Singular Lagrangians, Constrained Hamiltonian Systems and Gauge Invariance: An Example of the Dirac–Bergmann Algorithm

Universe

J. David Brown¹ 

[1] Department of Physics, North Carolina State University, Raleigh, NC 27695, USA;

关键词: gauge theories;    general relativity;    classical mechanics;    Hamiltonian formalism;   

DOI  :  10.3390/universe8030171

来源: DOAJ

【 摘 要 】

The Dirac–Bergmann algorithm is a recipe for converting a theory with a singular Lagrangian into a constrained Hamiltonian system. Constrained Hamiltonian systems include gauge theories—general relativity, electromagnetism, Yang–Mills, string theory, etc. The Dirac–Bergmann algorithm is elegant but at the same time rather complicated. It consists of a large number of logical steps linked together by a subtle chain of reasoning. Examples of the Dirac–Bergmann algorithm found in the literature are designed to isolate and illustrate just one or two of those logical steps. In this paper, I analyze a finite-dimensional system that exhibits all of the major steps in the algorithm. The system includes primary and secondary constraints, first and second class constraints, restrictions on Lagrange multipliers, and both physical and gauge degrees of freedom. This relatively simple system provides a platform for discussing the Dirac conjecture, constructing Dirac brackets, and applying gauge conditions.

【 授权许可】

Unknown