# Hidden Variable Interpretation

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What are advantages and disadvantages of hidden variable interpretation of quantum mechanics?

Demystifier
Gold Member
Advantage: Gives you a picture of what is going on irrespective of whether the system is measured or not.

It means that hidden variable interpretation is physically more right than the other interpretations, am I wrong?

Demystifier
Gold Member
It means that hidden variable interpretation is physically more right than the other interpretations, am I wrong?
If it is the right hidden variable interpretation, then it is more right than other interpretations. Unfortunately, there is more than one hidden variable interpretation in the literature, and we don't know which one, if any, is the right one.

Could you mention those hidden variable interpretations, please?
Actually I want to study QM from hidden variable interpretation, which I think it give us more clear physical interpretation of QM than the other interpretations...

bhobba
Mentor
It means that hidden variable interpretation is physically more right than the other interpretations, am I wrong?

There is no right or wrong here. Its purely what floats your boat.

Thanks
Bill

Nugatory
Mentor
Could you mention those hidden variable interpretations, please?
Bohmian mechanics is the most carefully worked out hidden variable interpretation that is also consistent with experiment. However....
Actually I want to study QM from hidden variable interpretation, which I think it give us more clear physical interpretation of QM than the other interpretations...
It will be more effective to learn the mathematical formalism through the minimal statistical interpretation first. Once you have a clear idea of exactly what that formalism does and doesn't do, you'll be better able to make sense of the various interpretations. One reasonable paths is to work through a first-year QM textbook, and then get started with Ballentine; there are more mathematical approaches as well, but others here will comment more sensibly on them than I can.

Demystifier and bhobba
bhobba
Mentor
One reasonable paths is to work through a first-year QM textbook, and then get started with Ballentine; there are more mathematical approaches as well, but others here will comment more sensibly on them than I can.

My suggestion is the following sequence:
https://www.amazon.com/dp/0465075681/?tag=pfamazon01-20
https://www.amazon.com/dp/0465062903/?tag=pfamazon01-20
https://www.amazon.com/dp/0674843924/?tag=pfamazon01-20
https://www.amazon.com/dp/0071765638/?tag=pfamazon01-20
https://www.amazon.com/dp/0306447908/?tag=pfamazon01-20
https://www.amazon.com/dp/9810241054/?tag=pfamazon01-20

It starts from the basics and gradually gets more advanced.. Don't skip the early on books - they may be basic but develop valuable intuition.

Thanks
Bill

Demystifier
There is no right or wrong here. Its purely what floats your boat.

Thanks
Bill

That's why I used "more right" term. What I mean by "more right" is that hidden variable interpretation gives us more physical reason behind the strange of QM. I think the collapse wave function, for example, is like a magic and therefore we should search for more physical interpretation, i.e. via hidden variable.

Bohmian mechanics is the most carefully worked out hidden variable interpretation that is also consistent with experiment. However....

It will be more effective to learn the mathematical formalism through the minimal statistical interpretation first. Once you have a clear idea of exactly what that formalism does and doesn't do, you'll be better able to make sense of the various interpretations. One reasonable paths is to work through a first-year QM textbook, and then get started with Ballentine; there are more mathematical approaches as well, but others here will comment more sensibly on them than I can.

I've just finished my course on "usual" elementary QM. My lecturer used Griffiths and I studied Bluumel for my own addition. How about that?

Demystifier
Gold Member
I've just finished my course on "usual" elementary QM. My lecturer used Griffiths and I studied Bluumel for my own addition. How about that?
With that background, you are certainly ready to study the Bohmian hidden-variable approach. I recommend
https://www.amazon.com/dp/0521485436/?tag=pfamazon01-20
a large part of which can also be taken as a textbook for standard QM.

Nugatory
stevendaryl
Staff Emeritus
Bohmian mechanics is the most carefully worked out hidden variable interpretation that is also consistent with experiment.

Is there any other at all?

Demystifier
Gold Member
Is there any other at all?
Many-world interpretation and dynamic-collapse interpretations (e.g. GRW) are also sometimes classified as hidden-variable interpretations. Namely, these interpretations assert that wave function is an ontological real stuff existing even when we don't measure it, which is what makes it hidden. One should distinguish hidden variables from additional variables, where additional means something added to the wave function. Bohmian theory has additional variables, while MWI and GRW do not have additional variables.

The Bohmian theory is not the only theory with additional hidden variables, but others are not widely known. One example is my solipsistic hidden variables, which is mathematically very similar to Bohmian theory but philosophically very different from it.

Minnesota Joe
Nugatory
Mentor
Is there any other at all?
I refuse to answer that question, on the grounds that by limiting myself to the most uncontroversial example I hope to avoid becoming embroiled in a discussion of classification of interpretations.

Last edited:
Is there any other at all?
Nelsonian stochastics, which appears in many variants. The main point is that they derive the equations instead of postulating them. The clearest derivation I have seen comes from Caticha's entropic dynamics.

Caticha, A. (2011). Entropic Dynamics, Time and Quantum Theory, J. Phys. A44:225303, arxiv:1005.2357

Could you mention those hidden variable interpretations, please?
Actually I want to study QM from hidden variable interpretation, which I think it give us more clear physical interpretation of QM than the other interpretations...

The most well-known one is de Broglie-Bohm theory, also known as Bohmian mechanics. It is even deterministic.

