Cosmological Zitterbewegung

We describe a new phenomenon of zitterbewegung of a free Dira c particle in cosmological spacetimes. Unlike the similar effect theori zed by Schrödinger in 1930, the cosmological zitterbewegung is a real, physica lly attainable effect, which originates from the mixing of positive and negat ive frequency modes of a field operator in cosmological spacetimes. We brie fly discuss the potential for observing this effect in laboratory exper iments with trapped ions. Essay received an Honorable Mention in the Gravity Research Foundation essay contest 2013 1pgir1104@uni.sydney. edu.au 2archilk@physics.usyd. edu.au A consistent merger of relativity and quantum mechanics bri ngs in full flavour of interesting, qualitatively new physical phenomena. The great example of this is Dirac’s relativistic theory of electron [1], which not on ly successfully explained the origin of the electron’s spin and its magnetic moment, bu t also introduced the notion of antiparticles and laid down a foundation for moder n quantum-mechanical description of vacuum. It took quite sometime for these revolutionary insights to b e properly understood within the framework of quantum field theory. In the ear ly days, misinterpretations of negative frequency solutions of the Dirac equation often lead to physically controversial conclusions. In 1930 E. Schrödi nger published a paper [2] where, by allowing a linear superposition of positive an d negative frequency electron wavefunctions, he demonstrated that a free relati vistic electron undergoes a trembling motion. This phenomenon of zitterbewegung was in odds with Ehrenfest’s theorem [3], according to which the expectatio n value of the velocity operator of a free electron should behave as the classica l velocity, i.e. it must be constant. As it was understood subsequently, Schrodinge r’s derivation of the zitterbewegung effect was based on an incorrect treatment o f negative frequency modes. With the correct approach both within the first-quant ized Dirac equation in the Foldy-Wouthuysen representation [4] and within the s econd-quantized theory one can demonstrate that zitterbewegung is an unphysica l effect, unobservable by any inertial observer. Even more unusual phenomena emerge when one considers quant um fields in curved spacetimes, e.g. within the framework of semiclas sical gravity, where inertial reference frames are defined only locally. Since an inertial frame is not available globally, the separation of negative and positiv e frequency modes is typically impossible globally, and thus the standard flat space time notions of particle and anti-particle become ambiguous. This lead to a particle creation phenomenon in curved spacetime which is manifested in the Hawking radia tion of black holes [5], the Unruh radiation in an uniformly accelerated refere nc frame [6], the cosmological particle production [7], which is believed to be r esponsible for primordial anisotropies in the cosmic microwave background radia tion generated during inflation, and other related phenomena. In this essay we would like to describe a new phenomenon of cos mological zitterbewegung, which unlike the effect originally propos ed by Schrodinger in flat spacetime [2], is a physical effect precisely because of int rinsic mixing of positive and negative modes of a free Dirac field in cosmological space times. Similar effect has been recently considered by one of us for the Dirac equation in an uniformly accelerated reference frame [8]. In what follows we consider a toy cosmological model in (1+1) -dimensions described by the metric: ds = dt − a(t)dx , (1)