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Quantization of massive vector fields in curved space–timeEdward P. Furlani Citation: J. Math. Phys. 40, 2611 (1999); doi: 10.1063/1.532718 View online: http://dx.doi.org/10.1063/1.532718 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v40/i6 Published by the American Institute of Physics. Related ArticlesNew proofs for the two Barnes lemmas and an additional lemma J. Math. Phys. 54, 012304 (2013) Non-relativistic Lee model in two-dimensional Riemannian manifolds J. Math. Phys. 53, 053501 (2012) Coherences of accelerated detectors and the local character of the Unruh effect J. Math. Phys. 53, 012107 (2012) A Goldstone theorem in thermal relativistic quantum field theory J. Math. Phys. 52, 012302 (2011) A connection between spin and statistics based on Galilean invariant delta functions J. Math. Phys. 51, 083522 (2010) Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors
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JOURNAL OF MATHEMATICAL PHYSICS VOLUME 40, NUMBER 6 JUNE 1999
Quantization of massive vector fields in curvedspace–time
Edward P. FurlaniDepartment of Physics, State University of New York at Buffalo, Buffalo, New York 14260
~Received 7 December 1998; accepted for publication 12 March 1999!
We develop a cannonical quantization for massive vector fields on a globally hy-perbolic Lorentzian manifold. ©1999 American Institute of Physics.@S0022-2488~99!04506-5#
I. INTRODUCTION
Rigorous theories have been developed for scalar, Dirac, and electromagnetic quantumon globally hyperbolic manifolds.1–6 However, apparently few rigorous results exist for massvector fields on such manifolds.7 A particle described by such a field is thev-meson with a massmv5783 MeV.8 In this article, we develop a canonical quantization for massive vector fielda globally hyperbolic Lorentzian manifold. The analysis is divided into two parts, the clasproblem and the quantum problem.
For the classical problem, we start with the equations of motion (d (4)d(4)1m2)A50 whichare the curved space–time generalizations of Proca’s equations. We reduce these to a hysystem (h1m2)A50, and a constraintd (4)A50. The hyperbolic system has global fundamen
solutionsEm6(1)
and a propagatorEm(1)
5Em1(1)
2Em2(1)
.9–12 We introduce a series of operatorsr (0) ,r (d) , r (d) , and r (n) that map a solutionA to its Cauchy dataA(0) , A(d) , A(d) , and A(n) ,respectively. These operators have transposesr (0)8 , r (d)8 , r (d)8 , andr (n)8 . We construct a series ooperatorsEm
(1)r (0)8 , Em
(1)r (d)8 , Em
(1)r (d)8 , andEm
(1)r (n)8 and show that these operators collective
map Cauchy data to a unique field solution. We apply these operators to obtain a global sfor Proca’s equations wherein we satisfy the constraint by restricting the Cauchy data. Ouresentation of the solution is apparently new, and especially useful for field quantizationdemonstrate. This method also applies to Maxwell’s equations. Since these equations areinterest, but not needed for our quantum problem, we treat them separately in an appendi
For the quantum problem, we start with a representation (f,p,H) of the CCRs on an arbitraryCauchy surface. We construct a space–time field operatorA in terms of the data (f,p) inaccordance with the classical initial value problem. We then pass to the Weyl formW of the CCRsand study theC* algebra of observablesA generated byW. We show thatA is independent of therepresentation on a given Cauchy surface, and also independent of the Cauchy surface. Thgeneralizes Dimock’s treatment of scalar fields.3
II. PRELIMINARY CONCEPTS
Let (M,g) be a globally hyperbolic, orientable, time-orientable, space–time consistingsmooth four-dimensional manifoldM endowed with a smooth Lorentzian metricg with signature(21,1,1,1). As a consequence of global hyperbolicity, there is a~nonunique! smooth time coor-dinatet, and (M,g) can be foliated by a one-parameter family of Cauchy surfaces$t%3 S t givingit the topologyM'R3S.2,13 Each Cauchy surfaceS inherits a smooth, proper Riemanniametric g. We label events inM using (t,x) wherexP S, and adopt the standard conventionwhich Greek subscripts apply to (M,g) taking values from 0 to 4, and Latin subscripts apply(S,g) and range from 1 to 3.
Let E (p)(M) denote the space of smooth, real-valuedp-forms on (M,g) and D(p)(M)specify such forms with compact support. These spaces have dualsE 8(p)(M) andD8(p)(M) with
26110022-2488/99/40(6)/2611/16/$15.00 © 1999 American Institute of Physics
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2612 J. Math. Phys., Vol. 40, No. 6, June 1999 Edward P. Furlani
topologies similar to those defined for the corresponding spaces onRn.10–12 We use the standardnotation^T,F& to denote the action of a distributionT on a test functionFPD(p)(M). Special-izing to TPD(p)(M),D8(p)(M) we have^T,F&5^T,F&M , where
^T,F&M[EMT`* (4)F, ~1!
