26
G. RIDEAU COY ARIANT QUANTIZATIONS OF THE MAXWELL FIELD Cet article ne saurait commencer sans que j'evoque la memoire de Bernard Jouvet. a qui il est dedie. Parmi ceux qui ont collabore a ce volume, je me trouve sans doute etre un des plus anciens a l'avoir connu. Nos premieres rencontres se firent au cours des annees cinquante dans la frequentation commune du seminaire Proca, dont Ie role fut si determinant pour Ie renouveau de la physique theorique franc;:aise. Nous y decouvrions, non sans echanger parfois nos perplexites,les travaux alors nouveaux de Feynman, de Schwinger, de Tomonaga et de bien d'autres. Nons y primes l'un et l'autre ce qui cor- respondait a nos gouts respectifs, plus pragmatiques, plus instrumentaux sans doute chez Bernard Jouvet que chez moi, plus attire par l'elucidation du jeu mathematique. Ce fut l'objet de discussions nombreuses et passion- nees, tout au long de ces presque trente annees qui nous procurerent souvent l'occasion de confronter nos options et nos resultats dans des conversations dont les detours ultimes nous entrainaient invariablement vers des domaines vastes et varies, bien eloignes en tout cas de l'initiale motivation scientifique. Bernard Jouvet n'aura rien connu du present travail, complete, pour sa plus grande part, depuis sa disparition. II n'aurait sans doute pas manque de Ie dissequer, d'en critiquer les aspects qui lui auraient paru incompatibles avec Ie sens profound qu'il avait de ce qui etait physique ou non physique. Nous aurions noue a nouveau Ie fi1 de nos ami cales querelles, dans la fougue ct l'elan qui etaient siens. C'est avec beaucoup d'emotion et de tristesse que j'evoque ici son absence et que je salue une derniere fois celui qui fut, pour moi, un chercheur exemplaire et un ami. ABSTRACT. The basic axioms of a covariant quantization of the Maxwell field are reviewed, and the general form of the resulting two-points function for the vector potential field is given. Cohomological considerations are decisive for this derivation and allow a significant weakness of the axioms. An application is made in the particular case of linear conformal invariant gauge. Explicit constructions are obtained. The main tool is the theory of extensions of mass-zero representations of the Poincare group. Under a supplementary condition concerning the indecomposability of the mono-particle states, it is shown that the linear conformal invariant gauges are identical with the so-called generalized Lorentz gauges. If this condition is removed, one is faced with a field theory where the spectral condition is no longer valid except for the physical states. E. Tirapegui (Ed.), Field Theory, QuanNzarion and Statistical Physics, 201-226. Copyright © 1981 by D. Reidel Publishing Company. 201

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G. RIDEAU

COY ARIANT QUANTIZATIONS OF

THE MAXWELL FIELD

Cet article ne saurait commencer sans que j'evoque la memoire de Bernard Jouvet. a qui il est dedie. Parmi ceux qui ont collabore a ce volume, je me trouve sans doute etre un des plus anciens a l'avoir connu. Nos premieres rencontres se firent au cours des annees cinquante dans la frequentation commune du seminaire Proca, dont Ie role fut si determinant pour Ie renouveau de la physique theorique franc;:aise. Nous y decouvrions, non sans echanger parfois nos perplexites,les travaux alors nouveaux de Feynman, de Schwinger, de Tomonaga et de bien d'autres. Nons y primes l'un et l'autre ce qui cor­respondait a nos gouts respectifs, plus pragmatiques, plus instrumentaux sans doute chez Bernard Jouvet que chez moi, plus attire par l'elucidation du jeu mathematique. Ce fut l'objet de discussions nombreuses et passion­nees, tout au long de ces presque trente annees qui nous procurerent souvent l'occasion de confronter nos options et nos resultats dans des conversations dont les detours ultimes nous entrainaient invariablement vers des domaines vastes et varies, bien eloignes en tout cas de l'initiale motivation scientifique.

Bernard Jouvet n'aura rien connu du present travail, complete, pour sa plus grande part, depuis sa disparition. II n'aurait sans doute pas manque de Ie dissequer, d'en critiquer les aspects qui lui auraient paru incompatibles avec Ie sens profound qu'il avait de ce qui etait physique ou non physique. Nous aurions noue a nouveau Ie fi1 de nos ami cales querelles, dans la fougue ct l'elan qui etaient siens. C'est avec beaucoup d'emotion et de tristesse que j'evoque ici son absence et que je salue une derniere fois celui qui fut, pour moi, un chercheur exemplaire et un ami.

ABSTRACT. The basic axioms of a covariant quantization of the Maxwell field are reviewed, and the general form of the resulting two-points function for the vector potential field is given. Cohomological considerations are decisive for this derivation and allow a significant weakness of the axioms. An application is made in the particular case of linear conformal invariant gauge. Explicit constructions are obtained. The main tool is the theory of extensions of mass-zero representations of the Poincare group. Under a supplementary condition concerning the indecomposability of the mono-particle states, it is shown that the linear conformal invariant gauges are identical with the so-called generalized Lorentz gauges. If this condition is removed, one is faced with a field theory where the spectral condition is no longer valid except for the physical states.

E. Tirapegui (Ed.), Field Theory, QuanNzarion and Statistical Physics, 201-226. Copyright © 1981 by D. Reidel Publishing Company.

201

202 G. RIDEAU

INTRODUCTION

It is well known that there arise special problems in the quantization of the Maxwell field. Two main ways are open in order to get a consistent answer. In the first one, the redundant variables are systematically elimi­nated and the remaining transverse components are quantized. Then, we are concerned with physically meaningful quantities only, but the vector potential field does not transform like a true vector under the Lorentz group. This becomes cumbersome when this quantization is used in quantum electrodynamics, for then the renormalization programme exhibits ambiguities which can be overcome only if we have learnt from covariant renormalization which diagramms must be put together.

