Question

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.

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