is the global inner product, and* (4):D(p)(M)→D(42p)(M) is the Hodge star operator witrespect tog ((* (4))25(21)p11). On (M,g) we have an exterior derivatived(4) and codifferen-tial d (4)5* (4)d4* (4) which are the formal adjoints of one another^d(4)A,F&M5^A,d (4)F&M ,where APD(p21)(M) and FPD(p)(M). There is also the D’Alembertianh5(d (4)d(4)
1d(4)d (4)) which is formally self-adjoint~on Minkowski spaceh5]022¹2).14,15
Let E (k)(S), D(k)(S), E 8(k)(S) andD8(k)(S) denote the corresponding spaces on (S,g). ForTP D(k)(S),D8(k)(S) we have^T,F&5^T,F&S , where
^T,F&S[EST`* (3)F, ~2!
is the global inner product on (S,g), and* (3): D(k)(S)→D(32k)(S) is the Hodge star operatowith respect to g ((* (3))251). The exterior derivatived(3) and codifferential d (3)
5(21)k* (3)d(3)* (3) on (S,g) are formal adjoints of one another^d(3)A,F&S5^A,d (3)F&S , withAPD(k21)(S) and FPD(k)(S). The Laplace–Beltrami operatorn5d (3)d(3)1d(3)d (3) is for-mally self-adjoint.14,15 Finally, we work in units in whichc51.
III. PROCA’S EQUATIONS
We study the initial value problem for a vector fieldA of massmP(0, ) satisfying thecurved space–time generalization of Proca’s equations
~d (4)d(4)1m2!A50. ~3!
These equations reduce to the hyperbolic system
~h1m2!A50, ~4!
and a constraint
d (4)A50. ~5!
The system ~4! has unique fundamental solutionsEm6(1)
(p,q)PD8(1)(Mp3Mq) where
Em6(1)
(p,q):D(1)(Mq)→E (1)(Mp) (p,qPM represent space–time events andMp , Mq identify
the action ofEm6(1)
(p,q)).9,10,16These solutions satisfy
~hp1m2!Em6(1)
~p,q!5d (1)~p,q!, ~6!
where d (1)(p,q) is the Dirac 1-tensor kernel, and suppEm6(1)
(•,q),J6(q). Also A6(p)
5^Em6(1)
(p,q),F(q)&Mqsatisfies (h1m2)A65F, with
supp~A6!,J6~supp~F!!, ~7!
whereJ6(S) is the set of point in (M,g) that can be reached from the setS,M by a future/pastdirected causal curve.10
The kernelsEm6(1)
(p,q) are identified with operatorsEm6(1)
:D(1)(M)→E (1)(M) and from~6!,
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2613J. Math. Phys., Vol. 40, No. 6, June 1999 Edward P. Furlani
Em6(1)
~h1m2!5~h1m2!Em6(1)
5I . ~8!
The Em6(1)
, which are linear and continuous, give rise to transpose operatorsEm68(1)
:E 8(1)(M)
→D8(1)(M) which are also continuous.17 Sinceh1m2 is formally self-adjointEm68(1)
5Em7(1)
on
D(1)(M). Moreover, sinceEm68(1)
are linear and continuous onE 8(1)(M) we extendEm6(1)
fromD(1)(M),E8(1)(M) to E8(1)(M), i.e.,
Em7(1)
5Em68(1)
:E 8(1)~Mp!→D8(1)~Mq!. ~9!
Lastly, we introduce the propagator
Em(1)
5Em1(1)
2Em2(1)
, ~propagator!,
where (h1m2)Em(1)
50. This operator has a transposeEm8(1)
5Em18(1)
2Em28(1)
and from~9! we have
Em8(1)
52Em(1)
:E 8(1)~Mp!→D8(1)~Mq!. ~10!
Next, we introduce a series of operators that collectively map a solution of~4! to its data.Choose any Cauchy surface say$0%3S, and leti :S→M be the inclusion operator with pullbaci * . We define the operators
r (0)[ i * ~pullback!
r (d)[2* (3)i * * (4)d(4) ~ forward normal derivative!~11!
r (d)[ i * d (4) ~pullback of divergence!
r (n)[2* (3)i * * (4) ~ forward normal!,
wherer (0) , r (d) :E (1)(M)→E (1)(S), andr (d) , r (n) :E (1)(M)→E (0)(S). These operators can bapplied to any smoothp-form. The motivation for~11! comes from an analysis of Green’s identifor h1m2 ~Appendix A!.
Next, letAPE (1)(M) and define
A(0)[r (0)A, ~12!
A(d)[r (d)A, ~13!
A(n)[r (n)A, ~14!
and
A(d)[r (d)A, ~15!
with A(0) , A(d)PE (1)(S) and A(n) , A(d)PE (0)(S). Specifying A(0) , A(d) , A(n) , and A(d) isequivalent to specifying the Cauchy data forA. To see this, let (n,ei) be a surface normareference frame for (S,g), then~12! and~14! specifyAm(0,x) from which we obtain,kAm(0,x).Given,kAm(0,x), ~13! specifies,0Ak(0,x) and finally,~15! gives,0A0(0,x). Thus, the opera-tors~11! collectively map a solutionA to its Cauchy data (Am , na,aAm). Alternatively, we viewA(0) , A(d) , A(d) andA(n) as Cauchy data for~4!.
