Entropic dynamics

A realist subquantum theory with epistemic interpretation of the wave function

Entropic dynamics was proposed by Caticha 2011 as an interpretation of quantum theory. It is in my opinion the simplest and internally most consistent one among the realistic interpretations. It is, in particular, in complete agreement with classical common sense. I present here a slightly modified variant (it identifies an unspecified set \(Y\) of additional hidden variables with the configuration of the universe which is external to the system under consideration).

An additional condition:
No zeros of the wave function \(\psi(q)\)

The interpretation uses the polar decomposition of the wave function \[ \psi(q) = \sqrt{\rho(q)} \exp({\frac{i}{\hbar}\phi(q)}) \] and has to presume that the phase \(\phi(q)\) is a well-defined global function. This would not be possible for wave functions which have zeros in the configuration space representation. Thus, we have to exclude this possibility explicitly: \[ (\forall q\in Q) \quad |\psi(q)|^2 > 0. \] As a consequence, at least in principle the interpretation differs from from quantum theory, that means, it is a different, subquantum theory. It could be empirically falsified by preparing a quantum state which violates the condition. Once the condition holds only for the configuration space representation, different choices of the configuration space define different subquantum theories. The most reasonable choice for a quantum field theory (that means, also, for the Standard Model of particle physics) is the field ontology. In this paper I argue that with this choice the theory remains viable.

The definition of the wave function as an epistemic function

The state of reality is defined by a continuous trajectory in the configuration space \(q(t) \in Q\). The difference to the classical Lagrange formalism is only that the trajectory should not be differentiable.

A quantum system is not completely independent of the external world, so we consider, together with the configuration of the system \(Q_{sys}\) also the configuration of the external world \(Q_{ext}\). Together they define the configuration of the whole universe: \(Q_{univ}\cong Q_{sys} \times Q_{ext}\). (In Caticha's variant, we have instead of \(Q_{ext}\) a completely unspecified space of some other variables \(Y\). I have not been able to identify any reason which would prevent using \(Q_{ext}\) as this space of other variables. As the result, we obtain a much simpler ontology for the whole universe.)

We have only incomplete knowledge about the universe. This incomplete knowledge defines, following the logic of plausible reasoning, a probability distribution over the configuration space of the whole universe: \[\rho(q_{univ})d q_{univ} = \rho(q_{sys},q_{ext})dq_{sys}dq_{ext}.\]

We can use this global probability distribution to define two important functions on the configuration space of the quantum system \(Q_{sys}\) taken alone.

First, we define the probability distribution \(\rho(q_{sys}) d q_{sys}\) on the configuration space \(Q_{sys}\) of the quantum system by simple integration over \(Q_{ext}\): \[\rho(q_{sys}) = \int_{Q_{ext}} \rho(q_{sys},q_{ext}) dq_{ext}.\]

Then, we define the entropy \(S(q_{sys})\) of the probability distribution of the external variables given a fixed configuration of the system \(q_{sys}\in Q_{sys}\). This entropy is \[ S(q_{sys}) = \int_{Q_{ext}} -\ln\rho(q_{sys},q_{ext})\rho(q_{sys},q_{ext}) dq_{ext}. \]

These two functions \(\rho(q_{sys})\) and \(S(q_{sys})\) defined on the configuration space of the quantum system \(Q_{sys}\) are, then, used to define the wave function \(\psi(q_{sys})\) as \[ \psi = \sqrt{\rho} \exp({\frac{i}{\hbar}\phi}) \quad \text{ with the phase defined by } \quad \phi = S - \ln\sqrt{\rho}.\]

The evolution equations

The evolution equations for \(\rho\) and \(\phi\) are the same as in the minimal realistic interpretation: \[\partial_t \rho + \nabla \left(\rho \nabla \phi \right) = 0\] \[\partial_t \phi + \frac{1}{2} (\nabla \phi)^2 + V -\frac{\hbar^2}{2} \frac{\Delta \sqrt{\rho}}{\sqrt{\rho}} = 0\]

The interesting advantage of Caticha's interpretation is that they are both derived from sufficiently simple assumptions, together with the completely general method of entropic inference, a method which is itself a consequence of the logic of plausible reasoning.

Unfortunately, the method of entropic inference itself is something which requires some mathematical apparatus, too much to give here a simple introduction in a few lines. So, let's mention here only what is required, beyond the logic of plausible reasoning:

The continuity equation in terms of probability \(\rho(q)\) and entropy \(S(q)\) is: \[\partial_t \rho + \partial_i\left(\rho \partial_i S\right) - \Delta \rho = 0.\]

This is a combination of a simple diffusion with a drift term in the direction of increasing entropy. So, we have here a very simple equation, which can also be easily derived and understood. The only additional structure which is necessary to derive this equation is a metric \(\delta^{ij}\) on the configuration space \(Q_{sys}\). Without it, we would be unable to define the Laplace operator \(\Delta = \partial_i \delta^{ij}\partial_j\).

To derive the equation for the entropy \(S(q)\), we have to postulate that the evolution preserves something which one could name energy. This energy is defined by \[ E = \int \left(\frac12 \left( (\partial_i\phi)^2 + (\hbar\partial_i \ln \sqrt{\rho})^2\right) + V \right) \rho d q. \]

But what about the \(\psi\)-ontology theorems?

Once in entropic dynamics the wave function is completely epistemic, this seems to be in contradiction with the so-called \(\psi\)-ontology theorems. The most famous one is the PBR theorem, but there are variants, for example by Hardy 2013 or Gao 2016, see Leifer 2014 for a review article.

Fortunately, there is none. These theorems simply use a misleading naming. According to the definition of a \(\psi\)-ontological interpretation, entropic dynamics is (formally) a \(\psi\)-ontological interpretation, that's all. That there has been used a defintion of being \(\psi\)-ontological so that a certainly epistemic interpretation will be \(\psi\)-ontological too is the problem of those who have formulated the theorems, not of the epistemic interpretations.

What creates the confusion is quite simple: In \(\psi\)-epistemic interpretations, the wave function is defined by incomplete knowledge about the quantum state. We know what defines this knowledge: It is the knowledge about the preparation process, which consists of a preparation measurement and its result. But this preparation process is part of reality too. It is not part of the system - it is part of the external world. But the external world is also objective, real. In this sense, the wave function is, indeed, defined by the state of reality. But only if one includes the whole world - or at least the preparation device - into the reality which is considered.

Now, nothing in the theorem forbids to do this. So, nothing forbids that the reality considered does not explicitly contain the incomplete knowledge - the knowledge about the real preparation procedure - too. For some more details, see this paper.


Ideal compatibility with classical common sense

To summarize, let's note that entropic dynamics is completely compatible with classical common sense assumptions about our universe:

So, the compatibility with common sense is given, and in some aspects it is even better compatible with common sense than classical mechanics.