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Journal of Physics Conference Series

Symmetry in finite phase plane

J. Žák

Published Mar 1, 2010• 1 citation
Publisher

Abstract

The known symmetries in one-dimensional systems are inversion and translations. These symmetries persist in finite phase plane, but a novel symmetry arises in view of the discrete nature of the coordinate x i and the momentum p i : x i and p i can undergo permutations. Thus, if x i assumes M discrete values, i = 0, 1,2,..., M – 1, a permutation will change the order of the set x 0 , x 1 ,..., x M −1 into a new ordered set. Such a symmetry element does not exist for a continuous x -coordinate in an infinite phase plane. Thus, in a finite phase plane, translations can be replaced by permutations. This is also true for the inversion operator. The new permutation symmetry has been used for the construction of conjugate representations and for the splitting of the M -dimensional vector space into independent subspaces. This splitting is exhaustive in the sense that if M = i M i with M i being prime numbers, the M -dimensional space splits into M 1 , M 2 ,... M n -dimensional independent subspaces. It is shown that following this splitting one can design new potentials with appropriate constants of motion. A related problem is the Weyl-Heisenberg group in the M-dimensional space which turns into a direct product of its subgroups in the M i -dimensional subspaces. As an example we consider the case of M = 8.

DOI
10.1088/1742-6596/213/1/012023

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The known symmetries in one-dimensional systems are inversion and translations. These symmetries persist in finite phase plane, but a novel symmetry arises in view of the discrete nature of the coordinate x i and the momentum p i : x i and p i can undergo permutations. Thus, if x i assumes M discrete values, i = 0, 1,2,..., M – 1, a permutation will change the order of the set x 0 , x 1 ,..., x M −1 into a new ordered set. Such a symmetry element does not exist for a continuous x -coordinate in an infinite phase plane. Thus, in a finite phase plane, translations can be replaced by permutations. This is also true for the inversion operator. The new permutation symmetry has been used for the construction of conjugate representations and for the splitting of the M -dimensional vector space into independent subspaces. This splitting is exhaustive in the sense that if M = i M i with M i being prime numbers, the M -dimensional space splits into M 1 , M 2 ,... M n -dimensional independent subspaces. It is shown that following this splitting one can design new potentials with appropriate constants of motion. A related problem is the Weyl-Heisenberg group in the M-dimensional space which turns into a direct product of its subgroups in the M i -dimensional subspaces. As an example we consider the case of M = 8.
The structures are physical systems and/or their mathematical descriptions consisting of elements such as particles, fields, observables, operators etc and of relations between these elements: the equations of motion and the rules of quantum mechanics. Examples of symmetries are reflections, permutations of particles, time reversal. The physical system has a symmetry group means, then, that there is a group of transformations leaving the equations of motion and the rules of quantum mechanics invariant. The discrete symmetries lead to conservation laws frequently expressible in terms of quantum numbers, but sometimes requiring a more elaborate description. The observables corresponding to the super-selection quantum numbers can be measured without disturbing any other measurement; their values are always sharp. The chapter shows that the invariant subspace may be labelled by continuous quantum numbers (for instance momentum) when the direct sum becomes a direct integral.
In recent years the finite phase plane has been used to accommodate spin 1/2 particles which play a decisive role as a tool in quantum computing [1]. For this, one often uses conjugate operators in the framework of the Wigner function approach. Conjugate variables are important both in classical and quantum mechanics. The best known example of such operators are the coordinate x and the momentu p [2]. In elementary quantum mechanics there are very few such pairs of conjugate operators. One of the striking results in a finite phase plane of dimension M is that there exist more than one pair of conjugate operators in the xp-plane. In this talk we show how to build 2 conjugate kq-representations (k-quasimomentum, q- quasicoordinte) [3]. These conjugate representations are then applied to Harper-like Hamiltonians, and it's shown how to design physical systems with energy spectra with any desired number of discrete energy levels, and with prescribed degeneracy. A general theory of quantum mechanics in finite phase plane was developed by Schwinger [4].
The Weyl-Heisenberg symmetries originate from translation invariances of various manifolds viewed as phase spaces, e.g. Euclidean plane, semi-discrete cylinder, torus, in the two-dimensional case, and higher-dimensional generalisations. In this review we describe, on an elementary level, how this symmetry emerges through displacement operators and standard Fourier analysis, and how their unitary representations are used both in Signal Analysis (time-frequency techniques, Gabor transform) and in quantum formalism (covariant integral quantizations and semi-classical portraits). An example of application of the formalism to the Majorana stellar constellation in the plane is presented.
The Weyl–Heisenberg symmetries originate from translation invariances of various manifolds viewed as phase spaces, e.g. Euclidean plane, semi-discrete cylinder, torus, in the two-dimensional case, and higher-dimensional generalizations. In this paper, we describe, on an elementary level, how this symmetry emerges through displacement operators and standard Fourier analysis, and how their unitary representations are used both in Signal Analysis (time–frequency techniques, Gabor transform) and in quantum formalism (covariant integral quantizations and semi-classical portraits). An example of application of the formalism to the Majorana stellar constellation in the plane is presented.
It is well-known (see [1]) that for a Heisenberg magnet symmetry operators and symmetry classes can be defined in a very similar way as for tensors (see e.g. [2, 3, 4]). Newer papers which consider the action of permutations on the Hilbert space Ti of the Heisenberg magnet are [5, 6, 7, 8]. We define symmetry classes and commutation symmetries in the Hilbert space H of the 1D spin-1/2 Heisenberg magnetic ring with N sites and investigate them by means of tools from the representation theory of symmetric groups C[ S N ] such as decompositions of ideals of the group ring C[ S N ], idempotents of C[ S N ], discrete Fourier transforms of S N , Littlewood-Richardson products. In particular, we determine smallest symmetry classes and stability subgroups of both single eigenvectors v and subspaces U of eigenvectors of the Hamiltonian H of the magnet. Expectedly, the symmetry classes defined by stability subgroups of v or U are bigger than the corresponding smallest symmetry classes of v or U , respectively. The determination of the smallest symmetry class for U bases on an algorithm which calculates explicitely a generating idempotent for a non-direct sum of right ideals of C[ S N ]. Let U (r1,r2) μ be a subspace of eigenvectors of a fixed eigenvalue μ of H with weight ( r 1 , r 2 ). If one determines the smallest symmetry class for every v U (r1,r2) μ then one can observe jumps of the symmetry behaviour. For "generic" v U (r1,r2) μ all smallest symmetry classes have the same maximal dimension d and 'structure'. But U (r1,r2) μ can contain linear subspaces on which the dimension of the smallest symmetry class of v jumps to a value smaller than d. Then the stability subgroup of v can increase. We can calculate such jumps explicitely. In our investigations we use computer calculations by means of the Mathematica packages PERMS and HRing.
The concepts of von Neumann lattices and tight frames are used for defining discrete quantum mechanical transforms in the phase plane. These transforms are obtained by finite shifts la and mb in the coordinate x and momentum p, respectively, of the Weyl–Heisenberg group, and they are called the discrete Weyl–Heisenberg transforms ψ(la,mb). Here ab=h/N with h the Planck constant, l and m integers, and N a positive integer. A construction is carried out of ψ(la,mb) for a general Weyl–Heisenberg set by using the kq-representation, in which a useful formula is established for the frame operator. The construction is illustrated on an example of the ground state of a harmonic oscillator. It is shown that any physical quantity can be described by the discrete Weyl–Heisenberg transform. Connections are established between ψ(la,mb), the Bargmann representation, and the Husimi distribution function.