We now turn our attention to the local Hamiltonian generators $H_{Ax}$, and more specifically, we look to provide explicit expressions for these generators in terms of the canonical variables. This problem was solved in the context of a purely classical geometrodynamics (with or without sources) by HKT in [82]. Here we aim to apply their techniques and methodology to a different problem: a geometrodynamics driven by “quantum" sources. Fortunately, a considerable portion of the HKT formalism can be directly adopted for our purposes.

§.§ The super-momentum

To determine the generators $H_{Ax}[\rho,\Phi;g_{ij},\pi^{ij}]$ it is easiest to begin with the so-called super-momentum $H_{ix}[\rho,\Phi;g_{ij},\pi^{ij}]$. This is largely because the function of this generator is well understood: it pushes the canonical 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 [82], consider an infinitesimal tangential deformation such that a point originally labeled by $x^{i}$ is carried to the point previously labeled by $x^{i} + \delta\xi^{i}$. This will induce a corresponding change in any dynamical variables $F$ defined on that surface, $F\to F+\delta F$. This change $\delta F$ can then be computed in two distinct ways, which, of course, must agree. One is by calculating the Lie derivative along $\delta\xi$

\begin{eqnarray}
\delta F = \pounds_{\delta\xi} F~.\notag
\end{eqnarray}

and the other is using the super-momentum $H_{ix}$, so that

\begin{eqnarray}
\delta F = \int dx \left\{ F, H_{ix}\right \}\delta\xi^{i}_{x}~.\notag
\end{eqnarray}

*Gravitational super-momentum
A straightforward example of this is shown for the metric $g_{ij}(x)$, which is a rank $(0,2)$ tensor. Its Lie derivative is given by [119]

\begin{eqnarray}
\pounds_{\delta\xi} g_{ij} = \partial_{k}g_{ij}\delta \xi^{k}_{x} + g_{ik}\partial_{j}\delta\xi^{k}_{x} + g_{kj}\partial_{i}\delta\xi^{k}_{x}~.\label{6 Lie derivative metric}
\end{eqnarray}

Alternatively, by using the Poisson brackets we obtain

\begin{eqnarray}
\delta g_{ijx} = \int dx^{\prime}\left\{g_{ijx}, H_{kx^{\prime}}\right \}\delta\xi_{x^{\prime}}^{k} = \int dx^{\prime}\frac{\delta H_{kx^{\prime}}}{\delta\pi^{ij}_{x}}\delta\xi_{x^{\prime}}^{k}~.\notag
\end{eqnarray}

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

\begin{eqnarray}
\frac{\delta H_{kx^{\prime}}}{\delta\pi^{ij}_{x}} = \partial_{kx}g_{ijx}\delta( x,x^{\prime})+ g_{ikx}\partial_{jx}\delta( x,x^{\prime}) + g_{kjx}\partial_{ix}\delta( x,x^{\prime}).\label{6 Tangential deformation metric}
\end{eqnarray}

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 $\pi^{ij}$ is a rank $(2,0)$ tensor density of weight one, we find that

\begin{eqnarray}
\pounds_{\delta\xi} \pi^{ij} = \partial_{kx}\left(\pi^{ij}\delta\xi^{k}_{x}\right) - \pi^{ik}\partial_{kx}\delta\xi^{j}_{x} - \pi^{kj}\partial_{kx}\delta\xi^{i}_{x}.\label{6 Tangential deformation momentum}
\end{eqnarray}

The same equation can be obtained through the use of a Hamiltonian,

\begin{eqnarray}
\delta\pi^{ij}_{x} = \int dx^{\prime}\left\{\pi^{ij}_{x},H_{kx^{\prime}}\right\}\delta\xi^{k}_{x^{\prime}} = -\int dx^{\prime}\frac{\delta H_{kx^{\prime}}}{\delta g_{ijx}}\delta\xi^{k}_{x^{\prime}} ~,\notag
\end{eqnarray}

so long as $H_{ix}$ satisfies

\begin{eqnarray}
-\frac{\delta H_{kx^{\prime}}}{\delta g_{ijx}} = \partial_{kx}\left(\pi^{ij}_{x}\delta(x,x^{\prime})\right) - \pi^{il}_{x}\partial_{lx}\delta(x,x^{\prime}) \delta^{j}_{k}- \pi^{lj}_{x}\partial_{lx}\delta(x,x^{\prime})\delta^{i}_{k}~.\label{6 Tangential deformation momentum}
\end{eqnarray}

Integrating these equations for $H_{ix}$ yields

\begin{eqnarray}
H_{ix} =H_{ix}^{G} + \tilde{H}_{ix} ~,\label{6 Super-momentum grav + H}
\end{eqnarray}

\begin{eqnarray}
H_{ix}^{G} = -2 \partial_{jx}\left(\pi^{jk}\, g_{ik}\right) + \pi^{jk}\partial_{ix}g_{jk}~, \label{6 Super momentum grav}
\end{eqnarray}

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

\begin{eqnarray}
\pounds_{\delta\xi} g_{ijx} = g_{jk}\nabla_{i}\delta\xi_{x}^{k} + g_{ik}\nabla_{j}\delta\xi_{x}^{k}~\label{6 Tangential deformation metric cov}
\end{eqnarray}

\begin{eqnarray}
\pounds_{\delta\xi} \pi^{ij}_{x} = \nabla_{k}\left(\pi^{ij}\delta\xi^{k}_{x}\right) - \pi^{ik}\nabla_{k}\delta\xi^{j}_{x} - \pi^{kj}\nabla_{k}\delta\xi^{i}_{x}~.\label{6 Tangential deformation momentum cov}
\end{eqnarray}

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

\begin{eqnarray}
H_{ix}^{G} = -2g_{ik} \nabla_{j}\pi^{jk}= -2\nabla_{j}\left(\pi^{jk}g_{ik}\right) = -2 \nabla_{j}\pi^{j}_{i}~.\label{6 Super momentum grav cov}
\end{eqnarray}

*The “matter" super-momentum—
Next, we turn to the response of the variables $\rho_{\sigma}[\chi]$ and $\Phi_{\sigma}[\chi]$ under a relabeling of the surface coordinates $x^{i} \to x^{i}+\delta\xi^{i}_{x}$. As the coordinates are shifted, so too are the fields defined upon that surface so that

\begin{eqnarray}
\chi_{x} \to \chi_{x} + \pounds_{\delta\xi}\chi_{x}\quad\text{where}\quad \pounds_{\delta\xi}\chi_{x} = \partial_{ix}\chi_{x}\delta\xi^{i}_{x}\notag
\end{eqnarray}

is the Lie derivative for a scalar $\chi_{x}$. This induces a change in the probability, given by

\begin{eqnarray}
\delta\rho_{\sigma}[\chi] \equiv \rho_{\sigma}[\chi + \pounds_{\delta\xi}\chi_{x}] - \rho_{\sigma}[\chi] = \int dx \frac{\delta\rho_{\sigma}[\chi]}{\delta\chi_{x}}\partial_{ix}\chi_{x}\delta\xi^{i}_{x}~.
\end{eqnarray}

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

\begin{eqnarray}
\delta\rho_{\sigma}[\chi] = \int dx \left\{\rho_{\sigma}[\chi],H_{ix}\right\}\delta\xi^{i}_{x}~. \label{Tangential deformation rho a}
\end{eqnarray}

If we insert the super-momentum $H_{ix}$ given in eq.(<ref>), we notice that only the $\tilde{H}_{ix}$ piece will contribute. And so we have

\begin{eqnarray}
\delta\rho_{\sigma}[\chi] = \int dx \left\{\rho_{\sigma}[\chi],\tilde{H}_{ix}\right\}\delta\xi^{i}_{x} = \int dx \frac{\tilde{\delta}\tilde{H}_{ix}}{\tilde{\delta}\Phi_{\sigma}[\chi]}\delta\xi_{x}^{i}~,\label{6 Tangential deformation rho b}
\end{eqnarray}

which for arbitrary $\delta\xi^{i}_{x}$ requires that

\begin{eqnarray}
\frac{\tilde{\delta}\tilde{H}_{ix}}{\tilde{\delta}\Phi_{\sigma}[\chi]} = \frac{\delta\rho_{\sigma}[\chi]}{\delta\chi_{x}}\partial_{ix}\chi_{x}~.\label{6 Tangential deformation rho c}
\end{eqnarray}

A similar argument for $\Phi_{\sigma}[\chi]$ shows that we must also have

\begin{eqnarray}
-\frac{\tilde{\delta}\tilde{H}_{ix}}{\tilde{\delta}\rho_{\sigma}[\chi]} = \frac{\delta\Phi_{\sigma}[\chi]}{\delta\chi_{x}}\partial_{ix}\chi_{x}~,\label{6 Tangential deformation Phi a}
\end{eqnarray}

so that

\begin{eqnarray}
\tilde{H}_{ix} = - \int D\chi \rho_{\sigma}\frac{\delta\Phi_{\sigma}}{\delta\chi_{x}}\partial_{ix}\chi_{x}~.\label{6 Super momentum ensemble}
\end{eqnarray}

Thus the total super-momentum, eq.(<ref>),

\begin{eqnarray}
H_{ix} = -2 \nabla_{j}\pi^{j}_{i} - \int D\chi \rho_{\sigma}\frac{\delta\Phi_{\sigma}}{\delta\chi_{x}}\partial_{ix}\chi_{x}\label{6 Super momentum}
\end{eqnarray}

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.] Note that there is no gravitational dependence in the “matter" side, or “matter" dependence on the gravitational side, that is,

\begin{eqnarray}
H_{ix} = H_{ix}^{G}[g_{ij},\pi^{ij}] + \tilde{H}_{ix}[\rho,\Phi]~.\label{6 Super momentum split}
\end{eqnarray}

By equation (<ref>) it is, of course, also understood that $H_{ix}$ is subject to the constraint

\begin{eqnarray}
H_{ix}\approx 0~.\label{6 Super momentum constraint}
\end{eqnarray}

Finally, although $H_{ix}$ was obtained, in essence, independently of the Poisson bracket (<ref>), relating two tangential deformations, it nonetheless satisfies it automatically.[More technically, $H_{ix}$ is defined by our procedure up to an overall “constant" vector density $f_{i}(x)$, which is independent of any canonical variables. This, it turns out, is required to vanish by eq.(<ref>). See e.g., [82].] It is therefore appropriate to view $H_{ix}$ as being completely determined; this is important as it allows us to treat equation (<ref>) as a set of equations for $H_{\perp x}$ in terms of the known $H_{ix}$.

§.§ The super-Hamiltonian

We now turn out attention to the generator $H_{\perp x}$ of local time evolution. Following Teitelboim [80], it is useful to decompose $H_{\perp x}$ into two distinct pieces

\begin{eqnarray}
H_{\perp x} = H_{\perp x}^{G}[g_{ij},\pi^{ij}] + \tilde{H}_{\perp x}[\rho,\Phi;g_{ij}, \pi^{ij}]~,\label{6 Super Hamiltonian split a}
\end{eqnarray}

consisting of a gravitational piece $H_{\perp x}^{G}$ depending only on the gravitational variables, and a “matter" contribution that we suggestively denote by $\tilde{H}_{\perp x}$. It will sometimes be convenient to refer to these as the gravitational and “matter" super-Hamiltonians, respectively.

As noted by Teitelboim, we make no assumptions in writing $H_{\perp x}$ in this way.[By construction, we can always identify a piece of the super-Hamiltonian that updates the geometry alone. The “matter" super-Hamiltonian is then just defined as the difference between the total and gravitational pieces.] Given this splitting, however, we do make the following simplifying assumption: we require the “matter" super-Hamiltonian $\tilde{H}_{\perp x}$ to be independent of the gravitational momentum $\pi^{ij}$, i.e.,

\begin{eqnarray}
\tilde{H}_{\perp x} = \tilde{H}_{\perp x}[\rho,\Phi;g_{ij}]~\notag
\end{eqnarray}

so that the super-Hamiltonian takes the form

\begin{eqnarray}
H_{\perp x} = H_{\perp x}^{G}[g_{ij},\pi^{ij}] + \tilde{H}_{\perp x}[\rho,\Phi;g_{ij}]~.\label{6 Super Hamiltonian split}
\end{eqnarray}

With this assumption in hand, it is possible to prove [80] that the metric appears in $\tilde{H}_{\perp x}$ only as a local function of $g_{ij}$. That is, no derivatives of the metric are allowed, nor any other complicated functional dependencies; due to this fact, this was referred to as the  non-derivative coupling assumption by Teitelboim.

*Modified Poisson brackets—
A particularly appealing aspect of this separation into gravitational and “matter" pieces is that the Poisson bracket relations (<ref>)-(<ref>) also split along similar lines. In fact, insert the decomposed generators $H_{Ax}$ from eqns.(<ref>) and (<ref>) into the Poisson bracket relations (<ref>)-(<ref>). We find that the gravitational generators $H_{Ax}^{G} = (H_{\perp x}^{G},H_{ix}^{G})$ must satisfy a set of Poisson brackets

\begin{eqnarray}
\left\{ H_{\bot x}^{G},H_{\bot x^{\prime }}^{G}\right\} &=&(g_{x}^{ij}H_{jx}^{G}+g_{x^{\prime
}}^{ij}H_{jx^{\prime }}^{G})\partial _{ix}\delta (x,x^{\prime })~,  \label{PB 1 G}
\\
\left\{ H_{ix}^{G},H_{\bot x^{\prime }}^{G}\right \} &=&H_{\bot x}^{G}\partial _{ix}\delta
(x,x^{\prime })~,  \label{PB 2 G} \\
\left\{ H_{ix}^{G},H_{jx^{\prime }}^{G}\right \} &=&H_{ix^{\prime }}^{G}\,\partial _{jx}\delta
(x,x^{\prime })+H_{jx}^{G}\,\partial _{ix}\delta (x,x^{\prime })~,  \label{PB 3 G}
\end{eqnarray}

which have closing relations identical to those of (<ref>)-(<ref>).

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

\begin{eqnarray}
\left\{ \tilde{H}_{\bot x},\tilde{H}_{\bot x^{\prime }}\right\} &=&(g_{x}^{ij}\tilde{H}_{jx}+g_{x^{\prime
}}^{ij}\tilde{H}_{jx^{\prime }})\partial _{ix}\delta (x,x^{\prime })~,  \label{6 PB 1 matter}
\\
\left\{ H_{ix},\tilde{H}_{\bot x^{\prime }}\right \} &=&\tilde{H}_{\bot x}\partial _{ix}\delta
(x,x^{\prime })~,  \label{6 PB 2 matter} \\
\left\{ \tilde{H}_{ix},\tilde{H}_{jx^{\prime }}\right \} &=&\tilde{H}_{ix^{\prime }}\,\partial _{jx}\delta
(x,x^{\prime })+\tilde{H}_{jx}\,\partial _{ix}\delta (x,x^{\prime })~.  \label{6 PB 3 matter}
\end{eqnarray}

Here we call attention to the fact that eq.(<ref>) 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 eq.(<ref>), 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.

*The “matter" super-Hamiltonian—
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 (<ref>)-(<ref>). 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 (<ref>). Moreover, the relation (<ref>) 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 (<ref>) 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 (<ref>).

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

\begin{eqnarray}
\big\{ \rho_{\sigma}[\chi],H_{\perp x} \big\} =  \frac{\delta\rho_{\sigma}[\chi]}{\delta\xi^{\perp}_{x}}~, \label{FP equation H a}
\end{eqnarray}

which translates to

\begin{eqnarray}
\frac{\tilde{\delta}\tilde{H}_{\perp x}}{\tilde{\delta}\Phi_{\sigma}[\chi]}= -\frac{1}{g_{x}^{1/2}}\frac{\delta}{\delta\chi_{x}}\left (\rho_{\sigma}[\chi]\frac{\delta\Phi_{\sigma}[\chi]}{\delta\chi_{x}}    \right )~.\label{FP equation H b}
\end{eqnarray}

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

\begin{eqnarray}
\tilde{H}_{\perp } = \int D\chi \rho_{\sigma} \frac{1}{2g_{x}^{1/2}}\left (  \frac{\delta\Phi_{\sigma}}{\delta\chi_{x}}\right )^{2} +F_{x}[\rho;g_{ij}]~.\label{e Hamiltonian a}
\end{eqnarray}

The first term in (<ref>) 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, eq.(<ref>).

Before proceeding, note that, up until this point, our discussion of path independence has been developed on a formal level, largely independent 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 (<ref>), which leads to the LTFP equations. In principle, this has nothing to do with abstract parameters $\delta\xi^{A}_{x}$ 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 (<ref>), that the two notions coincide. In ED, the clock that measures the local proper time $\delta\xi^{\perp}_{x}$ is nothing but the field fluctuations themselves.

