CHAPTER: AN INTRODUCTION

Quantum mechanics is weird. This much we can all agree on. Nonetheless, despite this, we can also agree that the past century has been nothing short of a marvelous success story for quantum theory:

“And what a triumph it was, in the old sense of the word: a glorious victory parade, full of wonderful things brought back from far places to make the spectator gasp with awe and laugh with joy."— , [2]

That is, on one hand, quantum theory has allowed us to make remarkable progress in physics — from particle physics and the standard model, to optics and materials physics. But on the other hand, there are still unresolved issues needing to be addressed — what does it mean to `quantize'? How is `the measurement problem' resolved? What is the significance of the quantum state $\Psi$?

“Ask anyone today working on foundational questions in quantum theory and you are likely to hear that there is still no consensus on many of these questions - all the while, of course, everybody seems to be in perfect agreement on how to apply the quantum formalism when it comes to making experimental predictions."— , [3]

The situation is itself a contradiction: although there is no uniquely compelling `story' or `interpretation' that we can tell to make sense of quantum theory,[The dominant interpretations are, of course: the so-called Copenhagen interpretation [5], the De Broglie-Bohm theory [6] (related to this, see [7][8]), and the `many-worlds' or relative state formulation [9]. Informal polling (see e.g., [3][4]) seems to imply that none of these interpretations are widely favored above the others.] physicists seem to be making do with it all the same. Indeed, the simple fact is that, if pushed, many would likely admit that [10] “quantum theory needs no `interpretation'." It is justified on empirical grounds. Or, to put it even more bluntly, we should all just: “shut up and calculate!"

This state of affairs indicates a rather nihilistic view of physicists towards `interpretations' of quantum theory, and `quantum foundations' more generally. If the goal has been to convince physicists of a particular interpretation, then this enterprise has largely been a failure [11]. But this does not mean that understanding quantum theory is a pursuit that should be abandoned. Not in the least. What it does suggest, however, is that new methods are necessary for clarifying these issues. Interpretations are, after all, largely post hoc efforts — from the answer, retroactively attach a meaning. The approach ventures on philosophical means rather than constructive and scientific ones.

§.§.§ Entropic dynamics and quantum reconstruction—

“...quantum mechanics will cease to look puzzling only when we will be able to derive the formalism of the theory from a set of simple physical assertions (“postulates”, “principles”) about the world. Therefore, we should not try to append a reasonable interpretation to the quantum mechanics formalism, but rather to derive the formalism from a set of experimentally motivated postulates."— , [12]

The view that we take here is that the proper means for understanding quantum theory is to first construct it from some basic a priori defined principles and concepts. This is the basis of the quantum reconstruction program.

In the last half century or so, there has been a surge of interest in this type of endeavor, coming from a variety of directions. Some proposals, for example, entertain the notion that quantum theory might emerge from some (presumably more palatable) sub-quantum theory (see e.g., [13]-[20]). More recently, a wave of efforts have featured information theoretic concepts as unifying principles for quantum reconstruction [21]-[29]. In any case, a crucial choice must be made as to which principles, postulates, ideas, etc., must be regarded as sacred and immutable — this is the definitive step on the basis of which a reconstruction will live or die.

Enter Entropic Dynamics (ED), which is a reconstruction of quantum theory in which the principles of inference and information processing are held as sacrosanct. In ED, probabilities are Bayesian [30][31], they are manipulated using the rules of probability theory [32][33], and they are updated by the Maximum Entropy (ME) method [34]-[36]. End of story. The notion of `quantum probabilities' [37][38] or `generalized probabilities' [39][40] does not enter. Indeed, it makes no sense in ED to derive the `Born rule' from the notion of an `amplitude' $\Psi$ (see e.g., [22][26]) — our goal is, rather, to derive the notion of an amplitude from the more fundamental concept of probability.

More specifically, ED is a framework for constructing dynamical theories of probability. A challenging aspect of this approach is that a fully epistemic interpretation of the wave function $\Psi $ is not achieved by merely declaring that the square of a wave function $|\Psi |^{2}$ yields a Bayesian probability. Rather, to insure consistency, one must also show that the rules for updating those probabilities — which include both the unitary evolution of the wave function $\Psi $ and its “collapse” during a measurement — are in strict accord with Bayesian and entropic methods.[In [41]-[43] wave function collapse was demonstrated to be an instance of discrete entropic updating via Bayes' rule.]

This viewpoint highlights serious differences between ED and other probabilistic reconstructions (see e.g., [27][44][45]). To start, it is not clear that these approaches take probability seriously enough: if probabilities are being updated, then this should be accomplished within the framework of probability theory and entropic inference. It is not obvious, for instance, that the operational language of `amplitudes', `experiments', `transformations', etc., is entirely adherent to these more fundamental principles. Another key distinction arises in how probabilities are utilized. Let us explain. An important task in any probabilistic theory is identifying the propositions, or microstates, that are the subject of inquiry.[Are we uncertain, for instance, of the rise or fall of a stock index price, which side of a coin has landed face up, or perhaps, the location of a particle?] Some approaches, such as quantum Bayesianism [46] (or `QBism'), for example, assign probabilities to the outcomes of measurements. And certainly, there is nothing strictly speaking `wrong' about this; probabilities are, by design, flexible in this regard. Nevertheless, in doing so, QBists and others (see e.g., [45]) are aligning themselves to many of the philosophical and (indefinite) ontological commitments of complementarity, and of the Copenhagen interpretation, more generally. How much “physical" insight, after all, can possibly be gained from such a viewpoint?

By contrast, ED commits to a definite ontology — if we are talking particles, then the positions of those particles are real; if we are talking fields, then the values of the fields are real! In the language of Kochen and Specker [47]: in ED we commit to a set of (abelian) non-contextual variables, similar to those appearing in the Bohmian approach [6][48][49]. In fact, at first glance there seems to be a significant amount of formal overlap between ED and the de Broglie-Bohm theory.[It is, of course, quite characteristic of many reconstructions that they converge upon a particular formulation of standard quantum theory — whether it be Hilbert spaces, Feynman path integrals, etc. Depending on the design of your reconstruction, one or another formulation may be more natural to obtain.] But it is important to stress that they are quite different. For instance, we do not subscribe to the belief, as the Bohmians do, that $\Psi$ has ontic status. The state $\Psi$ is epistemic. Moreover, the Bohmian approach is completely deterministic at the microscopic level: the particles, fields, etc., follow smooth trajectories. ED, on the other hand, is completely indeterministic from the outset: our goal is only to infer this underlying behavior on the basis of some limited information. (It may even be, in fact, that ED contains the formalism of the Bohmian theory as a special case! See e.g., [50].)

Phrased a bit differently, our aim in ED is to obtain the probability $\rho$ that the microstates achieve certain values, and to then update that probability according to the available information. Thus, in an entropic dynamics the evolution must be driven by information codified into constraints; it is through these constraints that the “physics” is introduced. And, as one might suspect, significant complications lie in identifying those constraints.

In recent years, much work has been done [51]-[54] to determine the constraints leading to a quantum dynamics, and they appear in stages.[There is no way around the fact that inference is a messy business, just as science, in general, is a messy business — we start with some constraints, modify them, and modify them some more until we begin to converge on a theory that is more appealing, in that it better explains, predicts, and controls the observed phenomenon.] A preliminary constraint involves the introduction [52][53] of an auxiliary variable $\phi$, the drift potential, which guides the probabilities; in a more familiar sense, $\phi$ is the mathematical manifestation of the `wave' part of De Broglie's famous `wave-particle' duality, or the pilot wave in Bohmian mechanics.[Interestingly, the ED approach helps to clarify this duality: the particle aspect is real, the wave aspect is an epistemic property of the ensemble, not an ontological property of the particle.] A secondary constraint emerges once we recognize that it is not just the evolution of $\rho$, but the coupled evolution of $\rho$ and $\phi$ that is important — namely, concerns arise as to how one updates the drift potential $\phi$ in response to the evolving probability $\rho$. Addressing this issue is one of the topics of this thesis.

*Wait, hold on a second!—
Before proceeding to develop the specific contents of this work, it is important to preemptively counter some concerns about our methodology. The issue is the following. In the program of quantum reconstruction, we know the answer — linear, unitary time evolution, Hilbert spaces, Feynman rules, all of that. We know it. No doubt, when the answer is known, it is very easy — too easy, in fact — to cook up axioms, postulates, principles, or in our case, constraints, that produce these correct answers. So how do you know that you've succeeded in your reconstruction? Simple: if you've recovered the quantum formalism from your first principles, then clearly it succeeded. The issue of whether it's a compelling reconstruction, however, is an entirely different matter; surely what is compelling to one physicist may not be to another, i.e. some subjectivity is involved.

One (mildly objective) criterion for success that we offer here is that any compelling reconstruction should also provide a new perspective on quantum theory.

“Therefore psychologically we must keep all the theories in our heads, and every theoretical physicist who is any good knows six or seven theoretical representations for exactly the same physics. He knows that they are equivalent, and nobody is ever going to be able to decide which one is right at that level, but he keeps them in his head, hoping that they will give him different ideas for guessing."— , The Character of Physical Law [1]

Put another way, a worthwhile reconstruction is one that equips us with new and “different ideas for guessing." But, for guessing what, exactly? Quantum theory, as is, already correctly accounts for much of what is known. (And with a couple new ingenious models, it'll describe the rest, right?) There is, however, still one domain where there is still plenty of room left for guessing: gravity. That is, if your reconstruction paves the way for a novel view on a quantum theory of gravity, then this is successful; in our view, at least.

§.§.§ Quantum field theory on curved space-time

The study of quantum field theory on a curved space-time (QFTCS) deals with the interplay of quantum field theory (QFT) together with classical general relativity (GR) (see [55][56] and the recent review [57]). In the past few decades interest in QFTCS has been driven by a variety of developments. Foremost among these was the recognition by Bekenstein [58], Hawking [59][60][61], Unruh [62], and others [63][64] in the 1970's that the intersection of gravitation and quantum theory yields novel features of which information and entropy are inseparable ingredients.

Since that time, QFTCS has reached a rather mature stage of development; having been accepted as a reasonable effective theory, valid wherever the effects of gravity are not too strong; or more importantly, as a necessary step intermediate to a full theory of interacting matter and gravity. Because of this, it is not uncommon to look upon QFTCS as a harbinger of a quantum gravity yet to come — that is, any such theory must, in some way or another, touch upon QFTCS and its quintessential information-based features.[A prominent example of this is, of course, the development of the holographic principle [65][66] and correspondingly, the ADS/CFT correspondence [67].] This makes QFTCS a worthy topic for foundational study and reconstruction.

*What's new?—
The objective of our thesis is a formulation of QFTCS from the viewpoint of ED. This contribution builds off of a period of sustained effort within the ED program in which several aspects of the quantum formalism have been reconstructed — including theories of non-relativistic scalar particles [53][54]; particles with half-integer spin [68]; and, relativistic scalar fields [69][70]. But the singular focus on flat space-time that dominates these developments stands in some contrast to the rather more ambitious goals of the ED program: to ultimately formulate quantum gravity from within the framework of inference and information processing. Here we make inroads on these loftier aims by re-examining and revising the constraints and assumptions of ED in flat space-time in light of the transition to more general, curved space-times.

More to the point: here we propose a model for a manifestly relativistic ED, based on work [71][72] in collaboration with M. Abedi and A. Caticha, that is valid for arbitrarily curved space-times with a globally hyperbolic topology; which, of course, includes Minkowski space as a special case. The scheme requires a couple of conceptual innovations with respect to older versions of ED; one involving the notion of time, the other dealing with the update of  $\phi$, the drift potential. The essential issue is that in a flat space-time with translational symmetry, it is convenient to introduce a global notion of time in which all degrees of freedom are updated in a uniform fashion; and in such instances, a natural constraint is one where $\phi$ evolves such that a global energy functional be conserved. However both of these ingredients are inadequate in a generic space-time where it is better to proceed by abandoning such global notions in favor of a more local description. This takes place in two steps. One is the adoption of a local notion of time, in the spirit of Dirac [73][74], Tomonaga [75], and Schwinger [76], where time evolution is allowed to vary across spatial locations. Correspondingly, the update of $\phi$ is determined, not by conserving a global energy, but by introducing a set of local Hamiltonian generators, four per spatial point, to perform the updating.

Embedded in this last statement is the central assumption of this reconstruction, which is the adoption of a Hamiltonian framework for evolving $\phi$ — or more technically, the introduction of a symplectic structure on the joint space of $\rho$ and $\phi$. An advantage of this approach is that it allows us to borrow from schemes developed for classical physics in pursuit of a manifestly relativistic ED that is both statistical and quantum. Of particular relevance to us are the covariant Hamiltonian methods pioneered by Dirac [77][78], Kuchař [79], Teitelboim [80][81], and others [82]. These approaches are, in short, characterized by expressing the local Lorentz invariance of space-time in terms of a new symmetry — foliation invariance. In turn, this requirement is implemented by a consistency condition called path independence: the evolution of an initial to a final state is independent of the intervening choice of foliation. The result is a collection of Poisson brackets and constraints that are to be satisfied by the aforementioned local Hamiltonian generators.

One feature of these developments is that there is very little friction in transitioning from a fixed background geometry to a dynamical one that reacts in response to the evolving quantum state. We demonstrate that this is essentially what is called “semi-classical" gravity (see e.g., [55][56]) in the literature, and it is obtained in our framework in a very natural way. And, although such an approach has long been outright dismissed as a viable theory of quantum gravity (see e.g., [83]), more recent work [84][85] suggests that semi-classical theories of gravity cannot be totally ruled out. Put another way, the gravity theory we put forth blends together some intriguing features — quantum fields, entropy, and geometry — we ask whether this could be quantum gravity.

The layout of this thesis is as follows. Chapter <ref> proceeds with a short overview of the framework of entropic inference, borrowing heavily from many influential sources. We then develop the ED of non-relativistic particles in chapter <ref> as a way of introducing the key features of an ED model. Next, we build off of these developments in chapter <ref> to address the topic of relativistic fields in Minkowski space. This is followed by our main contributions in chapters <ref> and <ref> where we develop the framework of covariant ED in curved space-time, first in the context of a fixed background in chapter <ref>, then in a dynamical background in chapter <ref>. In chapter <ref> we make some concluding remarks and analyze the model developed in the previous chapter.

CHAPTER: ENTROPIC INFERENCE

“But in this world, nothing can be said to be certain, except death and taxes."— , 1789

Uncertainty is a fact of life; it is the rule, not the exception. This is a viewpoint that we feel is so uncontroversial that it serves as the bedrock perspective from which we move forward in this dissertation (and if you don't take this author's humble word for it, perhaps the above words of Benjamin Franklin will be more convincing!).

