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.

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.

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