Your assumption that negative and positive energy states could be orthogonal with respect to each other intrigues me, and I wonder how you could come to this conclusion.

In electromagnetism, electric and magnetic fields are orthogonal to each other with respect to the direction of motion in space of a charged particle, irrespective of whether the particle is negatively or positively charged.

In QED, the signs of the charges of electron and positron  are hypothesized as being a relative property as they define  the electron as energy propagating forward in space-time and a positron as an electron with energy moving backward in space-time.

This can be seen as an opposite-directions relation, not an orthogonal relation..

Andre -- thanks for your response. If I am right about the orthogonality, will I be right to say that this would mean that negative and positive energy matter can not exist in the same universe?

I don't think so. If you really wish to become familiar with accepted views on negative and positive energy, I suggest you dig into quantum electrodynamics.

This textbook is a possibility, but many other good refs can be used:

Walter Greiner and Joachim Reinhardt. Quantum Electrodynamics. Springer-Verlag. Second Edition, 1994. On page  41 for example, you will find mention of the timewise hypothesis on the opposite directions of positively and negatively signed particles.

The idea of having orthonormal solutions is so they span the whole space, so we have a complete set of solutions for the system. I.e. you can define your solution to your equation, lets call it F, as a linear combination of solutions F=sum{an * fn} where an are coefficients i.e. F= a1 f1 + a2 f2+ ... where they are orthogonal < fm | fn > = 0 and also orthonormal meaning < fm | fn > = \deltamn  as well complete sum{fn (x) fn(x')} = \delta(x,x').  We always try to define our space of solutions in Quantum Mechanics because it makes things easy, i.e. Hilbert space. When you talk about states, you talk about the Fock space which is defined by the creation and annihilaton relationships.

I cannot understand how spin is considered if there are only two solutions one for the positive energy state and the other for the negative energy state, as  said by the researcher who formulates this question. As explained in standad texts (for instance the book of  W.Greiner, RELATIVISTIC QUANTUM MECHANICS)  for a free particle there are four solutions of Dirac's equation (four spinors): two correspond to the same value of positive energy and the difference between them is associated to the orientation of the spin (+z or -z); similarly the other two solutions corresponds to the same value of negative energy . This four solutions are orthogonal.

I would like to sudy the article of this resercher for giving a more detailed opinion

Hi Patricio

You are right. The issue of spin does introduce two more solutions. They are however linked to the magnetic aspect of charged particles, which is different from their electric aspect.

This is not covered in QM except under the guise of a quantum number linked to the magnetic moment of the particle.

HI André

Thanks for your clarification.

Golden> Am I right to say that this means that it must be impossible for a positive energy state to be found in a negative energy state?

Yes (with the caveat spelled out below). If you have found the eigenstates of a Hamiltonian, states with different energies are orthogonal. Which implies exactly what you write.

However, you should be aware that one almost never can find the exact eigenstates of an Hamiltonian.  F.i., the positive (or negative) energy eigenstates of the Dirac hamiltonian are eigenstates only when one ignore interactions with the electromagnetic field. When you include such interactions, then according to the original view of Dirac, the

positive-energy Dirac state + no photons

is not longer orthogonal to a

negative-energy Dirac-state + (one or more photons with sufficient energy)

Golden> am I right to say the Pauli Exclusion Principle (as used by Dirac himself) is not necessary in-order to prevent the positive energy states from cascading down into the negative energy?

No, because they would not be cascading down to negative energy: They would be cascading down to a state with one "negative energy electron", plus one or more positive energy photons, in such a way that the total energy was positive.

I should warn you that the picture I have used above, in terms of a filled Dirac sea, is somewhat outdated. But only because we insist that Dirac particles must obey the Pauli principle (Fermi-Dirac statistics).

This is a standard homework assignment and described in all textbooks. There aren't any ``versions of Dirac theory'', there's the Dirac equation and it's uniquely defined. That states of positive and negative energy are orthogonal to each other doesn't make sense absent further  qualification. They are, for the free Hamiltonian, for which a global choice of positive and negative frequencies  can be made-that defines the vacuum state. That this  state is, indeed, stable,  means that it is annihilated by the annihilation operators of *both* signs, which occurs only if they anticommute-that's how the Pauli principle enters the picture. If these commuted, then the vacuum would have been unstable. The creation and annihilation operators of each sign already must anticommute, anyway. 

Once this is established, it follows that positive and negative energy states don't mix. In the presence of interactions, however, they can mix (cf. fuller explanation below).

Thanks all for your inputs, I will soon post my paper and you can comment on it

Indeed, in the presence of interactions, the positive and negative energy states of the free Hamiltonian  do mix-only they mix in such a way that the vacuum isn't rendered unstable-that's what vacuum polarization means. If they don't mix, that means that the interaction, in fact, is a coordinate artifact, it can be eliminated by redefining the fields. 

They can mix in a way that renders the vacuum unstable-which can happen when an electric field is switched on.

All this is in Weinberg's textbook, vol I, so it might be useful to consult it-or any book on quantum field theory for that matter. It's not recommended to write papers on topics, whose solutions are learned in university courses already-but to learn them and write papers on problems whose solution isn't, already,  learned by everybody that is studying physics.

In general, quantum states are orthogonal when they belong to different coherent subspaces of the Hilbert space.

When I asked this question, this is what I had in mind:

https://www.researchgate.net/publication/299597834_Pauli_Exclusion_Principle_the_Dirac_Void_and_the_Preponderance_of_Matter_over_Antimatter

Will be glad to hear from on the matter based on the above paper!

Article Pauli Exclusion Principle, the Dirac Void and the Prepondera...