Baby Universes and Worldline Field Theories
Eduardo Casali, Donald Marolf, Henry Maxfield, Mukund Rangamani
PPrepared for submission to JHEP
Baby Universes and Worldline Field Theories
Eduardo Casali, a Donald Marolf, b Henry Maxfield, b Mukund Rangamani a a Center for Quantum Mathematics and Physics (QMAP)Department of Physics & Astronomy, University of California, Davis, CA 95616 USA b Department of Physics, University of California, Santa Barbara, CA 93106, USA
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
The quantum gravity path integral involves a sum over topologies that invitescomparisons to worldsheet string theory and to Feynman diagrams of quantum field theory.However, the latter are naturally associated with the non-abelian algebra of quantum fields,while the former has been argued to define an abelian algebra of superselected observablesassociated with partition-function-like quantities at an asymptotic boundary. We resolvethis apparent tension by pointing out a variety of discrete choices that must be made inconstructing a Hilbert space from such path integrals, and arguing that the natural choicesfor quantum gravity differ from those used to construct QFTs. We focus on one-dimensionalmodels of quantum gravity in order to make direct comparisons with worldline QFT. a r X i v : . [ h e p - t h ] F e b ontents Recent work has revived ideas from the late 1980’s and early 1990’s [1–4] (based in turn onthe earlier refs. [5–8]) suggesting that spacetime wormholes elucidate the quantum physicsof black holes [9–17], and other issues involving gravitational entropy [18]. Here the termspacetime wormhole is used to denote a configuration in the quantum gravity path integral for– 1 –hich two separate connected components of the spacetime boundary are connected throughthe bulk of spacetime; see Fig. 1.Furthermore, because boundary partition functions tend not to factorize in the presenceof spacetime wormholes (see e.g., [19]), the above suggestions have led to renewed discussionof the nature of the AdS/CFT correspondence. Indeed, as described in the above referencesit may be that a given bulk theory is dual to an ensemble of field theories – though see alsocomments in [20], [21], [22] and [23]. The physics of spacetime wormholes and their possibleimplications are thus very much a part of current research.
Figure 1 : Schematic illustration of a spacetime wormhole connecting three boundaries.
However, attempting to understand issues involving spacetime wormholes brings oneface-to-face with the absence of a fully developed and universally accepted set of rules formanipulating and interpreting quantum gravity path integrals. This deficit can lead to muchconfusion in both the technical investigation of such path integrals and in communicating theresults. Our work below seeks to aid both tasks by pointing out certain discrete choices thatmust be made in order to define a theory of quantum gravity from diagrams like those inFig. 1, and by exploring certain implications of such choices.In particular, one may note that the spacetimes of Fig. 1 are similar to the diagrams ofworldsheet string theory and to the Feynman diagrams of QFT. Taken together with otherparallels between free particles in Minkowski space and simple models of quantum gravity,this feature is often taken to suggest that the full structure of quantum gravity will againresemble that of string theory or QFT [3, 24–31], in which context the process of buildingthe corresponding theory of quantum gravity is often called ‘third quantization’. However,we avoid using this term below due its status as a work-in-progress and the resulting lack ofa clear definition in the literature. We emphasize below that such QFT-like approaches correspond only to certain possiblechoices that might be made in interpreting the diagrams of Fig. 1. In contrast, distinctly For similar reasons, we also avoid use of the terms which have previously been applied, eg., “multiversefield theory” parenthetically mentioned in [3], “universal field theory” of [28], and “universe field theory”coined in [32]. – 2 –ifferent choices were (perhaps implicitly) made in recent comparisons of Jackiw-Teitelboimquantum gravity with ensembles of matrix models [9–13, 15], and (more explicitly) in generalarguments [14] that spacetime wormholes lead to superselection sectors for boundary partitionfunctions associated with states of a so-called ‘baby universe’ Hilbert space; see also [33, 34]and the Lorentz-signature discussion of baby universes in [35].Indeed, a key feature of the worldline formulation of QFT [36] is that the boundary con-ditions on worldline path integrals are associated with the arguments of correlation functions,and thus with the non-abelian algebra of quantum fields (see e.g., [37, 38] for applications).But the arguments of [14] for baby-universe superselection sectors imply that boundary condi-tions of quantum gravity path integrals correspond to the arguments of correlation functionsassociated with an abelian algebra of operators that can be simultaneously diagonalized onthe baby universe Hilbert space H BU . This apparent tension was highlighted in [32]. Toresolve this, below we focus on carefully identifying the steps in the QFT-like constructionsthat deviate from the framework defined in [14]. In particular, we will see that this issue is notrelated to any choice of spacetime signature, as both Lorentz- and Euclidean-signature con-structions can in principle lead to either sort of algebra. Instead, the critical issue is whetherthe inner product on the quantum gravity Hilbert space is constructed from an adjoint (or CPT conjugation) operation that leaves the set of allowed boundary conditions invariant.We will proceed by example, exploring a series of constructions one might use to relatepath integral amplitudes to some quantum gravity inner product in various simple models.Our main goal is to illustrate some key places where choices must be made, and where QFT-like approaches deviate from the framework of [14]. But this is only one step in analyzingthe treatment of quantum gravity path integrals. We will thus not concern ourselves withmaking the models particularly realistic. In particular, we will mostly study models whichdo not allow universes to split and join, so that our path integrals reduce to a collection ofcylinders.In fact, we will consider models of quantum gravity in which spacetime is one-dimensional(so that the above cylinders degenerate to become just line segments). Such models may thusbe called “worldline theories”. This choice was made for simplicity and also for ease ofcomparison with QFT. Since there is no concept of spatial boundary for one-dimensionalLorentzian spacetimes, our models are most analogous to studies of closed universes in higherdimensions. In particular, we encourage the reader to think of the quantum gravity Hilbertspaces described below as analogues of the ‘baby universe sectors’ of higher-dimensional mod-els discussed in [1–4, 14]. We will thus often refer to them as baby universe Hilbert spacesbelow. Comments on higher dimensional cases are interspersed throughout the text, but a fulltreatment of higher dimensional cases may require additional inputs beyond those discussedhere. Similar-but-different conclusions were reached in [1–3] by following the QFT-like 3rd quantization paradigmtogether and using an additional ‘locality’ assumption that restricts attention to an abelian sub-algebra, thoughthis assumption can then be questioned as in [3]. See also further comments in § – 3 –hoices ESFTs § §§ §
61) Proper Time (Lapse) Range ( R ± , R ) R + R + R
2) Spacetime (Worldline) Signature (E,L) E E/L L3) Restricted Boundary Conditions No Yes No4)
CPT requires extra Z ? No For E target No5) Target Space Signature E E/L any but E Table 1 : The choices explored below associated with transforming a worldline path integral into a candidatequantum gravity Hilbert space, and the options chosen to define so-called Group Averaged Theories (GATs), QFT-like theories, and Euclidean Statistical Field Theories (ESFTs). E and L denote Euclidean and Lorentzian signaturesrespectively. Other terminology will be explained in the sections below.
The set of options we explore is enumerated in Table 1, though we defer a full explanationof the terms used there to the sections below. We make no claim that this list is exhaustive.In particular, we primarily consider unoriented spacetimes, only occasionally commentingon the possibility of including an orientation (e.g., in parallel with the treatment of [39] forJackiw-Teitelboim gravity).We furthermore make no claim that any of the options described below correspond pre-cisely to the way that specific models were studied in [14] or in [9–13, 15]. We thus defer anydiscussion of the detailed connection between those references and the approaches below tothe discussion in § § §
3. We then proceed by examining various types ofconstructions in turn in §§ § To set the stage for our discussion we begin with a recapitulating the essence of the argumentfrom [14]. The path integral for any quantum system defines a map from a set of boundaryconditions to numbers, the amplitudes. The amplitudes depend on a prescription for thedynamics (the set of configurations to be summed over in the path integral and the corre-sponding weights, typically specified by an action), as well as the allowed set of boundaryconditions. We may superpose boundary conditions, turning the set of allowed boundaryconditions into a vector space on which the map to amplitudes acts linearly.– 4 – Φ || Φ (cid:105)(cid:104) Φ | Φ (cid:105) Figure 2 : Slicing open a quantum amplitude to reveal the bra and ket components.
