Quantum gravity in timeless configuration space
On the path towards quantum gravity we find friction between temporal relations in quantum mechanics (QM) (where they are fixed and field-independent), and in general relativity (where they are field-dependent and dynamic). This paper aims to attenuate that friction, by encoding gravity in the timeless configuration space of spatial fields with dynamics given by a path integral. The framework demands that boundary conditions for this path integral be uniquely given, but unlike other approaches where they are prescribed — such as the no-boundary and the tunneling proposals — here I postulate basic principles to identify boundary conditions in a large class of theories. Uniqueness arises only if a reduced configuration space can be defined and if it has a profoundly asymmetric fundamental structure. These requirements place strong restrictions on the field and symmetry content of theories encompassed here; shape dynamics is one such theory. When these constraints are met, any emerging theory will have a Born rule given merely by a particular volume element built from the path integral in (reduced) configuration space. Also as in other boundary proposals, Time, including space-time, emerges as an effective concept; valid for certain curves in configuration space but not assumed from the start. When some such notion of time becomes available, conservation of (positive) probability currents ensues. I show that, in the appropriate limits, a Schroedinger equation dictates the evolution of weakly coupled source fields on a classical gravitational background. Due to the asymmetry of reduced configuration space, these probabilities and currents avoid a known difficulty of standard WKB approximations for Wheeler DeWitt in minisuperspace: the selection of a unique Hamilton-Jacobi solution to serve as background. I illustrate these constructions with a simple example of a full quantum gravitational theory (i.e. not in minisuperspace) for which the formalism is applicable, and give a formula for calculating gravitational semi-classical relative probabilities in it.
- 1 Introduction.
- 2 Axioms
- 3 Path integrals in configuration space
- 4 Records
- 5 Applications to quantum gravity
- 6 Conclusions
- A The Wheeler DeWitt equation
- B Connections on principal fiber bundles
- C The timeless transition amplitude in quantum mechanics
- D Timeless semi-classical composition law.
- E Probability currents for weakly coupled systems.
- F Jacobian
- G The full Van-Vleck determinant
- H Locality
Conventional approaches to quantum gravity suffer from serious conceptual and technical problems. On the technical side, most focus has been concentrated on the perturbative regime and on extending the theory to higher energy domains while maintaining some semblance of unitarity. Conceptual problems on the other hand, are starker when trying to make sense of the non-perturbative theory. There we witness a violent clash between, on the one hand, the standard quantum mechanical properties of:
the quantum superposition principle
the non-locality of instantaneous measurements;
and, on the other, the general relativistic properties of
a fixed causal structure
These two pairs still leave out difficulties quantum cosmology faces arising directly from the measurement problem in the foundations of quantum mechanics, which the present work also aims to address. Of course, the non-perturbative regime poses infinitely more challenging obstacles to quantitative treatment. Nonetheless, resolving a conceptual incongruence between full quantum mechanics and general relativity might point us to different approaches, friendlier to quantum gravity. The aim of this paper is indeed to point out a framework in which such incongruences are resolved. The framework suggests the adoption of alternative formulations of gravity at a fundamental level, such as shape dynamics  (see also  and references therein).
The first clash, between 1a and 1b, arises because, intuitively, a quantum superposition makes sense for space-like separated components of a system – i.e. belonging to a single constant-time hypersurface. But, to be meaningful, the label ‘space-like’ already requires a fixed, unique,222Unique up to conformal transformations. gravitational field.
Accounting for back-reaction, it becomes obvious a problem lurks here. We understand how non-back-reacting matter degrees of freedom can be in states of superposition in a background spacetime, e.g. what we usually mean when we write something of the sort in the position space representation. But due to the universality of the gravitational interactions, back-reaction would necessarily put the gravitational field in a state of superposition as well. How does a superposed state gravitate – and therefore affect the causal structure? The problem is also present in covariant axiomatic QFT language, where one demands that operators with space-like separated support, commute, . Again, ‘space-like’ presumes a definite property for the very field we would like to be able to superpose.333For QFT in curved spacetimes, different “time” problems are present. There, the issue rears its head in the absence of Killing vector fields. The issue is that one should find solutions of the dynamical equations , which are also eigenfunctions of the time derivative (i.e. along the Killing vector), . One then uses this eigenfunction split into positive and negative frequencies to construct associated raising and lowering operators, a Fock space, etc. In the absence of a preferred time-like direction in space-time, raising and lowering operators become less meaningful (see  for a more complete treatment of this point). What is the meaning of this demand when the operators are acting on states representing superpositions of gravitational fields? It is hard to tell, and it seems to me there are very few ways forward in thinking about these problems.444See  for a notable exception, using the formalism of partial and complete observables  (see [6, 7] for reviews).
Further tension lies between 2a and 2b. It stems from distinct roles of (global) time in quantum theory and in general relativity. In quantum mechanics, all measurements are made at “instants of time”, and in this sense only quantities referring to the instantaneous state of a system have physical meaning. Dynamics is most naturally understood as an evolution from one instantaneous state to another. On the other hand, in general relativity, a global “time” is an arbitrary label assigned to spacelike hypersurfaces, and physically meaningful quantities are required to be independent of such labels. In other words, in GR, only the spacetime geometry is measurable, and thus only the histories of the Universe have physical meaning. The dynamics of the gravitational field must be carved out of the theory by subjecting it to an initial value formulation, an unnatural procedure from the covariant standpoint.
When (naively) merging quantum theory and general relativity, it should cause no surprise that the only physically meaningful surviving quantities are those which are both: 1) instantaneously measurable (i.e. refer to quantities defined on a spacelike hypersurface) and 2) depend only on the spacetime geometry (i.e. are independent of a choice of spacelike hypersurface) (see e.g. [8, 9]). In a full geometrodynamical theory, this creates great constraints on what constitute reasonable quantum observables [10, 11].
