Induced Maxwell–Chern–Simons effective action in very special relativity

In this paper, we study the one-loop induced photon’s effective action in the very special relativity electrodynamics in $$(2+1)$$ ( 2 + 1 ) spacetime ($$\hbox {VSR}$$ VSR –$$\hbox {QED}_{3}$$ QED 3 ). Due to the presence of new nonlocal couplings resulting from the VSR gauge symmetry, we have additional graphs contributing to the $$\langle AA\rangle $$ ⟨ A A ⟩ and $$\langle AAA \rangle $$ ⟨ A A A ⟩ amplitudes. From these contributions, we discuss the VSR generalization of the Abelian Maxwell–Chern–Simons Lagrangian, consisting in the dynamical part and the Chern–Simons-like self-couplings, respectively. We use the VSR–Chern–Simons electrodynamics to discuss some non-Ohmic behavior on topological materials, in particular VSR effects on Hall’s conductivity. In the dynamical part of the effective action, we observe the presence of a UV/IR mixing, due to the entanglement of the VSR nonlocal effects to the quantum higher-derivative terms. Furthermore, in the self-coupling aspect, we verify the validity of the Furry’s theorem in the $${\hbox {VSR}}$$ VSR –$$\hbox {QED}_{3}$$ QED 3 explicitly.

Stability of viscous flow between rotating cylinders. II. Cylinders rotating in opposite directions

In a previous paper (Meksyn 1946; it will be referred to as Part I) the problem of stability of viscous fluid between coaxial cylinders, rotating in the same direction, was solved by expanding the integrals in inverse powers of a large parameter. The question was first considered theoretically and experimentally by Taylor (1923), the problem being solved by expanding the integrals in orthogonal Bessel functions. The aim of the present paper is to extend the solution to the case when the cylinders rotate.in opposite directions. The important difference between the two cases in its mathematical aspect consists of the following. It is necessary to find the asymptotic integrals of a certain linear differential equation. In the case when the cylinders rotate in opposite directions, these integrals become infinite within the range under consideration; namely, approximately at the point where the mean velocity of rotation is equal to zero, and the asymptotic expansions change their form in passing through this point. It is, therefore, necessary to find the law of transformation of these integrals; that requires a rather extensive mathematical investigation. For the sake of convenience the work is divided into two separate parts, hydro - dynamical (Part II) and mathematical (Part III). In the present paper (Part II) the transformations are only quoted, whereas in the mathematical part the detailed solution of the equations is developed. The main results obtained are as follows.

The invariant theory of isotropic turbulence

The statistical theory of isotropic turbulence, initiated by Taylor (3) and extended by de Kármán and Howarth (2), has proved of value in attacking problems associated with the decay of turbulence. In its application to such hydro-dynamical problems, the theory falls into two parts, a kinematical part and a dynamical part. The kinematical aspect consists in setting up correlations between velocity components, or their derivatives, at two arbitrary points in the fluid, and reducing the form of the tensor thus obtained in accordance with the severely restrictive assumption of isotropic turbulence; the success of de Kármán and Howarth's investigations is largely attributable to their improved treatment of this purely kinematical problem. The dynamical part then consists in applying the implications of the equations of continuity and motion to the functions defining the correlation tensors, in order to obtain information concerning their functional dependence on time and on the displacement between the two points for which the correlations are computed.