IL NUOVO CIMENTO VoL 23 A, N. 4 21 Ottobre 1974

Poinear4 Covariance and ~ u a n t i z a t i o n o f Zero-Mass Fields.

IX: The Weak Gravitational Field.

J. BErTRAnD (*) Universitd de Paris V I I - Paris

(ricevuto il 27 Settembre 1973; manoscritto revisionato ricevuto l ' l Marzo 1974)

S u m m a r y . - - Following the work done in a previous paper on the quan- t ization of a potential vector A~, we find out the consequences of general requirements such as Poincar6 covariance and basic quan tum mechanics (i.e. existence of a Hilbert space of one-particle states support ing a uni ta ry representation of Poincar6 group ~) on the quant izat ion of a symmetric tensor field h ~v. While in the case of the electromagnetic potential we had a 2-parameter family of metrics all equivalent to Gupta 's and corresponding to a transverse photon, we obtain here a wide variety of descriptions for the graviton. First we must choose between a graviton possessing helicities 0. :~ 2 and a transverse one. Then, we have two different ways of getting rid of the hclicities 0: we may suppress them covariantly or <~ gauge them away ~. Previously we had obtained dircctly a quantif ication of Maxwell's equations while, here. we mast postulate Einstein linearized equations as an addit ional requirement which only mildly narrows the above choice.

1 . - I n t r o d u c t i o n .

I n a first pape r (1) we showed how the q u a n t i z a t i o n of a 4 -vec to r p o t e n t i a l

A " desc r ib ing mass-zero par t ic les w i th t he r e q u i r e m e n t of Po inca r6 i n v a r i a n c e

led necessa r i ly to t he desc r ip t ion of a t r a n s v e r s e p h o t o n (**). Moreover , all t h e

q u a n t u m field theor ies o b t a i n e d t h e r e a f t e r were e q u i v a l e n t to the G u p t ~ -

B leu le r (5) q u a n t i z a t i o n of Maxwel l ' s equa t ions . Here , we are go ing to a p p l y t he

(*) Postal address: Universit6 Paris VII, Tour 33/43-2, place Jussieu, 75005 Paris. (1) J. BERTRAND: ~UOVO Cimento, 1A, 1 (1971), referred to as I. (**) If one insists on the appearance of maximal helicities (4-1). (e) S . N . GUPTA: Prec. Phys. See., A63, 681 (1950); K. BL]~ULER: Helv. Phys. Acta, 23, 567 (1950).

703

704 j. BERTRAND

same principles to the quant izat ion of a symmetr ic tensor field An" describing

mass-zero, helicity ± 2 particles and eventual ly lower helieities. We shall then ask whether we have obtained a quant izat ion of Einstein linearized equations.

The notat ions are the same as in I , unless otherwise stated.

2. - Q u a n t i z a t i o n o f a s y m m e t r i c traceless field An~.

Suppose we are given a space H (*) of functions A "~, # , v = 0, ..., 3, defined on C t and such tha t

{ A ~ ' = A~n ,

(1) A n = 0 . /z

H is a carrier space for the nonuni ta ry representat ion U t~ of the Poinear6

group ~ given by

(2) U11(a, A):A"'(p) --> exp [ia .p],~, 3;, A~"" (A-~p) ,

where ( a , A ) ~ , p ~ C + and A~, is the element of S03,~ corresponding to

A ~ SL2. c. Our problem now is to find the minimal conditions to impose on A n" in order

to make U ~ un i ta ry and containing at least helicity ± 2 ~-representat ions

(or (, e lementary particles ~)). Using the decomposit ion (I.3)-(I.6) and defining

(3) v~'~(p) = A;~,A;~A~(p) ,

we obtain an equivalent form V of (2):

(4) V~(a, A) :v"~(p) --> exp lip. a] J~, t~, v#" (A-~p) ,

where B ---- A AA74!,~ E E2. We shall write

B = cxp [iqq 0

exp [-- i~] exp [-- i~] , ~ )eC, 0<q~ < 2~r.

