Bouncing Unitary Cosmology I. Mini-Superspace General Solution

Sean Gryb1, 2, ∗ and Karim P. Y. Thébault1, † 1Department of Philosophy, University of Bristol 2H. H. Wills Physics Laboratory, University of Bristol (Dated: January 17, 2018) We offer a new proposal for cosmic singularity resolution based upon a quantum cosmology with a unitary bounce. This proposal is illustrated via a novel quantization of a mini-superspace model in which there can be superpositions of the cosmological constant. This possibility leads to a finite, bouncing unitary cosmology. Whereas the usual Wheeler–DeWitt cosmology generically displays pathological behaviour in terms of non-finite expectation values and non-unitary dynamics, the finiteness and unitarity of our model are formally guaranteed. For classically singular models with a massless scalar field and cosmological constant, we show that well-behaved quantum observables can be constructed and generic solutions to the universal Schrödingerequation are singularity-free. Key features of the solutions of our model include a cosmic bounce due to quantum gravitational effects, a well-defined FLRW limit far from the bounce, and a super-inflationary epoch in the intermediate region. Furthermore, our model displays novel features including: i) superpositions of values of the cosmological constant; ii) a non-zero scattering length around the big bounce; and iii) bound ‘Efimov universe’ states for negative cosmological constant. The last feature provides a new platform for quantum simulation of the early universe. A companion paper provides detailed interpretation and analysis of particular cosmological solutions.

PACS numbers: 04.20.Cv Keywords: quantum cosmology, singularity resolution, problem of time

CONTENTS I. INTRODUCTION

I. Introduction1 The ‘big-bang’ cosmic singularity can be characterised classically in terms of both the incompleteness of causal II. Relational Quantization3 (i.e., non-spacelike) past-directed curves and the existence of a curvature pathology [1–4]. That the pathology III. Classical Mini-Superspace Cosmology4 in question is physically problematic is unambiguously A. Configuration Space Geometry4 demonstrated by the existence of scalar curvature invari- B. Classical Singular Behaviour5 ants that grow without bound in finite proper time along C. General Explicit Solutions7 the incomplete curves in question [5,6]. Such patholog- IV. Quantum Mini-Superspace Cosmology9 ical behaviour in observable classical quantities can be A. Algebra of Observables9 understood to signal the breakdown of general relativ- B. Hamiltonian 11 ity and, thus, the requirement for new theoretical tools. 1. ‘Bound States’ (Λ < 0) 12 A reasonable hope is that the cosmic singularity prob- 2. ‘Unbound States’ (Λ > 0) 13 lem might be resolved via the introduction of inflationary 3. Critical Case (Λ = 0) 15 mechanisms [7,8]. In particular, one might expect that C. Bouncing Unitary Cosmology 15 eternal inflation models could resolve the initial singular- D. Efimov Analogue Cosmology 16 ity [9]. However, the Penrose-Hawking singularity theorems can, in fact, be extended to show that a broad range V. Singularity Resolution by Quantum Evolution 16 of ‘physically reasonable’ eternal inflationary spacetimes are necessarily geodesically past incomplete and there- VI. Conclusions 18 fore singular in the relevant sense [10]. There are thus reasons to expect a ‘big-bang’ singularity to exist prior Funding 18 to the inflationary phase of such models. A complete description of the early universe requires us to introduce Acknowledgments 18 new physics that goes beyond inflation. References 18 Most contemporary approaches to resolving the cosmic singularity problem are based upon cosmic bounce scenarios. In such models the classical singularity is replaced by a new pre-big bang t → −∞ epoch. Big bounce cos- ∗ [email protected] mologies have hitherto been proposed based upon stringy † [email protected] effects [11–15], path integral techniques [16, 17], loop ap- 2 proaches [18–21], and group field theory [22–24]. Whilst, the universe [34]. An equation of the same form was in bouncing cosmologies can be combined with inflation – fact derived some time ago in the context of unimodu- giving mixed scenarios – a particular attraction is that lar gravity [35]. That earlier treatment did not include the big bounce can potentially replace inflation as the a detailed analysis of generic or specific cosmological so- mechanism for solving many of the problems of standard lutions, an explicit construction of the observable oper- big bang cosmology [25]. The horizon problem, in par- ators, or an investigation of the fate of the singularity. ticular, is automatically solved in bouncing cosmologies, More problematically, the quantization did not include since the observed isotropy of the cosmic microwave back- the self-adjoint representation of the Hamiltonian nec- ground can be explained simply by causal interactions in essary to guarantee unitarity via appeal to Stone’s the- the pre-big bang epoch. Bouncing cosmologies are thus orem. In our approach, the Hamiltonian will be given attractive as either a supplement to or replacement of the an explicit self-adjoint representation. Furthermore, for inflationary cosmological paradigm. classically singular models with a massless scalar field In this paper we propose an alternative quantization and cosmological constant Λ, well-behaved quantum ob- procedure for cosmology in which superpositions of the servables will be constructed and generic solutions to cosmological constant are allowed. This possibility al- our ‘universal Schrödingerequation’ will be shown to lows for a new model of bouncing cosmology: a bouncing be singularity-free. Key features of the solutions of our unitary cosmology. Our approach is based upon canon- model include a cosmic bounce due to quantum gravi- ical quantization of isotropic and homogeneous mini- tational effects, a well-defined FLRW limit far from the superspace models but differers significantly from the bounce, and a super-inflationary epoch in the intermedi- two standard canonical treatments of such models. The ate region. Our bounce scenario is phenomenologically first, older approach is based upon a Dirac quantization distinct form those studied in the literature – e.g. those of mini-superspace expressed in terms of ADM variables found in LQC [18]. In particular, our model displays [26, 27], and leads to a Wheeler–DeWitt mini-superspace novel features including: i) superpositions of values of quantum cosmology. Simple models, where one consid- the cosmological constant; ii) a non-zero scattering length ers a massless scalar field with zero spatial curvature and around the big bounce; and iii) bound ‘Efimov universe’ non-negative cosmological constant, can be shown ex- states for negative cosmological constant. The last fea- plicitly to contain observable operators with divergent ture provides a new platform for a ‘few-body’ quantum expectation values [28]. In this strong sense, the big simulation [36, 37] of the early universe. bang singularity is not resolved in even the most basic The remainder of the paper is organised as follows. 1 class of mini-superspace quantum cosmologies. The sec- Section §II provides a brief overview of the alternative ond, newer approach is inspired by the techniques used relational quantization procedure upon which our treat- in Loop Quantum Gravity and is known as Loop Quan- ment of quantum mini-superspace is based. Section §III tum Cosmology [18–21]. Loop Quantum Cosmology represents the main formal details of the classical mini- lies upon a ‘polymer’ representation of the observable superspace model. Our initial focus, in Section §IIIA, is algebra of mini-superspace and involves the introduction upon developing a coordinate-free representation of the of a Planck-scale cutoff for the problematic operators of configuration space and its boundary. In Section §IIIB, the Wheeler–DeWitt theory. The same models that ex- we then consider the nature of the classical singularity hibit pathological behaviour under the Wheeler–DeWitt and introduce the ‘tortoise’ coordinate chart that pro- treatment can be shown to feature operators with finite vides a conformal completion of the configuration space. expectation values when treated in Loop Quantum Cos- Finally, in Section §IIIC, we give an explicit analysis of mology [28]. In this precise sense, the big bang singular- the general solutions in order to both isolate their patho- ity is resolved in Loop Quantum Cosmologies for models logical features and offer suggestions as to why one might where the Wheeler–DeWitt treatment fails. expect such features to persist in the quantum treatment. In this paper, we will offer a new proposal for singu- The considerable care taken in setting up the classical larity resolution in mini-superspace quantum cosmology analysis will reap benefits in the quantum theory. In that is based upon evolution. Our method does not make particular, our classical analysis will be instrumental for use of polymer methods. Rather, the observable opera- isolating and resolving the various quantum pathologies tors in our theory evolve unitarily and remain finite be- that are encountered. cause they are ‘protected’ by the uncertainty principle. The unitary evolution of operators in our model is based We start, in Section §IV A, by solving the problem upon a novel approach to the quantization of ADM mini- of defining an algebra of self-adjoint quantum observ- superspace that leads to a Schrödinger-type equation for ables. Then, as detailed in Section §IV B, we construct a self-adjoint ‘Wheeler–DeWitt’ Hamiltonian operator and analyse three families of eigenfunctions that are distin- guished based upon the value of the cosmological con- 1 Although there are good general reasons to expect singularities to persist in Wheeler–DeWitt mini-superspace cosmologies, forms stant. Particularly noteworthy is the Λ < 0 family which of singularity resolution have been shown to hold for models with has a mathematical form that mirrors that of bound Efi- Brown-Kuchaˇrdust fields [29, 30]. See [31–33] and §V for further mov states [38] found in few-body quantum physics. Pro- discussion. ceeding beyond the standard Dirac analysis, in Section 3

