dorsal/arxiv
View SchemaThe C Operator in PT-Symmetric Quantum Theories
| Authors | Carl M. Bender, Joachim Brod, Andre Refig, Moritz Reuter |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0402026 |
| URL | https://arxiv.org/abs/quant-ph/0402026 |
| DOI | 10.1088/0305-4470/37/43/009 |
| Journal | J. Phys. A: Math. Gen. 37 (2004) 10139 |
Abstract
The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). This paper investigates an alternative way to construct quantum theories in which the conventional requirement of Hermiticity (combined transpose and complex conjugate) is replaced by the more physically transparent condition of space-time reflection (PT) symmetry. It is shown that if the PT symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitian quantum-mechanical Hamiltonians are H=p^2+ix^3 and H=p^2-x^4. The crucial question is whether PT-symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken PT symmetry also possesses a physical symmetry represented by a linear operator called C. Using C it is shown how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables are defined, probabilities are positive, and the dynamics is governed by unitary time evolution. After a review of PT-symmetric quantum mechanics, new results are presented here in which the C operator is calculated perturbatively in quantum mechanical theories having several degrees of freedom.
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"abstract": "The Hamiltonian H specifies the energy levels and the time evolution of a\nquantum theory. It is an axiom of quantum mechanics that H be Hermitian because\nHermiticity guarantees that the energy spectrum is real and that the time\nevolution is unitary (probability preserving). This paper investigates an\nalternative way to construct quantum theories in which the conventional\nrequirement of Hermiticity (combined transpose and complex conjugate) is\nreplaced by the more physically transparent condition of space-time reflection\n(PT) symmetry. It is shown that if the PT symmetry of a Hamiltonian H is not\nbroken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitian\nquantum-mechanical Hamiltonians are H=p^2+ix^3 and H=p^2-x^4. The crucial\nquestion is whether PT-symmetric Hamiltonians specify physically acceptable\nquantum theories in which the norms of states are positive and the time\nevolution is unitary. The answer is that a Hamiltonian that has an unbroken PT\nsymmetry also possesses a physical symmetry represented by a linear operator\ncalled C. Using C it is shown how to construct an inner product whose\nassociated norm is positive definite. The result is a new class of fully\nconsistent complex quantum theories. Observables are defined, probabilities are\npositive, and the dynamics is governed by unitary time evolution. After a\nreview of PT-symmetric quantum mechanics, new results are presented here in\nwhich the C operator is calculated perturbatively in quantum mechanical\ntheories having several degrees of freedom.",
"arxiv_id": "quant-ph/0402026",
"authors": [
"Carl M. Bender",
"Joachim Brod",
"Andre Refig",
"Moritz Reuter"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/37/43/009",
"journal_ref": "J. Phys. A: Math. Gen. 37 (2004) 10139",
"title": "The C Operator in PT-Symmetric Quantum Theories",
"url": "https://arxiv.org/abs/quant-ph/0402026"
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