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arXiv:quant-ph/0106079 · Journal of Modern Optics

Lorentz transformations of open systems

Asher Peres, Daniel R. Terno

Published Jul 1, 2002• 17 citations
arXivHTMLPDF

Abstract

We consider open dynamical systems, subject to external interventions by agents that are not completely described by the theory (classical or quantal). These interventions are localized in regions that are relatively spacelike. Under these circumstances, no relativistic transformation law exists which relates the descriptions of the physical system by observers in relative motion. Still, physical laws are the same in all Lorentz frames.

DOI
10.1080/09500340110108602

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Reply to “Comment on ‘Classical interventions in quantum systems. II. Relativistic Invariance’ ”

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Contrary to the claim in the preceding Comment, the operators that describe the complete time evolution of a quantum system have a covariant transformation law, even though they are built from expressions that belong to separate subsystems and do not obey Lorentz transformations. The reason is that these are open subsystems, and different parts of the universe are excluded in their description. However, the complete system still obeys Lorentz covariance.

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We consider open dynamical systems, subject to external interventions by agents that are not completely described by the theory (classical or quantal). These interventions are localized in regions that are relatively spacelike. Under these circumstances, no relativistic transformation law exists which relates the descriptions of the physical system by observers in relative motion. Still, physical laws are the same in all Lorentz frames.
If several interventions performed on a quantum system are localized in mutually spacelike regions, they will be recorded as a sequence of ``quantum jumps'' in one Lorentz frame, and as a different sequence of jumps in another Lorentz frame. Conditions are specified that must be obeyed by the various operators involved in the calculations, so that these two different sequences lead to the same observable results. These conditions are similar to the equal-time commutation relations in quantum field theory. They are sufficient to prevent superluminal signaling. (The derivation of these results does not require most of the contents of the preceding article [Peres, Phys. Rev. A 61, 022116 (2000)]. What is needed is briefly summarized here, so that the present article is essentially self-contained.)
To investigate how quantum effects might modify special relativity, we will study a Lorentz transformation between classical and quantum reference frames and express it in terms of the four-dimensional (4D) momentum of the quantum reference frame. The transition from the classical expression of the Lorentz transformation to a quantum-mechanical one requires us to symmetrize the expression and replace all its dynamical variables with the corresponding operators, from which we can obtain the same conclusion as that from quantum field theory (given by Weinberg's formula): owing to the Heisenberg's uncertainty relation, a particle (as a quantum reference frame) can propagate over a spacelike interval.
The dynamics of a system of particles acting on one another at a distance can be relativistically invariant if the assumption of invariant world-lines is given up. This is shown by constructing a particular dynamics in which invariance over the homogeneous Lorentz group is trivial, but space as well as time displacement requires the solution of equations of motion or of a Schroedinger equation. This particular dynamics reduces in the nonrelativistic limit to the most general dynamics of a system of interacting particles admitting the Newtonian group.
Irreversibility implies a preferred flow of time, yet special relativity denies the existence of a preferred clock. This tension has long obstructed the formulation of a relativistic master equation: standard Markovian approximations either break Lorentz covariance, trigger catastrophic vacuum heating, or depend arbitrarily on the observer's foliation. In this work, we derive a Lorentz-invariant description of irreversibility for quantum fields. We take an approach that explicitly models the measurements required to observe irreversible dynamics. Instead of evolving the system along an abstract geometric time parameter, we anchor the dynamics to a physical, relational scalar clock field. Using a relational Tomonaga-Schwinger framework, we derive a local, non-Markovian master equation that is manifestly covariant and completely positive. We show that the finite resolution of the physical clock acts as a covariant regulator, preventing the vacuum instability that plagues white-noise models. This framework demonstrates that a consistent relativistic theory of decay exists, provided the reference frame is treated as a dynamical quantum resource rather than a gauge choice. In a gravitating context, the resulting dynamics is described by a hybrid classical-quantum (CQ) evolution that remains completely positive and trace preserving (CPTP).
In most approaches to fundamental physics, spacetime symmetries are postulated a priori and then explicitly implemented in the theory. This includes Lorentz covariance in quantum field theory and diffeomorphism invariance in quantum gravity, which are seen as fundamental principles to which the final theory has to be adjusted. In this paper, we suggest, within a much simpler setting, that this kind of reasoning can actually be reversed, by taking an operational approach inspired by quantum information theory. We consider observers in distinct laboratories, with local physics described by the laws of abstract quantum theory, and without presupposing a particular spacetime structure. We ask what information-theoretic effort the observers have to spend to synchronize their descriptions of local physics. If there are 'enough' observables that can be measured universally on several different quantum systems, we show that the observers' descriptions are related by an element of the orthochronous Lorentz group , together with a global scaling factor. Not only does this operational approach predict the Lorentz transformations, but it also accurately describes the behavior of relativistic Stern–Gerlach devices in the WKB approximation, and it correctly predicts that quantum systems carry Lorentz group representations of different spin. This result thus hints at a novel information-theoretic perspective on spacetime.