Making a Quantum Universe: Symmetry and Gravity
AArticle
Making a Quantum Universe: Symmetry and Gravity
Houri Ziaeepour Institut UTINAM, CNRS UMR 6213, Observatoire de Besançon, Université de Franche Compté, 41 bis ave. del’Observatoire, BP 1615, 25010 Besançon, France; Mullard Space Science Laboratory, University College London, Holmbury St. Mary, GU5 6NT, Dorking, UK;[email protected] October 14, 2020 submitted to Universe
Abstract:
So far none of attempts to quantize gravity has led to a satisfactory model that not onlydescribe gravity in the realm of a quantum world, but also its relation to elementary particles andother fundamental forces. Here we outline preliminary results for a model of quantum universe, inwhich gravity is fundamentally and by construction quantic. The model is based on 3 well motivatedassumptions with compelling observational and theoretical evidence: quantum mechanics is valid at allscales; quantum systems are described by their symmetries; Universe has infinite independent degreesof freedom. The last assumption means that the Hilbert space of the Universe has SU p N Ñ 8q – area preserving Diff. p S q symmetry, which is parameterized by two angular variables. We show that inabsence of a background spacetime, this Universe is trivial and static. Nonetheless, quantum fluctuationsbreak the symmetry and divide the Universe to subsystems. When a subsystem is singled out as reference- observer - and another as clock , two more continuous parameters arise, which can be interpreted asdistance and time. We identify the classical spacetime with parameter space of the Hilbert space ofthe Universe. Therefore, its quantization is meaningless. In this view, the Einstein equation presentsthe projection of quantum dynamics in the Hilbert space into its parameter space. Finite dimensionalsymmetries of elementary particles emerge as a consequence of symmetry breaking when the Universe isdivided to subsystems/particles without having any implication for the infinite dimensional symmetryand its associated interaction percived as gravity. This explains why gravity is a universal force. Keywords: quantum gravity; quantum mechanics; symmetry; quantum cosmology
1. Introduction and summary of results
More than a century after discovery of general relativity and description of gravitational force as themodification of spacetime geometry by matter and energy, we still lack a convincing model for explainingthese processes in the framework of quantum mechanics. Appendix A briefly reviews the history of effortsfor finding a consistent Quantum Gravity (QGR) model. Despite tremendous effort of generations ofscientists, at present none of proposed models seems fully satisfactory.Quantization of gravity is inevitable. Examples of inconsistencies in a universe where matter is ruledby quantum mechanics but gravity is classical are well known [1,2]. In addition, in [2] it is argued thatthere must be an inherent relation between gravity and quantum mechanics. Otherwise, universalityof Planck constant ¯ h as quantization scale cannot be explained . Aside from these arguments, the fact Nonetheless, Ref. [3] advocates a context dependent Planck constant.Submitted to
Universe a r X i v : . [ g r- q c ] O c t ersion October 14, 2020 submitted to Universe that there is no fundamental mass/energy scale in quantum mechanics means that it has to have a closerelation with gravity that provides a dimensionful fundamental constant, namely the Newton gravitationalconstant G N (or equivalently the Planck mass M P ” a ¯ hc { G N , where c is the speed of light in vacuum(or equivalently Planck length scale L P ” ¯ h { cM P ). We should remind that a dimensionful scale does notarise in conformal or scale independent models. Indeed, conformal symmetry is broken by gravity, whichprovides the only fundamental dimensionful constant to play the role of a ruler and make distance andmass measurements meaningful .In what concerns the subject of this volume, namely representations of inhomogeneous Lorentzsymmetry (called also Poincaré group), they were under special interest since decades ago, hoping theyhelp formulate gravity as a renormalizable quantum field. The similarity of the compact group of localLorentz transformations to Yang-Mills gauge symmetry has encouraged quantum gravity models basedon the first order formulation of general relativity. These models use vierbein formalism and extension ofgauge group of elementary particles to accommodate Poincaré group [5,6]. However, Coleman-Madulatheorem [7] on S -matrix symmetries - local transformations of interacting fields that asymptoticallyapproach Poincaré symmetry at infinity - invalidates any model in which Poincaré and internal symmetriesare not factorized. According to this theorem total symmetry of a grand unification model includinggravity must be a tensor product of spacetime and internal symmetries. Otherwise, the model must besupersymmetric [8]. However, we know that even if supersymmetry is present at M P scale, it is brokenat low energies. Moreover, any violation of Coleman-Madula theorem and Lorentz symmetry at highenergies can be convoyed to low energies [10] and violate e.g. equivalence principle and other testedpredictions of general relativity [11,12]. For these reasons modern approaches to unification of gravity as agauge field with other interactions consider the two sector as separate gauge field models. In addition, inthese models gravity sector usually has topological action to make the formulation independent of thegeometry of underlying spacetime, see e.g. [13–15]. However, like other quantum gravity candidates thesemodels suffer from various issues. Separation of internal and gravitational gauge sectors means that thesemodels are not properly speaking a grand unification. Moreover, similar to other approaches to QGR,these models do not clarify the nature of spacetime, its dimensionality, and relation between gravity andinternal symmetries.In addition to consistency with general relativity, cosmology and particle physics, a quantum modelunifying gravity with other forces is expected to solve well known problems related to gravity andspacetime such as: physical origin of the arrow of time; apparent information loss in black holes; and UVand IR singularities in Quantum Field Theory (QFT) and general gravity . There are also other issues thata priori should be addressed by a QGR model, but are less discussed in the literature:1. Should spacetime be considered as a physical entity similar to quantum fields associated toparticles, or rather it presents a configuration space ?
General relativity changed spacetime from a rigid entity to a deformable media. But it doesnot specify whether spacetime is a physical reality or a property of matter, which ultimatelydetermines its geometry and topology. We remind that in the framework of QFT vacuum is In addition to M P , we need two other fundamental constants to describe physics and cosmology: the Planck constant ¯ h andmaximum speed of information transfer that experiments show to be the speed of light in classical vacuum. We remind thattriplet constants p ¯ h , c , M P q are arbitrary and can be any nonzero value. Selection of their values amounts to definition of a systemof units for measuring other physical quantities. In QFT literature usually ¯ h “ c “ h and c are dimensionless. Some quantum gravity models such as loop quantum gravity emphasize on the quantization of gravity alone. However, givingthe fact that gravity is a universal force and interacts with matter and other forces, its quantization necessarily has impact onthem. Therefore, any quantum gravity only model would be at best incomplete.ersion October 14, 2020 submitted to
Universe not the empty space of classical physics, see e.g. [17,18]. In particular, in presence of gravitythe naive definition of quantum vacuum is frame dependent. A frame-independent definitionexists [19] and it is very far from classical concept of an empty space. Explicitly or implicitlysome of models reviewed in Appendix A address this question.2.
Is there any relation between matter and spacetime ?
In general relativity matter modifies geometry of spacetime, but the two entities are considered asseparate and stand alone. In string theory spacetime and matter fields - compactified internalspace - are considered and treated together, and spacetime has a physical reality similar to matter.By contrast, many other QGR candidates concentrate their effort only on the quantization ofspacetime and gravitational interaction. Matter is usually added as an external ingredient anddoes not intertwine in the construction of quantum gravity and spacetime.3.
Why do we perceive the Universe as a 3D space (plus time) ?
None of extensively studied quantum gravity models discussed in Appendix A answer thisquestion, despite the fact that it is the origin of many troubles for them. For instance, theenormous number of possible models in string theory is due to the inevitable compactificationof extra-dimensions to reduce the dimension of space to the observed 3+1. In backgroundindependent models the dimension of space is a fundamental assumption and essential for manytechnical aspects of their construction. In particular, the definition of Ashtekar variables [16]for SU p q – SO p q symmetry and its relation with spin foam description of loop quantumgravity [20] are based on the assumption of a 3D real space. On the other hand, according toholography principle the maximum amount of information containable in a quantum system isproportional to its area rather than volume. If the information is projected and available on theboundary, it is puzzling why we should perceive the volume.In a previous work [21] we advocated foundational role of symmetries in quantum mechanics andreformulated its axioms accordingly, see Appendix B for a summary. Of course the crucial role ofsymmetries in quantum systems is well known. But axioms of quantum mechanics à la Dirac and vonNeumann consider an abstract Hilbert space and do not specify its relation with symmetries of quantumsystems. In addition to symmetries of their classical Lagrangian, Hilbert space of quantum systemsrepresents SU p N q group, called state symmetry, see Appendix C for more details. Transformation of statesby this group modifies their coherence, and recently quantification of this property and its usefulness as aresource has become a subject of interest in quantum information theory literature [22,23].Inspired by these developments, in this work we study a standalone quantum system, considered tobe the Universe. The model assumes infinite number of independent and simultaneously commuting observablesin the Universe. But, it does not consider any background spacetime. Hilbert space H U of such systemrepresents SU p N Ñ 8q symmetry. However, in absence of a background spacetime its dynamics is trivialand its Lagrangian is defined on the group manifold of SU p8q symmetry. Therefore, states are pure gauge.The vector space of gauge transformations, corresponding to linear transformation of the Hilbert spaceis B r H s – SU p8q . On the other hand quantum fluctuations break the state symmetry and factorize theHilbert space to blocks of tensor product of subspaces according to criteria studied in [24,25]. For eachsubsystem the rest of the Universe plays the role of a background parameterized by 3 continuous quantitiesthat can be identified with the classical space. Moreover, division of the Universe to subsystems leads ersion October 14, 2020 submitted to Universe to emergence of time and its arrow à la Page & Wootters [26] or similar methods [27]. We show thatthe 3+1 dimensional parameter space is in general curved and invariant under inhomogeneous Lorentztransformations and its curvature is determined by quantum states of the subsystems. We also commenton the signature of parameter space metric. Based on these observations we interpret SU p8q sector of themodel as Quantum Gravity . The finite rank factorized symmetries become local gauge fields acting on aHilbert space that presents matter fields.These results demonstrate the importance of the division of Universe to subsystems and distinctionof observer and clock from the rest. Nonetheless, in contrast to Copenhagen interpretation of quantummechanics, the absence of observer does not make the model meaningless, but trivial and static. Thismodel answers some of issues raised in questions 1-3. In particular it clarifies the nature of spacetime andits dimensionality and provides an explanation for universality of gravitational force.A crucial proposal of the model is that what we perceive as classical spacetime is the configuration(parameter) space of its content. In other words, rather than saying particles/objects (such as strings) live ina 3+1 dimensional space, according to this model we can say ensemble of abstract objects with SU p8q ˆ G symmetry look like a 3+1 dimensional infinite curved spacetime with gravity, where subsystems arefields representing group G as a local gauge symmetry. Thus, we can completely neglect the geometricinterpretation and just consider the Universe as an infinite tensor product. This aspect of the model issimilar to the approach of [28]. However, their model is somehow inverse of that studied here. They usetensor product and quantum entanglement to make a symplectic geometry that at infinite limit becomesa continuous curved spacetime. The drawback is that symplectic geometries defined by graphs canbe embedded in any space of dimension D ě
2. Consequently, they cannot explain dimension of thespacetime.Axioms and structure of the model is discussed in Sec. 2. Lagrangian of the system before its divisionis described in Sec. 3. Properties of the model after symmetry breaking and division of the Universe arestudied in Sec. 4. A brief comparison of this model with string and loop quantum gravity is presentedin Sec. 5. Outlines and prospective for future investigations are discussed in Sec. 6. Accompanyingappendices contain technical details and review of previous results. Appendix A gives a short recountof the history of quantum gravity models. Appendix B summaries axioms of quantum mechanics insymmetry language. State space and its associated symmetry is reviewed in Appendix C. Properties of SU p8q and its representations are summarized in Appendix D and its Cartan decomposition in AppendixE.
