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A Super-Version of Quasi-Free Second Quantization. I. Charged Particles

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A Super-Version of Quasi-Free Second Quantization. I. Charged Particles AMC Oi-ibi UWThPh-1990-39 November 6, 1990 A Super-version of Quasi-free Second Quantization. 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 field problem, bosons and fermions, 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 fermion. 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 quantum 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 Fock space 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 boson 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 quantum mechanics 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<j (B hj where h§ and \v\ correspond to the one-particle Hubert spaces of the bosons and fermions respectively. The CCR and CAR of the boson and fermion field operators iß \f) (/ € hö) and aF (s) (9 £ ^l) are then combined to operators 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 operator 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.
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