Given the quantum path integral, we can extract from it a Hilbert space by cutting it openalong a codimension-1 slice. This follows from the convolutional property of path integrals;each cut corresponds to a resolution of the identity. The two halves of the path integralproduced by the cut each use the boundary conditions appropriate to that part of the pathintegral to construct a state in the Hilbert space or in its dual; i.e., a ket-vector or a bra-vector.The full path integral constructed by sewing them back together computes then the innerproduct between these bra- and ket- states, see Fig. 2. In other words, we split the boundaryconditions into ‘bra’ and ‘ket’ pieces (typically corresponding to future and past, respectively,when our cut is at some fixed time in QM or QFT), and the amplitudes define a bilinearproduct between these pieces. To obtain a Hilbert space (with an inner product between twoket-vectors, say) we must have an anti-linear map turning a ket boundary condition for thepath integral into a bra boundary condition, which squares to the identity. We may think ofthis as prescribing the action of a
CPT map on the space of allowed boundary conditions forthe path integral.Now, if the quantum gravity path integral is a sum over all topologies, then it naturallyallows topologies with an arbitrary number of boundaries. As a result, if BC is the space ofallowed boundary conditions at a single boundary, then any list b , . . . b n of elements b i ∈ BC (of any length n ) will define an allowed boundary condition for our quantum gravity theory.Furthermore, the path integral is to be computed by summing over all spacetimes withboundaries matching b , . . . , b n . Thus the ordering of the boundary conditions plays no role,and two lists that differ by a permutation should be viewed as defining the same boundaryconditions. Using (cid:104) b , . . . , b n (cid:105) to denote the path integral with boundary conditions b , . . . , b n ,we may then write (cid:104) b , b , . . . , b n (cid:105) = (cid:10) b σ (1) , b σ (2) , . . . , b σ ( n ) (cid:11) , (2.1)where σ ∈ S n is a permutation of the boundary conditions. This turns the vector space ofboundary conditions into an abelian algebra, with the product defined by disjoint union of– 5 –oundaries.As noted above, we should be able to cut open such a path integral to define a state. Dueto the presence of boundaries, there are various ways in which we could introduce such a cut.For simplicity let us introduce a cut that does not intersect any of the existing boundaries b , b , . . . , b n , but merely partitions them into two disjoint subsets. Since each piece of theresulting path integral should define a state on the cut, there should be states | a , . . . , a m (cid:105) associated with arbitrary lists of boundary conditions, where the symmetry (2.1) means thatwe must identify states that differ only by the ordering of the boundary conditions a , . . . , a m .And since the above cuts are closed surfaces, we should think of these states as describingclosed universes without boundary. As a result, it is traditional to call this the Hilbertspace H BU of ‘baby universes,’ with the idea that such closed universes may have somehowbeen ‘produced’ by some larger (infinite) ‘parent universe’ having a non-trivial asymptoticboundary.For each allowed boundary condition b ∈ BC , it is then natural to define an operator ˆ b on H BU that simply inserts an additional boundary with the stated boundary condition; i.e.,ˆ b | a , . . . , a m (cid:105) = | b, a , . . . , a m (cid:105) . (2.2)Since the ordering of the boundary conditions is unimportant, it is manifest that any twosuch operators commute:ˆ b ˆ b | a , . . . , a m (cid:105) = | b , b , a , . . . , a m (cid:105) = | b , b , a , . . . , a m (cid:105) = ˆ b ˆ b | a , . . . , a m (cid:105) . (2.3)Finally, there is one special state | HH (cid:105) which corresponds to the absence of boundaries, m = 0.All the states of H BU are then generated by the action of the algebra of boundary-insertingoperators ˆ b acting on | HH (cid:105) , | b , . . . , b n (cid:105) = ˆ b · · · ˆ b n | HH (cid:105) , (2.4)and linear combinations.As noted above, this is not quite enough to define the inner product on H BU , sincein addition we must choose an anti-linear ‘ CPT ’ operation acting on boundary conditions.Indeed, we will see below that a single set of amplitudes may be associated with severaldifferent Hilbert spaces, by making a different choice of this conjugation operation. Thischoice is equivalent to defining the adjoint of the boundary-inserting operators ˆ b (and theconjugate to the no-boundary state). We then have (cid:104) a , . . . , a m | b , . . . , b n (cid:105) = (cid:104) HH | ˆ a † · · · ˆ a † m ˆ b · · · ˆ b n | HH (cid:105) = (cid:68) a † , . . . , a † m , b , . . . , b n (cid:69) , (2.5)where we have assumed that the application of CPT to a list of boundary conditions a , . . . a m is given by applying CPT to the individual members of the list. This means that for any b ∈ BC , ˆ b † acts by inserting some boundary b † ∈ BC (reusing the adjoint notation from– 6 –he operator interpretation), so † acts on the space of connected boundaries and extends tomultiple boundaries in the simplest possible manner. This defines a sesquilinear product on the space of boundaries. For this to give a sensibleHilbert space, we must require that it is positive semi-definite; that is, the norm of any statethus computed is nonnegative. Under that assumption, we can define H BU as the completionof the span of states | b , . . . , b n (cid:105) in the given inner product. Since the inner product is requiredonly to be positive semi -definite, nontrivial linear combinations of the states | b , . . . , b n (cid:105) canbe ‘null’: they have zero norm, and hence are equal to the zero state.However, there are two more issues that we should consider. The first is to show thatour boundary-inserting operators ˆ b are truly well-defined on H BU . The potential issue herearises due to null states, since ˆ b is a well-defined operator on H BU only if it preserves thespace of null states. But this is straightforward to show from (2.5). The key observation isthat ˆ b acting to the right is equivalent to ˆ b † acting to the left, and have assumed that ˆ b † actsby adding some boundary b † : (cid:104) a , . . . , a m | ˆ b | b , . . . , b n (cid:105) = (cid:104) a , . . . , a m | (ˆ b † ) † | b , . . . , b n (cid:105) = (cid:68) a , . . . , a m , b † (cid:12)(cid:12)(cid:12) b , . . . , b n (cid:69) . (2.6)As a result, for any null state | N (cid:105) and any boundaries a , . . . , a m we have (cid:104) a , . . . , a m | ˆ b | N (cid:105) = (cid:68) a , . . . , a m , b † (cid:12)(cid:12)(cid:12) N (cid:69) = 0 , (2.7)where the last equality holds because | N (cid:105) is null. This means that the overlap of ˆ b | N (cid:105) withany state is zero, so ˆ b | N (cid:105) is also null, and thus ˆ b preserves the null space as desired.Secondly, we would also like to show that the ˆ b can be simultaneously diagonalized. Ingeneral these operators are not Hermitian, but they are normal , meaning that ˆ b commuteswith its adjoint ˆ b † , (cid:104) ˆ b, ˆ b † (cid:105) = 0. This follows from the fact that ˆ b † also acts by inserting aboundary b † , so we can apply (2.3) with b = b and b = b † . This means that we can applythe spectral theorem, so ˆ b is diagonalizable. In fact, all of the operators ˆ b and ˆ b † commute,and hence all ˆ b can be simultaneously diagonalized as desired. The baby universe Hilbertspace H BU has a basis of simultaneous eigenvectors | α (cid:105) for all boundary-inserting operatorsˆ b , labeled by some (continuous or discrete) parameters α :ˆ b | α (cid:105) = b α | α (cid:105) (2.8)for some b α ∈ C , for all b ∈ BC . These α -states give superselection sectors for the commu-tative algebra generated by boundary-inserting operators. In particular, this means that the no-boundary condition is left invariant, so the norm of | HH (cid:105) is given bythe path integral over closed spacetimes with no boundary whatsoever. We may also choose to normalize | HH (cid:105) ,which means defining the amplitudes to include a denominator of the path integral over closed spacetimes.Equivalently, we can integrate only over spacetimes without closed components, in the same way that vacuumdiagrams are removed by normalization in QFT. Defining H BU from the amplitudes of the algebra BC in the Hartle-Hawking state in this way is closelyanalogous to the GNS construction [40, 41] (see also [32, 34]). – 7 –t is clear that the above argument is very general. The key point is simply that quantumgravity inner products are given by path integral amplitudes as in (2.5) for some set of single-boundary boundary conditions BC that is invariant under the action of † . While this appearsto us to be a natural condition to impose on theories of quantum gravity, as we review belowit is certainly not the case for the construction of QFT from worldline path integrals. Thisis illustrated by the examples below in which we also discuss certain other choices that mustbe made to define a Hilbert space from quantum-gravity-like path integrals. We now describe a general framework that forms the backbone of our one-dimensional quan-tum gravity models. Thinking of a gravitational theory as a path integral over spacetimes,we first describe the amplitudes resulting from the sum over one-dimensional manifolds. Forthe sake of simplicity we will focus on the non-interacting limit (the free theory), where ourworldlines do not intersect with each other. We then discuss the choice of ‘matter’ degrees offreedom which lives on these spacetimes, giving the prominent example of a minisuperspacemodel obtained as a dimensional reduction of a theory in higher dimensions.These ingredients will however not suffice to characterize our quantum gravitational the-ories completely: there will be additional discrete choices that need to be made in order tofully specify the model as summarized in Table 1. We will explore these choices in detailsubsequently in §§ For given boundary conditions, our one-dimensional gravity amplitudes will be defined by anintegral over the compatible one-dimensional manifolds, perhaps with some ‘matter’ quantummechanics living on those manifolds. Fortunately one-dimensional manifolds and metrics aresimple: we have only intervals or circles, parameterized by their total proper length
T > T → ∞ limit of an interval.Since we are interested in the dependence on boundary conditions, we may ignore thecircles (which in any case only contribute an overall normalization; see footnote 3) and restrictto intervals. For the most part we take the spacetimes we sum over to be simply a union ofintervals, though we will take occasion to comment on the generalization to graphs, where weallow several intervals to be sewn together at their boundaries. Throughout our discussion, wewill use quantum gravity terminology so that the one-dimensional manifold is a ‘spacetime,’and a single such manifold represents a ‘universe’ (and not a ‘particle’). It will, however,sometimes be convenient to also refer to the spacetime as the ‘worldline’ of the universe.The boundary of a one-dimensional spacetime is a (zero-dimensional) collection of points.As above, we use BC to denote the set of allowed boundary conditions at any single suchpoint. Note that in one-dimensional gravity theories these will be conditions on the ‘matter’ If we were to sum over oriented spacetimes, we should also assign an orientation to the boundary, whichmeans a choice of sign for each point. – 8 –elds alone as the space of zero-dimensional metrics is trivial. For example, when one definesa one-dimensional quantum gravity model by Kaluza-Klein reduction of a higher-dimensionaltheory, most of the features the gravitational theory in fact become part of the Kaluza-Kleinmatter sector, with only an overall notion of proper time left to be treated as one-dimensionalgravity.Following the discussion of §
