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An integral transform related to quantization I. Daubechies and A. Grossmann Citation: J. Math. Phys. 21, 2080 (1980); doi: 10.1063/1.524702 View online: http://dx.doi.org/10.1063/1.524702 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v21/i8 Published by the American Institute of Physics.

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An integral transform related to quantization I. Daubechies a) Theoretische Natuurkunde, Vrije Universiteit Brussel, Pleinlaan 2, B-J050 Brussel Belgium

A. Grossmann CPT I/, Centre de Physique TMorique, Centre de Luminy, Case 907, 70 Route Leon Lachamp, F-13288 Marseille Cedex 2 France

(Received 18 December 1978; accepted for publication 39 August 1979) We study in some detail the correspondence between a functionJ on phase space and the matrix elements (Qj)(a, b) of its quantized Qj between the coherent states la) and Ib). It is an integral transform: Qj(a, b) = S{ a, b Iv J J(v) dv which resembles in many ways the integral transform of Bargmann. We obtain the matrix elements of Qj between harmonic oscillator states as the Fourier coefficients ofJwith respect to an explicit orthonormal system.

I. INTRODUCTION Quantization is a word which should be used with caution, since it means many things to many people. We understand it here in the sense first sketched by Weyl,1 where it describes a "harmonic analysis" procedure. It consists in Fourier analyzing a (fairly arbitrary) function on phase space, and then replacing the "elementary building blocks" (i.e., exponentials on phase space) by appropriate operators (which have since been known as Weyl operators, and are exponentials of linear combinations of the operators X and P). A satisfactory and intrinsic description of the procedure became possible when von Neumann z proved the uniqueness theorem (Steps towards the theorem can be found in Weyl's book. I) which states that for a given (finite) number of degrees offreedom there exists-up to unitary equivalence--essentially only one irreducible family ofWeyl operators in Hilbert space. This theorem is a cornerstone of quantum mechanics for a finite number of degrees of freedom. It seems however to have appeared too late to be fully incorporated in the mainstream of texbooks on the subject. The intrinsic and symplectic formulation of quantum mechanics, made possible by von Neumann's theorem, was developed by Segal 3 ,4 and Kastler,5 largely as a by-product of work aimed at systems with infinitely many degrees of freedom (The MIT thesis of R. Lavine 6 is devoted to finite numbers of degrees offreedom). The ingredients are as follows: (1) A phase space E which is defined (without any "a priori" decomposition into position and momentum) as an even-dimensional vector space (dim E = 2v) with an antisymmetric nondegenerate bilinear form fT. (2) A Weyl system W which is a family of unitary operators, labeled by points in phase space, acting irreducibly on a Hilbert space ~ and satisfying (1.1) Given E and fT, von Neumann guarantees the existence and

uniqueness (up to unitary equivalence) of W, but does not commit us to any concrete realization ofW. The Weyl quantization procedure is then a two-step affair: (a) Fourier analysis:J(v) is written as J(v) = 2 -

'J

eia(v,v'>](v')dv';

(1.2)

(b) substitution of W( - v/2) for eia(v.l, giving Q(f)=2-

v

Jw( ~v

)!(V)dV

(1.3)

as the definition of the "quantized" off I t was shown in Ref. 3 that the correspondence !-Q (f) is inverted by !(V)=2=

2-

V

tr(w( ~ v)Q(f»)

v( (W ( ~ v ),

Q (f»)

ts '

(1.4)

where ('»HS is the inner product in the Hilbert space if HS of Hilbert-Schmidt operators in ~, and that the map J-Q (f) is unitary from L Z(E) onto if HS' Consequently, (fIJZ)L'(El = «Q (f1),Q (fZ)))HS'

(1.5)

If e a denotes the function e O(v) = eia(o.v) we have Q (e Q) = W ( - a12), and so, by extension of Eq. (1.5) «Q (eQ),Q (eb»)HS =

J

e - Q(v)e b(v)dv = 2 zv D(a - b),

in a sense to be made precise (see, e.g., Ref. 7). This map is discussed in more detail by Pool. 8 In Ref. 9, one of us made the remark that step (a) of the quantization procedure (Fourier analysis) can be avoided at the price of replacing the Weyl operators W(v) by Wigner operators II (v) which are simply Weyl operators multiplied by parity, i.e., if II (0) is the parity operator (which can be defined intrinsically up to a sign in any Hilbert space that carries a Weyl system), and if ll(v) is defined by ll(v) = W(2v)ll(0) = W(v)llW( - v),

then Q (J) can be written directly as "Scientific collaborator at the Interuniversity Institute for Nuclear sciences (In the framework of research Project 21 EN). 2080

J. Math. Phys. 21(8), August 1980

Q (f) = 2"f J(v)ll (v) dv

0022·2488/801082080-11 $1,00

@ 1980 American Institute of Physics

Downloaded 25 Apr 2012 to 147.65.105.210. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/about/rights_and_permissions

(1.6)

2080

and we do not have to consider the Fourier transform!of the function! The reason for calling II (v) a Wigner operator is that the Wigner quasiprobability density p",(v) corresponding to a pure state tP is just the expectation value of II (v): p",(v) = 2V (tP,ll(v)tP)

(see Ref. 7). Equation (1.6) expresses Q (f) as a superposition of Wigner operators, which are in some ways simpler than Weyl operators, namely, (i) Every Wigner operator II (v), in addition to being unitary [ll *(v) = (ll (V»-I] is also selfadjoint [ll *(v) = II (v)]. Consequently, II (v) is involutive [(ll (vW = 1] and its spectrum consists of the numbers + 1 and -1. (ii) The relationship (1.1) for Weyl operators is replaced by II (v I)ll (v 2 )ll (v 3) = e