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arXiv:1706.03557 · Journal of Mathematical Physics

The groupoid of bifractional transformations

S. Agyo, C. Lei, Apostol Vourdas

Published May 1, 2017• 0 citations
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Abstract

Bifractional transformations which lead to quantities that interpolate between other known quantities are considered. They do not form a group, and groupoids are used to describe their mathematical structure. Bifractional coherent states and bifractional Wigner functions are also defined. The properties of the bifractional coherent states are studied. The bifractional Wigner functions are used in generalizations of the Moyal star formalism. A generalized Berezin formalism in this context is also studied.

DOI
10.1063/1.4983917

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Bifractional transformations which lead to quantities that interpolate between other known quantities are considered. They do not form a group, and groupoids are used to describe their mathematical structure. Bifractional coherent states and bifractional Wigner functions are also defined. The properties of the bifractional coherent states are studied. The bifractional Wigner functions are used in generalizations of the Moyal star formalism. A generalized Berezin formalism in this context is also studied.
The displacement operator is related to the displaced parity operator through a two dimensional \nFourier transform. Both operators are important operators in phase space \nand the trace of both with respect to the density operator gives the Wigner functions \n(displaced parity operator) and Weyl functions (displacement operator). The generalisation \nof the parity-displacement operator relationship considered here is called \nthe bi-fractional displacement operator, O(α, β; θα, θβ). Additionally, the bi-fractional \ndisplacement operators lead to the novel concept of bi-fractional coherent states. \nThe generalisation from Fourier transform to fractional Fourier transform can be \napplied to other phase space functions. The case of the Wigner-Weyl function is considered \nand a generalisation is given, which is called the bi-fractional Wigner functions, \nH(α, β; θα, θβ). Furthermore, the Q−function and P−function are also generalised to \ngive the bi-fractional Q−functions and bi-fractional P−functions respectively. The \ngeneralisation is likewise applied to the Moyal star product and Berezin formalism for \nproducts of non-commutating operators. These are called the bi-fractional Moyal star \nproduct and bi-fractional Berezin formalism. \nFinally, analysis, applications and implications of these bi-fractional transforms \nto the Heisenberg uncertainty principle, photon statistics and future applications are \ndiscussed.
The aim of this paper is to give a self-contained and unified presentation of a fermionic coherent state theory with the necessary mathematical details, discussing their definition, properties and some applications. After defining Grassmann algebras, it is possible to get a classical analog for the fermionic degrees of freedom in a quantum system. Following the basic work of Berezin (1966 The Method of Second Quantization (New York: Academic); 1987 Introduction to Superanalysis (Dordrecht: Reidel Publishing Company)), we show that we can compute with Grassmann numbers as we do with complex numbers: derivation, integration, Fourier transform. After that we show that we have quantization formulas for fermionic observables. In particular, there exists a Moyal product formula. As an application, we consider explicit computations for propagators with quadratic Hamiltonians in annihilation and creation operators. We prove a Mehler formula for the propagator and Mehlig–Wilkinson-type formulas for the covariant and contravariant symbols of ‘metaplectic’ transformations for fermionic states. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.
Schwinger’s algebra of selective measurements has a natural interpretation in terms of groupoids. This approach is pushed forward in this paper to show that the theory of coherent states has a natural setting in the framework of groupoids. Thus given a quantum mechanical system with associated Hilbert space determined by a representation of a groupoid, it is shown that any invariant subset of the group of invertible elements in the groupoid algebra determines a family of generalized coherent states provided that a completeness condition is satisfied. The standard coherent states for the harmonic oscillator as well as generalized coherent states for f-oscillators are exemplified in this picture.
The Heisenberg-Weyl group HW(d)HW(d)HW(d) related to a ddd-dimensional Hilbert space H(d)H(d)H(d), is enlarged into the Heisenberg-Weyl-parity group HWP(d)HWP(d)HWP(d) that incorporates parity transformations. It consists of 2d32d^32d3 elements, of which d3d^3d3 elements belong to the HW(d)HW(d)HW(d) subgroup, and extra d3d^3d3 elements which are related through a Fourier transform with the former ones. It is shown that HWP(d)HWP(d)HWP(d) is a generalised version of the dihedral group. The properties of operators that combine displacements and parity, are discussed. HWP(d)HWP(d)HWP(d) is shown to be a solvable group, and commutators of its elements perform displacement and parity transformations of quantum states, along loops in the discrete phase space.2d22d^22d2 coherent states related to the HWP(d)HWP(d)HWP(d) group are introduced, which consist of d2d^2d2 coherent states related to the HW(d)HW(d)HW(d) subgroup, and extra d2d^2d2 coherent states which are related through a Fourier transform with the former ones. In noisy cases, expansion of an arbitrary state in terms of the 2d22d^22d2 coherent states with Bargmann coefficients, is advantageous in comparison to expansion in terms of the d2d^2d2 coherent states related to HW(d)HW(d)HW(d). One of the consequences of the HWP(d)HWP(d)HWP(d) group, is a natural unification of the Wigner and Weyl functions. The properties of the unified Wigner-Weyl function are discussed.