PT Symmetric

If you don't mind a rather unique sense of humour, check out the nice new seminar by Carl Bender on PT symmetric quantum mechanics, with non Hermitian Hamiltonians such as $H=p^2+i{x}^3$.

These Hamiltonians have many real eigenvalues. In one of Bender's plots (below) we see the real eigenvalues for a typical family of Hamiltonians, defined by varying a parameter $\epsilon$. Consider the example $H=p^2+x^2(ix{)}^{\epsilon }$, where $\epsilon$ goes from $-1$ to infinity. At $\epsilon =0$ we get back the ordinary harmonic oscillator. When $\epsilon$ is negative PT symmetry is broken, so the line $\epsilon =0$ marks a phase transition, which has now been measured in the laboratory using classical waveguides. Instead of $†$ Hermiticity defining duals for states, we use the CPT operation. This brings charge naturally into the picture, for negative $\epsilon$, since it fixes the PT problem of negative probabilities. Hilbert spaces are defined dynamically by this canonical choice of inner product.