Hi Steve,

In quantum mechanics, for any observable a, that matches an experimental result, there is an operator A which acts on the wavefunction of the system, so that, if the system is in a quantum state described by |ψ>, the expectation value of is = = a. Every operator corresponding to an observable is both linear and Hermitian: that is, for any two wavefunctions |ψ> and |φ>, and any two complex numbers α and β, linearity implies that A(α|ψ> + β|φ>) = α(A|ψ>) + β(A|φ>). Unitary operators corresponds to transformations which preserve the scalar or inner product

( and therefore all lengths and distances) of two wave functions, .

We can associate a Hermitian operator to any one-parameter family of unitary operators near the identity operator, and we can associate a family of unitary operators to any Hermitian operator by exponentiation.

Now, in quantum mechanics, all experiments that determine observables are in tune with the Hermitian character of the operators associated, in quantum theory, to those observables. Hermitian character can not be measured, because Hermiticity does not correspond to an observable, but, in measuring observables we are confirming their Hermiticity character.

Exs. of Hermitian operators: position operator, x, linear momentum operator, p, angular momentum operators, Lx,Ly,Lz.

Thanks Cesar.

I think you are giving me what might be a textbook answer. I am challenging that. I want to know where measurement on a pure state contradicts non-unitary, non-Hermitian mathematics.

Hi Steve,

But if experimental evidence does not prevent us from using self-adjoint operators, then why shouldn't we? Mathematically, they are very convenient thanks to the spectral theorem. Note that observables are postulated to be represented by self-adjoint operators in quantum mechanics.

There are certainly many rational matrices with irrational (and possibly non-real) eigenvalues, so I do not understand what you mean by the last sentence.

Hi Steve

Self-adjoint operators are also Hermitian in bounded physical spaces so Pekko is right when he says: "Note that observables are postulated to be represented by self-adjoint operators in quantum mechanics." In this sense I agree with him. I did not get your point.

Thank you for your responses.

I'll try and give a little more clarity on the intentions behind my question.

1. My premise is that Quamtum Theory does not represent experiments completely, in the sense that measurements on pure states are deterministic, while measurements on mixed states are random.

2. Experiments conducted by a collaboration in Vienna, published in 2008, show that there is missing mathematics relating to this randomness: Logical Independence and Quantum Randomness, by Tomasz Paterek etal. http://iopscience.iop.org/article/10.1088/1367-2630/12/1/013019/meta

3. My view is, therefore, we must challenge Quantum Postulates of textbook theory.

4. There is no question that unitary, Hermitian and Hilbert space mathematics is necessary for the representation of mixed states. My question is: in not using these mathematical structures to describe pure states, where is there any contradiction?

The mathematics used to describe quantum mechanics was chosen to correspond to the observed physical realities of experiment. The reason we use Hermitian operators is because they have real eigenvalues, corresponding to the fact that the numbers produced by measurements are real. While you could certainly claim to "measure" a complex or other sort of number, ultimately that could be seen as measuring a pair of real numbers (the real and imaginary parts), or four real numbers for a quaternion, so we don't lose any possibilities by requiring real numbers.

As a further example of this, measurements DID force us to use spinors rather than just the single complex wave function; again the predictions are collections of real numbers.

The need for unitarity is a consequence of our interpretation of quantum mechanics as probabilistic and evolving via the Schrodinger equation. The wave function is unchanged by an overall constant multiple. By interpreting the magnitude of the wave function to be one, we normalize probabilities in the usual way using the real part of this number. The phase that remains is arbitrary (*relative* phases are not, just an overall one).

If we were to transform a wave function by a non-unitary transformation, it is easy to show (try it!) that the Schrodinger equation does not preserve total probability. This, with the standard interpretation, means that there is not probability one that the particle is *somewhere*. The experimental observation you are asking for is that we do not see particles spontaneously vanish or come into being.

Thank you James.

My question suggests the use of non-unitary quantum theory for representation of pure states. Only pure states. I seek contradiction, if any should exist, between such a representation and experiment. Your response suggests such a contradiction by saying that Schrodinger must live within a unitary environment for probabilistic reasons. But I am wondering if a non-unitary form of the Hamiltonian, with the imaginary unit replaced by any rational scalar, would be able to represent pure states. Is not the unitary environment needed only to preserve probability for mixed states?

Dear Steve

Probably you know, and maybe this is the origin of your concern, about the emergence on the last years of non-Hermitian quantum theories that are explicitly PT-symmetric and/or pseudo-Hermitian or quasi-Hermitian with applications to some interesting topics as nonlinear oligomers, quantum catastrophes, the Landau–Zener problem, pseudo-fermions and so on.

Thank you Cesar.

No, I don't know of those things. But thanks for telling me of them. I just notice that you can make quantum representations for pure states without unitary structures. But then you need to restrict those non-unitary structures to unitary ones in order to facilitate mixed states.

