Hilbert Space: Reply to James

I wanted a bit more space here, since you guys are teaching me a lot.

James wrote:

The most common states in your Hilbert spaces don't actually live there: such as position and momentum states. Rigged HS combines a nice space (physically realistic - nice and smooth), with your HS, and the nasty (generalised functions usually) dual of the nice guys.


James, I don’t dispute your mathematical point, and perhaps I misunderstood it, but when you said that position and momentum states don’t live in the Hilbert space, I assumed that you meant that it was position and momentum eigenstates that didn’t live in the HS. My point was that real physical systems can’t be prepared in position or momentum eigenstates (i.e., with infinitely precise position or momentum) – they are always superpositions with some uncertainty in both position and momentum. Those superpositions, I think, can and do live in the Hilbert space – or am I totally confused on this point?

I hadn’t heard of a “rigged Hilbert space” previously, but if I understood Wikipedia on the subject, it’s pretty much just Hilbert space augmented by the necessary mathematical machinery to implement Dirac delta functions and permit us to do calculus with them. Is that right?

The "rigging" (or additional structure) is made necessary by the bad behavior (not square integrable) of the Dirac delta functions corresponding to position and momentum eigenvalues, but these can't be prepared anyway - a precise position requires infinite momentum uncertainty and vice versa.

Nice Article: http://en.wikipedia.org/wiki/Quantum_mechanics

James later wrote:

Classical mechanics is a fibre bundle: the base maifold R^1 (time) and the fibre space (E^3). Or we could go Lagrangian with relaivity where a n-D system becomes 2D-d, (or we could Legendre Trans his and go H) but the point is that none of this fancy geometric-ballet will make classical stuff quantum-mechanical...


Consider the following:

Hermann Weyl, discussing the Hamiltonian formulation of quantum mechanics (The Theory of Groups and Quantum Mechanics, pg. 95, Dover edition):

It is a universal trait of quantum theory to retain all the relations of classical physics; but whereas the latter interpreted these relations as conditions to which the values of physical quantities were subject to in all individual cases, the former interprets them as conditions on the quantities themselves, or rather on the Hermitian matrices which represent them...

I think for our purposes we would replace “matrices” with “operators in Hilbert space,” but he is making a very specific claim about the relation between the classical formulation and the quantum. My interpretation: the symplectic manifold of state vectors in Hamiltonian classical mechanics becomes a structure on the operators in Hilbert space – does that make mathematical sense? If I understand correctly, that's why Lie groups are fundamental to QM.

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