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A C∗-algebra formulation of gauge transformations of the second kind for the electromagnetic field
Vol. 13 (1978)
REPORTS
oiv ~~ATHEMATI~AL
No. 3
PHYSICS
A C*-ALGEBRA FORMULATION OF GAUGE TRANSFORMATIONS OF THE SECOND KfND FOR THE ELECTROMAGNETIC FIELD
A. L. CAREY, J. M. GAFFNEY*, and C. A. HURST Department
of Mathematical
Physics, University
of Adelaide,
Adelaide,
S. Australia
(Received May 2, 1977) A definition of gauge transformations of the second kind for the quantized free electromagnetic field is proposed within a Weyl algebra framework. This paper is complementary to recent work of Strocchi and Wightman on gauge transformations for the electromagnetic field using the indefinite metric formalism. The algebraic approach accounts for the general features of the heuristic notion of gauge transformation in a natural way as well as clarifying some of the constructions of the indefinite metric approach.
1. Introduction
of this paper is to give an account of the notion of gauge transformation of the second kind for the electromagnetic field using a C*-algebra framework. The fact that there is a rigorous version of the Fermi method of quantizing the electromagnetic field which uses a Weyl system approach was the subject of two previous papers by the authors [I], [2]. The starting point of the present investigation was the paper of Strocchi and Wightman [3] which proposed a definition of gauge transformation within the indefinite metric formalism. The contents of our paper are largely complementary to the work of Strocchi and Wightman in the Sense that we clarify some of the constructions introduced in [3] by giving them a natural interpretation in the algebraic picture. We are also able to answer in part some of the questions posed in [3] about the algebraic side of the GuptaBleuler quantization. We believe that the various approaches to the quantized electromagnetic field (including the recent one of Bongaarts using the Borchers algebra [4]) are complementary rather than mutually exclusive. For this reason we have also been concerned here with understanding the connection between the Gupta-Bleuler and P-algebra approaches. This paper depends in part on our previous work and we refer the reader to [l] for a more leisurely introduction. The contents of the paper are organized as follows. In Sections 2 and 3 we define the field algebras and the algebra of observables for the electromagnetic field. Our approach here has been influenced by the ideas of Doplicher, Haag The
object
* Present address: Department U.S.A.
of Mathematics,
Michigan State University,
[4191
East Lansing, Michigan,
A. L. CAREY, J. M. GAFFNEY,
420
and C. A. HURST
and Roberts [5]. In Section 4 we give a general definition of gauge transformation of the second kind and illustrate it with the example of what we call a linear gauge transformation. In Section 5 we see that linear gauge transformations account for most of the commonly used gauges for the electromagnetic field with the exception of Landau gauge which remains an unsolved problem. We discuss briefly in Section 6 examples of gauge transformations which are automorphisms of the field algebra, namely c-number gauge transformations, Because we are unable to settle the question of what the most general gauge transformation looks like (in particular, Landau gauge suggests that linear gauge transformations are not sufficiently general), we have not investigated the question of when a gauge transformation is an automorphism of the field algebra. One might also ask to what extent the present formalism is peculiar to the electromagnetic field. We conclude with a discussion of the relationship between Gupta-Bleuler and the algebraic approach. 2. Local algebras Let X,+ = (k E R4/ kc, > 0, kZ = 0} and consider the real Hilbert space M defined from functions q: X,+ --+ C4 satisfying
(2.1) where k = lkj. The inner product on M is given byR= + j c (~,(k)~~(k)+~~(k)~;(k))d3k/2k. x0’ fi M becomes a complex Hilbert space if we define a complex structure JF by (JFq)(k) = - ig@)
(2.2)
(2.3)
where g = diag(- 1, - 1, - I, 1) is the metric tensor. The complex inner product on M is then (2.4) (~3 v’> = S vo(k)vb(k)+k vj(k)VJ(k)d3k/2k j=l X2 which has (2.2) as its real part. Unless we specify otherwise the symbol M will be interpreted as meaning a real Hilbert space. Each sufficiently smooth 9 satisfying (2.1) defines a real solution 6 of the wave equation lJ@ = 0 by taking the Fourier transform. Explicitly : i, = -&+ where k. x = kx,-k.
x.
S [ev(-Xi?
ik * 4plJk) + exp(ik *x)q+,(k)ld3k/2k
(2.5)
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Let 9 be the space of Cm functions of compact support in R3 taking values in R4. Each pair of functions cf, g) E 9 x 9 defines a function on X$ by
(2.6) Then q~satisfies (2.1) so that (f, g) defines a function $ with
ql@=
0 and it is not difficult
to see that $(r, 0) = f(x) and &(x, 0) = g(r). The subspace M,, of M defined by letting (f, g) range over 9 x 9 is dense in M. The PoincarC group acts on M,, by 6 -+ G’ where G’(x) = _,l@(/l-1(x-a))
(2.7)
where (a, A) (consisting of a translation a and Lorentz transformation il) is a typical PoincarC group element. It is well known that MO0 is invariant under the representation (by bounded, not unitary, operators) on A4 of the Poincare group defined via (2.7). We define two closed Poincart invariant subspaces of M as follows. Let N consist of those elements of M satisfying @ZJ&) = 0,
(k, k,) E x,+ f
and T consist of those elements of M of the form %(JV = k,X(k)Y
(k, k,) E x,+.
Let ,9: M,, x MO + R be a (not necessarily non-degenerate) symplectic form which is defined on a dense subspace iVfOof M. We restrict M,_, (and hence j3) by demanding that it be invariant under the (real) orthogonal projections PN and PT onto N and T respectively. Define the *-algebra IIo(MO) as the complex linear span of the functions S, on M, 9 E M,,, given by 0 if v # v’, 4&J’) = 1 if q3= pl’ with multiplication
and with the *-operation
By completing @(M,,)
in the norm:
we obtain a Banach algebra Of (MO). Let Pfi denote the non-degenerate strongly continuous *-representations 0 of &CM,) such that 2 + a(&)
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A. L. CAREY, J. M. GAFFNEX,
and C. A. HURST
is continuous in A E R for all 9 E M,,. Define the C*-algebra d$Vf,) tion of df( MO) in the norm
ll4l = ;s$ilc(‘4ll
to be the comple-
(G 11‘411)
(cf. Manuceau [6]). Although M,, is not invariant under PN and PT we can still define &?(MO,,) whenever MO > M,,. This is of interest because dz(M,,) has a local structure in the HaagKastler sense [7]. Namely, for each bounded open set Q c R we consider the algebra O@(L?)consisting of the linear span of those 6, such that for some spacelike hyperplane P the initial data for q on P have support in PnD. Then clearly d@(&f,,) = Ud@(Q) where D the union is over all bounded open sets Q c R4. Hence &!(M,,) = lJd,(Q), where n LIB,(~) denotes the closure of O@(Q) in /l~(M,,). Neither dc(PNMoO) nor il,(P,Mo,) (defined as the C?-subalgebras of @(MO) generated and (S,i v E PTh/loo) respectively) have a local structure. However by @,I cpE P,M,,) consider the closed subspace S of M defined from functions satisfying k *cp(k) = 0
((k, k,) E X,+)
and
v. 3 0.
We can write all elements of S in the form $44 = kx $(k) for some C-valued function $ or Xg . Hence a dense subspace So0 of S is spanned by those functions on Moo whose initial data cf, g) (at time zero) have the form “Ll = go = 0, f(x) = v xf ‘9
g(4 = v x g’b>
where f’ and g’ have compact support. We may form in the obvious way @(So,) and give this algebra a local structure by associating with each open bounded B c R4 those 8, such that 9 e So and 6, EL’I~(@. 3. Field algebras and the algebra of observables The algebras .lt(Mo) which we defined in the previous section are a little too genera1 to be regarded as field algebras for the electromagnetic field. We have already imposed (a) 8: MO x MO -+ R is a real bilinear symplectic form on MO (which is dense in M) and PN and PT. leave MO invariant. We now demand that (b) /? is invariant under the action of translations (equation (2.7)) and satisfies @(&MO, &MO) = 0. The origin of this constraint may be seen from [I], [2]. There we considered the symplectic form
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which, being the imaginary part of (2.4), is defined for all (p, @ EM. The algebra A:(M) we called the C*-algebra of the CCR for the electromagnetic vector potential. However B is just one choice among many and A:(M) just one of a large class of possible field algebras. The reason for this ambiguity in the choice of field algebras may be seen as follows. From [I] we know that Nis the Hilbert space direct sum of S and T and that the map 7~: 6, -+ d,s where v EN and ys is its S-component, extends to a homomorphism fiB of AZ(N) onto A:(S). The quotient algebra Af(N)/kernB we call the algebra of physical observables. We have the following isomorphisms 111: Af(N)/kerz,
z Af(N/T)
g At(S),
where Af(N/T) is defined using the symplectic form on N/T induced by B. These isomorphisms demonstrate that Af(N)/kernB is defined in terms of the transverse components of the one particle photon states and hence excludes the unphysical longitudinal and scalar components. Now A,B(N)/kern, has the important property of being Poincare invariant (under the automorphisms of A:(M) induced by the representation (2.7) of the PoincarC group on M). Thus it is only the algebraic structure of this quotient algebra which is of importance. Now a consequence of constraint (b) is that B(V, V’) = B(% >94)
(3.2)
where qs, 9; are the S-components of v, q~’EN respectively. Since AE(N)/kern, and are isomorphic it is sufficient therefore (in order to preserve the algebraic structure of Af(N)/kern,) to require
A:(S)
(c) B(vs, ~7:) = B(~s, q&) for all qk, q~,$E bogs. (Note that M,nS = (P,-P,)M, is dense in S.) DEFINITION. If the symplectic form 8: M,, x MO -+ R satisfies conditions (c) above we will call At(M,) a gauge algebra (for the electromagnetic field).
