly predicts the

results of gravitational lensing observations as well as the large scale

structure of the distribution of galaxies.

• In 1998 observations of type Ia supernovæ suggested that the universe

undergoes an accelerated cosmic expansion. These have been supported

by several experiments within the last few years, such as the observa-

tions of CMB and of the large scale structure.

Here comes our assumption: we assume that there exists a free neutral

massive spinor field which is in the vacuum state according to an inertial

observer. This field is free in the sense that it does not interact with other

fields in any of the ways prescribed by the Standard Model. This field,

however, does interact gravitationally with matter in general, within the

classical framework of General Relativity.

According to the theoretical predictions stemming from our previous

work, all the matter forming an accelerated galaxy should perceive the field

in a state that is different from the vacuum. Given our assumptions, the

only way the field can interact with ordinary matter is by virtue of Einstein

field equations, more precisely it enters as a correction to the total energy-

momentum tensor in the r.h.s. of

Gµν Λgµν 8πG

c4Tµν (3.1)

Our aim is to evaluate in what part our field can account for such energy

density, and under which assumptions. We are going to outline some well-

established results on the current models for dark matter distributions and

for its nature, we will then proceed with our estimate.

90

3.2. Dark matter models

3.2 Dark matter models

Dark Matter, together with the cosmological constant or vacuum energy

density are two key elements of modern cosmology. They enter directly into

the evolution equation for the universe as

9a

a H2

0

¸i

Ωi

a0

a

3p1ωiq k

a2(3.2)

where Ωiptq ρiptqρc are indeed the density parameters relative to bary-

onic matter, radiation, dark matter and dark energy. The density parameter

contributed today by visible, nonrelativistic baryonic matter in the universe

is ΩB 0.01 0.02, while the density parameter associated to radiation

is ΩR 2 105. Of course models of the universe with just these two

constituents are in violent disagreement with observations. The need to pos-

tulate (1) the existence of pressureless (non interacting, cold) nonbaryonic

dark matter and (2) an exotic form of matter called the vacuum-energy den-

sity both arise from the abovementioned considerations. Today’s models at-

tribute to dark matter a density of at least ΩDM 0.3 and to dark energy a

density around ΩV 0.7. Thus our universe seems to be largely dominated

by these two unknown constituents. The role of dark matter is especially

important for the formation of galaxies: today’s universe is formed by inho-

mogeneous structures which can be explained by assuming even just slight

inhomogeneities in the past. To see how this comes about in the simplest

context it is sufficient to consider the Friedmann equation in the equivalent

form

:a

2

9t20

1

a2(3.3)

and introduce a perturbation aptq Ñ aptq δaptq, in so doing one finds (for

more details see [44], chap.1) that the density perturbation grows with time.

When the perturbations have grown sufficiently, their self-gravity will start

dominating and the matter can collapse to form a gravitationally bound sys-

tem. The dark matter will form gravitationally bound structures with differ-

ent masses and radii, the barionic matter will then cool by radiating energy,

sink to the center of the dark matter halos and finally create galaxies.This is

how, schematically, dark matter halos are related to baryonic matter nowa-

days.

91

Chapter 3. Dark Matter

One of the most striking and reliable pieces of evidence for the presence

of dark matter comes from the study of the rotation curves of disk galaxies.

For a spiral galaxy it is expected, on the base of Newtonian dynamics, that

the rotation curve falls of sufficiently rapidly at large radii. Such a behavior

is not observed, we have data for over 70 spiral galaxies and in almost all

of them the rotation curve is nearly flat or slowly rising. A simple but

efficient hypothesis is that galaxies are embedded into a spherical halo of

dark matter, a constraint that such halos must satisfy is that the halo mass

increases linearly with the radius. For some lucky cases astronomers succeded

in retrieving different type of data which allowed them to attempt to model

the distribution of the dark halo. It is the case, for example, of galaxy NGC

3198. In this case the dark halo density profile is well described by

ρprq ρ01

ra

n (3.4)

with, in such specific case: ρ0 0.013h2Mdpc3, a 6.4h1kpc and n 2.1.

This model actually produces some reasonable estimates for the total matter

contained in such galaxy, however it is unfortunately not unique: different

arrangements of the matter between the halo and the disk can induce fair

variations in the parameters. Due to the nature of dark matter itself and to

the difficulties in astronomical measurements, it is actually very difficult to

achieve a reasonable confidence level for the halo structure models. As an

example, a good review of some of the most common models and comparison

with observational constraints for two galaxies in particular can be found

in [34]. One of the most commonly used model profiles for dark matter halos

is the Navarro-Frenk-White (NFW) profile (see e.g. [40], [9]), this is actually

an approximation to the equilibrium configuration of dark matter produced

in simulations of collisionless dark matter particles by numerous groups of

scientists. In the NFW profile the density of dark matter is given by

ρprq ρ0

rRs

1 r

Rs

2 (3.5)

where ρ0, Rs are parameters varying from case to case.

Dark matter also plays a major role in extragalactic dynamics, being

considered as the fundamental contribution to the large scale structure. There

is actually much work in cosmology attempting to model the large scale

92

3.2. Dark matter models

structure of the universe, most of the efforts consist in N-body simulations,

trying to fit the observed configuration of galaxies and galaxy clusters.

