The Entropic Dynamics of Quantum Scalar Fields Coupled to Gravity

We adopt a prior $Q\left[{\chi }^{\prime }|\chi \right]$ that incorporates the information that the fields change by infinitesimally small amounts, but is otherwise maximally uninformative. In particular, before the constraints are taken into account, knowledge of the dynamics at one point does not convey information about the dynamics at another point; i.e., the degrees of freedom are a priori uncorrelated.

Such a prior can itself be derived from the principle of maximum entropy. Indeed, maximize relative to the measure $\mu \left[{\chi }^{\prime }\right]$, which we assume to be uniform, and subject to appropriate constraints. (Since we deal with infinitesimally short steps, the prior Q turns out to be quite independent of the background measure $\mu$). The requirement that the field undergoes changes that are small and uncorrelated is implemented by imposing an infinite number of independent constraints, one per spatial point x, where $\Delta {\chi }_x={\chi }_x^{\prime }-{\chi }_x$ and the ${\kappa }_x$ are small quantities. The result of maximizing (2) subject to (3) and normalization is a product of Gaussians where ${\alpha }_x$ are the Lagrange multipliers associated to each constraint (3); the scalar density $g_x^{1/2}={\left(det{g}_{ij}\right)}^{1/2}$ is introduced so that ${\alpha }_x$ is a scalar field. For notational simplicity we write $d{x}^{\prime }$ instead of $d^3{x}^{\prime }$. To enforce the continuity of the motion we will eventually take the limit ${\kappa }_x\to 0$ which amounts to taking ${\alpha }_x\to \infty$.

The motion induced by the prior (4) leads to a rather simple diffusion process in the probabilities, in which the field variables evolve independently of each other. To model dynamics that exhibit correlations and are capable of demonstrating the full suite of quantum effects, such as the superposition of states, interference, and entanglement, however, we require additional structure. This is accomplished by imposing a single additional constraint that is non-local in space but local in configuration space, which involves the introduction of a drift potential $\varphi \left[\chi \right]$ which is a scalar-valued functional defined over the configuration space $\mathcal{C}$. More explicitly, we impose where we require ${\kappa }^{\prime }\to 0$. (Note that since ${\chi }_x$ and $\Delta {\chi }_x$ are scalars, in order that (5) be invariant under coordinate transformations of the surface, the derivative $\delta /\delta {\chi }_x$ must transform as a scalar density).

Before moving on, we discuss the central role of the drift potential in ED. As a set of dynamics of probabilities, ED brings together three ingredients. First, from an inference perspective, the dynamics consist of the updating of probabilities which must obey the rules of entropic inference. This requires information codified into constraints and the drift potential is the function that codifies such information. Second, in order to formulate a dynamical theory in which probabilities are generalized coordinates, it is only natural to attempt to identify the canonically conjugate generalized momenta and the corresponding symplectic structure. And third, from a purely geometric perspective, the trajectories we seek are curves on the space of probability distributions. For this statistical manifold it is natural to associate its tangent bundle. The fibers of such a bundle are spaces of velocity vectors tangent to all curves. It is also natural to consider the associated cotangent bundles of fibers as being the spaces of tangent covectors. As we shall see, ED brings together these three separate ingredients by identifying the constraint embodied in the drift potential functional with the momenta conjugate to the probabilities, with the cotangent vectors. More explicitly, we identify the three separate pairs: probability/constraint, coordinate/momentum, and point/covector. And as we shall see, eventually these three pairs are also identified with yet a fourth pair: the magnitude/phase of the wave function.

A straightforward example of this is shown for the metric $g_{ij}\left(x\right)=g_{ijx}$, which is a rank $(0,2)$ tensor. Its Lie derivative is given by [64]

Alternatively, by using the Poisson brackets we obtain

Comparing the two and using the fact that $\delta {\xi }_x^i$ is arbitrary yields

The delta functions imply that $H_{ix}$ is local in the momentum ${\pi }^{ij}$.

To fix the dependence on ${\pi }^{ij}$, in fact, we can use a similar argument as above. Recalling that ${\pi }^{ij}\left(x\right)={\pi }_x^{ij}$ is a rank $(2,0)$ tensor density of weight one, we find that

The same equation can be obtained through the use of a Hamiltonian, so long as $H_{ix}$ satisfies

Integrating these equations for $H_{ix}$ yields where is called the gravitational super-momentum, and the functional ${\tilde{H}}_{ix}={\tilde{H}}_{ix}[\rho ,\Phi ]$ is, at the moment, just an integration “constant.”