It is clear that A’(P,M,,)
= As(P,Mo) z:
f& + s,,,
(a), (b),
so that the map y E PlvMO
extends to a homomorphism of Ap(PNMO) onto A”(P,M,) (P, = PN-- PT). We note that as b coincides with B on PsMO and is therefore non-degenerate on Ps M,, , we have AS,(PsMo) = A,B(P, M,,) (there being just one C* norm on AB(Ps MO) (cf. [13]). THEOREM 3.1.
A:(Ps&).
z
extends
to a continuous homomorphism
tip of At(PNMO) onto
A. L. CAREY,
424 Proof:
As n is the identity
J. M. GAFFNEY,
on JB(PsMO)
and C. A. HURST
it is clear that the only difficulty
that ?c restricted to &(PzM,) is continuous. We introduce defined as the completion of df(Pz MO) in the norm II& where the supremum
= suPlle(~)Il,
e
is over all continuous
the
V-algebra
is to prove -__ 3(PTMo)
A. E~Im
representations
e of d, (PzMJ.
Then @(Pr MO)
E n(P,M,)/Z where Z is some ideal of d(P,M,) (cf. 1131). Let e be a representation of ds,(P, MO) such that I + @(a,& is strongly continuous from the real line into the unitary operators on the Hilbert space He of Q. Since @(A!(T)) 1s an Abelian c-algebra we may write H4 as a direct integral of Hilbert spaces over the spectrum of Q(A:(T)). If x is some element of the spectrum which occurs in this direct integral decomposition then 2 is a character of d!(T). Since this direct integral decomposition may also be regarded as a deEmposition of the representation ii -+ e(&,) (pu E P,M,) of the real line, we have ii + x(6,,) a continuous map into the complex numbers of modulus one. This implies that we may write x(6,) = exp(i&)) (y E PTMo) where i is a linear functional (not necessarily continuous) on PTMo. Now extend 2 to a linear functional on MO by setting it equal to zero on (1 - PTIMo . Define an automorphism of &(M,,)
We note that if x is a character
Furthermore
x(&
Ai 4,))
(cf. [6]) and hence
of dt(P,M,)
= X(X i&i&,J,
of @(PTMo),
by
then so too is % where
vi E P,M,.
Now x extends
to a character
of il (PT M,) (see [13]) and so will define a continuous character of @(P, MO) provided z(I) = 0. So let A. E Z and let (A,) be a sequence in AB(PrMo) converging to A in the topology of At(PTMo). Noting that a,! lS(PTM,,) also extends to an automorphism of _4,(P,Mo)
(see [I 31) we have a,(A,)
But x(~(A.n))
= +,)
and
-+ K~(A). Hence
x (Q(U)
+ ;c (x&0).
ti(A,) -+ n(A)
so that
we have
z’(A) = :(r,(A)).
Finally
we note that as CI~is an automorphism of both A(P,M,) and --Jt~(PTMO),it must leave Z invariant. Thus cc,(A) E Z so that 7 (cc,(A)) = 0. That is $I) = 0 completing the proof. We now have the isomorphisms
verifying our assertion that each gauge algebra determines a subalgebra of the algebra of observables. We will see below that different choices of ,5 correspond to different gauges and that gauge transformations can be defined as special maps between gauge algebras. Before
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taking up this discussion we make some remarks relating to Section 2. @(MOO) satisfies the Haag-Kastler axioms. Using the local structure on A,(&,,) we may give a local structure to a certain subalgebra of Af(l\r)/kerns. However, as would be expected, the axiom of relativistic covariance is violated for this algebra. Thus we have a local field algebra but a non-local algebra of observables (non-local, that is in the Haag-Kastler sense). Clearly therefore one valuable feature of the electromagnetic field example is that it may suggest modifications of the local algebra program [5] which will enable it to handle theories containing zero mass particles (cf. also [9]). 4. Gauge transformations Heuristically, a gauge transformation is a replacement AM(x)--+AL(x) of the vector potential A,,(x) by a new potential Ah(x) such that the observable fields E;,(x) = a, A,(x) - a&,(x) are in some sense unchanged. Now how one makes this precise will depend on the formalism adopted for making sense of A,(x), A.;(x) and FflY(x). Strocchi and Wightman have put forward one possibility [3] within the Gupta-Bleuler framework. A translation of their ideas into the algebraic picture together with some algebraic intuition leads us to the following be two gauge algebras for the electromagnetic DEFINITION. Let &!(M1) and &‘(M,) field. Let a be a homomorphism from @(Ml) to @‘(MJ which takes A$(PNM1) into Ac(P,M,) and which induces a homomorphism z : At(P,M,)/kerti;,
+ Ai’(P,M,)/keiq
(where n’r and 7~~are the extensions of JZ to At(P,M,) and Ac’(P,M,) respectively). If PN(MlnM,) is dense in N and for those 6, E AB,(P,M,)nA~(P,M,) we have z(6,+kerq)
= dq+ker?l,
then we call a a gauge transformation. The rest of *his section will be devoted to a special case of this definition, namely, what we will call linear gauge transformations into the Fermi gauge. We consider the special case where B(v, v’) = B(Zq, ZV’)
(4.1)
for v E MO = domain of Z (here Z is a real linear densely defined operator on M). Because we have demanded that B be translation invariant we impose the condition that Z have the form (4.2) Z@) = ~@Mk) where C(k) is a 4 x 4 matrix for each (k, k) E X,+ . (This condition is sufficient but probably not necessary.) Note that we cannot demand that /I be Lorentz invariant for, on physical grounds, it is only the induced form on N/T which need be invariant. The demand that /I be Lorentz invariant forces /I = B which fact distinguishes A:(M) from the other gauge algebras. We will call A:(M) the Fermi gauge.
A. L. CAREY, J. M. GAFFNEY,
426 Condition
and C. A. HURST
(c) on /? implies that B(Zv, Zpl’) = B(v, 91’)
for
9, P’ E PNMo,
from which it should be clear that there are many possible choices of Z which will give a form /I via (4.1) satisfying conditions (a), (b), (c) (for example Z need only be symplectic on M and leave N and T invariant). Most of these will not define what we would heuristically mean by a gauge transformation. To see what restrictions we need to place on Z we introduce the map: aZ: As(MO) + A:(M) defined by az(C
Ai a,,) = F k oz,, . From (4.1) it follows easily that tlz is an algebra
homomorphism. LEMMA 4.1.
Zf o E PB is a representation of A:(M) representation of Af(M,-,). Proof:
The norm on A$(MJ
then oz = u o uz extends to a
is defined by taking A E A&I4,).
sujlla(A)II9 UEP
Now if A = 7 li S,, E Af(M,) llCz(A)Vll
< C i
an d ZJ is in the representation lililllO(&q,)~lI
space of az we have
= C !&I llwll = llAlll Ilull* I
Thus ax extends to a representation of de(M) and by the properties of c, lies in Pp. Hence a~ extends to a representation of AS,(M,). COROLLARY 4.2.
We can extend aZ to a C*-homomorphism of At(M,)
into A:(M).
Proof Let IT be the Fock representation of A:(M) (i.e. the representation associated with the generating functional Q(V) = exp(-illy,ll’), v E M). Then l/uz(A)II < IIAII for A E A!!(M,). But u is faithful and so Ilaz(A)II = IlaZ(A)II. Hence Ib~b4Il Thus Q- is continuous LEMMA4.3. only if
G IMII
for
A E Af(M,).
and the rest follows.
The map uz is a gauge transformation between At(MO) and A:(M) if and
(a) Z maps PnM,, into N, (b) Z maps PTM,, into T, (c)for
vS E P,M,,
Zys+T
= qS+T.
(PS = Pn-Pr.)
Proof If tlz is a gauge transformation then (a) is immediate. For GZ to be defined clz must map kertiP into kern,. Now kerz$ is the closure in AZ(N) of the kernel of n’ restricted to A(N). (To see this let A E ker$ and {A,} be a sequence of elements of A(N)
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427
which converge to R. Then ~(4,) + 0. But n(A.) E d(S) E d(N) and so we may set As 7~acts as the identity on d(S) we have ti(B,) = 0 and B, --) A. Thus A is the limit of elements B. in the kernel of 7~ restricted to d(N)). Otherwise stated kernp
B, = A,-n(A,).
is generated by those Z’ai 8,: (vi E T) such that x/Ii
= 0. NOW if C&S,,
E kertiO (qi
i
i
E PTMo), then CRi SzViE kernB if and only if Zq+ E T for all QI~E P,MO. Hence (b) holds. Finally, (c) follows because for q~e PNMO
&(&+kera@
= &(6&+kerz,)
= ~&~,+kerza = StZ,,,-t-kerz8.
(Here the S-subscript refers to the S-component of the corresponding vector.) Now 6c~$, + kern, = S,, + kern, if and only if (Zq& = vs. That is Zvs + T = qs+ T. The converse follows by reversing the steps of the above arguments. Let 0 be defined by (4.1), then whenever 2 satisfies (a), (b), (c) of the above lemma we will call uz a linear gauge transformation into the Fermi gauge. Note that to date we have made no use of restriction (4.2). Using (4.2) we can connect our treatment with that of Strocchi-Wightman ([3], Proposition 2.2). LEMMA4.4.
(a) Z maps PNMo into N if and only if
Wheref(k) E C for each (k, k,) E X,+. (b) Z maps P,MO into T if and only if T(k),‘k, is proportional to kp for each (k, k,) EX$. (c) Z satisfies conditions (a), (b), (c) of Lemma 4.3 if and only if
P(k); = g,, + 5 r;lW + G,(k)k
(4.3)
where k + F(k) and k + G(k) are CQvaIued functions on X,+. Proof:
(a) If q~E PNMO and Z~J EN then a brief calculation gives kfi[(k);&(k)
= 0
which implies that k“t(k),,kt+k”C(k),k,
= 0.
