Jacobi's Action and the Density of States

Summary

The authors have introduced recently a “microcanonical functional integral” which yields directly the density of states as a function of energy. The phase of the functional integral is Jacobi's action, the extrema of which are classical solutions at a given energy. This approach is general but is especially well suited to gravitating systems because for them the total energy can be fixed simply as a boundary condition on the gravitational field. In this paper, however, we ignore gravity and illustrate the use of Jacobi's action by computing the density of states for a nonrelativistic harmonic oscillator.

DEDICATION

We dedicate this paper to Dieter Brill in honor of his sixtieth birthday. His continued fruitful research in physics and his personal kindness make him a model colleague. JWY would especially like to thank him for countless instructive discussions and for his friendship over the past twenty—five years.

INTRODUCTION

Jacobi's form of the action principle involves variations at fixed energy, rather than the variations at fixed time used in Hamilton's principle. The fixed time interval in Hamilton's action becomes fixed inverse temperature in the “periodic imaginary time” formulation, thus transforming Hamilton's action into the appropriate (imaginary) phase for a periodic path in computing the canonical partition function from a Feynman functional integral (Feynman and Hibbs 1965).