The Entropic Dynamics of Quantum Scalar Fields Coupled to Gravity
aa r X i v : . [ g r- q c ] J u l The Entropic Dynamics of Quantum ScalarFields Coupled to Gravity
Selman Ipek ∗ and Ariel Caticha † Physics Department, University at Albany-SUNY, Albany, NY 12222, USA.
Abstract
Entropic dynamics (ED) is a general framework for constructing inde-terministic dynamical models based on entropic methods. ED has beenused to derive or reconstruct both non-relativistic quantum mechanicsand quantum field theory in curved space-time. Here we propose a modelfor a quantum scalar field propagating in a dynamical space-time. Theapproach rests on a few key ingredients: (1) Rather than modelling thedynamics of the fields, ED models the dynamics of their probabilities. (2)In accordance with the standard entropic methods of inference the dy-namics is dictated by information encoded in constraints. (3) The choiceof the physically relevant constraints is dictated by principles of symme-try and invariance. The first such principle imposes the preservation ofa symplectic structure which leads to a Hamiltonian formalism with itsattendant Poisson brackets and action principle. The second symmetryprinciple is foliation invariance, which following earlier work by Hojman,Kuchaˇr, and Teitelboim, is implemented as a requirement of path inde-pendence. The result is a hybrid ED model that approaches quantum fieldtheory in one limit and classical general relativity in another, but is notfully described by either. A particularly significant prediction of this EDmodel is that the coupling of quantum fields to gravity implies violationsof the quantum superposition principle.
Without any empirical matter of fact, and none likely on the near horizon,quantum gravity (QG) research has largely split off into distinct channels,each reflecting a different set of attitudes, and yes, philosophies directedtowards the problem at hand. (See e.g., [1] for a recent overview of somefeasible experimental proposals.) But this state of affairs should not beentirely surprising. Solving the problem of QG should, after all, entailfirst addressing the issue of what QG even is — indeed, what is meant by quantum ? What is meant by gravity ? Which, if any, elements of Einstein’s ∗ [email protected] † [email protected] ravity, or the standard quantum formalism, should be abandoned in thetransition to QG?Pursuant to this, a common view is that the gravitational field itselfshould, in some way, be quantized. There are several routes to accomplish-ing this. Most typical, however, is through some manner of quantizationalgorithm: extending the existing quantum formalism to the gravitationaldomain by application of the standard ad hoc quantization rules to the ap-propriate gravitational degrees of freedom. Such is the tack taken by theprincipal QG candidates — string theory (ST) and loop quantum gravity(LQG). But such approaches should be met with some suspicion in the contextof QG. One issue is the quantization procedure itself. Take models withconstraints, such as general relativity (GR), for instance. Should we solvethe constraints then quantize, or quantize then solve the constraints? Thetwo methods are not, in general, the same (see e.g., [7]). Another issue ischoosing which degrees of freedom to quantize in the first place. Histor-ically this process has been guided by tight coordination between theoryand experiment, allowing trial and error to supplement an incomplete un-derstanding of the quantization process itself. But absent such help fromexperiment, a more fundamental understanding of quantization may benecessary to make progress.The lack of clarity around quantization is made particularly acutein the case of the gravitational field, which plays a dual role in GR asan object with genuine dynamical modes, but one that also serves toestablish spatial and temporal relationships. This calls into question theprecise nature of the gravitational field. Is it just another field to bequantized, along the lines of the gauge fields of Yang-Mills theories, or isit something else entirely? Canonical approaches to QG, such as LQG,have wagered, with varying degrees of success, that the former is true.But there are various clues (see e.g., [9]-[13]) suggesting that gravity is anemergent, statistical phenomenon, not unlike temperature and pressurein statistical physics. Would any physicist quantize a temperature field?Perhaps not, but this is exactly what we might be doing when we quantizethe gravitational field.All of this suggests that a different attitude with regards to QG maybe in order. In the past couple of decades there has been increased in-terest (see e.g., [14]-[19]) in developing approaches to QG that seek totreat gravitational considerations hand-in-hand with a more robust un-derstanding of quantum theory (QT). The goal in these cases, however, isnot just the development of specific models (although such models mustnecessarily follow), but the construction of entire frameworks that canreadily integrate salient features of gravitational and quantum physics Indeed, highlighting the interpretational difficulties of QG, is the way that notions of quantum can vary from model to model. A popular class of theories postulates that thecombination of quantum theory with gravity should give way to fundamentally discrete modelsof space and time, such as in the causal set [2] and causal dynamical triangulation [3] programs. To be sure, although there are vast differences between canonical approaches to QG,including LQG, and ST (see e.g., [4]), the adoption of a quantization algorithm is possiblyone of the few points of intersection. For some historical overviews in the development of STand LQG, see the reviews by S. Mukhi [5] and C. Rovelli [6], respectively. like. One such approach is afforded by entropic dynamics (ED), whichis a general framework for constructing indeterministic dynamical modelsbased on the principles of Bayesian probability and entropic inference.(For comprehensive overview of Bayesian and entropic methods, see e.g.,[20][21].)Among the successes of the ED framework are principled derivationsof several aspects of the quantum formalism [22]-[32], which have led tomany key insights. Foremost among these is the notion of probabilityitself. In ED, probabilities are Bayesian — that is, there are no classicalprobabilities or quantum probabilities, there are simply probabilities andthey follow the usual rules of probability theory [33]. But note that thisis not a trivial statement. Taking the structure of probability seriously is,in fact, highly constraining. For instance, once we adopt a probabilisticviewpoint, a natural question that follows is, probabilities of what? Are wedealing with the outcomes of an experimental device, the configurations offields, or the positions of particles? There is no flexibility here. We mustchoose; a notion that stands in stark contrast to the usual Copenhageninterpretation. Equally constraining is how probabilities are updated inED. Indeed, it is not enough to simply declare that | Ψ | yields a Bayesianprobability — one must also demonstrate that the rules for updatingthose probabilities are in strict accordance with the Bayesian and entropicmethods of inference. In short: ED is a dynamical framework driven byconstraints.One of the essential challenges in ED is therefore the appropriate iden-tification of constraints. There have been a series of developments in thisregard by appealing to principles of symmetry and invariance, which haverich traditions in both physics [34] and inference [35][36] alike. Earlyprogress in ED [22][23], aided by insights from Nelson’s stochastic mechan-ics [37], was made by recognizing that conservation of a suitable energyfunctional could be used as the main criterion for updating the evolv-ing constraints. This allowed ED to model many important aspects ofquantum mechanics, leading eventually to a fully Hamiltonian formalism,with its attendant action principle, a symplectic structure, and Poissonbrackets [24]. However, a sharper understanding of the deep role playedby isometries and symplectic symmetries in QT (see e.g., [38]-[42]) sug-gested another path where symplectic and metric structures take a morefundamental place in the ED approach [25]. Issues, such as the single-valued nature of the wave function Ψ, or more importantly, the linearityof quantum time evolution are clarified from this perspective as resultingfrom the marriage of symmetry principles with the probabilistic structureof ED. Building on these developments are the previous efforts by Ipek, Abedi,and Caticha (IAC) [30][31] to model the ED of quantum scalar fields on a fixed curved background. Of particular interest to us are the manifestlycovariant methods developed by IAC, which introduced several novel fea- This has been a crucial insight, for instance, for the development of a model for spin-1 / For a discussion the single-valuedness condition in quantum mechanics, see e.g., [44].This condition is particularly problematic in approaches to QT that resemble formal aspectsof Nelson’s stochastic mechanics (see e.g., [45]). See [25] for how this is resolved in ED. ures to ED. One such contribution pertains to the role of time. In ED,time, or better yet, entropic time is constructed as a scheme to keep trackof the accumulation of small changes [23]. In previous work, in the contextof a flat space-time [28][29], it was appropriate to introduce a global notionof time, in which all spatial points were updated uniformly. As discussedin [30][31], however, this assumption must be relaxed in a curved space-time in favor of a local notion of entropic time. This raises an importantchallenge, namely, the construction of an updating scheme that is local innature.To this end, the work done by IAC in [30][31] synthesized two devel-opments in ED: (I) the adoption of a symplectic structure, together withits accompanying Poisson bracket formalism and (II) an updating schemethat unfolds in local entropic time. The two pieces work in tandem. Theformer allows ED to marshal the full power of the canonical theory, which,in turn, facilitates the desired local time dynamics. In [30][31], this latteringredient is itself inspired by the seminal works of Dirac [46][47] as well asHojman et al. [48], Kuchaˇr [49], and Teitelboim [50][51] (DHKT) in theirdevelopment of covariant canonical methods in classical field theory. Drawing on the ideas of DHKT, the IAC framework proceeds as fol-lows. A chief concern is the notion of an instant, which is defined bya three-dimensional space-like surface embedded in space-time; plus thefields, probability distributions, and so on, that are defined on these sur-faces. It is then possible to slice or foliate space-time into a sequence ofsuch space-like surfaces. While the decomposition of space-time into spa-tial slices, indeed, obscures the local Lorentz symmetry of the full pseudo-Riemannian manifold, manifest covariance can be recovered by requiringthat the dynamics be unaffected by the particular choice of foliation. Thisfoliation invariance can itself be implemented by a consistency conditiondubbed path independence [49][50]: the evolution of all dynamical quan-tities from an initial to a final surface must be independent of the choiceof intermediate surfaces.The requirement of path independence, as laid out by DHKT, thusimposes certain conditions on the generators of the local time dynamics.Crucially important, however, is not simply the existence of such con-ditions, which merely reflects the constraints of path independence, butthat these conditions are of a universal nature. Put another way, any attempt to formulate a dynamics that unfolds in space-time must mirrorthis pattern, which is itself reflective of the structure of space-time defor-mations. Thus ED, which is designed as a dynamical scheme that evolvesstep-by-step from one instant to the next, can be made manifestly covari-ant by imposing just such a structure. This results in a relativistic EDwhere quantum fields evolve on a fixed classical background, generalizingthe pioneering efforts of DHKT which dealt solely with the evolution of classical fields.Here we model the indeterministic dynamics of a scalar field χ ( x ) inter-acting with classical gravity. Although there is a growing body of work on These approaches could themselves be traced to the earlier “many-time” efforts of Weiss[52], Tomonaga [54], Dirac [53], and Schwinger [55] in the context of relativistic quantumtheory. odifications of classical GR as a prelude to its eventual quantization, inthis paper we pursue a more modest goal: to explore the consequence of afirst principles derivation of quantum field theory (QFT) while maintain-ing classical gravity unmodified. Thus our current work, which expandson [32], extends the efforts of IAC in one crucial respect by allowing thebackground geometry itself to become dynamical . Following DHKT, thetransition to a dynamical background is not accomplished by altering theconditions of path independence, but rather, it is made by an appropri-ate choice of variables for describing the evolving geometry. The result isa hybrid ED model that approaches QFT in a fixed background in onelimit and approaches classical general relativity in another limit, but isnot fully described by either. In particular, the model shares some formalsimilarities with the so-called Semi-classical Einstein equations (SCEE)(see e.g., [56]), but here we model the fluctuations of the quantum fieldsand derive their coupling to classical gravity from first principles withoutthe ad hoc arguments typically used to justify the SCEE.To summarize, in a dynamics based on inference the relevant physi-cal information is supplied through constraints. In the present work theconstraints are chosen so as to enforce the symmetry and invariance prin-ciples that lie at the foundation of quantum theory and general relativity.More explicitly we impose that the dynamics be such as to preserve thesymplectic structure and to enforce path independence which amounts tofoliation invariance and local Lorentz invariance.An important feature of the ED derivation of nonrelativistic quantummechanics [25] is its analysis of the superposition principle which led tothe recognition that the linearity of quantum mechanics is a consequenceof the introduction of Hilbert spaces. After all, this is precisely the reasonwhy Hilbert spaces are introduced in the first place: while in principlethey are not needed for the formulation of quantum mechanics, their in-troduction is nevertheless a very convenient calculational trick becauseit allows one to make use of the calculational advantages of the linearitythey induce. A significant result of our present ED reconstruction of arelativistic QFT coupled to gravity is that the dynamics is fundamentallynonlinear. Not only does this imply violations of the quantum superpo-sition principle, but it brings into question the very reason for Hilbertspaces.Incidentally, this also makes explicit the fundamental disagreementbetween the ED approach and other approaches such as the orthodoxCopenhagen and the many-worlds interpretation. In these interpreta-tions the linear structure of the Hilbert space including the superpositionprinciple is given priority while the probabilistic structure is either a sec-ondary addition designed to how to handle those mysterious physical pro-cesses called measurements or, as in the many-worlds interpretation, it isavoided altogether. The unwelcome result is that the dynamical and theprobabilistic aspects of quantum theory are essentially incompatible witheach other. In contrast, ED resolves these problems by giving priority tothe probabilistic structure of QM which relegates Hilbert spaces to play Excellent reviews with extended references to the literature are given in e.g. [57][58]. Themany-worlds interpretation is discussed in [59]. secondary non-fundamental role. Indeed, in the particular problem dis-cussed here – quantum fields coupled to dynamical gravity – the dynamicsis intrinsically nonlinear, there is no superposition principle, and thereforeno reason for Hilbert spaces.The outline of the paper is the following. In section 2, we review theED of short steps. Following this, in section 3 we introduce some keynotation, which is useful in our development of entropic time in section4. Key concepts of space-time deformations and “embeddability” are in-troduced in section 5. In section 6 we introduce the canonical formalismin ED, which is a necessary step before we review the condition of pathindependence developed by DHKT in section 7. Section 8 outlines theconstruction of the local generators. In section 9 we describe the resultingdynamical equations, while in section 10 we apply these results to obtainan ostensibly quantum theory. We discuss our results in section 11. We present a short review of the ED of infinitesimal steps in curved space-time, adopting the notations and conventions of [30][31][32]. Here the ob-ject of analysis is a single scalar field χ ( x ) ≡ χ x that populates space andwhose values are posited to be definite, but unknown, and thus amenableto being described by probabilities. An entire field configuration, whichwe denote χ , lives on a 3-dimensional space σ , the points of which arelabeled by coordinates x i ( i = 1 , , σ is itself curved andcomes equipped with a metric g ij that is currently fixed, but that willlater become dynamical. A single field configuration χ is a point in an ∞ -dimensional configuration space C . Our uncertainty in the values ofthis field is therefore quantified by a probability distribution ρ [ χ ] over C ,so that the probability that the field attains a value ˆ χ in an infinitesimalregion of C is Prob[ χ < ˆ χ < χ + δχ ] = ρ [ χ ] Dχ , where Dχ is an integrationmeasure over C . On microstates—
In ED the field distributions χ x play a singularlyspecial role: they define the ontic state of the system. This ontologicalcommitment is in direct contrast with the usual Copenhagen interpre-tation in which such microscopic values become actualized only throughthe process of measurement. The Bohmian interpretation shares with ED A virtue of the ED approach is that it achieves ontological clarity by insisting on a sharpdistinction between its ontic and epistemic elements. A concept is ontic when it describessomething real that exists independently of any observer. In the words of John Bell onticvariables are beables. A concept is epistemic when it is related to the state of knowledge,opinion, or belief of an agent, albeit an ideally rational agent. Examples of epistemic quantitiesare probabilities, entropies, and wave functions. An important point is that the distinctionsontic/epistemic and objective/subjective are not the same. Probabilities are fully epistemicthey are tools for reasoning with incomplete information but they can lie anywhere in thespectrum from being completely subjective (two different agents can have different beliefs)to being completely objective. In QM, for example, probabilities are epistemic and objectivethere is such a thing as an objectively correct assignment of probabilities. We will say that thewave function Ψ, which is fully epistemic and objective, represents a “physical” state when itrepresents information about an actual “physical” situation. hat fact that in both the fields are ontic but the resemblance ends there;Bohmian wave functions are ontic while ED wave functions are fully epis-temic [25][60]. The metric g ij , on the other hand, in our approach is a tool whose purpose is to measure distances, areas, etc., and to character-ize the spatial relations between the physical degrees of freedom, the χ x .While the geometry may later become dynamical, we do not interpret thisto mean that g ij is itself an ontic variable; it is not. Put another way,the geometric variables enter much like parameters in a typical statisticalmodel. The value of those parameters are important in guiding the distri-bution of outcomes. However, unlike the ontic variables, their values arenot detected directly, but inferred from an ensemble of measurements.
