Amplitude paths and Markov processes

[wofsy]

In continuous Markov processes there is an idea of sample path. Starting at a point we follow the values of the process through time and find that almost surely the paths are continuous. In Brownian motion (Wiener process) the paths are crinkly curves, continuous paths of infinite variation.

The evolution of amplitudes seems to me to be a continuous Markov like process where complex amplitudes replace real probabilities. What are the sample paths for a free particle? Are they continuous? What is their variation? If these paths exist - why can't we define them as the paths of a quantum particle's state?

 

[mathman]

Your first paragraph is a discussion of mathematical concepts (Markov processes - brownian motion) which have physics applications, such as dust particles moving in air.

Your second paragraph seems to be a question of whether similar mathematical models can be used for quantum mechanical concepts. Personally I can't answer (I am a mathematician, not a physicist). However, it would be helpful if you could clarify your question, keeping in mind the difference between mathematics and physics.

 

[wofsy]

Your first paragraph is a discussion of mathematical concepts (Markov processes - brownian motion) which have physics applications, such as dust particles moving in air.

Your second paragraph seems to be a question of whether similar mathematical models can be used for quantum mechanical concepts. Personally I can't answer (I am a mathematician, not a physicist). However, it would be helpful if you could clarify your question, keeping in mind the difference between mathematics and physics.

my question was two fold part mathematical part physical.

Mathematical: Is there a notion of state path for a complex continuous Markov proces such as the state evolution of a free particle?

Physical: If so, do these paths have any physical meaning?

Since I posted this question I spoke with a physicist friend who said that in QM there really are no state paths as in the path of a dust particle in a fluid.,

On the other hand, it seems that Bohm's hidden variable theory reproduces the state process as paths of actual particles. so it seems that these paths can be given physical meaning.

I think Feynman may have some formulation interms of paths but I really don't know.

A comment on your cautionary suggestion that I keep in mind the difference between mathematics and physics. Often mathematical ideas have physical corrolaries whether or not they are directly related to a currently known theory. Chern Simons invariants,the theory of connections and the idea of a Riemannian manifold come to mind.I was asking about a mathematical feature of a known theory which has even a better chance of having a physical corollary.

I also have a philosophical interest in the structure of QM as a theory but this is a physics thread so i won't go into it.