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AMC Oi-ibi

UWThPh-1990-39 November 6, 1990

A Super-version of Quasi-free Second . I. Charged Particles

H. Grosse Institut für Theoretische Physik, Universität Wien

and

E. Langmann Institut für Theoretische Physik, Technische Universität Graz

PACS: 03.70.+k, 03.65Ca, 11.30Pb Keywords: External problem, and , current algebras, supersym- metry

Abstract

We present a formalism comprising and extending quasi-free second quan­ tization of charged bosons and fermicn: The second quantization of one- particle observables leads to current superalgebras and a super Schwinger term «hows up. We introduce anticommuting pa^meters in order to con­ struct super Bogoliubov transformations mixing bosons and . As an application, we give representations of Lie superalgebrae which are semidirect products of extensions of affine Kac-Moody algebras and the Virasoro algebra, and of the super Virasoro algebra. 1 Introduction

The interaction of particles — bosons or fermions — with external fields can be de­ scribed and studied rigorously within the framework of quasi-free second quantization (QFSQ). For fermions, QFSQ gives a precise mathematical meaning to the idea of 'filling up the Dirac-sea', and it is formulated very elegantly by means of C'-algebra theory (cf. e.g. [1] and [2]). This approach to external field problems of fermions is due to Bongaarts [3], and it was used to study e.g. spontaneous electron-positon production in strong external electromagnetic fields [4, 5, 6, 7j, anomalies [8], and models that determine certain numbers by themselves [9, 10, 11, 12, 13, 14] (and refe­ rences therein). Besides this aspect, QFSQ for fermions is rather important for the representation theory of certain infinite dimensional Lie algebras such as the affine Kac-Moody algebras and the Virasoro algebra (cf. e.g. [15], [16] and [17]). Let us give a short summary of the main points of QFSQ for charged fermions (an extensive treatment can be found in [18]): Given a one-particle Hubert space h, the fermion field algebra over h is generated by the particle operators a? (/) (/ 6 h) obeying canonical anticommutator relations (CAR). Quasi-free representations of this algebra are then constructed on the fermion over h from a given orthogonal projection T on h (in applications, T is usually the projection onto the negative energy states of the one-particle Hamiltonian). The important next step is the introduction of the socalled second quantization-map dlY(-) which has been studied, for example, by Lundberg [1]. By this map we assign 'charges' dtx(A) to one-particle observables A or operators on h. These quantities generate a current algebra: The commutator of two 'charges' dtr^A) and dIY(i9) is given by

[dfr(A), dtT(B)) = &tT{[A , B)) + CT(A ,B) (1) where the C-number Cr(- ,•) is the socalled Schwinger term. From a mathematical point of view, drV(«) gives a projective representation of a certain Lie algebra, and the Schwinger term is a 2-cocycle. We note that as far as we know, a general, rigorous construction of dTr(A) exists only for bounded operators A on h [1, 18], though in applications (e.g. external field problems or the construction of representations of the Virasoro algebra) its extension to unbounded operators is required. The work done on QFSQ for bosons (cf. e.g. [19]) is relatively small as compared to fermions. Probably one reason for this is that the field algebras generated by particle operators obeying canonical commutator relations (CCR) are not C~- algebras — in contrast to the fermion case — , and due to this fact the C"-algebra approach to QFSQ for bosons is more complicated (cf. e.g. [2]). But already Ruijsenaars, in his work [20] on Bogoliubov transformations within the framework of QFSQ for charged particles, showed that if one does not use C- algebra theory, it is possibel to treat bosons and fermions analogously. This suggests a unification of QFSQ for bosons and fermions. On the one hand, such a formalism is motivated by supersymmetric external field problems, i.e. the task of rigorously formulating and studying quantum field models describing bosons and fermions in

2 external fields which exhibit supersyrnmetry (cf. e.g. [21] or [22]). On the other hand, it is a natural generalization of supersymmetric studied rather intensively by Grosse and Pittner [23, 24, 25]. An outline of such a super-version of QFSQ was given in a recent letter [26]. In this paper we give a detailed, rather self-contained exposition of this formalism comprising QFSQ for charged fermions as described above, the boson analog thereof, and extending these formalisms in a nontrivial way. The paper is organized as follows: In Sect.2 we introduce and discuss certain *- algebras of — in general — unbounded Hubert space operators with common, dense, invariant domains of definition. We prove that operators in such an algebra have several nice properties. It will turn out that these results solve all technical problems assoziated with unbounded operators in QFSQ- In Sect.3 we demonstrate that the well-known fxmalims of free second quantization of bosons and fermions [2] can be unified elegantly by means of Z2-graded algebraic structures. The basic idea is as follows: We start with a Z2-graded Hubert space h - h

aH(/) = *$< *%/) + <£ W) V/ 6 h (2) which fulfill supercommutator relations (P^ and Pj are the projections onto h$ and hj respectively). Quasi-free representations U.T °f the superalgebra generated by the operators (2) are constructed in Sect.4 in one-to-one correspondence to orthogonal projections T on h leaving the Z^-gradation of h invariant. Furthermore we construct a super

second quantization-map dlY(-) thereby extending (1) to the 2Z2-graded case. This correspc nds to a construction of current superalgebras and super-Schwinger terms or — from a mathematical point of view — of projective representations of certain Lie

superalgebras and Z2-graded 2-cocycles respectively. The main steps of the general construction of dt^A) for bounded operators A on h were already outlined in [26]. We give here the details and extend the construction to unbounded operators A. In Sect.5 the formalism is extended by introducing anticommuting parameters; this allows to treat generalized Bogoliubov transformations 'mixing bosons und fer­ mions'. Examples of such 'super'-Bogoliubov transformations are the supers) mmetry transformations in supersymmetric external field problems. As an application of the super second quantization-map, we construct in Sect.6 representations of certain infinite dimensional Lie superalgebras extending the affine Kac-Moody algebras and the Virasoro algebra. For convenience of the reader, we summarize the basic definitions concerning 7L%- graded algebraic structures in Appendix A. Appendix B contains the proofs of some estimates needed. Besides QFSQ for charged particles, there is a corresponding formalism for neu­ tral ones (see e.g. [18] and [27]). A super-version of QFSQ for neutral partricles comprising and extending this will be given in a forthcomming paper [28].

