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American Journal of Physics

Fermion Operator Ordering and the Quantum c-Number Correspondence

C. L. Mehta, A. K. Jaiswal

Published Dec 1, 1969• 2 citations
Publisher

Abstract

The linearly ordered and the exponentially ordered forms of an arbitrary spin 12-Fermion operator are defined and evaluated. As an application an expression for exp (αb̂) exp (βb̂†) is evaluated which resembles the usual Baker-Hausdorff identity. It is shown that the trace of the product of two Fermion operators can be expressed as a three-dimensional integral over c-number functions. Further the quantum equations of motion are also expressible as equations involving c-number functions.

DOI
10.1119/1.1975287

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The linearly ordered and the exponentially ordered forms of an arbitrary spin 12-Fermion operator are defined and evaluated. As an application an expression for exp (αb̂) exp (βb̂†) is evaluated which resembles the usual Baker-Hausdorff identity. It is shown that the trace of the product of two Fermion operators can be expressed as a three-dimensional integral over c-number functions. Further the quantum equations of motion are also expressible as equations involving c-number functions.
A k-dependent state has been constructed from two pairs of mutually commuting Fermi-Dirac operators by replacing the Grassmann numbers in the conventional definition of a fermion coherent state. The Heisenberg time-dependent form of this state was then combined with a particular spin state to give a resultant state \G], the spin component containing a series of unknown k-dependent weights. A classical c-number field, phi(c),(r, t), was then defined by an expectation value of the quantum field, psi, using \G] and an energy and time-dependent phase. It is demonstrated that the form of the equation of motion for phi(c)(r, t) is identical to that for psi. (C) 1998 Elsevier Science B.V.
In this article a general linear quantum transformation U for the following fermion system with n modes, (b′+,b̃′)=U(b+,b̃)U−1=(b+,b̃)(B̃, A, C̃D), is studied. All above transformations in Fock space are proved to form a ray representation of a group which is isomorphic with O(2n,C). The explicit expressions of operator U in terms of the normal ordered product and non-normal ordered product are obtained. A series of auxiliary identities are given. It has been mentioned that the results are applicable to the case of fermion fields with interaction.
We define uncertainties in position and momentum for states of NNN identical fermions of spin sss. We prove that in a one-dimensional world the product of these uncertainties is greater than or equal to $[\frac{N}{(2s+1)}]\frac{\ensuremath{\hbar}}{2}$. We generalize this to arbitrary numbers of dimensions; in particular, the corresponding result in three dimensions has the factor $\frac{\ensuremath{\hbar}}{2}$ replaced by $(\frac{3}{4}){(\frac{\ensuremath{\hbar}}{2})}^{3}$.
Operator ordering for Fermi systems is formulated generally with the aid of the Grassmann numbers. The ordering schemes are restricted to a special class called s-ordering in contrast with those for Bose systems. A relation, similar to that for Bose systems, is shown to exist between different values of the equal-time contractions in the Dyson–Wick expansions of the S matrix and different operator orderings in the interaction Hamiltonian. The S matrix is independent of the ordering scheme chosen. In the functional integral context, different orderings correspond to different discretization prescriptions for the definition of the functional integrals in terms of coherent states for Fermi operators.