# Comparing Quantum Gravity Models: String Theory, Loop Quantum Gravity, and Entanglement Gravity versus SU(∞)-QGR

^{1}

^{2}

## Abstract

**:**

## 1. Introduction and Results

- It has been demonstrated [2] that Einstein’s equation can be obtained from the second law of thermodynamics and the holographic principle—that is, the proportionality of entropy inside a null (light-like) surface to its area rather than volume [3,4,5,6]. Holographic behavior has been also observed in many-body systems with negligible gravity [7,8]. These observations confirm the conclusion of [2] that Einstein’s equation should be considered an equation of state. This interpretation and universality of gravitational interaction imply that what is perceived as
`space`and its geometrical properties, such as distance and curvature, represent the state of its matter content. Thus, it seems that spacetime and matter are inseparable aspects of the same physical reality/entity. - Even without the holographic principle, the fact that the energy–momentum tensor of matter—the source of gravitational interaction—depends on the spacetime metric means that spacetime and matter are more intertwined than, for instance, bosonic gauge fields and their matter source in Yang–Mills models.
- In quantum field theory (QFT) spacetime or its dual energy–momentum mode space (but not both at the same time) are used as
`indices`to keep track of the continuum of matter and radiation. The fact that in a quantum realm the classical vacuum—the apparently empty space between particles—can be described as a sea of virtual—off-shell—quantum states [9] means that we could completely neglect the physical space—the perceived 3-dimensional space. This would be possible if we could identify, tag, and order all real and virtual particles, for instance, by using the strength of their mutual quantum entanglement [10,11,12,13,14] or interaction strength [15]. In this view, the classical Einstein equation could be interpreted as an equation of state, which dynamically modifies parameter (index) space according to variations of interactions and entanglement between particles, and with respect to a relational quantum clock [16]. - It is useful to remind that in most QGR models the dimension of spacetime is considered as a parameter and little attempt is made to explain why it has the observed value.

`Quantum First`models in the literature [17]. In addition, progress in quantum information has highlighted the crucial role of the division of the Universe in parts—subsystems—and thereby, the necessity for a proper mathematical definition of what can be considered as a distinguishable quantum (sub)system. This concept has special importance for gravity, because as far as we know from general relativity, it is a universal force, coupling everything to the rest of the Universe. Indeed, we will see later in this work that some QGR models struggle to find a naturally factorized—tensor product—Hilbert space in which each factor can be considered as presenting the Hilbert space of a subsystem. In QFTs without gravity, subsystems are particles /fields or their collections. A priori, the same concept can be applied to QGR. However, in the strong coupling limit of QGR spacetime/gravity and matter may be indistinguishable. Therefore, it is necessary to have a physically and mathematically well defined description of what may be called a distinguishable subsystem of the Universe. We also remind that the tensor product of Hilbert spaces of subsystems is not only important for gravitational interaction, but also for meaningful definition of locality, quantum clocks, quantum information flow and relative entropy, renormalization flow, and holographic properties of states. None of these concepts would make sense without mathematical and physical notion of distinguishable subsystems.

#### 1.1. Summary of Comparison Results

`similarity`or

`analogy`should not be interpreted as one-to-one correspondence. For instance, decomposition of $SU(\infty )$ to $SU(2)$ factors in $SU(\infty )$-QGR is not the same operation as discretizing space to tetrahedra weighed by spins on their edges. Nonetheless, they have analogous mathematical descriptions—in this case a spin network. If the QGR models reviewed in this work contain at least some of the features and properties of the true theory, they should be, most probably, reflected in these common or analogous characteristics.

#### 1.1.1. Presence of 2-Dimensional Spaces or Structures in the Construction of Models

`physical`object is rather philosophical. In practice, in mathematical formulations of physical phenomena, all entities are abstract but related to what can be measured. Thus, in this sense they can be considered as

`physical`.

#### 1.1.2. Decomposition to an Algebraic Tensor Product

`frozen`in a brane condensate in order to explain the observed (3+1)D spacetime. Their quantum fluctuations are treated similar to fields in QFTs, in which

`particles`(modes) are fundamental subsystems. In these models, the fundamental $D=10$ dimensional background space is static and unobservable.

#### 1.1.3. $SU(2)$ Group and Spin Network

#### 1.1.4. A Hidden or Explicit $SU(\infty )$ Symmetry

#### 1.1.5. Emergence of Time and Evolution as Relative and Relational Phenomenon

## 2. A Brief Review of $\mathit{SU}\mathbf{(}\mathbf{\infty}\mathbf{)}$-QGR

#### 2.1. Axioms and Algebra

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

`Universe`means the ensemble of everything causally or through its quantum correlations observable. Independent quantum observables correspond to mutually commuting hermitian operators applied to the Hilbert space, and their subspace is homomorphic to the Cartan subspace of the symmetry group of the quantum system [32].

- Axiom 1 is not trivial because some QGR models extend or restrict quantum mechanics and/or QFT in order to accommodate QGR; see Section 3.2.1 for a brief review of some of these models. As there is no spacetime in the above axioms, we also remind that QFT is not a model by itself and does not necessarily need to be defined in a spacetime. It is a formulation of quantum mechanics suitable for studying many-body systems parameterized by continuous variables, such as a Lorentz invariant spacetime.
- Axiom 2 is added to the above list because in postulates of quantum mechanics, as defined by Dirac [33] and von Neumann [34], the Hilbert space is an abstract Banach space and no relation to symmetries is explicitly mentioned. Axioms of quantum mechanics with symmetry as a foundational concept are described in [32]. Of course, in practice the Hilbert space is chosen such that it represents symmetries of the quantum system. However, this is due to the fact that the choice of Hilbert space is motivated by the configuration space of the classical limit and its symmetries. If we want to construct a fundamentally quantum model without referring to a corresponding classical system, we must specify how the Hilbert space should be defined.
- Axiom 3 defines the symmetry of the system—the Universe—which as explained above is the basis for determining other properties of the system. Of course, QFTs by definition have infinite number of observables/degrees of freedom, one or more at each point of the spacetime. However, in $SU(\infty )$-QGR there is no spacetime and the model is constructed as an abstractly and is defined exclusively by its symmetry and its representation by the Hilbert space. Therefore, this axiom is essential and far from being trivial.