3 2 On *- Algebras of Unbounded Operators

In this section we define and investigate a *-algebra of — in general — unbounded operators on a Hubert space h which can be assoziated naturally with a given self- adjoint H on h. Thinking of H as Hamiltonian of some quantum model, it is convenient to consider this algebra as observable algebra. (a) Let h be a separable Hubert space and £(h) the set of linear operators on h. For A 6 £(h), D\A) is the domain of A,

k D~(A) = DZ0D(A ) (3) and D»(A) = {/ 6 D-{A) | 3t > 0 : f) ^/li < oo} (4) is the set cf analytical vectors for A. We denote the closed, bounded, Hilbert-Schmidt, and trace-class operators in £(h) as C(h), 5(h), 02(h), and Bi(h) respectively. For A 6 B(h) we implicitely assume D(A) = h. For A,B e £(h) and V C D(A)f)D{B) we say 'A = B on V if Af = Bf holds for all / 6 V. (b) Let

H = /Rd^A (5) be a self-adjoint operator on h in the spectral representation. Then the family of projection operators

£n = s-lim /" ' dE* (6)

(En - 0 for n < 0) obeys

EnEm = EmEn = Em Vn>m

s-lim£n = I. (7)

We introduce the set of finite H-vectors

D'(H) =•- {/ e h I 3n < co : £„/ = /} . (8)

Note that D*(H) is a dense subspace of h for # = OO,Ü; and /,

/)>(//•) C £>"(#) C /?"(#), (9) and Dj{H) - h if and only if # G B(h). Introducing the self-adjoint operator

*»ir = i>(^-&-i) (10) we note the following simple relations frequently used in the sequel:

# D*(H) - D*(\H\) = £ (mH) (11)

4 for # = oo,u/ and /. We note that due to the closed graph theorem D*(H) C

D{A) 0 D{A-) if and only if A^En 6 5(h) Vn £ N {'A^.. .* stands for 'A... and .4"...'). (c) Let V be some dense subspace of h C±(V) is the «-algebra of all linear operators on h which have V as common, dense, invariant domain of definition, i.e.

C+{V) = {A€ £(h) | D(A) = VC D{A'), A^V C V) (12)

+ with involution A -» A = A' \v [29, 30]. Note that all A 6 C+{V) are closeable. An

Op'-algebra A on V is a *-subalgebra of C+(T>) containing the identity. It is called standard if each symmetric element in A is essentially self-adjoint or, equivalently, if 1+ = A' for all A € A {ibid.). (d) We define 0}(H) as set of all operators A 6 £+(£>'(#)) obeying

{m) Vn6N3/j

Furthermore we introduce the set Ou(H) of all operators A 6 0!(H) satisfying

(i) da < oo : ll^-^H < an Vn 6 N (14)

(Ü) 3K

Obviously Of(H) is a Op"-algebra on D}{H) and , to a locally convex space is given by the family of semi-norms {|-||n}^Li)

# \\A\\n~\\EnAEnl VnelMe0 (ff). (16)

The following lemma summarizes several nice properties of operators in Ou(H). Lemma 1 Let A,B £ 0"(H), and a, K < oo such that

\\AEn\

and similarly for B. Then exp (zA) and exp (zA)Bexp(-zA) (z € C) can be defined as absolute convergent power series on D*(H) for \z\ < 1/aK and \z\ < \j1aK respectively, and are analytic, TR -continuous one-parameter families of operators on Df(H). Hence Df{H) C D"{A) and

Ä+ = v4" VA£V(H). (18)

Furthermore,

Da>{H)CD(Ä), ÄD°°(H) C D°°(H) VAeO*{H). (19)

Proof: It follows from (17) that

k \\A En\\ = \AEnHk.i)KA • • • En+KAEn\

5 hence t £l^£»! * EH«*)* ("*/*) = (i - a^)-n/K for t < 1/aK which proves the statement for exp(zA). (18) follows with Nelson's analytic vector theorem. Similarly.

x(-aK)u+^(u + (i + l)l( "n/jft: J = an(l - 2aKt)-n/Kl finite for i < \j2aK and proving the statement for exp(.zA)5exp(-,2A).

For / G h, / = Y.%x f* with /, = Eift and /„ = (£„ - £„_!)/ for t/ > 1. Then for / e £°°(//),

i!™*/!|2 = Z ""IM* < «3 Vfc < OO , hence with (17)

k 2 2 \\m HAf\\ = £ I^IKS, - £UiM(£„+Jf - ^-K-i)/i

< 2a2 f) ^» ( jf |/j) < 2a' - (2* + l)2 f> + tf)"+2«/,||2 < oo for all k < oo, and (19) follows. D. It follows that every *-Lie superalgebra A contained in 0°(H) is a standard 0p"-Lie superalgebra, and the set

Ä = {Ä \ A <= A} is a *-Lie superalgebra with addition (A,P) —» A + B, inner product (A, B) —> [A ,B\ and involution A —• J4\ Moreover, D°°(H) is a common, dense, invariant domain for Ä In the following, if we define an operator A on some dense subset V of h with V C £'(#) and prove that ^")E„ is bounded for all n € N and that (13) holds on V then we do not distinguish A from its unique extension in Of(H). Similarly, if A € 0W{H) then we use the same symbol for A and its closure. Especially, v4+ = A" for all A G 0"(/J). Examples: (t) Obviously C?/(/7) = Ou{H) = 0(h) if and only if tf is bounded. If H is unbounded, then 3/ G h : / £ ^'(J/). Hence for 5 € I?/(H) the operator A = f(g , •) i« bounded but not in 0}{H). (ii) Let h = 12{TL) with the canonical basis {e,,}^ and

