Vectors, Hilbert Spaces, and Tensor Products

[Bashyboy]

If I ever say anything incorrect, please promptly correct me!

The state of a system in classical mechanics is specified by point in phase space, the point giving us the position and velocity at a given instance. Could we rephrase it by saying a vector in phase space specifies the system? If so, it would seem to make transition to QM slightly more natural.

My next question is, what is the motivation for thinking that a vector in a Hilbert space represents the state of a quantum? I would appreciate an explanation or a reference to some source that nicely answers this question.

My last question, related to the second, is, why is "The situation of two independent observers conducting measurements on a joint quantum system...usually modeled using a Hilbert space of tensor product form, each factor associated to one observer"? Again, is there motivation for this idea?

 

[vanhees71]

Take any quantum-mechanics textbook, e.g., Sakurai, Modern Quantum Mechanics. A good book will explain that it is actually a ray in Hilbert space that represents a (pure) state of the system. Most general is the representation of the system in terms of a statistical operator, which is a self-adjoint positive semidefinite operator of trace 1.

 

[andresB]

The state of a system in classical mechanics is specified by point in phase space, the point giving us the position and velocity at a given instance. Could we rephrase it by saying a vector in phase space specifies the system? If so, it would seem to make transition to QM slightly more natural.

Actually It is position and momentum. But in a sense yes, The state of a classical system can be represented by vector in an Hilbert space associated to the phase space

http://arxiv.org/abs/quant-ph/0301172

 

[Bashyboy]

The state of a classical system can be represented by vector in an Hilbert space associated to the phase space

http://arxiv.org/abs/quant-ph/0301172

@andresB You seem to be saying something slightly different than what I said in my question. Would you elaborate on the difference between a vector in phase space describing a classical system and a vector in an Hilbert space associated to the phase space?

 

[andresB]

In general, for the phase space you can't use vectors as in R^N (where there is an identification of a point with a vector) because the phase space is not necessarily an euclidean manifolds. I don't have enough mastery on the topic to give a clear picture but there are several good expositions like Arnold's mathematical methods of classical mechanics.