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Canonical quantization with indefinite inner product
ANNALS
OF PHYSICS
Canonical
161, 314-336 (1985)
Quantization
with
Indefinite
Inner
Product
LECH JAKOBCZYK Institute
of Theoretical Physics, Unillersity of Wroclaw, Cyhulskiego 36, 50-205 Wroclaw, Poland
Received June 12, 1984
In this paper an algebraic approach to canonical quantization on indefinite inner product space is presented. Concrete realization of such a quantization of a given classical system described by a symplectic space (M, a) is obtained by means of a so-called P-representation of CCR algebra d(M, u). So-called Fock-Krein representations of d(M, a) determined by some class of complex structures on (M, a) are studied in detail. It is shown that every Fock-Krein representation is unbounded. Starting with a fixed Fock-Krein representation of CCR sectors of non-Fock-Krein representations are constructed. ‘i-1 1985 Academic PRSS, hc.
I. INTRODUCTION
Recently, there is growing interest in the mathematical aspects of indefinite inner product quantum field theory (IIPQFT), especially in the context of a rigorous approach to the quantization of gauge fields. A pioneering work in this area is the paper of Strocchi and Wightman [I] concerning axiomatic formulation of quantum electrodynamics. Other papers by Strocchi and Morchio generalize the ideas formulated in Ref. [ 11 to the case of gauge fields and study such structural properties of the theory as: reconstruction of a field theory from a set of Wightman functions [2], PCT invariance, properties of local states and analytic continuation to the euclidean points [3], breaking of the symmetry and Higgs mechanism [4], etc. On the other hand, there are also interesting results concerning mathematical structure of general IIPQFT. It is worth mentioning the following: rigorous construction of free field theory on the Fock space with indefinite inner product [S-7], investigations of representations of the symmetry group on the Krein space [S, 93 and algebraic formulation of IIPQFT [ 10, 111. The present paper is also connected with this subject, but our intention is to develop the idea of (algebraic) canonical quantization with indefinite inner product (IIP) of linear systems. The starting point is the classical description of the system in terms of linear symplectic space (A4, a) and symplectic operators on it. In the case of a finite number of degrees of freedom M is a phase space of a mechanical system and 0 is a natural symplectic form on it [12]. For the system described by some (linear) differential equations (field equations), M is the set of real solutions to this equation and 0 is a non314 0003-4916/85 $7.50 CopyrIght (’ I985 by Academic Press. Inc. All nghls of reproductmn in any form reserved.
QUANTIZATION
WITH
INDEFINITE
PRODUCT
315
degenerate, antisymmetric, bilinear form on these solutions [13]. Over the symplectic space (M, a) we construct the abstract algebra d(M, a) containing all information about commutation relations of the elements of a quantum system corresponding to the classical one described by (M, a). From the algebraic point of view, the quantization of a classical description can be performed in purely algebraic terms, i.e., in terms of the algebra .4(M, a) and *-automorphisms of d(M, 0) representing the symmetries of the system [ 141. Concrete realization of this quantization procedure is obtained by the choice of (concrete) covariant representation of the algebra d(M, (z). If we take the standard representation on a Hilbert space X, we obtain (canonical) quantum field theory on X (or the ordinary quantum mechanics in the case of finite number of degrees of freedom). However, there is a possibility to take some “non-standard” representation of this algebra on the linear space with IIP, and in this way we can obtain a “canonical” field theory with IIP, or some “non-standard” quantum mechanics on the Krein space (quantization with IIP of some class of mechanical systems was recently discussed by Broadbridge [ 151, see also Section IV). The above idea is examined in detail in the present paper. After mathematical preliminaries (Section II) in Section III we formulate main ideas of our approach to quantization with IIP. First of all, we study general properties of so-called J*-representations of an *-algebra &. It is shown that such representations are determined by some non-positive functionals on the algebra d via the generalized version of Gelfand-Naimark-Segal construction (Theorem 111.1). Then we consider J*-representations of the algebra d(M, a). It turns out that we can characterize them by means of so-called K-positive functions on the symplectic space (M, a) (Theorem III.3). K-positive functions defined by some complex structures on (M, a) (complex structures of type K) are studied in Section IV. Such complex structures lead to so-called Fock-Krein representations of the algebra A(M, a). It is proved that the Fock-Krein representation is always unbounded (Proposition IV. 1). The existence of complex structures of type K is also investigated. We show that when dim M= 2n < co, there always exists a complex structure of type K. We analyse also the connection of our approach with results concerning quantization of general quadratic Hamiltonians [ 151 (Proposition IV.2). In the case of infinite-dimensional symplectic space, existence of a complex structure of type K requires some additional topological assumption, namely, we must assume the existence of a Krein space structure (M, $). At the end of this section we discuss a concrete example of quantization using complex structure of type K (quantization of electromagnetic field). In Section V we study equivalence of J*representations of CCR. Our definition of the notion of equivalence is a natural generalization of the notion of unitary equivalence in the standard case. Using the result of Ito [7] we show that Fock-Krein representations defined by two complex structures of type K, J, and Jz = TJ, T-’ (T is a symplectic operator commuting with K) are equivalent iff T- = $( T- J, TJ; ‘) is a Hilbert-Schmidt operator. Next we construct some kind of “coherent” representation starting from the Fock-Krein one. The classical result of Roepstorff concerning equivalence of such represen-
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LECHJAKOBCZYK
tations is generalized (Theorem V.l ), showing the existence of sectors of nonFock-Krein representations of CCR in quantum field theory with IIP.
II. MATHEMATICAL
PRELIMINARIES
1. Symplectic Space [16] Let M be a real vector space and o a symplectic form on M (i.e., o is a bilinear, antisymmetric, non-degenerate form on M). The pair (M, a) will be called a (uector) symplectic space. If dim M< GO (dim M= 2~) then there exists a symplectic base in M, i.e., a set of vectors (F, ,..., F,}, {E, ,..., E, j satisfying o(F,, Fk) = o(E,, Ek) = 0,
o(Fj, Ek) = & 3
j, k = 1,..., n.
(11.1)
A symplectic operator T on (M, a) is an operator from A4 onto M satisfying, fo every F, GE M, a( TF, TG) = a(F, G).
(II.2
(,SP(M, a) denotes the set of all symplectic operators on (M, a).) A complex structure J on (M, 0) is a symplectic operator satisfying J’=
-1,
4 F, JF) 3 0,
FEM.
(11.3)
2. The CCR Algebra A(M, g) [17]
Let (M, a) be the symplectic space. Let d(M, a) be the (complex) generated by functions 6,: M-t@,
FEM
S,(P) = 1
if F=p
=o
linear space
(11.4)
if F#i?
A(M, 0) equipped with the product
(11.5) and the involution (11.6)
QUANTIZATION
WITH
INDEFINITE
PRODUCT
317
becomes a *-algebra with identity 6,. Let I(. (1I denote the norm on d(M, 0) defined by (11.7) d,(M, a) := d(M, u)“‘“~ is a *-Banach algebra. For every T E SP(A4, a), the mapping (11.8)
T T . ‘6,-+6,,
can be extended to the unique *-automorphism of d(A4, a) (continuous *-automorphism of d ,(M, 6)). Let &I$ be the set of all functions p mapping M into @ and satisfying the condition
C~~‘~p(F,-F~)~-ia(~~.Fk)~o
(11.9)
j.k for every finite sequence ((z,, F,)}, (zi, F,) E @ x M. u is a positive linear form on d( M, a) iff the function p defined by FEM
p(F) = 46F),
(11.10)
belongs to %‘,. Under this condition o is denoted by o,, and the representation of d ,(M, Q) defined by w, through the construction of Gelfand, Naimark and Segal is denoted by (z,, X0). be the representation of the group G on the symplectic space Let {T,hEG (M, a), i.e., Gsg+
The mapping
T,ESP(M,
a).
(11.11)
g + tg defined by zg:= TTp,
gEG
(11.12)
is the representation of G on the algebra d(M, 0) (d,(M, a)). A representation (rc, X,) of d,(A4, a) is G-couariant if there exists a family {U;}REc of unitary operators on n such that for any A E A,(M, C) 7r(z,(A)) 3. indefinite
= U,“n(A)
vi*.
(11.13)
Inner Product Space [18, 191
Let E be a vector space with an inner product ( ., * ). The pair (E, ( ., . )) will be called an inner product space. If the quantity (e, e), e E E, may be positive,
318
LECH
JAKOBCZYK
negative or zero, (E, ( ., . )) is called indefinite inner product space. If ( ., . ) is nondegenerate and E=E+@E-;
E’=(eEE(e,e)?O]
(11.14)
then (E, ( ., . )) is called (non-degenerate) decomposable indefinite inner product space. Let r be a topology on (E, (., . )). We say that z is a majorant of the inner product ( ., . ) if: - r is locally convex, - ( ., . ) is jointly r-continuous. A topology z on (E, (., . )) is admissible if: - for every fixed e”E E the mapping e + (e, 0) is z-continuous, - for every r-continuous linear form L on E, there is an element e,, E E such that L,(e) = (5 e. >. Let Fc E be the subspace of E. Define F’ = {e E E: (e, e”) = 0 for every I?E F}. If r is an admissible topology on (E, ( ., . )) then z-closure of any subspace F c E coincides FL’. A majorant topology r defined by Hilbert space norm /I. IIH is called Hilbert majorant. Hilbert majorant topology is admissible. Let Fc E be such that (., . ) is definite on F. The equality IelF:=
I(e, e))“*
(11.15)
defines the norm on F. Topology induced by 1.1F is called intrinsic topology on F. If an indefinite inner product space (E, (. , . )) admits a decomposition of the form E = E+ @E- where the subspaces E+ and E- are intrinsically complete, then it is called a Krein space. Let P’E = E*. We set r=p+
-p-
and say that I is a fundamental symmetry of E (the set of all fundamental metries of E will be denoted by i(E)). Define I-inner product (e, ?), : = (e, ZZ).
(11.16) sym(11.17)
A decomposable, non-degenerate, indefinite inner product space (E, (., . )) is a Krein space iff for every ZE i(E), the Z-inner product turns E into a Hilbert space.
QUANTIZATION III.
REPRESENTATIONS
319
WITH INDEFINITE PRODUCT
OF ALGEBRA
OF COMMUTATION
RELATIONS
1. Segal Quantization [20) Let (M, (T) be the symplectic space describing the classical system. Let ( Tg}gE G be the representation of the symmetry group G on (M, c). G-covariant canonical quantization on a Hilbert space X is by definition a pair ( W, U) of mappings W: M -+ u(X)
(111.1)
U: M-u(X) (u(X)
denotes the group of unitary operators on a Hilbert
space 2)
for every F, FE M, W(F) W(F) = eeioCF,‘) W(F+ P) the mapping
t --+ W(tF) is weakly continuous
satisfying: (111.2)
and for every FE M, g E G, W( T, F) = U, W(F) Ut the mapping g -+ U, is weakly continuous.
