A question regarding tensor product of hilbert spaces

[azztech77]

So I've been reading Cohen-Tannoudji's "Quantum mechanics vol 1" and have understood the part that proves that the hilbert space of a 3-dimensional particle can be described/decomposed as a tensor product of hilbert spaces using position vectors (or analogously momentum vectors) in the x, y and z directions. This is really cool and neat. My question is: I know that x and p basis vectors cannot be mixed to form a complete system of orthogonal vectors making up a basis (this follows from the fact that they are the eigenvectors of the non-commuting observables/hermitian operators X and P) for the hilbert space. However, and this is just a curiosity - is there some other loosely ANALOGOUS way to write the hilbert space as some sort of (perhaps elaborate) tensorial product decomposition of x and p basis vectors to fully describe all the states? I'm guessing that if this were possible than the tensors would be anti-symmetric since the standard canonical relations apply. Just a thought - anyone know of anything like this?

 

[Bill_K]

Not exactly what you're asking, but -- in classical mechanics we do often use phase space, in which the axes are labeled by both x's and p's. There's a formulation of quantum mechanics which maps Hermitian operators in the Hilbert space into certain functions on phase space. This is called the Wigner-Weyl formalism. A good discussion of it can be found here.