(4.4)
To show that (a) holds we need only show that the vector u(k) with components a(k), = kpC(k)g is parallel to (k, k,). This follows directly from (4.4) which implies that the projection of w(k) onto vectors orthogonal to (k, k,) is zero. (b) This is immediate. (c) Let Z = 1+ Y where Z (and hence Y) satisfies conditions (a), (b), (c) of Lemma 4.3. Then we can write Y = YP,+ Y(l - PN) where because of condition (c) of Lemma 4.3 on Z, YP, must have range in T. Now PN is a multiplication operator [2] and so YPN is the operator of multiplication by q(k) say. But any multiplication operator with range in T must satisfy
428
A. L. CAREY, J. M. GAFFNEY,
v#),,
= &F,(k)
and C. A. HURST
for each (k, k,) E X,+.
For the other part Y(l -PN) is also a multiplication operator, say by t(k). As Y(1 - PN) has kernel No a direct calculation reveals that G(&, = G,(&)k, for some C4-valued function (k, k,) + G(k) on X,+. Thus T(k),‘, = g,Y+k~F,l(k)+G~,(k)k,,.
(4.5)
The converse is immediate. One should compare the matrix function T(k),, with the function f(&, of Proposition 2.2 of [3]. Our lemma appears at first sight to be more restrictive than the StrocchiWightman result, however there is an ambiguity in their proposition, which when clarified gives a result precisely analogous to that above [12]. Given a linear gauge transformation in the sense of our definition we would like to make contact with the heuristic notions. So let u E PB be a representation of d:(M) so that by Lemma 4.1 uz is a representation of @(M). By restriction 0 defines a representation of d:(N). Let H, be the representation space for U. As d:(T) is Abelian so FU = o(/$?(T))” is an Abelian von Neumann algebra. We can consider the direct integral decomposition of JV, = a(&!(N))”
t a(&T))’
induced by “diagonalizing” yO. Explicitly, we proceed by taking the spectrum B of u (O,(T)) Then there is an isomorphism of H, with the direct integral K = 1 HJ/&) I where ,.u is a measure on E satisfying the usual requirements [8], and H, carries some representation o, of d:(N) for each o E S’. The elements A E d:(N) are diagonal, that is, they act by @F)(o)
= &0(&F(@)
for FE K. Now a particular element of B is the character o,, which acts on d(T) by Wo:CliS,,
+C&
(pliET)*
(That this extends to a character on d,(T) is a corollary of Theorem 3.1.) Following Maurin [8] we may regard the elements of Ha, as distributions of the form v&,~ (?& is the Dirac d-function at wO) for each v E HmO. Furthermore we have LEMMA4.5.
The representation aZOof A:(N) is actually a representation of Af(N)/kerzcs.
Proof: It is sufficient to check that under uzO, kernB goes to zero. From the proof of Lemma 4.3 we know that kertiB is generated by those c Aidpi E A(T) such that c & = 0, whence we need only check that &,(d,) = 1 for each 13~E A(T) and this is clear because a;,,(&) = w,,(&J for v E T.
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Now consider a2 as a representation of @(MO). By restriction to @(P,Me) we obtain a representation of this algebra which is diagonalized by the above direct integral decomposition (as dF(PNMO) c @(IV)). Thus, as a2 maps kerzD into kerntl and Gz maps d~(P,MO)/kernp into Of(N)/ker& we have 0; = ooo o az, a representation of A<(P, MO)/kerns . Moreover as for all 9 EN, &z(891+kernP) = dP+kerna and L3g(P,VM,J/ker~fi E A:(PsMo) E 4:(s) we conclude that e$ regarded as a representation of Aff(PsMo) is just the restriction (to of co regarded as a representation of d:(S). Briefly stated all gauge algebras related to A:(M) by a linear gauge transformation give rise via the above procedure to the same representation of the algebra of physical observables. The above situation is summarized in the diagram Af!(P,M,))
/IQ&)
‘z
-1!(& $)lkernp
/IF(M)
2, B(H,)
G &?(~~/ker.z, ;; B;/H,J
which “commutes” if the vertical arrows are interpreted appropriately. Now we see that az and o act in the same space Ho and therefore we would expect the field operators to be related in some way. Writing Z = I+ Y as in Lemma 4.4 with Y acting as (Y&(k)
= k,F(k)“~,(k)+G,(k)k’~,,(~)
we define
so that R(ZP) = R(pl)+R(YP)* We call 9 + R(Yy) the gauge field. As we shall see in the next section it is only when F z 0 that R(Yv) has the form “8,x(x)“.
5. The Fock representation The Fock representation of @(M) is determined by the generating functional p -+ exp(-i]]q1)2), v E M. This representation has a number of peculiarities which were discussed in part in [l] and [2], the principal one being that the vacuum is not invariant under Lorentz boosts. Consequently we would expect there to be differences of detail between our approach and the Gupta-Bleuler method. Nevertheless as real Hilbert spaces the Fock space for d:(M) (which we showed in [l] to provide a rigorous version of the Fermi method of quantizing the four-vector potential) and the Gupta-Bleuler Fock space are identical. To see this we note that M, as defined in Section 2 is identical with the one particle space of Strocchi-Wightman [3] regarded as a real Hilbert space.
A. L. CAREY, J. M. GAFFNEY,
430
and C. A. HURST
Both Fock spaces therefore have the form
The important point to note is that the symmetrized tensor product of copies of M, being a tensor product of complex Hilbert spaces, will be defined differently in each case as the complex structure JF on M differs from that on the Strocchi-Wightman one particle space (being just multiplication by i). We will write Js for this Gupta-Bleuler complex structure. After some thought one sees that the tensor product MOM for the Fermi Fock space is defined by taking ql, 91~E M and writing %@%(kl> where
GMW
the annihilation
=
k2)
=
~lwo@2(~2)
and (@i)j(k) = (qi)j/(k) for j = 1,2, 3 and i = 1,2. - GCM~) and creation operators are defined by
NOW
a(pl)o,(yrO
... Oy,) = __l v
where o, is the projection of MO . . . @M onto MO s . . 0 sM (n-copies). Thus a*(q) creates on the vacuum Q, the state P, in contrast to the Gupta-Bleuler creation operator which gives v. The form of the gauge field R(Ygl) in this representation is most easily expressed by writing its action on the coherent states: Y(y) = exp(-$jjy/12)
$J (n!)-1’2a+(y)Q, n=O
Thus with (Yv)(k) = q(k)&) (~(yV)y)~!..,,cI,
we have
‘. .9 w
Thus with q(k&” = G,(k)k,. (that is P E 0) we have @(Y#qt:!..,“(k,
> . . . . k”)
REM.
(5.1)
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Now we introduce the scalar field:
where (G&(k)= (%f)W = G,W_fW and f is chosen so that Gf E M. Denote by P the map taking 9 to the function PP(k) = k’%(k). Then R(Yv) = x(Pv) so that the gauge field R(Yv) is heuristically, of the form 8,x(x). We take from [3] the following examples. Radiation gauge :
Set 2 = Ps so that M,, = M. Then it is straightforward that the commutation relations U’%), correspond
to check
R@‘)l E: iB(v, J’S@)
to the usual heuristic ones for radiation gauge as (QJ)O = 0
and (pSFh(k)
=
1
yj(k).
(d,-9)
i
Similarly, for the other “special gauges” listed in [3] we obtain the correct form of the commutation relations by setting C(k),, = g,g,+ G,(Q%,
(k, k,) E XL,
where G,(k) takes the form G,(k) = Mkp G,(k) = -n,/n
(Kallen-Rollnik-Stech-Nunneman -k
G,,(k) = (- n,/n - k + k,/(n - k)‘)
(Evans-Fulton
gauge),
gauge),
(Valatin gauge),
where np is a time-like vector. We note that the projections PT and PN have a form similar to (4.5) so that one can “cook-up” artificial gauges by playing with these operators. Now we have based our analysis on the form of the commutation relations (in contradistinction to [3] where the two-point function is used). This is because it is the commutation relations which determine the algebraic structure while the two point function specifies a representation of the algebra. Since Gupta-Bleuler is not a Weyl system we could not expect to obtain the two-point functions of [3] by taking a representation of the gauge algebra. To obtain two point functions which resemble those of [3] one must take the Fermi quantization of At(M) as the representation a2 where a is the Fock representation of A:(M). In general az may be reducible as a representation of A<(M) and so the
432
A. L. CAREY, J. M. GAFFNEY, and C. A. HURST
Fermi quantization is really the cyclic subrepresentation state D of the Fock representation of d:(M). In fact we compute that @, gz(S,)Qn> = exp(-$B(-%,
of oz generated by the vacuum
JFZcp)}
= exp{-iP(g,,
Jd+Wy,
J~y)fNg,,
JFYp))+B(Y~, JFYdl}.