2, a general quantum gravity boundary condition is anunordered list of elements of BC . The associated quantum gravity amplitude is to be definedby summing over all one-dimensional manifolds compatible with the boundary conditions.Since for now our one-dimensional manifolds are collections of n intervals, in the absence ofinter-universe interactions, the number of boundaries must be an even number 2 n for theamplitude to be nonzero. The topology of spacetime is specified by assigning the boundariesinto n pairs, where an interval connects the points in each pair. Each resulting pair ofboundary conditions then specifies a single-universe amplitude computed by integrating overmetrics and corresponding matter fields on the interval. The full quantum gravity amplitude isthen formed by multiplying together the n single-universe amplitudes defined by each pairingand then summing over pairings. For b , . . . , b n ∈ BC , the associated quantum gravityamplitude may thus be written (cid:104) b , . . . , b n (cid:105) = (cid:88) pairings of { b ,...,b n } (cid:89) pairs { b i ,b j } A SU ( b i , b j ) , (3.1)where A SU ( b i , b j ) = (cid:104) b i b j (cid:105) is the ‘single-universe amplitude’ associated with integrating overthe parameters of a single interval with the boundary conditions specified by the pair b i , b j .As in [14], we treat all boundaries as being distinguishable so that there are no additionalsymmetry factors in (3.1). Our non-interacting universe assumptions implies (3.1) has theform of Wick contractions, so that in simple cases the quantum gravity amplitudes can bewritten as Gaussian integrals over the space of boundary conditions with a covariance matrixspecified by the single-universe amplitudes A SU , and the baby universe Hilbert space will bea Fock space built on a single-universe Hilbert space.Note in particular that (3.1) is invariant under arbitrary permutations of the single-universe boundary conditions b , . . . , b n . This may remind the reader of bosonic quantumfield theory, and thus raise questions of whether other possibilities might be allowed as well.One might also ask about further modifications of (3.1). While such questions may be ofinterest, we will not explore them below. Instead, we take the structure embodied in (3.1) asgiven and consider additional choices that must be made in order to both define the single-universe amplitudes used in (3.1) (discussed below) and in order to construct the quantumgravity Hilbert space from the above quantum gravity amplitudes (discussed in §§ A SU ,which requires two ingredients. The first is a definition of the integral over spacetime (world- For oriented spacetimes, the endpoints of an interval must have opposite orientation. The spacetimetopologies are equivalent to maps from negatively oriented points to positively oriented points, which must beequal in number. – 9 –ine) metrics on each interval. In practice, this will involve specifying whether the signatureis Lorentzian or Euclidean and choosing a range of integration for the length (or proper time) T of each interval. The second is the specification of the ‘matter’ model and its associateddynamics and boundary conditions.Explicitly, we can write A SU with an integral over spacetimes (labeled by their length T ) and over matter fields, labeled x : A SU ( b , b ) = (cid:90) D dT (cid:90) b b D x e η S matter [ x ; T ] , (3.2)where η = + i and η = − b , b indicate some choice of boundary conditions for matter fields x at each end of the worldline.We may want to add a ‘gravitational action’ depending only on metrics, but the only localpossibility in one dimension is a ‘cosmological constant’ proportional to T , which we havechosen to absorb as a constant shift of the matter Lagrangian. The flat measure dT overmetrics is fixed by locality [42] (and may also be obtained by starting with a general metricand gauge-fixing one-dimensional diffeomorphisms by the Faddeev-Popov method). Finally,we must choose a range of integration D for the lapse or proper time T , with the only choicesrespecting locality being R , R + or R − .To define the amplitudes, the main choices open to us are explicit in (3.2): • The worldline signature η . • The range of proper time D . • The matter dynamics (fields x and action S matter ). • The allowed boundary conditions b for matter.In addition, to define the Hilbert space we must specify the CPT operation † . We will outlinethe class of matter theories we study below, and the remainder of the paper will be organizedby various permutations of the other choices, and devoted to discussing their consequencesfor the resulting baby universe Hilbert space.There are some additional choices that are less important for our purposes, and will onlybe mentioned parenthetically. First, we could choose our spacetimes to carry an orientation;we will mostly concentrate on unoriented worldlines, which requires restricting to mattertheories with time-reversal symmetry (c.f., the discussion of JT gravity in two dimensions[39]). Likewise, we could choose to have worldline supersymmetry, which for instance providesan example (once we make some of our discrete choices) of a topological sigma model [43](see [44] for the classic review of these developments). Finally, a more radical generalizationis to sum not only over disjoint unions of intervals but general graphs, which we will haveoccasion to comment on in various cases. – 10 – .2 The matter theory The main choice which determines the amplitudes A SU will be the matter theory. We candescribe this either by a path integral, specifying matter fields and Lagrangian (as we havedone above), or by a Hilbert space of matter states and a Hamiltonian H . In the Hamiltonianformulation, the contribution to A SU from an interval with proper length T will be givenby matrix elements of e − iHT or e − HT for Lorentzian or Euclidean spacetimes respectively,between states determined by the boundary conditions.In describing each class of examples below, we will focus mostly on a matter theorytaking the form of a sigma-model, so that the field content is a map from the one-dimensionalspacetime to some target space M target . We take this target space to be equipped with ametric g µν ( x ), and thus with a Laplacian (or wave operator) ∇ . The matter action takesthe form S matter = (cid:90) dτ (cid:20) g µν d x µ d τ d x ν d τ − U ( x ) (cid:21) , (3.3)allowing for the possibility of a potential U ( x ) on M target . Equivalently, the Hamiltonian is H = − ∇ + U ( x ) , (3.4)acting on the Hilbert space L ( M target ).The most general possible boundary condition is to give a wavefunction valued in M target ,that is a state in the matter Hilbert space L ( M target ). This is a linear combination ofboundary conditions that fix the fields to take a specific value x ∈ M target at the correspondingendpoint of spacetime, corresponding to delta-function wavefunctions | x (cid:105) matter , where we usethe matter label to avoid confusion with H BU . Thus, we will label our boundary conditionsas points x , so the amplitudes will be denoted (cid:104) x , . . . , x n (cid:105) for lists of points in M target .Models of this kind naturally arise in the so-called mini-superspace truncation of gravity[45, 46] in which one restricts a higher-dimensional model of gravity to e.g., homogeneousspacetimes. At this stage we will not restrict the signature of M target , so in particular theHamiltonian may not be bounded below. We will however require two properties that are not obviously natural in the most straight-forward construction of such minisuperspace models. In particular, we will take M target to begeodesically complete, so that ∇ defines an essentially self-adjoint operator on L ( M target ).If M target is Lorentzian then ∇ will have a continuous spectrum, while for Euclidean targetone may end up with a discrete spectrum if M target is compact. To provide additional context for this treatment of the one-dimensional gravitational sector,let us briefly discuss an example of a minisuperspace model (see e.g., [47, 48]). The dynamicsof spatially homogeneous 3 + 1 Lorentz signature Einstein-Hilbert gravity on a 3-torus withvanishing cosmological constant, the
Bianchi I model , is closely related to that of a free We will take the metric to have mostly positive signature for Lorentzian targets. – 11 –assless particle in 2+1 Minkowski space (say, with inertial coordinates x , x , x ), see [49].In particular, with Y i for i = 1 , , γ ab dX a dX b = − N ( t ) dt + e x ( t ) ( e X ( t ) ) ij dY i dY j (3.5)Here X ( t ) is a diagonal matrix after fixing the diffeomorphisms symmetries sans time repa-rameterization; specifically we take X ( t ) = diag { x ( t ) + √ x ( t ) , x ( t ) − √ x ( t ) , − x ( t ) } to describe the anisotropies. The overall scale-factor of the torus is given by e x . Usingthe standard lapse function N that measures proper time and momenta p , p , p conju-gate to x , x , x , viz., p µ = d x µ d t , the Einstein-Hilbert Lagrangian may be written as (nb: µ ∈ { , , } ) √− γ γ R = ˙ x µ p µ − N H . (3.6)Here we introduced, H , the Hamiltonian constraint, given by H = e − x
24 ( − p + p + p ) . (3.7)Now, H vanishes on-shell due to the equation of motion obtained by varying N . Owing tothe prefactor e − x , the constraint H tends to generate evolution that reaches a cosmologicalsingularity at x = −∞ . However, using a rescaled lapse N = N e − x and the associatedrescaled constraint H = 12 (cid:0) − p + p + p (cid:1) , (3.8)we may recast the dynamics in the advertised form of a standard massless particle in 2+1Minkowski space R , .In particular, reduction to 0+1 dimensions yields a ‘matter’ theory defined by (3.8), whichwe may think of a sigma-model with target space M target = R , . The gravitational sectorof the reduced theory naturally has Lorentz signature, and we define the ‘proper time’ T ofthe reduced theory by dT = N dt . Note that this differs from the natural notion of propertime N dt associated with the 3+1 geometry, though it corresponds to the usual notion for amassless particle on R , .We offer this model as an illustration of way to obtain Lorentz signature target spacesfor the worldline theory. Note that from the higher dimensional point of view we are onlyattempting to keep track of a subset of gravitational degrees of freedom in the minisuper-space approximation. One furthermore is also holding the higher dimensional topology fixed.Given the natural tendency of gravitational dynamics to lead to cosmological singularities, M target is typically not geodesically complete when the metric on it is defined by the con-straint associated with evolution in proper time. For now we note only that this difficulty cantypically be circumvented by using a rescaled ‘conformal time’ dynamics near the singularity(which amounts to the use of singularity-avoiding coordinates), so that we can readily con-struct gravity-inspired models of the mathematical form described here. But we will returnto discuss the physics of this rescaling in § Euclidean Statistical Theories
