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Journal of Mathematical Physics

Least uncertainty principle in deformation quantization

Murray Gerstenhaber

Published Feb 1, 2007• 4 citations
Publisher

Abstract

Deformation quantization generally produces families of cohomologically equivalent quantizations of a single physical system. We conjecture that the physically meaningful ones (i) allow enough observable energy distributions, i.e., ones for which no pure quantum state has negative probability, and (ii) reduce the uncertainty in the probability distribution of the resulting quantum states. For the simple harmonic oscillator this principle selects the classic Groenewold-Moyal (or Weyl) product on phase space while for the free particle, in which there is no real quantization, all cohomologically equivalent quantizations are equally good.

DOI
10.1063/1.2456311

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Deformation quantization generally produces families of cohomologically equivalent quantizations of a single physical system. We conjecture that the physically meaningful ones (i) allow enough observable energy distributions, i.e., ones for which no pure quantum state has negative probability, and (ii) reduce the uncertainty in the probability distribution of the resulting quantum states. For the simple harmonic oscillator this principle selects the classic Groenewold-Moyal (or Weyl) product on phase space while for the free particle, in which there is no real quantization, all cohomologically equivalent quantizations are equally good.
Deformation quantization produces families of mathematically equivalent quantization procedures from which one must select the physically meaningful ones. As a selection principle we propose that the procedure must allow enough `observable' energy distributions, i.e., ones for which no pure quantum state will appear with negative probability and must further have the property that for these the uncertainty in the probability distribution of the quantum states must not exceed that of the original distribution. For the simple harmonic oscillator we show that this allows only the classic Groenewold-Moyal (skew-symmetric) form.
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)
The paper develop the alternative formulation of quantum mechanics known as the phase space quantum mechanics or deformation quantization. It is shown that the quantization naturally arises as an appropriate deformation of the classical Hamiltonian mechanics. More precisely, the deformation of the point-wise product of observables to an appropriate noncommutative ⋆\star⋆-product and the deformation of the Poisson bracket to an appropriate Lie bracket is the key element in introducing the quantization of classical Hamiltonian systems. The considered class of deformations and the corresponding ⋆\star⋆-products contains as a special cases all deformations which can be found in the literature devoted to the subject of the phase space quantum mechanics. Fundamental properties of ⋆\star⋆-products of observables, associated with the considered deformations are presented as well. Moreover, a space of states containing all admissible states is introduced, where the admissible states are appropriate pseudo-probability distributions defined on the phase space. It is proved that the space of states is endowed with a structure of a Hilbert algebra with respect to the ⋆\star⋆-multiplication. The most important result of the paper shows that developed formalism is more fundamental then the axiomatic ordinary quantum mechanics which appears in the presented approach as the intrinsic element of the general formalism. The equivalence of two formulations of quantum mechanics is proved by observing that the Wigner-Moyal transform has all properties of the tensor product. This observation allows writing many previous results found in the literature in a transparent way, from which the equivalence of the two formulations of quantum mechanics follows naturally.
Motivated by some well-known results in the phase space description of quantum optics and quantum information theory, we aim to describe the formalism of quantum field theory by explicitly considering the holomorphic representation for a scalar field within the deformation quantization program. Notably, the symbol of a symmetric ordered operator in terms of holomorphic variables may be straightforwardly obtained by the quantum field analogue of the Husimi distribution associated with a normal ordered operator. This relation also allows to establish a [Formula: see text]-equivalence between the Moyal and the normal star-products. In addition, by writing the density operator in terms of coherent states we are able to directly introduce a series representation of the Wigner functional distribution, which may be convenient in order to calculate probability distributions of quantum field observables without performing formal phase space integrals at all.
We show that any rule for quantization of functions on the phase space of a dynamical system is equivalent to a deformation of the Poisson bracket, in the sense of Lie algebras. All Schrödinger-type quantizations are equivalent mathematically but not physically. We examine whether a quantization rule exists in which the Poisson bracket of the free Hamiltonian and any function not more than quadratic in momentum gets quantized into the commutator of the quantized versions of these two observables. We show that if the (Riemannian) configuration manifold is irreducible, this condition uniquely specifies the quantization of these functions, in the cases when it can be satisfied. However, it can only be satisfied for systems whose configuration manifolds are one-dimensional, or have vanishing Ricci tensor, or are spaces of constant curvature. Thus the condition will not serve as a general physical principle which could be invoked to fix a quantization rule.
Second quantization of a classical nonrelativistic one-particle system as a deformation quantization of the Schrodinger spinless field is considered. Under the assumption that the phase space of the Schrodinger field is CinftyC^{\\infty}Cinfty, both, the Weyl-Wigner-Moyal and Berezin deformation quantizations are discussed and compared. Then the geometric quantum mechanics is also quantized using the Berezin method under the assumption that the phase space is CPinftyCP^{\\infty}CPinfty endowed with the Fubini-Study Kahlerian metric. Finally, the Wigner function for an arbitrary particle state and its evolution equation are obtained. As is shown this new "second quantization" leads to essentially different results than the former one. For instance, each state is an eigenstate of the total number particle operator and the corresponding eigenvalue is always 1overhbar{1 \\over \\hbar}1overhbar.