(111.3)
Taking a concrete G-covariant representation (x,, ZO) (with the group { U;jgEG) the algebra A ,(M, a), defined by the function p E &?,,satisfying conditions: (A) for (B) for (C) for the pair ( W,
every F, FE M, the mapping t + p(tF+ F) is continuous, anygEG,poTg=p, every F, FE M, the mapping g --, p( T,F+ P) is continuous, we obtain U} of mappings (111.1) with properties (111.2) and (111.3), if we put
W(F)= ~,GM u,= 2. Quantization
of
with Indefinite
ug,
FEM gEG.
(111.4)
Inner Product
The extension of the Segal quantization procedure to the case of a theory with IIP can be obtained by means of so-called J*-representations of *-algebras. Since it is more natural to deal with the algebra A(M, a) (the Weyl operators in the field theory with IIP can be generally unbounded ES]), we ought to consider unbounded representations of (not-normed) *-algebras. Let us recall the definition and simple properties of J*-representations of (abstract) *-algebra & [ll, 211. If a$.(&) denotes the set of all * -automorphisms of d, let j(&)caut,(&) be the set of elements tl E aut*(&) satisfying CY’= id (a # id). 5951161:2-6
320
LECH
JAKOBCZYK
111.1. (x, D(z),j(n)) is called a .Z*-representation of G? if: (A) T: &’ -+ op(D(z)) is a mapping of ral into linear operators all defined on a linear space D(X) with an indefinite, non-degenerate inner product ( ., . ),; rr is such that: (Al) for any A E&, n(A) D(n) c D(n), (A2) for every A, BE d, z, w E C, YE D(x) DEFINITION
z(AB)
Y= x(A) n(B) Y
n(zA + wB) !P=zzn(A)
(A3)
!P+ MT(B) !P,
for every A E&‘, Y, @E D(n) (111.6)
(Y 4‘4) @>,= (n(A*) y, @>,; (B) j(n)cj(&‘) (Bl) properties:
(111.5)
is such that:
for any tx~j(rt)
there is a linear operator
Z,(E): D(x) -+ D(n) with
I”,(E) = 1 (Z,(a) K @>,=
(111.7)
(u: rn(~), @>,
and such that (., .), = ( ., Z,(U) @), is positive definite, (B2) for every cr~j(z), AE&, ~/ED(Z)
44A))
Y=ZI,(a) n(A) Z,(a).
(111.8)
Remarks. (A) Let Xz=D(n)““lz (j/.Ijz = (., .),) and (., . ), is the extension of (., * ), to X,, then (X,, (a, . >,) is the Krein space. (B) If CI,/I ~j(n) and CI# /I, the norms 11.jIW and I(. lIB are not generally equivalent. (C) z(d) = {rc(A): A E &> is the example of op J*-algebra discussed in [lo]. The representation (n, D(n),j(rr)) is a-cyclic (a~j(rr)) if there is a vector a,~D(n) such that:
(A)
n(zZ)Q,
is dense in X,,
(B)
Z,(a) Q, = Q,.
Suppose that there is defined the representation g --t tg E aut,(d). J*-representation (n, D(n),j(,)) is G-couariant linear operators with the following properties:
of a group G (111.9) if there exists a family { U;}BEG of
QUANTIZATION
(A) (B) (C)
WITH
INDEFINITE
for any gEG, U~:D(z)+D(~), for every gEG, Y, @ED(~), (U;Y, for every A E&, g E G, YE D(z) 7+,(A))
CD)
for every g,,g,EG,
(E)
the mapping
Vi@),=
Y = u; n(A) q*
y~D(n),
321
PRODUCT
(Y, @),,
Y,
(111.10)
U;, U;“, Y= lJ&+,,‘,
g + ( Y, U; @) rr is continuous.
Remark. Operators U; representing the symmetry group are generally unbounded (for example, it is the case of boosts in quantum electrodynamics [S]), therefore we assume that U; are defined only on the dense domain D(z). Let the representation (a, D(rc),j(rr)) be a-cyclic (for any a~j(rc)) with a-cyclic vector 52,. Let us define the functional
(III.1 1)
%(A) = (Q,, 64) Q,>,. The functional (A) (B)
(111.11) has the properties:
co,0 a = o,, a Ej(71), cu,(a(A*)
A) 2 0, a ~j(n), A EJZ’
It suggests the following general definition: DEFINITION 111.2. A functional w on d is a-positive if there exists a ~j(sxZ) such that o 0 a = o and for any A E SQ, w(a(A *) A) > 0. If o is a-positive, then we denote P(o) = (a Ej(&): 0 is a-positive}.
Suppose now that a-positive functional w on d is also G-invariant, geG UOTg=U for some representation
i.e., for any (111.12)
of G of type (111.9). Then for any a E P(o) ag=7gnaot1
(111.13)
R
belongs to the set cP(o). g(w) is in this case the sum of orbits of the form {a,:gEG,
Generalizing GNS construction o, one can obtain:
a is fixed}.
(111.14)
to the case of a-positive and G-invariant
THEOREM III.1 [ 11,211. Let w be a-positive Assume that for every A, B, C E -02, the mapping
g + wW,(B)
and G-invariant
C)
functional
functional
on d.
(111.15)
322
LECHJAKOBCZYK
is continuous. Then there exists G-covariant representation (z,, D,, j,) of d (with the symmetry group { U,“}g Eo). It is cr-cyclic for any LYEj,, Moreover, for any g E G
is the unitary operator between Hilbert spaces(Xg,
(., .),) and (X;,
(., ‘),,).
ProoJ: Let
(111.17)
Do,= d/L(w) with L(o)=
{AEs?:~(BA)=O
for any BE&‘}. (Yy,,
Y,),
Let us define (111.18)
:= o(A*B)
where Y, denotes the equivalence class in D, containing A E JZ?‘.It is easy to see that (D,, ( ., . ),) is an indefinite inner product space. If we put
%(A) y,:= ~A, ZoJ,(a)yy,:=
(111.19) a Ej, : = P(u)
ymy,,,,;
then (rr,, D,, j,) becomes a J*-representation x-cyclic vector
(111.20)
of d. It is cc-cyclic for any CIEj, with
sz:= Y,.
(III.21 )
Let us also define u,w!P,:=
It is easy to check that (n,, D,, j,,,) is G-covariant Besides
q%,
(111.22)
Y@). with the group
u,wy,),p=W(a~(ZR(A))*Zg(B))=w(a(A*)B)=(y’,,
Yu,),
{ UF}~~~. (111.23)
and for any g@, U,W can be extended3 the unitary operator between Hilbert spaces (X: = D, II 11=, (., .),) and (X;=DoI”I’u~, (a, .),). Remark. If we take CIE P(w) such that rg occ#aoz,andifthenorms IJ.JI.and 11.I(‘l;r are not equivalent, then U,” is the unbounded operator on Z’z. It is worth stressing that in the Fock space, the norms defined by different fundamental symmetries I and 7 are always inequivalent [22]. THEOREM 111.2. Let (x,, D,, j,) be a G-covariant J*-representation of d defined as in Theorem III. 1. For every c1Ej,,, there exists a J*-representation (XL, DL, (a}) (a-extension of (zW, D,, j,)) defined asfollows:
QUANTIZATION
WITH
D;:=
n
INDEFINITE
323
PRODUCT
(111.24)
D@,(A)“)
AEd
7qg.4) Y:= n,(A)* Y IL(a):=
if YYED;
(111.25)
Z,(a)
(111.26)
( T-” denotes the closure of T in the norm (1.(I.). J*-representation the foIlowing properties:
(A) (B)
For every A EZZZ, n;(A) 0:
(rc;, DE, {a} ) has
is closed in X;.
is complete in the topology defined by seminorms
IIYlla.s:=
(111.27)
(where S is a finite set of elements of d). (C) Let (~2, DF*gg,(a,}) g EG. Then we have
be un a,-extension of (n,,
D,,j,)
for
lJ”:D;+D”,s g
some fixed
(111.28)
and Uptpl)
Y = n~(z,(A))
u; Y,
YED:.
(111.29)
Proof. In the Krein space X any densely defined operator T is closable iff D(T*) is dense [18] (T* denotes the adjoint to T with respect to indefinite inner
product). For every J*-representation
(n, D(rc),j(rr))
D(n) = D(n(A*))c
we have by (111.6)
D(7t(A)*)
(111.30)
hence, D(n(A)*) is dense in Xz and for every A E S# rc(A) is closable in the norm 11.II=. Let n,(A)” be the closure of n,(A). Applying the result of Powers [23, Lemma 2.61 one can easily show that (n;, Dz,) defined by (111.24) and (111.25) is a representation of d. We show that (rc;, D;, (~3) is a J*-representation of d. Let Y,@ED; and YE D(q,,(A*)“), @E D(n,(A)*). There are sequences {Y,), { @,} c D, such that YY,-+ y,
n,“(A*) Yn + 7c,(A*)OL Y
@II-*@, T”(A)@n + %o I”@ in the norm I(. )I1. Thus we obtain the relation (Y,n~(A)~),=lim(y’,,7~,(A)~,),=lim(n,(A*)Y~,~,), n n = (n”(A*)
Y, @),.
324
LECH
JAKOBCZYK
Hence, for every A E .,&, Y, @E DE,
(n"(A) y, @>,= (Y, 71"(A*) @>, Next we show that Z,(a): 0: -+ 0;. Let Y’E 0;. There is a sequence ( Y,,} c D,,] such that Y,, --t Y and rc,(A) Y”-+ rt,(A)‘Y. Let @=Z,(cl) Y, then if @,:= Z,(a) vl,, @, + @ and
Thus, there exists the limit
Since rc,,(A) is closable, Z= rc,,(A)W and @E 0;. Similarly [23], one can show that for every A E&, Y, @E 0;
as in the standard case
Thus we have the relations
and
From this it follows that
Hence we obtain the result: (n;, D;, {a}) is a J*-representation. the points (A), (B) and (C): (A) (B) (C) U” Y’, + x:(A) is
Finally, we show
By construction every n;(A) is closed. It follows from the result of Powers [23]. Let YED; and ul, -+ Y, rc,(A) Yn+zr,(A)“Y in the norm I).IICL. Then U,” Y in the norm )I. )I*~. Moreover lim, rc,(A) U,” Y,, exists in Krg. Since closable in Xg 7c,,(A)‘x Up” Y= lim n,,(A) U; Y',
n
hence U; YE D",". Besides
U,"q%4 Y==~(TJA))U;Y,
YEED;.