Lastly lies a further distinction (but not necessarily a clash) between the notions of local time, or duration, in the two theories. In classical GR, there exists one notion of local time inherent in the theory, independently of the space-time metric: proper time. Proper time gives an approximate notion of duration along a world-line, it is a dimensionful quantity that is not given in relation to anything else. It has this role whether a space-time satisfies the Einstein equations or not, and is therefore a kinematical, general proxy for elapsed time. On the other hand, in modern relational approaches to quantum mechanics [12, 13], one can parse two notions of time: time as measured by relations between subsystems and just an external evolution time. One of these subsystems is usually called the “clock”; its emerging notion of local time is akin to duration, but only becomes available through relational dynamics. The other, global evolution time, is indeed not dynamic, but it also has no meaningful parallel in GR.
These issues surface on the technical front as well; efforts to obtain a viable notion of quantum evolution in quantum gravity – even at the semi-classical level – founder for a variety of reasons .555At least outside of drastic truncations of field space, such as minisuperspace. In many cases, such truncations do not commute with quantization . My claim is that these reasons have a common birthplace. They arise from the distinction between unitary evolution and reduction processes in quantum mechanics, and from the confluence of dynamics and kinematics in general relativity.
1.2 A diplomatic resolution
This paper outlines a proposal to address such questions. Although it is still of largely qualitative character, the aim of this first paper is to discuss the mechanisms by which one could implement said proposal.
My present attempt to untangle both the problem of the superposition of causal structures and the problems of time consists in basing gravitational theories, at the kinematical level, on fields for which there is no notion of causality. Causality emerges, but only dynamically. In this way, physics should be encoded in a path integral operating in timeless configuration space, embodying only compatible spatial symmetries and with prescribed boundary conditions. Such symmetries allow us to define and exploit a reduced space of physical configurations. No such reduced configuration space exists for theories with local refoliation invariance — such as GR — even abstractly.
The boundary conditions required for the constructions here are extremely special. They are based on a fundamental asymmetry of the reduced configuration space (configuration space modulo gauge-symmetries). Namely a reduced configuration space must not only exist, but also have a unique, most homogeneous element. This requirement places extreme constraints on the field content, on the symmetries, and on the topology of the manifold, but its satisfaction is what will allow most of the constructions here to be defined unambiguously. Anchoring the transition amplitude on this unique boundary,666Or more precisely, this unique ‘corner’, as we will see below. the present framework provides a ‘first principles’, non-covariant alternative to the Hartle-Hawking prescription of the global wavefunction of the Universe .
It also addresses questions that need to be answered in any fully relational, timeless theory referring to instantaneous states, such as Hartle-Hawking and Vilenkin’s tunneling proposals. These proposals yield a wave-function whose argument is an instantaneous field content, e.g. , where is the 3-metric and some spatial matter field. In this respect, such theories thus share many of the features of the framework constructed here. The difference, again, is that in such approaches, the wave-functions still need to correspond to covariant (i.e. space-time) quantities. Outside of minisuperpsace approximations, the requirement of covariance (at a fundamental level) does not allow such theories to be describable in a physical configuration space, even abstractly, and thus they do not assuage the conflicts between QM and GR I have outlined in the previous section.
As I will show, the present setup yields a single static wavefunction and a corresponding volume-form in physical configuration space. The passage of time is then abstracted from a particular notion of ‘records’ – subsets of configuration space where the volume form concentrates. Causality emerges thus as an approximate pattern encoded in the volume-form when it is well-described semi-classically. In this sense, no separate causal structures ‘superpose’ — the frozen patterns we associate with them can at most be said to mesh and interfere in configuration space. Unlike previous work in which one attempts to emulate the effects of wave-function collapse within standard unitary evolution, here both evolution and reduction should be emergent from a fundamentally timeless description of the entire Universe.
Here, space-times don’t exist a priori, but require a relational construction of duration – or definitions of clocks – on classical solutions. I.e. in general one must define clocks and an experienced duration through the classical evolution of relational observables; I call such an emergent notion of time duration-time. Indeed, the price to pay in the present, purely relational framework, is apparent in trying to recover, in the appropriate limits, the local, experienced time of GR. In general relativity, experienced time is directly related to proper time, but the gravitational theories compatible with the framework have no universal replacement for proper time. This feature might not be so damaging; it is also the case in relational approaches to quantum mechanics that one constructs clock-time from relations between subsystems , as discussed in the previous section.
The challenge to the sort of theories encompassed by the present work thus shifts, from the standard fundamental problems mentioned above — superposition of causal structures, untangling evolution and symmetries, and reduction vs unitary evolution — to one of recovering a smooth space-time description. While it may indeed be difficult to recognize classical relativity of simultaneity from the global evolution-time — i.e. the global time parametrization that will follow from the construction of records — it can be gleaned when using duration-time, as we will see.
In sum, the claim here is that, although unorthodox, this approach resolves issues both with the superposition of causal structures and the ‘evolution vs reduction’ dichotomy, while bringing the notion of duration in gravity closer to the non-unique versions of ‘clock systems’ in quantum mechanics. The constructions advocated in this paper can similarly work with other types of timeless configuration space; they do not limit themselves to the geometrodynamic setting in which they are mostly pursued here. Even the consequences of this approach to the construction of gravitational models will be only touched on here, being further pursued in an upcoming publication, as well as in .
I will now describe the main constructions of this paper and its relevance for quantum mechanics and quantum gravity. I will start by making clear what are the main assumptions, or axioms, of the work. This is done in section 2. The non-relativistic setting will in many ways resemble standard particle quantum mechanics. Leaving at most a single reparametrization constraint, the assumptions are then perfectly compatible with past work on relational dynamics and quantum mechanics, which I very briefly review in section 3. At the end of this section, I include a proof that the principles single out the Born rule as a measure in configuration space. Finally, I discuss another problem that is not fully addressed in relational quantum mechanical approaches within the context of path integrals: a replacement for the role of the epistemological updating of probabilities in the timeless context, through the notion of “record-holding submanifolds”, in section 4. Lastly, I sketch a gravitational dynamical toy model based on strong gravity, to illustrate the structures introduced here.
I would like to examine the conceptual picture that emerges from theories for which time plays no role at the kinematical level. I will first clarify the axioms I am led to adopt in order to have a geometric theory of gravity without such kinematical causal relations.
