For any diagonalisable non-Hermitian H with real spectrum, the biorthogonal Gibbs functional satisfies positivity of ω_bi(A†A) for all A if and only if H is quasi-Hermitian.
Making Sense of Non-Hermitian Hamiltonians
12 Pith papers cite this work. Polarity classification is still indexing.
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Pith papers citing it
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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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.
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.
The pseudo-hermitian scalar model exhibits a line of non-unitary 4D fixed points, massless flows between them, and cyclic RG flows, supported by three-loop beta functions and an all-order conjecture.
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.
Critical unstable qubits exhibit indefinite anharmonic oscillations and coherence-decoherence cycles in a co-decaying frame, with first-time explicit geometric constructions for pure and mixed state trajectories on the Bloch sphere.
Generalized quantum dimensions from SymTFTs classify massless and massive RG flows in pseudo-Hermitian systems and relate coset constructions to domain walls.
A two-Higgs-doublet model with SU(2)-based marginal operators produces unavoidable cyclic RG flows, pseudo-unitary behavior below pair-production threshold, and Russian Doll VEVs whose period is fixed by the Koide formula to yield three families.
A density matrix approach to non-Hermitian two-flavor neutrino oscillations shows steady-state probabilities not necessarily 1/2, indicating non-Markovian behavior.
In a BRST-symmetric theory with non-Hermitian fermion mass matrix, the one-loop contribution to the φ†φ two-point function becomes real for real external momentum after removing the i factor from the e^{iS} normalization, due to conjugate pole pairing.
The inverted harmonic oscillator and its dual are argued to underpin a unique unitary renormalizable quantum gravity in four dimensions, yielding a non-singular universe and Starobinsky inflation.
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