Diagonalizing Quadratic Bosonic Operators by Non-Autonomous Flow Equation

TL;DR

This paper introduces a non-autonomous, non-linear evolution equation derived from the Brocket-Wegner flow, providing a method to diagonalize quadratic bosonic Hamiltonians in quantum field theory.

Contribution

It establishes conditions for the global existence and asymptotic behavior of solutions to a novel non-autonomous flow equation, extending previous diagonalization techniques.

Findings

Proves global existence of solutions under optimal assumptions

Derives asymptotics at temporal infinity for the flow

Enables diagonalization of quadratic bosonic Hamiltonians

Abstract

We study a non-autonomous, non-linear evolution equation on the space of operators on a complex Hilbert space. We specify assumptions that ensure the global existence of its solutions and allow us to derive its asymptotics at temporal infinity. We demonstrate that these assumptions are optimal in a suitable sense and more general than those used before. The evolution equation derives from the Brocket-Wegner flow that was proposed to diagonalize matrices and operators by a strongly continuous unitary flow. In fact, the solution of the non-linear flow equation leads to a diagonalization of Hamiltonian operators in boson quantum field theory which are quadratic in the field.