# Making a Quantum Universe: Symmetry and Gravity

^{1}

^{2}

## Abstract

**:**

## 1. Introduction and Summary of Results

**Should spacetime be considered as a physical entity similar to quantum fields associated to particles, or rather it presents a configuration space ?**- General relativity changed spacetime from a rigid entity to a deformable media. However, it does not specify whether spacetime is a physical reality or a property of matter, which ultimately determines its geometry and topology. We remind that in the framework of QFT vacuum is not the empty space of classical physics, see e.g., [16,17]. In particular, in the presence of gravity the naive definition of quantum vacuum is frame dependent. A frame-independent definition exists [18] and it is very far from classical concept of an empty space. Explicitly or implicitly, some of models reviewed in Appendix A address this question.

**Is there any relation between matter and spacetime?**- In general relativity matter modifies the geometry of spacetime, but the two entities are considered as separate and stand alone. In string theory spacetime and matter fields—compactified internal space—are considered and treated together, and spacetime has a physical reality that is similar to matter. By contrast, many other QGR candidates only concentrate their effort on the quantization of spacetime and gravitational interaction. Matter is usually added as an external ingredient and it does not intertwine in the construction of quantum gravity and spacetime.

**Why do we perceive the Universe as a three-dimensional (3D) space (plus time)?**- None of extensively studied quantum gravity models discussed in Appendix A answer this question, despite the fact that it is the origin of many troubles for them. For instance, the enormous number of possible models in string theory is due to the inevitable compactification of extra-dimensions to reduce the dimension of space to the observed 3 + 1. In background independent models, the dimension of space is a fundamental assumption and essential for many technical aspects of their construction. In particular, the definition of Ashtekar variables [19] for $SU\left(2\right)\cong SO\left(3\right)$ symmetry and its relation with spin foam description of loop quantum gravity [20] are based on the assumption of a 3D real space. On the other hand, according to holography principle, the maximum amount of information that is containable in a quantum system is proportional to its area rather than volume. If the information is projected and available on the boundary, it is puzzling why we should perceive the volume.

#### 1.1. Summary of the Model and Results

## 2. An Infinite Quantum Universe

- Quantum mechanics is valid at all scales and applies to every entity, including the Universe as a whole;
- Any quantum system is described by its symmetries and its Hilbert space represents them;
- The Universe has an infinite number of independent degrees of freedom.

## 3. Lagrangian of the Universe

## 4. Division to Subsystems

- -
- There must exist sets of operators $\left\{{A}_{i}\right\}\subset \mathcal{B}\left[\mathcal{H}\right]$ such that $\forall \overline{a}\in \left\{{A}_{i}\right\}$ and $\forall \overline{b}\in \left\{{A}_{j}\right\},$ and $i\ne j,\phantom{\rule{3.33333pt}{0ex}}[\overline{a},\overline{b}]=0$;
- -
- Operators in each set $\left\{{A}_{i}\right\}$ must be local8;
- -
- $\left\{{A}_{i}\right\}$’s must be complementarity, which is ${\otimes}_{i}\left\{{A}_{i}\right\}\cong \mathrm{End}\left(\mathcal{B}\left[\mathcal{H}\right]\right)$.

#### 4.1. Properties of an Infinitely Divided Quantum Universe

#### 4.2. Parameterization of Subsystems

#### 4.3. Clocks and Dynamics

#### 4.4. Geometry of Parameter Space

#### 4.5. Metric Signature

#### 4.6. Lagrangian of Subsystems

## 5. Comparison with Other Quantum Gravity Models

## 6. Outline and Future Perspectives

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. A Very Brief Summary of the Best Studied Quantum Gravity Models

