Canonical Tensor Model Research Articles

The tensor of the exact circle: reconstructing geometry

Developing a theory for quantum gravity is one of the big open questions in theoretical high-energy physics. Recently, a tensor model approach has been considered that treats tensors as the generators of commutative non-associative algebras, which might be an appropriate interpretation of the canonical tensor model. In this approach, the non-associative algebra is assumed to be a low-energy description of the so-called associative closure, which gives the full description of spacetime including the high-energy modes. In the previous work it has been shown how to (re)construct a topological space with a measure on it, and one of the prominent examples that was used to develop the framework was the exact circle. In this work we will further investigate this example, and show that it is possible to reconstruct the full Riemannian geometry by reconstructing the metric tensor. Furthermore, it is demonstrated how diffeomorphisms behave in this formalism, firstly by considering a specific class of diffeomorphisms of the circle, namely the ellipses, and subsequently by performing an explicit diffeomorphism to ‘smoothen’ sets of points generated by the tensor rank decomposition.

Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models

The quest for a consistent theory for quantum gravity is one of the most challenging problems in theoretical high-energy physics. An often-used approach is to describe the gravitational degrees of freedom by the metric tensor or related variables, and finding a way to quantise this. In the canonical tensor model, the gravitational degrees of freedom are encoded in a tensorial quantity $P_{abc}$, and this quantity is subsequently quantised. This makes the quantisation much more straightforward mathematically, but the interpretation of this tensor as a spacetime is less evident. In this work we take a first step towards fully understanding the relationship to spacetime. By considering $P_{abc}$ as the generator of an algebra of functions, we first describe how we can recover the topology and the measure of a compact Riemannian manifold. Using the tensor rank decomposition, we then generalise this principle in order to have a well-defined notion of the topology and geometry for a large class of tensors $P_{abc}$. We provide some examples of the emergence of a topology and measure of both exact and perturbed Riemannian manifolds, and of a purely algebraically-defined space called the semi-local circle.

Emergence of Lie group symmetric classical spacetimes in the canonical tensor model

AbstractWe analyze a wave function of a tensor model in the canonical formalism, when the argument of the wave function takes Lie group invariant or nearby values. Numerical computations show that there are two phases, which we call the quantum and the classical phases. In the classical phase fluctuations are suppressed, and configurations emerge which are discretizations of classical geometric spaces invariant under Lie group symmetries. This is explicitly demonstrated for emergence of Sn (n = 1, 2, 3) for SO(n + 1) symmetries by checking the topological and geometric (Laplacian) properties of the emerging configurations. The transition between the two phases has the form of splitting/merging of distributions of variables, resembling a matrix model counterpart, namely the transition between one-cut and two-cut solutions. However, this resemblance is obscured by a difference in the mechanism of distribution in our setup from that in the matrix model. We also discuss this transition as a replica symmetry breaking. We perform various preliminary studies of the properties of the phases and the transition for such values of the argument.

Low-Rank Characteristic Tensor Density Estimation Part II: Compression and Latent Density Estimation

Learning generative probabilistic models is a core problem in machine learning, which presents significant challenges due to the curse of dimensionality. This paper proposes a joint dimensionality reduction and non-parametric density estimation framework, using a novel estimator that can explicitly capture the underlying distribution of appropriate reduced-dimension representations of the input data. The idea is to jointly design a nonlinear dimensionality reducing auto-encoder to model the training data in terms of a parsimonious set of latent random variables, and learn a canonical low-rank tensor model of the joint distribution of the latent variables in the Fourier domain. The proposed latent density model is non-parametric and universal, as opposed to the predefined prior that is assumed in variational auto-encoders. Joint optimization of the auto-encoder and the latent density estimator is pursued via a formulation which learns both by minimizing a combination of the negative log-likelihood in the latent domain and the auto-encoder reconstruction loss. We demonstrate that the proposed model achieves very promising results on toy, tabular, and image datasets on regression tasks, sampling, and anomaly detection.