Continuing with our task at hand, we seek a family of models that are consistent with eq.(<ref>). This is accomplished for suitable choices of $F_{x}$. We pursue this in manner similar to [72]. Begin by rewriting $\tilde{H}_{\perp x}$ as

\begin{eqnarray}
\tilde{H}_{\perp x} = \tilde{H}_{\perp x}^{0}+ F_{x}~, \label{6 e-H}
\end{eqnarray}

where we have introduced

\begin{eqnarray}
\tilde{H}_{\perp x}^{0} = \int D\chi \,\rho_{\sigma}\left ( \frac{1}{%
2g_{x}^{1/2}}\left( \frac{\delta \Phi_{\sigma} }{\delta \chi _{x}}\right) ^{2}+\frac{g^{1/2}_{x}}{2}g^{ij}\partial_{ix}\chi_{x}\partial_{jx}\chi_{x}\right ) ~.  \label{6 e-H b}
\end{eqnarray}

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

\begin{eqnarray}
\left\{ \tilde{H}^{0}_{\bot x},\tilde{H}^{0}_{\bot x^{\prime }}\right\} =(g_{x}^{ij}\tilde{H}_{jx}+g_{x^{\prime
}}^{ij}\tilde{H}_{jx^{\prime }})\partial _{ix}\delta (x,x^{\prime })~.  \label{6 PB 1 matter Ho}
\end{eqnarray}

Finding a suitable $F_{x}$ is therefore accomplished by satisfying[Since $F_{x}[\rho; g_{ij}]$ is independent of $\Phi$, we have that $\left\{ F_{x},F_{x^{\prime}}\right\} = 0 $, identically, from which eq.(<ref>) follows.]

\begin{eqnarray}
\left\{ \tilde{H}^{0}_{\bot x},F_{x^{\prime}}\right\} = \left\{ \tilde{H}^{0}_{\bot x^{\prime}},F_{x}\right\}~.\label{PB 1 matter necessary}
\end{eqnarray}

Clearly a necessary condition for an acceptable $F_{x}$ is that the Poisson bracket $\left\{ \tilde{H}^{0}_{\bot x},F_{x^{\prime}}\right\} $ must be symmetric upon exchange of $x$ and $x^{\prime}$. Since $\tilde{H}^{0}_{\perp x}$ must itself reproduce the LTFP equations, the condition (<ref>) translates to [71][72]

\begin{eqnarray}
\frac{1}{g_{x}^{1/2}}\frac{\delta}{\delta\chi_{x}}\left (\rho_{\sigma} \frac{\delta}{\delta\chi_{x}} \frac{\tilde{\delta}F_{x^{\prime}}}{\tilde{\delta}\rho_{\sigma}}   \right ) = \frac{1}{g_{x^{\prime}}^{1/2}}\frac{\delta}{\delta\chi_{x^{\prime}}}\left (\rho_{\sigma} \frac{\delta}{\delta\chi_{x^{\prime}}} \frac{\tilde{\delta}F_{x}}{\tilde{\delta}\rho_{\sigma}}   \right )~,\label{PB 1 matter necessary b}
\end{eqnarray}

which is an equation linear in $F_{x}$.

A complete description of solutions to eq.(<ref>) 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

\begin{eqnarray}
F_{x}[\rho] = \int D\chi f_{x}\left(\rho,\frac{\delta\rho}{\delta\chi_{x}}; g_{ijx}\right)~,\label{6 Local potentials}
\end{eqnarray}

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 (<ref>) that

\begin{eqnarray}
f_{x} \sim g_{x}^{1/2}\rho \chi_{x}^{n}\quad \text{(integer n)}\quad \text{and}\quad f_{x}\sim \frac{\rho}{g_{x}^{1/2}}\left (\frac{\delta\log\rho}{\delta\chi_{x}}\right )^{2}\notag
\end{eqnarray}

are acceptable solutions. Since eq.(<ref>) is linear in $F_{x}$, solutions can be superposed so that a suitable family of $\tilde{H}_{\perp x}$'s is given by

\begin{equation}
\tilde{H}_{\perp x} = \int D\chi \, \rho_{\sigma} \left ( \frac{1}{%
2g_{x}^{1/2}}\left( \frac{\delta \Phi_{\sigma} }{\delta \chi _{x}}\right) ^{2}+\frac{g^{1/2}_{x}}{2}g^{ij}\partial_{ix}\chi_{x}\partial_{jx}\chi_{x} + g_{x}^{1/2}V_{x}(\chi_{x})+ \frac{\lambda}{g_{x}^{1/2}}\left (\frac{\delta\log\rho_{\sigma}}{\delta\chi_{x}}\right )^{2}\right )~,\label{e Hamiltonian final}
\end{equation}

where $V_{x} = \sum_{n}\lambda_{n}\chi_{x}^{n}$ is a function that is polynomial in $\chi_{x}$.

As discussed in [72], 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 [70]

\begin{eqnarray}
Q =\int d^{3}x \int D\chi \rho \, \lambda \left (\frac{\delta\log\rho}{\delta\chi_{x}}\right )^{2}~.\label{Quantum Potential Flat}
\end{eqnarray}

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

\begin{eqnarray}
d^{3}x \to g_{x}^{1/2}d^{3}x\quad\text{and}\quad \frac{\delta}{\delta\chi_{x}}\to\frac{1}{g_{x}^{1/2}}\frac{\delta}{\delta\chi_{x}}~,\notag
\end{eqnarray}

yielding the result

\begin{eqnarray}
Q_{\sigma} = \int d^{3}x \int D\chi \rho  \frac{\lambda}{g_{x}^{1/2}} \left (\frac{\delta\log\rho}{\delta\chi_{x}}\right )^{2}~.\notag
\end{eqnarray}

The term inside the spatial integral is exactly what appears in (<ref>), 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.

*The gravitational super-Hamiltonian—
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 (<ref>)-(<ref>) 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 (<ref>)-(<ref>), 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 [82] — they proposed a solution to exactly those brackets that appear in (<ref>)-(<ref>). Their main result was an important one: the only time-reversible solution to equations (<ref>)-(<ref>) 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 [174] to be

\begin{eqnarray}
\delta g_{ij}(x) = \pounds_{\delta \xi^{\perp}}g_{ijx} = -2 K_{ijx}~,\label{Normal deformation metric a}
\end{eqnarray}

where $K_{ijx}$ 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 $\pi^{ij}_{x}$ — this must be derived.

Alternatively, the deformation in (<ref>) must also be attainable using the normal generator $H_{\perp x}^{G}$,

\begin{eqnarray}
\delta g_{ijx} = \int dx^{\prime}\left\{ g_{ijx},H_{\perp x^{\prime}}^{G} \right \}\delta\xi^{\perp}_{x^{\prime}}~.\notag
\end{eqnarray}

More explicitly $H_{\perp x}^{G}$ should satisfy

\begin{eqnarray}
\frac{\delta H_{\perp x^{\prime}}^{G}}{\delta\pi^{ij}_{x}} =-2 K_{ijx}\delta(x,x^{\prime})~. \label{6 Normal deformation metric}
\end{eqnarray}

The appearance of the Dirac delta in eq.(<ref>) implies that $H_{\perp x}^{G}$ is local in $\pi^{ij}_{x}$. Therefore $H_{\perp x}^{G}$ is a function, not a functional, of $\pi^{ij}_{x}$.

Such a simplification turns out to be important: once $H_{\perp x}^{G}$ is a function of $\pi^{ij}$, it is possible to produce an ansatz for $H_{\perp x}^{G}$ in terms of powers of $\pi^{ij}$. HKT then supplement this with an additional simplifying assumption: that geometrodynamics be time-reversible. This has the advantage of removing all terms in $H_{\perp x}^{G}$ that have odd powers of $\pi^{ij}_{x}$.

Incorporating these ingredients, it is then possible to consider $H_{\perp x}^{G}$'s of the form

\begin{eqnarray}
H_{\perp x}^{G} = \sum_{n = 0}^{\infty}G^{(2n)}_{i_{1}j_{1}i_{2}j_{2}\cdots i_{2n}j_{2n}x}\pi^{i_{1}j_{1}}_{x}\pi^{i_{2}j_{2}}_{x}\cdots\pi^{i_{2n}j_{2n}}_{x}~,\label{6 Super hamiltonian grav ansatz}
\end{eqnarray}

where the coefficients $G^{(2n)}_{ij\cdots}$ are functionals of the metric $g_{ij}$, but depend on the point $x$. Furthermore, since $H_{\perp x}^{G} $ is scalar density and $\pi^{ij}$ a tensor density, $G_{ij\cdots}^{(2n)}$ must transform as a tensor density of weight $1-2n$. Since $\pi^{ij}$ is a symmetric tensor we expect $G^{(2n)}_{ij\cdots}$ to be symmetric under exchange of indices $i_{a}j_{a}\leftrightarrow j_{a}i_{a}$ for any pair $a$; also, $G^{(2n)}_{ij\cdots}$ should be symmetric on interchange of any pair $i_{a}j_{a}\leftrightarrow i_{b}j_{b}$ as this just corresponds to exchanging the $\pi$'s.

Naturally, one narrows the allowable $H_{\perp x}^{G}$ by inserting the ansatz (<ref>) into the Poisson bracket (<ref>). Without delving too far into the details (which can be found in [82]) we quote their solution. Only the $n = 0, 1$ terms survive, giving

\begin{eqnarray}
H_{\perp x}^{G} = \kappa \, G_{ijkl}\pi^{ij}\pi^{kl} - \frac{g^{1/2}}{2\kappa}\left(R-\Lambda\right)~,
\label{6 Super hamiltonian grav cosmo}
\end{eqnarray}

where we have introduce the super metric

\begin{eqnarray}
G_{ijkl} =  \frac{1}{g_{x}^{1/2}}\left(g_{ik}g_{jl}+g_{il}g_{jk}-g_{ij}g_{kl}\right)~,
\end{eqnarray}

and where $R$ is the Ricci scalar for the metric $g_{ij}$.

The constant $\kappa$ is a coefficient that, eventually, determines the coupling to “matter". We follow standard convention in identifying it as $\kappa = 8\pi G$, where $G$ is Newton's constant. The other parameter $\Lambda$, of course, is the cosmological constant. For simplicity, going forward we set $\Lambda = 0$.

Putting everything together, we have that $H_{\perp x}^{G}$ takes the form

\begin{eqnarray}
H_{\perp x}^{G} = \frac{\kappa}{g_{x}^{1/2}}\left (2\pi^{ij}\pi_{ij}-\pi^{2}\right ) - \frac{g^{1/2}}{2\kappa}R~,\label{6 Super hamiltonian grav}
\end{eqnarray}

where $\pi = \pi^{ij}g_{ij} = \text{Tr}(\pi^{ij}) $. This is exactly the standard gravitational super-Hamiltonian obtained by Dirac [200] and ADM [201] by starting from the Einstein-Hilbert Lagrangian.

*Total super-Hamiltonian—
Putting together the ingredients of this section, the total super-Hamiltonian is thus

\begin{eqnarray}
H_{\perp x} = H^{G}_{\perp x}+\tilde{H}_{\perp x} ~,\label{6 Super-Hamiltonian Total}
\end{eqnarray}

where $H_{\perp x}^{G}$ is given above by eq.(<ref>) and where a suitable family of $\tilde{H}_{\perp x}$'s have been identified in eq.(<ref>). With this, the super-Hamiltonian constraint is then just

\begin{eqnarray}
H_{\perp x} = H^{G}_{\perp x}+\tilde{H}_{\perp x} \approx 0~.\label{6 Super-Hamiltonian constraint}
\end{eqnarray}

§ THE DYNAMICAL EQUATIONS

As argued by HKT, although the role of space-time was crucial to the developing the equations (<ref>)-(<ref>), one notices that all signs of the enveloping space-time have dropped out in the closing relations (<ref>)-(<ref>).[For instance, they depend on the surface metric $g_{ij}$, but not on its extrinsic curvature $K_{ij}$.] Thus in geometrodynamics we can dispense with the notion of an a priori given space-time and instead consider the three-dimensional Riemannian manifold $\sigma_{t}$ as primary.

The idea is a simple one. The canonical variables are evolved with the generators $H_{Ax}$, satisfying eqns.(<ref>)-(<ref>). Such an evolution will cause both the “matter" and geometry, to change. We might then give this manifold with updated intrinsic geometry a new name, $\sigma_{t'}$. Repeating this procedure results in an evolution of the dynamical variables, and consequently, what one might view as a succession of manifolds $\{\sigma_{t}\}$, parameterized by the label $t$. Thus this iterative process constructs a space-time, step by step. We now investigate the dynamical equations that result from this procedure.

§.§ Some formalism

Consider the evolution of an arbitrary functional $T_{t}$ of the dynamical variables defined on $\sigma_{t}$,

\begin{eqnarray}
\delta T_{t} = T_{t+dt} - T_{t} = \int dx \left \{T_{t}, H_{Ax} \right \}\delta\xi^{A}_{x} =  \int dx \, N^{A}_{xt} \left \{T_{t}, H_{Ax} \right \}dt~,\label{6 Evolution lapse shift}
\end{eqnarray}

where we have introduced four arbitrary functions

\begin{eqnarray}
N_{xt} = \frac{\delta\xi^{\perp}_{x}}{dt}\quad\text{and}\quad N^{i}_{xt} = \frac{\delta\xi^{i}_{x}}{dt}~,\label{6 Def Lapse Shift}
\end{eqnarray}

which are the lapse and vector shift, respectively. Just as the evolution parameters $\delta\xi^{A}_{x}$ were completely arbitrary, these functions can be freely specified, and amount eventually to picking a particular foliation of space-time.[At the moment, the lapse and shift have no definite geometrical meaning. But as is well known, we can eventually identify the lapse $N$ and shift $N^{i}$ as being related to components of the space-time metric $^{4}g_{\mu\nu} \equiv \gamma_{\mu\nu}$. (We changed the symbol temporarily to avoid confusion.) More specifically, we would have [82]

\begin{eqnarray}
N = \left(- \gamma^{00}\right)^{-1/2}\quad\text{and}\quad N_{i} = \gamma_{0i}~.\notag
\end{eqnarray}

As in the Dirac approach to geometrodynamics, the $N^{A}_{xt} = (N_{xt}, N^{i}_{xt})$ are not functionals of the canonical variables, therefore we can rewrite eq.(<ref>) as

\begin{eqnarray}
\delta T_{t}  = \left \{T_{t}, H[N,N^{i}] \right \}dt~,\label{6 Evolution smeared}
\end{eqnarray}

where we have introduced the notion of a “smeared" Hamiltonian

\begin{eqnarray}
H[N,N^{i}]=\int dx\,\left( N_{xt}H_{\bot x}+N_{xt}^{i}H_{ix}\right) \label{6 Smeared Hamiltonian}
\end{eqnarray}

that, for given $N^{A}_{xt}$, generates an evolution parameterized by $t$.[Had the $N^{A}$ been functionals of the canonical variables then the Poisson bracket in eq.(<ref>) would have generated extra terms from their action on $N^{A}$ and thus eq.(<ref>) would not have been equivalent to eq.(<ref>).] This global $H[N,N^{i}]$ (note the spatial integral) conforms more naturally to our typical notions of a Hamiltonian. Thus, the derivative with respect to the parameter $t$ is

\begin{eqnarray}
\partial_{t} T_{t} \equiv \frac{\delta T_{t}}{dt} = \left \{ T_{t}, H[N,N^{i}]    \right \}~,\label{6 Def time derivative}
\end{eqnarray}

or more explicitly by

\begin{eqnarray}
\partial_{t} T_{t} = \int dx\,\left( N_{xt}\left \{T_{t},H_{\perp x}\right \}+N_{xt}^{i}\left \{T_{t},H_{ix}\right \}\right)~,\label{6 Def time derivative b}
\end{eqnarray}

and more succinctly

\begin{eqnarray}
\partial_{t} T_{t} = \pounds_{m} T_{t} + \pounds_{N^{i}} T_{t}~, \label{6 Def time derivative c}
\end{eqnarray}

where the vector $m^{\mu}_{x} = N_{xt} n^{\mu}_{x}$ is the so-called evolution vector.

§.§ The evolution of the “matter" sector

The goal is to determine the evolution of the probability distribution $\rho_{t}[\chi]$ and phase functional $\Phi_{t}[\chi]$, given an initial state $(\rho_{t}, \Phi_{t}; g_{ij\, t}, \pi^{ij}_{t})$ that satisfies the initial value constraints (<ref>).

