Saturday, October 6, 2012

Fun with Feynman

Richard Feynman
Have you ever wondered what those diagrams in the above US commemorative postage stamp in honor of Richard Feynman   mean? Feynman Diagrams are used to compute the amazingly useful, but also notoriously
difficult  (among mathematicians at least!), Feynman integrals. Notorious because, among other things,  the integrals are badly divergent and  so far there is no rigorous mathematical definition of them! How is it possible that  a mathematically ill defined concept can accurately predict  the most precise measurements  that are done by mankind? Here we are talking about  an agreement to within ten parts in a billion with experiments! This is a  very interesting but  also very difficult question to which no one has been able to provide a satisfactory answer yet. But we won't deal with these issues in these talks.

While Feynman was mostly guided by  actual  physical  processes and saw the edges and vertices of his diagrams as paths of elementary particles  smashing with each other creating and annihilating new particles all the time,   it is possible to provided a purely mathematical justification for  these diagrams, by analogy with some finite dimensional integrals.

This year's Pizza Seminar starts with two talks where we probe some of the mathematical underpinnings  of Feynman Integrals. This story  starts with   the well  known Gaussian Integral.  You can see the mighty Gaussian lurking  in the background in  this old German 10 D Mark bill honoring Gauss (click on the image for better viewing).

Lord Kelvin, an eminent 19th century scientist,  is reputed  to have once written  the Gaussian  integral
$$ \int_{-\infty}^{\infty}e^{-x^2} dx = \sqrt{\pi}$$
on the blackboard, and then said  to his audience  "A mathematician is one to whom that is as obvious as that twice two makes four is to you. Liouville was a mathematician".

If Kelvin was alive today he would have perhaps used the following extension of the above integral, known as Feynman's Theorem,  to drive home his point:

$$ \frac{1}{Z_0} \int_{\mathbb{R}^n} e^{h S(x)}dx= \sum_{\gamma \in \Gamma} \frac{h^{\chi (\gamma)}}{|\text{Aut}\,\gamma |} w (\gamma) $$
The left hand side is the integral of an exponential type function while on the right  we have a sum over isomorphism classes of  all finite graphs! Using the coefficients of  the polynomial \( S(x) \),  to each graph  \( \gamma\) a number  \(w(\gamma)\),  its Feynman Weight,  is attached which appears on the right hand side. To get to know the other ingredients of this formula you can   attend the lectures!   It is remarkable that  this amazing result, which has many  applications in physics and maths,   is within the reach of an undergraduate math student.

What is required to understand these two lectures? For the first lecture just calculus, with a touch of class! and for the second:  calculus + very basic group theory + a lot of attention! In particular you don't need to know any physics to understand these lecture.

Lecture 1:   Wick's Theorem,  Tuesday, Oct. 16, MC 108, 4:30 PM. (Speaker: Masoud Khalkhali, UWO)
Abstract: Wick's theorem allows us to compute any exponential integral  by repeatedly differentiating a Gaussian function. It is a bridge that will take us from Gaussian integrals to Feynman integrals and their  evaluations in terms of graph sum, and of course to a delicious Pizza!

Lecture 2: Feynman's Theorem, Tuesday Oct. 23, MC 108, 4:30 PM.(Speaker: Travis Ens, UWO)
Abstract: Feynman's theorem, a fundamental mathematical tool in quantum field theory, provides a way to evaluate complicated integrals by summing over finite graphs. After exploring how to obtain this sum from an integral, we will reverse the correspondence and prove a famous result of Cayley, that the number of labelled trees with \(n \) vertices is \(n^{n-2}\), using known values of integrals.

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