2. An infinite quantum Universe
Our departure point for constructing a quantum universe consists of 3 well motivated assumptionswith compelling observational and theoretical evidence:I. Quantum mechanics is valid at all scales and applies to every entity, including the Universe as awhole;II. Any quantum system is described by its symmetries and its Hilbert space represents them;III. The Universe has infinite number of independent degrees of freedom.The last assumption means that the Hilbert space of the Universe H U is infinite dimensional and representsthe group SU p8q . There are sufficient evidence in favour of such assumption. For instance, thermaldistribution of photons at IR limit contains infinite number of quanta with energies approaching zeroand there is no minimum energy limit. For this reason, vacuum can be considered as superposition ofmulti-particle states of any type - not just photons - without any limit on their number [19]. In generalrelativity there is no upper limit for gravitons wavelength and thereby their number. Of course onemay argue that a lower limit on energy or spacetime volume may exist. Nonetheless, for any practical ersion October 14, 2020 submitted to Universe application the number of subsystems/quanta in the Universe can be considered to approach infinity.Indeed even in quantum gravity models that assume a symplectic structure for spacetime, such as spinfoam/loop quantum gravity and causal sets, there is not a fixed lattice of spacetime and number ofspacetime states is effectively infinite.The algebra associated to the SU p8q coherence (state) symmetry of the above model is defined as : r ˆ L a , ˆ L b s “ ¯ hcM P f cab ˆ L c “ L P f cab ˆ L c (1)where operators ˆ L α P B r H U s are generators of algebra su p8q and f cab are its structure coefficients. Theyare normalized such that the r.h.s. of (1) explicitly depends on the Planck constant ¯ h . If ¯ h Ñ
0, the r.h.s.becomes null and the algebra becomes abelian and homomorphic to  N Ñ8 U p q , in agreement withthe symmetry of configuration space of classical systems explained in Appendix C. The same happensif M P Ñ 8 , that is when Planck mass scale is much larger than scale of interest. In both cases L P Ñ SU p8q symmetry of the Universe can be associated to gravitational interaction - wewill provide more evidence in favour of this claim later - the above limits mean that in both cases gravitybecomes negligible .It is well known that B r H U s – SU p8q – area preserving Diff p S q [29,30], where S is 2D sphere.In fact SU p8q is homomorphic to area preserving diffeomorphism of any 2D Riemann surface [31–33].Therefore, here S indicates and 2D surface rather than just sphere. This theorem can be heuristicallyunderstood as the following: Any compact 2D Riemann surface can be obtained from sphere by removinga measure zero set of pair of points and sticking the rest of the surface pair-by-pair together. Althoughsurfaces with different genus are topologically different, they are homomorphic. This property may beimportant in presence of subsystems with singularity such as black holes, in which part of the parameterspace is inaccessible. From now on, we call a 2D surface that its diffeomorphism represents SU p8q a diffeo-surface .Homomorphism between SU p8q and Diff p S q makes it possible to expand ˆ L a ’s with respect tospherical harmonic functions depending on angular coordinates p θ , φ q on a sphere. Moreover, owing tothe Cartan decomposition, SU p8q generators can be described as tensor product of Pauli matrices [29,120]. In this case, indices in (1) consist of a pair p l , m q | l “ ¨ ¨ ¨ , ; ´ l ď m ď ` l . Appendix Ereviews decomposition and indexing of SU p8q generators. We continue to use single letters for indices ofgenerators when there is no need for their explicit description.The algebra (1) is not enough to make the system quantic and as usual ˆ L a ’s must respect Heisenbergcommutation relations: r ˆ L a , ˆ J b s “ ´ i δ ab ¯ h . (2)where ˆ J a P B r H ˚ U s is the dual of ˆ L a and H ˚ U is the dual Hilbert space of the Universe. As there is a one-to-onecorrespondence between ˆ L ’s and ˆ J ’s, they satisfy the same algebra, represent the same symmetry group,namely SU p8q , and have their own expansion to spherical harmonics. Owing to SU p8q – Diff p S q , vectorsof the Hilbert space are differentiable complex functions of angular coordinates p θ , φ q . Thus, sphericalharmonic functions constitute an orthogonal basis for H U . The Cartan subalgebra of B r H U s – SU p8q isinfinite dimensional. In this work all vector spaces and algebras are defined on complex number field C , unless explicitly mentioned otherwise. Although in (1) we show the dimensional scale ¯ h { M P in the definition of operators and their algebra, for the sake of conveniencein the rest of this work we include it in the operators, except when its explicit presentation is necessary for the discussion.ersion October 14, 2020 submitted to Universe
The quantum Universe defined here is static because there is no background space or time in themodel. Nonetheless, in Sec. 4 we show that continuous degrees of freedom similar to space and time arisenaturally when the Universe is divided to subsystems. The short argument goes as the following:We assume that eigen states of the Hilbert space of the Universe are not abstract objects and physicallyexist. This assumption is supported by the fact that in Standard model of particle physics states thatconstitute a basis for its Hilbert space and for its space of linear transformations are indeed observedparticles (fields). Then, taken into account the assumption that H U is infinite dimensional, we concludethat the Universe must consist of infinite number of particles/subsystems. Although subsystems may havesome common properties, which make them indistinguishable from each others, there are many otherdistinguishable aspects, which discriminate them from each others. This statement is in agreement withthe corollary in Appendix B about divisibility of a quantum Universe, derived from axioms of quantummechanics. Thus, this conclusion is independent of details of the model.Notice that without the assumption about physical existence of eigen states, an infinite dimensionalHilbert space does not necessarily mean that Universe must be infinitely divisible. Hilbert space of manyquantum systems have infinite number of states. But they are not necessarily occur in each instance(copy) of the system. The case of a Universe is different, because by definition there is only one copy of it.Therefore, every eigen state of a complete basis of its Hilbert space must physically exist. Otherwise, it canbe completely discarded,In the next sections we make this argument more rigorous and explain how it can lead to a 3+1dimensional spacetime and internal gauge symmetry of elementary particles. We begin with constructinga Lagrangian for this static model and show that it is trivial.