The tensorial E~ representat ion which appears here is not convenient. We shall ra ther use the Na imark realization ~1~ in the space of homogeneous

polynomials wri t ten 2

q(z,~)= ~ q . z ~ , i d = 0

(*) For the following we may take ~(C t) for example.

FOINCAR]~ COVARIANCE AND QUANTIZATION OF ZERO-MASS FIELDS - I I 7{)5

a n d r e n u m b e r t h e c o m p o n e n t s . T h e t r a n s f o r m a t i o n is e x p l i c i t l y g i v e n b y (*)

,W 0 ~ q00 = ~)11 -Jr- ~)22 .~_ 2V33 _~ 2•03 ,

wl = qlO = 2(vo2 + v2~) _ 2i(vOl + v13),

w2 ~ qOl = 2(vO2+ v23) + 2i(vOX + v ia ) ,

w 3 ~ q,1 = 4(v~1 ÷ v~),

(5) w 4 _~ q2O = v~2 __ v n __ 2 i v 10 . ,

w 5 ~ qO~ = v22 __ VlX _~_ 2 i v l ~ ,

w 6 ~_ q~2 = 2(vOZ __ v~3) _~_ 2 i ( v l a __ v O l ) ,

w 7 ~ q~ = 2(v o2 - - v23) - - 2i(vla - - vOi),

Ws ~_ q21 = v l l _~_ V22 ~ 2V33 __ 2~)o3 .

E q u a t i o n (4) l eads to

(6) W ( a , A ) : w ~ ( p ) -~ e x p l i p . (11) ] --1 a] ~ . . (B) w (A p ) .

F o r t h i s f o r m of t h e r e p r e s e n t a t i o n , i t is eas ie r t o f ind a l l t h e i n v a r i a n ~

s e s q u i l i n e a r fo rms ~ ( w , w'). B y a n a r g u m e n t s imi l a r t o I , we c o n c l u d e t h a t

t h e m o s t g e n e r a l f o r m m a y b e w r i t t e n

, ? d 3 p (7) ~ ( w , w ) = | ~ ~ ' ( p ) s~w~(p) ,

vJ~,l/~ I

w h e r e S is a c o n s t a n t 9 x 9 m a t r i x such t h a t

(8 ) 0~(1 a)(B)+ S -~- ~ q . ~ ( l ' l ) ( B ) - I (cf. 1.20).

(9)

T h e g e n e r a l so lu t ion

0 0

0 0

0 0

0 0

S = 0 0

0 0

0 - - 2 a

0 0

4 a 0

i s

0 0

0 0

0 0

0 a

0 0

0 0

0 0

~ 2 a 0

0 - - b

0 0 0 0 4a

0 0 ~ 2 a 0 0

0 0 0 - - 2 a 0

0 0 0 0 - - b

4a 0 0 0 0

0 4a 0 0 0

0 0 b 0 0

0 0 0 b 0

0 0 0 0 e

(*) ~Ve have taken into account relations (1).

7 0 6 J . BERTRAND

This does not lead to a posit ive-defini te fo rm ~ for any value of the cons tants a, b, e. According to *he general scheme (~.~.4), we look for the grea tes t invar ian t subspaee K of H on which ~ is posi t ive or zero and containing helicities =J= 2.

Proceeding step b y step, we first res t r ic t ourselves to the invar ian t sub- space defined b y

(10) wS= 0 ,

or, equivalent ly , b y

(11) p~ p , A ~" ---- 0 .

This amoun t s to the el iminat ion of one t y p e of scalar particles since the t rans- f o r m a t i o n law of w 8 is

W(a, A):wS(p) -+ exp [ip .a] w~(A-I p) .