Phase profile of the wave function of canonical tensor model and emergence of large space—times

In this paper, to understand space–time dynamics in the canonical tensor model of quantum gravity for the positive cosmological constant case, we analytically and numerically study the phase profile of its exact wave function in a coordinate representation, instead of the momentum representation analyzed so far. A saddle point analysis shows that Lie group symmetric space–times are strongly favored due to abundance of continuously existing saddle points, giving an emergent fluid picture. The phase profile suggests that spatial sizes grow in “time,” where sizes are measured by the tensor-geometry correspondence previously introduced using tensor rank decomposition. Monte Carlo simulations are also performed for a few small [Formula: see text] cases by applying a re-weighting procedure to an oscillatory integral which expresses the wave function. The results agree well with the saddle point analysis, but the phase profile is subject to disturbances in a large space–time region, suggesting existence of light modes there and motivating future computations of primordial fluctuations from the perspective of canonical tensor model.

Counting Tensor Rank Decompositions

Tensor rank decomposition is a useful tool for geometric interpretation of the tensors in the canonical tensor model (CTM) of quantum gravity. In order to understand the stability of this interpretation, it is important to be able to estimate how many tensor rank decompositions can approximate a given tensor. More precisely, finding an approximate symmetric tensor rank decomposition of a symmetric tensor Q with an error allowance Δ is to find vectors ϕi satisfying ∥Q−∑i=1Rϕi⊗ϕi⋯⊗ϕi∥2≤Δ. The volume of all such possible ϕi is an interesting quantity which measures the amount of possible decompositions for a tensor Q within an allowance. While it would be difficult to evaluate this quantity for each Q, we find an explicit formula for a similar quantity by integrating over all Q of unit norm. The expression as a function of Δ is given by the product of a hypergeometric function and a power function. By combining new numerical analysis and previous results, we conjecture a formula for the critical rank, yielding an estimate for the spacetime degrees of freedom of the CTM. We also extend the formula to generic decompositions of non-symmetric tensors in order to make our results more broadly applicable. Interestingly, the derivation depends on the existence (convergence) of the partition function of a matrix model which previously appeared in the context of the CTM.

Symmetry enhancement in a two-logarithm matrix model and the canonical tensor model

Abstract We study a one-matrix model of a real symmetric matrix with a potential which is a sum of two logarithmic functions and a harmonic one. This two-logarithm matrix model is the absolute square norm of a toy wave function which is obtained by replacing the tensor argument of the wave function of the canonical tensor model (CTM) with a matrix. We discuss a symmetry enhancement phenomenon in this matrix model and show that symmetries and dimensions of emergent spaces are stable only in a phase which exists exclusively for the positive cosmological constant case in the sense of CTM. This would imply the importance of the positivity of the cosmological constant in the emergence phenomena in CTM.

Phases of a matrix model with non-pairwise index contractions

Abstract Recently a matrix model with non-pairwise index contractions has been studied in the context of the canonical tensor model, a tensor model for quantum gravity in the canonical formalism. This matrix model also appears in the same form with different ranges of parameters and variables, when the replica trick is applied to the spherical $p$-spin model ($p=3$) in spin glass theory. Previous studies of this matrix model suggested the presence of a continuous phase transition around $R\sim N^2/2$, where $N$ and $R$ designate its matrix size $N\times R$. This relation between $N$ and $R$ intriguingly agrees with a consistency condition of the tensor model in the leading order of $N$, suggesting that the tensor model is located near or on the continuous phase transition point and therefore its continuum limit is automatically taken in the $N\rightarrow \infty$ limit. In the previous work, however, the evidence for the phase transition was not satisfactory due to the slowdown of the Monte Carlo simulations. In this work, we provide a new setup for Monte Carlo simulations by integrating out the radial direction of the matrix. This new strategy considerably improves the efficiency, and allows us to clearly show the existence of the phase transition. We also present various characteristics of the phases, such as dynamically generated dimensions of configurations, cascade symmetry breaking and a parameter zero limit, and discuss their implications for the canonical tensor model.

Canonical tensor model through data analysis: Dimensions, topologies, and geometries

The canonical tensor model (CTM) is a tensor model in Hamilton formalism and is studied as a model for gravity in both classical and quantum frameworks. Its dynamical variables are a canonical conjugate pair of real symmetric three-index tensors, and a question in this model was how to extract spacetime pictures from the tensors. We give such an extraction procedure by using two techniques widely known in data analysis. One is the tensor-rank (or CP, etc.) decomposition, which is a certain generalization of the singular value decomposition of a matrix and decomposes a tensor into a number of vectors. By regarding the vectors as points forming a space, topological properties can be extracted by using the other data analysis technique called persistent homology, and geometries by virtual diffusion processes over points. Thus, time evolutions of the tensors in the CTM can be interpreted as topological and geometric evolutions of spaces. We have performed some initial investigations of the classical equation of motion of the CTM in terms of these techniques for a homogeneous fuzzy circle and homogeneous two- and three-dimensional fuzzy spheres as spaces, and have obtained agreement with the general relativistic system obtained previously in a formal continuum limit of the CTM. It is also demonstrated by some concrete examples that the procedure is general for any dimensions and topologies, showing the generality of the CTM.

