Tensors and Algebras: An Algebraic Spacetime Interpretation for Tensor Models

Dennis Obster

Symmetry, integrability and geometry: methods and applications, Tome 19 (2023)

Résumé

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.

Détail

Keywords: algebraic tensor model, quantum gravity, canonical tensor model, interpretation.

Bibliographie

[1] Abbott B.P. et al., “Observation of gravitational waves from a binary black hole merger”, Phys. Rev. Lett., 116 (2016), 061102, 16 pp., arXiv: 1602.03837 | DOI | MR

[2] Ambjørn J., Durhuus B., Jónsson T., “Three-dimensional simplicial quantum gravity and generalized matrix models”, Modern Phys. Lett. A, 6 (1991), 1133–1146 | DOI | MR | Zbl

[3] Ambjørn J., Görlich A., Jurkiewicz J., Loll R., “Nonperturbative quantum gravity”, Phys. Rep., 519 (2012), 127–210, arXiv: 1203.3591 | DOI | MR

[4] Ambjørn J., Jurkiewicz J., Loll R., “Spectral dimension of the universe”, Phys. Rev. Lett., 95 (2005), 171301, 4 pp., arXiv: hep-th/0505113 | DOI

[5] Ambjørn J., Loll R., “Non-perturbative Lorentzian quantum gravity causality and topology change”, Nuclear Phys. B, 536 (1999), 407–434, arXiv: hep-th/9805108 | DOI | MR | Zbl

[6] Arnowitt R., Deser S., Misner C.W., “Dynamical structure and definition of energy in general relativity”, Phys. Rev., 116 (1959), 1322–1330 | DOI | MR | Zbl

[7] Barrett J.W., Druce P., Glaser L., “Spectral estimators for finite non-commutative geometries”, J. Phys. A, 52 (2019), 275203, 33 pp., arXiv: 1902.03590 | DOI | MR | Zbl

[8] Bonzom V., Gurau R., Riello A., Rivasseau V., “Critical behavior of colored tensor models in the large $N$ limit”, Nuclear Phys. B, 853 (2011), 174–195, arXiv: 1105.3122 | DOI | MR | Zbl

[9] Chen H., Sasakura N., Sato Y., “Equation of motion of canonical tensor model and Hamilton–Jacobi equation of general relativity”, Phys. Rev. D, 95 (2017), 066008, 23 pp., arXiv: 1609.01946 | DOI | MR

[10] Clemente G., D'Elia M., “Spectrum of the Laplace–Beltrami operator and the phase structure of causal dynamical triangulations”, Phys. Rev. D, 97 (2018), 124022, 21 pp., arXiv: 1804.02294 | DOI | MR

[11] Cohn D.L., Measure theory, Birkhäuser Adv. Texts Basler Lehrbücher, 2nd ed., Birkhäuser, New York, 2013 | DOI | MR | Zbl

[12] Connes A., “Geometry from the spectral point of view”, Lett. Math. Phys., 34 (1995), 203–238 | DOI | MR | Zbl

[13] Connes A., “On the spectral characterization of manifolds”, J. Noncommut. Geom., 7 (2013), 1–82, arXiv: 0810.2088 | DOI | MR | Zbl

[14] Conway J.B., A course in functional analysis, Grad. Texts in Math., 96, Springer, New York, 1985 | DOI | MR | Zbl

[15] Dyson F.W., Eddington A.S., Davidson C., “A determination of the deflection of light by the Sun's gravitational field, from observations made at the total eclipse of May 29, 1919”, Phil. Trans. Roy. Soc. Lond. A, 220 (1920), 291–333 | DOI

[16] Eichhorn A., Pereira A.D., Pithis A.G.A., “The phase diagram of the multi-matrix model with $ABAB$ interaction from functional renormalization”, J. High Energy Phys., 2020:12 (2020), 131, 33 pp., arXiv: 2009.05111 | DOI | MR | Zbl

[17] Einstein A., “Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie”, Sitzungsber. Preuss. Akad. Wiss., 47 (1915), 831–839

[18] Gelfand I., Neumark M., “On the imbedding of normed rings into the ring of operators in Hilbert space”, Sb. Math., 12 (1943), 197–213 | MR | Zbl

[19] Glaser L., Stern A.B., “Understanding truncated non-commutative geometries through computer simulations”, J. Math. Phys., 61 (2020), 033507, 11 pp., arXiv: 1909.08054 | DOI | MR | Zbl

[20] Glaser L., Stern A.B., “Reconstructing manifolds from truncations of spectral triples”, J. Geom. Phys., 159 (2021), 103921, 17 pp., arXiv: 1912.09227 | DOI | MR | Zbl

[21] Godfrey N., Gross M., “Simplicial quantum gravity in more than two dimensions”, Phys. Rev. D, 43 (1991), R1749–R1753 | DOI | MR