Then, diagonalizing S, we find t h a t K is defined b y

(]2)

or, equivalent ly , b y

(]3)

w°(p ) = w~(p) = w~(p) ~- o ,

p~, Al'~(p) = s(p) p" ,

where s is an a rb i t r a ry scalar funct ion belonging to ~ ( C ) . According to the t r ans fo rma t ion law of w

W(a,A,: (w' (p , l__ > ( exp[ip.a](exp[2iqJ]w6(A-lp,-~ - 2exp[2iq~]~w'(A-'p)) I

\wT(p)/ \ exp [ip. a] (exp [ - - 2ig~]wT(A-lp) ~- 2 exp [ - - 2i~] ew~(A-'p))/

(:12) means t h a t we have el iminated heliei ty -t- 1 particles as well as scalar ones. On K, the inva r i an t sesquilinear f o r m reads

~(w, w') = a(lwa(p)] ~ + 4[w,(p)l ~ + 4w~(p)[ ~) I

or, in t e rms of A ~v,

4 3 414) "CA, A')---- f ~o~ a(A'~(P)A~,(P)) "

F o r a ~ 0 we have ~ ( A , A ' ) > 0 . To obta in a posit ive-definite fo rm we need on ly to quot ient b y the subspace Ko on which ~ = 0. K ° is made of func-

(a) R. SHAW: Nuovo Cimento, 37, 1086 (1965). (4) A. McKERREL: Ann. of Phys., 40, 237 (1966).

POINCAR]~ COVARL4.NCE AND QUANTIZATION OF ZERO-MASS FIELDS - II 707

tions Wo such tha t

w:= w:= o,

or equivalent ly of functions A~" such t ha t

(15) A~'(p) = p~'f'(p) q- p']~'(p) ,

where ]" is an a rb i t ra ry vector funct ion satisfying

(16) p~, fJ'(p) = O .

Thus K/Ko is made of functions A ~" satisfying (1)1 (13) and the equivalence relat ion

ATV(p) .-. A~'(p) if and only if A • ' ( p ) - A~'(p) = A~'(p) .

I t can be completed into a I t i lber t space on which the representa t ion W be- come un i t a ry and is equivalent to the direct sum of three Wigner representa t ions (helicities 0, ~: 2) as can be seen from the W-transformat ion law:

w~(p) exp lip .a]w~(A-lp)

(17) W(a, A) @(p) --> exp [ip .a] exp [4iq;]w4(A-lp)

w~(p) exp [ip .a] exp [-- 4iq~]w~(A-ip)

Alternat ively, we may choose to suppress completely helicity-zero particles by sett ing

(18) w3(p) = O, i.e. s(p) = O,

and to consider the part icle described by the direct sum of helici ty ± 2 ~-representa t ions .

Thus we may state

Result I. The description of a mass-zero, spin-two particle, in relativist ic quan tum mechanics, using a symmetr ic traceless tensor wave funct ion A ~" , can be achieved in a t t i lber t space ~ f by two inequivalent schemes:

a) if the part icle may take helicities 0, ± 2, then ~%f is obta ined b y com- pletion of the space of functions A t" ~ ~ (C t) satisfying

(19) i) p~,A ~'" = sp", s ~ ~(C+),

(20) ii) f dap (A~'~(p)A~,,(p)) < + j [pO[ ,

7 ~ J, BERTRAND

iii) gauge invariance, i.e. equivalence between any function A~" and its

t r ans fo rm defined b y

(21) A~" ~-- A~"~- p~']"~- p"]~' for any ] ~ ~(C~),

such t h a t p~ ]s ~ 0 ;

b) if the particle m a y only take helicities ~= 2, then 5(( is obtained by

complet ion of the space of functions A s ' E ~ ( C t) satisfying conditions (20),

(21) and

(19') Ps Ag" = 0 .

I t is impor t an t to remark tha t we have wri t ten ~ in the form (20) only for simplicity. Actually, we should have wri t ten the most general form as obtained f rom (9), (7), (5) and (3): we thus have a whole family of metrics which all

coincide on K and lead to equivalent results for the physical particle.

:Now, if we give up the requirement t h a t the scalar product of ~ be invariant by the full spinorial representat ion (2), independent ly of condition (13), we m a y look directly for the invariant sesquilinear forms on K and find

(22) f d~P I~ ~(~' ~') = V~ (~ l~ (p ) + a,l~,(v)l ~ + ~[~(p) l~) ,

~>~0 when aa, a~, a s are arbi trary, positive or zero constants.

