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Foundations of Physics

Manifestly Covariant Quantum Theory with Invariant Evolution Parameter in Relativistic Dynamics

John R. Fanchi

Published Nov 12, 2009• 28 citations
Publisher
DOI
10.1007/s10701-009-9371-0

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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).
Entropic Dynamics (ED) is a framework for constructing dynamical theories of inference using the tools of inductive reasoning. A central feature of the ED framework is the special focus placed on time. In [2] a global entropic time was used to derive a quantum theory of relativistic scalar fields. This theory, however, suffered from a lack of explicit or manifest Lorentz symmetry. In this paper we explore an alternative formulation in which the relativistic aspects of the theory are manifest.The approach we pursue here is inspired by the methods of Dirac, Kuchař, and Teitelboim in their development of covariant Hamiltonian approaches. The key ingredient here is the adoption of a local notion of entropic time, which allows compatibility with an arbitrary notion of simultaneity. However, in order to ensure that the evolution does not depend on the particular sequence of hypersurfaces, we must impose a set of constraints that guarantee a consistent evolution.