Maximum Entropy—
Our goal is to predict the indeterministic dy-namics of the scalar field χ whose statistical features are captured by aprobabilistic model. To this end, we make one major assumption: in ED,the fields follow continuous trajectories such that finite changes can beanalyzed as an accumulation of many infinitesimally small ones. Such anassumption allows us to focus our interest on obtaining the probability P [ χ ′ | χ ] of a transition from an initial configuration χ to a neighboring χ ′ = χ + ∆ χ . This is accomplished via the Maximum Entropy (ME)method by maximizing the entropy functional, S [ P, Q ] = − Z Dχ ′ P (cid:2) χ ′ | χ (cid:3) log P [ χ ′ | χ ] Q [ χ ′ | χ ] , (1)relative to a prior Q [ χ ′ | χ ] and subject to appropriate constraints. The prior —
We adopt a prior Q [ χ ′ | χ ] that incorporates the infor-mation that the fields change by infinitesimally small amounts, but isotherwise maximally uninformative. In particular, before the constraintsare taken into account, knowledge of the dynamics at one point does notconvey information about the dynamics at another point, i.e. the degreesof freedom are a priori uncorrelated.Such a prior can itself be derived from the principle of maximum en-tropy. Indeed, maximize S [ Q, µ ] = − Z Dχ ′ Q (cid:2) χ ′ | χ (cid:3) log Q [ χ ′ | χ ] µ ( χ ′ ) , (2)relative to the measure µ ( χ ′ ), which we assume to be uniform, and sub-ject to appropriate constraints. The requirement that the field undergoeschanges that are small and uncorrelated is implemented by imposing an A model in which the metric tensor is itself of statistical origin is proposed in [12]. Here we adopt a view of entropy as a tool designed for updating probability distributionsfrom a prior to a posterior. (For an accessible introduction, see e.g., [20].) Such a perspectiveestablishes entropy, beyond just its role in thermodynamics, as an essential asset to inductiveinference writ large. Since we deal with infinitesimally short steps, the prior Q turns out to be quite independentof the background measure µ . nfinite number of independent constraints, one per spatial point x , h ∆ χ x i = Z Dχ ′ Q (cid:2) χ ′ | χ (cid:3) (∆ χ x ) = κ x , (3)where ∆ χ x = χ ′ x − χ x and the κ x are small quantities. The result ofmaximizing (2) subject to (3) and normalization is a product of Gaussians Q (cid:2) χ ′ | χ (cid:3) ∝ exp − Z dx g / x α x (∆ χ x ) , (4)where α x are the Lagrange multipliers associated to each constraint (3);the scalar density g / x = (det g ij ) / is introduced so that α x is a scalarfield. For notational simplicity we write dx ′ instead of d x ′ . To enforcethe continuity of the motion we will eventually take the limit κ x → α x → ∞ . The global constraint—
The motion induced by the prior (4) leadsto a rather simple diffusion process in the probabilities, in which the fieldvariables evolve independently of each other. To model a dynamics thatexhibits correlations and is capable of demonstrating the full suite of quan-tum effects, such as the superposition of states, interference, and entangle-ment, however, we require additional structure. This is accomplished byimposing a single additional constraint that is non-local in space but localin configuration space, which involves the introduction of a drift poten-tial φ [ χ ] which is a scalar-valued functional defined over the configurationspace C . More explicitly, we impose h ∆ φ i = Z Dχ ′ P (cid:2) χ ′ | χ (cid:3) Z dx ∆ χ x δφ [ χ ] δχ x = κ ′ , (5)where we require κ ′ →
0. (Note that since χ x and ∆ χ x are scalars, in orderthat (5) be invariant under coordinate transformations of the surface, thederivative δ/δχ x must transform as a scalar density.)Before moving on, we discuss the central role of the drift potential inED. As a dynamics of probabilities ED brings together three ingredients.First, from an inference perspective the dynamics consists in an updatingof probabilities which must obey the rules of entropic inference. This re-quires information codified into constraints and the drift potential is thefunction that codifies such information. Second, in order to formulatea dynamical theory in which probabilities are generalized coordinates itis only natural to attempt to identify the canonically conjugate general-ized momenta and the corresponding symplectic structure. And third,from a purely geometric perspective the trajectories we seek are curveson the space of probability distributions. To this statistical manifold itis natural to associate its tangent bundle. The fibers of such a bundleare spaces of velocity vectors tangent to all curves. It is also natural toconsider the associated cotangent bundle the fibers being the spaces oftangent covectors. As we shall see ED brings together these three sepa-rate ingredients by identifying the constraint embodied in the drift poten-tial functional with the momenta conjugate to the probabilities, with the otangent vectors. More explicitly, we identify the three separate pairs:probability/constraint, coordinate/momentum, point/covector. And aswe shall see, eventually these three pairs are also identified with yet afourth pair: the magnitude/phase of the wave function. The transition probability —
Next we maximize (1) subject to (5)and normalization. The multiplier α ′ associated to the global constraint(5) turns out to have no influence on the dynamics: it can be absorbed intothe yet undetermined drift potential α ′ φ → φ , effectively setting α ′ = 1.The result is a Gaussian transition probability distribution, P (cid:2) χ ′ | χ (cid:3) ∝ exp − Z dx g / x α x ∆ χ x − g / x α x δφ [ χ ] δχ x ! . (6)In previous work [28][29], α x was chosen to be a spatial constant α toreflect the translational symmetry of flat space. Such a requirement, how-ever, turns out to be inappropriate in the context of curved space-time.Instead, we follow [30][31] in allowing α x to remain a non-uniform spatialscalar. This will be a key element in developing our scheme for a localentropic time.The form of (6) allows us to present a generic change,∆ χ x = h ∆ χ x i + ∆ w x , as resulting from an expected drift h ∆ χ x i plus Gaussian fluctuations ∆ w x .Computing the expected short step for χ x gives h ∆ χ x i = 1 g / x α x δφ [ χ ] δχ x , (7)while the fluctuations ∆ w x satisfy, h ∆ w x i = 0 , and h ∆ w x ∆ w x ′ i = 1 g / x α x δ xx ′ . (8)Thus we see that while the expected step size is of order ∆ ¯ χ x ∼ /α x ,the fluctuations go as ∆ w x ∼ /α / x . Thus, for short steps, i.e. α x →∞ , the fluctuations overwhelm the drift, resulting in a trajectory that iscontinuous but not, in general, differentiable. Such a model describes aBrownian motion in the field variables χ x . The ED developed here deals with the coupled evolution of a quantumscalar field together with a classical dynamical background. The class oftheories that allow such evolving geometries are often called instances of As discussed in [25][60], with appropriate choices of the multipliers α and α ′ the fields orthe particles, as the case might be, can be made to follow paths typical of a Brownian motionor alternatively paths that are smooth and resemble a Bohmian motion. It is remarkable thatthese different paths at the sub-quantum level all lead to the same Schr¨odinger equation. eometrodynamics . In a typical geometrodynamics (see e.g., [61][62][63]),the primary object of interest is the evolving three-metric g ij ( x ), whosedynamics must be suitably constrained so that the time evolution sweeps afour-dimensional space-time with metric g µν ( µ, ν, ... = 0 , , , It is therefore possible to assign coordinates X µ to the space-time man-ifold. Furthermore, we deal with a space-time with globally hyperbolic topology, admitting a foliation by space-like surfaces { σ } . The embed-ding of such surfaces in space-time is defined by four scalar embeddingfunctions X µ (cid:0) x i (cid:1) = X µx . An infinitesimal deformation of the surface σ toa neighboring surface σ ′ is determined by the deformation vector, δξ µx = δξ ⊥ x n µx + δξ ix X µix . (9)Here we have introduced n µ , which is the unit normal to the surface thatis determined by the conditions n µ n µ = − n µ X µix = 0), and wherewe have introduced X µix = ∂ ix X µx , which are the space-time componentsof three-vectors tangent to σ . The normal and tangential componentsof δξ µ , also known as the infinitesimal lapse and shift, are collectivelydenoted δξ Ax = ( δξ ⊥ x , δξ ix ) and are given by δξ ⊥ x = − n µx δξ µx and δξ ix = X iµx δξ µx , (10)where X iµx = g µν g ij X νjx . Additionally, a particular deformation is de-fined by its components ξ Ax , which allows us to speak unambiguously aboutapplying the same deformations to different surfaces. Consequently, thevery same deformation ξ Ax applied to different surfaces with distinct nor-mal vectors n µx will generally yield different deformation vectors ξ µx , as pereq.(9). In ED, entropic time is introduced as a tool for keeping track of theaccumulation of many short steps. (For additional details on entropictime, see e.g., [20].) Here we introduce a manifestly covariant notion ofentropic time, along the lines of that in [30][31].
