# Subquantum theories which are compatible with common sense

All the interpretations of quantum theory I would classify as compatible with common sense essentially follow the same basic scheme. I have combined these elements shared by all those realistic interpretations into a minimal realist interpretation of quantum theory.

The interpretation which is most compatible with common sense is, actually, entropic dynamics, which has been proposed by Caticha 2011. I prefer a minor variant of Caticha's entropic dynamics. In this variant, the classical trajectory $$q(t)\in Q$$ of the universe is the complete description of reality. Instead, the wave function describes only incomplete knowledge about the trajectory of the quantum system.

To describe the complex wave function $$\psi(q,t)\in\mathbb{C}$$, realistic interpretations usually use the polar decomposition into the probability density $$\rho(q,t)>0$$ and the phase $$\phi(q,t)\in \mathbb{R}$$: $\psi = \sqrt{\rho} \exp({\frac{i}{\hbar}\phi}).$

This has been proposed by Madelung 1927 to give one-particle theory a hydrodynamical form, and therefore they have been (misleadingly, because the actual use has nothing to do with hydrodynamics) named "hydrodynamical variables". In these variables, the Schrödinger equation for $$\psi(q,t)$$ $i \hbar \partial_t \psi = -\frac{\hbar^2}{2} \Delta \psi + V \psi,$ (for simplicity we use here $$m=1$$) gives for $$\rho(q,t)$$ a continuity equation with the velocity $$\vec{v}(q) = \nabla \phi(q)$$: $\partial_t \rho + \nabla \left(\rho \nabla \phi \right) = 0$ and for the phase $$\phi(q,t)$$ the "quantum Hamilton-Jacobi equation": $\partial_t \phi + \frac{1}{2} (\nabla \phi)^2 + V -\frac{\hbar^2}{2} \frac{\Delta \sqrt{\rho}}{\sqrt{\rho}} = 0$ which differs from the classical Hamilton-Jacobi equation for $$\phi(q,t)$$ only by the so-called "quantum potential" $Q(q) = -\frac{\hbar^2}{2} \frac{\Delta \sqrt{\rho(q)}}{\sqrt{\rho(q)}}.$

Once there is a continuity equation for a probability density $$\rho(q)$$ of the configuration, we can assume that there exists a continuous trajectory $$q(t)$$ in configuration space $$Q$$. If this continuous trajectory is random or deterministic can be left open, in both cases the velocity $$\vec{v}(q) = \nabla \phi(q)$$ will be the average velocity of the trajectory.

The classical limit $$\hbar\to 0$$ of the equations gives simply the equations of classical mechanics in the Hamilton-Jacobi formalism. So, for $$\phi$$ we obtain the classical Hamilton-Jacobi equation $\partial_t \phi + \frac{1}{2} (\nabla \phi)^2 + V = 0,$ and the continuity equation is simply the densitized variant of the classical equation $$\dot{q} = \nabla \phi.$$ So, we have the same ontology as in classical mechanics and a completely unproblematic classical limit.

# The problem near the zeros of the wave function

Nonetheless, this interpretation leads to problems where the wave functions have zeros.

The first problem is that the phase $$\phi(q)$$ is undefined at the zeros, and near the zeros the velocity becomes arbitrary large: $\lim_{\rho(q)\to 0} |\nabla \phi| = \infty.$

In general, any infinity is a strong indication that something is wrong. That this singularity is quite harmless, given that the velocity is infinite only at the place where the density is zero, thus, there is nothing to move, and the resulting movement near the zeros is almost circular and essentially does not change the density does not change this: If one thinks that $$\nabla \phi$$ is really a velocity of something really existing, even if only an average velocity, something is wrong if it can approach arbitrary large numbers.

Of course, what I propose as the minimal realistic interpretation of quantum theory does not have to worry much about this. It is sufficient to recognize the fact that there are some infinities, and to make the necessary conclusion: Quantum theory is not the final, most fundamental theory, but an approximation of some different subquantum theory without infinities for physical variables.

The aim of this part is to consider the possibilities for such subquantum theories. In fact, we will present here a candidate for such a subquantum theory which is exceptionally simple and natural - a minor variant of Caticha's "entropic dynamics". We will as well consider other approaches, and in particular the problems which have to be solved in these other approaches.

