Non-Hermitian Quantum Mechanics

A fundamental assumption of quantum mechanics is that operators are represented by Hermitian matrices. This guarantees that observable quantities, which are given by the eigenvalues of these matrix operators, are real-valued (as opposed to complex), and that quantum mechanical systems evolve in a manner that conserves probability. In recent decades it has been shown that this assumption is sufficient but not necessary to ensure a valid quantum mechanical theory. (For a review see Bender, "Making sense of non-Hermitian Hamiltonians." Reports on Progress in Physics 70.6 (2007): 947. (2007).)   In PT quantum mechanics, the assumption of Hermitian operators is relaxed, and another set of assumptions is adopted, wherein the parity (P) and time-reversal (T) operators determine the specific properties required of matrix operators in a theory.  I have studied several interesting systems within PT quantum mechanics and non-Hermitian quantum mechanics more generally. 
 
 In [1] we extended the formalism of PT quantum mechanics to include systems that are odd under time reversal (i.e. T² = -1).  Odd time reversal applies to fermionic systems including quarks and leptons and a plethora of models in nuclear, atomic and condensed matter physics. A key result of [1] is the identification of  the PT analog of Kramer's degeneracy from Hermitian quantum mechanics. In [2] we use this formalism to construct a non-Hermitian, PT-symmetric version of the Dirac equation. We relax the assumption that the Dirac equation is constructed using only Hermitian matrices; we find that the ordinary solution to the Dirac equation (i.e. a 4-component wavefunction comprised of a pair of Weyl spinors coupled by a mass term) is identical in the non-Hermitian extension. Solutions to the Hermitian Dirac equation with 8 or 12 components trivially decouple into independent fermions, each described by a 4-component wavefunction.  Remarkably, by allowing the matrices in the Dirac equation to be non-Hermitian, the decoupling is prevented and we find the 8 and 12 component solutions represent new types of particles. This finding is significant as a new variant on the Dirac equation has not been found since the original work of Dirac and Majorana. 

I have also used non-Hermitian quantum mechanics to analyze problems in strongly correlated condensed matter physics. Doped cuprates are a very interesting class of materials that demonstrate high temperature superconductivity that is not explained by the conventional BCS theory.  Determining the underlying physics that gives rise to this high temperature superconductivity is one of the main goals of theoretical condensed matter physics. In 1956 Freeman Dyson used a non-Hermitian Hamiltonian to analyze excitations of a ferromagnet ("General theory of spin-wave interactions." Physical review 102.5 (1956): 1217.) In [3] I applied Dyson's non-Hermitian methods to doped antiferromagnets described by the t-J model, which is believed to capture the essential physics of the doped cuprates.  (The work presented in [1-3] is also described in detail my Ph.D thesis,  [4].) 

I have also analyzed several classic problems in the non-Hermitian realm, for example in [5] we show that the famous Hofstadter butterfly plot, which depicts the energy eigenvalues of electrons in a magnetic field, emerges from a beautiful "cocoon" pattern in the complex realm. I have also considered simple vector models in the PT realm [6] and the role of boundary conditions for the PT version of the classic particle in a box problem [7]. 

[1] Jones-Smith, Katherine, and Harsh Mathur. "Non-Hermitian quantum Hamiltonians with P T symmetry." Physical Review A 82.4 (2010): 042101. PDF available here.
[2] Jones-Smith, Katherine, and Harsh Mathur. "Relativistic non-Hermitian quantum mechanics." Physical Review D 89.12 (2014): 125014. PDF available here.
[3] Jones-Smith, Katherine. "A ‘Dysonization’scheme for identifying quasi-particles using non-Hermitian quantum mechanics." Phil. Trans. R. Soc. A 371.1989 (2013): 20120056.  PDF available here. 
[4]Jones-Smith, Katherine. "Non-Hermitian quantum mechanics." Doctoral dissertation, Case Western Reserve University, 2010. PDF available here.
[5] Jones-Smith, Katherine, and Connor Wallace. "Hofstadter’s Cocoon." International Journal of Theoretical Physics 54.1 (2015): 219-226. PDF available here.
[6] Jones-Smith, Katherine, and Rudolph Kalveks. "Vector Models in PT Quantum Mechanics." International Journal of Theoretical Physics 52.7 (2013): 2187-2195. PDF available here.
[7] Dasarathy, A., Isaacson, J. P., Jones-Smith, K., Tabachnik, J., & Mathur, H.  "Particle in a box in PT-symmetric quantum mechanics and an electromagnetic analog." Physical Review A, 87(6), 062111. (2013). PDF available here.