§IV C, we then apply the procedure of relational quan- (the final of these will be outlined shortly). All three tization leading to the general solution for the quantum quantizations rely on a basic observation that the inte- cosmological model. An interpretation of the general so- gral curves of the vector field generated by the Hamil- lution in terms of an experimentally realisable analogue tonian constraint in globally reparametrization invariant model is then provided in Section §IV D. theories should not be understood as representing equiv- Finally, in Section §V, we provide analysis and dis- alence classes of physically indistinguishable states since cussion of the meaning of cosmic singularity resolution. the standard Dirac analysis does not apply to these mod- A companion paper [39] provides detailed interpretation els [53–55]. On our view, successive points along a partic- and analysis of particular cosmological solutions. ular integral curve should be taken to represent physically distinct moments in time [34, 51, 52] and the vanishing of Hamiltonian does not mean that time evolution must II. RELATIONAL QUANTIZATION be, rather paradoxically, classified as the ‘unfolding of a gauge transformation’ [56]. Our approach is based upon an alternative prescrip- Relational quantization is based upon the exploitation tion for the quantization of globally reparametrization of an ambiguity in theories that are independent of global invariant models. Whereas the standard Dirac quanti- time parametrization. This ambiguity relates to the the zation approach leads to to a ‘frozen’ Wheeler–DeWitt- physical interpretation of the variable, −Π, that is canon- type formalism with the accompanying problem of time ically conjugate to the arbitrary parameter, t, associated [40–42], our ‘relational quantization’ approach leads to a with the Hamilton flow of the Hamiltonian vector field dynamical quantum formalism with unitary evolution of on phase space, XH . For all globally reparametrization the universal wavefunction. Key to this formalism is a invariant theories, the Hamiltonian is a constraint, and relational interpretation of the observables where, in the this implies that relevant conjugate variable is a constant. Heisenberg picture, observables operators can evolve ac- However, one is free to interpret this constant as either cording to an unobservable time label whose role in the an integral of motion – the value of which is determined formalism is to distinguish successive states of the uni- by initial conditions – or as a constant of Nature that verse. Our approach thus differs from the standard inter- takes a certain value fundamentally. nal time-type approaches (variously described as ‘evolv- In the context of an Hamilton-Jacobi analysis, this ‘in- ing constants of the motion’ or ‘complete observables’) to terpretational ambiguity’ leads to a ‘formal ambiguity’ in systems where the classical Hamiltonian is constrained how we treat the variation of Hamilton’s principle func- to be zero [43–50]. In all such approaches, the observ- tion, S, with the arbitrary parameter, t. The constant- ables evolve according to a (non-unique) time-dependent of-motion interpretation leads us to use the standard Hamiltonian on the physical Hilbert space as dictated Hamilton–Jacobi equation: by the choice of an internal clock parameter. Such evo- ∂S ∂S lution cannot be guaranteed to be unitary and, in fact, H q, = . (1) non-unitary quantum dynamics obtains for even simple ∂q ∂t implementations of the internal-time scheme [47]. In the The time independence of H, that is crucial for relational relational quantization approach detailed below, evolu- quantization, allows for the simple separation Ansatz: tion is guaranteed to be unitary and the physical Hilbert space can always be unambiguously defined. These two S(q, t) = Πt + W (q) , (2) formal advantages of our approach lead to the principal 2 result of this paper. Our proposal allows for a well- which leads to the reduced equation: defined quantum formalism that simultaneously meets various conditions for quantum singularity avoidance as ∂W H q, = Π . (3) defined in the literature. Most significantly, from a phys- ∂q ical perspective, we find that the expectation values of all observables in our theory are guaranteed to remain un- We can then solve the reduced equation and use (2) to problematic even when their classical counterparts break derive expressions for the flow of all phase space variables down. along XH as parametrised by t. Since the flow parameter In previous work, relational quantization has been is monotonically increasing, it acts as a book keeping motivated, in general, via independent analyses of the device to encode the ordering of the relative values of the Faddeev–Poppov path integral [51], constrained Hamil- physical variables. tonian methods [52] and the Hamilton–Jacobi [34] for- Under the alternative interpretation, we treat Π as a malism of globally reparametrization invariant theories constant of nature. This implies that S is time independent and, thus, that the reduced equation (3) is all we have to describe the physics of the system. In practice, this can be done by applying something like the partial 2 We defer detailed discussion of the basis for these advantages to and complete observable program [46, 48, 49] to describe §V. a set of relational observables for the theory defined by 4

(3). In this approach, the relative values of the physical gravitational degrees of freedom in terms of conformally variables are only ordered relative to an internal clock invariant three-geometries and then describes their evo- that need not be monotonically increasing. lution in terms of a global time. Understanding our mini- Classically, these two treatments are physically in- superspace model as an approximation to quantum shape distinguishable. However, they motivate very different dynamics rather than unimodular gravity lends plausibil- prescriptions for quantization and lead to quantum for- ity to the idea of a Schrödingerequation for the universe. malisms between which one could in principle empirically differentiate. If Π is understood as a constant of Nature, one follows the standard Dirac quantization route and III. CLASSICAL MINI-SUPERSPACE promotes the reduced equation to a Wheeler–DeWitt-like COSMOLOGY equation: h i A. Configuration Space Geometry Hˆ − Π Ψ = 0 . (4) The model we will consider is an homogeneous and If Π is understood as a integral of motion, it is natural isotropic FLRW universe with zero spatial curvature (k = to follow the relational quantization leading from (3) to 0) described by the scale factor, a, coupled to a massless a Schrödinger-type unitary evolution equation: free scalar field, φ. In terms of these variables the space- ∂Ψ time metric takes the form: Hˆ Ψ = i~ . (5) ∂t ds2 = −N(t)2dt2 + a(t)2 dx2 + dy2 + dz2 (6) In the quantum theory, the wavefunctions satisfying (4) and (5) are different rays in Hilbert space. More- with ∂iφ = 0. The symmetry reduced action becomes: over, the algebras of physical observables are not unitar- Z " 2 # ily equivalent and, in fact, will not even have the same 1 3 6 a˙ S = SEH + Sφ = − dt V0a + 2NΛ dimension. We can see this most clearly in terms of the 2κ R N a partial and complete observables program where there Z V a3 will not be an operator in the physical Hilbert space cor- + dt 0 φ˙2 , (7) responding to whichever classical variable is chosen as R 2N the internal clock. Contrastingly, in the relational quantization of theories with a global time parameter, the full where Λ is the cosmological constant, κ = 8πG and classical phase space is mapped to the quantum observ- Z 3 √ able algebra.3 V0 = d x g . (8) The purpose of the present paper is to implemented Σ0 relational quantization explicitly in the context of a cos- Here, gab is the intrinsic metric of a space-like Cauchy mological model. Our approach will rely on the rein- surface Σ and Σ0 ⊂ Σ is a ‘large’ space-like fuducial cell terpretation of the role of the cosmological constant, Λ. used to give meaning to relative spatial volumes. All Whereas, in most standard treatments, Λ is understood quantities are scalars on R, which labels successive in- as a constant of nature, in our approach it is reinter- stants of time. The Hamiltonian corresponding to (7) preted as a constant of motion in accordance with the is: role played −Π in the above discussion. This leads to a 3 quantum cosmological formalism where the wavefunction κ 2 1 2 V0a H = N − πa + 3 πφ + Λ , (9) of the universe can be in superpositions of eigenstates of 12V0a 2V0a κ Λ. Our approach is, thus, naturally connected to the unimodular approach to gravity [57]. As noted above, a while the symplectic form is such that πa and πφ are cosmological model of the same form as that studied here canonically conjugate to a and φ. The Lagrange multi- was derived in the context of that program [58]. Whilst plier N enforces the on-shell vanishing of the Hamilto- the unimodular approach to gravity was vulnerable to nian. The Hamiltonian induces a vector field on phase the objection that the the unimodular condition appears space Γ(a, πa; φ, πφ) via the canonical symplectic form, incompatible with foliation invariance [59], there is the and the integral curves of this vector field represent the potential for the application of our proposal to the quan- unparametrized classical solutions of the theory. As tization of gravity when understood in terms of the shape noted in §II, we believe that the integral curves in ques- dynamics formalism [60–62]. This is due to the fact that, tion should not be understood as representing equiva- within shape dynamics, foliation invariance is not man- lence classes of physically indistinguishable states accord- ifestly broken. Rather, shape dynamics re-encodes the ing to the standard Dirac analysis. To understand the general features of the classical solutions, and the particularities of the quantum theory, it will prove useful to develop a coordinate-free representa- 3 For a more detailed discussion of observables in relational quan- tion of this model in terms of the geometry of configu- tization and the comparison to Dirac quantization, see [34, 52]. ration space, C. Towards this end, we first express our 5 configuration variables in terms of relative spatial volume Geometrically, C is not the entire Minkowski plane be- (in the slicing selected by homogeneity and isotropy), v, cause v is restricted to R+. Rather, C can be identified and the dimensionless value of the scalar field, ϕ: with the upper Rindler wedge, which is defined to be such that the proper distance between the origin and arbitrary r2 r3κ points in the wedge is positive. This space has a bound- v = a3 ϕ = φ . (10) 3 2 ary, ∂C, on the light-cone centred on the origin where v = 0, and this boundary leads to the most physically im- Note that these definitions in terms of dimensionless portant properties, both classical and quantum, of this quantities imply that the dimension of all canonical mo- cosmological model. The geometry of Rindler space is menta will be that of angular momenta. This can be ab- well-known (see, for example, [63] for a review). We will sorbed into a single angular momentum scale, ~, which at now review some key features that will be useful in our the level of the classical theory merely serves as a book- construction. keeping device. If we conveniently identify this reference The boundary of Rindler can be identified in a scale with the Planck scale, we can use it to identify the coordinate-free way by using the global Killing vector regime where the classical formalism is expected to break fields of ηAB. Because ηAB is the flat Minkowski met- down. ric in 2D, its Killing vectors form an ISO(1,1) algebra Using ~, it is convenient to define the dimensionless that can be parametrized in terms of two translation lapse, N˜, and cosmological constant, Λ˜ as generators and one boost generator. The boundary is the union of the one dimensional surfaces spanned by r 2 2 ˜ 3 κ~ vN ˜ V0 the two translation generators where the boost genera- N = Λ = 2 2 Λ , (11) 2 V0 κ ~ tor becomes asymptotically null. Inspired by the form of the metric in (v, ϕ) coordinates, we can identify the so that the Hamiltonian, in this coordinate chart, be- boost parameter with ϕ. Our definition of ∂C is then comes clearly only compatible with the ISO(1,1) algebra when " ! # ϕ → ±∞, which is consistent with the claim above that π2 ˜ 1 2 ϕ ˜ ∂C is the null cone centered at the origin. The boost gen- H = N −πv + + Λ , (12) 2~2 v2 erator, which is hypersurface orthogonal to surfaces of constant ϕ, and the unique vector field orthogonal to it, q q 1 −2 2 which is hypersurface orthogonal to surfaces of constant where πv = a πa and πϕ = πφ are the con- 6 3κ v, provide a geometrically privileged set of coordinates jugate momenta to v and ϕ. This representation of – up to reparametrization and orientation – consisting the Hamiltonian generates evolution parametrized by the q of ϕ and v. Note that, while the integral curves of the ˜ 2 V0 rescaled time t = 2 t, which is proportional to stan- translation generators end on the part of the boundary 3 κ~ v dard cosmological time corresponding to N = 1. that is generated by the orthogonal translation, the in- After making these identifications, we find that the tegral curves of the boost generators can be continued natural metric on C relevant to the dynamical problem indefinitely. The boost, therefore, represents the only in consideration is the flat Minkowski metric in 2d: symmetry of C valid everywhere in the space, and will be useful for conveniently splitting the solution space in the 2 A B dsC = ηABq q , (13) quantum theory. where (A, B) = 1 and 2, the qA represent some arbitrary coordinate chart on C, and dsC is the dynamically rele- B. Classical Singular Behaviour vant proper distance element on C. That this is indeed the appropriate metric can be seen from the fact that the action (7) takes the form The existence of classical singularities can be understood in terms of the incompleteness of causal curves Z and the existence of some type of curvature pathology 1 A B ˜ ˜ S = dt ηABq˙ q˙ + NΛ , (14) [1–6, 64]. The physical significance of such features can 2N˜ be made apparent by considering the phenomenology of where qA = (v, ϕ) and the Minkowski metric is ex- a test particle traveling into a singular region. While the pressed in a hyperbolic polar coordinate chart where incompleteness of a causal curve corresponds to the test 2 2 particle traversing the complete extension of the curve in ηAB = ~ diag(−1, v ). In terms of this metric, the Hamiltonian constraint takes the form finite proper time, a curvature pathology corresponds to the particle experiencing unbounded tidal forces. Generic 1 statements about the precise character of spacetime sin- H = N˜ ηABp p + Λ˜ , (15) 2 A B gularities are difficult to come by, not least since curvature pathologies can be shown to be neither necessary nor which states that the generalized momenta, pA, are fixed- sufficient for curve incompleteness [64]. For our purposes, norm vectors on C. it will be sufficient to follow the logic of the Penrose- 6

