The paleoclassical interpretation
of quantum theory


The interpretation is presented in the article "The paleoclassical interpretation of quantum theory", arXiv:1103.3506 (submitted to "Foundations of Physics").

An important point of the interpretation is presented in An answer to the Wallstrom objection against Nelsonian stochastics, arXiv:1101.5774 in more detail.


In de Broglie-Bohm theory a classical trajectory q(t) is introduced as a hidden variable. But the wave function ψ(q) is also interpreted to be a really existing object (a "beable").

Instead, the paleoclassical interpretation is more radical, it revives classical ontology completely:

Only the classical trajectory q(t) in classical configuration space really exists.

The wave function decribes, instead, only some set of incomplete information about the real trajectory – the probability distribution ρ(q)=|ψ(q)|2 and the average velocity v(q)=∇ S(q) = ∇ Im ln ψ(q), which define the wave function by the polar decomposition

ψ(q) = ρ1/2(q) e i S(q).


There is a seemingly trivial objection: To describe the different quantum states of a single particle, we need more degrees of freedom - the different wave functions of a single particle appear as really different states, states which can be prepared differently, and lead to different probability distributions.

So the state of a single particle needs much more degrees of freedom than the position of a single particle.

But the interpretation is correct only for a closed system, and, from the point of view of the interpretation, a single particle does not define a closed system. In the quantum realm, there is always an interaction with the environment. It is the very interaction which causes the violation of Bell's inequality.

Because of these interactions with the environment, the only really closed system is the whole universe. This is, unfortunately, outside the domain of observation.

Instead, for a "closed system" in the usual, classical sense - the single particle - we need an object which describes the effective interaction of all the remaining parts of the universe. And this is, indeed, the wave function. So, for all subsystems, we have a description as usual, with configuration and wave function.



For the justification of this interpretation, I use a funny reformulation of a variant of classical mechanics, of Hamilton-Jacobi theory. In Hamilton-Jacobi theory, there is already an analogon of the wave function, namely the function S(q). It is unproblematic to add also a density ρ(q) which evolves following the continuity equation, and to combine them into a wave function using the same formula ψ(q) = ρ1/2(q) e i S(q). The resulting equation is very close to the Schrödinger equation. There is only one additional term in it – the quantum potential term. Once this is a non-linear term in itself, the Schrödinger equation itself appears as the linear approximation of this purely classical equation.

Now, to interpret this purely classical equation, we can use the exact equivalence of this wave function description with classical mechanics (which holds for short times, before caustics appear). This almost forces us to accept this interpretation. In fact, the wave function cannot define an additional reality which influences the trajectories, because the classical trajectories follow only classical equations, and no observation can falsify this.

But once we accept this interpretation for the classical variant, why not for the quantum one? There are not even new terms in the equation, it is only the linear approximation.


Quantum theory is only an approximation: The fundamental theory is different – it is a theory formulated completely in configuration variables.

But there is also another intermediate step between this fundamental theory and quantum theory: A theory formulated in terms of the incomplete information described by the probability flow variables ρ(q), vi(q), but without the special (non-fundamental) assumption of potentiality of the flow, which is violated around the zeros of the wave function anyway.

This intermediate theory is necessary to regularize the infinities of the velocity vi(q) near the zeros of ρ(q).

Predictions about subquantum theory

The development of these theories goes beyond the paleoclassical interpretation, which is only an interpretation of quantum theory, not a proposal for a subquantum theory. Nonetheless, it is remarkable that this interpretation already fixes essential parts of such a subquantum theory, namely its ontology (described fundamentally by the trajectory q(t), or by the probability flow defined by ρ(q), vi(q)), the equation which has to be given up (the condition of potentiality of the flow) and even the region where this becomes most important (near the zeros of the wave function).


The idea to use the decomposition ψ(q) = ρ1/2(q) e i S(q), with S(q) being a potential for velocities, is as old as quantum theory and goes back to Madelung and de Broglie. Madelung has presented his presentation as being "in hydrodynamic form". Given that a hydrodynamic interpretation is possible only for the single particle case, and, therefore, uninteresting, this may have been a cause for ignorance.

It has been used also by Bohm. The de Broglie-Bohm interpretation is currently recovering from its unjustified ignorance and becoming increasingly popular. But modern presentations of de Broglie-Bohm theory do not use this presentation and consider the wave function to be a really existing beable.

The decomposition is also a key ingredient of Nelsonian stochastics. Here the ontology is mixed: The density ψ(q) is interpreted as a probability distribution, thus, as incomplete information about the actual position. But the velocity field v(q) is, instead, assigned the character of a really existing field.

Different from the rejection of de Broglie-Bohm theory, there exists a serious objection against Nelsonian stochastics, as well as any other attempts to take the variables ρ(q), v(q) as more fundamental than the wave function: It is the Wallstrom objection (see below).

In this sense, it is not an accident that I have published this paleoclassical interpretation only after having found a way to solve the Wallstrom problem.


Physicists are not completely stupid, and if they do not follow an idea which is as old as quantum theory itself, they have some reasons. These reasons may be wrong, and sometimes influenced by various ideologies, group thinking, and simple ignorance, but one nonetheless has to take them seriously.

In case of the paleoclassical interpretation, there are three major objections to be discussed: Incompatibility with relativistic physics and metaphysics, the Wallstrom objection that such an interpretation cannot explain the quantization condition around zeros of the wave function, and the Pauli objection that it introduces an unnatural asymmetry between configuration and momentum variables.