Some of these expressions can be simplified by introducing the covariant derivative ${\nabla }_i$. For example, using ${\nabla }_k{g}_{ij}=0$ we have and

The gravitational super-momentum $H_{ix}^G$ also takes the particularly simple form

Next, we turn to the response of the variables ${\rho }_{\sigma }\left[\chi \right]$ and ${\Phi }_{\sigma }\left[\chi \right]$ under a relabeling of the surface coordinates $x^i\to x^i+\delta {\xi }_x^i$. As the coordinates are shifted, so too are the fields defined upon that surface so that is the Lie derivative for a scalar ${\chi }_x$. This induces a change in the probability, given by

Alternatively, this same variation can be computed using the canonical framework

If we insert the super-momentum $H_{ix}$ given in Equation (40), we notice that only the ${\tilde{H}}_{ix}$ piece will contribute. And so we have which for arbitrary $\delta {\xi }_x^i$ requires that

A similar argument for ${\Phi }_{\sigma }\left[\chi \right]$ shows that we must also have so that

Thus the total super-momentum, Equation (40), contains two pieces, which we refer to as the gravitational and “matter” contributions, respectively. (The division of generators into gravitational and “matter” pieces established by DHKT is, strictly speaking, an abuse of language. The variables ${\rho }_{\sigma }$ and ${\Phi }_{\sigma }$ that constitute “matter” are more properly understood as describing the statistical state of the material field ${\chi }_x$. Nonetheless, we stick with the convention as a useful shorthand). Inspection of (51) confirms that there is no gravitational dependence in the “matter” side, or “matter” dependence on the gravitational side, that is,

By Equation (35) it is, of course, also understood that $H_{ix}$ is subject to the constraint

Finally, although $H_{ix}$ was obtained, in essence, independently of the Poisson bracket (34) which relates two tangential deformations, it nonetheless satisfies it automatically. It is therefore appropriate to view $H_{ix}$ as being completely determined; this is important as it allows us to treat Equation (32) as a set of equations for $H_{\perp x}$ in terms of the known $H_{ix}$.

A particularly appealing aspect of this separation into gravitational and “matter” pieces is that the Poisson bracket relations (32)–(34) also split along similar lines. In fact, insert the decomposed generators $H_{Ax}$ from Equations (52) and (55) into the Poisson bracket relations (32)–(34). We find that the gravitational generators $H_{Ax}^G=(H_{\perp x}^G,H_{ix}^G)$ must satisfy a set of Poisson brackets which have closing relations identical to those of (32)–(34).

The “matter” generators, which we collectively denote by ${\tilde{H}}_{Ax}$, satisfy a somewhat modified set of brackets

Here we call attention to the fact that Equation (60) contains the total tangential generator $H_{ix}$, not just ${\tilde{H}}_{ix}$. This alteration occurs because the “matter” super-Hamiltonian depends on the metric. That is, if we want to shift ${\tilde{H}}_{\perp x}$ along a given surface, we must shift the variables $(\rho ,\Phi )$, as well as the metric $g_{ij}$; hence $H_{ix}$, not just ${\tilde{H}}_{ix}$, must appear.

Modulo the small distinction arising in Equation (60), we see that the gravitational and “matter” sectors decouple such that each piece forms an independent representation of the “algebra” of surface deformations. From a strategic point of view, the separation means that we can solve the Poisson bracket relations for geometry and “matter” independently of one another.

Our goal is to identify a family of ensemble super-Hamiltonians ${\tilde{H}}_{\perp x}[\rho ,\Phi ;g_{ij}]$ that are consistent with the Poisson brackets (59)–(61). Let us briefly outline our approach. Thus far, we have completely determined the correct form of the “matter” super-momentum ${\tilde{H}}_{ix}$ which is consistent with (61). Moreover, the relation (60) merely implies that ${\tilde{H}}_{\perp x}$ transforms as a scalar density under a spatial diffeomorphism. Thus we are left only to satisfy the first Poisson bracket (59) for the unknown ${\tilde{H}}_{\perp x}$.

In addition to these considerations, however, the ED approach itself imposes additional constraints of a fundamental nature on the allowed $H_{\perp x}$, or more specifically ${\tilde{H}}_{\perp x}$. This is because, in ED, the introduction of a symplectic structure and its corresponding Hamiltonian formalism is not meant to replace the entropic updating methods that yield the LTFP equations, but to augment them, appropriately. As a consequence, we demand, as a matter of principle, that the “matter” super-Hamiltonian ${\tilde{H}}_{\perp x}$ be defined so as to reproduce the LTFP equations of (15).