Therefore, a second way of quantization has been developed where the vector potential field is actually a four-vector in Minkowski space. This covariant quantization has been initiated in the 1950s by Gupta and Bleuler and has been applied with complete success to quantum electro­dynamics. But, the necessary use of redundant variables has three conse­quences: the Maxwell equations are no longer valid as field equations, there exists a subsidiary condition in order to eliminate the physically meaningless state vectors and, finally, the space of state vectors is provided with an invariant Hermitian form which is not a true scalar product.

Obviously, the usual axiomatic formulation of the quantum field theory in term of vacuum expectation functions (Wightman functions) must be accordingly modified. This has been done in [1] where there is provided a general framework well adjusted to the situation above. As a consequence, many other quantization processes as the Gupta-Bleuler one are a priori possible and can be consistently described. In the present paper, we shall study mainly in this framework the particular case of covariant quantization. Then it will appear that some basic hypotheses can be significantly weakened in a way which agrees at the best with the minimal physical needs. We shall show it in our first part which brings together some personal recent works yet unpublished or published only in short form. In the second part, we shall be concerned with the important case of quantization in a linear conformal invariant gauge and we shall be able to give a complete description of the quasi-free field situation. As we shall see, group theoretical considerations concerning the extension of mass zero representations of Poincare group will playa crucial and illuminating role. Finally, in the appendix, we give the proof of the chomo­logical theorem which is basic for the statements of the first part, but which has a too technical character for being incorporated easily in the main text.

COVARIANT QUANTIZATIONS OF THE MAXWELL FIELD 203

PART I: THE TWO-POINTS FUNCTION OF THE VECTOR

POTENTIAL FIELD

1.1. THE BASIC AXIOMS (SEE [1])

Let Yf be some Hilbert space provided with a non-degenerate sesquilinear Hermitian form < , ), non-negative on a subspace £', the so-called sub­space of the physical states.

Let U(a,A) be a representation of the Poincare group f!!> leaving in­variant the form < , ) (and the subspace £').

Let the vector potential field All (x) be a vector distribution on Y (1R4) with values in the set of unbounded operators of £ which are symmetric wi th respect to <,).

The electromagnetic field F IlV (x) is given by:

F,jx) = 0IlA)x) - 0v All (x)

and we require that £' is globally invariant under the Fllv's. We shall say we get a covariant quantization of the Maxwell field if the

following conditions are fulfilled:

1.1.1. Existence and Unicity of the Vacuum State

There exists ljJoE,y'f' such that:

(a,A)Ef!!>,

and IjJ 0 is the unique vector in £ invariant under the translation subgroup of f!!>.

1.1.2. Covariance

The representation U(a,A) and the field All (x) are related by:

U(a,A)AIl(x)U(a,A)-1 = A~Av(Ax + a)

1.1.3. Spectral Condition

For </JE£', the distribution:

< </J,Fllv (x)ljJo)

has Fourier transform with support in V +, the closed future cone.

204 G. RIDEAU

1.1.4. M axwe II Equation

The remaining Maxwell equation is satisfied in mean on the physical subspace Yf'. In other words, for ¢,I/1EYf':

<¢,Oll F Il,,(x)l/1) = o.

1.1.5. CPT Invariance

There exists an antilinear operator e, antiunitary with respect to < , ), leaving invariant 1/10 and such that, for ¢,I/1EYf':

<¢,e~(FIlV(x))e-ll/1) = <¢,~(F!,J - x))I/1)

where !?l' (F IlV (x)) denotes a polynomial in the F IlV'S with real coefficients.

Remark. For the sake of rigor, it would be necessary to assume there exists a dense domain in Yf on which would be defined all the unbounded smeared out operators A (f) and also the U(a,A) which are not necessarily bounded operators on Yf (see Ref. 2, for the Gupta- Bleuler case). It would be not difficult to introduce the corresponding restrictions in the statements above but we neglect to do it in so far as this does not playa role in the following.

1.2. THE TWO-POINTS FUNCTION OF THE ELECTROMAGNETIC

FIELD

Let us write:

(1.2.1) GapAIl(X - y) = <1/10.Fap(X)FAIl(y)1/10)'

Since the electromagnetic field is a covariant anti symmetrical tensor, we have:

and for any A in t.he Lorentz group:

(1.2.3) A~'ArA~'A~'Ga'P'A'Il.(X)= GapAIl(Ax).

Furthermore, from the global invariance of Yf' under the F IlV'S and the spectral condition, we deduce that Ga{lA!' (x) is the boundary value as Ij~O of a function GapAIl(Z) holomorphic in z=x-ilj for IJEV+. By analytic continuation, GapJ.Il(z) verifies also (1.2.2) and (1.2.3).

COVARIANT QUANTIZATIONS OF THE MAXWELL FIELD 205

Therefore, from ref. 3 and with the same standard trick as in ref. 1, we can write:

(I.2.4) G~PAIl(X) = (g~A8/)1l - gpA8A, - g~1l8p8A + gpll8~8JGl (x)

+ (8~"PAIlP8P - 8p "aAIlP 8P)G2 (x)

+ (8A"Il~pp8P - 81l"A~Pp8P)G3 (x)

+ (ga;..gPIl - gallgp).)G4 (x) + "~PAIlG s (x)

where the Gi(x), i = 1,2,3,4,5 are boundary values of holomorphic functions of Z2 for z in the domain above.

LEMMA I. 2.1 From the CPT invariance, we have:

G2 (x) = G3 (x).

Proof The CPT invariance implies:

(8a"PAllp 8P - 8p"aAllp 8P - 8 A"wlPp8P + 8 II "Aapp 8P )(G2 (x) - G3 (x)) = o. This is equivalent to:

i= 1,2,3.

Therefore, using the Lorentz invariance, we can write:

(1.2.5) G2 (x) = G3 (x) + a + -i-bx2 •

But, the constant a does not contribute to the right-hand side of (I.2.4) and the contribution from -i-bx 2 is absorbed in the last term if we add to G s(x) a suitable constant. Thus we can assume a = b = 0 in (I.2.5), hence G2 (x)= G3 (x).