QUANTIZATION
WITH
TNDEFINITE
Now we are prepared to consider J*-representations T(M, CT)= {KESP(M,
325
PRODUCT
of the algebra d(M, P). Let
a): K2= 1, K# +_I}
(111.31)
and j(r) = {zK: KE r(M, We consider such J*-representations
(ret,D(z),j(z))
4).
(111.32)
of d(M, U) for which (111.33)
An) CA0
(such a choice is suggested by the case of free field theory, see also [21]). Suppose now that the representation (x, D(x),j(z)) is G-covariant (with the symmetry group ( U; jgE G). Assume moreover that for any FE M the mapping (111.34)
t + JQJ,F) is weakly continuous.
For a fixed a = rK ~j(rc) a pair { W, U> of mappings (111.35)
w: M 4 op(D(n)) U: G + op(D(n)) defined by
W(F)= 46,) u*= u’g”
(111.36)
satisfies the following conditions: (A) (B) (C) (D) (E)
Forevery FEM,Y,@EE(~),(W(F)Y,@),=(Y, W(-F)@),. For every F, F’E M, YED(x), W(F) W(P) Y=e-i”‘cF) W(F+F) The mapping t -+ ( Y, W(tF) @), is continuous. For every FE&I, YeD(rc), W(KF) Y=Z,(?,) W(F) Z,JT~) Y. For every gE G, YE D(z), FE A4
!I?
W( T, F) Y = V, W(F) U; Y.
(F) The mapping g + ( Y, U,@ ), is continuous. A pair { W, Or} of mappings W: M+
op(D,)
(111.37)
U: G + op( D w)
(Dw is a dense subspace of a Krein space X) satisfying conditions analogous to (A)-(F)
will be called G-comriant
canonical quantization
on a Krein space K.
326
LECHJAKOBCZYK
Remark. If we consider an x-extension of J*-representation of the algebra d(M, a), then we obtain a pair { W, U) when W(F) := rP(6,). The pair { W”, U) satisfies all conditions of canonical quantization on the Krein space, but now Gcovariance means: U, Wa(F) Y = W”( T, F) U, !P for all YE D(z’). 3. Realization
of J*-Representations
of CCR: K-Positive Functions
If (71,D(~),j(n)) is an a-cyclic representation (a = r,), then the function p,(F):=
of d(M, a) with u-cyclic vector Q,
(Q,, 46,)
Q,),
(111.38)
has the properties: (A) (B)
in 0 K= pn, For every finite sequence {(z,, F,)), (zj, Fj) E C x M, (111.39)
A function p mapping M into @ and satisfying (A) and (B) will be called K-positive [21]. Let us introduce the following notation: .gK: = (p: M -+ @: p is K-positive ) .J$: =
u
c4!K
KS/-(A4.u)
l-(p):=
{KEZ-(M,a):p~&}.
Now we can formulate: THEOREM
III.3
[21].
Let p E W. Assume that:
(A) P(O)= 1. (B) The mapping t + p( tF + p) is continuous. (C) For anJ’ g E G, p 0 T, = p and the mapping g + p( T,F + P) is continuous. Then, for any CI= ~~ (KE T(p)) there exists G-covariant canonical quantization on the Krein space X,. Proof:
Similarly
as in [21] let us define the functional (111.40)
327
QUANTIZATION WITH INDEFINITE PRODUCT
w, is a-positive (CI = rK, K E T(p)) and o, 0 r, = op. By Theorem III.1 there exists a G-covariant representation of d(M, a), hence a G-covariant quantization on the Krein space Xa = E’l’ ‘12. Remarks. (A) The existence of a G-covariant canonical quantization on the Krein space is now equivalent to the existence of a K-positive function on the symplectic space (M, c). K-positive functions defined by some complex structure on (M, G) will be discussed in the next section. (B) If p is positive on (M, a), then it is known [24] that for any FE M IP(F)I G P(O).
The functional (111.40) is in this case continuous with respect to the norm 11.I/ 1 and can be extended to the continuous functional on the Banach algebra d I (M, B). If p ~9, then the above inequality is generally not satisfied, and, for example, the Fock-Krein functional (Section IV) is not continuous with respect to 1).Jji . Thus, the functional wp is generally defined only on (not-normed) algebra d(M, CT)and the operators rrp(6,) are unbounded.
IV
FOCK-KREIN
REPRESENTATIONS OF COMMUTATION RELATIONS
In this section we investigate the class of J*-representations of d(M, (r) defined by complex structures on (M, a). As we know, in the standard quantum field theory, definition of the complex structure on (M, a) determines the construction of free field operators (see, e.g., an interesting discussion of this point in [25]). In our case, the similar role is played by so-called complex structures of type K. 1. Complex Structures
of Type K on a Symplectic Space
DEFINITION IV. 1. A symplectic operator JE SP(M, CJ)is a complex structure of G)) if:
type K (KET(M, (A) (B)
J*= -1, [J, K] = 0 and for any FE M, G( F, JKF) 3 0.
The set of all complex structures of type K will be denoted by yK. If JELL functional p,(F)
: = exp - $(F,
JF)
the
(IV.1)
is a K-positive. A J*-representation of d(M, (r) defined by the function (IV.l) will be called a Fock-Krein representation of type K and we denote it by (rcJ, D,, {rK}). Remarks. (A) The standard Fock representation is defined also by the function (IV. 1), but the complex structure J is of type K= 1. (B) If JE#~ satisfies [T,, J] =0 for all gE G and if the mapping
328
LECHJAKOBCZYK
g + a( T,F, P) is continuous for every F, F’E M, then the complex structure J determines G-covariant canonical quantization { W,, U,} on the Krein space Sr, (CC= TK). PROPOSITION IV.1 [21]. If KET(M, a) and K# 1, then c.uJ(zj ~~6,) = cj z,p,(F,), where JE j$ and pJ is defined by (IV.l), with respect to the norm 1). I( , on d(M, a).
the functional is discontinuous
ProoJ: Let FeM be such that a(F, JF)
nEN(.
(IV.2)
We see that llA,il, -+ 0 when n + co, but 1 =- IpAW n
lw.0.N
(IV.3)
--f ~0.
2. Existence of Complex Structure of Type K Consider first the case of a finite-dimensional symplectic space. Let {EL,..., E,}, @ 1,..., ,!?,,} be the symplectic base in (M, a). Suppose that KE r(M, a) has a diagonal form on this base, i.e., (IV.4)
K: = diag(z, ,..., z,, 2, ,..., 2,). Since K* = 1 and KE SP(A4, a) we have zi= fl,
and
zi= +1
zjzi = 1,
j = l,..., n.
(IV.5)
Hence K = diag(z, ,..., z,, z, ,..., z,),
zi= fl.
(IV.6)
Let us define J=
(IV.7)
where Z = diag(z, ,..., z.). It is easy to check that: (A) (B) (C) When z, =
[J, K] = 0. JE SP(M, a). For every FE M, a( F, JKF) > 0.
. . = z, = 1 (K= 1) (IV.7) is the standard complex structure on (M, c). If n > 1, it is always possible to choose the sequence (ii,..., z,} in such a way that
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329
PRODUCT
K # f 1 and (IV.5) is satisfied. Hence, when dim M = 2n < 03 and n > 1, there exists a complex structure of type K. At this point it is worth stressing the connection between our approach to the canonical quantization and the recent results of Broadbridge [ 151 concerning quantization of quadratic Hamiltonians. In a series of earlier papers (see, e.g., [26-28-J) Broadbridge considered the quantization of linear dynamical systems with arbitrary quadratic Hamiltonian
H=I~=fjz 2
(IV.8)
with zT= (q, p), and 8= Z?’ a real symmetric 2n x 2n matrix. The dynamics in such a system is determined by a symplectic flow qSI= exp - Gi?t
(IV.9)
with G= (y ;l ). Such a system may be quantized provided there exists a complex structure J on a phase space, satisfying [J, GA] = 0.
(IV.10)
If in addition we assume that the complex structure J must be positive, the quantizable systems can be reduced, by a symplectic transformation, to a collection of independent harmonic oscillators [27]. The interesting possibility arises when we give up positivity of J. Then the condition (IV.10) can be fulfilled by a larger class of classical systems. In the paper [ 151 a complex structure J satisfying (IV.10) (and not necessarily positive) is called pseudo-unitarizing of a symplectic flow 4,. The connection between such complex structures and complex structures of type K defined in this section is described by the following PROPOSITION IV.2. Let J be a pseudo-unitarizing complex structure on (R2”, a). There exists KE SP(R2”, a), K2 = 1, such that J is a complex structure of type K.
Proof: Let J be a complex (M, a). Let us define
structure
on finite-dimensional
(F, F) : = 2a(F, JF) + 2ia(F, F).
symplectic
space (IV.1 1)
Since a is non-degenerate, (IV.1 1) defines a non-degenerate, indefinite inner product on M. On the other hand, every finite-dimensional indefinite inner product space is decomposable, hence there exists a decomposition M=Mf@iW.
(IV.12)
Let K be the fundamental symmetry corresponding to (IV.12). Since K is complex linear on M, [K, J] = 0. Moreover (KF, KF) = (F, F), i.e., KE SP((M, a)) and a(F, JKF)=$(F,
KF)=$(F+,
F’)-$(F-,F-)>O.
330
LECH
JAKOBCZYK
Remark. In the case of indefinite inner product space, decomposition (IV.12) in never unique [ 181, hence for a given complex structure J pseudo-unitarizing a symplectic flow d,, there exists many symplectic operators K satisfying [J, K] = 0 and cr(F, JKF) > 0. In the case of infinite-dimensional symplectic space (A4, a) determined by some field equation, the existence of a complex structure of type K depends on topological properties of the space M. In many cases there exists a Krein space &l and a mapping *:lW-+A?
(IV.13)
such that: (A) (B)
Ran $ is dense in li;r. Im(@(F), $(P)) = 2a(F, P), F, FEM.
On the Krein space &? we have a natural complex structure J (multiplication by 1 and a fundamental symmetry IE i(M). Suppose that d-1 - J, I: Ran $ + Ran $ (IV.14) then $ -’ o 70 @ is a complex structure of type K= Ic/~ ’ 0 10 $ on (M, 0). 3. Concluding Remarks As in the standard case, the Fock-Krein representation of commutation relations is determined by the choice of some complex structure on the symplectic space (M, a). Corresponding complex structures can be characterized by the operator KE SP(M, a) satisfying K2 = 1. When is the choice of complex structure of type K natural? The answer depends on the concrete realization of the symplectic space (M, a) and the representation of symmetry group on (M, a). Consider the following example (quantization of the electromagnetic field): Let M be the real linear space of (W4-valued solutions F to the wave equation 0 F = 0. (IV.15) Let the symplectic form 0 be defined by (IV.16) The Poincare group Pf+ acts on (M, cr) by L -+ TL E SP(M, a), TL F(x) = AF(A-‘(x
- a)),
LEPf, L = (A, a).
