Investigating Unitarity Violation of Lee–Wick’s Complex Ghost with Quantum Field Theory

Quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. It is an extremely successful theory that has explained and predicted many phenomena. Moreover, it forms the basis for the Standard Model of particle physics, which is our current best understanding of the fundamental forces and particles of the universe.

Accordingly, there have been many attempts to develop a consistent QFT of gravity. One potential candidate for this is quadratic gravity (QG). QG is a modification of Einstein’s field equations in general relativity by adding higher-order derivative terms that address the infinities arising during gravitational interactions at very high energies, making the theory renormalizable. However, this modification, similar to Lee–Wick’s quantum electrodynamics model, leads to the presence of “ghost particles” of negative norm, which endanger the unitarity of the theory.

Lee and Wick had noticed that the ghost acquires a complex mass by radiative corrections and had claimed that such complex ghosts would never be created during the collision of physical particles due to energy conservation, thus preserving the physical unitarity. If this claim is true, all theories, in principle, can be made renormalizable or even finite without violating unitarity by simply adding higher-order derivative terms. This would also make QG a viable theory.

In this study, we addressed the above unitarity problem based solely on QFT. To this end, we faithfully applied the operator formalism of indefinite metric QFT and calculated the amplitudes of the scattering process of physical particles.

Interestingly, our calculations showed that complex ghosts can indeed be created with a non-zero probability during the collisions of physical particles, thereby violating the physical unitarity contrary to Lee and Wick's claim. Even if one devises a clever method to modify scattering amplitudes to satisfy unitarity, it will no longer be a QFT. Notably, we identified an energy limit below which no complex ghosts are created and thus unitarity holds.

The complex delta function (a generalization of the Dirac delta function), which appears at each interaction vertex of the complex ghost, played a central role in our investigation. In particular, we found that the usual Feynman rule applied at each vertex with a complex energy conservation law assumed in advance is wrong and has no ground in QFT. To address this, we also gave the corrected starting expression.

In conclusion, this study addresses the unitarity problem in the context of QG and takes us a step closer to realizing a proper renormalizable QFT of gravity.