We restrict immedia te ly ourselves to a 4 --~ a s (i.e. same weight for and - - 2 helicities) and write ~ in the A s" space:

+ 2

(23) d Ip °]

F r o m (22) it is easy to see that , if ~3, a4V= 0~ ~ 0 on the same K0 as above. However , if a 4 =/: 0, a 3 = 0, ~ is zero on the subspace made of functions

w~ such tha t

(24) w:'(p) = w~(p) = o ,

or equivalent ly of funct ions A~ v" such tha t

(25) A,o~,~= p~,/,(p) + p , ] S ( p ) _ i g~,,pq/e(p)

for an a rb i t r a ry vector funct ion Is.

Thus we m a y state

Result I I . The description of a mass-zero, spin-2 particle in relativistic

q u a n t u m mechanics, using a symmetr ic traceless tensor wave funct ion A s"

POINCAR]~ COVAI~IA]~CE AND QUANTIZATION OF ZERO-MASS FIELDS - II 7 0 9

such tha t

p~, A ~" ~ s ( p ) p " (i.e. A ~" ~ K )

can be realized in a Hilbert space ~ ' in different ways.

a) I f we take as sesquilinear form

t ,n d 8 4 ( 4 ~ a - - ~ 4 ) [ s ( p ) [ u )

with ~3, ~ , ¢ 0, we get back Result I with the added possibility of giving different weights to the 0- and ~ 2-helicity particles.

b) I f we take ~3----0 in (23), 9~' is obtained by completion of the space of functions A~ '~ K s~tisfying

i) f d~p ( A " ' ( p ) A , , ( p ) - - 4 [ s ( p ) [ ~) < c~, JIp°l

if) gauge invariance, i.e. equivalence between any funct ion A~" and its t ransform A~" defined by

for any ] ' e ~(C*). Then, the scalar functions s~ and s~ corresponding to A~ and A~ sat isfy

s ~ ( p ) - s~(p) -~ ½pq/q(p) .

In this way we describe only helicity ± 2 particles. These are the only possibilities.

3. - Quant izat ion o f Einste in l inearized equations.

In general relativity, a weak gravitat ional field is described by a metr ic tensor g~,v~-ehf,,(xQ), e small, satisfying Einstein equations. Outside mat te r , these equations give, to first order,

where

[] ~ ~ ~ , h ~ h~.

7 1 0 j . B E R T R A N D

Moreover, in an infinitesimal co-ordinate t ransformat ion

(27) x ~ -+ x~ '= x ~ + e ~ ( x e)

h is t rans formed according to

(2s)

which leaves eq. (26) invariant . Thus, to solve the latter , one usually requires a co-ordinate condit ion in the form

(29) ~q h q = ½3~, h ,

which reduces (26) to

1 h . (30) [ ] ( h - - ~ g . . ) = 0

To preserve this form, the f reedom of co-ordinate t ransformations (27) has t he n to be narrowed by the conditions

(31) [] '~== 0.

Here, instead, we shall choose co-ordinates such tha t ~z', the 4-dimen- sional Four ier t r ans form of h z~, has support on C +. Equat ions (26) then become

(32) p~ p , ~ = 0 ,

(33) p . p ]i q+ p~,pq~q --p.p~,]i~ = O.

To quant ize eovar ian t ly these equations, we demand tha t )~"" become a quan tum p r oba b ih ty amph tude and belong to a carrier space for a uni ta ry representa t ion of ~ . Moreover, we want the highest possible helieities (i.e. ± 2) to appear . We m a y apply the results of the last Section to the traceless par t of ~"~ and add the un i t a ry scalar representa t ion of ~ corresponding to )~. We shall just have to s tudy the compat ib i l i ty of the results with eqs. (32) and (33). Fi rs t we look for the consequences of Result I. The Hi lber t space of the quant ized t he o r y is obta ined f rom functions ~z~ on C + satisfying the three conditions

(34) i)

o r

(34')

(35) ii)

where fl~) 1.

p . ~ " ' = ( s+ [)i)p ' , s # o ,

Ip°[ <cx3,

POINCARI~ COVARIANCE AND Q U A N T I Z A T I O N OF ZER0-MASS F I E L D S - I I 7 1 1

iii) gauge invariance, i.e. equivalence be tween any funct ion ~ " and its t ransform )~" defined by

(36) ~ ' - ~ ) ~ ' - d - p ~ ' / ' - - ~ p ' f l ' (when fl~ s ~ - - ¼ ) ( ' ) ,

where ]~ is a funct ion such t ha t ps ]~ = O. Such functions wiU be solutions of :Einstein equat ions (32) and (33) if and

only if

(37) s---- ¼)~,

o r

(37') ~ = 0 ,

respectively. They will describe particles of helicities

0, d-2 , --2 (case (34), (37)),

-4- 2 , - 2 (case (34'), (37')).