For our first two examples, discussed in this section and the next, we will choose the worldlinequantum gravity theory to have Euclidean signature. While this is perhaps not as interestingas the Lorentzian theories discussed later from the point of view of higher-dimensional modelsof quantum gravity, we present it first as a clean and familiar setting to illustrate the impactof certain choices made to define the Hilbert space.In fact, in both this section and in § H BU which is very natural if we interpret our field theory as a classical statistical model. A differentset of choices (designed to make contact with a quantum field theory Hilbert space) will bediscussed in § Taking our spacetimes to have Euclidean signature means that the matter amplitudes on aninterval of length T are given by the matrix elements (cid:104) x | e − HT | y (cid:105) matter . Our use of unorientedworldlines means that we have an x ↔ y symmetry, so these matrix elements are symmet-ric and real. After integrating over one-dimensional metrics, the single-universe amplitudes A SU ( x, y ) that enter the quantum gravity amplitudes (3.2) thus take the form A SU ( x, y ) = (cid:90) D dT (cid:104) x | e − HT | y (cid:105) matter (4.1)for some choice of integration domain D ⊂ R . The three local options are just the positivereals R + , the negative reals R − , or the entire real line. But for convergence, we require thatthe Hamiltonian H is positive, and D = R + (or H is negative, in which case we may replace H → − H w.l.o.g.). For a sigma-model as in (3.4), this means that the target space M target must have Euclidean signature. With this choice, the resulting amplitudes are given by thematrix elements of the inverse of the Hamiltonian: A SU ( x, y ) = (cid:90) R + dT (cid:104) x | e − HT | y (cid:105) matter = (cid:104) x | H | y (cid:105) matter . (4.2)In other words, A SU ( x, y ) is the Green’s function for the Hamiltonian H (with appropriatefall-off conditions if M target is non-compact).For general amplitudes, we recall that (3.1) is computed from (4.2) from Wick contrac-tions, and can thus be written in terms of an appropriate Gaussian path integral over a realscalar field Φ valued in M target : (cid:104) x , x , . . . , x n (cid:105) = N (cid:90) D Φ Φ( x ) · · · Φ( x n ) e − I [Φ] , where I [Φ] = (cid:90) M target dx (cid:20) ( ∂ Φ( x )) + U ( x )Φ( x ) (cid:21) , (4.3)– 13 –ith N a normalization constant. In other words, our quantum gravity amplitudes are thecorrelation functions of a free Euclidean field theory on M target , consisting of a scalar field Φwith action I . As already emphasized, the amplitudes alone do not provide sufficient information to con-struct the baby universe Hilbert space H BU . In particular, in order to define the innerproduct we must additionally choose a conjugation operation, ‘ † ’, acting on boundary condi-tions. Here, we will make the most obvious choice, that † acts trivially on the x boundaryconditions: x † = x. (4.4)From this, we can construct and interpret the baby universe Hilbert space. As notedearlier, since the general inner product is computed by Wick contractions, H BU is a Fockspace built on the single-universe Hilbert space, spanned by | x (cid:105) for x ∈ M target . A generalsingle universe state is a superposition (cid:82) dx F ( x ) | x (cid:105) for some complex-valued function F on M target . The inner product of two such states F , G is constructed from the single-universeamplitude (4.2), as (cid:82) dxdy G ∗ ( x ) A SU ( x, y ) F ( y ). This is positive-definite, which can be seenby decomposing F in a basis of eigenfunctions of H and using positivity of the correspondingeigenvalues. In particular, there are no nontrivial null (zero-norm) states.There is in fact a nicer characterization of the full Hilbert space H BU , as a space of(complex-valued) functionals F of our scalar field Φ taking values in M target . Formally, wemay write a functional F in terms of its Taylor expansion, F [Φ] = F + (cid:90) dx F ( x )Φ( x ) + (cid:90) dx dx F ( x , x )Φ( x )Φ( x ) + · · · , (4.5)where F n , the n th functional derivative of F , is a symmetric function M n target → C . Inparticular, the one-universe Hilbert space considered above corresponds to linear functionals,where all F n vanish for n (cid:54) = 1. We can then write any state of H BU in terms of a functional F , as | F (cid:105) = F | HH (cid:105) + (cid:90) dx F ( x ) | x (cid:105) + (cid:90) dx dx F ( x , x ) | x , x (cid:105) + · · · , (4.6)and the inner product of two such states is given by (cid:104) G | H (cid:105) = N (cid:90) D Φ G ∗ [Φ] F [Φ] e − I [Φ] . (4.7)This holds term-by-term in the Taylor expansion, since both Gaussian integrals and ourquantum gravity amplitudes (3.1) are computed by Wick contractions.In this form, we may see that H BU has a simple interpretation if we take the pathintegral to define a classical statistical system, such as the continuum limit of an Ising-like model. The path integral (cid:82) D Φ · · · e − I [Φ] defines a probability distribution (perhaps aBoltzmann distribution where I [Φ] is β times the energy of the given field configuration).– 14 –he functionals F are then the observables of such a model, which are random variablesdepending on the probability distribution. The inner product then gives us the covariancematrix of these random variables, (cid:104) G | F (cid:105) = Cov( F, G ) . (4.8)Thus, H BU is a standard construction in probability theory, the Hilbert space of randomvariables (of finite variance).Moreover, it is now simple to interpret the superselection sectors (or α -states) of H BU :they are states where the field Φ takes a definite value, and the eigenvalues of boundary oper-ators ˆ x are given by Φ( x ). The action e − I [Φ] gives the square of the overlap of (appropriatelynormalized) α states with the no-boundary state | HH (cid:105) , so the probability distribution of su-perselection sectors is precisely identified with the original distribution defining the classicalstatistical theory. One can exemplify the construction above with many familiar examples where M target iscompact. We could just as well also choose a non-compact target geometry. For definiteness,consider the Wick-rotation of the Bianchi I model introduced in § x → ix yields H = p + p + p (4.9)in terms of the Euclidean target space BC = M target = R .From (4.2), it is clear that we may generalize this approach to allow any matter theorywith a positive-definite Hamiltonian, a sufficiently well-defined resolvent operator, and time-reflection symmetry. Notably, this excludes the case of Lorentz-signature target spaces asnaturally occur in quantum gravity models. However, if there is an appropriate Z symmetryas above, one can Wick-rotate such models to Euclidean signature and then apply the aboveapproach. As before, the time-reflection symmetry is required due to our choice to sum overunoriented spacetimes.Considering instead oriented spacetimes removes this requirement (for example, allowingus to include a background magnetic field for our particle), both giving a time orientation forthe dynamics and an orientation (a sign) for boundary conditions. In that case, we denotea Dirichlet boundary condition with positive orientation by Φ( x ), and one with negativeorientation by ¯Φ( x ). Any spacetime is a union of intervals connecting a Φ boundary with a¯Φ boundary, so the amplitudes are (cid:68) Φ( x ) · · · Φ( x n ) ¯Φ( y ) · · · ¯Φ( y m ) (cid:69) = N (cid:90) D Φ D ¯Φ Φ( x ) · · · Φ( x n ) ¯Φ( y ) · · · ¯Φ( y m ) e − (cid:82) M target ( ∂ ¯Φ)( ∂ Φ) , (4.10) Here we again use the ‘rescaled lapse’ N = e − x N . – 15 –hich is just the path integral over a complex scalar field on M target . In this case we choose[Φ( x )] † = ¯Φ( x ) and [ ¯Φ( x )] † = Φ( x ) to make the quantum gravity inner product positive-definite.The generalization of this case to allow general graphs is now straightforward and familiar.We merely replace the quadratic action in either (4.3) or (4.10) with a more general functionalof Φ. Of course, the Euclidean fields Φ( x ) remain simultaneously diagonalizable. Such modelshave been discussed in the past, see e.g., [50, 51].We can arrive at interesting possibilities in some cases by restricting the allowed set ofboundary conditions. Consider a case where we take M target to be Euclidean hyperbolic space H d , and require boundary conditions for the worldlines to end on the asymptotic boundary S d − = ∂ H d . Take the non-interacting case (without vertices) and with constant potential U = m . The resulting quantum gravitational amplitudes are the conformally invariantcorrelators of Euclidean mean field theory (or a ‘generalized free theory’) defined on theconformal boundary ∂ M target [52]. We thus arrive at a critical Euclidean statistical theorydefined on the boundary of target space. This example is interesting primarily as the free limitof a gravitational theory with AdS asymptotics, for which the boundary Euclidean statisticaltheory has a local description by the AdS/CFT correspondence: see further discussion in § In § § § From the gravitational perspective, since we are taking the same amplitudes as in §
4, weshould ask what other choices were made to construct the Hilbert space, and consider otheroptions. The most obvious is our adjoint operation † . A simple possible generalization of thechoice made in § † to act locally on M target , so x † = σ ( x ) for some function σ : M target → M target . Since the adjoint must square to the identity, so must σ , so it isrequired to be an involution: σ ( σ ( x )) = x . This means that σ can be represented as a self-adjoint and unitary operator acting on the Hilbert space L ( M target ) of our matter theory.To make the inner product on the one-universe sector conjugate symmetric ( A SU ( σ ( x ) , y ) = A SU ( σ ( y ) , x ) ∗ ), we also require this operator to commute with the matter Hamiltonian H , This depends on a choice of conformal frame (i.e., choice of metric within the conformal class of theboundary S d − = ∂ H d ), related to how we regulate the infinite length of worldlines. – 16 –o σ should be a Z symmetry of M target (preserving the metric and potential). This is ofcourse a constraint on M target as well as σ , since not every matter theory will admit such asymmetry.The simplest example to have in mind is the product space M target = Σ × R (with aconstant potential U = m ), where σ acts trivially on Σ, and reflects the coordinate t E for the R factor: σ ( t E ) = − t E . We will see that this is the most important example forthe application to QFT, since the corresponding amplitudes are the Euclidean correlationfunctions for the vacuum state of a scalar field of mass m on the spatial manifold Σ, with t E interpreted as the Euclidean time. A specific example of this is Bianchi I model with M target = R , after Wick rotation x → − i t E .However, this presents an immediate problem for our inner product on the one-universeHilbert space. This inner product is computed by the matrix elements of σ H , but this willnever be a positive operator, so the inner product will not be positive-definite. The reasonis that we can diagonalize σ and H simultaneously, so we may choose eigenstates of H withdefinite parity ± σ . But H is a positive operator, so all the eigenstates with negativeparity will have negative norm. If all eigenstates have positive parity, it means that σ is theidentity and we return to the construction of § σ fixes a hypersurface Σ, and M target − Σ has two connected components M − target and M +target = σ ( M − target ). We then define our Hilbert space to be spanned by states | x , . . . , x n (cid:105) ,but now only allowing x i ∈ M − target . We illustrate this in Fig. 3. In the simple example M target = Σ × R , the hypersurface Σ lies at the moment of time-reflection symmetry t E = 0, M +target and M − target are the regions t E > t E < t E only.Note that with this restriction on boundary conditions, our adjoint operation x † = σ ( x )does not preserve the space M − target of single-universe boundary conditions to which (2.5)applies. This construction therefore violates an implicit assumption of [14]. We will later seesome implications for the operators ˆ x .To see that these choices