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arXiv:math-ph/9911015 · Journal of Mathematical Physics

The role of infrared divergence for decoherence

Joachim Kupsch

Published Sep 1, 2000• 11 citations
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Abstract

Continuous and discrete superselection rules induced by the interaction with the environment are investigated for a class of exactly soluble Hamiltonian models. The environment is given by a boson field. Stable superselection sectors can only emerge if the low frequences dominate and the ground state of the boson field disappears due to infrared divergence. The models allow uniform estimates of all transition matrix elements between different superselection sectors.

DOI
10.1063/1.1287642

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Continuous and discrete superselection rules induced by the interaction with the environment are investigated for a class of exactly soluble Hamiltonian models. The environment is given by a boson field. Stable superselection sectors can only emerge if the low frequences dominate and the ground state of the boson field disappears due to infrared divergence. The models allow uniform estimates of all transition matrix elements between different superselection sectors.
Superselection rules induced by the interaction with a mass zero Boson field are investigated for a class of exactly soluble Hamiltonian models. The calculations apply as well to discrete as to continuous superselection rules. The initial state (reference state) of the Boson field is either a normal state or a KMS state. The superselection sectors emerge if and only if the Boson field is infrared divergent, i. e. the bare photon number diverges and the ground state of the Boson field disappears in the continuum. The time scale of the decoherence depends on the strength of the infrared contributions of the interaction and on properties of the initial state of the Boson system. These results are first derived for a Hamiltonian with conservation laws. But in the most general case the Hamiltonian includes an additional scattering potential, and the only conserved quantity is the energy of the total system. The superselection sectors remain stable against the perturbation by the scattering processes.
Superselection rules induced by the interaction with the environment are a basis to understand the emergence of classical observables within quantum theory. The aim of this article is to investigate the decoherence effects, which lead to superselection sectors, with the help of exactly soluble Hamiltonian models. Starting from the examples of Araki and of Zurek more general models with scattering are presented for which the projection operators onto the induced superselection sectors do no longer commute with the Hamiltonian. The example of an environment given by a free quantum field indicates that infrared divergence plays an essential role for the emergence of induced superselection sectors. For all models the induced superselection sectors are uniquely determined by the Hamiltonian, whereas the time scale of the decoherence depends crucially on the initial state of the total system.
We examine two exactly solvable models of decoherence---a central spin-system, (i) with and (ii) without a self-Hamiltonian, interacting with a collection of environment spins. In the absence of a self-Hamiltonian we show that in this model (introduced some time ago to illustrate environment-induced superselection) generic assumptions about the coupling strengths can lead to a universal (Gaussian) suppression of coherence between pointer states. On the other hand, we show that when the dynamics of the central spin is dominant a different regime emerges, which is characterized by a non-Gaussian decay and a dramatically different set of pointer states. We explore the regimes of validity of the Gaussian decay and discuss its relation to the spectral features of the environment and to the Loschmidt echo (or fidelity).
We discuss the applicability of the programme of decoherence -- emergence of approximate classical behaviour through interaction with the environment -- to cases where it was suggested that the presence of symmetries would lead to exact superselection rules. For this discussion it is useful to make a distinction between pure symmetries and redundancies, which results from an investigation into the constraint equations of the corresponding theories. We discuss, in particular, superpositions of states with different charges, as well as with different masses, and suggest how the corresponding interference terms although they exist in principle, become inaccessible through decoherence.