Entropic Dynamics: Reconstructing Quantum Field Theory in Curved Space-time
aa r X i v : . [ g r- q c ] A ug Entropic Dynamics: Reconstructing QuantumField Theory in Curved Space-time
Selman Ipek, Mohammad Abedi, and Ariel Caticha
Physics Department, University at Albany-SUNY, Albany, NY 12222, USA.
Abstract
The Entropic Dynamics reconstruction of quantum mechanics is ex-tended to the quantum theory of scalar fields in curved space-time. TheEntropic Dynamics framework, which derives quantum theory as an ap-plication of the method of maximum entropy, is combined with the co-variant methods of Dirac, Hojman, Kuchaˇr, and Teitelboim, which theyused to develop a framework for classical covariant Hamiltonian theories.The goal is to formulate an information-based alternative to current ap-proaches based on algebraic quantum field theory. One key ingredient isthe adoption of a local notion of entropic time in which instants are definedon curved three-dimensional surfaces and time evolution consists of theaccumulation of changes induced by local deformations of these surfaces.The resulting dynamics is a non-dissipative diffusion that is constrainedby the requirements of foliation invariance and incorporates the necessarylocal quantum potentials. As applications of the formalism we derive theEhrenfest relations for fields in curved-spacetime and briefly discuss thenature of divergences in quantum field theory.
Research on quantum field theory in curved space-time (QFTCS) has been thesubject of a sustained effort for several decades. (See, for example, [1][2] and therecent review [3].) The motivation is twofold. There is an intrinsic interest inQFTCS as an effective theory that is reliable except in those regions of space-time where the curvature is extremely high. But there is also a widespreadbelief that QFTCS is a necessary intermediate step on the way towards a morefundamental theory of interacting quantum and gravitational fields.As discussed in [3] QFTCS is usually formulated by merging classical generalrelativity with quantum field theory. Identifying the principles that govern thegravity sector does not appear to be problematic. Unfortunately, the choice ofwhich principles of quantum field theory in flat space-time should be retainedin a curved space-time is not nearly as clear. The lack of Poincare invarianceimplies the absence of a preferred vacuum state and forces one to revisit thenotion of particle [4][5]. The representation of quantum fields as operators, or1etter, as operator-valued distributions, is problematic on many levels. Promi-nent among them is the issue of the Hilbert space itself. A representation ofthe relevant commutation relations and field equations requires operators thatact on a Hilbert space. The problem is that in curved space-time there is aninfinite number of such representations that are unitarily inequivalent. Anotherproblem is that representing fields as operators may itself be deeply misguided— it may turn out to be an attempt to preserve in curved space something thathas already turned out to be questionable at the level of flat space-time, namely,the standard quantum theory of measurement.But there is another direction from which this subject can be approached.The discoveries of black hole entropy and thermodynamics [6]-[8] suggest a deepconnection between the fundamental laws of physics and information. Indeed,in recent decades the subject of foundations of quantum mechanics (FQM) hasundergone a renaissance (for reviews see e.g. [9]-[11]) culminating in a varietyof proposals for the reconstruction of quantum mechanics (see e.g. , [12]-[18]).Notable among the latter are proposals that seek a deeper foundation based oninformation theory (see e.g. , [19]-[27]). To a large extent the two subjects —QFTCS and FQM — have developed independently. In fact, it is somewhatsurprising that progress in QFTCS has contributed so little towards clarifyingthe conceptual foundations of quantum mechanics and vice versa.The purpose of this paper is to contribute to bridge this gap by extendingone particular approach to the reconstruction of quantum mechanics — theentropic dynamics approach — to the reconstruction of quantum field theory.Entropic Dynamics (ED) is a framework for constructing dynamical theorieson the basis of Bayesian and entropic principles of inference [30][31]. (For arecent self-contained presentation and review see [32].) The goal is to developmodels for the time evolution of the probability distributions of the positions ofparticles or the values of fields. ED differs from other information approachesto quantum theory by strictly adhering to Bayesian and entropic methods ofinference, and by paying close attention to the notion of time.One challenge that ED was designed to address is that a fully epistemic interpretation of the wave function Ψ is not achieved by merely declaring thatthe square of a wave function | Ψ | yields a Bayesian probability. To insureconsistency one must also show that the rules for updating those probabilities— which include both the unitary evolution of the wave function Ψ and its“collapse” during a measurement — are in strict accord with Bayesian andentropic methods [33]-[35]. Thus, in an entropic dynamics the evolution mustbe driven by information codified into constraints; it is through these constraintsthat the “physics” is introduced.An important early insight in the context of Nelson’s stochastic mechanics,was his realization that important aspects of quantum mechanics could be mod-elled as a non-dissipative Brownian motion [36]. This idea was suitably adaptedto the ED setting and imposing the conservation of an appropriate energy func- Within a model such as ED a concept is referred as ‘epistemic’ when it is related to thestate of knowledge, opinion, or belief of a rational agent. For example, Bayesian probabilitiesare epistemic — they are tools for reasoning with incomplete information [35]. global energy is not in general available. An alternative local criterion forevolving constraints is needed.Another challenge that ED is designed to address concerns the concept oftime. Entropic time is constructed as a scheme to keep track of the accumula-tion of small changes [30]. This involves identifying a suitable notion of ordered“instants” and then introducing a convenient measure of the interval or sep-aration between them. In ED an instant is defined through the informationthat is necessary to predict or “construct” the next instant. This amounts tospecifying a state of knowledge together with a purely kinematic criterion ofsimultaneity. In [38][39] the concept of a global instant was used to generate anED of quantum scalar fields in Minkowski space-time. However, although themodel was fully relativistic, its relativistic invariance was not manifest, that is,the freedom to represent the relativity of simultaneity was not explicit.The goal of this paper is to formulate an inference-based approach to QFTCSin which quantum fields are not represented as operators; there is no mentionof a preferred vacuum state, nor of an associated Hilbert space. We constructa manifestly relativistic quantum ED in curved space-time incorporating ideasdeveloped in the classical field theories of Dirac, Hojman, Kuchaˇr, and Teitel-boim (DHKT) [41]-[46] which can themselves be traced to the earlier theoriesof Weiss, Tomonaga, Dirac, and Schwinger [47]-[50]. Drawing on the ideas ofDHKT, we relax the assumption of a global time in favor of a notion of localtime, and the non-covariant criterion of energy conservation is replaced by thecovariant requirement of foliation invariance. In this view of ED, an instant isdefined by a three-dimensional spacelike surface embedded in space-time plusthe fields and probability distributions defined on such surfaces. In a fully co-variant theory, such surfaces are constructed by slicing or foliating space-timeinto a sequence of spacelike surfaces. The freedom to choose the foliation, whichamounts to the local relativity of simultaneity, is implemented by a consistencyrequirement: the evolution of all dynamical quantities from an initial to a finalsurface must be independent of the choice of intermediate surfaces. We canrefer to foliation invariance either as a requirement of consistency or, followingKuchaˇr and Teitelboim, as a requirement of “path independence” — if there aretwo alternative ways to evolve from an initial to a final instant, then the twoways must lead to the same result. The formalism is developed for a scalar field χ ( x ). First we find the transitionprobability for an infinitesimal change — the effect of an infinitesimal localdeformation of the instant. The introduction of a local entropic time thenleads to a set of functional “local-time Fokker-Planck” diffusion equations forthe evolution of probability distribution ρ [ χ ]. Requiring that the evolution A brief presentation of some of these ideas appeared in [40]. Wheeler originally called this an “embeddability” condition — the requirement that allthese surfaces be embedded in the same space-time.
