Moyal-like form of the star product for generalized<i>SU</i>(2) Stratonovich-Weyl symbols

Klimov, A. B.; Espinoza, P.

doi:10.1088/0305-4470/35/40/305

Cited by 35 publications

(63 citation statements)

References 38 publications

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“…Examples could well include open quantum systems and quantum chemistry. We note that a number of authors including Moyal and Groenewold have produced similar arguments to the above although the presentation has tended to be in a more system-specific form [18,23,30,32].…”

Section: (S)mentioning

confidence: 74%

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General approach to quantum mechanics as a statistical theory

et al. 2019

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Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 and Phys. Rev. A 96, 022117] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function -the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics -extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. Firstly we provide a brief modernized discussion of the general framework of phasespace quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space -putting it for the first time on a truly equal footing to Schrödinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of "parity" and "displacement" in a natural broadening of previously developed phase space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.

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Section: (S)mentioning

confidence: 74%

“…This motivates an extension of Eq. (18) that allows us to perform a convolution of two functions, generating a Moyal star product kernel: (22) so that, by setting s i = s, we can define a generalization of the usual star product following similar arguments by Klimov [30] according to…”

Section: (S)mentioning

confidence: 99%

General approach to quantum mechanics as a statistical theory

et al. 2019

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“…For that we compute explicitely W (σ) S 3 ⋆ Y lm , using (21). We first notice that [19]: Finally, we obtain:…”

Section: Discussionmentioning

confidence: 99%

“…Recently Klimov and Espinoza [19] derived a differential form for the star product in the spin case. On the other hand, in the spinless case the evaluation of the star product becomes simpler with the use of Bopp operators [20].…”

Section: Introductionmentioning

confidence: 99%

Bopp operators and phase-space spin dynamics: application to rotational quantum Brownian motion

Zueco

Calvo²

2007

J. Phys. A: Math. Theor.

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Abstract. For non-relativistic spinless particles, Bopp operators give an elegant and simple way to compute the dynamics of quasiprobability distributions in the phase space formulation of Quantum Mechanics. In this work, we present a generalization of Bopp operators for spins and apply our results to the case of open spin systems. This approach allows to take the classical limit in a transparent way, recovering the corresponding Fokker-Planck equation.

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“…Expressed in terms of spherical harmonics the spinor amplitude has the form (29), with coefficients now given from (68) by a 00 = (ψ 1/2 ϕ 1/2 + ψ −1/2 ϕ −1/2 ) cos 1 …”

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confidence: 99%

Phase-space spinor amplitudes for spin-1/2systems

Watson

Bracken

2011

Phys. Rev. A

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The concept of phase-space amplitudes for systems with continuous degrees of freedom is generalized to finite-dimensional spin systems. Complex amplitudes are obtained on both a sphere and a finite lattice, in each case enabling a more fundamental description of pure spin states than that previously given by Wigner functions. In each case the Wigner function can be expressed as the star product of the amplitude and its conjugate, so providing a generalized Born interpretation of amplitudes that emphasizes their more fundamental status. The ordinary product of the amplitude and its conjugate produces a (generalized) spin Husimi function. The case of spin -1 2 is treated in detail, and it is shown that phase-space amplitudes on the sphere transform correctly as spinors under rotations, despite their expression in terms of spherical harmonics. Spin amplitudes on a lattice are also found to transform as spinors. Applications are given to the phase space description of state superposition, and to the evolution in phase space of the state of a spin-1 2 magnetic dipole in a time-dependent magnetic field.

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Moyal-like form of the star product for generalizedSU(2) Stratonovich-Weyl symbols

Cited by 35 publications

References 38 publications

General approach to quantum mechanics as a statistical theory

General approach to quantum mechanics as a statistical theory

Bopp operators and phase-space spin dynamics: application to rotational quantum Brownian motion

Phase-space spinor amplitudes for spin-1/2systems

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