# The Entropic Dynamics Approach to Quantum Mechanics

## Abstract

**:**

## 1. Introduction

## 2. The ED of Short Steps

**X**with metric ${\delta}_{ab}$. For N particles the configuration space is ${\mathbf{X}}_{N}=\mathbf{X}\times \dots \times \mathbf{X}$. We assume that the particles have definite positions ${x}_{n}^{a}$ and it is their unknown values that we wish to infer [82]. (The index $n=1,\dots ,N$ denotes the particle and $a=1,2,3$ the spatial coordinates.)

**The prior**. The choice of prior $Q\left({x}^{\prime}\right|x)$ must reflect the state of knowledge that is common to all short steps. (It is through the constraints that the information that is specific to any particular short step will be supplied.) We adopt a prior that carries the information that the particles take infinitesimally short steps and reflects the translational and rotational invariance of the Euclidean space

**X**but is otherwise uninformative. In particular, the prior expresses total ignorance about any correlations. Such a prior can itself be derived from the principle of maximum entropy. Indeed, maximize

**The drift potential constraint**. In Newtonian dynamics one does not need to explain why a particle perseveres in its motion in a straight line; what demands an explanation—that is, a force—is why the particle deviates from inertial motion. In ED one does not require an explanation for why the particles move; what requires an explanation is how the motion can be both directional and highly correlated. This physical information is introduced through one constraint that acts simultaneously on all particles. The constraint involves a function $\varphi \left(x\right)=\varphi ({x}_{1}\dots {x}_{N})$ on configuration space ${\mathbf{X}}_{N}$ that we call the “drift” potential. We impose that the displacements $\Delta {x}_{n}^{a}$ are such that the expected change of the drift potential $\left(\right)$ is constrained to be

**The gauge constraints**. The single constraint (5) already leads to a rich entropic dynamics but by imposing additional constraints we can construct even more realistic models. To incorporate the effect of an external electromagnetic field we impose that for each particle n the expected displacement $\langle \Delta {x}_{n}^{a}\rangle $ will satisfy

**The transition probability**. An important feature of the ED model can already be discerned. The central object of the discussion so far, the transition probability $P\left({x}^{\prime}\right|x)$, codifies information supplied through the prior and the constraints which makes no reference to anything earlier than the initial position x. Therefore ED must take the form of a Markov process.

## 3. Entropic Time

#### 3.1. Time as an Ordered Sequence of Instants

#### 3.2. The Arrow of Entropic Time

#### 3.3. Duration and the Sub-Quantum Motion

**On the nature of clocks**. In Newtonian mechanics time is defined to simplify the dynamics. The prototype of a clock is a free particle which moves equal distances in equal times. In ED time is also defined to simplify the dynamics of free particles (for sufficiently short times all particles are free) and the prototype of a clock is a free particle too: as we see in (23) the particle’s mean displacement increases by equal amounts in equal times.

**On the nature of mass**. In standard quantum mechanics, “what is mass?” and “why quantum fluctuations?” are two independent mysteries. In ED the mystery is somewhat alleviated: as we see in Equation (25) mass and fluctuations are two sides of the same coin. Mass is an inverse measure of the velocity fluctuations.

**The information metric of configuration space**. In addition to defining the dynamics the transition probability Equation (18) serves to define the geometry of the N-particle configuration space, ${\mathbf{X}}_{N}$. Since the physical single particle space $\mathbf{X}$ is described by the Euclidean metric ${\delta}_{ab}$ we can expect that the N-particle configuration space, ${\mathbf{X}}_{N}=\mathbf{X}\times \dots \times \mathbf{X}$, will also be flat, but for non-identical particles a question might be raised about the relative scales or weights associated with each $\mathbf{X}$ factor. Information geometry provides the answer.

**Invariance under gauge transformations**. The fact that constraints (5) and (6) are not independent—they are both linear in the same displacements $\langle \Delta {x}_{n}^{a}\rangle $—leads to a gauge symmetry. This is evident in Equation (7) where $\varphi $ and ${A}_{a}$ appear in the combination ${\partial}_{na}\varphi -{\beta}_{n}{A}_{a}$ which is invariant under the gauge transformations,

**Interpretation:**The drift potential $\varphi \left(x\right)=\varphi ({\overrightarrow{x}}_{1},{\overrightarrow{x}}_{2},\dots )$ is assumed to be an “angle”–$\varphi \left(x\right)$ and $\varphi \left(x\right)+2\pi $ are meant to describe the same angle. The angle at ${\overrightarrow{x}}_{1}$ depends on the values of all the other positions ${\overrightarrow{x}}_{2},{\overrightarrow{x}}_{3},\dots $, and the angle at ${\overrightarrow{x}}_{2}$ depends on the values of all the other positions ${\overrightarrow{x}}_{1},{\overrightarrow{x}}_{3},\dots $, and so on. The fact that the origins from which these angles are measured can be redefined by different amounts at different places gives rise to a local gauge symmetry. To compare angles at different locations one introduces a connection field, the vector potential ${A}_{a}\left(\overrightarrow{x}\right)$. It defines which origin at $\overrightarrow{x}+\Delta \overrightarrow{x}$ is the “same” as the origin at $\overrightarrow{x}$. This is implemented by imposing that as we change origins and $\mathsf{\Phi}\left(x\right)$ changes to $\mathsf{\Phi}+\overline{\chi}$ then the connection transforms as ${A}_{a}\to {A}_{a}+{\partial}_{a}\chi $ so that the quantity ${\partial}_{A}\mathsf{\Phi}-{\overline{A}}_{A}$ remains invariant.