The microstates—
In an inference scheme such as ED the physics is intro-duced through the choice of dynamical variables and of the constraints thatcapture the dynamically relevant information. Here we consider the dynam-ics of a scalar field χ ( x ). For notational convenience we will often write the x -dependence as a subscript, χ ( x ) = χ x . Unlike the standard Copenhageninterpretation of quantum theory where observables have definite values onlywhen elicited through an experiment, in the ED approach these fields have def-inite values at all times. However, these values are unknown and the dynamicsis indeterministic.The field χ lives on a 3-dimensional curved space σ , the points of which arelabeled by coordinates x i ( i = 1 , , σ is endowed with a metric g ij induced by the non-dynamical background space-time in which σ is embedded.Thus σ is an embedded hypersurface; for simplicity we shall refer to it as a“surface”. We assume that space-time is globally hyperbolic. It can be foliatedby Cauchy surfaces and its topology is R × Σ where Σ stands for any Cauchysurface σ [59]. The field χ x is a scalar with respect to 3-diffeomorphisms on thesurface σ . The ∞ -dimensional space of all possible field configurations is theconfiguration space C . A single field configuration, labelled χ , is representedas a point in C , and the uncertainty in the field is described by a probabilitydistribution ρ [ χ ] over C . Maximum Entropy—
Our goal here is to predict the evolution of the scalarfield χ . To this end we make one major assumption: in ED, motion is con-tinuous, the fields follow continuous trajectories in C . This implies that anyfinite change can be analyzed as the accumulation of many infinitesimally shortsteps. Therefore our first goal is to calculate the probability P [ χ ′ | χ ] that thefield undergoes a small change from an initial configuration χ to a neighboring χ ′ = χ + ∆ χ and later we calculate the probability of the finite change thatresults from a sequence of short steps. The transition probability P [ χ ′ | χ ] isfound by maximizing the entropy functional, S [ P, Q ] = − Z Dχ ′ P [ χ ′ | χ ] log P [ χ ′ | χ ] Q [ χ ′ | χ ] , (1)4elative to a prior Q [ χ ′ | χ ] and subject to appropriate constraints. It is throughthe prior and the constraints that the relevant physical information is intro-duced. The prior—
We adopt a prior Q [ χ ′ | χ ] that incorporates the information thatchange happens by infinitesimally small amounts, but is otherwise maximallyuninformative. In particular, as far as the prior is concerned, changes at differentpoints are uncorrelated. Such a prior can itself be derived from the principle ofmaximum entropy. Indeed, maximize S [ Q, µ ] = − Z dχ ′ Q [ χ ′ | χ ] log Q [ χ ′ | χ ] µ ( χ ′ ) , (2)relative to the measure µ ( χ ′ ) which we assume to be uniform and subject toappropriate constraints. The requirement that the field undergoes changes thatare small and uncorrelated is implemented by imposing an infinite number ofindependent constraints, one at each point x , h ∆ χ x i = Z Dχ ′ Q [ χ ′ | χ ] (∆ χ x ) = κ x , (3)where ∆ χ x = χ ′ x − χ x and, to enforce the continuity of the motion, we willeventually take the limit κ x →
0. The result of maximizing (2) subject to (3)and normalization is a product of Gaussians, Q [ χ ′ | χ ] ∝ exp − Z dx g / x α x (∆ χ x ) (4)where α x are the Lagrange multipliers associated to each constraint (3), andthe scalar density g / x = (det g ij ) / is introduced so that α x is a scalar field.The limit κ x → α x → ∞ . For notational simplicity wewrite dx instead of d x . The drift potential constraint—
The dynamics induced by the prior (4)is a diffusion that is isotropic in configuration space. In order to introduce cor-relations, directionality, and such quintessential quantum effects as interferenceand entanglement, we impose one additional single constraint that is non-local in space but local in configuration space. This single constraint involves a func-tional on configuration space, the “drift” potential φ [ χ ]. We impose that theexpectation of the change of the drift potential ∆ φ = φ [ χ ′ ] − φ [ χ ] is anothersmall quantity κ ′ that will eventually be taken to zero, h ∆ φ i = κ ′ or Z C Dχ ′ P [ χ ′ | χ ] Z dx ∆ χ x δφ [ χ ] δχ x = κ ′ . (5) Since we deal with infinitesimally short steps the prior Q turns to be quite independentof the choice of the underlying measure µ . χ x and ∆ χ x are scalars, in order for (5) to be invariant undercoordinate transformations of the surface the derivative δ/δχ x must transformas a scalar density.) The physical meaning of the drift potential φ [ χ ] will not bediscussed here. As in so many other situations in physics the mere identificationof forces and constraints can turn out to be useful even when their microscopicorigins is not yet fully understood. The transition probability—
Next we maximize (1) subject to (5) andnormalization. As discussed in [56] the multiplier α ′ associated to the globalconstraint (5) turns out to have no influence on the dynamics: it can be absorbedinto the drift potential α ′ φ → φ which means we can effectively set α ′ = 1.The resulting transition probability is a Gaussian distribution, P [ χ ′ | χ ] = 1 Z [ α x , g x ] exp − Z dx g / x α x ∆ χ x − g / x α x δφ [ χ ] δχ x ! , (6)where Z [ α x , g x ] is the normalization constant. In previous work [38][39] α x waschosen to be a spatial constant α to reflect the translational symmetry of flatspace. Here we make no such restriction and instead relax the global constant α in favor of a non-uniform spatial scalar α x which will be a key element inimplementing our scheme for a local entropic time.The Gaussian form of (6) allows us to present a generic change,∆ χ x = h ∆ χ x i + ∆ w x , (7)as resulting from an expected drift h ∆ χ x i plus fluctuations ∆ w x . At each x theexpected short step is h ∆ χ x i = 1 g / x α x δφ [ χ ] δχ x ≡ ∆ ¯ χ x , (8)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 ′ . (9)Thus we see that ∆ ¯ χ x ∼ /α x and ∆ w x ∼ /α / x , so that for short steps, α x → ∞ , the fluctuations dominate the motion. The resulting trajectory iscontinuous but non-differentiable — a Brownian motion. In ED the idea of time is derived from the idea of change. Time is introducedas a device to keep track of how the accumulation of many infinitesimal changes Elsewhere we show that in order to describe single particle states with nonzero angularmomentum φ needs to have the topological properties of an angle [60]. Additional evidencethat φ must be interpreted as an angle is provided in the ED of non-relativistic particles withspin [58]. Ordered instants—
Of particular importance is the notion of an instant,which in ED involves several ingredients: (1) A foliation of spacelike surfaces σ that codify spatial relations and provide a criterion of simultaneity and dura-tion. (2) We must specify the “epistemic contents” of the surfaces. This is aspecification of a statistical state that is sufficient for the prediction of futurestates. It is given by a probability distribution ρ [ χ ] and a drift potential φ [ χ ].And (3) an entropic step in which the statistical state at one instant is updatedto generate the state at the next instant. This is the requirement that generatesthe sequence of ordered instants which makes the dynamics come alive. Some space-time kinematics—