result in a positive definite inner product and to make contactwith QFT constructions, we return to the path integral expression (4.3) for the amplitudes.We may write our inner product as (cid:104) y , . . . , y n | x , . . . , x m (cid:105) = N (cid:90) D Φ Φ( σ ( y )) · · · Φ( σ ( y n ))Φ( x ) · · · Φ( x m ) e − I [Φ] , (5.1)where x , . . . , x m , y , . . . , y n are points in M − target . To consider general states (linear com-binations of states | x , . . . , x m (cid:105) ), we may denote them as functionals of fields as in (4.6),with (cid:104) G | F (cid:105) = N (cid:90) D Φ G ∗ [Φ ◦ σ ] F [Φ] e − I [Φ] , (5.2)but now F, G are functionals depending only on restriction of the field Φ to M − target . Since F [Φ] depends only on the field in M − target and G ∗ [Φ ◦ σ ] only on the field in M +target , we– 17 – M − target M +target σ (a) A target space M target with reflection symmetry σ . The reflection fixes the surface Σ , which splits the targetspace into two pieces M ± target . | x (cid:105) ∼ x × , | y (cid:105) ∼ y × −→ (cid:104) y | x (cid:105) ∼ x × σ ( y ) × (b) The single-universe states | x (cid:105) and | y (cid:105) are defined by a choice of points x and y (indicated by the crosses × ) in thehalf-space M − target . The inner product (cid:104) y | x (cid:105) is computed by the worldline path integral, with boundary conditionsspecifying that worldlines end at x and σ ( y ) = y † . More general states are defined by including multiple insertionsin M − target , and most generally by superpositions (both summing over the number of insertions and integrating overtheir locations with some weighting). Figure 3 may split the path integral into two pieces, integrating separately over fields Φ ± restrictedto the respective regions. These are only identified at the common boundary Σ, so we haveΦ ± | Σ = Φ Σ , and the inner product is written as the residual path integral on Σ: (cid:104) G | F (cid:105) = (cid:90) Σ D Φ Σ Ψ ∗ G [Φ Σ ]Ψ F [Φ Σ ] , (5.3)where Ψ F [Φ Σ ] = √N (cid:90) Φ − | Σ =Φ Σ D Φ − F [Φ − ] e − I − [Φ − ] . (5.4)The path integral defining Ψ F is performed over fields Φ − on M − target with the specified valueson Σ, and I − is the action in (4.3), but with the integration restricted to M − target . In thisform, the inner product is manifestly positive semidefinite, since the norm of the state F isgiven by integrating the positive functional | Ψ F (Φ Σ ) | with respect to a positive measure. Interestingly, it is more complicated to show positive-definiteness working directly within the worldlineformalism. This is related to the comment in [14] that it is unclear what conditions on the quantum gravity – 18 –ndeed, we can interpret (5.4) as the path integral computation of a Schr¨odinger-picturewavefunctional on the surface Σ, and (5.3) as the inner product of two such wavefunctionals.In the example M target = Σ × R , the no-boundary state | HH (cid:105) constructs the vacuum of theQFT at time t E = 0, and nontrivial boundary conditions for worldlines produce excited statesby inserting (Euclidean time-ordered) operators in the lower half-space t E <
0. Our choicesare essentially equivalent to the Osterwalder-Schrader construction of the QFT Hilbert spaceon the spatial slice Σ from Euclidean correlation functions.
Unlike in § H Φ( x ) = 0 where H is the matter Hamiltonian for our one-dimensional quantumgravity theory. Importantly, this holds only at separated points: when field insertions collidethis may be violated by contact terms. Indeed, these contact terms explain why the fieldequations did not give rise to null states in § x, y ∈ M − target , operators Φ( x ) and Φ( σ ( y )) can never collide, so inner products donot produce contact terms.More explicitly, let us focus on the one-universe Hilbert space, and consider states | f (cid:105) = (cid:82) dx f ( x ) | x (cid:105) , where f is has compact support contained in M − target . In particular, we maychoose a function f = Hh , where the support of h is contained in M − target . Now the innerproduct with another state | g (cid:105) = (cid:82) dx g ( x ) | x (cid:105) is given by (cid:104) g | f (cid:105) = (cid:104) g ◦ σ | H | f (cid:105) matter = (cid:104) g ◦ σ | h (cid:105) matter = 0 , (5.5)where | f (cid:105) matter is the state in the matter Hilbert space with wavefunction f (with an L innerproduct on M target ). The final inner product vanishes because the support of h and g ◦ σ are disjoint, lying within M − target and M +target respectively. Hence, | f (cid:105) has vanishing innerproduct with any state, so it must be null: | f (cid:105) = 0. To connect this with the field equations H Φ( x ) = 0, the state is associated with the functional (cid:82) dx Hh ( x )Φ( x ) = (cid:82) dx h ( x ) H Φ( x )(where we may ‘integrate by parts’ because H is a symmetric operator on L ( M target )).While at first the states of the one-universe Hilbert space would appear to be determinedby functions f on M − target , such null states mean that the independent states are in factdetermined by far less data: a function only on Σ (see appendix A for a non-redundantcharacterization of states in terms of a function Σ → C ). Thus, the QFT Hilbert spaceconstructed in this section is in a sense much smaller than that discussed in §
4, with one-universe states determined by a function on the submanifold Σ of M target . Finally, we briefly discuss the construction of operators in this formalism. From the generaldiscussion of §
2, one might expect to have boundary-inserting operators ˆ x acting on the baby path integral are required for positive-definiteness of the full quantum gravity inner product. We include aworldline argument for positive-definiteness of the inner product in Appendix A for the Gaussian case. – 19 –niverse Hilbert space. But in fact, these operators are not well-defined, since they do notpreserve the space of null states. For example, consider acting on the null state | f (cid:105) with f = Hh in (5.5) with ˆ x , inserting a boundary at a point x ∈ M − target , and take the overlapwith the no-boundary state: (cid:104) HH | ˆ x | f (cid:105) = (cid:104) x | H | f (cid:105) matter = (cid:104) x | h (cid:105) matter = h ( x ) . (5.6)This will be nonzero for some choice of h , so ˆ x | f (cid:105) is not a null state. As a result, the boundary-inserting operator ˆ x does not give a well-defined operator on the baby universe Hilbert space,where we have performed a quotient by null states.Such a result was possible only because our Hilbert space does not obey the axioms of[14]. The particular failure is the invariance of the set of allowed boundary conditions underthe CPT operation † , which means that the argument of (2.7) is inapplicable.This is in fact perfectly in line with expectations from QFT. We would expect theoperators ˆ x to be associated with the quantum fields ˆΦ( x ), but these do not give a well-defined operator algebra on a Euclidean space. For example, if M target = Σ × R and weuse the usual quantization with respect to Euclidean time t E , products of field operators aresensible only when they appear in Euclidean time order. This above discussion is readily generalized to any context where quantum field theory is well-defined and has the required Z reflection symmetry. For example, we may discuss complexscalar fields by using oriented worldlines, or generalize the amplitudes to general Feynmangraphs in the manner of interacting quantum field theory.Note, however, that gravitational models will not typically admit a time-reflection sym-metry (for example, the Kantowski-Sachs model introduced later in (7.3)). Indeed, a Z symmetry reversing time would usually be a relation between the physics of large universesand of small universes, so there are few semiclassical gravitational models to which this for-malism could apply. However, it might be interesting to consider whether — at least in somecases — string-inspired models with some suitable notion of T-duality (see e.g. [57]) couldprovide the required symmetry. This is associated with the well-known fact that while the Osterwalder-Schrader reconstruction directlyreconstructs the states of QFT, it does not provide a similarly direct construction of the operator algebra. More precisely, the domain of ˆΦ( x ) and the image of ˆΦ( y ) are disjoint if x is in the Euclidean future of y (except in the case when M target is one-dimensional). There are some boundary-inserting operators ˆ x that are well-defined (though unbounded) on the babyuniverse Hilbert space, namely when x lies on the σ -invariant slice Σ. Since this set is invariant underour CPT operation, the general argument of [14] applies. And indeed, these operators are self-adjoint andmutually commuting, hence simultaneously diagonalizable (with eigenstates corresponding to delta-functionwavefunctionals in (5.3)). – 20 – Group Averaged Theories
We now turn to the case where the worldline is taken to have Lorentz signature. We will beginby discussing a framework that may be unfamiliar to many practitioners of QFT or stringtheory. It is however inspired by a popular approach to studying single-universe quantumgravity models by treating them as constrained systems [58–65] and by treatments [66] oflinearization-instabilities in quantum gravity.The single-universe amplitudes A SU ( x, y ) that enter the quantum gravity amplitudes(3.1) take the form A SU ( x, y ) = (cid:90) D dT (cid:104) x | e − iHT | y (cid:105) matter (6.1)for some choice of integration domain D ⊂ R . The three options respecting worldline localityare the positive reals R + , the negative reals R − , or the full real line R . We here discuss thelatter choice ( D = R ), which fits into the so-called ‘group averaging’ paradigm discussed in[58–66] (sometimes under other names). With this choice, we may write A SU ( x, y ) = (cid:104) x | δ ( H ) | y (cid:105) matter , (6.2)which shows that choosing D = R imposes the constraint H = 0 in a strong sense.As an example, consider the mini-superspace truncation of the Bianchi I model discussedin §
