Are the Bohmian and Copenhagen interpretations of QM isomorphic?

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I'm writing an essay comparing Bohmian Mechanics to the standard Copenhagen interpretation and came across the notion that Bohmian mechanics implies a fundamental epistemic uncertainty to how we can measure reality, despite a fully deterministic ontology.

Although Copenhagen doesn't answer any ontological questions, can't we then view these two theories as "isomorphic" in some sense of the word, due to this "fundamental epistemic uncertainty"? If there's no difference between our measurements of the world under either interpretation, what's the difference?

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In Bohm's book Wholeness and the Implicate Order (which you can download for free here http://www.gci.org.uk/Documents/DavidBohm-WholenessAndTheImplicateOrder.pdf ) in chapter 8 "Steps Toward a More Detailed Theory of Hidden Variables" he puts forward a program for developing a theory of global hidden variables. His program aims at finding a sub-quantum level of knowledge that would account for the measurements we make, all of which are at the classical level.

I think the big difference between the two interpretations is that CI asserts that it is not possible in principle to find such a deeper level of knowledge. Bohm considers the possibility that there may be a way to account for the indeterminacy of QM by way of a theory of global hidden variables that would yield a verifiable understanding of an underlying relationship between measurements.

In as much as theory determines the kinds of experiments performed, which measurements we make depends on our theory. So, if the same measurements are made we would get the same results. However, given the different theories, we would not be making the same measurements.

As a case in point of theory leading to measurements particular to Bohm see this quote from the Wikipedia page for D. Bohm https://en.wikipedia.org/wiki/David_Bohm

In 1959, Bohm and Aharonov discovered the Aharonov–Bohm effect, showing how a magnetic field could affect a region of space in which the field had been shielded, but its vector potential did not vanish there. That showed for the first time that the magnetic vector potential, hitherto a mathematical convenience, could have real physical (quantum) effects.

There is on that page a schematic of an experiment that later confirmed his theory.

A very good explanation of the significance of theory to measurement (regarding entropy, but widely applicable) is given by E.T. Jaynes in The Gibbs Paradox http://www.damtp.cam.ac.uk/user/tong/statphys/jaynes.pdf section 9, bottom of page 13:

"...the theoretical explanation would predict generalizations; the range of possible nonequilibrium conditions is many orders of magnitude greater than that of equilibrium conditions, so if new reproducible connections exist they would be almost impossible to find without the guidance of a theory that tells the experimenter where to look."

Good point about the kind of measurements we would make depends on the theory we are trying to test. I formatted your answer into a few paragraphs for readability. You may roll this back if you feel it is inappropriate. – Frank Hubeny – 2018-10-21T15:22:25.693

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Wikipedia describes isomorphism as the following:

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism. Two mathematical objects are isomorphic if an isomorphism exists between them. An automorphism is an isomorphism whose source and target coincide. The interest of isomorphisms lies in the fact that two isomorphic objects cannot be distinguished by using only the properties used to define morphisms; thus isomorphic objects may be considered the same as long as one considers only these properties and their consequences.

To talk about an isomorphism one would need a reversible homomorphism, perhaps metaphorical, between the Bohmian interpretation and the Copenhagen interpretation. Such a morphism may be constructed to preserve the "measurements of the world" as the OP suggests. The isomorphism would then show that these two interpretations are the same with respect to these measurements.

The isomorphism would remove the differences between these two interpretations focusing on what they have in common.

However, it is the differences between these two interpretations that may be what is most interesting about them, not their similarities. An isomorphism would ignore those differences.

The OP mentioned some differences between these two interpretations. The "Bohmian mechanics implies a fundamental epistemic uncertainty to how we can measure reality, despite a fully deterministic ontology". The "Copenhagen doesn't answer any ontological questions". For a more detailed survey of the differences between the various interpretations see the Wikipedia article, "Interpretations of quantum mechanics".

Reference

Wikipedia, "Interpretations of quantum mechanics" https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics

Wikipedia, "Isomorphism" https://en.wikipedia.org/wiki/Isomorphism

I would suggest looking up also https://en.wikipedia.org/wiki/Quantum_non-equilibrium which seems to be a straight negative answer.

– sand1 – 2018-10-20T21:26:09.977

@sand1 Good point. Given this, there likely wouldn't be an isomorphism preserving measurement results although so far it does not seem there have been any differences between the measurement results. – Frank Hubeny – 2018-10-20T21:34:01.190

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Generally, When I read physics and ontology on the same paragraph I slightly cringe. Not that it is bad to use it, but the implicit mathematization of ontology is what takes me back.

Let me tell you why, Quantum Mechanics is undeterministic solely due to the fact that the solution to Schrodinger's equation is a wave-distribution. That is, we get a probably spread of postional coordinates for a particle. On the other hand, for Pilot-Wave we have a wave fuction, which deterministically evolves. That is, given the initial determined input one can "theoretically" get a fully deterministic output for it; however, the uncertainty in Pilot-Wave creeps in with the initial input because of heisenberg's uncertainty principle. So, inessence, Pilot-wave is as deterministic as it could get, but the only problem is the initial measurement. That is, can we get a precise location for a particles? According to expirements and heisenberg, we can not. As for Quantum Mechanics, the actual developement of a system is essentially undeterministic. That is, it is dynamically evolving-until you get a measurement and that is where the wave fuction collapses and we get an acrual value. So this probabilistic evolution of schrodinger's equation is what gave rise to these whimsical interpretations of Quantum Mechanics like Copenhagen Interpretation and Multi World Interpretation. As it is quite obvious, the only reason we have these vastly different intepretations of quantum phenomenon is the mathematics.