3. Lagrangian of the Universe
Although the infinite dimensional Universe described in the previous section is static, it has to satisfysome constraints imposed by symmetries that we associated to it. They are analogous to constraintsimposed on systems in thermodynamic equilibrium. Although there is no time variation in such systems,a priori small perturbations occur, for instance by absorption and emission of energy. They must be inbalance with each others, otherwise the system would lose its equilibrium. Therefore, it is useful to definea Lagrangian that quantifies these constraints. In the case of present model the Lagrangian should quantifythe SU p8q symmetry and its representation by H U .Lagrangian of a system must be invariant under transformations of fields by application of membersof its symmetry group. As there is no background spacetime in this model, the most appealing candidateis a Lagrangian similar to Yang-Mills but without a kinetic term. In such situation the only availablequantities are invariants of the symmetry group: L U “ ż d Ω b | g p q | „ ÿ a , b L ˚ a p θ , φ q L b p θ , φ q tr p ˆ L a ˆ L b q ` ÿ a L a tr p ˆ L a ρ p θ , φ qq , d Ω ” d p cos θ q d φ (3)where g is the determinant of 2D metric of the diffeo-surface. If we use description (A5) for ˆ L operators, a “ b “
1. If we use (A12) expression, a , b “ p l , m q , l “ ¨ ¨ ¨ , ; ´ m ď l ď ` m . The latter case explicitlydemonstrates the Cartan decomposition of SU p8q to SU p q factors, described in Appendix E. Notice that p θ , φ q are internal variables [30], reflecting the fact that vectors of the Hilbert space representing SU p8q arefunctions on a 2D Riemann surface. For the same reason, in contrast to usual Lagrangians in QFT, there isno term containing derivatives with respect to these parameters in L U , If we use differential representationof ˆ L lm defined in (A5) and apply it to amplitudes L lm p θ , φ q , the first term in the Lagrangian will depend onderivative of amplitudes, just like in the QFT. However, it is straightforward to see that derivatives withrespect to cos θ and φ will have different amplitudes and thereby the kinetic term will be unconventional ersion October 14, 2020 submitted to Universe and non-covariant, unless we consider amplitudes L lm p θ , φ q as functions of the metric of a deformedsphere. This is the explicit demonstration of SU p8q – Diff p S q invariance of this Lagrangian.Generators ˆ T a , ˆ T b P SU p N q , @ N can be normalized such that tr p T a T b q 9 δ ab , see e.g. [29]. In analogywith field strength in Yang-Mills theories, the function L a p θ , φ q can be interpreted as the amplitude of thecontribution of operator ˆ L a in the dynamics of the Universe. Due to global U p q symmetry of operatorsapplied to a quantum state, L a ’s are in general complex. On the other hand, considering the Cartandecomposition of SU p8q to tensor product of SU p q factors and the fact that σ : “ p σ ˚q t “ σ , we concludethat ˆ L : a “ ˆ L a , Similar to QFT, one can use L U to define a path integral. In absence of time, the path integralpresents the excursion of states in the Hilbert space by successive application of ˆ L a operators. Nonetheless,owing to SU p8q symmetry, variation of states is equivalent to gauge transformation and non-measurable.The analogy of L U with Yang-Mills theory has interesting consequences. For instance, differentialrepresentation of ˆ L lm defined in (A5) can be written as ˆ L lm “ b | g p q | (cid:101) µν pB µ Y lm qB ν . In classical limit onecan consider that ˆ L lm acts on the field amplitude L lm and the first term in the integrand of Lagrangian L U can be arranged such that it become proportional to Ricci scalar R p q . As the geometry of 2D diffeo-surfaceis arbitrary, for each set of L lm the metric g µν can be chosen such that L lm dependent part of the integrandbecomes proportional to Ricci scalar for that metric. Thus, in classical limit the first term is topological .We could arrive to this conclusion inversely. As SU p8q – Diff p S q , in the classical limit the Lagrangianshould be the same as Einstein gravity in a static 2D curve space. Thus, the first term in (3) can be replacedby ş d Ω b | g p q | R p q . Then, the definition of ˆ L lm operators in (A5) and amplitudes L lm can be used towrite R p q with respect to ˆ L lm and relate metric and connection of the 2D surface to amplitudes Ł lm . Weleave detailed demonstration of these relations to a future work. The relation between gauge field term in L U and Riemann curvature in classical limit is crucial for interpretation of this term as gravity when theUniverse is divided to subsystems.Notice that in both representations of SU p8q , namely Cartan decomposition to tensor product of SU p q factors and diffeomorphism of 2D surfaces, angular coordinates θ and φ play the role of parametersthat identify/index the members of the symmetry group. Consequently, their quantization is meaningless.Presuming the physical reality of Hilbert space and operators applied to it, as discussed in the previoussection and in Appendix C, we can interpret L lm as intensity of force mediator particles related to thesymmetry represented by operators ˆ L lm , and ρ in the second term of the Lagrangian L U as density matrixof matter .Although L U is static, we can apply variational principle with respect to amplitudes to obtainfield equations and find equilibrium values of L lm and ρ . However, it is easily seen that solutions ofthese equations are trivial. At equilibrium L lm Ñ ρ lm Ñ
0, see Appendix E.2 for the details. As SU p8q n – SU p8q @ n , this solution has properties of a frame independent vacuum of a many-particleUniverse defined using coherent states [19]. Their similarity implicitly implies that the Universe is divisibleand consists of infinite number of particles /subsystems interacting through mediator particles of SU p8q force, that is the action of ˆ L lm . We investigate this conclusion in more details in the next section.
4. Division to subsystems
There are many ways to see that the quantum vacuum (equilibrium) solution of a Universe with L U Lagrangian (3) is not stable. Of course there are quantum fluctuations. They are nothing else than We remind that ş M d Ω b | g p q | R p q “ πχ p M q , where χ is the Euler characteristic of the compact Riemann 2D surface M .Moreover, Ricci scalar alone does not determine Riemann curvature tensor R µν and only provides one constraint for threeindependent components of the metric tensor.ersion October 14, 2020 submitted to Universe random application of ˆ L lm operators, in other words random scattering of force mediator quanta by matter .They project the Hilbert space to itself. But owing to SU p8q symmetry of Lagrangian, states are globallyequivalent and the Universe maintain its equilibrium. Nonetheless, locally states are different and donot respond to ˆ L lm in the same manner. Here locality means restriction of Lagrangian and projectionsto a subspace of the Hilbert space. As state space is homomorphic to the space of smooth functions onthe sphere f p θ , φ q , restriction of transformations to a subspace is equivalent to a local deformation of thediffeo-surface. Moreover, difference between structure coefficients of SU p8q can be used to define a locality or closeness among operators belonging to B r H U s . These observations are additional evidence to theargument given at the end of Sec. 3 in favour of the divisibility of the quantum Universe introduced in Sec.2 to multi-particle/subsystems.A quantum system divisible to separate and distinguishable subsystems must fulfill 3 conditions [24]:- There must exist sets of operators t A i u Ă B r H s such that @ ¯ a P t A i u and @ ¯ b P t A j u , and i ‰ j , r ¯ a , ¯ b s “ t A i u must be local ;- t A i u ’s must be complementarity, that is b i t A i u – End p B r H sq .The most trivial way of fulfilling these conditions is a reducible representation of symmetries by B r H s . Inthe case of B r H U s – SU p8q , as: SU p8q n – SU p8q @ n (4)the above condition can be easily realized. Moreover, instabilities, quantum correlations, and entanglementmay create local symmetries among groups of states and/or operators. There are many examples of suchgrouping and induced symmetries in many-body systems, see e.g. [34] for a review. A hallmark ofinduced symmetry by quantum correlations is the formation of anyon quasi-particles having non-abeliansymmetry in fractional quantum Hall effect [35]. On the other hand, there is only one state in the infinitedimensional Hilbert space in which all pointer states have the same probability, namely the maximallycoherent state defined in (A2). Even if a many-body system begins in such a maximally symmetric state,quantum fluctuations rapidly change it to a less coherent and more asymmetric one. In addition, dueto (4), irreducible representations of SU p8q are partially entangled [25] and there is high probability ofclustering of subspaces in a randomly selected state.Lets assume that such groupings indeed have occurred in the early Universe and continue to occurat Planck scale. They provide necessary conditions for division of the Universe to parts or particles with SU p8q ˆ G – SU p8q as their symmetry. The local symmetry G is assumed to be a compact Lie group offinite rank and respected by all subsystems. Although different subsystems may have different internalsymmetries, without lack of generality we can assume that G is their tensor product, but some species ofparticles/subsystems are in singlet representation of some of the component groups.As the rank of G is assumed to be finite, complementarity condition dictates that the number ofsubsystems must be infinite to account for the infinite rank of H U . If states are in a finite dimensionalrepresentations of G , at least one the representations must have infinite multiplicity and their Hilbertspace would be infinite dimensional. Thus, despite division of B r H U s , SU p8q remains a symmetry ofsubsystems and t A i u Ă B r H i s – SU p8q , where H i is the Hilbert space of subsystem i . Clustering of statesand subsystems are usually the hallmark of strong interaction and quantum correlation [34]. Therefore, In statistical quantum or classical mechanics distinguishability of particles usually means being able to say, for instance, whetherit was particle 1 or particle 2 which was observed. Here by distinguishability we mean whether a particle/subsystem can beexperimentally detected, i.e. through application of ˆ L lm to a subspace of parameter space and identified in isolation from othersubsystems or the rest of the Universe. 