We understand the notion of uncertainty, colloquially at least, to be related to the fact that we have insufficient information to predict, or know, a particular something (a proposition, perhaps) with complete certainty. This is, indeed, usually the case; we do not posses and, in fact, will often never possess the information necessary to attain certainty. So what do we do then? How do we proceed? What conclusions can be made, what inferences can be drawn from the limited information that we do, in fact, possess?

These are, in fact, difficult questions to answer and a priori it is not obvious that we can answer any of these questions independent of the context in which they were asked. Indeed, situations of uncertainty constitute almost all the situations we can conceive of — from estimating the number of different species on earth, to whether there is any species at all on Mars, to whether the Dow Jones will go up or down tomorrow — all are situations where our information is incomplete. But in each case, the domain knowledge, the data collected, and the general sources of information are extremely disparate. From this one might conclude that the problem is not well posed, that we should give up! Let econometrics be unto the economists  and let biostatistics be unto the biologists and pray that they never need to walk into a bar together!

Luckily, we need not resort to praying in this instance; we can rely instead on the important contributions of many pioneers, such as the early progenitors of probabilistic thought J. Bernoulli [86] and P.S. Laplace, as well as the breakthrough 20$^{\text{th}}$ century insights of  R.T. Cox [32] and E.T. Jaynes [87], and the more recent works of A. Caticha [34], K. Vanslette [88][89], as well as many other fine contributions (see e.g., [36][90][91][92]).[See also the monograph of Caticha [35] and the thesis of A. Giffin [36] for a detailed discussion of these issues and review of the history of probability and entropy.]

The framework for handling incomplete information, based on these groundbreaking works, will be called entropic inference (for reasons that will become apparent) and can be summarized by two key ingredients. The first is that a partial state of knowledge — degrees of rational belief — can be quantified by a set of real numbers. These degrees of rational belief are in turn (subject to a consistency requirement, first due to Cox) manipulated by a set of rules that are in one-to-one correspondence with the rules of probability, thus prompting the controversial[It is controversial in the sense that there is an active debate surrounding the interpretation of probability with Bayesians and frequentists on opposing sides. And, even within the Bayesian camp, there are a wide spectrum of views ranging from the personalistic subjective de Finetti approach to the “logical" objective view espoused by Jaynes. See Jaynes [31] for a thorough discussion of this topic, albeit coming from a partisan actor.] statement:

\begin{equation*}
\text{Degrees of Rational Belief ~=~ Probabilities~.}
\end{equation*}

Once we have established degrees of rational belief as being quantified by probabilities, the second aspect of entropic inference concerns how one updates their beliefs, i.e. updates probabilities, when new information is made available. Indeed, such a procedure is not trivial. To what extent should we change our previously held, or prior beliefs on the basis of a single new experimental outcome, for example? Certainly new data should induce some change in our state of knowledge, but how much exactly? How about two, or three outcomes? What if, for instance, we learn that the average expected roll of a die was not 3.5, as is expected for a fair die, but actually 4.0? How does one incorporate such information into their state of knowledge? What are the optimal set of probabilities that satisfy this constraint? Amazingly, the universal solution to this problem is provided by the method of maximum (relative) entropy (ME). The scheme hinges on the basic principle that prior information is valuable. That is, we would like to incorporate any newly available information, but without sacrificing, to the greatest extent possible, the information held in our prior beliefs.

Together, probability theory and entropic updating form the foundation of inductive reasoning. And more importantly, they are universal in this regard; implying that these are the default tools for whenever we lack complete information. The universality is key since it justifies the use of these tools in any situation where uncertainty plays a factor. (This, we will find, is also applicable to quantum theory, the subject to which the majority of this thesis is devoted.)

The structure of this short chapter is as follows. Section <ref> introduces probability theory as a set of rules for consistently manipulating degrees of rational belief. Section <ref> introduces relative entropy as a tool for ranking probability distributions, and thus as a means for updating probabilities, according to a set of desideratum with wide appeal.

The treatment of probability theory and entropic updating that we include here is not meant to be exhaustive, nor is it an extensively detailed account. For further details please consult [35] by A. Caticha as well as the theses of A. Giffin [36] and K. Vanslette [89], and the references therein for a list of original works and a more detailed presentation.

§ PROBABILITY THEORY

In situations where complete conviction in an outcome is absent, we are left to grapple with the possibility that multiple different outcomes may be realized, and as such, we are left to deliberate on the relative merits of each outcome. That is to say, given limited (or no) knowledge of some assertion, we wish to gauge the truth content of the potential outcomes. This is the problem that we aim to address here. And faced with such a task, we set out to design a tool for aiding and facilitating our judgment in this regard.

§.§ The design of probability theory

The key idea in designing a tool for rational inductive reasoning is the following: we wish to formulate a quantitative measure of the degree to which we may rationally presume a particular assertion to be true. There are several components to this idea that require explanation. The first is that the object of interest here is our belief in the truth of a particular proposition; something we will later find can be identified with probabilities. Additionally we, by design, require that our beliefs be amenable to quantitative analysis; the purpose of any tool, after all, is to be useful in practice, which is exactly what a mathematical description allows for. The third, and most crucial factor, is that the rules for manipulating our beliefs must be rational. The difficulty in this requirement, however, is that we cannot easily identify a set of universal guidelines that lead to rational behavior in all circumstances. Instead, we settle for a much more modest goal: we aim to avoid those glaring cases that we can all agree are irrational. This implies that we hope to avoid obvious inconsistencies in our reasoning; the more we believe, for instance, that $A$ is true, then the less we ought to believe that its negation $\tilde{A}$ is true, and anything to the contrary, most would agree, would be irrational.

*Quantifying rational belief—
Concerning the mathematical description, the objective is to model the pure intensity with which we believe a particular proposition to be true. Two words here are operative: intensity and proposition. The latter is important since it makes clear that we are assigning degrees of rational belief to logical Boolean propositions; thus, as will be important later, more complex propositions may be built from simpler propositions using the AND and OR operations of Boolean logic.

The word intensity is also key: an intensity is defined simply as being either more or less intense. As such, our mathematical description should allow for intuitive notion that the intensity of rational beliefs is transitive: if we believe in $A$ is more than $B$, and believe in $B$ is more than $C$, then it should be that we believe in $A$ more than $C$. These criterion suggest the claim that

degrees of rational belief (which we can later identify with probabilities) can be represented by real numbers.

We establish here some standard notation which is consistent with that of [35]. We assume some basic familiarity with Boolean logic. For any proposition $a$, it has a negation, denoted $\tilde{a}$, such that if $a$ is true then $\tilde{a}$ is false. Sometimes we are interested in the assertion that $a$ is true subject to some background information $c$. This is denoted $a|c$ and read as “$a$ given $c$".

From any two propositions $a$ and $b$ we can construct more complicated composite propositions using the conjunction and disjunction operations. The conjunction, or AND operation will be denoted $ab$ and the disjunction of two propositions $a$ and $b$ will be denoted $a\vee b $. The truth value of the composite propositions $a\vee b$ and $ab$ that are conditional on some given background information $c$ will be denoted as $a\vee b|c$ and $ab|c$, respectively.

The degree to which we believe the proposition $a$ to be true is denoted $[a]$, and is such that $[a] \in \mathbb{R}$ is a real number. Eventually, we will find that there is a direct correspondence between $[a]$ and the probability $p(a)$. Finally, degrees of belief will range from some extreme lower value $v_{F}$ reflecting extreme disbelief in a proposition (such as $[a|\tilde{a}] = v_{F}$, for any $a$) to the extreme upper bound $v_{T}$ which reflects complete belief in a proposition (such as $[a|a] = v_{T}$).

The basic element that one has access to is the real number $[a|c]$ associated to any proposition $a$, subject to some background information denoted by $c$. (The truth value of any proposition is, of course, dependent on some context, which for conciseness we denote simply using $c$.) The question of how one assigns a value to $[a|c]$ in the first place is interesting, and is relevant for the subsequent section on entropic updating, but here we assume that such an assignment has already been made.

The problem that we would like to solve, on the other hand, is how to rationally assign values to the degrees of belief $[ab|c]$ and $[a\vee b|c]$ for the composite propositions $ab|c$ and $a\vee b|c$, respectively.

Assumption: in order to be rational, our beliefs in $a\vee b$ and $ab$ must somehow be related to our separate beliefs in $a$ and $b$.

A mathematical representation of this assumption would be that there exist some functions $F$ and $G$ that satisfy

\begin{equation}
[a\vee b|c] = F\left([a|c],[b|c],[a|bc],[b|ac]\right)
\label{Sum rule F}
\end{equation}

\begin{equation}
[ab|c] = G\left([a|c],[b|c],[a|bc],[b|ac]\right).
\label{Product rule G}
\end{equation}

Our goal is to determine the unknown functions $F$ and $G$ that must hold for any choice of propositions $a$, $b$, $c$.

The essential constraint is that the appropriate functions $F$ and $G$ must respect the associative and distributive properties of the Boolean AND and OR operations. This turns out to be extremely restrictive, resulting in a family of equivalent representations, of which standard probability theory is a member.

§.§ Sum rule

We aim here to outline the derivation of the sum rule of probability theory, which amounts to a determination of the function $F$ in (<ref>). We proceed in two steps. We first determine the form of $F$ for propositions that are mutually exclusive by imposing an associativity constraint. The general case is easily obtained from there.

*Special case: mutual exclusivity—
For the $F$ in eq.(<ref>), when $a$ and $b$ are mutually exclusive in the context of $d$ then $[a|bd] = [b|ad] = v_{F}$, and so $F$ is essentially a function of two arguments:

\begin{equation}
[a\vee b|d] = F\left([a|d],[b|d],v_{F},v_{F}\right) = F\left([a|d],[b|d]\right).\label{Sum rule F MEx}
\end{equation}

Of course the $F$ cannot be chosen in any haphazard fashion, it must reflect some kernel of rationality. The basic idea is consistency: if a degree of rational can be computed in two different ways, they must agree. To execute this idea it is useful to consider three mutually exclusive propositions $a$, $b$, $c$ in the context of some information $d$. The requirement of consistency follows from the associativity of the Boolean OR,

\begin{equation}
(a\vee b) \vee c = a\vee (b\vee c).
\label{associativity}
\end{equation}

The consequences of imposing such a constraint leads to a functional equation (see e.g., [33]) whose general solution was first found by Cox (see e.g., [32][93]). Unfortunately, for an arbitrary ranking scheme $[\, \cdot \, ]$, the general solution is quite complicated. Cox, however, demonstrated that one could always regraduate (or rescale) the degrees of belief

\begin{equation}
[\, \cdot \, ]\to\xi(\cdot),
\end{equation}

while maintaining the original ordering, the solution takes a very simple form:

\begin{equation}
\xi(a\vee b|d) = \xi(a|d)+\xi(b|d),
\label{Sum rule xi MEx}
\end{equation}

which is just a sum rule.

Utter disbelief:

In the previous ranking scheme using $[\,\cdot\,]$, the value $v_{T}$ reflected complete certainty, while $v_{F}$ demonstrated complete disbelief. In the modified ranking scheme using $\xi(\cdot)$, complete certainty is given, by definition, as $\xi_{T} = \phi(v_{T})+\beta$; its value cannot be determined exactly. On the other hand, it is a simple matter to show that (<ref>) determines the value of complete disbelief to be $\xi_{F} = 0$ in the new scale [35][33].

*General sum rule—
The mutual exclusivity assumption can be easily relaxed to obtain a general representation of the OR rule. Note that any pair of arbitrary propositions $a$ and $b$ can be written as $a = (ab)\vee(a\tilde{b})$ and $b = (ab)\vee(\tilde{a}b)$, respectively. Thus we have that

\begin{equation}
a\vee b = (a b)\vee (\tilde{a}b) \vee (a\tilde{b}) = a\vee(\tilde{a}b),
\end{equation}

which is composed of two mutually exclusive options. As such, the sum rule in (<ref>) applies, and we obtain

\begin{align}
\xi(a\vee b|d) &= \xi(a|d)+\xi(\tilde{a}b|d) + (\xi(ab|d)-\xi(ab|d))\notag\\
&= \xi(a|d) + \xi(\tilde{a}b\vee ab|d) - \xi(ab|d).
\end{align}

Using the fact that $\tilde{a}b\vee ab = b$ gives us

\begin{equation}
\xi(a\vee b|d) = \xi(a|d)+\xi(b|d) - \xi(ab|d),
\label{Sum rule general}
\end{equation}

which is the general sum rule.

§.§ Product rule

For arbitrary propositions $a$ and $b$ we would like to assign a degree of rational belief $\xi(ab|c)$ to the conjunction $ab$ in the context of some background information $c$. The idea is to identify the function $G$, which in its most general version depends on four inputs

\begin{equation}
\xi(ab|c) = G\left(\xi(a|c),\xi(b|c),\xi(a|bc),\xi(b|ac)\right).
\end{equation}

A lengthy argument due to Caticha [35][33], which is beyond the scope of this thesis, narrows the space of  functions $G$ to those of just two variables. In particular, $G = G(\xi(a|c),\xi(b|ac))$.[Essentially, one considers all possible choices of functions for $G$, ranging from two inputs to all four. By examining certain clever special choices of inputs, one can eliminate all but one of the functions as the others demonstrate clearly irrational results (such as, for example $G = $constant).]

*Consistency constraint—
To determine $G(\xi(a|c),\xi(b|ac))$ we impose consistency once again: whatever the function $G$ is, it must be consistent with the sum rule eq.(<ref>) and the distributive properties of the Boolean AND and OR operations. Our brief derivation follows Caticha [35].

