Aharonov–Bohm rings in de Sitter expanding universe

Abstract

The $(1+3)$-dimensional Dirac equation of the fermions moving in ideal Aharonov–Bohm rings in the de Sitter expanding universe is used for deriving the exact expressions of the general relativistic partial currents and corresponding energies. In the de Sitter geometry, these quantities depend on time but these are related each other just as in the non-relativistic case or in special relativity. A specific relativistic effect is the saturation of the partial currents for high values of the total angular momentum. The total relativistic persistent current at $T=0$ takes over this property even though it is evolving in time because of the de Sitter expansion.

A modified Bessel functions

According to the general properties of the modified Bessel functions, $I_{\nu }(z)$ and $K_{\nu }(z)=K_{-\nu }(z)$ [40], we deduce that those used here, $K_{\nu _{\pm }}(z)$, with $\nu _{\pm }=\frac{1}{2}\pm i \mu$ are related among themselves through

$$\begin{aligned} H^{(1,2)}_{\nu }(z)=\mp \frac{2i}{\pi }e^{\mp \frac{i}{2}\pi \nu }K_{\nu }(\mp iz), \quad z\in {\mathbb R}. \end{aligned}$$

(49)

The functions used here, $K_{\nu _{\pm }}(z)$ with $\nu _{\pm }=\frac{1}{2}\pm i \mu$ ($\mu \in {\mathbb R}$), are related among themselves through

$$\begin{aligned}{}[K_{\nu _{\pm }}(z)]^{*} =K_{\nu _{\mp }}(z^*),\quad \forall z \in {\mathbb C}, \end{aligned}$$

(50)

satisfy the equations

$$\begin{aligned} \left( \frac{d}{dz}+\frac{\nu _{\pm }}{z}\right) K_{\nu _{\pm }}(z)=-K_{\nu _{\mp }}(z), \end{aligned}$$

(51)

and the identities

$$\begin{aligned} K_{\nu _{\pm }}(i z)K_{\nu _{\mp }}(-i z)+ K_{\nu _{\pm }}(-i z)K_{\nu _{\mp }}(i z)=\frac{\pi }{| z|},\quad z\in {\mathbb R}. \end{aligned}$$

(52)

For $|z|\rightarrow \infty$ these functions behave as [40]

$$\begin{aligned} I_{\nu }(z) \rightarrow \sqrt{\frac{\pi }{2z}}e^{z}, \quad K_{\nu }(z) \rightarrow K_{\frac{1}{2}}(z)=\sqrt{\frac{\pi }{2z}}e^{-z}, \end{aligned}$$

(53)

regardless the index $\nu$.

The asymptotic approximation is rough since it looses the dependence on index of the functions $K_{\nu }$. For this reason we propose the pre-asymptotic approximation,

$$\begin{aligned} K_{\frac{1}{2}\pm i\nu }(i\nu z)\sim \sqrt{\frac{\pi }{2\nu z}}\frac{\sqrt{\sqrt{1+z^2}\pm 1}}{(1+z^2)^{\frac{1}{4}}}\,e^{-i\nu \sqrt{1+z^2}+i\Theta (z)},\quad z,\nu \in {\mathbb R}^+, \end{aligned}$$

(54)

where $\Theta (z)$ remains undefined. This is a numerically satisfactory approximation inspired by the uniform expansion of the modified Bessel functions which is proved only for real or pure imaginary indices [39, 40]. Moreover, assuming that $\Theta (z) \rightarrow \frac{\pi }{4}$ for very large $z'=\nu z$ we recover Eq. (53) with $z\rightarrow i z'$ .