Paul Balduf13h ago
Thursday's #paperOfTheDay is "Tropical Mathematics" from 2009.
I'm currently developing a version of #QuantumFieldTheory called #tropicalFieldTheory . The present article is background on what "tropical" means in #mathematics : This term has first appeared in the context of #computerScience in the 1970s, and it was coined in honor of the early work being done in São Paulo, Brasil. The basic idea is to consider a special type of (mathematical) ring: A typical example of a ring would be the real numbers, together with addition and multiplication. Now, the "tropical semiring" is the real numbers and infinity, but "addition" is replaced by "taking minimum", while "multiplication" is replaced by "addition". This strange object behaves well in many ways. For example, in the usual ring of real numbers one would have
7 + 2*3 = 7+6 = 13
in the tropical semiring, the same equation becomes
min{ 7, 2+3 } = min { 7, 5 }= 5.
The tropical semiring is only SEMI because taking minimum does not always have an inverse: There is no x such that min {x,5}=8 .
In the following decades, tropical arithmetics has been developed into a full mathematical theory. In particular, ome has tropical polynomials, where the conventional addition of monomials is replaced by taking minimums. This is exactly what we do in tropical field theory: The #FeynmanIntegral s are integrals over rational functions, and we replace their denominators and numerators by tropical polynomials.
Today's article was written before tropical field theory, but it discusses a nice application from #biology : One can compute phylogenetic trees with the help of tropical algebraic geometry.
https://arxiv.org/abs/math/0408099
Tropical Mathematics

These are the notes for the Clay Mathematics Institute Senior Scholar Lecture which was delivered by Bernd Sturmfels in Park City, Utah, on July 22, 2004. The topic of this lecture is the ``tropical approach'' in mathematics, which has gotten a lot of attention recently in combinatorics, algebraic geometry and related fields. It offers an an elementary introduction to this subject, touching upon Arithmetic, Polynomials, Curves, Phylogenetics and Linear Spaces. Each section ends with a suggestion for further research. The bibliography contains numerousreferences for further reading in this field.

arXiv.org
Show threadStephen Brooks 🦆3h ago

@paulbalduf If you replace min by max in that definition, what you have is arithmetic in the limit of a very fast log-scale.

e.g. 10^100 + 10^1000 ~= 10^1000
max{100,1000} = 1000

10^100 * 10^1000 = 10^1100
100+1000 = 1100

Show threadStephen Brooks 🦆3h ago
@paulbalduf Which means the tropical semiring is in some sense a limit of actual rings.
Show threadPaul Balduf

@sjb Yes exactly!
Taking max instead of min appears more natural to me, too, but I think they wanted min in order to have +infinity and not -infinity as a neutral element of their operation ( min{x,infinity}=x for all x, so +infinity is the identity for min).

There is by now quite a lot of work about this logarithmic perspective in computing Feynman integrals: The ordinary integrand is a polynomial in Schwinger parameters; this can be written as an exponential function in logarithmic coordinates. Then, the argument of the exponential function is linear (in several variables) and the inequalities between the terms in the tropical polynomial become linear inequalities, i.e. high-dimensional half-spaces. The tropical Feynman integral is then computing the volume of some polytope in this logarithmic space.
This is the connection to the whole "amplituhedron" world: Is a scattering amplitude secretly computing the volume of some geometric object?

I find this all very fascinating, but it's a bit too technical to fully discuss here. A precise description is e.g. in section 6 of https://arxiv.org/pdf/1908.09820 .

8:31pmWeb
Show threadStephen Brooks 🦆3h ago

@paulbalduf I found something like this in a much simpler setting: calculation the electric field inside a uniformly charged ellipsoid that has very different lengths (like 10^4 ratio) for its three axes.

The formula is an 1D integral that crosses all the scales involved, but the integral function ends up having approximate corners when the scales are far apart.