The five given structures that the present work is based on are the following:
A closed topological manifold, . Prior to defining our fields, we require some weak notion of locality, which I will take to mean ‘open neighborhoods’. In the gravitational case, I will take this to be provided by the closed topological manifold (i.e. compact without boundary), of dimension .777 The demand of being closed is a consequence of relationalism. It does not imply that through evolution we couldn’t get causal horizons, or that there are no meaningful definitions of effectively open subsystems within . If one wanted to apply the following constructions to a fundamentally discrete theory, could be replaced by a lattice, or piecewise linear manifold, on top of which the physical degrees of freedom live.
The kinematic field space . In the field theory case, this is the infinite-dimensional space of field configurations over . Each configuration – the ‘instantaneous’ field content of an entire Universe– is represented by a point of , which we denote by . These should correspond to relational data, on which the dynamical laws act. For defiteness, I will take field configurations to be sections of tensor bundles over . The example for pure gravity would be , the space of positive smooth sections on the symmetrized (0,2)-covariant tensor bundle. In principle the constructions here should apply for non-causally related observables of any kind, such as e.g. field values on a lattice.
“The Past Hypothesis” – boundary (or ‘initial’) conditions for the Universe: The requirement of such boundary conditions for the wave-function goes in line with the fact that, if time plays no fundamental role, the wave-function of the Universe must be given uniquely, and only once. Thus I require unique boundary conditions in order to anchor the construction of a unique transition amplitude. Unlike what is the case with the ‘no-boundary’ proposal , this will be a choice of ‘the most homogeneous’ configuration, , with respect to which the wave-function is defined, , see equation (1), below. As I will show, furthermore plays a fundamental part in the definition of ‘records’, and for ‘records’ to function in the capacity their name suggests, the boundary conditions should correspond to field configurations which are as ‘structureless as possible’. Below, I give a criterion that can, in some circumstances, select a unique such boundary state.
An action functional on curves on . I.e. , for , invariant wrt the gauge group acting on , , and (up to boundary terms) depending only on first derivatives in the time parametrization. The action should be invariant with respect to the given gauge symmetry group , defining , for . For this definition, the gauge-symmetry group needs to have a pointwise action on . Such an action functional will be used for the transition amplitude between two physical configurations, for , and :
To avoid cluttered notation, I will drop the square brackets in most of the paper, calling attention when the distinction between full configuration space and the reduced one becomes material.
A positive scalar function . I will call this function the pre-probability density, , as it will give a measure in configuration space. To serve our purposes, must preserve the multiplicative group structure,
For future reasons I will term (2) the factorization property of the density. It is a required condition for the definition of record that I introduce here to have physical significance.888And for it to have cluster decomposition properties ). It is the specific form of this obeying (2) which encodes the Born rule.
These axioms are quite standard implicit choices in most work related to field theory, here I am only making these choices explicit. Axiom 5 is implicit in the Born rule, and 3 is usually explicitly stated in the literature, either as boundary conditions, or as some initial condition.
The only structure required for doing physics, arising from the five premises above, is the density over , given by
where is restricted by axiom 4, is given by axiom 5, and is defined from the action functional given in axiom 3, and (12) and (14). This measure gives a volume form on configuration space. It gives a way to “count” configurations, , and thereby the likelihood of finding certain relational observables within . It is assumed to be a positive functional of the only non-trivial function we have defined pointwise on , namely, . I should stress that roughly the same or similar assumptions are implicit in both Hartle-Hawking and Vilenkin’s tunneling proposals.
Here, we have a pre-conceived notion of space, or rather, “of things which are not causally related”, but not necessarily of time, in either its absolute or relativistic forms. We have the space of relational objects on which dynamical laws should act.
At the end of the day, we have at our disposal the density over field space, . But without space-time, and without any notion of absolute time, can we still extract some physics from the formalism? I would like to show that there can still be enough structure in the timeless path integral in timeless configuration space for doing just that.999Moreover, this formalism could apply to any relational system with a specified configuration space, action over curves on it, and unique maximally homogeneous point. For example, certain types of tensor network models. This is what I will focus on for most of the paper.
2.1 Relationalism and the symmetry group .
Intimately related to our clash 2a and 2b is what is known as the ‘problem of time’. The GR Hamiltonian mingles local gauge symmetries and evolution [14, 8].101010I note that while a relativistic particle also has a Hamiltonian which generates a single reparametrization invariance, it does not mix this symmetry with local gauge transformations . Moreover, as a subsystem, it is easily describable by relational observables. I give a broader analysis of the time problems in appendix A, and related ones in their formulation of a transition amplitude in superspace, in appendix A.2. At least without the use of matter fields there is no preferred manner to split the Hamiltonian constraints such that all but one are fixed by a “definition of simultaneity”– a partial gauge-fixing of the Hamiltonian well defined everywhere in phase space.
Although the main constructions of the paper should be applicable in a more general setting, let’s investigate the standard case for gravity in closer detail.
The standard ADM action and symmetries.
Upon a Legendre transformation, the vacuum Einstein-Hilbert action yields primary and then secondary first-class constraints, . The action can then be put in the form
where are Lagrange multipliers and
with some algebra given by
It has been shown that the local constraints above are essentially unique if the algebra they generate is required to mimic the commutation algebra of vector fields orthogonally decomposed along a hypersurface of space-time (the hypersurface deformation algebra). Thus the Hamiltonian constraint, and therefore the local Wheeler-DeWitt equation, are intimately tied to a covariant picture of space-time.
Under the transformation generated by the flow of the constraints,
and under the more ad hoc
the action transforms by a boundary term:
which clearly vanishes for constraints linear in the momenta, or if the generator of refoliations, , vanishes at the initial and final surfaces.
has pointwise dependence in , the transformation
depends not only on the metric, but also on the momenta. We could also have obtained a similar result directly from the ADM decomposition (with no Legendre transformation): decomposing a vector field along the time direction and tangential components to the hypersurface, , one obtains (assuming zero shift):
where is the Lie derivative along the vector field tangential to the hypersurface. This part of the action is easy to make sense of: it is the infinitesimal action of a spatial diffeomorphism, acting on the spatial metric through pull-back: Diff.