The operators~11! which are continuous and linear, give rise to continuous transpose otors r (0)8 , r (d)8 :E 8(1)(S)→E 8(1)(M), andr (n)8 , r (d)8 :E 8(0)(S)→E 8(1)(M).17 We construct thefollowing continuous operatorsEm
(1)r (d)8 , Em
(1)r (n)8 , Em
(1)r (d)8 , andEm
(1)r (0)8 , where
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2614 J. Math. Phys., Vol. 40, No. 6, June 1999 Edward P. Furlani
Em(1)
r (0)8 ,Em(1)
r (d)8 :D(1)~S!,E 8(1)~S!→D8(1)~M!, ~16!
and
Em(1)
r (d)8 ,Em(1)
r (n)8 :D(0)~S!,E 8(0)~S!→D8(1)~M!. ~17!
These play a crucial role in the classical Cauchy problem as is now shown.Theorem 1: Let A(0) , A(d)PD(1)(S), and A(n) , A(d)PD(0)(S) specify Cauchy data on
(S,g), and let mP@0, ). Then,
A 852Em(1)
r (d)8 A(0)2Em(1)
r (n)8 A(d)1Em(1)
r (d)8 A(n)1Em(1)
r (0)8 A(d) ~18!
is the unique smooth solution of(h1m2)A 850 with these data. Moreover,A 8PE (1)(M) iscontinuously dependent on the data.
Proof: First, from ~16! and ~17! we know that~18! makes sense. LetFPD(1)(M) be any1-form test function, and consider
^A 8,F&5^A(0) ,r (d)Em(1)F&S1^A(d) ,r (n)Em
(1)F&S2^A(n) ,r (d)Em(1)F&S2^A(d) ,r (0)Em
(1)F&S .
From Theorem 5 we know that ifA is a smooth solution with data~12!–~15!, then ^A 8,F&5^A,F&M which implies thatA 85A in a distributional sense. ThusA 8 is identified with theunique smooth solutionAPE (1)(M).
It remains to show thatA 8 is continuously dependent on the data. For this result, we assthat S is compact. From the first part of the proof we haveEm
(1)r (0)8 , Em
(1)r (d)8 :D(1)(S)
→E (1)(M) and Em(1)
r (d)8 ,Em(1)
r (n)8 :D(0)(S)→E (1)(S). It suffices to show that these restrictionare continuous. The same analysis applies to both sets of operators so we need only consformer. Recall thatEm
(1)r (0)8 , Em
(1)r (d)8 :E 8(1)(S)→D8(1)(M) are continuous with respect to th
weak topologies of dual spaces. Thus, the graphs of the restrictionsEm(1)
r (0)8 , Em(1)
r (d)8 D(1)(S)→E (1)(M) are closed with respect to these weak topologies and it follows that they are cwith respect to the topologies ofD(1)(S) and E (1)(M) as well. Finally, sinceD(1)(S) andE (1)(M) are Frechet spaces~assumingS is compact! it follows from the closed graph theoremthat the restrictions ofEm
(1)r (0)8 andEm
(1)r (d)8 are continuous. As noted, a similar analysis applies
the restrictions ofEm(1)
r (d)8 andEm(1)
r (n)8 and thereforeA 8PE (1)(M) is continuously dependent othe data. This presumably holds for noncompactS as well. j
Thus, we obtain a global solution to~4! as a mapping of Cauchy data. Notice that Theoremapplies form50 which we use in our study of Maxwell’s equations~Appendix B!.
Corollary 1: On D(1)(S) we have
r (0)Em(1)
r (d)8 52I , r (d)Em(1)
r (d)8 50, r(d)Em(1)
r(d)8 50, r (n)Em(1)
r (d)8 50, ~19!
r (0)Em(1)
r (0)8 50, r (d)Em(1)
r (0)8 5I , r (d)Em(1)
r (0)8 50, r (n)Em(1)
r (0)8 50, ~20!
and onD(0)(S) we have
r (0)Em(1)
r (n)8 50, r (d)Em(1)
r (n)8 50, r(d)Em(1)
r(n)8 52I, r(n)Em(1)
r(n)8 50, ~21!
r (0)Em(1)
r (d)8 50, r (d)Em(1)
r (d)8 50, r(d)Em(1)
r(d)8 50, r (n)Em(1)
r (d)8 5I . ~22!
Proof: These identities follow from~18!. For example, the first identity in~19! is obtained byapplyingr (0) to ~18! with A(d) , A(n) , A(d)50. The remaining indentities are obtained in a simifashion. h
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alue
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2615J. Math. Phys., Vol. 40, No. 6, June 1999 Edward P. Furlani
We are finally ready for the main result of this section. Notice that Theorem~1! gives a globalsolution to~4!. We apply it to Proca’s equations and obtain a global solution to the initial vproblem by restricting the data.
