Abstract
We examine an Unruh-DeWitt particle detector that is coupled linearly to the scalar density of a massless Dirac field in Minkowski spacetimes of dimension $d\ge2$ and on the static Minkowski cylinder in spacetime dimension two, allowing the detector's motion to remain arbitrary and working to leading order in perturbation theory. In $d$-dimensional Minkowski, with the field in the usual Fock vacuum, we show that the detector's response is identical to that of a detector coupled linearly to a massless scalar field in $2d$-dimensional Minkowski. In the special case of uniform linear acceleration, the detector's response hence exhibits the Unruh effect with a Planckian factor in both even and odd dimensions, in contrast to the Rindler power spectrum of the Dirac field, which has a Planckian factor for odd $d$ but a Fermi-Dirac factor for even~$d$. On the two-dimensional cylinder, we set the oscillator modes in the usual Fock vacuum but allow an arbitrary state for the zero mode of the periodic spinor. We show that the detector's response distinguishes the periodic and antiperiodic spin structures, and the zero mode of the periodic spinor contributes to the response by a state-dependent but well defined amount. Explicit analytic and numerical results on the cylinder are obtained for inertial and uniformly accelerated trajectories, recovering the $d=2$ Minkowski results in the limit of large circumference. The detector's response has no infrared ambiguity for $d=2$, neither in Minkowski nor on the cylinder.
Highlights
In quantum field theory, the interaction between a scalar field and an observer is often studied by modelling the observer by a spatially pointlike system with discrete energy levels, an Unruh-DeWitt detector [1, 2]
In the special case of Minkowski vacuum, the divergent term is proportional to the mass of the field, and for a massless field a consistent regularisation is accomplished by dropping the additive term [22, 23]
Working within first-order perturbation theory, we regularised the interaction by dropping an additive term that is technically ill-defined but formally proportional to the field’s mass [22, 23]
Summary
The interaction between a scalar field and an observer is often studied by modelling the observer by a spatially pointlike system with discrete energy levels, an Unruh-DeWitt detector [1, 2]. In this paper we consider an Unruh-DeWitt detector coupled to a Dirac field, taking the interaction Hamiltonian to be linear in the Dirac field’s scalar density, ψψ [20, 21, 22, 23, 24]. Our first objective is to evaluate the detector’s response on an arbitrary trajectory in Minkowski spacetime of dimension d ≥ 2 when the field is initially prepared in Minkowski vacuum, working in linear perturbation theory and allowing the detector to be switched on and off in an arbitrary smooth way. Our results show that the detector’s response has no infrared ambiguity, neither in Minkowski nor on the cylinder In this respect the massless Dirac field differs from the massless scalar field, whose response in two-dimensional Minkowski vacuum is ambiguous due to the additive ambiguity in the Wightman function [26]. Overline on a scalar denotes the complex conjugate and overline on a spinor denotes the Dirac conjugate. o(1) denotes a quantity that tends to zero in the limit under consideration
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