Abstract
Canonical tensor model (CTM for short below) is a rank-three tensor model formulated as a totally constrained system in the canonical formalism. In the classical case, the constraints form a first-class constraint Poisson algebra with structures similar to that of the ADM formalism of general relativity, qualifying CTM as a possible discrete formalism for quantum gravity. In this paper, we show that, in a formal continuum limit, the constraint Poisson algebra of CTM with no cosmological constant exactly reproduces that of the ADM formalism. To this end, we obtain the expression of the metric tensor field in general relativity in terms of one of the dynamical rank-three tensors in CTM, and determine the correspondence between the constraints of CTM and those of the ADM formalism. On the other hand, the cosmological constant term of CTM seems to induce non-local dynamics, and is inconsistent with an assumption about locality of the continuum limit.
Highlights
JHEP10(2015)109 above basically deal with Euclidean signatures
We show that, in a formal continuum limit, the constraint Poisson algebra of canonical tensor model (CTM) with no cosmological constant exactly reproduces that of the ADM formalism
We obtain the expression of the metric tensor field in general relativity in terms of one of the dynamical rank-three tensors in CTM, and determine the correspondence between the constraints of CTM and those of the ADM formalism
Summary
We would like to stress that the form of the Hamiltonian constraint in (2.3) is the most general one under some physically reasonable assumptions [33], and we cannot ignore the cosmological constant term from the beggining.. It is important to note that the algebra (2.5) has a structure depending on P on the right-hand side in the first line, and it is not a genuine Lie algebra This is a similar situation as the constraint algebra [38,39,40] of general relativity in the ADM formalism, and will be essential for (2.5) to reproduce the constraint algebra of the ADM formalism in a formal continuum limit. The constraints (2.3), (2.4) and the first-class algebra (2.5) are unique under some physically reasonable assumptions [33]
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