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| Richard Feynman |
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.