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 B r H U s .ersion October 14, 2020 submitted to Universe interaction of subsystems through internal G symmetry is expected to be stronger than through SU p8q ,thereby the weak gravitation conjecture [36] is satisfied.We could formulate the above Universe in a bottom-up manner too. Consider ensemble of infinitenumber of quantum systems - particles - each having finite symmetry G and coherently mixed with others.Their ensemble generates a Universe with SU p8q ˆ G – SU p8q as symmetry represented by its Hilbertspace. Therefore, top-down or bottom-up approaches to an infinitely divisible Universe give the sameresult. The bottom-up view helps better understand the origin of SU p8q symmetry. It shows that for eachsubsystem it is the presence of other infinite number of subsystems and its own interaction with them thatis seen as a SU p8q symmetry. Division of the Universe to subsystems has several consequences. First of all the global U p q symmetryof H U becomes local, because Hilbert spaces of subsystems H i , @ i , where index i runs over all subsystems,acquire their own phase symmetry. Therefore, we expect that there is at least one unbroken U p q local -gauged - symmetry in nature. It may be identified as U p q symmetry of the Standard Model. From nowon we include this U p q to the internal symmetry of subsystems G . Additionally, the infinite numberof subsystems in the Universe means that each of them has its own representation of SU p8q symmetry.However, these representations are not isolated and are part of the SU p8q symmetry of the whole Universe.This property is similar to finite intervals on a line, which are homomorphic to R p q and at the same timepart of it and have the same algebra. Therefore, the memory of being part of the whole Universe is notwashed out with the division to subsystems. Otherwise, according to the corollary discussed in AppendixB subsystems could be considered as separate and isolated universes.The area of diffeo-surface is irrelevant when only one SU p8q is considered. But, it becomesrelevant and observable when it is compared with its counterparts for other subsystems. More precisely,homomorphism between Hilbert spaces of two subsystems s and s defined as: R ss : H s Ñ H s (5)can be considered as an additional parameter necessary for their identification and indexing. A morequalitative description of how a third continuous parameter emerges from division of Universe tosubsystem is given in the next subsection. There are various ways to see that division of the Universe to subsystems defined in Sec. 2 induces anew continuous parameter. As discussed in the previous subsection, each subsystem represent SU p8q ˆ G .When SU p8q representation of different subsystems are compared, e.g. through a morphism, the radius ofdiffeo-surface becomes relevant, because different radius means different area. This dependence allows toclassify subsystems according to a size scale. More precisely, in the definition of ˆ L lm in (A5), Y lm r l , where r is the distance to center in spherical coordinates when the 2D surface is embedded in R p q . If we factorize r -dependence part of Y lm , the algebra of ˆ L lm defined in (A4) becomes: r ˆ L lm , ˆ L l m s ˇˇˇˇ r “ “ r l ´ l ´ l f l ” m ” lm , l m ˆ L l ” m ” ˇˇˇˇ r “ (6)where all ˆ L lm operators are defined for r “ r can beinterpreted as a coupling which quantifies the strength of correlation between ˆ L lm operators. Moreover, as SU p8q n – SU p8q @ n , ˆ L lm ’s of subsystems are part of ˆ L lm ’s of the full system. Consequently, subsystems ersion October 14, 2020 submitted to Universe
10 of 28 are never completely isolated and interact through an algebra similar to (6), but their r factors can bedifferent: r ˆ L p r q lm , ˆ L p r q l m s ˇˇˇˇ r “ “ r “ l r l r ´ l f l ” m ” lm , l m ˆ L r l ” m ” ˇˇˇˇ r “ (7)where r indices on ˆ L lm operators are added to indicates that they may belong to different subsystems.Nonetheless, the algebra remains the same because operators ˆ L lm belong at the same time to theglobal SU p8q . On the other hand, nonlocality of this algebra in point of view of subsystems shouldinduce dependence on derivative with respect to parameters when infinitesimal transformations areconsidered, e.g. in the Lagrangian. Specifically, we expect a relation between r and p r , r q , determined byhomomorphism (5). In the infinitesimal limit, the r.h.s. of (7) becomes Lie derivative of ˆ L lm in the directionof ˆ L l m in the manifold defined by parameters p r , θ , φ , t q , where the last parameter is time with respect toan observer, as described in the next subsection.In summary, after the division of the Universe to subsystems, their SU p8q symmetries are indexed byangular parameters p θ , φ q and an additional continuous parameter r “ p . They share the algebra ofglobal SU p8q , but acquire a new index and in this sense their algebra becomes nonlocal. Notably, in theinfinitesimal limit the algebra can be considered as Lie derivative of ˆ L a P B r H s operators on the manifoldof parameter space p r , theta , φ , t q . Differential properties of the model need more investigation and will bereported elsewhere.Finally, we can define a conjugate set of parameters for the dual space of H U and dual operators ˆ J a defined in (2). Therefore, in contrast to some quantum gravity candidates this model does not have apreference for position or momentum spaces. The last step for construction of a dynamical quantum Universe is the introduction of a clock usingcomparison between variation of states of two subsystems, tagged as system and clock , under application ofoperators ˆ L α P SU p8q ˆ G by a third subsystem, tagged as observer who plays the role of a reference. Thenecessity of an observer/reference is consistent with the foundation of quantum mechanics as describedin [21]. In the context of the present model this discrimination can be understood as the following:Although the global SU p8q symmetry means that any variation of full state by application of ˆ L α is a gaugetransformation, variation of subsystems with respect to each others is meaningful and can be quantified.Technical details of introducing a clock and relative time in quantum mechanics is intensively studiedin the literature, see e.g. [27] for a review and proof of the equivalence of different approaches. Herewe describe this procedure through an example. Consider the application of operators ˆ L c P B r H C s andˆ L s P B r H s s to two subsystems called clock and system , respectively, such that:ˆ L c ρ c ˆ L : c “ ρ c ` d ρ c “ d ρ c , ˆ L s ρ s ˆ L : s “ ρ s ` d ρ s “ d ρ s (8)Because these operations are local and restricted to subsystems they are not gauged out. One wayof associating a c-number quantity to these variations is to define parameter t such that, for instance, dt ” | tr p ρ c ˆ O c q| where ˆ O c can be any observable varying during this operation. This quantity is positiveand by definition incremental. Hamiltonian operator of the system H s P B r H s s according to this clockwould be an operator for which d ρ s { dt “ ´ i { ¯ h r H s , ρ s s .More generally, defining a clock is equivalent to comparing excursion path of two subsystems in theirrespective Hilbert space under successive application of ˆ L c and ˆ L s to them, respectively. The arrow of timearises because through the common SU p8q symmetry any operation - even a local one - is communicatedto the whole Universe. Thus, inverting the arrow of time amounts to performing an inverse operation onall subsystems, which is extremely difficult. Therefore, although dynamical equation of one system may ersion October 14, 2020 submitted to Universe
11 of 28 be locally symmetric with respect to time reversal, due to global effect of every operation, its effect cannotbe easily reversed.
The final stride of time definition brings the dimension of continuous parameter space necessary fordescribing states and dynamics of an infinite dimensional divisible Universe and its symmetries to 3+1,namely p r , θ , φ , t q . Although these parameters arise from different properties of the Universe, namely p θ , φ q from SU p8q symmetry, r from division to infinite number of subsystems, and t from their relativevariation, the global SU p8q symmetry, arbitrariness of the choice of reference frame for parameters, andquantum superposition of states among subsystems mix them. Therefore, geometry of the parameter spaceis R p ` q .The 2D parameter space of the whole Universe is by definition diffeomorphism invariant as it is therepresentation of SU p8q . However, at this stage it is not clear whether the subdivided Universe is rigid,that is invariant only under global frame transformations of the (3+1)D parameter space, or deformableand invariant under diffeomorphism. Here we show that it is indeed diffeomorphism invariant. Moreover,its geometry is determined by states of subsystems .Consider a set of 2D diffeo-surfaces representing SU p8q symmetries of subsystem. They can beobtained from application of ˆ L lm P SU p8q operators to vacuum state of each subsystem, considered tobe a sphere. They are smooth functions of parameters p r , θ , φ q and can be identified with states of thesubsystems, which are also smooth functions of the same parameters. After ordering these surfaces - forinstance, according to their average distances - and defining a projection between neighbours suchthat if on i th surface the point p θ i , φ i , r i q is projected to p θ i ` , φ i ` , r i ` q on p i ` q th surface, the distancein R p q between points in an infinitesimal surface ∆Ω i ă (cid:101) containg p θ i , φ i , r i q and infinitesimal surface ∆Ω i ` ă (cid:101) containg p θ i ` , φ i ` , r i ` q approaches zero if (cid:101) , (cid:101) Ñ
0. The path connecting closest pointson ∆Ω i and ∆Ω i ` defines an orthogonal direction in a deformed S ˆ R p q– R p q and Riemann curvatureof this space can be determined from sectional curvature. Therefore, parameter space (or equivalentlyHilbert space) is curved. Moreover, as the projection between ∆Ω i and ∆Ω i ` used for this demonstrationis arbitrary, we conclude that the parameter space is not rigid and its diffeomorphism does not change thephysics. The same procedure can be applied when a clock is chosen. Therefore, the above conclusionsapplies to the full (3+1)D parameter space of the subdivided Universe.Finally, from homomorphism between diffeo-surfaces and states of subsystems we conclude that(3+1)D classical spacetime can be interpreted as parameter space of the Hilbert space of subsystems of theUniverse, and gravity as the interaction associated to SU p8q symmetry. Evidently, in addition to 3+1 external parameters each subsystem represents the internal symmetry G , which its representationshave their own parameters. Notice that even in classical general relativity diffeomorphism and relation between geometry and state of matter are independentconcepts. In particular, Einstein equation is not the only possible relation and a priori other diffeomorphism invariant relationsbetween geometry and matter are allowed - but constrained by experiments. More generally, any measure of difference between states, such as Fubini-Study metric or fidelity can be used to order states. AsHilbert spaces of quantum systems with SU p8q symmetry consist of continuous functions, we can use usual analytical tools fordefining a distance. But we should not forget that functions are vectors of a Hilbert space. Moreover, Hilbert space vectors are ingeneral complex functions and each projection between diffeo-surfaces corresponds to two projection in the Hilbert space, onefor real part and one for imaginary part of vectors. This projection is isomorphic to a homomorphism between B r H s s of subsystems.ersion October 14, 2020 submitted to Universe
