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What would be a recommendable path for me to study Hairer's theory of regularity structure?

+ 3 like - 0 dislike
306 views

Interest originally sparked by the discussion Discussion about Hairer's existence theory for stochastic differential equations (BTW, the discussion happpened days before Hairer was awarded the Fields prize!), and given that I'm still finding Hairer's own introduction to the subject "Introduction to regularity structures" incomprehensible (for one, I have zero knowledge on stochastic PDE), I'd like to ask in my situation what would be a recommendable learning path, to the end of understanding its relevance to QFT/renormalization?

Here's some of my relevant background:

1. Quite familiar with statistical mechanics/QFT/renormalization in the physicist style (but barely knows any operator product expansion technique), near zero exposure to constructive QFT in mathematics literatures.

2. Have some brief exposure to measure theory and functional analysis, i.e., I studied extensively the terse little book by Komolgorov and Fomin, very little distribution theory was included in there. 

3. Zero knowledge on probability/stochastic theories.

Again, life might be too short to fully master Hairer's theory as a physicist, for now I can only wish to study it to the extent of understanding its relevance to QFT.  Any advice or material recommendation is very welcomed.

And Merry Chirstmas to everyone!

asked Dec 24, 2015 in Resources and References by Jia Yiyang (2,465 points) [ no revision ]

If you want to learn Distribution theory with some connection to PDE then I have the following suggestions:

1. To learn Distribution theory from Fourier analysis the best guide is Trigonometric Series by Zygmund (I was recommended this book by a lecturer in a course in Harmonic Analysis for mathematicians that I took, he said that it covers much more than what was covered in the course, and in the course we covered introduction to distribution theory amognst other topics).

2. The book called Generalized Functions and PDE by Avner Friedman, I started reading this book also but stopped since I started reading other books on other topics.

Each book has its strengths and weaknesses.

I don't know how much these books are any use for your ultimate goal, but that's life with mathematics, a lot of stuff to master. Mathematicians should live long and prosper to appreciate these theories...

@MathematicalPhysicist,

Thanks. I'm feeling more and more that the future of math and physics depends on immortality technology....

:-D

You must heard of Shinchi Mochizuki's proof of ABC conjecture, what he wrote is a book for a proof of this claim.

I wonder if that what it takes to solve also the Millenium problems.

1 Answer

+ 3 like - 0 dislike

 Hairer's work is about classical stochastic partial differential equations. These describe random versions of classical field theories; e.g., stochastic hydromechanics for studying turbulence. To learn stochastic differential equations from a physicists point of view you should start with the 1D case - ordinary SDE. These are well-treated in books by Gardiner, van Kampen, and the like. This is close to statistical mechanics, and you should find it easy to follow. It is relevant for a deeper understanding of nonequilibrium thermodynamics; so you should read it anyway, independent of Hairer. (After having mastered that, you may reconsider which path to go to learn more. Life is typically longer than you may think now....)

A mathematical treatment of the same topic (ordinary SDE) requires the Ito integral, which needs quite some abstract measure theory - you can find expositions in books on stochastic processes from a mathematical perspective. One example is the book by Øksendal. Hairer's work is even more abstract than that.

Stochastic partial differential equations are also obtained from relativistic quantum field theories by analytic continuation to imaginary time. But to go back one needs an additional condition called reflection positivity (Osterwalder-Schrader theorem). Thus stochastic SDEs are a more general concept than Euclidean QFTs. 

for now I can only wish to study it to the extent of understanding its relevance to QFT. 

At present, none. What Hairer does is sort of the converse - he shows how to solve classical stochastic differential equations that informally correspond to superrenormalizable quantum field theories (for which existence proofs have been given around 1980 by Glimm and Jaffe). It is a step forward in classical stochastic equations; but the techniques are (presently) not strong enough to tackle the Euclidean versions of only renormalizable quantum field theories.

answered Dec 25, 2015 by Arnold Neumaier (12,355 points) [ no revision ]

Thanks!

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