The hypotheses about the nature of dark matter can be divided into three

main groups: baryonic, non-baryonic and axions.

Baryonic dark matter could come in many forms: for example interstellar

clouds with mass less than 0.8Md will not be able to reach the in-

ternal temperatures that are needed for nuclear ignition. Other forms

of baryonic dark matter could be white dwarfs, neutron stars or even

black holes, all of which are remnants of stellar evolution. There is a

constraint on the amount of DM that can be due to such remnants,

by limiting the contribution of thellar evolution to the background ra-

diation. Another possibility for the baryons is to exist in the form of

primordial black holes, which could be several hundreds times more

massive than the sun and which would be formed without expelling

matter such as contemporary black holes.

Non-baryonic dark matter is mainly thought in terms of massive neutral

fermions, belonging to one of the possible generalizations to the SM.

There are generalizations of the electroweak model that include the pos-

sibility of massive neutrinos, evidence for their mass has been indeed

provided by the observation of the oscillation phenomena. However the

bounds on the neutrino masses seem to rule them out as possible DM

candidates. Another possibility are WIMPS such as heavy neutrinos,

if such particle exists then its coupling to standard particles should be

very weak or its mass should be greater than 35 GeV. By considering

the constraints derived from different experiments it is actually possi-

ble to rule out all Dirac fermion candidates with masses in the range

30 eV 1 TeV. Another possibility is the neutralino which is thought

by many to be the LSP (lightest supersymmetric particle), its mass

could range from 30 GeV to 1 TeV. This candidate is quite popular

since there exists a variety of models leading to an abundancy of such

particles, a cosmologically favorable element. The main difficulty with

this candidate is that virtually nothing is known about SUSY breaking,

which must occur in order for such particles to exist. Actually, within

the hypothesis of non-baryonic dark matter, there are some recent and

93

Chapter 3. Dark Matter

quite remarkable results indicating a relation between the mass of dark

matter particles and the shape of the density profile. Such models pre-

dict a cored profile in the case of light dark matter particles with masses

of 1 2 eV; a cusped profile at the scale of 0.03 pc would instead mean

that the dark matter should be composed of very heavy particles with

masses around mwimp 100 GeV (for more details about these models

see [15]).

Axions are another possibility of dark matter particles. Axions arise within

the theory of QCD, they were proposed by F.Wilczek in the attempt to

justify the smallness of the θ parameter of QCD. In the latest models

such particles are supposed to interact with the Higgs fields of grand

unified interactions which means that they are practically invisible to

us. Axions are believed to be very light particles and in the Big Bang

models they would have been copiously produced in the early Universe,

in fact so copiously that they have become one of the most plausible

candidates of dark matter.

3.3 Heuristic evaluation of the energy density

Our derivation will be heuristic, since we want to roughly get an order of

magnitude for such energy density without dealing with the complicated reg-

ularization procedures involved with the energy momentum tensor in curved

spacetimes.

Recalling the canonical modes quantization scheme that we built in the

last chapter, the accelerated observer will see a population of quanta of the

field given by a Fermi Dirac distribution. More precisely: if we call |jy |k0 kK ry a pure state, representing a Rindler-Majorana particle with energy

k0, transverse momentum kK and a helicity-like quantum number in state

r 1 or 2, then the number of such quanta in the Minkowskian vacuum

would be

dnj gd3k dV

p2π~q31

e2πck0

a~ 1(3.6)

94

3.3. Heuristic evaluation of the energy density

while the energy fraction given by particles in that state would be

dEj gd3k dV

p2π~q3k0

e2πck0

a~ 1(3.7)

where g 2s 1 is the spin factor. If we employ the standard relation

pk0q2 pk cq2 pmc2q2 we may cast the total energy in the suitable form

E 2V 4π c

p2π~q3» 8

0

dkk2ak2 pmcq2

e2πc2

a~

?k2pmcq2 1

(3.8)

V c

π2 ~3pmcq4

» 8

0

dtt2?t2 1

e2πc pmc2q

a~?t21 1

pmc2q4 Vπ2 p~cq3

» 8

1

dxx2?x2 1

e2πc pmc2q

a~ x 1 pmc2q4 V

π2 p~cq3 Ipµq

where

µ 2πc pmc2qa~

(3.9)

thus, employing m 1keV (see [14]) and a 107 cm s2 (see [11]) it

turns out that µ 2.86 1036 (dimensionless). Then up to a very good

approximation we can proceed according to

ρE E

V pmc2q4π2 p~cq3

» 8

1

dx x2?x2 1 eµx

pmc2q4π2 p~cq3

d

dµ

» 8

1

dx x?x2 1 eµx

by virtue of equation 3.389 4. in [25] the latter expression takes the form

Ipµq d

dµ

2?π

Γ

3

2

1

µK2pµq

where K2 is a Basset-Mc Donald function. Now, by virtue of the asymptotic

expansion in eq. 8.451 6 in [25] it turns out that

K2pµq c

π

2µeµ

1O

1

z

ñ Ipµq

?2 Γ

3

2

µ32 eµ (3.10)

95

Chapter 3. Dark Matter

this implies a mass density of the order

ρE ?

