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Not even wrong blog
Not even wrong blog





The answer is exciting only in that it’s surprisingly dull: when we geometrically quantize we get back the Hilbert space So a space of quantum states is an example of a space of classical states-equipped with precisely all the complicated extra structure that lets us geometrically quantize it!Īt this point, if you don’t already know the answer, you should be asking: and what do we get when we geometrically quantize it?

not even wrong blog

To reach this realization, we must admit that quantum states are not really vectors in a Hilbert space from a certain point of view they are really 1-dimensonal subspaces of a Hilbert space, so the set of quantum states I’m talking about is the projective space But this projective space, at least when it’s finite-dimensional, turns out to be the simplest example of that complicated thing I mentioned: a Kähler manifold equipped with a holomorphic hermitian line bundle whose curvature is the imaginary part of the Kähler structure! It also reveals that a space of quantum states can be seen as a space of classical states! Geometric quantization is not just a procedure for converting a space of classical states into a space of quantum states. Victoria-Monge, Mathematical foundations of geometric quantization.īut there’s a flip side to this story which indicates that something big and mysterious is going on.

  • Matthias Blau, Symplectic geometry and geometric quantization.
  • Unfortunately I don’t have time, so try these:

    not even wrong blog

    That’s quite a mouthful-but it makes for such a nice story that I’d love to write a bunch of blog articles explaining it with lots of examples. Then the space of holomorphic sections of that line bundle gives the Hilbert space we seek. We learn that it works much better to start with a Kähler manifold equipped with a holomorphic hermitian line bundle with a connection whose curvature is the imaginary part of the Kähler structure. We soon learn that this procedure requires additional data as its input: a symplectic manifold is not enough. As beginners, we start by thinking of geometric quantization as a procedure for taking a symplectic manifold and constructing a Hilbert space: that is, taking a space of classical states and contructing the corresponding space of quantum states. I want to play my cards fairly close to my chest, because there are some interesting ideas I haven’t fully explored yet… but still, there are also plenty of ‘well-known’ clues that I can afford to explain. I feel it holds some lessons about the relation between classical and quantum mechanics that we haven’t fully absorbed yet.

    not even wrong blog

    I can’t help thinking about geometric quantization.







    Not even wrong blog