It's not so much an issue of experiment, but of theoretical consistency. If the Hamiltonian isn't a Hermitian operator, therefore the evolution operator isn't a unitary operator, then the system isn't closed--the states of the system aren't described correctly. The distinction between pure and mixed states doesn't make sense either, then, since it's not possible to define the linear combinations of states properly.

It doesn't make sense to distinguish pure states and mixed states, incidentally. While eigenstates of a specific Hamiltonian will not evolve to linear superpositions, not all states are eigenstates of all Hamiltonians.

Hi, dear Steve

thnaks to your answer and to Stanley's answer's I think I also understand now your question. Sorry, for taking so much time. I start to mention a statement made by Katherine Jones-Smith, in her thesis about non-Hermitian quantum mechanics published a few years ago,

"...about a decade ago, Carl Bender et al showed that the assumption that operators, most importantly the Hamiltonian operator, need not be Hermitian in order to construct a consistent theory of quantum mechanics..."

I am not an adept of non-Hermitian quantum mechanics but I have always been open as a researcher to new ideas. Recently, despite working in a consistent manner (at least I think so...) with general relativity, I have followed new perspectives to work with an extension of GR still in the conceptual context, developed by a long-dead friend Walter Greiner called pseudo-complex general relativity. Thanks to your question I went back to consult my list of references in the topic of non-Hermitian quantum mechanics. Thanks for this.

And she add in some point of her thesis:

..."And then there is the so-called totalitarian principle: “Anything which is not forbidden is compulsory”. First introduced in T.H. White’s on "The Once and Future King as the governing principle of a colony of ants", and later conjectured by Murray Gell-Mann to apply to the physical laws that govern the universe, the totalitarian principle in this case says that there needn’t be anything wrong with Hermitian quantum mechanics– but if we can do non-Hermitian quantum mechanics then we must..."

Well, let's see ... concerning pure states on non-Hermitian QM I have no defitive answer to your question, will think about. I promisse.

Dear Steve

What still puzzles me when I read texts on Non-Hermitian Quantic Mechanics, and this can be easily proven theoretically, is that if we relax the imposition of Hermitian symmetry but imposing instead physical requirement of space-time reflection symmetry (PT symmetry) to QM without losing any of the remaining essential physical principles of quantum mechanics, the Dirac equation, for example, remains completely intact in all its formal aspects, which could be an indication (I am not advocating this but I am really intrigued) that the condition of Hermiticity in quantum mechanics could be less crucial than one might think ....

Interesting.

Dear Steve

Many years ago, working in the context of the Bethe-Salpeter approach I have examined two-body and three-body solutions of the Bethe-Saltpeter equation (in the realm of nuclear physics) using the Wick-Cutkosky model and I got Nakanishi's type abnormal solutions with negative norm. These solutions are known as "ghost solutions" and are in fact related to quasi-particle solutions and excitations, and more precisely to non-Hermitian quantum mechanics.

I also remember to come across, when working with renormalization, with states with negative norm, because renormalization causes the Hamiltonian to become non-Hermitian.

Recently it was shown by Bender that the combination of non-Hermitian quantum mechanics with the Dirac equation reveals new particle states with zero mass even if the mass matrix is different from zero. There are interpretations of these solutions as representing quasi-particle states (neutrino field fluctuations).

And he also show that "the Dirac equation constructed according to the principles of PT quantum mechanics is identical to the Hermitian Dirac equation, thereby endowing non-Hermitian quantum mechanics with a host of observed phenomena; anything that is accurately described by the standard Dirac equation is also described by the PT Dirac equation. Even more remarkable is that higher dimensional representations of the non-Hermitian Dirac equation describe new particles, with properties forbidden

in ordinary quantum mechanics and the Standard Model."

He showed that all of the virtues of Hermiticity e.g. real eigenvalues, unitary time evolution, etc., can be obtained by adopting an alternative set of assumptions. Non-Hermitian quantum mechanics is only non-Hermitian with respect to the standard inner product; operators in Bender's theory are self-adjoint with respect to a different definition of the inner product compared to the conventional form adopted in Hermitian quantum mechanics.

Thank you again for the motivation to think about a theme that I had left thinking about for a long time and is a very interesting topic. Again, I am not advocating this theory, and even because I work with Hermitian quantum mechanics, but I must admit, as my Italian grandfather liked to say that "se non é vero, é bene trovato" (if it is not true, it is well found)...

Best Regards

It's the other way around: non-hermitian or non-unitary don't constrain much so they don't imply anything explicit to check, by themselves.

Let U is evolution operator.

Actually preservation of probability requires U^+ U=1, but it does not require U U^+=1. It is an additional technical assumption. If U^+ U=1 and U U^+\not=1, then U^-1 does not exist.

One could easily construct non-unitary evolution with such non-reversible evolution operators by adding some extra non-accessable states to Hilbert space of pure states.

Thank you Michail.

Does your non-unitary system support existence of mixed states? If so How?