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 $(P_{\textrm{inv}} ^\mu )$ 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).
Highlights
In 1918, Emmy Noether published her famous article on invariant variational problems in which she stated and proved the so-called Noether theorem(s) [1]
It is commonly believed that, within the Hamiltonian formulation of a classical field theory, the Noether charges generate the symmetry transformations of the phase space variables φ ∈ {φ, π ≡ ∂L/∂φ} by means of the Poisson brackets, e.g. for infinitesimal translations, δaφ(x) ≡ {φ(x), aμPμ} = aμ∂μφ(x). This is the case for matter fields, but, as we will discuss in detail in the present article, it is definitely more subtle for a gauge field (Aμ): This is due to the fact that the gauge invariance of the action functional S[A] implies the presence of constraints for the phase space variables
Conserved charges associated to conformal invariance: We note that, in four spacetime dimensions, pure Abelian or non-Abelian gauge field theory is invariant under the Poincaré group, and under the larger conformal group: The generators corresponding to dilatations and special conformal transformations can be treated along the same lines
Summary
In 1918, Emmy Noether published her famous article on invariant variational problems in which she stated and proved the so-called Noether theorem(s) [1]. It is commonly believed that, within the Hamiltonian formulation of a classical field theory, the Noether charges generate the symmetry transformations of the phase space variables φ ∈ {φ, π ≡ ∂L/∂φ} by means of the Poisson brackets, e.g. for infinitesimal translations, δaφ(x) ≡ {φ(x), aμPμ} = aμ∂μφ(x) This is the case for matter fields (scalar or Dirac fields), but, as we will discuss in detail in the present article, it is definitely more subtle for a gauge field (Aμ): This is due to the fact that the gauge invariance of the action functional S[A] implies the presence of constraints for the phase space variables (as was already noted by Heisenberg and Pauli for electrodynamics in their pioneering work).
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