Abstract
The Hamiltonian $H$ specifies the energy levels and the time evolution
of a quantum theory. It is an axiom of quantum mechanics that $H$ be Hermitian
because Hermiticity guarantees that the energy spectrum is real and that the
time evolution is unitary (probability preserving). In this talk we investigate
an alternative formulation of quantum mechanics in which the conventional
requirement of Hermiticity (transpose+complex conjugate) is replaced by the
more physically transparent condition of space-time reflection ($\mathcal{PT}$)
symmetry. We show that if the $\mathcal{PT}$ symmetry of a Hamiltonian $H$ is
unbroken, then the spectrum of $H$ is real. Examples of $\mathcal{PT}$-symmetric
non-Hermitian quantum-mechanical Hamiltonians are $H=p^2+ix^3$ and $H=p^2-x^4$.
Amazingly, the energy levels of these Hamiltonians are all real and positive!
The crucial question is whether $\mathcal{PT}$-symmetric Hamiltonians specify
physically acceptable quantum theories in which the norms of states are positive
and the time evolution is unitary. The answer is that a Hamiltonian that has an
unbroken $\mathcal{PT}$ symmetry also possesses a physical symmetry that we call
$\mathcal{C}$. Using $\mathcal{C}$, we show how to construct an inner product
whose associated norm {\it is} positive definite. The result is a new class of
fully consistent complex quantum theories. Observables are defined,
probabilities are positive, and the dynamics is governed by unitary time
evolution. Many examples of $\mathcal{PT}$-symmetric quantum mechanical and
quantum field theoretic Hamiltonians will be discussed.