Entanglement between the Future and the Past in the Quantum Vacuum

Abstract

We note that massless fields within the future and past light cone may be quantized as independent systems. We show that the vacuum is an entangled state of these systems, exactly mirroring the known entanglement between the spacelike separated Rindler wedges. We describe a detector which exhibits a thermal response to the vacuum when switched on at t = 0. The feasibility of experimentally detecting this effect is discussed.PACS numbers: 03.70.+k, 03.65.Ud A basic and far-reaching property of the quantum vacuum is that it is an entangled state -a fact underlying an impressive number of theoretical insights and predictions [1]. In the case of flat Minkowski spacetime, this is typically shown in the context of the Unruh effect [2,3]. There, the vacuum state of the field can be written as an entangled state between two sets of modes, respectively spanning two space-time wedges, known as the Rindler wedges (see Fig. 1). A uniformly accelerated observer sees only one set of Rindler modes. The tracing out of the unobserved Rindler modes leads to the prediction that such an accelerated observer sees a thermalized vacuum. Having been predicted over 30 years ago, the Unruh effect remains unobserved.Here, our main result is to demonstrate that precisely the same entanglement exists between the fields within the future and past light cone (F and P, respectively) as between the left and right Rindler wedges (L and R), and that the Unruh effect can be mapped onto an equivalent vacuum thermal effect for an inertial observer constrained to interact with the field in only the future or the past. We will demonstrate this result for a scalar field in 2-d spacetime, though the result is general for massless fields. Dimensional analysis suggests that observation of this effect may be within range of current technology. This paper is organized as follows: We first note that massless fields in F and P may be quantized as independent systems, and then describe our coordinatization of spacetime, and the mode functions living in each quadrant. We then express the state of the Minkowski vacuum restricted to F and P in terms of these modes, and note entanglement. An Unruh-DeWitt detector is then described, which shows a thermal response to these modes in F (or P), and a related prediction of Martinetti and Rovelli [4] is interpreted in light of this result. The feasibility of an experimental observation of this effect is discussed, based on dimensional analysis. We then offer some conclusions.Past/future commutator and independent systems: The concept of entanglement between the left and right Rindler wedges rests on the fact that the fields within may be quantized as independent systems. This is expressed through the vanishing of the Pauli-Jordan function, i∆(x − y) = [φ(x),φ † (y)] for spacelike intervals. This general feature holds for both massive and massless fields.In the case of massless fields, however, the PauliJordan function ∆(x − y) vanishes for all but lightlike intervals, (x − y) 2 = 0 [5]. In particular, it vanishes for...

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