2 pmc2q4π2 p~cq3 Γ

3

2

µ32 eµ

5.43 1042 e2.861036 101.301036

g cm3 (3.11)

such a negative exponent clearly denies any observable effect.

0.001 0.1 10

10-140

10-138

10-136

10-134

10-132

Figure 3.1: The value of ρEpµq (g cm3) within the range r105, 100s

Actually in our heuristic calculation we might take the limit m Ñ 0 by

claiming that the energy spectrum for the Rindler observer doesn’t have a

mass gap, just as in the massless case. This limit yields

E

VmÑ0Ñ ~ a4

18 π6 c7 I

I » 8

0

dxx3

ex 1

» 8

0

dx x38

k1

ekx

d

dk

3 » 8

0

dx8

k1

ekx d

dk

3 8

k1

p1qkk

3!8

k1

p1qk1

k4

given the definition of Riemann’s zeta function as a series ζpsq °8n1 n

s,

96

3.3. Heuristic evaluation of the energy density

one may notice that1

ζpsq 1 1

2s 1

3s 1

4s 1

5s

1 1

2s 1

3s 1

4s 1

5s 2

1

2s 1

4s 1

6s

1 1

2s 1

3s 1

4s 1

5s 2

2s

1 1

2s 1

3s

8

n1

p1qn1

ns 21sζpsq (3.12)

ñ8

n1

p1qn1

ns p1 21sqζpsq (3.13)

that readily fits into our calculus for I, bringing

I p1 23q 3! ζp4q 7

8 π

4

15 5.682 (3.14)

ñ ρE 1.67 10132 g cm3 (3.15)

which leaves, once again, the energy density far below the observational val-

ues. Indeed it is estimated that ΩDM 0.3 which brings ρDM ΩDM ρc 1029g cm3. Note from the graphic in fig. 3.1 that, however, the massless

case is the one for which the effect is most important.

Nonetheless, on account of our formulae, we may derive the acceleration

needed to reproduce such a density, and it turns out that an acceleration of

the order of

a 5 1018 cm s2 (3.16)

is needed to account for Dark Matter entirely in terms of Majorana-Unruh

fermions, corresponding to a heat bath of temperature T 192µK.

We stress that this derivation is actually just an approximation, indeed

by using the deBroglie postulate into equation 1.108, id est taking the full

Fourier transform one would get the following dispersion relation

pk0q2 iak0 paξωq2 ia2ξkξ pa2q2 0 (3.17)

1This is just a proof of relation 9.522 2. of [25], the impatient reader could skip it by

taking comfort in the mentioned reference.

97

Chapter 3. Dark Matter

where we understand ω aβ2 k2

x as the standard Minkowskian energy.

From this equation it is clear that the deBroglie hypothesis cannot hold true

in the Rindler frame since: on the one hand if k P R3 then =pk0q 0 , on

the other hand if we postulate k0 to be real (since it is one of our dynamical

variables) we get that =pkxq 0 which means that the integral in 3.8 would

not make sense. The only reliable way of evaluating the energy density

remains the regularization of the stress energy tensor, for the purposes of this

work we will however limit ourselves to this very rough estimate, pointing

out its limited validity.

3.4 Majorana-Unruh fermions in strong grav-

itational fields

So far we have reviewed the energy and mass densities generated by cos-

mic expansion, we have seen how small is the contribution given by the Unruh

effect. Nonetheless this depends on the intensity of the acceleration consid-

ered. One may wonder what happens when greater accelerations come into

play, such as in the vicinity of a Schwarzschild black hole, like in a typical

galactic centre.

Let us use polar spatial coordinates and consider the metric

gµν diag

1 2M

r

,

1 2M

r

1

,r2,r2 sin2 θ

(3.18)

given the spherical symmetry of the problem, we may consider a point-like

observer (or particle detector) placed somewhere onto a certain spherical

shell (outside the Schwarzschild radius). We shall use the notion of surface

gravity as a substitute for the proper acceleration of the Rindler observer,

there are at least two good reasons for doing this: the first is that κ (the

surface gravity) is defined as the acceleration needed to keep the observer at

the horizon (the shell, in our case); the second reason is that this is in perfect

agreement with the Hawking effect, which states that a Black Hole radiates

a thermal spectrum with temperature T pκ2πqp~2kBq 106pMdMqKupon the formal identification aÑ κ.

98

3.4. Majorana-Unruh fermions in strong gravitational fields

A mathematically meaningful definition for κ is

kµ∇µ kν κ kν (3.19)

where kα is a Killing vector that in the Schwarzschild case reads kµ p1, 0, 0, 0q. Turning to Eddington-Finkelstein coordinates the metric under-

goes the change

v t r 2M ln |r 2M |ds2

1 2M

r

dv2 2dvdr r2

dθ2 sin2 θ dφ2

(3.20)

such that the Killing vector now reads

k1µ1 p1, 0, 0, 0q k1µ1 p1 2Mr, 0, 0, 0q (3.21)

and equation 3.19 reads for ν v

κ 1

2

BBr

1 2M

r

M

r2

GM

r2(3.22)

where the symbol is to declare that we left Planck units and turned to

c.g.s.