We deal with a curved space-time; eventsare labeled by space-time coordinates X µ ; and the metric is g µν (cid:0) X β (cid:1) . Space-time is foliated by a sequence of space-like surfaces { σ } . Points on the surface σ are labeled by coordinates x i and the embedding of the surface within space-time is defined by four functions X µ = X µ (cid:0) x i (cid:1) . The metric induced on thesurface is g ij ( x ) = X µi X νj g µν where X µi = ∂X µ ∂x i . (10)The metric g ij will in general depend on the particular surface. In this workneither g µν ( X ) nor g ij ( x ) are themselves dynamical.Following Teitelboim and Kucha˘r, we consider an infinitesimal deformationof the surface σ to a neighboring surface σ ′ . This is specified by the deformationvector δξ µ which connects the point in σ with coordinates x i to the point in σ ′ with the same coordinates x i , δξ µ = δξ ⊥ n µ + δξ i X µi , (11)where n µ is the unit normal to the surface ( n µ n µ = − n µ X µi = 0). Thenormal and tangential components are given by δξ ⊥ x = − n µx δξ µx and δξ ix = X iµx δξ µx , (12)where X iµx = g ij g µν X νjx . They are known as lapse and shift respectively and arecollectively denoted ( δξ ⊥ , δξ i ) = δξ A . As a matter of convention a deformationis identified by its normal δξ ⊥ and tangential δξ i components independentlyof the surface upon which it acts ( i.e. , independently of the normal n µ ). Thisallows us to speak about applying the same deformation to different surfaces; auseful concept for our discussion of path independence. We use Greek indices ( µ, ν, ... = 0 , , ,
3) for space-time coordinates X µ and latin indices( a, b, ...i, j, ... = 1 , ,
3) for coordinates x i on the surface σ . The spacetime metric has signature( − + ++). uration— In ED time is defined so that motion looks simple. The specifi-cation of the time interval between two successive instants is dictated purely byconvenience, that is, the best choice is that which reflects the various symme-tries of the physical situation. For example, in a non-relativistic QM one wouldadopt a Newtonian time ∆ t that reflects the translational symmetries of spaceso that “time flows equably everywhere and everywhen”. In our case the mostconvenient choice is one that reflects the symmetries of the background curvedspace-time.Since for short steps the dynamics is dominated by fluctuations, eq.(9), thechoice of time interval is achieved through an appropriate choice of the multi-pliers α x . So far the present development of ED has followed closely along thelines of the non-covariant models discussed in [37] and [39]). The importantpoint of departure is that here we are concerned with instants defined on thecurved embedded surfaces σ and σ ′ . It is then natural to define a local notionof duration in terms of an invariant — the proper time. The idea is the famil-iar one: at the point x in σ draw a normal segment reaching out to σ ′ . Theproper time δξ ⊥ x along this normal segment provides us with the local measureof duration between σ and σ ′ at x . More specifically, let α x = 1 δξ ⊥ x so that h ∆ w x ∆ w x ′ i = δξ ⊥ x g / x δ xx ′ . (13) The statistical state and its evolution—
Entropic dynamics is generatedby the short-step transition probability P [ χ ′ | χ ]. In a generic short step boththe initial χ and the final χ ′ are unknown. Integrating the joint probability, P [ χ ′ , χ ], over χ gives P [ χ ′ ] = Z dχ P [ χ ′ , χ ] = Z dχP [ χ ′ | χ ] P [ χ ] . (14)These equations are true by virtue of the laws of probability; they involve noassumptions. However, if P [ χ ] happens to be the probability of χ at an “in-stant” labelled σ , then we can interpret P [ χ ′ ] as the probability of χ ′ at the“next instant,” which we will label σ ′ . Accordingly, we write P [ χ ] = ρ σ [ χ ] and P [ χ ′ ] = ρ σ ′ [ χ ′ ] so that ρ σ ′ [ χ ′ ] = Z Dχ P [ χ ′ | χ ] ρ σ [ χ ] . (15)This is the basic dynamical equation; it allows one to update the statisticalstate ρ σ [ χ ] from one instant to the next. Note that since P [ χ ′ | χ ] is found bymaximizing entropy not only are these instants ordered but there is a natural entropic arrow of time: σ ′ occurs after σ . But we are not done yet. Withthe definition (13) of duration, the dynamics given by (15) and (6) describes aWiener process evolving along a given foliation of space-time. To obtain a fullycovariant dynamics we require that the evolution of any dynamical quantity suchas ρ σ [ χ ] from an initial σ i to a final σ f must be independent of the intermediate8hoice of surfaces. This “foliation invariance” or “path independence”, whichamounts to the local relativity of simultaneity, is a consistency requirement:if there are different ways to evolve from a given initial instant into a givenfinal instant, then all these ways must agree. The conditions to implement thisconsistency are the subject of the next section. The local-time diffusion equations—
The dynamics expressed in integralform by (15) with (6) and (13) can be rewritten in differential form as 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) with Φ σ [ χ ] = φ σ [ χ ] − log ρ / σ [ χ ] . (16)(The derivation is given in Appendix A.) This set of equations describes the flowof the probability ρ σ [ χ ] in the configuration space C as the surface σ is deformed.More explicitly, the actual change in ρ [ χ ] as σ is infinitesimally deformed to σ ′ is δ ⊥ ρ σ [ χ ] = Z dx δρ σ [ χ ] δξ ⊥ x δξ ⊥ x = − Z dx δξ ⊥ x g / x δδχ x (cid:18) ρ σ [ χ ] δ Φ σ [ χ ] δχ x (cid:19) . (17)In the special case when both surfaces σ and σ ′ happen to be flat then g / x = 1and δξ ⊥ x = dt are constants and eq.(17) becomes ∂ρ t [ χ ] ∂t = − Z dx δδχ x (cid:18) ρ t [ χ ] δ Φ t [ χ ] δχ x (cid:19) , (18)which we recognize as a diffusion or Fokker-Planck equation written as a con-tinuity equation for the flow of probability in configuration space C . Accord-ingly we will refer to (16) as the “local-time Fokker-Planck” equations (LTFP).These equations describe the flow of probability with a current velocity v x [ χ ] = δ Φ /δχ x . Eventually, the functional Φ will be identified as the Hamilton-Jacobifunctional, or the phase of the wave functional in the quantum theory.Anticipating later developments we note that the LTFP eqs.(16) can berewritten in an alternative and very suggestive form involving the notion of anensemble functional or e-functional . Just as a regular functional such as ρ [ χ ]maps a field distribution χ into a real number (a probability in this case), ane-functional maps a functional, such as ρ [ χ ] or Φ [ χ ], into a real number. Then,just as one can define functional derivatives, one can also define e-functionalderivatives. Introduce an e-functional ˜ H ⊥ x [ ρ σ , Φ σ ] such that at each point xδρ σ [ χ ] δξ ⊥ x = ˜ δ ˜ H ⊥ x [ ρ σ , Φ σ ]˜ δ Φ σ [ χ ] (19)reproduces (16). In what follows we denote all ensemble quantities such as ˜ H ⊥ x with a tilde: ˜ δ/ ˜ δ Φ[ χ ] is the e-functional derivative with respect to Φ [ χ ]. We An excellent brief review of the ensemble calculus is given in the appendix of [61]. H ⊥ x can always be found. Indeed, substitute (16) into the leftof (19), then an easy integration gives˜ H ⊥ x [ ρ, Φ; σ, χ ] = Z Dχ ρ g / x (cid:18) δ Φ δχ x (cid:19) + F x [ ρ ; σ, χ ] , (20)where the integration constants F x = F x [ ρ ; σ, χ ] are independent of Φ; they maydepend on ρ , on the geometry of the surface σ , and also on the fields χ x . Inlater sections we will see that ˜ H ⊥ x captures dynamical information about theevolution of Φ as well as ρ and can be cast as a Hamiltonian generator. Dynamics in local time must reflect the kinematics of surface deformations, andthis kinematics can be studied independently of the particular dynamics beingconsidered. As a surface is deformed, its geometry and, more generally, thestatistical state associated with it is