On the other hand, equation (10) shows us that refoliations act on the metric in a manner also depending on its velocity.111111In fact, the strict relationship between the Hamiltonian constraint and refoliations holds only on-shell. It is the action of refoliations that makes it difficult to meaningfully define surfaces and points in superspace, the quotient space Diff. For suppose one has two different curves intersecting at , , , for a positive-definite symmetric tensor (they intersect at ). Since the action of the spatial diffeomorphisms depends solely on the metric, the two curves will still intersect after the action of (or ), as they will be jointly moved. But with the action of , the curves will be shifted: , and will not intersect anymore. Taking the quotient by (spatial) diffeomorphisms will not help since it is easy to find such that for all Diff.121212That this must be so is easy to ascertain from a simple degree of freedom count: the space of symmetric tensor has 6 degrees of freedom per space point, whereas the generators of spatial diffeomorphisms can only account for three.
Even though is an infinite-dimensional space, there is at least one situation in which such curves can still intersect at different . If the transformation corresponds merely to a reparametrization along each curve, , the transformed curves merely shift their time-parameters:131313This would happen order by order for higher contributions in powers of . , and thus the curves will still intersect, but now at . However, if the refoliation is not spatially constant, generically the curves will miss each other, also in superspace.
This fact makes it doubtful that one can implement boundary conditions on superspace for the path integral that are physically meaningful from a covariant space-time perspective.
The present requirement on symmetries.
The alternative explored here requires its fundamental symmetries to be generated by local constraints linear in the momenta. The treatment of such linear symmetries, including their quantization, is then much more straightforward than for local constraints quadratic in the momenta. I will show how this non-fundamentally-covariant setting still allows the emergence of an on-shell refoliation invariance.141414I should note that in the ADM Hamiltonian formalism for GR , refoliations only generate a symmetry on-shell in any case .
Although the principle of having a pointwise action in configuration space is generically applicable, in the case of gravity in metric variables, i.e. Riem(M), the symmetries which satisfy this criterion are found by requiring a phase space representation of first-class constraints linear in the momenta,
where is a constant in , and are appropriate (tensor) smearings of the local constraint densities, , e.g.
the conjugate momentum to the metric is , and curly brackets denote the Poisson bracket wrt this canonical relation. Linearity in the momenta implies that the symmetries have an intrinsic action on configuration space, as required, i.e.
i.e. its action on depends only on and the smearing. The first class property implies that these constraints correspond to symmetries. It turns out that, under some assumptions, that the allowed non-trivial symmetries that act as a group in the metric configuration space — and thus fulfill our axioms — are diffeomorphisms and scale transformations (see  for more details).
In the field theoretic framework proposed here, relational observables are those that live in the quotient – e.g.: those whose spatial position and scale are only defined relationally – and thus I will bypass a more complicated relational phrasing of the amplitudes by just assuming that all statements are suitably translated when necessary, from to . In the presence of this structure, boundary conditions for the path integral will have physical meaning.
The demands of our axioms thus severely restrict the form of the infinite-dimensional groups , and imply that configuration space form a principal fiber bundle ,151515This is not quite true, for the base space may not form a manifold, as it doesn’t for the spatial diffeomorphisms. However, insofar as I will use this structure for a field-space connection 1-form – which I associate with an abstract observer – the group can be restricted by a further condition: only allow the diffeomorphisms that maintain a given (observer location) and a observer frame at , fixed. Calling this restricted set of diffeomorphisms Diff, then the space Diff is indeed a manifold  and one can use its principal fiber bundle structure in the usual ways. making the quantum treatment of gauge-symmetries straightforward. For instance, unlike what is the case for ADM , in which one requires the more complicated use of the FBV formalism [22, 23], standard Fadeev-Popov is sufficient to give rise to a well-defined BRST charge; the Fadeev Popov determinant is a true determinant, which implements gauge-covariance of the gauge-fixed path integral. It is the principal fiber bundle structure that allows us to unambiguously consider individual points and dynamics in the quotient space , an impossibility when gauge symmetries become tangled with dynamics (as is the case with ADM).
In the 3+1 path integral setting here, one can make use of the principal fiber bundle structure for a straightforward treatment of gauge-symmetries, with the use of a a field-space connection 1-form, , as described in appendix B. This field-space connection 1-form selects a way to horizontally lift to a given curve in . For instance, take Diff and Riem(M), , the vector fields with the algebra being just the commutator algebra, acting on the metric with the Lie derivative. The horizontal projection of a field-space vector, is given by . I.e. it acts as a field-dependent shift in the ADM formalism.161616The connection 1-form itself can be related to a choice of equilocality and associated to abstract, non-back-reacting, observers  (see also footnote 15). In a completely gauge-covarint way, it selects a manner in which an observer dynamically distinguishes physical transformations from pure gauge transformations, along time. This is a generalization of Barbour’s concept of “best-matching coordinates’ . According to (88), under a time-dependent diffeomorphism, Diff, generated by the vector field ,171717Strictly speaking, we are only considering diffeomorphisms connected to the identity. Outside of this domain, many problems appear. For instance, there are always diffeomorphisms arbitrarily close to the identity which are not connected to it through the exponential flow of some vector field. This poses problems for the path integral. we have the following transformation of :
which is precisely the transformation required of the shift from (8) under time-dependent spatial diffeomorphisms.
Summary of this section.
In this subsection I have sketched how the local symmetry groups can be more or less uniquely selected in such a manner as to have a pointwise representation in configurational field space. In this manner, a well-defined reduced configuration space exists, and I guarantee that local symmetries will be distinguished from dynamics. Moreover, a gauge-invariant path integral may be constructed making use of the connection-form, possible from the emerging principal fiber bundle structure (see also [24, 17] for a more complete account).
