Monday, July 23, 2018

Math, beauty, and condensed matter physics

There is a lot of discussion these days about the beauty of mathematics in physics, and whether some ideas about mathematical elegance have led the high energy theory community down the wrong path.  And yet, despite that, high energy theory still seems like a very popular professed interest of graduating physics majors.  This has led me to identify what I think is another sociological challenge to be overcome by condensed matter in the broader consciousness. 

Physics is all about using mathematics to model the world around us, and experiments are one way we find or constrain the mathematical rules that govern the universe and everything in it.  When we are taught math in school, we end up being strongly biased by the methods we learn, so that we are trained to like exact analytical solutions and feel uncomfortable with approximations.  You remember back when you took algebra, and you had to solve quadratic equations?  We were taught how to factor polynomials as one way of finding the solution, and somehow if the solution didn’t work out to x being an integer, something felt wrong – the problems we’d been solving up to that point had integer solutions, and it was tempting to label problems that didn’t fit that mold as not really nicely solvable.  Then you were taught the quadratic formula, with its square root, and you eventually came to peace with the idea of irrational numbers, and eventually imaginary numbers.  In more advanced high school algebra courses, students run across so-called transcendental equations, like \( (x-3)ex + 3 = 0\).  There is no clean, algorithmic way to get an exact analytic solution to this.  Instead it has to be solved numerically, using some computational approach that can give you an approximate answer, good to as many digits as you care to grind through on your computer. 

The same sort of thing happens again when we learn calculus.  When we are taught how to differentiate and integrate, we are taught the definitions of those operations (roughly speaking, slope of a function and area under a function, respectively) and algorithmic rules to apply to comparatively simple functions.  There are tricks, variable changes, and substitutions, but in the end, we are first taught how to solve problems “in closed form” (with solutions comprising functions that are common enough to have defined names, like \(sin\) and \(cos\) on the simple end, and more specialized examples like error functions and gamma functions on the more exotic side).  However, it turns out that there are many, many integrals that don’t have closed form solutions, and instead can only be solved approximately, through numerical methods.  The exact same situation arises in solving differential equations.  Legendre, Laguerre, and Hermite polynomials, Bessel and Hankel functions, and my all-time favorite, the confluent hypergeometric function, can crop up, but generically, if you want to solve a complicated boundary value problem, you probably need to numerical methods rather than analytic solutions.  It can take years for people to become comfortable with the idea that numerical solutions have the same legitimacy as analytical solutions.

I think condensed matter suffers from a similar culturally acquired bias.  Somehow there is a subtext impression that high energy is clean and neat, with inherent mathematical elegance, thanks in part to (1) great marketing by high energy theorists, and (2) the fact that it deals with things that seem like they should be simple - fundamental particles and the vacuum.  At the same time, even high school chemistry students pick up pretty quickly that we actually can't solve many-electron quantum mechanics problems without a lot of approximations.  Condensed matter seems like it must be messy.  Our training, with its emphasis on exact analytic results, doesn't lay the groundwork for people to be receptive to condensed matter, even when it contains a lot of mathematical elegance and sometimes emergent exactitude.




10 comments:

David Brown said...

""... condensed matter ... emergent exactitude ..." Are many of the mathematical techniques involved in quantum gravity also applicable to condensed matter? If there is a string landscape, is the mathematics of the landscape's structure related to the mathematics of impurity-induced scattering?
"A Bose-Einstein condensate in an atom" by Ashley G. Smart, July 2018 Physics Today
R├Ątzel, Dennis, Richard Howl, Joel Lindkvist, and Ivette Fuentes. "Dynamical response of Bose-Einstein condensates to oscillating gravitational fields." arXiv preprint arXiv:1804.11122 (2018).

sylow said...

I think most physicists really misjudge the role of mathematics in physics. Yes, things should look rigorous mathematically but unless you can explain your results without any equations you do not understand what you are doing. That is what Feynman said decades ago and it is still very much true. I sometimes think that the reason why physicists are relying more and more on advanced mathematics is simply obfuscation. In other words, there aren't that many fundamental problems left to be solved anymore in physics. That means you have to invent new methods to solve the same problems over and over. It then requires new mathematical tools. I come across that type of papers every day. "A new method to solve the blah blah problem..." The results are already known so what is the point other than generating more papers that nobody reads?

Anonymous said...
This comment has been removed by a blog administrator.
Douglas Natelson said...

Anon@12:17, I deleted your comment b/c it looked a lot like spam. Though I'm not 100% certain, I'm not going to encourage people to click on a mysterious download link.

Anonymous said...

My bad, didn't search enough. Here is the link, https://arxiv.org/abs/1807.08572

Douglas Natelson said...

Yes, that's an odd one. It would be exciting if correct, but there is a long history of reports of superconductivity that don't hold up - see here and here. As far as we know, neither Ag nor Au superconduct down to sub-mK temperatures. That makes it extraordinarily unlikely that a Au/Ag nanoparticle composite would do so near (or above) room temperature....

gilroy0 said...

Maybe we need to approach math education differently. Instead of spending so much time on increasingly remote exactness, students should get used to "messy" (or at least, numerical and approximate) math. With cheap computing, you could allow kids to explore numerics while learning the basics. In addition, the beauty and rarity of the exact results then pop out. Right now we get a lot of "why can't I just use my calculator?" and not a lot of great replies.

Anonymous said...

In regards to gilroy0’s comment, I recommend the following TED talk: https://www.ted.com/talks/arthur_benjamin_s_formula_for_changing_math_education/up-next

suomynona said...

Surprised there's no mention of topological matter in this post. The modern field of topological condensed matter is extremely rich in exotic mathematics, including much of the same mathematical tools used in string theory. I think a lot of what drives physics students towards high energy physics is that the notion of reductionism still very much forms the core of a physicst's education, despite it being now over 45 years since Anderson's More is Different paper. I agree on the point that there is some preconceived notion that condensed matter is "messy" and particle physics is "clean". A counterexample might be to compare the calculation of the quantum Hall conductivity versus calculating the mass of the proton.

Anonymous said...

Suomynona, Doug in fact did (indirectly) mention the fractional quantum hall effect, via the link in his last sentence.