also subject to change. Consider a genericfunctional T [ X ( x )] that assigns a number to every surface X µ ( x ). The changein the functional δT resulting from an arbitrary deformation δξ Ax has the form δT = Z dx δξ µx δTδξ µx = Z dx (cid:0) δξ ⊥ x G ⊥ x + δξ ix G ix (cid:1) T , (21)where G ⊥ x = δδξ ⊥ x = n µx δδX µx and G ix = δδξ ix = X µix δδX µx (22)are the generators of normal and tangential deformations respectively. Unlikethe vectors, δ/δX µx , which form a coordinate basis in the space of hypersur-faces and therefore commute, the generators of deformations δ/δξ Ax form a non-holonomic basis. Their non-vanishing commutator is δδξ Ax δδξ Bx ′ − δδξ Bx ′ δδξ Ax = Z dx ′′ κ CBA ( x ′′ ; x ′ , x ) δδξ Cx ′′ (23)where κ CBA are the “structure constants” of the “group” of deformations.The calculation of κ C BA is given in [43][45]. The basic idea is embeddability :When we perform two successive infinitesimal deformations δξ A followed by δη B , σ δξ → σ δη → σ ′ , the three surfaces are all embedded in the same space-time.The same happens when we execute them in the opposite order, σ δη → σ δξ → σ ′′ . The notation F x [ ρ ; σ, χ ] indicates that F x is a functional of various dynamical and non-dynamical variables. To unclutter the notation some of these dependencies will not be explic-itly displayed. Eq.(19) provides the criterion that allows us to identify the momentum Φ[ χ ] that is canon-ically conjugated to the generalized coordinate ρ [ χ ]. σ ′ and σ ′′ are embedded in thesame space-time there must exist a third deformation δζ α that takes σ ′ to σ ′′ : σ ′ δζ → σ ′′ . Thus the deformation from σ to σ ′′ can be attained by following twodifferent paths: either we follow the direct path σ δη → σ δξ → σ ′′ or we follow theindirect path σ δξ → σ δη → σ ′ δζ → σ ′′ . Then, as shown in [43][45], eq.(23) leads tothe “algebra”,[ G ⊥ x , G ⊥ x ′ ] = − ( g ijx G jx + g ijx ′ G jx ′ ) ∂ ix δ ( x, x ′ ) , (24)[ G ix , G ⊥ x ′ ] = − G ⊥ x ∂ ix δ ( x, x ′ ) , (25)[ G ix , G jx ′ ] = − G ix ′ ∂ jx δ ( x, x ′ ) − G jx ∂ ix δ ( x, x ′ ) . (26)The previous quotes in “group” and “algebra” are a reminder that strictly,the set of deformations do not form a group. The composition of two successivedeformations is itself a deformation, of course, but it also depends on the surfaceto which the first deformation is applied. Thus, the “structure constants” onthe right hand sides of (24-26) are not constant, they depend on the surface σ through its metric g ij which appears explicitly on the right hand side of (24). To obtain a fully covariant dynamics we require that the evolution of any dy-namical quantity such as ρ σ [ χ ] from an initial σ i to a final σ f be consistent withthe kinematics of surface deformations. Thus, the requirement of embeddabilitytranslates into a consistency requirement of path independence : if there are dif-ferent paths to evolve from an initial instant into a final instant, then all thesepaths must lead to the same final values for all quantities.In ED the relevant physical information — supplied through the prior (4) andthe constraint (5) — have led us to a diffusive dynamics in which the probability ρ σ [ χ ] evolves under the action of the externally prescribed drift potential φ [ χ ].It is a curious diffusion in a curved background space-time, but it is a diffusionnonetheless. This, however, is not what we seek: it is not a quantum dynamics.A quantum dynamics requires a different choice of constraints. Specifically,in the ED developed in the previous sections there is one basic dynamical vari-able, the distribution ρ σ [ χ ]. The drift potential φ [ χ ], being externally pre-scribed, is not a dynamical variable. In contrast, in a quantum dynamics thereare two dynamical variables, the magnitude and the phase of the wave func-tion. An additional degree of freedom must be introduced into ED. Perhapsthe simplest way is to replace the fixed prescribed potential φ [ χ ] in constraint(5) by an evolving drift potential φ σ [ χ ] that is updated at every step in localtime in response to the evolving ρ σ [ χ ]. In such an ED every infinitesimal step inlocal time involves two updates: one is the entropic update of ρ σ [ χ ], the otheris updating the constraint represented by φ σ [ χ ]. The resulting ED describes thecoupled evolution of ρ σ [ χ ] and φ σ [ χ ] . φ σ [ χ ] be updated? TheLTFP equations, particularly when written in the form (19), suggest that therules for updating φ σ are more conveniently expressed in terms of the trans-formed variable Φ σ . In previous work on ED [30][37][39] the basic criterioninvolved imposing the conservation of a global energy functional. This non-dissipative diffusion led to a fully Hamiltonian formalism with ρ σ [ χ ] and Φ σ [ χ ]as conjugate variables: one Hamilton equation describes the entropic evolutionof ρ σ [ χ ], while the conjugate Hamilton equation describes the evolving con-straints through Φ σ [ χ ]. In a covariant ED involving local surface deformationsthis is not satisfactory because the notion of a global energy is not available.Here we propose instead that the update of Φ σ must reflect the kinematics ofsurface deformations. We require path independence; the update must be inde-pendent of the selected foliation. Next we tackle the problem of implementingthis proposal.We saw that in an inference-based framework such as ED the concept oftime is designed so that each instant — which includes a specification of boththe surface σ and the statistical state ρ σ [ χ ] and Φ σ [ χ ] — contains the relevantinformation to construct the next instant. This means that by design in EDtime is constructed so that “given the present, the future is independent of thepast.” Thus in ED the equations of motion will necessarily be first order in time.We have also seen that in the limit of flat space limit — whether relativistic [39],or not [37][32] — the non-dissipative ED is Hamiltonian. Therefore it is naturalto adopt a Hamiltonian formalism in which Φ σ is the momentum canonicallyconjugate to ρ σ .The assumption of an underlying symplectic structure is a strong one thatdemands justification. In the context of non-relativistic quantum mechanics thesymplectic and complex structures characteristic of quantum mechanics can bemotivated using arguments from information geometry [32].Once a Hamiltonian framework is adopted, we can follow Dirac and treatthe surface variables as if they were dynamical variables too. This allows aHamiltonian formalism that treats dynamical and kinematical variables in aunified way. To do this one formally introduces auxiliary variables π µ ( x ) = π µx that will play the role of momenta conjugate to X µx . These π ’s are definedthrough the Poisson bracket (PB) relations,[ X µx , X νx ′ ] = 0 , [ π µx , π νx ′ ] = 0 , [ X µx , π νx ′ ] = δ µν δ ( x, x ′ ) . (27)The canonical pairs ( X µx , π µx ) represent the geometry of the surfaces and howthey change along the foliation.The change of a generic functional T [ X, π, ρ,
Φ] resulting from an arbitrarydeformation δξ Ax = ( δξ ⊥ x , δξ ix ) is expressed in terms of PBs, δT = Z dx δξ µx [ T, H µx ] = Z dx (cid:0) δξ ⊥ x [ T, H ⊥ x ] + δξ ix [ T, H ix ] (cid:1) , (28)where H ⊥ x [ X, π, ρ,
Φ] and H ix [ X, π, ρ,
