1. 2012
Nov
22

B meson oscillations and the CPT theorem

As if last week’s announcements of new Higgs results, B-dimuon decay, the rediscovered Y(4140), and all sorts of other goodies at HCP 2012 weren’t enough, there’s more big news from the world of experimental particle physics this week. A paper published just a few days ago in Physical Review Letters (here’s the PDF, and the arXiv page) describes the first observation ever of actual time reversal asymmetry: a difference between the behavior of a particular physical process and the time-reversed version of the same process.

Lest you get too excited, though: this has nothing to do with actual reversing of time, so it doesn’t mean time travel is possible or anything like that. And in fact, nobody in physics is the least bit surprised that it worked out the way it did. There is a theorem in physics called the CPT theorem (or sometimes PCT, or TCP, but not that TCP) which basically guaranteed that time reversal asymmetry had to show up somewhere. The theorem is suddenly getting a lot of attention in the news coverage of the discovery, but it’s technical enough that most people aren’t bothering to explain it. I …

2. 2012
Feb
07

Scale invariance and the power law

With all the fuss about SOPA and PIPA, plus having actual work to do, I haven’t been able to write anything in a while. So I figured it’s time for some good old fashioned physics. Today I would like to introduce the concept of a scale invariant (or dilatation invariant) function. This special class of functions is defined by the property that when you scale (or dilate) the argument of the function by some factor, it’s equivalent to scaling the value of the function by some related factor.

$$f(\lambda x) = C(\lambda) f(x)$$

$$C(\lambda)$$ is something that depends on $$\lambda$$, but not on $$x$$.

To understand why physicists find scale invariant functions so fascinating, we have to go way back to the definition of an analytic function, the power series expansion. Pretty much every mathematical function used in physics can be expressed as some power series, like this:

$$f(x) = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots$$

Even if the function isn’t normally written like this, it’s enough that it can be. (The sine and cosine, for example.)

Now, what happens when you plug some value with units into …