- Hawking singularity theorems [1–3] and consider the be- ξ 4 haviour of the expansion parameter of a congruence of geodesics. That the expansion parameter becomes unbounded and negative somewhere along the congruence in a finite proper time is taken as an indication that 3 the relevant spacetime contains incomplete causal curves. Additionally, we will take curvature pathologies to be sig- 2 A nalled by a divergence in any Kretschmann invariant. A Λ>0 spacetime region containing both such features is unques- tionably a singular one. In §IIIC we will demonstrate Λ<0 explicitly that our model is singular in both the curve 1 incompleteness and curvature pathology sense.4 Classically, the action (14) enforces a geodesic princi- 5 ple on C in terms of the metric η. The classical solutions 0 ξ+ are geodesics of 2d Minkowski space: straight lines. By inspecting the Hamiltonian constraint and momentarily undoing the Legendre transformation, it is straightfor- 0 1 2 3 4 ward to see that Λ controls the sign (and magnitude) of infinitesimal proper-distance along classical curves in C. FIG. 1: Two examples of typical classical solutions with Thus, ‘time-like’ geodesics correspond to Λ > 0, ‘null’ positive (in blue) and negative (in red) cosmological geodesics correspond to Λ = 0, and ‘space-like’ geodesics constant as expressed in light-cone coordinates correspond to Λ < 0. ξ± = x0 ± x1, where xA are standard Cartesian The presence of the boundary ∂C implies that C coordinates in the upper Rindler wedge. Lines of is geodesically incomplete – unlike the full Minkowski constant a (the hyperbolas) and φ (the straight lines) plane. This can be deduced from the fact that, as men- are drawn in dashed lines. The solutions live in the tioned at the end of §IIIA, the translational Killing vec- restricted domain indicated by the shaded region. It is tors are orthogonal to the boundary – but it is perhaps clear that all solutions will cross the singular boundary, most easily seen by considering the space of solutions in ∂C, at least once. The units are arbitrary. a simple coordinate chart, say standard light-cone coordinates on R2+. In this chart, C is the upper-right quadrant and the geodesics are straight lines where the where, in the (v, ϕ) chart, Ω = 1/v. If we define the sign of the slope is proportional to the sign of the cosmo- tortoise coordinate µ such that logical constant. As illustrated in Figure1, any geodesic passing through an arbitrary point A on C that is not on µ = log v , (17) the boundary will cross the boundary at least once. This happens exactly once for non-negative Λ and twice for which respects the split suggested by the boost symmetry negative Λ. Because the metric is finite along the entire of C, then it is easy to see that, in the (µ, ϕ) coordinates, curve in this chart, the proper-distance will also be finite. η0 = diag(−1, 1), which is the Minkowski metric on This means that all geodesics passing through A will ter- AB the full R(1,1) plane. This conformal completion and the minate on the boundary in finite proper distance, thus associated tortoise coordinate will be essential tools in proving geodesic incompleteness (of C not spacetime). defining the eigenfunctions of the momentum operator While the set (C, η) itself is geodesically incomplete, it in the quantum theory §IV A. is possible to construct a conformal completion (C , η ) 0 0 It is important to distinguish between the geodesic in- that is geodesically complete and where C0 is globally isomorphic to C. This can be achieved by defining the completeness of C and the geodesic incompleteness of the reference metric space-time metric gµν . Although the former may be sufficient for the latter, the two senses of incompleteness are 0 2 ηAB = Ω ηAB , (16) logically distinct. In the context of the minisuperspace model, we will see that the boundary ∂C corresponds to the configuration space point where we have both that: 4 The singular nature of the FLRW model is treated explicitly in i) the expansion parameter of some congruence becomes [4] with Λ = 0. Here we are principally interested in the Λ > 0 negative and unbounded, implying that the space-time case. For this class of models the singularity theorems do not, geodesics terminate in finite proper time; and ii) there is a in fact, apply since the relevant energy conditions do not hold. curvature pathology. This can be seen straightforwardly For k = 0 we nevertheless recover singular behaviour for generic in our chosen coordinate chart: the condition v = 0 both classical solutions in the senses described above. 5 One way to see this is to integrate out N˜ by evaluating the action defines the boundary of C and can be connected to a sin- using the equations of motion for N˜. This shows that the action gularity in the senses i) and ii) above. This will be shown is proportional to the geodesic length of a curve on C using the explicitly in §IIIC below. All non-trivial cosmological so- metric η. lutions of our model thus contain at least one big bang or 7 big crunch singularity, and these occur precisely at the which represent the classical Big Bang and Crunch. boundary ∂C. That these times represent a classical singularity in the sense defined in §IIIB above can be seen as follows. Consider the time-like congruence that is hypersurface C. General Explicit Solutions orthogonal to the homogeneous and isotropic spatial geometries that define our constant-τ slices. By homogene- The geometric methods used in the previous sections, ity, the expansion parameter, θ, of this congruence is con- though valuable for highlighting generic features of the stant along these slices. Because these slices are the same solution space, do not provide the most physically en- as those used in our canonical split, the value of θ is equal ab lightening tools for investigation of certain features that to the trace of the extrinsic curvature tensor, g Kab, of will prove fundamental to the quantum analysis. Rather, our canonical slices. A straightforward canonical analysis to both connect our solutions to cosmological observa- then indicates that tions and to construct the quantum formalism, it is valu- ˜ able to describe general solutions in the observationally θ ∝ −πv/N. (22) relevant coordinate chart where the configuration vari- The form of the Hamiltonian constraint, ables are v and ϕ.6 In this section, we will restrict to Λ > 0 since, as will be shown explicitly in §IV B, only π2 these cases will have a well-defined and non-trivial semi- 2 ϕ 2 ˜ πv = 2 − 2~ Λ , (23) classical limit. v

