Don’t be scared of ghosts

Theories with higher derivatives result in quantum theories with negative norm states, called ghosts. Physically, ghosts imply negative probabilities which has led many to disregard higher derivative theories as unphysical. However, theories with ghosts are regularly used in physics under the name of gauge theories.

In this talk we will construct some indefinite representations of the Heisenberg algebra. The existence of ghosts will allow us to define a quantum gauge symmetry. Positivity is restored when we restrict to the gauge invariant physical content. We then add some dynamics by constructing a Hamiltonian that is compatible with the gauge structure. In the case of the simplest gauge theory, we will find that the quantum theory is trivial while the corresponding classical theory is a higher derivative harmonic oscillator with an abelian gauge symmetry.