Emergent symmetries in the canonical tensor model

The canonical tensor model (CTM) is a tensor model proposing a classically and quantum mechanically consistent model of gravity, formulated as a first-class constraint system with structural similarities to the ADM formalism of general relativity. A recent study on the formal continuum limit of the classical CTM has shown that it produces a general relativistic system. This formal continuum limit assumes the emergence of a continuous space, but ultimately continuous spaces should be obtained as preferred configurations of the quantum CTM. In this paper we study the symmetry properties of a wave function which exactly solves the quantum constraints of the CTM for general $N$. We have found that it has strong peaks at configurations invariant under some Lie-groups, as predicted by a mechanism described in our previous paper. A surprising result was the preference of configurations invariant not only under Lie-groups with positive signatures, but also with spacetime-like signatures, i.e., $SO(1,n)$. Such symmetries could characterize the global structures of spacetimes, and our results are encouraging towards showing spacetime emergence in the CTM. To verify the asymptotic convergence of the wave function we have also analyzed the asymptotic behaviour, which for the most part seems to be well under control.

Symmetric configurations highlighted by collective quantum coherence

Recent developments in quantum gravity have shown the Lorentzian treatment to be a fruitful approach towards the emergence of macroscopic space-times. In this paper, we discuss another related aspect of the Lorentzian treatment: we argue that collective quantum coherence may provide a simple mechanism for highlighting symmetric configurations over generic non-symmetric ones. After presenting the general framework of the mechanism, we show the phenomenon in some concrete simple examples in the randomly connected tensor network, which is tightly related to a certain model of quantum gravity, i.e., the canonical tensor model. We find large peaks at configurations invariant under Lie-group symmetries as well as a preference for charge quantization, even in the Abelian case. In future study, this simple mechanism may provide a way to analyze the emergence of macroscopic space-times with global symmetries as well as various other symmetries existing in nature, which are usually postulated.

Mother canonical tensor model

The canonical tensor model (CTM) is a tensor model formulated in the Hamilton formalism as a totally constrained system with first class constraints, the algebraic structure of which is very similar to that of the ADM formalism of general relativity. It has recently been shown that a formal continuum limit of the classical equation of motion of CTM in a derivative expansion of the tensor up to the fourth derivatives agrees with that of a coupled system of general relativity and a scalar field in the Hamilton–Jacobi formalism. This suggests the existence of a ‘mother’ tensor model which derives CTM through the Hamilton–Jacobi procedure, and we have successfully found such a ‘mother’ CTM (mCTM) in this paper. The quantization of the mCTM is as straightforward as the CTM. However, we have not been able to identify all the secondary constraints, and therefore the full structure of the model has been left for future study. Nonetheless, we have found some exact physical wave functions and classical phase spaces, which can be shown to solve the primary and all the (possibly infinite) secondary constraints in the quantum and classical cases, respectively, and have thereby proven the non-triviality of the model. It has also been shown that the mCTM has more interesting dynamics than the CTM from the perspective of randomly connected tensor networks.

An OSp extension of the canonical tensor model

Tensor models are generalizations of matrix models, and are studied as discrete models of quantum gravity for arbitrary dimensions. Among them, the canonical tensor model (CTM for short) is a rank-three tensor model formulated as a totally constrained system with a number of first-class constraints, which have a similar algebraic structure as the constraints of the ADM formalism of general relativity. In this paper, we formulate a super-extension of CTM as an attempt to incorporate fermionic degrees of freedom. The kinematical symmetry group is extended from $O(N)$ to $OSp(N,\tilde N)$, and the constraints are constructed so that they form a first-class constraint super-Poisson algebra. This is a straightforward super-extension, and the constraints and their algebraic structure are formally unchanged from the purely bosonic case, except for the additional signs associated to the order of the fermionic indices and dynamical variables. However, this extension of CTM leads to the existence of negative norm states in the quantized case, and requires some future improvements as quantum gravity with fermions. On the other hand, since this is a straightforward super-extension, various results obtained so far for the purely bosonic case are expected to have parallels also in the super-extended case, such as the exact physical wave functions and the connection to the dual statistical systems, i.e. randomly connected tensor networks.

