Examples of non-rigorous but efficient mathematical methods in physics - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T12:57:45Z http://mathoverflow.net/feeds/question/48671 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/48671/examples-of-non-rigorous-but-efficient-mathematical-methods-in-physics Examples of non-rigorous but efficient mathematical methods in physics Cristi Stoica 2010-12-08T20:30:42Z 2012-02-15T16:26:11Z <p>There are situations of applications of mathematics in physics which</p> <ul> <li>seem to work well enough for physicists (for example they agree with the experimental data)</li> <li>but are considered unacceptable or at least non-rigorous to mathematicians</li> </ul> <p>Please help me gather some examples. Which of these techniques were eventually made rigorous?</p> <p>Thank you. </p> <p>I apologize if this question may seem inappropriate for MO. I consider these examples a great source of research problems for mathematicians who are interested in mathematical physics. </p> http://mathoverflow.net/questions/48671/examples-of-non-rigorous-but-efficient-mathematical-methods-in-physics/48672#48672 Answer by Andrey Rekalo for Examples of non-rigorous but efficient mathematical methods in physics Andrey Rekalo 2010-12-08T20:35:37Z 2010-12-08T20:35:37Z <p><img src="http://upload.wikimedia.org/wikipedia/commons/b/b4/Dirac_function_approximation.gif" alt="alt text"></p> http://mathoverflow.net/questions/48671/examples-of-non-rigorous-but-efficient-mathematical-methods-in-physics/48673#48673 Answer by Laie for Examples of non-rigorous but efficient mathematical methods in physics Laie 2010-12-08T21:01:59Z 2010-12-08T21:01:59Z <p>Feynman's path integral in quantum field theory. It involves integration over spaces of fields, using measures that have not been made rigorous.</p> http://mathoverflow.net/questions/48671/examples-of-non-rigorous-but-efficient-mathematical-methods-in-physics/48684#48684 Answer by liuyao for Examples of non-rigorous but efficient mathematical methods in physics liuyao 2010-12-08T22:57:55Z 2010-12-08T22:57:55Z <p>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.</p> http://mathoverflow.net/questions/48671/examples-of-non-rigorous-but-efficient-mathematical-methods-in-physics/48686#48686 Answer by Jeff Harvey for Examples of non-rigorous but efficient mathematical methods in physics Jeff Harvey 2010-12-08T23:06:45Z 2010-12-08T23:06:45Z <p>Perhaps it would not be out of place to quote Miles Reid's Bourbaki seminar on the McKay correspondence here:</p> <p>"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,..."</p> http://mathoverflow.net/questions/48671/examples-of-non-rigorous-but-efficient-mathematical-methods-in-physics/48697#48697 Answer by Phil Wild for Examples of non-rigorous but efficient mathematical methods in physics Phil Wild 2010-12-09T00:46:49Z 2010-12-09T00:46:49Z <p>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.</p> http://mathoverflow.net/questions/48671/examples-of-non-rigorous-but-efficient-mathematical-methods-in-physics/48704#48704 Answer by Eric Zaslow for Examples of non-rigorous but efficient mathematical methods in physics Eric Zaslow 2010-12-09T03:28:18Z 2010-12-09T03:28:18Z <p>Finally, a Math Overflow question that addresses my specialty: non-rigor!</p> <p>Here are a few examples of non-rigor as applied to evidence for dualities:</p> <ol> <li><p>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.</p></li> <li><p>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).</p></li> <li><p>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.</p></li> <li><p>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.</p></li> <li><p>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 <em>physical</em> parameters [numbers are not necessarily physical quantities] is what underlies interesting explanations of wall-crossing.)</p></li> </ol> <p>I'm probably including too many that don't fit and excluding a lot that do. Very non-rigorous of me!</p> http://mathoverflow.net/questions/48671/examples-of-non-rigorous-but-efficient-mathematical-methods-in-physics/48895#48895 Answer by Gene S. Kopp for Examples of non-rigorous but efficient mathematical methods in physics Gene S. Kopp 2010-12-10T08:11:05Z 2010-12-10T08:11:05Z <p>The use of <a href="http://en.wikipedia.org/wiki/Random_matrix" rel="nofollow">random matrix theory</a> to model energy levels of heavy nuclei and other physical systems. See also the following <a href="http://www.williams.edu/go/math/sjmiller/public_html/ntrmt10/handouts/general/Hayes_spectrum_riemannium.pdf" rel="nofollow">historical piece</a> 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.</p> http://mathoverflow.net/questions/48671/examples-of-non-rigorous-but-efficient-mathematical-methods-in-physics/49207#49207 Answer by Peter Shor for Examples of non-rigorous but efficient mathematical methods in physics Peter Shor 2010-12-13T03:33:16Z 2010-12-13T03:33:16Z <p>The <a href="http://en.wikipedia.org/wiki/Replica_trick" rel="nofollow">replica method</a> and the <a href="http://en.wikipedia.org/wiki/Cavity_method" rel="nofollow">cavity method</a> 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 <a href="http://people.math.jussieu.fr/~talagran/spinglasses/" rel="nofollow">proven rigorously</a> some of the results that have been obtained by these methods.</p> http://mathoverflow.net/questions/48671/examples-of-non-rigorous-but-efficient-mathematical-methods-in-physics/73597#73597 Answer by jjcale for Examples of non-rigorous but efficient mathematical methods in physics jjcale 2011-08-24T18:56:36Z 2011-08-24T18:56:36Z <p>The Hypernetted-chain approximation used in statistical mechanics.</p> <p>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. </p> http://mathoverflow.net/questions/48671/examples-of-non-rigorous-but-efficient-mathematical-methods-in-physics/88526#88526 Answer by Jamahl Peavey for Examples of non-rigorous but efficient mathematical methods in physics Jamahl Peavey 2012-02-15T16:26:11Z 2012-02-15T16:26:11Z <p>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.</p>