H = £ «/ei/(e„ , •) (20)

6 e Then En = £"__„ e„( f , •) and mj = \H\. Let a : TL —» Z; 1/ -»/x(i/) be a bijection and /„ € C (y £ TL). Then the operator

A = $1 Ue^v)(ev , •) is in (^(tf). and it is in 0"(/f) only if

3a>/f

3 z2-graded Second Quantisation

(a) Let h = hö © hj be a Z2graded, separabel Hubert space and Pa the orthogonal projections onto h0:

h0 = Pah VQ 6 Z2 = {Ö, 1} (21) We introduce the Klein operator 7 = PQ — P\ and regard hg and hj as one-particle Hubert spaces of the bosons and fermions respectively. Note that every algebra ci operators on h containing the identity and the Klein operator 7 is naturally a »superalgebra with

deg {PaAPß) = a + ß Va, ß € Z2 (22)

for all A in this algebra. We define the super field algebra on h as the tensor product of the boson and the fermion field algebras on hö and hj together with a natural ZVgradation:

Definition 1 The super field algebra Ay(h) is the *-superalgebra with involution A —* A+, generated by elements o+(/), / € h such that the map h 3 f —» a+(f) is linear and grading preserving (i.e.,

dega<+>(/) = deg/ (23)

for all homogeneous / € h) and the following canonical supercommutator relations (CSR) are satisfied (a{f) = (o+(/))+)

W/),«W1 = 0 V/.yeh (24)

((• , •) is the scalar product in h).

(b) Let ^s(ho) and ^V"(hi) be the boson and fermion Fock fiaces over hfl and hj with particle operators a.g\f) (/ G hö) and ap(g) (g € hj) obeying the usual CCR and OAR, with vacua Q$ and Up, *°d the particle number operators NB and

Np respectively [2]. Obviously a natural representation n0 of the super field algebra

Ay(h) can be given on the super Fock space ^7(h)

7,(h) = rB(h)®?F(h) (25)

7 with the vacuum ft = Qg ® ftp as follows: a(+H/) = °i*W) + 4"W) /eh (26)

+ + (for simplicity we identify a* )(-) and no(a* )(-))). Note that

a(f)Q = 0 V/eh (27)

(c) The self-adjoint

N = NB ® 1 + 1 ® NF (28)

induces a l\l0-gradation in Ty(b):

*r00 = ©h? , h?> = (Pt - Pt~x)T,{h) (29) *=o (F_i = 0) with h^ the ^-particle subspace and Pi the projection onto the vectors with particle number less or equal to I:

N = Y.W-Pt~*)- (30)

Let V be a subset of h. We denote as V^(V) the set of vectors in J>(h) which are finite linear combinations of monomials 1 i /I»T/»«T"A//^II a+(/„)ft (31)

with /i, f2, • • • ,fi € V and £ € W0- It is easily seen that "Di,(V) is dense in T^{h) if and only if V is dense in h. Note that

+ a (/)/i ®, • • • ®T /* = v^Tl • / 8, /i ®, • • • ®T /<

d at/)/, ®T • • • ®, /, = (1 - *w) £(-) ««>ir>«><(/ , /„)

x/i®7,"®^»T",«t/< (32)

for all homogeneous /i, ••,/*€ h and I € Wo ('/./ means that jv is omited). In Appendix A (a) we prove the following estimate \P\m\ * VTTT«/» v/eh. (33)

As (cf. (32)) «(/)/* - ft-i«(/), a+(/)^/ = P/+ia+(/) (34) and a+(/)P« = *(/)'?,, a(/)ft = («+(/))"fl

for all / e h, t € W0 we obtain the

8 Proposition 1 The operators a^{f) are in (^{N) for all f 6 h, hence «(/) = («+(/)r v/eh. (35)

Moreover, the maps a^+\-) : h[||-j] —• Ow(N)[Tpi] are continuous.

(d) Second quantisation of bounded operators: Let A £ B(h) and A =

Af) •+• Ai with deg/la = a for a 6 2c2. Setting

5:: :,,def/ dr(i4)/,®7.. •»,/,= £ E(-)° ' '/i®T---®7>i«/l®7--®-r// (36) aeZZ, "=i

for all homogeneous /i, - ,/< € h and £ £ W0 defines a linear operator on Z>^(h). It is obvious that the map B(h) 3 A —* dT(A) is linear and grading preserving, and that the following relations are fulfilled on V^h) for all A, B € B{h), /eh:

dT(A)' = dT{A') (37) [dr^),a+(/)I = a+(Af) (38) [dr(A),dr(B)l = dT{[A,B)). (39)

In Appendix A (b) we prove the following estimate:

ldT(A)Pi\\ < 211 A\\ V£ € Wo, A G 5(h). (40)

Obviously (cf. (36)),

6T{A)Pt = PtdY{A) VfeWo-

Hence dr(yl) e CfJV) for all A 6 B(h), and we obtain the

Proposition 2 77ie map

dr(-) : B(h)[|.|] - 0«(N)[TN}; A ~ dr(i4) provides a continuous representation of the *-Lie superclgebra B(h).

(e) Second quantisation of unbounded operators: We introduce a self- adjoint operator H on h commuting with the Klein operator 7. Then Of{H) is natu­ rally a *-Lie superalgebra. If A € Of{H) then dl{A) can be denned on V{{DS(H)) by means of (36). It ie easily seen that the relations (37)-(39) are fulfilled for all

A,B e 0'{H) and / € D*(H) on £>>(£>(#)). Especially dT{H) and dT{mH) are symmetric operators. In Appendix A (c) we show that V$(Dt(H)) is a set of analyti­ cal vectors for dT(mjj) and dT(H), hence these operators are essentially self-adjoint, and we denote their closures by the same symbols. The spectrum of m# •* {1,2,..., N} with N < 00 (N = 00 if and only if H is unbounded). It follows easily that the spectrum of dT(mn) is W0, and there is a spectral representation

dV(mH) = f,n(En-En.l) (41) n=l

9 with {i?n}£l0 a familie of projection operators on J>(h) obeying (7). Note that the subspace En^Fy(h.) is spanned by vectors

£», A ®-y • • • ®7 Eni f,, r.i e No (42) with T.i=int S n, fi,. ..,fi (E h and I

EnPt = PtEn Vn,/6iM0 (43) and

dT(A)En = dT(AEn)En

EniT(A) = £„dr(£U) VneW (44) for all A 6 C?/(/f) (cf.(36)). It follows that the operators dr(y4), A € Of(H), can be continuously extended to D^(dT(mH)) such that the relations (37)-(39) are fulfilled.