There are two ways to get rid of these infinities:

## The first possibility: The wave function has to remain non-zero everywhere

The first way is to solve the problem is to add an explicit condition that in the configuration space $$Q$$ the wave function has to be non-zero everywhere:

$\quad\mathbf{(\forall q\in Q) \quad |\psi(q)|^2 > 0}$

This possibility has been considered in my paper The Wallstrom objection as a possibility to augment quantum theory.

This restriction defines itself an additional physical prediction - it is impossible to prepare, or to observe, a pure state such that its wave function $$\psi(q)$$ in the configuration space representation has zeros. This prediction may fail, it may be possible to prepare such states. If this would be done, subquantum theories following this way would be empirically falsified.

### Popper's criterion of empirical content makes this preferable in comparison with quantum theory

But as long as such an explicit empirical falsification has not been made, the modified theory is even preferable to quantum theory. Once it makes an additional prediction, its empirical content is higher than that of quantum theory without this restriction, and according to Popper's criterion of empirical content one has to prefer the theory with higher empirical content.

So, such an additional restriction is not a bug but a feature: It increases the empirical content of quantum theory.

This holds, of course, only as long as the additional empirical prediction survives the real empirical tests.

### Different choices of the configuration space define different theories

Before we start to look for such empirical tests, we have to recognize that the additional empirical prediction depends on something which up to now has been considered to be some purely metaphysical choice - the choice of the configuration space.

The same state $$|\psi\rangle$$ may have zeros in one representation $$\psi(q)\in\mathcal{L}^2(Q)$$ but no zeros in another one, $$\tilde{\psi}(\tilde{q})\in\mathcal{L}^2(\tilde{Q})$$. So, preparing this state will falsify the theory where the former, $$Q$$, is used as the configuration space, but not the theory with the latter one, $$\tilde{Q}$$, as the configuration space.

This seems to become relevant for the choice between particle ontology and field ontology in field theory. If one uses the particle ontology, then there are nice candidates for wave functions with stable zeros: States with non-zero angular momentum.

For the field ontology, the situation looks different. What one can reasonably expect to prepare in a pure form are only stable low energy states - moving states have some uncertainty in space and time, so what one could prepare would be, in fact, superpositions with plausibly quite large contributions of the vacuum state. Stable states in field ontology correspond to standing waves. Standing waves define, essentially, one-dimensional subspaces. For the one-dimensional energy eigenstates one can use real wave functions. Beyond the vacuum state, they all have zeros. But a minor imaginary distortion will be sufficient to get rid of them. And no real preparation procedure could completely avoid such small imaginary contributions.

So, it seems quite plausible that the theory may be empirically falsified if the particle ontology is used, but remains viable in the field ontology.

## The second possibility: The Schrödinger equation has to be modified near the zeros of the wave function

If one cannot avoid zeros of the wave function, the theory has to be modified at least near the zeros where the velocity becomes infinite. Then, the velocity is in this case no longer a potential flow $$v^i(q) = \partial_i \phi(q)$$. This may remain only an approximation far away from the zeros. The independent variables of a subquantum theory would have to be, together with the density $$\rho$$, the whole velocity vector field $$v^i(q)$$. The formula $$v^i(q) = \partial_i \phi(q)$$ can appear only in the quantum limit.

### The Wallstrom objection: A serious problem for a subquantum theory

But to recover quantum theory as some limit of a subquantum theory based on the variables $$\rho(q), v^i(q)$$ seems difficult if not impossible if one thinks about an objection against theories of this type proposed by Wallstrom [1989], [1994], known as the Wallstrom objection: For a loop around the zeros, the following quantization condition has to be fulfilled: $\oint v^i(q) dq^j = \oint \partial_j \phi(q) dq^j = 2\pi m\hbar, \qquad m\in\mathbb{Z}.$ with integer values of $$m$$. For a small loop around single zero, this integer has to be non-zero. What one would expect from the limit of some subquantum theory based on $$\rho(q), v^i(q)$$, which does not even have a phase function $$\phi(q)$$, one would expect either zero for all such integrals or arbitrary real values, but not such integer values.

