Ineffective Theory

Why privilege closed-form expressions?

If I say, “let $n$ be the smallest integer expressible as a sum of two positive integer cubes in two distinct ways”, I have unique identified an integer. Likewise, if I say “let $r$ be the ratio between the masses of the lightest vector and lightest scalar in $SU(3)$ Yang-Mills”, I have (under a reasonable conjecture) uniquely identified a positive real number. If you ask me for a positive real number, and I reply “$r$, where $r$ is the ratio…”, you will be rightly annoyed with me. You wanted a reply more like “1.86”.

But what if I reply with “$(5\pi + 1)/9$”? It’s not an immediately useful expression: you have to do a bit of work even to decide if it’s greater than or less than $2$. But at least it’s a “closed-form expression”. Aren’t I nice?

Closed-form expressions are exciting. We can calculate the ratio of masses $r$ above to a few digits of precision; we can get critical exponents of some CFTs quite precisely. But even ten digits is not as desirable as a closed-form expression. What gives?

First, the notion of a closed-form expression has no universally accepted, rigorous definition. It certainly excludes stuff like implicit definitions ($x = 1 + 1/x$, is not a closed-form expression). It definitely includes $(5\pi+1)/9$, which consists only of well-known quantities and a few simple arithmetic operations. Integration is more controversial: most people seem to hold that integration is not allowed, although I think low-dimensional integrals ought to be. Exponentiation is generally allowed; more exotic functions (the Bessels, say) are often allowed. (I’m comfortable including such functions only because they can be written as low-dimensional integrals.)

Roots of polynomials are a good example. Nobody would consider “the largest-magnitude root of $P(x)$” to be a closed-form expression. If $P$ is quadratic, cubic, or quartic, there are of course closed-form expressions available for $x$ in terms of nothing more complicated than radicals. Even for a quintic polynomial, though, a closed-form expression is available if you’re willing to admit Jacobi theta functions — and that’s perfectly satisfactory.

There’s a qualitative difference between (most) closed-form expressions and other types of specifications: closed-form expressions are algorithms. I don’t mean that algorithms exist for evaluating closed-form expressions, but rather that most closed-form expressions are, quite literally, the algorithms themselves. The expression $(5\pi + 1)/9$ specifies an algorithm: start with $5$, multiply by $\pi$, add $1$, finally divide by $9$. Low-dimensional integrals (of sufficiently smooth functions, I guess) specify approximation algorithms: the Riemann integral is defined as the output of an algorithm.

Expressions we’d acknowledge as closed-form are generally also efficient algorithms. Here’s a pathological example: $$ f(N) = \int \mathrm d^{N}\!x\; e^{-x^2} $$ Since this is an integral, as we’ve discussed, it constitutes an algorithm for evaluating $f(x)$. Alas, the algorithm requires time exponential in one of the parameters! This is not so useful.

So there is, after all, a nice unifying principle here. Closed-form expressions are those that describe efficient (polynomial in all relevant parameters) algorithms. And of course, that’s a moral lesson as well as a definition: for nearly all purposes, an efficient algorithm is as good as (and as hard to find as) a clean-looking expression.