## Appendix B. Quantum Mechanics Postulates in Symmetry Language

- A quantum system is defined by its symmetries. Its state is a vector belonging to a projective vector space called state space representing its symmetry group. Observables are associated to self-adjoint operators. The set of independent observables is isomorphic to subspace of commuting elements of the space of self-adjoint (Hermitian) operators acting on the state space and generates the maximal abelian subalgebra of the algebra associated to symmetry group.
- The state space of a composite system is homomorphic to the direct product of state spaces of its components.16 In the special case of separable components, this homomorphism becomes an isomorphism. Components may be separable-untangled—in some symmetries and inseparable—entangled—in others. The symmetry group of the states of a composite system is a subgroup of direct product of its components.
- Evolution of a system is unitary and is ruled by conservation laws imposed by its symmetries and their representation by the state space.
- Decomposition coefficients of a state to eigen vectors 17 of an observable presents the coherence/degeneracy of the system with respect to its environment according to that observable. Projective measurements
**by definition**correspond to complete breaking of coherence/degeneracy. The outcome of such measurements is the eigen value of the eigen state to which the symmetry is broken. This spontaneous decoherence (symmetry breaking) 18 reduces the state space to the subspace generated by other independent observables, which represent remaining symmetries/degeneracies. - A probability independent of measurement details is associated to eigen values of an observable as the outcome of a measurement. It presents the amount of coherence/degeneracy of the state before its breaking by a projective measurement. Physical processes that determine the probability of each outcome are collectively called preparation.19