*Dynamical equations for the probability and phase—
Given the basic dynamical law, eq.(<ref>), the time evolution of the variables $\rho_{t}$ and $\Phi_{t}$ are given by

\begin{eqnarray}
\partial_{t}\rho_{t} = \left \{\rho_{t}, H   \right \}\quad\text{and}\quad \partial_{t}\Phi_{t} = \left \{\rho_{t}, H   \right \}~,
\end{eqnarray}

where $H$ is the smeared Hamiltonian given above. Thus, in accordance with eq.(<ref>), we have the result

\begin{eqnarray}
\partial_{t}\rho_{t}  &=& \int dx \, \left ( N_{xt}\left \{\rho_{t}, H_{\perp x}\right \}+ N^{i}_{xt}\left \{\rho_{t}[\chi], H_{i x}\right \} \right ) \\
\partial_{t}\Phi_{t}  &=& \int dx \, \left ( N_{xt}\left \{\Phi_{t}, H_{\perp x}\right \}+ N^{i}_{xt}\left \{\Phi_{t}, H_{i x}\right \} \right )~. \label{6 Evolution Phi}
\end{eqnarray}

Using the family of $H_{\perp x}$'s that we identified in eq.(<ref>) and the super-momentum $H_{ix}$ in eq.(<ref>) we can compute all of the necessary Poisson brackets. From the super momentum in eq.(<ref>) we can compute the tangential pieces

\begin{eqnarray}
\left \{\rho_{t}[\chi], H_{i x}\right \} &=& \frac{\tilde{\delta}\tilde{H}_{ix}}{\tilde{\delta}\tilde{\Phi_{t}}} = \frac{\delta\rho_{t}}{\delta\chi_{x}}\partial_{ix}\chi_{x}~,\\
\left \{\Phi_{t}[\chi], H_{i x}\right \} &=& -\frac{\tilde{\delta}\tilde{H}_{ix}}{\tilde{\delta}\tilde{\rho_{t}}} = \frac{\delta\Phi_{t}}{\delta\chi_{x}}\partial_{ix}\chi_{x}~.\label{6 Tangential PB}
\end{eqnarray}

Of course, the entire family of $\tilde{H}_{\perp x}$'s were designed to reproduce the LTFP equations, thus by construction we have

\begin{eqnarray}
\left \{\rho_{t}[\chi], H_{\perp x}\right \} = \frac{\tilde{\delta}\tilde{H}_{\perp x}}{\tilde{\delta}\tilde{\Phi_{t}}} = -\frac{1}{g_{x}^{1/2}}\frac{\delta}{\delta\chi_{x}}\left ( \rho_{t}\frac{\delta\Phi_{t}}{\delta\chi_{x}}  \right )~,\label{FP equation final}
\end{eqnarray}

in agreement with eq.(<ref>). The remaining Poisson bracket determines the local time evolution of the phase functional $\Phi_{t}$ and is given by

\begin{align}
\left \{\Phi_{t}[\chi], H_{\perp x}\right \} = - \frac{\tilde{\delta}\tilde{H}_{\perp x}}{\tilde{\delta}\tilde{\rho_{t}}}  = \frac{1}{2 g_{x}^{1/2}}\left(\frac{\delta\Phi_{t}[\chi]}{\delta\chi_{x}}\right)^{2}+\frac{g_{x}^{1/2}}{2}g^{ij}\partial_{i}\chi_{x}\partial_{j}\chi_{x}+ \frac{\tilde{\delta}F_{x}[\rho]}{\tilde{\delta}\rho_{t}[\chi]}~,\label{6 LTHJ equations}
\end{align}

where $F_{x}$ is of the form specified by eq.(<ref>).

*The local time Hamilton-Jacobi equations—
To interpret the local equations (<ref>), we write the full time evolution for the phase functional by inserting eqns.(<ref>) and (<ref>) into eq.(<ref>), yielding

\begin{align}
-\partial_{t}\Phi_{t} =& \int dx \, \left [N_{xt} \left (\frac{1}{2 g_{x}^{1/2}}\left(\frac{\delta\Phi_{t}}{\delta\chi_{x}}\right)^{2} +\frac{g_{x}^{1/2}}{2}g^{ij}\partial_{i}\chi_{x}\partial_{j}\chi_{x} + \frac{\tilde{\delta}F_{x}}{\tilde{\delta}\rho_{t}}\right )+ N^{i}_{xt}\frac{\delta\Phi_{t}}{\delta\chi_{x}}\partial_{ix}\chi_{x}\right ]~.
\end{align}

To bring this equation into a more familiar form we consider the special case of flat space-time by setting the metric to be a Kroenecker delta $g_{ij} = \delta_{ij}$, so that  $g_{x}^{1/2} = 1$, and we let  $N = 1$, $N^{i} = 0$. Moreover, for simplicity we also make the assignment $F_{x} = 0$. This results in a time evolution for $\Phi_{t}$ that has the form

\begin{eqnarray}
-\partial_{t}\Phi_{t} = \int dx \left (\frac{1}{2}\left(\frac{\delta\Phi_{t}}{\delta\chi_{x}}\right)^{2} +\frac{1}{2}\delta^{ij}\partial_{i}\chi_{x}\partial_{j}\chi_{x} \right )~,\notag
\end{eqnarray}

which is exactly the classical Hamilton-Jacobi equation for a massless Klein-Gordon field in flat space-time. Thus, in analogy with the LTFP equations, we refer to eq.(<ref>) as the local time Hamilton-Jacobi (LTHJ) equations, as there is one equation of the Hamilton-Jacobi type for every spatial point.

The LTFP and LTHJ equations, with the tangential equations (<ref>), and the evolution equations (<ref>), give us the ability to evolve an appropriately chosen initial state $(\rho_{t},\Phi_{t})$. In general, this is a coupled non-linear evolution driven by a dependence on the metric $g_{ij}$. To a large extent, this completes our discussion of how the epistemic variables evolve. In a subsequent section, however, we discuss in some detail the dynamics of a specific 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 written within the canonical language. (A good review of these equations is given, for example, by ADM [205].)

*Evolution of metric—
The goal is to determine the evolution of the geometrical variables $(g_{ij},\pi^{ij})$. Beginning with the metric $g_{ij}$ defined on some initial three-space $\sigma_{t}$, we wish to determine how it evolves in response to the generators $H_{Ax}$. Applying  eq.(<ref>) for the time derivative, we have that

\begin{eqnarray}
\partial_{t} g_{ijx} = \int dx^{\prime} \left\{g_{ijx} , H_{Ax'} \right \}N^{A}_{x^{\prime}t} =  \int dx^{\prime} \frac{\delta H_{Ax'}}{\delta\pi^{ij}_{x}}N^{A}_{x^{\prime}t}  ~.\label{6 Metric time deriv a}
\end{eqnarray}

To compute this, recall that the tangential piece is known from equations (<ref>) and (<ref>). This gives us

\begin{eqnarray}
\pounds_{N^{i}}\, g_{ijx} =  \int dx^{\prime} \, \frac{\delta H_{ix'}}{\delta\pi^{ij}_{x}} \,  N^{i}_{x^{\prime}t} = \nabla_{i}N_{jxt}+\nabla_{j}N_{ixt}~, \label{6 Lie derivative shift  metric}
\end{eqnarray}

where $\pounds_{N^{i}} g_{ijx}$ is just the Lie derivative along the vector field defined by the shift $N^{i}$, and where $N_{ixt} = g_{ijx}N^{j}_{xt}$.

To obtain the remaining piece, first differentiate the $H_{\perp x}$ given in eq.(<ref>)

\begin{eqnarray}
\frac{\delta H_{\perp x^{\prime}}}{\delta\pi^{ij}_{x}} = \frac{4\kappa}{g_{x}^{1/2}}\left(\pi_{ijx} - \frac{1}{2}\pi_{x} g_{ijx}\right)\delta(x,x^{\prime})~,\notag%\label{6 Normal deformation metric b}
\end{eqnarray}

where $\pi_{ij}$ is the conjugate momentum with its indices lowered, and $\pi = g_{ij}\pi^{ij} = \text{Tr}(\pi^{ij})$ is the trace of the gravitational momentum. Putting these terms together, eq.(<ref>) reads as

\begin{eqnarray}
\partial_{t}g_{ijx} &=& \pounds_{m}g_{ijx} + \pounds_{N^{i}}g_{ijx} \label{6 Metric time deriv b}\\
\pounds_{m}g_{ijx} &=&  \frac{2\kappa}{g_{x}^{1/2}}\left(2\pi_{ijx} - \pi_{x} g_{ijx}\right)N_{xt} \\
\pounds_{N^{i}}g_{ijx} &=& \nabla_{i}N_{jxt}+\nabla_{j}N_{ixt}~.
\end{eqnarray}

This gives us the evolution of the metric with foliation parameter $t$.

We can also now identify the extrinsic curvature tensor

\begin{eqnarray}
K_{ij} = \frac{\kappa}{g^{1/2}_{x}}\left(\pi_{x} g_{ijx} - 2\pi_{ijx}\right) ~.
\end{eqnarray}

Inverting this relationship we get $\pi^{ij}$ in terms of $K^{ij}$, which yields

\begin{eqnarray}
\pi^{ij}_{x} = \frac{g^{1/2}_{x}}{2\kappa}\left (Kg^{ij}_{x} - K^{ij}_{x} \right )~. \label{6 Extrinsic curvature pi b}
\end{eqnarray}

where $K = K_{ij}g^{ij} = \text{Tr}(K_{ij})$ is the trace of $K_{ij}$. This is of some interest if one wishes to compare the canonical formulation to the standard so-called “Lagrangian" approach (see e.g., [206]).

*Evolution of conjugate momentum—
To this point, the dynamics of the geometry has not differed from a purely classical geometrodynamics, such as that of HKT. This is because the “matter" super-Hamiltonian and super-momentum were, by definition, completely independent of $\pi^{ij}$ (due, of course, to the non-derivative coupling assumption) therefore the evolution of $g_{ij}$ did not receive contributions from the “matter" sector. This changes when we consider the dynamics of the conjugate momentum $\pi^{ij}$.

The evolution of $\pi^{ij}$ is determined via the equation

\begin{eqnarray}
\partial_{t} \pi_{x}^{ij} = \int dx^{\prime} \left\{\pi^{ij}_{x} , H_{Ax'} \right \}N^{A}_{x^{\prime}t} =  - \int dx^{\prime} \frac{\delta H_{Ax'}}{\delta g_{ijx}}N^{A}_{x^{\prime}t}  ~.\label{6 Pi time deriv a}
\end{eqnarray}

Recalling now that both the gravitational and “matter" super-Hamiltonians $H_{\perp x} = H_{\perp x}^{G}[g_{ij},\pi^{ij}] + \tilde{H}_{\perp x}[\rho,\Phi;g_{ij}]$ depend explicitly on the metric, but that $\tilde{H}_{ix}$ does not, this expression slightly simplifies to

\begin{eqnarray}
\partial_{t}\pi^{ij}_{x}= -\int dx^{\prime}\left(N_{x^{\prime}t}\frac{\delta H^{G}_{\perp x^{\prime}}}{\delta g_{ijx}}+ N_{x^{\prime}t}\frac{\delta \tilde{H}_{\perp x^{\prime}}}{\delta g_{ijx}} +N^{i}_{x^{\prime}t}\frac{\delta H_{ix^{\prime}}^{G}}{\delta g_{ijx}}\right)~,
\end{eqnarray}

where we have have separated the contributions from the gravitational and “matter" sectors. In the notation of eq.(<ref>) we write this as

\begin{eqnarray}
\partial_{t}\pi^{ij}_{x} =  \pounds_{m}\pi^{ij}_{x} +\pounds_{N^{i}} \pi^{ij}_{x}~,
\end{eqnarray}

\begin{eqnarray}
\pounds_{m}\pi^{ij}_{x} = \pounds_{m}^{G}\pi^{ij}_{x}+ \pounds_{m}^{M}\pi^{ij}_{x}~,
\end{eqnarray}

\begin{eqnarray}
\pounds_{m}^{G}\pi^{ij}_{x} &=& -\int dx^{\prime}\left(N_{x^{\prime}t}\frac{\delta H^{G}_{\perp x^{\prime}}}{\delta g_{ijx}}\right)~,  \label{6 Pi Lie normal G} \\
\pounds_{m}^{M}\pi^{ij}_{x} &=& -\int dx^{\prime}\left(N_{x^{\prime}t}\frac{\delta \tilde{H}_{\perp x^{\prime}}}{\delta g_{ijx}}\right)~, \label{6 Pi Lie normal M} \\
\pounds_{N^{i}}\pi^{ij}_{x} &=& -\int dx^{\prime}\left(N^{i}_{x^{\prime}t}\frac{\delta H^{G}_{i x^{\prime}}}{\delta g_{ijx}}\right)~.\label{6 Pi Lie tangential G}
\end{eqnarray}

From (<ref>) the last term is fairly easy to determine,

\begin{eqnarray}
\pounds_{N^{i}}\pi^{ij}_{x} =  \nabla_{kx}\left(\pi^{ij}_{x}N^{k}_{x}\right) - \pi^{ik}_{x}\nabla_{kx}N^{j}_{x} - \pi^{kj}_{x}\nabla_{kx}N^{i}_{x}
~.\label{6 Pi Lie tangential G b}
\end{eqnarray}

The calculation of the other two terms is much more involved. Fortunately the expression for $\pounds_{m}^{G}\pi^{ij}$ is already well known [174][205] and we merely quote the result,

\begin{align}
\pounds_{m}^{G}\pi^{ij}_{x} &=-\frac{g^{1/2}}{2\kappa}\left(R^{ij}_{x}-\frac{1}{2}g^{ij}_{x}R_{x}\right )N_{xt}+ \frac{\kappa}{g^{1/2}_{x}} \, g^{ij}_{x}\left(\pi^{kl}_{x}\pi_{klx}-\frac{1}{2}\pi^{2}_{x}\right)N_{xt}\notag\\
&-\frac{4\kappa}{g^{1/2}_{x}}\left(\pi^{ik}_{x}\pi^{j}_{kx}-\frac{1}{2}\pi_{x}\pi^{ij}_{x}\right)N_{xt}+\frac{g^{1/2}_{x}}{2\kappa}\left(\nabla^{i}_{x}\nabla^{j}_{x}N_{xt} - g^{ij}_{x}\nabla^{k}_{x}\nabla_{kx}N_{xt}\right)~.\label{Lie derivative normal pi G}
\end{align}

To calculate the remaining piece $\pounds_{m}^{M}\pi^{ij}_{x}$ we first recall that having assumed a non-derivative coupling of gravity to matter, the metric $g_{ij}$ appears in $\tilde{H}_{\perp x}$ as an undifferentiated function (not a functional) and without any derivatives [80], which implies that

\begin{eqnarray}
\frac{\delta \tilde{H}_{\perp x^{\prime}}}{\delta g_{ijx}} =\frac{\partial \tilde{H}_{\perp x}}{\partial g_{ijx}} \,  \delta(x,x') ~,\notag
\end{eqnarray}

\begin{eqnarray}
\pounds_{m}^{M}\pi^{ij}_{x} = - \frac{\partial \tilde{H}_{\perp x}}{\partial g_{ijx}} N_{xt}~.\notag
\end{eqnarray}

Using eq.(<ref>), for an arbitrary choice of $F_{x}$, but one that still satisfies eq.(<ref>), the “matter" source has the form

\begin{eqnarray}
\pounds_{m}^{M}\pi^{ij}_{x}  = N_{xt} \left (  \frac{\partial \tilde{H}^{0}_{\perp x}}{\partial g_{ijx}} + \frac{\partial F_{x}}{\partial g_{ijx}}  \right )~,\label{6 Matter source}
\end{eqnarray}

\begin{align}
\frac{\partial \tilde{H}^{0}_{\perp x}}{\partial g_{ijx}} =& -\frac{1}{2}\int D\chi \rho \left [\left (\frac{1}{g_{x}^{1/2}}\left (\frac{\delta\Phi}{\delta\chi_{x}}\right )^{2}  -g^{1/2}_{x}g^{kl}_{x}\partial_{kx}\chi_{x}\partial_{lx}\chi_{x}    \right )g^{ij}_{x}- g_{x}^{1/2}\partial_{x}^{i}\chi_{x}\partial^{j}_{x}\chi_{x}\right ]~,\label{6 Matter source b}
\end{align}

where $\tilde{H}_{\perp x}^{0}$ was given in eq.(<ref>), and where $\partial^{i}_{x}\chi_{x} = g^{ij}_{x}\partial_{jx}\chi_{x}$.

In total, the equations of motion for the gravitational field follow Hamilton's equations for the gravitational variables $(g_{ijx},\pi^{ij}_{x})$, given by eq.(<ref>) for the evolution of the metric, and for the conjugate momentum $\pi^{ij}_{x}$ we have

\begin{eqnarray}
\partial_{t}\pi^{ij}_{x} - \pounds_{n}^{G}\pi^{ij}_{x} - \pounds_{N^{i}}\pi^{ij}_{x} = -N_{xt}\frac{\partial \tilde{H}_{\perp x}}{\partial g_{ijx}} ~.\label{6 Pi time deriv b}
\end{eqnarray}

This is an equation in which geometrical variables on the left-hand side are sourced by the variables $(\rho,\Phi)$, which contain all the information available about the field $\chi_{x}$ on the right-hand side. Below we will discuss this equation in the presence of “quantum matter."

§ QUANTUM SOURCES OF GRAVITATION

The transition to what may be termed a quantum form of dynamics amounts to an appropriate choice of the functional $F_{x}[\rho;g_{ij}]$ (see e.g., [70][71][72]). In particular, for $F_{x}[\rho;g_{ij}]$ we choose exactly the local quantum potential introduced as part of (<ref>). A convenient choice for the coupling constant is $\lambda = 1/8$.[As argued, for instance, in [53], there is no loss of generality in making this choice. For the case of nonrelativistic particles it can be proved [54] that an ED that preserves the appropriate symplectic and metric structures implies the presence of a quantum potential with the correct coefficient.] (This choice of $\lambda$ makes plain that we work with a system of units where $\hbar = c = 1$.)

The connection to conventional quantum theory is made explicit by a change of variables from the probability $\rho$ and phase $\Phi$ to the complex variables

\begin{eqnarray}
\Psi= \rho^{1/2}e^{i\Phi}\quad\text{and}\quad \Psi^{*} = \rho^{1/2}e^{-i\Phi}~.\label{6 Define Psi}
\end{eqnarray}

Such a change of variables is, in fact, a canonical transformation, and so, the new variables form a canonical pair given by $(\Psi, i\Psi^{*})$, which obey a natural generalization of the standard Poisson bracket relations

\begin{eqnarray}
\left \{ \Psi[\chi], i\Psi^{*}[\chi^{\prime}]  \right \} = \delta[\chi - \chi^{\prime}]~,
\end{eqnarray}

where $ \delta[\chi - \chi^{\prime}]$ is a Dirac delta functional.