+) y (Note that 4 (/) e O (dr(//)) for all / € D'{mH).) Moreover, (as £n = £nPn)

||dr(A)(->^„U < 2n||A("^„|| Vn6W (45) and

A(-^n = E„A^En <=> dT(A^)En = £„dr(i4<->)£„ (46)

/ / for fi > n, hence dr(A) G 0 (dr(mff)) for all A G 0 (mH). Hence we obtain the following generalisation of proposition 2:

Proposition 3 The map dr(.): 0'(m*)fr.»] - ^(dr^ff))^^)]; * - «UW provides a continuous representation of the *-Lie superalgebraO^m^).

4 Quasi-free Representations

In general, the free representation IIo of the CSR (24) considered in the last section is not adequate for studying quantum field models as it does not allow for constructing Hamiltonians bounded from below. Restricting oneselves to external field problems at zero temperature, one has to consider only irreducibel, quasi-free representatios which are constructed from no and an orthogonal projectios T on h leaving the TL2- gradation of h invariant. We note that reduzibel, quasi-free representations adequate for external field problems at finite temperature can be reduced to irreduzibel ones by doubling the degrees of freedom [26]. (a) Let T = T_ be an orthogonal projection on h commuting with 7, T+ = 1-1-, and J a conjugation in h commuting with T and 7. It is easily seen that the operators

+ + nT(a (/)) = *+(/) = a (T+f) + a{JTJ)

+ DT(a(/)) = M/) = a(T+/)-a (77T_/) V/

10 give a representaton Ux of the super field algebra Ay(h). Obviously ay (/) 6 Ou(N) V/ e h, and it follows from (33) and (47) that

l4+)(/)ftl < 2V7TTI/H V/GWo,/6h (48) and

{ + +) ä T \m = Pi+1ä!r (f)Pt V* € Wo, / € h - (49) In general (7 ^ —1) this representation is not unitary as

«r(/) = (#(*/))' (50)

with 9 = T+ -7T_. (b) A Bogohubov transformation (BT) is given by W)-M4(/))^J(t//) v/eh (51) with U a closed, invertibel operator on h obeying

U'qU = UqU" = g (52) and commuting with 7: It follows from (50)-(52) that

l MäT(f)) = äT(U~f) = Mä}{qf))' , (53) hence the operators äj '(f) = Tu(äj(f\) give a representation 01 Ay(h) obeying (50). The BT is called unitarily implementabel if there is a unitary operator TT(U) in ^7(h) obeying

tT(U)ä$(f)=äl(Uf)tT(U) V/€h. (54) It is well-known that the BT (51) is unitarily implementabel if and only if U obeys the Hilbert-Schmidt condition, i.e.

t/+_,£/_+e£2(h) (55)

[20] (we use the notation

A*=T.ATj Ve,e'G{+,-} (56) for A 6 £(h)).

(c) As 9+_ = g_+ = 0, q gives a unitarily implementabel BT, and there is a grading operator Q = Tr{q) in ^(h) suc.i that

4(f) = QäTV)-Q v/eh. (57)

Hence the representation IIx is unitary with respect to the »emidefinite bilinearform « v »1 «F,G» = VF.Ge^h), (58)

11 and one can regard the map A --* QA'Q as 'natural' involution in D"(N) for the representation UT (see also (b3) below). (d) Second quantization of bounded operators: Let A 6 ßi(h) and {/„}£Li,

{ffn}jJLi orthonormal systems in h and An € C such that

A'-EK/niUnf). (59) rvrl \Ve define the second quantization of A in the representation HT as operator

dfr(i4) =: QT(A): = QT(A)- < n , QT(A)SI > (60) with

QT(A) = £ K4(fn)äT(gn). (61) n=l

Note that dfT(A) € 0*(N) for all A e #i(h) as it follows from (60)-(61) (cf. (48), (49)) 7 ||dfT{ApPi\ < (4* + 9)tr(v/Z4 ) (tr(-) is the trace in h) and

dtriApPt = Pl+7dtT(ApPt W e W0

(cf. lemma 1). From (47), (24) and (27) we conclude that

= -jtr(r_j4) (62) with str(-) the super trace in h as usual (i.e.

stt(A) = ti(jA) Vyi€*i(h).)

It is easily seen that the map i?i(h) 3 A —• drr(j4) is linear and the relations

m (dtT(A)) = dtT(qA"q) = QdtT(A-)Q (63) and

[dtT{A),%(/)) = 4(Af) (64) are fulfilled for all A £ Bj(h) and / € h. Moreover, it follows from

[QT(A),QT(A)] = QT([A,B)) and (62) that

[dtT(A),dtT(B)} = dtT(lA,B})-CT(A,B) (65)

for all A, Be 5i(h) with CT(A ,B) = str(T.[A ,B)) the super-Schwnger term [26]. Using stT{AB) = {~f-8AdetBsti{BA)

12 for all homogeneous A,B£ #i(h), it follows that

CT{A , B) = str(4_+5+_ - A+_£_+) (66)

It is easy to set- that (cf. (36))

