ar X iv : 0 80 5 . 08 06 v 2 [ he p - th ] 1 7 Ju n 20 08 Modelling a Particle Detector in Field Theory

The Hamiltonian of traditionally adopted (“Unruh-DeWitt”) detector models features off diagonal elements between the vacuum and the one particle states of the field to be detected. We argue that in realistic detector models the configuration “detector in its ground state + vacuum of the field” should be an eigenstate of the Hamiltonian, because it generally corresponds to a stable bound state of the underlying fundamental theory (e.g. the ground state-hydrogen atom in a suitable QED with electrons and protons). As a concrete example, we study a local relativistic field theory where a stable particle can capture a light quantum and form a quasi-stable state. As expected, to such a stable particle correspond eigenstates of the full theory, as is shown explicitly by using a dressed particle formalism at first order in perturbation theory. We derive a model of detector (at rest) where the stable particle and the quasi-stable configurations correspond to the two internal levels, “ground” and “excited”, of the detector. Our analysis suggests that realistic detectors have no direct access to the local field degrees of freedom. As opposed to the Unruh-DeWitt detector, our model seems to show no response when forced along an accelerated trajectory. In order to produce operationally meaningful statements, physical theories are challenged by the problem of describing – within their own formalism – experiments and measurements. When field quantization is applied to general background spacetimes, particles are not universally defined [1]; still, with sound operational attitude, one can attempt to model a particle detector, calculate its response along some trajectory and associate a particle content to the corresponding observer/detector. A risk faced by this approach is that of modeling ad hoc measuring devices and ending up describing gedanken experiments that may eventually be contradicted by the responses of real apparatuses. For this reason, it is important to critically analyze the relation between model devices and the field theory that describes them at a more fundamental level. In this paper we highlight a generally expected property of particle detectors, namely, that of being described by bound state/eigenstate configurations of the underlying fundamental theory. We derive and discuss the implications that this property has on the construction of detector models. 1 Detector Models: a Critique A model detector [2, 3, 4] is a quantum system whose states live in a product Hilbert space HD⊗Hφ (i.e. detector and field) and provided with a Hamiltonian operator Hm = H m+H m+H m (suffix m stands for “model”). In the simplest scenario, φ is an – otherwise free – scalar field, i.e. H m = ∫ dkE(k)ckck, where E(k) = √ k2 +m2 and ck and ck are the usual creation and annihilation operators; the detector Hamiltonian H m accounts for at least two energy levels: unexcited, |0〉D, and excited, |E〉D; say that H m |E〉D = ∆E|E〉D, H m |0〉D = 0. Regardless of the choice of H m, the model state |0〉D ⊗ |0〉 is thus interpreted, by construction, as “the detector is in its ground state and the field is in its vacuum state”. The traditionally used Unruh–DeWitt detector features an interaction Hamiltonian of the type H m = σ φ(x(t), t), (1) where σ is a self adjoint operator acting onHD and containing off diagonal elements. For simplicity, assume that σ = σ↑ + σ↓, where σ↑ = |E〉DD〈0| and σ↓ = (σ↑). The above expression is meant to imply that the detector is a localized object following the trajectory x(t). A striking feature of this traditionally adopted detector is that the state |0〉D ⊗ |0〉 is not an eigenstate of (1), due to the presence of the creation operators ck inside φ. Accordingly, if the system is initially prepared in the configuration |0〉D ⊗ |0〉, there is always a non vanishing transition rate to a state of type |E〉D ⊗ |one particle〉 at finite times, regardless of the state of motion of the detector. In the interaction picture, and at first order in perturbation theory, the amplitude for this process reads