§.§ Quantum operators and geometrodynamics revisited

Having chosen an $F_{x}[\rho;g_{ij}]$ of the type described above, the ensemble generators $\tilde{H}_{Ax}$ take a particularly special form:

\begin{eqnarray}
\tilde{H}_{Ax} = \int D\chi \Psi^{*}\hat{H}_{Ax}\Psi = \left \langle \hat{H}_{Ax}  \right \rangle~, \label{6 Quantum Super Hamiltonian ensemble}
\end{eqnarray}

which is the expected value of the local Hamiltonian operators $\hat{H}_{Ax} = (\hat{H}_{\perp x},\hat{H}_{ix})$, given by

\begin{align}
\hat{H}_{\perp x} &= - \frac{1}{2g^{1/2}}\frac{\delta^{2}}{\delta\chi^{2}_{x}}+\frac{g^{1/2}}{2}g^{ij}\partial_{i}\chi_{x}\partial_{j}\chi_{x}+g^{1/2}V_{x}(\chi_{x};g_{ij}) \label{6 Quantum Super Hamiltonian operator} \\
\hat{H}_{i x} &= i\,\partial_{i}\chi_{x}\frac{\delta}{\delta\chi_{x}}~. \label{6 Quantum Super momentum operator}
\end{align}

To obtain the “matter" contribution in eq.(<ref>) for the conjugate momentum $\pi^{ij}_{x}$, note that the metric appears in $\tilde{H}_{\perp x}$ through the density $g_{x}^{1/2}$ and the inverse metric $g^{ij}_{x}$. Variations of these quantities with respect to the metric $g_{ijx}$ are given by [206]

\begin{eqnarray}
\delta g_{x} = g  \,g^{ij}_{x} \, \delta g_{ijx}\quad\text{and}\quad \delta g^{ij}_{x} = g^{ik}g^{jl}\delta g_{kl}~.\notag
\end{eqnarray}

\begin{eqnarray}
\frac{\partial \tilde{H}_{\perp x}}{\partial g_{ijx}} = \int D\chi \, \Psi^{*}  \frac{\partial \hat{H}_{\perp x}}{\partial g_{ijx}} \Psi~,\notag
\end{eqnarray}

where we have defined the operator

\begin{eqnarray}
\frac{\partial \hat{H}_{\perp x}}{\partial g_{ijx}} = \partial^{i}_{x}\chi_{x}\partial^{j}_{x}\chi_{x}+g^{ij}\left (\frac{1}{2g_{x}^{1/2}}\frac{\delta^{2}}{\delta\chi_{x}^{2}}+\frac{g^{1/2}_{x}}{2}g^{kl}\partial_{k}\chi_{x}\partial_{l}\chi_{x}+V_{x}(\chi_{x})\right ).\label{6 Quantum stress operator}
\end{eqnarray}

*Geometrodynamics with quantum sources—
Our goal here is to rewrite the main equations of geometrodynamics with sources given by quantum matter. We begin first with the constraint equations. With the aid of the local operators introduced in (<ref>) and (<ref>), the total Hamiltonian generators take the explicit form

\begin{align}
H_{\perp x} &= \frac{\kappa}{g_{x}^{1/2}}\left (2\pi^{ij}_{x}\pi_{ijx} -\pi^{2}_{x}\right) - \frac{g^{1/2}_{x}}{2\kappa}R_{x} + \int D\chi \, \Psi^{*} \hat{H}_{\perp x} \Psi , \label{6 Quantum Hamiltonian}\\
H_{i x} &= -2\nabla_{kx}\left (\pi^{kj}_{x}g_{ijx}\right )+ \int D\chi \, \Psi^{*} \hat{H}_{i x} \Psi ~,\label{6 Quantum momentum}
\end{align}

which are, of course, subject to the constraints

\begin{eqnarray}
H_{\perp x}\approx 0\quad \text{and}\quad H_{i x}\approx 0~. \label{Quantum constraints}
\end{eqnarray}

As relations (<ref>) and (<ref>) coupled together with (<ref>) make abundantly clear, the quantum state and the geometrical variables can no longer be treated as independent. This has important consequences for the time evolution of $\Psi$.

Moving on, note that the dynamical equation for the metric, given in eq.(<ref>), does not depend directly on the choice of $F_{x}$ and therefore remains unchanged in the quantum context. The dynamical equation for the conjugate momentum $\pi^{ij}_{x}$, however, is modified. Using the operator introduced in (<ref>), this becomes

\begin{eqnarray}
\partial_{t}\pi^{ij}_{x} - \pounds_{n}^{G}\pi^{ij}_{x} - \pounds_{N^{i}}\pi^{ij}_{x} = -N_{xt}\int D\chi \, \Psi^{*}\frac{\partial \hat{H}_{\perp x}}{\partial g_{ijx}}\Psi~,\label{6 Quantum Pi evolution}
\end{eqnarray}

where $\pounds_{m}^{G}\pi^{ij}_{x} = $ and $\pounds_{N^{i}}\pi^{ij}_{x}$ are given in eqns.(<ref>) and (<ref>), respectively. This is the crucial equation in which the dynamical geometry is itself affected by the epistemic state $\Psi$.

Putting it all together, the eqns.(<ref>)-(<ref>), and eq.(<ref>) for the metric, constitute a system of equations that are formally equivalent to the semi-classical Einstein equations (SCEE) (see e.g., [56]) put in the canonical form. However, that is where the similarities end. On several key issues of interpretation, in particular, the ED approach is vastly different from the SCEE as they are normally understood. Far from being trivial, these distinctions turn out to be quite important since many objections (see e.g., [207][83]) to the usual SCEE are, in fact, based on such considerations.

§.§ Quantum dynamics

One advantage of the complex variables $(\Psi,\Psi^{*})$ is that the dynamics takes a familiar form. Indeed, since $\Psi$ and $\Psi^{*}$ are just functions of our canonical variables $(\rho,\Phi)$ we can just use eq.(<ref>) to determine their evolution along a time parameter $t$, which gives

\begin{eqnarray}
\partial_{t}\Psi_{t}[\chi] =  \int dx \left \{\Psi_{t}[\chi],H_{Ax}\right \}N^{A}_{xt}~.\label{6 Quantum Evolution Psi a}
\end{eqnarray}

The tangential component is obtained in a straightforward fashion by

\begin{eqnarray}
\left \{\Psi_{t}[\chi],H_{ix}\right \} = \partial_{i}\chi_{x}\frac{\delta\Psi_{t}[\chi]}{\delta\chi_{x}}~,\label{6 Quantum Tangential Psi}
\end{eqnarray}

which is reasonable since this is just the Lie derivative of $\Psi[\chi]$ along the surface.[Compare, for example, to eqns.(<ref>) and (<ref>) for $\rho[\chi]$ and $\Phi[\chi]$.] The local normal evolution of $\Psi$, on the other hand, is given by

\begin{eqnarray}
\left \{\Psi_{t}[\chi],\tilde{H}_{\perp x}\right \} =-i \, \hat{H}_{\perp x}\Psi_{t}[\chi]~.
\end{eqnarray}

Inserting these results into eq.(<ref>) for a general evolution of $\Psi_{t}[\chi]$, we then have

\begin{eqnarray}
i \, \partial_{t}\Psi_{t}[\chi] = \int dx\left (N_{xt}\hat{H}_{\perp x}+N^{i}_{xt}\hat{H}_{ix}\right )\Psi_{t}[\chi]~.\label{6 Quantum Schrodinger equation}
\end{eqnarray}

Finally, substituting eqns.(<ref>) and (<ref>) in for $\hat{H}_{\perp x}$ and $\hat{H}_{i x}$, respectively, yields the equation

\begin{align}
i \, \partial_{t}\Psi_{t} &= \int dx \left [ N_{xt}\left(-\frac{1}{2g^{1/2}}\frac{\delta^{2}}{\delta\chi^{2}_{x}}+\frac{g^{1/2}}{2}g^{ij}\partial_{i}\chi_{x}\partial_{j}\chi_{x}+g^{1/2}V_{x}\right ) - N^{i}_{xt}\partial_{i}\chi_{x}\frac{\delta}{\delta\chi_{x}}\right ]\Psi_{t}
\end{align}

which is, ostensibly, just a linear differential operator for the complex variable $\Psi_{t}$, which suggests calling it a Schrödinger functional equation. We discuss below.

*But is it quantum?—
The question of which criteria a theory must obey to deserve being called a quantum theory is a matter of taste and of convention. If the criteria involve the presence of $\hbar$, of a wave equation for a complex wave function, with an uncertainty principle and non-local correlations, the reconstruction of the standard formalism of quantum field theory in the limit of say, a flat space-time, then the ED model formulated above definitely qualifies as being quantum. On the other hand, if the defining criterion is the existence of a superposition principle, then our ED model is not a quantum theory.

To see that the coupling to classical gravity implies violations of the superposition principle we note that geometrodynamics is a constrained dynamical system, where the operative constraints are given by the Hamiltonian constraints in (<ref>). Solving these constraint equations often involves solving for components of the metric in terms of the quantum sources, which then gets fed back into evolution equations for $\Psi$.[This dynamic can most readily be seen within the linearized gravity regime, or the weak field limit. In fact, there are some arguments (see e.g., [208]) that the so-called Newton-Schrödinger equation, which is a non-linear equation, results from the Newtonian limit of the semi-classical Einstein equations. Investigation of this result coming from ED is underway.] This feedback leads to a non-linear time evolution (see e.g., [207]) in which the $\Psi$ itself appears as a potential in eq.(<ref>).

§ CONCLUDING REMARKS

The ED developed here couples quantum “matter" to a dynamical background on the basis of three key principles: (1) A properly entropic setting wherein the dynamics of probability is driven by information encoded into constraints. (2) The preservation of a symplectic structure as a primary criterion for updating the evolving constraints. (3) Imposing the Poisson bracket “algebra" of DHKT as a representation of the kinematics of surface deformations.

Our approach results in several interesting features.
Although written in the relatively less common language of geometrodynamics,
the eqns.(<ref>) and
(<ref>)-(<ref>) are
formally equivalent to the so-called semi-classical Einstein equations

\begin{eqnarray}
G_{\mu\nu}=8\pi G\langle \Psi | \hat{T}_{\mu\nu}| \Psi\rangle
~,\label{6 Semi classical equations}%
\end{eqnarray}

with classical Einstein tensor $G_{\mu\nu}$ (see e.g., [206]),
but sourced by the expected value of the quantum stress-energy
tensor.[The components of the quantum stress-energy tensor are
related in a simple manner to the ensemble quantities in
eqns.(<ref>) and
(<ref>). The derivation of the SCEE from an action
by Kibble and Randjbar-Daemi in [207] makes this
relationship more precise.] Such a theory of gravity has long been seen as a
desirable step intermediate to a full theory of QG, in part because it
contains well-established physics — QFTCS and classical GR
— in the limiting cases where they are valid. But there has been much debate
(see e.g., [83][209][210][211]), on the other hand, as to the status of
semi-classical theories as true QG candidate; with many harboring a negative view. In the next chapter we conclude by tackling these criticisms head on, discussing what's good, what's bad, and what's ugly about the ED approach to semi-classical gravity.

CHAPTER: THE ENTROPIC DYNAMICS APPROACH TO SEMI-CLASSICAL GRAVITY — IS IT VIABLE?

“Today there is no interpretation of quantum mechanics that does not have serious flaws, and that we ought to take seriously the possibility of finding some more satisfactory other theory, to which quantum mechanics is merely a good approximation."—  [212]

Over the past century experimentalists and engineers have managed to extract a wide array of new predictions and technologies out of quantum theory. For all practical purposes, quantum theory, whatever that might mean, appears to be working just fine. Why is it then that a certain segment of theorists keep squawking about the problems of quantum theory, like canaries in a coal mine that never seems to collapse? Indeed, what is it about textbook quantum theory that is so difficult for some theorists to accept? Are they just being pedants with a penchant for intellectual tidiness, or is there something valid in their criticism of quantum theory? Is it rigor, or is it just many words and hot air?

Here we have taken the view that there are serious problems with quantum theory that cannot be ignored. The prevailing view that interpretations of quantum theory have no consequences for physics is not just wrong, you might say it is not even wrong. The standard Copenhagen approach, complete with its division of the universe into microscopic/macroscopic, quantum/classical, system/apparatus, readily admits that it cannot describe nature as an undivided whole. Surely seeking to bridge this gap is in the purview of physics. Indeed, what, after all, is the projection postulate? Is it a physical law? If it is, then it is the job of physicists to explain this.[If it is not, we have argued here, based on [41][42], that it is an instance of entropic updating in the form of Bayes rule.] From another point of view, if one adopts a Many Worlds, or Everettian [9] interpretation, then what is to be made of the apparent adherence of experimental outcomes to the Born rule? Can this be derived? If not, does it make sense to look for violations? Alternatively, postulating additional microscopic variables to make sense of quantum theory, such as in the de Broglie-Bohm approach, introduces new questions, but also new opportunities, that are not present in the other dominant interpretations (see e.g., [213]). The point is that understanding quantum theory, successful though it might be, is not an empty pursuit. Just the contrary, it is by truly fleshing out its conceptual framework that we gain the ability to ask those questions which may, in time, probe the blemishes of quantum theory and not just reify its virtues.

“Ultimately, we need not just to calculate a theory's predictions for familiar types of experiment, but to understand the theory's properties. The reason is that this is crucial in deciding the extent to which a theory has been tested, and hence in designing new tests."—  [214]

In this work we have advocated for using the Entropic Dynamics (ED) framework for the purpose of constructing the quantum formalism. The benefit of such an approach is that there is no need to interpret the theory, its interpretation is built into its construction. In ED, we presume the existence of an external world whose features are not always perfectly known, but are, in principle, knowable. Our incomplete information about nature is therefore codified in probabilities and in constraints. Thus in ED the quantum state $\Psi$ is fully epistemic in character, i.e. it is a tool for reasoning about nature. However, it is a tool for reasoning about nature on the basis of some very particular pieces of information. An advantage of the ED approach is that it makes explicit that information which does lead to a fully quantum dynamics, and hence also that which does not.

One of the chief developments in ED over the past few years, motivated in large part by the work in this thesis, has been the recognition that the constraints that result in a quantum theory are constraints based in symmetry principles. In the context of quantum scalar fields in curved space-time, which has been our primary focus here, there have been two constraints that have proved particularly crucial: (I) Conservation of a symplectic form, (II) Foliation invariance symmetry. A consequence of the ED approach is the insight that these constraints apply, not directly to the ontic microstates themselves, but rather, to the epistemic macrostates $\rho$ and $\Phi$. From this perspective, it is not quite appropriate to address whether the fields themselves propagate in accordance with relativity, instead it is the case that the statistical predictions made on the basis of these constraints will not violate relativity.

As mentioned previously, however, the constraints (I) and (II) do not quite fix the unitary time evolution characteristic of quantum dynamics. For this, some additional inputs are required; whether that be in the form of a quantum potential, as we have employed here, or through additional constraints besides that (see e.g.,  [54]). In any case, the ability in ED to separate models into quantum/non-quantum should be considered a feature of the approach, as this allows an assessment of the conditions under which that linear structure may emerge, and those situations in which it does not. To this end, among the central results of this thesis is that quantum field theory in a curved space-time emerges rather naturally on the basis of the constraints (I) and (II), plus a quantum potential, so long as the background is fixed. The situation of a dynamical background, on the other hand, is qualitatively different. The  ingredients that lead to a full quantum theory in the context of a fixed background, no longer do so when the background is allowed to evolve. It is, in fact, a fundamental aspect of the coupling of quantum scalar fields to a dynamical background that the theory will instead be nonlinear. While surely one expects a regime in which these nonlinearities can be neglected and we recover the standard quantum field theory in curved space-time, this must be regarded only as an approximation to the full dynamics. In this approach there is thus no unitary time evolution, nor any quantum superposition principle. Period.

What then should we make of this? Do we trust this model, or are these features so unsavory that the whole thing should be dispensed with? That is to say, is the ED approach to semi-classical gravity viable?

§ PROBLEMS AND NON-PROBLEMS IN SEMI-CLASSICAL GRAVITY

“On principle, it is quite wrong to try founding a theory on observable magnitudes alone. In reality the very opposite happens. It is the theory which decides what we can observe."— , [215]

Semi-classical gravity often refers to the coupling of relativistic quantum fields to a dynamical background that evolves classically and has typically been evoked phenomenologically, as a means to model processes where quantum effects are present, but gravity cannot be ignored. On a formal level there appear to be clear similarities between the usual semi-classical gravity considered in the literature (hereby simply semi-classical gravity) and the ED model developed here, right down to the semi-classical Einstein equations, which are the operative dynamical equations that are shared by both. There are therefore a whole host of mathematical features that are common to ED and semi-classical gravity alike. However, the conceptual underpinnings of semi-classical gravity, to the extent that they are ever seriously considered, deviate wildly from those of ED. Here we detail some of the standard charges leveled against semi-classical gravity and discuss whether similar criticisms apply to ED, pointing to future work in situations where such concerns are valid.