## Appendix C. State Space Symmetry and Coherence

## Appendix D. SU(∞) and Its Polynomial Representation

## Appendix E. Cartan Decomposition of SU(∞)

#### Appendix E.1. Eigen Functions of $\widehat{L}$(θ,ϕ) and ${\widehat{L}}_{lm}$

#### Appendix E.2. Dynamics Equations of the Universe before its Division to Subsystems

## References

- Eppley, K.; Hanna, E. The Necessity of Quantizing the Gravitational Field. Found. Phys.
**1977**, 7, 51. [Google Scholar] [CrossRef] - Ziaeepour, H. And what if gravity is intrinsically quantic? J. Phys. Conf. Ser.
**2009**, 174, 012027. [Google Scholar] [CrossRef] - Barrow, J.D.; Magueijo, J. A contextual Planck parameter and the classical limit in quantum cosmology. arXiv
**2020**, arXiv:2006.16036. [Google Scholar] - Kolb, E.W.; Turner, M.S. The Early Universe; Addison-Wesley Publishing Company: Boston, MA, USA, 1991. [Google Scholar]
- Utiyama, R. Invariant theoretical interpretation of interaction. Phys. Rev.
**1956**, 101, 1597. [Google Scholar] [CrossRef] - Kibble, T.W.B. Lorentz invariance and the gravitational field. J. Math. Phys.
**1961**, 2, 212. [Google Scholar] [CrossRef] [Green Version] - Coleman, S.; Mandula, J. All possible symmetries of the S matrix. Phys. Rev.
**1967**, 159, 1251. [Google Scholar] [CrossRef] - Haag, R.; Lopuszanski, J.T.; Sohnius, M. All possible generators of supersymmetries of the S matrix. Nucl. Phys. B
**1975**, 88, 257. [Google Scholar] [CrossRef] - Percacci, R. Mixing internal and spacetime transformations: Some examples and counterexamples. J. Phys. A
**2008**, 41, 335403. [Google Scholar] [CrossRef] - Collins, J.; Perez, A.; Sudarsky, D.; Urrutia, L.; Vucetich, H. Lorentz invariance and quantum gravity: An additional fine-tuning problem? Phys. Rev. Lett.
**2004**, 93, 191301. [Google Scholar] [CrossRef] [Green Version] - Asenbaum, P.; Overstreet, C.; Kim, M.; Curti, J.; Kasevich, M.A. Atom-Interferometric test of the equivalence principle at the 10
^{12}level. arXiv**2020**, arXiv:2005.11624. [Google Scholar] - Abdo, A.A.; Ackermann, M.; Ajello, M.; Asano, K.; Atwood, W.B.; Axelsson, M.; Baldini, L.; Ballet, J.; Barbiellini, G.; Baring, M.G.; et al. A limit on the variation of the speed of light arising from quantum gravity effects. Nature
**2009**, 462, 331. [Google Scholar] [CrossRef] [PubMed] - Wilczek, F. Riemann-Einstein Structure from Volume and Gauge Symmetry. Phys. Rev. Lett.
**1998**, 80, 4851. [Google Scholar] [CrossRef] [Green Version] - Torres-Gomez, A.; Krasnov, K. Gravity-Yang-Mills-Higgs unification by enlarging the gauge group. Phys. Rev. D
**2010**, 81, 085003. [Google Scholar] [CrossRef] [Green Version] - Barrett, J.W.; Kerr, S. Gauge gravity and discrete quantum models. arXiv
**2013**, arXiv:1309.1660. [Google Scholar] - Birrell, N.D.; Davies, P.C.W. Quantum Fields in Curved Space; Cambridge University Press: Cambridge, UK, 1982. [Google Scholar]
- Parker, L.; Toms, D. Quantum Field Theory in Curved Spacetime; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
- Ziaeepour, H. Issues with vacuum energy as the origin of dark energy. Mod. Phys. Lett. A
**2012**, 27, 1250154. [Google Scholar] [CrossRef] - Ashtekar, A. New Variables for Classical and Quantum Gravity. Phys. Rev. Lett.
**1986**, 57, 2244. [Google Scholar] [CrossRef] - Livine, E.R. Projected Spin Networks for Lorentz connection: Linking Spin Foams and Loop Gravity. Class. Quant. Grav.
**2002**, 19, 5525. [Google Scholar] [CrossRef] [Green Version] - Ziaeepour, H. Symmetry as a foundational concept in Quantum Mechanics. J. Phys. Conf. Ser.
**2015**, 626, 012074. [Google Scholar] [CrossRef] [Green Version] - Baumgratz, T.; Cramer, M.; Plenio, M.B. Quantifying Coherence. Phys. Rev. Lett.