This last expression should be compared with equation (2.58) of [3]. Finally we note that for radiation gauge
as JF and Js coincide on S. Thus the cyclic representation of the radiation gauge algebra d!(M) generated from 52 is, as expected, the Fock representation of n:(S) with the operator az(6J for q E S-i- acting as the identity. However the representation 02 is in this case also clearly the Fock representation of @(S)Z I~F(N/T)z &!(N)/kernB. We conclude from our previous analysis therefore that for any gauge algebra linearly related to the Fermi gauge the corresponding representation 02 of the algebra of observables (for (T the Fock representation of /l:(M)) is essentially the Fock representation of d:(s). 6. c-number gauge transformations The analysis of the preceding sections gives gauge transformations which are linear in the vector potential. On the other hand the gauge transformations which have been most studied are of the form A.,(x) + 4(x) + &X(4
(6.1)
where x is a real valued solution of qx = 0. Manuceau [6] gives an account of such c-number gauge transformations for general Weyl systems and we deduce from his work that (6.1) can be obtained by considering automorphisms of d,(M) of the form %’* S,+
exPiiB(y,
PI> 8,
(6.2)
where Y,~E 8,x, so that v E T. By taking the Fock representation c of &!(M) and regarding T as a gauge group we obtain some of the structure of the compact gauge theory of Doplicher, Haag and Roberts [5]. For us T is represented in the Fock representation by the operators 0(8,), y E T, and we have the field algebra as 0 (d:(M)). H ence in the sense of [5] the algebra of observables is a(Az(M))no(&!(T))’ = cr(df(N)). A n explicit calculation [lo] reveals that the representations of d:(N) which occur in the direct integral decomposition of c restricted to &Y(N) are the representations
where vs, IJ+ denote the S and T components of v EN and CT~is the Fock representation of &!(S) and VT’ ranges over the orthogonal complement T’ of N in M. NOW,
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Qw ** d, -+ d_,,, d, d,,, = clqeifcpT-prF) is an automorphism of d:(N). Furthermore T' may be thought of as the dual (in the group sense) of T.Thus we have shown that for each representation e of d:(N) occurring in the direct integral decomposition of 0, there is an automorphism c(,,.~,,labelled by an element vYTPof the dual of T such that (cf. Corollary 6.3 of [5]). We include here some speculations on further developments of this formalism. Firstly it would be useful to have an axiomatic formulation of the passage from a field algebra to an algebra of observables in the case where this passage is determined by gauge transformations of the second kind. Secondly an interesting test of this material would be to give an account of Landau gauge. We are not entirely happy with the discussion of [3] (cf. [ll]) while heuristic calculations indicate that one may obtain Landau gauge by considering a very singular operator Y on M. There remains some work to be done on this point. Finally we give an example of a linear gauge transformation which is an automorphism of the Fermi gauge algebra. If we set 2 = 1 + PTJF then we obtain B(o1, y) = B(V> V)+B(PJ@>
fu>+@,
PTJFV)+B(PT&F,
P,J#fJ).
Now B is degenerate on T so that
W’TJF~J, PTJF~) = 0. We also have PT as a real orthogonal
projection so that
B(v, JFPTY) = B&y,
JFY)
which implies That is B(pl, PTJFY) = -W,J,q,
WI.
Hence p(p), v) = B&J, 4y) implying that Z is symplectic. Consequently c(~ is an automorphism of the Fermi gauge algebra. We conclude therefore that c-number gauge transformations are not the only ones which are automorphisms. 7. The Fermi quantization
as a Strocchi-Wightman
gauge
While Gupta-Bleuler is not a Weyl system we would nevertheless expect there to be some connection between the algebraic approach and that of Strocchi-Wightman [3]. As we remarked in Section 5, the Fock spaces for the two quantizations (Gupta-Bleuler and Fermi) differ only in their structure as complex Hilbert spaces. Our object in this section is to show how the Gupta-Bleuler operators relate to those of the Weyl algt4bra and to show that there is a sense in which the Fermi quantization is a Strocchi-Wightman gauge.
434
A. L. CAREY, J. M. GAFFNEY,
Let o be the Fock representation
H’ = Ho
of d:(M)
and C. A. HURST
in H say, with vacuum Q. Define
(v(i':c",",)Q)-
,
= (9(&'(JF i;))Q)-0
(7.1)
Cl2
where JF is as before the Fermi complex structure, the bar denotes closure and the tilde is used as in Section 5 (i.e. f+ = {f$ E A4 91EN)). Form the space ~1’ = (G(@(J&))H~)-
and the quotient Hphys = H’/H”. multiplication by i. The form
(7.2)
Let Js denote the complex structure on M defined by
(pl>9’) = B(v,
Jse7’)+Wg,,$1
(7.3)
the usual Poincare invariant sesquilinear form on the one particle space. It extends to a form on the Fock space in the obvious way and we shall write ( , ) for this extension. Given 9~E N, ~(8_;2~,$2 E H’ has as its component in the n-particle space the function @(“) where Q’l:!..,&, . . . , 4 = y,,(k) . . . qp,(kd> is
from which it is clear that H’ coincides with the subspace of H given in [3] (equation 2.29). Hence on H’ the form (7.3) is positive semidefinite. We claim that H” consists of those @ E H’ such that (@, @) = 0 and thus H’, H” and H,,hys coincide with the subspaces introduced by Strocchi-Wightman to discuss the Gupta-Bleuler quantization. To see this we note from (7.2) that a dense subspace of H” is spanned by the vectors Y(q) = 5 (@“‘-&))/I/ n=O function
n ! where v E N, pls is its S-component
and c#“) denotes the
9$,!..fi (4, . . . . k) = pl,,(k,) . . . y&n). These clearly satisfy (Y(v), 6(q)) = 0. Let H,, denote the subspace of H’ consisting of vectors Y with (p’, Y) = 0 and PO the projection onto H,,. Now we know that the linear span of the coherent states Q(v) = ngO@“)/~~pl EN is dense in H’ so that PO acting on this linear span gives a dense subspace of Ho. But PO@@) = u’(p) which proves the result. Thus given the Fermi-Fock space we can construct a quotient space Hphys which is the space of Gupta-Bleuler physical states. Now H,,hys actually carries a representation of d,(N)/kern. To see this we use the isomorphism d,(S) = d,(N)/kerti and observe that as $& = ys for ys E S, so cr(6,J and 0(6,,)* = o(&,,) leave H’ and H” invariant. Hence a(&) induces a map on the quotient Hphys and therefore gives a representation of the algebra of observables Af(N)/kerfii,. We compute, using the indefinite form on H’ the generating functional : (Q+H”,
o(&,,>Q+H”)
= (Q, a(4&2)
= exp(-~lIysl12).
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435
Thus as expected we obtain the Fock representation of d:(S). We conclude that GuptaBleuler and Fermi lead to the same representation of d:(N)/ker7c,. In particular we can check that on Hphys, the Gupta-Bleuler annihilation and creation operators coincide with the Fermi annihilation and creation operators. The above analysis allows us to deduce that the Fermi method is essentially a GuptaBleuler quantization in the Strocchi-Wightman sense. To see this we take the covariant representation of the Fermi method described in [I], [2]. This consists of a direct integral K=
~K,M?9 X
where X is a mass shell (with mass = 1) with invariant measure dp and Kp (p E X) is the Fock space for the representation o, of d:(M) with generating functional cp + exp(-~ll~1~%l12) (AP is a Lorentz transformation
taking (0, 0, I, 1) E X, p = (‘JJ,pO)). Realise Kp as co~,o~*o,~,o
..‘,
(7.4)
where MP z A4 is the completion of M,,, in the inner product (91, Y’)p = NY, ‘l,,,JF(l(p)-l~‘)+iB(gl,
pl’).
That is MP is M regarded as a complex Hilbert space with complex structure given by
Jp = &,JF~;. On K we have a representation
of d:(M)
given by
(c(&) F)(P) = %0,)%)
f
(7.5)
So we can define H; = H,” = H;’ = H;i,,.s =
where Q, is the vacuum in Kp and the tilde takes account of the fact that the tensor product in (7.4) is not the usual one. The analysis of the first part of this section carries over to each Kp and on Hi/H;’ we have a Gupta-Bleuler quantization. It is easy to check that Js and Jp coincide on Hnqlys. If we form K’ = j H;, P
K” = j H;‘, P
then the quadruple (R,, K, <, >, 0
Kphys = K/K”,
A. L. CAREY,
436
J. M. GAFFNEY,
and C. A. HURST
where v -+ R,(yl) is the Wightman field defined by the representation (7.5) and (, ) is the inner product on K, forms a Gupta-Bleuler gauge in the Strocchi-Wightman sense except that the vacuum is degenerate. Nevertheless the above analysis shows that:
and so this degeneracy is trivial as we have a direct integral of identica1 Hilbert spaces. Since K’/K” we can define a Strocchi-Wightman
f@?) 3 ‘?,
ES L= (A-, ~JLL(~))QH’/H”
gauge transformation 0 E H’fH”,
by
J’E L’(X 44P)).
Acknowledgement One of us (A.L.C.) acknowledges the financial support of a Rothman’s Fellowship. REFERENCES Carey, A. L., Gaffney, J. M., and Hurst, C. A.: to appear, J. Math. PI7ys. 1977. Carey, A. L., and Hurst, C. A.: Adelaide preprint 1976. Strocchi, F., and Wightman, A. S.: J. Muth. Phw. 15 (1974) 2198. Bongaarts, P. J. M.: Leiden preprint 1976. Doplicher, S., Haag, R., and Roberts, J. E.: Con7777rtri. Math. Phys. 13 (1969), 1; fbid. 15 (1969), 173. [6] Manuceau, J.: Anft. bnt. Henri Poincnre’ 8 (1968), 139. ;7] Haag, R., and Kastler, D.: J. Muth. PZrys. 5 (1969), 848. [8] Maurin, K. : General eigetrfmction e.upm7siotr.s ad 7mifar.vrepreserttations o/‘ ~opologicol groups, PWN -Polish Scientific Publishers, Warsaw 1968. 191 Streater, R. F., and \I’iIde, I.: fv’wzlenr Physics B 24 (197b), 561. [lo] Gaffney, J. M.: Ph.D. thesis, Adelaide 1974 (unpublished). [ll] Strocchi, F., and Wightman, A. S.: Erratum, J. MntR. Phys. 17 (1976), 1930. [12] Wightman, A. S.: Private communication. 1131 Manuceau, J., Sirigue, M., Testard, D., Verbeure, A.: Commrm. Mmh. Pl7ys. 32 (1973), 231. [I] [2] [3] [4] [5]
REPORTS
oiv ~~ATHEMATI~AL
No. 3
PHYSICS
A C*-ALGEBRA FORMULATION OF GAUGE TRANSFORMATIONS OF THE SECOND KfND FOR THE ELECTROMAGNETIC FIELD
A. L. CAREY, J. M. GAFFNEY*, and C. A. HURST Department
of Mathematical
Physics, University
of Adelaide,
Adelaide,
S. Australia
(Received May 2, 1977) A definition of gauge transformations of the second kind for the quantized free electromagnetic field is proposed within a Weyl algebra framework. This paper is complementary to recent work of Strocchi and Wightman on gauge transformations for the electromagnetic field using the indefinite metric formalism. The algebraic approach accounts for the general features of the heuristic notion of gauge transformation in a natural way as well as clarifying some of the constructions of the indefinite metric approach.
