On The Classification Of Hidden Variable Models

On the classification of hidden variable models

Since I already posted several entries about Bohmian Mechanics, it's time to have a look on the classification of hidden variable theories in general. Though in the following we adopt the dogma of the wave function being a representative of physical reality (which can be denied of course), I hope you will get a more differentiated impression of possible approaches to quantum theory. The classification presented was developed by Nicholas Harrigan and Robert Spekkens [1]. Here we will reduce ourselfes to a simplified and non-mathematical overview.

What are the models of interest?

The categorization concerns ontological models. This means we assume that there is an ontic state of the physical system, denoted by λ, which describes the physical properties of the system (remember, this is a huge assumption that can be neglected completely in other approaches of quantum theory.) Then, a measurement reveals something about the ontic state (remember, the measurement does not create new facts in this view) which existed prior to the measurement. Our task is to find out how the ontic state λ is related to the wave function ψ in a given model. It's important to remark that any ontological model needs to reproduce the empirical results of textbook quantum theory - the Born rule. p=|ψ|^2

ψ-complete, ψ-supplemented and ψ-epistemic

The first class, the ψ-complete models are the most straightforward: Here we have a one-to-one-relation between the ontic state λ and the wave function ψ. A complete description of reality is provided by the wave function - that's it. Figure A shows the one dimensional ontic state space represented by ψ . A line which is basically the same as λ.

ψ-supplemented models have additional structure that completes the wave function. This structure can be regarded as the well-known 'hidden variable' since you cannot measure their values in an experimental setup. Figure B shows the ontic state space of such a model: The ontic state is not only provided by ψ, but also by the hidden variable ω. The ontic state space has therefore an additional dimension.

A ψ-epistemic model reduces the wave function ψ to be of epistemic character - it is not an element of the ontic state space (therefore the loop outside the one-dimensional state space denoted by λ in figure C).

Given these classes you can define two dichotomic classifications: ψ-complete v.s. ψ-incomplete and ψ-ontic v.s. ψ-epistemic. Those properties and the three classes are summarized in the following figure (Ask yourself, why there is no model that is ψ-complete and ψ-epistemic simultaneously!)

To which class does Bohmian Mechanics belong?

The ontology of Bohmian Mechanics is clear: The wave function ψ is part of the physical reality, though a complete description can only be achieved with additional variables: the positions of point particles. Therefore Bohmian mechanics falls in the class of ψ-supplemented ontological models.

Finally it's crucial to mention what all of these models have in common: their ontic character requires the acceptance of non-locality. A result which is of such fundamental character that it is not tolerated by the entirety of the physics community - therefore you can also find a lot of attempts of restoring locality in non-ontic-models. More on this will follow in the future.

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[1] Einstein, incompleteness, and the epistemic view of quantum states - Harrigan, Spekkens