Is the path integral well defined

[pellman]

From a QM (not QFT) context, one particle, we start with a hamiltonian H(q,p) and develop something like

$$\langle q'',T|q',0\rangle \approx \int e^{-i\sum_{l=0}^{N}[H(q_l,p_l)-p_l\dot{q}_l]\delta t}\prod_{j=1}^N{dq_j}\prod_{j=0}^N{\frac{dp_k}{2\pi}}$$

where $$\delta t = T/(N+1)$$ and $$\dot{q}_j \equiv (q_{j+1}-q_j)/\delta t$$ and where the approximation becomes equality for small delta-t or, equivalently, large N. In that case we write

$$e^{-i\sum_{l=0}^{N}[H(q_l,p_l)-p_l\dot{q}_l]\delta t}\rightarrow e^{-i\int_0^T{[H(q(t),p(t))-p(t)\dot{q}(t)}]dt}$$

Now generally when we look at a Riemann sum, we are dealing with a given function $$f(x)$$ and looking at a sum over $$f(x_i)\delta x$$ for finer and finer slices $$\delta x$$.

However, here the individual $$q_l$$'s are not part of a given function $$q(t)$$ nor do I see any reason to expect them to approach anything like an integrable function $$q(t)$$. Indeed, since each $$q_l$$ is a variable of integration over the entire real line, I can't see them settling down to anything like a piece-wise continuous function for large N.

For any discrete set of time slices for a free particle, given that a particle at $$q_j$$ at time $$t_j$$, then there is a non-zero probability that it be found at any other value of $$q$$ at time $$t_j + \delta t$$. The smaller we make $$\delta t$$, the more jagged most of the potential paths become. I don't see how they approach something we can integrate over.

All of my QFT books make no mention of this. Can someone recommend a link or text which covers this limit with more rigor?

 

[Haelfix]

I believe the mathematicians have made it well defined for nonrelativistic quantum mechanics through a variety of means (starting with the Wiener measure and working from there). The problem for relativistic QFT, amongst other things is the details of the analytic continuation from the Euclidean to the Minkowski PI (does it exist or not)

 

[strangerep]

[...] I don't see how they approach something we can integrate over.
All of my QFT books make no mention of this. Can someone recommend
a link or text which covers this limit with more rigor?

I'll add a little on top of what Haelfix said.

The deeper problem is this:

http://en.wikipedia.org/wiki/There_is_no_infinite-dimensional_Lebesgue_measure

It can be evaded by introducing a Wiener (or Gaussian) measure instead
of the usual translation-invariant Lebesgue measure.

In typical physical situations, the free part of the Hamiltonian is quadratic
and (by analytically-continuing to Euclidean space), one can use the free
part of the Hamiltonian to give you a de-facto Gaussian measure. (That's
what gets done in many intro texts of Feynman path integrals - they
get re-expressed as Gaussian integrals.)

If you really want a rigorous treatment, try the text of Glimm & Jaffe,
"Quantum Physics - A Functional Integral Point of View".
Warning: it's not an easy read and assumes quite a bit of pure math
background in functional analysis and abstract algebra.

 

[ExactlySolved]

I recommend "Quantum Physics: A Functional Integral Point of View" by Jaffe and Glibb (chapter 3 gives a rigorous treatment of the Feynman-Kac path integral formula for the propagator of a time independent Hamiltonian with a potential that depends on space but not velocity) or another book that is entirely about path integrals and goes at a slower pace (after all its a whole book) is "The Feynman Integral and Feynman's Operator Calculus" by Johnson and Lapidus. Read both if you can, Jaffe is more aimed towards pros while Johnson has a lot of historical anecdotal standard-Feynman material, as a bonus Ed Witten cites Jaffe's book as background material in rigorous field theory in the description of the Yang-Mills mass gap millenium prize problem.

 

[pellman]

Two votes for Quantum Physics: A Functional Integral Point of View. Thank you. I will check it out.

Thank the gods for inter-library loan.

 

[Haelfix]

You might want to Spires or google for Kleinerts work on path integrals in addition to checking the suggestions above. He's a physicist but he's probably one of the worlds foremost experts on the PI.

