How to Choose a Gauge? The Case of Hamiltonian Electromagnetism

Abstract

We develop some ideas about gauge symmetry in the context of Maxwell’s theory of electromagnetism in the Hamiltonian formalism. One great benefit of this formalism is that it pairs momentum and configurational degrees of freedom, so that a decomposition of one side into subsets can be translated into a decomposition of the other. In the case of electromagnetism, this enables us to pair degrees of freedom of the electric field with degrees of freedom of the vector potential. Another benefit is that the formalism algorithmically identifies subsets of the equations of motion that represent time-dependent symmetries. For electromagnetism, these two benefits allow us to define gauge-fixing in parallel to special decompositions of the electric field. More specifically, we apply the Helmholtz decomposition theorem to split the electric field into its Coulombic and radiative parts, and show how this gives a special role to the Coulomb gauge (i.e. div$$(\mathbf{A}) = 0$$ ( A ) = 0 ). We relate this argument to Maudlin’s (Entropy, 2018. https://doi.org/10.3390/e20060465) discussion, which advocated the Coulomb gauge.

The Hole Argument and Beyond: Part II: Treating Non-isomorphic Spacetimes

“…Namely, in the alternative symplectic framework, this same connection is also determined by symplectic orthogonality. Thus in the Hamiltonian formulation of electromagnetism, the part of the electric field that is determined by the instantaneous distribution of charges is encoded in a Coulombic potential, and the Singer-DeWitt connection can be obtained as a projector onto the symplectically orthogonal complement of this potential; (see [24] for a philosophical introduction). In the abelian Yang-Mills case, this connection is integrable; and it gives rise to the Coulomb gauge.…”

“…And these reasons need not be specific to a single problem or group of problems. For example: in the Hamiltonian treatment of electromagnetism, there are theoretical reasons to favour the Coulomb gauge that are wholly general, and so apply to any problem ( [24]).…”

“…Although we will stick to the more popular "physicists' " parametrization, i.e. A, our formalism can be readily adapted with minimal modifications to treating ω as the fundamental variable 24. For G g to be dimensionless (in units of ), it has to be multiplied by e−2 , where e is the Yang-Mills coupling constant.…”

The Gauge Argument: A Noether Reason

“…Each choice of physical coordinate system, or rather, of physical correlates of spacetime points, can be seen as a realiser of Einstein's 'point-coincidences' argument; each one answers the follow-up question that Einstein never pursued: point coincidences of what? 16 The similarities to gauge theory, and the relation between gauge-invariance and gauge-fixing, has been fleshed out in (Gomes, 2022;Gomes & Butterfield, 2022).…”