On the canonical formulation of gauge field theories and Poincare transformations

Abstract

We address the Hamiltonian formulation of classical gauge field theories while putting forward results some of which are not entirely new, though they do not appear to be well known. We refer in particular to the fact that neither the canonical energy momentum vector $(P^{\mu })$ nor the gauge invariant energy momentum vector $(P_{\text{inv}}^{\mu })$ do generate space-time translations of the gauge field by means of the Poisson brackets: In a general gauge, one has to consider the so-called kinematical energy momentum vector and, in a specific gauge (like the radiation gauge in electrodynamics), one has to consider the Dirac brackets rather than the Poisson brackets. Similar arguments apply to rotations and to Lorentz boosts and are of direct relevance to the "nucleon spin crisis" since the spin of the proton involves a contribution which is due to the angular momentum vector of gluons and thereby requires a proper treatment of the latter. We conclude with some comments on the relationships between the different approaches to quantization (canonical quantization based on the classical Hamiltonian formulation, Gupta-Bleuler, path integrals, BRST, covariant canonical approaches).

43 pages; Version accepted for publication in Nucl. Phys. B. Includes the interpretation of results on Hamiltonian symmetry generators as well as a concise and unified derivation of the gauge invariant currents associated to conformal invariance

References in corpus (7)

• Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism
• Reply to the Comment on "Spin and orbital angular momentum in gauge theories: Nucleon spin structure and multipole radiation revisited" [PRL 100:232002 (2008)]
• Spin and orbital angular momentum in gauge theories: Nucleon spin structure and multipole radiation revisited
• Introduction to the Gribov Ambiguities In Euclidean Yang-Mills Theories
• Tutorial on Scale and Conformal Symmetries in Diverse Dimensions
• Nucleon spin decomposition and differential geometry
• Boundary terms in quantum field theory and the spin structure of QCD

Cited by in corpus (5)

• Covariant canonical formulations of classical field theories
• Flux Quantization
• Flux Quantization on Phase Space
• Improvement of a conserved current density versus adding a total derivative to a Lagrangian density
• Impact of gauge fixing on angular momentum operators of the covariantly quantized electromagnetic field