Non-Hermitian quantum mechanics explained

In physics, non-Hermitian quantum mechanics, describes quantum mechanical systems where Hamiltonians are not Hermitian.

History

The first paper that has "non-Hermitian quantum mechanics" in the title was published in 1996 [1] by Naomichi Hatano and David R. Nelson. The authors mapped a classical statistical model of flux-line pinning by columnar defects in high-Tc superconductors to a quantum model by means of an inverse path-integral mapping and ended up with a non-Hermitian Hamiltonian with an imaginary vector potential in a random scalar potential. They further mapped this into a lattice model and came up with a tight-binding model with asymmetric hopping, which is now widely called the Hatano-Nelson model. The authors showed that there is a region where all eigenvalues are real despite the non-Hermiticity.

Parity–time (PT) symmetry was initially studied as a specific system in non-Hermitian quantum mechanics.[2] [3] In 1998, physicist Carl Bender and former graduate student Stefan Boettcher published a paper[4] where they found 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] Bender won the 2017 Dannie Heineman Prize for Mathematical Physics for his work.[6]

A closely related concept is that of pseudo-Hermitian operators, which were considered by physicists Paul Dirac,[7] Wolfgang Pauli,[8] and Tsung-Dao Lee and Gian Carlo Wick.[9] Pseudo-Hermitian operators were discovered (or rediscovered) almost simultaneously by mathematicians Mark Krein and collaborators[10] [11] [12] [13] as G-Hamiltonian in the study of linear dynamical systems. The equivalence between pseudo-Hermiticity and G-Hamiltonian is easy to establish.[14]

In 2002, Ali Mostafazadeh showed that every non-Hermitian Hamiltonian with a real spectrum is pseudo-Hermitian. He found that PT-symmetric non-Hermitian Hamiltonians that are diagonalizable belong to the class of pseudo-Hermitian Hamiltonians.[15] [16] [17] However, this result is not useful because essentially all interesting physics happens at the exception points where the systems are not diagonalizable. In 2020, it was proven that in finite dimensions PT-symmetry entails pseudo-Hermiticity regardless of diagonalizability, which indicates that the mechanism of PT-symmetry breaking at exception points, where the Hamiltionian is usually not diagonalizable, is the Krein collision between two eigenmodes with opposite signs of actions.

In 2005, PT symmetry was introduced to the field of optics by the research group of Gonzalo Muga by noting that PT symmetry corresponds to the presence of balanced gain and loss.[18] In 2007, the physicist Demetrios Christodoulides and his collaborators further studied the implications of PT symmetry in optics.[19] [20] The coming years saw the first experimental demonstrations of PT symmetry in passive and active systems.[21] [22] PT symmetry has also been applied to classical mechanics, metamaterials, electric circuits, and nuclear magnetic resonance.[23] In 2017, a non-Hermitian PT-symmetric Hamiltonian was proposed by Dorje Brody and Markus Müller that "formally satisfies the conditions of the Hilbert–Pólya conjecture."[24] [25]

