Canonical Analysis of Scalar Fields in Two Dimensional Curved Space

TL;DR

This paper analyzes the canonical structure of scalar fields in two-dimensional curved space using different formulations of the Einstein-Hilbert action, revealing unique gauge transformations involving the metric and scalar fields.

Contribution

It compares the first and second order Einstein-Hilbert actions in 2D and adapts the Dirac constraint formalism to identify gauge generators with novel metric-scalar interactions.

Findings

Identified gauge transformations mixing affine connection and scalar fields.

Compared inequivalent formulations of the Einstein-Hilbert action in 2D.

Modified HTZ formalism to accommodate unique gauge transformations.

Abstract

Scalar fields on a two dimensional curved surface are considered and the canonical structure of this theory analyzed. Both the first and second order forms of the Einstein-Hilbert (EH) action for the metric are used (these being inequivalent in two dimensions). The Dirac constraint formalism is used to find the generator of the gauge transformation, using the formalisms of Henneaux, Teitelboim and Zanelli (HTZ) and of Castellani (C). The HTZ formalism is slightly modified in the case of the first order EH action to accommodate the gauge transformation of the metric; this gauge transformation is unusual as it mixes the affine connection with the scalar field.