1. Introduction
of this paper is to give an account of the notion of gauge transformation of the second kind for the electromagnetic field using a C*-algebra framework. The fact that there is a rigorous version of the Fermi method of quantizing the electromagnetic field which uses a Weyl system approach was the subject of two previous papers by the authors [I], [2]. The starting point of the present investigation was the paper of Strocchi and Wightman [3] which proposed a definition of gauge transformation within the indefinite metric formalism. The contents of our paper are largely complementary to the work of Strocchi and Wightman in the Sense that we clarify some of the constructions introduced in [3] by giving them a natural interpretation in the algebraic picture. We are also able to answer in part some of the questions posed in [3] about the algebraic side of the GuptaBleuler quantization. We believe that the various approaches to the quantized electromagnetic field (including the recent one of Bongaarts using the Borchers algebra [4]) are complementary rather than mutually exclusive. For this reason we have also been concerned here with understanding the connection between the Gupta-Bleuler and P-algebra approaches. This paper depends in part on our previous work and we refer the reader to [l] for a more leisurely introduction. The contents of the paper are organized as follows. In Sections 2 and 3 we define the field algebras and the algebra of observables for the electromagnetic field. Our approach here has been influenced by the ideas of Doplicher, Haag The
object
* Present address: Department U.S.A.
of Mathematics,
Michigan State University,
[4191
East Lansing, Michigan,
A. L. CAREY, J. M. GAFFNEY,
420
and C. A. HURST
and Roberts [5]. In Section 4 we give a general definition of gauge transformation of the second kind and illustrate it with the example of what we call a linear gauge transformation. In Section 5 we see that linear gauge transformations account for most of the commonly used gauges for the electromagnetic field with the exception of Landau gauge which remains an unsolved problem. We discuss briefly in Section 6 examples of gauge transformations which are automorphisms of the field algebra, namely c-number gauge transformations, Because we are unable to settle the question of what the most general gauge transformation looks like (in particular, Landau gauge suggests that linear gauge transformations are not sufficiently general), we have not investigated the question of when a gauge transformation is an automorphism of the field algebra. One might also ask to what extent the present formalism is peculiar to the electromagnetic field. We conclude with a discussion of the relationship between Gupta-Bleuler and the algebraic approach. 2. Local algebras Let X,+ = (k E R4/ kc, > 0, kZ = 0} and consider the real Hilbert space M defined from functions q: X,+ --+ C4 satisfying
(2.1) where k = lkj. The inner product on M is given by
(2.2)
(2.3)
where g = diag(- 1, - 1, - I, 1) is the metric tensor. The complex inner product on M is then (2.4) (~3 v’> = S vo(k)vb(k)+k vj(k)VJ(k)d3k/2k j=l X2 which has (2.2) as its real part. Unless we specify otherwise the symbol M will be interpreted as meaning a real Hilbert space. Each sufficiently smooth 9 satisfying (2.1) defines a real solution 6 of the wave equation lJ@ = 0 by taking the Fourier transform. Explicitly : i,
x.
S [ev(-Xi?
ik * 4plJk) + exp(ik *x)q+,(k)ld3k/2k
(2.5)
A C*-ALGEBRA FORMULATION OF GAUGE TRANSFORMATIONS
421
Let 9 be the space of Cm functions of compact support in R3 taking values in R4. Each pair of functions cf, g) E 9 x 9 defines a function on X$ by
(2.6) Then q~satisfies (2.1) so that (f, g) defines a function $ with
ql@=
0 and it is not difficult
to see that $(r, 0) = f(x) and &(x, 0) = g(r). The subspace M,, of M defined by letting (f, g) range over 9 x 9 is dense in M. The PoincarC group acts on M,, by 6 -+ G’ where G’(x) = _,l@(/l-1(x-a))
(2.7)
where (a, A) (consisting of a translation a and Lorentz transformation il) is a typical PoincarC group element. It is well known that MO0 is invariant under the representation (by bounded, not unitary, operators) on A4 of the Poincare group defined via (2.7). We define two closed Poincart invariant subspaces of M as follows. Let N consist of those elements of M satisfying @ZJ&) = 0,
(k, k,) E x,+ f
and T consist of those elements of M of the form %(JV = k,X(k)Y
(k, k,) E x,+.
Let ,9: M,, x MO + R be a (not necessarily non-degenerate) symplectic form which is defined on a dense subspace iVfOof M. We restrict M,_, (and hence j3) by demanding that it be invariant under the (real) orthogonal projections PN and PT onto N and T respectively. Define the *-algebra IIo(MO) as the complex linear span of the functions S, on M, 9 E M,,, given by 0 if v # v’, 4&J’) = 1 if q3= pl’ with multiplication
and with the *-operation
By completing @(M,,)
in the norm:
we obtain a Banach algebra Of (MO). Let Pfi denote the non-degenerate strongly continuous *-representations 0 of &CM,) such that 2 + a(&)
422
A. L. CAREY, J. M. GAFFNEX,
and C. A. HURST
is continuous in A E R for all 9 E M,,. Define the C*-algebra d$Vf,) tion of df( MO) in the norm
ll4l = ;s$ilc(‘4ll
to be the comple-
(G 11‘411)
(cf. Manuceau [6]). Although M,, is not invariant under PN and PT we can still define &?(MO,,) whenever MO > M,,. This is of interest because dz(M,,) has a local structure in the HaagKastler sense [7]. Namely, for each bounded open set Q c R we consider the algebra O@(L?)consisting of the linear span of those 6, such that for some spacelike hyperplane P the initial data for q on P have support in PnD. Then clearly d@(&f,,) = Ud@(Q) where D the union is over all bounded open sets Q c R4. Hence &!(M,,) = lJd,(Q), where n LIB,(~) denotes the closure of O@(Q) in /l~(M,,). Neither dc(PNMoO) nor il,(P,Mo,) (defined as the C?-subalgebras of @(MO) generated and (S,i v E PTh/loo) respectively) have a local structure. However by @,I cpE P,M,,) consider the closed subspace S of M defined from functions satisfying k *cp(k) = 0
((k, k,) E X,+)
and
v. 3 0.
We can write all elements of S in the form $44 = kx $(k) for some C-valued function $ or Xg . Hence a dense subspace So0 of S is spanned by those functions on Moo whose initial data cf, g) (at time zero) have the form “Ll = go = 0, f(x) = v xf ‘9
g(4 = v x g’b>
where f’ and g’ have compact support. We may form in the obvious way @(So,) and give this algebra a local structure by associating with each open bounded B c R4 those 8, such that 9 e So and 6, EL’I~(@. 3. Field algebras and the algebra of observables The algebras .lt(Mo) which we defined in the previous section are a little too genera1 to be regarded as field algebras for the electromagnetic field. We have already imposed (a) 8: MO x MO -+ R is a real bilinear symplectic form on MO (which is dense in M) and PN and PT. leave MO invariant. We now demand that (b) /? is invariant under the action of translations (equation (2.7)) and satisfies @(&MO, &MO) = 0. The origin of this constraint may be seen from [I], [2]. There we considered the symplectic form
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423
which, being the imaginary part of (2.4), is defined for all (p, @ EM. The algebra A:(M) we called the C*-algebra of the CCR for the electromagnetic vector potential. However B is just one choice among many and A:(M) just one of a large class of possible field algebras. The reason for this ambiguity in the choice of field algebras may be seen as follows. From [I] we know that Nis the Hilbert space direct sum of S and T and that the map 7~: 6, -+ d,s where v EN and ys is its S-component, extends to a homomorphism fiB of AZ(N) onto A:(S). The quotient algebra Af(N)/kernB we call the algebra of physical observables. We have the following isomorphisms 111: Af(N)/kerz,
z Af(N/T)
g At(S),
where Af(N/T) is defined using the symplectic form on N/T induced by B. These isomorphisms demonstrate that Af(N)/kernB is defined in terms of the transverse components of the one particle photon states and hence excludes the unphysical longitudinal and scalar components. Now A,B(N)/kern, has the important property of being Poincare invariant (under the automorphisms of A:(M) induced by the representation (2.7) of the PoincarC group on M). Thus it is only the algebraic structure of this quotient algebra which is of importance. Now a consequence of constraint (b) is that B(V, V’) = B(% >94)
(3.2)
where qs, 9; are the S-components of v, q~’EN respectively. Since AE(N)/kern, and are isomorphic it is sufficient therefore (in order to preserve the algebraic structure of Af(N)/kern,) to require
A:(S)
(c) B(vs, ~7:) = B(~s, q&) for all qk, q~,$E bogs. (Note that M,nS = (P,-P,)M, is dense in S.) DEFINITION. If the symplectic form 8: M,, x MO -+ R satisfies conditions (c) above we will call At(M,) a gauge algebra (for the electromagnetic field).