Theorem 2: Let A(0) , A(d)PD(1)(S), and A(n) , A(d)PD(0)(S) specify Cauchy data onSand let mP@0, ). Set
A(d)50, ~23!
and
d (3)A(d)5m2A(n) . ~24!
Then,
A52Em(1)
r (d)8 A(0)1Em(1)
r (d)8 A(n)1Em(1)
r (0)8 A(d) ~25!
is the unique smooth solution of(d (4)d(4)1m2)A50 with these data. Moreover,APE (1)(M) iscontinuously dependent on the data. When m.0 ~Proca’s equations) we satisfy (24) with arbtrary A(d) and A(n)5@d (3)A(d) /m2#, thus
A52Em(1)
r (d)8 A(0)1Em(1)S d(4)d (4)
m21I D r (0)8 A(d) ~m.0!, ~26!
where we have usedr (d)8 d (3)5d(4)d (4)r (0)8 on D(1)(S). When m50 we satisfy (24) with arbitraryA(n) , and with A(d) satisfyingd (3)A(d)50 ~see Maxwell’s equations in Appendix B).
Proof: We show thatA is a smooth solution of (h1m2)A50 with d (4)A50. The formerfollows from Theorem 1. For the latter, it suffices to show that^d (4)A, f &M50, for all fPD(0)(M). Consider,
^d (4)A, f &M5^A,d(4)f &M ,
5^A(0) ,r (d)Em(1)
d(4)f &S2^A(n) ,r (d)Em(1)
d(4)f &S ,
2^A(d) ,r (0)Em(1)
d(4)f &S . ~27!
Now, sinceEm(1)
d(4)5d(4)Em(0)
on D(0)(M) we haver (d)Em(1)
d(4)f 50 and the first term on theright-hand side of~27! is zero. Consider the second term on the right-hand side of~27!,
r (d)Em(1)
d(4)f 5 i * d (4)Em(1)
d(4)f ,
5 i * d (4)d(4)Em(0)
f ,
5 i * ~h2d(4)d (4)!Em(0)
f ,
52r (0)m2Em
(0)f , ~28!
where we have used (h1m2)Em(0)
f 50, andd (4)Em(0)
f 50 (Em(0)
f is a function!. Substituting~28!
into ~27! and making use ofr (0)Em(1)
d(4)f 5d(3)r (0)Em(0)
f we have
^d (4)A, f &M5^A(n) ,r (0)m2Em
(0)f &S2^A(d) ,d(3)r (0)Em
(0)f &S
5^m2A(n)2d (3)A(d) ,r (0)Em(0)
f &S ,
50,
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2616 J. Math. Phys., Vol. 40, No. 6, June 1999 Edward P. Furlani
where, in the last step we have used~24!. Thusd (4)A50 which is compatible with~23!.Lastly, from Theorem 1 we know thatA is unique. We also have thatA is continuously
dependent on the data whenS is compact. j
From Theorem 2 we have the following additional result which is need for the quanproblem.
Corollary 2: The operator identity
S d(4)d (4)
m21I D Em
(1)52Em
(1)r (d)8 r (0)S d(4)d (4)
m21I D Em
(1)1Em
(1)S d(4)d (4)
m21I D r (0)8 r (d)Em
(1), ~29!
holds onD(1)(M) for m.0.Proof: Let FPD(1)(M). It is easy to check that (d(4)d (4)/m2 1I )Em
(1)F is a smooth solutionto Proca’s equations. Thus,~29! follows from Theorem 2 withA5@(d(4)d (4)/m2) 1I #Em
(1)F andthe fact thatr (d) @d(4)d (4)/m2# Em
(1)F50. j
This last result completes the prerequisite classical work. We proceed to the quantumlem.
IV. THE QUANTUM PROBLEM
Our approach to field quantization closely follows the work of Dimock.3 We start with arepresentation (f,P,H… of the CCRs on an arbitrary Cauchy surfaceS. Let h denote the comple-tion of smooth complex-valued 1-forms onS with respect to the normi•ih
25^•,•&h , where
^F,G&h5E ~ F,G!x dt, ~30!
and
~F,G!x5Fn~x!Gn~x!, ~n51,2,3! ~31!
with dt5Agdx1`dx2`dx3, wherex5(x1,x2,x2) are local coordinates for (S,g). Next, con-struct the Bose–Fock spaceH over h,
H5C% S %n51
`
h(n)D , ~32!
where h(n)5 ^ snh, and the subscripts denotes the symmetric tensor product. Leta(•), a* (•)
denote the usual creation and annihilation operators defined on finite particle vectors inH with@a(F),a* (G)#5^F,G&h .18 Let
f~F 
a
2617J. Math. Phys., Vol. 40, No. 6, June 1999 Edward P. Furlani
A[2Em(1)
r (d)8 f1Em(1)S d(4)d (4)
m21I D r (0)8 P , ~35!
in accordance with the classical initial value problem~Theorem 2 withm.0). This holds in adistributional sense, therefore,
A~F!5f~r (d)Em(1)F!2PS r (0)Fd(4)d (4)
m21I GEm
(1)FDfor FPD(1)(M). This makes sense becauseEm
(1)FPE (1)(M) andJ6(supp(F))ùS is compact.3
Theorem 3: Let (f,P,H) be a representation of the CCRs overD(1)(S), and letA be thefield operator (35) with test functionsFPD(1)(M). ThenA satisfies the Proca’s equations indistributional sense,
~d (4)d(4)1m2!A50, ~36!
and
@A~F!,A~F8!#52 i K F,S d(4)d (4)
m21I D Em
(1)F8LM
. ~37!