Φ] are the generators of normal and tan-gential deformations respectively, and the generic PB of two arbitrary function-12ls U and V is[ U, V ] = Z dx (cid:18) δUδX µx δVδπ µx − δUδπ µx δVδX µx (cid:19) + Z Dχ ˜ δU ˜ δρ ˜ δV ˜ δ Φ − ˜ δU ˜ δ Φ ˜ δV ˜ δρ ! . (29)Thus, the PBs perform a double duty: on one hand they reflect the kinematicsof deformations of surfaces embedded in a background space-time, and on theother hand they express the genuine entropic dynamics of ρ and Φ.To comply with the requirement of path independence we follow Teitelboimand Kucha˘r [43]-[45] to seek generators H ⊥ x and H ix that provide a canonicalrepresentation of the DHKT “algebra” of surface deformations. Unlike DHKTwho developed a classical formalism based on choosing the field χ ( x ) and itsmomentum as canonical variables, here we develop a quantum formalism. Wechoose the functionals ρ [ χ ] and Φ[ χ ] as the pair of canonical variables.The idea then is that in order for the dynamics to be consistent with thekinematics of surface deformations the PBs of H ⊥ x and H ix must close in thesame way as the “group” of deformations (24-26) — that is, they must providea “representation” involving the same “structure constants”, [ H ⊥ x , H ⊥ x ′ ] = ( g ijx H jx + g ijx ′ H jx ′ ) ∂ ix δ ( x, x ′ ) , (30)[ H ix , H ⊥ x ′ ] = H ⊥ x ∂ ix δ ( x, x ′ ) , (31)[ H ix , H jx ′ ] = H ix ′ ∂ jx δ ( x, x ′ ) + H jx ∂ ix δ ( x, x ′ ) . (32)It may be worth noting that these equations have not been derived; it is moreappropriate to say that imposing (30-32) as strong constraints constitutes our definition of what we mean by a “representation”. To complete the definition, weadd that, as shown in [43][44], the requirement that the evolution of an arbitraryfunctional T [ X, π, ρ,
Φ] satisfy path independence implies that the initial valuesof the canonical variables must be restricted to obey the weak constraints H ⊥ x ≈ H ix ≈ . (33)Furthermore, once satisfied on an initial surface σ the dynamics will be such asto preserve (33) for all subsequent surfaces of the foliation. Next we seek explicit expressions for H ⊥ x and H ix . A surface deformation isdescribed by (11), δX µx = δξ ⊥ x n µx + δξ ix X µix . (34)On the other hand, we can evaluate δX µx using (29), δX µx = Z dx ′ (cid:0) [ X µx , H ⊥ x ′ ] δξ ⊥ x ′ + [ X µx , H ix ′ ] δξ ix ′ (cid:1) . (35) The difference in sign in the Poisson brackets (30-32) relative to the Lie brackets (24-26)arises from the change δT in (28) being written in terms of [ T, H µx ] rather than [ H µx , T ]. X µx , H ⊥ x ′ ] = δH ⊥ x ′ δπ µx and [ X µx , H ix ′ ] = δH ix ′ δπ µx , (36)comparing (34) and (35) leads to δH ⊥ x ′ δπ µx = n µx δ ( x, x ′ ) and δH ix ′ δπ µx = X µix δ ( x, x ′ ) . (37)These equations can be integrated to give, H ⊥ x = π ⊥ x + ˜ H ⊥ and H ix = π ix + ˜ H ix , (38)where π ⊥ x = n µx π µx and π ix = X µix π µx , (39)and ˜ H ⊥ and ˜ H i are constants of integration that are independent of the surfacemomenta π µ but can in principle depend on the other canonical variables, X , ρ , and Φ.Thus, the generators H ⊥ x and H ix separate into two components: one pair, π ⊥ x and π ix , that acts only on the geometry and another pair, ˜ H ⊥ x and ˜ H ix ,that acts both on the matter variables and the geometry. The latter, ˜ H ⊥ x and ˜ H ix , are called the ensemble Hamiltonian and the ensemble momentum.In what follows these will be abbreviated to e-Hamiltonian and e-momentumrespectively.It is a lengthy but straightforward algebraic exercise to check that π ⊥ x and π ix satisfy the DHKT “algebra”, eqs.(30-32),[ π ⊥ x , π ⊥ x ′ ] = ( g ijx π jx + g ijx ′ π jx ′ ) ∂ ix δ ( x, x ′ ) , (40)[ π ix , π ⊥ x ′ ] = π ⊥ x ∂ ix δ ( x, x ′ ) , (41)[ π ix , π jx ′ ] = π ix ′ ∂ jx δ ( x, x ′ ) + π jx ∂ ix δ ( x, x ′ ) . (42) The generators of tangential deformations are the simpler ones: they inducetranslations of the dynamical variables parallel to the surface. The change in ρ and Φ (and functionals thereof) under a tangential deformation δξ ix is δρδξ ix = δρδχ x ( ∂ ix χ x ) and δ Φ δξ ix = δ Φ δχ x ( ∂ ix χ x ) . (43)This change is generated by the e-momentum ˜ H ix according to We call ‘matter’ any quantity that is not ‘geometry’. It is an abuse of language to referto the epistemic quantities ρ and Φ as ‘matter’ but it is nevertheless convenient to do so. We are comparing new χ ′ and old χ fields located at different points with the samecoordinate x . Under a tangential deformation δξ a the new field is χ ′ ( x ) = χ ( x + δξ ) so that δχ x = χ ( x + δχ ) − χ ( x ) = ( ∂ ix χ x ) δξ ix . ρδξ ix = [ ρ, ˜ H ix ] = ˜ δ ˜ H ix ˜ δ Φ , (44) δ Φ δξ ix = [Φ , ˜ H ix ] = − ˜ δ ˜ H ix ˜ δρ . (45)One can easily check that the required e-momentum is˜ H ix [ ρ, Φ] = − Z Dχ ρ [ χ ] δ Φ[ χ ] δχ x ∂ ix χ x , (46)which shows that ˜ H ix [ ρ, Φ; χ ] has an explicit dependence on χ x but is indepen-dent of the surface variables X µ . It is also straightforward to check that ˜ H ix satisfies the condition (32),[ ˜ H ix , ˜ H jx ′ ] = ˜ H ix ′ ∂ jx δ ( x, x ′ ) + ˜ H jx ∂ ix δ ( x, x ′ ) , (47)so that the tangential generators π ix and ˜ H ix satisfy (32) separately. The mixed PB relations, eq.(31), are the easiest to satisfy and therefore theleast informative. Using (41) and [ ˜ H ix , π ⊥ x ′ ] = 0 we have[ π ix + ˜ H ix , ˜ H ⊥ x ′ ] = ˜ H ⊥ x ∂ ix δ ( x, x ′ ) , (48)which tells us that ˜ H ⊥ x is a scalar density. In contrast, the normal PB relations,eq.(30) which we use (40) to re-write as[ ˜ H ⊥ x , ˜ H ⊥ x ′ ] + [ π ⊥ x , ˜ H ⊥ x ′ ] + [ ˜ H ⊥ x , π ⊥ x ′ ] = ( g ijx ˜ H jx + g ijx ′ ˜ H jx ′ ) ∂ ix δ ( x, x ′ ) , (49)are crucial. They provide the criteria for updating the constraints that definethe entropic dynamics. Thus, our goal is to find a functional ˜ H ⊥ x [ X, ρ, Φ; χ ]that generates a path-independent entropic dynamics — that is, it reproducesthe local time Fokker-Planck equations (16), while remaining consistent withthe algebra of deformations.The desired ˜ H ⊥ x is given by (20), and the relation (53) will serve to de-termine the so-far unknown e-functional F x [ X, ρ ; χ ]. Finding the most generalsolution of (49) lies beyond the scope of this paper; what we will do is to identifya sufficiently large class of solutions that proves to be of physical interest.We will restrict our search to situations where ˜ H ⊥ x (and F x ) depend on thegeometric variables X µx only through the metric g ijx and not through any of itsderivatives. Then, using δg ijx ′ δξ ⊥ x = n µx δg ijx ′ δX µx = 2 K ijx δ ( x ′ , x ) (50)15here K ijx is the extrinsic curvature, we find that[ π ⊥ x , ˜ H ⊥ x ′ ] = 2 K ijx ∂ ˜ H ⊥ x ∂g ijx δ ( x ′ , x ) (51)is symmetric in ( x ′ , x ). Therefore[ π ⊥ x , ˜ H ⊥ x ′ ] + [ ˜ H ⊥ x , π ⊥ x ′ ] = 0 , (52)and (49) simplifies to[ ˜ H ⊥ x , ˜ H ⊥ x ′ ] = ( g ijx ˜ H jx + g ijx ′ ˜ H jx ′ ) ∂ ix δ ( x, x ′ ) . (53)Thus, the normal generators π ⊥ x and ˜ H ⊥ x satisfy (30) separately [44].Next we turn out attention to finding a class of physically interesting e-functionals F x [ X, ρ ; χ ]. We proceed in steps. First we re-write (20) in the form,˜ H ⊥ x [ X, ρ, Φ; χ ] = ˜ H ⊥ x [ X, ρ, Φ; χ ] + F x [ X, ρ ; χ ] (54)where ˜ H ⊥ x = Z Dχ ( ρg / x (cid:18) δ Φ δχ x (cid:19) + ρ g / x g ij ∂ i χ x ∂ j χ x ) . (55)This amounts to a mere definition of a new F x in terms of the old F x so thereis no loss of generality. The reason that adding the second term in (55) turnsout to be convenient is that the new ˜ H ⊥ x satisfies (53),[ ˜ H ⊥ x , ˜ H ⊥ x ′ ] = ( g ijx ˜ H jx + g ijx ′ ˜ H jx ′ ) ∂ ix δ ( x, x ′ ) . (56)Then, substituting (54) into (53), and noting that[ F x , F x ′ ] = 0 , (57)leads to [ ˜ H ⊥ x , F x ′ ] = [ ˜ H ⊥ x ′ , F x ] , (58)which is a homogeneous and linear equation for F x . Thus, the condition for afunctional F x to be acceptable is that the PB [ ˜ H ⊥ x , F x ′ ] be symmetric underthe exchange of x and x ′ .The next step is to calculate the PB on the left,[ ˜ H ⊥ x , F x ′ ] = − Z Dχ ˜ δ ˜ H ⊥ x ˜ δ Φ ˜ δF x ′ ˜ δρ , (59)and note that ˜ δ ˜ H ⊥ x / ˜ δ Φ reproduces the LTFP eq.