We give an upper bound of the relative entanglement entropy of the ground state of a massive Dirac-Majorana field across two widely separated regions A and B in a static slice of an ultrastatic Lorentzian spacetime. Our bound decays exponentially in dist(A, B), at a rate set by the Compton wavelength and the spatial scalar curvature. The physical interpretation our result is that, on a manifold with positive spatial scalar curvature, one cannot use the entanglement of the vacuum state to teleport one classical bit from A to B if their distance is of the order of the maximum of the curvature radius and the Compton wave length or greater.

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A Duistermaat-Guillemin-Gutzwiller trace formula for Dirac-type operators on a globally hyperbolic spatially compact standard stationary spacetime is achieved by generalising the recent construction by A. Strohmaier and S. Zelditch [Adv. Math. 376, 107434 (2021)] to a vector bundle setting. We have analysed the spectrum of the Lie derivative with respect to the Killing vector field on the solution space of the Dirac equation and found that it consists of discrete real eigenvalues. The distributional trace of the Cauchy evolution operator has singularities at the periods of induced Killing flow on the space of lightlike geodesics. This gives rise to the Weyl law asymptotic at the vanishing period.1 We do not require Einstein's field equations to be satisfied.2 Global existence of a Lorentz metric depends on the topology of M . In particular, such a metric exists in all non-compact manifolds and compact manifolds with vanishing Euler characteristics. 3A spin-structure always exists locally but its global existence depends on some higher orientability property of the base manifold. For example, spin c and spin structures exist if and only if the third integral and the second Stiefel-Whitney class of spacetime vanish, respectively.

A Duistermaat–Guillemin–Gutzwiller trace formula for Dirac-type operators on a globally hyperbolic spatially compact stationary spacetime is achieved by generalising the recent construction by Strohmaier and Zelditch (Adv Math 376:107434, 2021) to a vector bundle setting. We have analysed the spectrum of the Lie derivative with respect to a global timelike Killing vector field on the solution space of the Dirac equation and found that it consists of discrete real eigenvalues. The distributional trace of the time evolution operator has singularities at the periods of induced Killing flow on the space of lightlike geodesics. This gives rise to the Weyl law asymptotic at the vanishing period.

Gunn oscillation can occur via the transferred electron effect or the negative differential mass effect. We have mathematically formulated the frequency response of a Gunn diode operating in the transit time mode independent of the mechanism responsible for Gunn oscillation in bulk semiconductors. Domain growth dynamics with space and time, variation of domain velocity, and frequency response have been simulated for Gallium Nitride (GaN) by using our mathematical equations. Our simulation shows that gallium nitride based Gunn diodes at an active length of 5 micrometer can produce frequency around 40 GHz for DC biasing 150 V.