# Non-Hermitian quantum mechanics

Non-Hermitian quantum mechanics[1][2] is the study of quantum-mechanical Hamiltonians that are not Hermitian. Notably, they appear in the study of dissipative systems. Also, non-Hermitian Hamiltonians with unbroken parity-time (PT) symmetry have all real eigenvalues.[3]

## Parity-time (PT) symmetry

In 1998, physicist Carl Bender and former graduate student Stefan Boettcher published in Physical Review Letters a landmark paper in quantum mechanics, "Real Spectra in non-Hermitian Hamiltonians Having PT Symmetry."[4] In this paper, the authors overturned a longstanding incorrect assumption of quantum mechanics that a Hamiltonian must be Hermitian in order to have all-real eigenvalues. Rather, non-Hermitian Hamiltonians endowed with an unbroken PT symmetry (invariance with respect to the simultaneous action of the parity-inversion and time reversal symmetry operators) also may possess a real spectrum. Under a correctly-defined inner product, a PT-symmetric Hamiltonian's eigenfunctions have positive norms and exhibit unitary time evolution, requirements for quantum theories.[5]

New insights continued to increasingly vindicate the mathematical relevance of PT symmetry,[5] though the topic did not reach full renown until nearly a decade after its discovery. In 2007, the physicist Demetrios Christodoulides and his collaborators noticed that PT symmetry corresponds to the presence of balanced gain and loss in optical systems.[6][7] The coming years saw the first experimental demonstrations of PT symmetry in passive and active systems,[8][9] followed by an explosion of papers demonstrating the potential for new optical applications and devices. PT symmetry is now also known to have uses in many other areas of physics, including but not limited to classical mechanics, metamaterials, electric circuits, and nuclear magnetic resonance.[10][6] Bender won the 2017 Dannie Heineman Prize for Mathematical Physics for his work.[11]

In 2017, PT-symmetric Hamiltonians drew attention in the mathematical community when Bender, Dorje Brody, and Markus Muller described a non-Hermitian Hamiltonian that "formally satisfies the conditions of the Hilbert–Pólya conjecture." [12][13]

## Non-Hermitian Hamiltonians

Non-Hermitian quantum mechanics deals with two types of physical phenomena. One type of phenomena cannot be described by the standard (Hermitian) quantum mechanics since the local potentials in the Hamiltonians are complex. The second type of phenomena are associated with local real potentials that support continuous spectra.

The second type of phenomena can be described only by the time dependent Schrödinger equation. The potentials can be complex because of different reasons, such as when complex absorbing potentials (CAPs) are introduced into the physical Hamiltonian in order to enable one to carry out long time wavepacket propagations by using finite grids or finte number of basis functions that impose box quantization boundary conditions on the solutions on the time-dependent and the time independent Schrödinger equations. The reflection free CAPs RFCAP suppress the artificial reflections of the tail of the wavepackets from the edge of the grid or from the use of box-quantization boundary conditions.

## References

1. ^ N. Moiseyev, "Non-Hermitian Quantum Mechanics", Cambridge University Press, Cambridge, 2011
2. ^ "Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects". Wiley.com. 2015-07-20. Retrieved 2018-06-12.
3. ^ Bender, Carl M. (2007-06-01). "Making Sense of Non-Hermitian Hamiltonians". Reports on Progress in Physics. 70 (6): 947–1018. arXiv:hep-th/0703096. Bibcode:2007RPPh...70..947B. doi:10.1088/0034-4885/70/6/R03. ISSN 0034-4885.
4. ^ Bender, Carl M.; Boettcher, Stefan (1998-06-15). "Real Spectra in Non-Hermitian Hamiltonians Having $\mathsc{P}\mathsc{T}$ Symmetry". Physical Review Letters. 80 (24): 5243–5246. arXiv:physics/9712001. Bibcode:1998PhRvL..80.5243B. doi:10.1103/PhysRevLett.80.5243.
5. ^ a b Bender, Carl M. (2007). "Making sense of non-Hermitian Hamiltonians". Reports on Progress in Physics. 70 (6): 947–1018. arXiv:hep-th/0703096. Bibcode:2007RPPh...70..947B. doi:10.1088/0034-4885/70/6/R03. ISSN 0034-4885.
6. ^ a b Bender, Carl (April 2016). "PT symmetry in quantum physics: from mathematical curiosity to optical experiments". Europhysics News. 47, 2: 17–20.
7. ^ Makris, K. G.; El-Ganainy, R.; Christodoulides, D. N.; Musslimani, Z. H. (2008-03-13). "Beam Dynamics in $\mathcal{P}\mathcal{T}$ Symmetric Optical Lattices". Physical Review Letters. 100 (10): 103904. Bibcode:2008PhRvL.100j3904M. doi:10.1103/PhysRevLett.100.103904.
8. ^ Guo, A.; Salamo, G. J.; Duchesne, D.; Morandotti, R.; Volatier-Ravat, M.; Aimez, V.; Siviloglou, G. A.; Christodoulides, D. N. (2009-08-27). "Observation of $\mathcal{P}\mathcal{T}$-Symmetry Breaking in Complex Optical Potentials". Physical Review Letters. 103 (9): 093902. Bibcode:2009PhRvL.103i3902G. doi:10.1103/PhysRevLett.103.093902. PMID 19792798.
9. ^ Rüter, Christian E.; Makris, Konstantinos G.; El-Ganainy, Ramy; Christodoulides, Demetrios N.; Segev, Mordechai; Kip, Detlef (March 2010). "Observation of parity–time symmetry in optics". Nature Physics. 6 (3): 192–195. Bibcode:2010NatPh...6..192R. doi:10.1038/nphys1515. ISSN 1745-2481.
10. ^ Miller, Johanna L. (October 2017). "Exceptional points make for exceptional sensors". Physics Today. 10, 23 (10): 23–26. doi:10.1063/PT.3.3717.
11. ^
12. ^ Bender, Carl M.; Brody, Dorje C.; Müller, Markus P. (2017-03-30). "Hamiltonian for the Zeros of the Riemann Zeta Function". Physical Review Letters. 118 (13): 130201. arXiv:1608.03679. Bibcode:2017PhRvL.118m0201B. doi:10.1103/PhysRevLett.118.130201. PMID 28409977.
13. ^ "Quantum Physicists Attack the Riemann Hypothesis | Quanta Magazine". Quanta Magazine. Retrieved 2018-06-12.