Proof: First we verify ~36!. Consider,
~d (4)d(4)1m2!A~F!5A~~d (4)d(4)1m2!F!,
5f~r (d)Em(1)
~d (4)d(4)1m2!F!
2PS r (0)Fd(4)d (4)
m21I G ~d (4)d(4)1m2!Em
(1)FD ,
52f~r (d)d(4)d (4)Em
(1)F!2P~r (0)@h1m2#Em(1)F!,
50,
where we have usedr (d)d(4)50 and (h1m2)Em
(1)50 on E (1)(M).
Next, we verify~37!. Consider,
@A~F!,A~F8!#52 i K r (d)Em(1)F,r (0)S d(4)d (4)
m21I D Em
(1)F 8LS
1 i K r (0)S d(4)d (4)
m21I D Em
(1)F,r (d)Em(1)F8L
S
,
5 i K F,Em(1)
r (d)8 r (0)S d(4)d (4)
m21I D Em
(1)F8LM
2 i K F,Em(1)S d(4)d (4)
m21I D r (0)8 r (d)Em
(1)F8LM
,
52 i K F,S d(4)d (4)
m21I D Em
(1)F8LM
,
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2618 J. Math. Phys., Vol. 40, No. 6, June 1999 Edward P. Furlani
where we have applied Corollary 2. j
Notice from ~7! that
suppS Fd(4)d (4)
m21I GEm
(1)F8D ,J1~supp~F8!!øJ2~supp~F8!!,
and therefore~37! implies causality. Also,~37! reduces to the usual relation on Minkowsspace.19
We now show that a representation (f,P,H) on S induces a representation on any othCauchy surface.
Corollary 3: Let (f,P,H) be a representation of the CCRs overD(1)(S), and letA be the
field operator (35). LetS be another Cauchy surface inM, and letf and P be data ofA on S,i.e.,
f[r (0)A,
and,
P[r (d)A.
Then(f,P,H) is a representation of the CCRs overD(1)(S). Furthermore, let
A[2Em(1)
r (d)8 f1Em(1)S d(4)d (4)
m21I D r (0)8 P, ~38!
then
A~F!5A~F!. ~39!
Proof: We first show that (f,P,H) is a representation. LetFPD(1)(S), then
f~F !5A~ r (0)8 F !,
5f~r (d)Em(1)
r (0)8 F !2PS r (0)Fd(4)d (4)
m21I GEm
(1)r (0)8 F D , ~40!
and
P~F !5A~ r (d)8 F !,
5f~r (d)Em(1)
r (d)8 F !2PS r (0)Fd(4)d (4)
m21I GEm
(1)r (d)8 F D . ~41!
These make sense becauseEm(1)
r (0)8 ,Em(1)
r (d)8 :D(1)(S)→ E (1)(M) ~Theorem 1! andJ6(supp(F))ùS is compact.3 Consider,
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red
ntation,
2619J. Math. Phys., Vol. 40, No. 6, June 1999 Edward P. Furlani
@f~F !,P~F8!#52Ff~r (d)Em(1)
r (0)8 F !,PS r (0)Fd(4)d (4)
m21I GEm
(1)r (d)8 F8D G
1Ff~r (d)Em(1)
r (d)8 F8!,PS r (0)Fd(4)d (4)
m21I GEm
(1)r (0)8 F D G
52 i ^r (d)F,r (0)F 8&S1 i ^r (0)F,r (d)F 8&S
[ iVS~F,F 8! ~42!
whereF[$@d(4)d (4)/m2# 1I %Em(1)
r (0)8 F andF 8[$@d(4)d (4)/m2# 1I %Em(1)
r (d)8 F8 are smooth solu-tions of Proca’s equations. It follows from Green’s identity thatVS(•,•) is independent of theCauchy surface for such solutions~see the vector potential formulation of Maxwell’s equations!.2,4
Thus,~42! is independent of the Cauchy surface and we have
@f~F !,P~F8!#5 iVS~F,F 8!
52 i ^r (d)F,r (0)F 8&S ~43!
1 i ^r (0)F,r (d)F 8&S
5 i ^F,F8&S , ~44!
where in the last step we have applied Corollary~1! to S. Thus (f,P,H) is a representation.We now verify ~39!. Substitute~40! and ~41! into ~38! and obtain
A~F
CCRs,ch
et
f all
s
2620 J. Math. Phys., Vol. 40, No. 6, June 1999 Edward P. Furlani
and independent of the Cauchy surface. To this end, we pass to the Weyl form of theconstruct an algebra of observablesA, and exploit known results on the equivalence of sualgebras.