**2014**, 113, 140401. [Google Scholar] [CrossRef] [Green Version] - Mondal, D.; Datta, C.; Sazim, S. Quantum Coherence Sets The Quantum Speed Limit For Mixed States. Phys. Lett. A
**2016**, 380, 689. [Google Scholar] [CrossRef] [Green Version] - Zanardi, P.; Lidar, D.; Lloyd, S. Quantum tensor product structures are observable-induced. Phys. Rev. Lett.
**2004**, 92, 060402. [Google Scholar] [CrossRef] [Green Version] - Mazenc, E.A.; Ranard, D. Target Space Entanglement Entropy. arXiv
**2019**, arXiv:1910.07449. [Google Scholar] - Page, D.N.; Wootters, W.K. Evolution without evolution: Dynamics described by stationary observables. Phys. Rev. D
**1983**, 27, 2885. [Google Scholar] [CrossRef] - Hoehn, P.A.; Smith, A.R.H.; Lock, M.P.E. The Trinity of Relational Quantum Dynamics. arXiv
**2019**, arXiv:1912.00033. [Google Scholar] - Cao, C.; Carroll, S.M.; Michalakis, S. Space from Hilbert Space: Recovering Geometry from Bulk Entanglement. Phys. Rev. D
**2017**, 95, 024031. [Google Scholar] [CrossRef] [Green Version] - Hoppe, J. Quantum Theory of a Massless Relativistic Surface and a Two-dimensional Bound State Problem. Ph.D. Thesis, MIT, Cambridge, MA, USA, 1982. [Google Scholar]
- Floratos, E.G.; Iliopoulos, J.; Tiktopoulos, G. A note on SU(∞) classical Yang-Mills theories. Phys. Lett. B
**1989**, 217, 285. [Google Scholar] [CrossRef] [Green Version] - Hoppe, J. Diffeomorphism, Group, Quantization, and SU(∞). Int. J. Mod. Phys. A
**1989**, 4, 5235. [Google Scholar] [CrossRef] - Hoppe, J.; Schaller, P. Infinitely Many Versions of SU(∞). Phys. Lett. B
**1990**, 237, 407. [Google Scholar] [CrossRef] - Zunger, Y. Why Matrix theory works for oddly shaped membranes. Phys. Rev. D
**2001**, 64, 086003. [Google Scholar] [CrossRef] [Green Version] - Su, Z.-Y. A Scheme of Cartan Decomposition for su(N). arXiv
**2006**, arXiv:quant-ph/0603190. [Google Scholar] - Jacobson, T. Thermodynamics of Spacetime: The Einstein Equation of State. Phys. Rev. Lett.
**1995**, 75, 1260. [Google Scholar] [CrossRef] [Green Version] - Kitaev, A. Anyons in an exactly solved model and beyond. Ann. Phys.
**2006**, 321, 2. [Google Scholar] [CrossRef] [Green Version] - Nakamura, J.; Liang, S.; Gardner, G.C.; Manfra, M.J. Direct observation of anyonic braiding statistics at the ν=1/3 fractional quantum Hall state. arXiv
**2006**, arXiv:2006.14115. [Google Scholar] - Arkani-Hamed, N.; Motl, L.; Nicolis, A.; Vafa, C. The String Landscape, Black Holes and Gravity as the Weakest Force. J. High Energy Phys.
**2007**, 06, 060. [Google Scholar] [CrossRef] - Mandelstam, L.; Tamm, I. The Uncertainty Relation Between Energy and Time in Non-relativistic Quantum Mechanics. J. Phys. (USSR)
**1945**, 9, 249. [Google Scholar] - Oppenheim, J.; Wehner, S. The Uncertainty Principle Determines Nonlocality of Quantum Mechanics. Science
**2010**, 330, 1072. [Google Scholar] [CrossRef] [Green Version] - Rosenfeld, L. Zur Quantelung der Wellenfelder. Annal der Physik
**1930**, 397, 113. [Google Scholar] [CrossRef] - Rocci, A. On first attempts to reconcile quantum principles with gravity. J. Phys. Conf. Ser.
**2013**, 470, 012004. [Google Scholar] [CrossRef] - Dewitt, B. Quantum Theory of Gravity. I. The Canonical Theory. Phys. Rev.
**1967**, 160, 1113. [Google Scholar] [CrossRef] [Green Version] - Hartle, J.B.; Hawking, S.W. Wave function of the Universe. Phys. Rev. D
**1983**, 28, 2960. [Google Scholar] [CrossRef] - Wheeler, J.A. On the nature of quantum geometrodynamics. Ann. Phys.
**1957**, 2, 604. [Google Scholar] [CrossRef] - Kiefer, C. Quantum geometrodynamics: Whence, whither? Gen. Rel. Grav
**2009**, 41, 877. [Google Scholar] [CrossRef] [Green Version] - Arnowitt, R.; Deser, S.; Misner, C. Dynamical Structure and Definition of Energy in General Relativity. Phys. Rev.