Hence the energy density (for a single mode of the field) predicted by our

heuristic model in the vicinity of a Black Hole of mass M depends on the

distance from the center r as

dE

dV g

d3k

p2π~q3k0

exp

2πc~GM r2 k0

1(3.23)

in order to obtain the total amount of energy carried by such field we must

integrate over momentum space and over configuration space outside the

Schwarzschild radius RS 2GMc2. We will need to use a cutoff for mo-

99

Chapter 3. Dark Matter

mentum space, let us employ the Planck scale Λ EPlanckc

EΛtot g

p4πq2cp2π~q3

» 8

RS

dr r2

» Λ

0

dk k2

ak2 pmcq2

exp

2πc2

~GM r2ak2 pmcq2

1

g2cpmcq4π~3

» 8

RS

dr r2

» Λmc

0

dx x2

?x2 1

exp

2πmc3

~GM r2?x2 1

1

g2cpmcq4π~3

» 8

RS

dr r2

» bp Λmcq21

1

dyy2

ay2 1

exp

2πmc3

~GM r2 y 1

g2cpmcq4π~3

~GM2πmc3

32» 8

ZS

dz z2

» Λmc

1

dyy2

ay2 1

exp pz2 yq 1

g2cpmcq4π~3

~GM2πmc3

32

I pZS,Λ,mq (3.24)

where we used the approximation that ZSΛ " mc and that the last integrand

is exponentially suppressed for large values of y. Taking a black hole mass of

the order of M 10Md, we understand

ZS RS

2πmc3

~GM

12

8π GM m

~ c

12

108 (3.25)

and is dimensionless, in such a way that I ultimately depends on Λ,m,M .

Again we are looking for an order of magnitude and if we note that

Λmc 1025 we can proceed quantitavely in evaluating I:

I pZs,Λ,mq » 8

108

dz z2

» 1025

1

dy y2ay2 1 ey z

2

» 8

108

dz z2

d

dpz2q» 8

1

dy yay2 1 ey z

2

2?π

Γ

3

2

» 8

108

dz z2

d

dpz2qez

2

pz2q32

?

2 Γ

3

2

Eipz2q

z8z108

(3.26)

where use has been made of formulæ 3.389 4, 8.451 6 and 8.451 6 of [25].

If we employ eq. 8.215 of the abovementioned reference we eventually come

to the value of this dimensionless quantity

I pZs,Λ,mq cπ

810161016log10peq 101016

(3.27)

100

3.4. Majorana-Unruh fermions in strong gravitational fields

now, if we insert this rough value into equation (3.24) it becomes evident

how the amount of energy produced by this mechanism is far below any

observable quantity

EΛtot 8.40 1010 101016 101016

g (3.28)

We finally came to the conclusion that, even in extremely strong gravita-

tional fields, the energy produced by means of the Unruh mechanism wouldn’t

be sufficient to account for Dark Matter. Moreover, its amount is too small

even to be observed.

101

CHAPTER 4

Conclusions

The significant problems we face cannot be solved at the same level

of thinking we were at when we created them

Albert Einstein

Our investigation of the Unruh effect and of its possible role within the

Dark Matter problem has been conducted from the grounds and has gone

through three main steps. These are: the explicit study of the classical

and quantum field theories of neutral fields with spin 0 and 1/2 in Rindler

spacetime, a study of the Unruh effect for both theories, an estimate of the

energy density that is due to this mechanism within the frameworks of cosmic

expansion and of the Schwarzschild gravitational field.

In the first part we studied the field theories of accelerated neutral scalar

and Majorana fields in four dimensions. While the theory of an accelerated

scalar field has been widely studied since the celebrated articles by Unruh,

Fulling et al., we found few works investigating on fermionic fields. One of

the purposes of this work has been to fill this gap. We decided to follow the

approach of canonical quantization: first solving the classical field equations

and then applying the Dirac principle. Among the advantages of this scheme,

we felt that having an insight on the normal modes of the two fields has been

103

Chapter 4. Conclusions

important for our understanding of the Unruh effect. We focused mainly on

the Majorana field and we proceeded to study the analytical and algebraic

structure of its normal modes; one peculiar feature that we found is that

the only modes that can have a definite helicity are those which momentum

and spin lie along the direction af acceleration. We developed two different

quantization schemes, based on two different sets of normal modes. A first

set of modes, which we called helicity-eingenstate modes, have the advantage

of providing an intuitive physical interpretation, since they reduce to helicity

eingestates for vanishing transverse momentum. On the other hand these are

not at all convenient when one wants to perform canonical quantization, the

reason is that one must introduce as many operators as for an ordinary Dirac

field. In turn these would depend on each other due to the self conjuation

property of a Majorana field that follows directly form the field equations.

The second set of modes, which we called canonical modes, enjoys the so-

called reality condition and expanding the field over this set results in a

manifestly self conjugated expression, this allowed for a cleaner quantization

procedure, exploiting the nature of the Majorana field as a Dirac field with

half of its degrees of freedom. We proved that the two sets are mathematically

equivalent, one being useful to perform quantization while the other provides

a meaningful physical interpretation.

The second part of this work is devoted to the study of the Unruh effect

for both the scalar and the Majorana fields. We followed the approach of the

Bogolyubov transformation. For the scalar field, we decided to derive the

Unruh effect by evaluating the amplitude between Rindler and Minkowskian

modes taking advantage of Takagi’s trick, rather than involving Unruh modes.

