Making sense of nonHermitian Hamiltonians Speaker(s): Carl Bender
Abstract: It is a standard axiom of quantum mechanics that the Hamiltonian H must be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary. In this talk we examine an alternative formulation of quantum mechanics in which the conventional requirement of Hermiticity is replaced by the more general and physical condition of space time reflection (PT) symmetry. We show that if the PT symmetry of H is unbroken, Then the spectrum of H is real. Examples of PTsymmetric nonHermitian Hamiltonians are $H=p^2+ix^3$ and $H=p^2x^4$. Amazingly, the energy levels of these Hamiltonians are all real and positive despite the ``wrong'' sign in the $x^4$ potential! We show that such PTsymmetric Hamiltonians specify physically acceptable quantummechanical theories in which the norms of states are positive and time evolution is unitary. To do so we demonstrate that a Hamiltonian that has an unbroken PT symmetry also possesses a new physical symmetry that we call C. Using C, we construct an inner product whose associated norm is positive definite. The result is a new class of consistent complex quantum theories. In effect, we have extended and generalized quantum mechanics into the complex domain. We then discuss PTsymmetric quantum field theories. PTsymmetric scalar fieldtheoretic Hamiltonians corresponding to the above quantummechanical Hamiltonains have interaction terms $igphi^3$ and $gphi^4$. The latter Theory is interesting because (1) it is asymptotically free and (2) the expectation value of $phi$ is nonzero. (Thus, such a theory might be useful in describing the Higgs sector.) PT symmetry resolves the longstanding problem of ghosts in the Lee model. When the renormalized coupling constant in this model increases past a critical value, the Hamiltonian ceases to be Hermitian and a negativenorm ghost state appears. At this transition the Hamiltonian becomes PTsymmetric, and the ghost is a physical particle. PTsymmetric QED and the PTsymmetric massive Thirring model will also be discussed. Finally, we mention recent papers which suggest that PTsymmetry may provide insight into cosmological problems.
Date: 13/04/2006  11:00 am
Series: Quantum Gravity