**1959**, 116, 1322. [Google Scholar] [CrossRef] - Dirac, P. Generalized Hamiltonian dynamics. Proc. Roy. Soc. Lond. A
**1958**, 246, 326. [Google Scholar] [CrossRef] - Rovelli, C. Quantum Gravity; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Ashtekar, A.; Lewandowski, J. Background Independent Quantum Gravity: A Status Report. Class. Quant. Grav.
**2004**, 21, R53. [Google Scholar] [CrossRef] - Wang, C.H.; Rodrigues, D.P.F. Closing the gaps in quantum space and time: Conformally augmented gauge structure of gravitation. Phys. Rev. D
**2018**, 98, 124041. [Google Scholar] [CrossRef] [Green Version] - Crnkovic, C. Symplectic geometry of covariant phase space. Class. Quant. Grav.
**1988**, 5, 1557. [Google Scholar] [CrossRef] - Migdal, A.A. Quantum Gravity as Dynamical Triangulation. In Two Dimensional Quantum Gravity and Random Surfaces; Gross, D.J., Piran, T., Weinberg, S., Eds.; World Scientific: Singapore, 1991; pp. 41–79. [Google Scholar]
- Ambjorn, J.; Carfora, M.; Marzuoli, A. The Geometry of Dynamical Triangulations; Springer: Berlin/Heidelberg, Germany, 1997. [Google Scholar]
- Ambjørn, J.; Görlich, A.; Jurkiewicz, J.; Loll, R. Quantum Gravity via Causal Dynamical Triangulations. In Springer Handbook of Spacetime; Ashtekar, A., Petkov, V., Eds.; Springer Handbooks; Springer: Berlin/Heidelberg, Germany, 2014. [Google Scholar]
- Ambjorn, J.; Gizbert-Studnicki, J.; Goerlich, A.; Jurkiewicz, J.; Loll, R. Renormalization in quantum theories of geometry. arXiv
**2002**, arXiv:2002.01693. [Google Scholar] [CrossRef] - Sorkin, R.D. A Unitary Substitute for Continuous Topology. Int. J. Theor. Phys.
**1991**, 30, 923. [Google Scholar] [CrossRef] - Dowker, F. Causal sets and the deep structure of spacetime. In 100 Years of Relativity: Space-Time Structure-Einstein and Beyond; Ashtekar, A., Ed.; World Scientific: Singapore, 2005; p. 445. [Google Scholar]
- Mattingly, D. Causal sets and conservation laws in tests of Lorentz symmetry. Phys. Rev. D
**2008**, 77, 125021. [Google Scholar] [CrossRef] [Green Version] - Dowker, F.; Sorkin, R.D. Symmetry-breaking and zero-one law. Class. Quant. Grav.
**2020**, 37, 15. [Google Scholar] [CrossRef] [Green Version] - Snyder, H.S. Quantized Space-Time. Phys. Rev.
**1947**, 71, 38. [Google Scholar] [CrossRef] - Yang, C.N. On quantized space-time. Phys. Rev.
**1947**, 72, 874. [Google Scholar] [CrossRef] - Seiberg, N.; Witten, E. String Theory and Noncommutative Geometry. J. High Energy Phys.
**1999**, 9909, 032. [Google Scholar] [CrossRef] [Green Version] - Steinacker, H. Emergent Gravity from Noncommutative Gauge Theory. J. High Energy Phys.
**2007**, 0712, 049. [Google Scholar] [CrossRef] - Amelino-Camelia, G.; Mandanici, G.; Yoshida, K. On the IR/UV mixing and experimental limits on the parameters of canonical noncommutative spacetimes. J. High Energy Phys.
**2004**, 0401, 037. [Google Scholar] [CrossRef] - Hogan, C. Quantum Geometry and Interferometry. In Proceedings of the 9th LISA Symposium; Binétruy, P., Plagnol, E., Eds.; Astronomical Society of the Pacific: San Francisco, CA, USA, 2013. [Google Scholar]
- Polyakov, A.M. Quantum geometry of bosonic string. Phys. Lett. B
**1981**, 103, 207. [Google Scholar] [CrossRef] - Polyakov, A.M. Quantum geometry of fermionic string. Phys. Lett. B
**1981**, 103, 211. [Google Scholar] [CrossRef] - Green, M.B.; Schwarz, J.H.; Witten, E. Superstring Theory I & II; Cambridge University Press: Cambridge, UK, 1987. [Google Scholar]
- Polchinski, J. String Theory I & II; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Kaluza, T. Zum Unitätsproblem der Physik. Sitz. Preuss. Akad. Wiss. Phys. Math.
