Stochastic differential equations for models of non-relativistic matter interacting with quantized radiation fields

Abstract

We discuss Hilbert space-valued stochastic differential equations associated with the heat semi-groups of the standard model of non-relativistic quantum electrodynamics and of corresponding fiber Hamiltonians for translation invariant systems. In particular, we prove the existence of a stochastic flow satisfying the strong Markov property and the Feller property. To this end we employ an explicit solution ansatz. In the matrix-valued case, i.e., if the electron spin is taken into account, it is given by a series of operator-valued time-ordered integrals, whose integrands are factorized into annihilation, preservation, creation, and scalar parts. The Feynman-Kac formula implied by these results is new in the matrix-valued case. Furthermore, we discuss stochastic differential equations and Feynman-Kac representations for an operator-valued integral kernel of the semi-group. As a byproduct we obtain analogous results for Nelson's model.

Continuity properties of the semi-group and its integral kernel in non-relativistic QED

“…While FK representations have been available in non-relativistic QED for quite some time [17,22,25], the latter SDE has been derived and investigated only recently in our earlier work together with B. Güneysu and J.S. Møller [12]. (Our article [12] also provides a new version of the FK formula with spin, which will be employed in our examples; compare Rem.…”

“…Møller [12]. (Our article [12] also provides a new version of the FK formula with spin, which will be employed in our examples; compare Rem. 2.18 below.…”

“…The well-known positivity improvement by the corresponding semi-group [19] is an immediate corollary of this result. The relatively simple proof given in this section is based on a novel factorization of the FK integrand found in [12] and traditional ideas associated with Perron-Frobenius type arguments in mathematical quantum field theory; see, e.g., [7,34].…”

Stability of Ferromagnetism in Many-Electron Systems

“…L is essentially anti-self-adjoint. 11 We denote its closure by the same symbol. A unitary operator e L is called the Lang-Firsov transformation [23].…”

“…11 The Coulomb interactions in H H and H HH can be chosen, independently. ♦We will provide a sketch of proof of Theorem 4.10 in Appendix D.…”