Determine whether the LMG action can become negative

Determine whether the transient action S_α(x'',t|x',0) for the driven-dissipative Lipkin–Meshkov–Glick model with nonconvex Husimi or P-representation auxiliary Hamiltonians can become negative, thereby violating the nonnegativity property that holds for convex auxiliary Hamiltonians.

Background

The paper compares quantum quasiprobability-based auxiliary Hamiltonians with the convex Hamiltonians arising in classical stochastic systems. For convex Hamiltonians, the finite-time action is nonnegative, nondecreasing, and related to a minimum-action principle. The Husimi and P representations of the driven-dissipative Lipkin–Meshkov–Glick model instead produce auxiliary Hamiltonians that are nonconvex in momentum.

Numerical results showed that the action can decrease transiently along an instanton, violating monotonicity. However, the authors did not observe negative action values and explicitly state that they cannot exclude this possibility. Establishing whether nonnegativity can fail would clarify the implications of Hamiltonian nonconvexity for the instanton framework and for the validity of minimum-action interpretations.