Wave Equations of Relativistic Quantum Mechanics
Outline
This lecture addresses the most important wave equations of relativistic quantum mechanics: the Klein-Gordon (KG) equation and the Dirac equation. Especially the latter is of great importance in particle physics as it describes all known fermions, e.g. electrons and quarks.
The emphasis of this lecture is on the most important conceptual aspects and on the rigorous analysis of the equations. The lecture will start from some basic representation theory of the Lorentz group and then deal with the theory of the existence and uniqueness of solutions of the Klein-Gordon and Dirac equations. Important properties of the equations such as the existence of a conserved probability current, as well a propagation speed smaller than the velocity of light shall be established. Finally, we will come to many-body extensions of these equations in the multi-time formalism of Dirac, Tomanaga and Schwinger.
Target audience:
The lecture might be interesting for students specializing in mathematical physics, mathematically inclinded students of physics and mathematics students interested in analysis, applied mathematics and partial differential equations.
Prerequisites:
basic courses in analysis and linear algebra, special relativity and quantum mechanics. Some knowledge of partial differential equations and functional analysis would be helpful but is not mandatory.
Recommended Literature:
- Thaller, The Dirac Equation, Springer (1992)
- Schweber, An Introduction to Relativistic Quantum Field Theory, Chap. 2-4, Dover Books (2005)
- Garabedian, Partial Differential Equations, AMS Chelsea Pub. (1998)
- Zauderer, Partial Differential Equations of Applied Mathematics, Wiley (2006)