It is clear that A’(P,M,,)
= As(P,Mo) z:
f& + s,,,
(a), (b),
so that the map y E PlvMO
extends to a homomorphism of Ap(PNMO) onto A”(P,M,) (P, = PN-- PT). We note that as b coincides with B on PsMO and is therefore non-degenerate on Ps M,, , we have AS,(PsMo) = A,B(P, M,,) (there being just one C* norm on AB(Ps MO) (cf. [13]). THEOREM 3.1.
A:(Ps&).
z
extends
to a continuous homomorphism
tip of At(PNMO) onto
A. L. CAREY,
424 Proof:
As n is the identity
J. M. GAFFNEY,
on JB(PsMO)
and C. A. HURST
it is clear that the only difficulty
that ?c restricted to &(PzM,) is continuous. We introduce defined as the completion of df(Pz MO) in the norm II& where the supremum
= suPlle(~)Il,
e
is over all continuous
the
V-algebra
is to prove -__ 3(PTMo)
A. E~Im
representations
e of d, (PzMJ.
Then @(Pr MO)
E n(P,M,)/Z where Z is some ideal of d(P,M,) (cf. 1131). Let e be a representation of ds,(P, MO) such that I + @(a,& is strongly continuous from the real line into the unitary operators on the Hilbert space He of Q. Since @(A!(T)) 1s an Abelian c-algebra we may write H4 as a direct integral of Hilbert spaces over the spectrum of Q(A:(T)). If x is some element of the spectrum which occurs in this direct integral decomposition then 2 is a character of d!(T). Since this direct integral decomposition may also be regarded as a deEmposition of the representation ii -+ e(&,) (pu E P,M,) of the real line, we have ii + x(6,,) a continuous map into the complex numbers of modulus one. This implies that we may write x(6,) = exp(i&)) (y E PTMo) where i is a linear functional (not necessarily continuous) on PTMo. Now extend 2 to a linear functional on MO by setting it equal to zero on (1 - PTIMo . Define an automorphism of &(M,,)
We note that if x is a character
Furthermore
x(&
Ai 4,))
(cf. [6]) and hence
of dt(P,M,)
= X(X i&i&,J,
of @(PTMo),
by
then so too is % where
vi E P,M,.
Now x extends
to a character
of il (PT M,) (see [13]) and so will define a continuous character of @(P, MO) provided z(I) = 0. So let A. E Z and let (A,) be a sequence in AB(PrMo) converging to A in the topology of At(PTMo). Noting that a,! lS(PTM,,) also extends to an automorphism of _4,(P,Mo)
(see [I 31) we have a,(A,)
But x(~(A.n))
= +,)
and
-+ K~(A). Hence
x (Q(U)
+ ;c (x&0).
ti(A,) -+ n(A)
so that
we have
z’(A) = :(r,(A)).
Finally
we note that as CI~is an automorphism of both A(P,M,) and --Jt~(PTMO),it must leave Z invariant. Thus cc,(A) E Z so that 7 (cc,(A)) = 0. That is $I) = 0 completing the proof. We now have the isomorphisms
verifying our assertion that each gauge algebra determines a subalgebra of the algebra of observables. We will see below that different choices of ,5 correspond to different gauges and that gauge transformations can be defined as special maps between gauge algebras. Before
A C*-ALGEBRA
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425
taking up this discussion we make some remarks relating to Section 2. @(MOO) satisfies the Haag-Kastler axioms. Using the local structure on A,(&,,) we may give a local structure to a certain subalgebra of Af(l\r)/kerns. However, as would be expected, the axiom of relativistic covariance is violated for this algebra. Thus we have a local field algebra but a non-local algebra of observables (non-local, that is in the Haag-Kastler sense). Clearly therefore one valuable feature of the electromagnetic field example is that it may suggest modifications of the local algebra program [5] which will enable it to handle theories containing zero mass particles (cf. also [9]). 4. Gauge transformations Heuristically, a gauge transformation is a replacement AM(x)--+AL(x) of the vector potential A,,(x) by a new potential Ah(x) such that the observable fields E;,(x) = a, A,(x) - a&,(x) are in some sense unchanged. Now how one makes this precise will depend on the formalism adopted for making sense of A,(x), A.;(x) and FflY(x). Strocchi and Wightman have put forward one possibility [3] within the Gupta-Bleuler framework. A translation of their ideas into the algebraic picture together with some algebraic intuition leads us to the following be two gauge algebras for the electromagnetic DEFINITION. Let &!(M1) and &‘(M,) field. Let a be a homomorphism from @(Ml) to @‘(MJ which takes A$(PNM1) into Ac(P,M,) and which induces a homomorphism z : At(P,M,)/kerti;,
+ Ai’(P,M,)/keiq
(where n’r and 7~~are the extensions of JZ to At(P,M,) and Ac’(P,M,) respectively). If PN(MlnM,) is dense in N and for those 6, E AB,(P,M,)nA~(P,M,) we have z(6,+kerq)
= dq+ker?l,
then we call a a gauge transformation. The rest of *his section will be devoted to a special case of this definition, namely, what we will call linear gauge transformations into the Fermi gauge. We consider the special case where B(v, v’) = B(Zq, ZV’)
(4.1)
for v E MO = domain of Z (here Z is a real linear densely defined operator on M). Because we have demanded that B be translation invariant we impose the condition that Z have the form (4.2) Z@) = ~@Mk) where C(k) is a 4 x 4 matrix for each (k, k) E X,+ . (This condition is sufficient but probably not necessary.) Note that we cannot demand that /I be Lorentz invariant for, on physical grounds, it is only the induced form on N/T which need be invariant. The demand that /I be Lorentz invariant forces /I = B which fact distinguishes A:(M) from the other gauge algebras. We will call A:(M) the Fermi gauge.
A. L. CAREY, J. M. GAFFNEY,
426 Condition
and C. A. HURST
(c) on /? implies that B(Zv, Zpl’) = B(v, 91’)
for
9, P’ E PNMo,
from which it should be clear that there are many possible choices of Z which will give a form /I via (4.1) satisfying conditions (a), (b), (c) (for example Z need only be symplectic on M and leave N and T invariant). Most of these will not define what we would heuristically mean by a gauge transformation. To see what restrictions we need to place on Z we introduce the map: aZ: As(MO) + A:(M) defined by az(C
Ai a,,) = F k oz,, . From (4.1) it follows easily that tlz is an algebra
homomorphism. LEMMA 4.1.
Zf o E PB is a representation of A:(M) representation of Af(M,-,). Proof:
The norm on A$(MJ
then oz = u o uz extends to a
is defined by taking A E A&I4,).
sujlla(A)II9 UEP
Now if A = 7 li S,, E Af(M,) llCz(A)Vll
< C i
an d ZJ is in the representation lililllO(&q,)~lI
space of az we have
= C !&I llwll = llAlll Ilull* I
Thus ax extends to a representation of de(M) and by the properties of c, lies in Pp. Hence a~ extends to a representation of AS,(M,). COROLLARY 4.2.
We can extend aZ to a C*-homomorphism of At(M,)
into A:(M).
Proof Let IT be the Fock representation of A:(M) (i.e. the representation associated with the generating functional Q(V) = exp(-illy,ll’), v E M). Then l/uz(A)II < IIAII for A E A!!(M,). But u is faithful and so Ilaz(A)II = IlaZ(A)II. Hence Ib~b4Il Thus Q- is continuous LEMMA4.3. only if
G IMII
for
A E Af(M,).
and the rest follows.
The map uz is a gauge transformation between At(MO) and A:(M) if and
(a) Z maps PnM,, into N, (b) Z maps PTM,, into T, (c)for
vS E P,M,,
Zys+T
= qS+T.
(PS = Pn-Pr.)
Proof If tlz is a gauge transformation then (a) is immediate. For GZ to be defined clz must map kertiP into kern,. Now kerz$ is the closure in AZ(N) of the kernel of n’ restricted to A(N). (To see this let A E ker$ and {A,} be a sequence of elements of A(N)
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which converge to R. Then ~(4,) + 0. But n(A.) E d(S) E d(N) and so we may set As 7~acts as the identity on d(S) we have ti(B,) = 0 and B, --) A. Thus A is the limit of elements B. in the kernel of 7~ restricted to d(N)). Otherwise stated kernp
B, = A,-n(A,).
is generated by those Z’ai 8,: (vi E T) such that x/Ii
= 0. NOW if C&S,,
E kertiO (qi
i
i
E PTMo), then CRi SzViE kernB if and only if Zq+ E T for all QI~E P,MO. Hence (b) holds. Finally, (c) follows because for q~e PNMO
&(&+kera@
= &(6&+kerz,)
= ~&~,+kerza = StZ,,,-t-kerz8.
(Here the S-subscript refers to the S-component of the corresponding vector.) Now 6c~$, + kern, = S,, + kern, if and only if (Zq& = vs. That is Zvs + T = qs+ T. The converse follows by reversing the steps of the above arguments. Let 0 be defined by (4.1), then whenever 2 satisfies (a), (b), (c) of the above lemma we will call uz a linear gauge transformation into the Fermi gauge. Note that to date we have made no use of restriction (4.2). Using (4.2) we can connect our treatment with that of Strocchi-Wightman ([3], Proposition 2.2). LEMMA4.4.
(a) Z maps PNMo into N if and only if
Wheref(k) E C for each (k, k,) E X,+. (b) Z maps P,MO into T if and only if T(k),‘k, is proportional to kp for each (k, k,) EX$. (c) Z satisfies conditions (a), (b), (c) of Lemma 4.3 if and only if
P(k); = g,, + 5 r;lW + G,(k)k
(4.3)
where k + F(k) and k + G(k) are CQvaIued functions on X,+. Proof:
(a) If q~E PNMO and Z~J EN then a brief calculation gives kfi[(k);&(k)
= 0
which implies that k“t(k),,kt+k”C(k),k,
= 0.
