Speaker
Prof.
Carl Bender
(Washington University in St Louis)
Description
The average quantum physicist on the street would say that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under combined matrix transposition and complex conjugation) in order to guarantee that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian $H=p^2+ix^3$, which is obviously not Dirac Hermitian, has a positive real discrete spectrum and generates unitary time evolution, and thus it defines a fully consistent and physical quantum theory!
Evidently, the axiom of Dirac Hermiticity is too restrictive. While $H=p^2+ix^3$ is not Dirac Hermitian, it is PT symmetric; that is, invariant under combined parity P (space reflection) and time reversal T. The quantum mechanics defined
by a PT-symmetric Hamiltonian is a complex generalization of ordinary quantum
mechanics. When quantum mechanics is extended into the complex domain, new kinds
of theories having strange and remarkable properties emerge. In the past few
years, some of these properties have been verified in laboratory experiments. A
particularly interesting PT-symmetric Hamiltonian is $H=p^2-x^4$, which contains
an upside-down potential. We will discuss this potential in detail, and explain
in intuitive as well as in rigorous terms why the energy levels of this
potential are real, positive, and discrete.
Applications of PT-symmetry in quantum field theory will be discussed.
Primary author
Prof.
Carl Bender
(Washington University in St Louis)