3, where the Hamiltonian (3.8) (in rescaled lapse variable) corresponds to that of a freeparticle motion in R , . The single universe amplitudes compute the matrix elements of theon-shell constraint δ ( p ). Any general linear combination (cid:82) R , f ( y ) | y (cid:105) of the allowed bound-ary conditions is effectively projected into the space of solution of the quantum Hamiltonianconstraint H | ψ (cid:105) = 0.From (6.2) it is manifest that our single-universe amplitudes are the matrix elementsof a positive operator on the matter Hilbert space. As a result, we can define a positive-definite quantum gravity inner product by taking CPT to act trivially on M target ; i.e., x † = x . In contrast, integrating T only over a half-line would generate complex single-universeamplitudes from which the construction of a good Hilbert space is more complicated (see § H | ψ (cid:105) = 0, we may obtain a moreexplicit description of our theory by working directly with such solutions. In particular, forthe Bianchi I model since δ ( p ) = 1 | p | δ (cid:18) p − (cid:113) p + p (cid:19) + 1 | p | δ (cid:18) p + (cid:113) p + p (cid:19) , (6.3)it suffices to use the plane wave solutions (cid:104) x | p , p ; η (cid:105) = e − i η x √ p + p e i ( x p + x p ) (6.4)for η = ± and to replace (6.2) by the ‘projected’ amplitudes (cid:101) A SU ( p (cid:48) , p ) = 1 | p | δ ( p − p (cid:48) ) δ ( p − p (cid:48) ) δ ηη (cid:48) , (6.5)– 21 –here on the left-hand side p = ( p , p , η ) and p (cid:48) = ( p (cid:48) , p (cid:48) , η (cid:48) ). In particular, we emphasizethat the group-averaging approach keeps both positive- and negative-frequency solutions tothe constraint and treats both on an equal footing.The full quantum gravity Hilbert space can now be described succinctly by using theobservation that (3.1) is just the result of performing a Gaussian integral over the space ofplane wave solutions p with covariance given by the right-hand side of (6.5). Consistentwith the general argument from [14], the allowed boundary conditions p then define a set ofsimultaneously-diagonalizable operators on this Hilbert space. The above construction can be used in great generality. As it clear from (6.2), it requiresonly a matter Hamiltonian H with continuous spectrum that includes zero and whose matrixelements are symmetric (so that A SU ( x, y ) = A SU ( y, x )). The latter requirement is tanta-mount to requiring the matter to have a time-reversal symmetry, and is a consequence of oursumming over unoriented spacetimes. This condition would be dropped if we instead summedover oriented spacetimes.In particular, the above conditions on H allow the case of sigma-models with general(geodesically complete) Lorentzian target space, or in fact any signature with additional ‘time’directions (or indeed none) so long as the sign of H remains indefinite. We are similarly freeto add a potential to H that preserves continuity of the spectrum and the inclusion of theeigenvalue zero. Using a rescaled notion of lapse as described in § H | ψ (cid:105) = 0 with some non-Gaussian integral over the same space of constraints.Similarly, one can allow the constraint to depend on the ‘coupling constants’ g that controlany non-Gaussianities, so that the space of random variables at each g is defined by solving anew constraint H g | ψ (cid:105) = 0. In each case, the allowed boundary conditions continue to definesimultaneously diagonalizable sets of operators. We leave for future investigation whether thediagrammatics of such theories matches expectations from the quantum gravity path integral,though we will return to comment further on the physics of quantum gravity constraints in § The construction of § R , in the Bianchi Icase, for example) as one might have expected form the worldline formalism of QFT [37, 38].In this section we examine the choices one might take (different from §
6) to pursue the analogywith QFT.We first note that the amplitudes defined from our GAT are not the usual correlationfunctions of QFT. The one-universe amplitudes (6.2) are the matrix elements of δ ( H ), whilethe usual two-point functions of free fields are Green’s functions of the matter Hamiltonian(inverses of H ). This is a result of integrating the lapse T over the entire real line, D = R .We can consider instead the choice of integrating only over a half-line, say D = R + : A SU ( x, y ) = (cid:90) R + dT (cid:104) x | e − iHT | y (cid:105) = (cid:104) x | i ( H − i(cid:15) ) | y (cid:105) , (7.1)Here the integral gives a well-defined a distribution (associated with the Fourier transformof a step function) which we have written in terms of in i(cid:15) prescription. This is in factgives a standard description of iG F where G F is the Feynman Green’s function (see e.g.,[68]). In particular, the path integral is well-defined for Hamiltonians with a continuousspectrum, which need not be bounded, and computes matrix elements between states withan appropriately smooth and rapidly-decaying energy representation. The result can also bewritten in the form 1 i ( H − i(cid:15) ) = P iH + π δ ( H ) , (7.2)where P denotes the principal value distribution.Due to the imaginary part of (7.2), using (2.5) as written, and taking CPT acting triviallyon M target , one no longer defines a real and positive quantum gravity inner product. YetQFTs do have a positive inner product as well as a notion of CPT -conjugation (e.g., for realscalar fields Φ( x ) † = Φ( x )). Furthermore, in the free case that is relevant when we excludenon-trivial graphs, it has precisely the structure described by (3.1).The explanation is that in QFT the amplitude (7.1) does not define an inner productbetween states Φ( x ) | Ω (cid:105) and Φ( y ) | Ω (cid:105) for some ‘vacuum’ | Ω (cid:105) (we discuss the state | Ω (cid:105) below).That inner product would be given by the expectation value of Φ( y )Φ( x ), with operatorsordered as written (a Wightman function). But (7.1) would instead be interpreted as a time-ordered correlation function: the expectation value of T { Φ( x )Φ( y ) } , where the ordering ofoperators depends on their order in Lorentzian time. Choosing the lapse to run over thenegative reals gives us the anti-time-ordered correlation function (the complex conjugate ofthe time-ordered correlator). In this QFT language, our choice of integrating over the entirereal line in § x )Φ( y ) + Φ( y )Φ( x ).To recover the QFT inner product, we must therefore apply (7.1) only when y lies to thefuture of x , and instead use its conjugate (integrate over T ∈ R − ) if x lies to the past of y .But this condition is not symmetric in x and y : the unordered list of boundary conditions x is– 23 –ot sufficient information to tell us to construct the correlation function, as in the formalismof § | Ω (cid:105) of the theory. For freeQFT on Minkowski space this | Ω (cid:105) is clearly the usual vacuum, though generalizations maybe interesting as will be briefly discussed below.With these choices, the Hilbert space becomes the usual bosonic Fock space associatedwith a real scalar fields on M target . We then find operators ˆ φ ( x ) associated with each bound-ary condition x ∈ BC = R , , but their algebra is famously non-abelian and the operatorscannot be simultaneously diagonalized. We see that this is a result of deviating from thestructure assumed in the arguments of [14]. Of course, one may nevertheless choose to focuson some abelian subalgebra, perhaps the one defined by choosing x = 0. This seems to bethe approach taken in [1, 2, 4], though the more general ‘3rd quantization’ described in [3]allows the algebra of operators on H BU associated with boundary quantities to be non-abelian.For an alternative approach that remains somewhat closer to the axiomatic framework of[14], one may assign an additional parameter to each boundary condition that determines theirrelative ordering. We must then define an adjoint operation † that reverses that ordering, andadditionally place a constraint on boundary conditions so that ‘bra’ boundaries are alwaysordered after ‘ket’ boundary conditions. One can think of this as formally assigning eachoperator with an ‘ i(cid:15) ’ deformation in imaginary time, with correlation functions always inEuclidean time order. The adjoint takes (cid:15) → − (cid:15) , and we restrict boundary conditions tonegative (cid:15) . The result is much like the Euclidean construction in §
5, and in particular thestructure of null states and the operator algebra is similar. Equivalently, we may think ofthe target space as having many ‘time-folds’ in a Schwinger-Keldysh type contour, and ourboundary conditions label which contour an operator insertion lies on (cf., [69]). Correlationfunctions are then always contour-ordered, the adjoint reverses the order of contours, and werestrict our states to be defined by insertions on the appropriate half of the contours.
Note first that interpreting our amplitudes (7.1) as time-ordered correlation functions uniquelyconstructs not only the Hilbert space, but also a particular state | Ω (cid:105) in the theory. It is naturalto expect that this | Ω (cid:105) is the Sorkin-Johnston state of [70–72], as that state is also definedcovariantly from the QFT equation of motion. However, we save investigation of this potentialconnection for future work.We expect that this approach can be generalized significantly. However, to maintaincontact with a worldline quantum gravity formalism one would like to continue to use (7.1)(or its conjugate) to define the single-universe amplitudes. An important question is thento understand the class of models for which the resulting inner product is positive definite. Note that restricting to a constant value of x means that one can work entirely with (7.1) withoutrequiring the complex conjugate. As a result, with this restriction the formalism fits within the framework of[14], and the resulting abelian algebra is consistent with the general argument of [14]. – 24 –hen this is the case, we expect that our procedure defines the usual Hilbert space for theassociated free quantum field theory. Nevertheless, at least a few minimal properties would seem to be required for success.The first is that there be some notion of target space M target , where in particular M target isa time-orientable Lorentz-signature manifold. While the definition of the amplitude in (7.1)makes sense in any signature, only time-orientable Lorentzian spacetimes have a partial orderimposed by the causal structure. This partial order is required to interpret the amplitudes asexpectation values of time-ordered products, and hence would appear to be essential to con-struct a QFT-like Hilbert space. In addition, in Minkowski space the discussion is simplifiedby the fact that H is an essentially self-adjoint operator on L ( M target ). When this is notthe case, the role of boundary conditions defined by the details of the matter path integralwill be more important, and we expect it to be necessary to choose boundary conditions thatmake H self-adjoint. Essential self-adjointness is to be expected when M target is both globallyhyperbolic and geodesically complete, though more generally it should be expected to fail.In addition to such mathematical questions, additional physical considerations may berelevant as well. For example, as pointed out long ago in [59], the physics appropriate for aQFT may not always agree with the physics appropriate for a theory of quantum gravity. Toillustrate this issue, let us follow [59] and consider a gravitational model in which universesstart from a big bang, expand to some maximum size, and then collapse to a big crunch.A simple example is given by the so-called Kantowski-Sachs model of anisotropic vacuumgravity on S × S × R , which in the end [74] differs from (3.8) only by using symmetry toset x = 0 and adding an external potential: H = − p + p − e x − x ) . (7.3)The overall classical dynamics is clear from the fact that any future-directed timelike curve on M target = R , has increasing 2 x − x . So classical solutions cannot be completely describedby such future-directed timelike curves, as in the far future this would require the spacelikecondition p − p >
0. Instead, while one might begin with such a curve (say, emerging from abig bang at x = −∞ ), the dynamics forces the trajectory to become spacelike at some point,and in fact to turn around so that x then begins to decrease. Eventually, the trajectorybecomes time-like again, but it is now past-directed on M target = R , and eventually resultsin a big crunch at x = −∞ .One might thus expect the quantum version of this model to display similar behavior.But the quantum field theory associated with (7.3) is very different. The scalar experiencesan external potential which becomes unboundedly negative at large positive x . This resultsin enormous particle creation, which in a gravitational interpretation would imply creationof a large number of universes at large x . Thus, rather than large x (scale factor) being The idea is that, so long as the metric and any potential are smooth, in the far UV our construction willagree with the case of Minkowski-space QFT. This is well-known to give the correct UV behavior for any QFT.And, in the absence of IR divergences, this condition determines a unique space of QFT states; see e.g [73].So if our procedure defines a Hilbert space, it must be the usual one of QFT. – 25 –orbidden as in the classical theory, large scale factor seems to be dynamically preferred inthe QFT-like quantum treatment.
