Making sense of non-Hermitian Hamiltonians
By C. Bender. 2007
2,691 citations recorded by the source corpus.
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 ( ) symmetry. If H has an unbroken symmetry, then the spectrum is real. Examples of -symmetric non-Hermitian quantum-mechanical Hamiltonians are and . Amazingly, the energy levels of these Hamiltonians are all real and positive! Does a -symmetric Hamiltonian H specify a physical quantum theory in which the norms of states are positive and time evolution is unitary? The answer is that if H has an unbroken symmetry, then it has another symmetry represented by a linear operator . In terms of , one can construct a time-independent inner product with a positive-definite norm. Thus, -symmetric Hamiltonians describe a new class of complex quantum theories having positive probabilities and unitary time evolution. The Lee model provides an excellent example of a -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. The correct interpretation of the ghost is simply that the non-Hermitian Lee-model Hamiltonian is -symmetric. The 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 symmetry are illustrated by using many quantum-mechanical and quantum-field-theoretic models.
Subfields: Atomic and Molecular Physics, and Optics
View Paper: https://www.semanticscholar.org/paper/ebd05647a038d5ec90af8f2b2c9c7071be0179d0
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In Atomic and Molecular Physics, and Optics
Citations in the corpus
- Biorthogonal quantum mechanics (2013, background): “A PT-symmetric Hamiltonian is in general not Hermitian, but if the corresponding eigenstates are also PT symmetric, then the eigenvalues are real and eigenstates may be complete, and can be used to describe quantum systems [3, 4, 5].”
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- Symmetry-protected nodal phases in non-Hermitian systems (2018, neutral): “[50] C.”
- Exceptional topology of non-Hermitian systems (2019, neutral): “…classical systems, including optical settings, electrical circuits, and mechanical systems, which may be mapped to an effective NH Schrödinger equation, all the way to quantum materials (Datta, 2005; Bender, 2007; Rotter, 2009; El-Ganainy et al. , 2018; Miri and Alù, 2019; Ozawa et al. , 2019).”
- Anti-parity–time symmetry with flying atoms (2015, background): “The extracted EIT spectra formed by these red dots are a linear superposition of the two eigen-EIT supermodes as given in eqn.(3).”
- Loss-induced suppression and revival of lasing (2014, neutral): “es a departure from conventionaof this situation l Hermitian models with real eigenvalues and orthogonal eigenstates to Hermitian models nonfeaturing complex eigenvalues and nonorthogonal eigenstates (1,2,3). When tuning the parameters of such a system, its dissipative complex eigenvalues and the co”