LEMMA 1. 2.2 Let us express that the field FIl)x) is a rotational field. Then we get:

(I.2.6) GapAIl(x) = (gaA8p81l - gPA8a81l- Yall8p8A + gPIl8a8A)G(x)

with G(x) = G1 (x) + -i-bx2 , where b is some constant. Proof Since the field F IlV (x) is a rotational, we have:

8a" GP" (x) = o. app" All

Applied to (1.2.4) where G 2 (x) = G 3 (x), this gives:

(gPIl8A - gPA8J(D G2 (x) - Gs(x)) + 8~"aPAIlG4(x) = 0

206 G. RIDEAU

with the general solution (c4 and Cs are some constants):

If we take into account the following identity:

we get:

(O,,BPAILp OP - OpBaAILPOP + 0ABJL(].PP OP - °ILBAaPpoP)Gz(x) + BaPAIL 0 G2 (x) = 0,

GapAIL(X) = (gaAopoIL - gPAoaoIL - gaILopoA + gPILOa0..)G(x) + BaPAILcS

with G (x) = G 1 (x) + iC4 X 2 • Let us consider now the following distribution:

GaAIL(X - y) = (t/Jo,A"Jx)FAIL(y)t/Jo)·

We can set down:

GaAIL(X) = (gaAoIL - gaILoA) + tCsBaPAILxP + Ha).IL(x)

where the distribution H aAIL (x) verifies:

0aHPAIL(x) - opHaA/X) = O.

Therefore, there exists a distribution HAIL (X) such that:

HaAIL(X) = °aHAIL(x).

Let us express again that FILv(x) is a rotational. We obtain:

3csgILa + OaOABAILPuHpa(x) = 0

which is equivalent to:

OABAILPaHPU(x) + 3csxIL = cIL

where the cIL are some constants. If we contract this last equation with OIL, we get:

12cs = O.

Therefore Cs = 0, and GapA,..(x) has well the form (1.2.6).

THEOREM I. 2.1 Under the hypothesis given in the previous section, we have:

(1.2.7) GapAIL(X) = y(ga).apo,.. - gpAoa0,.. - ga,..OPO A + gp,..OaOA)~ + (x)

COVARIANT QUANTIZATIONS OF THE MAXWELL FIELD 207

where y ~ 0 and the distribution L\ + (x) is given by:

f d3k L\ + (x) = lkf exp ( - ikx).

ko~ Ikl

Proal From the validity in mean on yt' of the equation Oil F Il V = 0 it results:

oaGap})x) = O.

With (1.2.6), this gives:

(1.2.8) D G = c.

But G is an invariant distribution and the limit as 1] --> 0 of a holomorphic function of z = x - i1], 1] E V +. The general solution of (1.2.8) which fits these conditions can be written:

G(x) = yL\ + (x) + icx2 .

After substitution into (1.2.6), we obtain:

(1.2.9) Gap»)x) = y(gaJo0poll - gPJo0/)1l - gallopoJo + gPlloa0Jo)L\ +(x)

+ 1c(gaAgPIl - gPJoga)'

Now, from the positivity of < , ) on yt', we must have for any antisym­metrical tensor f, with components in 9'([R4):

f d4xd4y!,,-P(x)GaPJoIl(x - y)fAIl(y) ~ O.

Let us specify f to be a tensor with only its Ol (or 23) component different from zero and equal to a function f(X)E9'([R4) such that Sf(x)d4x = 1. Then, we get from (1.2.9) the following inequality:

fd 3 k 2 2 lfl ~I ~ 12 . fd 3 k 2 2 I ~ I ~I ~ 12 (1.2.10) -y I kl(k2 +k3 ) (k,k) ~c/2~y I kl(k2 +k3 )f(k,k)

where j(k) is the Fourier transform of f(x). This implies readily y ~ O. Let us replace now in (1.2.10) j(k) by j(k/f.). The only change in (1.2.10) is to multiply the left- and right-hand sides by 3 4 . Therefore, they go to zero with f., so that necessarily c = 0, and (1.2.9) becomes (1.2.7).

Remark. The Maxwell equation Oil Fllv = 0 has been used only in the proof of the last theorem. As a consequence, we conclude that the form (1.2.6)

208 G. RIDEAU

for the two-points function of the electromagnetic field is also valid when the field interacts with charged matter.

1.3. THE FUNCTION G"'''Il(X)

We have already introduced G"'''Il(X) and have found for it the following form:

(1.3.1) G"'''Il(X) = y(ga .. all - galla..)t1. + (x) + a",H "1l(X)

where H"1l (x) is a tempered distribution such that:

(1.3.2) a"e"IlP"HP" = cll

LEMMA 1.3. The constants cll in (1.3.2) are necessarily equal to zero. Proof The general solution of (1.3.2) is given by:

H"1l = ie .. IlP"cpx" + a .. HIl(x) - aIlH .. (x)

where the H .. (x) are tempered distributions. Now, the covariance of the vector-potential field implies:

(1.3.3) a .. (A~' HIl,(Ax) - HIl(x» - all(A~'H .. ,(Ax) - H .. (x»

- ie"IlP,,(A~,cP' - cP)x" = y .. Il(A)

where the y(A) are indefinitely differentiable functions on SL(2, q (see Ref. I, p. 2208 for the argument). Let us express now that the first two terms in the left-hand side are the component of a rotational. We obtain, after some calculation:

A cP' - c = ° IlP' Il

i.e., cll = 0, since the D(~,~) representation of SL(2,q is irreducible.

THEOREM 1.3.1. Under the hypotheses of the first section, we have neces­sarily:

(1.3.4) GaJ.Il(X) = y(ga .. all - galla ;)t1. + (x).

Proof From (1.3.3), with cll = 0, we deduce easily:

y"Il(A I A2) = y"Il(A2) + (A2)~' (A2~' y"'Il,(AJ

so that y"ll (A - 1) is a I-cocycle defined on SL(2, q with respect to the anti symmetrical tensorial product of DH,~) by itself. Since SL(2,q is semi-simple and simply connected and the representation is finite-

COVARIANT QUANTIZATIONS OF THE MAXWELL FIELD 209

dimensional, such a I-cocycle is trivial. Therefore, we can find an antisym­metrical tensor Y AI' such that:

Yl'i (A) = YA" - A~' A;;' Y;I"

Let us write:

(1.3.5) H~(x)=H;,(x)+-txP;'P)'

There (1.3.3) becomes:

0Je (A~ H~(Ax) - H;,(x» - o,,(A1 H~,(Ax) - H~(x» = 0

Therefore, there exists a distribution H(A,x), which is a COO-function on SL{2, IC) and such that:

(1.3.6) A~'H~, (Ax) - H~(x) = o"H( A,x).