However, we must emphasize t h a t t ransformat ions (28) and (36) are iden- tical only if

(3s) p~ ~ = 0,

where ~ is the Fourier t ransform of ~'. ~ o w we investigate the consequences of Resul t IIb). There we have to

s ta r t f rom functions ]~"" on C t sat isfying

(34) p , )~u, = (s -~ ¼)~)p",

(39) ~(h, h / - (~;"'(p/g.,(p/-418(p)l ~ + flJi@/P) < ~ ,

where fl~> -- I . Such a form ~ is identically zero on the space of funct ions hi"" defined b y

J ~'o ~'" =P" I ' -FpV" -F yg"'pq]qd- g"'F, (40) l r l~rjQ~ t

where ]e F are funct ions on C t and ~ E R. There are t w o d i f fe rent cases

(*) For the special case f12 ~ - 3, see the recapitulating list.

712

TABLE I. -- Recapitulating list o/ the dif]erent models.

g. B E R T R A N D

The condit ions defining a subspace on which is nonnegat ive and invar ian t are

where

25(h, h) : ] ( dap/Ip°]) B(h, h), where B(h, h) =

Definition of gauge invariance. The kernel of ~ consists of functions

~o~ = p~/~ + p ' /~ + ÷ ggV(ypqfq + ~) , So = ½po/~, where

s # O , h # O ~'~(p)hz~(p)÷fl~lf~(p)l 2 f12>--¼ P~] " = / v = 0

s : o . h # O ~,~(p)h,~(p)÷fl21~(p)l 2 f l~>--¼ p , ] , : ~ = O

fl~ = - - ¼ p , / " = 0

s # 0, h = 0 ~ g ~ ( p ) ~ ( p ) p , ] , = F = 0

s # O , h # O ~ ' ( p ) ~ ( p ) - - 4 ] s ( p ) 1 2 + f l 2 ] h ( p ) ] 2 ~ = 0 , y - -½ Z~> - - ¼

s # O, h = 0 ~'~(p) h~(p ) - - 4Is(p){ 2 (1 + 2y)po]e ÷ 2F : 0

s # O, h # 0 ~i,~(p) h~,~(p) - - 4Is(p)] 2 - ¼ ]~(p)l 2

]~, F a n d y a r e a r b i t r a r y in (40). T h e r e p r e s e n t a t i o n o b t a i n e d is e q u i v a l e n t to

t h e s u m of t w o u n i t a r y i r r e d u c i b l e P - r e p r e s e n t a t i o n s c o r r e s p o n d i n g to he l ic i t i es

± 2. T h u s t h e s c a l a r f u n c t i o n s s a n d h, w h i c h desc r ibe he l ic i t ies 07 a c t u a l l y

do n o t a p p e a r in ~ a n d can b e m a d e t o s a t i s f y a n y s u p p l e m e n t a r y c o n d i t i o n

as l ong as i t is c o m p a t i b l e w i t h t h e r e l a t i o n s def in ing t h e k e r n e l of ~ .

F o r e x a m p l e , fo l l owing BI%ACCI a n d STlCOCCEr (5), we m a y w r i t e

(41) s = - - ( q + ¼ ) ) ~ ¢ O , q e R ,

w h i c h l e a d s to

(42) p~. ~'" ~- - - q~p"

(5) L. BI~ACCI: Nuovo Cimento, 8 A , 129 (1972); L. BlCACCI and F. STROCCHI: Journ. Math. Phys . , 13, 1151 (1972).

