Lorentz Symmetry and Non-Unitary Quantum Field Theories

Citation

Wang, Tian (2023) Lorentz Symmetry and Non-Unitary Quantum Field Theories. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/h2d6-8387. https://resolver.caltech.edu/CaltechTHESIS:06122023-221753249

Abstract

We know quantum field theory is unitary. However, since the early 1980s, there have been numerous attempts to construct quantum field theories where time evolution is non-unitary. Some of these endeavors aimed to address the issue of black hole information loss, as non-unitary evolution does not necessarily require information preservation. Some wanted to use it as a modification of quantum mechanics to allow objective collapse. Some wanted to construct classical-quantum gravity which could serve as an alternative to quantum gravity.

I embarked on a similar path, attempting to construct a Lorentz covariant non-unitary quantum field theory. At a certain point, I believed we were making significant progress. However, I gradually realized that our construction faced serious problems. We took a lot of assumptions and results from unitary quantum field theory for granted, and used them without justification. After struggling with it for a long time, I decide to make a complete reversal and prove that non-unitary quantum field theories fundamentally conflict with Lorentz covariance.

There are three approaches to constructing a Lorentz covariant non-unitary QFT. The first approach involves constructing a theory based on unitary quantum field theory, where a system is coupled to an environment. If we only consider the system and trace out the environment, the resulting equation of motion appears non-unitary. In this case, unitarity emerges as an emergent property. The second approach is to propose a theory from scratch where the time evolution is fundamentally non-unitary, described by the Lindblad master equation. Both in the emergent and fundamentally non-unitary theories, the dynamics are intended to be Lorentz covariant. The third approach employs the Schwinger-Keldysh formalism to construct a path integral, and examines the symmetry of the Keldysh action within the path integral. It is assumed that, similar to quantum field theory, the non-unitary theory will possess the same symmetry as the Keldysh action.

Regrettably, none of these three classes of theories prove successful. This thesis thoroughly analyzes the issues associated with these three constructions. The most significant problems include:

1, The fundamental assumption that the quantum fields (and their excitations) form a unitary representation of the Lorentz group is invalid, and they cannot form a non-unitary representation either.

2, The system Hamiltonian in the Lindblad equation is ill-defined and does not transform as the first component of a Lorentz four-vector.

3, Even if we overlook the aforementioned inconsistencies, the dynamics fail to produce expected results when applied to phenomena such as particle decay, as they exhibit a preferred reference frame.

4, The symmetry of the Keldysh action does not guarantee the corresponding symmetry in the dynamics. Invariant Keldysh actions can correspond to non-covariant equations of motion.

In conclusion, the Lorentz symmetry is incompatible with non-unitary quantum field theories.

Thesis Files