arXiv subjects

Naoki Sasakura

Publications and source records attributed to Naoki Sasakura.

At least 55 records · Page 3Linked to original sources

arXiv2013

A canonical rank-three tensor model with a scaling constraint

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.

hep-th↗

arXiv2013

Canonical Tensor Model with Local Time and its Uniqueness

A canonical formalism of the rank-three tensor model with the notion of local time is proposed. The consistency of the local time evolution is guaranteed by imposing that local Hamiltonians and the so(N) kinematical symmetry of the tensor model should form a first class constraint algebra. By imposing some physically reasonable assumptions, it is shown that there exist only two such local Hamiltonians with a slight difference in index contraction. The first class constraint algebra is shown to approach the DeWitt constraint algebra of the general relativity in a certain locality limit. Quantization of the system is briefly discussed.

hep-th↗

arXiv2013

Uniqueness of canonical tensor model with local time

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.

hep-th↗

arXiv2012

Fuzzy spaces from tensor models, cyclicity condition, and n-ary algebras

The rank-three tensor models, which have a rank-three tensor as their only dynamical variable, may be interpreted as models of dynamical fuzzy spaces. In this interpretation, the generalized Hermiticity condition on the rank-three tensor leads to a cyclic property of the algebra of functions on fuzzy spaces. The fuzzy spaces with the cyclic property are shown to have various physically interesting characteristics. (i) Although the function algebras of the kind are nonassociative in general, various properties analogous to quantum mechanics hold on the fuzzy spaces. (ii) The symmetry of the rank-three tensor models can be shown to be represented systematically by n-ary transformations on the fuzzy spaces. The transformations contain, for instance, diffeomorphism on fuzzy spaces. (iii) There exists a systematic procedure of truncating the function algebras of the kind, and it can be used to consider subspaces, compactifications, lattice theories, and coarsegraining procedures of fuzzy spaces in physical applications.

hep-th↗

arXiv2012

Canonical tensor models with local time

It is an intriguing question how local time can be introduced in the emergent picture of spacetime. 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.

hep-th↗

arXiv2011

Tensor models and 3-ary algebras

Tensor models are the generalization of matrix models, and are studied as models of quantum gravity in general dimensions. In this paper, I discuss the algebraic structure in the fuzzy space interpretation of the tensor models which have a tensor with three indices as its only dynamical variable. The algebraic structure is studied mainly from the perspective of 3-ary algebras. It is shown that the tensor models have algebraic expressions, and that their symmetries are represented by 3-ary algebras. It is also shown that the 3-ary algebras of coordinates, which appear in the nonassociative fuzzy flat spacetimes corresponding to a certain class of configurations with Gaussian functions in the tensor models, form Lie triple systems, and the associated Lie algebras are shown to agree with those of the Snyder's noncommutative spacetimes. The Poincare transformations on the fuzzy flat spacetimes are shown to be generated by 3-ary algebras.

hep-th↗

arXiv2011

Tensor models and hierarchy of n-ary algebras

Tensor models are generalization of matrix models, and are studied as models of quantum gravity. It is shown that the symmetry of the rank-three tensor models is generated by a hierarchy of n-ary algebras starting from the usual commutator, and the 3-ary algebra symmetry reported in the previous paper is just a single sector of the whole structure. The condition for the Leibnitz rules of the n-ary algebras is discussed from the perspective of the invariance of the underlying algebra under the n-ary transformations. It is shown that the n-ary transformations which keep the underlying algebraic structure invariant form closed finite n-ary Lie subalgebras. It is also shown that, in physical settings, the 3-ary transformation practically generates only local infinitesimal symmetry transformations, and the other more non-local infinitesimal symmetry transformations of the tensor models are generated by higher n-ary transformations.

hep-th↗

arXiv2011

Super tensor models, super fuzzy spaces and super n-ary transformations

By extending the algebraic description of the bosonic rank-three tensor models, a general framework for super rank-three tensor models and correspondence to super fuzzy spaces is proposed. The corresponding super fuzzy spaces must satisfy a certain cyclicity condition on the algebras of functions on them. Due to the cyclicity condition, the symmetry of the super rank-three tensor models are represented by super n-ary transformations. The Leibnitz rules and the fundamental identities for the super n-ary transformations are discussed from the perspective of the symmetry of the algebra of a fuzzy space. It is shown that the super n-ary transformations of finite orders which conserve the algebra of a fuzzy space form a finite closed n-ary super Lie algebra. Super rank-three tensor models would be of physical interest as background independent models for dynamical generation of supersymmetric fuzzy spaces, in which quantum corrections are under control.

hep-th↗

arXiv2010

A renormalization procedure for tensor models and scalar-tensor 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 parameterized 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.

hep-th↗

arXiv2010

Hopf algebra description of quantum circuits

The controlled-NOT gate of qubit quantum circuits is shown to be described in terms of a Hopf algebra. Accordingly, any qubit quantum circuit can be expressed as the Hopf algebraic computations and unitary transformations on one qubit.

quant-ph↗

arXiv2009

Emergent general relativity in the tensor models possessing Gaussian classical solutions

