2022/01/31 by J. Brown, J. David Brown
Mathematics · Physics and Astronomy · #Canonical quantum gravity #Classical mechanics #Dirac (video compression format) #Gauge theory #General relativity #Hamiltonian (control theory) #Hamiltonian system #Lagrange multiplier #Loop quantum gravity #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Physics #Quantum #Quantum Mechanics and Applications #Quantum Mechanics and Non-Hermitian Physics #Quantum gravity #Quantum mechanics #Theoretical physics #gr-qc #hep-th #math-ph #math.MP
paper · pdf · doi:10.3390/universe8030171
published as Universe 2022, 8(3), 171 · Final Version
arxiv created 2022/03/09 · openalex publication_date 2022/03/09 · arxiv updated 2022/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.