We did so in order to overcome some of the possible issues raised by certain

authors about the use of Unruh modes, nonetheless we finally obtained the

well known Planckian distribution. More interesting is the case of the Ma-

jorana field: we first derived the spinor transformation law under a change

from Minkowski to Rindler coordinates, which resulted in a time-varying

Lorentz boost, this important feature tells us that the Rindler modes are in-

deed eigenstates of the boost generator since they were derived as stationary

waves, actually in the literature they are sometimes referred to as boost modes

for this reason. Thereafter the construction of the Bogolyubov coefficients re-

quired us to search for a suitable expression for the Unruh modes. We found

104

such expressions both for the canonical modes and for the helicity-eigenstate

modes. The Unruh modes reduce exactly to Rindler modes in the corre-

sponding wedges and vanish elsewhere in RS, they are analytical over the

whole MS and they are written manifestly as a combination of positive and

negative-frequency Minkowskian modes. We then obtained the Bogolyubov

coefficients together with the distribution of Majorana fermions produced

by means of the Unruh mechanism, which resulted being a Fermi-Dirac dis-

tribution. Such result, however, did not tell us much about the details of

Unruh creation since it has been obtained via physically meaningless, al-

though mathematically-comfortable, modes. This led us, for completeness,

to make a new evaluation of the Bogolyubov coefficients as amplitudes be-

tween physically meaningful, i.e. observable-defined, modes.

In the third part of this work we abandoned the purely theoretical ap-

proach and searched for a connection between our results and the possibility

of a Dark Matter candidate. We made the assumption that there exists a

free neutral massive spinor field which is in the vacuum state according to an

inertial observer, such field is free in the sense that it does not interact with

other fields in any of the ways prescribed by the Standard Model, nonethe-

less it does interact gravitationally within the classical framework of General

Relativity. According to the theoretical predictions stemming from our pre-

vious work, all the matter forming an accelerated galaxy should perceive the

field in a state that is not the vacuum, i.e. it should perceive a non-vanishing

energy density. This lead us to explore the effects of cosmic expansion on

the fields by means of the Unruh mechanism. We chose not to dwelve into

the complicated regularization schemes of the energy-momentum tensor that

always occur in curved spacetimes, we preferred to proceed heuristically so

to obtain an order of magnitude for such energy density. It came out that the

energy density produced by this effect should be mostly important for small

values of the mass of the field, anyway its value is many orders of magnitude

below the current observational values of Dark Matter density. The hypo-

thetical value of a cosmic acceleration needed to account for the whole DM

density is, in turn, extremely high. Finally we took into consideration the

strongest non-inertial effects that occur in our universe by evaluating the Un-

ruh Majorana energy density produced by a black hole, once again we came

to the conclusion that the Unruh mechanism is not suitable for accounting

105

Chapter 4. Conclusions

for significant portions of Dark Matter.

106

Appendix

A.1 Orthonormality and completeness of Rindler

modes

a. A study of the scalar Rindler modes

Let’s put ourselves into the Right rindler wedge, we want to normalize the

Fulling modes defined in this region of spacetime. The following procedure

is essentially the one outlined in [52].

Let us first pass to the coordinate

ξ log rξλs (1)

where λ P R and rλs eV1; the line element is then

gµν diagpλ2e2ξ ,λ2e2ξ ,1,1q (2)

equation 1.21 then takes a Schrodinger-like form" d

dξ2 λ2β2e2ξ

*φk0,kK

k0

a

2

φk0,kK (3)

As |ξ| varies from 0 to 8, ξ varies over R; the effective potential of this

equation approaches zero for ξ Ñ 8 and rises sharply for ξ Ñ 8,

such that the eigenvalues k0a range continuously from 8 to 8, without

107

Appendix

the typical energy gap of massive Minkowski field theory. The solutions of

equation (3) can be so normalized as to satisfy» 8

8dξ φk0,kK

φk01,kK δpk0 k0

1q (4)

hence, it is possible to choose the coefficient C of the fulling modes (1.25) in

such a way that condition (1.28) holds. Next, note that, since the effective

potential vanishes at ξ Ñ 8, the form of all the φko,kK in that region is

φk0,kK p2πq12Ak0,kKpeiξpk0aq Rk0,kKeiξpk0aqq h (5)

where Rk0,kK is the phase factor representing the phase shift of the reflected

wave. Note that the reflection coefficient must be unity, as the potential

increases unboundedly. The normalization condition on these modes requires

that also Ak0,kK is a phase factor, which we will for now leave unspecified.

Now let us employ null coordinates

u aη ξ v aη ξ (6)

ds2 λ2epvuq du dv dxK2, gµν

0 1

2λ2epvuq 0 0

12λ2epvuq 0 0 0

0 0 1 0

0 0 0 1

(7)

Since the effective potential rises unboundedly at ξ Ñ 8, then the wave-

function φk0,kK must vanish at ξ 8, hence we can invoke Gauss’ theorem

and say that the spacelike hypersurface Σ used to calculate the Klein-Gordon

inner product pφk0,kK , φk01,kK

1q can be chosen at will (see [30]). In particular

one can choose to push Σ upwards until in nearly coincides with the null

hypersurface

H tpt, x, y, zq; t x ¥ 0u (8)

that is depicted in fig 1, actually we will only work in that limit, without

ever getting on H, since it must be a spacelike one. Notice that in this

hypersurface η 8 while ξ 0, hence u 8 and v varies from 8 to

8.

