Electromagnetism, axions, and topology: A first-order operator approach to constitutive responses provides greater freedom

Abstract

We show how the standard constitutive assumptions for the macroscopic Maxwell equations can be relaxed. This is done by arguing that the Maxwellian excitation fields (D, H) should be dispensed with, on the grounds that they (a) cannot be measured, and (b) act solely as gauge potentials for the charge and current. In the resulting theory, it is only the links between the fields (E, B) and the charge and current (ρ, J ) that matter; and so we introduce appropriate linear operator equations that combine the Gauss and Maxwell-Ampère equations with the constitutive relations, eliminating (D, H). The result is that we can admit more types of electromagnetic media -notably, the new relations can allow coupling in the bulk to a homogeneous axionic material; in contrast to standard EM where any homogeneous axion-like field is completely decoupled in the bulk, and only accessible at boundaries. We also consider a wider context, including the role of topology, extended non-axionic constitutive parameters, and treatment of Ohmic currents. A range of examples including an axionic response material is presented, including static electromagnetic scenarios, a possible metamaterial implementation, and how the transformation optics paradigm would be modified. Notably, these examples include one where topological considerations make it impossible to model using (D, H).

Temporary Singularities and Axions: An Analytic Solution that Challenges Charge Conservation

“…in ref. [12], this extension permits a new axionic type of interaction with the electromagnetic field. In addition to dispensing with D and H, and in the presence of a spacetime singularity, it has been shown that Maxwell's equations no longer need enforce global charge conservation [11] ; but although posing some intriguing possibilities, this proposal did not include any explicit examples of constitutive relations, material configurations, or field distributions.…”

“…In ref. [12] a minimal extension to Maxwell's equations was considered where D and H did not appear, and which provided an additional axionic term to the vacuum Maxwell's equations. Using this axionic term, we now give in this article an explicit construction of an electromagnetic field configuration which can lead to the breaking of global charge conservation.…”

“…[15][16][17] However, in these, the model corresponds to well defined D and H, in-effect also demanding they are measureable, when in fact this is not necessary. [18][19][20][21][22][23][24] To avoid this unnecessary assumption, an alternative approach based purely on the physically measurable fields E and B was developed, [12] and this is what we use here.…”

“…The extension to Maxwell's equations [12] considered the constitutive properties of a background medium on the basis that there may be an axion field that does not derive from an axion scalar field. In contrast, in this article we posit that such an axion field can exist independently in the vacuum, instead of just being a property of some background medium.…”

“…This may include cases where a black hole forms and then subsequently evaporates. [8,44] In Section 2 we summarize the key points of a Maxwellian electrodynamics based on first-order operators [12] that dispenses with D and H and admits the topological axion field. In Section 3 we define a temporary singularity and its future.…”