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arXiv:1811.01202

Asymmetry nature in symmetry breakdown: Anti-PT-symmetry

Biswanath Rath

Published Nov 3, 2018• 0 citations
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Abstract

We discover present anti-PT-symmetry operator is a problematic non-hermit operator. However the the usage of similarity transformation we seriously change to a new non-hermit operator underneath the identical commutating operator. Interestingly the use of new shape of PT-symmetry we reproduce preceding experimental outcomes and propose new nature of asymmetry in symmetry breakdown.

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This dissertation concerns the concept of PT symmetry as a replacement of the conventional symmetry of the Dirac Hermiticity of the operators, in particular, for the Hamiltonian. After solving a controversial issue surrounding the claimed violation of the special relativity by PT-symmetric quantum mechanics, we move further to discuss PT symmetry for the systems whose associated time reversal symmetry is odd, i.e., T^2 = −1. Our results, as published papers, are incorporated in the subsequent chapters. Then we conclude by suggestions for future work.

•Jan 1, 2019•0 citations
We discover present anti-PT-symmetry operator is a problematic non-hermit operator. However the the usage of similarity transformation we seriously change to a new non-hermit operator underneath the identical commutating operator. Interestingly the use of new shape of PT-symmetry we reproduce preceding experimental outcomes and propose new nature of asymmetry in symmetry breakdown.
Anti-PT\mathcal{PT}PT symmetry for a non-Hermitian Hamiltonian
Abstract Anti-PT\mathcal{PT}PT symmetry, (PT)H=−H(PT)(\mathcal{PT})H=-H(\mathcal{PT})(PT)H=−H(PT), is a plausible variant of PT\mathcal{PT}PT symmetry. Of particular interest is the situation when all the eigenstates of an anti-PT\mathcal{PT}PT-symmetric non-Hermitian Hamiltonian HHH are also eigenstates of the PT\mathcal{PT}PT operator; then, the quasi-energies are purely imaginary, which implies that the Hermitian conjugate H+=−HH^{+}=-HH+=−H, and thus they are connected via the relation (PT)H=H+PT(\mathcal{PT})H=H^{+}\mathcal{PT}(PT)H=H+PT, similar to the quasi-Hermiticity relation. Therefore, the eigenfunctions of the anti-PT\mathcal{PT}PT-symmetric HHH form a complete orthonormal set with positive definite norms, and moreover the time evolution is unitary.
Investigations during the last few years show that complex PT-symmetric or pseudo-Hermitian Hamiltonians possess a real discrete spectrum and several other features akin to a Hermitian Hamiltonian. These developments not only show us new directions but also demand a revisit to the conventional physics wherein PT-symmetry could have been invoked. After a review of some very interesting results, we present the super-barrier reflectionlessness and the cranking model of the nucleus. These two instances are being viewed as the meeting ground of the two: PT-symmetry and Hermiticity. The former presents an agreement and the latter is presently a dichotomy.
This dissertation concerns the concept of PT symmetry as a replacement of the conventional symmetry of the Dirac Hermiticity of the operators, in particular, for the Hamiltonian. After solving a controversial issue surrounding the claimed violation of the special relativity by PT-symmetric quantum mechanics, we move further to discuss PT symmetry for the systems whose associated time reversal symmetry is odd, i.e., T^2 = −1. Our results, as published papers, are incorporated in the subsequent chapters. Then we conclude by suggestions for future work.