**1921**, K1, 966. [Google Scholar] - Klein, O. Quantentheori und fünfdimensionale Relativistätstheori. Zeits Phys.
**1926**, 37, 895. [Google Scholar] [CrossRef] - Nakamura, S.; Yamaguchi, M. Gravitino Production from Heavy Moduli Decay and Cosmological Moduli Problem Revived. Phys. Lett. B
**2006**, 638, 389. [Google Scholar] [CrossRef] [Green Version] - Davis, S.C.; Postma, M. Successfully combining SUGRA hybrid inflation and moduli stabilisation. J. Cosmol. Astrop. Phys.
**2008**, 0804, 022. [Google Scholar] - Shih, D. Pseudomoduli Dark Matter. J. High Energy Phys.
**2009**, 0909, 046. [Google Scholar] [CrossRef] - Dimopoulos, K.; Axenides, M. Hybrid Inflation without Flat Directions and without Primordial Black Holes. J. Cosmol. Astrop. Phys.
**2005**, 0506, 008. [Google Scholar] [CrossRef] - Planck Collaboration. Planck 2018 results. IX. Constraints on primordial non-Gaussianity. Astron. Astrophys.
**2020**, 641, A9. [Google Scholar] [CrossRef] - Maldacena, J. Non-Gaussian features of primordial fluctuations in single field inflationary models. J. High Energy Phys.
**2003**, 0305, 013. [Google Scholar] [CrossRef] - Kinney, W.H.; Vagnozzi, S.; Visinelli, L. The zoo plot meets the swampland: Mutual (in)consistency of single-field inflation, string conjectures, and cosmological data. Class. Quant. Grav.
**2019**, 36, 117001. [Google Scholar] [CrossRef] [Green Version] - Vafa, C. The String Landscape and the Swampland. arXiv
**2005**, arXiv:hep-th/0509212. [Google Scholar] - Susskind, L. Dynamics of spontaneous symmetry breaking in the Weinberg-Salam theory. Phys. Rev. D
**1979**, 20, 2619. [Google Scholar] [CrossRef] [Green Version] - Bousso, R. The Cosmological Constant Problem, Dark Energy, and the Landscape of String Theory. In Proceedings of the Subnuclear Physics: Past, Present and Future, Vatican City, Vatican, 30 October–2 November 2011; Pontifical Academy of Sciences: Vatican City, Vatican, 2011. [Google Scholar]
- Kumar, J. A Review of Distributions on the String Landscape. Int. J. Mod. Phys. A
**2006**, 21, 3441. [Google Scholar] [CrossRef] [Green Version] - Banks, T.; Fischler, W.; Shenker, S.H.; Susskind, L. M Theory As A Matrix Model: A Conjecture. Phys. Rev. D
**1997**, 55, 5112. [Google Scholar] [CrossRef] [Green Version] - Dijkgraaf, R.; Verlinde, E.; Verlinde, H. Matrix String Theory. Nucl. Phys. B
**1997**, 500, 43. [Google Scholar] [CrossRef] [Green Version] - Banks, T.; Johnson, M.; Shomer, A. A note on Gauge Theories Coupled to Gravity. J. High Energy Phys.
**2006**, 0609, 049. [Google Scholar] [CrossRef] [Green Version] - Randall, L.; Sundrum, R. A Large Mass Hierarchy from a Small Extra Dimension. Phys. Rev. Lett.
**1999**, 83, 3370. [Google Scholar] [CrossRef] [Green Version] - Randall, L.; Sundrum, R. An Alternative to Compactification. Phys. Rev. Lett.
**1999**, 83, 4690. [Google Scholar] [CrossRef] [Green Version] - Steinhardt, P.J. General considerations of the cosmological constant and the stabilization of moduli in the brane-world picture. Phys. Lett. B
**1999**, 462, 41. [Google Scholar] [CrossRef] [Green Version] - Deffayet, C.; Dvali, G.; Gabadadze, G. Accelerated Universe from Gravity Leaking to Extra Dimensions. Phys. Rev. D
**2002**, 65, 044023. [Google Scholar] [CrossRef] [Green Version] - Franceschini, R.; Giudice, G.F.; Giardino, P.P.; Lodone, P.; Strumia, A. LHC bounds on large extra dimensions. J. High Energy Phys.
**2011**, 1105, 092. [Google Scholar] [CrossRef] [Green Version] - Alcaniz, J.S.; Zhu, Z.-H. Complementary Constraints on Brane Cosmology. Phys. Rev. D
**2005**, 71, 083513. [Google Scholar] [CrossRef] [Green Version] - Lazkoz, R.; Maartens, R.; Majerotto, E. Observational constraints on phantom-like braneworld cosmologies. Phys. Rev. D
**2006**, 74, 083510. [Google Scholar] [CrossRef] [Green Version] - Dubovsky, S.L.; Rubakov, V.A.; Tinyakov, P.G. Brane world: Disappearing massive matter. Phys. Rev. D
**2000**, 62, 105011. [Google Scholar] [CrossRef] [Green Version] - Dubovsky, S.L.; Rubakov, V.A.; Tinyakov, P.G. Is the electric charge conserved in brane world? J. High Energy Phys.
