of quantum theory

Do we really have to give up most of our common sense ideas about how the universe works because of quantum theory? **The minimal realist interpretation aims to answer this question** by trying to preserve as much of common sense as possible. It goes beyond the minimal interpretation of quantum theory, but only as far as this is necessary to preserve classical realism and common sense. To understand the reasoning which leads to the minimal realist interpretation, it seems useful to imagine a 19th century scientist appearing our world, who learns about the mathematical apparatus of quantum theory (as well as that of relativity), learns the minimal interpretation, and tries to make sense of all this, but completely ignores all the other interpretations of quantum theory. So, he has everything he needs to do the physics. But he is not ready to give up, without serious evidence, any part of classical common sense, in particular classical ideas about realism and causality.

The interpretation assumes that there exists configuration space \(q=(q^1,\ldots.q^N)\in Q\) and presupposes that the Schrödinger equation has the usual form: \[ i \hbar \frac{\partial}{\partial t} \psi(q,t) = \left(-\frac{\hbar^2}{2} \delta^{ij}\frac{\partial}{\partial q^i}\frac{\partial}{\partial q^j} + V(q)\right) \psi(q,t). \]

This is sufficient to handle relativistic field theories, in particular the standard model of particle physics.

The result is a realist interpretation where we have, as in classical Lagrange formalism, **a continuous trajectory in the configuration space \(\mathbf{q(t)\in Q}\).**

But we have only incomplete knowledge of this trajectory, and the **wave function \(\mathbf{\psi(q,t)\in \mathcal{L}^2(Q,\mathbb{C})}\) defines our incomplete knowledge** of the trajectory.

The knowledge defined by the wave function is defined by the Madelung variables, namely by the probability density \(\rho(q,t) = |\psi|^2\) and, where it is non-zero, also by the phase \(S(q) = \hbar \Im \ln \psi(q)\), which defines the potential for the average velocity of the probability flow: \[ v^i(q,t) = \delta^{ij} \frac{\partial}{\partial q^j} S(q,t).\]

For the probability density together with this velocity, a continuity equation follows from the Schrödinger equation: \[\partial_t \rho(q,t) + \frac{\partial}{\partial q^i}\left(\rho(q,t) v^i(q,t)\right) = 0\]

This continuity equation justifies the thesis that there exists a continuous trajectory in the configuration space.

The Schrödinger equation gives also a second equation for the phase function \(S(q)\) named the quantum Hamilton-Jacobi equation: \[-\partial_t S(q) = \frac{1}{2} \delta^{ij}\frac{\partial S(q)}{\partial q^i}\frac{\partial S(q)}{\partial q^j} + V(q) + Q(q).\]

It differs from the classical Hamilton-Jacobi equation by the a quantum potential \[Q(q) = -\frac{\hbar^2}{2} \frac{\Delta \sqrt{\rho(q)}}{\sqrt{\rho(q)}}. \]

This potential automatically disappears in the classical limit \(\hbar\to 0\). This makes the classical limit completely unproblematic.

- While the velocity is defined by the same formula as the "guiding equation" of de Broglie-Bohm (dBB) theory, it is interpreted only as an average velocity. This is all one can conclude from the continuity equation for the probability flow. That means, a minimal realist interpretation has to restrict itself to this. This does not exclude that this really is a deterministic velocity. But it is, in fact, not really plausible. Stochastic interpretations like Nelsonian stochastics are, in this point, much more plausible, in particular because the quantum potential appears there in a quite natural way. Moreover, interpreted as a deterministic velocity, the interpretation faces the "surrealistic trajectories" argument, because some trajectories would be quite implausible. For example, for a stationary state this would mean there is no movement at all.
- Another difference to dBB theory is that the wave function \(\psi(q)\) is interpreted epistemically, as describing our incomplete knowledge of the system. Instead, dBB theory postulates a really existing wave function, which leads to such artefacts as the wave function of the universe.
- As in the Copenhagen interpretation, we have a subdivision into classical and a quantum part. But it plays a very different and much more unproblematic role. The two parts simply correspond to different states of knowledge about the same reality, which is defined in both parts by a continuous trajectory \(q(t)\in Q\). In the classical part, we know the trajectory itself, which is approximately defined by the classical equations and therefore smooth. Instead, in the quantum part we know only the wave function, which gives us only a probability distribution and an average velocity.
- Different from other realist interpretations, no attempt is made to derive the Schrödinger equation from some different, more fundamental theory. In particular, no attempt is made to construct the wave function \(\psi(q,t)\) out of \(\rho(q,t)\) and \(S(q,t)\) by the polar decomposition formula \(\psi(q) = \sqrt{\rho(q)} \exp(\frac{i}{\hbar} S(q))\). The wave function, as it is, describes the knowledge which we can obtain from the preparation procedure of the quantum state. This avoids the Wallstrom objection that wave functions with zeros cannot be defined in such a way because there exists no global phase function \(S(q)\).
- Different quantum/classical splits may define the same state of reality, but describe different degrees of knowledge. They have to be compatible with each other, given that they describe the same reality. The different descriptions have to be compatible with each other. That means, if we have a common classical part defined by the configuration variables \(q_{classical}\), a common quantum part defined by \(q_{quantum}\), and an intermediate part defined by \(q_{intermediate}\), the two different wave functions \(\psi_{small},\psi_{big}\) and the trajectory \(q_{intermediate}(t)\) known in the variant with the greater classical part are related by \[ \psi_{small}(q_{quantum},t) = \psi_{big}(q_{intermediate}(t),q_{quantum},t).\] This formula solves the "measurement problem" in the same way as it is solved in dBB theory.
- How to handle the Wallstrom objection is a problem which has to be left to a more fundamental, subquantum theory. A proposal how this can be solved has been presented in my paper An answer to the Wallstrom objection against Nelsonian stochastics, arXiv:1101.5774. But the aim of the minimal realist interpretation is not the development of a subquantum theory, but the interpretation of quantum theory.
- The average speed of the configuration \(|\vec{v}(q,t)|\) becomes infinite at zeros of the wave function, thus, arbitrary large near these zeros. This strongly indicates that quantum theory is only an approximation, and suggests that the place to search for violations of quantum theory are the zeros of the wave function.

