On the Canonical Structure of First Order Einstein-Hilbert Action Coupled to Bosonic Matter

TL;DR

This paper develops a Hamiltonian formulation of the first order Einstein-Hilbert action coupled with bosonic matter, revealing the structure of constraints and their algebra, and comparing it with the ADM formalism.

Contribution

It introduces a novel Hamiltonian approach for Einstein-Hilbert action with bosonic matter, analyzing constraint algebra and its relation to ADM results.

Findings

Massive scalar fields do not alter the PB algebra of constraints.

Gauge fields introduce linear contributions to the PB algebra of the constraints.

The Hamiltonian is weakly zero on the constraint surface for closed spaces.

Abstract

A Dirac Hamiltonian formulation of d-dimensional Einstein-Hilbert action in first order form, has shown that as well as secondary first class constraints, tertiary first class constraints also arise, with an unusual nonlocal Poisson bracket (PB) algebra among first class constraints [8 ]. This approach is different from that of ADM in that of the equations of motion which are independent of the time derivative of fields only those which correspond to second class constraints (in the sense of the Dirac constraint formalism) are used to eliminate fields from the action. In this paper, we consider coupling of a cosmological term, massive scalar fields, Maxwell gauge fields and Yang-Mills fields to the first order EH action in this formalism, and show that in spite of the apparent differences with the ADM results in the Hamiltonian formulation of the first order EH action and its constraint structure, the generic properties of the ADM Hamiltonian formulation of the first order EH action in the presence of Bosonic matter are derivable from this novel Hamiltonian formulation. Addition of a massive scalar field to the EH action leaves the PB algebra of constraints unaltered, and when the Yang-Mills fields or Maxwell gauge fields are coupled to the EH action, the PB algebra of the constraints pertaining to the EH action receives linear contributions from the generator of the gauge transformations of the action for matter fields, and those generators form a closed algebra amongst themselves. Moreover, it is found that for closed spaces, the Hamiltonian of the EH action coupled to Bosonic matter is weakly zero on the constraint surface defined by the first class constraints, including the constraints arising from the matter fields.