*On divergences—
One area, in particular, where our development of ED has been rather silent pertains to the infrared and ultraviolet divergences that are prevalent in all local quantum field theories, and thus to semi-classical gravity, as well. To this end, ED can take advantage of the extensive work done in semi-classical gravity the past few decades by adapting to ED methods meant to treat technical issues encountered first in semi-classical gravity, such as those relating to regularization and subsequent renormalization. To the extent that these methods are unsatisfactory in semi-classical gravity, perhaps because they require particular symmetries or that the differing techniques give similarly differing answers, they will also likely fail to be optimal in ED, as well.

For ED there is also the added difficulty that it is formulated most naturally in the Schrödinger representation, which has not proven to be the most convenient setting for demonstrating renormalizability in many quantum field theory applications. While one may conceive of first transforming to the Heisenberg or interaction pictures for computational purposes, such a transformation would technically require a well-defined $ \hat{T}_{\mu\nu}$ to do so; nonetheless, an attempt in this direction for simple semi-classical models is given in [216], and for semi-classical gravity itself in [217]. Other schemes based firmly in the Schrödinger representation have also been explored (see e.g., [156][172][218]), but it would be rather misleading to suggest that the literature on this subject is nearly as rich as in the usual approaches.

Nevertheless, it is possible that what is needed to address the issue of divergences that plague relativistic quantum field theories is not necessarily a technical breakthrough, but some additional physical insights. One option in this direction is the introduction of a minimum length. Widely viewed as a plausible consequence of most quantum gravity candidates, such a notion would certainly remedy many of the issues with ultraviolet divergences. That being said, implementation of this notion often itself comes accompanied with several issues; either by requiring a fully quantized quantum gravity theory, such as in loop quantum gravity or string theory, which have their own obvious technical hurdles, or an explicitly discretized rendering of space-time, which consequently also breaks invariance under the full Lorentz group. Drawing on the work of R.L. Ingraham [219][220], and later A. Caticha [190][191], another possibility is that the statistical fluctuations of space itself may induce a minimum length that would still be fully compatible with the usual relativistic symmetries. Such an approach would clearly be within the spirit of an ED  driven by information codified into constraints, and thus suggests a promising avenue of future research.

*On the coupling to gravity—  Criticism of semi-classical models of gravity often, quite rightfully, point to the coupling of gravity to matter in (<ref>) as being rather ad hoc. Indeed, the usual argument for the expectation value $\langle \hat{T}_{\mu\nu} \rangle$ on the right hand side of eq.(<ref>) is at best a guess. In ED, on the other hand, the coupling of geometry to the expected value of matter is derived on the basis of well-defined assumptions and constraints. In other words, here we derive semi-classical gravity from first principles. Moreover, such principles have already been tested elsewhere: not only do they provide a reconstruction of general relativity through the work of Dirac, Hojman, Kuchař, and Teitelboim, but they also provide a reconstruction of non-relativistic quantum mechanics and relativistic quantum field theory.

Related to this is the issue of identifying an appropriate quantum stress energy tensor $\hat{T}_{\mu\nu}$ in the first place. Even in classical general relativity there are a couple of definitions one may choose for the stress-energy tensor $T_{\mu\nu}$; either by simple variation of the action, or possibly through the construction of symmetrized Noether currents. But this problem is made considerably worse in quantum field theory, especially in a curved space-time. One chief reason for this is that, in the standard approaches, the stress-energy operator $\hat{T}_{\mu\nu}$ consists of products of operators, or more precisely, operator-valued distributions. As is familiar from just ordinary quantum mechanics, there is an ambiguity in how to order these operators, as compared to their classical analogues, owing to their lack of commutativity. Besides this, however, products of distributions are also know to be poorly-defined mathematically, an issue that is compounded in curved space-time since the usual techniques for regularization available in Minkowski space are no longer available, and there is no suitable replacement valid for generically curved space-times.

Some of these challenges are avoided in ED. First, because the approach, inspired by Dirac, Hojman, Kuchař, and Teitelboim, relies from the outset on the canonical formalism there are no problems with identifying the sources of gravitation due to matter. Indeed, such sources are constructed explicitly through “algebraic" methods. Since ED also dispenses with the notion of fields as operators, and the Hilbert spaces on which they act, there are no issues with operator ordering, either; a marked improvement on the usual approaches. As remarked above, however, ED shares with standard approaches the complication of constructing a consistent, finite value of $\langle \hat{T}_{\mu\nu} \rangle$, readily definable in all space-times of interest. This is a matter of great importance and must absolutely be addressed in some way, shape, or form. Again, the work of A. Caticha in devising fully relativistic models with a minimum length appears to be a promising direction.

*On some features of semi-classical Einstein equations—
With the semi-classical equations (<ref>) in hand, it is reasonable to inspect the character of their solutions. We may be interested with, for example, whether the resulting space-times are similar to that of classical general relativity, that is, whether the solutions have some qualitatively new features. But a major difficulty is that the semi-classical Einstein equations are thoroughly daunting. One reason for this, as we discussed above, is that eqns.(<ref>) are not quite correct on their own, they need to be fixed to take care of the divergences implicit in $\hat{T}_{\mu\nu}$. Addressing this issue in the standard way induces additional terms in eqns.(<ref>) that are, for instance, quadratic in the Ricci curvature tensor $R_{\mu\nu}$. Thus while the classical Einstein equations, which contain derivatives of up second order, have a well-posed initial value formulation, the corresponding semi-classical equations may not owing to the counterterms introduced by renormalization. Such systems of equations are therefore often unstable [221]; and indeed, there are hints that the Minkowski vacuum state is not a stable solution to these equations (see e.g., [222][223]).

Related to this, in classical physics it is well understood that Einstein's equations do not have much content without imposing some reasonable conditions on the distribution of energy-momentum (see e.g., [206]). These are typically called energy conditions and they come in a variety of flavors; a common one is the so-called weak energy condition $T_{\mu\nu}u^{\mu}u^{\nu}\geq 0$, for time-like $u^{\mu}$, but there are other, such as the strong energy condition, null energy condition, dominant energy condition, and more besides these (see e.g., [224]). However many of these conditions appear to be violated by quantum field theory quite generically [225], and certainly therefore in semi-classical gravity as well (see e.g., [226]). On its face this appears problematic. First since many trusted results in gravitational physics, as well as black hole physics and thermodynamics, seem to rely on these conditions. But also because there are arguments that violations of these conditions lead to issues with causality (see e.g., [226][227][228]). Nevertheless, it is possible that these problems can be tamed somewhat by resorting to alternative energy conditions (see e.g., [229][230][231]).

It is not our goal here to address these issues in any significant way other than simply to acknowledge that they are there and need to be investigated. ED, of course, is not a cure all for problems such as these and we do not expect that the specific model laid out here will be particularly advantageous in settling all of these difficulties. What ED does provide, however, is the kind of conceptual clarity that may allow future progress. How, for instance, does a minimum length model change the issues of stability? Alternatively, below we discuss how ED avoids the common claim that semi-classical gravity results in superluminal signaling. Can the issues of causality resulting from energy condition violations be similarly avoided in ED? Indeed, in what sense are these conditions violated? Are these violations on the expected values of stress-energy? Or are they violations on the level of the eigenstate spectrum, and thus connected to measurement? Indeed, the ED perspective suggests we view the latter with a healthy dose of suspicion.

*On ontology/epistemology—
In ED issues of interpretation are carefully addressed, while a similar level of conceptual rigor is often severely lacking in most standard accounts of semi-classical gravity. Indeed, it is entirely plausible that the exact same mathematical framework may be wielded to produce entirely different answers to the very same question. Thus a direct comparison of the ED approach with the semi-classical gravity encountered in the literature can be somewhat misguided without additional considerations.

One area where the ED approach proves particularly valuable is in maintaining a clear distinction between the ontic/epsitemic aspects of semi-classical gravity. In ED the field variables are ontic, or “beable", whereas their probabilities are fully Bayesian, and thus epistemic. Somewhat less clear, however, is the status of the additional variables, which include the geometry and the drift potential. But here they too are taken to be of epistemic origin. From this viewpoint, while these variables do characterize information relevant for the prediction of the ontic microstates, they themselves are not subject to direct measurement in an experiment. Instead, it is rather more appropriate to claim that such variables are inferred from an ensemble of measurements, much like parameters entering into a statistical model, suggesting the name “inferrables".

On its face this is, of course, a rather shocking (and perhaps criminal) stance to take, given that it seems to grant the gravitational field a secondary physical role. But note crucially that the gravitational field being epistemic is not to say it lacks physical content. Indeed, similar concerns could be raised about the role of temperature in statistical physics. Is temperature physical? Clearly the notion of temperature enters in our description of physics, but we are under no illusions that it is temperature itself that we are observing in a column of mercury. Therefore, just as we may regard the notion of temperature as being “physical" but not ontic, a similar interpretation may be given to the gravitational field in the ED approach to semi-classical gravity.

This interpretation complicates the typical analysis of semi-classical gravity. In fact, even the name is severely misleading. It implies that there is something “classical" in the first place. What though is “classical" about Entropic Dynamics? One could, perhaps, point to the fact that gravity in our model shares with classical mechanics a common mathematical formalism. But this is hardly useful.  Indeed, in ED the quantum state $\Psi$ is similarly described within the canonical framework — is $\Psi$ classical, too? Clearly not. What truly distinguishes classical gravity from the gravitational field in our approach is that we do not adopt the, often tacit, interpretive postulate of classical mechanics, namely, that the points of phase space are ontic degrees of freedom. Thus the classification of variables into classical/quantum on the basis of their mathematical description appears unsatisfactory in ED. Likely better is the division of variables into ontic/epistemic, which is mirrored by the division into beable/inferrable.

*On superluminal signaling—
Owing to the absence of experiments that could test semi-classical gravity (or any other theory of quantum gravity) on an empirical level, considerations of interpretation have come to have great weight in assessing its viability. In the past few decades there have, in fact, been several influential arguments, first by Eppley and Hannah [209], later by Page and Geilker [210], as well as by Unruh [83], that sought to rule out semi-classical gravity on purely theoretical grounds.

The basic idea is that, while we tend to think of classical or quantum mechanics on their own as being, in some sense, self-consistent (a claim that is itself highly questionable), the status of hybrid models where gravity is classical and matter is quantum, are in comparison less understood. Thus it appears reasonable to investigate aspects of such theories, particularly with regards to measurement, to understand whether they conflict with well-established physical principles, such as energy-momentum conservation, the uncertainty principle, or even relativity. Since many objections to semi-classical gravity utilize a similar reasoning, it is useful to address their criticisms.

Take the argument of Eppley and Hannah (EH), for instance, whose work is frequently relied upon. We follow the treatment due to Huggett and Callender [175]. EH consider a gedanken experiment in which there is an electron in a box that is later partitioned by a barrier so that the electron is equally likely to be on either side of the divide. The state of the electron is therefore given by

\begin{equation}
\psi = \frac{1}{\sqrt{2}}(\psi_{L}+\psi_{R})~,\notag
\end{equation}

where $\psi_{L}$ and $\psi_{R}$ are wavefunctions of identical shape but with domains of support on opposite sides of the box. Each half of the box is then carried far apart by two friends, Armin and Bahar, who take the left and right box, respectively.

In a standard version of such an experiment, Bahar looks in her box and, for instance, finds it empty. According to the usual Copenhagen interpretation, Bahar has performed an experiment that “collapses" the wavefunction $\psi \to \psi_{L}$, which thereby instantaneously “produces" an electron in Armin's box. If quantum mechanics is a complete description of reality then this, in the words of Einstein, corresponds to some “spooky action at a distance". Nevertheless, as is well known, this non-locality cannot be used to send superluminal signals from Bahar to Armin. This is because when Armin performs a single trial he is not aware of whether Bahar has already performed her measurement or not, so finding (or not finding) an electron is not informative. Even in the long run, after Armin and Bahar have repeated many trials, each will see the electron roughly about half the time regardless of whether the other has looked in their box or not (see e.g., [84]).

In the analysis of EH things are changed by allowing gravity to interact with a quantum system. They consider two possibilities: either (a) the interaction will collapse the wavefunction, or (b) the interaction will not collapse the wavefunction. Before continuing, it is worthwhile to note that EH must proceed like this, by classifying modes of collapse, because they do not have a proper model of collapse themselves. This turns out to be problematic, first because ED is just such a theory that it does not fall into either of the categories laid out by EH, and thus evades their conclusions. But second, Albers et al [84] make the point that if the interaction does, in fact, collapse the wavefunction, corresponding to option (a), then the details of that interaction will be important in examining the behavior of the system. But there have been many attempts in recent years to model the collapse of the quantum state due to gravity, with varying degrees of success. (For recent reviews of the subject, see e.g., [232][233].) So long as such models are actively being improved, and not experimentally ruled out, the arguments by EH cannot be conclusive on this front.

We are thus left to consider option (b), that the gravitational interaction does not cause the state to collapse. Here more definite complications appear to arise. If gravity can interact with a quantum system without causing collapse then one could consider a situation where an incident gravitational wave “scatters" off of a quantum system. Presumably, in such a scenario, how the wave behaves depends on the state of the quantum system. But it is exactly at this point that one confronts problems with relativity. If Armin can always monitor his box, and therefore probe the quantum state, he can determine if and when Bahar has performed a measurement, and even perhaps the outcome of her measurement. However, if the collapse occurs instantaneously, as it is typically assumed, then Armin will likewise have instant access to the actions of Bahar, who we have already assumed is far away. Thus Huggett and Callender conclude that semi-classical gravity leads to superluminal signaling, subject to (i) the standard interpretation of quantum mechanics, (ii) the division of “the world", or ontology, into gravitational and quantum parts, and (iii) that gravity does not force a collapse.

ED avoids this conclusion in essence because it avoids all three of these criteria. First, as mentioned above, in ED there is a clear delineation between variables that are ontic and those that are epistemic. In the analysis of EH, they always assume that “semi-classical" means not just that the gravitational field has a certain mathematical form, but that it is a separate physical resource that can be controlled, manipulated, and detected in a measuring device. By contrast, in our model, while the gravitational field always has a precise value, this value cannot elicited in the process of measurement. It must instead be inferred from reading a material detector that is itself quantum mechanical. Thus the detector and the electron must be treated together by a joint quantum state. Furthermore, ED does have a clear interpretation, and it is certainly distinct from the usual Copenhagen interpretation. This allows ED to satisfactorily resolve the problem of measurement [41][43]. Indeed, in ED a measuring device is not described as a black box subject to rules that violate the Schrödinger equation, but as a physical process subject to the very same laws that describe the rest of the world. As a result, in a fully covariant ED, such as the model developed in this thesis, the process of measurement is described by the same causal flow of probability that characterizes any other physical process. Claims, like those of EH and others, that semi-classical gravity leads to superluminal signaling are thus, more properly, viewed as an artifact of a rather primitive interpretation of quantum theory. Given a satisfactory relativistic model of the measuring device itself, no such issues are expected occur in the ED approach.

*On macroscopic superpositions—
Another source of confusion with semi-classical gravity is the issue of what happens when the location of a macroscopic source is uncertain. (See e.g., [83][208].) Consider, for example, a
macroscopic mass $m$ that is equally likely to be at $x_{1}$ or at $x_{2}$.
Will the gravitational field itself be equally likely to point either towards
$x_{1}$ or towards $x_{2}$? Or, will the gravitational field be as if
generated by a mass $m$ located at the expected position $(x_{1}+x_{2})/2$? The resolution of this paradox in ED demonstrates the advantages of having a derivation from first principles. The $\langle\hat{T}_{\mu\nu}\rangle$ was derived, or better, it
was inferred from a very specific type of information that leads, by an
abuse of language, to what one might call a pure quantum state. The issue then is what
is the gravitational field generated by a pure state that happens to be a
macroscopic "Schrödinger cat"? To the extent that this is a
“pure” state then ED gives a sharp
prediction: the gravitational field is generated by an $\langle\hat{T}_{\mu
\nu}\rangle$ centered at the average position.

But as highlighted many years ago by Unruh [83], such a prediction seems to contradict our everyday expectation that the gravitational field is produced from a source at the location $x_{1}$ or $x_{2}$, not their average $(x_{1}+x_{2})/2$. It is worth revisiting Unruh's argument, albeit in a slightly modified form.[To save space, we adapt Unruh's argument to the situation described above. The key difference is that in Unruh's experiment $\langle\hat{T}_{\mu\nu}\rangle$ was time dependent, which is not a crucial fact in the argument.] He considers the situation where the macroscopic superposition above was prepared in a box with a Cavendish balance just outside of it which would swing depending on which side of the box the mass actually resided. Naturally, one would not expect to observe the balance and see it at the center, rather it should presumably swing to one side or the other. Since semi-classical gravity seems to predict the former, Unruh goes on to argue that it must be wrong, or in some sense incomplete. But this conclusion does not apply to the ED approach, so long as one is careful to include all available information into the analysis. The key insight is that the balance is itself a material detector which, if it is useful at all, will be correlated (through interactions) with the matter inside. Observing the position of the balance is therefore no passive activity since it gives you information about the state of matter inside the box.[In fact, one does not even need ED to resolve this issue. The notion that the measuring device should also be treated quantum mechanically is one that goes back to von Neumann and is a firmly established practice even in the Copenhagen interpretation.]