An instant—
Central to the formulation of entropic time is the notionof an instant which includes two main components. One is kinematicthe other informational. The former amounts to specifying a particularspace-like surface. The latter consists of specifying the contents of theinstant, namely, the information the relevant probability distributions,drift potentials, geometries, etc. that are necessary to generate the nextinstant. Remarkably, as discussed by Teitelboim [51], although certain aspects of the formalism,indeed, rely crucially on space-time itself, such notions are ultimately absent from the operativeequations of geometrodynamics. In view of this, we can rightfully regard three-dimensionalspace as taking a primary role, with space-time being taken as a secondary construct. rdered instants— Establishing the notion of an instant proves cru-cial because it supplies the structure necessary to inquire about the field χ x at a moment of time. Equivalently, such a notion allows us to assigna probability distribution ρ σ [ χ ] corresponding to the informational stateof the field χ x at an instant labeled by the surface σ .Dynamics in ED, on the other hand, is constructed step-by-step, asa sequence of instants. Thus it is appropriate to turn our attention tothe issue of updating from some distribution ρ σ [ χ ] at some initial instant,to another distribution ρ σ ′ [ χ ] at a subsequent instant. Such dynamicalinformation is encoded in the short-step transition probability from eq.(6),or better yet, the joint probability P [ χ ′ , χ ] = P [ χ ′ | χ ] ρ σ [ χ ].Application of the “sum rule” of probability theory to the joint distri-bution yields ρ σ ′ [ χ ′ ] = Z Dχ P (cid:2) χ ′ | χ (cid:3) ρ σ [ χ ] . (11)The structure of eq.(11) is highly suggestive: if we interpret ρ σ [ χ ] as beingan initial state, then we can interpret ρ σ ′ [ χ ′ ] as being posterior to it inthe sense that it has taken into account the new information captured bythe transition probability P [ χ ′ | χ ]. Here we take this hint seriously andadopt eq.(11) as the equation that we seek for updating probabilities. Duration—
A final aspect of time to be addressed is the durationbetween instants. In doing so, one must distinguish between two separateissues. On one hand there is a natural notion of time that can be inheritedfrom space-time itself; this being the local proper time δξ ⊥ x experiencedby an observer at the point x (see e.g., [64]). On the other hand, however,notions of time and duration cannot themselves be divorced from those ofdynamics and change. Indeed, the two are closely bound together.Our strategy can be summarized by Wheeler’s maxim [61]: “time isdefined so that motion looks simple.” In Newtonian mechanics, for ex-ample, time is defined so as to simplify the motion of free particles; theprototype of a clock is a free particle that moves equal distances in equaltimes. In ED for short steps the dynamics is dominated by fluctuations,eq.(9). Accordingly, the prototype of a clock is provided by the fluctua-tions of the field. Since the specification of the time interval is achievedby an appropriate choice of multipliers alpha we proceed by setting α x = 1 δξ ⊥ x so that h ∆ w x ∆ w x ′ i = δξ ⊥ x g / x δ xx ′ . (12)With this the transition probability eq.(6) resembles a Wiener process,albeit in a rather unfamiliar context involving the propagation of fields oncurved space. Equation (12) shows that at each point x time is definedby a local clock so that fluctuations as measured by the variance increaseby equal amounts in equal time delta ξ ⊥ x . The local-time diffusion equations—
The dynamics expressedin integral form by (11) and (12) can be rewritten in differential form. he result is [30][31], δρ σ [ χ ] = Z dx δρ σ [ χ ] δξ ⊥ x δξ ⊥ x = − Z dx g / x δδχ x (cid:18) ρ σ [ χ ] δ Φ σ [ χ ] δχ x (cid:19) δξ ⊥ x , (13)where we have introducedΦ σ [ χ ] = φ σ [ χ ] − log ρ / σ [ χ ] , which we refer to as the phase functional. For arbitrary choices of the infinitesimal lapse δξ ⊥ x we obtain an infiniteset of local equations, one for each spatial point δρ σ δξ ⊥ x = − g / x δδχ x (cid:18) ρ σ δ Φ σ δχ x (cid:19) . (14)To interpret these local equations, consider again the variation givenin eq.(13). In the special case where both surfaces σ and σ ′ happen to beflat then g / x = 1 and δξ ⊥ x = dt are both spatial constants and eq.(13)becomes equivalent to ∂ρ t [ χ ] ∂t = − Z dx δδχ x (cid:18) ρ t [ χ ] δ Φ t [ χ ] δχ x (cid:19) . (15)We recognize this [28][29] as a diffusion or Fokker-Planck equation writtenas a continuity equation for the flow of probability in configuration space C . This suggests identifying the V x = 1 g / x δ Φ σ [ χ ] δχ x that appears in eq.(14) as the velocity of the probability current which isvalid for curved and flat spaces alike. Accordingly we will refer to (14) asthe “local-time Fokker-Planck” equations (LTFP). Starting with Dirac [46][61] and developed more fully by Hojman, Kuchaˇr,and Teitelboim, a chief contribution of the DHKT program was the recog-nition that covariant dynamical theories had a rich structure that couldbe traced to the kinematics of surface deformations. Such structure can,however, itself be studied independently of any particular dynamics being A key aspect of ED is that we model the dynamics of probabilities, rather than thecorresponding microstates. Therefore, while we do introduce a symplectic structure and fullyHamiltonian formalism in ED, it is for the probability ρ σ and its eventual conjugate variableΦ σ , not for the ontic field χ x , which, strictly speaking, has no equivalent notion of a conjugatemomentum. Nevertheless, an analogous concept can be introduced in ED using the phasefunctional Φ σ through P x = δ Φ σ /δχ x . As shown in [31], the expected values of χ x and P x do, in fact, satisfy a Poisson bracket that is very reminiscent of the canonical Poisson bracketrelations of classical physics. This suggests identifying P x itself as a type of momentum, whichmuch resembles the notion of momentum familiar from the Hamilton-Jacobi formulation ofclassical physics. onsidered. We give a brief review of the subject following the presenta-tions of Kuchaˇr [49] and Teitelboim [50].For simplicity, we consider a generic functional T [ X ( x )] that assignsa real number to every surface defined by the four embedding variables X µ ( x ). The variation in the functional δT resulting from an arbitrarydeformation δξ Ax has the form δT = Z dx δξ µx δTδξ µx = Z dx (cid:16) δξ ⊥ x G ⊥ x + δξ ix G ix (cid:17) T , (16)where G ⊥ x = δδξ ⊥ x = n µx δδX µx and G ix = δδξ ix = X µix δδX µx (17)are the generators of normal and tangential deformations respectively.The generators of deformations δ/δξ Ax form a non-holonomic basis. Thus,unlike the vectors δ/δX µx , which form a coordinate basis and thereforecommute, the generators of deformations have a non-vanishing commuta-tor “algebra” given by δδξ Ax δδξ Bx ′ − δδξ Bx ′ δδξ Ax = Z dx ′′ κ CBA ( x ′′ ; x ′ , x ) δδξ Cx ′′ (18)where κ CBA are the “structure constants” of the “group” of deformations.The previous quotes in “group” and “algebra” are a reminder thatstrictly, the set of deformations do not form a group. The compositionof two successive deformations is itself a deformation, of course, but italso depends on the surface to which the first deformation is applied. Aswe will see below, the “structure constants” κ CBA are not constant, theydepend on the metric g ij of the initial surface. The calculation of κ CBA is given in [49][50]. The key idea is thatof embeddability , which proceeds as follows. Consider performing twosuccessive infinitesimal deformations δξ A followed by δη A on an initialsurface σ : σ δξ → σ δη → σ ′ . Performing now the very same deformations inthe opposite order σ δη → σ δξ → σ ′′ yields a final surface σ ′′ that, in general,differs from σ ′ . The key point is that since both σ ′ and σ ′′ are embeddedin the very same space-time, then there exist be a third deformation δζ A that relates the two: σ ′ δζ → σ ′′ .As shown by Teitelboim [50], however, the compensating deformationis not at all arbitrary, but can be determined entirely by geometricalarguments: δζ Cx ′′ = Z dx Z dx ′ κ CAB ( x ′′ ; x, x ′ ) δξ Ax δη Bx ′ , (19) In a dynamical approach to gravity the metric is itself a functional of the canonicalvariables. Thus its appearance in the “algebra” of deformations is in part responsible for therich structure of geometrodynamics. here the only non-vanishing κ ’s have the form κ ⊥ i ⊥ ( x ′′ ; x, x ′ ) = − κ ⊥⊥ i ( x ′′ ; x ′ , x ) = − δ ( x ′′ , x ) ∂ ix ′′ δ ( x ′′ , x ′ ) (20a) κ kij ( x ′′ ; x, x ′ ) = − κ kji ( x ′′ ; x ′ , x )= δ ( x ′′ , x ) ∂ ix ′′ δ ( x ′′ , x ′ ) δ kj − δ ( x ′′ , x ) ∂ jx ′′ δ ( x ′′ , x ′ ) δ ki (20b) κ i ⊥⊥ ( x ′′ ; x, x ′ ) = − κ i ⊥⊥ ( x ′′ ; x ′ , x )= − g ij ( x ′′ ) δ ( x ′′ , x ′ ) ∂ jx ′′ δ ( x ′′ , x )+ g ij ( x ′′ ) δ ( x ′′ , x ) ∂ jx ′′ δ ( x ′′ , x ′ ) (20c)Identification of the κ ’s implies that the commutator in eq.(18) satisfiesthe “algebra”[49][50],[ G Ax , G Bx ′ ] = Z dx ′′ κ CAB ( x ′′ ; x, x ′ ) G Cx ′′ , (21)which can now be written more explicitly as[ G ⊥ x , G ⊥ x ′ ] = − ( g ijx G jx + g ijx ′ G jx ′ ) ∂ ix δ ( x, x ′ ) , (22a)[ G ix , G ⊥ x ′ ] = − G ⊥ x ∂ ix δ ( x, x ′ ) , (22b)[ G ix , G jx ′ ] = − G ix ′ ∂ jx δ ( x, x ′ ) − G jx ∂ ix δ ( x, x ′ ) , (22c)with all other brackets vanishing. In an entropic dynamics, evolution is driven by information codified intoconstraints. An entropic geometrodynamics, it follows, consists of dy-namics driven by a specific choice of constraints, which we discuss here.In [30][31], quantum field theory in a curved space-time (QFTCS) wasderived under the assumption that the geometry remains fixed. But suchassumptions, we know, should break down when one considers states de-scribing a non-negligible concentration of energy and momentum. Thuswe must revise our constraints appropriately. A natural way to proceedis thus to allow the geometry itself to take part in the dynamical process:the geometry affects ρ σ [ χ ] and φ σ [ χ ], they then act back on the geometry,and so forth. Our goal here is to make this interplay concrete. The canonical updating scheme—
A natural question that arisesfrom the above discussion is how to implement the update of the geom-etry and drift potential φ σ [ χ ]. Such a task involves two steps. The firstis the proper identification of variables for describing the evolving geom-etry, while the other is the specific manner in which this joint system ofvariables, including the drift potential φ σ [ χ ], is updated. Fortunately, thetwo challenges can be dealt with quite independently of each other.In devising a covariant scheme for updating we draw primarily fromthe work of IAC [30][31]. To review briefly, a primary assumption in theIAC approach was the adoption of a canonical framework for governing he coupled dynamics of ρ σ and φ σ , expressed more conveniently throughthe transformed variable Φ σ . Although this is certainly a strong assump-tion, it is one that has some justification. On one front, it can be arguedthat canonical structures seem to have a rather natural place in ED; aris-ing from conservation laws in [24][29] and alternatively from symmetryconsiderations [25]. However, from another perspective entirely, the useof a canonical formalism in ED can also be traced to more pragmaticconcerns, as it allows one to borrow from a roster of covariant canonicaltechniques designed in the context of classical physics (see e.g., [47]), butdeployed for the purposes of ED. These together suggest that we view thecanonical setting, and the symplectic symmetries that undergird it, as acentral criterion for updating in ED. The canonical variables—