**2000**, 8, 041. [Google Scholar] [CrossRef] - Dubovsky, S.L.; Rubakov, V.A. On models of gauge field localization on a brane. Int. J. Mod. Phys. A
**2001**, 16, 4331. [Google Scholar] [CrossRef] [Green Version] - Dvali, G.; Gabadadze, G.; Shifman, M. (Quasi)Localized Gauge Field on a Brane: Dissipating Cosmic Radiation to Extra Dimensions? Phys. Lett. B
**2001**, 497, 271. [Google Scholar] [CrossRef] [Green Version] - Ziaeepour, H. Color Glass Condensate in Brane Models or Don’t Ultra High Energy Cosmic Rays Probe 1015eV Scale ? Mod. Phys. Lett. A
**2005**, 20, 419. [Google Scholar] [CrossRef] [Green Version] - Ziaeepour, H. QCD Color Glass Condensate Model in Warped Brane Models. Grav. Cosmol. Suppl.
**2005**, 11, 189. [Google Scholar] - Hong, D.-K.; Hsu, S.D.H. Holography, Entropy and Extra Dimensions. Phys. Lett. B
**2004**, 591, 208. [Google Scholar] [CrossRef] [Green Version] - Creek, S.; Gregory, R.; Kanti, P.; Mistry, B. Braneworld stars and black holes. Class. Quant. Grav.
**2006**, 23, 6633. [Google Scholar] [CrossRef] - CMS Collaboration. Search for microscopic black hole signatures at the Large Hadron Collider. Phys. Lett. B
**2011**, 697, 434. [Google Scholar] [CrossRef] - Dvali, G.; Gomez, C.; Mukhanov, S. Probing Quantum Geometry at LHC. J. High Energy Phys.
**2011**, 1102, 12. [Google Scholar] [CrossRef] [Green Version] - Dvali, G.; Folkerts, S.; Germani, C. Physics of Trans-Planckian Gravity. Phys. Rev. D
**2011**, 84, 024039. [Google Scholar] [CrossRef] [Green Version] - Dvali, G.; Giudice, G.F.; Gomez, C.; Kehagias, A. UV-Completion by Classicalization. J. High Energy Phys.
**2011**, 08, 108. [Google Scholar] [CrossRef] [Green Version] - Padmanabhan, T. Gravity and the Thermodynamics of Horizons. Phys. Rep.
**2005**, 406, 49. [Google Scholar] [CrossRef] [Green Version] - Verlinde, E.P. On the Origin of Gravity and the Laws of Newton. J. High Energy Phys.
**2011**, 1104, 029. [Google Scholar] [CrossRef] [Green Version] - Hu, B.L. Can Spacetime be a Condensate? Int. J. Theor. Phys.
**2005**, 44, 1785. [Google Scholar] [CrossRef] [Green Version] - Sindoni, L. Emergent gravitational dynamics from multi-BEC hydrodynamics? Phys. Rev. D
**2011**, 83, 024022. [Google Scholar] [CrossRef] [Green Version] - Hooft, G.T. Dimensional Reduction in Quantum Gravity. arXiv
**1993**, arXiv:gr-qc/9310026. [Google Scholar] - Van Raamsdonk, M. Building up spacetime with quantum entanglement. Gen. Rel. Grav
**2010**, 42, 2323–2329. [Google Scholar] [CrossRef] - Jacobson, T. Gravitation and vacuum entanglement entropy. Int. J. Mod. Phys. D
**2012**, 21, 1242006. [Google Scholar] [CrossRef] [Green Version] - Piazza, F. Glimmers of a pre-geometric perspective. Found. Phys.
**2010**, 40, 239. [Google Scholar] [CrossRef] [Green Version] - Eichhorn, A.; Koslowski, T.; Lumma, J.; Pereira, A.D. Towards background independent quantum gravity with tensor models. Class. Quant. Grav.
**2019**, 36, 15. [Google Scholar] [CrossRef] [Green Version] - Maldacena, J.M. The Large N Limit of Superconformal Field Theories and Supergravity. Adv. Theor. Math. Phys
**1998**, 2, 231. [Google Scholar] [CrossRef] - Donnelly, W.; Giddings, S.B. Diffeomorphism-invariant observables and their nonlocal algebra. Phys. Rev. D
**2016**, 93, 024030. [Google Scholar] [CrossRef] [Green Version] - Giddings, S.B. Quantum-first gravity. Found. Phys.
**2019**, 49, 177. [Google Scholar] [CrossRef] [Green Version] - Dirac, P.A.M. The Principles of Quantum Mechanics; Oxford University Press: Oxford, UK, 1958. [Google Scholar]
- Von Neumann, J. Mathematical Foundation of Quantum Theory; Princeton University Press: Princeton, NJ, USA, 1955. [Google Scholar]
- Kiefer, C. On the interpretation of quantum theory—from Copenhagen to the present day. In Time, Quantum and Information; Springer: Berlin/Heidelberg, Germany, 2003. [Google Scholar]

