Diagonal Representation for a Generic Matrix Valued Quantum Hamiltonian

TL;DR

The paper introduces a novel method to derive the diagonal representation of matrix-valued quantum Hamiltonians, utilizing new mathematical objects and expansions in Planck's constant, with applications to Dirac particles and neutrinos.

Contribution

It presents a general, compact formalism for diagonalizing matrix-valued quantum Hamiltonians, including explicit second-order expansions in Planck's constant.

Findings

Derived a formal expression for the diagonal Hamiltonian as a power series in Planck's constant.

Provided explicit second-order diagonal representations for generic Hamiltonians.

Applied the method to physical systems like Dirac electrons and neutrinos in external fields.

Abstract

A general method to derive the diagonal representation for a generic matrix valued quantum Hamiltonian is proposed. In this approach new mathematical objects like non-commuting operators evolving with the Planck constant promoted as a running variable are introduced. This method leads to a formal compact expression for the diagonal Hamiltonian which can be expanded in a power series of the Planck constant. In particular, we provide an explicit expression for the diagonal representation of a generic Hamiltonian to the second order in the Planck constant. This last result is applied, as a physical illustration, to Dirac electrons and neutrinos in external fields.