First-Quantized Fermions in Compact Dimensions

A. V. Marshakov<sup>ab

a</sup> P. N. Lebedev Physical Institute, Russian Academy of Sciences
^b Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)

Abstract: We discuss a path integral representation for fermionic particles and strings in the spirit of V. Ya. Fainberg and the author [1], [2]. We concentrate on the problems arising when some target-space dimensions are compact. We consider the partition function for a fermionic particle at a finite temperature or in compact time in detail as an example. We demonstrate that a self-consistent definition of the path integral generally requires introducing nonvanishing background Wilson loops and that modulo some common problems for real fermions in the Grassmannian formulation, these loops can be interpreted as condensates of world-line fermions. Properties of the corresponding string-theory path integrals are also discussed.

Received: 20.04.2001

DOI: 10.4213/tmf508

 English version: Theoretical and Mathematical Physics, 2001, 128:3, 1213–1224

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