Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1101.2495 · Journal of Physics A Mathematical and Theoretical

A unifying perspective on the Moyal and Voros products and their physical meanings

Prasad Basu, Biswajit Chakraborty, Frederik G. Scholtz

Published Jun 10, 2011• 30 citations
arXivHTMLPDF

Abstract

The Moyal and Voros formulations of non-commutative quantum field theory has been a point of controversy in the recent past. Here we address this issue in the context of non-commutative non-relativistic quantum mechanics. In particular we show that the two formulations simply correspond to two different representations associated with two different choices of basis on the quantum Hilbert space. From a mathematical perspective the two formulations are therefore completely equivalent, but we also argue that only the Voros formulaton admits a consistent physical interpretation. These considerations are elucidated by considering the free particle transition amplitude in the two representations.

DOI
10.1088/1751-8113/44/28/285204

Related papers

A GENERAL FORMULATION OF THE MOYAL AND VOROS PRODUCTS AND ITS PHYSICAL INTERPRETATION

Laure Gouba, DOMAGOJ KOVACEVIC, STJEPAN MELJANAC

A unifying perspective on the Moyal and Voros products and their physical meanings has been recently presented in the literature, where the Voros formulation admits a consistent physical interpretation. We define a star product ⋆, in terms of an antisymmetric fixed matrix Θ, and an arbitrary symmetric matrix Φ, that is a generalization of the Moyal and the Voros products. We discuss the quantum mechanics and the physical meaning of the generalized star product.

arXiv:1106.0469•Modern Physics Letters A•Dec 30, 2011•4 citations

Formulation, interpretation and application of non-commutative quantum mechanics

Frederik G. Scholtz, Laure Gouba, A Hafver +1 more

In analogy with conventional quantum mechanics, non-commutative quantum mechanics is formulated as a quantum system on the Hilbert space of Hilbert-Schmidt operators acting on non-commutative configuration space. It is argued that the standard quantum mechanical interpretation based on Positive Operator Valued Measures, provides a sufficient framework for the consistent interpretation of this quantum system. The implications of this formalism for rotational and time reversal symmetry are discussed. The formalism is applied to the free particle and harmonic oscillator in two dimensions and the physical signatures of non commutativity are identified.

arXiv:0812.2803•Journal of Physics A Mathematical and Theoretical•Apr 7, 2009•132 citations

A coherent-state-based path integral for quantum mechanics on the Moyal plane

Hai Siong Tan

Inspired by a recent work that proposes using coherent states to evaluate the Feynman kernel in noncommutative space, we provide an independent formulation of the path-integral approach for quantum mechanics on the Moyal plane, with the transition amplitude defined between two coherent states of mean position coordinates. In our approach, we invoke solely a representation of the noncommutative algebra in terms of commutative variables. The kernel expression for a general Hamiltonian was found to contain Gaussian-like damping terms, and it is non-perturbative in the sense that it does not reduce to the commutative theory in the limit of vanishing θ—the noncommutative parameter. As an example, we studied the free particle's propagator which turned out to be oscillating with period being the product of its mass and θ. Further, it satisfies the Pauli equation for a charged particle with its spin aligned to a constant, orthogonal B field in the ordinary Landau problem, thus providing an interesting evidence of how noncommutativity can induce spin-like effects at the quantum mechanical level.

arXiv:hep-th/0611254•Journal of Physics A Mathematical and General•Nov 21, 2006•9 citations

Non-relativistic quantum mechanics in a non-commutative space

S. Chaturvedi, R. Jagannathan, Radhakrishnan Sridhar +1 more

Non-relativistic quantum theory of a particle in a non-commutative space is formulated within the framework of the usual quantum mechanics itself. This is accomplished by showing explicitly that the Wess-Zumino formalism of deformed quantum mechanical phase space corresponding to a non-commutative space can be realized, in principle, in terms of the usual quantum mechanical phase space. The case of a slightly non-commutative space is considered in detail. A free particle in such a space is found to behave as if it is a charge particle moving in a self-generated magnetic field.

Journal of Physics A Mathematical and General•Feb 7, 1993•9 citations

From Quantum Field Theory to Quantum Mechanics

We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of N identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation between the position and momentum operators acting in the former spaces and the field operators acting on the latter.This is done in the context of the non-interacting Klein-Gordon field.It may not be possible to extend the procedure to interacting field theories since it relies crucially on particle number conservation.We find it nevertheless important that such an explicit relation can be found at least for free fields.It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations.The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton-Wigner position operator.We make some clarifications on the interpretation of Newton-Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations.