The classical analysis in these variables can be per- implies that |πv| → ∞ as v → 0 since πϕ is a con- formed by integrating Hamilton’s equations generated by stant of motion. This implies that θ → −∞ for the the Hamiltonian (12). Hamilton’s second equation for ϕ branch of solution we are interested in. The condition tells us that πϕ is a conserved quantity, which we will k = 0 means that the 3d Ricci scalar vanishes. Given call k0 in anticipation of the variables used to construct this, the divergence of the trace of Kab indicates (via our cosmological quantum state. Hamilton’s first equa- the Gauss-Codazzi relations) that the 4d Ricci scalar is tion forv ˙ can then be inverted and inserted into to the also divergent. This confirms that, when v = 0, we have Hamiltonian constraint leading to both incompleteness of causal curves, in the sense of the Penrose-Hawking singularity theorems, and a curvature 2 2 dτ dv pathology, in the sense that the Ricci scalar diverges. = 2 , (18) ~ ˜ 1 k0 The integration constants k and ω can be used to 2Λ + v2 0 0 ~ define a dimensionless parameter ˜ ˜ where dτ = Ndt (this is precisely Friedmann’s first equa- k0 tion for this system). Integration of this equation is s ≡ (24) straightforward, although care must be taken to interpret ω0 different branches of the solution in terms of expanding and/or contracting phases, which differ by a choice of that sets the (relative) scale where the dynamics of ϕ the arrow of time. Solutions with a contracting ‘crunch’ become relevant to the evolution of v. At late and early phase followed by an expanding ‘bang’ phase are obtained times, τ/τsing 1, the dynamics of v become indifferent by the branch whose integral of motion is given by to the value of s, and the system enters an approximate de Sitter expansion phase: k2 ω2 v(τ)2 = − 0 + 0 τ 2 , (19) 2 4 ω0 |τ| p ˜ |τ| ω0 ~ x(τ) → = 2Λ . (25) ~2 ~ where we have defined the quantity Figure 2a illustrates the general behaviour of this solu- p tion. ω ≡ 2 2Λ˜ , (20) 0 ~ It is now possible to integrate Hamilton’s first equation which will be useful later in our treatment of the quantum for ϕ(τ). Care must again be taken to set limits of inte- theory. In this solution, we have used the time trans- gration such that the resulting integration constants can lational invariance of the theory so that the classically be straightforwardly interpreted. A convenient choice singular points of the solution, where v = 0, occur sym- leads to metrically about τ = 0 at the singular times τ ϕ(τ) = ϕ + tanh−1 sing , (26) ∞ τ 2k ± τ ≡ ±~ 0 , (21) sing 2 where we interpret ϕ as the late and early asymptotic ω0 ∞ value of ϕ. The divergence of ϕ(τ) at τ = τsing represents a genuine physical divergence and is consistent with the singular behaviour observed above. Figure 2b illustrates 6 As we will see in §IV A, quantization is not chart dependent. the general behaviour. The translational invariance of ηab 8

v v 7 3.0 6 2.5 5 2.0 4 1.5 3 1.0 2

0.5 1

t -3 -2 -1 1 2 3 ϕ -2 -1 0 1 2 (a) Spatial volume v(τ). FIG. 3: Classical solutions on C in units where s = 1 v and ϕ∞ = 0. 8

6 some dimensionless version of ~ – labels the relative size 4 of quantum eﬀects.

2 It is instructive to consider at this point a very general argument as to why we might expect that the singularity t should not be resolved in the quantum theory. Consider -3 -2 -1 1 2 3 the generalised Ehrenfest theorem -2

-4 ∂ D E 1 D E D∂πˆv(t)E πˆv(t) = [ˆπv(t), Hˆ ] + (28) ∂t i~ ∂t (b) Scalar ﬁeld, ϕ(τ).

under the observation that θ ∝ −πv/N˜. Given that we FIG. 2: Classical solutions for small ϕ∞. Asymptotic have just shown that the classical value of πv is prob- values of the parameters are shown in yellow. Singular lematic, as indicated by the fact that v → 0, one might lines drawn in black. Temporal units are chosen so that expect Ehrenfest’s theorem to imply that the quantum τsing = 1. Units for v are chosen so that s = 1. expectations values should be equally problematic. More precisely, the logic of the Hawking–Penrose singularity theorem is to give very general conditions under which in ϕ (which are boosts in C) implies that the solutions the time derivative of the expansion parameter, θ, of themselves have symmetries under shifts in ϕ. some congruence of geodesics in the spacetime is always Reparametrization invariant curves on C can be ob- negative, and that the expansion parameter itself will tained by parametrizing the solution for v in terms of ϕ eventually go to minus inﬁnity. What we have shown by eliminating τ. This is possible because ϕ is mono- in this section is that the conditions for this theorem are tonic along our chosen branch and represents, therefore, met, in the case of minisuperspace models with vanishing a good clock. Branches that contract and then re-expand spatial curvature, even when the cosmological constant is in this way can be conveniently written as positive so that the strong energy condition is violated. It follows that the same logic can be straightforwardly v = s |cosech(ϕ − ϕ∞)| . (27) applied to argue that, given the Ehrenfest theorem and Figure (3) illustrate these curves on C. Note that the the requirement that the classical Poisson algebra should solution illustrated in Figure (3) characterizes all physi- be faithfully represented by the quantum operator alge- cally meaningful solutions to the system. This is because bra, the expectation value of θ˙ should be always negative, the arbitrary units of space and time can be used to ar- and, therefore, hθi → −∞. Thus, provided that the con- bitrarily rescale the two constants of motion Λ and k0. ditions for the relation (28) hold, one should expect a What is left is the un-parametrized dimensionless curve kind of quantum generalisation of the classical singular- of Figure (3). We will see that the time-dependent wave- ity theorem. Fortunately, as we will see in §IV A below, function that can be deﬁned over-top of this space can theπ ˆv operator fails to be self-adjoint, meaning that the have a meaningful size in relation to the classical solution. conditions for (28) are violated and a quantum version This implies that an additional parameter – representing of the singularity theorem can be avoided. 9

IV. QUANTUM MINI-SUPERSPACE Na¨ıvely, one would expect that these operators should be COSMOLOGY unitarily equivalent since they correspond to classically equivalent variables. However, there are obstructions to A. Algebra of Observables defining the self-adjoint shift operators of the coordinates of an arbitrary chart of C. A detailed analysis of these In the quantum theory, the existence of the classically issues in terms of the formal structure of quantization singular boundary ∂C leads to difficulties in finding self- has been given by Isham [68]. For our purposes, we will adjoint representations of symmetric operators in Hilbert see these obstructions arise through the geometric prop- space.7 Specifically, the presence of a boundary allows erties of the states and operators. In particular, there is for the possibility of boundary terms to contribute to the a mismatch between the condition for square integrabil- action of the adjoint, and self-adjointness requires that ity on states, which transforms like a scalar on C, and these boundary terms vanish. For a particular represen- the condition for the self-adjointness of the momentum tation of some operator, three distinct possibilities arise: operators, which transforms like a co-vector. This mismatch implies that the standard canonical quantization i) All states, which are square integrable on the ap- procedure leading to an algebra of self-adjoint operators propriate interval and measure, automatically sat- is not well-defined in all charts over T ∗C. isfy the conditions required for the boundary terms It is important to note that these obstructions are com- to vanish. In this case, the operator is termed essen- patible with the Stone–von Neumann theorem, which as- tially self-adjoint. sumes that the classical phase space is R2N , with N the ii) All square integrable functions are spanned by the number of Lagrangian degrees of freedom. This assump- solutions of a unitary family of boundary condition is violated in the problematic charts of C. For exam- tions that guarantee the vanishing of the boundary ple, the physically natural variable v has the phase space terms. This implies an underdetermination in the R+ × R, and will be seen to be problematic below. In self-adjoint representations of the operator in ques- particular, we will see that its momentum operator,π ˆv, tion, and the solution space of each member of the will not be self-adjoint. As discussed at the end of §IIIC unitary family defines the domain of a particular self- above, rather than being problematic, this obstruction is adjoint extension of the operator. important for evading a quantum generalisation of the singularity theorem, and is ultimately responsible for al- iii) There exist square integrable functions such that the lowing singularity resolution. boundary terms are non-zero. In this case, the oper- In order for the quantum observable algebras to reduce ator is not self-adjoint or, rather, is said to have no to the classical one in the appropriate limit, it is still self-adjoint extensions. necessary to find a well-defined momentum operator to A theorem, due to von Neumann, allows us to diagnose replaceπ ˆv. Fortunately, Darbaux’s theorem guarantees into which of the three categories a particular symmet- that there exists a coordinate chart where T ∗C = R2N in ric operator falls based upon the dimensionality of the some finite patch. Moreover, in our model, it is possible relevant ‘deficiency subspaces’ (see [66] theorem X.1). A to find a global set of coordinates that has this prop- further theorem implies that all real symmetric operators erty, and this set of coordinates will provide us with a fall in to category i or ii (see [66] theorem X.3). In our well-defined momentum operator. In the semi-classical analysis, we will focus on the behaviour of the relevant limit, the eigenvalues of this operator can be used to re- boundary terms, which have a more direct geometric sig- construct the classical observable algebra in all charts of nificance, to diagnose the self-adjointness of our operator T ∗C. algebra. This method can be shown to be equivalent to We will now give the details of an explicit geometric the von Neumann treatment [67]. construction of representations of the operator algebra of From now on, we will consider building representations our quantum theory. Given the geometric considerations of our quantum operator algebra using point-wise mul- of the previous√ section, we consider the Hilbert space, tiplication operators of the coordinates in a particular L2(C, d2q −η), of square integrable functions on C un- chart of C and shift operators of these coordinates in that 2 √ der the Borel measure dθ = d q −η, where η = det ηAB. chart (see the definitions (34) below). This represents an This space includes all complex functions (Φ, Ψ) that infinite family of conjugate operators corresponding to obey classical phase space functions, where each chart provides a pair of operators. Changes of coordinates on Z √ C induce transformations between pairs of operators.8 hΦ, Ψi ≡ d2q −η Φ†Ψ < ∞ . (29) C