Consider the propositions $a$, $b$, $c$, where $b$ and $c$ are mutually exclusive, in the context of a fourth assertion $d$. Using the distributivity of AND over OR we have

\begin{equation}
a(b\vee c) = ab\vee ac.
\end{equation}

This implies that we can compute the degree of belief $\xi(a(b\vee c) |d)$ in two ways,

\begin{equation}
\xi(a(b\vee c)|d) = \xi(ab\vee ac|d) = \xi(ab|d)+\xi(ac|d),
\end{equation}

where the second equality follows because $b$ and $c$ are mutually exclusive. Fulfilling this constraint requires that

\begin{equation}
\frac{\xi(ab|d)}{\xi_{T}} = \frac{\xi(a|d)}{\xi_{T}}\frac{\xi(b|ad)}{\xi_{T}}.
\label{product rule xi}
\end{equation}

*The main result—
Equation (<ref>) suggests introducing a new, more convenient, set of values, scaled by $\xi_{T}$:

\begin{equation}
p(\cdot) = \frac{\xi(\cdot)}{\xi_{T}}.
\end{equation}

In this regraduated scale, the representation of the AND operation of Boolean logic is a simple product rule,

\begin{equation}
p(ab|c) = p(a|c)\, p(b|ac).
\label{Product rule prob}
\end{equation}

On the other hand, in this new scale, the sum rule retains the same form

\begin{equation}
p(a\vee b|c) = p(a|c) + p(b|c) - p(ab|c).
\label{sum rule prob}
\end{equation}

The values $p(\cdot)$ takes values from $p_{F} = 0$ — complete disbelief — to $p_{T} = 1$ — absolute certainty. This justifies calling the values $p(\cdot)$ probabilities.

In short we conclude that:

Degrees of rational belief are represented by probabilities and those probabilities are manipulated by the rules of probability theory, i.e. the sum and product rules.

While other formal choices are possible, probability theory stands alone as a particularly convenient representation of the rules for manipulating degrees of plausibility. Once one obtains the standard sum and product rules, there is no impediment to obtaining all the standard results of probability theory — the central limit theorem, marginalization, etc.[There are many good books on probability theory. This author has gained much insight from the book by Caticha [35] and the book by Jaynes [31].]

The implications are also vast: a partial state of knowledge is to be represented by probabilities. Thus probability theory provides the framework for reasoning in the face of uncertainty. (The question, of course, is whether this applies to so-called “quantum" uncertainties as well.)

*A comment on continuous distributions
It is worth mentioning that we did not specify the features of the propositions $a$, $b$, etc. They could, for instance, be discrete or continuous (and indeed, our derivation relied on this generality). In the continuous case, however, plenty of subtleties arise owing to an infinite number of possible outcomes. In particular, for a variable $X$ taking on a continuous set of values $x$, the probability of a single point $p(X=x) = 0$.

A well defined notion, on the other hand, is that of the probability of an infinitesimal range of values

\begin{equation}
p(x\leq X \leq x+\delta x) = \rho(x)dx,
\end{equation}

where $\rho(x)$ is called the probability density function. And, while a true probability is only assigned to the product $\rho(x)dx$, it is common to also call the probability density $\rho(x)$ a probability, or probability distribution; although it is always the case that actual probabilities must come accompanied with some measure $dx$.

§ ENTROPIC UPDATING

That one can assign an interpretation (or any interpretation at all) to the notion of probability is important for the simple fact that knowing what a probability is suggests how probabilities should be used. Which is to say that, if probabilities are nothing more than a set of rational convictions, then we should treat them as such! And indeed, to this point, another common facet of rational beliefs is that they are not rigid: a rational agent changes her beliefs in light of new evidence. This suggests that, yes, probabilities too must be updated when new information becomes available. Here we introduce the universal tool for accomplishing just this, called the maximum entropy (ME) method.

Throughout this section we borrow heavily from [35] in crafting the motivation and rationale for the ME method, however, we have updated the assumptions and design specifications to reflect the work of K. Vanslette in [88][89], which improves on the treatment in [35] by way of reducing the number of necessary desiderata.

§.§ The design of entropic updating

The idea is the following. We are interested in predicting the values of a variable $x$ about which we are uncertain.[We assume $x$ is continuous throughout, but the simplification to discrete variables is straightforward.] This state of partial knowledge is encoded, naturally, by a probability distribution $q(x)$. The problem we would like to address is, given that we are presented with some new information about the $x$'s, how should we modify our current beliefs, i.e. $q(x)$, to accommodate this new information. That is, how do we update from a prior distribution $q(x)$ to a particular posterior distribution $p(x)$ that takes into account the new information that we have learned?

*An aside on information—
A chief concern in our task is an identification of what we mean by “information". To be sure, the meaning of the word “information" is quite contentious, and it is not our purpose here to present an exhaustive discourse on this. In the simplest sense, we define information operationally: information is whatever constrains the beliefs of a rational agent. That is to say, information takes the form of constraints on our state of knowledge, i.e. constraints on the allowed probability distributions. This is the stance that we take throughout this thesis. A more thorough discussion of our perspective is available in [35][94].

*The general design criteria—
The goal is to design a tool for updating probability distributions based on some design criteria that are sufficiently reasonable so as to garner wide appeal.

The hope is not to develop a tool that works in this problem or that problem, but a tool that is robust enough to function in any problem where inference is needed.

We take here a conservative viewpoint: prior information is valuable. Thus we should update our beliefs to the minimum extent required by the new information. We call this the principle of minimum updating (PMU). This has an appealing practical consequence: we do not proceed by instructing a rational agent how to change their beliefs (there are, of course, many ways to do this), but by detailing when not to update our beliefs.[For example, such a principle guards against a clear form of irrationality: we should not change our beliefs at all when no information has been provided. As usual, this idea is more eloquently and lucidly stated in [35].]

When doing science we always tacitly assume that we can identify a set of relevant variables which can be considered independently from all others — if we could not assume, for instance, that the global wind patterns on earth are independent of the behavior of the Andromeda galaxy, then science would be a hopeless endeavor. This principle will be implemented in two manners — subsystem independence and subdomain independence — which will we develop more rigorously later on.

We seek to update from a prior distribution $q(x)$ to a candidate posterior distribution $p(x)$

\begin{equation}
q(x) \longrightarrow p(x),
\end{equation}

where $p(x)$  captures some newly obtained information. (The notation convention follows [35]: $q(x)$ denotes priors, while $p(x)$ denotes a possible candidate posterior, and $P(x)$ signifies the optimal posterior.) The strategy we implement for accomplishing this is the following: we rank the possible posteriors $p(x)$ according to preference. (We will be more specific about what we mean by preference shortly.) That is, to each possible posterior distribution $p_{i}(x)$ we assign a single real number $S[p_{i},q]\equiv S[p_{i}]$ — the entropy functional — such that if we prefer $p_{1}$ to $p_{2}$ then $S[p_{1}] > S[p_{2}]$.[The ranking scheme that we utilize for updating probability distributions can traces its origins to the work of Shore and Johnson [95] as well as J. Skilling [96].] Naturally, such a ranking, if it is to be useful, should also be transitive: if $p_{1}$ is preferred to $p_{2}$ — $S[p_{1}] > S[p_{2}]$ — and $p_{2}$ preferred to $p_{3}$ — $S[p_{2}] > S[p_{3}]$ — then $p_{1}$ is preferred to $p_{3}$ — $S[p_{1}] > S[p_{3}]$.

The objective is then, obviously, to choose the most preferred distribution $\hat{p}$; this is the one that maximizes the entropy functional $S[p]$. Hence the scheme is called the method of Maximum Entropy (ME).

Eliminative induction:

The strategy for identifying an appropriate entropy functional is called eliminative induction. Briefly: once one identifies a set of reasonable criteria to be satisfied, one imposes these conditions on the allowed form of the entropy $S[p]$. As we shall see, the criteria chosen here are sufficiently constraining, such that only a single entropy functional is allowed.

§.§ The relative entropy functional

We consider here two specific design criteria (DC) that our entropy functional must satisfy. We follow the argument of K. Vanslette [88][89] whose method largely follows that of A. Caticha [35], but improves it by removing one superfluous ingredient. Here we simply state the criteria and their implications; proofs can be found in the associated references.

*DC1: Subdomain independence—
The requirement of subdomain independence is an instance of specifying when it is rational to not update our beliefs. Consider that the space $\mathcal{X}$ of $x$'s is divided into two domains such that $D \cup \tilde{D} = \mathcal{X}$, where $D$ and $\tilde{D}$ are complements. DC1 states that when a constraint $C$ refers only to the domain $\tilde{D}$, then the conditional probabilities $p(x|D)$ should not be revised, i.e. $p(x|D) = q(x|D)$. If one imposes this for every conceivable division $\mathcal{X}$ into sets of subdomains, then this implies that each subdomain contributes additively to the entropy:

\begin{equation}
S[p] = \int dx F\left(p(x),q(x),x\right),
\label{Entropy DC1 F}
\end{equation}

where $F\left(p(x),q(x)\right)$ is a regular function (not functional) of the probabilities $p(x)$ and $q(x)$, and may depend explicitly on the coordinates $x$.[The proof of this result can be found in [35] under the label of the “locality" DC.]

A special case of DC1 is that if we do not impose any constraints at all, then we should have that the optimal posterior is just the prior: $P(x) = q(x)$.

*DC2: Subsystem independence—
Consider the joint space of variables $\mathcal{X} =\mathcal{X}_{1}\cup \mathcal{X}_{2}$ where $x_{1}\in \mathcal{X}_{1}$ and $x_{2}\in \mathcal{X}_{2}$ are their respective coordinates, and moreover, consider in addition that the joint prior $q(x_{1},x_{2}) = q(x_{1})q(x_{2})$ reflects the property of statistical independence. DC2 states that if the new information contained in some constraint $C$ does not explicitly introduce correlations between the variables $x_{1}$ and $x_{2}$, then the optimal posterior $P(x_{1},x_{2}) = P(x_{1})P(x_{2})$ should also reflect the statistical independence properties of the prior $q(x_{1},x_{2}) = q(x_{1})q(x_{2})$. This is a clear instance of the PMU: if we believe the systems to be independent a prior and the new information does not explicitly introduce information to the contrary, then we should maintain the belief a posteriori that the system remain independent.

The implications of this DC are particularly constraining, requiring (subject to some caveats of convenience) that the entropy functional be of the form

\begin{equation}
S[p,q] = -A\int dx \, p(x)\log\frac{p(x)}{q(x)},
\label{Entropy DC2}
\end{equation}

which is the well-known relative entropy functional (see e.g., [31] and references therein).

§.§ The Maximum Entropy method

Once a suitable entropy functional has been designed, we can proceed to use it for inference. The method can be presented algorithmically as follows:

1. Establish the prior $q(x)$ that requires updating.

2. Obtain that information which is deemed relevant for updating: this obliges us to identify the family of posteriors $\left\{p(x)\right\}$ that are consistent with this information

\begin{equation*}
q(x)\quad \overset{\text{new information}}{\xrightarrow{\hspace*{2.5cm}}}\quad p(x).
\end{equation*}

3. Define a relative entropy $S[p,q]$ for ranking the posteriors $p(x)$ relative to the prior $q(x)$.

4. Select the optimal $p^{*}(x) = P(x)$; the one that maximizes the entropy subject to the constraints, $S^{*} = \text{max}_{p} \, S[p,q]$.

There are two types of information that are common enough to warrant specific mention: data and expected value constraints. An interesting aspect of the entropic inference framework is that both types of information can be handled using the ME method. Indeed, the ME method generalizes both Bayesian data analysis as well as the MaxEnt formalism of E.T. Jaynes [87].

On one hand, ME easily allows for processing some data that is not readily accommodated in the standard framework; such as if the data is itself a bit uncertain. On the other, the original MaxEnt of Jaynes was not designed for updating from an arbitrary prior to a posterior; its purpose was rather for assigning probabilities when the prior was a uniform distribution.

As it turns out, both types of updating are crucial for developing quantum theory from a perspective that takes entropic inference as the foundation. Since measurement is nothing but the collection of data, it is possible to model the infamous collapse of the wave function as nothing but an application of entropic updating using Bayes' rule. Moreover, entropic updating plays a crucial role in obtaining the Brownian stochastic processes that underlie all quantum systems.[Another aspect of entropic inference crucial to quantum theory is information geometry. While this aspect of entropic inference will be briefly reviewed within the course of this thesis, this does not do the subject justice. See the book of Caticha [35] for an accessible introduction and the book by Amari [97] for further details.]

CHAPTER: NON-RELATIVISTIC ENTROPIC DYNAMICS

The first instance of inferential methods playing a prominent role in fundamental physics occurred with the work of E.T. Jaynes in statistical mechanics [87]. Up to that point, the development of statistical mechanics — from Boltzmann to Maxwell to Gibbs and so on — had been plagued by general confusion, as was evident with Maxwell's demon, the Gibbs paradox, irreversibility, etc.[See, for example, the writings of Jaynes himself [98][99] as well as the monograph by A. Caticha [35] for an in-depth discussions of these issues and how the inference viewpoint casts light on them.] The beauty of Jaynes' approach was that by placing the notions of probability, entropy, and information at the forefront of the theory, many of these fundamental issues and paradoxes in statistical mechanics were clarified, while many of the enigmatic building blocks of the conventional formulation were left behind. Indeed, once one adopts, as Jaynes did, a Bayesian view of probability, conceptual issues that hindered statistical mechanics — such as the ergodic hypothesis, etc. —  were no longer relevant! The fuzzy picture of entropy as a measure of disorder, or chaos or (enter the interpretation du jour), of a system's physical state that was inherited from thermodynamics was replaced by entropy as a tool for processing information; a notion that began with the work of C. Shannon [101].

While attaining clarity in statistical mechanics was a significant achievement in its own right, Jaynes' statistical mechanics is important in another regard as well: it shows that statistical mechanics results from an application of inferential methods. From this viewpoint, statistical mechanics is not a theory that describes the mechanics of a physical system — it is a theory about processing the information we possess about a physical system. This begs the question of whether statistical mechanics is a unique theory in this regard, or are other theories of physics amenable to being treated within this framework? Are the laws of nature firm and mechanical, or is it as John Wheeler [102] once said:

“that every law of physics, pushed to the extreme, will be found statistical and approximate, not mathematically perfect and precise"?

§.§ An entropic view of quantum theory — why bother?

One might look at the state of quantum theory today and think that it is reminiscent of the status of statistical mechanics in the time of Jaynes, almost 70 years ago. Indeed, quantum mechanics is another theory of physics in which uncertainties and probabilities enter in an intrinsic way. And, not coincidentally, quantum theory is itself infamous for the conceptual issues that it poses — measurement and wavefunction collapse, non-locality and entanglement, and the list goes on.[That conventional quantum theory is chock-full of paradoxes that challenge the intuition we've gained from classical mechanics is well known and we do not attempt here to compile an exhaustive list. The textbook by L. Ballentine [103] is more sensitive to these issues than most and the papers by J.S. Bell [104] provide penetrating insight.]

It is natural to wonder, in light of Jaynes' work, if we could do for quantum mechanics, what Jaynes did for statistical mechanics? In [87], Jaynes said of statistical mechanics:

“it is possible to maintain a sharp distinction between its physical and statistical aspects. The former consists only of the enumeration of the correct states of the system and their properties; the latter is a straightforward example of statistical inference."