**A fractional Brownian motion**? The choices ${\alpha}_{n}\propto 1/\Delta t$ and ${\alpha}_{n}\propto 1/\Delta {t}^{3}$ lead to Einstein–Smoluchowski and Oernstein–Uhlenbeck processes, respectively. For definiteness throughout the rest of this paper we will assume that the sub-quantum motion is an OU process but more general fractional Brownian motions [89] are possible. Consider

## 4. The Evolution Equation in Differential Form

## 5. The Epistemic Phase Space

**P**of all normalized probabilities,

**P**the associated tangent and cotangent bundles, respectively $T\mathbf{P}$ and ${T}^{*}\mathbf{P}$, are geometric objects that are always available to us independently of any physical considerations. Both are manifolds in their own right but the cotangent bundle ${T}^{*}\mathbf{P}$—the space of all probabilities and all covectors—is of particular interest because it comes automatically endowed with a rich geometrical structure [56,57,58,59,60,61,62]. The point is that cotangent bundles are symplectic manifolds and this singles out as “natural” those dynamical laws that happen to preserve some privileged symplectic form. This observation will lead us to identify e-phase space $\{\rho ,\mathsf{\Phi}\}$ with the cotangent bundle ${T}^{*}\mathbf{P}$ and provides the natural criterion for updating constraints, that is, for updating the phase $\mathsf{\Phi}$ [94].

#### 5.1. Notation: Vectors, Covectors, Etc.

**P**at the point $\rho $. Curves in ${T}^{*}\mathbf{P}$ allow us to define vectors. Let $X=X\left(\lambda \right)$ be a curve parametrized by $\lambda $, then the vector $\overline{V}$ tangent to the curve at $X=(\rho ,\pi )$ has components $d{\rho}^{x}/d\lambda $ and $d{\pi}_{x}/d\lambda $, and is written

**P**is constrained to normalized probabilities means that the coordinates ${\rho}^{x}$ are not independent. This technical difficulty is handled by embedding the ∞-dimensional manifold $\mathbf{P}$ in a (∞ + 1)-dimensional manifold ${\mathbf{P}}^{+1}$ where the coordinates ${\rho}^{x}$ are unconstrained [95]. Thus, strictly, $\tilde{\mathsf{\nabla}}F$ is a covector on ${T}^{*}{\mathbf{P}}^{+1}$, i.e., $\tilde{\mathsf{\nabla}}F\in {T}^{*}{\left(\right)}_{{T}^{*}}X$ and $\tilde{\mathsf{\nabla}}{\rho}^{x}$ and $\tilde{\mathsf{\nabla}}{\pi}_{x}$ are the corresponding basis covectors. Nevertheless, the gradient $\tilde{\mathsf{\nabla}}F$ will yield the desired directional derivatives (54) on ${T}^{*}\mathbf{P}$ provided its action is restricted to vectors $\overline{V}$ that are tangent to the manifold $\mathbf{P}$. Such tangent vectors are constrained to obey

#### 5.2. The Symplectic Form in ED

#### 5.3. Hamiltonian Flows and Poisson Brackets

#### 5.4. The Normalization Constraint

## 6. The Information Geometry of E-Phase Space

#### 6.1. The Metric on the Embedding Space ${T}^{*}{\mathbf{P}}^{+1}$

- (a)
- the metric on ${T}^{*}{\Sigma}^{+1}$ be compatible with the metric on ${\Sigma}^{+1}$; and
- (b)
- that the spherical symmetry of the $(k+1)$-dimensional space ${\Sigma}^{+1}$ be enlarged to full spherical symmetry for the $2(k+1)$-dimensional space ${T}^{*}{\Sigma}^{+1}$.

#### 6.2. The Metric Induced on ${T}^{*}\mathbf{P}$

#### 6.3. A Complex Structure

#### 6.4. Complex Coordinates

## 7. Hamilton-Killing Flows

## 8. The E-Hamiltonian

## 9. Entropic Time, Physical Time, and Time Reversal

## 10. Linearity and the Superposition Principle

#### 10.1. The Single-Valuedness of Ψ

#### 10.2. Charge Quantization

## 11. The Classical Limits and the Bohmian Limit

#### 11.1. Classical Limits

#### 11.2. The Bohmian Limit

## 12. Hilbert Space

**A vector space**. As we saw above the infinite-dimensional e-phase space—the cotangent bundle ${T}^{*}\mathbf{P}$—is difficult to handle. The problem is that the natural coordinates are probabilities ${\rho}_{x}$ which, due to the normalization constraint, are not independent. In a discrete space one could single out one of the coordinates and its conjugate momentum and then proceed to remove them. Unfortunately, with a continuum of coordinates and momenta the removal is not feasible. The solution is to embed ${T}^{*}\mathbf{P}$ in a larger space ${T}^{*}{\mathbf{P}}^{+1}$. This move allows us to keep the natural coordinates ${\rho}_{x}$ but there is a price: we are forced to deal with a constrained system and its attendant gauge symmetry.