2.2 Uniqueness of preferred boundary conditions
The principle I will use to select boundary conditions is based, roughly, on information content. I would like to set the anchor of the transition amplitude (1) to be the configuration (or reduced configuration) that carries the least amount of information. What I mean by that is that it carries the most amount of symmetries, under my group of transformations. I.e. it is the most “homogeneous” configuration.
As mentioned in footnote 15, it is not true, for groups which don’t act freely on configuration space, that the quotient forms a manifold. Indeed, if there are elements of that remain fixed under the action of subgroups of , such as is the case with and Diff for instance, then the quotient can be at most a stratified manifold . A stratified manifold is basically a union of manifolds of different dimensions, with concatenated boundaries of boundaries. The standard example is a cube, with its bulk being of dimension 3, and its boundary being composed of a further union of manifolds; a face of dimension two, with its boundaries composed of a further union of manifolds; edges of dimension one, with its boundaries composed of a further union of (zero-dimensional) manifolds. The points with the highest isotropy group will correspond to the lowest dimensional boundaries (of all the other boundaries). In other words,
where each is a manifold with boundaries, such that for .
Given , let be the set composed of all the most homogeneous field configurations, i.e. the subset of elements with the largest dimensional isotropy subgroup of . This will stand in for the configurations being ‘as structureless as possible’. For example, let us take and , acting through pull-back . Then let
Allowing for degenerate metrics, we have a unique such point, , the completely degenerate metric, since it has the full Diff(M) as an isotropy group.
In accordance with the symmetry principles of the theory, as described above, one could extend the group Diff to the one given by the ‘maximum geometric group’, Diff, with the Weyl group of conformal transformations (through pointwise multiplication of the metric by positive scalar functions , and a semi-direct product between the two groups) [26, 17].
In fact, the case of Diff does not require further specification of the field space boundary at all (as is required, for example in Hartle-Hawking ). This property can be seen either by parametrizing physical space with unimodular metrics, or using the horizontal lifts of the previous section. In the extended case, the orbit in corresponding to the completely degenerate metric is not continuously path-connected to the rest of . For , this leaves only , the round metric, as generating the unique allowed past orbit.
In the first case, using unimodular metrics as the conformal section of the principal fiber bundle ,181818Note that since this group is Abelian, there is no Gribov problem . for a curve of metrics in to change signature, the determinant must become degenerate, which disconnects the physical spaces of positive definite signatures from those with other signatures. In other words, the ‘cone’191919Riem is a cone inside the affine space of sections of symmetric (0,2)-covariant tensors, , in that the sum of two metrics is still inside Riem, but not their difference . which makes up the boundary of Riem inside the affine space becomes unreachable coming from .
In the horizontal lift picture, the orbits are defined by vectors of the form . For any ultralocal supermetric in configuration space of the form
where where , and is a function of the metric and its (spatial) derivatives, the orthogonal vectors to the orbits are going to be of traceless form. Thus the standard horizontal lifts orthogonal to the Weyl orbits (through any standard canonical supermetric in Riem) imply a traceless velocity, , which does not change the volume-form . Thus such curves cannot reach metrics with different signatures than the initial ones, as this would require going through a zero in the determinant of the metric. We can thus formulate the path integral (see section below) in terms of horizontal lifts and without stipulating further boundary conditions.
Summary of this section.
In either the cases of Diff or Diff, the quotient space is only a stratified manifold, and the orbits corresponding to are the lowest strata, the ‘ultimate boundaries’, or the lowest dimensional corners, of , and the natural place to set up boundary conditions. When there is a unique least structured configuration, which is also the corner of corners in reduced configuration space, the boundary conditions can be uniquely specified. This is the case for Diff on . Moreover, no further boundary conditions for the path integral on field space need to be specified.
In other words, the specification of such a unique initial point in the amplitude kernel is sufficient to fully determine the entire wave-function of the Universe. Moreover, as mentioned above, the existence of such a preferred point will be fundamental in defining a ‘record’, which, in its turn, will replace standard notions of time.
3 Path integrals in configuration space
There are different ways of relating the standard relational setting for quantum mechanics to the path integral formalism. In the end, the constructed wave-function should obey some form of the reparametrization constraint. In appendix C I report on work of Chiou showing, for a relational particle model, how a rigorously defined timeless path integral regains this constraint, and how it reduces to the standard transition amplitude in the presence of a subsystems that behaves like a clock. The mechanisms used in this proof (e.g. the Riemann-Stieltjes integral in terms of a mesh) can be recycled for building the path integral with only global reparametrization constraints, as we have here.
Equation (94) guarantees that in the presence of a reliable clock in a given portion of configuration space, one can recover the standard quantum mechanics transition amplitude purely relationally. But it is silent in what regards the existence of such subsystems. As we will see, the asymmetry of reduced configuration space will give us enough structure to build such time functions in the appropriate approximations, even for the (infinite-dimensional) gravitational case.
3.1 The basic definitions
Given an action functional as above, a connection-form , and a preferred ‘in’ configuration (see The Past Hypothesis, below), the (timeless) transition amplitude (or propagator) to the orbit of the configuration is given by a timeless Feynman path integral in configuration space:202020Rigorously, I should start with a phase space action, and only if the momenta can be integrated out of the path integral – which up to the measure amounts to a Legendre transformation – move onto a configuration space action. Here I overlook these issues.
where here acts on an arbitrary representative of the final point of the transition amplitude, , and is integrated over with some measure. A Haar measure is not required here; unlike the standard case, we are not doing a group averaging procedure, each path on the base space corresponds to at most one . If the curvature of the field-space connection form is zero, there is no relative holonomy on the fiber for two paths , interpolating between and the orbit of . Thus all the lifts for the paths will end up in a single height of the orbit, let’s say . Then the path integral will acquire a functional delta: , cancelling the integral over .
