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Theoretical and Mathematical Physics

Star product, discrete Wigner functions, and spin-system tomograms

Петер Адам, V. A. Andreev, Аурелиан Исар, Vladimir I. Man’ko, Margarita A. Man’ko

Published Mar 1, 2016• 10 citations
Publisher

Abstract

We develop the star-product formalism for spin states and consider different methods for constructing operator systems forming sets of dequantizers and quantizers, establishing a relation between them. We study the physical meaning of the operator symbols related to them. Quantum tomograms can also serve as operator symbols. We show that the possibility to express discrete Wigner functions in terms of measurable quantities follows because these functions can be related to quantum tomograms. We investigate the physical meaning of tomograms and spin-system tomogram symbols, which they acquire in the framework of the star-product formalism. We study the structure of the sum kernels, which can be used to express the operator symbols, calculated using different sets of dequantizers and also arising in calculating the star product of operator symbols, in terms of one another.

DOI
10.1134/s0040577916030041

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We develop the star-product formalism for spin states and consider different methods for constructing operator systems forming sets of dequantizers and quantizers, establishing a relation between them. We study the physical meaning of the operator symbols related to them. Quantum tomograms can also serve as operator symbols. We show that the possibility to express discrete Wigner functions in terms of measurable quantities follows because these functions can be related to quantum tomograms. We investigate the physical meaning of tomograms and spin-system tomogram symbols, which they acquire in the framework of the star-product formalism. We study the structure of the sum kernels, which can be used to express the operator symbols, calculated using different sets of dequantizers and also arising in calculating the star product of operator symbols, in terms of one another.
The review of star-product formalism providing the possibility to describe quantum states and quantum observables by means of the functions called symbols of operators which are obtained by means of bijective maps of the operators acting in Hilbert space onto these functions is presented. Examples of the Wigner-Weyl symbols (like Wigner quasi-distributions) and tomographic probability distributions (symplectic, optical and photon-number tomograms) identified with the states of the quantum systems are discussed. Properties of quantizer-dequantizer operators which are needed to construct the bijective maps of two operators (quantum observables) onto the symbols of the operators are studied. The relation of the structure constants of the associative star-product of the operator symbols to the quantizer-dequantizer operators is reviewed.
Abstract This paper presents a review of star-product formalism. This formalism provides a description for quantum states and observables by means of the functions called’ symbols of operators’. Those functions are obtained via bijective maps of the operators acting in Hilbert space. Examples of the Wigner-Weyl symbols (Wigner quasi-distributions) and tomographic probability distributions (symplectic, optical and photon-number tomograms) identified for the states of the quantum systems are discussed. Properties of quantizer-dequantizer operators required for construction of bijective maps of two operators (quantum observables) onto the symbols of the operators are studied. The relationship between structure constants of associative star-product of operator symbols and quantizer-dequantizer operators is reviewed.
Considering physics as it is done today one notices that physics separates into two formally and conceptually different parts. There is on the one hand classical physics that deals with macroscopic phenomena and there is on the other hand quantum physics that deals with microscopic phenomena. Penrose [96] described this situation as a disturbing analogy to ancient Greece, where two different sets of laws for earth and heaven were applied. So immediately the question arises if one really has to use different formalisms on different scales or if it is possible to describe physics in a unified way. In order to investigate this question one first has to consider how classical and quantum physics are related. Quantum mechanics results from classical physics by a procedure called quantization and classical physics is reobtained by taking the classical limit. Both procedures are heavily plagued by problems [61, 100]. In the first chapter the mathematical problems of quantization will be addressed and it will be described how the star product formalism circumvents these problems. The star product formalism has the advantage that there is no formal break if one goes from classical physics to quantum physics. This formal advantage and the resulting beauty is then used as a guiding principle for the further development. The first question that arises is if the spin can be described in the star product formalism. That this is indeed possible is shown in the second chapter. As a starting point the spin description with pseudoclassical mechanics as it was developed by Berezin is used. One can then construct a fermionic star product and apply it for deformation quantization of pseudoclassical mechanics. In analogy to the bosonic star product formalism one obtains spin Wigner functions that act as spin projectors. Besides the nonrelativistic case it is also possible to formulate Dirac theory with star products. The Clifford algebra of the gamma matrices is hereby described as a deformed version of a four dimensional Grassmann algebra. The fermionic star product in combination with the bosonic Moyal product leads to a supersymmetric star product formalism that can be used to describe supersymmetric quantum mechanics and in the relativistic case to describe the supersymmetric structure of Dirac theory. The other direction of generalization of the star product formalism is the application of star products in quantum field theory that is described in chapter three. After constructing a suitable normal product it is shown that the algebraic structures of perturbative quantum field theory appear also in the star product formalism, which is an expression of the algebra morphism of the operator and the star product formalism. But moreover the quantum group structure that was recently found in perturbative quantum field theory is shown to be a natural algebraic structure of the star product. The essential advantage of deformation quantization is that the classical limit has a well defined meaning. In the context of the spin description with star products this leads to the question of the classical limit of spin, or equivalently to the question of the physical status of pseudoclassical mechanics. In chapter four it is shown that this question is solved if one realizes that the fermionic sector together with the fermionic star product describe the underlying geometric structure. The deformation of a Grassmann algebra leads to a Clifford calculus that is equivalent to geometric algebra. The formulation of geometric algebra in the star product formalism is given in chapter four. One sees there that geometric algebra as the most fundamental geometric formalism that unifies all geometric structures that appear in physics can be described in a supersymmetric manner that parallels the bosonic star product structures. Having obtained a formulation of geometry with fermionic star products this formalism is then applied in the fifth chapter to physical problems. As examples for the application of geometric algebra in classical physics the rigid body and the Kepler problem are considered. In both cases the formalism of geometric algebra gives the most elegant formulation of the problem. In the quantum case one can then combine the fermionic star product formalism that describes the underlying geometric structure with the bosonic star product that describes the noncommuative structure of quantum mechanics. The result is a noncommutative version of geometric algebra that leads to a natural appearance of spin terms. The same idea applied on the phase space leads to the split in supersymmetric partner systems. Geometric algebra gives in this way a natural geometric foundation of supersymmetric quantum mechanics. Similarly one can interpret the hidden BRST-structure of classical mechanics that was found by E. Gozzi and M. Reuter in the path integral formalism from a star product point of view.