Can non-commutative geometry describe phase space with Planck-scale uncertainty?

[mhill]

the question is if we have a classical phase space (p,q) the idea is using Heisenberg's uncertainty could we generalize the usual 'geometry' to a non-commutative phase space ?

for example we could impose the conditions $$[ x_i , x_j ]= iL_p \hbar$$

where L_p means Planck's Energy scale and the same for the momentum $$[ p_i , p_j ]= iL_p \hbar$$.

if someone could provide a good and comprehensible introduction to Non-commutative geometry book and how is used in physics (with examples) thanks a lot.

 

[atyy]

Hmmm, I'm not quite sure what you're asking. If it's about Alain Connes's stuff, I know nothing about it. As for normal quantum mechanics, the state space is usually the Hibert Space spanned by the eigenvectors of the Hamiltonian. The closest thing to classical phase space is the Wigner distribution.