Given a representation (f,P,H) let
W~F,F8!5exp~ i ~f~F !2P~F8!!!.
ThenW(•,•)is a map fromD(1)(S)3D(1)(S) to unitary operators onH satisfying
W~F,F8!W~G,G8!5W~F1G,F81G8!exp~2 i /2 sS~~F,F8!,~G,G8!!!, ~45!
where
sS~~F,F8!,~G,G8!
e
r fieldsnitial
eldntationis field
,tors,
2621J. Math. Phys., Vol. 40, No. 6, June 1999 Edward P. Furlani
From this last result, we see that the Fock representation (f,P,H) defined by~33! and ~34!gives rise to an algebra of observablesA that is unique up to* 2 isomorphism. This concludes thquantum problem.
V. CONCLUSION
We have obtained classical and quantum results for the propagation of massive vectoon a globally hyperbolic Lorentzian manifold. Our classical results include solutions of the ivalue problem for Proca’s equations~3!, the vector Klein Gordon equation~4!, and Maxwell’sequations~Appendix B!. The form of these solutions is apparently new and useful for fiquantization. Our quantum results include a causal field operator constructed from a represeof the CCRs on an arbitrary Cauchy surface. The algebra of observables generated by thoperator is independent of the representation and independent of the Cauchy surface.
ACKNOWLEDGMENT
The author would like to thank Professor J. Dimock for numerous helpful discussions.
APPENDIX A: GREEN’S IDENTITY
In this section we develop Green’s identity forh1m2. Start with Stoke’s Theorem,
EO
d(4)G5E]O
i *G, ~A1!
whereO,M, ]O is the boundary ofO, i :]O→O is the natural inclusion,i * is the pullback, andGPD(3)(M).20–22 Let G (1)5d (4)F`* (4)A2d (4)A`* (4)F and G (2)5A`* (4)d(4)F2F`* (4)d(4)A, with A, FPD(1)(M). We apply~A1! with G5G (1)1G (2) and obtain
EOA`* (4)hF2F`* (4)hA52E
]Oi * ~A`* (4)d(4)F1d (4)A`* (4)F!
1E]O
i * ~d (4)F`* (4)A1F`* (4)d(4)A!. ~A2!
Finally, addA`* (4)m2F2F`* (4)m2A to the left-hand side of~A2! and obtain Green’s identityfor (h1m2),
EOA`* (4)~h1m2!F2F`* (4)~h1m2!A52E
]Oi * ~A`* (4)d(4)F1d (4)A`* (4)F!
1E]O
i * ~d (4)F`* (4)A1F`* (4)d(4)A!.
~A3!
Next, apply~A3! to the regionsO5S6[J6(S)\S, with ]O5S and obtain
ES6A`* (4)~h1m2!F2F`* (4)~h1m2!A57$^r (0)A,r (d)F&1^r (d)A,r (n)F&S
2^r (n)A,r (d)F&S2^r (d)A,r (0)F&S , ~A4!
where r (0) , r (n), r (d) , and r (d) are as defined in~11!, and, for example,*Si * d (4)F`* (3)
(2* (3)i * * (4))A5^r (d)F,r (n)A&S ~the standard orientation is used for both regionsS6).The action ofr (0) andr (d) is obvious, however,r (n) andr (d) are more subtle. Specifically
r (n) and r (d) are the forward normal, and pullback of the forward normal derivative opera
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ent
ft
2622 J. Math. Phys., Vol. 40, No. 6, June 1999 Edward P. Furlani
respectively. To see this, letxm5(t,xi), thenem5]xm represents the standard basis for the tangspace of (M,g). Let (S,g) be a Cauchy surface with forward normaln. Following the presen-tation of Misner, Thorne and Wheeler we haven5nmem ,
n5N21~] t2Ni]xi !
where N5(2g00)2(1/2) is the lapse function, andNi5gikgok are the components of the shivector.14 Notice that n52N dt in covariant form, and recall thatAugu5NAg. Now, let A5Amdxm and consider,
r (n)A52* (3)i * * (4)A mdxm,
52* (3)i *1
3!AuguA memabhdxa`dxb`dxh,
52* (3)1
3!NAgA 0e0i jkdxi`dxj`dxk,
521
3!NgA 0e0lmne i jkg i l g jmgkn,
52NA 0,
5A~n!,
whereemabh is the Levi Cevita symbol.14
For the analysis ofr (d) it is convenient to work with a basisem wheree05n andei5]xi. Thedual for this basis isvm where v05Ndt and v i5dxi1Nidt.14 We also haveugu5g. Let A5Amvm, and F5dA5 (1/2!)Fmnvm`vn. It follows that F0s5F(e0 ,es)5F(nmem ,es)5nmFms and therefore,
nmF msdxs5r (0)F~n,• !. ~A5!
