Lindblad Quantum Dynamics as Euler-Poincar\'e Reduction on Adjoint-Coupled Semidirect Products

Abstract

We present a geometric and variational derivation of the Gorini--Kossakowski--Sudarshan--Lindblad equation from Euler--Poincar'e reduction on an adjoint--coupled semidirect product (ACSP). In this construction a Lie group $G$ acts on $V=\mathfrak{g}^{\oplus m}$ by the adjoint representation together with a second, adjointly compatible action whose failure to commute defines an adjoint torsion $K(\xi,v)$. This torsion generates a canonical quadratic curvature operator on $\mathfrak{g}^*$ that survives reduction and yields a metric double--bracket term. For $G=SU(n)$ the reduced Euler--Poincar'e equation reproduces exactly the GKSL generator: the Hamiltonian part arises from the coadjoint action, while the dissipator $-\tfrac{\gamma}{2}[L,[L,\rho]]$ appears as the torsion--induced metric component of an ACSP bracket. We prove a characterization theorem showing that any quadratic $SU(n)$--equivariant operator generated by torsion factorizes into a Lindblad double commutator; a uniqueness theorem establishing that, under natural structural assumptions, the only admissible dissipator is the Lindblad form; and an orbit--contraction theorem showing strict contraction toward the commutant of the Lindblad operators. For $SU(2)$ and $SU(3)$ the ACSP geometry yields explicit Bloch equations for representative dissipative channels. We also show that the ACSP bracket fits into a metriplectic and contact--geometric framework in which the Lindblad term is the metric component and the Reeb part of a contact Hamiltonian flow. In this picture, decoherence is a curvature--induced contraction generated by Euler--Poincar'e reduction rather than a phenomenological input.