FOINCA.R]~ COVAF~IANCE A.ND QUANTIZA.TION OF ZERO-MASS FIELDS - II 713

Helicities Supplementary conditions Helicities of particles for compatibility of particles described with Einstein equations described

Identi- fication number of the model

o, o, + 2 , - - 2 s = ~ f / , /~ # - - ½ o, + 2 , - - 2

o, + 2 , - - 2 s = ¼ g = o , / ~ = 0 +2 , - - 2 2

o, + 2 , - - 2 ~ = o, 2 ' = 0 + 2 , - - 2 2

+2 , - - 2

O, +2 , - - 2 s = O +2 , - - 2 2

O, +2 , - - 2 s = ~ , pl~]l,=O O, +2 , - - 2 1

+2 , - - 2 s = 0 , pof = 0 , 2 ~ = 0 +2 , - - 2 2

+ 2 , - - 2 s=¼/ , , F + ~ , p ~ / e 0 + 2 , - - 2 3

if we set

[1 ] (43) F = - - [ ~ ~ + r P° ]o .

There is also the possibi l i ty to h a v e

/ ~ = 0, (44) [ s :/: 0 ,

a n d this is compat ib le wi th (1 + 2:F)p~]q~ - 2 F -~ O,

(451 ~ # o

we h a v e to res t r ic t gauge changes b y the cond i t ion

(46)

B u t if we w a n t

(i.e. q = - ~ in (o)),

( respec t ive ly ~ ~ 0),

pQ/Q = 0 ( respect ively pQ 1e ~ F ---- 0).

7 1 4 j . B E R T R A N D

Finally, if we demand formal compatibi l i ty with Einstein equations, we must choose

( 4 7 ) s = 41 ) i ,

(48) ~ p ~ / q ÷ F = o .

Gauge invariance is then wri t ten

~ ~ if and only if

which leads to

)~: -- ~. __-- 2p~ ]Q.

This is the Fierz-Pauli (6) quant ized theory of the gravitou

2/Z~> - ~.

_ : F ~ 0. Einstein equations (which require The only possibility is ~ - - : ,

conditions (47)) can only be satisfied with the restricted gauge invariance de- fined by pQ]e~- O. The same is t rue of any subsidiary condition of the form (41) or (45).

l~ecapitulating, we draw up a list of the different descriptions of a mass- zero, helicity ± 2 particle t h a t can be obtained, using a symmetr ic tensor

field ~ " on the light-cone. I n each case we end up with a uni tary representat ion of ~ and we give explicitly the helicity labels of the irreducible representat ion

appear ing in the direct-sum decomposit ion ((< helicities of particles described >>). I n conclusion, we want to insist on the differences tha t have appeared (*)

in first quant iza t ion between the case of A ~ (4 -vec to r )and h "" (symmetric

second-rank tensor): the former can only describe helicity =L 1 particles and satisfy Maxwell equations while the lat ter can describe several types of helicity

2 pal~icles or mixtures of helicities 0 and ± 2 and does not have to satisfy :Einstein linearized equations. Thus we have to make a choice of what we shall call a graviton.

I n a for thcoming paper we shall t rea t in detail the problem of construct ing a Fock space f rom the above one-particle wave functions h "~. We shall ex- amine which models can lead to physical ly acceptable quan tum field theories

and in what sense t h e y describe a quant ized graviton.

The au thor would like to t h a n k Prof. G. I~I~)EAV for helpful discussions.

(e) M. FI~RZ and W. ZPAuLI: Prec. Roy. See., 173 A, 211 (1939). (*) This has been possible because we started from hypotheses different form those of A. O. BARum and R. ]~ACZKA (7). (~) A. O. BARUT and R. RACZKA: Ann. Inst. H. Poincar6, 17, 111 (1972).

