ua nt - p h / 96 08 01 8 v 1 1 3 A ug 1 99 6 Path Integral in Holomorphic Representation without Gauge Fixation ∗

A method of path integral construction without gauge fixing in the holomorphic representation is proposed for finite-dimensional gauge models. This path integral determines a manifestly gauge-invariant kernel of the evolution operator. 1. It is well known that a gauge symmetry leads to constraints on dynamical variables in the theory [1]. Therefore, the evolution of unphysical degrees of freedom should be given when working with gauge theories, which implies gauge fixing. Alternatively, one can go over to gauge-invariant variables by means of an appropriate canonical transformation. In the latter case constraints turns into some of the new canonical momenta. Gauge-invariant variables are, in general, described by curvilinear coordinates, and their configuration space differs from the Euclidean space [2], [3]. In other words, a physical coordinate may take its value not on the whole real axis but only on its part (a halfline or a segment). Moreover physical degrees of freedom can have a phase space which differs from a plane [4], [5]. It leads to a modification of PI [5], and as a result, the quasi-classical description is changed [6]. According to the above remarks the following question can be raised: is there any way to construct PI which does not require elimination of unphysical degrees of freedom, and the evolution operator determined by such PI would be manifestly gauge-invariant? It is shown below that for finite-dimensional models with a gauge group (including the Yang-Mills quantum mechanics [7]) this question is not deprived of sense, and the recipe of finding PI that involves no gauge condition is proposed. 2. We shall explain the main idea of the note by a simple example where there is only one physical degree of freedom. The Lagrangian of the model is [4] L = (ẋ− yaT x)/2− V (x) . (1) ∗JINR preprint E2-89-678, JINR, Dubna, 1989 (unpublished)