+ £ Ana (/n)a(<7n) = iT{A) and

+ :f)A>.o(J/n)a (.7Äl): = dr(A?) n=l wi»h AT ~ J A'J and A = A - 2PiAPi (67) for all .4 G £(h) Hence (60), (61) and (47) give

dfr(A) = dr(^++)-dr(7if_) +

+ + +A_+ao - A+-fa a (68) with (c.:. (£9))

+ + ,4a a = f^\na+(fn)a+(Jgn) n=l oo

yiaa = J^\na{Jfn)a{gn). (69) n=l In Appendix B (d) we prove the following bounds

\AaaPt[ < l\AU [Aa+a+Pt\ < (/ + 2)Mla IG Wo (70)

V4 G £i(h) (||-|2 is the Hilbert-Schmidt norm). With (68), (70) and (40) this leads to |dfrM)fi|<2(S*+l)|il| VfeWo (71)

M| = M++| + M„| + M-+li + IA+.I, (72) (note tha.t M..|<3M._| ). It follows from (71) that the map A ~> ATT(A) can be continuously extended from #i(h) to the Lie superalgebra

gj(h) = {Ae B(h)\A.+ ,A+. e B,(h)} (73) such that the operators dtriA) are in GU(N) for all A 6 #f(h) and the relations (63)—(66) remain true. Note that |-| makes g$(h) to a nnrmed algebra.

13 Proposition 4 The map

r tfr(-) • 52 (h)[H] - 0»(N)[TN}- A ~ dtT(A) provides a continuous, projective representation of the Lie superalgebra g% (b).

Remark: Introducing a matrix notation

with fa = Paf and Aaß - PaAPß for all ; e h, A € £(h), a.fl € Z2, and

4+)(/) = £ &L+,(/J <"V(A) = E «"*(**), (75)

it is easily seen that the supercommutator relations (64) and (65) give the follo­ wing (anti-) commutator relations (here [• , •] and {• , •} denote commutators and anticommutators respectively)

[dfM(4»),aJ(/9)] = OJ(4H»/Ö)

[dfIÖ(ATö),^(/ö)] - ä±{Al6f6) (76)

and

[dröö(>löö) ,dröö(j5öö)] = dröö([.4öö , -5»]) - Cöö(^öö i #öö) (drn(>ln),dfn(ßii)] - «tuiiAi^BnD + CuiAn.Bn)

[dföö(>löö),dföi(ßöi)] = dhi{A6bB6i)

[dfn(i4„),df»(BM)] = dtu(AuBn)

{df6i(4,i),df »(£,„)} = dtn(AnBn)-r dtn{BaAn)

-C:o(Aöi,Böi) (77)

with

Caß(Aaß ,B0a) = tT((Aa0)-+(Bßa)+„ - (A^)+_(B/fa).+) (78)

for all a,ß € 2Z2. Here the even df/ööC*) and drjj(-) are the second quantization-maps in the bos"»nic and fermionic sector respectively, whereas the odd dr$i(-) and drja(') 'mix' these iwo sectors. (e) If A C ^(h) is even and satisfies A = qA'q, then dtriA) is a even, self-adjoint operator (cf. (63), proposition 4 and lemma 1). Due to (t € F) ^tAy = q(eitA)-\

and iiA WM \\T±t T^7 < \t\iT±ATi\* \/t £ F

14 the operator exp(t'M) defines a unitanly implementabel BT for all t 6 R (cf. (52) and (55)), and it is easily seen that the one-parameter family of implementing operators is given by

fT(e«") = exp(i*drT(,4)) (t€R), (79) viz., iU e^riA^fyMtriA)= ä+(e /) (( 6 Rj (80) for all / € h: The l.h.s. and the r.h.s. of equ.(80) are strongly analytic, one parameter families of operators on D*(N) for respectively \t\ < l/8max(|/J,3|>l|) and t € C (cf. lemma 1 and (33)), and their power series in t coincide due to (64). Similarly one can prove (B

M M eitdrT{A)d^B)c-MtT(A) = dfr(e' i?e-' ) - br{tA ,B) {t e R) (81) with

UA UA bT{A ,B) = i f'dsCT{A , e Bc- ) (82) Jo (the l.h.s. of equ.(81) is strongly analytic for t < l/24max(||i4|, |£|), and — by using (65) -- it is straightforward to show that its Taylor series coincides with the one of the r.h.s. which is strongly analytic on Df{N) for i G C (cf. (71))). If also B is even and obeys B' = qBq, then (81) is a self-adjoint operator for allf 6 R| exponentiation for t — 1 gives the relation

u e.dfT(,t)eidfT(fl)e-idfT(>i) = e -*r(* ,B) eXpidfT(e'*fle- ). (83)

Proposition 5 Let A G gl{h) satisfy A' = qAq. Then (80) and (81) hold for all / G h and B G <7^(h), and (83) holds if B is even and satisfies B" = qBq.

Remark: bT(- , •) (82) is an integrated 2Z2-graded 2-cocycle obeying

l,A i,A bT{tA,e Be- ) + bT{sA,B) = bT((s + t)A,B) («,(eR) (84) due to the group properties of exp(itA) and exp (itdIY(.4)) for t € R (cf. also [1]) (f) Second Quantization of Unbounded Operators: Due to (70), the ope­ rators Aaa and Aa+a+ can be defined for all A G f?2(h) as operators in 0"(N). It is easy to see that these operators are no* in Of(dT(mH)) in general, but only if

3N < oo : A = ENAEN (85)

(cf. (69), (32), and the discussion following (41)). It follows with proposition 3 and equ.(68) that dtr{A) can be denned for all operators A G Of(rri,n) obeying

f (*) A++,ÄT eO(mH)

(ii) A-+,A+- € 5j(h)

(in) 3JV < oo : A+. = F,ffA+.EN,

>L+ = ENA-+EN . (86)

1!) We denote the set of all operators A G Of(mf{) obeying (86) as gj(h;m^). It is a locally convex vectorspace as the family of seminorms {('InlSLn

|i4|n = lEnAEnlt Vn € IM € gl(h,mu), separate points and provide a natural topology TU,T- Moreover,

