open access

https://doi.org/10.14943/83338

https://hdl.handle.net/2115/68940

Scaling limit of a model of quantum electrodynamics

英語

1993-04

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

194

1-53

This article gives a solution to an open problem in the paper: A.Arai; J.Math.Phys.31, 2653(1990). We present an abstract asymptotic theory of families of unitary operators {U(11:)},.>o and self-adjoint operators {H,.},.>o acting in the tensor product of two Hilbert spaces. We prove that H;en (V, 11: ), which represents a scaled total Hamiltonian of a coupled system of a one elec­tron atom and a quantized radiation :field, with parameters O 􀀖 e 􀀖 1, 11: > 0, and the electron mass renormlized, is unitarily equivalent to an operator fI;en (V, 11:), which can be regarded as a decoupled Hamiltonian. Applying the abstract asymptotic theory and the unitary equiva­lence, we prove that the resolvent of H;en (v, 11:) strongly converges as 11: -+ oo to an operator which de:fines an effective potential of the electron. We compare the effective potential with that obtained in the paper mentioned above.

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