Can we clarify and understand the foundations of quantum theory in a similar fashion by recognizing, in a clear-eyed manner, the probabilistic and statistical aspects of the theory as distinct from its physical features?

The program of entropic dynamics (ED) is an attempt to do just that — to derive quantum theory using the methods and tools typical of information processing and inference. This involves understanding the probabilities $\rho=|\Psi|^{2}$ that occur in quantum mechanics to be epistemic Bayesian probabilities and, accordingly, that those probabilities be updated by rules that are consistent with entropic and inference methods.

As one might expect, this has significant effects on how the theory is to be interpreted. Indeed, one question that is common when discussing the foundations of quantum mechanics is, what is the interpretation of the quantum state $\Psi$? Is the wave function an element of the ontology? That is to say, is it a “real" object, like a particle or a field? Or, is the wave function epistemic, related to our state of knowledge of the system? Clearly in ED the answer is the latter, and as a result, interpretations of quantum mechanics where the wave function is given an ontic status are dismissed outright; such interpretations include the Bohmian theory [6][105] and the provocative “Many-Worlds" interpretations first proposed by Everett [9].

This viewpoint also has utility beyond simply interpretation. Viewing probabilities in quantum theory as Bayesian goes a large way towards solving the measurement problem; perhaps the most infamous open problem in the foundations of quantum mechanics.[see [41][43] for a full discussion of how the problem is resolved in ED.] Once probabilities are Bayesian, one has access to the full toolkit of entropic and Bayesian inference; including the Bayesian and entropic modes of updating probabilities. As a consequence, in ED, an independent collapse postulate is not required — one can show [41] that the infamous collapse of the wave function is nothing but an application of Bayes rule!

But is ED truly necessary? Quantum mechanics is, after all, the most successful scientific theory that humankind is in possession of — time after time its predictions have been validated with great precision. The conventional viewpoint, it seems, is doing just fine! And to be absolutely clear, our goal is not to claim otherwise. However the goals of theoretical physics have expanded beyond describing tabletop and accelerator physics, and now include research into quantum gravity and quantum cosmology — situations describing the universe at large. It is pertinent to ask, how does one quantify the uncertainty of the conditions of the big bang? What is the significance of the entropy of a black hole? What does it mean to assign a probability to a space-time, as some quantum gravity approaches seem to do? Can we interpret (or even conceive of) the second law of thermodynamics in a cosmological setting?

These questions raise several issues. If the relationship between gravity, entropy, and quantum theory is indeed important, as we expect it is, then perhaps the entropic framework can shed some light on it. Moreover, as Ernst Mach astutely noted: the universe is not twice-given, but only once. Thus it is an open question as to how frequentist views of probability can handle questions at this scale. Is the Bayesian viewpoint not more coherent in this case? And if it is, then consistency suggests that we treat probabilities as Bayesian all the way down to microscopic scales.

This all implies that rethinking physics, in a philosophically sound manner, should be considered a necessary aspect of doing important theoretical work going forward. Feynman, it is said, would often comment that (paraphrasing) “it is more important for a physicist to be physically rigorous before being mathematically rigorous." It seems, however, that more and more, theorists must be “philosophically rigorous" as well.  As the ever quotable John Wheeler said [106],

“Philosophy is too important to be left to the philosophers!"

§ ENTROPIC DYNAMICS

Here we develop the ED approach to formulating a quantum theory. This task consists primarily of determining an evolution for the probability $\rho = |\Psi|^{2}$ and its complement, the drift potential $\phi$, that is in accordance with the rules for processing information. That information is then supplied, in turn, in the form of constraints. Thus ED is a dynamical scheme driven by constraints. A chief aim of the ED program is to identify those constraints that inevitably result in the Schrödinger equation and quantum theory. It is only once we have understood these constraints that we can hope to make progress in domains where our understanding is more incomplete, such as gravity.

ED as a dynamical scheme proceeds in stages; each stage being characterized by the corresponding constraints. Imposing constraints on the microscopic dynamics, for instance, leads to a Brownian motion, and a diffusion equation in the form of a Fokker-Planck equation for $\rho$. While this is a perfectly legitimate form of dynamics, it falls short of the intended aim of a quantum dynamics, which contains two dynamical degrees of freedom — a magnitude $\rho = |\Psi|^{2}$ and a phase. A natural way to proceed is then to promote the drift potential $\phi$ to a true dynamical variable that guides the probabilities, but which is itself then affected by the probabilities. The natural question that follows is, how does one determine the evolution of $\phi$? For this, we require, naturally, additional constraints.

In ED this choice of additional constraints has been pursued from various directions. An early hint was provided in Nelson's work on stochastic mechanics [108] — quantum theory can be formulated as a non-dissipative Brownian motion by imposing that $\phi$ evolve such that an “energy" functional be conserved. This insight was first adopted in ED [51][69] in a rather basic fashion: one assumes the existence of an energy functional $\tilde{H}$ and assumes entirely its functional form. This results in a set of coupled dynamical equations for $\rho$ and $\phi$ that are wholly equivalent to the Schrödinger equation, but which provides little in the way of clarifying the underlying apparatus of quantum mechanics — i.e. the Hilbert spaces, operators, Lie algebras, etc.

A similar, but more sophisticated, path was pursued in [53][70]. There, while the central focus was still on conserving an energy to update $\phi$, the  energy functional $\tilde{H}$ in question was now recognized as a true Hamiltonian functional with $\rho$ and $\phi$ playing the role of a canonical pair whose evolution was subject to Hamilton's equations. In other words, $\rho$ and $\phi$ formed a phase space, called the ensemble phase space (EPS), or e-phase space for short. This represented quite a breakthrough in ED. The canonical framework is extremely rich; coming equipped with Hamiltonian generators, Lie algebras, Noether theorems, and the like. Thus with this insight ED could be seen to be encroaching on the kind of structures — symmetries, algebras, conserved quantities — that seem to underpin all of modern theoretical physics, including quantum theory.

As the astute reader may notice, however, this flavor of updating scheme does not work in curved space-time where the notion of a globally conserved energy is not generally available. Thus it is desirable to pursue a more robust criterion for updating that survives, even when energy conservation as an updating principle does not. One such alternative strategy is to pursue a criterion for updating that is founded on geometric principles. For instance, in [54], A. Caticha models the e-phase space as a cotangent bundle, which comes naturally equipped with a symplectic structure and thus all the usual accoutrements of the canonical framework, including Hamiltonian generators, Poisson brackets, etc. However, while such an approach provides an interesting way forward in ED, there are still some lingering issues needing to be resolved; not least of which is whether the cotangent bundle is a geometric structure that is rich enough to model all physically relevant quantum states (see e.g., [107]).

To sidestep these issues, we pursue here a manner of updating that encompasses the various methods we have discussed thus far — energy conservation, cotangent bundles, etc. — but which is not bound to any one procedure, in particular. The basic idea we pursue here is inspired by the work of M. Hall and M. Reginatto (see e.g., [24]) and is as follows. Since quantum theory is obtained from ED when the probability $\rho$ and drift potential $\phi$ evolve according to Hamilton's equations, we directly impose this structure on ED. This is accomplished by requiring the e-phase space to, in fact, have the formal structure of a phase space, namely, by imposing the existence of a symplectic structure on our space; a geometrical object that allows for notions of Poisson brackets, Hamiltonian generators, and all the typical tools of the canonical formalism. This approach neither requires us to impose a conserved energy — which may or may not be reasonable — nor does it require us to introduce additional assumptions about the e-phase space, which may or may not not be warranted. Thus while this approach is quite versatile and useful in that it allows us to develop various aspects of ED, such as the theory of relativistic fields, it is also silent as to why such structures prove to be so useful in describing physics.

In the remainder of this chapter we deal with a system of non-relativistic quantum particles, the simplest ED model. We develop the theory of relativistic quantum scalar fields in subsequent chapters. In this section we introduce the subject matter of ED, i.e. the elements about which we make inferences, and introduce the information that guides these inferences. In <ref> we introduce entropic time as a scheme for organizing our succession of inferences. In section <ref> we use information geometry to assign a metric to the configuration space of particles. Section <ref> introduces the dynamical equation for the probability $\rho$ while section <ref> discusses how to integrate the drift potential $\phi$ into the dynamics, while in section <ref> we introduce the canonical framework in ED. In section <ref> we address how a quantum theory is obtained from these developments and in section <ref> we conclude with some remarks that are relevant for subsequent chapters.

*On subject matter—
Our goal is to build a model of physics from the viewpoint of inference and information processing. The first step in formulating an inference problem consists of a specification of the subject matter: what are the microstates about which we are inquiring? Without this basic (but difficult) step, we cannot proceed.

However, a daunting aspect of formulating quantum mechanics from our viewpoint is exactly the fact that capturing what the “is" in quantum mechanics is, has remained elusive. What, after all, are the objective building blocks of the theory, or in the words of J.S. Bell, what are its “beables"? In the conventional Copenhagen interpretation, no answer is given, or worse yet, no answer can even, in principle, be given. It is not at all uncommon, for example, in conventional quantum mechanics to speak of quantum “particles" that are neither particulate nor wave-like, and that have neither position nor momentum until measured; such particles are very strange indeed!

To move forward it would seem that such questions must, eventually, find a satisfactory resolution. Here the words of Jaynes [100] are particularly insightful:

“[Quantum mechanics] is not purely epistemological; it is a peculiar mixture describing in part realities of Nature, in part incomplete human information about Nature, all scrambled up by Heisenberg and Bohr into an omelette that nobody has seen how to unscramble. Yet we think that the unscrambling is a prerequisite for any further advance in basic physical theory. For, if we cannot separate the subjective and objective aspects of the formalism, we cannot know what we are talking about; it is just that simple.”

*On microstates— In ED we proceed by taking a definitive stance on the choice of microstates. Here we consider a system of $N$ point particles in a flat Euclidean space $\mathbf{X}$ with metric $\delta_{ab}$. The particles, by assumption, always have definite positions, but their exact locations are uncertain, and so our goal is to infer their unknown positions.

The positions of these particles can be given by their Cartesian coordinates $x_{n}^{a} \in \mathbf{X}$ ( with $n = 1 \cdots N$ denoting particle labels and $a = 1,2,3$ denoting spatial indices). The collective set of these positions $\{x_{n}^{a}\}$ can be denoted efficiently as just $x$, which is a point in the $3N$ dimensional configuration space $\mathbf{X}_{N} = \mathbf{X}\times \mathbf{X} \times \cdots \times \mathbf{X}$. (In the language of probability, the configuration space $\mathbf{X}_{N}$ would be referred to as the space of potential outcomes.) Our goal is to obtain the probability $\rho(x)$ that the system of particles forms a particular configuration $x$.

That this viewpoint, from the very outset, contradicts the Copenhagen interpretation, where observable quantities attain a value only when elicited during a measurement process, is clear. In ED, we may not know where the particles reside, but they do, in fact, reside somewhere. The particle positions $x$ are said, in other words, to represent what is called the ontic state of the system; they are the physical constituents of the theory. Other quantities that are typical of physics — such as energy, momentum, angular momentum, etc. — are not, however, given the same physical status. These quantities, rather, are statistical in nature; they are inferred from the process of measurement, and thus reflect properties of the epistemic state, quantified by the probability $\rho(x)$ (and later, the drift potential $\phi(x)$ as well).

*On notation—
For our treatment here of the ED of particles we use notation that is in agreement with [53] (and [54], where applicable). As we've seen so far, particle positions are denoted $x_{n}^{a}$ where $n = 1\cdots N$ denotes the particle label, and lower case Latin letters $a, b,c,$ etc. denote spatial indices, so that $a = 1,2,3$. A point in the configuration space $\mathbf{X}_{N}$ is given by $x$, while configuration space vectors are denoted $V^{A} = V_{n}^{a}$, where capital Latin letters $A, B, C,$ etc. denote the configuration space index, so that $A = (a,n)$.

Unless otherwise specified, we use the Einstein summation convention throughout. For spatial vectors $v^{a}$ and one-forms $u_{b}$ this implies that

\[v^{a} \, u_{a} = \sum_{a = 1}^{3} v^{a} \, u_{a},\quad\text{whereas}\quad V^{A} \, U_{A} = \sum_{n=1}^{N}\sum_{a=1}^{3} V^{a}_{n}\,U_{an}\]

is true for configuration space vectors $V^{A}$ and one-forms $U_{B}$. For convenience, we denote the $3N$-dimensional integration measure $d^{3N}x=dx$. Furthermore, as is standard, we use the notation $f(x): \mathbf{X}_{N}\to \mathbb{R}$ to denote functions, which map points on the configuration space $\mathbf{X}_{N}$ to a real number. On the other hand, square brackets, such as $g[\cdot]:\mathcal{F}\to\mathbb{R}$, denotes a functional dependence, which maps a point on a function space $\mathcal{F}$ to a real number. Additional notation will be introduced as the relevant material presents itself; we do not feel it is convenient to the reader to introduce new notation in a context completely divorced from the pertinent subject matter.

*The inference problem—
Having established the microstates of the system, we can proceed to considering the dynamics. ED has two primary assumptions in this regard. The first is that change happens; we do not ask why change happens, but given that it does, we wish to predict what changes are likely to occur. The second assumption is that motion of the particles is continuous such that finite changes result from the accumulation of many infinitesimally small steps. This latter assumption is crucial because it implies that to predict a trajectory we only need knowledge of an infinitesimal step; with finite changes occurring from the iteration of many small steps.

We deal with the following dynamical problem: given some initial state $x\in\mathbf{X}_{N}$ we look to determine the subsequent neighboring state $x^{\prime} = x + \Delta x \in \mathbf{X}_{N}^{\prime}$.[The question of whether the spaces $\mathbf{X}_{N}^{\prime}$ and $\mathbf{X}_{N}$ are the same dates back to Newton. Here we take the view that while $\mathbf{X}_{N}^{\prime}$ and $\mathbf{X}_{N}$ are necessarily isomorphic to one another, they are not necessarily the same. This is because they are answers to two distinct sets of questions — where is the particle now? Where is the particle later? We discuss this point further in the section on entropic time.] In the strictest sense, however, this problem is not well posed here because both $x$ and $x'$ are in general uncertain. The distribution that encodes this uncertainty in the initial state $x$ and final state $x'$ is given by the joint probability $\rho(x',x)$. Using the product rule we can rewrite this probability as $\rho(x',x) = P(x'|x)\, \rho(x)$. It is the probability of the infinitesimal step, given by the transition probability $P(x'|x)$, that determines the information relevant for determining a final state given an arbitrary initial state. Thus our goal at this juncture is to determine an appropriate transition probability $P(x'|x)$.

