Quantum mechanics of a twisted superparticle

Abstract

This paper is devoted to the quantization of the ten-dimensional superparticle in the formulation, due to Siegel, in which the local fermionic symmetry is implemented using explicit gauge fields and in which momentum coordinates conjugate to the fermionic superspace coordinates are introduced. Here, a truncation of the complete N = 1 superparticle is considered (the complete superparticle is analysed in a separate paper). This “truncated model” (which was the first modified superparticle considered by Siegel) is shown to be invariant under a “twisted” N = 2 space-time supersymmetry, and a light-cone gauge analysis shows that the physical spectrum is a representation of this algebra which includes negative norm states. The Batalin-Vilkovisky method is used to covariantly quantize the system in two ways, one with an infinite number of ghost fields and one with only a finite number. In both cases, the BRST cohomology is analysed in detail. Our results illustrate certain general features: (i) In a formulation with gauge fields for the fermionic symmetries the invariance can be fixed by imposing gauge conditions on the gauge fields. This results in a free quantum action, without the need for the problematic field redefinitions that arise in earlier formulations in which gauge conditions are imposed on the spinorial coordinates. (ii) There is an interesting pattern for the infinite number of ghost-for-ghosts, which naturally divide into two sets. The first is an infinite sequence of gauge fields while the second is a set of ghosts for the corresponding sequence of gauge invariances. (iii) The infinite set of quantum fields packages into simple infinite-dimensional spinor representations of the supergroup OSp(9, 1 | 2). The cohomology classes are particularly easy to describe in terms of OSp(9, 1 | 2) spinor notation.