(IV.17)
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331
PRODUCT
Let K E SP(M, rr) be of the form (IV.18)
K:F+eF
with G=diag(l,
-1, -1, -1). If (IV.19)
J: F+&lF
then JE yK. Moreover [T,, J] = 0 for every L E P’, . Thus the complex structure (IV.19) defines Pi-covariant canonical quantization on the Fock-Krein space. On the other hand, the complex structure Jo defined by (IV.20)
JG: F+neF
does not commute representation
with the whole family
{ TL}LEp~,
hence the standard
Fock
determined by Je is not Py-covariant.
Another example of the system for which complex structures of type K are very natural is the tensor field with zero mass (see, e.g., [29]). On the other hand, for the Klein-Gordon field, the most natural is the standard complex structure and standard Fock representation [30]. When J is a fixed complex structure of type K, then there exists a whole family K(J) of symplectic operators satisfying [K, J] = 0 and a(F, JKF) > 0. Moreover, for any K E K(J), Kg : = T,- l KT, E K(J) for every gE G since I$= 1, [Kg, J] =0 and o(F, K,JF)= o(T,F, JKT,F)>O. As we know from the construction of GNS representation defined by o,, the topology on DJ is determined by elements of the set K(J). Thus, the functional oJ does not uniquely determine the topology on the space of “local states” D,, since when K, RE K(J) and K # R the norms 1)’ II OKand /I. /I ‘x are not equivalent [22] (physical consequences of this fact will be discussed in another paper [31]).
V.
EQUIVALENCE
OF J*-REPRESENTATIONS
OF COMMUTATION
RELATIONS
1. A Notion of Equivalence of J*-Representations [ll, 211 A natural generalization of unitary equivalence of standard representations to be the following conception of equivalence of J*-representations: DEFINITION V.l. J*-representations (n, D(x),j(z)) to be equivalent iff: (A) There exists an isomorphic mapping
and (it, D(%),j(jt))
p: j(n) -j(E).
(B)
There exists a linear operator
(Bl) (B2)
U: D(n) -+ D(S). For every vl, Q1E D(z),
U satisfying:
(UY,
UQt), = (Y, di),
seems
are said
332
LECHJAKOBCZYK
(B3)
For every YE D(X), A E d(M, a) &(A)
(B4)
For every a~j(rc), U,(a)
Y = ii(A) UY.
= Z,(p(a)) U.
It is easy to prove the following: PROPOSITION V.l [21]. Let (n, D(z),j(z)) be an a-cyclic representation of A(M, a) (a Ej(n)). Suppose that there exists such an a-positive functional o that j(z) = (w) and for every A EA(M, a)
(Qm W)Q,>,=
(yl, n,(A) y,>,.
P.1)
Then representations (x, D(z), j(n)) and (rc,, D,, j,) are equivalent. Remarks. (A) In the case described in Proposition V.l the isomorphism p is trivial, since j(71) =j(c0). (B) From Proposition V.l we see that any a-positive functional w determines corresponding J*-representation up to equivalence. 2. Non-Equivalent Representations of CCR and Existence of non-Fock-Krein Representations
For technical reasons, we are now discussing the equivalence of J*-representations (rc, D(z), j(z)) and (E, D(5), j(iz)) only in the case when j(z) = j(5), i.e., the mapping p is trivial. Suppose that p E 9, is fixed. For any TE SP(M, a), define p,(F) : = ,o(T- ‘F).
When [T, K] =O, ~~~93~ since pT(KF)=p(T-‘KF)
(V.2) =p(KT-*F)
=pT(F)
and
~~j~kpT(F,-KFk)e~i”‘~~KF~‘=~~j~kp(T-’~-KT~1Fk)eiu’T~‘~~KT~’Fk)~O jk j.k
(when [T, K] #O, PTE%?KT with KT=TKT-‘). PROPOSITION V.2 [7]. Let (z,, DJ, {zK}) be the Fock-Krein representation of type K determined by the complex structure JE yK. The Fock-Krein representation determined by the complex structure J, = TJT-’ with TE SP(M, 6) and [K, T] = 0 is equivalent to (nJ, D,, {zK}) iff T_ =$(T- JTJ-‘) is the Hilbert-Schmidt operator on M,: = H”“‘K (JIFI(:, = 2o(F, KJF)).
Let now JE 2K be fixed. Define M+:=
;(l+K)M
(V.3)
333
QUANTIZATION WITH INDEFINITE PRODUCT
and for any FE M, denote by Ff the positive part of F, i.e., F+ = $(F+ KF). Suppose that &M+-tR is a linear functional
(V.4)
on M, . Put /?,(S,) := e-i@(F+) hF.
(V.5)
of the algebra d(M, a). Extension by linearity of (V-5) defines an *-automorphism Let (rrJ, D,, (rK)) be the Fock-Krein representation defined by this fixed complex structure and let (rr;, 07, {a}) be the a-extension of (rc,, D,, (TV)) (for a = rK). Let us deline 7-Q:= npfi,,
D,:=
0;.
(V.6)
One can check that (n+,, D,, (rcKj) is a J*-representation of d(M, a). Suppose now that (Q, D,, (~~1) and tq, Di, (~~1) are .I*-representations defined by (V.6) and corresponding to linear functionals 4 and & respectively. Adopting the method of Roepstorff [32] we are able to prove: THEOREM V.I. J*-representations (n@, D,, (zK)) and (ni, Dg, (~~1) equivalent iff there is a constant d > 0 such that for any F+ E M + IW’+)-&f’+)I
GdllF+II~.
For the proof we need some results concerning collect them in the following:
are
(V.7 )
representation
(npl, D;, {a}).
LEMMA V.l. (A) For any F+ EM, the linear operator n”,(S,+) to the unitary operator W(F’) on the Hilbert space (X^,, (., b),).
We
can be extended
(B) The mapping F+ --, W(F+)E ES(&) is continuous with respect to the strong operator topology for II%(&) and the norm ((*I(‘, = 2a(. , JK. ) on M + . (C)
There exists a unique extension of W” to a continuous map
@ has the following
properties:
(Cl)
p(F+)
(C2)
For every F+ E @\‘lK, GEM, p(F+
D;cD;,
) n;(6,)
F+ E@$“K. YE 0;
Y = e- 2fM+,G+) n;(JG)
p(F+)
y,
tV.8)
334
LECH
ProoJ:
(A) For F+ EM, (n;(d,+)
JAKOBCZYK
we have KF+ = F+, so
K n;(d,+)
@I, =
y, Z,(a) qxJ,+)
= (q(S,+)
y, $(J,+)
@>,
I,(cc) @>,
=(y,I,(~)@),=(y,@),.
Hence z;(SF+) can be extended to the unitary operator Wa(F+) on the Hilbert space (Xi, (., . ),I. (B) The mapping F-+ (!P, rr;(dF) O), is continuous with respect to the norm I(.lIK on M since the mapping F+ o(F, G) is continuous for every GEM. Since W’(F+) for Ff EM, is unitary, the mapping Ff + W(F+) E IB(&) is continuous with respect to the strong operator topology for B(XE) and the norm 11.)IK. (C) By (B) there exists a unique extension of W to the continuous map P: Let p(F+)
my + B(Xu)).
{F+ } c M,
be a sequence such that FT + F+ E J%~$~x and Y = lim, W”(F,f ) Y, YE 0;. Since there exists the limit lim rc;(bG) W”(F,+)
Y
n
and for every GE M the linear operator rc;(SG) is closed, p(F+ ~;(a~) p(F’)
Y= lim rr;(bC) W(F,+) n
= eW-+.G+)
p(F+
) !PE DpI and
Y
) n$(dG) y.
Proof of Theorem V.I. Let x(F+)
:= &F+)&F+). If (V.7) is satisfied, one can (similarly as in [32]) that there exists G+ E &$y 1~ such that x(F+) = 2a(G+, F+). By Lemma V.1 we have: show
F(G’)
n;(ti,)
Y=epiX’“+‘n;(G,)
P(G+).
Hence p(G+)
~~(6,) Y=~6(6F)
L$@(G+) Y.
Suppose now that for every FEM there is the linear operator (Bl)-(B3) (Definition V.l) and commuting with I,(r,). Thus eixcF+‘= (Un;(d,) If FEM,
U satisfying
Q,, 7-c;(hF) UQ,),.
(F= F+) then
tFF+)=
(Un;(d,+)
Q,, 7c;(dF+) us2,),,.
W.9)
QUANTIZATION
WITH INDEFINITE PRODUCT
335
The right-hand side of (V.9) is continuous in F+ with respect to the norm (1./IK. be such that F,‘-+O but Suppose that x is discontinuous. Let {E;:}cM, exp ix( F; ) -+ -1. The right-hand side of (V.9) tends to + 1, thus must be continuous [32]. Remark. Theorem V.l shows that every discontinuous functional on M, determines a J*-representation of canonical commutation relations not equivalent to the Fock-Krein one.
ACKNOWLEDGMENTS I would like to thank Professor W. Karwowski
for helpful discussions.
REFERENCES 1. 2. 3. 4. 5. 6.
F. STROCCHIAND A. S. WIGHTMAN, J. Math. Phys. 15 (1974), 2198. G. MORCHIO AND F. STROCCHI,Ann. Inst. H. PoincarP Sect. A 33 (1980), 251. F. STROCCHI,Phys. Rev. D 17 (19781, 2010. F. STROCCHI,Comm. Math. Phys. 56 (1977), 57. M. MINTCHEV, J. Phys. A 13 (1980), 1841. L. JAKOBCZYK,Bull. Acad. PO/. Sci. Ser. Phys. Astr. 28 (1980), 167. 7. K. ITO, Publ. Rex Inst. Maih. Sci. 14 (1978), No. 2. 8. J-P. ANTOINE, in “Mathematical Aspects of Quantum Field Theory” (A. Pekalski and T. Paszkiewicz, Eds.), Wroclaw Univ. Press, Wroclaw, 1979. 9. A. H. V~LKEL, Institut fur Mathematik III, Freie Universitat Berlin, preprint No. 150, 1983. 10. K. Yu. DADASHVAN AND S. S. HORUJY,Teor. Mar. Fiz. 54 (1983), 57. 11. L. JAKOBCZYK,Borcher’s algebra formulation of an indefinite inner product quantum field theory, J. Math. Phys. 25 (1984), 617. 12. V. I. ARNOLD, “Mathematical Methods of Classical Mechanics” (in Polish), PWN, Warsaw, 1981. 13. A. ASHTEKAR AND A. MAGNON-ASHTEKAR, Gen. Relativity Gravitation 12 (1980), 205. 14. F. GALLONE AND A. SPARZANI,J. Phys. A 14 (1981), 1341. 15. P. BROADBRIDGE,J. Phys. A 16 (1983), 3271. 16. N. WOODHOUSE,“Geometric Quantization,” Oxford Univ. Press (Clarendon), Oxford, 1980. 17. J. MANUCEAU AND A. VERBEURE,Comm. Math. Phys. 9 (1968), 293. 18. J. BOGNAR, “Indefinite Inner Product Spaces,” Springer-Verlag, Berlin, 1974. 19. A. Z. JADCZYK, Rep. Math. Phys. 2 (1971), 263. 20. J. E. SEGAL, “Mathematical Problems of Relativistic Physics,” Amer. Math. Sot., Providence, R.I., 1963. 21. L. JAKOBCZYK, On representations of CCR in indefinite metric quantum field theory, Rep. Math. Phys., in press. 22. L. JAKOBCZYK,Wroclaw University Preprint No. 553, 1982. 23. R. T. Powa~s., Comm. Math. Phys. 21 (1971), 85. 24. G. G. EMCH, “Algebraic Methods in Statistical Mechanics and Quantum Field Theory,” Wiley, New York, 1972. 25. A. ASHTEKAR AND A. MAGNON, Proc. Roy. Sot. London Ser. A 346 (1975), 375. 26. P. BROADBRIDGEAND C. A. HURST, Ann. Phys. (N. Y) 131 (1981), 104. 27. P. BROADBRIDGE,Hadronic J. 4 (1981) 899.
