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Journal of Physics A Mathematical and General

Closed-form relations for phase-space representatives of spin-J operators

Vitor Rocha Vieira, Pedro D Sacramento

Published Oct 21, 1994• 1 citation
Publisher

Abstract

By introducing the holomorphic representation we obtain, in closed form, relations between the c-number functions, which represent an operator in spin-J space, without recurring to expansions in spherical harmonics. In particular, we obtain the diagonal representative from the diagonal matrix element through an integral relation. New analogies between these and the spin (Bloch) coherent states are established which avoid the introduction of Schwinger bosons, and the consequent projection to the physical subspace, by introducing a generalized Fourier transformation in spin-J space in a way similar to the treatment by Cahill and Glauber (1969) for boson coherent states. Also, we study the generalization to the spin case of the s-ordering of Cahill and Glauber.

DOI
10.1088/0305-4470/27/20/005

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By introducing the holomorphic representation we obtain, in closed form, relations between the c-number functions, which represent an operator in spin-J space, without recurring to expansions in spherical harmonics. In particular, we obtain the diagonal representative from the diagonal matrix element through an integral relation. New analogies between these and the spin (Bloch) coherent states are established which avoid the introduction of Schwinger bosons, and the consequent projection to the physical subspace, by introducing a generalized Fourier transformation in spin-J space in a way similar to the treatment by Cahill and Glauber (1969) for boson coherent states. Also, we study the generalization to the spin case of the s-ordering of Cahill and Glauber.
We consider a spin coherent states description of a general quantum spin system. It is shown that it is possible to use the spin-1/2 representation to study the general spin-J case. We identify the 1/2 spinor components as the homogeneous coordinates of the projective space associated to the complex variable that labels the coherent states and establish a relation between the two-component spinor and the bosonic Schwinger representation of a spin operator. We rewrite the equations of motion, obtained from the path integral for the evolution operator or partition function, in terms of the 1/2 spinor and define the effective Hamiltonian of its evolution.
Quantum statistical mechanics of spin systems is formulated by the use of the representation of atomic coherent state of spin, which is specified by a complex number. After introducing a newly defined distribution function of spin, the Bloch equation for the equilibrium density matrix, the von Neumann equation for the nonequilibrium density matrix and the Heisenberg equation of motion are converted into partial differential equations of the type of separating variables respectively. Formal solutions of them are presented. Any operator which is composed of spins can be converted uniquely into a differential operator. Thus we succeed in formulating equilibrium and nonequilibrium quantum statistical mechanics in the complex variable space, which differs from the previous one in the diagonal representation of density matrix. We further introduce the interaction representation for the perturbational expansion. The linear response formula is written down in the present representation.
The path integral single spin partition function in the basis of boson coherent states, for a general normal ordered two-mode Schwinger boson Hamiltonian, has been computed. Within the initial phase of the calculational process, by a specific approach, the Schwinger boson constraint by means of the projection operator has been implemented.\nAfter, the case of the Zeeman Hamiltonian has been taken on. Using the expression for the partition function obtained for the general case, the appropriate single spin partition function has been produced. Additionally, partition functions have been computed for a few specific spin quantum numbers and then compared with those calculated by the straightforward means, which brought about great confidence in the projection operator implementation; both results match perfectly. \nFinally, the Heisenberg Model has been tackled, and the appropriate expressions for the partition functions have been computed. Their validity has not been verified for any specific cases, but the Zeeman Hamiltonian case dealt with earlier appears to be a strong indicator that these are indeed the correct expressions. \nMany open questions remain, however, for future research to address. All in all, it has been discovered that using the projection operator as a means to enforce the Schwinger boson constraint, veritably works for a Zeeman Hamiltonian problem for specific spin quantum numbers.
The Bloch coherent states for a spin or a system of spins and the Glauber coherent states for bosons are examined from the viewpoint of Lie algebras. It is pointed out that the Bloch coherent states are vectors in the space spanned by the basis functions for an irreducible representation of the unitary unimodular group SU(2), and that the Glauber coherent states are vectors in the space spanned by the basis functions for the infinite-dimensional irreducible representation of a contracted group of SU(2). A deeper understanding of many of the useful properties of these coherent states is gained.
A new method of treating problems of spin systems is developed, in which spin operators are transformed into two kinds of boson operators according to Schwinger's method. Usual treatment of coherent state representation is generalized to spin systems with a fixed magnitude S . Some basic properties and the relations to the atomic coherent state are investigated rather in details. The method is applied to spin relaxation process and a corresponding Fokker-Planck type equation is obtained and solved to give a multiple relaxation process due to nonlinearities. It is shown that our formulation has several advantages over other ones in some respects.