Managing Director, Digital Science; Visiting Professor, Department of Physics, Washington University in St Louis
PT Symmetry

When Dirac formulated the postulates of quantum theory, he required Hermiticity to be the fundamental symmetry for his equations. For Dirac, the requirement of Hermiticity was the mathematical device that he needed to ensure that all predictions for the outcomes of real-world measurements of quantum systems resulted in a real number. This is important since we only observe real outcomes in actual experimental observations. Dirac’s choice of Hermiticity as the fundamental symmetry of quantum theory was not seriously challenged for around seventy years.  

Hermiticity is a subtle and abstract symmetry that is mathematical in its origin. Broadly speaking, the requirement of Hermiticity imposes a boundary on a system. This is an idealization in which a system is isolated from any surrounding environment (and hence cannot be measured). While this gives a tractable mathematical framework for quantum theory, it is an unphysical requirement since all systems interact with their environment and if we wish to measure a system then such an interaction is required. 

In 1998, Carl Bender and Stefan Boettcher wrote a paper exploring the replacement of Hermiticity with another symmetry. They showed that they could replace the mathematically motivated symmetry of Dirac by a physically motivated symmetry preserving the reality of experimental outcomes. Their new theory, however, had interesting new features—it was not a like-for-like replacement. 

The underlying symmetry that Bender and Boettcher found was what they called “PT symmetry.” The symmetry here is geometric in nature and is hence closer to physics than is Hermiticity. The “P” stands for “parity” symmetry, sometimes called mirror symmetry. If a system respects “P” symmetry, then the evolution of the system would not change for a spatially reflected version of the system. The “T” stands for “time-reversal.” Time-reversal symmetry is just as it sounds—a physical system respecting this symmetry would evolve in the same way regardless whether time runs forward or backward. Some systems do individually exhibit P and T symmetries, but it is the combination of the two that seems to be fundamental to quantum theory.  

Instead of describing a system in isolation, PT symmetry describes a system that is in balance with its environment. Energy may flow in and out of the system, and hence measurements can be made within the theoretical framework of a system described by a PT symmetry. The requirement is that the same amount of energy that flows in must also flow out of the system.   

This subtler definition of a system’s relationship with its environment, provided by PT symmetry, has made it possible to describe a much wider class of systems in mathematical terms. This has led not only to an enhanced understanding of these systems but also to experimental results that support the choice of PT as the underlying symmetry in quantum mechanics. Several physical models for specific systems that had previously been studied and rejected, because they did not respect Hermiticity, have been re-examined and found to be PT symmetric.  

It is remarkable that the study of PT symmetry has progressed so rapidly. For many areas of theoretical physics, the time-lag between theory and experiment is now on the order of several decades. We may never be able to fully test string theory and experimental verification of the fifty-year-old theory of supersymmetry remains elusive. 

In the eighteen years since Bender and Boettcher’s 1998 paper, experimentalists have created PT lasers, PT superconducting wires, PT NMR and PT diffusion experiments to mention just a few validations of their theory.  As PT symmetry has matured, it has inspired the creation of exotic metamaterials that have properties that allow us to control light in new ways. The academic community, initially skeptical of such a fundamental change in quantum theory, has warmed to the idea of PT symmetry. Over 200 researchers from around the world have published scholarly papers on PT symmetry. The literature now extends to more than 2000 articles, many in top journals such as Nature, Science and Physical Review Letters.  

The future is bright for PT-symmetric quantum mechanics, but there is still work to be done. Many of the experiments mentioned have quantum mechanical aspects but are not full verifications of PT quantum mechanics. Nevertheless, existing experiments are already leading to exciting results. PT is a hot topic in the optics and graphene communities and the idea of creating a computer based on optical rather than electronic principles has recently been suggested. At the beginning of the 21st century, we are finding a new understanding of quantum theory that has the potential to unlock new technologies in the same way that semi-conductor physics was unlocked by the rise of quantum mechanics one hundred years ago.