Seminar 500 018   Physical Mathematics   Kevin Cahill

This one-credit seminar will focus, according to the wishes of the students, on one or more of these topics: group theory, tensors, special and general relativity, gauge theory, probability and statistics, Monte Carlo methods, functional derivatives, path integrals, the renormalization group, chaos and fractals, and string theory. On the first day of class, the students chose to spend three weeks on group theory and then start tensors and general relativity. The course meets on Wednesdays in room 5 from 4:30 to 5:20. We will use articles from the internet and arxiv.org as well as the textbook Physical Mathematics used in 466 during the fall.

I will make videos of the lectures for students who must miss classes and for people not at UNM.

Lecture 1, sections 10.1-10.3 and 10.14-10.15 of the book Physical Mathematics: What is a group? representations of groups, representations acting in Hilbert space, compact and noncompact Lie groups, Lie algebra. The video of this lecture also is on YouTube

Lecture 2, sections 10.15-10.16 and 10.20-10.21: Lie algebra, the rotation group, the Jacobi identity, and the adjoint representation. The video of this lecture also is on YouTube

Lecture 3, sections 10.17, 10.21-10.22, 10.24-10.32: physical demonstration of the commutation relations of the rotation group, the adjoint representation, Casimir operators, simple and semisimple Lie algebras, SU(3), SU(3) and quarks, Cartan subalgebra, quarternions, the symplectic group Sp(2n), compact simple Lie groups, group integration, the Lorentz group. The video of this lecture also is on YouTube

Lecture 4, sections 10.33-10.35 and 11.1-11.3: Two-dimensional representations of the Lorentz group, The Dirac representation of the Lorentz group, The Poincaré group and points and Points and coordinates, Scalars, Contravariant vectors, Covariant vectors. The video of this lecture also is on YouTube

Lecture 5, sections 11.2--11.24: Scalars, Contravariant vectors, Covariant vectors, Summation convention, Special relativity and electrodynamics, Tensors, Differential forms, Tensor equations, The quotient theorem, The metric tensor and tetrads, A basic axiom, The contravariant metric tensor, Raising and lowering indices, Sperical coordinates, The gradient of a scalar field. The video of this lecture also is on YouTube

Lecture 6, sections 11.25--11.31: Levi-Civita's tensor, The Hodge star, Derivatives and affine connections, Parallel transport, Notations for derivatives, Covariant derivatives, The covariant curl. The video of this lecture also is on YouTube

Lecture 7, sections 11.16--11.21, 11.29--11.35: Tetrads, The metric tensor, A basic axiom, The principle of equivalence, The contravariant metric tensor, Dual vectors and the raising and lowering of indices, Derivatives and affine connections, Parallel transport, Notations for derivatives, Covariant derivatives, The covariant curl, Covariant derivatives and antisymmetry, Affine connection and metric tensor. The video of this lecture also is on YouTube.

Lecture 8, sections 11.35-39: Covariant derivative of a tensor, Covariant derivatives and antisymmetry, Affine connection and metric tensor, Covariant derivative of the metric tensor, Divergence of a contravariant vector, The covariant laplacian, The principle of stationary action. The video of this lecture also is on YouTube.

Lecture 9, sections 11.38--11.43: A particle in a gravitational field, The principle of equivalence, Weak, static gravitational fields, Gravitational time dilation, Curvature, Einstein's equations. The video of this lecture also is on YouTube.

Lecture 10, sections 11.43--11.48: Einstein's equations, The action of general relativity, Standard form, Schwarzschild's solution, Black holes, Cosmology. The video of this lecture also is on YouTube.

Lecture 11, section 11.48: Cosmology. The video of this lecture also is on YouTube.

Lecture 12, sections 11.48 and 11.49: Cosmology and Model cosmologies. The video of this lecture also is on YouTube.

Lecture 13, sections 11.48--11.49 and 19.1--19.3: Cosmology, Model cosmologies and The infinities of quantum field theory, The Nambu-Goto string action, Regge trajectories. The video of this lecture also is on YouTube.

Lecture 14 was erased by an IT unprofessional.

Lecture 15, sections 19.4--19.7: light-cone coordinates, light-cone gauge, quantized strings, superstrings, covariant and Polyakov actions, D-branes or P-branes, string-string scattering. The video of this lecture also is on YouTube. In 1926, Joseph Ser analytically continued Riemann's zeta function $\zeta(s) = \sum_{n=0}^\infty \frac{1}{n^s} = {} \frac{1}{s-1} \sum_{n=0}^\infty \frac{1}{n+1} \sum_{k=0}^n {n \choose k} \frac{(-1)^k}{(k+1)^{s-1}} .$ String theory is a quantum field theory in two dimensions $$\tau$$ and $$\sigma$$. Like other quantum field theories, it has divergences which string theorists interpret by using Ser's analytic continuation of Riemann's zeta function $$\sum_{n=1}^\infty n ={} \zeta(-1) = - \frac{1}{2} \sum _{n=0}^{\infty} \frac{1}{n+1} \sum _{k=0}^n (-1)^k (k+1)^2 \binom{n}{k} = - \frac{1}{2} \sum _{n=0}^{2} \frac{1}{n+1} \sum _{k=0}^n (-1)^k (k+1)^2 \binom{n}{k} = - \frac{1}{12} .$$ The relation $${} -1 = - (D-2)/24$$, then says that open bosonic strings make sense in $$D = 26$$ dimensions. Adding fermions, brings $$D$$ down to 10.

Lecture 16, sections 17.1--17.3: The renormalization group in quantum field theory, The renormalization group in lattice field theory, The renormalization group in condensed-matter physics. The video of this lecture also is on YouTube.