Mathematics Asked by KeithMadison on December 24, 2020
I’m writing an essay for a popular (but mathematically mature) audience on the history of mathematical physics, wherein I have a section devoted to string theory. Unfortunately, neither I (nor my audience) have the background to understand the material, however (remaining in theme with the rest of the paper) I’d like to present something substantial and quantitative (i.e. be able to present some sort of hands-on mathematical concept(s)).
While I lack the necessary foundations (I’ve followed the traditional undergraduate Calculus sequence, and have some number theory and group theory) I feel that I have the mathematical maturity to understand the concepts. I keep coming across these vague references to "symmetries" and "Gauge symmetries", and the notation $SU(n)$. In my extensive search, I’ve come across only very advanced/terse material, or very watered down and vague material (which is to be expected, I suppose).
What are "symmetries" and "Gauge symmetries"? What is this notation (which I’ve seen extensively even in popular books for the general population, without much explanation): $SU(n)$? Why are these symmetries important (particularly as they relate to string theory?).
I recommed you first look a the simplest gauge theory, that of Maxwell theory, which is understandable in terms of vector calculus. This describes the electric field E and the magnetic field B in terms of a scalar potential $phi$ and a vector potential $A$.
In particular the magnetic field is given by $B=nabla times A$. We can see that if we change $A$ by a adding a term $nabla f$ the magnetic field remains the same. We call this a gauge symmetry and refer to $A$ as a gauge potential/field. This is a gauge thery based on the group $U(1) = SU(1)$ where we are free to change $A$ by adding a gradient of a real valued function $f$.
Theories of this form can then be generalized by instead of having a scalar field $f$ we can have matrices. In particular $SU(n)$ are a group consisting of nxn dimensional matrices of a particular form (unitary with determinant 1). In particular the theory of the electroweak force is a gauge theory described by $SU(2)$ while the strong force is described by $SU(3)$. The entire standard model of particle physics is a gauge theory based on the product group $U(1) times SU(2) times SU(3)$.
This can be further generalized to other continous groups (Lie groups) but this is less interesting to physicists. Note that gauge theories are important in physics in general and are by no means particular to string theory.
Answered by NIntegrate on December 24, 2020
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