#### 2.2. Representation of the $SU(\infty )$ Group and Hilbert Space

`diffeo-surface`for the surfaces whose area preserving diffeomorphism is homomorphic to $SU(\infty )$ of interest. Diffeo-surfaces with different genus correspond to non-equivalent (non-isometric) representations of $SU(\infty )$ [38,39]. These surfaces, and thereby $\mathcal{B}[{\mathcal{H}}_{U}]\cong SU(\infty )$, are parameterized by two angular parameters $(\theta ,\phantom{\rule{3.33333pt}{0ex}}\varphi )$. On the other hand, $su(\infty )$ algebra is homomorphic to Poisson bracket of spherical harmonic functions, which for $\hslash =1$ and dimensionless operators can be written as:

#### $SU(2)$ in $SU(\infty )$-QGR

#### 2.3. Subsystems of the Universe

#### 2.3.1. Parameter Space of Subsystems

`internal`symmetry, the division of this quantum Universe induces a size or more precisely an area scale. Indeed, although the preserved area of one diffeo-surface is irrelevant for its diffeomorphism as representation of $SU(\infty )$ group, it becomes important when parameter spaces of multiple systems with this symmetry, including the Universe as a whole, are compared. This is analogous to comparing finite intervals on a line with each other. An infinite line alone is scale invariant. However, lengths of finite intervals can be compared with each others. This operation induces a length scale for the finite intervals, and thereby for the whole line. Therefore, after division into subsystems, the parameter space of $SU(\infty )$ part of their Hilbert spaces will depend on a third dimensionful parameter that we call r. It is measured with respect to a reference subsystem. Diffeo-surfaces of subsystems can be considered to be embedded in this 3D space. Notice that quantum state of a subsystem does not necessarily have a fixed r, and can be a superposition of pointer states with fixed r.

`locally`to states $|t,r,\theta ,\varphi ;a\rangle $, where index a represents G symmetry. These properties are similar to those of a Yang–Mills gauge field defined on the classical spacetime. Thus, considering this analogy, we identify the parameter space of the $SU(\infty )$ symmetry with the classical spacetime.

#### 2.4. Relation to Classical Geometry and Einstein Equation

#### Lorentz Invariance of the Parameter Space

- -
- Choices of a quantum clock and a reference subsystem for comparisons between diffeo-surfaces are arbitrary. Changes in these choices amount to changing the corresponding parameters.
- -
- Division into subsystems is not rigid and may change with a change of clock and reference subsystem such that they respect necessary conditions defined in [28]. Thus, changing t and r in general lead to modification of $SU(\infty )$ parameters $(\theta ,\varphi )$, and each of the new parameters $({t}^{\prime},{r}^{\prime},{\theta}^{\prime},{\varphi}^{\prime})$ would be a function of old parameters $(t,r,\theta ,\varphi )$.
- -
- By definition, the ensemble of subsystems must generate the static 2D Universe irrespective of how subsystems are defined and parameterized. This condition imposes Lorentz and diffeomorphism invariance on the parameter space—the spacetime.

#### 2.5. Evolution

#### 2.5.1. The Whole Universe

`fields`defined on the 2D diffeo-surface.

`local`details of the Universe are not distinguishable, only its global—topological—properties may characterize its states. In the present model, the relevant global property is the topology of the diffeo-surface, which determines non-homomorphic representations of the $SU(\infty )$ symmetry [38,39] of the Universe. In other words, the only possible difference between whole Universes is the representation of the $SU(\infty )$ symmetry realized by their Hilbert spaces.

`dynamics`of the quantum Universe onto its parameter space.

#### 2.5.2. Evolution of Subsystems

`distance`from reference subsystem, and their relative evolution is measured by a clock parameter t. In addition, a full action must include terms invariant under the internal symmetry group of subsystems G. The formal description of this functional is [18]:

#### 2.5.3. Classical Limit

#### 2.5.4. Spin-1 Quantum Gravitational Interaction

`average`effect.

#### 2.6. Summary of $SU(\infty )$-QGR Model and Its Properties

- Assuming that Hilbert spaces of the Universe and its subsystems represent $SU(\infty )$ symmetry, we showed that the Hilbert space of the Universe as a whole can be parameterized by two continuous parameters. When the Universe is divided into subsystems presenting a finite rank symmetry group G, and a quantum reference subsystem and a quantum clock are chosen, two additional parameters arise: a relative distant and a relative time à la Page and Wootter or equivalent proposals.
- We interpreted the above 4D parameter space as the classical spacetime and demonstrated that its signature must be negative—i.e., it has a Lorentzian metric. Moreover, as the spacetime is a parameter space, its quantization is meaningless.
- The coordinate independent affine parameter of the spacetime is related to the variations in the quantum states of the subsystems.
- We defined two symmetry invariant functionals over the Hilbert space of the Universe as a whole, and over those of its subsystems. They play the role of action functional for the evolution of the Universe and its subsystems, respectively.
- The action for the subdivided Universe has the form of Yang–Mills gauge theories on the parameter space for both $SU(\infty )$ and subsystem specific
`(internal)`finite rank G symmetry. Thus, similar to other fundamental forces, at the quantum level the mediator boson of gravity is spin-1. - We showed that the action functional for the whole Universe is static. Moreover, its purely $SU(\infty )$ Yang–Mills part is topological and proportional to the Euler constant. Thereby, it is proportional to integral over the 2D Ricci scalar curvature. The constant of the proportionality is not an observable.
- When the Universe is divided into subsystems, in the classical limit when the quantum Yang–Mills vector field of the $SU(\infty )$ symmetry cannot be detected, the purely $SU(\infty )$ Yang–Mills part of the action functional will be proportional to the 4D Ricci scalar curvature. Therefore, the classical limit of $SU(\infty )$-QGR is the Einstein gravity and the observed spin-2 graviton is an effective classical field.
- This important prediction should be testable with future quantum experiments, for instance, those seeking decoherence or entanglement initiated by quantum gravity.

