Phys 450/550: Relativistic Quantum Mechanics

Spring 2024

Topics Covered in Lectures

Textbooks:

[BD] Bjorken and Drell, Relativistic Quantum Mechanics, 1964

[Ca] Capri, Relativistic Quantum Mechanics and Introduction to Quantum Field Theory, 2002

[Gr] Greiner, Relativistic Quantum Mechanics, 2000

[LL] Landau and Lifshits, The Classical Theory of Fields, 4^th edition, 2002

[Sc] Schwable, Advanced Quantum Mechanics, 2^nd edition, 2000

[We] Weinberg, The Quantum Theory of Fields I, 1995

[Wo] Woodhouse, Special Relativity, 2003

Lecture Number	Date	Content	Corresponding Reading material
1	Feb. 12	Frames of reference in classical mechanics, inertial frames, absolute nature of time in Newtonian mechanics, Galilean transformations and Galilean group	Wo: Chapter 1
2	Feb. 14	Galilean principle of relativity, infiniteness of the speed of propagation of interactions in Newtonian mechanics, Postulates of Einstein’s special theory of relativity, consequences of the inertial-frame-independence of maximum speed of propagation of interactions: Relativity of simultaneity, time dilation, length contraction; Lorentz transformation connecting two inertial frame with synchronized clocks and aligned coordinate axes	LL: Chapter 1, Sec. 1
3	Feb. 15	Relativistic law of addition of velocities, the scalar quantity that takes the same value at every inertial frame, Minkowski inner product and metric, the definition of a general Lorentz transformation	LL: Chapter 1, Secs. 2 & 6
4	Feb.26	General properties of the Lorentz and Poincare groups: Proper orthochronous Lorentz groups, time-reversal and parity transformations	We: Pages 55-58
5	Feb. 28	Lorentz boosts as spacetime rotations, degrees of freedom of the Poincare group; Contravariant and covariant 4-vectors, proper time of a moving clock, casual structure of Minkowski spacetime	LL: Chapter 1, Secs. 4 & 6
6	Mar. 06	4-velocity and 4-acceleration, action for a free relativistic particle, relativistic linear momentum and energy-momentum 4-vector, Lorentz transformation property and conservation of energy and momentum, particle decay	LL: Chapter 1, Secs. 8, 9 & 11
7	Mar. 11	Infinitesimal Lorentz transformations, Nöther’s theorem, conservation laws associated with space-time translation symmetry and Lorentz transformations; Basic ingredients of quantum mechanics	LL: Chapter 1, Secs. 14
8	Mar.13	Antilinear and anti-unitary operators, time-reversal operator, time-reversal-invariant systems, Wigner’s symmetry theorem	-
9	Mar. 18	Lie groups and Lie algebras: Group, Lie groups, Lie algebras, associative algebras, representation of a group, faithful, unitary, and projective representations, invariant subspaces and irreducible representations of a group; Wigner’s identification of elementary relativistic particles with irreducible projective unitary representations of the Poincare group.	Appendix A of the Book: The Geometric Phase in Quantum Systems
10	Mar. 20	Unitary representations of a Lie algebra, generators of a Lie group, structure constants of the corresponding Lie algebra, groups generators in a unitary representation, generators of Poincare group in a unitary representation	We: Pages 50-55 & 58-59
11	Mar. 25	Transformation properties of the generators of the Poincare group under Lorentz transformations and spacetime translations, Calculation of the structure constants of the Poincare Lie algebra	We: Pages 59-60 & 58-59
12	Mar. 27	Commutation relations for generators of rotations, boosts and space-time translations; unitary-equivalent unitary representations; basic idea of constructing the irreducible unitary representations of the Poincare algebra	We: Pages 60-63 & 58-59
13	Mar. 28	Casimir operators for the Poincare algebra: Mass and Pauli0-Lubanski 4-vector, Classification of the irreducible unitary representations of the Poincare algebra; Free massive spin-zero representations and the Klein-Gordon eq., scalar and pseudo-scalar fields/particles	Ca: Pages 4-9
14	Apr. 01	General solutions of the Klein-Gordon eq., Schrödinger equations for the positive- and negative-energy solutions of the Klein-Gordon eq., conserved current and the Klein-Gordon inner product	Ca: Pages 10-12
15	Apr. 03	Invariance of Klein-Gordon inner product under proper orthochronous Poincare transformations, massive scalar field minimally coupled to a background electromagnetic filed, charge conjugation and the interpretation of the conserved current as the charge current density.	Ca: Pages 12, 15-16
Spring Break
16	Apr. 22	Dirac Eq.: Conserved current density for solutions of the Dirac Eq., Gamma matrices, covariant form of the Dirac Eq. and the conserved current density, Gamma matrices as the generators of the Clifford algebra Cl1,3(R), Pauli’s Fundamental Theorem on irreducible representations of Cl1,3(R).	Ca: Pages 24-30
17	Apr. 24	Solutions of the Dirac Eq., spin and helicity operators	Ca: Pages 31-33 & Gr: Pages 107-111
Midterm Exam	Apr. 25
18	Apr. 29	Nonrelativistic limit of the Dirac Eq. corresponding to a Dirac field minimally coupled to a background EM field, Gamma matrices for different inertial observers	Sc: Pages 125-128 Gr: Pages 120-126
19	May 02	Covariance of the Dirac Eq., connection representations of the Poincare group, construction of the representation map.	Sc: Pages 135-138
20	May 06	Representation map for rotations and boosts, the identification of the space of solutions of the Dirac Eq. with the vector space furnishing the spin-1/2 representation of the Poincare group; Pseudo-Hermitian operators, pseudo-adjoint, and the pseudo-Hermiticity of the gamma matrices, relation between the adjoint and the inverse of the representation map for orthochronous and non- orthochronous Poincare transformations	Sc: Pages 138-143
21	May 08	Covariance of the conserved current density for free Dirac fields, Various properties of gamma matrices and their products, trace identities; Charge conjugation for Dirac fields minimally coupled to EM, Conserved current density for charge-conjugated Dirac fields	Sc: Pages 144 Gr: Pages 299-301
22	May 13	Parity, time-reversal, and CPT transformation of the Dirac fields	Ca: Pages 48-49 BD: Pages 66-70
23	May 15	Chirality operator and expressing Dirac fields minimally coupled to EM fields in terms of definite chirality field; Passive and active transformations of the Dirac fields, the total angular momentum operator as the generator of rotations.	Ca: Pages 54-58 Sc: Pages 155-158
24	May 20
25	May 22

Note: The pages from the textbook listed above may not include some of the material covered in the lectures.