Uniqueness of Canonical Tensor Model With Local Time

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. * sasakura@yukawa.kyoto-u.ac.jp * Noncommutative spaces [3,4] are the special classes of fuzzy spaces, which are described by noncommutative associative algebra of functions. Nonassociative spaces [5]-[8] are also of physical interest.

Numerical and analytical analyses of a matrix model with non-pairwise contracted indices

“…In this paper, however, we will take a different strategy. This is because the new strategy makes more transparent the rather complicated counting of combinatorics performed in [21], and make it straightforward to include the extra coupling λ d U d (φ) and also to consider the next order in t. For λ d = 0, the new strategy gives essentially the same result as [21] in the leading order, as commented below (17).…”

“…One of the present authors and his collaborators have been studying a tensor model in the canonical formalism, which we call canonical tensor model [16,17], as a model of quantum gravity. In [18], it has been shown that the exact wave function of the tensor model has peaks at the configurations that are invariant under Lie groups.…”

“…If the wave function has a peak at a configuration that can well be described by a macroscopic spacetime picture, then the model can be considered to be potentially successful. An indirect motivation for the present paper is to understand the properties of the wave function [15] that is an exact solution to a tensor model in the Hamilton formalism [16,17]. It has been argued and shown for some simple cases that the wave function has peaks at the tensor values that are invariant under Lie groups [18].…”

A random matrix model with non-pairwise contracted indices

“…In this subsection, we will apply the results of the previous subsections to study the integrability of the wavefunction of the model introduced in [25]. It is a toy model closely related to a tensor model in the Hamilton formalism, called the canonical tensor model [23,24], which is studied in a quantum gravity context. Let us consider the following wave function depending on a symmetric tensor P abc , where (a, b, c = 1, 2, .…”

“…One of our motivations to initiate the study of the model (1) is to investigate the properties of the wave function [18,19] of a tensor model [20,21,22] in the Hamilton formalism [23,24], which is studied in a quantum gravity context. The expression (1) can be obtained after integrating over the tensor argument of the wave function of the toy model introduced in [25], which is closely related to this tensor model.…”