More explicitly, we require ${\tilde{H}}_{\perp x}$ to be such that its action on ${\rho }_{\sigma }$ generates the LTFP equations which translates to

It is simple to check that the ${\tilde{H}}_{\perp x}$ that satisfies this condition is given by

The first term in (64) is fixed by virtue of consistency with the LTFP equations, whereas $F_x[\rho ;g_{ij}]$ is a yet undetermined “constant“ of integration, which may depend on $\rho$ as well as the metric. However, $F_x$ is not entirely arbitrary, its functional form is restricted by the Poisson bracket, Equation (59).

Before proceeding, note that, up until this point, our discussion of path independence has been developed on a formal level, largely independently of ED itself. Such a formalism on its own, however, is necessarily devoid of many crucial physical ingredients. For instance, in ED we define local time as a measure of the field fluctuations through (8), which leads to the LTFP equations. In principle, this has nothing to do with abstract parameters $\delta {\xi }_x^A$ introduced as part of local updating. It is only once we say that the entropic updating of ED must agree with the local time evolution generated by ${\tilde{H}}_{\perp x}$, made explicit in (13), that the two notions coincide. In ED, the clock that measures the local proper time $\delta {\xi }_x^{\perp }$ is nothing but the field fluctuations themselves.

Continuing with our task at hand, we seek a family of models that are consistent with Equation (59). This is accomplished for suitable choices of $F_x$. We pursue this in manner similar to [29]. Begin by rewriting ${\tilde{H}}_{\perp x}$ as where we have introduced

This amounts simply to a redefinition of the arbitrary $F_x$ in (64). An advantage of this definition, however, is that the newly defined ${\tilde{H}}_{\perp x}^0$ automatically satisfies

Since $F_x[\rho ;g_{ij}]$ is independent of $\Phi$, we have that $\left\{F_x,F_{x^{\prime }}\right\}=0$, identically. Thus, finding a suitable $F_x$ amounts to satisfying

Clearly a necessary condition for an acceptable $F_x$ is that the Poisson bracket $\left\{{\tilde{H}}_{\perp x}^0,F_{x^{\prime }}\right\}$ must be symmetric upon exchange of x and $x^{\prime }$. Since ${\tilde{H}}_{\perp x}^0$ must itself reproduce the LTFP equations, the condition (68) translates to [28,29] which is an equation linear in $F_x$.

A complete description of solutions to Equation (69) lies outside the scope of the current work. However, a restricted family of solutions, which are nonetheless of physical interest, can be found for $F_x$s of the form where $f_x$ is a function, not functional, of its arguments. For such a special type of $F_x$ one can check by substitution into (69) that are acceptable solutions. Since Equation (69) is linear in $F_x$, solutions can be superposed so that a suitable family of ${\tilde{H}}_{\perp x}$s is given by where $V_x={\sum }_n{\lambda }_n{\chi }_x^n$ is a function that is polynomial in ${\chi }_x$.

As discussed in [29], the last term can be interpreted as the ”local quantum potential.“ To see this we recall that in flat space-time the quantum potential is given by [27]

The transition to a curved space-time is made by making the substitutions yielding the result

The term inside the spatial integral is exactly what appears in (71), which justifies the name. The contribution of the local quantum potential to the energy is such that those states that are more smoothly spread out in configuration space tend to have lower energy. The corresponding coupling constant $\lambda >0$ controls the relative importance of the quantum potential; the case $\lambda <0$ is excluded because it leads to instabilities.

We now proceed to determining the last remaining element of our scheme, the gravitational super-Hamiltonian $H_{\perp x}^G$. Fortunately, under the assumption of non-derivative coupling mentioned above, it is possible to completely separate the Poisson bracket relations (32)–(34) into pieces that are purely gravitational and those that are pure “matter.” Consequently, to determine $H_{\perp x}^G$ it suffices merely to solve the Poisson bracket relations (56)–(58), which involve only the gravitational variables $g_{ij}$ and ${\pi }^{ij}$.

Such a task, however, is mathematically equivalent to determining the generators 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 [44]—they proposed a solution to exactly those brackets that appear in (56)–(58). Their main result was an important one: the only time-reversible solution to Equations (56)–(58) is nothing less than Einstein’s GR in vacuum. We briefly review their 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 behavior under a purely normal deformation is known [60] to be where $K_{ij}\left(x\right)=K_{ijx}$ is a symmetric $(0,2)$ tensor that is called the extrinsic curvature. Note that identifying $K_{ijx}$ with the response of $g_{ijx}$ under a 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 of space-time. That being said, $K_{ijx}$ is at this juncture an undetermined functional of the canonical variables, and more to the point, we do not assume at the moment any simple relationship with the momenta ${\pi }_x^{ij}$—this must be derived.