Path integrals and foundations of quantum mechanics

[Prathyush]

I don't know if was already pointed out in thread but PI is very natural to study non purturbative dynamics like instantons, solitons and phenomenon like vacuum tunneling . I am not aware of how to study these using canonical methods.

Yes you are definitely right.

I overstimated my ability to convey the reasoning. Sorry about that. I realize that we don't get much fruther here until I can be more explicit.

/Fredrik

 

[Demystifier]

I don't know if was already pointed out in thread but PI is very natural to study non purturbative dynamics like instantons, solitons and phenomenon like vacuum tunneling . I am not aware of how to study these using canonical methods.

In principle, canonical methods are certainly not restricted to perturbative methods, even if in practice it is not easy to include nonperturbative effects in a canonical framework.

But here we do not discuss which method is more practical. We discuss whether the two methods are equivalent IN PRINCIPLE.

 

[tom.stoer]

Instantons, theta-angle etc. can be studied in the canonical approach as well; the canonical approach is definately NOT restricted to perturbative treatment!

 

[muppet]

I've often wondered about this. A partial answer seems to be that we can regard the path integral as giving the transition between elements of the classical configuration space. The propagator for example is just the transition amplitude
\langle t_2,x_2| t_1,x_1 \rangle
To the extent that this is true, it's just a statement that QM borrows its conceptual framework from classical physics. What's not clear to me is

• how to incorporate the idea of a superposition of position states into our conceptual framework.
• How to formalise the idea of observables

One could try arguing as follows, although I'm throwing this out principally for discussion rather than as a claim that our theory is complete:

• In the absence of a position measurement, the indeterminacy of the time evolution of position implies that we should always begin with some "initial time smearing" of the original position, consistent with the complex exponential form of the contributions from each path. This should of course yield the wavefunction, which satisfies the Schroedinger equation and for which the propagator is a Green function.
• The question then would be how to justify the implementation of what we would usually think of as 'operators in the position basis' as tools by which we can extract information about observables. You could argue that energy and momentum generate translations in time and space, which would almost immediately identify the corresponding differential operators up to constant factors.

The hardest aspect of the state-vector formalism to incorporate in this framework seems to me to be spin. Classically, one can have "spin vectors" that identify some angular momentum intrinsic to a body as the translation-invariant part of of the total angular momentum; but how you could justify the introduction of spin-1/2 representations (without identifying the space of wavefunctions as a Hilbert space and rederiving the canonical framework) isn't clear to me at all.

 

[tom.stoer]

muppet, I think we agree that the hardest issue for the PI as a conceptual basis is to explain if and why something beyond the specific representation (usually position rep.) can work w/o referring to Hilbert states

 

[atyy]

What about http://arxiv.org/abs/1002.3917 ?

 

[tom.stoer]

What about http://arxiv.org/abs/1002.3917 ?

Seems that this doesn't solve the questions NOT related to the PI

 

[Demystifier]

What about http://arxiv.org/abs/1002.3917 ?

Interesting paper, but I don't see how is it related to the topic of this thread.