scholarly journals CANONICAL TENSOR MODELS WITH LOCAL TIME

2012 ◽  
Vol 27 (05) ◽  
pp. 1250020 ◽  
Author(s):  
NAOKI SASAKURA
Keyword(s):  
Field Theory ◽  
General Relativity ◽  
Local Time ◽  
String Field Theory ◽  
Dynamical Variable ◽  
Tensor Model ◽  
Canonical Variables ◽  
Hamilton Formalism ◽  
Tensor Models ◽  
Constraint Algebra

It is an intriguing question how local time can be introduced in the emergent picture of space–time. In this paper, this problem is discussed in the context of tensor models. To consistently incorporate local time into tensor models, a rank-three tensor model with first class constraints in Hamilton formalism is presented. In the limit of usual continuous spaces, the algebra of constraints reproduces that of general relativity in Hamilton formalism. While the momentum constraints can be realized rather easily by the symmetry of the tensor models, the form of the Hamiltonian constraints is strongly limited by the condition of the closure of the whole constraint algebra. Thus the Hamiltonian constraints have been determined on the assumption that they are local and at most cubic in canonical variables. The form of the Hamiltonian constraints has similarity with the Hamiltonian in the c < 1 string field theory, but it seems impossible to realize such a constraint algebras in the framework of vector or matrix models. Instead these models are rather useful as matter theories coupled with the tensor model. In this sense, a three-index tensor is the minimum-rank dynamical variable necessary to describe gravity in terms of tensor models.

scholarly journals UNIQUENESS OF CANONICAL TENSOR MODEL WITH LOCAL TIME

2012 ◽  
Vol 27 (18) ◽  
pp. 1250096 ◽  
Author(s):  
NAOKI SASAKURA
Keyword(s):  
Local Time ◽  
Time Reversal ◽  
Canonical Formalism ◽  
Hamiltonian Constraint ◽  
Tensor Model ◽  
Group Iii ◽  
Canonical Variables ◽  
Change Of Variables ◽  
Cubic Polynomial ◽  
Constraint Algebra

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.

scholarly journals THE FLUCTUATION SPECTRA AROUND A GAUSSIAN CLASSICAL SOLUTION OF A TENSOR MODEL AND THE GENERAL RELATIVITY

2008 ◽  
Vol 23 (05) ◽  
pp. 693-718 ◽  
Author(s):  
NAOKI SASAKURA
Keyword(s):  
General Relativity ◽  
Classical Solution ◽  
Flat Space ◽  
Effective Field ◽  
Tensor Model ◽  
Metric Tensor ◽  
Four Dimensions ◽  
Tensor Models ◽  
Momentum Spectra ◽  
General Coordinate Transformation

Tensor models can be interpreted as theory of dynamical fuzzy spaces. In this paper, I study numerically the fluctuation spectra around a Gaussian classical solution of a tensor model, which represents a fuzzy flat space in arbitrary dimensions. It is found that the momentum distribution of the low-lying low-momentum spectra is in agreement with that of the metric tensor modulo the general coordinate transformation in the general relativity at least in the dimensions studied numerically, i.e. one to four dimensions. This result suggests that the effective field theory around the solution is described in a similar manner as the general relativity.

scholarly journals A CANONICAL RANK-THREE TENSOR MODEL WITH A SCALING CONSTRAINT

2013 ◽  
Vol 28 (10) ◽  
pp. 1350030 ◽  
Author(s):  
NAOKI SASAKURA
Keyword(s):  
General Relativity ◽  
Fixed Points ◽  
Configuration Space ◽  
Canonical Formalism ◽  
Classical Solutions ◽  
Hamiltonian Constraint ◽  
Tensor Model ◽  
Class Constraint ◽  
Constraint Algebra ◽  
Group Symmetries

A rank-three tensor model in canonical formalism has recently been proposed. The model describes consistent local-time evolutions of fuzzy spaces through a set of first-class constraints which form an on-shell closed algebra with structure functions. In fact, the algebra provides an algebraically consistent discretization of the Dirac–DeWitt constraint algebra in the canonical formalism of general relativity. However, the configuration space of this model contains obvious degeneracies of representing identical fuzzy spaces. In this paper, to delete the degeneracies, another first-class constraint representing a scaling symmetry is added to propose a new canonical rank-three tensor model. A consequence is that, while classical solutions of the previous model have typically runaway or vanishing behaviors, the new model has a compact configuration space and its classical solutions asymptotically approach either fixed points or cyclic orbits in time evolution. Among others, fixed points contain configurations with group symmetries, and may represent stationary symmetric fuzzy spaces. Another consequence on the uniqueness of the local Hamiltonian constraint is also discussed, and a minimal canonical tensor model, which is unique, is given.

