Dirac fermion, cosmological event horizons, and quantum entanglement

Abstract

We discuss the field quantisation of a free massive Dirac fermion in the two causally disconnected static patches of the de Sitter spacetime, by using mode functions that are normalisable on the cosmological event horizon. Using this, we compute the entanglement entropy of the vacuum state corresponding to these two regions, for a given fermionic mode. Further extensions of this result to more general static spherically symmetric and stationary axisymmetric spacetimes are discussed. For the stationary axisymmetric Kerr-de Sitter spacetime in particular, the variations of the entanglement entropy with respect to various eigenvalues and spacetime parameters are depicted numerically. keywords : de Sitter, cosmological horizon, fermionic entanglement, stationary axisymmetric spacetimes * sbhatta@iitrpr.ac.in † s.chakrabortty@iitrpr.ac.in ‡ 2017phz0003@iitrpr.ac.inThere has been a tremendous effort over decades to explore various aspects of quantum fields living in a de Sitter universe. A complete review on this topic is far from the scope of this paper. We refer our reader to [2,3,4] for various aspects of particle creation and vacuum states in the cosmological de Sitter spacetime. See [5] and references therein for a study on the Schwinger effect in de Sitter. See e.g. [6,7,8,9,10,11] for aspects of particle creation and thermal effects in the static de Sitter or de Sitter black hole spacetimes. Further, we refer our reader to e.g. [12,13,14] (also references therein) for discussions on the late time non-perturbative infrared effects in the cosmological de Sitter spacetime.A natural and interesting aspect of the de Sitter space is the relativistic quantum entanglement of fields, which is the focus of this paper. If we consider an 'in' vacuum state in the cosmological de Sitter spacetime, due to the accelerated expansion, the state may evolve in the future to a different or 'out' vacuum state, indicating particle pair production. Such pairs turn out to be entangled. On the other hand, due to the accelerated expansion, all parts of the de Sitter space cannot be causally connected. Quantum fields living in various causally disconnected parts of de Sitter can show very non-trivial aspect of quantum entanglement. We refer our reader to [15]-[29] and references therein for a study of quantum field theoretic entanglement in the cosmological and hyperbolic coordinatisation of de Sitter. We further refer our reader to [30] and references therein for a study of holographic aspects of de Sitter entanglement entropy.The static coordinatisation of de Sitter is interesting in the sense that the cosmological event horizon is explicitly 'visible' in it and second, it is explicitly time translational invariant (within the cosmological event horizon), e.g. [6]. The maximal analytic extension of the spacetime across this horizon shows, alike that of the non-extremal black hole or the Rindler horizon, four causally disconnected spacetime regions, two of which are static, Section 2. A quantum field living in these two regions possesses...

Some measures for fermionic entanglement in the cosmological de Sitter spacetime

“…1-4) with a scalar field theory discussed respectively in the context of transition from de Sitter to radiation dominated era [42] and the hyperbolic de Sitter spacetime [41]. However, the chief qualitative difference for the cosmological de Sitter from them or the noninertial frame (e.g., [21]), or even the static de Sitter spacetime (e.g., [50]), is that the Bogoliubov coefficients for the "in" and "out" vacua in this case is independent of the spatial momentum or the total energy, Eq. (A12).…”

“…Another well motivated sector of the relativistic entanglement is the cosmological scenario, important chiefly because such study might provide us insight about the initial state as well as the geometry of the early inflationary universe. Such studies might involve investigations like the vacuum entanglement entropy, various other measures of quantum entanglement between initially entangled states for both bosonic and fermionic fields as well as the quantum decoherence of cosmological perturbations and their possible observational consequences, e.g., [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52] and references therein.…”