Another quite well-known one is Nelsonian stochastics. But my current favorite is Caticha's entropic dynamics, given that it is the closest one to classical common sense.

They all share some mathematics, namely that they use the Schrödinger equation in the configuration space Q as the starting point, that means, the Hamiltonian is assumed to have a quadratic dependence on the momentum variables ##H= \frac{1}{2m}p^2 + V(q)##. This suggests that it is applicable only to non-relativistic particles (and even some of those working with it seem to think so, given that they introduce it using many-particle theory), but it appears applicable to relativistic field theories too, the field-theoretic Hamiltonian is quadratic in the field-theoretic momentum variables ##\pi(x) = \dot{\phi}(x)## too.

Then, to get the point it is useful to consider the so-called "hydrodynamic variables" (proposed by Madelung for a hydrodynamic interpretation possible only for a single particle) ##\psi(q) = \sqrt{\rho(q)}e^{\frac{i}{\hbar}\phi(q)}## which splits the complex wave function into the probability density for the configuration ##\rho(q) = |\psi(q)|^2## and the (real) phase. In these variables, the Schrödinger equation splits into a continuity equation
$$\partial_t \rho(q,t) + \nabla(\rho(q,t)\vec{v}^i(q,t)) = 0.$$
and an equation for the phase ##\phi(q,t)## which looks like a quantum generalization of the classical Hamilton-Jacobi equation. The velocity ## \vec{v}^i = \partial_i \phi## is known as the Bohmian velocity, but used by all the other realist interpretations too.

The realist interpretations share also the idea that once we have a continuity equation for ##\rho(q,t)##, one can safely assume that there exists a continuous trajectory ##q(t)##. For all other observables, like momentum or energy or spin, there is no such continuity equation, and so it makes no sense to assume a continuous trajectory for them.

The realist interpretations can use what they share to solve some of the problems of quantum foundations. First, the classical limit becomes much simpler. The equation for ##\phi## becomes the classical Hamilton-Jacobi equation, the continuity equation is essentially already classical, job done. They all have the problem to explain the measurement results for variables other than the configuration space, but the solution for this which can be reused by all of them has been found by Bohm in his original paper. What we see in the classical part is simply the trajectory ##q(t)##, not the wave function or so. Mathematically we do the Schrödinger equation for the quantum part together with the measurement instrument, the quantum result is a superposition, like Schrödinger's cat, we observe not the wave function but the cat itself, and this additional knowledge reduces the wave function. This happens in a continuous measurement process. The discontinous thing is that at some moment we decide that to consider the wave function (showing Schrödinger's cat) is no longer reasonable and restrict ourselves to the further consideration of the quantum subsystem, by using the observable trajectory ##q_{dev}(t)## of the measurement device:
$$\psi_{eff}(q_{sys},t) = \psi_{full}(q_{sys},q_{dev}(t),t).$$
It is this reduced, effective wave function which, during the time of the measurement, does not follow a Schrödinger equation.

Then, the Bohmian velocity ## \vec{v}^i(q) = \partial_i \phi(q)## has, except for the single point particle case where ##q=\vec{x}\in\mathbb{R}^3##, a global character. Already for two particles ##q=(\vec{x}_1,\vec{x}_2)\in\mathbb{R}^6## the configuration, and therefore also the velocity, depends on the positions of both particles. This is possible in the relativistic context only if there is a preferred frame.

Now about the differences. First, the role of the Bohmian velocity ## \vec{v}^i = \partial_i \phi##. In de Broglie-Bohm theory (dBB) it defines the deterministic trajectory. In the other interpretations, it defines only the average velocity. The interpretation as the average velocity is, essentially, what follows from the continuity equation itself.

Then, dBB theory presumes that the wave function is something really existing too. Instead, Nelsonian stochastics has a mixed interpretation, the interpretation for ##\rho## is epistemic, it defines our incomplete knowledge about some Wiener process, while the interpretation of ##\phi## is less clear. Caticha's entropic dynamics is, instead, quite explicit, as ##\rho##, as ##\phi## are epistemic. In particular, ##\phi## is, modulo a ##\ln \rho## term, simply the entropy ##S(q)## of the rest of the world given our knowledge about the rest of the world and the particular configuration of the system.

The weakest place of the realist interpretations are the zeros of the wave function in the configuration space representation (in other representations they may exist), because for those zeros there exists no phase function, and if there are at least some zeros, the phase cannot be a global function. This is the Wallstrom objection. The radical solution is to introduce the additional condition that no such zeros exist. If there are none initially, they will not appear during the evolution, thanks to the action of the quantum potential. Wallstrom argued that it is not viable, but he had in mind only the particle configuration space, and in particular the eigenstates of the angular momentum operator, which have zeros, and where even small distortions will have zeros. Schmelzer argues that this may be a problem for particle ontology, but that the situation looks much more harmless in the field ontology. (While for QT the choice of the configuration space makes no difference, for this additional condition the difference becomes important.)

Ref:
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. A44:225303, arxiv:1005.2357

Madelung, E. (1926). Quantentheorie in hydrodynamischer Form. Z Phys 40, 322-326

Wallstrom, T.C. (1994). Inequivalence Between the Schrödinger Equation and the Madelung Hydrodynamic Equations, Phys. Rev. A 49, 1613-1617

Schmelzer, I. (2019). The Wallstrom objection as a possibility to augment quantum theory. arxiv:1905.03075

mattt