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Up to now we indicated the dimension of spacetime - parameter space of SU p8q symmetry - aftersubdivision of the Universe as 3+1. This implicitly means that we have considered a Lorentzian metricwith negative signature. In special and general relativity the signature of metric is dictated by observationof the constant speed of light in classical vacuum. Indeed, diffeomorphism invariance, Einstein equation,and interpretation of gravity as curvature of spacetime are independent of signature of the spacetimemetric.In quantum mechanics Heisenberg uncertainty relation imposes Mandelstam-Tamm constraint [37]on the minimum time necessary for transition of a quantum state ρ to another perfectly distinguishablestate ρ [23]: ∆ t ě ¯ h ? ´ A p ρ , ρ q b Q p ρ , ˆ H q , A p ρ , ρ q ” tr p? ρ ? ρ q , Q p ρ , ˆ H q ” | tr pr? ρ , ˆ H s q| (9)where ˆ H is the system’s Hamiltonian. Consider ρ as the state of Universe after selecting and separatingan observer and a clock and ρ as an infinitesimal variation of ρ , that is ρ “ ρ ` d ρ . We assume that theclock is chosen such that in (9) minimum time is achieved. Then, (9) becomes: Q p ˆ H , ρ q dt “ tr p a d ρ a d ρ : ” ds (10)This equation is similar to a Riemann metric for a system at rest with respect to the chosen coordinatesframe for the parameter space p r , θ , φ , t q . On the other hand, the r.h.s. of (10) depends only on the variationof state and independent of the chosen frame for parameters. Therefore, ds is similar to an infinitesimalseparation. A coordinate transformation, i.e. p r , θ , φ , t q Ñ p r , θ , φ , t q does not change state of the Universeand is equivalent to a basis transformation in the Hilbert space. On the other hand, the l.h.s. of (10) changesand considering the similarity of (10) to metric equation, we can write it as: g dt ˘ g ii dx i dx i “ ds (11)where we have used Cartesian coordinates in place of spherical. We have chosen parameter transformationsuch that g i “ g i “
0. We assume g ij ą t is ˆ H and Mandelstam-Tamm relation imposes: Q p ˆ H , ρ q dt ” g dt ě tr p a d ρ a d ρ : “ ds (12)For ds ě
0, constraint (12) is satisfied only if the sign of spatial part in (11) and thereby signature of themetric is negative. We remind that Mandelstam-Tamm constraint does not apply to states that do notfulfill distinguishability condition. In these cases ds ă ersion October 14, 2020 submitted to Universe
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Finally, the Lagrangian of the Universe after division to subsystems and selection of reference observerand clock takes the following form: L U s “ ż d x a ´ g „ π G N ¯ h ÿ l , m , l , m tr p L ˚ lm p x q L l m p x q ˆ L lm ˆ L l m q ` ÿ a , b tr p T ˚ a p x q T b p x q ˆ T a ˆ T b q ` p π G N q { ÿ l , m , a tr p L lm p x q T a p x q ˆ L lm b ˆ T a q ` ÿ lm L lm tr p ˆ L lm b G ρ p x qq ` ÿ a L a tr p SU p8q b ˆ L a ρ p x qq . (13)The terms of this Lagrangian can be interpreted as the following. The first term is the Lagrangian forensemble of SU p8q symmetries of all subsystems except observer and clock, and L l m p x q depend on full SU p8q parameter space, that is p r , θ , φ , t q . Due to the nonlocal algebra (7), we expect that L l m p x q ’s includederivative terms. We notice that if ¯ hG N ¯ h { M P Ñ G symmetry and will have the standard form of Yang-Mills models if T a p x q fields are 2-forms in the (3+1)Dparameter space. The last two terms present interaction of matter with gravitational and internal gaugefields, respectively.We leave explicit description of L l m p x q ’s and T a ’s as functionals of spacetime and determinationof its semi-classical limit of this Lagrangian for future works. Nonetheless, the Lagrangian (13) is notcompletely abstract. L lm operators can be expressed as tensor product of Pauli matrices and regroupedby r and t indices, which have no other role than associating a group of matrices to subsystems. This isbecause tensor product of SU p8q is homomorphic to itself. However, such expansion is not very usefuland practical for analytical calculations, in particular for finding semi-classical limit of the model.
5. Comparison with other quantum gravity models
It is useful to compare this model with string theory and Loop Quantum Gravity (LQG) - the twomost popular quantum gravity candidates,A common aspect of string/superstring theory with present model is the presence of a 2D manifoldin their foundation. However, in contrast to string theory in which a 2D world sheet is introduced as anaxiom without any observational support, the presence of a 2D manifold here is a consequence of theinfinite symmetry of the Universe, which has compelling observational support. Moreover, the 2D natureof the underlying Universe manifests itself only when the Universe is considered as a whole. Otherwise, itis always perceived as a (3+1)D continuum (plus parameters space of internal symmetry of subsystems).In string theory matter and spacetime are fields living on the 2D world sheet, or equivalently theworld sheet can be viewed as being embedded in a multi-dimensional space without any explanation forthe origin of such non-trivial structures. On the other hand, in present model the approach to matter israther bottom-up . The Cartan decomposition of SU p8q to smaller groups, in particular SU p q means thatthey can be easily break and separate from the pool of the SU p8q symmetry - for instance by quantumcorrelation between pair of subsystem- without affecting the infinite symmetry. And indeed it seemsto be the case because SU p q and SU p q Ă SU p q ˆ SU p q – SU p q are Standard Model symmetries.Additionally, string theory is fundamentally first quantized and string based field theories are consideredas low energy effective descriptions. But as explained in the previous sections, in the present model owing ersion October 14, 2020 submitted to Universe
14 of 28 to its infinite dimensional symmetry, Hilbert and Fock spaces are homomorphic and the model can bestraightforwardly considered as first or second quantized.Importance of SU p q symmetry in the construction of LQG and its presentation as spin foam [20]is shared with the present model. However, SU p q – SO p q manifold on which Ashtekar variables aredefined has its origin in the ADM (3+1)D formalism, based on the presumption that spacetime and therebyquantum gravity should be formulated in the physical spacetime. Moreover, LQG does not address theorigin of matter as the source of gravity, neither the origin of the Standard Model symmetries. The presentmodel explains both the dimension of spacetime and relation between quantum gravity, matter, and SMsymmetries.A concept that string theory and LQG does not consider - at least not in their foundation - is the factthat in quantum mechanics discrimination between observer and observered is essential and models whichdo not consider this concept in their construction - specially when the models is intended to be applied tothe whole Universe - are somehow metaphysical , because they implicitly consider that the observer is out ofthis Universe.
6. Outline and future perspectives
In this work we proposed a new approach to quantum gravity by constructing a Universe in whichgravity is fundamentally quantic and demonstrated how it may answer some of questions we raised inthe Introduction section about gravity and the nature of spacetime. As we have already summarized themodel and its results in Sec. 1.1, here we concentrate on perspectives for further studies.Understanding nonlocality and differential form of the algebra of subsystem defined by equation(7) is crucial for finding an algebraic expression for the Lagrangian (13), which at present is too abstract.This task is specially important for investigating semi-classical limit of the model. On the other hand,this Lagrangian describes an open system, because state of the observer and probably some of degrees offreedom of the clock are traced out. Formulation of the subdivided Universe as an open system shouldhelp application of the model to black hole physics and cosmology.We discussed a bottom-up procedure for emergence of internal symmetries in Sec. 4. In particular,we concluded that they should generate stronger couplings between particles/subsystems than gravity.However, it does not explain how the hierarchy of couplings arises. We conjecture that clustering ofsubsystems, which leads to emergence of internal symmetries, determines also their couplings, probablythrough processes analogous to the formation of moiré super-lattice and strong correlation betweenelectrons in 2D materials. The fact that in this model both the Universe as a whole and its subsystemshave SU p8q symmetry, which is represented by diffeomorphism of 2D surfaces, means that the necessaryingredients for formation of moiré-like structures are readily available.In absence of experimental quantum gravity tests ability of models to solve theoretical issues hasprominent importance. Among topics that must be addressed black holes and puzzles of information lossin semi-classical approaches have high priority. As the model studied here is inherently quantic, the firsttask is finding a purely quantic definition for black holes. Naively, a quantum black hole may be definedas a many particle system in a quantum well in real space. However, we know that quantum field theoryin curved spacetime background of black holes leads to Hawking radiation and extraction of energy fromblack hole. Consequently, in the realm of quantum mechanics black holes are not really contained in alimited region of space. Their potential well is not perfect and their matter content extends to infinity.Other interesting issues which should be investigated in the context of this model are inflation anddark energy. Notably, it would be interesting to see whether the topological nature of 2D Lagrangian ofthe whole Universe can have observable consequences, for instance as a small but nonzero vacuum energy. ersion October 14, 2020 submitted to Universe
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As for inflation, an exponential decoupling and decoherence of particles/subsystems in the early universemay be interpreted as inflation and extension of spacetime. These possibilities need detail investigation.In conclusion, the inhomogeneous Lorentz transformation may be the classical interface of a muchdeeper and global realm of a quantum Universe.
Funding:
This research received no external funding.
Acknowledgments:
The author thanks Institut Henry Poincaré for hospitality and bibliographic assistance duringaccomplishment of this work.
Conflicts of Interest:
The author declares no conflict of interest.
