Examples of non-rigorous but efficient mathematical methods in physics

Perhaps it would not be out of place to quote Miles Reid's Bourbaki seminar on the McKay correspondence here:

"The physicists want to do path integrals, that is, they want to integrate some "Action Man functional" over the space of all paths or loops $\gamma : [0; 1] \rightarrow Y$. This impossibly large integral is one of the major schisms between math and fizz. The physicists learn a number of computations in finite terms that approximate their path integrals, and when sufficiently skilled and imaginative, can use these to derive marvellous consequences; whereas the mathematicians give up on making sense of the space of paths, and not infrequently derive satisfaction or a misplaced sense of superiority from pointing out that the physicists' calculations can equally well be used (or abused!) to prove 0 = 1. Maybe it's time some of us also evolved some skill and imagination. The motivic integration treated in the next section builds a miniature model of the physicists' path integral,..."

Feynman's path integral in quantum field theory. It involves integration over spaces of fields, using measures that have not been made rigorous.

Finally, a Math Overflow question that addresses my specialty: non-rigor!

Here are a few examples of non-rigor as applied to evidence for dualities:

1. Heterotic-Type II. In earlier times, the best evidence for heterotic-Type-II duality was a) counting the number of supersymmetries of the theory, and (b) comparing the moduli spaces.

2. AdS-CFT. For AdS-CFT the earliest and best comparisons were counting the so-called anomalous dimensions of various operators. To date, I think the tests are far from rigorized (and yes, this would be a great problem to make mathematically precise).

3. Mirror Symmetry, early days. Recall that mirror symmetry in CY moduli space came from constructing a chart of the Euler characteristics of CY complete intersections and noticing the symmetry of the chart about zero. Other non-rigorous arguments involve counting the dimensions (just the dimensions) of the moduli of purportedly mirror objects. Then there's the old compute-on-flat-space-and-let-supersymmetry-take-care-of-the-rest trick.

4. Low energy effective field theory. The "fact" that string theory reduces to an oft-identifiable QFT in a low energy limit is a huge source of argumentation/inspiration in string theory. Accounting for (effective) black holes helped lead to M-theory in one context, and to the microscopic description of black-hole entropy in another. One can also argue for dualities by identifying equivalent field contents in two different models. This brings up another point.

5. Invariance of BPS states under perturbation. It is great to take a quantity that does not vary and evaluate it in a limit where it is easy to compute. This argument appears again and again in physics -- and also in math, of course (e.g. in the heat-kernel proof of the index theorem). BPS numbers are just that. (Of course, they do vary, and the continuity of the relevant physical parameters [numbers are not necessarily physical quantities] is what underlies interesting explanations of wall-crossing.)

I'm probably including too many that don't fit and excluding a lot that do. Very non-rigorous of me!

The replica method and the cavity method have been used by physicists to calculate thermodynamic quantities in various statistical mechanics settings (including quite a few classes of random combinatorial objects). The results are often exactly right, even though the method is not at all rigorous. Michel Talagrand has recently proven rigorously some of the results that have been obtained by these methods.

The use of random matrix theory to model energy levels of heavy nuclei and other physical systems. See also the following historical piece and the pictures therein: There is striking statistical evidence that the eigenvalues of large random self-adjoint matrices, the energy levels of heavy nuclei, and the normalized zeros of $L$-functions (!) are all spaced about the same.

In boundary value problems, physicists consider the infinity (in space and in time) to be part of the boundary. Mathematicians know there's a distinction between compact and non-compact spaces.

Another example from theoretical high-energy physics I've encountered: sometimes when physicists have some equation of motion for an arbitrary number $N$ of particles with positions $x_i$, e.g. something of the form $\frac{1}{N}\sum_i f(x_i) + \frac{1}{N^2}\sum_{ij} g(x_i, x_j) = 0$, they wish to know what the solutions to this equation look like for large $N$. A technique they use is to replace the variables $x_i$ with a probability measure $\mu$ on the space of their possible values, which is supposed to represent the number of $x_i$'s in a given region in the large $N$ limit, and instead of solving the original equation they solve the analogous equation in $\mu$, e.g. $\int f(x) \mathrm{d}\mu(x) + \int g(x, y) \mathrm{d}(\mu \times \mu) (x, y) = 0$. In fact it's not hard to come up with a toy example where the original equation can be solved exactly for all $N$ and the solutions "look like" a particular probability distribution in the large $N$ limit, but that probability distribution fails to satisfy the corresponding equation, and for that reason I have some doubt that this method can be turned into something rigorous.

The Hypernetted-chain approximation used in statistical mechanics.

Was for instance used in the theory of the fractional quantum hall effect by Laughlin in order to estimate the energies of elementary excitations of Laughlins wave function.

Yang-Mills Equations are experimentally proven but have no strong mathematical foundations. In the Clay Mathematics Institute the mass gap problem is worth one million dollars.