From this, we deduce easily the following equation:

H(A; 1 A; 1,X) - H(A; 1 ,x) - H(A; 1 ,A 1 x) = y(Al'Az),

Al'AZ ESL(2,1C)

where the COO-function y(Al'Az) is a 2-cocycle with respect to the trivial representation of SL(2, IC). Therefore it is trivial ~md we can find a COO_ function deAl such that:

y(Al'Az) = d(A 1) + d(A z ) - d(A1 A2 )

But then H(A- 1,x)+d(A) is finally a l-cocycle with value in Y"(~4) with respect to the following representation V(Al of SL(2, C):

T(x) ~ V(A)T(x) = T(A -1 x).

It will be proved in the appendix to the present paper that such a l-cocycle is indeed trivial, so that there exists a distribution H (x) with the property:

H(A,x) + d(A -1) = H(x) - H(Ax)

After substitution in (1.3.6) we have:

A~' (H~,(Ax) + o",H(Ax» = H;,(x) + 0I'H(x).

But since an invariant vectorial distribution is a gradient, we can find a distribution T(x) such that:

HA(x) = -~XPYpA +o"T(x)

210 G. RIDEAU

This implies first that:

HAl' (x) = YAI'

and consequently we have H,J.I' (x) = 0 in (1.3.1).

1.4. THE TWO-POINTS FUNCTION OF THE VECTOR POTENTIAL

FIELD

Let us set down:

G,p(x - y) = <l/Jo,Aa (x)Ap (Y)I/J o)'

We can state the following theorem:

THEOREM 1.4.1 Under the hypotheses of thefirst section, we have:

(1.4.1) Gap (x) = - YYap,1 +(x) + oaopH(x)

where H(x) is a Lorentz invariant tempered distribution. Proof Let us write:

(1.4.2) G,p(x) = -YYap,1+(x)+hap (x).

The distribution h,p(x) is such that:

0J.ha)x) - 0l'h'A(x) = 0

In other words, there exists ha(x) such that:

(1.4.3) haP (x) = oph,(x)

From the covariance of the vector-potential field, we get:

(1.4.4) A~'ha' (Ax) - h,(x) = c,(A -1)

where the ca(A) are COO-function on SL(2,q. It results from (1.4.4) that they are the components of an indefinitely differentiable 1-cocycle with respect to the D(~,!) representation: therefore it is trivial and we can find constants c, such that:

ca (A - 1) = A~' Ca ' - c,

After substitution in (1.4.2), we find that h, (x) - ca is a vectorial invariant distribution; therefore it is a gradient and there exists a Lorentz invariant distribution H(x) such that:

ha(x) = ca + °aH(x).

But, from (I.4.3) and (1.4.2), this implies (1.4.1).

COVARIANT QUANTIZATIONS OF THE MAXWELL FIELD 211

Remark. In the proof of Theorem 1.3.1, we have used only the covariance of the vector-potential field and a general result on the triviality of some l-cocycle. In other words, if we started with the form (1.2.6) for the two­points function of the electromagnetic field, we should be able to prove with the same methods that:

G. (x)=(g.D -g D.)G(x). u~ u ~ .~ A

Then, proceeding as above, we get readily:

G~fJ(x)= -gafJG(x)+DaDfJH(x).

This appears to be the general expression of the two-points function for the vector potential. Here G(x) is the boundary value of a holomorphic function of Z2 when Z = x - i'1, '1 E V +, such that the associated kernel Ga{u)x - y) is the kernel of a non-negative sesquilinear Hermitian form.

PAR TIl: THE LINEAR CONFORMAL INVARIANT GA UGES

II. I. GENERALITIES

In all this part, we add to the hypotheses of Section 1.1, the following ones:

11.1.1. Gauge Condition

We select among the various covariant quantizations described above those which verify the following equation:

(II.l.l) DD~A~(x)=O.

This equation characterizes the linear conformal invariant gauges (cf. Ref. 4) and it appears particularly suitable for quantization since it allows to speak unambiguously of the positive or negative frequency parts of DI' AI'(x) (see Ref. 5)

11.1.2. Quasi1ree Field Condition

We assume now that the n-points truncated functions of the quantum field AI'(x) are identically zero for n > 2.

Moreover, we require that the finite linear complex combinations of

212

the vectors:

AUj ) '" AU,.)I/Jo,

from a total set in .Ye.

G. RIDEAU

n=O,I,2, ...

From the first condition, we deduce readily the following equation:

(fl.l.2) 0/i02H(x)=O.

Its general Lorentz invariant solution is given by:

(fl.1.3) H(x)=d+G+(x)+d_G-(x)+c 1 +C2X 2+C3 (X 2)2

+M+L'\+(x)+M_L'\-(x)

where G± (x) are those particular solutions of the equations:

D G± (x) = L'\± (x)

which are defined by:

G±(x) = f d4ke+ikXe(ko)b'(k2).

If we remark that:

e -, 2 1 0 2 kp (ko)6 (k )=2okpe(ko)b(k )

and if we take into account the following identity for any F(k)E.9"([R4):

of 1 ~I ~ k. of 1 ~I ~ D 1 ~ ~ oki(± k,±k)=rtroko(± k,±k)±ok;F(± kl,±k)

we get after a suitable integration by parts:

(1I.1.4) f d4xd4yr(x)f'P(y)oaopG± (x - y)

=~fd~k[_~ O(± k)O'(± k)] 2 Ikl oko ko ko=lkl

with O(k) = kaJa(k),O'(k) = kaJ~(k) where Ja(k) and J~(k) are the Fourier transforms of the test-functions in .9"([R4),j~(X) and f~(x). Obviously, the hermicity of < , > implies that the constants in (11.1.3) are real constants.

