of quantum mechanics

Do we really have to give up most of our common sense ideas about how the universe works because of quantum theory? **The minimal realistic interpretation aims to answer this question**.

There have been different proposals for interpretations of quantum mechanics. Among them we have deterministic (de Broglie-Bohm theory, also named "Bohmian mechanics") as well as stochastic (Nelsonian stochastics). The author favors entropic dynamics proposed by Caticha 2011.

Nonetheless, these realistic interpretations have a weak point: They go beyond quantum mechanics, modify it, even if only in a minor way. This opens a possibility to criticize and reject them not as interpretations of quantum theory, but because of these differences. That's why it seems useful to have also an interpretation which restricts itself to a pure interpretation of quantum mechanics, without any attempt to go beyond it toward some subquantum theory.

With "quantum mechanics" we mean essentially Schrödinger's wave mechanics, that means, quantum theory in the configuration space representation \(\psi(q)\in\mathcal{L}^2(Q,\mathbb{C})\), with the Schrödinger equation \[ i \hbar \partial_t \psi = -\frac{\hbar^2}{2} \Delta \psi + V \psi. \] This is not at all a restriction to non-relativistic theory: In relativistic quantum field theory we have a similar (only infinite-dimensional) quadratic dependence of the energy on the momentum operators. If we regularize them (something we have to do anyway) using the straightforward lattice regulatization, we obtain a Schrödinger equation of this type.

- While we are free to use the Madelung variables - the density \(\rho(q,t) = |\psi(q,t)|^2\) and (where \(\rho\neq 0\)) the phase \(phi(q) = \hbar \Im \ln \psi(q)\) - we do not attempt to construct the wave function out of density and phase. In this way, we avoid the Wallstrom objection.
- The Schrödinger equation defines in the region where \(\rho \neq 0\) a continuity equation for \(\rho\) with the velocity \( v^i(q,t) = \partial_i \phi(q,t)\): \[\partial_t \rho + \partial_i \left(\rho v^i\right) = 0.\] This continuity equation is used as the justification that there exists a continuous trajectory in the configuration space \(q(t)\in Q\).
- The fact that the velocity becomes infinite at the zeros of the wave function is taken as a fact which indicates that quantum mechanics becomes wrong near the zeros of the wave function. As a consequence, quantum mechanics cannot be the most fundamental theory. It has to be replaced by some subquantum theory which does not have such infinities. But no particular proposal for such a subquantum theory is made - this is not the job of an interpretation. For an interpretation it is sufficient to define the physical meaning of the mathematical expressions of the theory.
- The velocity \(v^i(q,t)\) is interpreted as an average velocity. This does not predecide the question if the theory is deterministic or stochastic - in both cases, there exists an average velocity, only in the deterministic case it is also the actual velocity.
- The Schrödinger equation gives also a second equation for the phase function \(\phi(q)\) named the quantum Hamilton-Jacobi equation: \[\partial_t \phi + \frac{1}{2} (\nabla \phi)^2 + V -\frac{\hbar^2}{2} \frac{\Delta \sqrt{\rho}}{\sqrt{\rho}} = 0\] 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.
- The wave function is interpreted as epistemic. It describes the incomplete knowledge about the quantum system, as defined by the preparation procedure. Once the information about preparation device and the result of the preparation measurement objectively exist - else we cannot have this information - the wave function is also defined uniquely by what really exists. So, we do not have to argue about "epistemic vs. ontological interpretation of the wave function".
- Once we have, together with the wave function, also a trajectory of the configuration, the collapse of the wave function can be described by the solution of the Schrödinger equation of the system together with the measurement device, \(\psi(q_{sys},q_{dev},t)\), the trajectory of the measurement device \(q_{dev}(t)\) which is visible because the device is macroscopic, and the formula which defines the effective wave function of the system by \[ \psi_{sys}(q_{sys},t) = \psi(q_{sys},q_{dev}(t),t). \] So, there is no measurement problem at all.

Given that the minimal realistic interpretation reuses some well-known formulas from other realistic interpretions, it is quite obvious what could be objected against it: The objections applied to those other realistic interpretations 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 realistic 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 realistic 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 realistic 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 realistic 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 realistic 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 \(\hat{H} = \frac12 \hat{p}^2 + V(\hat{q})\), 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 realistic interpretation.

- Madelung, E. (1927). Quantentheorie in hydrodynamischer Form. Z Phys 40, 322-326
- Bohm, D. (1952). A suggested interpretation of the quantum theory in terms of ``hidden'' variables, Phys. Rev. 85, 166-193.
- 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.
- 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.