*Maximum Entropy Method—
The procedure for selecting the appropriate transition probability is, of course, provided by the method of Maximum Entropy (ME). (The details of this method have been discussed elsewhere in this thesis; see Ch. 2 for details.) That is, the optimal $P(x'|x)$ is that which maximizes the entropy

\begin{equation}
S[P,Q] = - \int dx' P(x'|x)\ln\frac{P(x'|x)}{Q(x'|x)},
\label{Particles Entropy Transition probability}
\end{equation}

relative to a prior distribution $Q(x'|x)$ and subject to the appropriate constraints.

A crucial part of the ME method is the choice of prior. Here the prior distribution $Q(x'|x)$ reflects the information that the motion unfolds in infinitesimal steps, but which is otherwise maximally uninformative. Such an extreme state of ignorance is one that ignores possible correlations between degrees of freedom

\begin{equation}
Q(x^{\prime}|x) = \prod_{n} Q(x_{n}^{\prime}|x_{n}).
\end{equation}

Moreover, when confronted with a complete lack of information, it is natural to appeal to symmetry; a principle dating back to the time of Laplace (see e.g. [110] or [111]) and his principle of insufficient reason. Here, for instance, we can take advantage of the rotational and translational symmetries of the three-dimensional Euclidean space $\mathbf{X}$, which the particles reside in, so that

\begin{equation}
Q(x'|x) = \prod_{n} Q(x_{n}'|x_{n}) = \prod_{n} Q(|\Delta x_{n}|).
\end{equation}

The prior $Q(x'|x)$ that reflects this state of affairs can itself be derived as a result of the ME method. Consider the entropy

\begin{equation}
S[Q,\mu] = - \int dx' Q(x'|x)\ln\frac{Q(x'|x)}{\mu(x'|x)}
\label{Particles Relative Entropy Priors}
\end{equation}

relative to the uniform measure $\mu(x'|x)$. Now maximize $S[Q,\mu]$ subject to normalization and the $N$ constraints[We use a notation here where $ \left\langle A(x') \right\rangle_{Q} = \int dx' Q(x'|x) A(x')$. If the subscript $Q$ is not written, then it means we perform an expectation value with respect to the posterior $P(x'|x)$ such that $ \left\langle A(x') \right\rangle = \int dx' P(x'|x) A(x')$. We suspect the convention will be obvious from context.]

\begin{equation}
\left\langle \Delta x_{n}^{a}\Delta x_{n'}^{b}\right\rangle_{Q} = \int dx^{\prime} \, Q(x^{\prime}|x) \, \delta_{ab} \Delta x_{n}^{a}\, \Delta x_{n}^{b} = \Delta\kappa_{n},
\label{Particles Continuity Constraint}
\end{equation}

which number one per particle, and which respect the rotational and translational symmetries of the underlying space. Continuity is enforced by requiring the constants $\Delta\kappa_{n}$ to be small, i.e. $\Delta\kappa_{n}\to 0$. The result is a product of Gaussians,

\begin{equation}
Q(x'|x) \propto \exp - \frac{1}{2}\sum_{n} \alpha_{n} \Delta x_{n}^{a}\Delta x_{n}^{b}\delta_{ab}.
\label{Particles Prior Q}
\end{equation}

The $N$ Lagrange multipliers $\alpha_{n}$ are a set of particle dependent parameters, which we will later take to be spatially constant. The common practice for determining these multipliers is familiar from statistical mechanics: we compute the constraints (<ref>) using the transition probability (<ref>) and solve for the multipliers $\alpha_{n}$ in terms of the constraints. The same procedure applies here as well.

To determine these multipliers in the standard style we must compute the expected Euclidean distance $ \delta_{ab}\left\langle \Delta x_{n}^{a} \Delta x_{n}^{a} \right\rangle_{Q}$ in (<ref>). To do this we exploit the fact that we deal with a Gaussian distribution which has mean

\begin{equation}
\left\langle \Delta x_{n}^{a}\right\rangle_{Q} = 0 \quad\text{and variance}\quad \left\langle \Delta x_{n}^{a}\Delta x_{n'}^{b}\right\rangle_{Q} = \frac{1}{\alpha_{n}}\delta_{nn'}\delta^{ab}.
\end{equation}

Thus we can easily compute (<ref>) to obtain

\begin{equation}
\delta_{ab}\left\langle \Delta x_{n}^{a} \Delta x_{n}^{a} \right\rangle_{Q} = \frac{3}{\alpha_{n}} = \Delta\kappa_{n},
\end{equation}

where $\Delta\kappa_{n}$ tends to zero. This implies that $1/\alpha_{n}$ must be small, or rather, that $\alpha_{n}\to \infty$.

Alternatively, inspection of (<ref>) also shows that the probability of finite steps is vanishingly small only in the limit $\alpha_{n}\to \infty$. This, of course, amounts to setting the width of the Gaussian to be infinitely small so that (<ref>) is, for all intents and purposes, a delta function. (The matter of choosing an explicit form for these multipliers will be dealt with later on, within the context of constructing time in ED.) This result is unaltered as we update to the posterior distribution $P(x'|x)$.

*Drift constraint—
On their own, the $N$ constraints eqns.(<ref>) lead to a rather simple dynamics; an isotropic diffusion that demonstrates no correlations among the particles. To go further towards a richer dynamics that exhibits such correlations, we require a device that allows for interdependence between the particles. This is achieved by introducing a global scalar function $\phi(x)$ (i.e. a function defined on the configuration space $\mathbf{X}_{N}$, rather than real space $\mathbf{X}$), which we refer to as the drift potential and that mediates the inter-particle relationships in our system.[Alternative formulations of ED (see e.g., [51][69]) favor a different strategy, introducing some additional y-variables whose entropy $S_{y}(x)$ assumes the role normally reserved for the drift potential. However, these models run into some trouble, since entropy must necessarily be single-valued function, whereas the phase of the quantum state $\Psi(x)$ must, in fact, be multi-valued to accommodate, for example, states of non-zero angular momentum. For this reason we work exclusively with the drift potential $\phi(x)$.]

We include this information into the dynamics of the particles by imposing an additional global constraint on the change of the drift potential $\phi(x)$:

\begin{equation}
\Delta \phi(x) = \phi(x^{\prime}) - \phi(x) = \frac{\partial\phi(x)}{\partial x_{n}^{a}} \Delta x_{n}^{a},
\label{Particles phi taylor expansion}
\end{equation}

so that

\begin{equation}
\left\langle \Delta \phi(x) \right\rangle = \int dx^{\prime} \, P(x^{\prime}|x) \, \frac{\partial\phi(x)}{\partial x_{n}^{a}} \Delta x_{n}^{a} = \Delta\kappa^{\prime},
\label{Particles Drift Constraint}
\end{equation}

where $\Delta\kappa^{\prime}$ is a small number. (Additional terms higher than first order in the Taylor expansion (<ref>) contribute negligibly as $\kappa_{n}\to 0$.)

The net result of imposing this constraint is that the particles are guided by the gradient of the drift potential, and as such, $\phi(x)$ plays the role of a pilot wave, similar in spirit to that of Bohmian mechanics. Eventually $\phi(x)$ will be responsible for many quintessential quantum phenomenon, such as entanglement and interference. A recent improvement in ED [54] has been to recognize that the introduction of such a function is not as ad hoc as it might, nevertheless, first appear. Note that in the ED we develop here there are two important pairings. The first pertains to the entropic origins of ED, and consists of probabilities and their constraints. Another important pairing occurs when we adopt a Hamiltonian formalism to update the evolving constraints. It is here that we must introduce a canonical coordinate, the probability $\rho(x)$, and its conjugate momentum. In an ED driven by constraints and by Hamiltonians it is therefore exceedingly natural that the two pairings may have some intimate connection. This is accomplished by assigning to the drift potential $\phi(x)$ a dual role, first as encoding the information in the form of constraints, and second, by identifying it as the momentum conjugate to $\rho(x)$.

Some properties of the drift potential $\phi(x)$ are inherited from its role as the momentum conjugate to $\rho(x)$, while others must be imposed for our model to be sufficiently flexible. To this end, we claim that $\phi(x)$ is a multi-valued function, such that $\phi(x)$ and $\phi(x)+2\pi n \beta$ represent the same constraint, where $n$ is an integer and $\beta$ is a dimensionless constant. This will be discussed more thoroughly later in the chapter.

*The posterior distribution—
To select a particular posterior $P(x^{\prime}|x)$ that satisfies the constraint (<ref>), we again resort to the ME method. The appropriate criterion for ranking the posteriors $P(x'|x)$, which satisfy (<ref>) relative to the prior $Q(x'|x)$ is the relative entropy $S[P,Q]$ in (<ref>). The most preferred distribution $P(x^{\prime}|x)$ is the one that maximizes $S[P,Q]$. Choosing the optimal $P(x^{\prime}|x)$ is, as usual, implemented by employing the method of Lagrange multipliers. Maximizing the entropy $S[P,Q]$ subject to (<ref>), and normalization, we have

\begin{equation}
P(x^{\prime}|x) = \frac{1}{\zeta(\alpha_{n},\alpha^{\prime})} \, \exp -\sum_{n} \left( \frac{\alpha_{n}}{2} \delta_{ab} \Delta x_{n}^{a}\, \Delta x_{n}^{b} - \alpha^{\prime}\,\frac{\partial\phi(x)}{\partial x_{n}^{a}} \Delta x_{n}^{a}\right),
\end{equation}

where $\zeta$ is a normalization constant, and $\alpha^{\prime}$ is a single multiplier for the global constraint (<ref>). Since the distribution $P(x^{\prime}|x)$ consists only of terms linear and quadratic in $x^{\prime}$, it is a Gaussian,

\begin{equation}
P(x^{\prime}|x) = \frac{1}{Z(\alpha_{n})} \, \exp -\sum_{n} \, \frac{\alpha_{n}}{2} \left( \Delta x_{n}^{a}- \frac{\alpha^{\prime}}{\alpha_{n}}\,\frac{\partial\phi(x)}{\partial x_{na}}\right)\left( \Delta x_{na}- \frac{\alpha^{\prime}}{\alpha_{n}}\,\frac{\partial\phi(x)}{\partial x_{n}^{a}}\right),
\label{Particles Transition probability}
\end{equation}

where, for convenience, we have raised and lowered indices appropriately using the Euclidean metric $\delta_{ab}$.

Concerning the remaining multiplier $\alpha^{\prime}$, we have no strong restrictions here on what its value must be. Work done by D. Bartolomeo and A. Caticha in [50] suggests that this freedom in the choice of $\alpha^{\prime}$ reflects a universality class of microscopic models; which includes the smooth trajectories of Bohmian mechanics as a special case. More succinctly: different choices of $\alpha^{\prime}$ lead to different microscopic dynamics, but all of which result in the same quantum theory. For our work here, we continue along the lines of [53][70] by absorbing, without loss of generality, $\alpha^{\prime}$ into the undetermined $\phi$, such that $\alpha^{\prime} \, \phi = \phi^{\prime}$ and then redefine $\phi^{\prime}\to \phi$; or, equivalently, we can simply set $\alpha^{\prime} = 1$.

*Particle motion—
We are now in a position to analyze what microscopic dynamics results from the the transition probability $P(x'|x)$. Since $P(x^{\prime}|x)$ is a Gaussian, it is appropriate to write a typical step as consisting of an expected drift with additional fluctuations, $\Delta x_{n}^{a} = \Delta \bar{x}_{n}^{a} + \Delta w_{n}^{a}$. This allows us to easily compute the moments of $P(x^{\prime}|x)$. For example, the expected step is given by

\begin{equation}
\left\langle \Delta x_{n}^{a} \right\rangle = \frac{1}{\alpha_{n}}\delta^{ab} \frac{\partial\phi(x)}{\partial x_{n}^{b}} = \Delta\bar{x}_{n}^{a}.
\label{Particles Expected Step}
\end{equation}

The fluctuations, on the other hand, are given by,

\begin{equation}
\left\langle \Delta w^{a}_{n} \right\rangle = 0 \quad\text{and}\quad \left\langle \Delta w^{a}_{n} \, w^{b}_{n'} \right\rangle = \frac{1}{
\alpha_{n}}\delta_{ab}\delta_{nn'}.
\label{Particles Fluctutions}
\end{equation}

(Additional moments can be computed using the standard techniques for computing Gaussian integrals.) As anticipated, the expected drift $\Delta \bar{x}_{n}^{a}$ and fluctuations $\Delta w_{n}^{a}$ do, indeed, tend towards small values in the limit $\alpha_{n}\to \infty$; which confirms that the particles in our model do, in fact, travel along continuous paths.

To tease out additional insight, let us inspect the behavior of these trajectories in the small step limit, $\alpha_{n}\to \infty$. From equations (<ref>) and (<ref>) we see there that, while the fluctuations are of order $O(\alpha_{n}^{-1/2})$, the expected steps are of order $O(\alpha_{n}^{-1})$. Thus, as $\alpha_{n}$ tends to large values, the fluctuations dominate; the resulting motion is thus continuous, but not differentiable. In short: the particles undergo a Brownian motion.

It is worth pointing out that this simple result has quite wide implications. Brownian motions are ubiquitous; with applications in physics, biology, finance, etc. That such models are easily obtained within the ED framework suggests that it may be possible to apply the notions developed here in other fields as well.[See e.g., the work by M. Abedi and D. Bartolomeo  [112][113] in using ED for developing financial models.]

§ ENTROPIC TIME

In an inference-based theory, our knowledge of the relevant physics is implemented through constraints. The exact nature of these constraints depends on what information one is trying to incorporate into the model. For ED, a dynamical theory, one place where such considerations arise is at the level of time. While time is intimately related with dynamics and vice versa, neither are necessary ingredients of inference; these features must be supplemented to the theory. Here we introduce a scheme for incorporating time into ED that we call entropic time.[Entropic time, as a scheme for introducing time into a theory of inference, was first introduced by A. Caticha [51]. The current scheme mirrors very closely this original viewpoint by Caticha, but contributes a slight twist in perspective that makes the framework more amenable to future compatibility with relativistic theories.]

§.§.§ Entropic instants

In ED, change happens, and it happens in a way such that motion occurs in a continuous sequence of steps. Or rather, we might say that motion unfolds as a succession of states. It is these states that we are uncertain of, but wish to infer, prompting for each such instance the singular question: “What is the state of our system?" In fact, one might turn this on its head and instead claim that ED unfolds as a sequence of questions; a perspective that makes clear the fundamental role of information, or lack thereof.

