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The lecture notes appeared in the second volume of "Quantum Fields and Strings, a course for mathematicians". I would like to understand the derivation of (1.3), the 2-point correlation function:

$$ \int_{C_{\rm{per}}([0,L])} \phi(x_1) \phi(x_2) d\mu_G(\phi) = \text{tr } e^{-x_1 H} \phi e^{(x_2 - x_1)H} / \text{tr } e^{-LH} $$

which is listed as a problem, under the heading of Feynman kac's formula. Specifically I would like to know how exactly $\mu_G$ is defined and how it relates to Wiener measure.

On a more practical note, I would like to know what's a good source (if not here) for obtaining solutions to these sporadic exercises in high level lecture notes. Since time is limited, those of us who do not want to specialize in an area, but only want to get a taste of a subject, but do not want to sacrifice rigor, might find this kind of information very useful.

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1 Answer 1

As explained in Gawedzki's lectures, in the line preceding equation (3), the measure $d\mu_G$ differs from the Wiener measure by the multiplicative density $e{-\frac{\beta m^2}{2\pi}\int_0^L\phi(x)^2 dx}$. The exponent is proportional to the Potential energy of the Harmonic oscillator (This example treats the case of the harmonic oscillator). The equation of the correlation functions is sometimes called the Feynman-Kac-Nelson formula. It describes a probabilistic evaluation of the trace formula of the correlators in Hilbert space .This is similar to the usual Feynman-Kac formula which is a probabilistic description of the solution of the diffusion equation. A full proof of the Feynman-Kac-Nelson formula can be found in appendix D of Arai's paper

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Thanks for the great reference. I guess what I really would like to understand is how to use fourier transform to rewrite the left hand side: $$\int_{C_{\rm{per}[0,L]}} \phi(x_1) \phi(x_2) d\mu_(\phi)$$ Also are there good references that systematically explore the connection between Gaussian processes and Feynman path integrals accessible to probabilists? –  John Jiang Jun 7 '10 at 16:30
    
The answer is given in two comments: The following two lecture notes by Mikko Lane contain a heuristic description the use of Fourier analysis to evaluate the Harmonic oscillator path integral. The required two point function is given only as an exercise, but I think that it is clear how to apply the same method for its evaluation. physik.uni-bielefeld.de/~laine/thermal/lec01.pdf physik.uni-bielefeld.de/~laine/thermal/lec02.pdf –  David Bar Moshe Jun 8 '10 at 10:59
    
In the following lecture notes in math. by Albeverio, Høegh-Krohn, Mazzucchi, chapter 10.4.2 treats the path integrls within (Gaussian) white noise analysis. In addition in section 10.4.4 there is an approach based on Poisson processes. books.google.com/… –  David Bar Moshe Jun 8 '10 at 11:00
    
Thanks for the lecture notes! –  John Jiang Jun 9 '10 at 19:45

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