Appendix A. A very brief summary of the best studied quantum gravity models
Introduction of quantum mechanical concepts to general relativity was first mentioned by Einsteinhimself in his famous 1916 paper. The first detailed work on the topic was by Léon Rosenfeld in 1930 [40],in which the action of Einstein-Hilbert model with matter is quantized by replacing classical variables withhermitian operators, see e.g. [41] for the history of early approaches to quantum gravity. This canonicalapproach and its modern variants based on the quantization of 3+1 dimensional Hamiltonian descriptionof dynamics, notably Wheeler-DeWitt (WD) formalism [42,43] and quantum hydrodynamics [45] lead tononrenormalizable models . See e.g. [46] for review of other issues of these approaches and their currentstatus.Another model, inspired by the ADM Hamiltonian formulation of GR [44], the Dirac Hamiltoniandescription of quantum mechanics [47], and the WD approach to QGR is Loop Quantum Gravity (LQG),see e.g. [48,49] and references therein. In this approach, triads defined on a patch of the 3D space - whatis called Ashtekar variables [16] - replace spatial coordinates and are considered as Hermitian operatorsacting on the Hilbert space of the Universe. Their conjugate operators form a SU p q Yang-Mills theory andprovide a connection - up to an undefined constant called Immirzi parameter - for the quantized 3D space.However, to implement diffeomorphism of general relativity without referring to a fixed background, thephysical quantized entities are holonomies - gauge invariant nonlocal fluxes and Wilson loops definedon 2D surfaces and their boundaries, respectively. Similar to WD formalism in LQG Hamiltonian is aconstraint, and thereby there is no explicit time in the model [48]. Recently it is shown that a conformalversion of LQG has an explicit time parameter [50], but the conformal symmetry must be broken to inducea mass or distance scale in the model. Other issues in LQG are lack of explicit global Lorentz invariance,absence of any direct connection to matter, and most importantly quantization of space, that violatesLorentz invariance even when the absence of time parameter in the model is neglected.Regrading the violation of Lorentz invariance, even if discretization is restricted to distances closeto the Planck scale, matter interaction propagates it to larger distances [10]. This issue is also presentin other background independent approaches to quantum gravity, in which in one way or another thespacetime is discretized. Examples of such models are symplectic quantum geometry [51] and dynamicaltriangulations, in which space is assumed to consist of a dynamical lattice [52,53]. See also [54] for a recentreview of these approaches and [55] for some of their issues, in particular a likely absence of a UV fixedpoint, which is necessary for renormalizabilty of these models. Therefore, the claimed quantization ofspace volume or in other words emergence of a fundamental length scale in UV limit of these models is stilluncertain. Another example is causal sets - a discretization approach with causally ordered structures [56],see e.g. [57] for a review. They probably suffer from the same issue as other discretization models, notably We should emphasize that references given in this appendix are only examples of works on the subjects on which tens or evenhundreds of articles can be found in the literature.ersion October 14, 2020 submitted to
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16 of 28 breaking of Lorentz symmetries, see e.g [58], but also [59] for counter-arguments. We should remind thatall quantum gravity models depend on a length (or equivalent mass) scale, namely the Planck length L P (ormass M P ). Dimensionful quantities need a unit , which does not arise from dimensionless or scale invariantquantities. Therefore, discretization is not a replacement for a dimensionful fundamental constant inquantum gravity models.Another way of quantizing spacetime without discretization is consideration of a noncommutativespacetime [60,61]. This formalism is in fact one of the earliest proposals for a quantum gravity. Morerecently this approach is studied in conjunction with other QGR models such as string theory [62] andmatrix models [63]. An essential issue of this class of models is their inherent nonlocality that leads tomixing of low and high energy scales [64]. On the other hand, this characteristic might be useful forconstraining them, and thereby related QGR models [65].In early 1980’s the discovery of both spin-1 and spin-2 fields in 2D conformal quantum field theoriesembedded in a D -dimensional spacetime - called string models - opened a new era and discipline forseeking a reliable quantum model for gravity, and ultimately unifying all fundamental forces in a GreatUnified Theory(GUT) . Nambu-Goto and Polyakov string theories were studied in 1970’s as candidates fordescribing strong interaction of hadrons. Although with the establishment of Quantum Chromo-Dynamics(QCD) as the true description of strong nuclear force string theories seemed irrelevant, their potential forquantizing spacetime [66,67] gave them a new role in fundamental particle physics. String and superstringtheories became and continue to be by far the most extensively studied candidates of quantum gravityand GUT .Quantized strings/superstring models are finite and meaningful only for special values of spacetimedimension D . For these cases, the central charge of Virasoro algebra or its generalization to affine Liealgebra vanishes when the contribution of all fields, including ghosts of the conformal theory on the 2Dworld-sheet are taken into account. Without this restriction the theory is infested by anomalies, singularities,and misbehaviour. The allowed dimension is D “
26 for bosonic string theories and D “
10 forsuperstrings. The group manifold on which a viable string model can live is restricted as well. For instance,the allowed symmetry in heterotic Polyakov model is SO p q or E ˆ E . Wess-Zumino-Novikov-Witten(WZNW) models with 2D affine Lie algebras provide more variety of symmetries, including coset groups.However, restriction on dimension/rank of symmetry groups remains the same. Therefore, to makecontact with real world, which has 3+1 dimensions, the remaining dimensions must be compactified.Initially the inevitable compactification of fields in string models was welcomed because it mightexplain internal global and local (gauge) symmetries of elementary particles, in a similar manner as inKaluza-Klein unification of gravity and electromagnetism [70,71]. However, intensive investigations of thetopic showed that compactification generates a plethora of possible models. Some of these models maybe considered more realistic than others based on the criteria of having a low energy limit containing theStandard Model symmetries. But, unobserved massless moduli, which may make the Universe overdenseif they acquire a mass at string or even lower scales, strongly constrain many of string models. Therefore,moduli must be stabilized [72,73]. For instance, they should acquire just enough effective mass to makethem a good candidate for dark matter [74]. Moreover, in string theories there is no natural inflationcandidate satisfying cosmological observations without fine-tuning. Although moduli are considered aspotential candidates for inflation [75], small non-Gaussianity of Cosmic Microwave Background (CMB)anisotropies [76] seems to prefer single field inflation [77]. In addition, single field slow roll inflation maybe inconsistent [78] with constraints to be imposed on a scalar field interacting with quantum gravity in A textbook description and references to original works can be found in textbooks such as [68,69].ersion October 14, 2020 submitted to
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17 of 28 the framework of swampland extension of string models landscape [79]. Some researchers still believethat a genuinely non-perturbative formulation of superstring theories may solve many of these issues .However, the absence of any evidence of supersymmetry up to „ TeV energies at LHC - where it wasexpected, such that it could solve Higgs hierarchy problem [80] - is another disappointing result for stringmodels.Observation of accelerating expansion of the Universe due to a mysterious dark energy with propertiesvery similar to a cosmological constant - presumably a nonzero but very small vacuum energy - seems tobe another big obstacle for string theory [81] as the only quantum gravity candidate including both matterand gravity in its construction. The landscape of string vacua has Á ´ minima - depending on howmodels are counted [82]. But there is no rule to determine which one is more likely and why the observeddensity of dark energy - if it is the vacuum energy - is „ fold less than its expected value, namely M P . To tackle and solve some of these issues, extensions and/or reformulations of string theories haveled to their variants such as matrix models [83,84], M-theory, F-theory, and more recently swampland [79]and weak gravity conjecture [36,85], and models constructed based on them.In early 1999 Randall-Sundrum brane models [86,87] and their variants - inspired by D-branes intoroidal compactification of open strings and propagation of graviton closed strings in the bulk of oneor two non-compactified warped extra dimensions - generated a great amount of excitation and weresubject of intensive investigations. By confining all fields except gravitons on 4D branes these modelsare able to lower the fundamental scale of quantum gravity to TeV energies - presumably the scale ofweak interaction - and explain the apparent weakness of gravitational coupling and high value of Planckmass. Thus, a priori brane models solve the problem of coupling hierarchy in Standard Model of particlephysics. In addition, an effective small cosmological constant on the visible brane may be achievable [88,89].However, in general brane models have a modified Friedmann equation, which is strongly constrainedby observations [90–92]. Moreover, it is shown that the confinement of gauge bosons on the brane(s)violates gauge symmetries, and if gauge fields propagate to the bulk, so do the matter [93,94]. Nonetheless,some methods for their localization on the brane are suggested [95,96]. On the other hand, observation ofultra high energy cosmic rays constrains the scale of quantum gravity and characteristic scale of warpedextra-dimension to ą
100 TeV [97,98]. This constraint is consistent with other theoretical and experimentalissues of brane models, specially in the context of black hole physics, that is instability of macroscopic blackholes, nonexistence of an asymptotically Minkowski solution [99,100], and observational constraint [101]on the formation of microscopic black holes in colliders at TeV energies [102].In the view of these difficulties more drastic ideas have emerged. Some authors suggests UV/IRcorrespondence of gravity. They propose that at UV scales graviton quantum condensate behavesasymptotically similar to classical gravity [103,104]. Other proposals attracting some interest include theemergence of classical gravity and spacetime from thermodynamics and entropy [105,106] or condensationof more fundamental fields [107,108].More recently, the development of quantum information theory and its close relation withentanglement of quantum systems, their entropy and the puzzle of information loss in Hawking radiationof black holes have promoted models that interpret gravity and spacetime as an emergent effect ofentanglement [109–111] and tensor networks [112,113]. These ideas are in one way or another related toholography principle and Ads/CFT equivalence conjecture [114]. In these models spacetime metric andgeometry emerge from tensor decomposition of the Hilbert space of the Universe to entangled subspaces.The resulted structures are interpreted as graphs and a symplectic geometry is associated to them. In Non-supersymmetric string models may have no non-perturbative formulation and should be considered as part of asupersymmetric model, see e.g. Chapter 8 of [69].ersion October 14, 2020 submitted to
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18 of 28 the continuum limit the space of graphs can be considered as a quantum spacetime. In a somehowdifferent approach in the same category of models the concept of locality specified by subalgebras isused to decompose the Universe. Local observables belong to spacelike subspaces in a given referenceframe/basis [115,116]. This means that in these models a background spacetime is implicitly postulatedwithout being precise about its origin and nature. In addition to spacetime, subsystems/subalgebras shouldsomehow present matter. But, it is not clear how they are related. Moreover, the problem of the spacetimedimensionality and how it acquires its observed value is not discussed. In any case, investigation of theseapproaches to quantum gravity is still in its infancy and their theoretical and observational consistency arenot fully worked out.