Constraint algebra of open string field theory in midpoint coordinates

1987 ◽  
Vol 198 (2) ◽  
pp. 184-188 ◽  
Author(s):  
Robertus Potting ◽  
Cyrus Taylor ◽  
Boris Velikson
Keyword(s):  
Field Theory ◽  
String Field Theory ◽  
Open String ◽  
Constraint Algebra

scholarly journals THE LOWEST MODES AROUND GAUSSIAN SOLUTIONS OF TENSOR MODELS AND THE GENERAL RELATIVITY

2008 ◽  
Vol 23 (24) ◽  
pp. 3863-3890 ◽  
Author(s):  
NAOKI SASAKURA
Keyword(s):  
General Relativity ◽  
Two Dimensions ◽  
Tensor Model ◽  
Momentum Dependence ◽  
Two Dimensional ◽  
Renormalization Procedure ◽  
Metric Tensor ◽  
Tensor Models ◽  
General Coordinate Transformation ◽  
Metric Fluctuations

In the paper arXiv:0706.1618[hep-th], the number distribution of the low-lying spectra around Gaussian solutions representing various dimensional fuzzy tori of a tensor model was numerically shown to be in accordance with the general relativity on tori. In this paper, I perform more detailed numerical analysis of the properties of the modes for two-dimensional fuzzy tori, and obtain conclusive evidences for the agreement. Under a proposed correspondence between the rank-3 tensor in tensor models and the metric tensor in the general relativity, conclusive agreement is obtained between the profiles of the low-lying modes in a tensor model and the metric modes transverse to the general coordinate transformation. Moreover, the low-lying modes are shown to be well on a massless trajectory with quartic momentum dependence in the tensor model. This is in agreement with that the lowest momentum dependence of metric fluctuations in the general relativity will come from the R2-term, since the R-term is topological in two dimensions. These evidences support the idea that the low-lying low-momentum dynamics around the Gaussian solutions of tensor models is described by the general relativity. I also propose a renormalization procedure for tensor models. A classical application of the procedure makes the patterns of the low-lying spectra drastically clearer, and suggests also the existence of massive trajectories.

scholarly journals A RENORMALIZATION PROCEDURE FOR TENSOR MODELS AND SCALAR-TENSOR THEORIES OF GRAVITY

2010 ◽  
Vol 25 (23) ◽  
pp. 4475-4492 ◽  
Author(s):  
NAOKI SASAKURA
Keyword(s):  
General Relativity ◽  
Models Of Quantum Gravity ◽  
Quantum Gravity ◽  
Matrix Models ◽  
Dynamical Variable ◽  
Renormalization Procedure ◽  
Emergent Phenomena ◽  
Tensor Models ◽  
Theories Of Gravity

Tensor models are more-index generalizations of the so-called matrix models, and provide models of quantum gravity with the idea that spaces and general relativity are emergent phenomena. In this paper, a renormalization procedure for the tensor models whose dynamical variable is a totally symmetric real three-tensor is discussed. It is proven that configurations with certain Gaussian forms are the attractors of the three-tensor under the renormalization procedure. Since these Gaussian configurations are parametrized by a scalar and a symmetric two-tensor, it is argued that, in general situations, the infrared dynamics of the tensor models should be described by scalar-tensor theories of gravity.

scholarly journals QUANTUM CANONICAL TENSOR MODEL AND AN EXACT WAVE FUNCTION

2013 ◽  
Vol 28 (21) ◽  
pp. 1350111 ◽  
Author(s):  
NAOKI SASAKURA
Keyword(s):  
Wave Function ◽  
Poisson Algebra ◽  
Wave Functions ◽  
Tensor Model ◽  
Class Constraint ◽  
Constraint System ◽  
Canonical Pair ◽  
Tensor Models ◽  
Constraint Algebra ◽  
Space Wave

Tensor models in various forms are being studied as models of quantum gravity. Among them the canonical tensor model has a canonical pair of rank-three tensors as dynamical variables, and is a pure constraint system with first-class constraints. The Poisson algebra of the first-class constraints has structure functions, and provides an algebraically consistent way of discretizing the Dirac first-class constraint algebra for general relativity. This paper successfully formulates the Wheeler–DeWitt scheme of quantization of the canonical tensor model; the ordering of operators in the constraints is determined without ambiguity by imposing Hermiticity and covariance on the constraints, and the commutation algebra of constraints takes essentially the same form as the classical Poisson algebra, i.e. is first-class. Thus one could consistently obtain, at least locally in the configuration space, wave functions of "universe" by solving the partial differential equations representing the constraints, i.e. the Wheeler–DeWitt equations for the quantum canonical tensor model. The unique wave function for the simplest nontrivial case is exactly and globally obtained. Although this case is far from being realistic, the wave function has a few physically interesting features; it shows that locality is favored, and that there exists a locus of configurations with features of beginning of universe.

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