Compare this now with the structure of dynamical theories: dynamics evolves as a series of instants: ED unfolds as a sequence of questions. The language itself is suggestive: in ED, a question is an instant. The construction of this question relies on a few important, but distinct, elements:

1. A formal tool for labeling the succession of queries;

2. A specification of the subject matter to which our question pertains; i.e. the propositions $x\in\mathbf{X}_{N}$ about which we are uncertain;

3. The variables $\rho(x)$ and $\phi(x)$ that captures the informational state.

Let us explain. On one hand, a question is itself an admission of uncertainty. Why else, after all, would we ask a question, if not for the fact that we lack complete information? This, in turn, evokes the use of probabilities for quantifying this uncertainty, hence $\rho(x)$. On the other hand, in order to even begin an inquiry, we must supply enough structure to phrase the question in the first place. This amounts to specifying not only which question, in particular, that we are asking, but also what the possible answers to the question could be; what we might call the kinematic structure of the problem.[The relation between questions, inquiry, and probability is an interesting one that dates back to the work of R.T. Cox [115]. Some more recent research that explores this issue along the lines developed by Cox was done by K. Knuth, and can be found in [116][117]. The viewpoint that a probability distributions and questions are intimately linked is explored by A. Caticha in [118]. Our colorful exposition on instants in ED was inspired heavily by these considerations.]

The first ingredient is crucial in regards to this latter point as it is through the introduction of just such a tool that we can even keep track of which question we are entertaining. Indeed, “where are the particles now?" and “where are the particles later?" are different questions because the information available for answering these questions through $\rho(x)$ and $\phi(x)$, has changed. Thus we require a scheme that allows us maintain these distinctions. In a flat space-time it is reasonable to adopt a notion of time and of instant that is in the Newtonian spirit. This means that a single parameter will be sufficient for labeling the succession of queries and amounts to a unique and global notion of simultaneity.

Let us highlight some interesting points. When dealing with a single particle, for example, we desire to know its location, which is labeled by three values $\vec{x} = (x^{1},x^{2},x^{3})$. It comes as no surprise here that the components $(x^{1},x^{2},x^{3})$ of a single particle occur at an instant; this is, after all, the definition of a point particle. Things become more peculiar when we consider multiple particles such that a configuration is defined by $x\in\mathbf{X}_{N}$. The implication is that the point $x = (\vec{x}_{1},\cdots,\vec{x}_{N})$, and consequently all its components — the locations of each particle — occur at the very same instant.

However our construction is weirder still — the set of simultaneous “events" is the entire configuration space $\mathbf{X}_{N}$, not just a single point $x\in\mathbf{X}_{N}$. This latter statement is closely related to the probabilistic nature of our theory: although a particular configuration of particles $x$ occurs at an instant, we are uncertain of which $x$ is the correct one. Thus any $x$ is, in principle, possible (with some probability $\rho(x)$), and so, all must be treated equally at the kinematic level. This justifies calling $\mathbf{X}_{N}$ the set of simultaneous events.

Things are even more interesting in a relativistic theory. In that scenario, there is greater flexibility in the notion of an instant, owing to the relativity of simultaneity. In order to accommodate this situation, we will require an additional construction, that of a hypersurface in space-time. It is this additional kinematic structure that allows us to even phrase the question — “what is the state of the system now?" — by identifying a notion of “now" that is compatible with a relativistic theory. Addressing this issue will be the topic of subsequent chapters.

§.§.§ Ordered instants

The goal is to predict, given what we know now, what will likely occur next. That is, once supplied with some information $\rho(x)$ about where the particles currently reside $x\in\mathbf{X}_{N}$, we wish to determine the probability $\rho^{\prime}(x^{\prime})$ that they are located at some $x'\in\mathbf{X}^{\prime}_{N}$ at a subsequent instant. It helps to be explicit with the notation. The proposition that the particles are found at the state $x_{i}$ at the $i^{\text{th}}$ step will be denoted $s_{i} = x_{i}$; for the $(i+1)^{\text{th}}$ step, $s_{i+1} = x_{i+1}$, and so on, with each instant labeled by the index $i$.

To make predictions about where the particles are at the step $s_{i+1}$ we require information both about where the particles are, and where they will be going. This information is captured by the joint probability distribution introduced earlier

\begin{equation}
\rho(s_{i+1} = x_{i+1}, s_{i} = x_{i}) = P(s_{i+1} = x_{i+1}| s_{i} = x_{i})\rho(s_{i} = x_{i}),
\label{Particles joint distribution}
\end{equation}

which gives the probability that the particles begin in a configuration $x_{i}\in\mathbf{X}_{N}^{(i)}$ and end at the configuration $x_{i+1}\in\mathbf{X}^{(i+1)}_{N}$. When the configurations $x_{i}$ and $x_{i+1}$ correspond to successive states (as the labels $i$ and $i+1$ imply) the transition probability $P(s_{i+1} = x_{i+1}| s_{i} = x_{i})$ is given precisely by the ME distribution (<ref>).

To determine the probability of this later position $x_{i+1}$ we apply the “sum rule" of probability theory to the joint distribution (<ref>) to obtain

\begin{equation}
\rho(s_{i+1} = x_{i+1}) = \int dx_{i}\, P(s_{i+1} = x_{i+1}| s_{i} = x_{i})\rho(s_{i} = x_{i}).
\label{Particles stepping equation}
\end{equation}

Just as $\rho(s_{i} = x_{i})$ reflects the informational state at a prior instant, the distribution $\rho(s_{i+1} = x_{i+1})$ reflects the posterior statistical state corresponding to the next instant.

The temporal ordering can be made even more apparent by introducing a label $t\in\mathbb{R}$, which we call time. This allows us to write $\rho(s_{i} = x_{i})=\rho_{t}(x)$ and $\rho(s_{i+1} = x_{i+1}) = \rho_{t'}(x')$, where $t'$ is a time that is slightly later than $t$. The result is an iterative process: from an instant $\rho_{t}(x)$ we update to a subsequent instant $\rho_{t'}(x')$ using the maximum entropy distribution $P(x'|x)$, and from $\rho_{t'}(x')$ to another later instant $\rho_{t''}(x'')$, etc. The equation (<ref>) can now be rewritten as

\begin{equation}
\rho_{t'}(x') = \int dx\, P(x'| x)\rho_{t}(x),
\label{Particles Chapman kolmogorov}
\end{equation}

which is the dynamical equation we seek. In short: dynamics is composed of a succession of instants. Each posterior $\rho_{t'}(x')$ is obtained from a prior $\rho_{t}(x)$ by way of maximizing an entropy to obtain $P(x'|x)$; we call this an entropic updating scheme.

*The arrow of entropic time—
The task of dynamics is, given an arbitrary initial state, determine the final state that follows. Viewed from the perspective purely of probability, this task is symmetric: we can decompose the joint distribution $\rho(x',x)$ into either

\begin{equation}
\rho(x',x) = P(x'|x)\,\rho_{t}(x)\quad\text{or}\quad\rho(x',x) = \rho_{t'}(x')P(x|x')
\end{equation}

using the product rule. Indeed, there is nothing in the rules of probability that suggests that we update from an initial $\rho_{t}(x)$ to a final $\rho_{t^{\prime}}(x')$ or vice versa.

It is the action of obtaining $P(x'|x)$ rather than $P(x|x')$ via the ME method that breaks this symmetry. That is, while $\rho_{t'}(x')$ is obtained from $\rho_{t}(x)$ using (<ref>) and the ME distribution $P(x'|x)$, the “time-reversed" equation

\begin{equation}
\rho_{t}(x) = \int dx' \, P(x|x') \,\rho_{t'}(x')
\label{Particles time reversed CK equation}
\end{equation}

is acquired using $P(x|x')$, which must be determined through Bayes' theorem

\begin{equation}
P(x|x') = \frac{\rho_{t}(x)P(x'|x)}{\rho_{t'}(x')}.
\end{equation}

The “time reversed" dynamics determined by $P(x|x')$ follows only after we have computed $\rho_{t'}(x')$ using the forward equation (<ref>).

The consequence of this is that there is a intrinsic directionality to the flow of entropic time: we update from an initial $\rho_{t}(x)$ to a final $\rho_{t'}(x')$ using the ME distribution $P(x'|x)$. Moreover, this asymmetry is inferential in nature; resulting from an asymmetry in what is known, rather than of anything mechanical.

The final ingredient to a notion of time, is that of the duration, i.e. the separation $\Delta t$ between instants. The basic criterion here is that of convenience: in ED, time is designed such that the dynamics is as simple as possible. For short steps, eqns.(<ref>) and (<ref>) reveal that the dynamics is dominated by fluctuations. The dynamics then is most simplified by choosing a measure of duration $\Delta t$ defined in terms of the fluctuations, such that

\begin{equation}
\alpha_{n} = \frac{m_{n}}{\eta}\frac{1}{\Delta t}\quad\text{so that}\quad\left\langle \Delta w^{a}_{n} \, w^{b}_{n'} \right\rangle = \frac{\eta}{
m_{n}}\delta_{ab}\delta_{nn'} \Delta t,
\label{Particles Duration}
\end{equation}

where the proportionality is given by some particle dependent constants $m_{n}$, and a constant $\eta$, which sets the units of length to that of time. While the constants $m_{n}$ will be later identified with the masses of the particles, the constant $\eta$ will be related to Planck's constant $\hbar$.

Regarding the choice of $\Delta t$, a natural option presents itself in the context of flat space and Newtonian-style time: we choose $\Delta t$ to be a spatial constant that is independent of time so that entropic time flows “equably everywhere and every-when." With this choice of $\alpha_{n}$ in eq.(<ref>), the transition probability $P(x'|x)$ takes the form of a standard Wiener process.

Typically in physics, time is defined such that motion is as simple as possible. In Newtonian mechanics, for example, we define time such that the motion of free particles is as simple as possible: free particles travel equal distances in equal times. The free particle provides a clock in ED as well: a particle undergoes equal fluctuations in equal times.

§.§.§ Formal structure of entropic time

So far we have eschewed a strictly formal introduction of entropic time in favor of an approach that stresses the subtle interpretational aspects of the formalism. However, it is also, of course, important to be clear about the kind of mathematical structures that this construction implies. It is our goal here to touch on these matters so as to draw contrast to the situation in curved space-time.

We consider here a system of non-relativistic particles. It follows with little surprise then that the notion of time that we adopt here shares some formal similarity with the Newtonian view of time. For instance, it is well known (see e.g., [119]) that a Newtonian theory can be formulated as a fiber bundle. The construction is the following: we can take the time $t\in\mathbb{R}$ to be the base manifold and take each fiber to be a copy of real space $\mathbb{R}^{3}$. (In addition, each fiber is also endowed with a Galilean symmetry group.) Looked at from this angle, it is clear that a single number $t\in\mathbb{R}$ specifies an instant of time to which we associate a copy of three-dimensional Euclidean space; this, in turn, makes it explicit that each point $\vec{x}\in\mathbb{R}^{3}$ is simultaneous with one another in Newtonian physics.

A similar kinematic setup can be attributed to ED as well (although, to be clear, we do not assume Newton's laws of motion). In particular, the design of entropic time too results in a fiber bundle structure, where $t\in\mathbb{R}$ is again the base manifold, but where each fiber is a copy of the configuration space $\mathbf{X}_{N}$ (again, by construction, each $x\in\mathbf{X}_{N}$ is simultaneous with one another, analogous to the role of $\vec{x}\in\mathbf{X}$ in the Newtonian theory).

In addition to this, however, in ED we associate to each instant a probability $\rho(x)$, which is a distribution over the fibers. It is this latter ingredient that is of crucial importance: at the kinematical level, each instant parameterized by $t$ is identical, it is the inclusion of the probability $\rho_{t}(x)$ that truly distinguishes each instant from the next. It is this character of entropic time, for instance, that introduces an asymmetry into the flow of time.

§ THE INFORMATION GEOMETRY OF CONFIGURATION SPACE

Before proceeding, it is convenient to consider the geometry of the $3N$-dimensional configuration space $\mathbf{X}_{N}$. One might anticipate that $\mathbf{X}_{N} = \mathbf{X}\otimes\cdots\otimes\mathbf{X}$, being $N$ copies of flat Euclidean space $\mathbf{X}$,  that it too might be flat. Considerations from information geometry, in fact, confirm this intuition.

The transition probability $P(x'|x)$ is one probability distribution for every point $x\in\mathbf{X}_{N}$. This endows the configuration space $\mathbf{X}_{N}$ with the structure of a statistical manifold (see e.g. [35][97]). The benefit of this insight is that there is a natural metric associated to a statistical manifold, the information metric, which is unique (see e.g. [120] and [121]) up to an overall scale factor.

The information metric that is appropriate to the transition probability $P(x'|x)$ is given by

\begin{equation}
\gamma_{AB} = C\int dx' \, P(x'|x)\frac{\partial\log P(x'|x)}{\partial x^{A}}\frac{\partial\log P(x'|x)}{\partial x^{B}},
\label{Particles info_metric}
\end{equation}

where we have used the notation that capital Latin indices $A = (a,n)$ denote both spatial component and particle index, e.g. $x^{A} = x_{n}^{a}$, and where $C$ is an arbitrary positive constant. Using for $P(x'|x)$ the transition probability (<ref>) and making use of equations (<ref>) and (<ref>), we have for short steps ($\Delta t\to 0$) that

\begin{equation}
\gamma_{AB} = \frac{C m_{n}}{\eta\Delta t}\delta_{nn'}\delta_{ab} \equiv \frac{C m_{n}}{\eta\Delta t}\delta_{AB}.
\label{Particles info_metric config}
\end{equation}

The divergence as $\Delta t\to 0$ occurs because $\gamma_{AB}$ measures the distinguishability between neighboring probability distributions. For small $\Delta t$ the width of the Gaussian in $P(x'|x)$ becomes sharper, and thus it becomes easier to distinguish it from a distribution $P(x'|x+\delta x)$ in its vicinity. Thus $\gamma_{AB}$ diverges.

To define a geometry that remains finite, even for short steps, we can choose the constant $C\propto \Delta t$. Better still, we can define the “mass" tensor

\begin{equation}
m_{AB} = \gamma_{AB}\frac{\eta\Delta t}{C} = m_{n}\delta_{AB},
\label{Particles mass tensor}
\end{equation}

since it is always this combination that appears in our equations. Moreover, from the inverse metric $\gamma^{AB}$ we can introduce the inverse mass tensor,

\begin{equation}
m^{AB} = \frac{C}{\eta\Delta t}\gamma^{AB} = \frac{1}{m_{n}}\delta^{AB}.
\label{Particles fluctuation tensor}
\end{equation}

Note that it has been common practice in physics (see e.g., [122]) to attribute to the $N$-particle configuration space a geometry, where the mass tensor plays the role of a metric. It is therefore interesting to see that here that the mass tensor truly is, up to some scale factors, the metric of configuration space, and that this results directly from considerations of information geometry!