7 See [65] for a discussion of the connection between self- To convert this into a condition on states, we make use 0 0 adjointness and singular spacetimes. of the conformal completion (C , η ) of (C, η) discussed 8 Note that this is different from considering different representa- in §IIIB. In terms of the tortoise coordinate µ and ϕ, a tions of the same operators in different charts. well-known result, [67, Lemma 2.13], establishes that the 10 condition same is not true for scalar operators, whose boundary term will transform in the same way as the square inte- Z p hΦ, Ψi ≡ dµdϕ −η0 Φ†Ψ < ∞ , (30) grability condition. The Hamiltonian operator presented R2 below in §IV B (see equation (47)) is an example of such combined with the requirement that the momentum a scalar function and exists in all possible coordinate rep- eigenstates are also in the Hilbert space, implies resentations. A particularly convenient coordinate split can be Φ†Ψ(±∞, ϕ) → 0 Φ†Ψ(µ, ±∞) → 0 . (31) achieved by taking advantage of the global Killing vector field provided by the boosts. As discussed in §IIIA, the The falloff conditions above can be adopted to an arbi- boost symmetry of ηAB suggests a coordinatization of C, trary measure by multiplying and dividing the integrand up to reparametrization, in terms of the boost parame- √ 9 of (30) by −η0. The condition (31) then becomes ter ϕ and the orthogonal coordinate v. The ϕ-dependent part of the quantization procedure is rather unproblem- r η † atic, and the self-adjoint momentum shift operator, Φ Ψ → 0 , (32) η0 ∂C ∂ πˆ = −i , (37) which, for an arbitrary state, converts to the necessary ϕ ~∂ϕ condition: conjugate to the point-wise multiplication operator,ϕ ˆ =

η 1/4 ϕ, can readily be constructed. This can be done because |Ψ| → 0 (33) η0 the ϕ-dependence of ηAB is identical to that of the refer- ∂C 0 ence metric ηAB on the conformal completion. Thus, as for square integrability. This scalar condition on the far as the ϕ-dependence of the wavefunction is concerned, states is not compatible with the condition for the self- the condition (32) reduces to adjointness of all momentum operators. To see this explicitly, consider the following represen- Ψ(ϕ → ±∞) → 0 , (38) tation of the infinite family of point-wise multiplication which is the standard condition for L2(R, 1) functions. and shift operators discussed above: Moreover, the self-adjointness condition (36) takes exactly the same form, and confirms the usual result that A A −1/4 ∂ h 1/4 i qˆ Ψ = q Ψp ÂΨ = −i (−η) (−η) Ψ . the translation operator on R is self-adjoint. ~ ∂qA (34) For the directions orthogonal to the boosts, the situation is more subtle. We consider first a parametrization The ordering is chosen to be symmetric under the chosen of this direction given by the volume v. For the momen- measure. Integration by parts tells us that tum operator, I 1 ∂ √ √ † πˆvΨ ≡ √ vΨ , (39) hΦ, pÂΨi = hpÂΦ, Ψi − nAdl χΦ Ψ , (35) v ∂v ∂C √ conjugate to the point-wise multiplication operatorv ˆ = where dl is an integration measure on ∂C, χ is the pull- √ v, the condition (36) takes the form back of −η onto ∂C and n is a unit normal to the √ A boundary. Because χ splits into a volume form on ∂C † † vΦ Ψ = 0 vΦ Ψ → 0 , (40) and a 1-form in C, the condition v=0 v→∞ √ √ because, −η = v. This implies, in particular, that all n dl χ Φ†Ψ → 0 , (36) A ∂C wavefunctions, Ψ, satisfying the self-adjointness require- transforms, as expected, as a co-vector and is, therefore, ment can have a divergence of order up to (but not in- −1/2 √ chart-dependent. This chart dependence implies that not cluding) v at v = 0 because ( v|Ψ|)|v=0 → 0. On all square integrable functions will satisfy the condition the other hand, the condition for square integrability (32) of self-adjointness for every choice of coordinates. can be easily computed in this chart by noting that Any canonical quantization procedure leading to an al- η0 = v−2η . (41) gebra of self-adjoint operators will, therefore, break the AB AB elegant coordinate-free construction presented thus far. Using this, we find It will then be necessary below to consider specific coordinate representations of our operators. Note that the (v|Ψ|)|v=0 → 0 . (42) This condition is weaker than the self-adjointness requirement for the momentum operator conjugate to this 9 Note that this follows because the limits µ → ±∞ and ϕ → ±∞ coordinate because divergences of up to (but not includ- correspond to the boundary ∂C. −1 ing) order v are allowed. Thus,π ˆv is not self-adjoint. 11

A well-defined momentum operator can be found by is the standard Laplace–Berltrami operator on C. This making use of the tortoise coordinate, µ. In these coor- construction is similar to the procedure proposed in [69]. dinates, the self-adjointness requirement takes the form By virtue of being a real scalar function on C, the Hamil- tonian operator enjoys certain desirable properties not µ (e |Ψ|)|µ→−∞ → 0 , (43) shared by the momentum operators of the previous sec- √ tion. Being scalar means that the self-adjointness con- because −η = e2µ. This is identical to the condition for dition will transform on C as a scalar in the same way square integrability. Thus, the momentum operator,π ˆµ, as the square integrability condition. This further im- is essentially self-adjoint. This establishes the existence plies that different charts of C correspond to unitarily of a well-defined momentum operator conjugate to the equivalent representations of the same operator. More- tortoise operator,µ ˆ = µ. An orthonormal basis for the over, being real means that the Hamiltonian is guaran- spectrum of all momentum operators that fall into this teed to have self-adjoint extensions – although these are class can be constructed using the reference metric via not guaranteed to be unique. We will find below that the spectrum of the Hamiltonian operator splits into two 1/4 i A 1 η0 − q kA qualitatively different branches: unbound states, which ψkA = e ~ . (44) 2π~ η behave like bounce cosmologies and dominate early- and late-time solutions, and bound states, which are only im- Note that the momentum of all the independent coordi- portant near the bounce and provide the potential for nates of this chart are included in this definition. One further phenomenological investigations for near-bounce can easily verify that the functions above will span the physics. space of functions that obey both the condition (32) and The condition for self-adjointness of the Hamiltonian the eigenvalue equation can be obtained by computing the boundary term in the integral below via two applications of integration by parts pÂψkA = kAψkA . (45) D E D E 1 I √ For the tortoise coordinates, (µ, ϕ), for which the mo- Φ, Hˆ Ψ = Hˆ Φ, Ψ − dl χ ηAB 2 mentum operators are self-adjoint, these eigenfunctions ∂C † † take the explicit form × n(A Φ ∂B)Ψ − Ψ∂B)Φ , (49)

i 1 −µ+ (kϕ+rµ) where round brackets indicate symmetrization of indicies. ψ(k,r) ≡ e ~ , (46) 2π~ Because this is a scalar condition, it is sufficient to work out the conditions for its vanishing in a global coordinate where (r, k) are the eigenvalues of (ˆπµ, πˆϕ) respectively. chart. The general conditions can be worked out in any For the configuration operators that are represented by chart via diffeomorphism. For a convenient chart, we point-wise multiplication operators, self-adjointness is will once again take advantage of the boost symmetry manifest and the spectrum is straightforwardly evalu- of the theory using the boost parameter, ϕ. It will turn ated. out that the physically relevant variable v will be most convenient for parametrizing the direction orthogonal to the boosts. In these variables, the metric is diagonal and B. Hamiltonian the boundary term above splits into two pieces. The first piece is the boundary contribution in the limit where |ϕ| Our next task is to construct self-adjoint represen- grows unboundedly. In this limit, we have tations of our Hamiltonian operator. A coordinate- † † invariant and symmetric representation of the Hamilto- Φ ∂ϕΨ − Ψ∂ϕΦ ϕ→±∞ → 0 , (50) nian operator10 corresponding to (15) can be written as: which is automatically satisfied for all L2(R, 1) functions 1 Hˆ = − , (47) according to the condition (38). As this corresponds to 2 the free-particle Hamiltonian on R, it is straightforward to see that the ϕ-part of the Hamiltonian is essentially where self-adjoint. 1 √ The second contribution to the boundary term in (49) = √ ∂ ηAB −η∂ . (48) −η A B comes from the classically singular boundary ∂C at v = 0. This contribution takes the form