### A step toward solving this problem

The situation is, nonetheless, not hopeless.

A way to solve it could be based on the following additional restriction, which could be possibly derived in such a limit:

Almost everywhere where $$\quad\mathbf{\rho(q)=0}\quad$$ we have $$\quad\mathbf{0 < \Delta \rho(q) < \infty}\quad$$

This possibility I have proposed and considered in more detail in my paper A solution for the Wallstrom problem of Nelsonian stochastics.

This restriction allows some zeros of the wave function, thus, is less restrictive than the one considered above. And it appears that this condition identifies exactly those zeros where the integral above has $$m = \pm 1$$, that means, first order zeros, like those of the complex function $$\psi(z)=z$$ or $$\psi(z)=\bar{z}$$. If the value of $$|m|$$ is greater than 1, we obtain $$\Delta \rho(q) = 0$$, if it is lower than 1, we obtain $$\Delta \rho(q) = \infty$$.

There is no necessity to obtain higher values of $$m>1$$ for a isolated zeros, which would correspond to higher order zeros $$\psi(z)=z^m$$ or $$\psi(z)=\bar{z}^m$$. These would be forbidden solutions, but are nonetheless important? No problem, we can add a small distortion $$\psi(z)=z^m+\varepsilon$$ and we have already m separate first order zeros, and no experiment could distinguish them with certainty.

So, every solution of the Schrödinger equation has also some solution as close to it as one likes which fulfills the condition above. This makes the theory effectively equivalent to quantum theory. (This is a standard mathematical technique named "general position": Often one does not have to care about exceptional cases, and it is sufficient to consider only general, typical cases, if there are always enough typical cases near the exceptional ones.)

But with this condition, the situation looks already much simpler for the subquantum theory. The probability density $$\rho$$ is already part of the theory itself, the term $$\Delta \rho(q)$$ is anyway important, given that it is used in the quantum potential, and has to be non-negative at the minimum anyway. So, the subquantum theory will provide some expression for $$\Delta \rho(q)$$, and all one has to prove is that one can exclude the extremal values $$0$$ and $$\infty$$ in the quantum limit.

But up to now there is no necessity to use this emergency possibility. It seems much more reasonable to hope that the theory which excludes zeros completely is viable. In this case, we have much simpler and much more natural interpretations and we don't even have to change the equations of quantum theory. There are several variants, a variant which is widely known is Nelsonian stochastics, but I prefer another one, namely the following:

# 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. I present here a slightly modified variant.

To avoid the inifinite velocity near the zeros, one has to use the first possibility, that means, we introduce the condition $(\forall q\in Q) \quad |\psi(q)|^2 > 0.$

This is necessary because the interpretation prescribes that the phase $$\phi(q)$$ has to be a global function, defined everywhere. If the theory would be empirically falsified for a field ontology (something I would not expect to happen), it would have to be given up, and one would have to search for another subquantum theory.

## 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 $$Q_{ext}$$ which together 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 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. 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. 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 or Gao, see here 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: The wave function is defined by incomplete knowledge about the quantum state, namely the knowledge about the preparation process. 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:

• First of all, the ontology is exactly the same as in classical mechanics in the conceptually simplest form, the Lagrange formalism: Reality is described by a continuous trajectory $$q(t) \in Q$$ in some fixed configuration space.
• The wave function is epistemic, it describes only our incomplete knowledge of the configuration of the system. So, no wave function of the universe has to be postulated to exist, the classical configuration $$q(t)\in Q$$ is completely sufficient.
• The evolution is not deterministic, as in classical mechanics, but stochastic. This is nicely compatible with common sense ideas about free will, and does not have fatalism as a consequence, as determinism. So, it is even better compatible with common sense than classical mechanics.
• The classical limit appears conceptually completely unproblematic, in the limit $$\hbar\to 0$$ the quantum potential disappears, and we obtain the classical equations in the Hamilton-Jacobi formalism.
• The evolution equation for $$\rho(q)$$ describes a random process based on a completely classical form - a simple diffusion term defined by $$\Delta \rho$$, as in a Brownian motion, combined with a drift with velocity $$v^i=\partial_i S(q)$$ toward increasing entropy: $\partial_t \rho = \Delta \rho - \partial_i\left(\rho \partial_i S\right) = 0.$
• The equation which describes the change of the phase $$\phi(q)$$ is a generalization of the classical Hamilton-Jacobi equation. While the original equation has problems with caustics, which would lead to infinities for a probability density $$\rho$$, the quantum potential prevents such singularities.