## 3. Comparison with Other Quantum Gravity Proposals

#### 3.1. Background Independent Models

#### 3.1.1. Regge Discrete Geometry

#### 3.1.2. Ponzano–Regge 3D QGR

#### 3.1.3. Ashtekar Variables and Loop Quantum Gravity

#### 3.1.4. $SU(2)$ Symmetry, Degeneracies, and Observables in LQG

`Loop`QGR, and one of its most remarkable predictions is the quantization of area [58]. This feature establishes the relation between LQG formulation using continuous Ashtekar variables, a spin network as its approximation, and the symplectic geometry of Ponzano–Regge: Quantized surfaces have non-trivial $SU(2)$ holonomy, and triangulated 3D space à la Regge becomes a manageable approximation, including essential properties of a quantized curved space with a meaningful continuum limit.

#### 3.1.5. Analogies between the Foundations of LQG and Related Models and $SU(\infty )$-QGR

`internal`$SU(2)$ symmetry of triads. However, in contrast to Ashtekar variables, their values are obtained from ensembles of representations of $SU(2)$ factors in the decomposition of $SU(\infty )$ in Equation (3). This property is similar to Ponzano–Regge and spin networks, where edges of tetrahedra are weighed by spins. However, in $SU(\infty )$-QGR, both l and m quantum numbers of $SU(2)$ representations are involved in the action of the model, and they are not constrained. The reason is that in contrast to LQG and Ponzano–Regge models, in $SU(\infty )$-QGR the Hilbert space does not represent a real space geometry.

#### 3.1.6. Hilbert Spaces of LQG and Related Models

`adjacent`spin states.

`atom of space`—is generated by application of these operators to a vacuum state, such that the projection (amplitude) of the total spin of the tetrahedron is equal to its associated 6j symbol. This procedure can be extended to ensemble of $N\to \infty $ tetrahedra content of space, which can be also considered as spin-weighted graphs [60]. Thus, we conclude that state generator operators, and thereby Hilbert spaces of discrete QGR (DQGR) models such as Ponzano–Regge and LQG models, which we collectively call ${\mathcal{H}}_{DQGR}$, are subspaces of the Hilbert space of a quantum system with $SU(\infty )$ symmetry, such as $SU(\infty )$-QGR.

#### 3.1.7. Kinematical and Physical Hilbert Spaces and Reality Conditions

`physical`, but

`kinematical`[60]. The Hilbert space of physical states ${\mathcal{H}}_{phys}$ containing quantized background independent geometries is a subspace of ${\mathcal{H}}_{DQGR}$; that is, ${\mathcal{H}}_{DQGR}\supset {\mathcal{H}}_{phys}$. However, it is in general difficult to construct ${\mathcal{H}}_{phys}$ explicitly [60]. In addition, demonstration of diffeomorphism and Lorentz invariance of the physical states is not straightforward, and one might expect violations of Lorentz invariance in QGR models with discretized space [62]. Indeed, the diffeomorphism invariance of DQGR is explicitly shown only for special cases [63,64].

`fifth force`-type effect on matter. This effect is also constrained by various tests of gravity [71].

#### 3.1.8. Time and Matter in LQG

`historian`—a reference subsystem with respect to which histories are defined. However, construction of background independent QGR models does not clarify how to satisfy necessary conditions for division of a quantum system [28]. In fact, kinematical Hilbert space ${H}_{DQGR}$ seems to be inseparable [60]. Specifically, the division of Hilbert space to orthogonal blocks of subsystems needs an additional symmetry. The $SU(2)$ symmetry in these models cannot used for this purpose, because it is inherently related to the construction of space and gravitational interaction. We might consider tetrahedra as the most fundamental

`atomic`subsystems [75]. However, to discriminate one tetrahedron as a reference, there must be a selection criterion—another symmetry and its observables (charges). This issue is directly related to the fact that LQG and related models do not consider matter fields—a symmetry orthogonal to space—in their foundations. Although a time parameter and matter fields can be easily added to the Lagrangian of gravity sector, described with respect to Ashtekar variables and their duals (see, e.g., [76]), the foundational issue of time definition in LQG and related models is not fully solved. Attempts to solve this problem, for instance, through quantization of phase space [77,78,79], indeed include matter and/or symmetries orthogonal to the diffeomorphism symmetry.

#### 3.1.9. The Non-Perturbative Characteristic of LQG and Related Models

#### 3.1.10. Outline of Comparison between Background Independent Models and $SU(\infty )$-QGR

`raisons d’être`in these models are very different. Notably, in LQG, GFT, and other symplectic QGR models it is strictly related to the assumption of a physical 3D quantum space. Nonetheless, spin network realization of LQG can be considered as a subspace of the Hilbert space of $SU(\infty )$-QGR.

#### 3.2. Quantum Approaches to QGR

`Quantum First`by some authors [15]—are relatively recent arrivals into the jungle of QGR proposals, and $SU(\infty )$-QGR can be classified in this group. For this reason it is crucial to investigate its similarities with and differences from other models in this category.