(16) and (19),˜ δ ˜ H ⊥ x ˜ δ Φ = − g − / x δδχ x (cid:18) ρ δ Φ δχ x (cid:19) . (60)16e restrict our search further by considering e-functionals F x [ X, ρ ; χ ] of theform, F x [ X, ρ ; χ ] = Z Dχ f x (cid:18) X x , ρ, δρδχ x ; χ x , ∂χ x (cid:19) , (61)where f x is a function (not a functional) of its arguments. For such a special F x we have ˜ δF x ′ ˜ δρ = ∂f x ′ ∂ρ − δδχ x ′ ∂f x ′ ∂ ( δρ/δχ x ′ ) . (62)Substituting (60) and (62) into (59) gives[ ˜ H ⊥ x , F x ′ ] = Z Dχ g / x δδχ x (cid:18) ρ δ Φ δχ x (cid:19) (cid:18) ∂f x ′ ∂ρ − δδχ x ′ ∂f x ′ ∂ ( δρ/δχ x ′ ) (cid:19) . (63)Any F x , whether of type (61) or not, must be a scalar density which means that f x must be a scalar density too. Since the available scalar densities are g / x and δ/δχ x , some natural proposals are f x ∼ g / x ρχ nx (integer n ) and f x ∼ g / x ρ (cid:18) δρδχ x (cid:19) . (64)A straightforward substitution into (63) shows that all these trials satisfy (58)— indeed, the PB [ ˜ H ⊥ x , F x ′ ] is symmetric in ( x, x ′ ). Finally, since (58) is lin-ear we can also consider linear combinations of these trial forms which leads toa generic potential describing self-interactions and interactions with the back-ground geometry, V ( χ x , X x ) = X nℓ λ n χ nx , (65)where λ n are coupling constants. We have therefore shown that the family ofHamiltonians˜ H ⊥ x = Z Dχρ ( g / x (cid:18) δ Φ δχ x (cid:19) + g / x g ij ∂ i χ x ∂ j χ x + g / x V ( χ x ) + λg / x (cid:18) δ log ρδχ x (cid:19) ) , (66)generates a path-independent entropic dynamics. To interpret the last term in(66) we recall that in flat space-time the quantum potential is given by [39] Q = Z Dχ Z d x λρ (cid:18) δρδχ x (cid:19) . (67) A more systematic study, carried out in forthcoming work, shows that trials of the form f x ∼ g / x g / x δρδχ x ! k ρ l χ mx ( ∂χ x ) n , where k , ℓ , m , and n are integers, are ruled out except for (64). d x → g / x d x and δδχ x → g − / x δδχ x (68)which gives Q σ = Z Dχ Z d x λg / x ρ (cid:18) δρδχ x (cid:19) . (69)Therefore the last term in (66) may be called the “local quantum potential.”Its contribution to the energy is such that those states that are more smoothlyspread out in configuration space tend to have lower energy. The correspond-ing coupling constant λ > λ < We will now summarize the main results of the previous sections by writingdown the equations that describe how the probability distribution ρ [ χ ] evolvesin a curved space-time. Given a space-time with metric g µν ( X ) we start by specifying a foliation ofsurfaces σ t labeled by a time parameter t , X µ = X µ ( x, t ), where x i are coordi-nates on the surface. The metric induced on the surface is given by (10). Thedeformation of σ t to σ t + dt is given by (11), δξ µ = δξ ⊥ n µ + δξ i X µi = [ N xt n µ + N ixt X µi ] dt , (70)where we introduced the scalar lapse , N xt = δξ ⊥ x /dt , and the vector shift , N ixt = δξ ix /dt .The goal is to determine the evolution of the probability distribution ρ t [ χ ]with time t . This requires finding the evolution of the phase functional, Φ t [ χ ].The evolution of ρ t and Φ t is given by (28), ∂ρ t ∂t = [ ρ t , H ] and ∂ Φ t ∂t = [Φ t , H ] , (71)where H is the smeared Hamiltonian, H [ N, N i ] = Z dx (cid:0) N xt H ⊥ x + N ixt H ix (cid:1) . (72)The result is ∂ρ t ∂t = Z d x (cid:18) δρ t δξ ⊥ x N xt + δρ t δξ ix N ixt (cid:19) (73)18nd ∂ Φ t ∂t = Z d x (cid:18) δ Φ t δξ ⊥ x N xt + δ Φ t δξ ix N ixt (cid:19) . (74)The tangential derivatives, δρ t /δξ ix and δ Φ t /δξ ix , are given by eqs.(43-45). Thenormal derivatives, δρ t /δξ ⊥ x and δ Φ t /δξ ⊥ x , are given by δρ t δξ ⊥ x = [ ρ t , ˜ H ⊥ x ] = ˜ δ ˜ H ⊥ x ˜ δ Φ t , (75) δ Φ t δξ ⊥ x = [Φ t , ˜ H ⊥ x ] = − ˜ δ ˜ H ⊥ x ˜ δρ t . (76)Substituting ˜ H ⊥ x from (66) gives the local-time Fokker-Planck equations (16), δρ t δξ ⊥ x = − g / x δδχ x (cid:18) ρ t δ Φ t δχ x (cid:19) , (77)and the local time generalization of the Hamilton-Jacobi equations, − δ Φ t δξ ⊥ x = 12 g / x (cid:18) δ Φ t δχ x (cid:19) + g / x g ij ∂ i χ x ∂ j χ x + g / x V ( χ x , X x ) − λg / x ρ / δ ρ / t δχ x . (78)The formulation of the ED of fields in curved space-time is thus completed.However, it may not yet be obvious that this is a quantum theory; to make itexplicit is the next task. The relation of the ED formalism to quantum theory can made explicit bymaking a canonical transformation (often called a
Madelung transformation)from the dynamical variables ρ and Φ into a pair of complex variables, Ψ[ χ ] = ρ / e i Φ . (79)The equation of evolution for the new variable Ψ t [ χ ] is then given by ∂ Ψ t ∂t = Z d x (cid:18) δ Ψ t δξ ⊥ x N xt + δ Ψ t δξ ix N ixt (cid:19) . (80)The tangential derivative, δ Ψ t /δξ ix , is obtained from eq.(43), δ Ψ t δξ ix = ( ∂ ix χ x ) δ Ψ t δχ x . (81)Using (66) the normal derivative, δ Ψ t δξ ⊥ x = [Ψ t , ˜ H ⊥ x ] , (82) For non-relativistic particles the underlying symplectic and complex structures and theirrelation to information geometry are derived in [32]. t [ χ ], i ~ δ Ψ t δξ ⊥ x = − ~ g / x δ Ψ t δχ x + g / x g ij ∂ i χ x ∂ j χ x Ψ t + g / x V Ψ t , (83)where we have introduced ~ by setting λ = ~ / t → Φ t / ~ , sothat Φ t has units of action. The limit of flat space-time is obtained setting g / x = 1, δξ ⊥ x = dt , N =1, and N i = 0. Then eqs.(80) and (83) become the Schr¨odinger functionalequation, i ~ ∂ Ψ t ∂t = Z d x (cid:26) − ~ δ Ψ t δχ x + 12 g ij ∂ i χ x ∂ j χ x Ψ t + V Ψ t (cid:27) . (84)This is quantum field theory in the Schr¨odinger functional representation [64].From here one can proceed to introduce a Hilbert space, operators, and thestandard machinery of quantum field theory. Eq.(84) justifies identifying theexpression (8 λ ) / with Planck’s constant ~ . We thus see that the coupling λ = ~ / ~ andsets the scale that separates quantum from classical regimes. As laid out above, the ED that we have developed here is formally identical tothe standard quantum field theory in the Schr¨odinger functional representation.This means that predictions made on the basis of equations (80)-(83) are iden-tical to those obtained using the standard methods. And indeed, there havebeen a wide range of topics pursued within this formalism; including, studiesof vacuum states in curved space-time [62], research on the Hawking effect [65],applications in cosmology [66], and investigations into symmetries [67][68], justto name a few. In other words: the ED developed here, while being fully consis-tent with all of these developments, it does not go beyond them. And, indeed,the purpose of the ED developed here is not to generate better techniques forcalculation, but to put QFTCS on a firm conceptual foundation.The ED formulation of QFTCS, nonetheless, is sufficiently different from theusual approaches that a demonstration of the framework is, in fact, warranted.We do this with two examples. One example illustrates the formal flexibilityof the ED formalism, while the other demonstrates the conceptual clarity thatan entropic framework supplies to QFTCS. Both insights may have importantimplications as ED moves beyond QFTCS and begins to incorporate dynamicalgravity. Note also that this modification means that we have adjusted the units of χ so that[ χ ] = [ ~ ] / / length. .1 The Ehrenfest equations in Entropic Dynamics We begin by obtaining the Ehrenfest equations for a quantum scalar field inED. The derivation highlights many of the novel features of the ED approach;in particular, the utilization of Hamiltonians, Poisson brackets, and so on, ratherthan the conventional quantum tools.