Crucial to our updating scheme is anappropriate choice of variables. We pursue a conservative approach inwhich ρ σ and Φ σ are packaged together as canonically conjugate variablesfollowing the prescription detailed in IAC [31]. The nontrivial task ofchoosing the geometric variables has long been the subject of a livelydebate. Among these various approaches, however, the pioneering effortsof Hojman, Kuchaˇr, and Teitelboim (HKT) [48] prove to be of specialinterest, as their work centers a purely canonical approach to the dynamicsof geometry. Following HKT, it is possible to take the six components ofthe metric g ij to be the starting point for a geometrodynamics. Theargument for this is grounded in simplicity. If one assumes, as they do,that evolution in local time should mirror the structure of space-timedeformations, then the metric appearing on the right hand side of (20c)for the “structure constant” κ i ⊥⊥ must necessarily be a functional of thecanonical variables. Clearly this is most easily satisfied if the metric isitself a canonical variable.To complete the canonical framework, however, we must also introducesix variables π ij that are canonically conjugate to the g ij . These variables,the conjugate momenta, are themselves defined through the canonicalPoisson bracket relations, (cid:8) g ijx , g klx ′ (cid:9) = n π ijx , π klx ′ o = 0 (23a) n g ijx , π klx ′ o = δ klij δ ( x, x ′ ) = 12 (cid:16) δ ki δ lj + δ kj δ li (cid:17) δ ( x, x ′ ) . (23b)That such relations are satisfied is made manifest by writing the Poisson Just to name a few approaches, there are, of course, the original attempts at geometro-dynamics from Dirac [61] as well as Arnowitt, Deser, and Misner (ADM) [62] that start fromthe Einstein-Hilbert action and take the metric g ij as the fundamental building block. Some-what more recently, due in part to the modern success of gauge theories, there has been someinterest in taking, not the metric g ij , but the Levi-Civita connection Γ ijk , as the fundamentalgravitational object (see e.g., [66]). In a similar spirit is the well-known discovery of Ashtekar[67], which uses triads and spin connections to rewrite GR. In particular, we do not assume the existence of a Legendre transformation from π ij tothe “velocity” of the metric g ij . rackets in local coordinates { F, G } = Z dx (cid:18) δFδg ijx δGδπ ijx − δGδg ijx δFδπ ijx (cid:19) + Z Dχ (cid:18) ˜ δF ˜ δρ [ χ ] ˜ δG ˜ δ Φ[ χ ] − ˜ δG ˜ δρ [ χ ] ˜ δF ˜ δ Φ[ χ ] (cid:19) , (24)where F and G are some arbitrary functionals of the phase space vari-ables. (Note that π ijx must be a tensor density for the Poisson bracket totransform appropriately under a change of variables of the surface.) A fully covariant dynamics requires that the updating in local time ofall dynamical variables be consistent with the kinematics of surface de-formations. Thus, the requirement that the deformed surfaces remainembedded in space-time, which amounts to imposing foliation invariance,translates into a consistency requirement of path independence: if theevolution from an initial instant into a final instant can occur along dif-ferent paths, then all these paths must lead to the same final values for alldynamical quantities. The approach we adopt for quantum fields coupledto dynamical classical gravity builds on previous work by HKT [48] forclassical geometrodynamics, and by IAC [30][31] for quantum field theoryin a non-dynamical space-time.Within this scheme the evolution of an arbitrary functional F of thecanonical variables is generated by application of a set of local Hamilto-nians according to δF = Z dx { F, H Ax } δξ Ax = Z dx (cid:16) { F, H ⊥ x } δξ ⊥ x + { F, H ix } δξ ix (cid:17) , (25)where parameters δξ Ax with A = ( ⊥ , i = 1 , ,
3) describe an infinitesimaldeformation, as per eqns.(9) and (10), and H Ax are the correspondinggenerators. (Defined in this way, the H Ax turn out to be tensor densities.) Path independence—
The implementation of path independence[49][50] then rests on the idea that the Poisson brackets of the generators H Ax form an “algebra” that closes in the same way, that is, with the same“structure constants”, as the “algebra” of deformations in eqns.(22a)-(22c), { H ⊥ x , H ⊥ x ′ } = ( g ijx H jx + g ijx ′ H jx ′ ) ∂ ix δ ( x, x ′ ) , (26a) { H ix , H ⊥ x ′ } = H ⊥ x ∂ ix δ ( x, x ′ ) , (26b) { H ix , H jx ′ } = H ix ′ ∂ jx δ ( x, x ′ ) + H jx ∂ ix δ ( x, x ′ ) . (26c)We conclude this section with two remarks. First, we note that theseequations have not been derived. Indeed, imposing (22a)-(22c) as strongconstraints constitutes the definition of what we mean by imposing con-sistency between the updating of dynamical variables and the kinematicsof surface deformations. But this is not enough. As discussed in detail y Teitelboim [50] and HKT [48], one achieves path independence by re-quiring that the initial values of the canonical variables be restricted tosatisfy the weak constraints H ⊥ x ≈ H ix ≈ . (27)While it is beyond the scope of this paper to explore the full consequencesof these weak constraints we merely point out that formally their origin istraced to the fact that κ i ⊥⊥ in eq.(20c) depends on the metric and there-fore it is not a true structure “constant”. More physically the constraintsrepresent the fact that the canonical variables g ij and π ij are redundantbecause they represent the true dynamical degrees of freedom plus addi-tional kinematical variables that allow us the freedom to choose space-timecoordinates; and, as discussed by Kuchaˇr [49], separating these dynamicaland kinematical variables is not an easy task. Furthermore, once satis-fied on an initial surface σ the dynamics will be such as to preserve theseconstraints for all subsequent surfaces of the foliation.We also note that it is only by virtue of relating the conditions of “in-tegrability” to those of “embeddability” that we can interpret the role ofthe local Hamiltonians H ⊥ x and H ix in relation to space and time. This isa crucial and highly non-trivial step. It is only once this is established thatwe can interpret H ⊥ x as a scalar density that is responsible for genuinedynamical evolution and H ix as a vector density that generates spatialdiffeomorphisms; the former is called the super-Hamiltonian, while thelatter is the so-called super-momentum. We now turn our attention to the local Hamiltonian generators H Ax , andmore specifically, we look to provide explicit expressions for these gener-ators in terms of the canonical variables. This problem was solved in thecontext of a purely classical geometrodynamics (with or without sources)by HKT in [48]. Here we aim to apply their techniques and methodologyto a different problem: a geometrodynamics driven by “quantum” sources.Fortunately, a considerable portion of the HKT formalism can be directlyadopted for our purposes. The super-momentum
To determine the generators H Ax [ ρ, Φ; g ij , π ij ] it is easiest to begin withthe so-called super-momentum H ix [ ρ, Φ; g ij , π ij ]. This is largely becausethe function of this generator is well understood: it pushes the canoni-cal variables along the surface they reside on. Since there is no motion“normal” to the surface, the action of this generator is purely kinematical.Following HKT [48], consider an infinitesimal tangential deformationsuch that a point originally labeled by x i is carried to the point previ-ously labeled by x i + δξ i . This will induce a corresponding change in anydynamical variables F defined on that surface, F → F + δF . This change δF can then be computed in two distinct ways, which, of course, must gree. One is by calculating the Lie derivative along δξδF = £ δξ F and the other is using the super-momentum H ix , so that δF = Z dx { F, H ix } δξ ix . Gravitational super-momentum
A straightforward example ofthis is shown for the metric g ij ( x ) = g ijx , which is a rank (0 ,
2) tensor.Its Lie derivative is given by [69] £ δξ g ij = ∂ k g ij δξ kx + g ik ∂ j δξ kx + g kj ∂ i δξ kx . (28)Alternatively, by using the Poisson brackets we obtain δg ijx = Z dx ′ { g ijx , H kx ′ } δξ kx ′ = Z dx ′ δH kx ′ δπ ijx δξ kx ′ . Comparing the two and using the fact that δξ ix is arbitrary yields δH kx ′ δπ ijx = ∂ kx g ijx δ ( x, x ′ ) + g ikx ∂ jx δ ( x, x ′ ) + g kjx ∂ ix δ ( x, x ′ ) . (29)The delta functions imply that H ix is local in the momentum π ij .To fix the dependence on π ij , in fact, we can use a similar argumentas above. Recalling that π ij ( x ) = π ijx is a rank (2 ,
0) tensor density ofweight one, we find that £ δξ π ij = ∂ kx (cid:16) π ij δξ kx (cid:17) − π ik ∂ kx δξ jx − π kj ∂ kx δξ ix . (30)The same equation can be obtained through the use of a Hamiltonian, δπ ijx = Z dx ′ n π ijx , H kx ′ o δξ kx ′ = − Z dx ′ δH kx ′ δg ijx δξ kx ′ , so long as H ix satisfies − δH kx ′ δg ijx = ∂ kx (cid:16) π ijx δ ( x, x ′ ) (cid:17) − π ilx ∂ lx δ ( x, x ′ ) δ jk − π ljx ∂ lx δ ( x, x ′ ) δ ik . (31)Integrating these equations for H ix yields H ix = H Gix + ˜ H ix , (32)where H Gix = − ∂ jx (cid:16) π jk g ik (cid:17) + π jk ∂ ix g jk , (33)is called the gravitational super-momentum, and the functional ˜ H ix =˜ H ix [ ρ, Φ] is, at the moment, just an integration “constant”.Some of these expressions can be simplified by introducing the covari-ant derivative ∇ i . For example, using ∇ k g ij = 0 we have £ δξ g ijx = g jk ∇ i δξ kx + g ik ∇ j δξ kx (34) nd £ δξ π ijx = ∇ k (cid:16) π ij δξ kx (cid:17) − π ik ∇ k δξ jx − π kj ∇ k δξ ix . (35)The gravitational super-momentum H Gix also takes the particularly simpleform H Gix = − g ik ∇ j π jk = − ∇ j (cid:16) π jk g ik (cid:17) = − ∇ j π ji . (36) The “matter” super-momentum—
Next, we turn to the responseof the variables ρ σ [ χ ] and Φ σ [ χ ] under a relabeling of the surface coordi-nates x i → x i + δξ ix . As the coordinates are shifted, so too are the fieldsdefined upon that surface so that χ x → χ x + £ δξ χ x where £ δξ χ x = ∂ ix χ x δξ ix is the Lie derivative for a scalar χ x . This induces a change in the proba-bility, given by δρ σ [ χ ] ≡ ρ σ [ χ + £ δξ χ x ] − ρ σ [ χ ] = Z dx δρ σ [ χ ] δχ x ∂ ix χ x δξ ix . (37)Alternatively, this same variation can be computed using the canonicalframework δρ σ [ χ ] = Z dx { ρ σ [ χ ] , H ix } δξ ix . (38)If we insert the super-momentum H ix given in eq.(32), we notice that onlythe ˜ H ix piece will contribute. And so we have δρ σ [ χ ] = Z dx n ρ σ [ χ ] , ˜ H ix o δξ ix = Z dx ˜ δ ˜ H ix ˜ δ Φ σ [ χ ] δξ ix , (39)which for arbitrary δξ ix requires that˜ δ ˜ H ix ˜ δ Φ σ [ χ ] = δρ σ [ χ ] δχ x ∂ ix χ x . (40)A similar argument for Φ σ [ χ ] shows that we must also have − ˜ δ ˜ H ix ˜ δρ σ [ χ ] = δ Φ σ [ χ ] δχ x ∂ ix χ x , (41)so that ˜ H ix = − Z Dχρ σ δ Φ σ δχ x ∂ ix χ x . (42)Thus the total super-momentum, eq.(32), H ix = − ∇ j π ji − Z Dχρ σ δ Φ σ δχ x ∂ ix χ x (43) ontains two pieces, which we refer to as the gravitational and “matter”contributions, respectively. Note that there is no gravitational depen-dence in the “matter” side, or “matter” dependence on the gravitationalside, that is, H ix = H Gix [ g ij , π ij ] + ˜ H ix [ ρ, Φ] . (44)By equation (27) it is, of course, also understood that H ix is subjectto the constraint H ix ≈ . (45)Finally, although H ix was obtained, in essence, independently of the Pois-son bracket (26c), relating two tangential deformations, it nonetheless sat-isfies it automatically. It is therefore appropriate to view H ix as beingcompletely determined; this is important as it allows us to treat equation(26a) as a set of equations for H ⊥ x in terms of the known H ix . The super-Hamiltonian