595;161'2-I
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28. P. BROADBRIDGE,J. Austral.
Math. Sm. Ser. B 24 (1983), 439. 29. A. L. CAREY AND C. A. HURST, J. Math. Phys. 18 (1977), 1553. 30. P. J. M. BANGAARTS, in “Mathematics of Contemporary Physics”
(R. F. Streater, Ed.), Academic Press, New York/London, 1972. 31. L. JAKOBCZYK, Properties of scattering operator in quantum field theory with indefinite inner product, in preparation. 32. G. ROEPSTORFF,Comm. Math. Phys. 19 (1980), 301.
OF PHYSICS
Canonical
161, 314-336 (1985)
Quantization
with
Indefinite
Inner
Product
LECH JAKOBCZYK Institute
of Theoretical Physics, Unillersity of Wroclaw, Cyhulskiego 36, 50-205 Wroclaw, Poland
Received June 12, 1984
In this paper an algebraic approach to canonical quantization on indefinite inner product space is presented. Concrete realization of such a quantization of a given classical system described by a symplectic space (M, a) is obtained by means of a so-called P-representation of CCR algebra d(M, u). So-called Fock-Krein representations of d(M, a) determined by some class of complex structures on (M, a) are studied in detail. It is shown that every Fock-Krein representation is unbounded. Starting with a fixed Fock-Krein representation of CCR sectors of non-Fock-Krein representations are constructed. ‘i-1 1985 Academic PRSS, hc.
I. INTRODUCTION
Recently, there is growing interest in the mathematical aspects of indefinite inner product quantum field theory (IIPQFT), especially in the context of a rigorous approach to the quantization of gauge fields. A pioneering work in this area is the paper of Strocchi and Wightman [I] concerning axiomatic formulation of quantum electrodynamics. Other papers by Strocchi and Morchio generalize the ideas formulated in Ref. [ 11 to the case of gauge fields and study such structural properties of the theory as: reconstruction of a field theory from a set of Wightman functions [2], PCT invariance, properties of local states and analytic continuation to the euclidean points [3], breaking of the symmetry and Higgs mechanism [4], etc. On the other hand, there are also interesting results concerning mathematical structure of general IIPQFT. It is worth mentioning the following: rigorous construction of free field theory on the Fock space with indefinite inner product [S-7], investigations of representations of the symmetry group on the Krein space [S, 93 and algebraic formulation of IIPQFT [ 10, 111. The present paper is also connected with this subject, but our intention is to develop the idea of (algebraic) canonical quantization with indefinite inner product (IIP) of linear systems. The starting point is the classical description of the system in terms of linear symplectic space (A4, a) and symplectic operators on it. In the case of a finite number of degrees of freedom M is a phase space of a mechanical system and 0 is a natural symplectic form on it [12]. For the system described by some (linear) differential equations (field equations), M is the set of real solutions to this equation and 0 is a non314 0003-4916/85 $7.50 CopyrIght (’ I985 by Academic Press. Inc. All nghls of reproductmn in any form reserved.
QUANTIZATION
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315
degenerate, antisymmetric, bilinear form on these solutions [13]. Over the symplectic space (M, a) we construct the abstract algebra d(M, a) containing all information about commutation relations of the elements of a quantum system corresponding to the classical one described by (M, a). From the algebraic point of view, the quantization of a classical description can be performed in purely algebraic terms, i.e., in terms of the algebra .4(M, a) and *-automorphisms of d(M, 0) representing the symmetries of the system [ 141. Concrete realization of this quantization procedure is obtained by the choice of (concrete) covariant representation of the algebra d(M, (z). If we take the standard representation on a Hilbert space X, we obtain (canonical) quantum field theory on X (or the ordinary quantum mechanics in the case of finite number of degrees of freedom). However, there is a possibility to take some “non-standard” representation of this algebra on the linear space with IIP, and in this way we can obtain a “canonical” field theory with IIP, or some “non-standard” quantum mechanics on the Krein space (quantization with IIP of some class of mechanical systems was recently discussed by Broadbridge [ 151, see also Section IV). The above idea is examined in detail in the present paper. After mathematical preliminaries (Section II) in Section III we formulate main ideas of our approach to quantization with IIP. First of all, we study general properties of so-called J*-representations of an *-algebra &. It is shown that such representations are determined by some non-positive functionals on the algebra d via the generalized version of Gelfand-Naimark-Segal construction (Theorem 111.1). Then we consider J*-representations of the algebra d(M, a). It turns out that we can characterize them by means of so-called K-positive functions on the symplectic space (M, a) (Theorem III.3). K-positive functions defined by some complex structures on (M, a) (complex structures of type K) are studied in Section IV. Such complex structures lead to so-called Fock-Krein representations of the algebra A(M, a). It is proved that the Fock-Krein representation is always unbounded (Proposition IV. 1). The existence of complex structures of type K is also investigated. We show that when dim M= 2n < co, there always exists a complex structure of type K. We analyse also the connection of our approach with results concerning quantization of general quadratic Hamiltonians [ 151 (Proposition IV.2). In the case of infinite-dimensional symplectic space, existence of a complex structure of type K requires some additional topological assumption, namely, we must assume the existence of a Krein space structure (M, $). At the end of this section we discuss a concrete example of quantization using complex structure of type K (quantization of electromagnetic field). In Section V we study equivalence of J*representations of CCR. Our definition of the notion of equivalence is a natural generalization of the notion of unitary equivalence in the standard case. Using the result of Ito [7] we show that Fock-Krein representations defined by two complex structures of type K, J, and Jz = TJ, T-’ (T is a symplectic operator commuting with K) are equivalent iff T- = $( T- J, TJ; ‘) is a Hilbert-Schmidt operator. Next we construct some kind of “coherent” representation starting from the Fock-Krein one. The classical result of Roepstorff concerning equivalence of such represen-
316
LECHJAKOBCZYK
tations is generalized (Theorem V.l ), showing the existence of sectors of nonFock-Krein representations of CCR in quantum field theory with IIP.
II. MATHEMATICAL
PRELIMINARIES
1. Symplectic Space [16] Let M be a real vector space and o a symplectic form on M (i.e., o is a bilinear, antisymmetric, non-degenerate form on M). The pair (M, a) will be called a (uector) symplectic space. If dim M< GO (dim M= 2~) then there exists a symplectic base in M, i.e., a set of vectors (F, ,..., F,}, {E, ,..., E, j satisfying o(F,, Fk) = o(E,, Ek) = 0,
o(Fj, Ek) = & 3
j, k = 1,..., n.
(11.1)
A symplectic operator T on (M, a) is an operator from A4 onto M satisfying, fo every F, GE M, a( TF, TG) = a(F, G).
(II.2
(,SP(M, a) denotes the set of all symplectic operators on (M, a).) A complex structure J on (M, 0) is a symplectic operator satisfying J’=
-1,
4 F, JF) 3 0,
FEM.
(11.3)
2. The CCR Algebra A(M, g) [17]
Let (M, a) be the symplectic space. Let d(M, a) be the (complex) generated by functions 6,: M-t@,
FEM
S,(P) = 1
if F=p
=o
linear space
(11.4)
if F#i?
A(M, 0) equipped with the product
(11.5) and the involution (11.6)
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317
becomes a *-algebra with identity 6,. Let I(. (1I denote the norm on d(M, 0) defined by (11.7) d,(M, a) := d(M, u)“‘“~ is a *-Banach algebra. For every T E SP(A4, a), the mapping (11.8)
T T . ‘6,-+6,,
can be extended to the unique *-automorphism of d(A4, a) (continuous *-automorphism of d ,(M, 6)). Let &I$ be the set of all functions p mapping M into @ and satisfying the condition
C~~‘~p(F,-F~)~-ia(~~.Fk)~o
(11.9)
j.k for every finite sequence ((z,, F,)}, (zi, F,) E @ x M. u is a positive linear form on d( M, a) iff the function p defined by FEM
p(F) = 46F),
(11.10)
belongs to %‘,. Under this condition o is denoted by o,, and the representation of d ,(M, Q) defined by w, through the construction of Gelfand, Naimark and Segal is denoted by (z,, X0). be the representation of the group G on the symplectic space Let {T,hEG (M, a), i.e., Gsg+
The mapping
T,ESP(M,
a).
(11.11)
g + tg defined by zg:= TTp,
gEG
(11.12)
is the representation of G on the algebra d(M, 0) (d,(M, a)). A representation (rc, X,) of d,(A4, a) is G-couariant if there exists a family {U;}REc of unitary operators on n such that for any A E A,(M, C) 7r(z,(A)) 3. indefinite
= U,“n(A)
vi*.
(11.13)
Inner Product Space [18, 191
Let E be a vector space with an inner product ( ., * ). The pair (E, ( ., . )) will be called an inner product space. If the quantity (e, e), e E E, may be positive,
318
LECH
JAKOBCZYK
negative or zero, (E, ( ., . )) is called indefinite inner product space. If ( ., . ) is nondegenerate and E=E+@E-;
E’=(eEE(e,e)?O]
(11.14)
then (E, ( ., . )) is called (non-degenerate) decomposable indefinite inner product space. Let r be a topology on (E, (., . )). We say that z is a majorant of the inner product ( ., . ) if: - r is locally convex, - ( ., . ) is jointly r-continuous. A topology z on (E, (., . )) is admissible if: - for every fixed e”E E the mapping e + (e, 0) is z-continuous, - for every r-continuous linear form L on E, there is an element e,, E E such that L,(e) = (5 e. >. Let Fc E be the subspace of E. Define F’ = {e E E: (e, e”) = 0 for every I?E F}. If r is an admissible topology on (E, ( ., . )) then z-closure of any subspace F c E coincides FL’. A majorant topology r defined by Hilbert space norm /I. IIH is called Hilbert majorant. Hilbert majorant topology is admissible. Let Fc E be such that (., . ) is definite on F. The equality IelF:=
I(e, e))“*
(11.15)
defines the norm on F. Topology induced by 1.1F is called intrinsic topology on F. If an indefinite inner product space (E, (. , . )) admits a decomposition of the form E = E+ @E- where the subspaces E+ and E- are intrinsically complete, then it is called a Krein space. Let P’E = E*. We set r=p+
-p-
and say that I is a fundamental symmetry of E (the set of all fundamental metries of E will be denoted by i(E)). Define I-inner product (e, ?), : = (e, ZZ).