Appendix B. Quantum mechanics postulates in symmetry language
In this appendix we reformulate axioms of quantum mechanics à la Dirac [117] and vonNeumann [118] with symmetry as a foundational concept:i. A quantum system is defined by its symmetries. Its state is a vector belonging to a projective vectorspace called state space representing its symmetry group. Observables are associated to self-adjointoperators. The set of independent observables is isomorphic to subspace of commuting elements ofthe space of self-adjoint (Hermitian) operators acting on the state space and generates the maximalabelian subalgebra of the algebra associated to symmetry group.ii. The state space of a composite system is homomorphic to the direct product of state spaces ofits components . In the special case of separable components, this homomorphism becomes anisomorphism. Components may be separable - untangled - in some symmetries and inseparable -entangled - in others. The symmetry group of the states of composite system is a subgroup of directproduct of components.iii. Evolution of a system is unitary and is ruled by conservation laws imposed by its symmetries andtheir representation by the state space.iv. Decomposition coefficients of a state to eigen vectors of an observable presents thecoherence/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 isbroken. This spontaneous decoherence (symmetry breaking) reduces the state space to the subspacegenerated by other independent observables, which represent remaining symmetries/degeneracies.v. A probability independent of measurement details is associated to eigen values of an observable asthe outcome of a measurement. It presents the amount of coherence/degeneracy of the state beforeits breaking by a projective measurement. Physical processes that determine the probability of eachoutcome are collectively called preparation .These axioms are very similar to their analogues in the standard quantum mechanics, except that we donot assume an abstract Hilbert space. The Born rule and classification of the state space as a Hilbert spacecan be demonstrated using axioms (i) and (v), and properties of statistical distributions [21]. We remind Notice that this axiom differentiates between possible states of a composite system, which is the direct product of those ofsubsystems, and what is actually realized, which can be limited to a subspace of the direct product of individual componentsand have reduced symmetry. More precisely rays because state vectors differing by a constant are equivalent. Ref. [21] explains why decoherence should be considered as a spontaneous symmetry breaking similar to a phase transition. Literature on foundation of quantum mechanics consider an intermediate step called transition between preparation andmeasurement. Here we include this step to preparation or measurement operations and do not consider it as a separate physicaloperation.ersion October 14, 2020 submitted to
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19 of 28 that the symmetry represented by the Hilbert space of a quantum system is in addition to the global U p q symmetry of states, which leaves probability of outcomes in a projective measurement unchanged.When system is divided to subsystems that can be approximately considered as non-interacting, eachsubsystem acquire its own local U p q symmetry. Even in presence of interaction between subsystems,a local U p q symmetry can be considered, as long as the interaction does not change the Hilbert spaceof subsystems. We notice that axiom ii slightly diverges from its analogue in the standard quantummechanics. It emphasizes on the fact that the symmetry group represented by a composite system can besmaller than the tensor product of those of its components. In particular, entanglement may reduce thedimension of Hilbert space and thereby the rank of symmetry group that it represents.A corollary of these axioms is that without division of the Universe to system(s) and observer(s) the process of measurement is meaningless. In another word, an indivisible universe is trivial andhomomorphic to an empty set. In standard quantum mechanics the necessity of the division of theUniverse to subsystems arises in the Copenhagen interpretation, which has many issues, see e.g. [119] fora review. In covariant quantum models and ADM canonical quantization of gravity, in which Hamiltonianis always null and naively the Universe seems to be static, relational definitions of time is based on thedivision of the Universe to subsystems, see e.g. [27]. Therefore, we conclude that division to subsystems isfundamental concept and must be explicitly included in the construction of quantum cosmology models. Appendix C. State space symmetry and coherence
The choice of a Hilbert space H to present possible states of a system is usually based on thesymmetries of its classical Lagrangian. Although these symmetries have usually a finite rank - thenumber of simultaneously measurable observables - the Hilbert space presenting them may be infinitedimensional. For example, translation symmetry in a 3D space is homomorphic to U p q ˆ U p q ˆ U p q and has a global SU p q – SO p q symmetry under rotation of coordinates. They can be presented by 6parameters/observables. Thus, the rank of the symmetry is finite. Nonetheless, due to the abelian natureof U p q group, the Hilbert space of position operator H X is infinite dimensional. More generally, thedimension of the Hilbert space depends on the dimension of the representation of the symmetry groupof Lagrangian and its reducibility. The Hilbert space of a multi-particle system can be considered as areducible representation of the symmetry, even if single particles are in an irreducible representation. Inparticular, Fock space of a many-particle system can be presented as an infinite dimensional Hilbert spacerepresenting symmetries of the Lagrangian in a reducible manner. This property is important for theconstruction of the quantum Universe model studied here, because it demonstrates that the infinite size ofthe physical space can be equally interpreted as manifestation of infinite number of particles/subsystemsin a composite Universe.Ensemble of linear operators acting on a Hilbert space B r H s represents SU p N q group where N isthe dimension of the Hilbert space H and can be infinite. As discussed in details in [21] configurationspace of classical (statistical) systems have  N U p q symmetry where each U p q is isomorphic to thecontinuous range of values that an observable may have. Thus, quantization extends the symmetry ofclassical configuration space to B r H s “ SU p N q xU p q – U p N q Ą  N U p q , where here we have alsoconsidered the global U p q symmetry of the Hilbert space.Application of linear operators can be interpreted interaction with another system or more generallywith the rest of the Universe. The change of state can be also considered as Positive Operator ValuedMeasurement (POVM). In particular, a projective measurement and decoherence makes the statecompletely incoherent ˆ ρ inc : ˆ B ˆ ρ c ˆ B : Ñ ˆ ρ inc “ ÿ i ρ i ˆ ρ i , ˆ ρ i ” | i yx i | (A1) ersion October 14, 2020 submitted to Universe
20 of 28 where ˆ B P B r H s , | i y is an eigen basis for the measured observable, and subscript inc means incoherent. Weremind that the space of simultaneously observable operators corresponds to the Cartan subalgebra of B r H s . Coefficients ρ i are probability of occurrence of eigen value of | i y as outcome of the measurement.Because ˆ ρ inc is diagonal, completely incoherent states ˆ ρ inc represent the Cartan subgroup of B r H s . Amaximally coherent state in the above basis is defined as:ˆ ρ maxc ÿ i , j | i yx j | (A2)This is a pure state in which all eigen states have the same occurrence probability in a projectivemeasurement. Notice that due to the projectivity of Hilbert space ˆ ρ maxc is unique and application of anyother member of B r H s reduces its coherence, quantified for instance by fidelity or Fubini-Study metric [22].More generally, action of B r H s members changes coherence of any state which is not completely incoherent.For this reason, we call SU p N q symmetry of B r H s the coherence symmetry .It is useful to remind that in particle physics generators of B r H s space physically exist and are notabstract operation of an apparatus controlled by an experimenter. In the Standard Model (SM) B r H s isgenerated by vector boson gauge fields in fundamental representation of SM symmetry group. They acton the Hilbert space generated by matter fields. If gravity, which is the only known universal interactionfollows the same rule, we should be able to define a Hilbert space for matter on which linear operatorsrepresenting gravity act.Regarding the example of translational and rotational symmetries of the physical space mentionedearlier, despite the fact that the dimension of the Cartan subalgebra of B r H X s – SU p N Ñ 8q is infinite,and a priori there must be infinite simultaneously observable quantities in the physical space, in quantummechanics only one vector observable is associated to B r H X s , namely the position of a particle/system.QFTs define field operators at every point of the space and assume that at equal time operators at differentpositions commute (or in the case of fermions anti-commute). However, in the formulation of QFT modelsposition is a parameter not an operator. These different interpretations of spacetime highlight the ambiguityof its nature in quantum contexts - as described in question 2 in the Introduction section. Appendix D. SU p8q and its polynomial representation Special unitary group SU p N q can be considered as N -dimensional representation of SU p q . For thisreason generators T p N q lm of the associated Lie algebra su p N q can be expanded as a matrix polynomialof N -dimensional generators of SU p q . Indices p l , m q in these generators are the same as in SU p q representations: l “ ¨ ¨ ¨ , N ´ m “ ´ l . . . , ` l . Lie bracket of generators T p N q lm is defined as: r ˆ T p N q lm , ˆ T p N q l m s “ f p N q l ” m ” lm , l m ˆ T p N q l ” m ” (A3)Structure coefficients f p N q l ” m ” lm , l m of su p N q can be written with respect to 3j and 6j symbols, see e.g. [29] fortheir explicit expression. For N Ñ 8 , after rescaling these generators ˆ T p N q lm Ñ p N { i q { ˆ T p N q lm , they satisfythe following Lie brackets: r ˆ L lm , ˆ L l m s “ f l ” m ” lm , l m ˆ L l ” m ” (A4) In some quantum information literature coherence symmetry is called asymmetry [23]. In this work we call it coherence symmetry orsimply coherence to remind that its origin is quantum degeneracy and indistinguishability/symmetry of states before a projectiveobservation.ersion October 14, 2020 submitted to