We now turn our attention to the generator H ⊥ x of local time evolution.Following Teitelboim [51], it is useful to decompose H ⊥ x into two distinctpieces H ⊥ x = H G ⊥ x [ g ij , π ij ] + ˜ H ⊥ x [ ρ, Φ; g ij , π ij ] , (46)consisting of a gravitational piece H G ⊥ x depending only on the gravita-tional variables, and a “matter” contribution that we suggestively denoteby ˜ H ⊥ x , called the gravitational and “matter” super-Hamiltonians, re-spectively.As noted by Teitelboim, we make no assumptions in writing H ⊥ x inthis way. Given this splitting, however, we do make the following sim-plifying assumption: we require the “matter” super-Hamiltonian ˜ H ⊥ x tobe independent of the gravitational momentum π ij , i.e.,˜ H ⊥ x = ˜ H ⊥ x [ ρ, Φ; g ij ]so that the super-Hamiltonian takes the form H ⊥ x = H G ⊥ x [ g ij , π ij ] + ˜ H ⊥ x [ ρ, Φ; g ij ] . (47)With this assumption in hand, it is possible to prove [51] that the met-ric appears in ˜ H ⊥ x only as a local function of g ij . That is, no derivativesof the metric are allowed, nor any other complicated functional dependen-cies; due to this fact, this was referred to as the non-derivative couplingassumption by Teitelboim. The division of generators into gravitational and “matter” pieces established by DHKTis, strictly speaking, an abuse of language. The variables ρ σ and Φ σ that constitute “matter”are more properly understood as describing the statistical state of the material field χ x .Nonetheless, we stick with the convention as a useful shorthand. More technically, H ix is defined by our procedure up to an overall “constant” vectordensity f i ( x ), which is independent of any canonical variables. This, it turns out, is requiredto vanish by eq.(26c). See e.g., [48]. By construction, we can always identify a piece of the super-Hamiltonian that updatesthe geometry alone. The “matter” super-Hamiltonian is then just defined as the differencebetween the total and gravitational pieces. odified Poisson brackets— A particularly appealing aspect ofthis separation into gravitational and “matter” pieces is that the Poissonbracket relations (26a)-(26c) also split along similar lines. In fact, insertthe decomposed generators H Ax from eqns.(44) and (47) into the Poissonbracket relations (26a)-(26c). We find that the gravitational generators H GAx = ( H G ⊥ x , H Gix ) must satisfy a set of Poisson brackets n H G ⊥ x , H G ⊥ x ′ o = ( g ijx H Gjx + g ijx ′ H Gjx ′ ) ∂ ix δ ( x, x ′ ) , (48a) n H Gix , H G ⊥ x ′ o = H G ⊥ x ∂ ix δ ( x, x ′ ) , (48b) n H Gix , H
Gjx ′ o = H Gix ′ ∂ jx δ ( x, x ′ ) + H Gjx ∂ ix δ ( x, x ′ ) , (48c)which have closing relations identical to those of (26a)-(26c).The “matter” generators, which we collectively denote by ˜ H Ax , satisfya somewhat modified set of brackets n ˜ H ⊥ x , ˜ H ⊥ x ′ o = ( g ijx ˜ H jx + g ijx ′ ˜ H jx ′ ) ∂ ix δ ( x, x ′ ) , (49a) n H ix , ˜ H ⊥ x ′ o = ˜ H ⊥ x ∂ ix δ ( x, x ′ ) , (49b) n ˜ H ix , ˜ H jx ′ o = ˜ H ix ′ ∂ jx δ ( x, x ′ ) + ˜ H jx ∂ ix δ ( x, x ′ ) . (49c)Here we call attention to the fact that eq.(49b) contains the total tan-gential generator H ix , not just ˜ H ix . This alteration occurs because the“matter” super-Hamiltonian depends on the metric. That is, if we wantto shift ˜ H ⊥ x along a given surface, we must shift the variables ( ρ, Φ), aswell as the metric g ij ; hence H ix , not just ˜ H ix , must appear.Modulo the small distinction arising in eq.(49b), we see that the grav-itational and “matter” sectors decouple such that each piece forms an in-dependent representation of the “algebra” of surface deformations. Froma strategic point of view, the separation means that we can solve thePoisson bracket relations for geometry and “matter” independently of oneanother. The “matter” super-Hamiltonian—
Our goal is to identify afamily of ensemble super-Hamiltonians ˜ H ⊥ x [ ρ, Φ; g ij ] that are consistentwith the Poisson brackets (49a)-(49c). Let us briefly outline our approach.Thus far, we have completely determined the correct form of the “matter”super-momentum ˜ H ix which is consistent with (49c). Moreover, the rela-tion (49b) merely implies that ˜ H ⊥ x transforms as a scalar density under aspatial diffeomorphism. Thus we are left only to satisfy the first Poissonbracket (49a) for the unknown ˜ H ⊥ x .In addition to these considerations, however, the ED approach itselfimposes additional constraints of a fundamental nature on the allowed H ⊥ x , or more specifically ˜ H ⊥ x . This is because, in ED, the introductionof a symplectic structure and its corresponding Hamiltonian formalism isnot meant to replace the entropic updating methods that yield the LTFPequations, but to augment them, appropriately. As a consequence, wedemand, as a matter of principle, that the “matter” super-Hamiltonian˜ H ⊥ x be defined so as to reproduce the LTFP equations of (14). ore explicitly, we require ˜ H ⊥ x to be such that its action on ρ σ gen-erates the LTFP equations (cid:8) ρ σ [ χ ] , H ⊥ x (cid:9) = δρ σ [ χ ] δξ ⊥ x , (50)which translates to˜ δ ˜ H ⊥ x ˜ δ Φ σ [ χ ] = − g / x δδχ x (cid:18) ρ σ [ χ ] δ Φ σ [ χ ] δχ x (cid:19) . (51)It is simple to check that the ˜ H ⊥ x that satisfies this condition is given by˜ H ⊥ = Z Dχρ σ g / x (cid:18) δ Φ σ δχ x (cid:19) + F x [ ρ ; g ij ] . (52)The first term in (52) is fixed by virtue of consistency with the LTFPequations, whereas F x [ ρ ; g ij ] is a yet undetermined “constant” of integra-tion, which may depend on ρ as well as the metric. However, F x is notentirely arbitrary, its functional form is restricted by the Poisson bracket,eq.(49a).Before proceeding, note that, up until this point, our discussion of pathindependence has been developed on a formal level, largely independentof ED itself. Such a formalism on its own, however, is necessarily devoidof many crucial physical ingredients. For instance, in ED we define localtime as a measure of the field fluctuations through (8), which leads tothe LTFP equations. In principle, this has nothing to do with abstractparameters δξ Ax introduced as part of local updating. It is only once we saythat the entropic updating of ED must agree with the local time evolutiongenerated by ˜ H ⊥ x , made explicit in (13), that the two notions coincide.In ED, the clock that measures the local proper time δξ ⊥ x is nothing butthe field fluctuations themselves.Continuing with our task at hand, we seek a family of models that areconsistent with eq.(49a). This is accomplished for suitable choices of F x .We pursue this in manner similar to [31]. Begin by rewriting ˜ H ⊥ x as˜ H ⊥ x = ˜ H ⊥ x + F x , (53)where we have introduced˜ H ⊥ x = Z Dχ ρ σ g / x (cid:18) δ Φ σ δχ x (cid:19) + g / x g ij ∂ ix χ x ∂ jx χ x ! . (54)This amounts simply to a redefinition of the arbitrary F x in (52). Anadvantage of this definition, however, is that the newly defined ˜ H ⊥ x au-tomatically satisfies n ˜ H ⊥ x , ˜ H ⊥ x ′ o = ( g ijx ˜ H jx + g ijx ′ ˜ H jx ′ ) ∂ ix δ ( x, x ′ ) . (55)Finding a suitable F x is therefore accomplished by satisfying n ˜ H ⊥ x , F x ′ o = n ˜ H ⊥ x ′ , F x o . (56) Since F x [ ρ ; g ij ] is independent of Φ, we have that { F x , F x ′ } = 0, identically, from whicheq.(56) follows. learly a necessary condition for an acceptable F x is that the Poissonbracket n ˜ H ⊥ x , F x ′ o must be symmetric upon exchange of x and x ′ . Since˜ H ⊥ x must itself reproduce the LTFP equations, the condition (56) trans-lates to [30][31]1 g / x δδχ x (cid:18) ρ σ δδχ x ˜ δF x ′ ˜ δρ σ (cid:19) = 1 g / x ′ δδχ x ′ (cid:18) ρ σ δδχ x ′ ˜ δF x ˜ δρ σ (cid:19) , (57)which is an equation linear in F x .A complete description of solutions to eq.(57) lies outside the scopeof the current work. However, a restricted family of solutions, which arenonetheless of physical interest, can be found for F x ’s of the form F x [ ρ ] = Z Dχf x (cid:18) ρ, δρδχ x ; g ijx (cid:19) , (58)where f x is a function , not functional, of its arguments. For such a specialtype of F x one can check by substitution into (57) that f x ∼ g / x ρχ nx (integer n) and f x ∼ ρg / x (cid:18) δ log ρδχ x (cid:19) are acceptable solutions. Since eq.(57) is linear in F x , solutions can besuperposed so that a suitable family of ˜ H ⊥ x ’s is given by˜ H ⊥ x = Z Dχ ρ σ g / x (cid:18) δ Φ σ δχ x (cid:19) + g / x g ij ∂ ix χ x ∂ jx χ x + g / x V x ( χ x ) + λg / x (cid:18) δ log ρ σ δχ x (cid:19) ! , (59)where V x = P n λ n χ nx is a function that is polynomial in χ x .As discussed in [31], the last term can be interpreted as the “localquantum potential”. To see this we recall that in flat space-time thequantum potential is given by [29] Q = Z d x Z Dχρ λ (cid:18) δ log ρδχ x (cid:19) . (60)The transition to a curved space-time is made by making the substitutions d x → g / x d x and δδχ x → g / x δδχ x , yielding the result Q σ = Z d x Z Dχρ λg / x (cid:18) δ log ρδχ x (cid:19) . The term inside the spatial integral is exactly what appears in (59), whichjustifies the name. The contribution of the local quantum potential tothe energy is such that those states that are more smoothly spread out inconfiguration space tend to have lower energy. The corresponding couplingconstant λ > λ < he gravitational super-Hamiltonian— We now proceed to de-termining the last remaining element of our scheme, the gravitationalsuper-Hamiltonian H G ⊥ x . Fortunately, under the assumption of non-derivativecoupling mentioned above, it is possible to completely separate the Pois-son bracket relations (26a)-(26c) into pieces that are purely gravitationaland those that are pure “matter”. Consequently, to determine H G ⊥ x itsuffices merely to solve the Poisson bracket relations (48a)-(48c), whichinvolve only the gravitational variables g ij and π ij .Such a task, however, is mathematically equivalent to determining thegenerators of pure geometrodynamics, in which the coupling to “matter”is absent. But it is precisely this latter challenge which was, in fact,addressed by the efforts of HKT in [48] — they proposed a solution toexactly those brackets that appear in (48a)-(48c). Their main result wasan important one: the only time-reversible solution to equations (48a)-(48c) is nothing less than Einstein’s GR in vacuum. We briefly reviewtheir argument here, and adapt their solution to our current work in ED.To begin, since g ij is the intrinsic metric of the surface, its behaviorunder a purely normal deformation is known [64] to be δg ij ( x ) = £ δξ ⊥ g ijx = − K ijx , (61)where K ij ( x ) = K ijx is a symmetric (0 ,
2) tensor that is called the extrin-sic curvature. Note that identifying K ijx with the response of g ijx undera normal deformation is a geometric requirement, not a dynamical one;without this, we could not interpret g ij as residing on a space-like cut ofspace-time. That being said, K ijx is at this juncture an undeterminedfunctional of the canonical variables, and more to the point, we do notassume at the moment any simple relationship with the momenta π ijx —this must be derived.Alternatively, the deformation in (61) must also be attainable usingthe normal generator H G ⊥ x , δg ijx = Z dx ′ n g ijx , H G ⊥ x ′ o δξ ⊥ x ′ . More explicitly H G ⊥ x should satisfy δH G ⊥ x ′ δπ ijx = − K ijx δ ( x, x ′ ) . (62)The appearance of the Dirac delta in eq.(62) implies that H G ⊥ x is local in π ijx . Therefore H G ⊥ x is a function , not a functional, of π ijx .Such a simplification turns out to be important: once H G ⊥ x is a functionof π ij , it is possible to produce an ansatz for H G ⊥ x in terms of powers of π ij .HKT then supplement this with an additional simplifying assumption:that geometrodynamics be time-reversible . This has the advantage ofremoving all terms in H G ⊥ x that have odd powers of π ijx .Incorporating these ingredients, it is then possible to consider H G ⊥ x ’sof the form H G ⊥ x = ∞ X n =0 G (2 n ) i j i j ··· i n j n x π i j x π i j x · · · π i n j n x , (63) here the coefficients G (2 n ) ij ··· are functionals of the metric g ij , but dependon the point x . Furthermore, since H G ⊥ x is scalar density and π ij a tensordensity, G (2 n ) ij ··· must transform as a tensor density of weight 1 − n . Since π ij is a symmetric tensor we expect G (2 n ) ij ··· to be symmetric under exchangeof indices i a j a ↔ j a i a for any pair a ; also, G (2 n ) ij ··· should be symmetric oninterchange of any pair i a j a ↔ i b j b as this just corresponds to exchangingthe π ’s.Naturally, one narrows the allowable H G ⊥ x by inserting the ansatz (63)into the Poisson bracket (48a). Without delving too far into the details(which can be found in [48]) we quote their solution. Only the n = 0 , H G ⊥ x = κ G ijkl π ij π kl − g / κ ( R − Λ) , (64)where we have introduce the super metric G ijkl = 1 g / x ( g ik g jl + g il g jk − g ij g kl ) , (65)and where R is the Ricci scalar for the metric g ij .The constant κ is a coefficient that, eventually, determines the couplingto “matter”. We follow standard convention in identifying it as κ = 8 πG ,where G is Newton’s constant. The other parameter Λ, of course, is thecosmological constant. For simplicity, going forward we set Λ = 0.Putting everything together, we have that H G ⊥ x takes the form H G ⊥ x = κg / x (cid:16) π ij π ij − π (cid:17) − g / κ R , (66)where π = π ij g ij = Tr( π ij ). This is exactly the standard gravitationalsuper-Hamiltonian obtained by Dirac [61] and ADM [62] by starting fromthe Einstein-Hilbert Lagrangian. Total super-Hamiltonian—
