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

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.