(11.16) sym(11.17)
A decomposable, non-degenerate, indefinite inner product space (E, (., . )) is a Krein space iff for every ZE i(E), the Z-inner product turns E into a Hilbert space.
QUANTIZATION III.
REPRESENTATIONS
319
WITH INDEFINITE PRODUCT
OF ALGEBRA
OF COMMUTATION
RELATIONS
1. Segal Quantization [20) Let (M, (T) be the symplectic space describing the classical system. Let ( Tg}gE G be the representation of the symmetry group G on (M, c). G-covariant canonical quantization on a Hilbert space X is by definition a pair ( W, U) of mappings W: M -+ u(X)
(111.1)
U: M-u(X) (u(X)
denotes the group of unitary operators on a Hilbert
space 2)
for every F, FE M, W(F) W(F) = eeioCF,‘) W(F+ P) the mapping
t --+ W(tF) is weakly continuous
satisfying: (111.2)
and for every FE M, g E G, W( T, F) = U, W(F) Ut the mapping g -+ U, is weakly continuous.
(111.3)
Taking a concrete G-covariant representation (x,, ZO) (with the group { U;jgEG) the algebra A ,(M, a), defined by the function p E &?,,satisfying conditions: (A) for (B) for (C) for the pair ( W,
every F, FE M, the mapping t + p(tF+ F) is continuous, anygEG,poTg=p, every F, FE M, the mapping g --, p( T,F+ P) is continuous, we obtain U} of mappings (111.1) with properties (111.2) and (111.3), if we put
W(F)= ~,GM u,= 2. Quantization
of
with Indefinite
ug,
FEM gEG.
(111.4)
Inner Product
The extension of the Segal quantization procedure to the case of a theory with IIP can be obtained by means of so-called J*-representations of *-algebras. Since it is more natural to deal with the algebra A(M, a) (the Weyl operators in the field theory with IIP can be generally unbounded ES]), we ought to consider unbounded representations of (not-normed) *-algebras. Let us recall the definition and simple properties of J*-representations of (abstract) *-algebra & [ll, 211. If a$.(&) denotes the set of all * -automorphisms of d, let j(&)caut,(&) be the set of elements tl E aut*(&) satisfying CY’= id (a # id). 5951161:2-6
320
LECH
JAKOBCZYK
111.1. (x, D(z),j(n)) is called a .Z*-representation of G? if: (A) T: &’ -+ op(D(z)) is a mapping of ral into linear operators all defined on a linear space D(X) with an indefinite, non-degenerate inner product ( ., . ),; rr is such that: (Al) for any A E&, n(A) D(n) c D(n), (A2) for every A, BE d, z, w E C, YE D(x) DEFINITION
z(AB)
Y= x(A) n(B) Y
n(zA + wB) !P=zzn(A)
(A3)
!P+ MT(B) !P,
for every A E&‘, Y, @E D(n) (111.6)
(Y 4‘4) @>,= (n(A*) y, @>,; (B) j(n)cj(&‘) (Bl) properties:
(111.5)
is such that:
for any tx~j(rt)
there is a linear operator
Z,(E): D(x) -+ D(n) with
I”,(E) = 1 (Z,(a) K @>,=
(111.7)
(u: rn(~), @>,
and such that (., .), = ( ., Z,(U) @), is positive definite, (B2) for every cr~j(z), AE&, ~/ED(Z)
44A))
Y=ZI,(a) n(A) Z,(a).
(111.8)
Remarks. (A) Let Xz=D(n)““lz (j/.Ijz = (., .),) and (., . ), is the extension of (., * ), to X,, then (X,, (a, . >,) is the Krein space. (B) If CI,/I ~j(n) and CI# /I, the norms 11.jIW and I(. lIB are not generally equivalent. (C) z(d) = {rc(A): A E &> is the example of op J*-algebra discussed in [lo]. The representation (n, D(n),j(rr)) is a-cyclic (a~j(rr)) if there is a vector a,~D(n) such that:
(A)
n(zZ)Q,
is dense in X,,
(B)
Z,(a) Q, = Q,.
Suppose that there is defined the representation g --t tg E aut,(d). J*-representation (n, D(n),j(,)) is G-couariant linear operators with the following properties:
of a group G (111.9) if there exists a family { U;}BEG of
QUANTIZATION
(A) (B) (C)
WITH
INDEFINITE
for any gEG, U~:D(z)+D(~), for every gEG, Y, @ED(~), (U;Y, for every A E&, g E G, YE D(z) 7+,(A))
CD)
for every g,,g,EG,
(E)
the mapping
Vi@),=
Y = u; n(A) q*
y~D(n),
321
PRODUCT
(Y, @),,
Y,
(111.10)
U;, U;“, Y= lJ&+,,‘,
g + ( Y, U; @) rr is continuous.
Remark. Operators U; representing the symmetry group are generally unbounded (for example, it is the case of boosts in quantum electrodynamics [S]), therefore we assume that U; are defined only on the dense domain D(z). Let the representation (a, D(rc),j(rr)) be a-cyclic (for any a~j(rc)) with a-cyclic vector 52,. Let us define the functional
(III.1 1)
%(A) = (Q,, 64) Q,>,. The functional (A) (B)
(111.11) has the properties:
co,0 a = o,, a Ej(71), cu,(a(A*)
A) 2 0, a ~j(n), A EJZ’
It suggests the following general definition: DEFINITION 111.2. A functional w on d is a-positive if there exists a ~j(sxZ) such that o 0 a = o and for any A E SQ, w(a(A *) A) > 0. If o is a-positive, then we denote P(o) = (a Ej(&): 0 is a-positive}.
Suppose now that a-positive functional w on d is also G-invariant, geG UOTg=U for some representation
i.e., for any (111.12)
of G of type (111.9). Then for any a E P(o) ag=7gnaot1
(111.13)
R
belongs to the set cP(o). g(w) is in this case the sum of orbits of the form {a,:gEG,
Generalizing GNS construction o, one can obtain:
a is fixed}.
(111.14)
to the case of a-positive and G-invariant
THEOREM III.1 [ 11,211. Let w be a-positive Assume that for every A, B, C E -02, the mapping
g + wW,(B)
and G-invariant
C)
functional
functional
on d.
(111.15)
322
LECHJAKOBCZYK
is continuous. Then there exists G-covariant representation (z,, D,, j,) of d (with the symmetry group { U,“}g Eo). It is cr-cyclic for any LYEj,, Moreover, for any g E G
is the unitary operator between Hilbert spaces(Xg,
(., .),) and (X;,
(., ‘),,).
ProoJ: Let
(111.17)
Do,= d/L(w) with L(o)=
{AEs?:~(BA)=O
for any BE&‘}. (Yy,,
Y,),
Let us define (111.18)
:= o(A*B)
where Y, denotes the equivalence class in D, containing A E JZ?‘.It is easy to see that (D,, ( ., . ),) is an indefinite inner product space. If we put
%(A) y,:= ~A, ZoJ,(a)yy,:=
(111.19) a Ej, : = P(u)
ymy,,,,;
then (rr,, D,, j,) becomes a J*-representation x-cyclic vector
(111.20)
of d. It is cc-cyclic for any CIEj, with
sz:= Y,.
(III.21 )
Let us also define u,w!P,:=
It is easy to check that (n,, D,, j,,,) is G-covariant Besides
q%,
(111.22)
Y@). with the group
u,wy,),p=W(a~(ZR(A))*Zg(B))=w(a(A*)B)=(y’,,
Yu,),
{ UF}~~~. (111.23)
and for any g@, U,W can be extended3 the unitary operator between Hilbert spaces (X: = D, II 11=, (., .),) and (X;=DoI”I’u~, (a, .),). Remark. If we take CIE P(w) such that rg occ#aoz,andifthenorms IJ.JI.and 11.I(‘l;r are not equivalent, then U,” is the unbounded operator on Z’z. It is worth stressing that in the Fock space, the norms defined by different fundamental symmetries I and 7 are always inequivalent [22]. THEOREM 111.2. Let (x,, D,, j,) be a G-covariant J*-representation of d defined as in Theorem III. 1. For every c1Ej,,, there exists a J*-representation (XL, DL, (a}) (a-extension of (zW, D,, j,)) defined asfollows:
QUANTIZATION
WITH
D;:=
n
INDEFINITE
323
PRODUCT
(111.24)
D@,(A)“)
AEd
7qg.4) Y:= n,(A)* Y IL(a):=
if YYED;
(111.25)
Z,(a)
(111.26)
( T-” denotes the closure of T in the norm (1.(I.). J*-representation the foIlowing properties:
(A) (B)
For every A EZZZ, n;(A) 0:
(rc;, DE, {a} ) has
is closed in X;.
is complete in the topology defined by seminorms
IIYlla.s:=
(111.27)
(where S is a finite set of elements of d). (C) Let (~2, DF*gg,(a,}) g EG. Then we have
be un a,-extension of (n,,
D,,j,)
for
lJ”:D;+D”,s g
some fixed
(111.28)
and Uptpl)
Y = n~(z,(A))
u; Y,
YED:.
(111.29)
Proof. In the Krein space X any densely defined operator T is closable iff D(T*) is dense [18] (T* denotes the adjoint to T with respect to indefinite inner
product). For every J*-representation
(n, D(rc),j(rr))
D(n) = D(n(A*))c
we have by (111.6)
D(7t(A)*)
(111.30)
hence, D(n(A)*) is dense in Xz and for every A E S# rc(A) is closable in the norm 11.II=. Let n,(A)” be the closure of n,(A). Applying the result of Powers [23, Lemma 2.61 one can easily show that (n;, Dz,) defined by (111.24) and (111.25) is a representation of d. We show that (rc;, D;, (~3) is a J*-representation of d. Let Y,@ED; and YE D(q,,(A*)“), @E D(n,(A)*). There are sequences {Y,), { @,} c D, such that YY,-+ y,
n,“(A*) Yn + 7c,(A*)OL Y
@II-*@, T”(A)@n + %o I”@ in the norm I(. )I1. Thus we obtain the relation (Y,n~(A)~),=lim(y’,,7~,(A)~,),=lim(n,(A*)Y~,~,), n n = (n”(A*)
Y, @),.
