gravita' quantistica

8/13/2019 gravita' quantistica

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T H E P R O C E E D I N G S O FT H E P H Y S I C A L S O C I E T Y

Section AVOL. 65, PART 1March 1952 No. 387 A

Quantization of Einsteins Gravitational Field:Linear Approximation

BY SURAJ N. G U P T ACavendish Laboratory Cambridge*

Communicated by L Rosenfeld;M S . recezved 3rd Octobev 1951*&LSTRACT. The approximate linear form of Einsteins gravitational field is quantizedby using an indefinite metric. It is shown that only two types of gravitons can be observed

though many more can exist in virtual states n the presence of interaction. Theobservable gravitons are shown to be particles of spin 2. Using the interactionrepresentation the interaction of the gravitational field with the matter field is brieflydiscussed.

1 . INTRODUCTIONH E present investigation is undertaken to carry out the quantization ofEinsteins gravitational field, and to investigate the interaction ofT ravitational quanta or grav itons with othe r elementary particles. Fo rsimplicity we shall first quantize the approximate linear form of Einsteins

field, while th e exact t rea tment of th e non-linear gravitational field will be givenin a subsequen t paper. Som e work on the quantization of the gravitationalfield has been carried out in earlier days by Rosenfeld (1930), but since thenvery little progress has been made in this direction.For the quantization of the approximate linear gravitational field we shallfollow the same treatment as has been applied to the radiation field in anearlier paper (Gupta 1950, to be referred to as A). T h us we shall use anindefinite metric for the components of the gravitational field with a negativecommutator, and then the gravitons corresponding to these components willbe made unobservable by means of supplementary conditions. I n this waywe shall find that only two types of gravitons can be observed, though manymore can exist in virtual states. Since the gravitational field has a large numberof components, the present treatment is necessarily more involved than thetreatment for the radiation field, bu t we shall not come across any new difficulty.It must be observed that the supplementary conditions play a vital role inour treatment. Fo r, if there were no supplementary conditions, the presenttheory would involve observable states with negative probabilities, which wouldbe physically meaningless, It appears, therefore, that the role of thesupplementary conditions (or the coordinate conditions) is more fundamentalthan was originally intended by Einstein.

* Now I .C.I . Research Fellow at the Unlverslw of ManchesterPROC PHYS. OC. LXV, 3-A I1


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I62 Suraj N . Gupta5 2. L I N E A R A P P R O X I M A T I O N F O R THE G R A V I T A T I O N A L F I E L D

T h e fundam ental equa tion for Einstein's gravitational field is given by

where the symbols gp , R pv ,and R have the usual meaning, K is a constant,a n d TPv is the energy mo men tum tensor of the ' m at te r ' field, where matterinc lud es eve ryth ing except the gravitational field. Since th e covariantdivergence of the left-hand side of (1) is known to vanish, it follows that thecovariant divergence of TW"will also vanish, whence one obtains

RFv- p g p v = ~ K ~ T P , ..... 1 )

. , . . .(2)where amp=ab( -g)lI2 is the energy momentum tensor density for the matterfield. Also. in ord er to exp ress th e conservation of energy and m omentum, we

......( 3write aaxy a;+ 1 = 0 ;here'f ' is the energy mom entum pseudo-tensor d ensity for the gravitationalfield, which satisfies the equation

... 4)Following Einstein (1918), we now obtain the linear approximation for theabove field equations by putting

where E,, are the constant Minkowskian values for g,,,, and the second andhigher powers of K are to be neglected. Us ing th e usual flat space-time tensornotation*, we can then w rite eqns. (1)and 4) s

g,v = + K S , 9 . ......( )

. . . (7)Ac cordin g to E instein , we can also choose coordinate conditions or supplementaryconditions given bv

so that ( 6 ) reduces toI t is convenient to pu tan d thu. we can write 9), (8) and (7) as

0 2 h p v - a p v0 2 h ; , i .= ~ T p , , .h p y y p v - ~ , ,VY i . i ,

. . . . 9)......10)O2YpV=KT,, # ..... (1 l )

..... 12 )

* We take the space-tlme coordmates as xll x2 , x l , zt.


