the possibility of studying new kinds of quantum theories that have heretofore never been studied because they have been thought to be physically unacceptable.

Let us review the properties of the space reflection (parity) operator $\mathcal
{P}$ and the time-reflection operator $\mathcal{T}$: $\mathcal{P}$ is a [*linear*]{} operator with the property that $\mathcal{P}^2=1$ and has the effect $p
\to-p$ and $x\to-x$; $\mathcal{T}$ is an antilinear operator with the property that $\mathcal{T}^2=1$ and has the effect $p\to-p$, $x\to x$, and $i\to-i$. The operator $\mathcal{T}$ is called [*antilinear*]{} because it changes the sign of $i$. We know that it reverses the sign of $i$ because, like $\mathcal{P}$, this operator preserves the fundamental commutation relation of quantum mechanics, $[x,p]=i$, known as the Heisenberg algebra.

It is easy to construct Hamiltonians of the form $H=p^2+V(x)$ that are not Hermitian but do possess $\mathcal{PT}$ symmetry. The trick is to take the potential to be a function of $ix$: $V=V(ix)$. We also impose a general condition that has not been widely emphasized in the literature; namely, we require that $H$ be symmetric: $H=H^{\rm T}$, where ${\rm T}$ represents the transpose. The reason for this symmetry condition will become clear later on. For example, consider the one-parameter family 