Thus a full accounting of the available information, including the state of the balance and its correlations with the matter inside the box, leads to an objective assignment for $\Psi$ where $ \langle \hat{T}_{\mu\nu} \rangle$ is centered on either $x_{1}$ or $x_{2}$. On the other hand, if you include the system and detector into the analysis such that the state of neither is known with certainty, then semi-classical gravity might predict a gravitational field sourced at the center position (or something else entirely, depending on the detector itself). Therefore we conclude that there is no paradox. What we have are two predictions drawn on the basis of two distinct informational states, each of which is entirely compatible with the ED approach to semi-classical gravity.

*On quantum coherence—

Following the above discussion, a more problematic issue for semi-classical gravity, as we have described it thus far, is the extent to which it is even possible to incorporate all of the relevant information. After all, real experimental setups are rather complicated and messy. Just take the scenario with the macroscopic “Schrödinger cat" states described above and place it in a terrestrial laboratory, for example. The matter in our “cat" state is correlated with the balance, and both are correlated with the matter that makes up the lab and planet it resides on. What is the gravitational field, now?

Echoing our earlier position, in ED we recognize the role of information. In practice we cannot keep track of all the variables affecting our system, and this is particularly true when the system in question reaches macroscopic proportions. Thus such a Schrödinger cat state could not be physically realized anyway: it would immediately suffer decoherence. To analyze such a situation the ED framework would need to be extended to incorporate information (i.e., additional constraints) that describes additional sources of uncertainty. In this extended ED it is conceivable that the gravitational field itself would be uncertain. This is an avenue of research that is currently under investigation.

CHAPTER: FORMALISM AND NOTATION

We deal here with a set of dynamical variables $\rho(x)$ and $\Phi(x)$ that are fields over a configuration space $x\in\mathbf{X}_{N}$, such that

\begin{equation}
\rho(x):\mathbf{X}_{N}\to\mathbb{R}\quad\text{and}\quad\Phi(x):\mathbf{X}_{N}\to\mathbb{R}.
\end{equation}

The notation that we introduce here deals with the particle theory introduced in chapter 3 to maintain brevity. However, the shift to fields does not introduce any major changes. Indeed, the major distinction is that integrals for fields will (often) take the form,

\begin{equation}
\int_{\mathbf{X}_{N}} d^{3N}x \left(\cdot\right)\quad\longleftrightarrow\quad \int_{\mathcal{C}} D\chi \, \int_{\mathbb{R}^{3}}d^{3}x \left(\cdot\right).
\end{equation}

This is true unless otherwise noted.

Let the space of probabilities $\rho(x)$ be $\mathcal{E}_{\rho}$ and the space of phase functions $\Phi(x)$ be $\mathcal{E}_{\Phi}$ so that the joint space can be written as[We are not concerned here with the detailed structure of $\mathcal{E}_{\rho}$ and $\mathcal{E}_{\Phi}$, however, it is interesting to note that the space $\mathcal{E}_{\rho}$ does have some interesting properties. For one, it has a Riemannian metric, the Fisher-Rao metric,

\begin{equation}
dl^{2} = C \int dx\frac{\delta\rho_{x}^{2}}{\rho_{x}} = 4C\int dx \left(\delta\rho^{1/2}_{x}\right)^{2}~.
\end{equation}

This structure turns out to be valuable for reconstructing the full formalism of quantum theory (see e.g., [54]).]

\begin{equation}
\Gamma = \mathcal{E}_{\rho}\otimes\mathcal{E}_{\Phi}.
\label{Particles Gamma defined}
\end{equation}

The space $\Gamma$ will be called the ensemble phase space, or e-phase space for short. We will see shortly that this is an appropriate name since the pair $\rho(x)$ and $\Phi(x)$ will form a canonical pair with $\rho(x)$ being the canonical coordinate and $\Phi(x)$ the conjugate momentum.

Continuing on, a point $X\in\Gamma$ will be given by coordinates $X^{\alpha x} = \left(\rho(x),\Phi(x)\right)$, where $\alpha$ takes on either of two values, $\rho$ or $\Phi$. (We use notation throughout that is consistent with [54].) Moreover, it is clear from the notation that $\Gamma$ is a doubly infinite-dimensional manifold: two-dimensions — one for $\rho(x)$ and another for $\Phi(x)$ — for each point $x\in\mathbf{X}_{N}$. As such, the dynamical system in question is also infinite-dimensional. This fact brings with it certain technical niceties (see e.g., [125]), but by-in-large, we will assume that things are well-defined for the objects we are interested; when such subtleties are, in fact, relevant, we will be sure to mention them.

*Ensemble functionals
Perhaps the simplest object that can be defined on $\Gamma$ is an ensemble functional, or e-functional, which is formally a rank-zero tensor. Such tensors $\tilde{F}[\rho,\Phi]=\tilde{F}[X]$ are mappings satisfying

\begin{equation}
\tilde{F}[\rho,\Phi]:\Gamma \to \mathbb{R},
\label{Particles e-functional mapping}
\end{equation}

which produce a real number at every point $X\in\Gamma$. Related to this is the notion of a functional derivative. Consider an infinitesimal variation of an e-functional

\begin{equation}
\delta\tilde{F}[\rho,\Phi] = \int dx \left(\frac{\delta\tilde{F}}{\delta\rho(x)}\delta\rho(x)+\frac{\delta\tilde{F}}{\delta\Phi(x)}\delta\Phi(x)\right),
\label{Particles e-functional variation}
\end{equation}

which follows from the rules of calculus. The coefficients of the variations of $\delta\rho(x)$ and $\delta\Phi(x)$ are then identified as the functional derivatives with respect to $\rho(x)$ and $\Phi(x)$, respectively; one can think of these as the generalization of a partial derivative to spaces of infinite dimensions (see e.g., ch.2 of [129] for an introduction).

Continuing on, vectors $\bar{\mathbf{V}}$ are rank $(0,1)$ tensors that are elements of $T_{X}$, the tangent space of $\Gamma$ at the point $X$. (We use the notation that all tensors, other than e-functionals, are identified with bold-face font. The components of tensors, however, do not since the index placement makes the tensor stucture clear. Also, vectors and one-forms being special, receive a “bar" and “tilde" overtop, respectively.) These tensors have components given by $V^{\alpha x}$, using a notation analogous to that used for $X^{\alpha x}$.

The coordinates $X^{\alpha x}$ also establish a set of coordinate basis vectors $\bar{\mathbf{e}}_{\alpha x}$ on $\Gamma$ (see e.g., [119]), given by[Although it is unusual to think of basis vectors $\bar{e}_{\alpha x}$ as linear differential operators $\delta/\delta X^{\alpha x}$, the index notation makes it clear that they do, in fact, transform in exactly the same way. And, this is nonetheless a common convention, so we employ it here as well.]

\begin{equation}
\bar{\mathbf{e}}_{\alpha x} = \frac{\delta}{\delta X^{\alpha x}} = \left(\frac{\delta}{\delta \rho(x)},\frac{\delta}{\delta \Phi(x)}\right).
\label{Particles coordinate basis}
\end{equation}

In this coordinate basis $\bar{\mathbf{e}}_{\alpha x}$, given by the variables $\rho(x)$ and $\Phi(x)$, a generic vector $\bar{\mathbf{V}}$ reads

\begin{equation}
\bar{\mathbf{V}} = V^{\alpha x}\bar{\mathbf{e}}_{\alpha x} = \int dx \left(V^{\rho x}\frac{\delta}{\delta\rho(x)} + V^{\Phi x} \frac{\delta}{\delta\Phi(x)}\right),
\label{Particles Vectors Gamma}
\end{equation}

where we have introduced the convention that repeated indices $\alpha$ and $x$ are summed and integrated over, respectively. We will use this notation for other operations as well.

A vector field is then a rule for assigning a vector $\bar{\mathbf{V}}$ to every point $X\in\Gamma$. That is, to each point $X\in\Gamma$ there is a tangent space $T_{X}$, and a vector field (or any tensor field, for that matter) is a rule for choosing one element of $T_{X}$ for each point $X$. Conversely, at each point along a curve in $\Gamma$ there is a tangent to that curve. For a given vector field $\bar{\mathbf{V}}$, the set of curves for which their tangents correspond to $\bar{\mathbf{V}}$ at each point $X$ are called the integral curves of that vector field. This is important because for a certain distinguished vector field, the Hamiltonian vector field for dynamics, the integral curves correspond the trajectories of $\rho(x)$ and $\Phi(x)$ through $\Gamma$.

A one-form is then defined in terms of vectors and e-functionals as that linear mapping that takes a vector as an input and produces an e-functional. That is, a one form $\tilde{\mathbf{B}}$ is a $(1,0)$ rank tensor,[Note that we have abused the “tilde" notation since it used both for one-forms $\tilde{\mathbf{B}}$ and e-functionals $\tilde{F}$. However while e-functionals receive plain font, the one-forms are bold-faced. The “tilde" notation is standard for one-forms in differential geometry, but it is also standard in ED to use them for e-functionals. We think the current notation is able to satisfy both conventions.] with components $B_{\alpha x}$, such that for any vector $\bar{\mathbf{V}}\in T_{X}$ we have

\begin{equation}
\tilde{\mathbf{B}}\left(\bar{\mathbf{V}}\right):T_{X}\to\mathbb{R}\quad\text{where}\quad \tilde{\mathbf{B}}\left(\bar{\mathbf{V}}\right)  =V^{\alpha x}B_{\alpha x}.
\label{Particles one-form vector map}
\end{equation}

Conversely, as the notation in (<ref>) suggests, we may also view a vector $\bar{\mathbf{V}}$ itself as a linear mapping of one-forms to real numbers: $\tilde{\mathbf{B}}\left(\bar{\mathbf{V}}\right) = \bar{\mathbf{V}}\left(\tilde{\mathbf{B}}\right) = V^{\alpha x}B_{\alpha x}$. Vectors are then said to be dual to one-forms, and vice versa. One forms are thus elements of $T^{*}_{X}$, the cotangent space at the point $X$, i.e. the space dual to the tangent space $T_{X}$.

A well-known example of a one-form is the gradient of a function, here the gradient of an e-functional. For an arbitrary e-functional $\tilde{F}[X]$ this is obtained by application of the exterior derivative,

\begin{equation}
\tilde{\mathbf{d}}\tilde{F}[X] = \int dx \frac{\delta\tilde{F}[X]}{\delta X^{\alpha x}}\tilde{\mathbf{d}}X^{\alpha x} = \int dx \left(\frac{\delta\tilde{F}[X]}{\delta \rho(x)}\tilde{\mathbf{d}}\rho(x)+\frac{\delta\tilde{F}[X]}{\delta \Phi(x)}\tilde{\mathbf{d}}\Phi(x)\right).
\label{Particles gradient one-form}
\end{equation}

Let us explain this a bit. First of all, the notation is, perhaps, misleading. The exterior derivative $\tilde{\mathbf{d}}$ is not itself a tensor, it is a tensor operation valid on most “nice" manifolds with the property that $\tilde{\mathbf{d}}^{2} = 0$. The true tensor here is the object $\left(\tilde{\mathbf{d}}\tilde{F}\right)$ since it combines a valid tensor $\tilde{F}$ with a valid tensor operation $\tilde{\mathbf{d}}$; in fact, $\left(\tilde{\mathbf{d}}\tilde{F}\right)$ is an exact one-form.

From (<ref>) it is possible to identify the set of coordinate basis one-forms as being

\begin{equation}
\tilde{\mathbf{b}}^{\alpha x} = \tilde{\mathbf{d}}X^{\alpha x} = \left(\tilde{\mathbf{d}}\rho(x),\tilde{\mathbf{d}}\Phi(x)\right),
\end{equation}

which satisfy

\begin{equation}
\tilde{\mathbf{b}}^{\alpha x}\left(\bar{\mathbf{e}}_{\beta x'}\right) = \delta^{\alpha}_{\beta}\delta(x,x'),
\end{equation}

for the basis vectors $\bar{\mathbf{e}}_{\beta x'}$ given in (<ref>). The components of the gradient one-form are then given by

\begin{equation}
\left(\tilde{\mathbf{d}}\tilde{F}\right)_{\alpha x} =\frac{\delta\tilde{F}[X]}{\delta X^{\alpha x}} = \left(\frac{\delta\tilde{F}[X]}{\delta \rho(x)},\frac{\delta\tilde{F}[X]}{\delta \Phi(x)}\right),
\end{equation}

so that

\begin{equation}
\tilde{\mathbf{d}}\tilde{F} =\left(\tilde{\mathbf{d}}\tilde{F}\right)_{\alpha x}\tilde{\mathbf{b}}^{\alpha x} = \left(\tilde{\mathbf{d}}\tilde{F}\right)_{\alpha x}\tilde{\mathbf{d}}X^{\alpha x}.
\end{equation}

Similarly, a generic one-form $\tilde{\mathbf{B}}$ is written as

\begin{equation}
\tilde{\mathbf{B}} = B_{\alpha x} \tilde{\mathbf{b}}^{\alpha x}.
\end{equation}

*Higher rank tensors
Additional tensors can also be introduced on $\Gamma$. For example, a $(0,2)$ rank tensor $\mathbf{G}:\bar{\mathbf{V}},\bar{\mathbf{U}}\to\mathbb{R}$, takes two vectors $\bar{\mathbf{V}}$ and $\bar{\mathbf{U}}$ and produces a real number. Such a tensor can be written in a coordinate free notation as $\mathbf{G}\left(\cdot,\cdot\right)$, or in a coordinate dependent notation with components $G_{\alpha x,\beta x'}$ such that

\begin{equation}
\mathbf{G}\left(\bar{\mathbf{U}},\bar{\mathbf{V}}\right) = G_{\alpha x,\beta x'}V^{\alpha x}U^{\beta x'}.
\label{Particles Tensors Gamma a}
\end{equation}

We will also sometimes need a $(2,0)$ rank tensor $\mathbf{C}(\cdot,\cdot)$, with components $C^{\alpha x,\beta x'}$. This object takes two one-forms $\tilde{\mathbf{A}}$ and $\tilde{\mathbf{B}}$ as inputs such that

\begin{equation}
\mathbf{C}\left(\tilde{\mathbf{A}},\tilde{\mathbf{B}}\right) = C^{\alpha x,\beta x'}A_{\alpha x}B_{\beta x'}.
\label{Particles Tensors Gamma b}
\end{equation}

Higher rank tensors may be introduced in analogous ways, however this is not necessary for our purposes.

CHAPTER: THE LOCAL-TIME FOKKER-PLANCK EQUATIONS

To rewrite the dynamical equation (<ref>) in differential form consider the probability $P[\chi ,\sigma
|\chi _{0},\sigma _{0}]$ of a finite transition from a field
configuration $\chi _{0}$ at some early surface $\sigma _{0}$ to a
configuration $\chi $ at a later $\sigma $. The result of a further
evolution from $\sigma $ to a neighboring $\sigma ^{\prime }$ obtained from $%
\sigma $ by an infinitesimal normal deformation $\delta \xi _{x}^{\bot }$ is
given by (<ref>),

\begin{equation}
P\left[ \chi ^{\prime },\sigma ^{\prime }|\chi _{0},\sigma _{0}\right] =\int
D\chi \,P\left[ \chi ^{\prime },\sigma ^{\prime }|\chi ,\sigma \right] P%
\left[ \chi ,\sigma |\chi _{0},\sigma _{0}\right] ~.
\end{equation}

To obtain a differential equation one cannot just Taylor expand as $\delta
\xi _{x}^{\bot }\rightarrow 0$ because $P\left[ \chi ^{\prime },\sigma
^{\prime }|\chi ,\sigma \right] $ becomes a very singular object — a delta
functional. Instead, we multiply by an arbitrary smooth test functional $T%
\left[ \chi ^{\prime }\right] $ and integrate

\begin{eqnarray}
\int D\chi ^{\prime }\,P\left[ \chi ^{\prime },\sigma ^{\prime }|\chi
_{0},\sigma _{0}\right] T\left[ \chi ^{\prime }\right]
= \int D\chi \,P\left[ \chi ,\sigma |\chi _{0},\sigma _{0}\right] \int
D\chi ^{\prime }\,T\left[ \chi ^{\prime }\right] \,P\left[ \chi ^{\prime
},\sigma ^{\prime }|\chi ,\sigma \right] \,.  \label{Test Function a}
\end{eqnarray}

Next expand the test function $T\left[ \chi ^{\prime }\right] =T[\chi
+\Delta \chi ]$ in powers of $\Delta \chi =\chi ^{\prime }-\chi $. Since $%
\chi $ is Brownian to obtain $T\left[ \chi ^{\prime }\right] $ to first
order in $\delta \xi _{x}^{\bot }$ we need to keep second order in $\Delta
\chi _{x}$,

\begin{eqnarray}
T\left[ \chi ^{\prime }\right] =T\left[ \chi \right] +\int dx\frac{\delta T%
\left[ \chi \right] }{\delta \chi _{x}}\Delta \chi _{x}
+\frac{1}{2}\int dx\,dx^{\prime }\frac{\delta ^{2}T\left[ \chi \right] }{%
\delta \chi _{x}\delta \chi _{x^{\prime }}}\Delta \chi _{x}\Delta \chi
_{x^{\prime }}+\cdots .  \label{Test Function b}
\end{eqnarray}