This paper gives a summary of the author's works concerning the emergent general relativity in a particular class of tensor models, which possess Gaussian classical solutions. In general, a classical solution in a tensor model may be physically regarded as a background space, and small fluctuations about the solution as emergent fields on the space. The numerical analyses of the tensor models possessing Gaussian classical background solutions have shown that the low-lying long-wavelength fluctuations around the backgrounds are in one-to-one correspondence with the geometric fluctuations on flat spaces in the general relativity. It has also been shown that part of the orthogonal symmetry of the tensor model spontaneously broken by the backgrounds can be identified with the local translation symmetry of the general relativity. Thus the tensor model provides an interesting model of simultaneous emergence of space, the general relativity, and its local gauge symmetry of translation.

hep-th↗

arXiv2009

Massive particles coupled with 2+1 dimensional gravity and noncommutative field theory

Recently, it has been shown that the effective field theory of the Ponzano-Regge model with which spinless massive particles are coupled is given by three dimensional Euclidean noncommutative scalar field theory in the Lie algebraic noncommutative space [x^i, x^j]=2i kappa epsilon^{ijk}x_k (i,j,k=1,2,3) with kappa=4 pi G, where G is a gravitational constant. We examine whether there exists the relation between spinless massive particles coupled with 2+1 dimensional Einstein gravity and the Lorentzian version of the noncommutative field theory. Then, we point out that the momentum space of the spinless massive particles in 2+1 dimensional Einstein gravity is generally different from that of the noncommutative field theory, which is given by SL(2,R)/Z_2 group space.

hep-th↗

arXiv2009

The Cutkosky rule of three dimensional noncommutative field theory in Lie algebraic noncommutative spacetime

We investigate the unitarity of three dimensional noncommutative scalar field theory in the Lie algebraic noncommutative spacetime [x^i,x^j]=2i kappa epsilon^{ijk}x_k. This noncommutative field theory possesses a SL(2,R)/Z_2 group momentum space, which leads to a Hopf algebraic translational symmetry. We check the Cutkosky rule of the one-loop self-energy diagrams in the noncommutative phi^3 theory when we include a braiding, which is necessary for the noncommutative field theory to possess the Hopf algebraic translational symmetry at quantum level. Then, we find that the Cutkosky rule is satisfied if the mass is less than 1/(2^(1/2)kappa).

hep-th↗

arXiv2009

Gauge fixing in the tensor model and emergence of local gauge symmetries

The tensor model can be regarded as theory of dynamical fuzzy spaces, and gives a way to formulate gravity on fuzzy spaces. It has recently been shown that the low-lying fluctuations around the Gaussian background solutions in the tensor model agree correctly with the metric fluctuations on the flat spaces with general dimensions in the general relativity. This suggests that the local gauge symmetry (the symmetry of local translations) is also emergent around these solutions. To systematically study this possibility, I apply the BRS gauge fixing procedure to the tensor model. The ghost kinetic term is numerically analyzed, and it has been found that there exist some massless trajectories of ghost modes, which are clearly separated from the other higher ghost modes. Comparing with the corresponding BRS gauge fixing in the general relativity, these ghost modes forming the massless trajectories in the tensor model are shown to be identical to the reparametrization ghosts in the general relativity.

hep-th↗

arXiv2008

Emergent general relativity in fuzzy spaces from tensor models

Tensor models can be regarded as theories of dynamical fuzzy spaces, and provide background independent theories of space. Their classical solutions correspond to classical background spaces, and the small fluctuations around them can be regarded as fluctuations of fields on them. In this paper, I numerically study a tensor model around its classical backgrounds of two-, three- and four-dimensional fuzzy flat tori and show that the properties of low-lying low-momentum modes are in clear agreement with general relativity. Numerical analysis also suggests that the lowest-order effective action is composed of curvature square terms, which is consistent with general relativity in view of the form of the considered action of the tensor model.

gr-qc↗

arXiv2007

Domain wall solitons and Hopf algebraic translational symmetries in noncommutative field theories

Domain wall solitons are the simplest topological objects in field theories. The conventional translational symmetry in a field theory is the generator of a one-parameter family of domain wall solutions, and induces a massless moduli field which propagates along a domain wall. We study similar issues in braided noncommutative field theories possessing Hopf algebraic translational symmetries. As a concrete example, we discuss a domain wall soliton in the scalar phi^4 braided noncommutative field theory in Lie-algebraic noncommutative spacetime, [x^i,x^j]=2i kappa epsilon^{ijk}x_k (i,j,k=1,2,3), which has a Hopf algebraic translational symmetry. We first discuss the existence of a domain wall soliton in view of Derrick's theorem, and construct explicitly a one-parameter family of solutions in perturbation of the noncommutativity parameter kappa. We then find the massless moduli field which propagates on the domain wall soliton. We further extend our analysis to the general Hopf algebraic translational symmetry.

hep-th↗

arXiv2007

The lowest modes around Gaussian solutions of tensor models and the general relativity

In the previous paper, 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-three 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 R^2-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.

hep-th↗

arXiv2007

Braided quantum field theories and their symmetries

Braided quantum field theories proposed by Oeckl can provide a framework for defining quantum field theories having Hopf algebra symmetries. In quantum field theories, symmetries lead to non-perturbative relations among correlation functions. We discuss Hopf algebra symmetries and such relations in braided quantum field theories. We give the four algebraic conditions between Hopf algebra symmetries and braided quantum field theories, which are required for the relations to hold. As concrete examples, we apply our discussions to the Poincare symmetries of two examples of noncommutative field theories. One is the effective quantum field theory of three-dimensional quantum gravity coupled with spinless particles given by Freidel and Livine, and the other is noncommutative field theory on Moyal plane. We also comment on quantum field theory on kappa-Minkowski spacetime.

hep-th↗