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

## References

• Madelung, E. (1927). Quantentheorie in hydrodynamischer Form. Z Phys 40, 322-326
• Nelson, E. (1966). Derivation of the Schrödinger Equation from Newtonian Mechanics, Phys.Rev. 150, 1079-1085
• Caticha, A. (2011). Entropic Dynamics, Time and Quantum Theory, J Phys A 44:225303, arxiv:1005.2357.
• Caticha, A. (2012). Entropic Inference and the Foundations of Physics. USP Press, Sao Paulo, Brazil, www.albany.edu/physics/ACaticha-EIFP-book.pdf
• Wallstrom, T.C. (1989). On the derivation of the Schrödinger equation from stochastic mechanics, Found. Phys. Letters 2(2), 113-126
• Wallstrom, T.C. (1994). Inequivalence between the Schrödinger equation and the Madelung hydrodynamic equations, Phys Rev A 49(3):1613-1617
• Schmelzer, I. (2011). A solution for the Wallstrom problem of Nelsonian stochastics. arXiv:1101.5774
• Schmelzer, I. (2019). The Wallstrom objection as a possibility to augment quantum theory. arxiv:1905.03075
• Pusey, M., Barrett, J., Rudolph, T. (2012). On the reality of the quantum state. Nature Phys. 8, 475-478.
• Hardy, L. (2013). Are quantum states real? International Journal of Modern Physics B 27, 1345012
• Gao, Sh. (2016). The Meaning of the Wave Function: In Search of the Ontology of Quantum Mechanics. Cambridge University Press.
• Leifer, M.S. (2014). Is the Quantum State Real? An Extended Review of $$\psi$$-ontology Theorems. Quanta 3(1), 67-155 arxiv:1409.1570.
• Schmelzer, I. (2019). Do psi-ontology theorems prove that the wave function is not epistemic? arxiv:1906.00956

# Popular explanation

If you have heard something about quantum theory, you have probably understood nothing, except that it is impossible to understand it even for professionals. Everything is completely mysterious. Uncertainly relations, strange wave-particle dualities, spooky actions at a distance. And in the "many worlds interpretation" everything becomes even more strange - all the other possible outcomes of our experiments exist too, in other, parallel universes. So, in the domain of quantum theory everything seems to be in deep contradiction with common sense. There are, of course, a lot of strange things in relativity too. But in quantum theory, it seems, common sense has to be rejected almost completely.

What is the physical base for such a rejection of common sense? Are there experiments which cannot be explained in a way compatible with common sense? Extraordinary claims require extraordinary evidence, and that something cannot be explained in a way compatible with common sense is a very extraordinary claim. So, where is the corresponding extraordinary evidence? This is a question which I think modern physics is obliged to answer.

Here we present the answer, and the answer is quite unexpected: There is no physical evidence at all which is in contradiction with common sense. The whole of quantum theory can be interpreted in such a way that everything is compatible with common sense.

In particular, in the interpretation preferred by the author, the real world can be completely described by a continuous trajectory of the classical configuration in absolute time. The wave function describes only our incomplete knowledge of the configuration of the quantum system.

## What a layman can check: It is published in a sufficiently good peer-reviewed mainstream journal

And this interpretation is not a nonsensical fantasy of some freak, but it has been published in a peer-reviewed journal of mainstream physics. To find the full name given the abbreviation "J Phys A" the search engine of your choice will be sufficient. In this case, the name of the journal is Journal of Physics A.

Unfortunately, you have to be aware that there are also some freak journals known to publish papers which would be rejected even without reviewing them by mainstream journals (here Galilean Electrodynamics would be an example). As well, recently a lot of Open Source journals have been founded, where the authors have to pay for publishing. These journals claim to be peer-reviewed, but in fact they publish everything authors are ready to pay for, and in modern "publish or perish" science many scientists are ready to pay something for publishing even in such worthless journals. How to distinguish good scientific journals from such junk journals?

