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
\begin{equation}
\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} .
\end{equation}
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.