† † v Φ ∂vΨ − Ψ∂vΦ v=0 = 0 . (51) 10 As was emphasised in §II, this Hamiltonian will be the appropriate one to use even when the cosmological constant is non-zero We see that, provided the v-derivatives are regular at because Λ can be interpreted as a separation constant obtained v = 0, all functions satisfying the square integrabil- from solving our evolution equation similar to the total energy ity condition (42) will automatically satisfy this self- in a time-dependent Schrödingerequation. adjointness condition. Unlike the boundary contributions 12 in the ϕ → ±∞ limit, the solutions of (51) are only mu- 1. ‘Bound States’ (Λ < 0) tually orthogonal when they satisfy a unitary family of boundary conditions, and, thus, particular solutions are When the cosmological constant is negative, the lin- required to satisfy one member of these boundary condi- early independent solutions to (55) are the modified tions. A variety of techniques can be used to study these p Bessel functions of the first, I ( 2Λ˜v), and second solutions and the nature of the boundary conditions in ik/~ p ˜ question. We will explore an avenue particularly catered kind, Kik/ ( 2Λv). The asymptotic form of these func- ~ p to our problem in the sub-sections below. First, however, tions for large x = 2Λ˜v is given by we will deal with the ϕ-dependence of the wavefunction. The boost symmetry of Hˆ can be used to motivate a ex I (x) ∼ √ (1 + O(1/x)) (57) separation Ansatz for computing the square integrable ik/~ x ˆ eigenstates of H and its corresponding spectrum. If we e−x postulate K (x) ∼ √ (1 + O(1/x)) . (58) ik/~ x ± ± ΨΛ (v, ϕ) = ψΛ,k(v)νk (k) (52) The exponential growth of Iik/~(x) is not compatible as the linearly independent states satisfying the eigen- with the square integrability requirement in the v → ∞ value equation limit. Thus, the only potentially square integrable solution is Kik/~(x), which decays exponentially in v and, Hˆ Ψ±(v, ϕ) = Λ˜Ψ˜ ±(v, ϕ) , (53) thus, represents a state ‘bound’ to the singular region near v = 0. We will henceforth refer to such states as then the ϕ-dependence of this equation can be found to ‘bound states’. To examine the square integrability and satisfy self-adjointness of Kik/~(x), we need to examine its behaviour near the regular singularity at v = 0. This is d2 k2 ± ± given by νk = − νk , (54) dϕ2 ~2 p ˜ k k Λ˜ Kik/ 2Λv ∼ sin log v + log − θk , while the v-dependence leads to ~ ~ 2~ 2 (59) d d k2 2 where θk = Arg(Γ(ik/ + 1)) and Γ is the Gamma- v v ψΛ,k + 2Λ˜v + ψΛ,k = 0 . (55) ~ dv dv ~2 function. The limit is not well-defined at v = 0, but the result is bounded and, therefore, square integrable in Equation (54) is the wave-equation equation whose solu- accordance with (42). Moreover, it is possible to assess tions are the in- and out-going waves whether these solutions are in the self-adjoint domain of i Hˆ by considering their behaviour in (51) for different val- ± 1 ± kϕ ν (ϕ) = √ e ~ . (56) ues of the cosmological constant – say Λ and Λ . After k 2π a b ~ inserting the asymptotic expansion above into (51) for The square integrability condition (38) is satisfied iff k is p ˜ p ˜ Φ = Kik/~( 2Λav) and Ψ = Kik/~( 2Λbv), we obtain real, and, therefore, the deficiency subspace has dimen- k sion zero, and the operator is essentially self-adjoint in sin 2 log (Λa/Λb) = 0 . (60) line with our previous expectations. ~ Equation (55) is Bessel’s differential equation for The zeros of this function occur, for integer n, when purely imaginary orders, ik, given the restriction to real Λa k. This equation has a regular singularity at v = 0, = e2nπ~/k . (61) Λ where we find physically interesting contributions to the b boundary term (51). Unlike the case of real or vanishing This implies that the bound eigenstates11 orders, none of the linearly independent solutions of (55) s for any eigenvalue is recessive at the origin. This means ˜ bound 4~|Λ| sinh (πk/~) p that there is no preferred choice of self-adjoint extension ψ = K ( 2Λ˜v) (62) Λ,k πk ik/~ for Hˆ . The character of the solutions to (55) and the nature of the self-adjoint extensions differ significantly belong to the self-adjoint domain of Hˆ provided the depending on the sign of the cosmological constant. We spectrum is restricted to discrete values of Λ that dif- therefore treat each case separately below. The proper- fer by factors of e2nπ~/k. The particular set of dis- ties of the solutions used in our discussion, including the crete values to be used represents a particular choice of relevant asymptotic expansions, are described in detail in [70], which we will reference in the following sub-sections. Alternatively, one can exploit the mathematical equiva- lence of our model with certain limiting regimes of the 11 The normalization below is computed by requiring that the 1/r2 potential. Complimentary treatments of such po- states be orthonormal and by making use of the integral: R ∞ πk dx xKik(uax)Kik(ubx) = δab. tentials can be found, for example, in [71–73]. 0 2uaub 13 self-adjoint extension of the domain of Hˆ . This choice onality relations can be parametrized by a choice of reference scale Λ , ref Z ∞ which is under-determined up to a log-periodicity given dvv ψ† , ψ = δ (63) Λa,k Λb,k ab by log Λref → log Λref +2π~/k. Within a particular choice 0 of self-adjoint extension, it is straightforward to verify that, given the normalization defined above, the orthog- hold in accordance with the observation that Bessel’s equation, (55), is of Sturm–Liouville form. These can be used to construct general states. Combining these eigenfunctions with the arbitrary linear combinations of the k-space eigenfunctions, we can construct a general Λ-eigenstate in terms of

s r Z ∞ ˜ q bound 2 kϕ kϕ |Λ| sinh(πk/~) ˜ ΨΛ (v, ϕ) = 2 dk (A(k) cos( ) + B(k) sin( )) Kik/~( 2|Λ|v) , (64) π ~ 0 ~ ~ k

where the normalization R ∞ dk |A(k)|2 = For sufficiently large v, these functions behave like oscil- 0 ~ √ R ∞ dk |B(k)|2 = 1 guarantees lating functions with a 1/ v decay envelope: 0 ~ 1/2 bound bound 2 Ψ , Ψ = δab . (65) Fk(x) ≈ [cos (x − π/4) + O(1/x)] (68) Λa Λb πx 1/2 The oddness of K in terms of k projects out the even 2 ik/~ Gk(x) ≈ [sin (x − π/4) + O(1/x)] . (69) part of the k-space solution and allows (64) to be written πx as an integral in k from 0 to ∞. Note the k-independence of the leading order behaviour, This spectrum is not bounded from below and has an which has resulted from the choice of normalization of accumulation point at Λ = 0. Nevertheless, states of these functions. These eigenfunctions satisfy the square this kind can be constructed as effective field theories in integrability and self-adjointness requirements for v → atomic 3-body systems, and the resulting phenomenon is ∞. This is not automatically the case, however, on the known as the Efimov effect [38, 71, 74]. The literature on other branch of the boundary at the regular singularity the Efimov effect is vast. However, this is the first time, at v = 0. to our knowledge, that a relation has been pointed out In this case, the Bessel functions behave as between this effect and the potential physics of the early Universe. The implications for this will be discussed in p 2 tanh(πk/2 )1/2 Section §IV D. F 2Λ˜v ≈ ~ ~ k πk h k k Λ˜ i 2 × cos log v + log − θk + O(v ) (70) ~ 2~ 2 2. ‘Unbound States’ (Λ > 0) and

When the cosmological constant is positive, the lin- p 2 coth(πk/2 )1/2 G 2Λ˜v ≈ ~ ~ early independent solutions to (55) are the Bessel k πk p ˜ functions of the first, Jik/ ( 2Λv), and second, h k k Λ˜ i 2 ~ × sin log v + log − θk + O(v ) . (71) p ˜ ~ 2~ 2 Yik/~( 2Λv), kind. In the case of purely imaginary order, these functions need to be rescaled so that they are As in the case of the bound states, the v → 0 limit is numerically well-behaved and suitably normalized func- not well-defined. Nevertheless, the functions are bounded tions of k. A choice of normalization that is both real and, therefore, the square integrability condition is sat- and well-behaved is given by (see [70]) isfied. To solve the self-adjointness condition, we first note that real functions automatically obey (51) when 1 πk Φ = Ψ. Since we have already constructed two real, Fk(x) ≡ sech Jik/ (x) + J−ik/ (x) (66) linearly independent solutions we can construct a new 2 2~ ~ ~ solution, for a single value of Λ, by adding them in a 1 πk Gk(x) ≡ cosech Jik/ (x) − J−ik/ (x) . (67) unitary combination such as ψθ = cos θψ1 + sin θψ2. To 2i 2~ ~ ~ get the full solution, we need to find states such as these 14

that independently solve the boundary conditions for dif- where ferent values of Λ. To deal with this, it is sufficient to k Λ note that the logarithmic dependence of the arguments θ = log . (75) of the trigonometric functions above makes it possible to 2~ Λref subtract the Λ-dependence of these functions near the origin by inserting the correct relative phase between the This satisfies the boundary condition (51) by construc- two linearly independent solutions. To this end, it is tion. The U(1) self-adjoint extension parameter θ gives a more precise representation of the physics of the di- convenient to take linear combinations of the Fk’s and mensionful reference scale Λref. In particular, we note the Gk’s to form Jik/~’s and note that the combination ik/2 that the periodicity θ → θ + nπ (for integer n) implies (Λ/Λref) ~ is sufficient to cancel the Λ-dependence of ψΛref (v) → −ψΛref (v), which is not a linearly indepen- the J ’s near the v = 0 boundary for any value of Λ,k Λ,k ik/~ dent function. This means that the reference scale is the reference scale Λref. The most general solution is to take unitary linear combinations of real and imaginary only defined up to parts. A convenient way to keep track of this freedom is Λ → e2nπ~/kΛ , (76) expressed through the use of the k-dependent reference ref ref scale Λref(k). Different choices of Λref(k) can be made as was the the case for Λ < 0. to correspond to different phases between real and imag- The physical interpretation of θ can be seen by evalu- inary pieces and can, therefore, be used to parameterize ating the eigenfunction ψΛref in the large v limit. In this the different possible choices of self-adjoint extensions. Λ,k limit, the asymptotic forms (68) and (69) can be used to The most general solution, therefore, is given by:12 show that p Λref ˜ π πk ψΛ,k ∝ cos 2Λv − 4 − arctan tanh 2 tan θ −ik/2~ ~ Λ p ˜ Re Jik/ ( 2Λv) p Λref(k) ~ ˜ Λref + O 1/( 2Λv) . (77) ψΛ,k = h i . (72) cosh πk + i k log Λ 2~ 2~ Λref(k) Since the cosine function is a linear combination of in- and out-going modes with opposite phases, this implies a phase shift of The normalization is chosen so that the eigenstates sat- π h i isfy ∆ ≡ + 2 arctan tanh πk tan k log Λ (78) 2 2~ 2~ Λref between asymptotic planes waves as they scatter in and ∞ Z † out of the bounce regime. Note that the log periodicity dvv ψθ ψθ = δ(Λ˜ − Λ˜ ) , (73) Λa,k Λb,k a b of Λref implies that the Λref → 0 and Λref → ∞ limits are 0 not well-defined. This means that ∆, according to (78), cannot take the value zero and, therefore, that there is no preferred choice of self-adjoint extension. The phase shift which can be verified by inserting the large v behaviour ∆ also defines, via the moments of the wavefunction, a of the J ’s. Note that simple choices for Λref could be the preferred notation of temporal units for the system via a ˆ mean of Λ or a constant phase, α, such that Λref(k) = characteristic time for deep scattering. These consider- e−αk/2~. Note that, the imaginary piece can be obtained ations will be crucial for interpreting the physics of the through a suitable redefinition of α. cosmological solutions studied in the companion paper [39]. With this choice of wavefunction, the near boundary If Λref(k) where independent of k (or even in k), then eigenstate takes the form the unbound wavefunctions would be even in k in contrast to the bound wavefunctions that were odd in k. The most general case, however, for the bound Λ-eigenstates " ˜ ! # involves arbitrary superpositions of positive and negative θ k k Λ ψ (v) ∝ cos log v + log − θk − θ , (74) values of k: Λ,k ~ 2~ 2

