For a first slogan (see here for slogan zero) I’ve chosen:
Quantum theory is representation theory.
One aspect of what I’m referring to is explained in detail in chapter 14 of these notes. Whenever you have a classical phase space (symplectic manifold to mathematicians), functions on the phase space give an infinite dimensional Lie algebra, with Poisson bracket the Lie bracket. Dirac’s basic insight about quantization (“Poisson bracket goes to commutators”) was just that a quantum theory is supposed to be a unitary representation of this Lie algebra.
For a general symplectic manifold, how to produce such a representation is a complicated story (see the theory of “geometric quantization”). For a finite-dimensional linear phase space, the story is given in detail in the notes: it turns out that there’s only one interesting irreducible representation (Stone-von Neumann theorem), it’s determined by how you quantize linear functions, and you can’t extend it to functions beyond quadratic ones (Groenewold-van Hove no-go theorem). This is the basic story of canonical quantization.
For the infinite-dimensional linear phase spaces of quantum field theory, Stone-von Neumann is no longer true, and the fact that knowing the operator commutation relations no longer determines the state space is one source of the much greater complexity of QFT.
Something that isn’t covered in the notes is how to go the other way: given a unitary representation, how do you get a symplectic manifold? This is part of the still somewhat mysterious “orbit method” story, which associates co-adjoint orbits to representations. The center of the universal enveloping algebra (the Casimir operators) acts as specific scalars on an irreducible representation. Going from the universal enveloping algebra to the polynomial algebra on the Lie algebra, fixing these scalars fixes the orbit.
Note that slogan one is somewhat in tension with slogan zero, since it claims that classical physics is basically about a Lie algebra (with quantum physics a representation of the Lie algebra). From the slogan zero point of view of classical physics as hard to understand emergent behavior from quantum physics, there seems no reason for the tight link between classical phase spaces and representations given by the orbit method.
For me, one aspect of the significance of this slogan is that it makes me suspicious of all attempts to derive quantum mechanics from some other supposedly more fundamental theory (see for instance here). In our modern understanding of mathematics, Lie groups and their representations are unifying, fundamental objects that occur throughout different parts of the subject. Absent some dramatic experimental evidence, the claim that quantum mechanics needs to be understood in terms of some very different objects and concepts seems to me plausible only if such concepts are as mathematically deep and powerful as Lie groups and their representations.
For more about this, wait for the next slogan, which I’ll try and write about next week, when I’ll be visiting the Bay area, partly on vacation, but partly to learn some more mathematics.
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