open access

https://doi.org/10.14943/84185

https://hdl.handle.net/2115/69845

Construction of dynamics and time-ordered exponential for unbounded non-symmetric Hamiltonians

英語

2013-10-15

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

1041

1-41

We prove under certain assumptions that there exists a solution of the Schr¨odinger or the Heisenberg equation of motion generated by a linear operator H acting in some complex Hilbert space H, which may be unbounded, not symmetric, or not normal. We also prove that, under the same assumptions, there exists a time evolution operator in the interaction picture and that the evolution operator enjoys a useful series expansion formula. This expansion is considered to be one of the mathematically rigorous realizations of so called “time-ordered exponential”, which is familiar in the physics literature. We apply the general theory to prove the existence of dynamics for the mathematical model of Quantum Electrodynamics (QED) quantized in the Lorenz gauge, the interaction Hamiltonian of which is not even symmetric or normal.

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