TY - JOUR

T1 - Functional integral representations of the Pauli-Fierz model with spin 1/2

AU - Hiroshima, Fumio

AU - Lorinczi, József

N1 - Funding Information:
We thank V. Betz, M. Gubinelli and I. Sasaki for useful discussions. This work was partially done at Zentrum Mathematik, Technical University Munich, Warwick University, Coventry, and at Erwin Schrödinger Institute, Vienna, all of whom we thank for kind hospitality. J.L. is grateful to Kyushu University for a travel grant and warm hospitality. This work is financially supported by Grant-in-Aid for Science Research (C) 17540181 from JSPS.

PY - 2008/4/15

Y1 - 2008/4/15

N2 - A Feynman-Kac-type formula for a Lévy and an infinite-dimensional Gaussian random process associated with a quantized radiation field is derived. In particular, a functional integral representation of e- t HPF generated by the Pauli-Fierz Hamiltonian with spin 1/2 in non-relativistic quantum electrodynamics is constructed. When no external potential is applied HPF turns translation-invariant and it is decomposed as a direct integral HPF = ∫R3⊕ HPF (P) d P. The functional integral representation of e- t HPF (P) is also given. Although all these Hamiltonians include spin, nevertheless the kernels obtained for the path measures are scalar rather than matrix expressions. As an application of the functional integral representations energy comparison inequalities are derived.

AB - A Feynman-Kac-type formula for a Lévy and an infinite-dimensional Gaussian random process associated with a quantized radiation field is derived. In particular, a functional integral representation of e- t HPF generated by the Pauli-Fierz Hamiltonian with spin 1/2 in non-relativistic quantum electrodynamics is constructed. When no external potential is applied HPF turns translation-invariant and it is decomposed as a direct integral HPF = ∫R3⊕ HPF (P) d P. The functional integral representation of e- t HPF (P) is also given. Although all these Hamiltonians include spin, nevertheless the kernels obtained for the path measures are scalar rather than matrix expressions. As an application of the functional integral representations energy comparison inequalities are derived.

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U2 - 10.1016/j.jfa.2008.01.002

DO - 10.1016/j.jfa.2008.01.002

M3 - Article

AN - SCOPUS:39849110864

VL - 254

SP - 2127

EP - 2185

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 8

ER -