His treatment might lack mathematical sophistication, but he does give very strong, technically precise physics based arguments that are rather hard to refute. He fought quite a war with some mathematicians for decades on various technicalities of existence.

 

[pellman]

You might want to Spires or google for Kleinerts work on path integrals

Thanks!

 

[George Jones]

pellman, you might be interested in Chapter 8

http://books.google.ca/books?id=3JX...=&as_maxm_is=0&as_maxy_is=&as_brr=0#PPA257,M1

from the book Quantum Field Theory: A Tourist Guide for Mathematicians. Unfortunately, section 8.5 does not appear to be available for preview.

Actually, you might be interested in the whole book. I think it's an amazing book that is a very useful addition to set of quantum field theory texts.

 

[kexue]

However, here the individual LaTeX Code: q_l 's are not part of a given function LaTeX Code: q(t) nor do I see any reason to expect them to approach anything like an integrable function LaTeX Code: q(t) . Indeed, since each LaTeX Code: q_l is a variable of integration over the entire real line, I can't see them settling down to anything like a piece-wise continuous function for large N.

I really do not see your problem here. You can get any continuous path you like by this procedure.

For any discrete set of time slices for a free particle, given that a particle at LaTeX Code: q_j at time LaTeX Code: t_j , then there is a non-zero probability that it be found at any other value of LaTeX Code: q at time LaTeX Code: t_j + \\delta t . The smaller we make LaTeX Code: \\delta t , the more jagged most of the potential paths become. I don't see how they approach something we can integrate over.

Probabilities? Each path, including the classical path,gets the same weight, a number of unit modulus. Then sum all paths over.

 

[Hans de Vries]

You might want to Spires or google for Kleinerts work on path integrals in addition to checking the suggestions above. He's a physicist but he's probably one of the worlds foremost experts on the PI.

Thanks!

His book is partly online here:

http://users.physik.fu-berlin.de/~kleinert/public_html/kleiner_reb3/psfiles/


Regards, Hans

 

[pellman]

I really do not see your problem here. You can get any continuous path you like by this procedure.

In the presentations with which I am familiiar the limit

$$\sum_{l=0}^{N}[H(q_l,p_l)-p_l\dot{q}_l]\delta t\rightarrow \int_0^T{[H(q(t),p(t))-p(t)\dot{q}(t)}]dt$$

is taken independently of the integrations over the q's and p's. All the time integration limit depends on is the limit of small delta-t. Ok. Imagine this limiting procedure for a moment. Start with N time slices with a q to go with each time slice. You can imagine that there is some (integrable) q(t) connecting these dots, but the values of this q(t) between the time slices are arbitrary.

Let us do the limit in this fashion: N, 2N, 4N, etc. so that with each increment we fill in a value of q for some t between each the previous step's slices. But each of these q's can take on any value of the real line. Any value. Every single one of the them. For N slices our imaginary q(t) with which we connect the dots might look smooth, but then with 2N values we find the between-qs are all over the place. So we modify our imaginary q(t) to connect those dots. We go to 4N, and the new q's are again all over the place, having nothing to do with the imaginary q(t) we used to connect the 2N dots. So again we modify our q(t) to connect 4N dots. With each increase of time-slices it gets worse and worse, not smoother and smoother.

The Riemann sum can only approach an integral if the function being summed is integrable.

 

[pellman]

George and Hans, those texts both look very promising. Thank you very much.

 

[kexue]

The path integral is an infinite product of integrals, each of these integral is taken over all q, each interval is taking at one point in time t.

Before you go infinite, when you slice time in intervals you already integrate over all q (at each interval of time) and then take the product of all those integrals.

By making the time slices narrower (by increasing N) and finally taking N to infinity, you approximating more and more all of the infinite paths, which have been only approximated discretely, while N was small.

 

[zetafunction]

Could, one define these kind of Path integral as infinite dimensional distributions ?

we are dealing with some kind of infinite-dimensional Fourier transform , many Fourier integrals exists of course, but only in the sense of distribution theory, so why not extend this definition to deal with infinite dimensional integrals.