I haven't read this yet, but I'm putting it up here for discussion as it seems so fascinating:(adsbygoogle = window.adsbygoogle || []).push({});

PT-Symmetric Versus Hermitian Formulations of Quantum Mechanics

Carl M. Bender, Jun-Hua Chen, Kimball A. Milton

A non-Hermitian Hamiltonian that has an unbroken PT symmetry can be converted by means of a similarity transformation to a physically equivalent Hermitian Hamiltonian. This raises the following question: In which form of the quantum theory, the non-Hermitian or the Hermitian one, is it easier to perform calculations? This paper compares both forms of a non-Hermitian $ix^3$ quantum-mechanical Hamiltonian and demonstrates that it is much harder to perform calculations in the Hermitian theory because the perturbation series for the Hermitian Hamiltonian is constructed from divergent Feynman graphs. For the Hermitian version of the theory, dimensional continuation is used to regulate the divergent graphs that contribute to the ground-state energy and the one-point Green's function. The results that are obtained are identical to those found much more simply and without divergences in the non-Hermitian PT-symmetric Hamiltonian. The $\mathcal{O}(g^4)$ contribution to the ground-state energy of the Hermitian version of the theory involves graphs with overlapping divergences, and these graphs are extremely difficult to regulate. In contrast, the graphs for the non-Hermitian version of the theory are finite to all orders and they are very easy to evaluate.

http://www.arxiv.org/abs/hep-th/0511229

Any comments?

[edit] At a quick glance, it appears to be a more natural way of including a Wick rotation. My hope would be that something like this would be a particularly natural way of implementing a theory where proper time is promoted to be an element of the geometry of space time, as I've proposed in the past. Along that line, it's interesting that, it's interesting that Kimball is a follower of Schwinger in Schwinger's later years which rejected the existence of the vacuum, which I also agree with. For more on this see:

http://www.arxiv.org/abs/hep-th/9901011

http://www.arxiv.org/abs/hep-th/9811054

[/edit]

Carl

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# PT symmetric non Hermitian formulation of QM

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