In a few videos I've seen where he discusses the new approach to calculating the super Yang Mills scattering amplitudes, Nima Arkani-Hamed sometimes alludes to the use of Motivic methods as being relevant. (For example in the last few seconds of this presentation).

I would be interested if someone could give even a superficial hint of what motivic mathematics is and how it is applied in physical problems.

  • $\begingroup$ The Wikipedia article has a bit of information, but not something I can make sense of... good question. $\endgroup$ – David Z Jun 3 '12 at 19:13
  • $\begingroup$ @DavidZaslavsky yes, I couldn't understand that either. I found this reference with some physics examples included. However, it seems inescapable that there's a lot of unfamiliar mathematics to wade through to understand this. $\endgroup$ – twistor59 Jun 4 '12 at 8:02
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    $\begingroup$ see the answers in physicsoverflow.org/8964 $\endgroup$ – Arnold Neumaier Sep 11 '14 at 7:42

Motives are a conjectural generalisation of two very important, but concrete, developments in 20th century number theory.

Number Theory studies zeta functions (and zeta functions turn out to be useful in Physics as a way to regularise a divergent formula, aka re-normalisation). "The" zeta function is Riemann's, which is, for any complex number s, the sum of the reciprocals of the integers raised to the s-th power, $$\sum _{n=1}^\infty {1 \over n^x}.$$ This diverges for s=1. But the function is complex-analytic and has an analytic continuation to the entire complex plane, denoted by $\zeta (s)$. For example, $\zeta (-1) = - \frac 1{12}$ or something like that. So it provides a way of "regularising" 1+2+3+4+5+6+7+...+n+ .... as equal to -1/12.

In the twentieth century, generalisations of the Riemann zeta function were formulated, first for number fields, and then for elliptic curves $E$, and then for any algebraic curve, or even surface, etc.. (Hasse and Weil then proved that these satisfied "functional equations" relating $\zeta_E(s)$ to $\zeta_E(1-s)$ and so these had analytical continuations to the entire complex s-plane even though their defining formulas diverged. So they provide different ways of regularising different things. In particular, zeta functions are used to regularise infinite determinants.

Grothendieck and Deligne proved the Weil conjectures, which said that these zeta functions of algebraic surfaces could be interpreted homologically and would always satisfy a functional equation etc. etc. analogous to the Riemann zeta function. The Riemann hypothesis for these geometric zeta functions has already been proved even though the Riemann hyptohesis for the original Riemann zeta function still resists all efforts to prove it.

A "motive" is a conjectural generalisation of the idea of an algebraic surface, to which one can attach an appropriate cohomology theory which will be a vast generalisation of what we are all used to (de Rham's cohomology theory for algebraic manifolds) but provide us with more and more zeta functions which will possess analytic continuations to the entire complex plane because they satisfy functional equations like that for the usual zeta functions. Then there will be more ways to regularise more things.

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  • $\begingroup$ I recommend giving this whole thing a miss, as far as Physics is concerned. $\endgroup$ – joseph f. johnson Dec 8 '15 at 15:47

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