108

A.1. Orthonormality and completeness of Rindler modes

x

c t

ξ=const.

H+

u=+∞

η1

η2

Figure 1: The bold black line is the event horizon H, on which u 8. In order to

perform our calculation we need to push the Cauchy hypersurface upwards to this limit.

Notice that η increases from η2 towards η1

The KG inner product between two Fulling modes becomes thenuk0,kK , uk0

1,kK1

i

»dΣv lim

u0Ñ8ruk0,kKsÐÑBv

uk0

1,kK1

uu0

with dΣv ?g

3!εvµνλ dxµdxνdxλ

?gεvuyz dxudxydxz

?gεvvyz dvdydz

?ggvuεuvyz dvdydz

dvdydz

ñ i

» 8

8dv

»dxdy lim

uÑ8uk0,kK

ÐÑBv uk01,kK

1

i CC 1p2πq1δpkK kK1q

» 8

8dv

"lim

uÑ8Kik0apβλepvuq2q eik0puvqp2aqÐÑBv

Kik01apβ1λepvuq2qeik0

1puvqp2aq)

(9)

109

Appendix

both hpk0,kKq (as given by (5)) and Kk0,kK satisfy (3), therefore the first

must be the asymptotic expansion of the second for suitable values of the

coefficients R,A. Indeed we know from [25] (see eq.s 8.407 1. and 8.405 1.)

that

Kνpzq iπ

2eiν

π2 Hp1q

ν pizq (10)

Hp1qν pzq Jνpzq iNνpzq (11)

then, by employing 8.403 1. and 8.406 1. of the abovementioned reference

we can infer

Kνpzq iπ

2 1

sinπνpIνpzq Iνpzqq (12)

where the symmetry under ν Ñ ν is manifest. Employing the exact series

expansion of eq. 8.445 for Iνpzq we obtain the behavior for Kνpzq when z Ó 0

Kνpzq zÓ0 iπ

2 sinπν

1

Γpν 1qz

2

ν 1

Γpν 1qz

2

ν(13)

consequently

Kik0apβξq Kik0apβλeξq

iπ r2 sinhpπk0aqΓp1 ik0aqs1

βλeξ

2

ik0a c.c.

αeipkoaqξ αeipkoaqξ (14)

with α iπ r2 sinhpπk0aqΓp1 ik0aqs1

βλ

2

ik0a(15)

Ak0,kK

Kik0apβξq α

eipk0aqξ α

αeipk0aqξ

(16)

by comparing the last line with (5) we obtain the reflection coefficient, that

is Rk0,kK pααq, as we already expected it is a phase constant. Moreover

we get thatuk0,kK , uk0

1,kK1

CC 1 αα1 δpkK kK1q

Ik0,k01 (17)

110

A.1. Orthonormality and completeness of Rindler modes

where I reads

I p2πq1

» 8

8dv lim

uÑ8

RR1pk0 k0

1qa1 eipk0k01qa1v

Rpk0aq eia1pk0vk0

1uq R1pk01aq eia

1pk0uk01vq

(18)

formally speaking, this inner product is not a function of k0, k01, more pre-

cisely it is to be intended as a distribution, hence we should not be surprised

that the Fulling modes are actually out of Hilbert space, just as for the

case of plane waves in MS. However, considering this as a distribution to be

smeared over a wave packet, we may drop the last two terms, by virtue of

the Riemann-Lebesgue lemma, it follows immediately that

I p2k0qδpk0 k01q

Finally, in order to satisfy (1.28), and by virtue of eq. 8.332 1. of [25]

we must take

C d

sinhπk0

a

π a

(19)

where we have chosen C to be real-valued.

b. Proof: orthonormality of Majorana Rindler modes

in MS

In this subsection we demonstrate (2.85) and (2.84): given the standard

MS scalar product for spinors

pψ1pXq, ψ2pXqqMS »R3

dX ψ1pXq γ0 ψ2pXqX00

(20)