324
LECH
JAKOBCZYK
Hence, for every A E .,&, Y, @E DE,
(n"(A) y, @>,= (Y, 71"(A*) @>, Next we show that Z,(a): 0: -+ 0;. Let Y’E 0;. There is a sequence ( Y,,} c D,,] such that Y,, --t Y and rc,(A) Y”-+ rt,(A)‘Y. Let @=Z,(cl) Y, then if @,:= Z,(a) vl,, @, + @ and
Thus, there exists the limit
Since rc,,(A) is closable, Z= rc,,(A)W and @E 0;. Similarly [23], one can show that for every A E&, Y, @E 0;
as in the standard case
Thus we have the relations
and
From this it follows that
Hence we obtain the result: (n;, D;, {a}) is a J*-representation. the points (A), (B) and (C): (A) (B) (C) U” Y’, + x:(A) is
Finally, we show
By construction every n;(A) is closed. It follows from the result of Powers [23]. Let YED; and ul, -+ Y, rc,(A) Yn+zr,(A)“Y in the norm I).IICL. Then U,” Y in the norm )I. )I*~. Moreover lim, rc,(A) U,” Y,, exists in Krg. Since closable in Xg 7c,,(A)‘x Up” Y= lim n,,(A) U; Y',
n
hence U; YE D",". Besides
U,"q%4 Y==~(TJA))U;Y,
YEED;.
QUANTIZATION
WITH
TNDEFINITE
Now we are prepared to consider J*-representations T(M, CT)= {KESP(M,
325
PRODUCT
of the algebra d(M, P). Let
a): K2= 1, K# +_I}
(111.31)
and j(r) = {zK: KE r(M, We consider such J*-representations
(ret,D(z),j(z))
4).
(111.32)
of d(M, U) for which (111.33)
An) CA0
(such a choice is suggested by the case of free field theory, see also [21]). Suppose now that the representation (x, D(x),j(z)) is G-covariant (with the symmetry group ( U; jgE G). Assume moreover that for any FE M the mapping (111.34)
t + JQJ,F) is weakly continuous.
For a fixed a = rK ~j(rc) a pair { W, U> of mappings (111.35)
w: M 4 op(D(n)) U: G + op(D(n)) defined by
W(F)= 46,) u*= u’g”
(111.36)
satisfies the following conditions: (A) (B) (C) (D) (E)
Forevery FEM,Y,@EE(~),(W(F)Y,@),=(Y, W(-F)@),. For every F, F’E M, YED(x), W(F) W(P) Y=e-i”‘cF) W(F+F) The mapping t -+ ( Y, W(tF) @), is continuous. For every FE&I, YeD(rc), W(KF) Y=Z,(?,) W(F) Z,JT~) Y. For every gE G, YE D(z), FE A4
!I?
W( T, F) Y = V, W(F) U; Y.
(F) The mapping g + ( Y, U,@ ), is continuous. A pair { W, Or} of mappings W: M+
op(D,)
(111.37)
U: G + op( D w)
(Dw is a dense subspace of a Krein space X) satisfying conditions analogous to (A)-(F)
will be called G-comriant
canonical quantization
on a Krein space K.
326
LECHJAKOBCZYK
Remark. If we consider an x-extension of J*-representation of the algebra d(M, a), then we obtain a pair { W, U) when W(F) := rP(6,). The pair { W”, U) satisfies all conditions of canonical quantization on the Krein space, but now Gcovariance means: U, Wa(F) Y = W”( T, F) U, !P for all YE D(z’). 3. Realization
of J*-Representations
of CCR: K-Positive Functions
If (71,D(~),j(n)) is an a-cyclic representation (a = r,), then the function p,(F):=
of d(M, a) with u-cyclic vector Q,
(Q,, 46,)
Q,),
(111.38)
has the properties: (A) (B)
in 0 K= pn, For every finite sequence {(z,, F,)), (zj, Fj) E C x M, (111.39)
A function p mapping M into @ and satisfying (A) and (B) will be called K-positive [21]. Let us introduce the following notation: .gK: = (p: M -+ @: p is K-positive ) .J$: =
u
c4!K
KS/-(A4.u)
l-(p):=
{KEZ-(M,a):p~&}.
Now we can formulate: THEOREM
III.3
[21].
Let p E W. Assume that:
(A) P(O)= 1. (B) The mapping t + p( tF + p) is continuous. (C) For anJ’ g E G, p 0 T, = p and the mapping g + p( T,F + P) is continuous. Then, for any CI= ~~ (KE T(p)) there exists G-covariant canonical quantization on the Krein space X,. Proof:
Similarly
as in [21] let us define the functional (111.40)
327
QUANTIZATION WITH INDEFINITE PRODUCT
w, is a-positive (CI = rK, K E T(p)) and o, 0 r, = op. By Theorem III.1 there exists a G-covariant representation of d(M, a), hence a G-covariant quantization on the Krein space Xa = E’l’ ‘12. Remarks. (A) The existence of a G-covariant canonical quantization on the Krein space is now equivalent to the existence of a K-positive function on the symplectic space (M, c). K-positive functions defined by some complex structure on (M, G) will be discussed in the next section. (B) If p is positive on (M, a), then it is known [24] that for any FE M IP(F)I G P(O).
The functional (111.40) is in this case continuous with respect to the norm 11.I/ 1 and can be extended to the continuous functional on the Banach algebra d I (M, B). If p ~9, then the above inequality is generally not satisfied, and, for example, the Fock-Krein functional (Section IV) is not continuous with respect to 1).Jji . Thus, the functional wp is generally defined only on (not-normed) algebra d(M, CT)and the operators rrp(6,) are unbounded.
IV
FOCK-KREIN
REPRESENTATIONS OF COMMUTATION RELATIONS
In this section we investigate the class of J*-representations of d(M, (r) defined by complex structures on (M, a). As we know, in the standard quantum field theory, definition of the complex structure on (M, a) determines the construction of free field operators (see, e.g., an interesting discussion of this point in [25]). In our case, the similar role is played by so-called complex structures of type K. 1. Complex Structures
of Type K on a Symplectic Space
DEFINITION IV. 1. A symplectic operator JE SP(M, CJ)is a complex structure of G)) if:
type K (KET(M, (A) (B)
J*= -1, [J, K] = 0 and for any FE M, G( F, JKF) 3 0.
The set of all complex structures of type K will be denoted by yK. If JELL functional p,(F)
: = exp - $(F,
JF)
the
(IV.1)
is a K-positive. A J*-representation of d(M, (r) defined by the function (IV.l) will be called a Fock-Krein representation of type K and we denote it by (rcJ, D,, {rK}). Remarks. (A) The standard Fock representation is defined also by the function (IV. 1), but the complex structure J is of type K= 1. (B) If JE#~ satisfies [T,, J] =0 for all gE G and if the mapping
328
LECHJAKOBCZYK
g + a( T,F, P) is continuous for every F, F’E M, then the complex structure J determines G-covariant canonical quantization { W,, U,} on the Krein space Sr, (CC= TK). PROPOSITION IV.1 [21]. If KET(M, a) and K# 1, then c.uJ(zj ~~6,) = cj z,p,(F,), where JE j$ and pJ is defined by (IV.l), with respect to the norm 1). I( , on d(M, a).
the functional is discontinuous
ProoJ: Let FeM be such that a(F, JF)
nEN(.
(IV.2)
We see that llA,il, -+ 0 when n + co, but 1 =- IpAW n
lw.0.N
(IV.3)
--f ~0.
2. Existence of Complex Structure of Type K Consider first the case of a finite-dimensional symplectic space. Let {EL,..., E,}, @ 1,..., ,!?,,} be the symplectic base in (M, a). Suppose that KE r(M, a) has a diagonal form on this base, i.e., (IV.4)
K: = diag(z, ,..., z,, 2, ,..., 2,). Since K* = 1 and KE SP(A4, a) we have zi= fl,
and
zi= +1
zjzi = 1,
j = l,..., n.
(IV.5)
Hence K = diag(z, ,..., z,, z, ,..., z,),
zi= fl.
(IV.6)
Let us define J=
(IV.7)
where Z = diag(z, ,..., z.). It is easy to check that: (A) (B) (C) When z, =
[J, K] = 0. JE SP(M, a). For every FE M, a( F, JKF) > 0.
. . = z, = 1 (K= 1) (IV.7) is the standard complex structure on (M, c). If n > 1, it is always possible to choose the sequence (ii,..., z,} in such a way that
QUANTIZATION
WITH
INDEFINITE
329
PRODUCT
K # f 1 and (IV.5) is satisfied. Hence, when dim M = 2n < 03 and n > 1, there exists a complex structure of type K. At this point it is worth stressing the connection between our approach to the canonical quantization and the recent results of Broadbridge [ 151 concerning quantization of quadratic Hamiltonians. In a series of earlier papers (see, e.g., [26-28-J) Broadbridge considered the quantization of linear dynamical systems with arbitrary quadratic Hamiltonian
H=I~=fjz 2
(IV.8)
with zT= (q, p), and 8= Z?’ a real symmetric 2n x 2n matrix. The dynamics in such a system is determined by a symplectic flow qSI= exp - Gi?t
(IV.9)
with G= (y ;l ). Such a system may be quantized provided there exists a complex structure J on a phase space, satisfying [J, GA] = 0.
(IV.10)
If in addition we assume that the complex structure J must be positive, the quantizable systems can be reduced, by a symplectic transformation, to a collection of independent harmonic oscillators [27]. The interesting possibility arises when we give up positivity of J. Then the condition (IV.10) can be fulfilled by a larger class of classical systems. In the paper [ 151 a complex structure J satisfying (IV.10) (and not necessarily positive) is called pseudo-unitarizing of a symplectic flow 4,. The connection between such complex structures and complex structures of type K defined in this section is described by the following PROPOSITION IV.2. Let J be a pseudo-unitarizing complex structure on (R2”, a). There exists KE SP(R2”, a), K2 = 1, such that J is a complex structure of type K.
Proof: Let J be a complex (M, a). Let us define
structure
on finite-dimensional
(F, F) : = 2a(F, JF) + 2ia(F, F).
symplectic
space (IV.1 1)
Since a is non-degenerate, (IV.1 1) defines a non-degenerate, indefinite inner product on M. On the other hand, every finite-dimensional indefinite inner product space is decomposable, hence there exists a decomposition M=Mf@iW.
(IV.12)
Let K be the fundamental symmetry corresponding to (IV.12). Since K is complex linear on M, [K, J] = 0. Moreover (KF, KF) = (F, F), i.e., KE SP((M, a)) and a(F, JKF)=$(F,
KF)=$(F+,
F’)-$(F-,F-)>O.
