Parity anomaly in a \mathcal{P}\mathcal{T}-symmetric quartic Hamiltonian |
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Authors: | Carl M Bender |
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Institution: | (1) Physics Department, Washington University, St. Louis, MO 63130, USA |
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Abstract: | In this paper, two independent methods are used to show that the non-Hermitian
-symmetric wrong-sign quartic Hamiltonian H = (1/2m)p
2 − gx
4 is exactly equivalent to the conventional Hermitian Hamiltonian
. First, this equivalence is demonstrated by using elementary differential-equation techniques and second, it is demonstrated
by using functional-integration methods. As the linear term in the Hermitian Hamiltonian
is proportional to ℏ, this term is anomalous; that is, the linear term in the potential has no classical analog. The anomaly
is a consequence of the broken parity symmetry of the original non-Hermitian
-symmetric Hamiltonian. The anomaly term in
remains unchanged if an x
2 term is introduced into H. When such a quadratic term is present in H, this Hamiltonian possesses bound states. The corresponding bound states in
are a direct physical measure of the anomaly. If there were no anomaly term, there would be no bound states. |
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Keywords: | " target="_blank"> gif" alt="
$$\mathcal{P}\mathcal{T}$$
" target="_blank">" align="middle" border="0"> symmetry anomaly non-Hermitian parity reflection time reversal |
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