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Quantum Deformed Canonical Transformations, W {\Infty UTS − DFT − 93 − 11 QUANTUM DEFORMED CANONICAL TRANSFORMATIONS, W∞-ALGEBRAS AND UNITARY TRANSFORMATIONS E.Gozzi♭ and M.Reuter♯ ♭ Dipartimento di Fisica Teorica, Universit`adi Trieste, Strada Costiera 11, P.O.Box 586, Trieste, Italy and INFN, Sezione di Trieste. ♯ Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, W-2000 Hamburg 52, Germany ABSTRACT We investigate the algebraic properties of the quantum counterpart of the classical canon- ical transformations using the symbol-calculus approach to quantum mechanics. In this framework we construct a set of pseudo-differential operators which act on the symbols of arXiv:hep-th/0306221v1 23 Jun 2003 operators, i.e., on functions defined over phase-space. They act as operatorial left- and right- multiplication and form a W∞ × W∞- algebra which contracts to its diagonal subalgebra in the classical limit. We also describe the Gel’fand-Naimark-Segal (GNS) construction in this language and show that the GNS representation-space (a doubled Hilbert space) is closely related to the algebra of functions over phase-space equipped with the star-product of the symbol-calculus. 1. INTRODUCTION The Dirac canonical quantization [1] can be considered as a map [2] [3]. ”D” sending real functions defined on phase space f1, f2, ··· to hermitian operators f1, f2 ··· which act on [2,3] some appropriate Hilbert space According to Dirac, this map shouldb satisfyb the following requirements (a) D λ1f1 + λ2f2 = λ1f1 + λ2f2, λ1,2 ∈ R 1 (b) D f1, f2 = f1b, f2 b pb ih¯ (1.1) (c) D 1 = I b b (d) q, p are represented irreducibly. b b Here ·, · pb denotes the Poisson bracket, and I is the unit operator. Groenwald and van Hove [4]. showed that it is impossible to satisfy these conditions for all functions. They can be established only for functions which are at most quadratic in p and q. This obstruction to a complete quantization implies that not all canonical transformation generated by an arbitrary real function f, i.e., δ(·)= ·, f pb can be implemented as unitary transformations 1 δ(·)= [(·), f] (1.2) ih¯ b on the Hilbert space. In order to avoid the consequences of this ”no-go” theorem one has to relax some of the conditions of (1.1). In geometric quantization [5],for instance, condition (d) is abandoned, and in Moyal quantization [6] condition (b) is modified. Moyal quantization involves replacing the Poisson bracket in (b) by a new,h ¯-dependent bracket, the Moyal bracket. It is a ”deformation” [7] of the Poisson bracket to which it reduces in the classical limith ¯ → 0. The formal structure underlying the Moyal quantization is the ”symbol calculus” [8]. Generally speaking it deals with the representation of operators acting on some Hilbert space V in terms of functions defined on a suitable manifold M. The ”symbol map”, symb, is a linear one-to-one map from the space of operators on V to F un(M) , the space of complex functions on M. We write f = symb(f) for the function f representing the operator f. The inverse of the symbol map, the ”operatorb map”, op, associates a unique operator tob any complex f ∈ F un(M): op(f) = f. On F un(M) one introduces the so- called star product ∗ which expresses the operator productb f1f2 in terms of the symbols of b b 2 f1 and f2: b b symb(f1f2)= symb(f1) ∗ symb(f2) (1.3) b b b b Like the operator product the star product is non-commutative but associative. Let us now return to quantum mechanics and let us assume that V is the state space of some quantum mechanical system with N degrees of freedom and that M≡M2N is its 2N phase-space. For simplicity we assume that M2N is the flat ℜ . A particularly important type of symbol is the Weyl symbol [9]. It is defined as 2N a d φ0 i a b f(φ )= exp φ ωabφ Tr T (φ0)f (1.4) Z (2πh¯)N h¯ 0 b b Its inverse is: 2N 2N d φ d φ0 i a b f = f(φ) exp φ ωabφ T (φ0) (1.5) Z (2πh¯)2N h¯ 0 b b a 1 N 1 N Here φ = (p , ··· p , q , ··· q ), a = 1 ··· 2N, are canonical coordinates on M2N , and ωab are the coefficients of the corresponding symplectic two-form. The operators T (φ0), [10] sort of translation operators on phase-space , are defined in ref.[15]. For the caseb when the phase-space is not flat see ref.