1. | Nonetheless, Ref. [3] advocates a context dependent Planck constant. |

2. | In addition to ${M}_{P}$, we need two other fundamental constants to describe physics and cosmology: the Planck constant ℏ and maximum speed of information transfer that experiments show to be the speed of light in classical vacuum. We remind that triplet constants $(\hslash ,c,{M}_{P})$ are arbitrary and can take any nonzero positive value. The selection of their values amounts to the definition of a system of units for measuring other physical quantities. In QFT literature usually $\hslash =1$ and $c=1$ are used. In this system of units—called high energy physics units [4]—ℏ and c are dimensionless. |

3. | Some quantum gravity models such as loop quantum gravity emphasize the quantization of gravity alone. However, giving the fact that gravity is a universal force and interacts with matter and other forces, its quantization necessarily has impact on them. Therefore, any quantum gravity only model would be, at best, incomplete. |

4. | In this work, all vector spaces and algebras are defined on complex number field $\mathbb{C}$, unless explicitly mentioned otherwise. |

5. | Although in (1) we show the dimensional scale $\hslash /{M}_{P}$ in the definition of operators and their algebra, for the sake of convenience in the rest of this work, we include it in the operators, except when its explicit presentation is necessary for the discussion. |

6. | We remind that ${\int}_{\mathcal{M}}{d}^{2}\mathsf{\Omega}\sqrt{|{g}^{\left(2\right)}|}{R}^{\left(2\right)}=4\pi \chi \left(\mathcal{M}\right)$, where $\chi $ is the Euler characteristic of the compact Riemann 2D surface $\mathcal{M}$. Moreover, Ricci scalar alone does not determine Riemann curvature tensor ${R}_{\mu \nu}$ and only provides one constraint for three independent components of the metric tensor. |

7. | In statistical quantum or classical mechanics distinguishability of particles usually means being able to say, for instance, whether it was particle 1 or particle 2 which was observed. Here by distinguishability we mean whether a particle/subsystem can be experimentally detected, i.e., through application of ${\widehat{L}}_{lm}$ to a subspace of parameter space and identified in isolation from other subsystems or the rest of the Universe. |

8. | This condition is defined for quantum systems in a background spacetime. In the present model there is not such a background. Nonetheless, as explained earlier, locality on the diffeo-surface can be projected to $\mathcal{B}\left[{\mathcal{H}}_{U}\right]$. |

9. | Evidently, in addition to 3 + 1 external parameters each subsystem represents the internal symmetry G, where its representations have their own parameters. |

10. | Notice that even in classical general relativity diffeomorphism and relation between geometry and state of matter are independent concepts. In particular, Einstein equation is not the only possible relation and a priori other diffeomorphism invariant relations between geometry and matter are allowed—but constrained by experiments. |

11. | More generally, any measure of difference between states, such as Fubini–Study metric or fidelity can be used to order states. As Hilbert spaces of quantum systems with $SU(\infty )$ symmetry consist of continuous functions, we can use usual analytical tools for defining a distance. However, we should not forget that functions are vectors of a Hilbert space. Moreover, Hilbert space vectors are, in general, complex functions and each projection between diffeo-surfaces corresponds to two projection in the Hilbert space, one for real part and one for imaginary part of vectors. |

12. | This projection is isomorphic to a homomorphism between $\mathcal{B}\left[{\mathcal{H}}_{s}\right]$ of subsystems. |

13. | We should emphasize that references given in this appendix are only examples of works on the subjects on which tens or even hundreds of articles can be found in the literature. |

14. | |

15. | Non-supersymmetric string models may have no non-perturbative formulation and should be considered as part of a supersymmetric model, see e.g., Chapter 8 of [70]. |

16. | Notice that this axiom differentiates between possible states of a composite system, which is the direct product of those of subsystems, and what is actually realized, which can be limited to a subspace of the direct product of individual components and have reduced symmetry. |

17. | More precisely rays because state vectors differing by a constant are equivalent. |

18. | Ref. [21] explains why decoherence should be considered as a spontaneous symmetry breaking similar to a phase transition. |

19. | Literature on the foundation of quantum mechanics consider an intermediate step called transition between preparation and measurement. Here we include this step to preparation or measurement operations and do not consider it as a separate physical operation. |

20. | In some quantum information literature coherence symmetry is called asymmetry [23]. In this work we call it coherence symmetry or simply coherence to remind that its origin is quantum degeneracy and indistinguishability/symmetry of states before a projective observation. |

21. | A priori N and ${N}^{\prime}$ can depend on $(\theta ,\varphi )$. However, their dependence on angular parameters can be included in $\eta $. Therefore, only constant eigen values matter. |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ziaeepour, H.
Making a Quantum Universe: Symmetry and Gravity. *Universe* **2020**, *6*, 194.
https://doi.org/10.3390/universe6110194

**AMA Style**

Ziaeepour H.
Making a Quantum Universe: Symmetry and Gravity. *Universe*. 2020; 6(11):194.
https://doi.org/10.3390/universe6110194

**Chicago/Turabian Style**

Ziaeepour, Houri.
2020. "Making a Quantum Universe: Symmetry and Gravity" *Universe* 6, no. 11: 194.
https://doi.org/10.3390/universe6110194