Constraint algebra of general relativity from a formal continuum limit of canonical tensor model

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.

Renormalization procedure for random tensor networks and the canonical tensor model

We discuss a renormalization procedure for random tensor networks, and show that the corresponding renormalization-group flow is given by the Hamiltonian vector flow of the canonical tensor model, which is a discretized model of quantum gravity. The result is the generalization of the previous one concerning the relation between the Ising model on random networks and the canonical tensor model with N=2. We also prove a general theorem which relates discontinuity of the renormalization-group flow and the phase transitions of random tensor networks.

Nonnegative Mixture for Underdetermined Blind Source Separation Based on a Tensor Algorithm

In this study, a tensor algorithm is proposed to blindly separate an instantaneous linear underdetermined mixture with non-stationary sources and nonnegative mixing matrix. It proceeds in two steps: 1) estimating the mixing matrix and 2) recovering the source signals. First, a canonical tensor model is constructed using a fourth-order cumulant tensor of the observed signals to estimate the mixing matrix. Then, an improved hierarchical alternating least squares algorithm is used to decompose the canonical tensor model, which ensures that all elements of the mixing matrix are positive. Finally, the sources are recovered using a minimum mean-squared error beamformer approach without any hypothetical limitation. We apply two classes of data (speech signals and biomedical signals) to substantiate the effectiveness of the proposed algorithm for underdetermined blind source separation.

Physical states in the canonical tensor model from the perspective of random tensor networks

Tensor models, generalization of matrix models, are studied aiming for quantum gravity in dimensions larger than two. Among them, the canonical tensor model is formulated as a totally constrained system with first-class constraints, the algebra of which resembles the Dirac algebra of general relativity. When quantized, the physical states are defined to be vanished by the quantized constraints. In explicit representations, the constraint equations are a set of partial differential equations for the physical wave-functions, which do not seem straightforward to be solved due to their non-linear character. In this paper, after providing some explicit solutions for N = 2, 3, we show that certain scale-free integration of partition functions of statistical systems on random networks (or random tensor networks more generally) provides a series of solutions for general N. Then, by generalizing this form, we also obtain various solutions for general N. Moreover, we show that the solutions for the cases with a cosmological constant can be obtained from those with no cosmological constant for increased N. This would imply the interesting possibility that a cosmological constant can always be absorbed into the dynamics and is not an input parameter in the canonical tensor model. We also observe the possibility of symmetry enhancement in N = 3, and comment on an extension of Airy function related to the solutions.

Ising model on random networks and the canonical tensor model

We introduce a statistical system on random networks of trivalent vertices for the purpose of studying the canonical tensor model, which is a rank-three tensor model in the canonical formalism. The partition function of the statistical system has a concise expression in terms of integrals, and has the same symmetries as the kinematical ones of the canonical tensor model. We consider the simplest non-trivial case of the statistical system corresponding to the Ising model on random networks, and find that its phase diagram agrees with what is implied by regrading the Hamiltonian vector field of the canonical tensor model with N = 2 as a renormalization group flow. Along the way, we obtain an explicit exact expression of the free energy of the Ising model on random networks in the thermodynamic limit by the Laplace method. This paper provides a new example connecting a model of quantum gravity and a random statistical system.

Interpreting canonical tensor model in minisuperspace

Canonical tensor model is a theory of dynamical fuzzy spaces in arbitrary space–time dimensions. Examining its simplest case, we find a connection to a special case of minisuperspace model of general relativity in arbitrary dimensions. This is a first step in interpreting variables in canonical tensor model based on the known language of general relativity.

Quantization of the canonical tensor model and an exact wave function

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 provides an algebraically consistent way of discretizing the Dirac algebra for general relativity. This paper successfully formulates the Wheeler-DeWitt quantization of the canonical tensor model. Formally one can obtain wave functions of the "universe" by solving the partial differential equations representing the constraints. For the simplest non-trivial case, the unique wave function is exactly and globally obtained. Although this case is far from being realistic, the wave function is physically interesting; locality is favored, and there exists a locus of configurations with features of the beginning of the universe.