Room P3.10, Mathematics Building

Carl Bender

Carl Bender, Washington University
PT symmetry and the taming of instabilities

Theoretical advances in PT symmetry have led to beautiful experimental results in optics, lasers, superconducting wires, atomic diffusion, NMR, microwave cavities, and electronic and mechanical simulations. Experiments on BECs are being planned and new PT-symmetric synthetic metamaterials are being developed. At an abstract level, PT-symmetric classical and quantum mechanics are complex generalizations of conventional classical and quantum mechanics. The advantage of extending physics into the complex domain is that theories that are unstable and physically unacceptable from the narrow perspective of real analysis may become stable and physically viable in the complex domain. This is because the conventional way to determine instability is based on energy inequalities; a theory is thought to be unstable if the potential is unbounded below. In generalizing from the real axis to the complex plane, the notion of ordering is lost; one can say that $x < y$ if $x$ and $y$ are real but not if they are complex. Thus, potentials that are unbounded below on the real axis may not be a problem.

A classical theory that is unstable on the real axis but stable in the complex domain is the $-x^4$ upside-down potential. Quantum theories such as the Lee model, the Pais-Uhlenbeck model, the double-scaling limit of $\phi^4$ theory, and time-like Liouville theory, were believed for many decades to suffer from instabilities (which give rise to ghost states and nonunitarity). These are all physically acceptable PT-symmetric theories. The powerful methods of PT-symmetric quantum theory may resolve even more timely problems such as the stability the Higgs vacuum in the Standard Model.