Given that the minimal realist interpretation reuses some well-known formulas from other realist interpretions, it is quite obvious what could be objected against it: The objections applied to those other realist interpretation may be tried here too.

Some of them (namely incompatibility with relativistic symmetry and the Pauli objection that it introduces an unnatural asymmetry between configuration and momentum variables) have to be answered, but this is, fortunately, not a big problem.

Other arguments (like the Wallstrom objection, the surrealistic trajectories argument, and objections against a wave function of the universe) are not valid objections against the minimal realist interpretation, because we have avoided to make any claims, reasonable or not, which go toward a more fundamental, subquantum theory.

Once we give a physical interpretation to the "Bohmian velocity" \(\vec{v}(q)\), we have to face the objection that this violates fundamental Lorentz invariance. While the velocity itself is not observable, it appears in the Schrödinger equation, and is interpreted as a physical velocity - the average velocity of the configuration. It depends on the complete configuration, thus, on the complete state of the universe, without taking into account restrictions created by Einstein causality.

**This objection is unavoidable for any realist interpretation**, given the violation of Bell's inequality by quantum theory. This was the very point for Bell to prove his theorem - to show that the problem is unavoidable for every realist interpretation.

But is this a serious problem? Here it is important to distinguish two problems – compatibility with relativistic physics, and compatibility with relativistic metaphysics. Of course, relativists like to think that they don't accept any metaphysics, but this is only a nonsensical remnant of positivism - in reality, they like to reject theories for purely metaphysical reasons.

With Lorentz symmetry for observables there is no problem at all. The minimal realist interpretation is compatible with relativistic QFT, and for all the observables Lorentz invariance holds. But Lorentz invariance is not fundamental - the interpretation has a hidden preferred frame. The time coordinate \(t\) of the Schrödinger equation is the preferred absolute time, and it is a hidden variable. (Note that even in non-relativistic quantum theory \(t\) is a hidden variable, there exists no operator for time measurement which could measure \(t\), and every physical clock goes, with some nonzero probability, sometimes even backward in time.)

If we restrict ourselves to physics, to observable effects, then there is not much base for argumentation against interpretations which use a preferred frame.

Let's note here that it is not at all a serious objection against a hidden variable theory that among its hidden variables there is also a hidden preferred frame.

This objection is about the asymmetry between configuration and momentum variables in the interpretation. It was part of Pauli's rejection of Bohm's causal interpretation. In Pauli's words, "the artificial asymmetry introduced in the treatment of the two variables of a canonically conjugated pair characterizes this form of theory as artificial metaphysics" (Pauli 1953). The minimal realist interpretation also prefers the configuration space, thus, the objection could be applied here too.

But, first of all, there are important parts of physics which do not show this symmetry at all. So, in particular, the Hamilton operator has the form \(\hat{H} = \hat{p}^2 + V(\hat{q})\), thus, a quite different dependence on both variables.

Then, as I have shown in (Schmelzer 2009), even if we restrict ourselves to Hamilton operators which have the form \(\hat{H} = \hat{p}^2 + V(\hat{q})\), one can, for a given \(\hat{H}\), find different pairs of conjugate operators \(\hat{p},\hat{q}\) so that the same \(\hat{H}\) has the same form \(\hat{H} = \hat{p}^2 + V(\hat{q})\) but with different potentials \(V(\hat{q}\), and, as a consequence, with different physical predictions. So, the physical predictions of the theory depend on the choice of the operators \(\hat{p},\hat{q}\).

This is already sufficient to reject the Pauli objection.

There have been objections against de Broglie-Bohm theory that the trajectories it predicts are quite surrealistic.

Given that we use the same dBB formula for the velocity, one could think that this argument may be applicable here too. But there is a difference, namely the dBB velocity is deterministic, while we interpret it only as an average velocity.

This makes a difference, as can be seen in the most simple case where the objection has been made - stable states the discrete eigenstates of one-dimensional theory. Once they are stable states, the probability distribution does not change in time. So, whatever the real velocities, if the flow is irrotational, the average velocity has to be zero. So, to object that these trajectories would be surrealistic makes no sense. The situation is different in dBB theory, where the velocity is deterministic. This deterministic velocity has to be zero too. That means, the electron in an atom does not move at all, even if it is attracted by the nucleus. This is already much more surrealistic.

So, avoiding to postulate that the velocity is deterministic also avoids the surrealistic trajectories argument.

An earlier and already outdated version of the interpretation named "paleoclassical" is presented in the article Schmelzer, I. (2015) The paleocclassical interpretation of quantum theory, **arXiv:1103.3506**, published in Reimer, A. (ed.), Horizons in World Physics, Volume 284, Nova Science Publishers.

I'm actually yet in the process of writing a paper about the minimal realist interpretation.

- Pauli, W. (1953). Remarques sur le problème des paramètres cachés dans la mécanique quantique et sur la théorie de l’onde pilote, in André George, ed., Louis de Broglie – physicien et penseur (Paris, 1953), 33-42
- Schmelzer, I. (2009). Why the Hamilton operator alone is not enough, arXiv:0901.3262, Found. Phys. vol.39, p. 486-498, DOI 10.1007/s10701-009-9299-4.