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163uantization oj the Gravitational FieldIt follows from the above equations that

In particular, we obta in fo r the Hamiltonian density

Th e eqns. ( l l ) , (12) and 13), describing a linear gravitational field, are sosimple that it may seem surprising that they have to be regarded only as anapproximation to a more involved theory. I t should, therefore, be noted thatthe supplementary conditions (12) are compatible with the field eqn. (11) onlyin an approximate sense, for (11) and (12) give respectively

which agree only in the first approximation. On the oth er hand, we shall showin the sub seq uen t paper th at f or the exact non-linear gravitational field thesupplementary conditions are exactly compatible with the field equations.3 Q U A N T I Z A T I O N O F T H E G R A V I T A T I O N A L F I E L DLagrangian density for the linear gravitational field may be taken as

where y,,,, and y are to be treated as independent variables. Th,is gives in theusual way for the field equations and th e Hamiltonian density0 2 Y p y= 0, OY = 0, . . . . . . (17)

In order that (18) may agree with (15), we shall ensure by means of asupplementary con dition, to be discussed in the next section, that the expectationvalue of y is equal to that of ypp. But for the purpose of quantization we regard yas an independent field quantity, so tha t we may easily sp lit th e contributionsof the various com ponents in the Hamiltonian (18).Now we have to obtain the comm utation relations for y p v and y , keepingin view the fact that Y ~ , , = Y , , ~ ~ .or this we note that the canonical conjugateaL 1 8711 . . . . . (19)of yI1 is given byTherefore the commutation relation [yl1(r, t , Tl1(r, t ] ia(r-r) gives[yll(r, ) , ( a y l l / a t )(r, t ] 2 i 8 ( r - ) ; or, using Schwingers notation (Schwinger1948), we haveOn the other hand , since y12= y21,we get

71 - -= - -11 a(ayll /at) 2 at *

[Yll(X), l l(X)I = 2 i W - L . . (20).... (21)

SO that the commutation relation in this case is given by[YlZ(X), YlZ(X)l=w ). (22)

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164 Suraj N . GuptaAccording to (20) and (22), the commutation relations for yI Iymay be written a8Fu rth er, t he canonical conjugate of y is given by

[ Y J ~ ) , A , ~ ) I = i ( S v , + S p , S v J D ( x - x ) - ...... 23)

an d hence we obtain[y(x),y(x)] = - 4 i D ( x - x ) . . . . . . . 24)

Following A, we now regard yPyand y as self-adjoint quantities, and expandthem asy p y (2k)-1/2{aP,( ) * r - k t ) +utV (k) - k t ) } , ...... 25). . . . . . 26)k=x2(2k)-1/2{a(k) ez ( k . r - k t ) + at(k) e-l(k*r--%t)},k

where a dagger denotes an adjoint. Substituting th e above Fou rier expansionsin (23) and (24) respectively, we get[apv(k), aX,(k)I = S v e + Sp$v I . . . . . . . 27)

[a(k ), ~ ( k ) ] - 1. ...... 28)Again, substituting (25) and (26) in (18), using the commutation relations (27)a n d 28), and om itting the zero-point energy, we o btain for the Hamiltonian ofthe gravitational field

J Hd v = ~ k { j u ~ v ( k ) u p y ( k )at(k)a(k)). ...... 29)azz ), a33 ) and aoo( ) by ail(k), aL2(k), a&( k ) and aAo(k),which are given byFor the present purpose it is convenient to replace the operators a,,(k),

Xciording to (27) and (30), the commutation relations fo r the new operators are. . . . . . 31)aldk), (k) I = 1,[a;&), k)l= 1, [a& (k), aL6(k)I= 1,bJO(k ), .66(k)1= 1,

and the Hamiltonian (29) may be written asHdv=~k{a1Z(k)a12(k)f aidk)a23(k) + a~l(k)a31(k)

lo(k)a,o(k) aSo(k)a,o(k) - o(k)a,,(k)+ all(k)al,(k) +@ ; m 4 2 ( k )+ ai;(k)a;,(k)+aAJ(k)aCxk) t(k)a(k)>, ...... 32)

where we have used the fact that aJk) = avF(k). Thus we have, in all, eleventypes of gravitons corresponding to the eleven independent components ofy p yand y . We shall refer to the gravitons corresponding to the operator a, )as the a,,-gravitons, and so on.

