Noncovariant gauge fixing in the quantum Dirac field theory of atoms and molecules

Abstract

The formalism of quantum mechanical gauge fixing in quantum electrodynamics (QED) is extended using techniques from non-relativistic QED. This involves expressing the redundant gauge degrees of freedom through an arbitrary functional of the gauge invariant transverse degrees of freedom. Particular choices of functional can be made to yield the Coulomb or Poincaré gauge representation of the Hamiltonian. The Hamiltonian we derive therefore serves as a good starting point for the relativistic description of atoms and molecules. Important Implications of the gauge freedom present in the Hamiltonian with regards to the ontology of QED in general are discussed.

A Physically-Motivated Quantisation of the Electromagnetic Field on Curved Spacetimes

“…For example, in the canonical approach, standard commutation relations cannot be satisfied. One can get around this by either breaking Lorentz invariance in intermediate steps of calculations, or by considering excess degrees of freedom with negative norms that do not contribute physically [ 37 ]. Standard path integral quantisation fails for electromagnetism because the resultant propagator is divergent.…”

“…However, the dynamics of subsystem observables O can depend on the concrete choice of U , since and O are in general not the same. For example, atom–field interactions depend on the gauge-dependent vector potential for most subsystem decompositions [ 36 , 37 ]. Hence it is important here to formulate quantum electrodynamics in an entirely arbitrary gauge and to maintain ambiguity as long as possible, thereby retaining the ability to later choose a gauge which does not result in the prediction of spurious effects [ 38 ].…”

“…Retaining gauge-independence is important when modelling the interaction of the electromagnetic field with another quantum system, like an atom. In this case, different choices of gauge correspond to different subsystem decompositions, thereby affecting our notion of what is ‘atom’ and what is ‘field’ [ 36 , 37 ]. Composite quantum systems can be decomposed into subsystems in many different ways.…”

A master equation for strongly interacting dipoles

“…Letting α k =1 in equation (2) yields the well-known Power-Zienau-Woolley (PZW) transformation that relates the Coulomb and multipolar gauges. While the relation between the Coulomb and multipolar gauge has been discussed extensively [27][28][29][30][31], the PZW transformation is in fact a special case of a broader class of unitary gauge-fixing transformations [31][32][33]. More generally still, the freedom to choose the α k within the canonical transformation (2) implies redundancy within our mathematical description and is henceforth referred to as generalised gauge-freedom.…”

“…The Γ μν (ω) are symmetric w w G = G mn nm ( ) ( ) and can be written In deriving equation (33) we have not yet performed a secular approximation, in contrast to the derivation of equation (20). However, naively applying a secular approximation that neglects off-diagonal terms for which z z ¹ ¢ in the summand in equation (33) would not be appropriate, because this would eliminate terms that are resonant in the limit  C 0. Instead we perform a partial secular approximation which eliminates off-diagonal terms for which ζ and ζ′ have opposite sign.…”