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Proceedings of the Royal Society of London A Mathematical and Physical Sciences

The Feynman principle for a Fermi system

J. L. Martin

Published Jun 23, 1959• 82 citations
Publisher

Abstract

Abstract A scheme for representing vectors and matrices as functions of a certain abstract symbol λ is set up: though λ has no numerical significance, it is found to behave as if it were an eigenvalue of a certain singular matrix A. The resulting ‘eigensymbol’ theory is developed and applied to the quantum theory of Fermi systems. It is shown how a Feynman principle for such systems may be formulated in analogy with the familiar Feynman principle for a system with canonical p and q. The results are illustrated by the case of a simple Fermi oscillator.

DOI
10.1098/rspa.1959.0127

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Inspired by the formalism that relates the star-exponential with the quantum propagator for bosonic systems, in this work we introduce the analogous extension for the fermionic case. In particular, we analyse the problem of calculating the star-exponential (i.e., the symbol of the evolution operator) for Fermi systems within the deformation quantization program. Grassmann variables and coherent states are considered in order to obtain a closed-form expression for the fermionic star-exponential in terms of its associated propagator. As a primary application, a fermionic version of the Feynman-Kac formula is derived within this formalism, thus allowing a straightforward calculation of the ground state energy in phase space. Finally, the method is validated by successfully applying it to the simple harmonic and driven Fermi oscillators, for which the results developed here provide a powerful alternative computational tool for the study of fermionic systems.
This chapter illustrates how one can obtain a matrix representation of a state vector, an operator, and an eigenvalue equation in a basis of orthonormal states. It helps to learn that matrix representations are very useful in calculations of the properties of operators, because algebra of matrices is very simple and completely developed. Readers familiar with the algebra of matrices agree that mathematical operations on matrices such as multiplication by a scalar, addition, subtraction, multiplication and diagonalization are simple and easy to perform. In fact, the application of the algebra of matrices to quantum physics results from the direct relations between operators and matrices that any mathematical relation that holds between operators also holds between the corresponding matrices. The eigenvalues and eigenvectors of the matrix representing a given operator  in the basis of orthonormal states are the same as the eigenvalues and eigenvectors of Â.
Abstract The general representation theory of quantum mechanics requires one to represent the states of a dynamical system by functions of a set of real variables q'r, each of which has a domain consisting of either discrete points or a continuous range of points, or possibly both together. A dynamical variable is represented by a function of two such sets of real variables q'r and q"r forming a generalized “matrix". In this paper we shall show that in certain cases it is advantageous to consider some of our variables q'r as complex variables and to suppose the representatives of states and dynamical variables to depend on them in accordance with the theory of functions of a complex variable. In the usual theory the domains of the s are the eigenvalues of certain observables q"r. This significance of the q"s of course gets lost when we consider them as complex variables, but we have, however, some beautiful mathematical features appearing instead, and we gain a considerable amount of mathematical power for the working out of particular examples.
Abstract The source-function technique of Schwinger is used to express any transformation matrix element in a form involving a functional derivative operator. An elementary algebra of such expressions is developed, and in the case of quantum mechanics is found to lead to a natural evaluation procedure; the result expresses the matrix element as an indefinitely repeated integral, with the integrations arranged in order of importance. Discrete as well as continuous eigenvalues can be dealt with by this method. In so far as the procedure is a kind of ‘sum-over-histories’, the result is related to the Feynman principle. The field-theoretic situation requires further developments in technique.