Some background
In ED, the focus of our inquiries are the field variables χ x which have definite, but unknown values. Thus, in the absence of such definiteinformation, our goal is to obtain an estimate of these values. One such estimateis provided by the expected value of the field variables˜ χ x = Z Dχ ρ χ x . (85)This, in turn, defines a functional ˜ χ x on the ensemble phase space, which makesit amenable to treatment through the canonical formalism.However, this expected value is not static and we wish to know how itchanges in time. To determine this evolution, we carry over the formalism of theprevious section and introduce a Hamiltonian H [ N, N i ], as in eq.(72), adaptedto a particular foliation with lapse N , shift N i , and parameter t . Moreover, weare concerned with a quantum dynamics so we choose H ⊥ x and H ix in eq.(72)to match that of eqns.(46) and (66). Evolution of the expected values
Since ˜ χ x = ˜ χ x [ ρ ] is a Hamiltonian func-tional, its update can be obtained by taking the appropriate Poisson bracketsfor a suitably chosen Hamiltonian H . Indeed, the velocity of ˜ χ x is given by ∂ t ˜ χ x = Z dx ′ (cid:0) N x ′ t { ˜ χ x , H ⊥ x ′ } + N ix ′ t { ˜ χ x , H ix ′ } (cid:1) (86)with { ˜ χ x , H ⊥ x ′ } = δ ( x, x ′ ) Z Dχ ρ g x ′ δ Φ δχ x ′ (87)and { ˜ χ x , H ix ′ } = − δ ( x, x ′ ) Z Dχ ρ ∂ ix ′ χ x ′ . (88)Taken together, eqns.(86)-(88) result in ∂ t ˜ χ x = Z Dχ ρ Ng / x δ Φ δχ x − N i ∂ ix χ x ! , (89)which contains two contributions. The latter contribution in eq.(89) is justdue to the shift, while the former is due to the flow of probability, which ischaracterized by the appearance of the current velocity v x = 1 g / x δ Φ δχ x , current momentum and its expectation,the ensemble current momentum, P x ≡ δ Φ δχ x = g / x v x and ˜ P x = Z Dχ ρ P x , (90)respectively. We can now conveniently rewrite the velocity ∂ t ˜ χ x in terms ofthe current momentum, yielding ∂ t ˜ χ x = Ng / x ˜ P x − N i h ∂ ix χ x i , (91)where we have used the notation h A i = R ρA to denote expectation.The advantage of introducing the current momentum (as opposed to thecurrent velocity) is that the expected current momentum ˜ P x , together with theexpected field value ˜ χ x , satisfy the canonical Poisson bracket relations n ˜ χ x , ˜ P x ′ o = δ ( x, x ′ ) . (92)This seems to suggest that ˜ P x plays the role of a momentum conjugate to ˜ χ x .Indeed, let us take this hint seriously and compute the corresponding Hamil-ton’s equations for this canonical pair. The time derivative of ˜ χ x was providedearlier, in eq.(91). The velocity of ˜ P x , on the other hand, is given by ∂ t ˜ P x = Z dx ′ (cid:16) N x ′ t n ˜ P x , H ⊥ x ′ o + N ix ′ t n ˜ P x , H ix ′ o(cid:17) . (93)To compute this we need the two Poisson brackets in eq.(93). A quick calculationgives n ˜ P x , H ⊥ x ′ o = − Z Dχ ρ g / x ′ (cid:18) δ ( x, x ′ ) ∂V x ′ ∂χ x ′ + g ijx ′ ∂ ix ′ χ x ′ ∂ jx ′ δ ( x ′ , x ) (cid:19) (94)and n ˜ P x , H ix ′ o = Z Dχ ρ δ Φ δχ x ′ ∂ ix ′ δ ( x ′ , x ) (95)so that from eq.(93) we obtain ∂ t ˜ P x = ∂ i (cid:16) N g / x g ij ∂ j ˜ χ x (cid:17) − ∂ i (cid:16) N i ˜ P x (cid:17) − N g / x (cid:28) ∂V∂χ x (cid:29) . (96)Thus an initial assignment of ˜ χ x , ˜ P x , and higher statistical moments of χ x , willbe sufficient to determine the evolution of ˜ χ x , i.e. no further derivatives arerequired. Note that while the current velocity is a scalar valued quantity, the current momentum is a scalar density of weight one. Alternatively, introduce the differential operator ˆ P x = − iδ/δχ x . Then the ensemble cur-rent momentum translates in the conventional language to the expected value of this operator.That is, ˜ P x = R Ψ ∗ ˆ P x Ψ, using the complex functionals (Ψ ∗ , Ψ) introduced above. hrenfest equations The equations (89) and (96) taken together have thecharacter of classical field equations for some “classical” field variables ˜ χ x and˜ P x . In fact, it is not difficult to show that these are nothing but the Ehrenfestequations (see e.g.,[69]).To see this, note that the velocity ∂ t ˜ χ x is linear in the current momentum˜ P x . Now, invert this relation for ˜ P x in terms of the velocity ∂ t ˜ χ x and substituteinto eq.(96). The result is that ˜ χ x evolves according to the equationˆ (cid:3) ˜ χ x = (cid:28) ∂V ( χ x ) ∂χ x (cid:29) , (97)whereˆ (cid:3) = − N g / ¯ ∂ t (cid:20)(cid:18) g / N (cid:19) ¯ ∂ t (cid:21) − ∂ i N i N ¯ ∂ t + 1 N g / ∂ i h N g / g ij ∂ j i (98)is the wave operator in curved space-time in foliation-adapted coordinates [62][63],and where we have introduced the operator ¯ ∂ t = ∂ t + N i ∂ i .Equation (97) comprises the Ehrenfest relations that we seek. An appealingfeature of such relations is that they are exact , and thus contain complete in-formation about the underlying quantum dynamics. This makes the Ehrenfestrelations ideal for probing the behavior of quantum fields and their deviationsfrom classical behavior. For instance, it is not difficult to see that ˜ χ x follows aclassical evolution only when (cid:28) ∂V ( χ x ) ∂χ x (cid:29) = ∂V ( h χ x i ) ∂χ x . (99)But this, of course, only occurs when the potential is itself quadratic in the field.Indeed, choose for V x the potential V x = m χ x so that ∂V /∂χ x = m χ x .Equation (97) then reduces to (cid:16) ˆ (cid:3) − m (cid:17) ˜ χ x = 0 , (100)which is a classical Klein-Gordon equation in curved space-time (see e.g., [3])for the expected field configuration. Thus ˜ χ x follows — exactly — the classicalequations of motion, which is precisely the content of Ehrenfest’s theorem, fa-miliar from non-relativistic quantum mechanics (see e.g., [69]). For potentialsthat are not quadratic, however, we can expect deviations from classical be-havior and it is legitimate to obtain quantum corrections via an approximationscheme. Comments
As opposed to standard formulations of equations of this type,our derivation of the Ehrenfest relations was performed entirely within theframework of ED, using Hamiltonians, Poisson brackets, etc., rather than com-mutators and the standard quantum machinery. These methods, which are Our approach to the Ehrenfest, or Ehrenfest-Heisenberg equations, follows closely that ofAshtekar and Schilling [70]. linearity of quantum theory. Indeed, whilethe assumption of linearity has thus far proved quite robust, it is not immedi-ately obvious that quantum gravity needs to follow suit (see e.g., [71]-[75]). Insuch cases the ED approach might provide a viable alternative framework.
Some of the principal benefits of the ED approach are conceptual. For instance,a central difficulty of any quantum field theory, one that is also shared by the EDformalism, is the problem of infinities. The nature of the infinities is, however,very different [39]. To see this consider the limit of flat space-time, i.e. N = 1, N i = 0, g / = 1. Setting V = m χ / e.g. , [62]). For example, in units suchthat ~ = c = 1, the wave functional of the ground state isΨ [ χ ] = 1 Z / e − iE t exp (cid:20) − Z d x Z d y χ x G xy χ y (cid:21) , (101)where G xy = Z d k (2 π ) ω k e i~k · ( ~x − ~y ) , with ω k = ( ~k + m ) / , (102)and the energy of the ground state, E = Z d x ˜ H ⊥ x [Ψ ] , (103)is obtained from (66) and (101),˜ H ⊥ x [Ψ ] = 12 Z Dχρ "(cid:18) δ Φ δχ x (cid:19) + g ij ∂ i χ x ∂ j χ x + m χ x + (cid:18) ρ δρ δχ x (cid:19) . (104)The result, E = 12 Z d x G xx = Z d x Z d k (2 π ) ω k , (105)is both infrared and ultraviolet divergent. Similarly, the expected value of thefield at any point ~x vanishes but its variance diverges, h χ x i = 0 and Var [ χ x ] = h χ x i = Z d k (2 π ) ω k . (106)The point we want to stress in repeating these well-known results is that inthe ED framework the divergent quantities in (105) and (106) are not physical,ontic quantities. Both the e-Hamiltonian ˜ H ⊥ x [Ψ ] in (104) and the variance in(106) are expected values. The infinities are not real; they are epistemic.
