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Wed, 02 Feb 2005 15:39:30 +0100Wed, 02 Feb 2005 15:39:30 +0100Ultracoherence and Canonical Transformations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1605
The symplectic group of homogeneous canonical transformations is represented in the bosonic Fock space by the action of the group on the ultracoherent vectors, which are generalizations of the coherent states. The intertwining relations between this representation and the algebra of Weyl operators are derived. They confirm the identification of this representation with Bogoliubov transformations.Joachim Kupsch; Subhashish Banerjeeworkingpaperhttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1605Wed, 02 Feb 2005 15:39:30 +0100Superselection rules induced by infrared divergence
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1606
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.Joachim Kupschworkingpaperhttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1606Wed, 02 Feb 2005 15:39:11 +0100Hilbert Norms For Graded Algebras
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1307
Abstract: This paper presents a solution to a problem from superanalysis about the existence of Hilbert-Banach superalgebras. Two main results are derived: 1) There exist Hilbert norms on some graded algebras (infinite-dimensional superalgebras included) with respect to which the multiplication is continuous. 2) Such norms cannot be chosen to be submultiplicative and equal to one on the unit of the algebra.Joachim Kupsch; Oleg G. Smolyanovpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1307Tue, 03 Jul 2001 00:00:00 +0200Mathematical Aspects of Decoherence
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1187
Abstract: The aim of this lecture is to investigate the mathematical frame for environment induced superselection rules. Some exactly soluble models are dicussed in detail and norm estimates on decoherence effects are derived.Joachim Kupschpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1187Thu, 05 Apr 2001 00:00:00 +0200States of quantum systems and their liftings
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1188
Abstract: Let H_1 , H_2 be complex Hilbert spaces, H be their Hilbert tensor product and let tr_2 be the operator of taking the partial trace of trace class operators in H with respect to the space H_2 . The operation tr_2 maps states in H (i.e. positive trace class operators in H with trace equal to one) into states in H_1 . In this paper we give the full description of mappings that are linear right inverse to tr_2 . More precisely, we prove that any affine mapping F(W) of the convex set of states in H_1 into the states in H that is right inverse to tr_2 is given by W -> W x D for some state D in H_2 . In addition we investigate a representation of the quantum mechanical state space by probability measures on the set of pure states and a representation - used in the theory of stochastic SchrÃ¶dinger equations - by probability measures on the Hilbert space. We prove that there are no affine mappings from the state space of quantum mechanics into these spaces of probability measures.Joachim Kupschpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1188Thu, 05 Apr 2001 00:00:00 +0200The role of infrared divergence for decoherence
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1166
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.Joachim Kupschpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1166Mon, 25 Sep 2000 00:00:00 +0200Exactly soluble models of decoherence
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1167
Superselection rules induced by the interaction with the environment are investigated 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.Joachim Kupschpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1167Mon, 25 Sep 2000 00:00:00 +0200Functional Representations for Fock Superalgebras
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1168
The Fock space of bosons and fermions and its underlying superalgebra are represented by algebras of functions on a superspace. We define Gaussian integration on infinite dimensional superspaces, and construct superanalogs of the classical function spaces with a reproducing kernel - including the Bargmann-Fock representation - and of the Wiener-Segal representation. The latter representation requires the investigation of Wick ordering on Z 2 -graded algebras. As application we derive a Mehler formula for the Ornstein-Uhlenbeck semigroup on the Fock space.Joachim Kupsch; Oleg G. Smolyanovpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1168Mon, 25 Sep 2000 00:00:00 +0200The structure of the quantum mechanical state space and induced superselection rules
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1169
The role of superselection rules for the derivation of classical probability within quantum mechanics is investigated and examples of superselection rules induced by the environment are discussed.Joachim Kupschpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1169Mon, 25 Sep 2000 00:00:00 +0200