In the above, we have enumerated various choices that could be made in order to build atheory of one-dimensional quantum gravity from worldline path integrals. We concentratedon situations where splitting and joining of universes is forbidden, so that the final quantumgravity inner product could be represented by a Gaussian integral. However, we also indicatedsuitable generalizations to the non-Gaussian case in the course of our discussion.The various choices that we examined are summarized in § In contrast, in either signature, the QFT-like ap-proaches relate the final inner product to path integral amplitudes only with some restrictionon the possible boundary conditions which is not invariant under the CPT operation used todefine the inner product. Since this restriction violates the framework described in [14], forsuch cases path integral boundary conditions need not define simultaneously-diagonalizableoperators and hence also do not lead to superselection sectors.Of course, given a non-abelian algebra of operators, it is possible to select an abeliansub-algebra. As a result, even using a QFT-like approach, one could attempt to claim thatnatural boundary objects correspond to such a sub-algebra of the possible baby universeoperators; one may thus consider them to be superselected. This appears to be the approachthat was taken in [1, 2]. However, as emphasized in [3], such approaches involve assumptionsabout the nature of the final results – in particular, a certain form of locality was assumed in[1, 2]. Reasoning of this kind thus lead [3] to question the supposed locality, and to suggestthat strict superselection may not hold. This stands in sharp contrast with the treatment of[14] reviewed in §
2, which argued that properties of the quantum gravity path integral require all boundary quantities to be simultaneously diagonalizable in H BU .By describing the above choices, we hope to reduce confusion in the literature. Forexample, the comments of [21] concerning 2d theories of gravity arising from Wilson loops inYang-Mills theories appear to be couched in the EST framework; it is unclear to us whetheranalogous comments hold in GAT-like constructions. Implications for quantum gravity:
In comparing the options described here, the ideathat general path integral boundary conditions define the quantum gravity inner productappears natural from many perspectives and is related to historic discussions of quantum See also the recent Lorentz-signature discussion of baby universes and superselection in [35]. – 26 –ravity and constrained systems [58–66, 75, 76]; see also [77]. This was the point of viewtaken in [14], and was used as the rationale there to justify the assumptions made in thatwork. However, we also noted that one may adapt arguments from [59] to show that – at leastfor certain models of gravitational physics – applying a QFT-like approach to quantum gravityleads to physics very different from the classical theory, even in an apparently semiclassicaldomain. In particular, for any cosmological model in which the universe always reaches amaximum size and then recollapses, QFT-like approaches lead to divergent ‘pair production’ ofuniverses with very large size. Thus, while the classical physics forbids universes of arbitrarilylarge size, the physics of QFT-like quantum gravity would be dominated by arbitrarily largeuniverses. In contrast, the physics of GAT quantum gravity for those models again forbidslarge universes and is thus consistent with the classical results. We take this as a furtherargument against the use of QFT-like constructions in quantum gravity.
Other discrete choices:
While we have explored several important discrete choices in theconstruction of candidate quantum gravity theories, it is important to emphasize that thereare many other places at which further choices could be introduced and whose consequencesremain to be explored. For example, as described in § H to have some explicit dependence on the ‘coupling constants’ g associated with the graph vertices, or perhaps to further generalize the structures discussedabove. Interpretation of low dimensional topological models:
However, the most intriguingquestions that stem from this work revolve around the relationship between the models dis-cussed here and the topological model of [14], the treatments of JT gravity in [9–17], andhigher dimensional gravity more generally. For example, having noted that our EST andGAT constructions both fall within the general framework described in [14], one might won-der which best corresponds to the way in which the topological model of [14] was explicitlysolved in that work. However, one should recall that the main differences between ESTs andGATs involved the treatment of the constraint H and the associated integration domain forthe proper time T , and that neither H nor T appears at all in a topological model like thatconsidered in [14]. As a result, it is far from clear whether such models can be meaningfullyassociated with either construction. – 27 –ow, in considering either the topological model of [14] or the treatments of JT gravityin [9–17], one might note that all of these works focus on Euclidean path integrals and thus betempted to associate them with the construction of ESTs in § §
6. However, in § §
6, the terms Lorentzian and Euclidean wereused to describe the natural contours of integration that define the desired path integral,while in many other contexts one uses the term to describe the sorts of spacetimes that oneuses to evaluate the path integral. This distinction is illustrated by standard non-relativisticquantum mechanics, which one may choose to think is fundamentally defined by a real-time(’Lorentz signature’) path integral, but which is usefully evaluated using Euclidean methodsin contexts that involve quantum tunneling under a classical barrier. In essence, the point isthat one may typically deform the contour of integration into the complex plane to rewrite anoriginally-Lorentzian path integral in a Euclidean form, or to rewrite an originally-Euclideanpath integral in a Lorentzian form. And it was shown in [61] that this could be done forthe GAT path integral by taking the contour to rotate in different directions depending onthe particular matter boundary conditions chosen.As a result, the mere use of Euclidean techniques in the above references is not sufficientto conclude that their treatment is more closely related to our ESTs than to our GATs.Indeed, we note that at least one element of the treatment in [9–17] appears to resemble theGAT construction. Namely, motivated by the fact that the dilaton φ appears linearly in theJT gravity action, the above references take functional integration over the dilaton to exactlyenforce the metric equation of motion R = −
2. Thus they choose an integration contour for φ much like the GAT contour for the proper time T (which resulted in exactly enforcing theequation of motion H = 0 by producing a δ ( H )). However, we leave for future work any moredetailed analysis of this JT path integral prescription and possible connection with GATs (orwith generalizations thereof). Minisuperspace models of quantum gravity:
Indeed, an important question is the ex-tent to which physics of higher-dimensional quantum gravity is in fact similar to any of themodels described here. We consider this to be an open question, with much to be investigated.In particular, in our work above, we took the matter Hamiltonian H to be a self-adjoint op-erator. While this seems natural from the perspective of familiar matter quantum mechanics,we believe it to be rather less obvious from the perspective of higher-dimensional quantumgravity. To this end we remind the reader that when one defines a one-dimensional quantumgravity model by Kaluza-Klein reduction of a higher-dimensional theory, most of the featuresand complications of the gravitational theory then become part of the Kaluza-Klein mattersector, with only some overall notion of proper time left to be treated as one-dimensionalgravity.Let us thus consider again the diagrams of Fig. 1 associated with the splitting and joiningof universes. From the higher-dimensional perspective, these are smooth Euclidean geome- One may also note that, even in standard quantum mechanics, one is free to study ‘Euclidean’ quantitieslike e − HT in the ‘Lorentzian’ theory. – 28 –ries. And in general, one expects smooth Euclidean solutions to be associated with tunnelingamplitudes between e.g., classically-allowed Lorentzian configurations.We can discuss this in more detail in a simple (and famous) minisuperspace model asso-ciated with the Hartle-Hawking wavefunction of the universe. To this end, consider spatiallycompact homogeneous isotropic universes with topology S × R in the presence of a pos-itive cosmological constant, but with no explicit matter. Such cosmologies are describedby a minisuperspace model having a single degree of freedom a (‘the scale factor’), whichupon Kaluza-Klein reduction becomes our matter sector. The scale factor takes values in R + , and the corresponding ‘matter’ Hamiltonian generating evolution in proper time is then H = p a + V ( a ), with V ( a ) = 1 − a . Note that H is not essentially self-adjoint, as classicalsolutions associated with energies greater than unity reach the boundary at a = 0 at finitetime, and more generally quantum wavefunctions have a finite probability to tunnel underthe potential barrier to reach a = 0. Indeed, this phenomenon is associated with the Hartle-Hawking no-boundary proposal for the wavefunction of the universe [79], or perhaps moredirectly with the Vilenkin tunneling-from-nothing wavefunction [80, 81].Of course, in the strict semiclassical limit (cid:96) p → H . Since the above tunneling turns off in this limit, this limiting matterHamiltonian should in fact be essentially self-adjoint. In the example above, this might bebecause the semiclassical limit is associated with some preferred boundary condition that de-fines the essentially self-adjoint H from H . And it may be useful to explore the perturbativeexpansion in (cid:96) p , where a similar structure should arise at all orders.However, at the non-perturbative level any notion of an essentially self-adjoint constraintdefined on a single-universe Hilbert space may cease to be relevant. It would be extremelyinteresting to understand the structure that replaces it, as well as the associated physics thatresults. Higher-dimensional generalizations:
A natural higher dimensional generalization of theideas discussed above would be to consider two dimensional spacetimes, where the allowedboundaries are line segments or closed loops. As a prominent example, we may regard stringtheory as a two-dimensional theory of gravity, taking a perspective where we regard theworldsheet as spacetime. From this perspective, target space is merely a manifold for matterfields. Boundary conditions correspond to asymptotic states of open or closed strings. Manyof the above constructions generalize naturally to such cases.Such a two-dimensional ‘quantum gravity’ could arise even for rather prosaic systems bycharacterizing the dynamics in terms of surface-like degrees of freedom. For instance, Ising-likemodels with Z -valued spins may described terms of domain wall variables (the surfaces acrosswhich spins flip), cf., [82–84]. Likewise, the effective dynamics of QCD flux tubes is capturedby a two-dimensional theory of the confining string [85, 86]. These theories are gravitationalin the sense that they sum over two-manifolds modulo diffeomorphisms (though the fields– 29 –iving on the manifolds do not include a dynamical metric). In particular, the dynamics ofthe domain walls typically involves a sum over topologies. If interpreted as a two-dimensionaltheory of gravity living on the domain walls, these theories clearly fit very naturally into theEST paradigm of §
4. We can then give boundary conditions by specifying loops in targetspace on which domain walls end. The amplitudes correspond in the statistical theory tocorrelation functions of ‘defect’ loop operators (’t Hooft loops for the Z spin symmetry).This is analogous to the worldline description of a theory with particle-like excitations as aone-dimensional theory of gravity, where we integrate in the worldline metric. Integratingin two dimensional gravitational dynamics is significantly more involved. In the case of theeffective QCD string, the Nambu-Goto dynamics may be recast as a T T -deformed free bosontheory, which could be interpreted as a two dimensional model coupled to gravity [87–89].Another example in this vein is given by the genus expansion of large- N gauge theo-ries, which one might hope to describe as a two-dimensional theory of gravity. This hopeis realized concretely for N = 4 Yang-Mills via the AdS/CFT correspondence: we may de-scribe this theory as two-dimensional gravity (a string theory), though the target space is M target = AdS × S rather than simply the four-dimensional spacetime on which the gaugetheory resides. Since the target space contains dynamical gravity, gauge-invariant boundaryconditions are associated with strings ending on the (fixed) asymptotic boundary of M target .For example, we have boundary conditions given by vertex operators corresponding to localoperator insertions in the boundary CFT, and by loops on which strings end on the bound-ary corresponding to Wilson loops. This situation was described in [21]. In particular, theyinterpreted JT gravity as a description of worldsheets in a topological string theory, and thecorresponding amplitudes as correlation functions of Wilson loops in the dual Chern-Simonsgauge theory. Again, this example fits naturally with the EST considerations of §
4, and inparticular as a higher-dimensional generalization of worldline descriptions of AdS/CFT notedat the end of that section. As presented there, the restriction of boundary conditions to theasymptotic boundary of AdS was rather artificial, but this restriction is expected to become arequirement of gauge invariance once our worldline particles include an interacting graviton.Having noted the similarity between one-dimensional (worldline) and string theories, weshould point out a notable difference. With the exception of the GAT theories in §
6, ourconstructions above were sensitive to the off-shell content of the matter theory: that is, wewere not limited to states on the worldline satisfying the constraint H = 0. This can be tracedback to the boundary of the integral over the lapse T , corresponding to ‘short’ worldlines with T = 0. However, string theory is different in this respect: amplitudes are sensible only withon-shell string states corresponding to physical vertex operators. This is ultimately due to thegauging of Weyl symmetry, via which any string state can be thought of as specified at infinitedistance on the worldsheet. One can attempt to eschew stringy constructions and attempt toeither quantize a particular class of diffeomorphism invariant two dimensional gravitationaldynamics on two-surfaces, or quantize a gauge fixed system where we only consider rigidgeometric structures. For example, we can take two-geometries to be cylinders and imposeas in the GAT construction independent delta function constrains for time translations and– 30 –patial rotations (this bears some resemblance to ambitwistor string constructions [90, 91]).Whether such models make sense, and how one might interpret off-shell string field theory inthe language described above, deserve further investigation. Acknowledgments
We thank Tarek Anous, Steve Giddings, Seth Koren, Jorrit Kruthoff, and Raghu Mahajan forconversations motivating much of this work. EC and MR were supported by U.S. Departmentof Energy grant DE-SC0009999 and by funds from the University of California. DM and HMwere supported by NSF grant PHY1801805 and by funds from the University of California.H.M. was also supported in part by a DeBenedictis Postdoctoral Fellowship.
A QFT positivity from Euclidean worldlines
As noted in §
5, positivity of the quantum field theory inner product is manifest from the pathintegral over fields, but it is less obvious from the worldline formalism. As noted in [14] onecan attribute this to the fact that it is unclear what conditions one ought to impose on thequantum gravity path integral for a positive-definite inner product. For a Gaussian field, onecan however demonstrate the positive-definiteness of the full QFT Hilbert space directly fromthe worldline formalism. We outline an argument below, in case it is helpful in providing aninsight into more general issues in quantum gravity.Let us first rewrite the inner product in a slightly different way, using functions F thatsolve the Klein-Gordon equation sourced by f : F ( x ) = (cid:90) M target dy G ( x, y ) f ( y ) ⇐⇒ ( m − ∇ ) F = f . (A.1)We have (with the Z involution σ ) (cid:104) f | f (cid:105) = (cid:90) M target dx F ∗ ( σ ( x ))( m − ∇ ) F ( x ) . (A.2)We now note that if F is positive frequency, meaning that f = ( m − ∇ ) F vanishes on M − target , this inner product can be written as an integral over M +target only. But we can thenintegrate by parts, and if F is positive frequency it means that F ∗ ◦ σ is negative frequency;the resulting integrand then vanishes on M +target . All that remains are boundary terms, andwe can write the inner product as an integral on Σ (the fixed point locus of σ ).To see this in more detail, let us introduce some notation. First, define a ‘symplecticform’ Ω on the space of functions on M target (or perhaps the space of functions that satisfythe Klein-Gordon equation ( m − ∇ ) F = 0 in a neighborhood of Σ):Ω( F , F ) = i (cid:90) Σ ( F ∇ n F − F ∇ n F ) . (A.3)– 31 –ere ∇ n is the normal derivative at Σ, pointing out of M +target and into M − target . The factorof i is a nice convention because it disappears if we ‘Wick rotate’ to Lorentzian signature, andthis becomes the usual symplectic form on Cauchy data for the wave equation. From Stokes’theorem, we haveΩ( F , F ) = i (cid:90) M +target ( F ∇ F − F ∇ F ) (A.4)= − i (cid:90) M +target F ( m − ∇ ) F ( F negative frequency) . (A.5)In the last line we assume that F is negative frequency, which means that ( m − ∇ ) F = 0on M +target . This now resembles our inner product (A.2).We can in fact write the precise relation in terms of the ‘complex structure’ J acting onour space of functions by J F ( X ) = i F ∗ ( σ ( X )) . (A.6)This operator satisfies J = − (cid:104) f | f (cid:105) = Ω( F , J F ) . (A.7)Writing this out explicitly gives us our inner product in terms of data on Σ as (cid:104) f | f (cid:105) = − (cid:90) Σ ∇ n ( F ∗ F ) , (A.8)where we have used the fact that Σ is fixed by σ , and ∇ n ( J F ) = − J ∇ n F .We can now use Stokes’ theorem again to write this as an integral on M − target : (cid:104) f | f (cid:105) = (cid:90) M − target ∇ ( F ∗ F ) (A.9)= 2 (cid:90) M − target (cid:2) ( ∇ F ∗ ) · ( ∇ F ) + m F ∗ F (cid:3) , (A.10)where we have used the fact that F , F are positive frequency again to write ∇ = m . Inthis form, the inner product is manifestly positive semi-definite: (cid:104) f | f (cid:105) = 2 (cid:90) M − target (cid:0) |∇ F | + m | F | (cid:1) ≥ . (A.11)Now that we have a Hilbert space, let’s understand it a little better. First, we can thinkabout it in terms of positive frequency functions F , satisfying ( m − ∇ ) F = 0 on M − target .This constraint determines F from its values on M +target (existence and uniqueness for theDirichlet problem given data on Σ), so the space of positive frequency F is roughly the spaceof all functions on M +target . But we have seen that the inner product can be written in terms– 32 –f data on Σ only, so the Hilbert space is much smaller: we have ‘null states’ corresponding tofunctions F which are nonzero only in the interior of Σ. From (A.8) it looks like the relevantdata is the value of F and its normal derivative on Σ. But that’s still too much data: F and ∇ n F on Σ are related (nonlocally) by the positive frequency condition. We should thereforethink of the states in the single-universe Hilbert space as depending on a single (complex)function of Σ, which we might think of as the wavefunction of a single particle.In terms of the source f , (a dense set of) the Hilbert space consists of functions withcompact support supp( f ) ⊂ M +target , modulo the image of ( m − ∇ ) among such functions.The latter give null states, since functions f = ( m − ∇ ) F with F vanishing in a neighbour-hood of Σ have zero inner product with any state. Integrating by parts, we can see this as astatement of the Hamiltonian constraint or Wheeler-deWitt equation:( m − ∇ ) | Φ( x ) (cid:105) = 0 . (A.12)That is, 0 = (cid:12)(cid:12) ( m − ∇ ) F (cid:11) = (cid:82) M +target (cid:0) ( m − ∇ ) F (cid:1) Φ = (cid:82) M +target F ( m − ∇ )Φ. 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