the norm we must evaluate can be computed as followspMqψ

ÒÓ,µpX|aq , pMqψ

òó,µ1pX|aq

MS

AaA

1a

4p2πq2 δpkK kK

1q δÒÓ,òó

»R

dx1

!β2 e

π2pµµ1q

»R2

dθ dθ1 eix1pk1k11qiµθiµ1θ1 θ2 θ1

2

β2 eπ2pµµ1q

»R2

dθ dθ1 eix1pk1k11qiµθiµ1θ1 θ2 θ1

2

)111

Appendix

by recalling that in this integral representation we set

k1 β sinh θ

one can proceed as follows

AaA

1a 2π3β2 e

π2pµµ1q δpkK kK

1q δÒÓ,òó

»R2

dθ dθ1δpθ θ1qβ cosh θ1

eiµθiµ

1θ1 θ2 θ1

2 eiµθiµ1θ1 θ

2 θ1

2

A

aA1a 2π3β e

π2pµµ1q δpkK kK

1q δÒÓ,òó

»R

dθ1

cosh θ

eiθpµµ

1qθ eiθpµµ1qθ

|Aa|2 8π4β eπµ δpµ µ1q δpkK kK

1q δÒÓ,òó

In the same manner one easily obtains thatpMqψ

ÒÓ,µpX|`q , pMqψ

òó,µ1pX|`q

MS

|A`|2 8π4β eπµ δpµ µ1q δpkK kK

1q δÒÓ,òó

Orthogonality is also easy to check:pMqψ

ÒÓ,µpX|aq , pMqψ

òó,µ1pX|`q

MS

AaA

1`

4p2πq2 δpkK kK

1q δÒÓ,òó

»R

dx1

!β2 e

iπ2piµiµ11q

»R2

dθ dθ1 eix1pk1k11qiµθiµ1θ1 θ2 θ1

2

β2 eiπ2piµiµ11q

»R2

dθ dθ1 eix1pk1k11qiµθiµ1θ1 θ2 θ1

2

) A

aA1`2β2π3 δpkK kK

1q δÒÓ,òó

»R2

dθ dθ1eiπ2piµiµ11q eiµθiµ

1θ1 θ2 θ1

2 eiπ2piµiµ11q eiµθiµ

1θ1 θ2 θ1

2

δpθ θ1qβ cosh θ1

AaA

1`2βπ3 δpkK kK

1q δÒÓ,òó eiπ2piµiµ1q

»R

dθeipµµ

1qθ

cosh θpi iq 0

which completes the proof.

112

A.1. Orthonormality and completeness of Rindler modes

c. Proof: completeness of Majorana Rindler modes in

MS

In this subsection we give the demonstration of (2.119), those for (2.120)

and for helicity-eigenstate modes being similar.

¸a,µ,kK

pMqUa,µ,kKpX|`q b pMqU :a,µ,kKpX 1|`q

X0X01

»

dµ dkKeπµ

64π4βeikKpXKXK

1q»

dθ dθ1 eipωω1qX0ikXik1X 1

¸a

β2 Θ

a bΘ:a eπµ eiµpθθ

1q θθ1

2 Θa b Θ:

a eπµ eiµpθθ1q θθ1

2

β Θa b Θ:

a eπµiπ2 eiµpθθ

1q θθ1

2 Θa bΘ:

a eπµiπ2 eiµpθθ

1q θθ1

2

»

dkK dθ1

8πβ

eikKpXKXK1q

p2πq2 ei pβ sinh θq pXX 1q

¸a

β2 Θ

a bΘ:a eθ Θ

a b Θ:a eθ iβ

Θa b Θ:

a Θa bΘ:

a

similarly, it can be checked that

¸a,µ,kK

pMqUa,µ,kKpX|aq b pMqU :a,µ,kKpX 1|aq

X0X01

»

dkK dθ1

8πβ

eikKpXKXK1q

p2πq2 ei pβ sinh θq pXX 1q

¸a

β2 Θ

a bΘ:a eθ Θ

a b Θ:a eθ iβ

Θa b Θ:

a Θa bΘ:

a

which yields

¸a,µ,kK

pMqUa,µ,kKpX|aq b pMqU :a,µ,kKpX 1|aq

pMqUa,µ,kKpX|`q b pMqU :a,µ,kKpX 1|`qX0X01

113

Appendix

»

dkK dθ1

4πβ

eikKpXKXK1q

p2πq2 ei pβ sinh θq pXX 1q cosh θ

¸a

β2 Θ

a bΘ:a Θ

a b Θ:a

»dkK

eikKpXKXK1q

p2πq21

2π

»dpβ sinh θq ei pβ sinh θq pXX 1q I4

δpXX1q (21)

which completes the proof.

d. Proof: completeness of Unruh modes in MS

Here we show that completeness of the ψÒÓ,µ,kKpX|aq , ψÒÓ,µ,kKpX|`q ac-

tually entails completeness of the RÒÓ,µ,kKpXq , LÒÓ,µ,kKpXq modes. The same

procedure can be applied to the canonical modes counterparts.

We shall first recall a general result of elementary algebra: given a com-

plete and orthonormal basis txiui of RN and a linear transformation T , if the

transformation is orthogonal then the set of vectors tyiui

yi N

j1

Tij xj (22)

is complete. The proof is simple, let pxiqa be the a-th component of the

vector xi, then:¸i

yi b yi

ab

¸i

pyiqa pyiqb

¸i

¸j

Tij pxjqa ¸

j

Tik pxkqb

¸j,k

pT T T qkj pxjqa pxkqb ¸j,k

δi,j pxjqa pxkqb

¸j

pxjqa pxjqb ¸

j

xj b xj

ab

δa,b (23)

Let us now turn to the case of Unruh modes: we shall compactify our notation

on the indices and adopt κ pµ,kK, ÒÓq, and we’ll rename

ψÒÓ,µ,kKpX|aq Ñ ψaκ ψÒÓ,µ,kKpX|`q Ñ ψ`κRÒÓ,µ,kKpXq Ñ Rκ LÒÓ,µ,kKpXq Ñ Lκ

114

A.2. Alternative derivation of the Unruh effect

let us cluster symbolically these Hilbert-space vectors as follows

Ψ

...

ψaκ...

ψ`κ...

X

...

Rκ

...

