Information Transfer Time: The Role of Holomorphism, Stationary Phase, and Noise

Abstract

In this paper we present an analysis of information transfer time based on holomorphism, causality and the classical principle of stationary phase. We also make a preliminary study of the effect of noise on information transfer time, and find that noise tends to increase transfer times. Noise and information signals are both essentially acausal, such that analytic continuation (i.e. prediction) is impossible, which also implies that their frequency spectra cannot be holomorphic. This leads to the paradox of a non-holomorphic information-bearing light signal, yet whose underlying Maxwell equations governing the propagation of the EM wave describe a holomorphic function in spacetime. We find that application of stationary phase and entropy arguments circumvents this difficulty, with stationary phase only suggesting the most likely transfer times of an information signal in the presence of noise. Faster transit times are not excluded, but are highly improbable. Stationary phase solutions, by definition, do not include signal forerunners, whose detection in the presence of noise is also unreliable. Hence a finite information capacity ensues, as expected from Shannon's law, and information cannot be transferred faster than c. We also find that the method of stationary phase implies complex transfer times. However, by considering spacetime to be isomorphic with the complex temporal plane, we find that an imaginary time is equivalent to a real distance, and can be interpreted as the uncertainty in the spatial position of the information pulse. Finally, we apply our theory to a photonic band gap crystal, and find that information transfer speed and tunneling is always subluminal.