Putting together the ingredients ofthis section, the total super-Hamiltonian is thus H ⊥ x = H G ⊥ x + ˜ H ⊥ x , (67)where H G ⊥ x is given above by eq.(66) and where a suitable family of ˜ H ⊥ x ’shave been identified in eq.(53). With this, the super-Hamiltonian con-straint is then just H ⊥ x = H G ⊥ x + ˜ H ⊥ x ≈ . (68) As argued by HKT, although the role of space-time was crucial to thedeveloping the equations (26a)-(27), one notices that all signs of the en-veloping space-time have dropped out in the closing relations (26a)-(26c). For instance, they depend on the surface metric g ij , but not on its extrinsic curvature K ij . hus in geometrodynamics we can dispense with the notion of an a priori given space-time and instead consider the three-dimensional Riemannianmanifold σ t as primary.The idea is a simple one. The canonical variables are evolved with thegenerators H Ax , satisfying eqns.(26a)-(27). Such an evolution will causeboth the “matter” and geometry, to change. We might then give thismanifold with updated intrinsic geometry a new name, σ t ′ . Repeatingthis procedure results in an evolution of the dynamical variables, andconsequently, what one might view as a succession of manifolds { σ t } ,parameterized by the label t . Thus this iterative process constructs aspace-time, step by step. We now investigate the dynamical equationsthat result from this procedure. Some formalism
Consider the evolution of an arbitrary functional T t of the dynamicalvariables defined on σ t , δT t = T t + dt − T t = Z dx { T t , H Ax } δξ Ax = Z dx N Axt { T t , H Ax } dt , (69)where we have introduced four arbitrary functions N xt = δξ ⊥ x dt and N ixt = δξ ix dt , (70)which are the lapse and vector shift , respectively. Just as the evolutionparameters δξ Ax were completely arbitrary, these functions can be freelyspecified, and amount eventually to picking a particular foliation of space-time. As in the Dirac approach to geometrodynamics, the N Axt = ( N xt , N ixt )are not functionals of the canonical variables, therefore we can rewriteeq.(69) as δT t = n T t , H [ N, N i ] o dt , (71)where we have introduced the notion of an “integrated”, or “smeared”Hamiltonian H [ N, N i ] = Z dx (cid:16) N xt H ⊥ x + N ixt H ix (cid:17) (72)that, for given N Axt , generates an evolution parameterized by t . This global H [ N, N i ] (note the spatial integral) conforms more naturally to At the moment, the lapse and shift have no definite geometrical meaning. But as is wellknown, we can eventually identify the lapse N and shift N i as being related to components ofthe space-time metric g µν ≡ γ µν . (We changed the symbol temporarily to avoid confusion.)More specifically, we would have [48] N = (cid:0) − γ (cid:1) − / and N i = γ i . Had the N A been functionals of the canonical variables then the Poisson bracket in eq.(71)would have generated extra terms from their action on N A and thus eq.(71) would not havebeen equivalent to eq.(69). ur typical notions of a Hamiltonian. Thus, the derivative with respectto the parameter t is ∂ t T t ≡ δT t dt = n T t , H [ N, N i ] o , (73)or more explicitly by ∂ t T t = Z dx (cid:16) N xt { T t , H ⊥ x } + N ixt { T t , H ix } (cid:17) , (74)and more succinctly ∂ t T t = £ m T t + £ N i T t , (75)where the vector m µx = N xt n µx is the so-called evolution vector. The evolution of the “matter” sector
The goal is to determine the evolution of the probability distribution ρ t [ χ ]and phase functional Φ t [ χ ], given an initial state ( ρ t , Φ t ; g ij t , π ijt ) thatsatisfies the initial value constraints (27). Dynamical equations for the probability and phase—
Giventhe basic dynamical law, eq.(73), the time evolution of the variables ρ t and Φ t are given by ∂ t ρ t = { ρ t , H } and ∂ t Φ t = { ρ t , H } , (76)where H is the smeared Hamiltonian given above. Thus, in accordancewith eq.(74), we have the result ∂ t ρ t = Z dx (cid:16) N xt { ρ t , H ⊥ x } + N ixt { ρ t [ χ ] , H ix } (cid:17) (77a) ∂ t Φ t = Z dx (cid:16) N xt { Φ t , H ⊥ x } + N ixt { Φ t , H ix } (cid:17) . (77b)Using the family of H ⊥ x ’s that we identified in eq.(67) and the super-momentum H ix in eq.(43) we can compute all of the necessary Poissonbrackets. From the super momentum in eq.(43) we can compute thetangential pieces { ρ t [ χ ] , H ix } = ˜ δ ˜ H ix ˜ δ ˜Φ t = δρ t δχ x ∂ ix χ x , (78a) { Φ t [ χ ] , H ix } = − ˜ δ ˜ H ix ˜ δ ˜ ρ t = δ Φ t δχ x ∂ ix χ x . (78b)Of course, the entire family of ˜ H ⊥ x ’s were designed to reproduce the LTFPequations, thus by construction we have { ρ t [ χ ] , H ⊥ x } = ˜ δ ˜ H ⊥ x ˜ δ ˜Φ t = − g / x δδχ x (cid:18) ρ t δ Φ t δχ x (cid:19) , (79) n agreement with eq.(14). The remaining Poisson bracket determines thelocal time evolution of the phase functional Φ t and is given by { Φ t [ χ ] , H ⊥ x } = − ˜ δ ˜ H ⊥ x ˜ δ ˜ ρ t = 12 g / x (cid:18) δ Φ t [ χ ] δχ x (cid:19) + g / x g ij ∂ i χ x ∂ j χ x + ˜ δF x [ ρ ]˜ δρ t [ χ ] , (80)where F x is of the form specified by eq.(53). The local time Hamilton-Jacobi equations—
To interpret thelocal equations (80), we write the full time evolution for the phase func-tional by inserting eqns.(80) and (78b) into eq.(77b), yielding − ∂ t Φ t = Z dx N xt g / x (cid:18) δ Φ t δχ x (cid:19) + g / x g ij ∂ i χ x ∂ j χ x + ˜ δF x ˜ δρ t ! + Z dx N ixt δ Φ t δχ x ∂ ix χ x . (81)To bring this equation into a more familiar form we consider the specialcase of flat space-time by setting the metric to be a Kroenecker delta g ij = δ ij , so that g / x = 1, and we let N = 1, N i = 0. Moreover, forsimplicity we also make the assignment F x = 0. This results in a timeevolution for Φ t that has the form − ∂ t Φ t = Z dx (cid:18) δ Φ t δχ x (cid:19) + 12 δ ij ∂ i χ x ∂ j χ x ! , which is exactly the classical Hamilton-Jacobi equation for a masslessKlein-Gordon field in flat space-time. Thus, in analogy with the LTFPequations, we refer to eq.(80) as the local time Hamilton-Jacobi (LTHJ)equations, as there is one equation of the Hamilton-Jacobi type for everyspatial point.The LTFP and LTHJ equations, with the tangential equations (78b),and the evolution equations (77b), give us the ability to evolve an appro-priately chosen initial state ( ρ t , Φ t ). In general, this is a coupled non-linearevolution driven by a dependence on the metric g ij . To a large extent,this completes our discussion of how the epistemic variables evolve. In asubsequent section, however, we discuss in some detail the dynamics of aspecific class of models, those that involve the local quantum potential. The evolution of the geometrical variables
We now review the content of the well-known Einstein’s equations writtenwithin the canonical language. (A good review of these equations is given,for example, by ADM [63].)
Evolution of metric—
The goal is to determine the evolution ofthe geometrical variables ( g ij , π ij ). Beginning with the metric g ij definedon some initial three-space σ t , we wish to determine how it evolves in esponse to the generators H Ax . Applying eq.(74) for the time derivative,we have that ∂ t g ijx = Z dx ′ { g ijx , H Ax ′ } N Ax ′ t = Z dx ′ δH Ax ′ δπ ijx N Ax ′ t . (82)To compute this, recall that the tangential piece is known from equations(29) and (34). This gives us £ N i g ijx = R dx ′ δH ix ′ δπ ijx N ix ′ t = ∇ i N jxt + ∇ j N ixt , (83)where £ N i g ijx is just the Lie derivative along the vector field defined bythe shift N i , and where N ixt = g ijx N jxt .To obtain the remaining piece, first differentiate the H ⊥ x given ineq.(66) δH ⊥ x ′ δπ ijx = 4 κg / x (cid:18) π ijx − π x g ijx (cid:19) δ ( x, x ′ ) , where π ij is the conjugate momentum with its indices lowered, and π = g ij π ij = Tr( π ij ) is the trace of the gravitational momentum. Puttingthese terms together, eq.(82) reads as ∂ t g ijx = £ m g ijx + £ N i g ijx (84a) £ m g ijx = 2 κg / x (2 π ijx − π x g ijx ) N xt (84b) £ N i g ijx = ∇ i N jxt + ∇ j N ixt . (84c)This gives us the evolution of the metric with foliation parameter t .We can also now identify the extrinsic curvature tensor K ij = κg / x ( π x g ijx − π ijx ) . (85)Inverting this relationship we get π ij in terms of K ij , which yields π ijx = g / x κ (cid:16) Kg ijx − K ijx (cid:17) . (86)where K = K ij g ij = Tr( K ij ) is the trace of K ij . This is of some interest ifone wishes to compare the canonical formulation to the standard so-called“Lagrangian” approach (see e.g., [70]). Evolution of conjugate momentum—
To this point, the dynam-ics of the geometry has not differed from a purely classical geometro-dynamics, such as that of HKT. This is because the “matter” super-Hamiltonian and super-momentum were, by definition, completely inde-pendent of π ij (due, of course, to the non-derivative coupling assumption)therefore the evolution of g ij did not receive contributions from the “mat-ter” sector. This changes when we consider the dynamics of the conjugatemomentum π ij . he evolution of π ij is determined via the equation ∂ t π ijx = Z dx ′ n π ijx , H Ax ′ o N Ax ′ t = − Z dx ′ δH Ax ′ δg ijx N Ax ′ t . (87)Recalling now that both the gravitational and “matter” super-Hamiltonians H ⊥ x = H G ⊥ x [ g ij , π ij ] + ˜ H ⊥ x [ ρ, Φ; g ij ] depend explicitly on the metric, butthat ˜ H ix does not, this expression slightly simplifies to ∂ t π ijx = − Z dx ′ (cid:18) N x ′ t δH G ⊥ x ′ δg ijx + N x ′ t δ ˜ H ⊥ x ′ δg ijx + N ix ′ t δH Gix ′ δg ijx (cid:19) , (88)where we have have separated the contributions from the gravitationaland “matter” sectors. In the notation of eq.(75) we write this as ∂ t π ijx = £ m π ijx + £ N i π ijx , (89)where £ m π ijx = £ Gm π ijx + £ Mm π ijx , (90)and £ Gm π ijx = − Z dx ′ (cid:18) N x ′ t δH G ⊥ x ′ δg ijx (cid:19) , (91a) £ Mm π ijx = − Z dx ′ (cid:18) N x ′ t δ ˜ H ⊥ x ′ δg ijx (cid:19) , (91b) £ N i π ijx = − Z dx ′ (cid:18) N ix ′ t δH Gix ′ δg ijx (cid:19) . (91c)From (35) the last term is fairly easy to determine, £ N i π ijx = ∇ kx (cid:16) π ijx N kx (cid:17) − π ikx ∇ kx N jx − π kjx ∇ kx N ix . (92)The calculation of the other two terms is much more involved. For-tunately the expression for £ Gm π ij is already well known [64][63] and wemerely quote the result, £ Gm π ijx = − g / κ (cid:18) R ijx − g ijx R x (cid:19) N xt + κg / x g ijx (cid:18) π klx π klx − π x (cid:19) N xt − κg / x (cid:18) π ikx π jkx − π x π ijx (cid:19) N xt + g / x κ (cid:16) ∇ ix ∇ jx N xt − g ijx ∇ kx ∇ kx N xt (cid:17) . (93)To calculate the remaining piece £ Mm π ijx we first recall that havingassumed a non-derivative coupling of gravity to matter, the metric g ij appears in ˜ H ⊥ x as an undifferentiated function (not a functional) andwithout any derivatives [51], which implies that δ ˜ H ⊥ x ′ δg ijx = ∂ ˜ H ⊥ x ∂g ijx δ ( x, x ′ ) , btaining £ Mm π ijx = − ∂ ˜ H ⊥ x ∂g ijx N xt . Using eq.(53), for an arbitrary choice of F x , but one that still satisfieseq.(49a), the “matter” source has the form £ Mm π ijx = N xt (cid:18) ∂ ˜ H ⊥ x ∂g ijx + ∂F x ∂g ijx (cid:19) , (94)with ∂ ˜ H ⊥ x ∂g ijx = − Z Dχρ g / x (cid:18) δ Φ δχ x (cid:19) − g / x g klx ∂ kx χ x ∂ lx χ x ! g ijx + 12 Z Dχρ g / x ∂ ix χ x ∂ jx χ x , (95)where ˜ H ⊥ x was given in eq.(54), and where ∂ ix χ x = g ijx ∂ jx χ x .In total, the equations of motion for the gravitational field followHamilton’s equations for the gravitational variables ( g ijx , π ijx ), given byeq.(84a) for the evolution of the metric, and for the conjugate momentum π ijx we have ∂ t π ijx − £ Gn π ijx − £ N i π ijx = − N xt ∂ ˜ H ⊥ x ∂g ijx . (96)This is an equation in which geometrical variables on the left-hand sideare sourced by the variables ( ρ, Φ), which contain all the informationavailable about the field χ x on the right-hand side. Below we will discussthis equation in the presence of “quantum matter.”
