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 applying the Dirac constraint formalism to the Einstein–Hilbert action and the action of the bosonic scalar field separately, we apply it to these actions when they are combined. Only in the latter case does a dynamical degree of freedom emerge.
In this paper, we study the Dirichlet problem for a singular Monge-Ampere type equation on unbounded domains. For a few special kinds of unbounded convex domains,we find the explicit formulas of the solutions to the problem. For general unbounded convex domain $\Om$, we prove the existence for solutions to the problem in the space$C^{\infty}(\Om)\cap~C~(\overline{\Om})$. We also obtain the local $C^{\frac{1}{2}}$-estimate up to the $\partial~\Omega$ and the estimate for the lower bound of the solutions.