Canonical Formulation of A Bosonic Matter Field in 1+1 Dimensional Curved Space

TL;DR

This paper develops a canonical formulation for a bosonic scalar field coupled to gravity in 1+1 dimensions, revealing the emergence of a dynamical degree of freedom when both are considered together.

Contribution

It provides a canonical analysis of a bosonic scalar in curved space coupled to gravity, highlighting the dynamical degrees of freedom in this simplified model.

Findings

Dynamical degree of freedom appears when scalar and gravity are coupled.

Canonical formulation clarifies constraints in 1+1 dimensional models.

Application of Dirac constraint analysis to coupled scalar-gravity system.

Abstract

We study a Bosonic scalar in 1+1 dimensional curved space that is coupled to a dynamical metric field. This metric, along with the affine connection, also appears in the Einstein-Hilbert action when written in first order form. After illustrating the Dirac constraint analysis in Yang-Mills theory, we apply this formulation to the Einstein-Hilbert action and the action of the Bosonic scalar field, first separately and then together. Only in the latter case does a dynamical degree of freedom emerge.