Consider,
r (d)A52* (3)i * * (4)d(4)Amvm,
52* (3)i * * (4)1
2!Fmnvm`vn,
52* (3)i *1
2!Augu
1
2!gmagnbFabemnstv
s`vt,
52* (3)Agg0agibFabe0i jkdxj`dxk,
5g1
2!F0sg
isg jmgkne0imne jkpdxp,
51
2!F0ses jke jkpdxp,
5F0sdxs,
5naF asdxs
5r (0)~d(4)A!~n,• !,
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ifically,
on
al
m-ion for
2623J. Math. Phys., Vol. 40, No. 6, June 1999 Edward P. Furlani
where in the last step we have used~A5!.We are finally ready to proveTheorem 5: Let A be a smooth solution of(h1m2)A50 with Cauchy data A(0)[r (0)A,
A(d)[r (d)APD(1)(S), and A(n)[r (n)A, A(d)[r (d)APD(0)(S). Then,
^A,F&M5^A(0) ,r (d)Em(1)F&S1^A(d) ,r (n)Em
(1)F&S2^A(n) ,r (d)Em(1)F&S2^A(d) ,r (0)Em
(1)F&S ,~A6!
for any test functionFPD(1)(M).Proof: Since (h1m2)A50, the second term on the left-hand side of~A4! is zero. For the
remaining integrals, substituteF5Em7(1)F 8 for the regionsS6, respectively. All integrals are
well-defined because they entail integrations of smooth functions over compact sets. Specfor the left-hand side of~A4! we have
supp~Em7(1)F 8!,J7~supp~F 8!!,
with J7(supp(F 8))ùJ6(S) compact (M is globally hyperbolic!, and for the right-hand side,Sis compact by assumption.3 Next, sum the integrations over theS6 regions substituting (h
1m2)Em6(1)
5I in S6 integrals andEm(1)
5Em1(1)
2Em2(1)
in S integrals. The sum of theS6 inte-grals gives an integral overM (S constitutes a set of measure zero relative to this integrati!.Finally, relabelF 8→F and obtain~A6!. This completes the proof. j
APPENDIX B: MAXWELL’S EQUATIONS
In this section we study Maxwell’s equations,
d(4)F50 ~B1!
and
d (4)F50, ~B2!
whereF is the field strength 2-form~not a test function as above!. We pose an initial valueproblem for these equations following the presentation in Wald.13 Specifically, we specify theinitial data forF in terms of the electric and magnetic fieldsE[r (n)F andB[r (0)F, respectively,which are regarded as a 1-form and 2-form onS, respectively. This data satisfies additionconstraintsd (3)E50 andd(3)B50. Given these data, we obtain a field solutionF that satisfies~B1! and ~B2! onM. Our approach is similar to Dimock’s; the differences being that our ephasis is on the fields rather than the vector potential, and we give an explicit representatF.4
Before we proceed, we make a further restriction onM. Specifically, we assume thatS iscompact and contractible. IfS is contractible then any closed p-form onS is exact, that is,KPD(p)(S) with d(3)K50⇒K5d(3)H for someHPD(p21)(S) (p.0).20,21
Theorem 6: Let EPD(1)(S) and BPD(2)(S) be data for the field strengthF, i.e.,
r (n)F5E, ~B3!
and
r (0)F5B, ~B4!
where
d (3)E50, ~B5!
and
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ret
ique
2624 J. Math. Phys., Vol. 40, No. 6, June 1999 Edward P. Furlani
d(3)B50. ~B6!
Then, given these data, there is a smooth potentialA such thatF5d(4)A satisfies Maxwell’sequations d(4)F50 and d (4)F50, as well as (B3)–(B6). Moreover, any two such potentials agauge equivalent i.e., they differ by the exterior derivative of a scalar. Here we assume thaS iscompact and contractible.
Proof: Existence:We choose data forA as follows:
A(d)5E, ~B7!
d(3)A(0)5B, ~B8!
with
d (3)A(d)50, ~B9!
A(d)50, andA(n) arbitrary. These choices of data are compatible with the field constraints~B5!,and~B6! in that ~B7! and~B9! imply ~B5!, and~B8! implies ~B6!. Regarding~B8!, we know thatsuch anA(0) exists becauseS is contractible andB is exact. These choices of data satisfy~23! and~24! of Theorem 2 for them50 case. Consequently, there is a unique smoothA that satisfies
d (4)d(4)A50, ~B10!
andF5d(4)A is our desired field strength. Next we show thatF renders the data~B3! and~B4!.From Theorem 2 we have
A52E0(1)
r (d)8 A(0)1E0(1)
r (d)8 A(n)1E0(1)
r (0)8 A(d) . ~B11!
Consider,
r (n)F52r (n)d(4)E0
(1)r (d)8 A(0)1r (n)d
(4)E0(1)
r (d)8 A(n)1r (n)d(4)E0
(1)r (0)8 A(d) ,
52r (d)E0(1)
r (d)8 A(0)1r (d)E0(1)
r (d)8 A(n)1r (d)E0(1)
r (0)8 A(d) ,
5A(d) ,
5E,
and
r (0)F52r (0)d(4)E0
(1)r (d)8 A(0)1r (0)d
(4)E0(1)
r (d)8 A(n)1r (0)d(4)E0
(1)r (0)8 A(d) ,
52d(3)r (0)E0(1)
r (d)8 A(0)1d(3)r (0)E0(1)
r (d)8 A(n)1d(3)r (0)E0(1)
r (0)8 A(d) ,
5d(3)A(0) ,
5B,
where we have used the results of Corollary 1.Uniqueness: We want to show that any two potentialsA 8 andA that satisfy~B10! with data
~B7!–~B9! are gauge equivalent, i.e.,
A 85A1d(4)f ,
where f PD(0)(M).4 It suffices to show that any solution is gauge equivalent to the unsolutionA above. Consider a solutionA 8, we want to show thatf 8 exists such that
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.
