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 \( (x-3)e^{x} + 3 = 0\).  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.