[22] Goroff M.H., Sagnotti A., “The ultraviolet divergences of gravity theories”, Nuclear Phys. B, 266 (1986), 709–736 | DOI | MR

[23] Gurau R., “Colored group field theory”, Comm. Math. Phys., 304 (2011), 69–93, arXiv: 0907.2582 | DOI | MR | Zbl

[24] Gurau R., Ryan J.P., “Melons are branched polymers”, Ann. Henri Poincaré, 15 (2014), 2085–2131, arXiv: 1302.4386 | DOI | MR | Zbl

[25] Jercher A.F., Oriti D., Pithis A.G.A., “Complete Barrett–Crane model and its causal structure”, Phys. Rev. D, 106 (2022), 066019, 32 pp., arXiv: 2206.15442 | DOI | MR

[26] Kawano T., Obster D., Sasakura N., “Canonical tensor model through data analysis: dimensions, topologies, and geometries”, Phys. Rev. D, 97 (2018), 124061, 25 pp., arXiv: 1805.04800 | DOI | MR

[27] Kawano T., Sasakura N., “Emergence of Lie group symmetric classical spacetimes in the canonical tensor model”, PTEP. Prog. Theor. Exp. Phys., 2022 (2022), 43A01, 27 pp., arXiv: 2109.09896 | DOI | MR

[28] Lionni L., Sasakura N., “A random matrix model with non-pairwise contracted indices”, PTEP. Prog. Theor. Exp. Phys., 2019 (2019), 073A01, 39 pp., arXiv: 1903.05944 | DOI | MR | Zbl

[29] Loll R., “Quantum gravity from causal dynamical triangulations: a review”, Classical Quantum Gravity, 37 (2020), 013002, 50 pp., arXiv: 1905.08669 | DOI | MR | Zbl

[30] Narain G., Sasakura N., “An OSp extension of the canonical tensor model”, PTEP. Prog. Theor. Exp. Phys., 2015 (2015), 123A05, 21 pp., arXiv: 1509.01432 | DOI | MR | Zbl

[31] Narain G., Sasakura N., “Mother canonical tensor model”, Classical Quantum Gravity, 34 (2017), 145009, 27 pp., arXiv: 1612.04938 | DOI | MR | Zbl

[32] Narain G., Sasakura N., Yuki S., “Physical states in the canonical tensor model from the perspective of random tensor networks”, J. High Energy Phys., 2015:1 (2015), 010, 39 pp., arXiv: 1410.2683 | DOI

[33] Nestruev J., Smooth manifolds and observables, Grad. Texts in Math., 220, Springer, Cham, 2020 | DOI | MR | Zbl

[34] Obster D., Sasakura N., “Symmetric configurations highlighted by collective quantum coherence”, Eur. Phys. J. C, 77 (2017), 783, 11 pp., arXiv: 1704.02113 | DOI

[35] Obster D., Sasakura N., “Emergent symmetries in the canonical tensor model”, PTEP. Prog. Theor. Exp. Phys., 2018 (2018), 043A01, 36 pp., arXiv: 1710.07449 | DOI | MR | Zbl

[36] Obster D., Sasakura N., “Phases of a matrix model with non-pairwise index contractions”, PTEP. Prog. Theor. Exp. Phys., 2020 (2020), 073B06, 30 pp., arXiv: 2004.03152 | DOI | MR | Zbl

[37] Obster D., Sasakura N., “Counting Tensor Rank Decompositions”, Universe, 7 (2021), 302, 26 pp., arXiv: 2107.10237 | DOI

[38] Qi L., “Eigenvalues of a real supersymmetric tensor”, J. Symbolic Comput., 40 (2005), 1302–1324 | DOI | MR | Zbl

[39] Reuter M., “Nonperturbative evolution equation for quantum gravity”, Phys. Rev. D, 57 (1998), 971–985, arXiv: hep-th/9605030 | DOI | MR

[40] Reuter M., Saueressig F., “Quantum Einstein gravity”, New J. Phys., 14 (2012), 055022, 56 pp., arXiv: 1202.2274 | DOI | MR | Zbl

[41] Rosenberg S., The Laplacian on a Riemannian manifold. An introduction to analysis on manifolds, London Math. Soc. Stud. Texts, 31, Cambridge University Press, Cambridge, 1997 | DOI | MR | Zbl

[42] Rovelli C., Smolin L., “A new approach to quantum gravity based on loop variables”, International Conference on Gravitation and Cosmology, Cambridge University Press, Cambridge, 1988, 14–19

[43] Sasakura N., “Tensor model for gravity and orientability of manifold”, Modern Phys. Lett. A, 6 (1991), 2613–2623 | DOI | MR | Zbl

[44] Sasakura N., “An invariant approach to dynamical fuzzy spaces with a three-index variable”, Modern Phys. Lett. A, 21 (2006), 1017–1028, arXiv: hep-th/0506192 | DOI | MR | Zbl

