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arXiv:math/9807137 · Journal of Physics A Mathematical and General

Dynamicalq-deformation in quantum theory and the stochastic limit

Luigi Accardi, Sergei Vladimirovich Kozyrev, Igor Vasil'evich Volovich

Published Jan 1, 1999• 17 citations
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Abstract

We show that in a model of a particle interacting with a quantum field, the field operators rescaled according to the prescriptions of the stochastic limit, obey q -commutational relations with q depending on time. After the stochastic limit, due to the nonlinearity, the particle and field degrees of freedom become entangled even at a kinematical level in the sense that the field and the atomic variables no longer commute but give rise to a new algebra with new commutation relations replacing the boson ones. This allows to give a simple proof of the fact that the non-crossing half-planar diagrams give the dominating contribution in a weak-coupling regime and to calculate explicitly the correlations associated with the new algebra.

DOI
10.1088/0305-4470/32/19/302

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We show that in a model of a particle interacting with a quantum field, the field operators rescaled according to the prescriptions of the stochastic limit, obey q -commutational relations with q depending on time. After the stochastic limit, due to the nonlinearity, the particle and field degrees of freedom become entangled even at a kinematical level in the sense that the field and the atomic variables no longer commute but give rise to a new algebra with new commutation relations replacing the boson ones. This allows to give a simple proof of the fact that the non-crossing half-planar diagrams give the dominating contribution in a weak-coupling regime and to calculate explicitly the correlations associated with the new algebra.
A procedure of bosonization of Fermions in an arbitrary dimension is suggested. It is shown that a quadratic expression in the fermionic fields after rescaling time t→t/λ2t\to t/λ^2t→t/λ2 and performing the limit λ→0λ\to0λ→0 (stochastic limit), gives rise to a bosonic operator satisfying the boson canonical commutation relations. This stochastic bosonization of Fermions is considered first for free fields and then for a model with three--linear couplings. The limiting dynamics of the bosonic theory turns out to be described by means of a quantum stochastic differential equations.
We introduce a Feynman diagram calculus for quantum stochastic fields and show that from the weak coupling limit of a Bose quantum field reservoir in a Gaussian state one obtains a quantum stochastic field process which has extended Wigner statistics. That is, we calculate explicitly the moments using the diagram rules and show that only moments corresponding to non-crossing diagrams are non-trivial. We further give the physical mechanism underlying this fact; namely the combination of momentum conservation (arising from the correct treatment of the full responsive interaction) and the mass-shell energy conservation for the virtual (stochastic) reservoir quanta. In our analysis we introduce a volume cut-off initially for the system particle to discretize its energy spectrum and then take the weak coupling limit. The individual noise fields associated with energy state transitions were Gaussian however, when the volume cut-off is then removed and we recover the noise correlations. Thus for this problem the removal of the cut-off and the weak coupling limit are in fact interchangeable.
In this paper we investigate a quantum stochastic calculus build of creation, annihilation and number of particles operators which fulfill some deformed commutation relations. Namely, we introduce a deformation of a number of particles operator which has simple commutation relations with well known q-deformed creation and annihilation operators. Since all operators considered in this theory are bounded we do not deal with some difficulties of a non deformed theory of Hudson and Parthasarathy. We define stochastic integrals and estimate them in the operator norm. We prove Ito's formula as well.
The density matrix and the Wigner function formalism requires the doubling of the degrees of freedom in quantum mechanics (QM) and quantum field theory (QFT). The doubled degrees of freedom play the role of the thermal bath or environment degrees of freedom and are entangled with the system degrees of freedom. They also account for quantum noise in the fluctuating random forces in the system-environment coupling. The algebraic structure of QFT turns out to be the one of the deformed Hopf algebra. In such a frame, the trajectories in the space of the unitarily inequivalent representations of the canonical commutation relations turn out to be classical trajectories and, under convenient conditions, they may exhibit properties typical of classical chaotic trajectories in nonlinear dynamics. The quantum Brownian motion and the two-slit experiment in QM are discussed in connection with the doubling of the degrees of freedom.
Abstract Emerging of free (or quantum Boltzmann) statistics for a model of quantum particle interacting with quantum field is described in the stochastic limit without dipole approximation. The quantum field is considered in a Gaussian (for example temperature) state. Entangled operators which describe interaction of the field and the particle satisfy the qqq -deformed relations which in the stochastic limit generate free statistics.
The Quantum Stochastic Limit of a quantum mechanical particle coupled to a quantum field without the neglect of the response details of the interaction (i.e., not making the dipole approximation) is made following the scheme of Accardi and Lu and the corresponding Quantum Stochastic Structure is derived. The stochastic sector for the noise is constructed and is shown to be of a qualitatively new type. We also include a physical discussion on the limit noise which obeys Interacting-Free statistics and include a new shorter proof of the noise convergence and also a new construction of Interacting-Free Fock Space.