10 Quantum sources of gravitation
The transition to what may be termed a quantum form of dynamicsamounts to an appropriate choice of the functional F x [ ρ ; g ij ] (see e.g.,[29][30][31]). In particular, for F x [ ρ ; g ij ] we choose exactly the local quan-tum potential introduced as part of (59). A convenient choice for thecoupling constant is λ = 1 / (This choice of λ makes plain that wework with a system of units where ~ = c = 1.)The connection to conventional quantum theory is made explicit by achange of variables from the probability ρ and phase Φ to the complexvariables Ψ = ρ / e i Φ and Ψ ∗ = ρ / e − i Φ . (97) As argued, for instance, in [24], there is no loss of generality in making this choice. For thecase of nonrelativistic particles it can be proved [25] that an ED that preserves the appropriatesymplectic and metric structures implies the presence of a quantum potential with the correctcoefficient. uch a change of variables is, in fact, a canonical transformation, and so,the new variables form a canonical pair given by (Ψ , i Ψ ∗ ), which obey anatural generalization of the standard Poisson bracket relations (cid:8) Ψ[ χ ] , i Ψ ∗ [ χ ′ ] (cid:9) = δ [ χ − χ ′ ] , (98)where δ [ χ − χ ′ ] is a Dirac delta functional . Quantum operators and geometrodynamics revis-ited
Having chosen an F x [ ρ ; g ij ] of the type described above, the ensemblegenerators ˜ H Ax take a particularly special form˜ H Ax = Z Dχ Ψ ∗ ˆ H Ax Ψ = D ˆ H Ax E , (99)which can be viewed as both the expected value with respect to the prob-ability ρ of a Hamiltonian density, as in (59), as well as the expectationvalue of the local Hamiltonian operatorsˆ H ⊥ x = − g / δ δχ x + g / g ij ∂ i χ x ∂ j χ x + g / V x ( χ x ; g ij )(100a)ˆ H ix = i ∂ i χ x δδχ x , (100b)with respect to the quantum state Ψ. To obtain the “matter” contributionin eq.(96) for the conjugate momentum π ijx , note that the metric appearsin ˜ H ⊥ x through the density g / x and the inverse metric g ijx . Variations ofthese quantities with respect to the metric g ijx are given by [70] δg x = g g ijx δg ijx and δg ijx = g ik g jl δg kl . Then ∂ ˜ H ⊥ x ∂g ijx = Z Dχ Ψ ∗ ∂ ˆ H ⊥ x ∂g ijx Ψ , where we have defined the operator ∂ ˆ H ⊥ x ∂g ijx = ∂ ix χ x ∂ jx χ x + g ij g / x δ δχ x + g / x g kl ∂ k χ x ∂ l χ x + V x ( χ x ) ! . (101) Geometrodynamics with quantum sources—
Our goal here isto rewrite the main equations of geometrodynamics with sources givenby quantum matter. We begin first with the constraint equations. Withthe aid of the local operators introduced in (100a) and (100b), the totalHamiltonian generators take the explicit form H ⊥ x = κg / x (cid:16) π ijx π ijx − π x (cid:17) − g / x κ R x + Z Dχ Ψ ∗ ˆ H ⊥ x Ψ(102a) H ix = − ∇ kx (cid:16) π kjx g ijx (cid:17) + Z Dχ Ψ ∗ ˆ H ix Ψ , (102b) hich are, of course, subject to the constraints H ⊥ x ≈ H ix ≈ . (103)As relations (102a) and (102b) coupled together with (103) make abun-dantly clear, the quantum state and the geometrical variables can nolonger be treated as independent. This has important consequences forthe time evolution of Ψ.Moving on, note that the dynamical equation for the metric, givenin eq.(84a), does not depend directly on the choice of F x and thereforeremains unchanged in the quantum context. The dynamical equation forthe conjugate momentum π ijx , however, is modified. Using the operatorintroduced in (101), this becomes ∂ t π ijx − £ Gn π ijx − £ N i π ijx = − N xt Z Dχ Ψ ∗ ∂ ˆ H ⊥ x ∂g ijx Ψ , (104)where £ Gm π ijx = and £ N i π ijx are given in eqns.(91a) and (92), respectively.This is the crucial equation in which the dynamical geometry is itselfaffected by the epistemic state Ψ.Putting it all together, the eqns.(102a)-(104), and eq.(84a) for themetric, constitute a system of equations that are formally equivalent tothe SCEE put in the canonical form. However, that is where the sim-ilarities end. On several key issues of interpretation, in particular, theED approach is vastly different from the SCEE as they are normally un-derstood. Far from being trivial, these distinctions turn out to be quiteimportant since many objections (see e.g., [71][72]) to the usual SCEEare, in fact, based on such considerations. Quantum dynamics
One advantage of the complex variables (Ψ , Ψ ∗ ) is that the dynamics takesa familiar form. Indeed, since Ψ and Ψ ∗ are just functions of our canonicalvariables ( ρ, Φ) we can just use eq.(75) to determine their evolution alonga time parameter t , which gives ∂ t Ψ t [ χ ] = Z dx { Ψ t [ χ ] , H Ax } N Axt . (105)The tangential component is obtained in a straightforward fashion by { Ψ t [ χ ] , H ix } = ∂ i χ x δ Ψ t [ χ ] δχ x , (106)which is reasonable since this is just the Lie derivative of Ψ[ χ ] along thesurface. The local normal evolution of Ψ, on the other hand, is given by n Ψ t [ χ ] , ˜ H ⊥ x o = − i ˆ H ⊥ x Ψ t [ χ ] . (107)Inserting these results into eq.(105) for a general evolution of Ψ t [ χ ], wethen have i ∂ t Ψ t [ χ ] = Z dx (cid:16) N xt ˆ H ⊥ x + N ixt ˆ H ix (cid:17) Ψ t [ χ ] . (108) Compare, for example, to eqns.(40) and (41) for ρ [ χ ] and Φ[ χ ]. inally, substituting eqns.(100a) and (100b) in for ˆ H ⊥ x and ˆ H ix , respec-tively, yields the equation i ~ ∂ t Ψ t = Z dxN xt (cid:18) − ~ g / δ δχ x + c g / x g ij ∂ i χ x ∂ j χ x + g / V x (cid:19) − ~ c Z dxN ixt ∂ i χ x δδχ x Ψ t , (109)where we have reinstated the constants ~ and c , as appropriate. Hereequation (109) is ostensibly just a linear equation for the complex variableΨ t , which suggests calling it a Schr¨odinger functional equation. We discussbelow. But is it quantum?—
The coupling to classical gravity describedabove implies violations of the superposition principle. To see this, wenote that geometrodynamics is a constrained dynamical system, wherethe operative equations are given by the Hamiltonian constraints in (103).Solving these constraints often involves determining the components ofthe metric in terms of the quantum sources, which then gets fed backinto evolution equations for Ψ. This feedback leads to a non-linear timeevolution (see e.g., [71]) in which the Ψ itself appears as a potential ineq.(109).A natural question that arises is whether such a dynamics can right-fully be called “quantum” or not. But this line of inquiry is rather mis-guided because the search for QG is, in many respects, the search forwhich criteria, in fact, constitute a quantum theory in the first place.Since the ED model formulated here involves the presence of ~ , a waveequation for a complex wave function, an uncertainty principle and non-local correlations, and limits to the standard quantum formalism in a flatspace-time, we claim that it is these ingredients that are fundamental fordefining a quantum theory. Additional features, such as the superpositionprinciple, emerge here only as an effect confined to a limiting regime.
11 Concluding remarks
The ED developed here couples quantum “matter” to classical gravity onthe basis of three key principles: (1) A properly entropic setting whereinthe dynamics of probability is driven by information encoded into con-straints . (2) The preservation of a symplectic structure as a primarycriterion for updating the evolving constraints. (3) Foliation invariancesymmetry, enacted by representing the DHKT “algebra” in terms of therelevant Poisson brackets.Our approach results in several interesting features. Although writ-ten in the relatively less common language of geometrodynamics, the This dynamic can most readily be seen within the linearized gravity regime, or the weakfield limit. In fact, there are some arguments (see e.g., [73]) that the so-called Newton-Schr¨odinger equation, which is a non-linear equation, results from the Newtonian limit of thesemi-classical Einstein equations. Investigation of this result coming from ED is underway. qns.(84a) and (103)-(104) are formally equivalent to the so-called semi-classical Einstein equations (SCEE) G µν = 8 πG h ˆ T µν i , (110)with classical Einstein tensor G µν (see e.g., [70]), but sourced by theexpected value of the quantum stress-energy tensor. Such a theory ofgravity has long been seen as a desirable step intermediate to a full theoryof QG, in part because it contains well-established physics — QFTCS andclassical GR — in the limiting cases where they are valid. But there hasbeen much debate (see e.g., [72][74][75][76]), on the other hand, as tothe status of semi-classical theories as true QG candidate; with manyharboring a negative view.Here we do not propose a definitive rebuttal to those critics, but notethat the ED formulation of SCEE has certain features that allow it toevade the most cogent criticisms. For one, a problem that is often raisedagainst the SCEE is that it is proposed in a rather ad hoc manner, basedon heuristic arguments. Indeed, the usual argument for the expectedvalue on the right hand side of eq.(110) is at best a guess. In ED, on theother hand, the coupling of geometry to the expected value of quantumoperators — made explicit in eqns.(102a)-(104) — is derived on the basisof well-defined assumptions and constraints. In other words, we derive theSCEE in ED from first principles . Indeed, these principles have alreadybeen tested elsewhere: not only do they provide a reconstruction of GRthrough the work of DHKT, but they also provide a reconstruction ofboth nonrelativistic quantum mechanics and relativistic quantum fieldtheory. Furthermore, while there is no guarantee that such constraints andassumptions are adequate for a full QG theory, ED provides a frameworkwherein additional information can be incorporated as needed.One source of confusion with the SCEE is the issue of what happenswhen the location of a macroscopic source is uncertain. (See e.g. , [72][73].)Consider, for example, a macroscopic mass m that is equally likely to beat x or at x . Will the gravitational field itself be equally likely to pointeither towards x or towards x ? Or, will the gravitational field be as ifgenerated by a mass m located at the expected position ( x + x ) /
2? TheED resolution of this paradox shows the advantage of having a derivationfrom first principles. The h ˆ T µν i on the right of the SCEE is not the ex-pectation value taken over any arbitrary source of uncertainty. The h ˆ T µν i was derived, or better, it was inferred from a very specific type of infor-mation that leads, by an abuse of language, to what one might call a purequantum state. The issue then is what is the gravitational field generatedby a pure state that happens to be a macroscopic ”Schr¨odinger cat”? Tothe extent that this is a “pure” state then ED gives a sharp prediction:the gravitational field is generated by an h ˆ T µν i centered at the averageposition. But such a Schr¨odinger cat state cannot be physically realized:it would immediately suffer decoherence. To analyze such a situation theED framework would need to be extended to incorporate information ( i.e. , The components of the quantum stress-energy tensor are related in a simple manner tothe ensemble quantities in eqns.(99) and (101). The derivation of the SCEE from an actionby Kibble and Randjbar-Daemi in [71] makes this relationship more precise. dditional constraints) that describes additional sources of uncertainty. Inthis extended ED it is conceivable that the gravitational field itself wouldbe uncertain. So the conclusion is that there is no paradox; the two dif-ferent predictions correspond to two different inferences arising from twodifferent pieces of information.Another family of objections revolve around paradoxes related to theproblem of quantum measurement. As pointed out long ago by Kibble [77]these criticisms are premature as long as the interpretation of quantummechanics is problematic and the problem of the collapse of the wavefunction remains unsolved. The issue is whether a measurement thatinduces a collapse of the wave function would result in a discontinuouschange in the expected h ˆ T µν i with its attendant violations of causality.Within the ED approach such objections do not arise. ED solvesthe problem of interpretation by starting with a clear definition of theontology, which eliminates the interpretation problem. Furthermore, byincluding in its very foundation the epistemic tools for inference, ED alsosolves the problem of measurement [26][27]. In ED a measument device isnot described as a black box subject to rules that violate the Schr¨odingerequation but as a physical process subject to the very same laws thatdescribe the rest of the world. As a result in a fully covariant ED such asthe model developed in this paper the process of measurement is describedby the same causal flow of probability that characterizes any other physicalprocess.Yet another argument that has been raised against the SCEE is thatthe left hand side, featuring the gravitational field, is a “physical” onticfield, while the right hand side contains the quantum state Ψ, which isepistemic. Within the ED approach, however, the fact that h ˆ T µν i on theright hand side of (110) is epistemic necessarily forces the left hand side,the geometry, to be epistemic too. Indeed, a specific proposal alongthese lines has been offered in [12][13], suggesting that the geometry ofspace-time itself can be given an entropic underpinning.Finally, the Schr¨odinger equation derived here is quite unorthodoxin that the dynamics of Ψ follows a non-linear equation. This is quiteproblematic in the standard view of QT, where linearity is held as sacro-sanct. But, as mentioned in the introduction, in the ED approach toquantum theory Hilbert spaces are not fundamental; they are introducedas a convenient trick precisely because of the calculational advantage ofthe linearity they induce. In the model developed here such a trick cannotbe carried out and the superposition principle becomes the first casualtyin a successful coupling of quantum matter with dynamical gravity.The nonlinearity of the SCEEs does extreme violence to our under-standing of quantum theory and raises many questions. Is the introduc-tion of Hilbert spaces at all justified? Are density matrices at all useful?Can we expect something like the evolution from pure to mixed states?Or, what seems more likely, the very concepts of pure and mixed states areso linked to the concept of Hilbert spaces that, absent the latter, the more Of course, other theories will lead to other interpretations. If one were to adopt alternativenon-Bayesian interpretations of probability (say, frequentist or propensity) or of QM (e.g., aBohmian ontic wave function), then the geometry of spacetime would not need to be epistemic. seful description is just in terms of probabilities ρ and phase fields Φ.What is the generalization of von Neumann’s entropy for such states? Cannon-orthogonal states be distinguished? Can quantum states be cloned?The point of even raising such questions is precisely to emphasize the verydifferent research directions one is led to once the standard tools of quan-tum mechanics are no longer fundamental and/or available. We must beopen to the possibility that the proper way to model gravity might be asneither a quantum nor a classical theory but something else altogether —perhaps an inferential model based on information geometry in the spiritof the ED approach to QM. Acknowledgments
We would like to thank M. Abedi, D. Bartolomeo, N. Carrara, N. Caticha,F. Costa, S. DiFranzo, K. Knuth, S. Nawaz, P. Pessoa, and K. Vanslettefor many valuable discussions on entropy, inference, quantum theory, andmuch more. We would also like to thank the physics department of Uni-versity at Albany—SUNY for their continued support.
References [1] Carney, D.; Stamp, P.C.E.; Taylor, J.M. “Tabletop experiments forquantum gravity: a users manual.” Class. Quantum Gravity ,36.3. arXiv: 1807.11494.[2] Surya, S. “The causal set approach to quantum gravity.” LivingRev. Relativ. , 22.1, pp. 5.[3] Loll, R. “Quantum gravity from causal dynamical triangula-tions: a review.” Class. and Quantum Gravity , 37.1. arXiv:1905.08669.[4] Butterfield, J.; Isham, C. “Spacetime and the philosophical chal-lenge of quantum gravity.” In
Physics meets philosophy at the Planckscale.
Eds., C. Callender, N. Huggett; Cambridge University Press,2001; pp. 33-89. arXiv: 9903072.[5] Mukhi, S. “String theory: a perspective over the last 25 years.”Class. and Quantum Gravity , 28.15. arXiv: 1110.2569.[6] Rovelli, C. “Loop quantum gravity: the first 25 years.” Class. andQuantum Gravity , 28.15. arXiv: 1012.4707.[7] Henneaux, M.; Teitelboim, C.
