In the Wigner-Moyal approach to quantum mechanics, we show that Moyal's starting point, the characteristic function $M(τ,θ)=\int ψ^{*}(x)e^{i(τ{\hat p}+θ{\hat x})}ψ(x)dx$ , is essentially the primitive idempotent used by von Neumann in his classic paper "Die Eindeutigkeit der Schrödingerschen Operatoren". This paper provides the original proof of the Stone-von Neumann equation. Thus the mathematical structure Moyal develops is simply a re-expression of what is at the heart of quantum mechanics and reproduces exactly the results of the quantum formalism. The "distribution function" $F(X,P,t)$ is simply the quantum mechanical density matrix expressed in an $( X,P)$ -representation, where $X$ and $P$ are the mean co-ordinates of a cell structure in phase space. The whole approach therefore clearly has little to do with classical statistical theories but is a consequence of a non-commutative nature of the theory.In the Wigner-Moyal approach to quantum mechanics, we show that Moyal's starting point, the characteristic function $M(τ,θ)=\int ψ^{*}(x)e^{i(τ{\hat p}+θ{\hat x})}ψ(x)dx$, is essentially the primitive idempotent used by von Neumann in his classic paper "Die Eindeutigkeit der Schrödingerschen Operatoren". This paper provides the original proof of the Stone-von Neumann equation. Thus the mathematical structure Moyal develops is simply a re-expression of what is at the heart of quantum mechanics and reproduces exactly the results of the quantum formalism. The "distribution function" $F(X,P,t)$ is simply the quantum mechanical density matrix expressed in an $( X,P)$-representation, where $X$ and $P$ are the mean co-ordinates of a cell structure in phase space. The whole approach therefore clearly has little to do with classical statistical theories but is a consequence of a non-commutative nature of the theory.

arXiv:1408.5680Aug 25, 20144 citations

Representations of Quantum Mechanics

John C. Inkson

Springer eBooksJan 1, 19840 citations

Quantum mechanics in phase space: the Schrodinger and the Moyal representations

Nuno Costa Dias, Maurice A. de Gosson, Franz Luef

We present a phase space formulation of quantum mechanics in the Schrödinger representation and derive the associated Weyl pseudo-differential calculus. We prove that the resulting theory is unitarily equivalent to the standard "configuration space" formulation and show that it allows for a uniform treatment of both pure and mixed quantum states. In the second part of the paper we determine the unitary transformation (and its infinitesimal generator) that maps the phase space Schrödinger representation into another (called Moyal) representation, where the wave function is the cross-Wigner function familiar from deformation quantization. Some features of this representation are studied, namely the associated pseudo-differential calculus and the main spectral and dynamical results. Finally, the relation with deformation quantization is discussed.We present a phase space formulation of quantum mechanics in the Schrödinger representation and derive the associated Weyl pseudo-differential calculus. We prove that the resulting theory is unitarily equivalent to the standard "configuration space" formulation and show that it allows for a uniform treatment of both pure and mixed quantum states. In the second part of the paper we determine the unitary transformation (and its infinitesimal generator) that maps the phase space Schrödinger representation into another (called Moyal) representation, where the wave function is the cross-Wigner function familiar from deformation quantization. Some features of this representation are studied, namely the associated pseudo-differential calculus and the main spectral and dynamical results. Finally, the relation with deformation quantization is discussed.

Jan 1, 201212 citations

Displaced and squeezed parity operator: Its role in classical mappings of quantum theories

R. F. Bishop, Apostol Vourdas

The displaced parity operators are shown to have properties that bear a deep relationship with those of the Wigner functions. By exploiting these properties we show that these operators play an important role in linking together many of the aspects of the various exact phase-space mappings of quantum mechanics. These include the Wigner and Weyl representations, coherent states and the Bargmann representation, the P and Q representations, the Weyl correspondence, and the Moyal star product formalism. We also introduce corresponding displaced Fourier operators and show that their squares are just the displaced parity operators. The formalism is extended to squeezed and displaced parity operators, and their corresponding central role in the theory of squeezed coherent states and general squeezing is explained. We also elucidate the part played by the displaced parity operators in the Moyal star product and its extensions, as a first step towards a potential application of these operators in such modern developments as deformation theory and quantum groups. Finally, we indicate how the apparatus developed might also find applications in other recent exact classical mappings of many-particle quantum mechanics or quantum field theory, which are not special cases of deformation theory. Prime examples here include the powerful so-called independent-cluster method techniques, which incorporate the coupled-cluster method formalism with its inbuilt supercoherent states. Throughout the work we stress the central and unifying role played by the displaced parity operator and its generalizations.The displaced parity operators are shown to have properties that bear a deep relationship with those of the Wigner functions. By exploiting these properties we show that these operators play an important role in linking together many of the aspects of the various exact phase-space mappings of quantum mechanics. These include the Wigner and Weyl representations, coherent states and the Bargmann representation, the P and Q representations, the Weyl correspondence, and the Moyal star product formalism. We also introduce corresponding displaced Fourier operators and show that their squares are just the displaced parity operators. The formalism is extended to squeezed and displaced parity operators, and their corresponding central role in the theory of squeezed coherent states and general squeezing is explained. We also elucidate the part played by the displaced parity operators in the Moyal star product and its extensions, as a first step towards a potential application of these operators in such modern developments as deformation theory and quantum groups. Finally, we indicate how the apparatus developed might also find applications in other recent exact classical mappings of many-particle quantum mechanics or quantum field theory, which are not special cases of deformation theory. Prime examples here include the powerful so-called independent-cluster method techniques, which incorporate the coupled-cluster method formalism with its inbuilt supercoherent states. Throughout the work we stress the central and unifying role played by the displaced parity operator and its generalizations.

Physical Review ADec 1, 199479 citations

Representation in Quantum Mechanics

Richard Healey

Elsevier eBooksJan 1, 20260 citations

In the Wigner-Moyal approach to quantum mechanics, we show that Moyal's starting point, the characteristic function

M(τ,θ)=\int ψ^{*}(x)e^{i(τ{\hat p}+θ{\hat x})}ψ(x)dx

, is essentially the primitive idempotent used by von Neumann in his classic paper "Die Eindeutigkeit der Schrödingerschen Operatoren". This paper provides the original proof of the Stone-von Neumann equation. Thus the mathematical structure Moyal develops is simply a re-expression of what is at the heart of quantum mechanics and reproduces exactly the results of the quantum formalism. The "distribution function"

F(X,P,t)

is simply the quantum mechanical density matrix expressed in an

( X,P)

-representation, where

X

and

P

are the mean co-ordinates of a cell structure in phase space. The whole approach therefore clearly has little to do with classical statistical theories but is a consequence of a non-commutative nature of the theory.