- -
- Models that consider locality and causality as indispensable for QGR: some of these models require modifications of standard quantum mechanics;
- -
- Models inspired by black hole entropy and its relationship with the Yang–Mills and AdS/CFT duality.

#### 3.2.1. Modified Quantum Mechanics and Locality

`Locality`is considered to be crucial for describing black holes, their thermodynamics [3,20], and its puzzles [21,88]. More generally, causality and observed finite speed of information propagation in both classical general relativity and QFT imply some degree of locality in any interaction, including QGR. For these reasons, locality and its close relationship with the definition of subsystems as

`localized`entities in the Universe have been the motivation of authors of [15,89,90] for proposing a

`generalized`quantum mechanics. Specifically, a history description of quantum mechanics [91,92] is generalized in [89] to define

`coarse-grained`histories as a bundle of

`fine-grained`histories (path integrals). They replace the Hilbert space of quantum mechanics, which in a QGR framework corresponds to a spacelike surface during an infinitesimal time interval, defined with respect to a reference clock. In addition, in this modified quantum mechanics, projection operators to eigen states of position are time-dependent, and during each time interval they project states to a different set of histories. In turn, sets of histories present subspaces of the bundle. Presumably, in this model not only the state of a system, but also its whole Hilbert space, changes with time.

`universal`quantum mechanics. In analogy with the bundle space of [89], it extends the space of physical states to provide additional labeling, such as

`in and out`states in curved spacetimes [93]. In addition, these labels can be interpreted as time or labels of states in a multiverse, as needed. Physical states can be considered as

`local`in this extended state space.

#### 3.2.2. QGR from Locality and Causality

#### 3.2.3. Comparison of LQFT-QGR with $SU(\infty )$-QGR

#### Type III Algebra in LQFT-QGR and $SU(\infty )$-QGR

`tagging`and fulfilling conditions necessary for defining quantum subsystems [28].

`internal`symmetry. By contrast, LQFT-QGR considers strict locality as a foundational concept and tries to use nontrivial topological structures as a replacement for internal symmetries in order to tag and identify subsystems. So far, quantum field solutions with such property are obtained only in the weak coupling regime of semi-classical gravity [97], and at present there is no evidence that such algebraic structures can exist in the general setup of QFTs.

#### 3.2.4. QGR and Emergent Spacetime from Entropy and Holography

#### 3.2.5. Entanglement-Based Models (EBM) of Quantum Gravity

- A preferred tensor decomposition of the Hilbert space $\mathcal{H}$ [of the Universe], where each factor ${\mathcal{H}}_{i}$ presents the Hilbert space of a point or a small space around a point of space:$$\mathcal{H}=\underset{i}{\u2a02}{\mathcal{H}}_{i}$$
- There is what is called redundancy constrained (RC) states for each subset of the Hilbert spaces $B\subset \mathcal{H}$, considered to be a subspace of the physical space. Its entropy is defined as:$$\begin{array}{ccc}\hfill S(B)& \equiv & \frac{1}{2}\sum _{i\in B,j\in \overline{B}}I(i:j)\hfill \end{array}$$$$\begin{array}{ccc}\hfill I(i:j)& \equiv & S(i)+S(j)-S(i\cup j)\hfill \end{array}$$
- It is assumed that the system is in an
`entanglement equilibrium`state, when subsystems are in RC states. Under small perturbations the entropy of B is assumed to be conserved. This means that the total entropy is conserved. Moreover, when states deviate from RC, their entropy can be decomposed to the entropy of a fiducial RC state and a subleading component, interpreted as an effective field theory. The two components cancel each other to preserve the total entropy.

`preferred`. However, the model does not specify what are the criteria for its selection.

`background`around which a perturbation is performed. Indeed, the model does not consider highly non-RC states and studies only the case of weak gravity interaction [14].

`clock`with which a Hamiltonian and an operator analogous to energy–momentum can be associated. The latter can be considered as an effective field theory generating subleading entropy of states, which are perturbatively deviated from RC states. Finally, by comparing this formulation with general relativity and by using Radon transform, reference [14] argues that Einstein’s equation can be concluded.

#### 3.2.6. Comparison of EBM with $SU(\infty )$-QGR

#### `Factorization of the Hilbert Space and Division to Subsystems`

`preferred`. This is in strict opposition to the approach of $SU(\infty )$-QGR. The reason behind the special factorization is again the absence of a concrete criterion to discriminate between factors-subsystems.

#### `Geometry and Classical Gravity`

#### `Analogy between Distance and Entanglement`

#### 3.3. String Theory, M-Theory, and AdS/CFT Duality in Three or More Dimensions

#### 3.3.1. Perturbative String Theories and Their Comparison with $SU(\infty )$-QGR

#### `2D Surfaces in String Theory and $SU(\infty )$-QGR`

#### `String Sigma Model`

#### `Curved Spacetime and Gravity in String Theory`

`gas`. In particular, This approach is used for the purpose of describing cosmological perturbations in the framework of string theory [113]. However, the inherently intertwined nature of spacetime and strings may make it impossible to consider their evolution separately. There are, nonetheless, exceptions. AdS/CFT duality conjecture, discussed in more detail in Section 3.3.3, is proved for ${\mathrm{AdS}}_{3}$ space, and is considered as to be the evidence of consistency of perturbative string theory, at least in some curved background spaces.

#### 3.3.2. M-Theory and Matrix Theories

`field`equations for ${X}^{a}$ and ${\psi}_{\alpha}$. In particular, considering only bosonic Yang–Mills sector, the field equation for ${X}^{a}$ is:

#### `Comparison of Matrix Models with $SU(\infty )$-QGR`

`internal`finite rank symmetry. By contrast, $SU(\infty )$-QGR is constructed from a Hilbert space and includes both square and column matrices as primary entities, in adjoint and fundamental representations of both $SU(\infty )$ and internal symmetries.