11.2. INDECOMPOSABLE REPRESENTATIONS OF THE POINCARE

GROUP AND EXPLICIT CONSTR UCTIONS

In this section, we introduce some non-unitary representations of the Poincare group f!J which belong to the rather large family of in de compos-

COVARIANT QUANTIZATIONS OF THE MAXWELL FIELD 213

able representations. As we shall see, they playa basic role in the realization of a quantum theoretical model which fits the two-points function corres­ponding to the value (11.1.3) of H(x). We delay up to the next section the detailed investigation of their mathematical properties. At the present time, we give only the basic definitions and the ensuing applications.

11.2.1. Let H + be the Hilbert space of four-components functions <PI' (k), .u = 0,1,2,3, such that:

(II.2.l) f w( tl<PJi(kW + Iw(kW + * I:; (k)12) < 00

where w(k) = k"<p,,(k),kO = Ikl. For real d+, we define in H + the following indecomposable representa­

tion, Ud + (a,A), of the group f!J:

(11.2.2) Ud + (a,A)<p,,(k) = eia-k(A~<p)A -1 k)

-td+m( A~~~y\-lk)+iaoW(A-lk))). It can be shown that the representations indexed by different value of d +

are inequivalent representations and there exists for each d+ a one­parameter family of invariant non-degenera te Hermitian forms given by:

(11.2.3) B~+ (<p,<p') = - f~;~(<P)k)<P'I1(k) + M +w(k)w'(k))

-ld f d3kW(k)W'(k) Z + If I Ikl z

where M + is a real constant. The proofs are given in Ref. 6 for functions <p,,(k) equal to zero in

a neighbourhood of the origin, but they can be extended easily to H +

by a density argument.

1I.2~ Let H _ be the Hilbert space of two-components functions WI (k), W z (k), such that:

d k k - Z 1 + k - Z OWz -f Z ( I-Iz I-IZ 31 IZ) (11.2.4) If I 1 + Iklzlwl (k)1 + Ikl z IWz(k)1 + i~1 8ki (k) < 00.

We define in H __ the following indecomposable representation U _ (a,A)

214 G. RIDEAU

of the group :!J>: ~ . ~

U _ (a,A)w 1 (k) = e-Wk(w1 (A -I k)

I (AOOW2 ~ . A~k ») +lkf pok/(A k)-zaOw2 ( )

- . ---> (II.2.5) U _ (a,A)w/k) = e-Wkwp (A -I k).

It can be shown that there exists a one-parameter family of invariant non-degenerate Hermitian forms given by:

(11.2.6) m- , fd 3k --,- -~ W 2(k)-;;;;;t)) B_ (W,w)= W(wl(k)wz(F)+Wz(k)wl(k)= Iklz

fd 3k - -- m_ . Ikl Wz (k)w~ (k)

where m is a real constant.

11.2.3 Let us denote by V(a,A) the representation of [1jJ defined in the test-function space by:

(H.2.7) V(a,A)flJ2) = A;fv(A -lex - I».

Then we can state the following propositions, the proofs of which will be omitted owing to their purely technical character:

PROPOSITION 112.1. There exists an intertwining operator lId between V(a, A) and Ud + (a,A).lf 11' (k), Jl = 0,1,2,3, are the Fourier transfo;ms of the components of the vector test1unction fl' (x), we have:

~ - I ~I ~ I k an ~ -(11.2.8) (IId+f)I'(k) = f/ k ,k) - ld+ m ako (Ikl,k)

where n(k) = kl'll'(k),kE~.4

PROPOSITION 112.2 There exists an intertwining operator 11_ between V(a,A) and U _ (a,A). With the same notations as in the previous proposition, we have:

(II.2.9)

COVARIANT QUANTIZATIONS OF THE MAXWELL FIELD 215

Let us consider now the Hilbertian sum H = H + EB H _ the elements of which will be denoted by F = (q>,w). In H, is defined a representation of &, U(a,A), direct sum of Ud+ (a,A) and U _ (a,A). The operator II = ll+ EBll_ is obviously an intertwining operator between V(a, A) and U(a,A). Among the set of invariant non-degenerate Hermitian forms, let us choose the following:

(1I.2.10) B(F,Fl) = B'f/ (q>,q>') + d_ B'!!.- (w,w')

where M + ,d+ ,d_ are as in (11.1.3) and m_ = M _/d_. Suppose, for the moment c1 = c2 = c3 = 0 in (11.1.3). Then it is readily

seen that:

(11.2.11) B(llj,llj') = f Gap (x - y)f"(x)f'P (y)d4 xd4y

for test functionsj andf'. Let now :Ye be the symmetrical Fock space built on H. Let Olt(a,A)

be the representation of & induced in :Ye by U(a,A) and let fJ6(cjJ,cjJ') be the invariant non-degenerate Hermitian form induced by B(F,F'). We introduce the creation operator a+ (F) in the usual way and we obtain the annihilation operator a(F) by adjunction with respect to fJ6(cjJ,cjJ'):

(11.2.12) fJ6(a(F)cjJ,cjJ') = fJ6(cjJ,a+ (F)cjJ').

We define now the vector potential field by:

(11.2.13) A(f) = a+(llf) + a(llf)

PROPOSITION 1I.2.3. The field defined by (11.2.13) is a covariant field. Proof First we have, since II is an intertwining operator:

A(V(a,A)f) = a+(U(a,A)llf) + a(U(a,A)llf).

But, by construction:

Olt(a,A)a+ (F)Olt(a,A)-l = a+ (U(a,A)F).

Therefore, from (1I.2.12):

Olt(a,A)a(F)Olt(a,A)-l = a(U(a,A)F).

Consequently:

A( V(a,A)f) = Olt(a,A)A(f)Olt(a,A)-l.

This is just the covariance property. Moreover, if we choose on :Ye the sesquilinear form < , > to be identical

216 G. RIDEAU

with PA(CP,CP'), then it results from (11.2.11) that A (f) has just the wanted two-points function. In other words, at least for c1 = c2 = c3 = 0, we have given an explicit construction for all cases a priori possible.