(4.4)
To show that (a) holds we need only show that the vector u(k) with components a(k), = kpC(k)g is parallel to (k, k,). This follows directly from (4.4) which implies that the projection of w(k) onto vectors orthogonal to (k, k,) is zero. (b) This is immediate. (c) Let Z = 1+ Y where Z (and hence Y) satisfies conditions (a), (b), (c) of Lemma 4.3. Then we can write Y = YP,+ Y(l - PN) where because of condition (c) of Lemma 4.3 on Z, YP, must have range in T. Now PN is a multiplication operator [2] and so YPN is the operator of multiplication by q(k) say. But any multiplication operator with range in T must satisfy
428
A. L. CAREY, J. M. GAFFNEY,
v#),,
= &F,(k)
and C. A. HURST
for each (k, k,) E X,+.
For the other part Y(l -PN) is also a multiplication operator, say by t(k). As Y(1 - PN) has kernel No a direct calculation reveals that G(&, = G,(&)k, for some C4-valued function (k, k,) + G(k) on X,+. Thus T(k),‘, = g,Y+k~F,l(k)+G~,(k)k,,.
(4.5)
The converse is immediate. One should compare the matrix function T(k),, with the function f(&, of Proposition 2.2 of [3]. Our lemma appears at first sight to be more restrictive than the StrocchiWightman result, however there is an ambiguity in their proposition, which when clarified gives a result precisely analogous to that above [12]. Given a linear gauge transformation in the sense of our definition we would like to make contact with the heuristic notions. So let u E PB be a representation of d:(M) so that by Lemma 4.1 uz is a representation of @(M). By restriction 0 defines a representation of d:(N). Let H, be the representation space for U. As d:(T) is Abelian so FU = o(/$?(T))” is an Abelian von Neumann algebra. We can consider the direct integral decomposition of JV, = a(&!(N))”
t a(&T))’
induced by “diagonalizing” yO. Explicitly, we proceed by taking the spectrum B of u (O,(T)) Then there is an isomorphism of H, with the direct integral K = 1 HJ/&) I where ,.u is a measure on E satisfying the usual requirements [8], and H, carries some representation o, of d:(N) for each o E S’. The elements A E d:(N) are diagonal, that is, they act by @F)(o)
= &0(&F(@)
for FE K. Now a particular element of B is the character o,, which acts on d(T) by Wo:CliS,,
+C&
(pliET)*
(That this extends to a character on d,(T) is a corollary of Theorem 3.1.) Following Maurin [8] we may regard the elements of Ha, as distributions of the form v&,~ (?& is the Dirac d-function at wO) for each v E HmO. Furthermore we have LEMMA4.5.
The representation aZOof A:(N) is actually a representation of Af(N)/kerzcs.
Proof: It is sufficient to check that under uzO, kernB goes to zero. From the proof of Lemma 4.3 we know that kertiB is generated by those c Aidpi E A(T) such that c & = 0, whence we need only check that &,(d,) = 1 for each 13~E A(T) and this is clear because a;,,(&) = w,,(&J for v E T.
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Now consider a2 as a representation of @(MO). By restriction to @(P,Me) we obtain a representation of this algebra which is diagonalized by the above direct integral decomposition (as dF(PNMO) c @(IV)). Thus, as a2 maps kerzD into kerntl and Gz maps d~(P,MO)/kernp into Of(N)/ker& we have 0; = ooo o az, a representation of A<(P, MO)/kerns . Moreover as for all 9 EN, &z(891+kernP) = dP+kerna and L3g(P,VM,J/ker~fi E A:(PsMo) E 4:(s) we conclude that e$ regarded as a representation of Aff(PsMo) is just the restriction (to of co regarded as a representation of d:(S). Briefly stated all gauge algebras related to A:(M) by a linear gauge transformation give rise via the above procedure to the same representation of the algebra of physical observables. The above situation is summarized in the diagram Af!(P,M,))
/IQ&)
‘z
-1!(& $)lkernp
/IF(M)
2, B(H,)
G &?(~~/ker.z, ;; B;/H,J
which “commutes” if the vertical arrows are interpreted appropriately. Now we see that az and o act in the same space Ho and therefore we would expect the field operators to be related in some way. Writing Z = I+ Y as in Lemma 4.4 with Y acting as (Y&(k)
= k,F(k)“~,(k)+G,(k)k’~,,(~)
we define
so that R(ZP) = R(pl)+R(YP)* We call 9 + R(Yy) the gauge field. As we shall see in the next section it is only when F z 0 that R(Yv) has the form “8,x(x)“.
5. The Fock representation The Fock representation of @(M) is determined by the generating functional p -+ exp(-i]]q1)2), v E M. This representation has a number of peculiarities which were discussed in part in [l] and [2], the principal one being that the vacuum is not invariant under Lorentz boosts. Consequently we would expect there to be differences of detail between our approach and the Gupta-Bleuler method. Nevertheless as real Hilbert spaces the Fock space for d:(M) (which we showed in [l] to provide a rigorous version of the Fermi method of quantizing the four-vector potential) and the Gupta-Bleuler Fock space are identical. To see this we note that M, as defined in Section 2 is identical with the one particle space of Strocchi-Wightman [3] regarded as a real Hilbert space.
A. L. CAREY, J. M. GAFFNEY,
430
and C. A. HURST
Both Fock spaces therefore have the form
The important point to note is that the symmetrized tensor product of copies of M, being a tensor product of complex Hilbert spaces, will be defined differently in each case as the complex structure JF on M differs from that on the Strocchi-Wightman one particle space (being just multiplication by i). We will write Js for this Gupta-Bleuler complex structure. After some thought one sees that the tensor product MOM for the Fermi Fock space is defined by taking ql, 91~E M and writing %@%(kl> where
GMW
the annihilation
=
k2)
=
~lwo@2(~2)
and (@i)j(k) = (qi)j/(k) for j = 1,2, 3 and i = 1,2. - GCM~) and creation operators are defined by
NOW
a(pl)o,(yrO
... Oy,) = __l v
where o, is the projection of MO . . . @M onto MO s . . 0 sM (n-copies). Thus a*(q) creates on the vacuum Q, the state P, in contrast to the Gupta-Bleuler creation operator which gives v. The form of the gauge field R(Ygl) in this representation is most easily expressed by writing its action on the coherent states: Y(y) = exp(-$jjy/12)
$J (n!)-1’2a+(y)Q, n=O
Thus with (Yv)(k) = q(k)&) (~(yV)y)~!..,,cI,
we have
‘. .9 w
Thus with q(k&” = G,(k)k,. (that is P E 0) we have @(Y#qt:!..,“(k,
> . . . . k”)
REM.
(5.1)
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Now we introduce the scalar field:
where (G&(k)= (%f)W = G,W_fW and f is chosen so that Gf E M. Denote by P the map taking 9 to the function PP(k) = k’%(k). Then R(Yv) = x(Pv) so that the gauge field R(Yv) is heuristically, of the form 8,x(x). We take from [3] the following examples. Radiation gauge :
Set 2 = Ps so that M,, = M. Then it is straightforward that the commutation relations U’%), correspond
to check
R@‘)l E: iB(v, J’S@)
to the usual heuristic ones for radiation gauge as (QJ)O = 0
and (pSFh(k)
=
1
yj(k).
(d,-9)
i
Similarly, for the other “special gauges” listed in [3] we obtain the correct form of the commutation relations by setting C(k),, = g,g,+ G,(Q%,
(k, k,) E XL,
where G,(k) takes the form G,(k) = Mkp G,(k) = -n,/n
(Kallen-Rollnik-Stech-Nunneman -k
G,,(k) = (- n,/n - k + k,/(n - k)‘)
(Evans-Fulton
gauge),
gauge),
(Valatin gauge),
where np is a time-like vector. We note that the projections PT and PN have a form similar to (4.5) so that one can “cook-up” artificial gauges by playing with these operators. Now we have based our analysis on the form of the commutation relations (in contradistinction to [3] where the two-point function is used). This is because it is the commutation relations which determine the algebraic structure while the two point function specifies a representation of the algebra. Since Gupta-Bleuler is not a Weyl system we could not expect to obtain the two-point functions of [3] by taking a representation of the gauge algebra. To obtain two point functions which resemble those of [3] one must take the Fermi quantization of At(M) as the representation a2 where a is the Fock representation of A:(M). In general az may be reducible as a representation of A<(M) and so the
432
A. L. CAREY, J. M. GAFFNEY, and C. A. HURST
Fermi quantization is really the cyclic subrepresentation state D of the Fock representation of d:(M). In fact we compute that @, gz(S,)Qn> = exp(-$B(-%,
of oz generated by the vacuum
JFZcp)}
= exp{-iP(g,,
Jd+Wy,
J~y)fNg,,
JFYp))+B(Y~, JFYdl}.
