One theory to the tune of another

My second favourite type of question in physics, after ‘what’s the simplest non-trivial example of this thing?’, is probably ‘how can I write these two things in the same formalism, so that the differences stand out more clearly?’

This may look like an odd choice, given that all I ever do here is grumble about how crap I am at picking up new formal techniques. But actually that’s part of why I like it!

Writing two theories in the same language is like putting two similar transparencies on top of each other, and holding them up to the light. Suddenly the genuine conceptual differences pop out visibly, freed from the distraction of all the tedious extraneous machinery that surrounds them.

Or at least that’s always the hope – it’s actually pretty hard work to do this.

There are two maps between classical and quantum physics that I’m interested in learning, and should probably have included in my crackpot grand plan. (I guess they can be shoved into the quantum foundations grab bag.)

One is the phase space reformulation of quantum mechanics. This is sort of a standard technique, but I still managed to avoid hearing about it until quite recently. Some subfields apparently use it a lot, but you’re unlikely to see it in any standard quantum course. It also has a weird lack of decent introductory texts. I met someone at the workshop I went to who uses it in their research and asked what I should read, and he just looked pained and said ‘My thesis, maybe? When I write it?’ So learning it may not be especially fun.

It looks really interesting though! You can dump all the operators and use something that looks very like a normal probability distribution, so the parallels with classical statistical mechanics are much more explicit. There are obviously differences – this distribution can be negative, for a start. (It’s known as a quasidistribution.) Ideally, I’d like to be able to hold them both up to the light and see exactly where all the differences are.

It’s less well known that you can also do classical mechanics on Hilbert space! It’s called Koopman – von Neumann theory. If you ever thought ‘what classical mechanics is really missing is a load of complex wavefunctions on configuration space’, then this is the formalism for you.

In this case, I ought to be luckier with the notes, because Frank Wilczek wrote some a couple of years ago.

I’m not so clear on exactly what this thing is and what I’d get out of learning it, but the novelty value of a Born rule in classical mechanics is high enough that I can’t resist giving it a go. And I’d have a new pair of formalisms to hold up to the light.

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2 thoughts on “One theory to the tune of another

  1. I’ve been cruising along this same trajectory, and I can recommend Structure and Interpretation of Quantum Mechanics (RIG Hugues). For me, it has mostly obviated the interest I initially had in the phase space formulation by describing very exactly what QM is, as a mathematical theory. Once the theory is demystified some, the interest in a direct analogy to classical mechanics appears less like a fruitful path towards a deeper understanding.

    As I’m sure you know, there is a direct transformation between phase space and regular QM. The former isn’t quite as obscure as you make it out to be – we covered phase space formulation briefly in both undergraduate and graduate quantum mechanics. As I recall, it was around the discussion about coherent states (naturally).

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    1. Thanks very much for the book recommendation, I’ve now had time to look it up and it looks good.

      > For me, it has mostly obviated the interest I initially had in the phase space formulation by describing very exactly what QM is, as a mathematical theory.

      Any chance of expanding on what you mean by ‘describing very exactly what QM is’? Is there a particular formalism, say, that made things clear to you? Why does the phase space formulation now look uninteresting?

      > The former isn’t quite as obscure as you make it out to be

      Yeah, this makes sense! I genuinely never came across it though, and this guy I talked to was genuinely despairing about decent references (do you know any good ones?). I had a bit of an odd physics education (maths degree first, then also did my phd in the maths department even though it was very much physics), so sometimes I have these weird gaps. I had at least *heard* of Wigner-Weyl transforms though so I suppose I must have known it can’t be that obscure.

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