Lκ...

X M Ψ (24)

where M reads

M

eπµ2?2 coshπµ

eπµ2?2 coshπµ

. . . . . .eπµ2?

2 coshπµeπµ2?2 coshπµ

eπµ2?2 coshπµ

eπµ2?2 coshπµ

. . . . . .eπµ2?2 coshπµ

eπµ2?2 coshπµ

(25)

clearly, this notation is improper since κ is a continuous index, however this

isn’t a problem since M only mixes discrete indexes leaving a Dirac-delta

over the continuous ones, indeed one may cast this matrix into the form

M 1?2 coshπµ

eπµ2 b I eπµ2 b I

eπµ2 b I eπµ2 b I

1?2 coshπµ

eπµ2 eπµ2

eπµ2 eπµ2

b I (26)

whereas I accounts for δÒÓ,òó δpµ µ1q δpkK kK1q. It is straightforward to

check that M is orthogonal, which means that tRκ , Lκuk is a complete set.

A.2 Alternative derivation of the Unruh ef-

fect

Here we present a different way to obtain the β Bogolyubov coefficient,

this is the original procedure proposed by Unruh.

Let us recall the canonical normal modes in Minkowski’s spacetime

ukpt,xq p2πq32p2ωkq12 eiωktikx (27)

115

Appendix

we may find a new, useful set of equivalent modes by performing a Bo-

golyubov transformation

ψpxq ¸µ,kK

ψµ,kKpxqbµ,kK ψµ,kKpxqb:µ,kK

(28)

ψpxq Bpφpxqq (29)

where B is a Bogolyubov transformation which coefficients read

αµ,kK;k1 p2πωq12ω k

ω k

iµ2

δpkK kK1q βµ,kK;k1 0 (30)

in which k1 pk,kK1q.By recalling that k2 β2 ω2 with β2 kK

2 m2 (not a Bogolyubov

coefficient!), and by using the substitution k β sinh θ, which implies ω β cosh θ, then the new normal modes can be easily obtained as

ψµ,kKpxq p2q52π2

» 8

8dθ eiβpt cosh θx sinh θqiµθikKxK (31)

these modes obey

pψµ,kKpt,xq, ψν,pKpt,xqq δpµ νqδpkK pKq 8 µ, ν 8pψµ,kKpt,xq, ψµ,kKpt,x1qq δpx x1q

It can be checked by direct inspection that within the right-Rindler wedge

the following identity holds

uk0,kK eπµ2ψµ,kK eπµ2ψµ,kK?

2 sinhπµµ k0

a(32)

in such a way that we can define a new Bogolyubov transformation between

the tψµ.kKu and the Fulling modes up0,pK which coefficients read

αµ,kK|p0,pK eπp0

2aa2 sinhπp0a δpµ p0aqδpkK pKq (33)

βµ,kK|p0,pK eπp0

2aa2 sinhπp0a δpµ p0aqδpkK pKq (34)

Finally the spectrum of detected particles turns out to be¸p0pK

|βµ,kK|p0,pK |2 1

e2πp0a 1(35)

116

A.2. Alternative derivation of the Unruh effect

in accordance with (2.41).

Nearly all the abovementioned passages are straightforward, except for

eq (32). Then let us give a brief demonstration of this key identity.

Let us use the following representation (see [25]) for Kiµpzq

Kiµpzq 1

cosh πµ2

» 8

0

dr cospz sinh rq cospµrq (36)

Then

ψµ 252π2

» 8

8dθ eiβpt cosh θx sinh θqiµθikKxK (37)

252π2

» 8

8dθ eiβξ sinh paηθqiµθikKxK

252π2eiµaη» 8

8dθ eiβξ sinh θiµθikKxK

232π2eiµaηikKxK» 8

0

dθ rcos pβξ sinh θq cospµθq sin pβξ sinh θq sinpµθqs

however the last integral is vanishing, to see this it is sufficient to split it into

two integrals and use the tool of analytical continuation by performing two

opposite rotations of π2:» 8

0

sinpz sinh θqeiµθ » 8

0

sinpz sinh θqeiµθ

» 0

8dy sinpz sin yqeµy

» 0

8dy sinpz sin yqeµy

0

hence we may write

ψµ,kKpxq ψµ,kKpxq

232π2eiµaηikKxK» 8

0

dθ cos pβξ sinh θq cospµθq

232π2eiµaηikKxK coshπµ

2

Kiµpβξq

(38)

which brings

eπµ2ψµ,kK eπµ2ψµ,kK

212 π2 sinh pπµq eiaµηikKxK Kiµpβξq

117

Table of constants

which completes the proof.

Table of constants

Here is a list of the constants involved in this work, they are expressed in

c.g.s. units

Symbol Value Units Description

c 3 1010 cm s1 speed of light in vacuum

kB 1.38 1016 erg K1 Boltzmann’s constant

mc2 1.6 109 erg rest energy of Majorana DM can-

didate (see [14])

a 1 107 cm s2 cosmic acceleration (see [11])

~ 1 1027 erg s Planck reduced constant

H 2.29 1018 s1 Hubble constant (current value)

G 6.67 108 dyn cm2 g2 Newton constant

ρc 1.88 1029 g cm3 cosmic critical density (current

value)

118

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