12 Note that another useful relation for writing the r h i normalization is: cosh πk + cos k log Λ = ~ ~ Λref √ h i 2 cosh πk + i k log Λ . 2~ 2~ Λref 15

kϕ kϕ " # Z ∞ C(k) cos + D(k) sin Λ −ik/2~ p Ψunbound = dk ~ ~ Re J ( 2Λ˜v) , (79) Λ,Λref √ h i ik/~ −∞ 2π cosh πk + i k log Λ Λref ~ 2~ 2~ Λref

∞ ∞ R dk 2 R dk 2 iΛt/~ where the normalization −∞ |C| = −∞ |D| = 1 phase e = 1. We, therefore, do not regard this as guarantees ~ ~ a physically interesting cosmological solution in the relational quantization paradigm. Ψunbound, Ψunbound = δ(Λ˜ − Λ˜ ) . (80) Λa,Λref Λb,Λref a b

As a final comment on the interpretation of Λref, C. Bouncing Unitary Cosmology we note that the vanishing of the boundary term in (49) is a conformally invariant equation because the Application of the relational quantization procedure of d’Alembertian is conformally covariant in 2d. It should, §II leads directly to a Schrödinger-type equation of the thus, be no surprise that the solutions to this equation form: take the form of the eigenfunctions of in the conformal completion (C , g ). One can, therefore, understand our ∂Ψ 0 0 Hˆ Ψ = i , (81) procedure as a way of anchoring the Bessel functions us- ~ ∂t ing the known solutions in on the conformally completed manifold. The introduction of the dimensionful parame- where we use the Hamiltonian operator (47). Since the Hamiltonian is self-adjoint, the time evolution is guaran- ter Λref can thus be interpreted as a necessary ingredient for removing the dependence of the bulk eigenfunctions teed to be unitary by Stone’s theorem [75, p.264]. on the dimensionful parameters of the theory so that the It is important to note that in this formalism, t is bulk theory can be compatible with the conformal bound- treated as an arbitrary label that orders successive inary condition. stants. It is not a physical observable. Rather, all observations can be transcribed purely in terms of the relative changes in correlations between operators. In this case, a 3. Critical Case (Λ = 0) complete set of self-adjoint operators that generalize the A full classical phase space are the operatorsq ˆ , pÂ. As outlined in §IV A, a self-adjoint representation of these The case where Λ is precisely vanishing is a criti- operators can be given explicitly in terms ofµ, ˆ ϕ,ˆ πˆ , and cal point13 of our model. The eigenvalue equation for µ πˆ as in §IV A. Although, t is needed as a bookkeeping the Hamiltonian becomes conformally invariant since the ϕ device to label how the observables change relative to Laplacian is conformally covariant in 2d. The flat refer- each other, the actual value of t is never required to be ence metric η on the conformal completion of C can then 0 directly or indirectly measured. be equivalently used to construct the quantum theory. In Having constructed an explicit representation for our terms of the conformal completion, the Hamiltonian be- Hilbert space and computed the self-adjoint spectrum for comes the flat metric on R1,1. The degenerate spectrum, our Hamiltonian, it remains to explicitly write the most in terms of the tortoise coordinates µ and ϕ, consists of general solution to (81). A separation Ansatz in- and out-going plane waves on the µϕ-plane satisfying 2 2 a dispersion relation r = k (in terms of the eigenvalues Z ∞ i A ˜ ˜ − Λ˜t defined in (46)). The domain of the Hamiltonian for any Ψ(q , t) = dΛ E(Λ)e ~ ΨΛ (82) −∞ particular choice of Λref, which is essentially self-adjoint in this representation, is then given by standard L(R2, 1) reduces (81) to the eigenvalue equation: functions. Although the Hamiltonian is well-defined and Hˆ ΨΛ = ΛΨ˜ Λ , (83) anomaly-free, the quantization in this regime is discon- nected from the quantum solutions given above for non- which we have already solved in detail in §IV B. The gen- zero Λ. This is because the solutions of the former cannot eral solution can constitute a superposition of negative be obtained as a continuous limit of the latter since the bound Λ-eigenstates and positive unbound Λ-eigenstates. Bessel functions are not defined in this limit. Hence, the However, because the bound states decay exponentially Λ = 0 case should be regarded as a separate quantization away from the bounce region at v = 0, they necessar- confined to the single eigenvalue Λ = 0. This solution, ily represent re-collapsing cosmologies and, therefore, ap- however, leads to a timeless wavefunction because the pear to be in contradiction with observations. Contrast- ingly, the unbound states have an oscillating norm at large v and therefore have the potential to reproduce a late-time expanding FLRW universe with positive cosmo- 13 Discussions of the link to critical phenomena in the context of logical constant in line with current observations. More- 1/r2 potentials can be found, for example, in [71–73]. over, while the unbound states admit continuous values of 16

Λ and have well-defined semi-classical limit for all choices black holes [78, 79] and FLRW [80] or inflationary cos- of the parameter space, the bound states are discrete in mologies [81, 82]. There has been exciting recent experi- Λ and only admit a semi-classical limit when the param- mental progress in using analogue models to simulate as- eter space is tuned in such a way that the wavefunction pects of Hawking radiation in the lab [83–85]. Analogue can have support in regions of comparatively large v and models for bouncing cosmologies have been suggested be- ϕ. Nevertheless, the bound wavefunction could have im- fore [24]. However, our proposed Efimov analogue cos- portant contributions in the deep quantum regime when mology is entirely novel. The particular importance of the bound and unbound wavefunctions have significant the analogy for the present work is the interpretation of overlap. In the companion paper [39], we explore the po- the self-adjoint extension parameters. tential implications of this potential period of overlap for Consider first the atomic system in the context of a cosmological observations. scattering experiment where a plane wave is scattered off In terms of these considerations, the general solution an Efimov state. Here the self-adjoint extension param- can then be taken to be eter is fixed by the fundamental physics of the bound state. That is, the true potential that governs the trimer Z ∞ i 2 A 1 ˜ Λ˜t − ˜ bound atomic system in the regime when the effective 1/r Efi- Ψ(q , t) = √ dΛ e ~ E (Λ)Ψ−Λ 2 0 mov model breaks down. Applying the self-adjoint ex- i ˜ tension procedure to the relevant Hamiltonian operator +e− ΛtE+(Λ)Ψ˜ unbound , (84) ~ Λ,Λref introduces a dimensionful parameter into the effective physics of the trimer sate. The value of this parameter where the bound and unbound eigenfunctions are respec- can then be measured by its effect on the plane wave tively given by (64) and (79) and we take the functions during the scattering. In particular, the extension pa- ± R ∞ ˜ ± 2 rameter fixes the magnitude of a phase shift in the plane E to be normalized such that 0 dΛ|E | = 1. Given our discussion in §IV B 1, the bound eigenstates admit wave during the scattering process. Thus we see that only a discrete spectrum specified by the choice of self- in the atomic Efimov effect the extension parameter is adjoint extension parameter, which can be parameter- a genuine physical magnitude that: i) has its origin in the fundamental 3-body atomic physics; and ii) can be ized by Λref. We, therefore, find that the Λ-dependence of E−(Λ)˜ can only take discrete values, and the above measured via the phase shift experienced by an scat- integral reduces to the sum tered plane wave. Given this, it is natural to consider an analogue version of this ‘experiment’ for the Efimov " ∞ cosmology. That is, we can consider a ‘scattering’ of the 1 X ˜ Ψ(qA, t) = √ eiΛnt/~E Ψbound unbounded states off the bound states in the deep quan- n −Λn 2 n=−∞ tum early universe regime. In this sense appeal to Efimov ∞ Z ˜ analogue cosmology provides a very natural strategy for + dΛ e−iΛt/~E+(Λ)Ψ˜ unbound , (85) Λ,Λref the physical interpretation of the self-adjoint extension 0 parameter, θ. As noted above, θ is more conveniently Z P 2 parametrised in terms of a reference value of the cosmo- where n ∈ , n |En| = 1 and logical constant Λref. In (78) we explicitly expressed the Λ = Λ2nπ~/k . (86) phase shift, ∆, by considering the difference between the n ref asymptotic forms of the constant Λ-eigenstates when v is large, and where the amplitudes of these eigenstates D. Efimov Analogue Cosmology behave like cosines. In keeping with the atomic analogy, this phase shift can now be interpreted as a giving a particular scattering length that parametrises the size of the As was noted above, fascinating connections exists be- region in which the model is no longer accurate. In the tween the physics of our cosmological model and that of companion paper we provide an explicit analysis of the atomic 3-body systems. In particular, the bound eigen- unbound states with the self-adjoint extension parameter states (64) have a mathematical form that mirrors that of constrained by the semi-classical value of the cosmologi- certain bound atomic trimer states, the physics of which cal constant [39]. is described by an effective 1/r2 potential. These states are known in the literature as Efimov states [38, 71, 74]. Given the experimental realisation and manipulation of Efimov states using ultra-cold atoms [76], our proposal V. SINGULARITY RESOLUTION BY thus presents the opportunity for a novel and experi- QUANTUM EVOLUTION mentally viable platform for analogue quantum cosmology. In general terms, analogue approaches to grav- What does it mean for a classical cosmological singu- ity [77] are based upon a formal correspondence be- larity to be ‘resolved’ in the quantum theory? One influ- tween the equations describing particular condensed mat- ential suggestion is that non-singular quantum behaviour ter systems, such as Bose-Einstein condensates, and the is indicated by the boundedness of the inverse 3-volume semi-classical description of gravitational systems such as operator [86, p.296], which indicates a potential degener- 17