Here, to look for a Wikipedia page for the journal may be a first indication (here this would be Journal of Physics A). Then there is this list which gives the actual evaluation of the actual impact (the number of citations per publication) of various physics journals, which counts as the most important quality criterion of a journal.

Of course, if one wants to evaluate positions of outsiders, one cannot expect that they succeed in publishing their papers in the leading top journals. If the journal is accessible at all in this list, this already means that it is not completely off, but a sufficiently serious journal with real peer review. But in this particular case the result looks very good, good enough to classify this even as a publication in a top journal:

Such top papers certainly will not publish nonsense. Ok, one cannot be completely sure about this, even top papers have published some junk papers. But the nonsense published by top journals will be fashionable nonsense, something compatible with the latest fads of the mainstream, and therefore reviewed less rigorously than submissions by outsiders. So, if something out of the mainstream is published in a sufficiently good mainstream journal, one can be sure that the content has been seriously checked.

## A preference for an interpretation is a personal metaphysical preference

Once there exists such an interpretation of quantum physics compatible with common sense, it follows automatically that there is no conflict between quantum theory and common sense at all.

This holds even if the mainstream of physics prefers, for whatever reasons, some other interpretations. Indeed, all imaginable other reasons are, by their nature, metaphysical preferences. They are not, and even in principle cannot be, backed up by any empirical, experimental evidence. So, if some (even if unfortunately most) scientists prefer some mystical interpretation, this is nothing but their own metaphysical preference. It does not show in any way that other interpretations, in particular those compatible with common sense, are wrong or even in a minor conflict with anything seen in any experiment or observation. The preference for particular interpretations is a purely personal choice.

As a consequence, the rejection of common sense based on quantum theory is not based on physics, but on arbitrary metaphysical preferences of particular physicists.

## The role of metaphysical concepts in fundamental physics

But let's note here: Metaphysical arguments are much weaker than physical, empirical evidence. But, despite various popular claims that they have no scientific value at all, they have scientific value. So, it makes sense for the physicists to consider arguments in favor of different interpretations, and to favor some particular interpretation oneself.

The most important place where the metaphysical choice between different interpretations matters is that such interpretations are closely connected with particular reserch programs for more fundamental physical theories. So, in particular, once we favor a common sense compatible interpretation of quantum physics, as well as a common sense compatible interpretation of relativity, this essentially predefines our own research program for quantum gravity: We will look for theories of quantum gravity among theories compatible with common sense.

Instead, a mystic who prefers the most counterintuitive interpretation of quantum physics - say, many worlds - as well as a counterintuitive interpretation of relativity like the curved spacetime interpretation, will not even look at a theory of quantum gravity which would be compatible with common sense. In his opinion, all those common sense principles which hold in this proposal would be already in conflict with established quantum theory (in his preferred interpretation) and relativity (in his preferred interpretation).

Who is right? This is something we cannot answer today, given that these are only programs for future theory development, not actual theories, and that we have no quantum gravity experiments yet which would allow to make a physical, empical choice between them. But there cannot be any doubt that thinking about such research programs is a legitimate and important part of the job of modern physicists working in the domain of the foundations of physics.

## Are there important arguments against common sense interpretations of quantum theory?

There is a quite simple one, namely that the average velocity increases very much near the zeros of the wave function, where it in the limit becomes infinite. The common sense conclusion is simple: Quantum theory cannot be the ultimate most fundamental theory, there has to be some different, subquantum theory without such infinities.

The Wallstrom objection, proposed by Wallstrom in [1989], [1994], is also about what happens near the zeros, but more technical. The point is, roughly, that there is no chance to derive the Schrödinger equation from some sort of subquantum theory which does not have a wave function. So, such subquantum theories would fail to derive quantum theory, but define some different theory, and this different theory plausibly fails.