`color`and loop number limit of QFT [118], conjectured to present a strong coupling regime. In $SU(\infty )$-QGR, the motivation is rather cosmological and based on the observed large number of degrees of freedom in the Universe. These apparently different motivations converge to each other, because for a perturbative estimation of observables, for instance an S-matrix, up to a given degree of precision, one has to take into account more loops, virtual particles, and their degrees of freedom for stronger couplings. The assumption of $SU(\infty )$-QGR that every subsystem of the Universe represents infinite degrees of freedom is an explicit realization of the above concept.

`internal`symmetries of subsystems (particles or fields), which are not constrained by the model. In fact, considering that when all constituents interact with each other, the symmetry is $G\to SU(\infty )$, one expects that many other smaller rank symmetries should have a nonzero probability arise in intermediate states, where the number of effectively coupled or entangled subsystems is finite but large.

#### 3.3.3. Anti-de Sitter–Conformal Field Theory (AdS-CFT) Duality

## 4. Outline

- Hawking radiation;
- Information loss paradox of black holes;
- Particle physics at Planck scale;
- Spacetime singularities;
- Topology of parameter space of the model identified as the classical spacetime and whether it can be changed dynamically.

`phase transitions`due to symmetry transition at high energies in (astro-)particle physics experiments or cosmological observations.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Classical Limit of the SU(∞) Yang–Mills Model of Subsystems

