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Journal of Mathematical Physics

Continuity statements and counterintuitive examples in connection with Weyl quantization

Ingrid C. Daubechies

Published Jun 1, 1983• 31 citations
Publisher

Abstract

We use the properties of an integral transform relating a classical function f with the matrix elements between coherent states of its quantal counterpart Q f, to derive continuity properties of the Weyl transform from classes of distributions to classes of quadratic forms. We also give examples of pathological behavior of the Weyl transform with respect to other topologies (e.g., bounded functions leading to unbounded operators).

DOI
10.1063/1.525882

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We use the properties of an integral transform relating a classical function f with the matrix elements between coherent states of its quantal counterpart Q f, to derive continuity properties of the Weyl transform from classes of distributions to classes of quadratic forms. We also give examples of pathological behavior of the Weyl transform with respect to other topologies (e.g., bounded functions leading to unbounded operators).
Any quantization maps linearly functions on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all ressources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. In this paper we emphasize the deep connection between Fourier transform and covariant integral quantization when the Weyl-Heisenberg and affine groups are involved. We show with our generalisations of the Wigner-Weyl transform that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations.
The Weyl transform is applied in quantum dynamics to derive and extend Moyal's statistical theory of phase-space distributions for noncommuting coordinate and momentum operators. The distinction is made between Weyl transforms in Schrödinger and Heisenberg pictures; the general case of time-dependent Hamiltonians is considered. The Wigner function for the probability distribution in a phase space of Cartesian coordinates Q and momenta K propagates according to a conditional probability P(t, Q, K | t0, Q0, K0), which is exhibited as a Feynman path integral in phase space. Properties of P(t, Q, K | t0, Q0, K0) are developed; it is expressed in terms of the quantum generalization of the classical Liouville operator. The Weyl transform of a Heisenberg operator propagates according to P(t, Q0, K0 | t0, Q, K) which is also given as a Feynman path integral. An equation for the time evolution of Weyl transforms of Heisenberg operators is obtained, according to which the transform of Heisenberg coordinate and momentum operators obey a quantum form of Hamilton's equations of motion. If the initial density operator of the system commutes with the coordinate operator, then the state of the system is a mixture of pure coordinate states; the spectrum of the density operator in this case is continuous. For a Heisenberg operator AH(t) with Weyl transform AH(t, Q, K), the function A(t,Q)=∫dKAH(t,Q,K) is the expectation at time t of the dynamical property for a quantum system initially in a pure state of coordinate Q; it is the quantum-mechanical generalization of the dynamical property of the system along the classical trajectory in configuration space at time t. The probability amplitude for the time dependence of A(t,Q) can be expressed as a Feynman path integral with a Heisenberg Lagrangian. The amplitude of the conditional probability P(t, Q | t0, Q0) considered by Feynman is expressed as a path integral with a Schrödinger Lagrangian. The velocity in the Heisenberg Lagrangian is the negative of that in the Schrödinger Lagrangian of Feynman, but it agrees with the velocity appearing in the Hamiltonian equations. It is the Heisenberg Lagrangian that is the Lagrangian of classical dynamics. For a particle whose potential energy is a function of position, a quantum form of Newton's second law is obtained. An extension of the formalism to non-Cartesian coordinate systems is given.
A concise derivation of all uncertainty relations is given entirely within the context of phase-space quantization, without recourse to operator methods, to the direct use of Weyl’s correspondence, or to marginal distributions of x and p. Phase-space quantization is the third autonomous and logically complete formulation of quantum mechanics beyond the conventional ones based on operators in Hilbert space or path integrals [1, 2, 3]. It is free of operators and wavefunctions: observables and matrix elements are computed through phasespace integrals of c-number functions (“classical kernels”) weighted by a Wigner function (WF) [4, 3]. This is a phase-space distribution function which is not positive semi-definite, and constitutes the Weyl correspondent [5] of the density matrix in the conventional formulation, fmn(x,p) = 1 2π dye −iyp ψ ∗ m(x − � 2 y)ψn(x + � 2 y) = f ∗ nm(x,p). (1)
We point out that there is a close connection between the Weyl correspondence and coherent state correspondence. By virtue of differential and integral method within normal ordered products we demonstrate that the essence of the Weyl correspondence is just coherent state correspondence in both boson and fermion cases. Furthermore, coherent state form and explicit operator form for the Wigner function are given. With these forms we can derive some important operator formulas from the Weyl correspondence. The Wigner function's zero-point value is also discussed.