A multi-Hamiltonian approach to higher time-derivative theories: An interacting Pais–Uhlenbeck model

Despite appealing theoretical features, higher time-derivative theories (HTDTs) are notoriously plagued by issues of instability—most prominently the emergence of ghost states, which signal the presence of unbounded-from-below Hamiltonians or non-normalisable states, that threaten the physical viability of these theories. Consequently, the construction of ghost-free representations is a central aim.

The Pais–Uhlenbeck (PU) model, a canonical example of a fourth-order differential system, is a paradigmatic example of an HTDT and encapsulates their essential challenges and features. Using its multi-Hamiltonian structures in conjunction with the Lie symmetries of the dynamical equation, one can construct distinct, but compatible, Poisson bracket formulations that preserve the system’s dynamics. Amongst other possibilities, this framework allows the recasting of models in a positive definite manner while leaving the dynamical flow unchanged, thus resolving the ghost problem. The application of the outlined approach has successfully demonstrated the construction of ghost-free representatives for the PU theory.

Introducing interactions to HTDTs generally challenges their stability further and poses a difficult problem. Going beyond the PU model, we here analyse an interacting extension of the system that admits closed-form solutions. This model suggests promising directions for the systematic construction of stable interacting HTDTs, future generalisations to field-theoretic settings, and further investigation into the quantisation of positive-definite PU models. A connection to an integrable realisation of the generalised Hénon–Heiles system (tied to Lax’s fifth-order KdV flow) is discussed.