**Dirac notation**. We can at this point introduce the Dirac notation to represent the wave functions ${\mathsf{\Psi}}_{x}$ as vectors $|\mathsf{\Psi}\rangle $ in a Hilbert space. The scalar product $\langle {\mathsf{\Psi}}_{1}|{\mathsf{\Psi}}_{2}\rangle $ is defined using the metric G and the symplectic form Ω,

**Hermitian and unitary operators**. The bilinear Hamilton functionals $\tilde{Q}[\mathsf{\Psi},{\mathsf{\Psi}}^{*}]$ with kernel $\hat{Q}(x,{x}^{\prime})$ in Equation (121) can now be written in terms of a Hermitian operator $\hat{Q}$ and its matrix elements,

**Commutators**. The Poisson bracket of two Hamiltonian functionals $\tilde{U}[\mathsf{\Psi},{\mathsf{\Psi}}^{*}]$ and $\tilde{V}[\mathsf{\Psi},{\mathsf{\Psi}}^{*}]$,

## 13. Remarks on ED and Quantum Bayesianism

- (a)
- QB adopts a personalistic de Finetti type of Bayesian interpretation while ED adopts an impersonal entropic Bayesian interpretation somewhat closer but not identical to Jaynes’ [15,16,17,18]. In ED, the probabilities do not reflect the subjective beliefs of any particular person. They are tools designed to assist us in those all too common situations in which are confused and due to insufficient information we do not know what to believe. The probabilities will then provide guidance as to what agents ought to believe if only they were ideally rational. More explicitly, probabilities in ED describe the objective degrees of belief of ideally rational agents who have been supplied with the maximal allowed information about a particular quantum system.
- (b)
- ED derives or reconstructs the mathematical framework of QM—it explains where the symplectic, metric, and complex structures, including Hilbert spaces and time evolution come from. In contrast, at its current stage of development QB consists of appending a Bayesian interpretation to an already existing mathematical framework. Indeed, assumptions and concepts from quantum information are central to QB and are implicitly adopted from the start. For example, a major QB concern is the justification of the Born rule starting from the Hilbert space framework while ED starts from probabilities and its goal is to justify the construction of wave functions; the Born rule follows as a trivial consequence.
- (c)
- ED is an application of entropic/Bayesian inference. Of course, the choices of variables and of the constraints that happen to be physically relevant are specific to our subject matter—quantum mechanics—but the inference method itself is of universal applicability. It applies to electrons just as well as to the stock market or to medical trials. In contrast, in QB the personalistic Bayesian framework is not of universal validity. For those special systems that we call ‘quantum’ the inference framework is itself modified into a new “Quantum-Bayesian coherence” in which the standard Bayesian inference must be supplemented with concepts from quantum information theory. The additional technical ingredient is a hypothetical structure called a “symmetric informationally complete positive-operator-valued measure”. In short, in QB Born’s Rule is not derived but constitutes an addition beyond the raw probability theory.
- (d)
- QB is an anti-realist neo-Copenhagen interpretation; it accepts complementarity. (Here complementarity is taken to be the common thread that runs through all Copenhagen interpretations.) Probabilities in QB refer to the outcomes of experiments and not to ontic pre-existing values. In contrast, in ED probabilities refer to ontic positions—including the ontic positions of pointer variables. In the end, this is what solves the problem of quantum measurement (see [70,71]).

## 14. Some Final Remarks

- Particles have definite but unknown positions and follow continuous trajectories.
- The probability of a short step is given by the method of maximum entropy subject to a drift potential constraint that introduces directionality and correlations, plus gauge constraints that account for external electromagnetic fields.
- The accumulation of short steps requires a notion of time as a book-keeping device. This involves the introduction of the concept of an instant and a convenient definition of the duration between successive instants.
- The e-phase space $\{\rho ,\mathsf{\Phi}\}$ has a natural symplectic geometry that results from treating the pair $({\rho}_{x},{\mathsf{\Phi}}_{x})$ as canonically conjugate variables.
- The information geometry of the space of probabilities is extended to the full e-phase space by imposing the latter be spherically symmetric.
- The drift potential constraint is updated instant by instant in such a way as to preserve both the symplectic and metric geometries of the e-phase space.

## Funding

## Acknowledgments

## Conflicts of Interest

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Caticha, A.
The Entropic Dynamics Approach to Quantum Mechanics. *Entropy* **2019**, *21*, 943.
https://doi.org/10.3390/e21100943

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Caticha A.
The Entropic Dynamics Approach to Quantum Mechanics. *Entropy*. 2019; 21(10):943.
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Caticha, Ariel.
2019. "The Entropic Dynamics Approach to Quantum Mechanics" *Entropy* 21, no. 10: 943.
https://doi.org/10.3390/e21100943