Quantum Bayesian Networks

April 16, 2011

Honey, I shrunk the theory

Filed under: Uncategorized — rrtucci @ 4:11 am

I recently gave a talk at my local “physics club”. The talk was about renormalization group (RG) theory. It was just a brief introduction to the subject. With the help of LaTex, it was an easy and relatively painless exercise to write-up the talk as a pedagogical article. My pedagogical article is about as significant to the RG literature as a mosquito turd is to the New York city sewer system. Nevertheless, I’m posting it here, as a sort of extended blog post, hoping that a few people will find it useful. As I’m sure someone must have said before, “Those who can’t,… teach, and those who can’t teach, write blog posts”. (Also, those who can’t write, Tweet).

You can find the article in pdf format here.

It contains 19 short sections (a nice prime number). The titles of the sections are:

  1. Introduction
  2. Books and other references on RG
  3. RG theory- a big tent
  4. What is renormalization?
  5. A whiff of thermodynamics
  6. An essence of field theory
  7. Naive versus fractal “scaling” dimensions
  8. Real and momentum space RG
  9. Correlations rule the world
  10. Renormalization (semi-)group
  11. RG streamlines
  12. Fixed points of the trivial and critical kind
  13. Critical exponents and universality classes
  14. Relevant, marginal and irrelevant operators
  15. Self-similar coupling constants and beta functions
  16. The regulator and the fiducial mass scale
  17. RG theory has its pi-groups too! Callan-Symanzik type equations
  18. Forgetting initial conditions. Are we cheating with infinities? Where did the infinities go?
  19. The many faces of a renormalizable theory

Here is section 1 of the article:

1 Introduction
In this blog post, I will give a very brief introduction to Renormalization Group (RG) theory.

This blog is about quantum computing and more generally about quantum information science (QIS). So why should a person working in QIS be interested in RG theory? One reason is that RG theory describes how correlation functions scale, and correlation functions are crucial in: (1)the study of quantum entanglement (2)both classical and quantum Shannon information theory.

Physicists like to study how a theory transforms under a family of operations. Such families of operations usually constitute a mathematical group. The operations might be discrete, as with PTC (P=parity, T=time reversal, C=charge conjugation) or continuous (continuous transformations are a type of generalized rotation). In the case of the renormalization group, physicists consider how a theory transforms under an operation that “scales” the unit of length.

It’s useful (at least to me) to think of such scaling as a type of lossy data-compression or smoothing. Accordingly, RG theory can be viewed as a meta-theory that describes how theories change under lossy data-compression. Hence, a more precise but less catchy title for this blog post would have been “Honey, I applied lossy data-compression to the theory (and the kids).”

P.S. While looking for a picture for this blog post, I came across a webpage (for a plastic surgery practice?) that had the following image with the caption “Honey, I shrunk the kids”


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