According to (27) and (28), the com mutators involving a,,(k) and a(k) havea negative sign. Therefo re, as explained in A , we have to use an indefinitemetric for the components ytoand y of th e gravita tiona l field, while the remainingcomponents are t o be treated in the usual way. T h us, all the operators @12(k),


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165uantization of the Gravitational Fieldo(k), o(k), a30(k), &(k), ai&), aho(k) and a (k ) appearas absorption operators, while their adjoints are all emission operators. It isto be noted that though now all the gravitons have positive energies, some states

will occur with negative probabilities. Fo r, due to the use of an indefinitemetric, we have to normalize any state Y s

where n n2,, n 3 0 and n are the numbers of the a - azo- uao- and a-gravitonsrespectively in the state Y. This occurrence of negative probabilities, however,does not raise any difficulty of physical interpretation, for we shall show belowthat for all real states we have n = n 2 0 = n 3 0 = n = 0 .

* . . I .(33)t y q - ) n , o + A s o + % 8 0 t %

9 4 . THE S U P P L E M E N T A R Y C O N D I T I O N SIn the absence of interaction the supplementary conditions for thegravitational field may be taken as

(34) and (yL:)-y(+))Y =0, . . . . . (35)where yht and y(+)are the positive frequency parts of y a p and y respectively.It follows from (34), (35) and their adjoint equations that ( a yp y / 8 xp ) 0 ,(yMp) ( y ) . T h u s the supplementary conditions (34) agree with (12) in theclassical limit, while (35) ensures that the Hamiltonian (18) is equivalent to (15).Substituting the Fourier expansions (25) in (34), and choosing the x3 axisalong k, we obtain {~3i(k)-aoi(k)}Y=O, . . . . . - ( 6 ~ )

{~3z(k)-~o,(k)}Y=O, . . . . . . 36b){ a33 ( k ) - - a , 3 ( k ) }Y= {1 / 2a~3 ( k ) - a , , ( k ) } F=0 , ..... (36c){ ~ ~ o ( k ) - ~ ~ ~ ( k ) } = { ~ ~ ~ ( k ) - ~ 2 ~ ~ ~ ( k ) } YO ......(36d)

As shown in A, a solution of (36 a) consists of a normalized state containing npa3,- and a,,-gravitons, and a superposition of a series of redundant states witharbitrary coefficients and zero normalization. These red undant states, however,do not contribute to any observable effect, Moreover, since the normalizationis conserved in course of time, a redundant state will always remain redundant.Thus for all practical purposes we may ignore such states, and regard theaal- and a,,-gravitons as entirely absent in a non-interacting gravitational field.Similarly, (36b) implies the absence of u3,- and a,,-gravitons in the absence ofinteraction, T h e situation regarding eqns. (36 c) and (36 d ) is slightlysimpler, for bo th of these equations involve the operator a,,(k). It can be easilyseen that th e only states satisfying (36c) and (36 d) are those which do not containany ~ 0 3 - a&- and a&gravitons. Substituting theFourier expansions (25) and (26) in ( 3 9 , we getUsing (30), (36c) and (36d), we obtain from (37)

Again, as explained above in the case of eqn. (36a), (38) implies the absenceof U& and a-gravitons in a pure gravitational field.

We now consider the supplementary condition (35).{a,,(k) - 2a(k)}Y =O

{&(k) - a(k)}Y? O s...... 37)...... 38)


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I66 Suraj N . GuptaWe have seen that the supplementary conditions (34) and (35) enable us toelim inate nine types of gravitons, so that for a pure gravitational field we may take

n2, = n31= nol= no2= no3 = n22= nys=no,= n = 0,f f . . . 39)where nZ3 is the nu mbe r of a,,-gravitons, an d so on. Therefo re, in a puregravitational field, only aI2- and a;,-gravitons can exist. I t follows that thestate of vacuum for the gravitational field may be defined as that containing nouI2- and &-gravitons, because the other types of gravitons will be excludedby the sup plem enta ry conditions. Moreover, since according to ou r treatmentall the components of yk:' and y(+) contain the absorption operators, the stateof vacuum will satisfy the relations

y y r o o, y'+'Yo=o. .. . 40)5 . S P I N O T H E G R A V I T O N S

In the last section it was shown that only two types of gravitons can existin a pu re gravitational field. We shall now discuss the sp in of these gravitons.From the Lagrangian density (16), one can obtain a the canonical energymomentum tensor tlly or the gravitational field as