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21 of 28 where ˆ L lm ” ˆ T p N Ñ8q lm and coefficients f l ” m ” lm , l m are N Ñ 8 limit of f p N q l ” m ” lm , l m . In addition, it is shown [29]that ˆ L lm can be expanded with to spherical harmonic functions Y lm p θ , φ q defined on a sphere, i.e. themanifold associated to SU p q :ˆ L lm “ B Y lm B cos θ BB φ ´ B Y lm B φ BB cos θ (A5)ˆ L lm Y l m “ ´t Y lm , Y l m u “ ´ f l ” m ” lm , l m Y l ” m ” (A6) t f , g u ” B f B cos θ B g B φ ´ B f B φ B g B cos θ , @ f , g defined on the sphere (A7)where θ “ r π s and φ “ r
0, 2 π s are angular coordinates and t f , g u is the Poisson bracket of continuousfunctions f and g on the sphere. We notice that although generators ˆ L lm are linear combination of B{Bp cos θ q and B{B φ , the latter operators cannot be considered as generators of SU p N Ñ 8q . because they commutewith each other and generate only the abelian subspace of SU p8q group.Using (A5) to (A7), coefficients f l ” m ” lm , l m can be determined: f l ” m ” lm , l m “ p l ” ` q π ż d Ω Y ˚ l ” m ” t Y lm , Y l m u , Y ˚ lm “ Y l , ´ m d Ω ” d p cos θ q d φ (A8)Here we normalize Y lm such that: ż d Ω Y ˚ l m Y lm “ π p l ` q δ ll δ mm (A9)Although ˆ L lm is defined in discrete p l , m q space - analogous to a discrete Fourier mode - we can use inverseexpansion to define operators which depend only on continuous angular coordinates:ˆ L p θ , φ q ” ÿ l , m Y ˚ lm ˆ L lm (A10)As t ˆ L p θ , φ qu are linear in ˆ L lm and contain all these generators, they are also generators of SU p N Ñ 8q –
Diff p S q and coefficients in their Lie bracket can be expressed with respect to θ and φ : f pp θ , φ q , p θ , φ q ; p θ ”, φ ” qq “ ÿ lm , l m , l ” m ” Y ˚ lm p θ , φ q Y ˚ l m p θ , φ q Y l ” m ” p θ ”, φ q f l ” m ” lm , l m (A11)Coefficients f are anti-symmetric with respect to the first two sets of parameters and can be considered asa 2-form on the sphere. Lie algebra of ˆ L p θ , φ q operators can be written as: r ˆ L p θ , φ q , ˆ L p θ , φ qs “ ż d Ω f pp θ , φ q , p θ , φ q ; p θ , φ qq ˆ L p θ , φ q (A12)Operators ˆ L p θ , φ q can be considered as continuous limit of ˆ L lm ’s and both set of generators are vectors andlive on the tangent space of the sphere. Appendix E. Cartan decomposition of SU p8q Representations of su p N q algebra can be decomposed to direct sum of smaller su algebras, seee.g. [120] and references therein. In the case of SU p8q the fact that its algebra is homomorphic to Poissonbrackets of spherical harmonic functions, which in turn correspond to representations of SU p q – SO p q ,means that su p8q algebra should be expandable as direct sum of representations of SU p q , see e.g. [29,30] ersion October 14, 2020 submitted to Universe
22 of 28 for the proof. Thus, up to a normalization factor depending only on l , generators of su p8q algebra ˆ L lm canbe expanded as: ˆ L lm “ R ÿ i α “
1, 2, 3, α “ ¨¨¨ , l a p m q i , ¨¨¨ i l σ i ¨ ¨ ¨ σ i l (A13)where σ i α ’s are N Ñ 8 representation of Pauli matrices [29]. Coefficients a p m q are determined fromexpansion of spherical harmonic functions with respect to spherical description of Cartesian coordinates,see [29] for details. This explicit description shows that up to a constant factor ˆ L lm operators can beconsidered as tensor product of 2 ˆ SU p8q – SU p q b S p q b . . .. This relation canbe understood from properties of SU p N q group. Specifically, SU p N q Ě SU p N ´ K q b SU p K q . For N Ñ 8 and finite K , SU p N ´ K Ñ 8q – SU p8q . Therefore SU p8q is homomorphic to infinite tensor productof SU groups of finite rank, in particular SU p q - the smallest non-abelian SU group. This shows that SU p q group, which has a key role in some quantum gravity models, notably in LQG, simply presents amathematical description rather than a fundamental physical entity. The description of SU p8q as tensorproduct of SU p q is comparable with Fourier transform, which presents the simplest decomposition toorthogonal functions, but can be replaced by another orthogonal function. It is only the application thatdetermines which one is more suitable. Appendix E.1. Eigen functions of ˆ L p θ , φ q and ˆ L lm We define eigen functions of ˆ L p θ , φ q and ˆ L lm operators as the followings:ˆ L p θ , φ q η p θ , φ q “ N η p θ , φ q (A14)ˆ L lm ζ lm “ N ζ lm (A15)where N and N are constants . N may depend on p l , m q . Using definition of ˆ L p θ , φ q and ˆ L lm andproperties of spherical harmonic functions, solutions of equations (A14) and (A15) are: $&% η p θ , φ q “ iN ř lm p l ` m q ! mA l p l ´ m q ! r F lm p cos θ q ´ F lm p cos θ p t qqs ` η p θ p t qq φ ` H p cos θ q “ ´r H p cos θ p t qq ´ φ p t qs (A16) A l ” c π l ` F lm ” ż d p cos θ q| P lm p cos θ q| ´ , (A17) H p cos θ q ” ż d p cos θ q ř lm A l p l ´ m q ! p l ` m q ! B| P lm cos θ | B cos θ ř l m im A l p l ´ m q ! p l ` m q ! | P l m p cos θ q| (A18)where t parameterizes tangent surface at initial point p θ , φ q . Elimination of this parameter from twoequations in (A16) determines η p θ , φ q for a set of initial conditions. Because the second equation does notdepend on N , without loss of generality we can scale initial value η p θ q Ñ iN η p θ q . With this choice theeigen value N can be factorized, and because Hilbert space is projective, N can be considered as an overall A priori N and N can depend on p θ , φ q . However, their dependence on angular parameters can be included in η . Therefore, onlyconstant eigen values matter.ersion October 14, 2020 submitted to Universe
23 of 28 normalization factor and irrelevant for physics. Therefore, each set of parameters p θ , φ q present a uniquepointer state for the Hilbert space.In the same way we can calculate eigen functions of ˆ L lm as a parametric function: $’&’% ζ lm p θ q “ ´ N e m W lm p θ q c p l ` m q ! p l ´ m q ! r Z lm p θ q ´ Z lm p θ p t qqs ` ζ lm p θ p t qq φ ´ imW lm p θ q “ φ p t q ´ imW lm p θ p t qq (A19) W lm ” ż d p cos θ q p ´ cos θ q P lm p cos θ qp l ´ m ` q P p l ` q m p cos θ q ´ p l ` q P lm p cos θ q (A20) Z lm ” ż d p cos θ q e ´ m W lm p cos θ q p l ´ m ` q P p l ` q m p cos θ q ´ p l ` q P lm p cos θ q (A21)Similar to η p θ , φ q , redefinition of initial value Z lm p θ p t qq Ñ Z lm p θ p t qq N e m W lm p θ q leads to a unique eigenfunction for ˆ L lm .Considering diffeomorphism invariance of the model, it is always possible to redefine coordinatessuch that θ “ const . and φ “ const . constitute a basis and any state can be written as: | ψ y “ ż d Ω ψ p θ , φ q| θ , φ y (A22)Thus, as explained in the main text, vectors of the Hilbert space representing SU p8q are complex functionson 2D surfaces. As SU p8q – Diff p S q , the range of p θ , φ q is θ “ r π s and φ “ r π q . However, SU p8q may be represented by diffeo-surfaces of higher genus. In this case | θ ` n π , φ ` n π y for any integer n and n may present different states. States can be also expanded with respect to | l , m y [29]. Appendix E.2. Dynamics equations of the Universe before its division to subsystems
The equilibrium solution for Lagrangian L U in (3) can be determined by variation with respect to L lm and components of the state ρ in an orthogonal basis of the Hilbert space. In absence of environment for thewhole Universe, ρ is pure and can be written as ρ “ | ψ yx ψ | , where ψ y is an arbitrary vector in the Hilbertspace H U . As discussed in Sec. 2, vectors of H U correspond to complex functions of angular coordinates p θ , φ q of the diffeo-surface. They can be expanded with respect to spherical harmonic functions. Herewe follow the usual bracket notation of quantum mechanics and call states of this orthogonal basis | l , m y ,where l P Z { ´ l ď m ď ` l . In this basis | ψ y “ ř l , m ψ lm | l , m y and ρ “ ř l , m , l , m ψ l , m ψ ˚ l , m | l , m yx l , m | .After this decomposition dynamics equations are expressed as: B L U B ψ lm “ ÿ l , m , l ”, m ” L l ” m ” ψ ˚ l m x l m | ˆ L l ” m ” | l , m y (A23) B L U B L lm “ ÿ l , m , l ”, m ” ψ ˚ l ” m ” ψ l ” m ” x l , m | ˆ L lm | l ” m ” y ` L lm tr p ˆ L lm ˆ L lm q (A24)Because ˆ L lm is a generator of SU p8q , the last term in (A24) is a constant depending only on l andnormalization of generators. Thus, we define C l ” tr p ˆ L lm ˆ L lm q . Using description of ˆ L lm in (A13) to tensorproduct of Pauli matrices, we conclude that ˆ L lm | l , m y ‰ l ě l and consists of linear combination ersion October 14, 2020 submitted to Universe
24 of 28 of | l , m y states. On the other hand x l , m | l , m y “ δ ll δ mm . Thus, x l , m | ˆ L lm | l ” m ” y is nonzero only forterms with equal l indices and we can solve (A24) for L lm as the following: L lm “ ´ C l ÿ | m | , | m |ď l , m ` m ` m “ ψ ˚ lm ψ lm x l , m | ˆ L lm | lm ” y (A25)By applying this solution to (A23) and using properties of ˆ L lm and | l , m y we find: ÿ m , m ψ ˚ l , ´p m ` m q ψ ˚ l , ´p m ` m q ψ lm x l , ´p m ` m q| ˆ L lm | l , m yx l , ´p m ` m q| ˆ L lm | l , m y “ | m | , | m | , | m ` m | , | m ` m | , ď l , @ l P Z { | l , m y states, this equation is satisfied only if ψ lm “ | m | ď l , @ l . Thus, equilibrium solution of the Lagrangian L U is a trivial vacuum. References
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