## References

- Eppley, K.; Hanna, E. The Necessity of Quantizing the Gravitational Field. Found. Phys.
**1977**, 7, 51. [Google Scholar] [CrossRef] - Jacobson, T. Thermodynamics of Spacetime: The Einstein Equation of State. Phys. Rev. Lett.
**1995**, 75, 1260. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bekenstein, J.D. Statistical black-hole thermodynamics. Phys. Rev. D
**1975**, 12, 3077. [Google Scholar] [CrossRef] - t’Hooft, G. Dimensional Reduction in Quantum Gravity. arXiv
**1993**, arXiv:9310026. [Google Scholar] - Susskin, L. The World as a Hologram. J. Math. Phys.
**1995**, 36, 6377. [Google Scholar] [CrossRef] [Green Version] - Bousso, R. The holographic principle. Rep. Mod. Phys.
**2002**, 74, 825. [Google Scholar] [CrossRef] [Green Version] - Holzhey, C.; Larsen, F.; Wilczek, F. Geometric and Renormalized Entropy in Conformal Field Theory. Nucl. Phys. B
**1994**, 424, 443. [Google Scholar] [CrossRef] [Green Version] - Calabrese, P.; Cardy, J. Entanglement Entropy and Quantum Field Theory. J. Stat. Mech.
**2004**, 0406, P06002. [Google Scholar] [CrossRef] [Green Version] - Ziaeepour, H. Issues with vacuum energy as the origin of dark energy. Mod. Phys. Lett. A
**2012**, 27, 1250154. [Google Scholar] [CrossRef] - Rovelli, C. Quantum mechanics without time: A model. Phys. Rev. D
**1990**, 42, 2638. [Google Scholar] [CrossRef] - Markopoulou, F. Space does not exist, so time can. arXiv
**2009**, arXiv:0909.1861. [Google Scholar] - Van Raamsdonk, M. Building up spacetime with quantum entanglement. Gen. Rel. Grav.
**2010**, 42, 2323, reprint in Int. J. Mod. Phys. D**2010**, 19, 2429. [Google Scholar] [CrossRef] - 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] - Cao, C.; Carroll, S.M. Bulk Entanglement Gravity without a Boundary: Towards Finding Einstein’s Equation in Hilbert Space. Phys. Rev. D
**2018**, 97, 086003. [Google Scholar] [CrossRef] [Green Version] - Giddings, S.B. Hilbert space structure in quantum gravity: An algebraic perspective. J. High Energy Phys.
**2015**, 2015, 1. [Google Scholar] [CrossRef] [Green Version] - Page, D.N.; Wootters, W.K. Evolution without evolution: Dynamics described by stationary observables. Phys. Rev. D
**1983**, 27, 2885. [Google Scholar] [CrossRef] - Giddings, S.B. Quantum-first gravity. Found. Phys.
**2019**, 49, 177. [Google Scholar] [CrossRef] [Green Version] - Ziaeepour, H. Making a Quantum Universe: Symmetry and Gravity. Universe
**2020**, 6, 194. [Google Scholar] [CrossRef] - Ziaeepour, H. SU(∞)-QGR: Emergence of Quantum Gravity in an Infinitely Divisible Quantum Universe. 2022; In preparation. [Google Scholar]
- Bekenstein, J.D. Black Holes and Entropy. Phys. Rev. D
**1973**, 7, 2333. [Google Scholar] [CrossRef] - Hawking, S. Break of predictability in gravitational collapse. Phys. Rev. D
**1976**, 14, 246. [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] - Rocci, A. On first attempts to reconcile quantum principles with gravity. J. Phys. Conf. Ser.
**2013**, 470, 012004. [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] - Zanardi, P.; Lidar, D.; Lloyd, S. Quantum tensor product structures are observable-induced. Phys. Rev. Lett.
**2004**, 92, 060402. [Google Scholar] [CrossRef] [PubMed] [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] - Witten, E. Anti De Sitter Space Additionally, Holography. Adv. Theor. Math. Phys.
**1998**, 2, 253. [Google Scholar] [CrossRef] - Aharony, O.; Gubser, S.S.; Maldacena, J.; Ooguri, H.; Oz, Y. Large N Field Theories, String Theory and Gravity. Phys. Rep.
**2000**, 323, 183. [Google Scholar] [CrossRef] [Green Version] - Ziaeepour, H. Foundational role of symmetry in Quantum Mechanics and Quantum Gravity. In Quantum Mechanics: Theory, Analysis, and Applications; Nova Science Publishers Inc.: New York, NY, USA, 2019. [Google Scholar]
- 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]
- 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 Groups, 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] - Ziaeepour, H. Furthermore, what if gravity is intrinsically quantic ? J. Phys. Conf. Ser.
**2009**, 174, 012027. [Google Scholar] [CrossRef] - Su, Z.-Y. A Scheme of Cartan Decomposition for su(N). arXiv
**2006**, arXiv:0603190. [Google Scholar] - Ziaeepour, H. Symmetry as a foundational concept in Quantum Mechanics. J. Phys. Conf. Ser.
**2015**, 626, 012074. [Google Scholar] [CrossRef] [Green Version] - Hoehn, P.A.; Smith, A.R.H.; Lock, M.P.E. The Trinity of Relational Quantum Dynamics. arXiv
**2019**, arXiv:1912.00033. [Google Scholar] - Mandelstam, L.; Tamm, I. The Uncertainty Relation Between Energy and Time in Non-relativistic Quantum Mechanics. J. Phys. (USSR)
**1945**, 9, 249. [Google Scholar] - Hoehen, P.H.; Lock, M.P.E.; Ali Ahmad, S.; Smith, A.R.H.; Galley, T.D. Quantum Relativity of Subsystems. arXiv
**2021**, arXiv:2103.01232. [Google Scholar] - Rosenfeld, L. Zur Quantelung der Wellenfelder. Annal der Physik
**1930**, 397, 113. [Google Scholar] [CrossRef] - Regge, T. General Relativity without Coordinates. Nuovo C.
**1961**, 19, 558. [Google Scholar] [CrossRef] - Gambini, R.; Pullin, J. Consistent discretization and canonical classical and quantum Regge calculus. Int. J. Mod. Phys. D
**2006**, 15, 1699. [Google Scholar] [CrossRef] [Green Version] - Ponzano, G.; Regge, T. Semiclassical limit of Racah coefficients. p1–58. In Spectroscopic and Group Theoretical Methods in Physics; Bloch, F., Ed.; North-Holland Publ. Co.: Amsterdam, The Netherlands, 1968. [Google Scholar]
- Ashtekar, A. New Variables for Classical and Quantum Gravity. Phys. Rev. Lett.