This energy momentum tensor, however, has to be modified according to thetreatment of Belinfante (see Wentzel 1949). For this we consider aninfinitesimal Loren tz transfo rmation~ X , = ~ W , , , X , wlth 80 = -&U,,,,, . . . . . 42)

an d find a tensor f p , given by

Then the modified energy momentum tensor 0, will be.. . 44)ae f i v = t p v - 2 G ( f p v , Q + f w , v + f e v , g ) *

and the angular mome ntum density is given byM p , = $ u e - v e p e

= X p t v e - v t p , + pv , I . 45)T hu s we obtain for th e components of the angular momentum

P , k = -i\ M , k , 4 d v = Ptb Plh, . . 46)Ptb= - i d V ( X , T k , - X k q 4 ) , ......( 7)Iith


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167uantization of the Gravitational Fieldwhere PZ is the orbital angular momentum, and Plk is the contribution dueto the spin. Substitu ting the Fourier expansions (25) in 48), e get

r-h t)l1x {a',(k') et(k'*r-k'O-ai,(k') e-t(k'

(a,,(k) e2(k -kt)+ut?(k) e-z(k.r-kt)x { a,,(kf) r - k ' t ) aj,(k') e-z(k'. r - h ' t ) } ]

= f [at,(k)aE,(k) - (k)a,,(k) - ,,(k)aL(k) + aX,(k)a,,(k)l= G&(k)%q(k)- t,(k)a,,(k)1. ......49)kSince the real gravitons correspond to the operators

andwe have to consider only that part of Plk which involves the operatorsa,@), a,,(k), a,&) and their adjoints. These operators occur only in thecomponent Pi2, and their contribution is given by

a m S{a,,(k) - 4 k ) h

p:2= E i [a l l (k)al l (k)+ 42(k)@l,(k)- all(k)azl(k)- 2(k)a,z(k)lk= X 2i[af2(k)all(k) {f(k)a,Z(k)]. ...... 50)

kIn order to separate the contributions of the two independent components inthe above equation, we introduce the operators1 1u+(k)= --qj{~&(k)-~&(k)}, a- (k )= - {&(k)+k i&) } , .... . S I )4 2where, according to (27) and (31), ,we have

......( 52)a+(k), aL(k)l =E+), .t_(k)I= 1,[a+(k), aL(k )] =O.In terms of these new operators we obtain

Pi2 = E 2 [ ~ y ( k ) ~ + ( k )U- (k)a-(k)lk= E 2[n+(k)- ( k ) ] , ...... 5 3 )k

where n,(k) and n-(k) give the numbers of the gravitons corresponding ZO theoperators a+(k) and a-(k).This shows that the gravitons are particles of spin 2 and they have twoindependent spin states with the spin axis parallel or antiparallel t o the directionof motion of the gravitons.

6 . T R E A T M E N T O F I N T E R A C T I O NI t is evident from (11) tha t the interaction of the gravitational field withthe matter field may be taken into account by modifying the Lagrangiandensity (16) as

which gives for the field equations0 2 ~ p v = ~ T N n0 ~~ T p p * *. . 55)


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I 68 &raj N . Guptahave also to be modified.conditions in th e Heisenberg representation as

I n th e presence of interaction, th e supplementary conditions (34) and (35)Following A, we may state the supplementary

rg] Y=o, [ y f l f l - y ]+ Y = O, . . . .. 56)where [ ]+ denotes the positive frequency part. These conditions are,of course, identical with (34) and (35) in th e absence of interaction. But thenecessity of writing them in the form (56) arises from the fact that in theHeisenberg representation y f l yand y cannot be split into the positive and thenegative frequen cy parts in th e presence of interaction. However, such aspl ittin g is still possible for ay,,,/ax,, and y f l p -y . For, according to ( 5 5 ) ,ayPy/ax,and ypp- still satisfy the wave equation for plane waves :

where we have taken into account the approximation used in $ 2 .When we pass over from the Heisenberg representation to the interactionrepresentation, the required supplementary conditions may be obtained in thesame way as in the case of the radiation field. Following the treatment givenfor the radiation field (Gupta 1951)*, we easily find in the interactionrepresentation

. .( y p - y ( * ) ) Y = o ,where D(+) (x -x ) is the positive frequency part of D ( x - x ) . It should beobserved tha t, due to the occurrence of the T,,-term in (57), hose gravitons whichcould not exist in a pure gravitational field can now appear in virtual states inthe presence of interaction.