Assume now the constants ci different from zero. Our description can be easily adapted: it is sufficient to start with a Hilbert space H' which is direct sum of H and of a finite-dimensional Hilbert space. The representa­tion U(a,A) is changed in a representation U'(a,A) which is the direct sum of U(a,A) and of a reducible but indecomposable finite dimensional representation of f!jJ. As a matter of fact, there exists always a vector of H' invariant with respect to the action of the translation subgroup of f!jJ.

But this is inconsistent with the unicity property which we have explicitly required from the vacuum state. Therefore, we have necessarily c1 = c2 = c3 = ° and the construction given above covers finally all cases.

As usual in quantum field theory we introduce the annihilation (creation) operators at the point k by the formal writing

a(F) = f d3 k(a"(k)<fJI'(k) + bi(k)wi(k)),

(II.2.14) f a+(F) = d3 k(a + I'(k) <fJ1' (k) + b+i(k)w;Ck)).

Then, we deduce easily the following commutation relations:

[al'(k),a+ V(1)] = - mJ(k - [)(gI'V + td+ ~~~: + M + kl'kV}

1 ~ +2 ~ d_ ~ ~ [b (k),b (k]=mD(k-l),

(11.2.15) [b2 (k), b + 1 (k)] = fkl b(k -1),

2 ~ +2 ~ 1 ~ ~ I ) [b (k),b (k)] = -fklb(k-l)(M_+d_lkI2 '

all other commutators being equal to zero. As to the supplementary condi­tions which define the subspace ,YP' of physical states, it can now be written in the following form:

(11.2.16) k'''a.f(k)cp = 0, b l (k)cp = 0.

Finally, we summarize this discussion in the following statement:

COVARIANT QUANTIZATIONS OF THE MAXWELL FIELD 217

THEOREM 11.2.1. The most general two-points function of a vector poten­tial field All (x) satisfying to the gauge condition (11.1.1) and to the hypotheses of Part I is given by:

<l/Jo,A)x)A)y}ljJo> = (- gllv + M + o/3.)!!. + (x - y) + d+ 0IlOVG+(X - y) + M _ Oil 0v !!. - (x - y) + d_ollovG-(x - y),

where the various constants are real and where the distribution !!. ± (x) and G± (x) are given by: .

!!. ± (x) = f d4ke+ikx(}(ko)(j(k2),

G±(x)= f d4ke+ ikX(}(ko)C5'(k2).

An explicit construction of All (x) as a quasi-free field corresponding to the two-points function above can be given for each set of real constants d+,d_,M +,M _. It is realized in a Fock space X where the six creation (a+ll(k),b+i(k» and the six annihilation operators (all(k),bi(k» verify the relations (II .2.15). The vector potential field itself is defined by:

AU) = a+(nf) + a(Of).

Here, we use the n?tation (1I.2.14) and, for the test function fll (x) with Fourier transform r(k); the various components of Of are given by:

- ~ I-I - ! k on -I -(Of)Il(k) = f) k ,k) - zd + rtI oko (k ,k)

- I on 1-1-(Of)! (k) = -m oko (- k ,k)

(Of)2(k) = n( -Ikl, - k)

where n(k) = kll!ll(k). Finally, the physical subspace X' is the set of elements in X which

satisfy the supplementary condition (II.2.16). The mean values of the physical quantities have to be evaluated by means of a sesquilinear Poin­care invariant hermitian form !!j(cjJ,cjJ') given in the one-particle

218 G. RIDEAU

space by:

, _ fd 3k( " - -,- -~ I W(k)W'(k)) B(F,F)- lkT -cP (k)cp,,(k)+M+w(k)w(k)-Zd+ Ikl2

+ d _ f m «(I); (k)w2 (k) + W z (k)w'l (k)

_ w2 (k);;;W _ M - (k-)--'-(k)) Ikl2 d_ W z W 2 .

It remains now to deduce the field equations and to precise the actual meaning of the construction above. This will be done in the last section, after we have afforded a deeper knowledge of the representations Vd+ (a,A) and V _ (a,A).

11.3. THE MATHEMATICAL STRUCTURES OF Vd+(a,A)AND V_(a,A)

The representations Vd+ (a,A) and V _ (a,A) belong to the set of represent a­tions of £!J> which are extensions of unitary representations of £!J>, in the present case, of mass zero unitary representations.

Let us recall first some definitions. Let G be a group and let V I' V z be continuous representations of G in topological spaces EI'E2. Then, in the topological sum E = E I + E2 we call extension of V z by V I a representation V given by:

(I1.3.l) V(g)= 1~I(g)

where Z(g) is a continuous mapping from E2 to E 1 , verifying the following equation:

(I1.3.2) Z(glg2)=Z(gI)+ V 1(gl)Z(g2)U;I(gJ

Z (g) is called a l-cocyde of extension and it is trivial if there exists ZE2(E2,E1 ) such that;

Z(g)=Z- V 1(g)ZV2(g)-I.

We are not actually interested by these l-cocydes since the representa­tion they define is equivalent to the direct sum V I + V 2' Similarly, when two l-cocydes Z I (g), Zz (g) differ by a triviall-cocyde, it is easily shown that they define equivalent representations. Therefore, we are interested with the solutions of (II.3.l) up to trivial ones. But it may appear that two

COVARIANT QUANTIZATIONS OF THE MAXWELL FIELD 219

inequivalent I-cocycles correspond in fact to equivalent representations. In particular, it is the case when the classes of I-cocycles form a one­dimensional vectorial space.

PROPOSITION II.3.1. The representation U _ (a,A) is a non-trivial extension by itself of a representation with mass zero, helicity zero, and negative energy. Proof U _ (a,A) has clearly the structure (II.3'!). The representations

w4)-->e-iakwi(A-lk), ko=lkl, i=1,2,

are well unitary representations of f!jJ with mass zero, helicity zero, and negative energy. As to the non-triviality of the extension, it proceeds from general results proved in Ref. 7.

