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

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arxiv 2201.06558 v3 pith:GOHMH5W5 submitted 2022-01-17 gr-qc hep-thmath-phmath.MP

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

classification gr-qc hep-thmath-phmath.MP
keywords algorithmdirac-bergmanngaugesystemconstrainedhamiltonianstepsconstraints
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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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.

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