This last expression should be compared with equation (2.58) of [3]. Finally we note that for radiation gauge
as JF and Js coincide on S. Thus the cyclic representation of the radiation gauge algebra d!(M) generated from 52 is, as expected, the Fock representation of n:(S) with the operator az(6J for q E S-i- acting as the identity. However the representation 02 is in this case also clearly the Fock representation of @(S)Z I~F(N/T)z &!(N)/kernB. We conclude from our previous analysis therefore that for any gauge algebra linearly related to the Fermi gauge the corresponding representation 02 of the algebra of observables (for (T the Fock representation of /l:(M)) is essentially the Fock representation of d:(s). 6. c-number gauge transformations The analysis of the preceding sections gives gauge transformations which are linear in the vector potential. On the other hand the gauge transformations which have been most studied are of the form A.,(x) + 4(x) + &X(4
(6.1)
where x is a real valued solution of qx = 0. Manuceau [6] gives an account of such c-number gauge transformations for general Weyl systems and we deduce from his work that (6.1) can be obtained by considering automorphisms of d,(M) of the form %’* S,+
exPiiB(y,
PI> 8,
(6.2)
where Y,~E 8,x, so that v E T. By taking the Fock representation c of &!(M) and regarding T as a gauge group we obtain some of the structure of the compact gauge theory of Doplicher, Haag and Roberts [5]. For us T is represented in the Fock representation by the operators 0(8,), y E T, and we have the field algebra as 0 (d:(M)). H ence in the sense of [5] the algebra of observables is a(Az(M))no(&!(T))’ = cr(df(N)). A n explicit calculation [lo] reveals that the representations of d:(N) which occur in the direct integral decomposition of c restricted to &Y(N) are the representations
where vs, IJ+ denote the S and T components of v EN and CT~is the Fock representation of &!(S) and VT’ ranges over the orthogonal complement T’ of N in M. NOW,
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Qw ** d, -+ d_,,, d, d,,, = clqeifcpT-prF) is an automorphism of d:(N). Furthermore T' may be thought of as the dual (in the group sense) of T.Thus we have shown that for each representation e of d:(N) occurring in the direct integral decomposition of 0, there is an automorphism c(,,.~,,labelled by an element vYTPof the dual of T such that (cf. Corollary 6.3 of [5]). We include here some speculations on further developments of this formalism. Firstly it would be useful to have an axiomatic formulation of the passage from a field algebra to an algebra of observables in the case where this passage is determined by gauge transformations of the second kind. Secondly an interesting test of this material would be to give an account of Landau gauge. We are not entirely happy with the discussion of [3] (cf. [ll]) while heuristic calculations indicate that one may obtain Landau gauge by considering a very singular operator Y on M. There remains some work to be done on this point. Finally we give an example of a linear gauge transformation which is an automorphism of the Fermi gauge algebra. If we set 2 = 1 + PTJF then we obtain B(o1, y) = B(V> V)+B(PJ@>
fu>+@,
PTJFV)+B(PT&F,
P,J#fJ).
Now B is degenerate on T so that
W’TJF~J, PTJF~) = 0. We also have PT as a real orthogonal
projection so that
B(v, JFPTY) = B&y,
JFY)
which implies That is B(pl, PTJFY) = -W,J,q,
WI.
Hence p(p), v) = B&J, 4y) implying that Z is symplectic. Consequently c(~ is an automorphism of the Fermi gauge algebra. We conclude therefore that c-number gauge transformations are not the only ones which are automorphisms. 7. The Fermi quantization
as a Strocchi-Wightman
gauge
While Gupta-Bleuler is not a Weyl system we would nevertheless expect there to be some connection between the algebraic approach and that of Strocchi-Wightman [3]. As we remarked in Section 5, the Fock spaces for the two quantizations (Gupta-Bleuler and Fermi) differ only in their structure as complex Hilbert spaces. Our object in this section is to show how the Gupta-Bleuler operators relate to those of the Weyl algt4bra and to show that there is a sense in which the Fermi quantization is a Strocchi-Wightman gauge.
434
A. L. CAREY, J. M. GAFFNEY,
Let o be the Fock representation
H’ = Ho
of d:(M)
and C. A. HURST
in H say, with vacuum Q. Define
(v(i':c",",)Q)-
,
= (9(&'(JF i;))Q)-0
(7.1)
Cl2
where JF is as before the Fermi complex structure, the bar denotes closure and the tilde is used as in Section 5 (i.e. f+ = {f$ E A4 91EN)). Form the space ~1’ = (G(@(J&))H~)-
and the quotient Hphys = H’/H”. multiplication by i. The form
(7.2)
Let Js denote the complex structure on M defined by
(pl>9’) = B(v,
Jse7’)+Wg,,$1
(7.3)
the usual Poincare invariant sesquilinear form on the one particle space. It extends to a form on the Fock space in the obvious way and we shall write ( , ) for this extension. Given 9~E N, ~(8_;2~,$2 E H’ has as its component in the n-particle space the function @(“) where Q’l:!..,&, . . . , 4 = y,,(k) . . . qp,(kd> is
from which it is clear that H’ coincides with the subspace of H given in [3] (equation 2.29). Hence on H’ the form (7.3) is positive semidefinite. We claim that H” consists of those @ E H’ such that (@, @) = 0 and thus H’, H” and H,,hys coincide with the subspaces introduced by Strocchi-Wightman to discuss the Gupta-Bleuler quantization. To see this we note from (7.2) that a dense subspace of H” is spanned by the vectors Y(q) = 5 (@“‘-&))/I/ n=O function
n ! where v E N, pls is its S-component
and c#“) denotes the
9$,!..fi (4, . . . . k) = pl,,(k,) . . . y&n). These clearly satisfy (Y(v), 6(q)) = 0. Let H,, denote the subspace of H’ consisting of vectors Y with (p’, Y) = 0 and PO the projection onto H,,. Now we know that the linear span of the coherent states Q(v) = ngO@“)/~~pl EN is dense in H’ so that PO acting on this linear span gives a dense subspace of Ho. But PO@@) = u’(p) which proves the result. Thus given the Fermi-Fock space we can construct a quotient space Hphys which is the space of Gupta-Bleuler physical states. Now H,,hys actually carries a representation of d,(N)/kern. To see this we use the isomorphism d,(S) = d,(N)/kerti and observe that as $& = ys for ys E S, so cr(6,J and 0(6,,)* = o(&,,) leave H’ and H” invariant. Hence a(&) induces a map on the quotient Hphys and therefore gives a representation of the algebra of observables Af(N)/kerfii,. We compute, using the indefinite form on H’ the generating functional : (Q+H”,
o(&,,>Q+H”)
= (Q, a(4&2)
= exp(-~lIysl12).
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Thus as expected we obtain the Fock representation of d:(S). We conclude that GuptaBleuler and Fermi lead to the same representation of d:(N)/ker7c,. In particular we can check that on Hphys, the Gupta-Bleuler annihilation and creation operators coincide with the Fermi annihilation and creation operators. The above analysis allows us to deduce that the Fermi method is essentially a GuptaBleuler quantization in the Strocchi-Wightman sense. To see this we take the covariant representation of the Fermi method described in [I], [2]. This consists of a direct integral K=
~K,M?9 X
where X is a mass shell (with mass = 1) with invariant measure dp and Kp (p E X) is the Fock space for the representation o, of d:(M) with generating functional cp + exp(-~ll~1~%l12) (AP is a Lorentz transformation
taking (0, 0, I, 1) E X, p = (‘JJ,pO)). Realise Kp as co~,o~*o,~,o
..‘,
(7.4)
where MP z A4 is the completion of M,,, in the inner product (91, Y’)p = NY, ‘l,,,JF(l(p)-l~‘)+iB(gl,
pl’).
That is MP is M regarded as a complex Hilbert space with complex structure given by
Jp = &,JF~;. On K we have a representation
of d:(M)
given by
(c(&) F)(P) = %0,)%)
f
(7.5)
So we can define H; = H,” = H;’ = H;i,,.s =
where Q, is the vacuum in Kp and the tilde takes account of the fact that the tensor product in (7.4) is not the usual one. The analysis of the first part of this section carries over to each Kp and on Hi/H;’ we have a Gupta-Bleuler quantization. It is easy to check that Js and Jp coincide on Hnqlys. If we form K’ = j H;, P
K” = j H;‘, P
then the quadruple (R,, K, <, >, 0
Kphys = K/K”,
A. L. CAREY,
436
J. M. GAFFNEY,
and C. A. HURST
where v -+ R,(yl) is the Wightman field defined by the representation (7.5) and (, ) is the inner product on K, forms a Gupta-Bleuler gauge in the Strocchi-Wightman sense except that the vacuum is degenerate. Nevertheless the above analysis shows that:
and so this degeneracy is trivial as we have a direct integral of identica1 Hilbert spaces. Since K’/K” we can define a Strocchi-Wightman
f@?) 3 ‘?,
ES L= (A-, ~JLL(~))QH’/H”
gauge transformation 0 E H’fH”,
by
J’E L’(X 44P)).
Acknowledgement One of us (A.L.C.) acknowledges the financial support of a Rothman’s Fellowship. REFERENCES Carey, A. L., Gaffney, J. M., and Hurst, C. A.: to appear, J. Math. PI7ys. 1977. Carey, A. L., and Hurst, C. A.: Adelaide preprint 1976. Strocchi, F., and Wightman, A. S.: J. Muth. Phw. 15 (1974) 2198. Bongaarts, P. J. M.: Leiden preprint 1976. Doplicher, S., Haag, R., and Roberts, J. E.: Con7777rtri. Math. Phys. 13 (1969), 1; fbid. 15 (1969), 173. [6] Manuceau, J.: Anft. bnt. Henri Poincnre’ 8 (1968), 139. ;7] Haag, R., and Kastler, D.: J. Muth. PZrys. 5 (1969), 848. [8] Maurin, K. : General eigetrfmction e.upm7siotr.s ad 7mifar.vrepreserttations o/‘ ~opologicol groups, PWN -Polish Scientific Publishers, Warsaw 1968. 191 Streater, R. F., and \I’iIde, I.: fv’wzlenr Physics B 24 (197b), 561. [lo] Gaffney, J. M.: Ph.D. thesis, Adelaide 1974 (unpublished). [ll] Strocchi, F., and Wightman, A. S.: Erratum, J. MntR. Phys. 17 (1976), 1930. [12] Wightman, A. S.: Private communication. 1131 Manuceau, J., Sirigue, M., Testard, D., Verbeure, A.: Commrm. Mmh. Pl7ys. 32 (1973), 231. [I] [2] [3] [4] [5]