‘Hidden variable solutions’ are theories of fundamental physics that propose that particles (e.g. the electron) have inner structure. By comparison quantum mechanics (QM) and the Standard Model are based on the premise that a particle is a zero-dimensional (0D) point.

However it is known from experiment that particles have many properties, that make them different to other particles. For example the electron has negative charge, a certain mass, and a spin.

How does QM explain these? It doesn’t. Instead it proposes that these are ‘intrinsic’ properties. i.e. disembodied attributes. How might these properties arise then? According to QM, that is not a meaningful question to ask. The mathematics simply requires these attributes, and QM pointedly rejects the notion that there might be natural explanations at a deeper level of physics. Consequently the more extremist interpretations of QM would have us believe that reality is fundamentally mathematical, and that the probabilistic nature of superposition is simply all there is, that there is nothing deeper (Copenhagen interpretation). Hence an assumption that indeterminism is fundamental (Born & Heisenberg).

The hidden variable theories propose that there is a deeper inner structure to a particle. This physical structure then causes the properties of charge, mass, spin (etc). These internal structures are ‘hidden’ to external inspection, hence the name. According to this perspective, the probabilistic equations of QM are approximations to a deeper mechanics. Einstein believed that QM was fundamentally incomplete (EPR, 1935) and suspected the existence of hidden variables. However the hidden sector has historically failed to live up to expectations, the main difficulty being the sheer lack of specific solutions. It is all very well to say that in principle a particle might have inner structure, but to do anything useful one has to propose a specific internal design. That’s where things have failed to progress. There is no obvious hidden variable solution, and very few candidate designs.

To make things harder, one whole category of possibilities, the *local *hidden-variable designs, have been eliminated by the Bell type inequalities (Bell, 1964). The other category, the *non-local *hidden variable (NLHV) designs, is also under theoretical siege (Leggett, 2013)(Groblacher 2007) such that the remaining solution space is limited. As those authors have commented, if a NLHV solution exists at all, it must be counter-intuitive.

The candidate hidden-variable designs are as follow:

- The de-Broglie-Bohm theory (Bohm, 1966), also called the ‘pilot-wave’ theory. (See wikipedia). This has not done well, though there are still scientists who are progressing the idea and seeking to extend it. However in its present state it is not able to explain a diverse range of other fundamental phenomena, and hence is not yet as extensive as QM. There are many things it cannot explain. Some have even suggested it is merely another interpretation of QM, but I think that’s taking it a bit too far.
- Others?At the present time, if you search for NLHV solutions there is not much more than de-Broglie-Bohm. This goes to show how hard it has been to come up with candidates that can evade the Bell-type inequalities. Here are a few more ideas. These are mostly mathematical treatments rather than specific proposals for natural structures, so are difficult to interpret or apply, and their ontological explanatory power is weak, but they show that people are still chipping away at the problem in creative ways.
- Lokajícek’s hidden variable theory.
- Bach’s theory, here or here, with a critique here.
- Others?

- The Cordus theory (covered elsewhere on this site) can be considered a NLHV design. Unusually, it has been developed using a systems engineering design methodology, as opposed to the mathematical theory building that every other attempt has used. Consequently it is descriptive theory, rather than a mathematical formalism. Nonetheless it has good ontological explanatory power, arguably better than QM. All that weirdness of quantum mechanics gets washed away in natural explanations involving the deeper sub-components of the particle. We think it can explain, in an ontological sense, anything that quantum mechanics purports to explain (which is not always a lot). But it doesn’t do the quantitative formalism as well as QM, so is limited in that regard.

Most physicists believe that quantum mechanics is a complete description of reality, and only needs extending. They are generally dismissive of hidden variable designs. However NLHV designs are not dead, just incredibly hard to find. It’s not impossible that a new physics could be found in the hidden sector.

Dirk Pons, 29 March 2014