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American Journal of Physics

Schwinger’s method and the computation of determinants

Luiz C. de Albuquerque, Carlos Farina, Silvio Rabello

Published Jun 1, 1998• 7 citations
Publisher

Abstract

We show how to use Schwinger’s method to evaluate determinants of differential operators that appear quite often in ordinary quantum mechanics. Although this method was introduced by Schwinger in the early fifties to compute effective actions for quantum electrodynamics and was recently applied by him in the Casimir effect, it has never been used in nonrelativistic problems. As explicit examples we compute the partition functions for both the bosonic and the fermionic harmonic oscillator.

DOI
10.1119/1.18894

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We show how to use Schwinger’s method to evaluate determinants of differential operators that appear quite often in ordinary quantum mechanics. Although this method was introduced by Schwinger in the early fifties to compute effective actions for quantum electrodynamics and was recently applied by him in the Casimir effect, it has never been used in nonrelativistic problems. As explicit examples we compute the partition functions for both the bosonic and the fermionic harmonic oscillator.
In this article, we explore the inconsistencies in the physics of fermionic oscillators and propose potential solutions to address them. By rigorously deriving the Hamiltonian and Lagrangian from first principles, we aim to provide a comprehensive and fundamental understanding of the system. Furthermore, we calculate the partition function for a system of fermionic oscillators by drawing a direct analogy to Planck's treatment of energy distribution in bosonic oscillators, offering a parallel approach to this well-established method. Our study extends beyond the conventional framework by investigating the generalized angular momentum algebra within the context of Schwinger's oscillator model. This includes a detailed examination of the algebraic structures for combinations of bosonic-bosonic, bosonic-fermionic, and fermionic-fermionic oscillators. Through this, we delve into these systems' underlying symmetries and algebraic richness, shedding light on the intricate relationships between these different types of oscillators. In addition to these foundational aspects, we explore the broader implications of this generalized Schwinger approach. Our analysis touches upon potential applications and consequences of this formalism, offering insights that could be relevant to various areas of theoretical physics. This work paves the way for a deeper understanding of quantum oscillators and their role in modern physics by bridging the gap between bosonic and fermionic oscillators.
In many situations, one can approximate the behavior of a quantum system, i.e. a wave function subject to a partial differential equation, by effective classical equations which are ordinary differential equations. A general method and geometrical picture are developed and shown to agree with effective action results, commonly derived through path integration, for perturbations around a harmonic oscillator ground state. The same methods are used to describe dynamical coherent states, which in turn provide means to compute quantum corrections to the symplectic structure of an effective system.
In many situations, one can approximate the behavior of a quantum system, i.e. a wave function subject to a partial differential equation, by effective classical equations which are ordinary differential equations. A general method and geometrical picture is developed and shown to agree with effective action results, commonly derived through path integration, for perturbations around a harmonic oscillator ground state. The same methods are used to describe dynamical coherent states, which in turn provide means to compute quantum corrections to the symplectic structure of an effective system.
In order to popularize the so called Schwinger's method we reconsider the Feynman propagator of two non-relativistic systems: a charged particle in a uniform magnetic field and a charged harmonic oscillator in a uniform magnetic field. Instead of solving the Heisenberg equations for the position and the canonical momentum operators, R and P, we apply this method by solving the Heisenberg equations for the gauge invariant operators R and pi = P-eA, the latter being the mechanical momentum operator. In our procedure we avoid fixing the gauge from the beginning and the result thus obtained shows explicitly the gauge dependence of the Feynman propagator.