2625J. Math. Phys., Vol. 40, No. 6, June 1999 Edward P. Furlani
A5A 81d(4)f 8.
SinceA is unique, it suffices to constructf 8PD(0)(M) such thatA 81d(4)f 8 satisfies
h~A 81d(4)f 8!50 ~B12!
onM with the same data asA on S. First, notice that~B12! is equivalent to
d (4)d(4)A 81d(4)d (4)A 81d(4)d (4)d(4)f 850. ~B13!
The first term in~B13! is zero becauseA 8 satisfies~B10!. Therefore~B12! reduces to
h f 852d (4)A 8. ~B14!
Next, we study the data. Recall from Theorem 2 that the unique solutionA has dataA(d) , A(n) ,A(0) , andA(d) , where
A(d)50, ~B15!
d (3)A(d)50, ~B16!
andA(0) andA(n) are arbitrary. We need to constructf 8 so thatA 81d(4)f 8 renders the same dataTo this end, we specify
r (d)~A 81d(4)f 8!50, ~B17!
r (n)A5r (n)~A 81d(4)f 8!, ~B18!
r (0)A5r (0)~A 81d(4)f 8!, ~B19!
and
r (d)A5r (d)~A 81d(4)f 8!. ~B20!
To satisfy~B17! we impose~B14!. The conditions~B18! and ~B19! are satisfied when
r (d) f 85r (n)~A2A 8!, ~B21!
and
d(3)r (0)f 85r (0)~A2A 8!, ~B22!
respectively. Regarding~B22!, we know that such ar (0)f 8 exists becauseS is contractible and, byassumption,r (0)(A2A 8) is exact, i.e.,d(3)r (0)(A2A 8)5B2B50. Finally, ~B20! is satisfiedbecause, by assumption,A andA8 satisfy ~B7!, and we know thatr (d)d
(4)f 850.Now, by assumption, we are givenA andA8 so we view~B14!, ~B21! and ~B22! as speci-
fying a Cauchy problem for the scalar fieldf 8. That is,~B21! and~B22! specify the Cauchy datar (d) f 8 andr (0)f 8 for the nonhomogeneous linear hyperbolic equation~B14!. A unique solution tothis problem is known to exist which gives us the desiredf 8.13 This shows that any solutionA 8
is gauge equivalent to the uniqueA and therefore, given two different solutionsA 8 and A wehaveA5A 81d(4)f 8, andA5A1d(4) f which shows thatA 85A1d(4)f , wheref 5 f 2 f 8. Thusany two such solutions give rise to the same field strengthF. This completes the proof. j
We obtain an explicit expression forF as follows:Corollary 4: LetA be a vector potential with data A(0) , A(d) , A(n) , and A(d) satisfying the
conditions of Theorem 6. The field strength is given by
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l
t
,
charge,
No. 10,
-
2626 J. Math. Phys., Vol. 40, No. 6, June 1999 Edward P. Furlani
F52d(4)E0(1)
r (d)8 A(0)1d(4)E0(1)
r (0)8 A(d) , ~B23!
whereFPE (2)(M) is continuously dependent on A(0) , A(d)PD(1)(S).Proof: From Theorem 2 we have
A52E0(1)
r (d)8 A(0)1E0(1)
r (d)8 A(n)1E0(1)
r (0)8 A(d) ~B24!
with
d (3)A(d)50. ~B25!
We show that~B23! equalsd(4)A. Let GPD(2)(M) be a 2-form test function, and consider,
^d(4)A,G&M5^A,d (4)G&M ,
5^A(0) ,r (d)E0(1)
d (4)G&S2^A(n) ,r (d)E0(1)
d (4)G&S
2^A(d) ,r (0)E0(1)
d (4)G&S
5^2d(4)E0(1)
r (d)8 A(0)1d(4)E0(1)
r (0)8 A(d) ,G&S ,
where, in the last step we have usedr (d)E0(1)
d (4)G50. Thus,~B23! is satisfied in a distributionasense. From Theorem 1 we know thatE0
(1)r (0)8 , E0
(1)r (d)8 :D(1)(S)→E (1)(M) are continuous, and
we also know thatd(4):E (1)(M)→E (2)(M) is continuous, thereforeFPE (2)(M) is continuouslydependent onA(0) andA(d) . j
From this final result, we see thatF depends only on the dataA(0) andA(d) . Thus, we can seA(n)50 in ~B24!. This choice of data is useful when quantizing the electromagnetic field.4
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1983!.
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