Use this expansion together with (<ref>) and (<ref>)
to obtain

\begin{eqnarray*}
\int D\chi ^{\prime }\,T\left[ \chi ^{\prime }\right] \,P\left[ \chi
^{\prime },\sigma ^{\prime }|\chi ,\sigma \right]
= T\left[ \chi \right] +\int dx\,\frac{\eta \delta \xi _{x}^{\bot }}{%
g_{x}^{1/2}}\left\{ \frac{\delta T\left[ \chi \right] }{\delta \chi _{x}}\,%
\frac{\delta \phi \left[ \chi \right] }{\delta \chi _{x}}+\frac{1}{2}\frac{%
\delta ^{2}T\left[ \chi \right] }{\delta \chi _{x}^{2}}\right\} \,.
\end{eqnarray*}

Substituting back into eq.(<ref>), leads to

\begin{eqnarray}
&&\int D\chi \,\left\{ P\left[ \chi ,\sigma ^{\prime }|\chi _{0},\sigma _{0}%
\right] -P\left[ \chi ,\sigma |\chi _{0},\sigma _{0}\right] \right\} T\left[
\chi \right]  \notag \\
&=&\int dx\,\frac{\eta \delta \xi _{x}^{\bot }}{g_{x}^{1/2}}\int D\chi P%
\left[ \chi ,\sigma |\chi _{0},\sigma _{0}\right] \left\{ \frac{\delta T%
\left[ \chi \right] }{\delta \chi _{x}}\,\frac{\delta \phi \left[ \chi %
\right] }{\delta \chi _{x}}+\frac{1}{2}\frac{\delta ^{2}T\left[ \chi \right]
}{\delta \chi _{x}^{2}}\right\}  \label{Test Function c}
\end{eqnarray}

Since $T[\chi ]$ is arbitrary, after some integrations by parts we get

\begin{eqnarray}
&&P\left[ \chi ,\sigma ^{\prime }|\chi _{0},\sigma _{0}\right] -P\left[ \chi
,\sigma |\chi _{0},\sigma _{0}\right] =\int dx\frac{\delta P\left[ \chi
,\sigma |\chi _{0},t_{0}\right] }{\delta \xi _{x}^{\bot }}\delta \xi
_{x}^{\bot }  \notag \\
&=&\int dx\,\frac{\eta \delta \xi _{x}^{\bot }}{g_{x}^{1/2}}\left\{ -\frac{%
\delta }{\delta \chi _{x}}\left( \,P\left[ \chi ,\sigma |\chi _{0},\sigma
_{0}\right] \frac{\delta \phi \left[ \chi \right] }{\delta \chi _{x}}\right)
+\frac{1}{2}\frac{\delta ^{2}}{\delta \chi _{x}^{2}}P\left[ \chi ,\sigma
|\chi _{0},\sigma _{0}\right] \right\} ~.  \notag \\
&&  \label{Evol eq b}
\end{eqnarray}

Finally, for a finite evolution from $\sigma _{0}$ to $\sigma $, (<ref>) reads,

\begin{equation}
\rho _{\sigma }\left[ \chi \right] =\int D\chi _{0}\,P\left[ \chi ,\sigma
|\chi _{0},\sigma _{0}\right] \rho _{\sigma _{0}}\left[ \chi _{0}\right] ~.
\end{equation}

A further infinitesimal normal deformation $\sigma \rightarrow \sigma
^{\prime }$ by $\delta \xi _{x}^{\bot }$ gives

\begin{eqnarray}
\rho _{\sigma ^{\prime }}\left[ \chi \right] -\rho _{\sigma }\left[ \chi %
\right] = \int dx\frac{\delta \rho _{\sigma }\left[ \chi \right] }{\delta
\xi _{x}^{\bot }}\delta \xi _{x}^{\bot }
= \int D\chi _{0}\,\left( \int dx\frac{\delta P\left[ \chi ,\sigma |\chi
_{0},t_{0}\right] }{\delta \xi _{x}^{\bot }}\delta \xi _{x}^{\bot }\right)
\rho _{\sigma _{0}}\left[ \chi _{0}\right]
\end{eqnarray}

which, using (<ref>) and the fact that $\sigma ^{\prime }$ (or $%
\delta \xi _{x}^{\bot }$) can be freely chosen leads to the local
Fokker-Planck equations,

\begin{equation}
\frac{\delta \rho _{\sigma }\left[ \chi \right] }{\delta \xi _{x}^{\bot }}=%
\frac{\eta }{g_{x}^{1/2}}\left\{ -\frac{\delta }{\delta \chi _{x}}\left(
\,\rho _{\sigma }\left[ \chi \right] \frac{\delta \phi \left[ \chi \right] }{%
\delta \chi _{x}}\right) +\frac{1}{2}\frac{\delta ^{2}}{\delta \chi _{x}^{2}}%
\rho _{\sigma }\left[ \chi \right] \right\} ~,  \label{FP 3}
\end{equation}

which is the LTFP equations (<ref>) we set out to derive.

CHAPTER: SOLUTIONS TO THE NORMAL-NORMAL POISSON BRACKET

§.§.§ Setup

We consider solutions to the Poisson bracket in (<ref>). By virtue of the assumption we made in section <ref>, this means we have already restricted ourselves to solutions where the surface variables $X^{\mu}_{x}$ appear through functions of the metric $g_{ij}$.

The allowed solutions for the bracket eq.(<ref>) are most conveniently examined by considering instead the Jacobi identity

\begin{equation}
\left\{\left\{\rho ,\tilde{H}_{\perp x}\right\},\tilde{H}_{\perp y}\right\}-\left\{\left\{\rho ,\tilde{H}_{\perp y}\right\},\tilde{H}_{\perp x}\right\}=\left\{\rho,\left\{\tilde{H}_{\perp y} ,\tilde{H}_{\perp x}\right\}\right\}~.
\end{equation}

Using the Poisson bracket relation (<ref>) we have

\begin{equation}
\left\{\left\{\rho ,\tilde{H}_{\perp x}\right\},\tilde{H}_{\perp y}\right\}-\left\{\left\{\rho ,\tilde{H}_{\perp y}\right\},\tilde{H}_{\perp x}\right\}=\left\{\rho,\left(\tilde{H}_{x}^{i}+\tilde{H}_{y}^{i}\right)\partial_{ix}\delta\left(x,y\right)\right\},
\label{5Necessary condition}
\end{equation}

\begin{equation}
\tilde{H}_{x}^{i}=g_{x}^{ij}\tilde{H}_{jx}\hspace{.75 cm}\text{and}\hspace{.75 cm}\tilde{H}_{y}^{i}=g_{y}^{ij}\tilde{H}_{jy}
\end{equation}

are the contravariant tangential generators that are given by eq.(<ref>). If we split $\tilde{H}_{x}$ as in (<ref>) then eq.(<ref>) implies that $F_{x}^{0}[X,\rho;\chi]$ must satisfy

\begin{equation}
\frac{1}{g_{x'}^{1/2}}\frac{\delta}{\delta\chi_{x'}}\left(\rho\,\frac{\delta}{\delta\chi_{x'}} \, \frac{\tilde{\delta}F_{x}^{0}}{\tilde{\delta}\rho}\right)=\frac{1}{g_{x}^{1/2}}\frac{\delta}{\delta\chi_{x}}\left(\rho\,\frac{\delta}{\delta\chi_{x}} \, \frac{\tilde{\delta}F_{x^{\prime}}^{0}}{\tilde{\delta}\rho}\right),
\label{5Necessary condition b}
\end{equation}

which, again, is an equation linear in $F_{x}^{0}$. This implies that a general solution is simply the sum of individual solutions. For brevity and conciseness we denote the linear functional differential operator in eq.(<ref>) to be

\begin{equation}
\hat{O}_{x} = \frac{1}{g_{x}^{1/2}}\frac{\delta}{\delta\chi_{x}}\left(\rho\,\frac{\delta}{\delta\chi_{x}} \right) = \frac{1}{g_{x}^{1/2}}\left(\frac{\delta\rho}{\delta\chi_{x}}\frac{\delta}{\delta\chi_{x}} +\rho \frac{\delta^{2}}{\delta\chi_{x}^{2}} \right).
\label{5Necessary condition Op}
\end{equation}

*On Notation:
For conciseness, we will sometimes write the domain for these e-functionals, consisting  of the variables $\rho$, $X^{\mu}_{x}$, and indirectly, $\chi_{x}$, as just $D$ so that $D = (\rho, X^{\mu}_{x}, \chi)$. Thus we may rewrite these e-functionals as just $F_{x}^{0}[D]$ or $f_{x}^{(n)}(D)$, respectively, when there is no risk of confusion.

We consider here a family of possible e-functionals for the unknown $F_{x}^{0}$. The subsequent solutions will not therefore be the most general solution possible, but hopefully the family comprises a physically reasonable subset of solutions. In particular, consider $F_{x}^{0}$ of the form

\begin{equation}
F_{x}^{0}[\rho, X;\chi] = \prod_{n = 1}^{N} \mathcal{F}^{(n)}_{x}[\rho, X;\chi],
\label{5Fx ansatz}
\end{equation}

which is an $N$-fold product ($N$ is as large as desired) of e-functionals that are integrals over functions $f^{(n)}_{x}$ of the arguments $\rho$, $\chi$, $X^{\mu}$, and their derivatives:

\begin{equation}
\mathcal{F}_{x}^{(n)}[\rho, X;\chi] = \int D\chi_{n} f_{x}^{(n)}\left(\rho,\frac{\delta^{(m)}\rho}{\delta\chi_{nx}^{m}}; X^{\mu}_{x},\chi_{nx}\right).
\label{5Fx Local Definition}
\end{equation}

The dependence of these functions $f_{x}^{(n)}$ can be as complicated as desired, so long as $F_{x}^{0}$ is a scalar density; in particular, dependence on $\rho$ could include derivatives up to any order, the field $\chi_{x}$ can appear as a polynomial to arbitrary order, and the geometry can be any complicated scalar function of the metric components.[What does not appear, for instance, are the analogue of kernel operators inside the integrals or non polynomial functions of $\chi_{x}$.]

A couple of identities will prove useful in our analysis here. The first takes advantages of the peculiar pathological nature of the Dirac delta

\begin{equation}
a(x)\delta(x,x')b(x') = a(x')\delta(x,x')b(x') = a(x)\delta(x,x')b(x).
\label{5Dirac identity}
\end{equation}

Essentially, once a Dirac delta is present we can freely switch the spatial dependences of the fields, functionals, etc.

Another useful fact is that the field functional derivative of an e-functional vanishes

\begin{equation}
\frac{\delta}{\delta\chi_{x}} M_{x}[\rho, X; \chi] = 0.
\label{5Functional deriv identity}
\end{equation}

We can see this result most clearly by noting that, strictly speaking, writing these functionals as $M_{x}[\rho, X; \chi]$ is an abuse of notation since we have actually integrated over all $\chi$. When we compute such a derivative, we let $\chi\to\chi+\epsilon$ so that $M_{x}'= M_{x}[\rho,\chi+\epsilon]$. But we could always change variables in the integral from $\chi\to\chi+\epsilon$ to obtain the original expression; thus the difference $M_{x}'-M_{x} = 0$, always.

§.§.§ Solution: $N\geq 2$

Now we look to insert the ansatz (<ref>) into the condition eq.(<ref>) and deduce the restrictions on $F_{x}^{0}$. Before doing this, let us compute

\begin{align}
\frac{\tilde{\delta}F_{x}^{0}}{\tilde{\delta}\rho} &= \frac{\tilde{\delta}\mathcal{F}_{x}^{(1)}}{\tilde{\delta}\rho}\prod_{n \neq 1}\mathcal{F}^{(n)}_{x}+\frac{\tilde{\delta}\mathcal{F}_{x}^{(2)}}{\tilde{\delta}\rho}\prod_{n \neq 2}\mathcal{F}^{(n)}_{x}+\cdots \notag\\
&= \sum_{m}\frac{\tilde{\delta}\mathcal{F}_{x}^{(m)}}{\tilde{\delta}\rho}\prod_{n \neq m}\mathcal{F}^{(n)}_{x}.
\label{5Fx fderiv rho}
\end{align}

Note that the e-functional derivatives acting on the “local" e-functionals $\mathcal{F}_{x}^{(m)}$, as defined in eq.(<ref>), produces a function

\begin{equation}
\frac{\tilde{\delta}\mathcal{F}_{x}^{(m)}[\rho, X;\chi]}{\tilde{\delta}\rho[\chi]} = a_{x}^{(m)}\left (\rho, \frac{\delta^{(n)}\rho}{\delta\chi_{x}^{n}}, X_{x}^{\mu}, \chi_{x}\right )\equiv a_{x}^{(m)}(D_{x})\label{5ax definition}
\end{equation}

of the arguments, not another e-functional; thus it depends explicitly on $\chi_{x}$.

Feeding (<ref>) into eq.(<ref>), and noting that the operator $\hat{O}_{x}$ passes through the $\mathcal{F}_{x}^{(n)}[D]$ due to the identity eq.(<ref>), we have that

\begin{equation}
\hat{O}_{x^{\prime}}\frac{\tilde{\delta}F_{x}^{0}[D]}{\tilde{\delta}\rho[\chi]} = \hat{O}_{x}\frac{\tilde{\delta}F_{x^{\prime}}^{0}[D]}{\tilde{\delta}\rho[\chi]}
\end{equation}

\begin{equation}
\sum_{m}\prod_{n\neq m}\mathcal{F}_{x}^{(n)}[D]\,\hat{O}_{x^{\prime}}\,a_{x}^{(m)}(D_{x}) = \sum_{m}\prod_{n\neq m}\mathcal{F}_{x^{\prime}}^{(n)}[D]\,\hat{O}_{x}\,a_{x^{\prime}}^{(m)}(D_{x^{\prime}}).
\end{equation}

Introduce now the compact notation

\begin{equation}
\prod_{n\neq m}\mathcal{F}_{x}^{(n)}[D] \equiv A_{x}^{(m)}[D]
\label{5Ax definition}
\end{equation}

so that we have the condition above rewritten as

\begin{equation}
\sum_{m}A_{x}^{(m)}[D] \, \hat{O}_{x^{\prime}}\, a_{x}^{(m)}(D_{x}) = \sum_{m}A_{x^{\prime}}^{(m)}[D] \, \hat{O}_{x}\, a_{x}^{(m)}(D_{x^{\prime}}).\label{5Necessary condition c}
\end{equation}

As was apparent in eq.(<ref>), for a particular $F_{x}^{0}[D]$ to work it is necessary that each side of eq.(<ref>) be symmetric in $x$ and $x^{\prime}$.

This is not, however, easily accomplished. One route to satisfying this condition is for $\hat{O}_{x} a_{x^{\prime}}^{(m)}(D_{x^{\prime}})$ to introduce a Dirac delta; this could then be used via the identity eq.(<ref>) to manipulate the spatial dependence and make both sides equal. This suggests that any acceptable function $a_{x}^{(m)}(D_{x})$ must have an explicit $\chi_{x}$ dependence since the only way for a Dirac delta to appear is from the identity: $\delta\chi_{x}/\delta\chi_{x^{\prime}} = \delta(x,x^{\prime})$. (Note that for arbitrary choices of $\rho[\chi]$, successive applications of the field functional derivatives on $\rho[\chi]$ need not produce a Dirac delta, and so, such situations are not considered.) But, without this mechanism, it is impossible for the left hand and right hand sides to agree since the e-functionals $A_{x}^{(m)}$ and $A_{x^{\prime}}^{(m)}$ have a different spatial dependence. This implies that we cannot have a dependence $a_{x}^{(m)} = a_{x}^{(m)}(\rho,X^{\mu})$ that does not expressly include the fields. (There is an exception to this conclusion that we will get to shortly.)[To clarify things, consider, for example, the case $N = 2$. Using the definition (<ref>), we then have

\begin{align}
&\mathcal{F}_{x}^{(2)}[D]\hat{O}_{x^{\prime}} a_{x}^{(1)}(D_{x}) + \mathcal{F}_{x}^{(1)}[D]\hat{O}_{x^{\prime}} a_{x}^{(2)}(D_{x})\notag\\
= & \mathcal{F}_{x^{\prime}}^{(2)}[D]\hat{O}_{x} a_{x^{\prime}}^{(1)}(D_{x^{\prime}}) + \mathcal{F}_{x^{\prime}}^{(1)}[D]\hat{O}_{x} a_{x^{\prime}}^{(2)}(D_{x^{\prime}}).
\end{align}

Clearly, no choice of $a$'s or $F$'s (or equivalently, $A$'s) can make this work. Even if, for example, we choose the $a$'s so that the $\hat{O}_{x}a_{x^{\prime}}$'s are symmetric in $x$ and $x^{\prime}$, we have no way to make the $A$'s match. ]

More exactly, a guaranteed solution of $F_{x}^{0}$ includes terms like

\begin{equation}
F_{x}^{0} = \prod_{n}\int D\chi_{n}\rho[\chi_{n}] f_{x}(\chi_{nx}; X_{x}^{\mu})~,\label{5Fx gen solution n>= 2}
\end{equation}

which takes the form of products of expected values of the field $\chi_{x}$, and linear combinations there of. This can easily be checked substituting into eq.(<ref>).