The availability of a metric on the configuration space $\mathbf{X}_{N}$ allows us to rewrite our main results thus far. A generic displacement can be written as a drift plus a fluctuation $\Delta x^{A} = b^{A}\Delta t + \Delta w^{A}$. The drift is given by

\begin{equation}
b^{A} = \frac{\left\langle\Delta x^{A}\right\rangle}{\Delta t} = m^{AB}\partial_{B}\phi,
\label{Particles drift velocity}
\end{equation}

where $\partial_{A} = \partial/\partial x_{n}^{a}$, and where the fluctuations yield

\begin{equation}
\left\langle \Delta w^{A} \right\rangle = 0 \quad\text{and}\quad \left\langle \Delta w^{A} \, w^{B} \right\rangle = \eta\, m^{AB}\Delta t.
\label{Particles Fluctution tensor}
\end{equation}

As seen in eq.(<ref>), the expected square fluctuations are thus proportional to the inverse mass tensor, for this reason it is appropriate to also call this the “fluctuation" tensor.

§ ON MULTI-VALUEDNESS
We previously asserted that $\phi(x)$ must be a multi-valued function. Here we demonstrate this feature by appealing to a special case, one where rotational symmetries are present. Crucially, we do not claim to have derived this property of the drift potential here. Thus ultimately the multi-valuedness of $\phi(x)$ must be regarded as an assumption in this version of ED. While our argument here provides some evidence for this, it does not account for why such a structure is present even in cases where no periodicity is present at all, such as with the case of the free particle. It appears likely that a resolution to this issue will require a deeper analysis of ED and its geometrical structure.

*Uniform circular motion—
Consider a single particle whose position, although uncertain, can be described by a point $\vec{x} \in \mathbb{R}^{3}$. The expected infinitesimal displacement of this particle is given by eq.(<ref>)

\begin{equation}
\left \langle \Delta \vec{x}\right \rangle = \frac{\eta}{\mu}\vec{\nabla}\phi(\vec{x})\Delta t ~,\label{Particles multi-valued expected step}
\end{equation}

where we have used $\alpha = \mu/\eta \Delta t$, as above, with $\mu$ to be interpreted as the particle mass. The periodic motions we are interested in are most easily expressed in spherical coordinates $\vec{x} = (r,\theta,\varphi)$, where $r$ is the radial distance, $\theta$ is the polar angle, and $\varphi$ is the azimuthal angle. In these coordinates, eq.(<ref>) takes the form

\begin{equation}
\left \langle \Delta \vec{x}\right \rangle  = \frac{\eta}{\mu}\,\vec{\nabla}\phi(\vec{x})\,\Delta t  =  \frac{\eta}{\mu} \left (\frac{\partial \phi}{\partial r}\hat{r}+\frac{1}{r}\frac{\partial \phi}{\partial \theta}\hat{\theta}+\frac{1}{r\sin\theta}\frac{\partial \phi}{\partial \varphi}\hat{\varphi}\right )\Delta t~.
\end{equation}

Furthermore, we consider a situation of extreme symmetry, one in which the expected particle trajectory passes through equal angles in equal times, i.e. uniform circular motion. We choose coordinates so that the motion takes place in the azimuthal plane (i.e. $\theta = \pi )$, leading to

\begin{equation}
\left \langle \Delta \vec{x}\right \rangle  = \frac{\eta}{\mu}\vec{\nabla}\phi(\vec{x})\Delta t = v\, \Delta t \,\hat{\varphi}~,\label{Particles multi-valued circular a}
\end{equation}

where $v$ is a constant parameter with dimensions of length per unit time. Of course, with this the radial and polar angle velocities vanish, which implies that the drift potential is itself independent of the radial length $r$ and polar angle $\theta$, so that $\phi(\vec{x})= \phi(\varphi)$. From this, eq.(<ref>) directly implies that $ \phi(\varphi)$ satisfies

\begin{equation}
\frac{d\phi(\varphi)}{d\varphi} = \frac{\mu\, R \, v}{\eta}~,\label{Particles multi-valued circular b}
\end{equation}

where we have changed from partial to ordinary derivatives, and where $R$ is a fixed length that gives the radius of the orbit. We can easily solve eq.(<ref>) for $\phi(\varphi)$, yielding

\begin{equation}
\phi(\varphi) = \frac{\mu R  v}{\eta}\, \varphi + c~,
\end{equation}

up to an overall integration constant $c$.

The solution shows that $\phi = \beta \, \varphi$, where $\beta = \mu R v/\eta$. Since $\varphi$ and $\varphi + 2\pi$ represent the exact same angle, then for fixed constant $c$ the drift potentials $\phi$ and $\phi + 2\pi \beta$ correspond to the same constraint. In quantum mechanics we also have that $\beta$ is integer valued, in which case the equivalence between $\phi $ and $ \phi + 2\pi \beta$ is precisely that of an angular variable. In other words, for the drift potential to generate motions consistent with the rotational symmetries of flat space, we must allow it to be a multi-valued function.

Of course, the requirement that $\beta$ is integer valued is just a restatement of the ad hoc quantization conditions familiar from the old quantum mechanics of Bohr and Sommerfeld. At this stage of the ED approach, there are no such restrictions on $\beta$, nor should they be expected. This, instead, comes from the requirement that the wave function $\Psi$ be single valued. A more complete discussion of this issue in ED is given in [54].

As discussed above, once it has been established that certain physical solutions of $\phi(x)$ must be multi-valued, it is reasonable to then allow that $\phi(x)$ may be multi-valued in a more general sense. Consider a situation where the motion is not periodic, such as the free particle, which will have $\vec{\nabla}\phi(\vec{x}) = \vec{k}$, where $\vec{k}$ is an arbitrary constant vector.[One can determine this by requiring the drift velocity be translation invariant, as is appropriate for a free particle, leading to a constant vector.] Therefore up to an additive constant we have that $\phi(\vec{x}) = \vec{k}\cdot\vec{x}$. While in principle the vector $\vec{k}$ is just a collection of constant parameters that determine the (expected) velocity of the particle, once $\phi(x)$ is a multi-valued function the vector $\vec{k}$ must assume an additional interpretation as a wave vector. To see this, consider a shift $\vec{x} \to \vec{x}+\vec{\lambda}$, for some vector $\vec{\lambda}$, so that $\phi(\vec{x}) \to \phi(\vec{x}) + \vec{k}\cdot\vec{\lambda}$. If the drift potential is multi-valued then for any shift $\vec{\lambda}$ there will be a corresponding $\vec{k}$ such that $\phi(\vec{x}) \to \phi(\vec{x}) + 2\pi \, n$, where $n$ is an integer.

§ DIFFUSIVE DYNAMICS

The evolution of the probability $\rho_{t}(x)$ is captured by eq.(<ref>). A more convenient description of the same dynamics is given by converting that integral equation into a differential equation, known as the Fokker-Planck (FP) equation (see Appendix <ref>):

\begin{equation}
\partial_{t}\rho_{t} = -\partial_{A}\left(\rho_{t}\, b^{A}\right)+\frac{1}{2}\eta\, m^{AB}\partial_{A}\partial_{B}\rho_{t}~.
\label{Particles FP equation}
\end{equation}

The FP equation can, in turn, be cast as a continuity equation,

\begin{equation}
\partial_{t}\rho_{t} = -\partial_{A}\left(\rho_{t}\, v^{A}\right),
\label{Particles Continuity equation}
\end{equation}

where we have introduced $v^{A}$, the current velocity, which describes the flow of probability.

The current velocity consists of two separate contributions,

\begin{equation}
v^{A} = b^{A} + u^{A},
\end{equation}

where the osmotic velocity $u^{A}$ is given by

\begin{equation}
u^{A} = -\eta\, m^{AB}\partial_{B}\log\rho^{1/2}.
\label{Particles osmotic velocity}
\end{equation}

The effect of the osmotic velocity is for $\rho(x)$ to flow down the probability gradient, resulting in a diffusion, which justifies the name “osmotic".

Owing to the form of the drift velocity and osmotic velocity, eqns.(<ref>) and (<ref>), respectively, the current velocity can itself be written as

\begin{equation}
v^{A} = m^{AB}\partial_{B}\Phi\quad\text{where}\quad\Phi = \eta \phi -\eta \log\rho^{1/2}.
\label{Particles def Phi}
\end{equation}

The variable $\Phi$ is particularly interesting. It will eventually be identified as a Hamilton-Jacobi-type function, and later, as the phase of the wave function $\Psi$. (From this point forward, it is more convenient to proceed by dealing directly with $\Phi$, rather than the drift potential $\phi$; although, of course, no results depend on this choice.)

The FP equation (or continuity equation) eq.(<ref>) can be rewritten as

\begin{equation}
\partial_{t}\rho_{t}(x) = -\partial_{A}\left(\rho_{t}\, v^{A}\right) = \frac{\delta\tilde{H}[\rho,\Phi]}{\delta\Phi(x)}
\label{Particles FP H}
\end{equation}

for a suitably chosen functional $\tilde{H}[\rho,\Phi]$. In fact, one can check that

\begin{equation}
\tilde{H}[\rho,\Phi] = \int dx \rho\frac{1}{2}m^{AB}\partial_{A}\Phi\,\partial_{B}\Phi + \tilde{F}[\rho],
\label{Particles Hamiltonian F}
\end{equation}

where $F[\rho]$ is an “integration constant", does reproduce the eq.(<ref>).

The notation in eq.(<ref>) is, by design, suggestive: the FP equation, written this way, looks very much like one half of Hamilton's equations. And, while this will eventually be the case, it is important to guard against the conclusion that the FP equation was derived from an action principle. It was not. The evolution of $\rho$ has been derived, through and through, from a completely inferential framework; it is only once we consider the full joint dynamics of $\rho$ and $\Phi$ together that $\tilde{H}[\rho,\Phi]$ obtains its interpretation as a true Hamiltonian functional.[The observant among us will have noticed that there was a bit of subjectivity in choosing the FP equation to be written as in (<ref>) since it automatically assumes $\rho(x)$ to be the canonical coordinate rather than the conjugate momentum. The choice is not, however, unreasonable. For example, this follows naturally when the e-phase space is modeled as a cotangent bundle, as in [54]. Alternatively, had we made the opposite choice, the e-functional $\tilde{H}$ would've been negative, and thus we could not have identified the generator $\tilde{H}$ as the “energy". A convenient convention is then to choose the generator of time translations to coincide with the energy, as the definition eq.(<ref>) then guarantees.]

§ NON-DISSIPATIVE DYNAMICS

The dynamical equation, eq.(<ref>), describes a standard diffusion process for the probability $\rho(x)$ under the influence of an externally prescribed drift potential $\phi(x)$. The role of $\phi(x)$ is particularly important: different assignments of an external potential $\phi(x)$ imply different flavors of dynamics, but each of which is inevitably a diffusion process. While such a scenario may be desirable for a variety of systems (see e.g., [123][124] for some standard applications in physics, finance, and otherwise), it is not accurate for the purposes of predicting the behavior of atoms and subatomic particles. Such microscopic systems display a host of interesting phenomena — bound states, oscillations, etc. — which cannot be described by eq.(<ref>) alone. In short: a richer dynamics is needed.

An early insight into what such a dynamics could possibly look like was provided by the work of E. Nelson [108] in stochastic mechanics. The idea is a non-dissipative diffusion: while the drift potential $\phi(x)$ guides the probability $\rho(x)$, now it too responds, and in turn reacts back on $\phi(x)$, resulting in a coupled dynamics between $\rho(x)$ and $\phi(x)$. This scheme amounts to updating the constraint eq.(<ref>) suitably at each instant, and thereby producing a Brownian dynamics capable of more interesting and diverse motions than was previously available.

*A canonical choice—
What remains to be done then, is to implement this idea. And here a variety of options are available. As mentioned above, one approach would be to follow Nelson, and earlier iterations of ED [51][53], by imposing that $\phi(x)$, or more conveniently $\Phi(x)$, be updated such that a global “energy" functional be conserved. The resulting dynamics was Hamiltonian, conferring the joint space, $\left(\rho(x),\Phi(x)\right)$ of dynamical variables with a canonical structure, i.e. a conserved symplectic form, Hamiltonian generators, Poisson brackets, etc. From here, one can go further towards obtaining a quantum dynamics by choosing an appropriate set of potentials in the Hamiltonian; namely, the quantum potential, familiar from Bohmian mechanics (see e.g. [6]). But also, as previously mentioned, the reliance on a conserved energy is not conducive to further development of the theory, and thus this criterion fails.

Instead, we follow the strategy outlined above: we take serious the hints provided in [53] that the probability $\rho(x)$ and phase function $\Phi(x)$ form a canonical pair. But rather than requiring a conserved energy to accomplish this, we impose this structure on $\rho(x)$ and $\Phi(x)$ outright, as has been considered by Reginatto and Hall [24]. This is accomplished by imposing that the our space of states be equipped with a symplectic structure; one that is compatible with a notion of Poisson brackets, Hamiltonian generators, and all the tools typical of the canonical framework. A primary advantage of the canonical toolkit, of course, is that it affords us the means to update the constraint eq.(<ref>) without reference to energy conservation, thus making the whole formalism more robust.

§ THE CANONICAL FORMALISM IN ED

The goal is to update the phase function $\Phi(x)$ in response to the changing probability $\rho(x)$. That is, we seek a dynamical flow for the pair of variables $\rho(x)$ and $\Phi(x)$. While the dynamics of $\rho(x)$ is fixed by the FP equation, the set of evolutions for $\Phi(x)$ have no restrictions. The goal is to identify a criterion for selecting a desirable subset. In the following we use a notation introduced in the appendix. In particular, the space $\Gamma$ will be called the ensemble phase space, or e-phase space for short, in which a point $X\in\Gamma$ denotes a pair of functions $\left ( \rho(x),\Phi(x)\right )$. That is, coordinates in $\Gamma$ will be given by $X^{\alpha x} = \left(\rho(x),\Phi(x)\right)$, where $\alpha$ takes on either of two values, $\rho$ or $\Phi$.