In the case of gravity, we would have:
The class of paths under which this is integrated over are the horizontal lifts of paths in , as explained in the previous section. I.e. is a path that has a horizontal velocity (i.e. it is a horizontal lift through ), ending at . The implementation of horizontality will in general incur a Jacobian, which substitutes the standard Fadeev-Popov determinant. For the purposes of this paper, the specification is not necessary.212121In the interest of completeness, horizontality by being orthogonal to the fibers generated by the group of conformal diffeomorphisms, Diff, as discussed below, for the standard supermetric in , will implement transverse and traceless conditions on the metric velocities, . The appropriate Jacobian is calculated in appendix F. We note that in the case of odd-dimensional Weyl symmetries, there is no conformal anomaly, and thus the measure can be suitably made Weyl-invariant in conjunction with the action. This is also true in the Hamiltonian setting if the anomalies have a local representation . This procedure eliminates the degenerate directions of the action functional — and makes the projection of the Liouville measure non-degenerate — and does not suffer from Gribov ambiguities (see  for more details).
Regarding the measure , Barvinsky (see , Sec II) has shown how to split such functional measures into a lower dimensional functional integral over spatial fields, and then another integral over parametrizations. The first order part is just the projected Liouville measure:
where we are using DeWitt notation, so that here runs over both the tensor indexes and the spatial continuous ones, and
and is the functional partial derivative. This is nothing but the invertibility matrix between velocity and momenta of course (and it is only non-degenerate in the absence of gauge symmetries). Of course, by using the horizontality conditions, here one would replace . Suppose some gravitational action is given (again) by (11)
which has non-zero determinant. As mentioned above, a choice of a connection form is largely equivalent to a choice of gauge, but more suited to the context we are applying here. For a very simple connection form for the diffeomorphisms, one would get that horizontal vectors are those for which , i.e. the transverse ones .
Going back to the more general case, when combining the measure with the one-loop determinant integrated over time parametrizations, one finds the amplitude: , which is only a determinant over the spatial fields, with no time integration, but for the on-shell action (equation 3.24 in ). The semi-classical approximation will be recapitulated in section 3.
The measure for the spatial diffeomorphisms (connected to the identity), are given by first replacing the diffeomorphisms by the path-ordered exponential of vector fields. I.e. by vector fields such that (here on the rhs actually indicates the coordinates of the image of under ), and such that , i.e. . Calling again the structure constants of the commutator algebra of diffeomorphisms, given implicitly in e.g. (8), and introducing the continuous matrix , Teitelboim has shown that : . Finally, if the Weyl group is part of , as it is in the example given, Diff(M), the integration over the conformal factor cancels out with a functional delta, because the conformal field-space connection-form is flat; all lifted paths end up at the same height in the conformal orbit, as mentioned above (between eqs (12)-(13)).
For more on the folding properties of the path integral, and other technical details, we can follow Teitelboim in the procedure given in , where only the conditions on the shift are local. These details, for a specific conformally invariant model, are made explicit in .
The horizontal lift and integration over the final point guarantees that the action functional is invariant wrt to gauge transformations everywhere but the initial point (which can be fixed by boundary conditions). Note also that there is no redundancy in this integral. Each path in the reduced configuration space is lifted uniquely. If the field-space connection had zero associated curvature, no integration over would be necessary, for all horizontal lifts would end up at the same height in the fiber over the final point .
Summary of this subsection:
In this subsection, we have briefly and formally sketched how the field space path integral can be defined in the timeless context. We have explored the fact that the local gauge groups allow the construction of a reduced configuration space, to formulate a natural “particle-like” path integral formally defining a gauge-invariant wave-function. We have largely used the example of gravity with spatial diffeomorphisms and Weyl transformations. From now, unless otherwise specified, I will keep a more abstract approach that ignores the presence of gauge-symmetry.
3.2 The semi-classical transition amplitude
Now we move back to the more general field configuration space specified above. First, I should note the ubiquity of interference experiments relying solely on multiple extremal paths; double slit and all sorts of interferometers rely on no dynamical information besides a semi-classical approximation with multiple extremal paths interpolating between initial and final configurations.
Indeed, here I will mostly study transition amplitudes between configurations that have at least one extremal (classical) path interpolating between them, and only briefly touch on more general cases in the accompanying .222222In the usual quantum mechanics setting, if the energy of the particle is lower than a potential barrier, then the fixed energy transition amplitude from one side to the other is exponentially decaying in the phenomenon known as tunneling. One can model the transition using imaginary time, or an Euclidean version of the path integral. Having said this, much progress has been made in explaining tunneling in the usual (or real time) path integral [33, 34]. Moreover, we can separate fields, as I will discuss in section 4.2. One field can be in a semi-classical approximation and serve as background, while the other undergoes evolution through the Schroedinger equation in this background. In the context of path integrals in configuration space, I will be in the semi-classical (or WKB, or saddle point), approximation (in the oscillatory domain).
Explicitly, the setting is given by a path integral in configuration space, (12), for (locally) extremal paths parametrized by the set , , between an initial and a final field configuration , where here stands for both the tensorial and continuous indices. I will denote the on-shell action for these paths as . The expansion, which is accurate for in arc-length parametrization, is then:
where is field-independent, and the Van Vleck determinant has been defined as
We here write the action as a functional of its initial and final points along . To clarify the notation, if we wanted to use continuous indices explicitly, we would have e.g.: the on-shell momenta defined as
I do not concern myself with the normalization factor for the moment, nor with the effect of the Maslov index (which emerges after focusing points only).232323Equation (16) is valid up to the point where the first eigenvalue of reaches an isolated zero, which is a focal point of the classical paths. At such points the approximation momentarily breaks down. However, it becomes again valid after the focal point, acquiring the phase factor known as the Maslov index , which is basically the Morse index (given by the signature of the Hessian) of the action. The field determinant contained in the Van Vleck factor is the only element that would require regularization. As previously mentioned, this Van-Vleck determinant arises from the combination of the projected Liouville measure and the one-loop determinant. In more generality, these semi-classical prefactors can be translated into each other also in the field theory setting using what are known as “reduction methods for functional determinants” , for which standard regularization methods can be applied.
