Local quanta, unitary inequivalence, and vacuum entanglement

Abstract

In this work we develop a formalism for describing localised quanta for a real-valued Klein-Gordon field in a one-dimensional box [0, R]. We quantise the field using non-stationary local modes which, at some arbitrarily chosen initial time, are completely localised within the left or the right side of the box. In this concrete set-up we directly face the problems inherent to a notion of local field excitations, usually thought of as elementary particles. Specifically, by computing the Bogoliubov coefficients relating local and standard (global) quantizations, we show that the local quantisation yields a Fock space F L which is unitarily inequivalent to the standard one F G . In spite of this, we find that the local creators and annihilators remain well defined in the global Fock space F G , and so do the local number operators associated to the left and right partitions of the box. We end up with a useful mathematical toolbox to analyse and characterise local features of quantum states in F G . Specifically, an analysis of the global vacuum state |0 G ∈ F G in terms of local number operators shows, as expected, the existence of entanglement between the left and right regions of the box. The local vacuum |0 L ∈ F L , on the contrary, has a very different character. It is neither cyclic nor separating and displays no entanglement. Further analysis shows that the global vacuum also exhibits a distribution of local excitations reminiscent, in some respects, of a thermal bath. We discuss how the mathematical tools developed herein may open new ways for the analysis of fundamental problems in local quantum field theory.

What does it mean for half of an empty cavity to be full?

“…The reduced state at time t ¼ 0 of, say, the left side of the cavity can then be represented with respect to the Fock basis corresponding to fâ m ;â † m g. As extensively discussed in [1], this provides a well-defined notion of the reduced state within a localized region. Indeed it is equivalent, up to a change of basis, to any other well-formulated notion of spatial reduced state (for example, by taking the continuum limit of a discretized lattice).…”

“…The obvious way of doing this is to define these modes to have support at a certain time t ¼ 0 only over their corresponding subregions. As pointed out in [1], however, one must be careful that the new basis modes still satisfy the correct boundary conditions of the cavity (and, in particular, not extra ones). This requirement immediately implies that if, say, the set fu m g are supported only in the left region at t ¼ 0, then their support will necessarily exceed this region for later times (u m satisfies the wave equation and we have not placed an extra boundary condition between the two regions).…”

“…These modes satisfy the proper boundary conditions and constitute a complete and orthonormal basis for the whole cavity [1], and thus form a proper expansion of the field. For our purposes in this section, however, we need only consider the instant t ¼ 0 at which they are localized to their respective sides of the cavity.…”

“…In short, powerful as it is, this positive energy representation provides only a feeble scaffolding for digging into issues pertaining to the localization of quanta. Here we will use an alternative local Fock space representation [1] that enables us to address these questions.…”

“…We begin by discussing the difficulties that appear when we intend to measure local vacuum excitations and the different scenarios that could allow us to circumvent them. We will be using the formalism of local quantization introduced in [1], which allows us to characterize the reduced state of a subregion in the cavity and study its local properties formally. We will utilize Gaussian quantum mechanics [16] in order to easily compute and characterize the reduced states of subcavity regions and the correlations between them, explaining how this equivalently describes the physics of slamming mirror(s) into the cavity.…”