[11]. For real functions f the operator f is hermitian. ∗ If f = symb(f) is complex, then the complex-conjugate function f is theb symbol of the hermitian-adjointb operator: op(f ∗)= f† , f ∗ = symb(f†) (1.6) b b Arbitrary (mixed) states of the system are characterized by ̺ = symb(̺) where ̺ is the ̺ ψ ψ [9] density operator. In particular, for pure states = | ih | one obtains the familiarb Wignerb function b N i 1 ∗ 1 ̺ψ(p, q)= d x exp − px ψ(q + x)ψ (q − x) (1.7) Z h¯ 2 2 Using the symbols of density operators ̺ and of observables O it is possible to reformulate the full machinery of quantum mechanicsb and in particular tob write down the expectation values of observables and so on. Details about this are provided in ref.[15]. The star product 3 for the Weyl symbol is given by an infinite-order differential operator: ← → ← → ih¯ ∂ ∂ ∂ ∂ f ∗ g (φ) = f(p, q) exp − g(p, q) 2 ∂qi ∂p ∂p ∂qi i i ← → h¯ ab = f(φ) exp i ∂a ω ∂ g(φ) (1.8) 2 b ∞ ih 1 ¯ n a1b1 anbn = ( ) ω ··· ω (∂a ··· ∂a f)(∂ ··· ∂ g) n! 2 1 n b1 bn nX=0 ab ab a Here ω is the inverse of ωab: ω ωbc = δc . In the classical limit the star-product becomes the ordinary pointwise product: (f ∗ g)(φ) = f(φ)g(φ)+ O(¯h). By expressing either f or g in terms of its Fourier transform, it is easy to derive other representations of the star [8] product . The Moyal bracket ·, · mb is defined as the commutator with respect to this star multiplication: 1 f,g mb = f ∗ g − g ∗ f ih¯ (1.9) 1 = symb [f, g] ih¯ b b 2 Forh ¯ → 0 the Moyal bracket approaches the Poisson bracket: {f,g}mb = f,g pb + O(¯h ). ab In our notation the Poisson bracket reads f,g pb ≡ ∂af ω ∂bg. From the fact that the star-multiplication is associative it follows that the Moyal bracket fulfills the Jacobi identity and that f,g1 ∗ g2 mb = f,g1 mb ∗ g2 + g1 ∗ f,g2 mb (1.10) [2] This means that f, · mb with f ∈ F un(M) fixed, is a derivation on the algebra F un(M), ∗ . For other properties regarding the ”derivation” property of the Moyal struc- ture, we refer the reader to ref.[7]. Let us now replace the Poisson bracket on the LHS of (b) by the Moyal bracket. We can then find a quantization map D which satisfies all the requirements (a) − (d) of (1.1) for −1 all functions f1,2 ∈ F un(M2N ), namely the operator map op = symb : 1 op f , f = f , f (1.11) 1 2 mb ih¯ 1 2 b b 1 The unitary transformation δf (·)= i¯h f, (·) are now in a one-to-one correspondence with the following transformations acting on symbolsb δf (·)= f, · mb = f, · pb +higher deriva- tives. These transformations could be called ”quantum deformed canonical transformations”. 4 They act on c-number functions, as in classical mechanics, but two consecutive transfor- mations are composed by forming the Moyal bracket f1, f2 mb instead of the Poisson bracket.The purpose of this paper is to study the covariance prop erties of quantum mechanics by using the notion of quantum deformed canonical transformations. Our emphasis is on the relation between the phase-space formulation and the standard Hilbert space formulation. In [12] Section-2 we construct a set of pseudo-differential operators on F un(M2N ) correspond- ing to operatorial left and right multiplication. They form a kind of W∞ × W∞ algebra. We give a careful discussion of its classical limit, in which it contracts to the diagonal sub- diag group W∞ . In Section 3 we consider the deformed canonical transformations from the point of view of the Gel’fand-Naimark-Segal (GNS) construction [13] which is at the heart of Thermo Field Dynamics (TFD) [14], for instance. In this construction the doubled Hilbert space V⊗V plays a central role. We show explicitly that V⊗V may be identified with the space of symbols, F un(M2N ). In Section 4 we realize the W∞ × W∞ algebra in the recently proposed [15] extended Moyal formalism which was used to formulate a tensor calculus on quantum phase-space. 2. THE W∞ × W∞ GENERATORS For each symbol f = symb(f) we define the operators Lf and Rf acting on the space of symbols as b Lf g = f ∗ g = symb(f g) (2.1) Rf g = g ∗ f = symb(gbfb) for all g = symb(g). The operators Lf and Rf implementb b on F un(M2N ) the left and right multiplication with f, respectively. By employing the form (1.8) of the star-product, b we can represent Lf and bRf by the following pseudo-differential operators a ih¯ ab Lf = : f(φ + ω ∂b): 2 (2.2) ih¯ R = : f(φa − ωab∂ ): f 2 b Here the normal ordering symbol : (···) : means that all derivatives ∂a should be placed to the right of all φ’s.
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