There are three ways to meet these objections. The simplest one is, roughly, a "so what". Once we cannot derive the Schrödinger equation from something more fundamental, we can nonetheless use the Schrödinger equation as it is, together with the wave function. The Schrödinger equation defines not only the probability density following the Born rule, but also a continuity equation for this density, so that we can nonetheless assume that a continuous trajectory exists, even if it is not observed. So, we get nonetheless rid of the most counterintuitive aspect of usual interpretations of quantum theory, the rejection of existence of such a continuous trajectory. This restriction to a pure interpretation of quantum theory I have named minimal realistic interpretation. That the average velocity becomes arbitrary large near the zeros of the wave function simply means that quantum theory is not the final theory, it has to be replaced by a theory without such infinities.

The next answer, proposed in this paper, is more about the technical problems how one could derive quantum theory, even near the zeros, from some subquantum theory.

The third one, proposed in this paper, is, instead, a quite radical one. We accept that the subquantum theory is a physically different theory, and argue that this theory is viable and even better than quantum theory. We do this not by changing the Schrödinger equation, but by adding an additional restriction, namely that the probability of every imaginable real configuration of a quantum system is non-zero. Once this holds for initial values, it follows from the Schrödinger equation that it will hold forever, so there is no conflict with the Schrödinger equation and defines a consistent theory.

The modified theory can possibly fail for empirical reasons. Once we observe a state so that the wave function describing this state would be forbidden, the theory would be falsified empirically. From the point of view of the scientific method, this adds empirical content and therefore improves the theory - as long as it has not been really falsified.

## "Wave or particle" may become an empirical question

A very interesting point is that for this possibility theories with different assumptions of what defines reality become physically different theories: An observation which could falsify one of them may not falsify another one. And this difference becomes possibly important if we want to decide a very old question, namely if light a sort of particles, photons, or a wave of the EM field. In particular, Wallstrom considered only the case of electrons as particles, and argued that this theory would not be viable, because states with non-zero angular momentum be exactly zero at the axis of rotation, and that such states would be too important in quantum theory to exclude them.

But if reality is defined by a field configuration instead of particle positions, this argument no longer holds. If photons, as well as electrons, are only pseudoparticles, similar to phonons ("sound particles") known from quantum condensed matter theory, without any fundamental importance, then what happens with the wave function as depending on their particle positions does not matter at all. Forbidden states would be only those where some particular field configuration of the EM field or some Dirac field of the electron would have a probability which is exactly zero.

So the most plausible answer to the "wave or particle" question is that for all those fields we have observed - the gravitational field as well as all the fields of the "standard model of particle physics", it is the field ontology which is the correct one, and all the particles - as photons, as electrons and all the other "particles" - are only pseudoparticles like phonons, quantum effects without any fundamental relevance.

## Entropic dynamics

I favor the last possibility because it is compatible with Caticha's "entropic dynamics" - an interpretation which allows to give a derivation of the equations from simple first principles, a derivation similar to classical statistical mechanics. In this interpretation (to be accurate, in a variant of the interpretation I prefer) the classical configuration (with a trajectory in the configuration space) already describes the complete ontology - there would be nothing beyond the trajectory of the configuration in reality.

Instead, the wave function would describe only our incomplete knowledge about the quantum system.

Not all realistic interpretations have this property. What they share is that there is a classical configuration which has a continous trajectory - a consequence of the continuity equation which follows from the Schrödinger equation. But there are possibly other parts of reality - in particular those described by the wave function. The typical example is de Broglie-Bohm theory, which postulates that the wave function is, together with the configuration, also a really existing object. In Nelsonian stochastics, the probability density describes only our incomplete knowledge about the actual configuration, but the phase of the wave function is another objectively existing object. But in entropic dynamics, the wave function is completely epistemic: The phase is defined by the entropy of the rest of the world if the quantum system is in a given configuration.

So what is, in this case, non-classical? First of all, there is some randomness. This leads to some diffusion, which disappears in the classical limit. This is, of course, nothing which would be in conflict with common sense. As a consequence, there is no deterministic velocity, there is only a statistical average velocity.

Then, all the other variables - momentum, energy, angular momentum, and whatever else - are not defined by the configuration of the system, but depend as well on the configuration of the measurement device. Such a dependence is named "contextual". That means, there is no actual value of, say, the momentum of a quantum system. There is, in fact, nothing very counterintuitive there, all what is confusing is an inappropriate naming convention: What is named "measurement" would be, in this cases, better named "interaction".