[45] Sasakura N., “Tensor model and dynamical generation of commutative non-associative fuzzy spaces”, Classical Quantum Gravity, 23 (2006), 5397–5416, arXiv: hep-th/0606066 | DOI | MR | Zbl

[46] Sasakura N., “Emergent general relativity on fuzzy spaces from tensor models”, Prog. Theor. Phys., 119 (2008), 1029–1040, arXiv: 0803.1717 | DOI | Zbl

[47] Sasakura N., “The fluctuation spectra around a Gaussian classical solution of a tensor model and the general relativity”, Internat. J. Modern Phys. A, 23 (2008), 693–718, arXiv: 0706.1618 | DOI | MR | Zbl

[48] Sasakura N., “Super tensor models, super fuzzy spaces and super $n$-ary transformations”, Internat. J. Modern Phys. A, 26 (2011), 4203–4216, arXiv: 1106.0379 | DOI | MR | Zbl

[49] Sasakura N., “Tensor models and 3-ary algebras”, J. Math. Phys., 52 (2011), 103510, 11 pp., arXiv: 1104.1463 | DOI | MR | Zbl

[50] Sasakura N., “Tensor models and hierarchy of $n$-ary algebras”, Internat. J. Modern Phys. A, 26 (2011), 3249–3258, arXiv: 1104.5312 | DOI | MR | Zbl

[51] Sasakura N., “Canonical tensor models with local time”, Internat. J. Modern Phys. A, 27 (2012), 1250020, 21 pp., arXiv: 1111.2790 | DOI | MR | Zbl

[52] Sasakura N., “Uniqueness of canonical tensor model with local time”, Internat. J. Modern Phys. A, 27 (2012), 1250096, 17 pp., arXiv: 1203.0421 | DOI | MR | Zbl

[53] Sasakura N., “A canonical rank-three tensor model with a scaling constraint”, Internat. J. Modern Phys. A, 28 (2013), 1350030, 11 pp., arXiv: 1302.1656 | DOI | MR

[54] Sasakura N., “Quantum canonical tensor model and an exact wave function”, Internat. J. Modern Phys. A, 28 (2013), 1350111, 17 pp., arXiv: 1305.6389 | DOI | MR | Zbl

[55] Sasakura N., “Phase profile of the wave function of canonical tensor model and emergence of large space-times”, Internat. J. Modern Phys. A, 36 (2021), 2150222, 22 pp., arXiv: 2104.11845 | DOI | MR

[56] Sasakura N., Sato Y., “Exact free energies of statistical systems on random networks”, SIGMA, 10 (2014), 087, 7 pp., arXiv: 1402.0740 | DOI | MR | Zbl

[57] Sasakura N., Sato Y., “Interpreting canonical tensor model in minisuperspace”, Phys. Lett. B, 732 (2014), 32–35, arXiv: 1401.2062 | DOI | Zbl

[58] Sasakura N., Sato Y., “Ising model on random networks and the canonical tensor model”, PTEP. Prog. Theor. Exp. Phys., 2014 (2014), 053B03, 15 pp., arXiv: 1401.7806 | DOI | MR | Zbl

[59] Sasakura N., Sato Y., “Constraint algebra of general relativity from a formal continuum limit of canonical tensor model”, J. High Energy Phys., 2015:10 (2015), 109, 19 pp., arXiv: 1506.04872 | DOI | MR | Zbl

[60] Sasakura N., Sato Y., “Renormalization procedure for random tensor networks and the canonical tensor model”, PTEP. Prog. Theor. Exp. Phys., 2015 (2015), 043B09, 19 pp., arXiv: 1501.05078 | DOI | MR | Zbl

[61] Sasakura N., Takeuchi S., “Numerical and analytical analyses of a matrix model with non-pairwise contracted indices”, Eur. Phys. J. C, 80 (2020), 118, arXiv: 1907.06137 | DOI | MR

[62] Strichartz R.S., “Analysis of the Laplacian on the complete Riemannian manifold”, J. Funct. Anal., 52 (1983), 48–79 | DOI | MR | Zbl

[63] 't Hooft G., Veltman M., “One-loop divergencies in the theory of gravitation”, Ann. Inst. H. Poincaré Sect. A (N.S.), 20 (1974), 69–94 | MR | Zbl

[64] Thiemann T., Modern canonical quantum general relativity, Cambridge Monogr. Math. Phys., Cambridge University Press, Cambridge, 2007 | DOI | MR | Zbl

[65] van Suijlekom W.D., Noncommutative geometry and particle physics, Math. Phys. Stud., Springer, Dordrecht, 2015 | DOI | MR | Zbl

[66] Weinberg S., “Ultraviolet divergences in quantum theories of gravitation”, General Relativity: An Einstein Centenary Survey, Cambridge University Press, Cambridge, 1980, 790–831

[67] Will C.M., “The confrontation between general relativity and experiment”, Living Rev. Relativ., 4 (2001), 2001–4, 97 pp., arXiv: 1403.7377 | DOI | MR