Tsinghua report and review in physics•Feb 1, 2014•0 citations

Classical and quantum mechanics across representations: an operational reading of the Wigner Weyl correspondence

Samuel Schlegel, Borivoje Dakić, Flavio Del Santo

The physical content of a theory is not intrinsically tied to any single mathematical formalism. Both classical and quantum mechanics admit equivalent representations, notably in phase space and in Hilbert space, related by the Wigner-Weyl correspondence. While this correspondence has long been studied in mathematical physics, its foundational and operational implications are often left implicit. Here we give a systematic account of what changes, and what does not, when classical and quantum theories are expressed in each other's native language. This representational viewpoint separates artifacts (such as the appearance of non-positivity or negativity under certain maps) from robust structural distinctions that persist across representations, in particular noncommutativity and its $\hbar$-dependent $\star$-deformation of the classical algebra. We develop the comparison at the level of states, kinematics, and dynamics, and extend it to measurement by formulating both outcome statistics and state-update rules within the same framework.

quant-pharXiv:2607.13135•Physical Review A•Jul 16, 2026•0 citations
The Moyal and Voros formulations of non-commutative quantum field theory has been a point of controversy in the recent past. Here we address this issue in the context of non-commutative non-relativistic quantum mechanics. In particular we show that the two formulations simply correspond to two different representations associated with two different choices of basis on the quantum Hilbert space. From a mathematical perspective the two formulations are therefore completely equivalent, but we also argue that only the Voros formulaton admits a consistent physical interpretation. These considerations are elucidated by considering the free particle transition amplitude in the two representations.
In analogy with conventional quantum mechanics, non-commutative quantum mechanics is formulated as a quantum system on the Hilbert space of Hilbert-Schmidt operators acting on non-commutative configuration space. It is argued that the standard quantum mechanical interpretation based on Positive Operator Valued Measures, provides a sufficient framework for the consistent interpretation of this quantum system. The implications of this formalism for rotational and time reversal symmetry are discussed. The formalism is applied to the free particle and harmonic oscillator in two dimensions and the physical signatures of non commutativity are identified.
Inspired by a recent work that proposes using coherent states to evaluate the Feynman kernel in noncommutative space, we provide an independent formulation of the path-integral approach for quantum mechanics on the Moyal plane, with the transition amplitude defined between two coherent states of mean position coordinates. In our approach, we invoke solely a representation of the noncommutative algebra in terms of commutative variables. The kernel expression for a general Hamiltonian was found to contain Gaussian-like damping terms, and it is non-perturbative in the sense that it does not reduce to the commutative theory in the limit of vanishing θ—the noncommutative parameter. As an example, we studied the free particle's propagator which turned out to be oscillating with period being the product of its mass and θ. Further, it satisfies the Pauli equation for a charged particle with its spin aligned to a constant, orthogonal B field in the ordinary Landau problem, thus providing an interesting evidence of how noncommutativity can induce spin-like effects at the quantum mechanical level.
Non-relativistic quantum theory of a particle in a non-commutative space is formulated within the framework of the usual quantum mechanics itself. This is accomplished by showing explicitly that the Wess-Zumino formalism of deformed quantum mechanical phase space corresponding to a non-commutative space can be realized, in principle, in terms of the usual quantum mechanical phase space. The case of a slightly non-commutative space is considered in detail. A free particle in such a space is found to behave as if it is a charge particle moving in a self-generated magnetic field.
We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of N identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation between the position and momentum operators acting in the former spaces and the field operators acting on the latter.This is done in the context of the non-interacting Klein-Gordon field.It may not be possible to extend the procedure to interacting field theories since it relies crucially on particle number conservation.We find it nevertheless important that such an explicit relation can be found at least for free fields.It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations.The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton-Wigner position operator.We make some clarifications on the interpretation of Newton-Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations.
The physical content of a theory is not intrinsically tied to any single mathematical formalism. Both classical and quantum mechanics admit equivalent representations, notably in phase space and in Hilbert space, related by the Wigner-Weyl correspondence. While this correspondence has long been studied in mathematical physics, its foundational and operational implications are often left implicit. Here we give a systematic account of what changes, and what does not, when classical and quantum theories are expressed in each other's native language. This representational viewpoint separates artifacts (such as the appearance of non-positivity or negativity under certain maps) from robust structural distinctions that persist across representations, in particular noncommutativity and its ℏ\hbarℏ-dependent ⋆\star⋆-deformation of the classical algebra. We develop the comparison at the level of states, kinematics, and dynamics, and extend it to measurement by formulating both outcome statistics and state-update rules within the same framework.