A PT-symmetric non-Hermitian free-fermion field theory realizes logarithmic conformal field theory with central charge c=-2 via a biorthogonal Virasoro algebra construction.
Bender,Making sense of non-Hermitian Hamiltonians,Rept
8 Pith papers cite this work. Polarity classification is still indexing.
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abstract
The Hamiltonian H specifies the energy levels and time evolution of a quantum theory. A standard axiom of quantum mechanics requires that H be Hermitian because Hermiticity guarantees that the energy spectrum is real and that time evolution is unitary (probability-preserving). This paper describes an alternative formulation of quantum mechanics in which the mathematical axiom of Hermiticity (transpose + complex conjugate) is replaced by the physically transparent condition of space-time reflection (PT) symmetry. If H has an unbroken PT symmetry, then the spectrum is real. Examples of PT-symmetric non-Hermitian quantum-mechanical Hamiltonians are H=p^2+ix^3 and H=p^2-x^4. Amazingly, the energy levels of these Hamiltonians are all real and positive! In general, if H has an unbroken PT symmetry, then it has another symmetry represented by a linear operator C. Using C, one can construct a time-independent inner product with a positive-definite norm. Thus, PT-symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution. The Lee Model is an example of a PT-symmetric Hamiltonian. The renormalized Lee-model Hamiltonian has a negative-norm "ghost" state because renormalization causes the Hamiltonian to become non-Hermitian. For the past 50 years there have been many attempts to find a physical interpretation for the ghost, but all such attempts failed. Our interpretation of the ghost is simply that the non-Hermitian Lee Model Hamiltonian is PT-symmetric. The C operator for the Lee Model is calculated exactly and in closed form and the ghost is shown to be a physical state having a positive norm. The ideas of PT symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.
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Non-Hermitian knot topology exhibits first-order transitions that mirror Hermitian topological phase transitions when singular values are matched to Hermitian eigenvalues, without exceptional points.
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The index of non-Hermitian Dirac operators that anticommute with a chirality operator is topologically protected when the operators are diagonalizable and elliptic.
Exact WKB analysis produces median-summed spectra and an algebraic equation for the exceptional point of PT-symmetry breaking in the inverted triple-well system.
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A PT-symmetric non-Hermitian free-fermion field theory realizes logarithmic conformal field theory with central charge c=-2 via a biorthogonal Virasoro algebra construction.
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The index of non-Hermitian Dirac operators that anticommute with a chirality operator is topologically protected when the operators are diagonalizable and elliptic.
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