PHY4404 Relativistic Quantum Mechanics and Gauge Theories

1999-2000

Code: PHY4404
Title: Relativistic Quantum Mechanics and Gauge Theories
Instructors: Dr P.J. Shepherd
HE credits: 10
ECTS credits: 5
Availability: unrestricted
Level: 4
Prerequisites: Classical Theory of Fields (PHY2214)
Corequisites: none
Background Assumed: Quantum Mechanics I (PHY2202) and Quantum Mechanics II (PHY3130)
Duration: Semester I
Directed Study: 22 lectures
Private Study: 78 hours
Supports Programme Aims: 1, 2, 5, 6, 8 and 11
Supports Programme Objectives: none

Assessment Methods

One 90-minute examination

Rationale

Quantum theory as formulated before Dirac was inconsistent with special relativity. Dirac discovered a wave equation that is not only consistent with relativity but also explains particle spin, predicts the existence of antiparticles, and provides us with a deep insight into such symmetries of nature as spatial inversion (parity), charge conjugation and time reversal. The first part of this module, which is a core module for students of Theoretical Physics and an option for other fourth-year MPhys students, covers Dirac's great synthesis of quantum mechanics, already covered in Quantum Mechanics I, II and III (PHY2202, PHY3130 and PHY3131), and Einstein's special theory of relativity, which students have studied previously in Vector Mechanics and Relativity (PHY1005), Classical Mechanics and Relativity (PHY2207) and the Classical Theory of Fields (PHY2214).

The "holy grail" of physics is undoubtedly a grand unified theory of all the forces of nature. After Maxwell's great unification of electricity and magnetism in the nineteenth century, no further unification was achieved until Weinberg and Salam devised their standard model in the 1970's - a theory that unifies the weak force and electromagnetism. The modern approach to this problem, namely, the use of the assumption of local gauge symmetries to derive the necessity of the existence of the fundamental forces of nature, has been extremely successful in the explanation and unification of the strong, weak and electromagnetic forces, and promises, via superstring theory, to provide a means of including gravity as well in a final "theory of everything". The second section of this module, which treats with greater rigour some of the topics the students will have encountered in High Energy Physics (PHY3120) and goes far beyond what is taught in most undergraduate physics degree programmes, gives an in-depth introduction to gauge theories of the fundamental interactions, describing, in particular, the Weinberg-Salam unified theory of the electroweak interaction and the role of the Higgs field in this theory.

Intended Learning Outcomes

Student will be able to:

• explain the concept of Lorentz covariance and the need for a quantum-mechanical wave equation consistent with special relativity
• outline the significance of the Dirac wave equation and its free-particle solutions
• describe how the electromagnetic field may be introduced into the Dirac equation, and explain the origin of particle spin
• calculate the effects on a Dirac wave function of spatial inversion, charge conjugation and time reversal, and explain how these concepts are used in the modern (Feynman) description of antiparticles
• explain how the existence of electromagnetism can be regarded as a consequence of a local gauge symmetry
• describe the fundamental gauge principles of the origin of the weak force, and also of the strong force between quarks (quantum chromodynamics)
• explain the role of the Higgs field in the acquisition of mass by the carriers of the weak force (the weak gauge bosons), and the Weinberg-Salam unification of the electromagnetic and weak interactions

Teaching and Learning Methods

Lectures, problems classes and on-line teaching resources.

Transferable Skills

Problem solving, ability to make connections with previous knowledge, application of new and unfamiliar concepts in a number of different situations, use of IT learning resources, presentational skills (in tutorials)

Assignments

Problems are set and discussed in a supporting problems class/tutorial given by the lecturer, and tutors also set related problems and assignments

Module Text

Not applicable

Supplementary Reading

Not applicable

Syllabus Plan and Content

1. Relativistic Quantum Mechanics
   1. Free-Particle Relativistic Wave Equations
      1. The Klein-Gordon equation
      2. Dirac matrices
      3. The Dirac equation
   2. Lorentz and Rotational Covariance of the Dirac equation
      1. Behaviour of Dirac spinor wave functions under a Lorentz transformation
      2. Behaviour of Dirac spinor wave functions under rotation; spin
   3. The Current Four-Vector
      1. The adjoint spinor
      2. The four-current and continuity equation
   4. Dirac Wave Function of a Free Particle at Rest
      1. Positive- and negative-energy solutions and their spins
   5. Dirac Wave Function of a Free Particle in Motion
   6. Motion of an Electron in an Electromagnetic Field
      1. Dirac equation for an electron in an electromagnetic field
      2. Large and small components
      3. The nonrelativistic limit and the Pauli equation
      4. Anomalous magnetic moment and gyromagnetic ratio
   7. Relativistic Covariance
      1. Behaviour of spinors under spatial inversion; parity
      2. Unitarity properties of spinor-transformation matrices
      3. Proof that the four-current is a four-vector
      4. Transformation property of the four-current
   8. Particles and Antiparticles
      1. Dirac interpretation of the negative-energy solutions
      2. Charge conjugation
      3. Time reversal
      4. Feynman interpretation of the negative-energy solutions
      5. PCT
      6. Parity violation, helicity and the weak interaction
2. Gauge Theories of the Fundamental Interactions
   1. Abelian Gauge Theories (Global and Local)
      1. Global phase invariance in quantum mechanics
      2. Local phase invariance and the necessity of the existence of the four-potential
   2. Nonabelian Gauge Theories (Global)
      1. The nucleon, and charge invariance of the nucleon-nucleon force; global phase invariance and SU(2); isospin-1/2 operators as generators of SU(2)
      2. Generalization to higher-dimensional representations of SU(2)
      3. Global SU(3) transformations; quarks; colour symmetry (exact) and flavour symmetry (approximate) of the strong interaction
   3. Nonabelian Gauge Theories (Local)
      1. SU(2) local phase invariance; covariant derivatives and the three four-vector gauge fields (Yang-Mills fields) of SU(2)
      2. Gauge transformation of the gauge fields; the gauge fields as an isospin triplet
      3. The "gauging" of colour SU(3) - quantum chromodynamics; the eight four-vector gauge fields (Yang-Mills fields) of SU(3)
   4. The Weak Interaction
      1. Two-current processes; transition currents and "weak-isospin" doublets; hidden weak-isospin symmetry
      2. Generations of leptons and quarks; the Cabibbo angle
      3. Gauging of the hidden weak-isospin symmetry, and the three four-vector gauge fields (Yang-Mills fields) of weak-isospin SU(2)
      4. The Weinberg-Salam SU(2) x U(1) model ("standard model") of the electroweak interaction
   5. The Higgs Mechanism of Spontaneous Symmetry Breaking
      1. Acquisition of mass by the four-potential in a superconductor; screening currents; the Meissner effect
      2. The Higgs field, and the acquisition of mass by the three gauge fields of the weak interaction
      3. The nonzero masses of the W and Z bosons, and zero mass of the photon

Feedback to Students

The problems that students are set on this module are marked and discussed in detail with the lecturer in the problems classes.

Feedback from Students

Feedback from students on the module is gathered via the standard student representation mechanisms.