The Wallstrom objection

The most serious problem of a quantum interpretation based on flow variables ρ(q) and v(q)=∇ instead of the wave function ψ(q) has been made by Wallstrom. He has objected that in quantum mechanics the integral over a closed path around the zeros of the wave function is quantized:

∫ vi d qi = ∫ ∂i S(q) d qi = 2 π m

The value m has to be an integer. In fact, only if this value is really an integer, the wave function would be a uniquely defined function on the whole plane.

To better understand the problem, one has to recognize that the equations in terms of ρ(q) and S(q) are equivalent to the Schrödinger equation only where the wave function is non-zero. In fact, these equations

-∂tS = 1/2 (∇S)2 + V - 1/2(Δ ρ1/2)/ ρ1/2

tρ= -∂i(ρ∂iS)

are obtained from the Schrödinger equation as their real and imaginary parts divided by ψ. At the zeros of the wave function S(q) is simply not defined. But it is clear that if one removes one point where the equations do not have to hold, nonsensical solutions may appear.

The solution for this problem

Once there is no equation for points with ρ(q)=0, I propose to impose for such points the following regularity postulate:

If ρ(q)=0 then almost everywhere Δρ(q) has to be defined and positive.

This excludes only ∞ and 0 as the extremal values. (If ρ(q)=0, Δρ(q) in non-negative anyway.)

The justification for this postulate has to consider properties of the more fundamental, subquantum theory. I have given it in detail in Schmelzer, I.: A solution for the Wallstrom problem of Nelsonian stochastics, arXiv:1101.5774v3.

Why this works is easy to explain: The solution of the equations outside zero is C(x±iy)|m|. For non-integer m, this is locally well-defined, but does not give a global wave function. But in this case, we obtain ρ(r,φ)=Cr2|m|. So Δρ(0) = C for m=±1. But for |m|<1 we obtain ∞ and for m>1 we obtain 0, thus, the extremal values excluded by our postulate.

But doesn't this exclude too much? It excludes, in fact, multiple zeros, which are quite legitimate solutions of quantum theory. But this does not matter, because a minimal distortion of such solutions with multiple zeros (usually adding a small constant is sufficient, for purely real solutions an imaginary one) gives approximate solutions with simple zeros. This is sufficient for empirical viability.

The Pauli objection

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".

I have been able to answer this argument in the paper Schmelzer, I.: Why the Hamilton operator alone is not enough, arXiv:0901.3262, Found. Phys. vol.39, p. 486 – 498 (2009), DOI 10.1007/s10701-009-9299-4. I have constructed there, for a fixed quantum Hamilton operator H, a series of conjugate operators p, q, such that the Hamilton operator has for all of them the same nice form H = p2 + V(q), but with different potentials V(q), and, therefore, giving different physical predictions. As a consequence, the complete definition of a quantum theory has to specify which operator is p and which is q, and to make a difference between them.

So the Pauli objection does not survive.

Incompatibility with relativistic physics

It is important to distinguish two problems – compatibility with relativistic physics, and 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 fact, they reject theories with a hidden preferred frame not because of any inconsistency with observation, thus, for purely metaphysical reasons.

If we restrict ourself to physics, then there is not much base for argumentation against interpretations using a preferred frame.

In itself, relativity is not a problem at all. One may be misguided by the fact that the description of de Broglie-Bohm theory or Nelsonian stochastics is often given in terms of many particle physics with non-relativistic Hamiltonian, quadratic in the momentum variables.

It would be, indeed, much better to present such theories as general theories defined on an abstract configuration space Q. This configuration space can be, in particular, also a function space for field theories or a lattice space for their lattice regularizations.

The funny point is that for relativistic field theories a Hamiltonian quadratic in the momentum variables is completely sufficient. In fact, the Lagrange density of a relativistic Klein-Gordon field is

L = 1/2((∂tφ)2 – (∂iφ)2) – V(φ),

which gives the momentum variables π=∂tφ and a Hamiltonian quadratic in them

H = 1/2(π2 + (∂iφ)2) + V(φ).

So it is clearly not relativity in itself which is problematic.

Problems with fermions and gauge fields

But what about fermions and gauge fields? Here, I have developed nice condensed matter (or ether) interpretation. An article about it, Schmelzer, I.: A Condensed Matter Interpretation of SM Fermions and Gauge Fields, arXiv:0908.0591, has been published in Foundations of Physics, vol. 39, nr. 1, p. 73 – 107 (2009), DOI: 10.1007/s10701-008-9262-9.

Problems with relativistic gravity

For relativistic gravity also exists a condensed matter (ether) interpretation – the General Lorentz Ether Theory. The article "A generalization of the Lorentz ether to gravity with general-relativistic limit" has been accepted now by the journal "Advances in Applied Clifford Algebras".

Incompatibility with relativistic metaphysics

So there are no physical objections left: For all experimentally established parts of modern physics there exist theories with preferred frame.

There remains the ideological part – relativistic metaphysics, which requires that relativistic symmetry cannot be simply an approximate symmetry, but has to be of fundamental character.

This position was a quite reasonable one at Einstein's time. But after Bell's theorem and the experimental confirmation of the violations of Bell's inequality it is no longer reasonable. To reject realism out of some metaphysical preference for theories with nice symmetry comes close to absurdity. I explain this in more detail in my defense of realism pages.