+ + {dtT(A)) = df T{qA q) (87) and

lEndtT{A}En\ < 2ntEnAEn\\ eventually Vn £ N . (88)

Hence we obtain the

Proposition 6 The map

T ATT(-) • <72 (h; H)\THJ) -

Remark: Usinf, the 2 x 2-matrix notation (74), the super-Schwinger term (66) can be written as

a CT(A , B) = £ (-) U((Aa0U(B0a)+_ - (Aaß)+.(B0a).+) (89)

a,0€Z3

Hence in t'.ie case that

hö=hTl TPö = TPit (90)

AM~MU Aöi=Aiö (91) and similarly for B, the super-Schwinger term Cj{A , B) vanishes as the contributions from the bosonic and the fermionic sectors cancel. This is a generalisation of the well- known cancellation due to supersymmetrie. In the case that (90) is fulfilled, we denote the set of all operators A in gj{h) resp. <7^(h;mtf) obeying (91) as 55(h) resp. g^(h;mn)- Obviously these two sets are Lie superalgebras and dfj(') provides representations of them. (g) It follows from (87) that dtr{A) (A £ <7^(h;m/f)) is symmetric if and only if

A* = qAq . (92)

In the following we provide several sufficient conditios for dT^A) to be essentially self-adjoint and generalize proposition 5.

Proposition 7 Let A £ <7^(h;m/f) satisfy (92) and

i4++f>T_€

16 Then dTr{A) is essentially self-adjoint on D*(dT(m.g)) if at least one of the following two conditions hold

(i) A.+ = A+- = 0

(Ü) 3£< oo : i4+J£„ = EnA++En

Ä^-En = EnÄZ-En Vn>L. (94)

In this case exp (it A) is a unitarily implementabel Bogoliubov transformation for all t £ F, and (79) holds (we denote the closure of dlVf-A) by the same symbol).

+ Proof: Let Al = dT(il++) - dT{-)AT.), A2 = A_+oo - A+-ia a+, and 0 <

Qi,a2, K, L < oo such that

\AxEnPt\ < axtn AtEn - En+KA,En

\A2Pt\ < a2l A2En = EnA2En \/n>L,teti.

Note that A2Pt = Pt+2A2Pt and AtP( = P,/li hold for alU G N0. Then for n > L,l£h (cf. (68))

fc k k v v fldfr(A) £nP,|| < £ [ \iAiEnHk^)KPl+2v\\ - • \\A2En+{k^)KPl+2A

< E (*) Mi + 2")(* + (* ~ ")^)]fc_" • M* + 21/)]" • (95)

If (94) (i) holds then .42 = 0> hence a2 - 0 and

k k ldtT(A) EnPll

It follows with the ratio test that

£ ^UtriAfEM < oo (96) for t < l/atitKe, hence

f D(dT(mH)) D £>>(#) = D'{dr(mH)) is a dense set of analytical vectors for dtr(A), and dfr(i4) is essentially self-adjoint due to Nelson's analytical vector theorem. If (94) fit J holds then

AxEn= EnA^En

for n > Lt hence K = 0 in (95) for n > L, and

h k v tdtT(A) EnPt\\ < £ (*)[<*,(/ + 2v)n\ - \a2{l + 2i/)]" < (a,n + a,)*(£ + 2*)* .

17 From the ratio test we conclude that (96) holds for all* < l/(ain -f a2)2e, and the essential seL'-adjointness of dry(.4) on Z)'(dr(m#)) follows as above. In both cases the validity of (79) follows from the fact that there is an R > 0 such that for / € D*{mH) the l.h.s. and the r.h.s. of equ.(80) are strongly analytical on D^(dT(mH)) for \t\ < R and t £ C respectively, and that their Taylor series in t coincide. 0 The following theorem is useful for proving the existence of second quantised Hamiltonians in external field problems.

Proposition 8 Let A £ gJ{h;H) satisfy (92) and (93). Then dtT{A) is essentially self-adjoint if it is semidefinite.

Proof: Let a3, a2> F, L be as in the proof of proposition 7. Using

(2M + 2i/)! - (2/i)!(2«/)! ~ \4/ \/i/ ^"' for all (i, v G IMo (the last estimate fol'.ows from Stirling's formula) we obtain from (95) for n > L, £ 6 l\l (ft - k ~ v)

which is obviously finite for all t £ R. Hence Df(dY(m.n)) is a set of semianalytical vectors for dIY(.4) and it follows form an extension of Nelson's theorem ([31], p206, Theorem X.40) that dTriA) is essentially self-adjoint if it is semibounded. D

5 Anticommuting Parameters and Super Bogoli- ubov Transformations

(a) Let N £ l\l0 and By the commutative »-superalgebra generated by elements

9{ = 6; with degöi = i for i = 1,2,... N (i.e., 0^ = -Ofr for all i,j = 1,2,... N) !32]. Obviously B^ is a 2/v-dimensional vectorspace and it is convenient to introduce

1 as an algebraic basis {&}*=o follows: Let

"=!;<««') •*. di(i,) €{Q,1) (97) j=o

be the base 2 expansion of v £ W0. Then

{v) ,{u) d N ,{v) ßl/ = ef' 8i ---6 N - (98)

N for 0 < v < 2 - 1. Note that 0O - 1, the ßv are homogeneous with deg/3^ = 'EjLö,^(")l»od(2)and

18 with z(i/) = 0 resp. 1 for [Y,*Jo <*j(f)]mod(4) = 0,1 repp. 2,3. Moreover,

ßißu - e(M. ")/W withe(/i,i/)e {0,±1}.

