Although $H=H^\dagger$ is sufficient to guarantee these properties, it is not necessary. Indeed, we believe that this condition of Hermiticity is a mathematical requirement whose physical basis is somewhat obscure. Recently, a more physical alternative axiom called space-time reflection symmetry ($\mathcal
{PT}$ symmetry), $H=H^\mathcal{PT}$, has been investigated. This symmetry allows for the possibility of complex non-Hermitian Hamiltonians but still leads to a consistent theory of quantum mechanics.

Because $\mathcal{PT}$ symmetry is an alternative condition to conventional Hermiticity, it is now possible to construct infinitely many new Hamiltonians that would have been rejected in the past because they are not Hermitian in the usual sense. One example of such a Hamiltonian is $H={\mbox{$\textstyle\frac{1}{2}$}}p^2+{\mbox{$\textstyle\frac{1}{2}$}}\mu^2x^2+i
\epsilon x^3$, which is the quantum mechanical analog of the quantum field theoretic Hamiltonian $H=\int d{\bf x}\left[{\mbox{$\textstyle\frac{1}{2}$}}\pi^2+{\mbox{$\textstyle\frac{1}{2}$}}(\nabla\varphi)^2+
{\mbox{$\textstyle\frac{1}{2}$}}\mu^2\varphi^2+i\epsilon\varphi^3\right]$. Another example of a $\mathcal{P
T}$-symmetric non-Hermitian Hamiltonian is $H={\mbox{$\textstyle\frac{1}{2}$}}p^2+{\mbox{$\textstyle\frac{1}{2}$}}\mu^2 x^2-\epsilon
x^4$, which is the $\mathcal{PT}$-symmetric analog of the quantum field theoretic Hamiltonian $H=\int d{\bf x}\left[{\mbox{$\textstyle\frac{1}{2}$}}\pi^2+{\mbox{$\textstyle\frac{1}{2}$}}(\nabla\varphi)^2+
{\mbox{$\textstyle\frac{1}{2}$}}\mu^2\varphi^2-\epsilon\varphi^4\right]$. This latter Hamiltonian could be an interesting candidate for describing the Higgs sector of the standard model. It should be emphasized that we do not regard the condition of conventional Hermiticity as wrong. Rather, we view the condition of $\mathcal{PT}$ symmetry as offering 