For practical purposes it is most convenient t o use the invariant perturbationtheo ry based on the interaction representation. Fo r this we requi re, in additionto the commutation relations (23) and (24), the vacuum expectation values ofth e anticomm utators. Using the vacuum conditions (40), we easily find, byfollowing Schwinger (1949),({Y,Y(x), Y&)I >a = 4A18*p +G w w x ), ] . .. . . 5 8 )( { y ( x ) , = -4D(1)(x - .

It should be noted that the use of an indefinite metric does not make anydifference to the calculation of the vacuum expectation values, for theindefinite metric operator 7 defined in A, is jus t equal to unity for the state ofvacuum, which does not contain any gravitons.

7. L O R E N T Z - I N V A R I A N C EIn the present paper we have treated the components yo f yp,, by means ofan indefinite metric, while the other components have been treated in the usualway. T hu s, thou gh the expectation value of y f l y is always real, we have toregard y,,, as a complex tensor with yo, antiHerm itian and the other componentsHerm itian. I n this way the Lorentz-invariance of our treatment is not qyite* The letter referred to above contalns an error.read [8D(8-d) /8 tI t -t =- X, -21) S X, - -x , )~(x , - 8 8 ) .There the relatlon precedlng eqn. (9) should


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Quantization o j the Gravitational Field 169obvious. However, th e Lorentz-invariance of the present theory can be easilyestablished by splitting up yPy into the Hermitian and the antiHermitiancomponents in a more general way, as we have done in A.

For this we take yP v to be of the formy p y =Njl)N l)n(ll)+N(1)N z)A(12)NjI)N J)A(lB)N(1)N O)n(lo)+NLz)N l)A(21)N(B)N Z)A(22)NfJN(3)R(23)N(2)N(O)A(20)+ $Tj3)N l)n(W Nj3)N 2)A(32)N(3)N 3)A(38)N(3)N O)A(30)

Njo)N(I)A(O1) N(O)N 2)A(02)N(o)NL3)A(03)N(O)N O)A(OO),P V

. . . 59),where, as in A, N f ) ,N r ) ,Nj3) nd N F ) are a set of orthogonal unit four-vectors,Njl), NL3)being space-like, and N,? time-like. Further, A lI),A(12), . .are scalars, of which A(1o),A(zo),A(30), 01),(o2)nd A(0a) re.a&iHermitian, whilethe rest are Hermitian. Th us , according to (59), we have split y p v nto sixteencomponents in a formally Lorentz-invariant way. We can now follow th eentire treatment of the present paper with two differences: Firstly, insteadof yPv ,we have to expand the A's into Fourier expansions of the form (25).Secondly, instead of taking the x3 axis along k, we have t o choose, as in A,Nf , j2) ,N;") n such a way that

kPN/13) - P PO .The treatment for y , of course, remains unchanged, because y is a scalarindependent of yP,. In this way the vacuum conditions (40) and all the otherresults remain unchanged.Since (59) is a tensor equation, all our results will now be formallyLorentz-invariant. Still, in order to establish the relativistic invariance of ourtreatment one fu rthe r point remains to be clarified. According to (59), if wechoose a fram e of reference in which N F)=(O, 0, 0, i , i.e. the time-axiscoincides with NLO), the components yo%of yPv will be antiHermitian whilethe other com ponents will be Hermitian. Since this property of yP y is notpreserved in all fram es of reference, our treatment has singled out those framesof reference in which the time-axis coincides with Njo) . Thus, in the presenttheory N j 0 )appears as a preferred direction. However, this does not matter,for NF ) can be chosen in an arbitrary way, and the results of physical interestare ind ependent of its direction. I n fact, after the vacuum conditions (40) havebeen established, N F ) never occurs explicitly in the interaction representation.

kPNjl'= kPNj2'= 0, . . .. . 60)

R E F E R E N C E SEINSTEIN,., 918, erlzner Berzchte, p. 154.GUPTA, . N., 950, roc. Phys. Soc. A, 3,681 1951,bzd., 64,850.ROSENFELD, 930,Z. hys., 65,589.SCHWINGER,., 1948, hys. Rev., 4, 1439; 949, btd., 75,651.WENTZEL,., 949, uantum Theory of Fzelds (New York Intersclence), p . 2 1 7 .

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gravita' quantistica