So long as the number of products $N\geq 2$, then eq.(<ref>) is the only possible form for $\mathcal{F}_{x}^{(n)}$.

The goal is to determine the allowed $a$'s (although, recall that the $a$'s are related to the integrands of the $F$'s in a complicated way). A general $a_{x}^{(m)}(D_{x})$ can always be written in powers of the field $\chi_{x}$ as

\begin{equation}
a_{x}^{(m)}(D_{x}) = \sum_{n} \chi_{x}^{n} \, \alpha_{x}^{(mn)}\left(\rho,\frac{\delta^{(k)}\rho}{\delta\chi_{x}^{k}}, X_{x}^{\mu}\right ),
\label{5ax ansatz}
\end{equation}

where the $\alpha_{x}^{(mn)}$ is a generic function of its inputs for every $m$ and $n$ that can be as complicated and general as desired. If we act the operator $\hat{O}_{x^{\prime}}$ on this, then some of the derivatives in $\hat{O}_{x^{\prime}}$ will hit the field $\chi_{x}$ and produce a $\delta(x,x')$. However, due to the product rule, the derivatives will also hit the $\alpha_{x}^{(mn)}$, and no Dirac delta will be produced. (Again, this analysis should hold for arbitrary choices of $\rho[\chi]$.) We write this symbolically as

\begin{equation}
\hat{O}_{x^{\prime}}a_{x}^{(m)}(D_{x}) = \delta(x,x')\beta_{x}^{(m)}(D_{x}) + B_{xx^{\prime}}^{(m)}
\end{equation}

\[B_{xx'} = \sum_{n}\chi_{x}^{n}\,\hat{O}_{x^{\prime}}\alpha_{x}^{(mn)}(D_{x})\]

(and the form of $\beta_{x}$ doesn't matter) so that the condition eq.(<ref>) has the structure

\begin{equation}
\sum_{m}A_{x}^{(m)}[D] \, B_{xx^{\prime}}^{(m)} = \sum_{m}A_{x^{\prime}}^{(m)}[D] \, B_{x^{\prime}x}^{(m)}.\label{5Necessary condition d}
\end{equation}

Again, no suitable choice of $B_{xx^{\prime}}^{(m)}$ — symmetric in $x$, $x^{\prime}$, or not — will make the two sides equal, owing to the e-functionals $A_{x}^{(m)}$ and $A_{x^{\prime}}^{(m)}$. In fact, eq.(<ref>), together with the ansatz eq.(<ref>) imply that the condition

\begin{equation}
\sum_{m}A_{x}^{(m)}[D]\sum_{n}\chi_{x}^{n}\,\hat{O}_{x^{\prime}}\alpha_{x}^{(mn)}(D_{x})=\sum_{m}A_{x^{\prime}}^{(m)}[D]\sum_{n}\chi_{x^{\prime}}^{n}\,\hat{O}_{x}\alpha_{x^{\prime}}^{(mn)}(D_{x^{\prime}})
\end{equation}

should be met, after accounting for the delta functions. Since no further delta's will be produced by $\hat{O}\alpha$, each side is only equal if they are each equal to a constant

\begin{equation}
\sum_{n}\chi_{x}^{n}\left[\sum_{m}A_{x}^{(m)}[D]\,\hat{O}_{x^{\prime}}\alpha_{x}^{(mn)}(D_{x})\right] = c,
\end{equation}

where $c$ is independent of any spatial dependence, and hence doesn't depend at all on the fields or geometry (but might depend, perhaps, on $\rho[\chi]$, for instance). The term in brackets, however, has no explicit $\chi_{x}$ dependence,[It depends on the fields only indirectly through functional derivatives of $\rho[\chi]$, but we only care about the explicit form of the potential $F_{x}^{0}$ for arbitrary choices of $\rho[\chi]$.] so the only way to maintain this constraint, without completely discarding all dependence on $\chi_{x}$ in $F_{x}^{0}$ — which is absolutely required for a non-trivial $F_{x}^{0}$ — is for the term in brackets to vanish. This, in turn, implies that[Consider a special case. If all the $A_{x}^{(m)}$ are equal, then $A_{x}^{(m)} = \prod_{n\neq m}\mathcal{F}_{x}^{(n)} = \prod_{n\neq m'}\mathcal{F}_{x}^{(n)} \equiv \left(\mathcal{F}_{x}\right)^{N-1} $, for all $m$, $m^{\prime}$. This then requires that the $\alpha$'s are related by $\alpha_{x}^{(mn)} = \alpha_{x}^{(m^{\prime}n)}$. We then have $\sum_{m}\hat{O}_{x^{\prime}}\alpha_{x}^{(mn)} = 0 \longrightarrow \hat{O}_{x^{\prime}}\alpha_{x}^{(mn)} = 0$. In the general case, if $A_{x}^{(m)}\neq A_{x}^{(m')}$ then $\sum_{m}A_{x}^{(m)}\hat{O}_{x^{\prime}}\alpha_{x}^{(mn)} = 0$. We can always rearrange this condition so that $\hat{O}_{x^{\prime}}\alpha_{x}^{(in)} = -(A_{x}^{(i)})^{-1}\sum_{m\neq i} A_{x}^{(m)}\hat{O}_{x^{\prime}}\alpha_{x}^{(mn)}$. For example, with $N=2$ we have

\begin{equation}
\hat{O}_{x^{\prime}}\alpha_{x}^{(1n)} = - \frac{A_{x}^{(2)}}{A_{x}^{(1)}}\hat{O}_{x^{\prime}}\alpha_{x}^{(2n)}.
\end{equation}

While the LHS has a function-like (i.e. “local") dependence on $\rho[\chi]$, etc., the RHS has an e-functional and function dependence. Now, unless the $A$'s are constant e-functionals (which would defeat our purpose) then an expression like this only makes sense if the $\hat{O}\alpha = 0$. A similar argument then holds for arbitrary $N$, and thus eq.(<ref>).]

\begin{equation}
\hat{O}_{x^{\prime}}\alpha_{x}^{(mn)} = \frac{1}{g_{x^{\prime}}}\frac{\delta}{\delta\chi_{x^{\prime}}}\left(\rho\frac{\delta\alpha_{x}^{(mn)}}{\delta\chi_{x^{\prime}}}\right) = 0~.\label{5O alpha = 0}
\end{equation}

For this to hold, however, it must mean that the term in parentheses is independent of $\chi_{x^{\prime}}$. But $\delta\alpha_{x}^{(mn)}/\delta\chi_{x^{\prime}}$ can only be independent of $\chi_{x^{\prime}}$ if $\alpha_{x}^{(mn)}$ doesn't depend at all on $\rho[\chi]$ since, if it did, it would be guaranteed to introduce such a dependence through

\begin{equation}
\frac{\delta\rho[\chi]}{\delta\chi_{x^{\prime}}} = g[\chi;\chi_{x^{\prime}})
\end{equation}

and related relations for higher derivatives. Thus,

\begin{equation}
\alpha_{x}^{(mn)} = c_{x}^{(mn)}(X_{x}^{\mu})
\end{equation}

where $c_{x}^{(mn)}$ depends only on the geometry, so that

\begin{equation}
a_{x}^{(m)} = \sum_{n}c_{x}^{(mn)}(X_{x}^{\mu})\,\chi_{x}^{n}.
\end{equation}

Furthermore, using the definition of $a_{x}^{(m)}$ from eq.(<ref>) we have that

\begin{equation}
a_{x}^{(m)} = \frac{\tilde{\delta}\mathcal{F}_{x}^{(m)}}{\tilde{\delta}\rho} = \sum_{n}c_{x}^{(mn)}(X_{x}^{\mu})\,\chi_{x}^{n}
\end{equation}

which can be easily integrated to yield

\begin{equation}
\mathcal{F}_{x}^{(m)} =\int D\chi_{m}\,\rho[\chi_{m}]\sum_{n}c_{x}^{(mn)}(X_{x}^{\mu})\,(\chi_{mx})^{n} .\label{5Fx gen solution}
\end{equation}

This is exactly the same form as eq.(<ref>) and concludes our proof.

*Some comments
Clearly the $\mathcal{F}_{x}^{(m)}$ can be interpreted as expected values of the field $\chi$, with some additional contributions from the geometrical variables. From this point on, however, we can offer no further restriction on the appearance of the $\chi$'s and geometrical quantities that appear. In principle, almost anything is possible; again, so long as $\mathcal{F}_{x}^{(m)}$ is a scalar density.

For completeness, we write down the expression for $F_{x}^{0}$. Since it must be a scalar density, it should be $\propto g_{x}^{1/2}$. Other than that, we define
\[V_{x}^{(m)}(\chi_{mx},X_{x}^{\mu}) = \sum_{n}c_{x}^{(mn)}(X_{x}^{\mu})\,(\chi_{mx})^{n}\]
so that $F_{x}^{0}$ takes the form

\begin{equation}
F_{x}^{0} = g_{x}^{1/2} \prod_{m}\int D\chi \, \rho[\chi ]\, V_{x}^{(m)}(\chi_{x},X_{x}^{\mu}),
\end{equation}

where it is assumed that the $V_{x}^{(m)}$'s should be scalar valued.

§.§.§ Solutions: $N = 1$

Something that perhaps avoided attention in our treatment above was a certain special case that requires additional care. In the ansatz for $F_{x}^{0}$ we considered this “potential" term to be a product of e-functionals $\mathcal{F}_{x}^{(m)}$. The analysis above appears correct so long as the number of products is $N\geq 2$. For just a single e-functional, we require a separate argument to determine its general form.

In the argument above, when $F_{x}^{0}$ is a product of e-functionals, then the $\tilde{\delta}F_{x}^{0}/\tilde{\delta}\rho$ introduces a sum of terms; each of these terms must be symmetric in $x$, $x^{\prime}$, but each of which also comes preceded by a product of e-functionals that depend on $x$ (or $x^{\prime}$) exclusively, and one requires a $\delta(x,x^{\prime})$ to appear — in every term — and thus we are led to the $\mathcal{F}_{x}^{(m)}$ in eq.(<ref>). When there is only one e-functional, on the other hand, this situation does not arise and so more analysis is needed.

Let $F_{x}^{0}[\rho]$ be just a single e-functional

\begin{equation}
F_{x}^{0}[\rho] = \int D\chi f_{x}(D_{x}).
\end{equation}

Inserting this term into eq.(<ref>) yields

\begin{equation}
\hat{O}_{x^{\prime}}a_{x}(D_{x}) = \hat{O}_{x}a_{x^{\prime}}(D_{x})\quad\text{where}\quad a_{x}(D_{x}) = \frac{\tilde{\delta}F_{x}^{0}}{\tilde{\delta}\rho}.\label{5Necessary condition local a}
\end{equation}

Now, $a_{x}(D_{x})$ is a function of $\rho$ (and its derivatives), $\chi$, and the geometry. We provide an ansatz for $a_{x}$

\begin{equation}
a_{x}(D_{x}) = \sum_{n}\chi_{x}^{n}\alpha_{x}^{(n)}(\rho,X_{x}^{\mu}),
\end{equation}

which is exactly as in eq.(<ref>); here $\alpha_{x}^{(n)}$ is a distinct function of $\rho$ (and its functional derivatives with respect to $\chi_{x}$, to any order) and the geometry $X_{x}^{\mu}$ for each $n$. And, again, as the operator $\hat{O}_{x^{\prime}}$ acts on $a_{x}$ we have that a Dirac delta is produced for some terms and not for others so that there is the general form

\begin{equation}
\hat{O}_{x^{\prime}}a_{x} = b_{x} \, \delta(x,x^{\prime}) + \sum_{n}\chi_{x}^{n}\hat{O}_{x^{\prime}}\alpha_{x}^{(n)}.
\end{equation}

The terms with the delta function vanish from (<ref>). What remains is

\begin{equation}
\sum_{n}\chi_{x}^{n}\hat{O}_{x^{\prime}}\alpha_{x}^{(n)} = \sum_{n}\chi_{x^{\prime}}^{n}\hat{O}_{x}\alpha_{x^{\prime}}^{(n)}.
\end{equation}

Since $\hat{O}_{x^{\prime}}$ acts on $\alpha_{x}(\rho,X_{x}^{\mu})$ but produces no delta functions, the $\chi_{x}$ (or $\chi_{x^{\prime}}$) on each side present an obstacle to symmetrizing this equation in $x$, $x^{\prime}$. This implies two options: (1) functions with no $\chi_{x}$ dependence at all, (2) each side identically vanishes since the functions $\alpha_{x}^{(n)}=\alpha_{x}^{(n)}(X_{x}^{\mu})$ depend only on the geometry, that is, they are constants with respect to $\chi$ (i.e. $\hat{O}\alpha = 0$) so that
\[a_{x} = a_{x}(\chi_{x},X_{x}^{\mu})\equiv g_{x}^{1/2}\, V_{x}(\chi_{x},X_{x}^{\mu}).\]
Using the definition in (<ref>), this latter option implies
\[F_{x}^{0}[\rho] = g_{x}^{1/2}\int D\chi \,\rho \, V_{x}(\chi_{x},X_{x}^{\mu}).\]
This is just the expected value of a “local potential" $V_{x}(\chi_{x}, X_{x}^{\mu})$ and no further restrictions can be imposed on functions of this type (other than they be scalar valued). Thus this $V_{x}$ can be identified with the one introduced in section <ref>, in eq.(<ref>).

On the other hand, the first option implies functions
\[a_{x} = \alpha_{x}\left(\rho,\frac{\delta^{(m)}\rho}{\delta\chi_{x}^{m}}, X_{m}^{\mu}\right)\]

which satisfy $\hat{O}_{x^{\prime}}\alpha_{x} = \hat{O}_{x}\alpha_{x^{\prime}}$. Or more explicitly,

\begin{equation}
\frac{1}{g_{x^{\prime}}^{1/2}}\left(\frac{\delta\rho}{\delta\chi_{x^{\prime}}}\frac{\delta\alpha_{x}}{\delta\chi_{x^{\prime}}} +\rho \frac{\delta^{2}\alpha_{x}}{\delta\chi_{x^{\prime}}^{2}} \right) = \frac{1}{g_{x}^{1/2}}\left(\frac{\delta\rho}{\delta\chi_{x}}\frac{\delta\alpha_{x^{\prime}}}{\delta\chi_{x}} +\rho \frac{\delta^{2}\alpha_{x^{\prime}}}{\delta\chi_{x}^{2}} \right).
\label{5Necessary condition local b}
\end{equation}

How can this condition be satisfied? First, recall that our objective is to determine a functional form for $\alpha_{x}$ such that both sides match. In other words, an $\alpha_{x}$ is acceptable when both sides of this equation “look the same". Next, notice that $\alpha_{x}$ depends on $\rho$, its functional derivatives $\delta^{(m)}\rho/\delta\chi_{x}^{m}$, and the geometry.

Inspecting eq.(<ref>), we see that $\hat{O}_{x^{\prime}}$ on the LHS will introduce derivatives of $\rho[\chi]$ with respect to $\chi_{x^{\prime}}$ of up to second order, while the RHS will do the same but for $\chi_{x}$. This requires us to consider only those $\alpha_{x}$ that are at most second order in the functional derivatives, i.e.
\[\alpha_{x} = \alpha_{x}\left(\rho,\frac{\delta\rho}{\delta\chi_{x}},\frac{\delta^{2}\rho}{\delta\chi_{x}^{2}}; X_{x}^{\mu}\right).\]
And, moreover, each term in the function must have only two functional derivatives. The most general such function can be written as[We do not need to begin with this truncated ansatz, in fact. We can begin with arbitrarily high orders of derivatives of $\rho$, but it should be apparent that they will never produce an $\alpha_{x}$ such that the functional form of each side in (<ref>) are the same.]

\begin{equation}
\alpha_{x} = \lambda_{x}^{(1)} \, m_{1}(\rho)\left(\frac{\delta\rho}{\delta\chi_{x}}\right)^{2} + \lambda_{x}^{(2)} \, m_{2}(\rho)\left(\frac{\delta^{2}\rho}{\delta\chi_{x}^{2}}\right)
\end{equation}

where $\lambda_{x}^{(1)}$, $\lambda_{x}^{(2)}$ are “constants" that depend on the geometry, but not $\chi$ or $\rho$. The functions $m_{i}(\rho)$ depend only on $\rho[\chi]$ (i.e. the probability at a single point in configuration space) and are not yet determined; indeed, it is our goal to find their functional form.

Another restriction comes from requiring that $F_{x}^{0}$ be a scalar density (since it is a piece of the local e-Hamiltonian $\tilde{H}_{x}$), which implies that $\alpha_{x}$ similarly be a scalar density. However, because each functional derivative itself is a scalar density, and there are two such derivatives in each term of $\alpha_{x}$, then $\lambda_{x}^{(i)}$ must be an inverse scalar density $\propto g_{x}^{-1/2}$ for $\alpha_{x}$ to have the correct tensor weight. To account for this, we redefine the $\lambda_{x}^{(i)}$'s so that
\[\lambda_{x}^{(i)} = \frac{\lambda^{(i)}}{g_{x}^{1/2}},\]