acy in the spacetime metric. Such a kinematic criterion finiteness of expectation values we are not implicitly rely- is prima facie well motivated on the basis of the classical ing upon extending the correspondence principle into the relation between curvature divergence and the existence quantum bounce regime, but rather simply insisting that of the inverse metric components. However, the bound- a physically reasonable quantum theory can be defined edness of kinematical operators, such as the inverse of the at all times. 3-volume, is not alone sufficient to establish non-singular That our model satisfies the finite-expectation-values behaviour [31, 87]. Rather, one needs to additionally con- condition for singularity avoidance is straightforward to sider dynamics in order to relate bounded operators to demonstrate. Given that the Hamiltonian is self-adjoint 14 non-divergent curvature. and we have a unitary Schrödingerequation, the gener- An alternative dynamical criterion is in terms of Bo- alised Ehrenfest’s theorem applies: jowald’s notion of quantum hyperbolicity [31]: a non- singular quantum spacetime obtains when a state can ∂ D E 1 D E D∂Aˆ(t)E be uniquely extended across or around all submanifolds Aˆ(t) = [Aˆ(t), Hˆ ] + . (87) of classically singular configurations.15 Although well ∂t i~ ∂t motivated for the analysis of quantum singularities in Wheeler–DeWitt theories with internal times, quantum Provided the quantum observable algebra of bounded hyperbolicity is not particularly relevant when applied operators is well-defined, the commutator on the RHS to the analysis of singularities in the context of quantum is also bounded and the evolution of all observable ex- cosmological models with unitary Schrödinger-type evo- pectation values will be well-behaved. Thus, for unitary lution. In the Wheeler–DeWitt case, the physical Hilbert quantum cosmology, the condition for singularity avoid- space on which evolution takes place is represented in ance ultimately amounts to the usual requirement for a terms of operators acting on functions whose domains well-defined quantum theory. are ever changing submanifolds, parametrized by inter- We can now explicitly contrast the fate of the big bang nal time, of the configuration space. Here, the dynamics singularity in mini-superspace models under Wheeler– can force the system directly through a classically singu- DeWitt-type and Schrödinger-type quantizations. For lar submanifold, and this can lead to potential problems. the Wheeler–DeWitt case, the relevant expectation val- In contrast, in the unitary case, the domain of the rele- ues can be been computed explicitly under the assump- vant functions remains fixed to the entirety of the origi- tions of a massless scalar field, zero spatial curvature, nal configuration space. Because the classically singular and zero cosmological constant [28]. Given the choice of region is a set of measure zero on this space, a represen- φ as the internal clock, one can construct a one param- ˆ tation for bounded operators can always be found using eter family of self-adjoint Dirac observables V |φ0 each the prescriptions developed in §IV A. of which have the physical interpretation of the (spa- In our view, the basic condition for non-singular be- tial) volume operator at ‘time’ φ = φ0. In [28, IV] it is haviour is that the expectation value of an observable shown that for left moving states in the domain of the ˆ operator, as evaluated on all possible state in Hilbert volume operator, the expectation value V |φ0 → ∞ as space, always remains finite. Given a classical theory ˆ φ → ∞ and V |φ0 → 0 as φ → −∞. The situation is in which some classical observable diverges it is neces- reversed for the right moving sector. Thus, for a dense sary for singularity resolution that the expectation value subset of physical states, Wheeler–DeWitt quantization of the corresponding quantum operator is always finite. fails our necessary condition to resolve the cosmic singu- This finite expectation value criterion is applicable both larity. This explicitly illustrates what can go wrong in to Schrödinger-type models, where the expectation val- the WDW formalism according to our general discussion ues are considered at all values of the external time pa- above: the deparametrization with respect to an inter- rameter, and to Wheeler–DeWitt-type models, where the nal clock, which must take explicit values, can force the expectation values are projected onto a physical Hilbert dynamics of the system directly into the classically sin- space and are considered at all values of some internal gular region where things can easily go wrong quantum time. Finiteness of expectation values can understood mechanically. What is lacking in the Wheeler–DeWitt as a special case of a more general requirement for the approach is an evolution with respect to a self-adjoint existence of well defined effective equations of motion as (or even symmetric) Hamiltonian. This fact bars the use given by quantum corrections to Hamilton’s equations of the Ehrenfest theorem, which guarantees that all op- 16 for the expectation values [89, 90]. Thus, in requiring erators remain bounded in the unitary approach.

14 The need to consider dynamics is made abundantly clear by the case of anisotropic cosmological models where non-singular evolution exists despite the relevant curvatures being not necessarily quantum moments associated with the system. For analysis bounded [31]. of Wheeler-DeWitt type quantum cosmology and Loop Quan- 15 Bojowald’s criterion is in the same spirit as the dynamical sin- tum cosmology within the framework of eﬀective theory based gularly resolution proposals outlined in [32, 88] upon moment expansion see [91–93]. We leave application of 16 These corrections can be explicitly characterised in terms of the this framework to Schr¨odinger-type model for future work. 18

VI. CONCLUSIONS a quantum theory of shape cosmology. In this respect, one might plausibly take the following The core result of this paper is the construction of novel features of our model to be persist in a full-fledged a bouncing unitary cosmology from a classically singular quantum cosmological treatment based upon shape de- mini-superspace model. We have shown that solutions to grees of freedom. First, the superpositions of values of our universal Schrödingerequation are generically singu- the cosmological constant as are consistent with the close larity free and a set of well-defined quantum observables connection between our approach and unimodular grav- can be constructed with the corresponding expectation ity [35]. Second, a non-zero scattering length around our values guaranteed to be finite. Key features of the solu- big bounce upon which a characteristic scale for the ‘deep tions that are consequences of the general model outlined quantum’ regime of the universe would be set. Finally, here include a cosmic bounce due to quantum gravita- and perhaps most fascinatingly, one should expect the tional effects, a well-defined FLRW limit for Λ > 0 far existence of bound ‘Efimov universe’ states for negative from the bounce, and an inflationary epoch in the inter- cosmological constant also be present in the full theory. mediate region. Such features are, to our knowledge, entirely novel, and In more general terms, from our perspective, the quan- also open a new path towards quantum simulation of the tization of singular cosmological models under the stan- early universe. A further paper, [39], will contain detailed dard Wheeler-DeWitt approach is doubly problematic. interpretation and analysis of particular cosmological so- First, there is the lack of unitary time evolution and sec- lutions. ond there is the persistence of pathological features, as indicated by non-finite expectation values for classically singular quantities. The alternative approach outlined FUNDING is this paper is guaranteed to resolve both issues for any model to which it can be consistently applied. By requir- We are very grateful for the support from the Insti- ing unitary Schrödingerevolution of the universal wave- tute for Advanced Studies and the School of Arts at the function, as generated by a self-adjoint Hamiltonian op- University of Bristol and to the Arts and Humanities Re- erator, we provide a generic mechanism for singularity search Council (Grant Ref. AH/P004415/1). S.G. would avoidance in quantum cosmology. like to acknowledge support from the Netherlands Or- This said, there are important limitations to our ap- ganisation for Scientific Research (NWO) (Project No. proach: in particular, the relational quantization pro- 620.01.784) and Radboud University. K.T. would like to cedure appealed to here is in its present form only ap- thank the Alexander von Humboldt Foundation and the plicable to models with a single Hamiltonian constraint, Munich Center for Mathematical Philosophy (Ludwig- and is thus not readily extendable to models with spa- Maximilians-Universität München) for supporting the tial inhomogeneities. The prospects for extension of our early stages of work on this project. approach to more realistic cosmological models thus de- pends upon approaches to gravity in which the constraint structure of the Dirac’s hyper-surface deformation alge- ACKNOWLEDGMENTS broid is exchanged for a simpler structure featuring a single Hamiltonian constraint. An example of such an We are appreciative to audiences in Bristol, Berlin, approach is given by the Shape Dynamics formulation Geneva, Harvard, Hannover, Nottingham and the of gravity [60, 61, 94, 95]. As noted in [34, 51, 52], the Perimeter Institute for comments. We also thank Hen- techniques of relational quantization are in principal ap- rique Gomes for useful discussions on an important plicable to the Shape Dynamics. Our mini-superspace technical point regarding square integrability and David model could thus be considered as an approximation to Sloan for helpful comments.

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