24D recognizes the role of incomplete information: the interpretation of thediverging Var[ χ x ] is not that the field χ x undergoes fluctuations of infinite mag-nitude, but rather that with the information that is available to us we arecompletely unable to predict the value of the field χ x at the sharply localizedpoint ~x . The infinities are epistemic: what diverges are not physical quantitiesbut our uncertainty about them. However, this does not mean the theory isuseless. It may be incapable of predicting some quantities but it can provideuseful predictions for many others. For example, the equal time correlationsbetween two field variables at different locations are perfectly finite, h χ x χ y i = Z d k (2 π ) e i~k · ( ~x − ~y ) ω k = m π | ~x − ~y | K ( m | ~x − ~y | ) , (107)where K is a modified Bessel function [62]. Entropic dynamics provides an inferential alternative to the standard methodsof quantization. The ED approach avoids a representation of fields as operatorsand any reference to the Hilbert or Fock spaces on which they presumably act.In effect this eliminates the issue of choosing among many inequivalent represen-tations. Consequently, the operator ordering ambiguities that are characteristicof conventional quantization methods are avoided too. Indeed, many of theproblems associated with the Dirac quantization method [42] and the laborioustechniques necessary to implement it (such as the identification and eliminationof second-class constraints etc.) are completely sidestepped.The ED approach to QFTCS, however, also offers new insights to problemssuch as the Unruh effect at the interface between QFTCS and the quantummeasurement problem. Indeed, the fact that in the ED approach fields arephysical entities which at all times have definite but possibly unknown values,while “particles are whatever particle detectors detect,” immediately raises thequestion of what is a particle within the ED framework.Moreover, the ED approach to QFTCS leads to a theory that is Hamiltonianin character, thus retaining the powerful tools and intuitive appeal of the classi-cal Hamiltonian framework but now in the context of a fully quantum theory.This is particularly attractive for a couple reasons. First is that the ED approachproceeds without ever invoking the use of linearity or of Hilbert spaces, thusleaving open the possibility that deeper theories will introduce non-linearities— something the standard approaches cannot readily account for. Another isthat the scheme introduced here allows us to borrow the methods of DHKT Being an inference theory, ED is particularly well suited to tackling the quantum problemof measurement [51][52]. That quantum theory can be formulated as a Hamiltonian theory has been exploredby many [26][27][70][76][77]. The connection between the ED framework and informationgeometry with the symplectic/Hamiltonian structure is explored in [32]. classical covariant Hamiltonian theories, butinstead apply them to a theory that is inherently statistical and quantum.This is not insignificant. A primary difficulty in formulating a theory ofquantum gravity is that general relativity and quantum field theory are couchedin completely different formal languages. The ED that we have developed here,however, actually helps to bridge this divide. Indeed, while a common approachto quantum gravity is to incorporate gravitation into the linear, algebraic frame-work of quantum theory, ED opens the door to an alternative approach whereinquantum theory more resembles general relativity. The key is to recognize thecentral role played by the Hamiltonian formalism, not only in the ED formu-lation of QFTCS here, but also in the geometrodynamics approach to generalrelativity. Therefore, in contrast to many other approaches to QFTCS, EDseems to offer the possibility of extending the scheme developed here towardsderiving, from first principles , a fully dynamical theory of quantum fields inter-acting with classical gravity [78].
Acknowledgements
We would like to thank D. Bartolomeo, N. Carrara, N. Caticha, S. DiFranzo,K. Knuth, P. Pessoa, and K. Vanslette for valuable discussions on entropicdynamics.
A The local-time Fokker-Planck equations
To rewrite the dynamical equation (15) in differential form consider the proba-bility P [ χ, σ | χ , σ ] of a finite transition from a field configuration χ at someearly surface σ to a configuration χ at a later σ . The result of a further evo-lution from σ to a neighboring σ ′ obtained from σ by an infinitesimal normaldeformation δξ ⊥ x is given by (15), P [ χ ′ , σ ′ | χ , σ ] = Z Dχ P [ χ ′ , σ ′ | χ, σ ] P [ χ, σ | χ , σ ] . (A.1)To obtain a differential equation one cannot just Taylor expand as δξ ⊥ x → P [ χ ′ , σ ′ | χ, σ ] becomes a very singular object — a delta functional.Instead, we multiply by an arbitrary smooth test functional T [ χ ′ ] and integrate Z Dχ ′ P [ χ ′ , σ ′ | χ , σ ] T [ χ ′ ]= Z Dχ P [ χ, σ | χ , σ ] Z Dχ ′ T [ χ ′ ] P [ χ ′ , σ ′ | χ, σ ] . (A.2)Next expand the test function T [ χ ′ ] = T [ χ + ∆ χ ] in powers of ∆ χ = χ ′ − χ .Since χ is Brownian to obtain T [ χ ′ ] to first order in δξ ⊥ x we need to keep second26rder in ∆ χ x , T [ χ ′ ] = T [ χ ] + Z dx δT [ χ ] δχ x ∆ χ x + 12 Z dx dx ′ δ T [ χ ] δχ x δχ x ′ ∆ χ x ∆ χ x ′ + · · · . (A.3)Use this expansion together with (8) and (9) to obtain Z Dχ ′ T [ χ ′ ] P [ χ ′ , σ ′ | χ, σ ]= T [ χ ] + Z dx ηδξ ⊥ x g / x (cid:26) δT [ χ ] δχ x δφ [ χ ] δχ x + 12 δ T [ χ ] δχ x (cid:27) . Substituting back into eq.(A.2), leads to Z Dχ { P [ χ, σ ′ | χ , σ ] − P [ χ, σ | χ , σ ] } T [ χ ]= Z dx ηδξ ⊥ x g / x Z DχP [ χ, σ | χ , σ ] (cid:26) δT [ χ ] δχ x δφ [ χ ] δχ x + 12 δ T [ χ ] δχ x (cid:27) (A.4)Since T [ χ ] is arbitrary, after some integrations by parts we get P [ χ, σ ′ | χ , σ ] − P [ χ, σ | χ , σ ] = Z dx δP [ χ, σ | χ , t ] δξ ⊥ x δξ ⊥ x = Z dx ηδξ ⊥ x g / x (cid:26) − δδχ x (cid:18) P [ χ, σ | χ , σ ] δφ [ χ ] δχ x (cid:19) + 12 δ δχ x P [ χ, σ | χ , σ ] (cid:27) . (A.5)Finally, for a finite evolution from σ to σ , (15) reads, ρ σ [ χ ] = Z Dχ P [ χ, σ | χ , σ ] ρ σ [ χ ] . (A.6)A further infinitesimal normal deformation σ → σ ′ by δξ ⊥ x gives ρ σ ′ [ χ ] − ρ σ [ χ ] = Z dx δρ σ [ χ ] δξ ⊥ x δξ ⊥ x = Z Dχ (cid:18)Z dx δP [ χ, σ | χ , t ] δξ ⊥ x δξ ⊥ x (cid:19) ρ σ [ χ ] (A.7)which, using (A.5) and the fact that σ ′ (or δξ ⊥ x ) can be freely chosen leads tothe local Fokker-Planck equations, δρ σ [ χ ] δξ ⊥ x = ηg / x (cid:26) − δδχ x (cid:18) ρ σ [ χ ] δφ [ χ ] δχ x (cid:19) + 12 δ δχ x ρ σ [ χ ] (cid:27) , (A.8)which is equation (16). 27 eferences [1] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (Cam-bridge U.P., Cambridge 1984).[2] R. M. Wald,
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