*Symplectic form—  The criterion that we seek is to only consider those flows such that a symplectic form $\mathbf{\Omega}\left(\cdot,\cdot\right)$ is preserved by the dynamical evolution. By symplectic form we mean here a two-form that satisfies [125]:

(a) $\mathbf{\Omega}$ is closed, i.e. $\tilde{\mathbf{d}}\mathbf{\Omega} = 0$,

(b) for every $X\in\Gamma$, $\mathbf{\Omega}_{X}:T_{X}\otimes T_{X}\to\mathbb{R}$ is non-degenerate.

(For a review of notation, see appendix <ref>.)

The first condition is necessary if our dynamical flows are to leave $\mathbf{\Omega}$ invariant. (More technically, if the Lie derivative of $\mathbf{\Omega}$ along the flow vanishes.) Eventually it is also this condition that allows the Poisson brackets to satisfy the Jacobi identity; this allows for the notion of an algebra. The second condition is also interesting: it says that $\mathbf{\Omega}$ can be viewed as a mapping from the tangent to cotangent spaces of $\Gamma$. Consider, for instance, inserting a vector $\bar{\mathbf{V}}$ into $\mathbf{\Omega}$ such that we have $\mathbf{\Omega}\left(\bar{\mathbf{V}},\cdot\right)$. Since $\mathbf{\Omega}$ is a two-form that requires two vectors as input to form a scalar, $\mathbf{\Omega}\left(\bar{\mathbf{V}},\cdot\right)$ now requires us to input an additional vector to produce a real number. That is, this object has all the properties of a one-form. This suggests the notation

\begin{equation}
\tilde{\mathbf{V}} = \mathbf{\Omega}\left(\bar{\mathbf{V}},\cdot\right)\quad\text{where}\quad \tilde{\mathbf{V}}:T_{X}\to\mathbb{R},
\end{equation}

so that $\tilde{\mathbf{V}}$ is the image of the vector $\bar{\mathbf{V}}$ in the cotangent space, hence it is a mapping.

Moreover, a manifold (such as the EPS) that has such a symplectic structure is called a symplectic manifold. And since, in addition, $\mathbf{\Omega}$ is non-degenerate, the mapping induced by $\mathbf{\Omega}$ is one-to-one (wherever it is defined), which implies that it has an inverse $\mathbf{\Pi} = \mathbf{\Omega}^{-1}$, which we call the Poisson tensor, and that we denote (in component form) $\Omega^{\alpha x,\beta x'}$, which satisfies

\begin{equation}
\Omega_{\alpha x,\beta x'}\Omega^{\beta x', \gamma x''} = \delta_{\alpha x}^{\gamma x''}.
\end{equation}

Also, since $\mathbf{\Omega}$, a two-form, is a $(0,2)$ rank tensor that is completely anti-symmetric it must be that $\mathbf{\Omega}\left(\bar{\mathbf{V}},\bar{\mathbf{V}}\right)=0$; of course, the inverse $\mathbf{\Omega}^{-1}$ is then also anti-symmetric.[Technically, when we deal with infinite dimensional spaces, it is often the case that we deal with so-called weakly non-degenerate symplectic forms, i.e. there may be some vectors which do not have an image in the cotangent space or vice versa (see e.g., [125]). That is, the mapping is one-to-one — read: non-degenerate — but it may not be onto. Interestingly, this subtlety does not appear in quantum theory: the symplectic structure is strongly non-degenerate — it is an isomorphism between the tangent and cotangent spaces of $\Gamma$ at every point. This is a direct result of the fact that the complex Hilbert space is a kind of Kähler manifold.]

The criterion of a conserved symplectic form seems a bit arbitrary. Why a symplectic structure? Why not something else? To this, no simple answers present themselves. We proceed simply by assuming this as a necessary ingredient for our theory to resemble physics; it is the symplectic structure, after all, that pervades both classical and quantum physics. The hope is, however, that a suitably developed ED will discard with this assumption.

*Hamiltonian generators—
The flows, or trajectories of interest are those along which the symplectic form $\mathbf{\Omega}$ remains unchanged (such flows are called symplectomorphisms). As was previously mentioned, such trajectories can be identified with the integral curves of some set of vector fields $\{\bar{\mathbf{V}}\}$. To show that the symplectic stucture is preserved along these trajectories it is sufficient, locally at least, for any vector field $\bar{\mathbf{V}}$ in this set to satisfy

\begin{equation}
\pounds_{\bar{\mathbf{V}}}\,\mathbf{\Omega} = 0.
\end{equation}

Computing this Lie derivative more explicitly [119], we have that

\begin{equation}
\pounds_{\bar{\mathbf{V}}}\,\mathbf{\Omega} = \left(\tilde{\mathbf{d}}\mathbf{\Omega}\right)\left(\bar{\mathbf{V}}\right)+\tilde{\mathbf{d}}\left(\mathbf{\Omega}\left(\bar{\mathbf{V}}\right)\right) = \tilde{\mathbf{d}}\left(\mathbf{\Omega}\left(\bar{\mathbf{V}}\right)\right)=0,
\end{equation}

where the second equality follows from the requirement (a) that $\mathbf{\Omega}$ be closed. This condition implies that the one-form $\tilde{\mathbf{V}} = \mathbf{\Omega}\left(\bar{\mathbf{V}}\right)$ is also closed, i.e. $\tilde{\mathbf{d}}\tilde{\mathbf{V}} = 0$. Clearly this is satisfied if $\tilde{\mathbf{V}}$ is exact, that is, $\tilde{\mathbf{V}} = \tilde{\mathbf{d}}\tilde{G}[X]$, for some e-functional $\tilde{G}[X]$ since $\tilde{\mathbf{d}}^{2}\tilde{G}=0$ identically. As is known, however, a closed form need not be exact; this is always true locally, but the global statement depends on the topology of $\Gamma$. Here we go further and assume that the topology of $\Gamma$ is such that it allows the identification of $\tilde{\mathbf{V}}=\tilde{\mathbf{d}}\tilde{G}[X]$ always.[This is consistent, for example, with the fact that the space of pure states in quantum theory is a simply connected Kähler manifold. See e.g., [126].] As one might expect, the e-functionals $\tilde{G}$ are special, they are the Hamiltonian generators.

A vector field $\bar{\mathbf{V}}_{\tilde{G}}\in T_{X}$ is a Hamiltonian vector field (HVF) if

\begin{equation}
\mathbf{\Omega}\left(\bar{\mathbf{V}}_{\tilde{G}}\right) = \tilde{\mathbf{d}}\tilde{G}.
\label{Particles Definition HVF}
\end{equation}

In the coordinates defined by $\rho(x)$ and $\Phi(x)$ the HVFs have the explicit form

\begin{equation}
V^{\alpha x}_{\tilde{G}} = \Omega^{\alpha x\beta x'}\frac{\delta\tilde{G}}{\delta X^{\beta x'}} = \left(\frac{\delta\tilde{G}}{\delta\Phi(x)},-\frac{\delta\tilde{G}}{\delta\rho(x)}\right)
\label{Particles Definition HVF Components}
\end{equation}

so that

\begin{equation}
\bar{\mathbf{V}}_{\tilde{G}} = \int dx\left(\frac{\delta\tilde{G}}{\delta\Phi(x)}\frac{\delta}{\delta\rho(x)}-\frac{\delta\tilde{G}}{\delta\rho(x)}\frac{\delta}{\delta\Phi(x)}\right).
\label{Particles Definition HVF Vector}
\end{equation}

By definition, since such flows leave $\mathbf{\Omega}$ invariant, it is these flows that we are interested in. The problem of finding a joint evolution for the variables $\rho(x)$ and $\Phi(x)$ thus reduces to finding an appropriate $\bar{\mathbf{V}}_{\tilde{H}}$ that generates a dynamical flow, i.e. the flow whose integral curves are labeled by the distinguished entropic time parameter $t$. And, since these HVFs are in one-to-one correspondence with the Hamiltonian generators, this task amounts to just finding an appropriate Hamiltonian e-functional $\tilde{H}[\rho,\Phi]$ .

The natural question that follows is, what e-functional $\tilde{H}$ is suitable for updating the variables $\rho(x)$ and $\Phi(x)$? Indeed, at first the problem seems quite severe since any e-functional generates a flow that leaves $\mathbf{\Omega}$ invariant. The situation is, however, not quite so dire since $\tilde{H}[\rho,\Phi]$ cannot be completely arbitrary: it must reproduce the FP equation, eq.(<ref>). Recalling the rewritten FP equation in (<ref>), this implies taking as our dynamical generator the e-functional $\tilde{H}[\rho,\Phi]$, introduced first in (<ref>).

*On Lagrangians—
The developments described above put ED in an interesting position. While in classical physics it is often difficult to construct a Hamiltonian directly, without first appealing to the notion of a Lagrangian,[Indeed, in [80][81], for example, C. Teitelboim sets out to determine a set of covariant dynamical theories without first appealing to a Lagrangian. His method is successful, but his method of constructing the Hamiltonian is much more tedious than if he had just began with a Lagrangian. And Dirac, who was a major proponent of the Hamiltonian viewpoint, in [77] himself advocated beginning from a Lagrangian.] in ED much of the Hamiltonian is fixed simply by requiring that it reproduce the FP equation. That is, the FP equation, which results directly from the rules of inference, fixes a good portion of the eventual quantum dynamics.

Interestingly, ED therefore has a problem that is somewhat inverted: how does one construct a Lagrangian formulation given a Hamiltonian? This, it seems is, in fact, not straightforward, or potentially, not possible at all. For example, Reginatto and Hall [127] suggest that a Lagrangian formulation for the variables $\rho(x)$ and $\Phi(x)$ would require the introduction of two additional variables to play the role of “velocities" corresponding to the “coordinates" $\rho(x)$ and $\Phi(x)$.

But why should we need such an indirect route to a Lagrangian formulation? Recall that in classical mechanics the Hamiltonian and Lagrangian viewpoints are often used interchangeably, largely because it is often easy to perform the Legendre transform which transitions between the Hamiltonian picture based on the cotangent bundle and the Lagrangian picture based on the tangent bundle (see e.g., [128]). In [54], however, A. Caticha does, in fact, model the e-phase space as a cotangent bundle. Should this not suggest a corresponding Lagrangian formulation of ED? In short, yes. Nevertheless, while this route may be available in principle, in practice obtaining such a Lagrangian picture would require us to invert the local differential FP equation to obtain the equivalent of $p = \partial L /\partial \dot{q}$ in classical mechanics, which would here be a non-local integral equation. Thus suggests that the Hamiltonian viewpoint, being more simple and compact, should actually be considered the more fundamental one.

*Poisson brackets—
As we mentioned in passing above, the symplectic form $\mathbf{\Omega}$ is a skew-symmetric bilinear product. The fact that it is skew-symmetric means that it is completely anti-symmetric on its components $\Omega_{\alpha x,\beta x'}$, while it is bilinear because it requires two vectors as inputs.

Having introduced the notion of a HVF $\bar{\mathbf{V}}_{\tilde{G}}$, it is interesting to consider supplying two such vectors to $\mathbf{\Omega}$. The result is, of course, a real number (an e-functional), but owing to the anti-symmetry of $\mathbf{\Omega}$, the resulting object also retains an anti-symmetric character. It is, in fact, nothing but the Poisson bracket between two e-functionals

\begin{equation}
\mathbf{\Omega}\left(\bar{\mathbf{V}}_{\tilde{F}},\bar{\mathbf{V}}_{\tilde{G}}\right) = \Omega_{\alpha x,\beta x'}V_{\tilde{F}}^{\alpha x}V_{\tilde{G}}^{\beta x'}\equiv \left\{\tilde{F},\tilde{G}\right\}.
\end{equation}

Using the inverse symplectic form $\mathbf{\Omega}^{-1}$ the Poisson bracket can be written as

\begin{equation}
\left\{\tilde{F},\tilde{G}\right\} = \Omega^{\alpha x,\beta x'}\frac{\delta\tilde{F}}{\delta X^{\alpha x}}\frac{\delta\tilde{G}}{\delta X^{\beta x'}},
\end{equation}

or more familiarly, it can be put (locally at least) in the canonical form

\begin{equation}
\left\{\tilde{F},\tilde{G}\right\} = \int dx\left(\frac{\delta\tilde{F}}{\delta\rho(x)}\frac{\delta\tilde{G}}{\delta\Phi(x)}-\frac{\delta\tilde{G}}{\delta\rho(x)}\frac{\delta\tilde{F}}{\delta\Phi(x)}\right).\label{Particles Poisson bracket}
\end{equation}

The Poisson bracket has a few interesting properties.

(a)It is skew-symmetric:

\begin{equation}
\left\{\tilde{F},\tilde{G}\right\}=-\left\{\tilde{G},\tilde{F}\right\},
\end{equation}

(b) It satisfies the Jacobi identity:

\begin{equation}
\left\{\tilde{F},\left\{\tilde{G},\tilde{H}\right\}\right\}+\left\{\tilde{G},\left\{\tilde{H},\tilde{F}\right\}\right\}+\left\{\tilde{H},\left\{\tilde{F},\tilde{G}\right\}\right\}=0,\label{Particles Jacobi Identity}
\end{equation}

(c) It also satisfies the Leibniz property:

\begin{equation}
\left\{\tilde{F}\tilde{G},\tilde{H}\right\} = \tilde{F}\left\{\tilde{G},\tilde{H}\right\} + \tilde{G}\left\{\tilde{F},\tilde{H}\right\}.
\end{equation}

The first two properties endow the Poisson bracket with the properties of a Lie bracket, such that it allows for a notion of an algebra between e-functionals. The last property allows for a notion of derivation, i.e. $\left\{\tilde{F},\tilde{G}\right\} $ can be viewed as the derivative of $\tilde{F}$ with respect to the flow generated by $\tilde{G}$. This is made explicit by the relation

\begin{equation}
\left\{\tilde{F},\tilde{G}\right\} = \bar{\mathbf{V}}_{\tilde{G}}\left(\tilde{F}\right),
\end{equation}

which uses the form of $\bar{\mathbf{V}}_{\tilde{G}}$ as a “differential operator", which is explicit in (<ref>).

Pertaining to the notion of algebra, consider two HVFs, $\bar{\mathbf{V}}_{\tilde{F}}$ and $\bar{\mathbf{V}}_{\tilde{G}}$. Clearly the flows generated by $\tilde{F}$ and $\tilde{G}$ both preserve the symplectic form, but what of the flow generated by their Poisson bracket $\left\{\tilde{F},\tilde{G}\right\} $? In the generic case this too leaves $\mathbf{\Omega}$ unchanged

\begin{equation}
\pounds_{\bar{\mathbf{V}}_{\{\tilde{F},\tilde{G}\}}}\mathbf{\Omega} = 0.
\end{equation}