As far as I know, the reason for using non-hermitian formalism is analytical simplicity for certain class of problems. Even when we use non-hermitian QM the eigenvalues still turn out to be real. If the Hamiltonian (H) has an unbroken space-time (PT) symmetry, then the eigenvalue spectrum is real for non-hermitian Hamiltonians. A good account of non-hermitian QM can be found in the textbook by Nimrod Moiseyev (Non-Hermitian Quantum Mechanics). 

A Hermitian operator has mutually orthogonal eigenvectors and so their eigenstates are distinguishable. Even though the non-Hermitian Hamiltonian has real eigenvalues in the situation mentioned by Naqib, it does not have distinguishable eigenvectors. Then what is the physical picture of systems having such Hamiltonians?

As I said it is mathematically convenient for certain class of problems. See for example arXiv:0703096. As long as it gives real answers to eigenvalue problems, its use can be justified. I am not aware of any physical significance of non-hermitian Hamiltonian. Though attempts have been made to relate such Hamiltonians for dissipative problems. Such interpretations are questionable. 

Dear Sumanth Kumar,

In quantum mechanics, the Hamiltonian H specifies the energy levels and time evolution of a quantum system. A standard mathematical, not physical axiom of quantum mechanics requires that H be Hermitian; then the energy spectrum is real and the time evolution is unitary (probability-preserving). 

In an alternative formulation of quantum mechanics, 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! 

These PT symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space. PT-symmetric systems have attracted lately a lot of attention; a search in the arXiv will return hundreds of papers. I would also suggest you to see the arXiv:hep-th/0703096 paper and other papers by Carl Bender.

Note that the notions and concepts of PT-symmetric quantum mechanics have been extended to optical systems as well as nonlinear classical lattices, where the PT symmetry relies on gain and loss. The variation of the gain/loss parameter makes such a system to undergo phase transition(s) from the exact PT symmetry phase to the broken PT symmetry phase, where no stable solutions exist.

Best,

Nikos Lazarides 

This means that the operator averages are unphysical or can't be observed

THE EIGENVALUES OF NON-HERMITIAN OPERATORS ARE COMPLEX.  MODULUS CAN HAVE REAL MEANING.

Dear Sumanth,

I think the previous comments on PT symmetry point in the right direction, but I have to disagree with the statement that eigenvalues non-Hermitian operators are unphysical. As has been said already, for PT symmetric operators the eigenvalues are real, despite the operator non being Hermitian. There are of course differences, for non-Hermitian operators we have distinguish between left and right eigenvectors to given eigenvalue. Adding to the list of references here is a example of a non-Hermitian operator in classical optics which has real and very much observable eigenvalues.

Article Eigenpolarizations for Giant Transverse Optical Beam Shifts

Dear colleagues, I should reply that any physical system is a dynamical one, following the Poincare definition. This one transforms one system state to another with the help of the group parameters (that plays as the time in an ordinary case). Is the state a classic Hamiltonian vector or a wave functions - it doesn't matter. The operator of a dynamical system is always Hermitian, else, say, the mean value of energy or momentum can be complex, in absurd. Evident examples of non hermitian: creation and annihilation operators are introduced for simplicity in manipulations, and are expressed only by allowed pair through the hermitian operators. So that, if there is no dynamical system in the case, then the kind of operator is what you want. Therefore, the latter case cover just a formal mathematical transform that can be arbitrary interpreted, maybe with plus for physics. Thanks.

Prof. Nikos Lazarides states that  "PT-symmetric quantum mechanics have been extended to optical systems" (....), where the PT symmetry relies on gain and loss." I would appreciate if he were more specific about a good reference. I have heard Prof. Carl Bender speak on the mathematical extension to non-hermitian Hamiltonians. If loss or gain is intended, complex canonical transformations can take that system to a Bargmann-type Hilbert space where one integrates over the complex plane (which Bargmann introduced to have the oscillator wavefunctions represented by  z^n, for raising and lowering operatores).  Otherwise, I have not read of such complex extensions for PT-invariant optics. I'm not disputing the mathematical consistency of the theory, but only its applicability to geometric and wave-optical systems. 

Non-hermitian operators are sometimes useful, e.g., in the scattering theory at describing an effect of common non-elastic processes on the elastic channel (the optic potential approach). The other example - in a number of pure theoretical investigations of different relations between the physical quantities (e.g., at the conclusion of  the Heisenberg uncerntainty relations from the basic notions of QM). 

To answer Professor Wolf's question on references to PT symmetry in optical systems, I would think that Professor Lazarides's own work could be a starting point, another name that springs to mind is Demetrios Christodoulides, who has published several papers on this topic.

Dear all,

I think that non-hermitian quantum mechanics, and  in particular pt- symmetric quantum mechanics, is a generalization of the ordinary quantum mechanics, very much in the same way complex number generalize real ones and used in physics. Since we use complex numbers quite comfortably,  we should not be hesitant about non-hermitian Hamiltonians in QM. I think that Carl Bender's work demonstrated that there is no physical reason to restrict QM to hermitian Hamiltonian only, as long as there are real eigenvalues for the observables, at least in certain parts of the parameter space. 

I am more familiar with the classical aspects of the pt-symmetric systems, where most of what is expected from a PT-symmetric material has actually been observed.

In optics, in particular, the subject was started indeed by Christodoulides' group, whose predictions were verified a few years later. Please see below refs(1,2)

The first experiments are reported in (3), (4). There is also a very interesting realization of a PT-system using electronic circuits (5). The latter inspired my collaborator and me to work on a simple model with PT-symmetry relying on gain and loss, aiming to connect the notions of PT-symmetric systems with that of metamaterials (PRL 110, 053901 2013). At this time there is a vast literature on PT-symmetric systems that makes it not so easy to keep track of everything that is going on!!

Best regards,

Nikos Lazarides

(1) El-Ganainy et. al.,   Opt. Lett. 32, 2632 (2007)

(2) Makris et al., Phys. Rev. Lett.  100, 103904 (2008)

(3) Guo et al, Phys. Rev. Lett. 103, 093902 (2009)

(4) Rutter et al., Nat. Phys. 6, 192 (2010).

(5) Schindler et al., Phys. Rev. A 84, 040101(R) (2011).

There are two ways to interpret this question. On the one hand there are bounded non Hermitian operators A. Even such an operator still defines a flow on the projectivised Hilbert space by

i \frac{\partial}{\partial t} \psi = A \psi. 

The eigenvectors are the stationary points of this flow, and the eigenvalues (up to a common constant) tell you how the flow approaches the stationary point. In the non Hermitian case the probability need not remain preserved however. 

In the other interpretation there is the can of worms that is the difference between an unbounded symmetric and an unbounded self adjoint  operator. That has everything to do with boundary values of which there can be many. For example the operator  p = i \frac{\partial}{\partial x} has no  self adjoint extension  on L_2[0,\infty) whereas p^2 on the half line has several different.  

Dear Sumanta

You first define (i) Non-Hermitian (PT-symmetry ) or (ii) Non-Hermitian  (Non-PT symmetry).For expt work on PT symmetry see the paper in NATURE(PHYSICS).

For non-Pt symmetry it will have large applications in future ,while making communications ,n Quantum Optics.

As non-Hermitian  ,is at preliminary stage of development , we should understand its theory first . You will come across some papers in which eigenvalues are (complex,partially real and partially complex,real) .As long as we do not understand this

break down ,i believe many people will not show any  interest in experimental work.

So I hope  your question  will have  a far reaching consequences in future. But do not bother for  expts but give some input for understanding  if you are really interested in complex Q M.

With best wishes 

Prof B.Rath