Higher time derivatives, stability and Fermi Statistics

TL;DR

This paper demonstrates that Fermi statistics ensures the boundedness of the Hamiltonian and normalizability of states in higher time derivative systems, affecting stability and unitarity considerations.

Contribution

It reveals the crucial role of Fermi statistics in stabilizing higher derivative theories, contrasting previous claims of inherent instability.

Findings

Fermi systems have well-bounded Hamiltonians despite higher derivatives.

Ghost states exist but do not necessarily violate unitarity under certain conditions.

Analysis extends from Grassman variables to Dirac fields.

Abstract

We show that statistics is crucial for the instability problem derived from higher time derivatives. In fact, and contrary to previous statements, we check that when dealing with Fermi systems, the Hamiltonian is well bounded and the quantum states are normalizable. Although, ghost states are still present, they do not affect unitarity under certain conditions. We first analyze a quantum oscillator involving Grassman variables and then we generalize it to a Dirac field. Finally, we discuss some physical implications