Feynman integral over histories= over paths?

[nomadreid]

In computing the integral over past histories, does one restrict oneself to continuous paths, or does one take all possible combinations of points between the beginning and the end? If one sticks to the continuous paths, what justification is there in that?

 

[arkajad]

Continuous, perhaps even with some conditions on derivatives that should exist almost everywhere. See for instance Albeverio and Hogh-Krohn, "Mathematical Theory of Feynman Path Integrals".

 

[nomadreid]

Thank you, arkajad, for the answer and the recommendation.

Since I presently do not have access to a scientific library, and the book does not seem to be easily downloadable, and rather expensive to order, I cannot follow up on your book recommendation. If you have any recommendations from the Internet, I would be happy to follow them up.

 

[arkajad]

For a starter you can check this: http://www.scholarpedia.org/article/Path_integral" , but there are several different approaches. The class of admissible paths depends on what you want to integrate. For instance you will find a sentence like this:

"This has the effect of restricting the integration to paths that satisfy a Hölder condition of order 3/2 and are thus differentiable, in such a way that expectations values with dq/dt are defined."

 

[nomadreid]

Thank you, arkajad. Upon a first reading the link appears quite useful, and I shall be working through it.

 

[lucid]

For a starter you can check this: http://www.scholarpedia.org/article/Path_integral" , but there are several different approaches. The class of admissible paths depends on what you want to integrate. For instance you will find a sentence like this:

"This has the effect of restricting the integration to paths that satisfy a Hölder condition of order 3/2 and are thus differentiable, in such a way that expectations values with dq/dt are defined."

there is no reason paths have to be differentiable. in fact the stochastic approach to QM gives inherently non-differentiable paths.

a path which is not continuous would mean a particle that starts at some point and disappears into the vacuum an then reappears and then makes it out to some final endpoint.

the book Path Integral Methods in QFT by Rivers explains this stuff. it is easily downloadable.

 

[nomadreid]

Thanks for the lucid account, lucid. The idea of making histories continuous by definition is an interesting idea,although I need to do some more reading to understand how the infinite number of allowable quantum discontinuities in a path are factored into the final integral. As far as that book being easily downloadable, I suppose if I sign up for a paid downloading site such as usenet or give one of these "14-day trials" my credit card number (something I am loathe to do), I might be able to download it, but I could not find any easy downloading otherwise.