Entropy and correlators in quantum field theory

Abstract

It is well known that loss of information about a system, for some observer, leads to an increase in entropy as perceived by this observer. We use this to propose an alternative approach to decoherence in quantum field theory in which the machinery of renormalisation can systematically be implemented: neglecting observationally inaccessible correlators will give rise to an increase in entropy of the system. As an example we calculate the entropy of a general Gaussian state and, assuming the observer's ability to probe this information experimentally, we also calculate the correction to the Gaussian entropy for two specific non-Gaussian states. PACS numbers: 03.65.Yz, 03.70.+k, 03.67.-a, 98.80.-k I. INTRODUCTIONThe most natural way of defining the entropy of a quantum system is to use the von Neumann entropy:whereρ denotes the density operator which in the Schrödinger picture satisfies the von Neumann equation:whereĤ is the Hamiltonian. Since quantum mechanics and quantum field theory are unitary theories, the von Neumann entropy is conserved, albeit in general not zero. The decoherence program [1, 2] usually assumes the existence of some environment that is barely observable to some observer. This allows us to construct a reduced density operatorρ red characterising the system S, which is obtained by tracing over the unobservable environmental degrees of freedom E:ρ red = Tr E [ρ]. This is a non-unitary process which consequently generates entropy. This operation is justified by the observer's inability to see the environmental degrees of freedom (or better: access the information stored in the ES-correlators). Yet, tracing out some degrees of freedom results in complications. The simple looking von Neumann equation (2) is replaced by an equation for the reduced density operator. This "master equation" [3,4] can be solved for only in very simple situations, which has hampered progress in decoherence studies in the context of interacting quantum field theories. In particular, we are not aware of any known solution of the master equation that would include perturbative corrections and implement the program of renormalisation. Also, one has no control of the error in a calculation in the perturbative sense induced by neglecting non-Gaussian corrections.Here we propose a decoherence program that can be implemented in a field theoretical setting and that also allows us to incorporate renormalisation procedures. The idea is very simple, and we present it for a real scalar field φ(x). A generalisation to other types of fields, e.g. gauge and fermionic fields, should be quite straightforward.The density matrixρ(t) contains all information about the (possibly mixed) state a quantum system is in. From the density matrix one can, in principle, calculate various correlators. For example, the three Gaussian correlators * J.F.Koksma@uu.nl, T.Prokopec@uu.nl, M.G.Schmidt@thphys.uni-heidelberg.de 1 As a simple example, consider the temperature correlations induced by scalar field perturbations from inflation: while the amplitude ...

Langevin description of nonequilibrium quantum fields

“…We end this paper with a simple application of the previous considerations to the physics of quantum decoherence, entropy production and thermalization in quantum field theory in the context of the so-called incomplete description picture, recently advocated in [58][59][60][61][62][63][64][65]. This is based on the observation that one's ability to measure the state of a quantum field is limited because of a restricted access to the infinite tower of n-point correlation functions: One typically has only access to low order correlators, most often to the subset of one-and two-point functions.…”

“…This is based on the observation that one's ability to measure the state of a quantum field is limited because of a restricted access to the infinite tower of n-point correlation functions: One typically has only access to low order correlators, most often to the subset of one-and two-point functions. The lack of knowledge of higher-order correlators may result, from the point of view of the observer, in effective loss of quantum purity and/or coherence and associated entropy production, even to effective thermalization [59,60].…”

“…From this knowledge one can reconstruct the least biased quantum state compatible with the measured correlators and infer its quantum properties. The corresponding density operator is a Gaussian in the field operators, characterized by the intrinsic -canonically invariant -occupation number [58][59][60] …”

“…As an illustration, we end this paper with an application of such local description to the physics of decoherence and entropy production in QFT in the so-called incomplete description picture recently advocated in [58][59][60][61][62][63][64][65].…”

Cosmic fluctuations from a quantum effective action

“…(investigations of the effects of interactions in a non-flat geometry can be found in refs. [27,[30][31][32][33][34][35][36][37][38][39][40][41].) In our simple approximation interactions appear in the form of the inflaton potential.…”