POINCAR~ COVARIANCE 2~ND QU&NTIZiTION OF ZERO-M&SS FIELDS - II 715

• R I A S S U N T O (')

P roseguendo nel l avoro ogge t to di un p receden te ar t icolo sul la quant izzaz ione di un ve t t o r e po tenz ia le A u, si t r o v a n o le eonseguenze di condiz ioni genera l i come la cova r i anza di Po incar4 e la meccan ica quan t i s t i ca di base (clog l ' es i s tenza di uno spazio di Hi lbc r t di s t a t i di una pa r t i ce l l a come base di una rappresen taz ione un i t a r i a del g ruppo di Po incar4 ~) sulla quan t izzaz ionc di un eampo tensor ia le s immet r i co h~,. Ment re pe r il po tenz ia le e l e t t romagne t i co si a v e v a una famig l ia di me t r i che d ipenden t i da 2 para- me t r i t u t t e cqu iva l en t i alla m e t r i c a di Gup ta e co r r i sponden t i a un fo tone t rasversa le , qui si o t t i ene una g rande va r i e t~ di deser iz ioni per il gravi~one. I n n a n z i t u t t o si pub scegliere f ra un g rav i tonc con el ic i t£ O, ~= 2 e uno t rasversa le . Quindi , si hanno due diverse man ie re per l iberarsi delle e l ic i th nul le : si possono soppr imere in fo rma cova- r i an t e o p p u r e imponendo l ' i n v a r i a n z a di gauge. Infine, m e n , r e si era o t t e n u t a d i re t t a - men te la quan t izzaz ione delle equaz ion i di Maxwell , qui si debbano pos tu la rc equa- zioni l inear izza te di E ins te in come condiz ione a g g i u n t i v a che res t r inge solo l i e v e m e n t e la scel ta precedente .

(*) Traduzione a cura della Redazione.

[~oBapHaHTHOCTb IIyatmape u KBaHTOBaHHe flo~e~t c Hy-rteBOfi Maeeofi.

Pe3mMe (*). - - Cneaya MeTo~y, pa3BUTOMy B npeabt~yme~ pa6oTe, nocBameHuo~ KBaHTO- BaHHtO BeKTOpHOFO lIOTeHHI4ana, A ~, MbI BblBO~HM CJle~tCTBH~ O6~t~nX Tpe6oBaHH~, TaKnx raK KOBApHaHTHOCTb FlyanKape n ncxo~Han KBaHTOBafl MexanHKa (T.e. cymecTBoBanHe npocTpaHCTBa Fnnb6epTa aa~ O~HoqacTI~qHBIX COCTO~IHPIffl, n o ~ e p m a a a ~ o m n x yn~Tapnoe npe~cTaBneHne rpynrmi VlyaHrape, ~ ) , ~na KBaHTOBaHHn CHMMeTpHqHOFO xen3opnoro non~ h ~. B TO BpeMa, rUE B cnyqae 3neKTpoMarnt4TnOrO I1OTeHIIHaJ]a, MBI HMeJIH )iByx- tiapaMeTpgqecKoe ceMe~ca~o Mexpnr, 3KBnBanenTn~tx MeTpnKe F y n x a n MeTpnKe, COOXBeT- ca~ytome~ nonepeunoMy qbOTOHy, 3~ecb MBI nonyqaeM mHpoKoe MHoroo6paane onncann~ rpaBHTOHa. CHaqana MbI ~onxrr~i BfiI6paTb Mex~y rpaBHTOHOM, o6na~]aro~nM cnnpanb- HogT~IMH 0, ~ 2 I4 n o n e p e q n o ~ CI1HpanbHOCTblO. 3aTeM MBI HMeeM ~/Ba pa3nHqHI, IX cnoco6a , qTO6bI OCBO60~HTbCJI OT cnnpam,nocTe~ 0: MbI MOdeM l'Io~aYlHTb HX roBap~aaTao n a n

(( rann6poBo~rtbIM o6pa30M >>. Ta~nM 06pa3oM, ecn~ B nep~o~ pa6oTe M/aI IIOJIyqHIIFl Henocpe)ICTBeHHO ~BaHTOBaHne ypaBaennfi Ma~caenna, To a 3TOkI pa6oxe M~ ~ o n ~ n ~ nocTya~tpoBaTb aaneapH3oBa~ab~e y p a a a e H ~ 9~amre f iHa xa~ ~o~o~H~Tenb~oe Tpe6o- BaHne, ~oTopoe nnmb cna6o cy~aeT a ~ i m e y n o M n n y T ~ B~i6op.

(*) Hepeeei)eno pec)a~quefi.