# Geometric Quantum Mechanics

1. May 3, 2010

I ran across a paper today that I found rather interesting. The idea is that "there exists a geometry description other than the conventional description in a Hilbert space...". The gist of the paper is that the quantum phase space can be viewed as a complex projective space if the dimensions of the Hilbert space is finite. The paper can be found here:

http://arxiv.org/ftp/math-ph/papers/0701/0701011.pdf

There is another paper I found that makes roughly the same argument, stating that "the manifold of pure quantum states is a complex projective space with the unitary-invariant geometry of Fubini and Study...the detailed physical characteristics of a given quantum system can be represented by specific geometrical features that are selected and preferentially identified in this complex manifold."

The first paper gives an example with the superposition principle. The second paper gives examples of geometrical features that arise in the state spaces for spin -1/2 systems. There are many other papers that talk about this idea, which could be called geometric QM. The argument that there is a corresponding relation between quantum states and points of complex projective space that is a sort of alternative to the conventional description of quantum mechanics in a Hilbert space. What do you think of this idea? Do you think it is valid?

2. May 3, 2010

### genneth

It is purely an equivalent mathematical description, nothing physical in it. As to whether it provides deeper insights or cleaner calculations depends on how familiar users are with it. As such, it is clearly correct, but of questionable use to a physicist. Mathematicians are, of course, not constrained by usefulness.

3. May 3, 2010