330
LECH
JAKOBCZYK
Remark. In the case of indefinite inner product space, decomposition (IV.12) in never unique [ 181, hence for a given complex structure J pseudo-unitarizing a symplectic flow d,, there exists many symplectic operators K satisfying [J, K] = 0 and cr(F, JKF) > 0. In the case of infinite-dimensional symplectic space (A4, a) determined by some field equation, the existence of a complex structure of type K depends on topological properties of the space M. In many cases there exists a Krein space &l and a mapping *:lW-+A?
(IV.13)
such that: (A) (B)
Ran $ is dense in li;r. Im(@(F), $(P)) = 2a(F, P), F, FEM.
On the Krein space &? we have a natural complex structure J (multiplication by 1 and a fundamental symmetry IE i(M). Suppose that d-1 - J, I: Ran $ + Ran $ (IV.14) then $ -’ o 70 @ is a complex structure of type K= Ic/~ ’ 0 10 $ on (M, 0). 3. Concluding Remarks As in the standard case, the Fock-Krein representation of commutation relations is determined by the choice of some complex structure on the symplectic space (M, a). Corresponding complex structures can be characterized by the operator KE SP(M, a) satisfying K2 = 1. When is the choice of complex structure of type K natural? The answer depends on the concrete realization of the symplectic space (M, a) and the representation of symmetry group on (M, a). Consider the following example (quantization of the electromagnetic field): Let M be the real linear space of (W4-valued solutions F to the wave equation 0 F = 0. (IV.15) Let the symplectic form 0 be defined by (IV.16) The Poincare group Pf+ acts on (M, cr) by L -+ TL E SP(M, a), TL F(x) = AF(A-‘(x
- a)),
LEPf, L = (A, a).
(IV.17)
QUANTIZATION
WITH
INDEFINITE
331
PRODUCT
Let K E SP(M, rr) be of the form (IV.18)
K:F+eF
with G=diag(l,
-1, -1, -1). If (IV.19)
J: F+&lF
then JE yK. Moreover [T,, J] = 0 for every L E P’, . Thus the complex structure (IV.19) defines Pi-covariant canonical quantization on the Fock-Krein space. On the other hand, the complex structure Jo defined by (IV.20)
JG: F+neF
does not commute representation
with the whole family
{ TL}LEp~,
hence the standard
Fock
determined by Je is not Py-covariant.
Another example of the system for which complex structures of type K are very natural is the tensor field with zero mass (see, e.g., [29]). On the other hand, for the Klein-Gordon field, the most natural is the standard complex structure and standard Fock representation [30]. When J is a fixed complex structure of type K, then there exists a whole family K(J) of symplectic operators satisfying [K, J] = 0 and a(F, JKF) > 0. Moreover, for any K E K(J), Kg : = T,- l KT, E K(J) for every gE G since I$= 1, [Kg, J] =0 and o(F, K,JF)= o(T,F, JKT,F)>O. As we know from the construction of GNS representation defined by o,, the topology on DJ is determined by elements of the set K(J). Thus, the functional oJ does not uniquely determine the topology on the space of “local states” D,, since when K, RE K(J) and K # R the norms 1)’ II OKand /I. /I ‘x are not equivalent [22] (physical consequences of this fact will be discussed in another paper [31]).
V.
EQUIVALENCE
OF J*-REPRESENTATIONS
OF COMMUTATION
RELATIONS
1. A Notion of Equivalence of J*-Representations [ll, 211 A natural generalization of unitary equivalence of standard representations to be the following conception of equivalence of J*-representations: DEFINITION V.l. J*-representations (n, D(x),j(z)) to be equivalent iff: (A) There exists an isomorphic mapping
and (it, D(%),j(jt))
p: j(n) -j(E).
(B)
There exists a linear operator
(Bl) (B2)
U: D(n) -+ D(S). For every vl, Q1E D(z),
U satisfying:
(UY,
UQt), = (Y, di),
seems
are said
332
LECHJAKOBCZYK
(B3)
For every YE D(X), A E d(M, a) &(A)
(B4)
For every a~j(rc), U,(a)
Y = ii(A) UY.
= Z,(p(a)) U.
It is easy to prove the following: PROPOSITION V.l [21]. Let (n, D(z),j(z)) be an a-cyclic representation of A(M, a) (a Ej(n)). Suppose that there exists such an a-positive functional o that j(z) = (w) and for every A EA(M, a)
(Qm W)Q,>,=
(yl, n,(A) y,>,.
P.1)
Then representations (x, D(z), j(n)) and (rc,, D,, j,) are equivalent. Remarks. (A) In the case described in Proposition V.l the isomorphism p is trivial, since j(71) =j(c0). (B) From Proposition V.l we see that any a-positive functional w determines corresponding J*-representation up to equivalence. 2. Non-Equivalent Representations of CCR and Existence of non-Fock-Krein Representations
For technical reasons, we are now discussing the equivalence of J*-representations (rc, D(z), j(z)) and (E, D(5), j(iz)) only in the case when j(z) = j(5), i.e., the mapping p is trivial. Suppose that p E 9, is fixed. For any TE SP(M, a), define p,(F) : = ,o(T- ‘F).
When [T, K] =O, ~~~93~ since pT(KF)=p(T-‘KF)
(V.2) =p(KT-*F)
=pT(F)
and
~~j~kpT(F,-KFk)e~i”‘~~KF~‘=~~j~kp(T-’~-KT~1Fk)eiu’T~‘~~KT~’Fk)~O jk j.k
(when [T, K] #O, PTE%?KT with KT=TKT-‘). PROPOSITION V.2 [7]. Let (z,, DJ, {zK}) be the Fock-Krein representation of type K determined by the complex structure JE yK. The Fock-Krein representation determined by the complex structure J, = TJT-’ with TE SP(M, 6) and [K, T] = 0 is equivalent to (nJ, D,, {zK}) iff T_ =$(T- JTJ-‘) is the Hilbert-Schmidt operator on M,: = H”“‘K (JIFI(:, = 2o(F, KJF)).
Let now JE 2K be fixed. Define M+:=
;(l+K)M
(V.3)
333
QUANTIZATION WITH INDEFINITE PRODUCT
and for any FE M, denote by Ff the positive part of F, i.e., F+ = $(F+ KF). Suppose that &M+-tR is a linear functional
(V.4)
on M, . Put /?,(S,) := e-i@(F+) hF.
(V.5)
of the algebra d(M, a). Extension by linearity of (V-5) defines an *-automorphism Let (rrJ, D,, (rK)) be the Fock-Krein representation defined by this fixed complex structure and let (rr;, 07, {a}) be the a-extension of (rc,, D,, (TV)) (for a = rK). Let us deline 7-Q:= npfi,,
D,:=
0;.
(V.6)
One can check that (n+,, D,, (rcKj) is a J*-representation of d(M, a). Suppose now that (Q, D,, (~~1) and tq, Di, (~~1) are .I*-representations defined by (V.6) and corresponding to linear functionals 4 and & respectively. Adopting the method of Roepstorff [32] we are able to prove: THEOREM V.I. J*-representations (n@, D,, (zK)) and (ni, Dg, (~~1) equivalent iff there is a constant d > 0 such that for any F+ E M + IW’+)-&f’+)I
GdllF+II~.
For the proof we need some results concerning collect them in the following:
are
(V.7 )
representation
(npl, D;, {a}).
LEMMA V.l. (A) For any F+ EM, the linear operator n”,(S,+) to the unitary operator W(F’) on the Hilbert space (X^,, (., b),).
We
can be extended
(B) The mapping F+ --, W(F+)E ES(&) is continuous with respect to the strong operator topology for II%(&) and the norm ((*I(‘, = 2a(. , JK. ) on M + . (C)
There exists a unique extension of W” to a continuous map
@ has the following
properties:
(Cl)
p(F+)
(C2)
For every F+ E @\‘lK, GEM, p(F+
D;cD;,
) n;(6,)
F+ E@$“K. YE 0;
Y = e- 2fM+,G+) n;(JG)
p(F+)
y,
tV.8)
334
LECH
ProoJ:
(A) For F+ EM, (n;(d,+)
JAKOBCZYK
we have KF+ = F+, so
K n;(d,+)
@I, =
y, Z,(a) qxJ,+)
= (q(S,+)
y, $(J,+)
@>,
I,(cc) @>,
=(y,I,(~)@),=(y,@),.
Hence z;(SF+) can be extended to the unitary operator Wa(F+) on the Hilbert space (Xi, (., . ),I. (B) The mapping F-+ (!P, rr;(dF) O), is continuous with respect to the norm I(.lIK on M since the mapping F+ o(F, G) is continuous for every GEM. Since W’(F+) for Ff EM, is unitary, the mapping Ff + W(F+) E IB(&) is continuous with respect to the strong operator topology for B(XE) and the norm 11.)IK. (C) By (B) there exists a unique extension of W to the continuous map P: Let p(F+)
my + B(Xu)).
{F+ } c M,
be a sequence such that FT + F+ E J%~$~x and Y = lim, W”(F,f ) Y, YE 0;. Since there exists the limit lim rc;(bG) W”(F,+)
Y
n
and for every GE M the linear operator rc;(SG) is closed, p(F+ ~;(a~) p(F’)
Y= lim rr;(bC) W(F,+) n
= eW-+.G+)
p(F+
) !PE DpI and
Y
) n$(dG) y.
Proof of Theorem V.I. Let x(F+)
:= &F+)&F+). If (V.7) is satisfied, one can (similarly as in [32]) that there exists G+ E &$y 1~ such that x(F+) = 2a(G+, F+). By Lemma V.1 we have: show
F(G’)
n;(ti,)
Y=epiX’“+‘n;(G,)
P(G+).
Hence p(G+)
~~(6,) Y=~6(6F)
L$@(G+) Y.
Suppose now that for every FEM there is the linear operator (Bl)-(B3) (Definition V.l) and commuting with I,(r,). Thus eixcF+‘= (Un;(d,) If FEM,
U satisfying
Q,, 7-c;(hF) UQ,),.
(F= F+) then
tFF+)=
(Un;(d,+)
Q,, 7c;(dF+) us2,),,.
W.9)
QUANTIZATION
WITH INDEFINITE PRODUCT
335
The right-hand side of (V.9) is continuous in F+ with respect to the norm (1./IK. be such that F,‘-+O but Suppose that x is discontinuous. Let {E;:}cM, exp ix( F; ) -+ -1. The right-hand side of (V.9) tends to + 1, thus must be continuous [32]. Remark. Theorem V.l shows that every discontinuous functional on M, determines a J*-representation of canonical commutation relations not equivalent to the Fock-Krein one.
ACKNOWLEDGMENTS I would like to thank Professor W. Karwowski
for helpful discussions.
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