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 $\dagger$ 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.

6 years ago

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