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Analytic Representation of Finite Quantum Systems

S. Zhang, Apostol Vourdas

Published Apr 4, 2005• 20 citations
OpenAlex

Abstract

A transform between functions in R and functions in Zd is used to define the analogue of number and coherent states in the context of finite d-dimensional quantum systems. The coherent states are used to define an analytic representation in terms of theta functions. All states are represented by entire functions with growth of order 2, which have exactly d zeros in each cell. The analytic function of a state is constructed from its zeros. Results about the completeness of finite sets of coherent states within a cell are derived.

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A transform between functions in R and functions in Zd is used to define the analogue of number and coherent states in the context of finite d-dimensional quantum systems. The coherent states are used to define an analytic representation in terms of theta functions. All states are represented by entire functions with growth of order 2, which have exactly d zeros in each cell. The analytic function of a state is constructed from its zeros. Results about the completeness of finite sets of coherent states within a cell are derived.
A transform between functions in and functions in is used to define the analogue of number and coherent states in the context of finite d -dimensional quantum systems. The coherent states are used to define an analytic representation in terms of theta functions. All states are represented by entire functions with growth of order 2, which have exactly d zeros in each cell. The analytic function of a state is constructed from its zeros. Results about the completeness of finite sets of coherent states within a cell are derived.
Quantum systems in a d-dimensional Hilbert space are considered, where \nthe phase spase is Z(d) x Z(d). An analytic representation in a cell S in the \ncomplex plane using Theta functions, is defined. The analytic functions have \nexactly d zeros in a cell S. The reproducing kernel plays a central role in \nthis formalism. Wigner and Weyl functions are also studied. \nQuantum systems with positions in a circle S and momenta in Z are also \nstudied. An analytic representation in a strip A in the complex plane is also \ndefined. Coherent states on a circle are studied. The reproducing kernel is \ngiven. Wigner and Weyl functions are considered.
Finite quantum systems with d-dimension Hilbert space, where position x and \nmomentum p take values in Zd(the integers modulo d) are studied. An analytic \nrepresentation of finite quantum systems, using Theta function is considered. \nThe analytic function has exactly d zeros. The d paths of these zeros on the \ntorus describe the time evolution of the systems. The calculation of these \npaths of zeros, is studied. The concepts of path multiplicity, and path winding \nnumber, are introduced. Special cases where two paths join together, are also \nconsidered. A periodic system which has the displacement operator to real \npower t, as time evolution is also studied. \nThe Bargmann analytic representation for infinite dimension systems, with \nvariables in R, is also studied. Mittag-Leffler function are used as examples of \nBargmann function with arbitrary order of growth. The zeros of polynomial \napproximations of the Mittag-Leffler function are studied.
Various Euclidean, hyperbolic and elliptic analytic representations are introduced and relations among them are discussed. The Bargmann analytic representation in the complex plane is considered and its relation to other phase-space methods for the harmonic oscillator is reviewed. The general theory that relates the growth of analytic functions with the density of their zeros is applied to Bargmann functions and it leads to theorems on the completeness of sequences of Glauber coherent states. Two hyperbolic analytic representations in the unit disc, based on SU (1, 1) coherent states and also on phase states are introduced. A third analytic representation in the complex plane based on Barut–Girardello states is also considered and transformations which relate it to the other ones are studied. In the case of systems with finite-dimensional Hilbert space, an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced. The Berezin formalism in the Euclidean, hyperbolic and elliptic cases is discussed. Contour analytic representations in these three cases are also presented.
The paper contains an investigation on basic problems about the zeros of analytic theta functions. A brief introduction to analytic representation of finite quantum systems is given. The zeros of this function and there evolution time are discussed. Two open problems are introduced. The first problem discusses the cases when the zeros follow the same path. As the basis change the quantum state |f transforms into different quantum state. The second problem is to define a map between two toruses where the domain and the range of this map are the analytic functions on toruses.