If I understand correctly--a big if--the path integration method, at least when applied to plain old QM, is described as (1) every possible path the particle could take is assigned an amplitude, (2) sum up (integrate over) these amplitudes for all possible paths.(adsbygoogle = window.adsbygoogle || []).push({});

The problem I have with this is when you look at the actual math, there seem to me be contributions from discontinuous paths. And I don't just mean pointwise continuous but radically discontinuous.

To actually derive the path integral expression, you slice up the time between the initial position and final position into N time intervals, and then integrate over all possible positions at each time interval. As you increase the number of slices, you increase the number of position integrations, until the number of position integrations supposedly "goes over" into a continuum limit as N goes to infinity (which doesn't sound all that convincing to me).

But for each N, the integrations over position are all independent of each other. Of course, for any wild set of N positions, we can always connect them with a continuous function q(t). But as we increase N, we increase the number of positions being integrated over, which necessarily include contributions which are wilder and wilder. I can't see any reason why they would all settle down into continuous functions as N goes to infinity.

I am not just talking about very wild continuous paths. What I mean is if we look at two close time-slices LaTeX Code: q_j,t_j and LaTeX Code: q_{j+1},t_{j}+\\delta t and we integrate over LaTeX Code: q_j and LaTeX Code: q_{j+1} independently, over the whole real line, then there is a lot of contributions from amplitudes associated with large values of LaTeX Code: |q_{j+1}-q_j| . That's not going to settle down into anything like continuous paths. LaTeX Code: |q_{j+1}-q_j| is not going to get any smaller as LaTeX Code: \\delta t goes to zero.

However, the author I am reading (MacKenzie http://xxx.lanl.gov/abs/quant-ph/0004090 ) uses arguments based on variations in the action integral from the particle's classical path (cf section 2.2.2 in which MacKenzie calculates the harmonic oscillator propagator) which, I would think, would only work if we are restricted to nice continous variations in the path.

I don't have any problem with the strict definition of the path integral in terms of a limit. But after that the conceptual exposition sounds pretty shaky.

What's my question? I guess it is, do I understand the above correctly?

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# Path integrals and supposed sum over paths

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