Quantization of gauge systems.
Princeton university press, 1994.[8] Isham, C.J. “Canonical quantum gravity and the problem of time.”In
Integrable systems, quantum groups, and quantum field theories.
Eds., L. A. Ibort, M. A. Rodr´ıguez; Springer: Dordrecht, Nether-lands, 1993; pp. 157-287. arXiv: 9210011.[9] Jacobson, T. “Thermodynamics of spacetime: the Einstein equationof state.” Phys. Rev. Lett , 75.7. arXiv: 9504004.[10] Verlinde, E. P. “On the Origin of Gravity and the Laws of Newton.”J. High Energy Phys , 2011.4.
11] Verlinde, E.P. “Emergent gravity and the dark universe.” SciPostPhys , 2.3. arXiv: 1611.02269.[12] Caticha, A. “Geometry from information geometry.” In Proceed-ings of the 35th International Workshop on Bayesian Inference andMaximum Entropy Methods in Science and Engineering , Potsdam,USA, 1924 July 2015; Eds., A. Giffin, K. H. Knuth; AIP Conf. Proc,2016, Vol. 1757.1. arXiv: 1512.09076.[13] Caticha, A. “The information geometry of space-time.” In Proceed-ings of the 39th International Workshop on Bayesian Inference andMaximum Entropy Methods in Science and Engineering , Garching,Germany, 30 June 5 July 2019; Eds. U. von Toussaint, R. Preuss;MDPI Conf. Proc., 2019, Vol. 33.1. arXiv: 1909.09657.[14] Hall, M.J.W. “Exact uncertainty approach in quantum mechanicsand quantum gravity.” Gen. Relativ. Gravit , 37.9, 1505-1515.arXiv: 0408098.[15] Reginatto, M. “Exact uncertainty principle and quantization: im-plications for the gravitational field.” Braz. J. Phys , 35.2B,476-480.[16] Hardy, L. “Towards quantum gravity: a framework for probabilistictheories with non-fixed causal structure.” J. Phys. A , 40.12.arXiv: 0608043.[17] Hardy, L. “The Construction Interpretation: Conceptual Roads toQuantum Gravity.” arXiv preprint . arXiv:1807.10980.[18] D¨oring, A.; Isham, C.J. “A topos foundation for theories of physics:I. Formal languages for physics.” J. Math. Phys , 49.5. arXiv:0703060.[19] D¨oring, A.; Isham, C.J. “A topos foundation for theories of physics:II. Daseinisation and the liberation of quantum theory.” J. Math.Phys , 49.5. arXiv: 0703062.[20] Caticha, A.
Entropic Physics: Probability, En-tropy, and the Foundations of Physics
ET Jaynes: Papers on probability, statistics and statistical physics the 29th International Workshop on Bayesian Inferenceand Maximum Entropy Methods in Science and Engineering , Ox-ford, Mississippi, 510 July 2009; Eds. P. M. Goggans, CY. Chan;AIP Conf. Proc., 2009, Vol. 1193.1. arXiv: 0907.4335.[23] Caticha, A. “ Entropic dynamics, time, and quantum theory.” J.Phys. A , 44. arXiv: 1005.2357.[24] Caticha, A.; Bartolomeo, D.; Reginatto, M. “Entropic Dynamics:from entropy and information geometry to Hamiltonians and quan-tum mechanics.” In Proceedings of the 34th International Workshop n Bayesian Inference and Maximum Entropy Methods in Scienceand Engineering , Amboise, France, 2126 September 2014; Eds. A.Mohammad-Djafari, F. Barbaresco; AIP Conf. Proc., 2015, Vol.1641.1. arXiv: 1412.5629.[25] Caticha, A. “The Entropic Dynamics approach to Quantum Me-chanics.” Entropy , 21.10. arXiv: 1711.02538.[26] Johnson, D. T.; Caticha, A. “Entropic dynamics and the quantummeasurement problem.” In Proceedings of The 31st InternationalWorkshop on Bayesian Inference and Maximum Entropy Methodsin Science and Engineering , Ontario, Canada, 916 July 2011; Eds. P.Goyal, et al. ; AIP Conf. Proc., 2012, Vol. 1443.1. arXiv: 1108.2550.[27] Vanslette, K.; Caticha, A. “Quantum measurement and weak valuesin entropic dynamics.” In the Proceedings of
The 36th InternationalWorkshop on Bayesian Inference and Maximum Entropy Methods inScience and Engineering , Ghent, Belgium, 1015 July 2016; Eds. G.Verdoolaege; AIP Conf. Proc., 2017, Vol. 1853.1. arXiv: 1701.00781.[28] Caticha, A. “The entropic dynamics of relativistic quantum fields.”In the Proceedings of the 32nd International Workshop on BayesianInference and Maximum Entropy Methods in Science and Engineer-ing , Garching, Germany, 1520 July 2012; Eds., U. von Toussaint;AIP Conf. Proc.., 2013, Vol. 1553.1. arXiv: 1212.6946.[29] Caticha, A.; Ipek, S. “Entropic Quantization of Relativistic ScalarFields”, in Proceedings of , Amboise, France, 2126 September 2014; Eds. A. Mohammad-Djafari, F. Barbaresco; AIP Conf. Proc., 2015, Vol. 1641.1. arXiv:1412.5637.[30] Ipek, S.; Abedi, M.; Caticha, A. ”A covariant approach to entropicdynamics.” In the Proceedings of
The 36th International Workshopon Bayesian Inference and Maximum Entropy Methods in Scienceand Engineering , Ghent, Belgium, 1015 July 2016; Eds. G. Ver-doolaege; AIP Conf. Proc., 2017, Vol. 1853.1. arXiv: 1803.06327.[31] Ipek, S.; Abedi, M.; Caticha, A. “Entropic dynamics: reconstructingquantum field theory in curved space-time.” Class. and QuantumGravity , 36.20. arXiv: 1711.02538.[32] Ipek, S.; Caticha, A. “An Entropic Dynamics Approach to Ge-ometrodynamics.” In Proceedings of the 39th International Work-shop on Bayesian Inference and Maximum Entropy Methods in Sci-ence and Engineering , Garching, Germany, 30 June 5 July 2019;Eds. U. von Toussaint, R. Preuss; MDPI Conf. Proc, 2019, Vol.33.1. arXiv: 1910.01188.[33] Cox, R.T. ”Probability, frequency and reasonable expectation.” Am.J. Phys , 14.1, 1-13.[34] Wigner, E.P.
Symmetries and reflections: Scientific essays.
IndianaUniversity Press: Bloomington, Indiana, 1967.[35] Jeffreys, H. “An invariant form for the prior probability in estima-tion problems.” Proc. R. Soc. Lond A , 186.1007, 453-461.
36] Jaynes, E.T. “Prior probabilities.” Syst. Sci. Control. Eng. ,4.3, 227-241.[37] Nelson, E. “Connection between Brownian motion and quantummechanics.” Einstein Symposium Berlin , 168-179.[38] Kibble, T.W.B. “Geometrization of quantum mechanics.” Commun.Math. Phys , 65.2, 189-201.[39] Heslot, A. “Quantum mechanics as a classical theory.” Phys. Rev.D , 31.6, 1341.[40] Cirelli, R.; Mania, A; Pizzocchero, L. “Quantum mechanics as aninfinitedimensional Hamiltonian system with uncertainty structure:Part I.” J. Math. Phys , 31.12, 2891-2897.[41] Cirelli, R.; Mania, A; Pizzocchero, L. “Quantum mechanics as aninfinitedimensional Hamiltonian system with uncertainty structure:Part II.” J. Math. Phys , 31.12 (1990), 2898-2903.[42] Ashtekar, A.; Schilling, T.A. ”Geometrical formulation of quantummechanics.” In
On Einsteins Path: Essays in Honor of EngelbertSch¨ucking ; Editor A. Harvey; Springer: New York, NY, 1999. 23-65.[43] Carrara, N.; Caticha, A. “The Entropic Dynamics of Spin-1/2.”Manuscript in preparation .[44] Merzbacher, E. “Single valuedness of wave functions.” Am. J. Phys , 30.4, 237-247.[45] Wallstrom, T.C. “Inequivalence between the Schrdinger equationand the Madelung hydrodynamic equations.” Phys. Rev. A ,49.3.[46] Dirac, P.A.M. “The Hamiltonian form of field dynamics.” Can. J.Math , 3.1, 1-23.[47] Dirac, P.A.M.
Lectures on quantum mechanics , 2nd ed.; CourierCorporation: 2013.[48] Hojman, S.A.; Kuchaˇr, K.; Teitelboim, C. “Geometrodynamics re-gained.” Ann. Phys. , 96.1, 88-135.[49] Kuchaˇr, K. “A Bubble-Time Canonical Formalism for Geometrody-namics.” J. Math. Phys , 13.5, 768-781.[50] Teitelboim, C. “How commutators of constraints reflect the space-time structure.” Ann. Phys. , 79.2, 542-557.[51] Weitzman, C.T. “The Hamiltonian Structure of Spacetime.” PhDthesis, Princeton, 1973.[52] Weiss, P. “On the Hamilton-Jacobi theory and quantization of adynamical continuum.” Proc. R. Soc. Lond A , 169.936.[53] Dirac, P.A.M. “Relativistic quantum mechanics.” Proc. R. Soc.Lond A , 136.829.[54] Tomonaga, S. “On a relativistically invariant formulation of thequantum theory of wave fields.” Prog. Theor. Exp. Phys , 1.2,27-42.
55] Schwinger, J. “Quantum electrodynamics. I. A covariant formula-tion.” Phys. Rev , 74.10, 1439.[56] Wald, R.M.
Quantum field theory in curved spacetime and black holethermodynamics.
University of Chicago Press, 1994.[57] Stapp, H. P. “The copenhagen interpretation.” Am. J. Phys ,40.8, 1098-1116.[58] Leifer, M. S. “Is the quantum state real? An extended review of ψ -ontology theorems.” arXiv preprint , arXiv:1409.1570.[59] Everett III, H. “Relative state” formulation of quantum mechanics.”Rev. Mod. Phys , 29.3, 454.[60] Bartolomeo, D.; Caticha, A. “Trading drift and fluctuations in en-tropic dynamics: quantum dynamics as an emergent universalityclass.” In the Proceedings of Emergent Quantum Mechanics 2015 ,Vienna, Austria, 2325 October 2015; J. Phys. Conf. Ser. Vol. 701.1.[61] Dirac, P.A.M. “The theory of gravitation in Hamiltonian form.”Proc. R. Soc. Lond A , 246.1246, pp. 333-343.[62] Arnowitt, R.; Deser, S.; Misner, C.W. “Canonical variables for gen-eral relativity.” Phys. Rev , 117.6, pp. 1595.[63] Arnowitt, R.; Deser, S.; Misner, C.W. “Republication of: The dy-namics of general relativity.” Gen. Relativ. Gravit , 40.9, pp.1997-2027. arXiv: 0405109.[64] Gourgoulhon, E. “3+ 1 formalism and bases of numerical relativity.”arXiv preprint , arXiv: gr-qc/0703035.[65] Misner, C.W.; Thorne, K.S.; Wheeler, J.A.
Gravitation.
Macmillan,1973.[66] Ferraris, M.; Kijowski, J. “General relativity is a gauge type theory.”Lett. Math. Phys. , 5.2, 127-135.[67] Ashtekar, A. ”New variables for classical and quantum gravity.”Phys. Rev. Lett , 57.18, pp. 2244.[68] Teitelboim, C. “The Hamiltonian structure of space-time.” In
Gen-eral Relativity and Gravitation. Vol. 1. One hundred years after thebirth of Albert Einstein,
Vol. 1; Ed., A. Held; Plenum Press: NewYork, NY, 1980; pp. 195.[69] Schutz, Bernard F.
Geometrical methods of mathematical physics.
Cambridge university press, 1980.[70] Carroll, S.M. “Spacetime and geometry: An introduction to generalrelativity.” Addison Wesley, 2004. arXiv: gr-qc/9712019.[71] Kibble, T. W. B.; Randjbar-Daemi, S. “Non-linear coupling of quan-tum theory and classical gravity.” J. Phys. A , 13.1, pp. 141.[72] Unruh, W. G. “Steps towards a quantum theory of gravity.” In
Quantum theory of gravity. Essays in honor of the 60th birthdayof Bryce S. DeWitt ; Ed., B.S. DeWitt; Adam Hilger Ltd.: Bristol,England, 1984; pp. 234-242
73] Bahrami, M., et al. “The SchrdingerNewton equation and its foun-dations.” New J. Phys , 16.11, pp. 115007. arXiv: 1407.4370.[74] Eppley, K.; Hannah, E. “The necessity of quantizing the gravita-tional field.” Found. Phys , 7.1-2, pp. 51-68.[75] Page, D. N.; Geilker, C. D. “Indirect evidence for quantum gravity.”Phys. Rev. Lett , 47.14, pp. 979.[76] Duff, M.J. In
Quantum Gravity 2: A Second Oxford Symposium.
Eds., C.J. Isham et al. ; Clarendon Press: Oxford, England, 1981.[77] Kibble, T. W. B. “Relativistic models of nonlinear quantum me-chanics.” Commun. Math. Phys , 64.1, pp. 73-82., 64.1, pp. 73-82.