Notes and References

  1. Hatano. Naomichi . Nelson. David R.. 1996-07-15. Localization Transitions in Non-Hermitian Quantum Mechanics. Physical Review Letters. 77. 3. 570–573. cond-mat/9603165. 1996PhRvL..77..570H. 10.1103/PhysRevLett.77.570. 43569614.
  2. N. Moiseyev, "Non-Hermitian Quantum Mechanics", Cambridge University Press, Cambridge, 2011
  3. Web site: 2015-07-20. Non-Selfadjoint Operators in Quantum Physics: Mathematical Aspects. 2018-06-12. Wiley.com. en-us.
  4. Bender. Carl M.. Boettcher. Stefan. 1998-06-15. Real Spectra in Non-Hermitian Hamiltonians Having $\mathsc\mathsc$ Symmetry. Physical Review Letters. 80. 24. 5243–5246. physics/9712001. 1998PhRvL..80.5243B. 10.1103/PhysRevLett.80.5243. 16705013 .
  5. Bender. Carl M.. 2007. Making sense of non-Hermitian Hamiltonians. Reports on Progress in Physics. 70. 6. 947–1018. hep-th/0703096. 2007RPPh...70..947B. 10.1088/0034-4885/70/6/R03. 119009206 . 0034-4885.
  6. Web site: Dannie Heineman Prize for Mathematical Physics.
  7. Bakerian Lecture - The physical interpretation of quantum mechanics . Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences . 18 March 1942 . 180 . 980 . 1–40 . 10.1098/rspa.1942.0023. 1942RSPSA.180....1D . free . Dirac . P. A. M. .
  8. Pauli . W. . On Dirac's New Method of Field Quantization . Reviews of Modern Physics . 1 July 1943 . 15 . 3 . 175–207 . 10.1103/revmodphys.15.175. 1943RvMP...15..175P .
  9. Lee . T.D. . Wick . G.C. . Negative metric and the unitarity of the S-matrix . Nuclear Physics B . February 1969 . 9 . 2 . 209–243 . 10.1016/0550-3213(69)90098-4. 1969NuPhB...9..209L .
  10. M. G. Krein, “A generalization of some investigations of A. M. Lyapunov on linear differential equations with periodic coefficients,” Dokl. Akad. Nauk SSSR N.S. 73, 445 (1950) (Russian).
  11. M. G. Krein, Topics in Differential and Integral Equations and Operator Theory (Birkhauser, 1983).
  12. I. M. Gel’fand and V. B. Lidskii, “On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients,” Usp. Mat. Nauk 10:1(63), 3−40 (1955) (Russian).
  13. V. Yakubovich and V. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Wiley, 1975), Vol. I.
  14. Zhang. Ruili. Qin. Hong. Xiao. Jianyuan. 2020-01-01. PT-symmetry entails pseudo-Hermiticity regardless of diagonalizability. Journal of Mathematical Physics. en. 61. 1. 012101. 10.1063/1.5117211. 0022-2488. 1904.01967. 2020JMP....61a2101Z . 102483351 .
  15. Mostafazadeh. Ali. 2002. Pseudo-Hermiticity versus symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian. Journal of Mathematical Physics. 43. 1. 205–214. math-ph/0107001. 10.1063/1.1418246. 2002JMP....43..205M. 15239201 . 0022-2488.
  16. Mostafazadeh. Ali. 2002. Pseudo-Hermiticity versus PT-symmetry. II. A complete characterization of non-Hermitian Hamiltonians with a real spectrum. Journal of Mathematical Physics. 43. 5. 2814–2816. math-ph/0110016. 10.1063/1.1461427. 2002JMP....43.2814M . 17077142 . 0022-2488.
  17. Mostafazadeh. Ali. 2002. Pseudo-Hermiticity versus PT-symmetry III: Equivalence of pseudo-Hermiticity and the presence of antilinear symmetries. Journal of Mathematical Physics. 43. 8. 3944–3951. math-ph/0107001. 10.1063/1.1489072. 2002JMP....43.3944M . 7096321 . 0022-2488.
  18. Ruschhaupt. A. Delgado. F. Muga. J G. 2005-03-04. Physical realization of -symmetric potential scattering in a planar slab waveguide. Journal of Physics A: Mathematical and General. 38. 9. L171–L176. 10.1088/0305-4470/38/9/L03. 0305-4470. 1706.04056. 118099017.
  19. Bender. Carl. April 2016. PT symmetry in quantum physics: from mathematical curiosity to optical experiments. Europhysics News. 47, 2. 2. 17–20. 10.1051/epn/2016201. 2016ENews..47b..17B. free.
  20. Makris. K. G.. El-Ganainy. R.. Christodoulides. D. N.. Musslimani. Z. H.. 2008-03-13. Beam Dynamics in $\mathcal\mathcal$ Symmetric Optical Lattices. Physical Review Letters. 100. 10. 103904. 2008PhRvL.100j3904M. 10.1103/PhysRevLett.100.103904. 18352189.
  21. Guo. A.. Salamo. G. J.. Duchesne. D.. Morandotti. R.. Roberto Morandotti . Volatier-Ravat. M.. Aimez. V.. Siviloglou. G. A.. Christodoulides. D. N.. 2009-08-27. Observation of $\mathcal\mathcal$-Symmetry Breaking in Complex Optical Potentials. Physical Review Letters. 103. 9. 093902. 2009PhRvL.103i3902G. 10.1103/PhysRevLett.103.093902. 19792798.
  22. 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. 2010NatPh...6..192R. 10.1038/nphys1515. 1745-2481. free.
  23. Miller. Johanna L.. October 2017. Exceptional points make for exceptional sensors. Physics Today. 10, 23. 10. 23–26. 10.1063/PT.3.3717. 2017PhT....70j..23M. free.
  24. 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. 10.1103/PhysRevLett.118.130201. 28409977. 1608.03679. 2017PhRvL.118m0201B. 46816531 .
  25. News: Quantum Physicists Attack the Riemann Hypothesis Quanta Magazine. Quanta Magazine. 2018-06-12.

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