The structure of Ud + (a,A) is more involved. For the sake of simplicity, we introduce on C+ the coordinates tEIR,(d:: defined by:

[I-I k2 + ik[ (II.3.3) t= -zlog2 k, (= -lkl-k

3

Furthermore, at each point k on the cone C +, we associate a set of four independent vectors: the vector k itself, the vector with components ko = ±Ikl,ki = - kj4lkl 2 , and the complex conjugate vectors X!,(k),X!,(k) given by:

Xo(k) - X 3 (k) = 1 + IW'

I Xz(k)+iX1(k)= I +1(1 2 '

Let us set down (kEC+):

( X O(k)+X 3 (k)= -1 +IW'

X Z (k) - iX I (k) = - 1 :~W·

z(A k) = _ e2t(~)(a - y() + (13( + (5)(13 - (5() , laJ + Yl2 + 113( + (5l z '

( 13(+(5)-2 w(A,k) = 113( + (51 '

(II.3.4)

where the matrix I ~ ~ IE SL(2, q is associated to the element A of the

Lorentz group.

220 G.RIDEAU

Then, by direct calculations, we can prove the following useful relations: ~ ~

k/1 - A;' (A -I k)v = klliz(A,k)j2 + z(A,k)XIl(k) + z(A,k)XIl(k),

(JI.3.5) XIl(k) - kllz(A,k) = w(A,k)A;'Xv(A -Ik).

Let us write now:

cp)k) = bl (k)kll + b2 (k)XIl (k) + b3 (k)XIl (k) + b4 (k)k ll .

From (II.2.2) and (II.3.5), we deduce readily the following law of transformation of the bJk), i = 1,2,3,4:

(II.3.6)

On this reduced form the structure of U d+ (a,A) becomes manifest. Indeed it appears as the extension of four irreducible unitary mass zero representations of &; two representations have their helicity equal to zero, and correspond to the components b l (k) and b4 (k); the two last representations have their helicity equal to ± I and correspond to the components b3 (k) and b2 (k) respectively. We have studied elsewhere (see Ref. 8) these extensions and we have shown that their equivalence classes form a one-parameter family; as a matter of fact the representation (11.3.6) can be chosen as a representative in each class. It is worth noticing that the indexing parameter arises actually from the existence of a one-parameter family of non trivial l-cocycles of extension by itself of the representation of '§ with mass zero and helicity zero. This gives the mathematical meaning of the constant d + in (11.2.2).

II. 4. FIELD EQUA TrONS

From (1.3.4), we deduce:

<l/Io,A)x) D FA/1 (y)l/Io> = 0

COVARIANT QUANTIZATIONS OF THE MAXWELL FIELD 221

Therefore, by the completeness hypothesis of Section II.1, we must have:

DFA/l(x)=O

But this is equivalent to:

DA/l(x)=o/lB(x)

where B(x) is a scalar field. To determine it, we apply the definition (II.2.13) with a test function of the form Dg/l(x), g/l(X)EY'(~4). Let us set down:

(11.4.1)' B+ (x) = ~ f d3k(k/la;(k)eikX - kl"a)k)e- ikX ),

ko=lkl

(1I.4.2) B_(X)=~ f d3k(b7(k)e-ikX-bl(k)eikX).

ko = Ikl

Then we get:

(II.4.3) 0 A/l(x) = d+ 0 /lB + (x) - 0 /lB- (x).

Similarly if we apply (1I.2.13) with a test function of the form ik j(k), ~ /l fEY' ~4), we obtain:

(II.4.4) 3/lA/l(x)=(d+ -l)B+(x)-B_(x).

From this, we deduce the quantum Maxwell field equation in the following form:

(II.4.S) o/lF/lv(x) = 0vB+ (x)

Thus the field B __ (x) does not appear in the operatorial equation which must replace the classical Maxwell field equation. In this respect the field B _ (x) appears as a spurious field. It can be easily eliminated if we add to our set of postulates the following one:

INDECOMPOSABILITY CONDITION. The representation of & restricted to the one-particle space must be an indecomposable representation.

With this postulate, the representation U _ (a,A) disappears, we must have d _ = M _ = 0 in (1I.1.3) and we find a two-points function character­istic of the generalized Lorentz gauges (see Ref. 6 for a detailed study) Therefore we have proved the following theorem:

222 G. RIDEAU

THEOREM IIA.l. Under the hypotheses of Part I and of Section II.l and with the indecomposability condition, the linear conformal covariant gauges are identical with the generalized Lorentz gauges. In this case, the spectral condition is satisfied by the vector potential field itself.

But in the general case, there is no reason to drop out the pathological field B _ (x) so that the two-points function cannot be the boundary value of some analytical function. Such a strange situation appears obviously as a direct consequence of the non-physical nature of the field All (x) implied by the gauge invariance of the theory. Nevertheless, it is worth noticing that in any case the vector potential field is a local field.

ApPENDIX: COMOHOLOGICAL RESULTS

Let us consider in [j/' (1R4) the representation V(A) of SL(2, C) defined by:

T(x) --+ T(A -I x), T(X)E[j/' (1R4), AESL(2, C).

An indefinitely differentiable l-cocyc1e with respect to V is an indefinitely differentiable mapping H(x,A) from SL(2,C) to [j/'(1R4) such that:

(A.I) H(x,A 1 A2 ) - H(x,A1 ) - H(A; 1 x,A2 ) = 0, AI'A2 ESL(2,C).

We prove here the following theorem:

THEOREM A.I. The general solution of (A .1) is given by

(A.2) H(x,A)=H(x)-H(A-1x),

where H(X)E9"'(1R4)

AESL(2,C)

In other words, each l-cocyc1e with respect to V is a coboundary.

Proof We can always assume that (A.2) is true when A is restricted to the compact group SU (2). Then, following Ref. 9, for each X Esl(2, C), the Lie algebra of SL(2, C), we introduce:

d (A.3) h (x, X) = dtH(x,exptX)lt=o.

