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Theoretical and Mathematical Physics

A new algebra in the stochastic approximation for the model of a particle interacting with a quantum field

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

Published Sep 1, 1998• 1 citation
Publisher
DOI
10.1007/bf02557146

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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 new kinetic master equation for the generalized Langevin dynamics of a mechanical system undergoing radiative interaction with a broadband electromagnetic field in a squeezed state has been derived from a quantum stochastic differential equation constructed using the algebra of generators of two-quantum stochastic processes. The latter express the Hamiltonian of the radiative interaction in terms of the Ito increments of these processes. The motion of a harmonic oscillator and a free particle has been analyzed using the equation for the Wigner quasiprobability, which is next in the harmonic approximation of the original equation for an open system. Features of the system dynamics in a broadband squeezed field and a broadband field in a classical state have been revealed.
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.