**1986**, 57, 2244. [Google Scholar] [CrossRef] [PubMed] - Immirzi, G. Quantum Gravity and Regge Calculus. Nucl. Phys. B Proc. Suppl.
**1997**, 57, 65. [Google Scholar] [CrossRef] [Green Version] - 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] - Immirzi, G. Real and complex connections for canonical gravity. Class. Quant. Grav.
**1997**, 14, L177. [Google Scholar] [CrossRef] [Green Version] - Barrett, J.W.; Crane, L. Relativistic spin networks and quantum gravity. J. Math. Phys.
**1998**, 39, 3296. [Google Scholar] [CrossRef] [Green Version] - Barrett, J.W.; Crane, L. A Lorentzian Signature Model for Quantum General Relativity. Class. Quant. Grav.
**2000**, 17, 3101. [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] - Ashtekar, A.; Rovelli, C.; Smolin, L. Weaving a Classical Metric with Quantum Threads. Phys. Rev. Lett.
**1992**, 69, 237. [Google Scholar] [CrossRef] [Green Version] - Rovelli, C.; Colosi, D.; Doplicher, L.; Fairbairn, W.; Modesto, L.; Noui, K. Background independence in a nutshell. Class. Quant. Grav.
**2005**, 22, 2971. [Google Scholar] - Smolin, L. An invitation to loop quantum gravity. In Quantum Theory and Symmetries; Argyres, P.C., Hodges, T.J., Mansouri, F., Scanio, J.J., Suranyi, P., Wijewardhana, L.C.R., Eds.; World Scientific: London, UK, 2004. [Google Scholar]
- Maran, S.K. Reality Conditions for Spin Foams. arXiv
**2005**, arXiv:0511014. [Google Scholar] - 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] [PubMed] [Green Version] - Gambini, R.; Pullin, J. Emergent diffeomorphism invariance in a discrete loop quantum gravity model. Class. Quant. Grav.
**2009**, 26, 035002. [Google Scholar] [CrossRef] [Green Version] - Ashtekar, A. Some surprising implications of background independence in canonical quantum gravity. Gen. Rel. Grav.
**2009**, 41, 1927. [Google Scholar] [CrossRef] [Green Version] - Bojowald, M.; Mortuza Hossain, G. Loop quantum gravity corrections to gravitational wave dispersion. Phys. Rev. D
**2008**, 77, 023508. [Google Scholar] [CrossRef] [Green Version] - Girelli, F.; Hinterleitner, F.; Major, S.A. Loop Quantum Gravity Phenomenology: Linking Loops to Observational Physics. SIGMA
**2012**, 8, 098. [Google Scholar] [CrossRef] [Green Version] - 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] - The LIGO Scientific Collaboration. Tests of General Relativity with GW170817. Phys. Rev. Lett.
**2019**, 123, 011102. [Google Scholar] [CrossRef] [Green Version] - The LIGO Scientific Collaboration. Tests of General Relativity with the Binary Black Hole Signals from the LIGO-Virgo Catalog GWTC-1. Phys. Rev. D
**2019**, 100, 104036. [Google Scholar] [CrossRef] [Green Version] - Perez, A.; Rovelli, C. Physical effects of the Immirzi parameter. Phys. Rev. D
**2006**, 73, 044013. [Google Scholar] [CrossRef] [Green Version] - Bergé, J.; Pernot-Borràs, M.; Uzan, J.-P.; Brax, P.; Chhun, R.; Métris, G.; Rodrigues, M.; Touboul, P. MICROSCOPE’s constraint on a short-range fifth force. arXiv
**2021**, arXiv:2102.00022. [Google Scholar] - Gaul, M.; Rovelli, C. Loop Quantum Gravity and the Meaning of Diffeomorphism Invariance. Lect. Notes Phys.
**2000**, 541, 277. [Google Scholar] - Halliwell, J.J.; Wallden, P. Invariant Class Operators in the Decoherent Histories Analysis of Timeless Quantum Theories. Phys. Rev. D
**2006**, 73, 024011. [Google Scholar] [CrossRef] [Green Version] - Reisenberger, M.; Rovelli, C. Spacetime states and covariant quantum theory. Phys. Rev. D
**2002**, 65, 125016. [Google Scholar] [CrossRef] [Green Version] - Terno, D.R. Quantum information in loop quantum gravity. J. Phys. Conf. Ser.
**2006**, 33, 469. [Google Scholar] [CrossRef] [Green Version] - Gielen, S.; Oriti, D. Cosmological perturbations from full quantum gravity. Phys. Rev. D
**2018**, 98, 106019. [Google Scholar] [CrossRef] [Green Version] - Giesel, K.; Thiemann, T. Algebraic Quantum Gravity (AQG) IV. Reduced Phase Space Quantisation of Loop Quantum Gravity. Class. Quant. Grav.
**2010**, 27, 175009. [Google Scholar] [CrossRef] [Green Version] - Husain, V.; Pawlowski, T. Time and a physical Hamiltonian for quantum gravity. Phys. Rev. Lett.
**2012**, 108, 141301. [Google Scholar] [CrossRef] [Green Version] - Giesel, K.; Vetter, A. Reduced Loop Quantization with four Klein-Gordon Scalar Fields as Reference Matter. arXiv
**2016**, arXiv:1610.07422. [Google Scholar] [CrossRef] [Green Version] - Wilczek, F. Riemann–Einstein Structure from Volume and Gauge Symmetry. Phys. Rev. Lett.
**1998**, 80, 4851. [Google Scholar] [CrossRef] [Green Version] - Seiberg, N. Emergent Spacetime. In The Quantum Structure of Space and Time; Harvey, J., Ed.; World Scientific: London, UK, 2007; p. 163. [Google Scholar]
- Westman, H.; Sonego, S. Coordinates, observables and symmetry in relativity. Ann. Phys.
**2009**, 324, 1585. [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] - Padmanabhan, T. Gravity and the Thermodynamics of Horizons. Phys. Rep.
**2005**, 406, 49. [Google Scholar] [CrossRef] [Green Version] - Padmanabhan, T. Gravity as an emergent phenomenon: A conceptual description. AIP Conf. Proc.
**2007**, 939, 114. [Google Scholar] - 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] - Zurek, W.H. Entropy Evaporated by a Black Hole. Phys. Rev. Lett.
**1982**, 49, 1683. [Google Scholar] [CrossRef] - Hartle, J.B. Generalizing Quantum Mechanics for Quantum Spacetime. In The Quantum Structure of Space and Time; Gross, D., Henneaux, M., Sevrin, A., Eds.; World Scientific: Singapore, 2007. [Google Scholar]
- Giddings, S.B. Universal quantum mechanics. Phys. Rev. D
**2008**, 78, 084004. [Google Scholar] [CrossRef] [Green Version] - Griffiths, R.B. Consistent Histories and the Interpretation of Quantum Mechanics. J. Stat. Phys.
**1984**, 36, 219. [Google Scholar] [CrossRef] - Isham, C.J. Quantum Logic and the Histories Approach to Quantum Theory. J. Math. Phys.
**1994**, 35, 2157. [Google Scholar] [CrossRef] [Green Version] - Birrell, N.D.; Davies, P.C.W. Quantum Fields in Curved Space; Cambridge University Press: Cambridge, UK, 1982. [Google Scholar]
- Henson, J. Quantum Histories and Quantum Gravity. J. Phys. Conf. Ser.