It is fruitful to present this object in the context of Riemannian geometry, for later purposes. There, its given by a time integral of the expansion scalar along a geodesic congruence. I.e. let the Lagrangian be just the infinitesimal line element of a curve, in a -dimensional (semi) Riemannian manifold, and the action is the length of the curve, . Let then be a geodesic congruence, defining the expansion as , the Van-Vleck can be written as:
where is the proper distance along between and , and is the arc-length parameter along . The geometric version of the Van-Vleck can be seen as quantifying the focussing or defocussing of classical trajectories interpolating between and and around . It provides a useful analogy to dynamical systems in configuration space, and a bridge between classical and quantum behavior, which we will explore later in section 5.
Indeed, the standard interpretation of (17) is as an indicator of the spread of classical trajectories in configuration space. That is, the Van Vleck determinant measures how the dynamics expand or contract extremal paths along configuration space, i.e. how densely the initial configurations are transported by the equations of motion to the final configurations.242424The interpretation of this fact is quite ubiquitous, for instance, the Raychaudhuri equation – which describes the spreading of geodesic congruences – has a compact formulation in terms of the Van Vleck determinant . The same is true of the Jacobi fields used to perform the semi-classical expansion of the path integral . As in the particle case, I assume that the classical field history maps an infinitesimal configuration space density to an infinitesimal configuration space density , and the Van Vleck determinant gives a ratio of these densities as propagated by paths around :
Under the assumption that there exists at least one classical (locally extremal) path between the two configurations, we find for the absolute value squared of the amplitude:
Here, to unclutter notation, I have omitted the dependence of the Van-Vleck on the configurations. Interference terms can be clearly identified from (20).
The most appropriate way of extending equation (16)-(20) to higher order of approximations was described in . This is still based on extremal paths, and it can be incorporated with a piecewise approximation as well. Although I will not technically require or use these higher approximations, they provide – at least in principle – a way of extending the analysis done here to a more general context.
Summary of this subsection.
I have here summarized necessary concepts regarding semi-classical approximations for oscillatory path integrals in configuration space. Most important is the appearance in second order of the Van-Vleck determinant; it has the interpretation of a ‘focussing’ of extremal trajectories.
3.3 The Born rule
In the absence of a separation into system and apparatus, the meaning of measurements and of the Born rule become more nebulous. Since my ultimate aim is to apply the constructions here to the whole Universe and to cosmology, I need to address the emergence and meaning of the Born amplitude in the present context.
I have left the form of the function , which “counts” the number of configurations in small regions undetermined. We can associate the Born rule with a relative ‘density of observers’ if we interpret the likelihood of finding oneself e.g. in a region around configuration relative to configuration as the relative volume of these two regions. I now turn to this.
Decoherence is a form of dynamical diagonalization of the (reduced) density matrix. I will use a form of diagonalization appropriate to timeless configuration space, assuming there is no significant interference between different extremal trajectories.252525This is the description of decoherence used extensively in the consistent histories formulation . In this regime, using (20), we can compare the density function to the classically propagated volume elements in configuration space.
Where there is only one extremal path connecting and , namely . More generally, one would have multiple extremal paths between and . This fact forbid us to take to define the densities, since it is path dependent, giving only , not – which needs to take into account interference from other paths. Nonetheless, for the semi-classical kernel, from the no-interference terms in (20), we also have
Thus if we demand that our density functional gives the classically propagated densities in the no-interference limit, i.e.
we find that from (22)
Since we will have a unique choice of , given by the preferred ‘vacuum’, or in-state , we can absorb into the normalization . This proves that, semi-classically, the Born rule emerges from the classical propagation of volumes in configuration space.
But even outside of the semi-classical regime, demanding positivity of , it is easy to see that if one expands the function into polynomials of , i.e. , one would obtain:
and, on the other hand
Alternatively, taking , by differentiating both sides wrt and setting we obtain:
and a similar equation for the complex conjugates, , which means that the function is homogeneous: , and demanding positivity, .262626I thank W. Wieland for these remarks. By the above classical limit, we must finally have that .
Therefore, extending (23) to the full volume form, from the static wave-function over configuration space ,272727Alternatively, we could have defined a “vacuum state” existing over a trivial (one complex dimension) Hilbert space with the usual complex space inner product, over , and then taking as an operator such that . gives
Equation (26) – i.e. the Born rule – is the extension of to arbitrary points .
Note that no mention of “measurements”, or “observations” need to be made. Volumes in configuration space suffice.
Summary of this subsection.
The decomposition property of the function is necessary if we want to translate certain statements about the amplitude to statements about probability. Here I have shown that given just this property and implementing a semi-classical limit, one can uniquely derive the Born rule as giving the volume-form in configuration space.
Given the Past Hypothesis, even allowing for an objective meaning to a relational transition amplitude, , we should be able to do science from assumptions about relative number of configurations in . It is true that while examining the results of an experiment, all we have access to for comparison are the memories, or records, of the setup of the experiment.
Since we are in a context that cannot rely on absolute time, the meaning of measurements and the updating of probabilities needs to be significantly modified. Here, I will give a semi-classical definition of records. This definition should be seen as a mathematical pre-requisite for any functioning role for records; they do not, however, pinpoint the physical encoding of information in physical structures.
4.1 Semi-classical records
I will denote a recorded configuration as , and the manifold whose elements have as a record, as . If these represent experiments, coexisting with the given record, each copy of “the experimenter” will find itself in one specific configuration, . The manifold , consisting of all those configurations with the same records, then represents the setup of the experiment. In the cosmological setting this could be, for instance, all of the configurations that contain a ‘record’ (as defined below) of the surface of last scattering. The post-selection is locating your configuration within .
Loosely, a configuration will be defined to hold a record of (the recorded) configuration if, for a given ‘in’ configuration , all extremal paths from to go through . This definition is meant to embody the idea that ’s “happening” is encoded in the state . At least semi-classically, ‘branches’ of the wavefunction contributing to the amplitude of have “gone through” .