(b) We consider the B^-modules BN ® h and Bn ® £(h). Note that each element in Bfi ® h resp. Bn ® £(h) can be represented as

7N-\ x=Y, 3»x" (") with x„ € h resp. x„ G £(h). The scalarproduct in h, the involution A —» J4+ in C7'(/f) and the super trace / -tr(-) in Bi{h) can be naturally extended to BN ® h, BN ® C (/T) and 5jv ® ^(h) by (anti-) linear extensions of

{ß®f,ß'®9)=ß*ßV,yd'*ß0'9) v/,jeh, (loo) {ß®A)+~ß-®id<*ßA+~fd'*0 VAeöf{H), (101) and str(/? ® A) = £str(7dt84U) V4 £ 5,(h) (102) for all homogeneous ß,ß' € B^. Note that Bs ®h and J3j\r® B(h) are Banach spaces with the norms WHE'ÄAI^'IW (103) I/=0 K=0 for all x in Bfi ® h resp. 5# ® B(h). Moreover, the topology TH extends natarally from 0#(H) to Bn®0#(H) for # = / and w. Similarly the topologies ||-| and THJ extend from gj{h) and 40 = 0 is called soul-like.. Obviously if J4 is soul-like then AN+1 = 0, and exp(zA) can be defined as power series on Bs ® Df(H) for all z 6 C and is a r^-continuous, analytic, one-parameter group of operators on Bn ® Df(H). We denote a soul-like, even operator A 6 By®Of{H) obeying A = A* as auperself-adjoint. Obviously if A is superself-adjoint then exp(itA) (t € F) is a r#-continuous, one-parameter group of superunitary opeiators (see also [24]). It is straightforward to extend the formalism of QFSQ from Hubert spaces h to B/v-modules Bjf ® h: The super Fock space extends to Bn ® ^>(h), and by (anti-) linear extensions of «f(0®/) = 0®&r(/) v/^h. (104)

dts0 äT(ß ®f) = 0"9 ä${t f) V/ € h (105) and

dtT(08A) = ߮dtT(A) VAtgJ(h;mH) (106)

19 for all homogeneous ß € BN, the operators öj (/) and dtr(A) are defined for all / G BN ® h and A 6 B^r ® gj(h;m#). Moreover, it is easily seen that the relations (63)-(66) and the estimate (81) remain true for all A, B € BN ®5^{h;m#) and / € By ® D*(mff). Hence we obtain the following generalisation of proposition 6:

Proposition 9 The map

f dfr(.) : BN ® gl(h;mH)\M] -+BN® O(dT{mB))[T^mH)\; A ~ dtT(A) provides a continuous, projective representation of the Lie superalgebra

BN®gZ{h;mH).

Remark: Note that dlY(-) can also be regarded as representation of a central extension of the Lie superalgebra i?yv<8><;J(h; mit} with the commutative superalgebra BN as center.

As drV(/l) is soul-like for soul-like operators A £ By ® <7j(h,f7tH), it is trivial to prove the following 'super-analog' of proposition 8:

Proposition 10 Fcr A G BN <8> gj(h; TUM) even, soxil-lik'. and obeying A+ = qAq, the operator dTr(A) is tuperself-adjoint and the relations (80}-(82) are fulfilled for

f all f 6 BN ® D(mH) and B £ BN ® glfamn)- If also B is even, souUike, and obeys B+ = qBq, then (83) holds.

In this section we demonstrate that the formalism developed in this paper provides a natural framework for constructing representations of certain infinite dimensional Lie superalgebras. At hrst Lie superalgebras are considered which are semidirect pro­ ducts of afHne Kac-Moody superalgebras [26] and the Virasoro algebra and generalize algebras of Wess-Zumino-Witter, models (cf. e.g. [33], equ.(2.18)). Then a representation of the super Virasoro algebra used [34] is given.

M/N M+JV (a) For M, iVeW» let € be the Z2-graded Hubert space C with the Klein operator -(o -I) (IM is the M x M-unit matrix), and gliu/N the Lie superalgebra of all (M + N) x (M + iV)-matnces acting on CM^N. Let {e^^jZ be the canonical basis in £2(2Z) i nd 2 the operators An and t„ on ^ (Z) defined by

*n — 2~i ef(ef+n ) ') *e2Z £n = £("+?-iM

We consider the Zj-graded Hubert space h = f*(Z) <8> CM,N with the Klein opertor 7 = 1 ® k and the quasi-free representation IIp_ of the super field algebra A,{b) with

P4e»®v)--=®(v-\K®v Vi/ € 2Z,i/ € tM/N (1.08)

20 and the self-adjoint operator H = IQ 0 1. Obviously

[Am®.4,An®£] = Am+n®[A,ß]

[lm , A„ ® A) = -nAm+n ® A

[

and the operators An ® A and l„ ® 1 are in sr2" (h; mj) for all n € Z and A G g^M/N Using (66), a straightforward calculation gives the super-Schwinger terms

Cp_{\m®A,\r.®B) = mÄmi_nstr(,4£)

C/>_(*m®l,An®5) = 0

i ü Cp_(C®Mn®l) = =if dm.-„(M - JV) Vm,ne7L,A,B€gtM/N.

It follows from proposition 5 that the operators

\n(A) = dtP_(\n®A) Vne7L,A£glM/f( (110)

and

Ln = drP_(4,®i) VneZ (in) exist, are in 0'(dr(m/j)) and obey the relations

[Am(A),\n(B)} = \m+n([A,B\)-m6m,.nstr(AB)

[Lni,\n{A)\ = -nAm+n(>l)

[Im, i»] = (m-n)Lm+n-te±^6mi-n(M-N)

Vm,ne7L,A,BeglM/lf. (112)

Remark. The supercommutator relations in the first and third line of (112) represent the Kac-Moody superalgebra gt-M/N [26] afld the Virasoro algebra respectively, hence (112) gives a representation of the semidirect product of these two algebras. (b) We consider the Hubert space h = 12{7L)® C1/1 and represent operators on h as 2 x 2-marices. Then the operators

with

*? = £ sgn(„ - i)^-Ti|7+ n - jMe^ , •)

l i = E("+i-lKK+n,') /„Öi = Eogn^-lK/i^Ti^e^,.) t>e2Z /" = E^ + »" JM^«.') VneZ (114) vs2Z

21 obey the relations

[*m . 'n] = (TO — n)lm4.n Tn [lm ,/n] = (y -n)/m+n

{/,»,/„} = 2*ro+n Vm,ir6Z (115) with q = P+ - 7P- and P_ (108) (A/ = N = 1). Then H = /* is self-adjoint, and obviously /„i/n € <72-(h;mjf) for all n 6 Z (cf. example (ii) on p6). Using (66) a straightforward calculation gives the super-Schwinge« terms

Cp-Cm./n) = 0

Cp.(fm,fn) = n'Snr.n Vm,n€Z.

