Physics 467 001

In the first class, on 16 January, I used Dirac notation without explaining it. One does not need Dirac notation to understand group theory, but in case you want to learn about Dirac notation, here are the sections of chapter 1 of Physical Mathematics that deal with it: Dirac notation.

Here is chapter 1 on linear algebra. Grassmann variables are in section 1.1 and in example 1.3; jacobians are in section1.21.
Here is chapter 10 on group theory.
Here is chapter 11 on tensors and local symmetries.
Here is chapter 13 (now 14) on probability and statistics.
Here is chapter 14 (now 15) on Monte Carlo methods.
Here is chapter 16 on artificial intelligence.
Here is chapter 17 on order, chaos, and fractals.

Video of lecture of 16 Jan. Definition of a group. Examples of groups: O(n), SO(n), U(n), SU(n), Sp(2n), the Lorentz and Poincaré groups. Representations of groups. Equivalent representations.

Video of lecture of 18 Jan. Groups and their representations, subgroups, normal subgroups, cosets, quotient groups, morphisms, Schur's lemma, characters, direct products, finite groups, continuous groups, generators of continuous groups.

Video of lecture of 23 Jan. Lie algebra, structure constants, the rotation group, demonstration of the commutation relations of angular momentum.

Video of lecture of 25 Jan. The groups \(O(3)\), \(SO(3)\), and \(SU(2) \); their structure constants \( \epsilon_{abc} \), and their representations. Direct products. Pauli's matrices. Dirac's belt demonstration. Casimir operators. Spin and statistics. Jacobi's identity. The adjoint representation.

Video of lecture of 30 Jan. Global and local symmetries. The standard model. Abelian and nonabelian gauge invariance. The Higgs mechanism. Invariant subalgebras, simple groups, semisimple groups. SU(3) of light quarks and the eight-fold way. The SU(3) of color. Grand unification. Gell-Mann's matrices. Quark masses. The symplectic group as the invariance group of the q's and p's of quantum mechanics. Squeezed states.

Video of lecture of 1 Feb. Cartan's classification of compact simple Lie groups: SU(n+1), SO(2n+1), USp(2n), SO(2n), and the five exceptional groups G_2, F_4, E_6, E_7, and E_8. Quarternions. Invariant integration over a group manifold. The Lorentz group and its representations. Two-dimensional representations of the Lorentz group. The Dirac representation of the Lorentz group. The Poincare group.

Video of lecture of 6 Feb. Points and coordinates. Scalars. Contravariant vectors. Covariant vectors. Euclidian space in euclidian coordinates. Summation conventions. Minkowski space. Lorentz transformations.

Video of lecture of 8 Feb. The metric \(\eta\) of flat space. Basis vectors. Their inner products. Dual vectors. Lorentz transformations. Special relativity. Time dilation. Kinematics. Electrodynamics. Tensors. Tensor equations. Principle of stationary action.

Video of lecture of 13 Feb. Differential forms, forms are scalars, wedge products, the exterior derivative, the curl as the exterior derivative of a 1-form, the Maxwell tensor F = dA, the square of the exterior derivative vanishes d^2 = 0, the homogeneous Maxwell equations are dF = 0, the quotient theorem, basis vectors.

Video of lecture of 15 Feb. Basis vectors in an embedding spacetime, metric of spacetime, tangent vectors, the metric of the sphere \(S^2\) , the metric of the hyperboloid \(H^2\), the metric of the sphere \(S^3\), the principle of equivalence, the metric of spacetime and its inverse, dual vectors, Cartan's moving frame, Levi-Civita's symbols and pseudotensors, the gradient, covariant derivatives.

Video of lecture of 20 Feb. Maximally symmetric spaces, hyperboloids in \(R^3\) one of which is maximally symmetric, the spheres \(S^2\) and \(S^3\), the maximally symmetric hyperboloid in \(R^4\), Killing vectors, Robinson-Walker cosmologies are the real line for time and a maximally symmetric space for space which on large scales is homogeneous and isotropic, notations for derivatives, covariant derivatives, parallel transport, Dirac notation, Christoffel symbols and their relation to the metric tensor, the vanishing of the covariant derivative of the metric tensor.

Video of lecture of 27 Feb. Divergence of a contravariant vector, covariant laplacian, principle of stationary action for a particle in a gravitational field, for a particle in a gravitational field and an electromagnetic field, and for the gravitational and electromagnetic fields in the presence of a charged current. The geodesic equation and its form when the gravitational field is weak and particles move slowly. Gravitational time dilation and the movie Interstellar. The gravitational red shift as measured by Pound and Rebka. Parallel transport and curvature. Einstein's equations. The energy-momentum tensor. Perfect fluids. Schwarzschild's solution. Black holes.

Video of lecture of 1 March Cosmology: inflation, Big Bang, era of radiation, era of matter, era of dark energy. Maximally symmetric expanding 3-spaces. Einstein's equations, energy-momentum tensor, perfect fluids. Three spacetime metrics. Friedmann's equations. Integration of Friedmann's equations for our universe.

Video of lecture of 6 March. Probability of union and intersection of sets of events, Bayes's theorem, the low-base-rate problem, the three-door problem, polling, mean and variance, moments, the binomial distribution.

Video of lecture of 8 March. The binomial distribution, its mean and variance, 28 grams of air in a box, coping with big factorials, Poisson's approximation to the binomial distribution, its mean is its variance, its accuracy, Gauss's approximation to the binomial distribution, its mean is its variance, its accuracy, the error function, polling.

Video of lecture of 20 March. Error analysis, Maxwell-Boltzmann distribution, Fermi-Dirac and Bose-Einstein statistics, diffusion, Langevin's theory of Brownian motion.

Video of lecture of 22 March. The Einstein-Nernst relation, fluctuation and dissipation, autocorrelation functions, characteristic functions, moment-generating functions, cumulants, fat tails, the central-limit theorem.

Video of lecture of 27 March. Review of central-limit theorem, examples of the theorem, random-number generators, generating random numbers that follow an arbitrary distribution, examples, pseudorandom and quasirandom numbers, consistent estimators, biased estimators, Bessel's correction to the estimator of the variance.

Video of lecture of 29 March. More about the central-limit theorem and about examples of it, the Fisher information matrix, the Cramer-Rao lower bound on the covariance, the Fisher information matrix of the gaussian distribution, the Cramer-Rao lower bound on the covariance of the mean and variance of the gaussian distribution, Kolmogorov's way to tell if experimental data come from a given theoretical distribution.

A video was not made of the lecture of 3 April which was on the Monte Carlo method. A link to a pdf of the class notes is posted near the top of this webpage.

Video of lecture of 5 April. A brief history of artificial intelligence, Slagle's symbolic automatic integrator, neural networks.

Video of lecture of 10 April. Neural networks, a linear unbiased neural network that identifies 82.16% of the handwritten numbers 0-9 corectly. Lei Ma's Python code for converting the MNIST files into soemthing usable. Fortran95 code tenvectors.f95 that makes 10 vectors that are the averages of all the teaining vectors that respectively match the integers 0-9. Fortran95 code testvectors.f95 that makes the vectors used to test the linear unbiased neural network. Fortran95 code labels.f95 that converts the double-precision training labels into integer labels. Fortran95 code testlabels.f95 that converts the double-precision testlabels into integer testlabels. Fortran95 code howwellitworks.f95 that tests how well the linear unbiased neural network works.

Undergraduates may want to apply for McNair summer research scholarships; direct your questions here.