One note of caution: in the oscillatory semi-classical regime, when one speaks of the major contribution to the amplitude as coming from extremal paths, what is really meant is that there is a coarse-graining of paths, seeded by the extremal ones, in which the paths close to the extremal ones enjoy constructive interference, whereas ones that deviate much have their interference wash out. To be rigorous, in the companion paper  I have formulated the constructions here directly from specific types of coarse-grainings of paths. The results below go through without problems.282828 In that more rigorous setting, for the semi-classical record to be of order , the length of the extremal paths that seed the coarse-graining, , need to obey , and, up to this order of approximation, the preferred extremal coarse-graining defined in  cannot resolve if lies along the actual extremal paths, or just around them. Moreover, to extend this notion to that of piece-wise extremal paths, I need to use that notion of extremal coarse-grainings as well.
Given , the action on curves in configuration space, and the collection of parametrized extremal paths such that , , then is said to have a semi-classical record of if for each , there exists a , such that and .
And with this definition we can prove the following
Given a configuration with a semi-classical record of , then
Combined with property (2) for the density functional, this means that the equation for the probability of automatically becomes an equation for conditional probability on ,
Using the semi-classical approximation (16),
If the system is deparametrizable, this means that one of the configuration variables can be used as “time”, and extremal trajectories can be monotonically parametrized by this variable. Then we could use (94), and the semi-classical composition law for the semi-classical transition amplitude (see )292929Here we are crucially assuming the ‘folding property’ for the path integral. See . to write:
for a given intermediary time . Choosing , there is a unique configuration through which all of the paths go through, . The integral gains a , since extremal paths pass only through that point at , and thus:
In the more general case however, there is more work to be done.
where are the sets of extremal paths interpolating between and and those between and . What we want to show is that to order , the contributing paths will be those that are continuous at . After this is done, we need to show that the Van Vleck determinants have the right composition law. This is done in appendix D.
Strings of records
For more than one recorded configurations, say , definition 1 demands that each extremal path go through in one order or another. Let contain multiple semi-classical records , i.e. . Then an ordering of the records must exist for each extremal curve. For each , the set is ordered: .
In the presence of a single element ,303030And remembering that the on-shell action of between and must be much larger than for records to be defined. we can concatenate . The decomposition of the density follows:
In other words, if there is one single (no interference), and the semi-classical limit holds (i.e. the action spacing between the records is larger than ), then the records in yield a coarse-grained classical history of the field. In this case we recover a (granular) notion of classical Time.313131In a context of consistent histories, the separation of order greater than between records will avoid the argument of Halliwell concerning the quantum Zeno effect .
We can still obtain a consistent string of records for many interfering branches if the ordering given for each , coincide, i.e. if for all and any , we have . Then kernel still can be decomposed:
This matches the analysis performed by Halliwell, in which he recovers “time” from a simple Hamiltonian Mott bubble chamber ansatz,323232Where spontaneous emission of -particles from a source ionize bubbles in a chamber of water vapor . finding that the amplitude for -bubbles to be excited, with the -th bubble configuration being , is given by:
where are Green’s functions (for the free Hamiltonian), are projections onto small regions of configuration space surrounding the -th configuration, and is the dimension of configuration space. For this derivation, Halliwell notes that it is essential that there is some asymmetry in configuration space, marked by the source of the -particles. It is the same here: we require axiom 5 of an’origin’ of configuration space (see also  for an extended version of this relationship).
The definition implies that a configuration can hold many records, and an ordering among these, with earlier recorded configurations being themselves recorded in later recorded configurations. Through this ordering, a semblance of global time emerges from a fundamentally timeless theory. Again, this is in line with Halliwell’s reconstruction of time in the Mott chamber context . Configurations that are far from the given initial configuration – but still connected to it by extremal paths – will in general have concentrated amplitude, and hold more records.
Relative probabilities of regions in with the same records.
The setup of an experiment implies the fixing of a submanifold in configuration space, , characterized by its points all possessing the same records. In this abstract simple example, the submanifold is characterized by a single configuration , which could represent for instance, “the whole laboratory setup of a double slit experiment and the firing of the electron gun”, or, “the cosmological surface of last scattering”.
We can compare transition amplitudes for configurations with the same records in the following way, given , for any initial :
This ratio gets rid of a common quantity to both density functions.
In most experimental settings then, equation (31) justifies the practical use of the record configuration as the effective initial point in the kernel. It is as if a measurement occurred, determining . While it is true that the theory is timeless, we can still give a meaning to active verbs such as “updating” (e.g. of our confidence level): to the extent that humans are classical systems, extremal paths in configuration will reflect anything that the equations of motion predict, including rational (and irrational) “updating” of our theories.
Summary of this subsection.
I have here defined records. These are structures that can arise given our volume-form on , and which have many interesting properties. They yield conditional probabilities (in much the same way that the Mott bubbles do); as correlated volumes in configuration space that emulate a notion of “causation”. I have only discussed these structures in the semi-classical regime, whence one obtains at most a ‘granular’ history 333333With records separated by action of order greater than . In the arc-length parametrization of the following sections, this is a separation in superspace.
It is important to note here that our arguments from section 2.1, regarding the gauge-dependent character of intersecting curves in superspace if refoliations are allowed as a local symmetry. I.e. two curves might intersect before, but not after the action of a refoliation. This is not the case if one allows only the sort of local symmetries we have considered here, and/or global reparametrizations. This is the reason why the relativistic particle and minisuperspace models would not present such a problem. But it shows a disconnect, in so far as the concept of records explored here is concerned, between minisuperspace and the full theory with local refoliations.
4.2 Records and conservation of probability
The generic case should be one of redundancy of records in .343434 I mean this in two ways: first, that many different subsystems will have redundant records of the same “event” (subset of a configuration, in the sense of ). For example, all the photons released by an electron hitting a fluorescent screen. By comparing the consistency of these records, one can formulate theories about the configuration space action and the probability amplitude. This “consistency of records” approach, has been recently emphasized in the context of selecting the basis for branch selection in decoherence . Since in this paper I’m not explicitly dealing with subsystems – a topic I have dealt with in  – I will ignore this type of redundancy; although it is very important for doing science. In other words, within one could have a polygamy of record relations; e.g.: with . This happens for example in the case of strings of records, discussed above. Ideally, when discussing conservation of probabilities, we would be able to discard such redundancy.
A choice of subset of for which there is no such redundancy will be called a screen, written as