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Register a new userDeformed Hopfion-Ra~nada Knots in ModMax Electrodynamics
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Source-free so-called ModMax theories of nonlinear electrodynamics in the four dimensional Minkowski spacetime vacuum are the only possible continuous deformations -- and as a function of a single real and positive parameter -- of source-free Maxwell linear electrodynamics in the same vacuum, which preserve all the same Poincare and conformal spacetime symmetries as well as the continuous duality invariance of Maxwells theory. Null field configurations of the latter however, including null electromagnetic knots, are singular for the Lagrangian formulation of any spacetime Poincare and conformal invariant theory of nonlinear electrodynamics. In particular null hopfion-Ra~nada knots are a distinguished and fascinating class on their own of topologically nontrivial solutions to Maxwells equations. This work addresses the fate of these configurations within ModMax theories. A doubled class of ModMax deformed hopfion-Ra~nada knots is thereby identified, each of which coalescing back in a continuous fashion to the original hopfion-Ra~nada knot when the nonlinear deformation parameter is turned off.
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The aims of this letter are three-fold: First is to show that nonlinear generalizations of electrodynamics support various types of knotted solutions in vacuum. The solutions are universal in the sense that they do not depend on the specific Lagrangian density, at least if the latter gives rise to a well-posed theory. Second is to describe the interaction between probe waves and knotted background configurations. We show that the qualitative behaviour of this interaction may be described in terms of Robinson congruences, which appear explicitly in the causal structure of the theory. Finally, we argue that optical arrangements endowed with intense background fields could be the natural place to look for the knots experimentally.
The ether concept -- abandoned for a long time but reinstated by Dirac in 1951-1953 -- has in recent years emerged into a fashionable subject in theoretical physics, now usually with the name of the Einstein-Dirac ether. It means that one special inertial frame is singled out, as the rest frame. What is emphasized in the present note, is that the idea is a natural example of the covariant theory of quantum electrodynamics in media if the refractive index is set equal to unity. A treatise on this case of quantum electrodynamics was given by the present author back in 1971, published then only within a preprint series. The present version is a brief summary of that formalism, with a link to the original paper. We think it is one of the first treatises on modern ether theory.
In this paper, a formulation, which is completely established on a quantum ground, is presented for basic contents of quantum electrodynamics (QED). This is done by moving away, from the fundamental level, the assumption that the spin space of bare photons should (effectively) possess the same properties as those of free photons observed experimentally. Within this formulation, bare photons with zero momentum can not be neglected when constructing the photon field; and an explicit expression for the related part of the photon field is derived. When a local gauge transformation is performed on the electron field, this expression predicts a change that turns out to be equal to what the gauge symmetry requires for the gauge field. This gives an explicit mechanism, by which the photon field may change under gauge transformations in QED.
We construct a new family of null solutions to Maxwells equations in free space whose field lines encode all torus knots and links. The evolution of these null fields, analogous to a compressible flow along the Poynting vector that is both geodesic and shear-free, preserves the topology of the knots and links. Our approach combines the Bateman and spinor formalisms for the construction of null fields with complex polynomials on $mathbb{S}^3$. We examine and illustrate the geometry and evolution of the solutions, making manifest the structure of nested knotted tori filled by the field lines.
Gauge invariance, a core principle in electrodynamics, has two separate meanings, only one of which is robust. The reliable concept treats the photon as the gauge field for electrodynamics. It is based on symmetries of the Lagrangian, and requires no mention of electric or magnetic fields. The other depends directly on the electric and magnetic fields, and how they can be represented by potential functions that are not unique. The first gauge concept has been fruitful, whereas the second has the defect that there exist gauge transformations from physical to unphysical states. The fields are unchanged by the gauge transformation, so that potentials are the necessary guides to correctness.
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