**2009**, 174, 012020. [Google Scholar] [CrossRef] - Hartle, J.B. Quantum Multiverses. arXiv
**2018**, arXiv:1801.08631. [Google Scholar] - Donnelly, W.; Giddings, S.B. How is quantum information localized in gravity? Phys. Rev. D
**2017**, 96, 086013. [Google Scholar] [CrossRef] [Green Version] - Donnelly, W.; Giddings, S.B. Gravitational splitting at first order: Quantum information localization in gravity. Phys. Rev. D
**2018**, 98, 086006. [Google Scholar] [CrossRef] [Green Version] - Von Neumann, J. Mathematische Grundlagen der Quantunmechanik; Springer: Berlin, Germany, 1932. [Google Scholar]
- Yngvason, J. The Role of Type III Factors in Quantum Field Theory. Rept. Math. Phys.
**2005**, 55, 135. [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] - Ryu, S.; Takayanagi, T. Holographic Derivation of Entanglement Entropy from AdS/CFT. Phys. Rev. Lett.
**2006**, 96, 181602. [Google Scholar] [CrossRef] [Green Version] - Green, M.B.; Schwarz, J.H.; Witten, E. Superstring Theory I & II; Cambridge University Press: Cambridge, UK, 1987. [Google Scholar]
- Polchinski, J. TASI lecture on D-branes. arXiv
**1996**, arXiv:9611050. [Google Scholar] - Adams, A.; Polchinski, J.; Silverstein, E. Do not Panic! Closed String Tachyons in ALE Spacetimes. J. High Energy Phys.
**2001**, 0110, 029. [Google Scholar] [CrossRef] - Karczmarek, J.L.; Strominger, A. Closed String Tachyon Condensation at c = 1. J. High Energy Phys.
**2004**, 0405, 062. [Google Scholar] [CrossRef] - Green, D. Nothing for Branes. J. High Energy Phys.
**2007**, 0704, 025. [Google Scholar] [CrossRef] [Green Version] - Maldacena, J. The gauge/gravity duality. In Black Holes in Higher Dimensions; Horowitz, G., Ed.; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Gubser, S.S.; Klebanov, I.R.; Polyakov, A.M. Gauge Theory Correlators from Non-Critical String Theory. Phys. Lett. B
**1998**, 428, 105. [Google Scholar] [CrossRef] [Green Version] - Maldacena, J.M.; Strominger, A. Semiclassical decay of near extremal fivebranes. J. High Energy Phys.
**1997**, 9712, 008. [Google Scholar] [CrossRef] [Green Version] - Polchinski, J. String Theory I & II; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Floratos, E.F.; Iliopoulos, J. A Note on the Classical Symmetries of the Closed Bosonic Membranes. Phys. Lett. B
**1988**, 201, 237. [Google Scholar] [CrossRef] - Antoniadis, I.; Ditsas, P.; Floratos, E.F.; Iliopoulos, J. New Realizations of the Virasoro Algebra as Membrane Symmetries. Nucl. Phys. B
**1988**, 300, 549. [Google Scholar] [CrossRef] [Green Version] - Nayeri, A.; Brandenberger, R.H.; Vafa, C. Producing a Scale-Invariant Spectrum of Perturbations in a Hagedorn Phase of String Cosmology. Phys. Rev. Lett.
**2006**, 97, 021302. [Google Scholar] [CrossRef] [Green Version] - Ishibashi, N.; Kawai, H.; Kitazawa, Y.; Tsuchiya, A. A Large-N Reduced Model as Superstring. Nucl. Phys. B
**1997**, 498, 467. [Google Scholar] [CrossRef] [Green Version] - Konechny, A.; Schwarz, A. Introduction to M(atrix) theory and noncommutative geometry. Phys. Rep.
**2002**, 360, 353. [Google Scholar] [CrossRef] [Green Version] - Bousso, R.; Mints, A.L. Holography and entropy bounds in the plane wave matrix model. Phys. Rev. D
**2006**, 73, 126005. [Google Scholar] [CrossRef] [Green Version] - Steinacker, H. Emergent Geometry and Gravity from Matrix Models: An Introduction. Class. Quant. Grav.
**2010**, 27, 133001. [Google Scholar] [CrossRef] - ’t Hooft, G. A Planar Diagram Theory for Strong Interactions. Nucl. Phys. B
**1974**, 72, 461. [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] - Kawahara, N.; Nishimura, J.; Takeuchi, S. High temperature expansion in supersymmetric matrix quantum mechanics. J. High Energy Phys.
**2007**, 0712, 103. [Google Scholar] [CrossRef] - Schild, A. Classical null strings. Phys. Rev. D
**1977**, 16, 1722. [Google Scholar] [CrossRef] - Connes, A.; Douglas, M.R.; Schwarz, A. Noncommutative Geometry and Matrix Theory: Compactification on Tori. J. High Energy Phys.
**1998**, 02, 003. [Google Scholar] [CrossRef] [Green Version] - Steinacker, H. Covariant Field Equations, Gauge Fields and Conservation Laws from Yang–Mills Matrix Models. J. High Energy Phys.
**2009**, 02, 044. [Google Scholar] [CrossRef] [Green Version] - Kumar, J. A Review of Distributions on the String Landscape. Int. J. Mod. Phys. A
**2006**, 21, 3441. [Google Scholar] [CrossRef] [Green Version] - Brahma, S.; Brandenberger, R.; Laliberte, S. Emergent Cosmology from Matrix Theory. arXiv
**2021**, arXiv:2107.11512. [Google Scholar] - Steinacker, H. Gravity as a Quantum Effect on Quantum Space-Time. arXiv
**2021**, arXiv:2110.03936. [Google Scholar] - The LIGO-Virgo Collaboration. Tests of General Relativity with Binary Black Holes from the second LIGO-Virgo Gravitational-Wave Transient Catalog. Phys. Rev. D
**2021**, 103, 122002. [Google Scholar] [CrossRef] - Anderson, P.W. Absence of Diffusion in Certain Random Lattices. Phys. Rev.
**1958**, 109, 1492. [Google Scholar] [CrossRef] - Koma, T.; Tasak, H. Symmetry Breaking and Finite-Size Effects in Quantum Many-Body Systems. J. Stat. Phys.
**1994**, 76, 745. [Google Scholar] [CrossRef] [Green Version] - Ziaeepour, H. QCD Color Glass Condensate Model in Warped Brane Models. Grav. Cosmol. Suppl.
**2005**, 11, 189. [Google Scholar] - Kühnel, W. Differential Geometry, 3rd ed.; AMS: Rhode Island, RI, USA, 2010. [Google Scholar]
- Gallier, J. Differential Geometry and Lie Groups, Volume I; Springer: Berlin/Heidelberg, Germany, 2020. [Google Scholar]

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

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ziaeepour, H.
Comparing Quantum Gravity Models: String Theory, Loop Quantum Gravity, and Entanglement Gravity versus *SU*(∞)-QGR. *Symmetry* **2022**, *14*, 58.
https://doi.org/10.3390/sym14010058

**AMA Style**

Ziaeepour H.
Comparing Quantum Gravity Models: String Theory, Loop Quantum Gravity, and Entanglement Gravity versus *SU*(∞)-QGR. *Symmetry*. 2022; 14(1):58.
https://doi.org/10.3390/sym14010058

**Chicago/Turabian Style**

Ziaeepour, Houri.
2022. "Comparing Quantum Gravity Models: String Theory, Loop Quantum Gravity, and Entanglement Gravity versus *SU*(∞)-QGR" *Symmetry* 14, no. 1: 58.
https://doi.org/10.3390/sym14010058