It follows with proposition 5 that the operators

IB = dfp_(£n), Fn = dfP.(Fn) VnGW (116) exist, are in 0^(dT(mg)) and obey the relations

m m [•^m 1 L„] — (m ~ n)Lm+n + ( 7 '6m^n m

[Lm,Fn] = (y-n)Fm.n

2 {^m,^} = 2Lm+n + m 5m,.n

Vm}n£7Z.. (117)

Remark: We note that this representation of the super Virasoro algebra is similar to the one well-known in the Ramond sector of string theory (see [34] and [26)), and it is easily seen that this representation is unitary as L+ = £_„ and F+ — F-n for all n € TL.

Appendix A

Z2-graded algebraic structures

(cf. e.g. [35],[36]). A 1L% graded vector space V is the direct sum of two subspaces: V = Vjj © V\, Elements of V which are either in Vg or in V\ are called homogeneous, and the degree of homogeneous element / 6 V is

deg/=a V/G V;, a e Z2 (118) Homogeneous elements / with deg/ = Ö resp. I are called even resp. o

A swperalgebra A is an assoziative, Z2-graded algebra (A — A§ © A\) such that

VoG>la,fce^=> o6eAQ+fl Va./JeZj. (119)

22 The bilinear supercommutator [•, •] is defined in a superalgebra A as

p [a ,b] = ab - (-)" ba Va G Aa,b G Aß, a,ß G Z2 . (120)

Note that it satisfies the following relations

[a , [6 ,c]J = [[a ,b},c} + (-)*••*•*[* , [a ,c]] (121)

for all homogeneous a,b,c G A The second relation in (121) is the 2Z2-graded Jakoby identity. A superalgebra A is called commuiative if [a , 6] = 0 for all o, 6 G A. With [• , •] as inner product a superalgebra A becomes a Lie superalgebra. A *-(Lie) superalgebra A is a (Lie) superalgebra and a »algebra such that the in­ volution * leaves the Z2-gradation invariant (i.e., dega = dego" for all homogeneous o€ A).

Appendix B Estimates

For proving estimates it is convenient to introduce a complete, orthonormal bases

(ONB) in .F7(h) as follows: Let {e„}£L0 be a homogeneous ONB in h (i.e. all e„ are homogeneous). Then the vectors

00 fflf.'e V" n 122 |n1,n2!...>=n^4- ( )

with n„ G Wo resp. nu G {0,1} for dege„ = Ö resp. 1 and

00 £n„ = *

dege ln,,legei a{eu)\nun2, >= (-) "^=« v/^:|nI,n2,... ,n„ - 1,... > (124)

(cf (32)).

(a) Proof of (33): Let / = £~ t fvev G h and

tf = £ V'n1,„„...|n1,n2,...>G^(h). (125) ni,nj,., It follows from (124) that

|a(e„)P^|»< £ nJVn.n, „„-.,...|2 , (126)

23 henc

Hf)Prtt < E IM • K*,Wfl

< (E IM2) (E M^UWI') < i/« (wr)1" and

»a(/)/><» < v^|/| V/€h,*tN0 (127) follows. Using that, (a+(/)F*)" = Pta(f), and (34) we obtain

+ l|a (/)PJ = KW+1I

(b) Proof of (40): Let A = ££,=1 A^fa , •) G 5(h). Then obviously

dT(A) = E A^a+(eJa(eJ on Z>'(h), hence

|< V ,dT(A)P^ >i = I E *»* < a(<^W .«KW >l

<\A\f,Hev)P^f <\A\.lUf where |tr(4F)| < |>l||tt(|fl|) and (126) was used. If A € B(h) is self-adjoint then dr(.4)P/ is self-adjoint for all £ G IM0, hence

m \dT{A)Ptl

It follows that

1 \\dT(A)Pt\\ < -{IA + A-\ -\A + A'\\) < 2£|[A|| VA € B(h)ti G W0 . (129)

w (c) Proof of V^D^mu)) C Z? (dr(mH)): Let V = /,«,.••/, with /lf...,/, G

D*{mH) and £ G N0. Then it follows by induction from (36) that VJt G N0

hence

E Ty«diWv

24 which is finite for some t > 0. Similarly one can show that

u Vi(IT{mH)) C D (dT(H)).

(d) Proof of (70): Let ij>,n e ^(h) (cf. (125)). For A € #i(h) we have

oo

Aaa = £ (AJ)^a(e„)a(eM)

with (J4J)„M = (e„ , i4JeM), hence with (124)

|< *P,AaaPtT, >| < £ \(AJ)UJ • |< ^ , a(c„)a(e„)if >| < f] K^UI

X £ !V'n„„2....lv^H;l'/n1.„J „„-, „„-1.J < |4|,-/tylM

n,+n3+...

where Schwarz's inequality was used twice (||-|j2 is the Hilbert-Schmidt norm in h). It follows that

\\AaaPt\

+ + m and with Aa a Pt = (PtA aa)"

+ + \Aa a Pl\ = \A"aaPMl<(t + 2)\A\2 VI € N0 (131)

Acknowledgement

One of us (E.L.) was supported by "Fonds zur Förderung der wissenschaftlichen Forschung in Österreich", Project Nr.P5588 and would like to thank Dr. W. Bulla, G. Kelnhofer, and T. Trenkler for helpful discussion.

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27