Let d V denote the representation of sl (2, C) derived from V. The linear functional h (x, X) verifies:

(A.4) h(x, [X, Y]) = d V(X)h(x, Y) - d V(Y)h(x,X)

COVARIANT QUANTIZATIONS OF THE MAXWELL FIELD 223

with the supplementary condition:

(A.5) h(x,X) = dV(X)h(x), XEsu(2),

for some h(x)=Y"((R4). The theorem will be proved if we can prove that h(x,X) has the form (A.5) for all X E sf (2, IC). It is obviously sufficient to prove it for a basis of sl (2, IC). We choose this basis as in Ref. 10. The commutation relations of sf (2, IC) and (A.4) give:

(A.6) h(x,F±)= ±dV(H±)h(x,F3 )±dV(F3)dV(H±)h(x).

Let us set down:

(A.7) t(x) = h (x,F 3) - d V(F 3)h (x)

The distribution t (x) verifies the following set of equations:

d V(H +)2 t(x) = 0,

(A.8) d V(H _)2 t(x) = 0, dV(H3)t(x) = 0,

(dV(H +)d V(H _) - 2)t(x) = 0,

and:

(A.9) dV(F+) + dV(F3)dV(H+)t(x) = 0, dV(F _) - d V(F3)dV(H _ )t(x) = O.

The first set means that:

(A.lO) t6(x) = t(x), I

t1 (x) =J_dV(H + )t(x), ± 1 2 -

form a canonical basis of the three dimensional representation of SU(2) Therefore for CPEY'((R4) and CPR (x) = cp(R;I),RESO(3) we have:

(t6, cp) = D6,1 (R)(t~' CPR) + Dbo(R)(t6,CPR) + D6, - 1 (R)(t ~ 1 ' CPR)'

(t; I'CP)= D; 1,I(R)(t~,CPR) + D; l,o(R)(t6,CPR)

+ DL.-l (R)(t~ l' CPR)'

After integration on SO(3), we get:

(A.1I) (t6,CP) = (S,CP6), (tL,cp) = (S,cpL)

where S(xo,r) is a tempered distribution and CP6 (xo,r), cP; 1 (xo,r) are

224 G. RIDEAU

fastly decreasing indefinitely differentiable functions, uneven with respect to r, equal to the components of (fJ on the usual spherical harmonics Y~ and Y!l'

Now writing the set of equation (A.9) in terms of distributions (A.lO), we get;

With (A.II) we have, after some lengthy calculations:

where (fJi 1 p·o,r) are the components of (fJ on the spherical harmonics Y; l' Therefore, we can set down:

(fJi(xo,r) + (fJ~ 1 (xo,r) = rljJ(xo,r)

where IjJ (xo,r) is some quickly decreasing function uneven with respect to r. Then, if we discard quantities which do not contribute to (A.lI), we derive from (A.l2) the following equation:

(A.l3) (3xo -r(xo: r +ra~J)s=o To solve it, we write:

(A.14) S=r3 U and we deduce from (A.l3) the following equation:

(A.IS) (Xo :r + r a~J U = to(xo)<5(r) + t1 (xoW(r) + tz(xoW'(r)

where to(xo)' t 1 (xo)' t z (xo) are some tempered distributions. But we are interested only by the solutions of (A.l4) which are even distributions with respect to r. Therefore we can assume to (xo) = t2 (xo) = 0 in (A.lS) and the general solution of the resulting equation can be written:

(A.l6) U = u(xo)<5(r) + U 0

where U 0 verifies the following equation:

COVARIANT QUANTIZATIONS OF THE MAXWELL FIELD 225

Now the first term of the right-hand side of(A.16) does not contribute to S, so that finally we have:

S=r3 U o

where U 0 is a solution of (A.17) which is an even distribution with respect to r.

We define now a distribution ZE.'I"([R4) by:

where <Pg is the component of <P on Yg. Suppose now that Z is a Lorentz invariant distribution.

Then our theorem is proved: indeed, since (x3 <p)g = r<P6, (A. I I) takes the form:

I.e.

t(x) = X3Z

But, from dV(F3)Z = 0 and [dV(F3),xoJ = X3 we get easily:

t(x) = dV(F3)xoZ

where the distribution Xo Z IS invariant by rotation. Then, we get from (A.7):

h (x,F 3) = d V(F 3)(h (x) + xoZ).

Similarly, (A.6) and the commutation relations of sl(2, q imply:

h(x,F ±) = d V(F ± )(h(x), + xoZ).

Finally, the rotational invariance of Xo Z allows (A.5) to be written in the form:

h (x, X) = dV(X)(h(x) + xoZ), XEsu(2)

Therefore it remains to prove that Z is a Lorentz invariant distribution. Z is obviously invariant by rotation. Thus we have only to prove that d V(F 3)Z = O. But, we have:

226 G. RIDEAU

so that:

. 12( 2 (0 a xo) 1) =1~3 r V o' XOor +raxo +2~ <Po

. 12(( Xo (0 a )) 2 1) =1~3 2~- XOor+ro-xo r Vo,<Po =0

since, by (A.17), the first term in the bracket is equal to zero.

Vniversite Paris VII

REFERENCES

[1] Strocchi, F., and Wightman, A. S., J. Math. Phys. 15,2198 (1974). [2] Rideau, G., in H. Bacry and A. Grossmann (eds.), Proc. 3rd Int. Coil. on Group Theoreti-

cal Methods in Physics, Marseille, 1974, pp. 210-216. [3] Hepp, K., Helv. Phys. Acta 36,355 (1963). [4] Bayen, F., and Flato, M., J. Math. Phys. 17, 112, (1976). [5] Tomczak, S. P., and Haller, K., Nuovo Cimento 8B, 1 (1972). [6] Rideau, G., J. Math. Phys. 19, 1627 (1978). [7] Rideau, G., 'Extension of representations of Poincare group', to appear in Rep. Math.

Phys. [8] Rideau, G., Letters Math. Phys. 2, 529 (1978). [9] Pinczon, G., and Simon, J., Letters Math. Phys. 1, 83 (1975).

PO] Naimark, M. A., Representations lineaires du groupe de Lorentz, Dunod, Paris.