Uniqueness of canonical tensor model with local time

TL;DR

This paper proves that, under specific assumptions, only two forms of Hamiltonian constraints exist in the canonical tensor model with local time, demonstrating a unique structure up to certain ambiguities.

Contribution

It establishes the uniqueness of the Hamiltonian constraint forms in the canonical tensor model with local time under a set of detailed assumptions.

Findings

Only two forms of Hamiltonian constraints satisfy the assumptions.

The previously obtained Hamiltonian can be transformed into one of these forms.

The uniqueness holds up to terms vanishing in pure gravitational physics.

Abstract

Canonical formalism of the rank-three tensor model has recently been proposed, in which "local" time is consistently incorporated by a set of first class constraints. By brute-force analysis, this paper shows that there exist only two forms of a Hamiltonian constraint which satisfies the following assumptions: (i) A Hamiltonian constraint has one index. (ii) The kinematical symmetry is given by an orthogonal group. (iii) A consistent first class constraint algebra is formed by a Hamiltonian constraint and the generators of the kinematical symmetry. (iv) A Hamiltonian constraint is invariant under time reversal transformation. (v) A Hamiltonian constraint is an at most cubic polynomial function of canonical variables. (vi) There are no disconnected terms in a constraint algebra. The two forms are the same except for a slight difference in index contractions. The Hamiltonian constraint which was obtained in the previous paper and behaved oddly under time reversal symmetry can actually be transformed to one of them by a canonical change of variables. The two-fold uniqueness is shown up to the potential ambiguity of adding terms which vanish in the limit of pure gravitational physics.