Does the same reasoning apply to Klein-Gordon equation? 

This problem is both experimental and theoretical. Experiments show that the mass energy of a positron is positive, since twice +511kev are necessary to create an electron-positron pair. The theory has difficulties to put together the Tetrode's energy-momentum tensor of the Dirac wave, non-symmetric, and the symmetric Ricci tensor of the General Relativity.

Nevertheless I have another solution with a generalisation of the Dirac wave allowing to describe all fermions of the first generation. The mass term allows both solutions with +E and -E energies, but the Tetrode's density of energy is always positive and is then compatible with experiments.

Federico, no, all KG solutions have the same positive energy.

Claude, any reference we can have a look at for more information about your solution?

For free : C. Daviau and J. Bertrand, (2014c) New Insights in the Standard Model

of Quantum Physics in Clifford Algebra (Je Publie, Pouillé-les-coteaux)

and http ://hal.archives-ouvertes.fr/hal-00907848

chapter 3

Your answer to Federico seems inappropriate : the problem of negative energies first arose  with the Klein-Gordon equation.

Yes, we discussed it in private: the point is that there is a distinction between positive/negative frequencies (plane-wave solutions of the free field equations) and energies, and for energies one has to further distinguish between the momentum vector (which is the current vector but without the charge factor) and the energy tensor (the variation of the Lagrangian with respect to the metric of the space-time): both KG and Dirac field have both frequencies, with the positivity/negativity of frequencies transferred onto the positivity/negativity of the momentum vector of KG and the energy tensor of Dirac field; but (the temporal component of) the energy tensor of KG and the momentum vector of Dirac fields is always positive. When I talk about energy I do not use anything but the quantity that is the variation of the L with respect to g because this is the source of the gravitational field equations: and for this quantity the KG field is always positive (it is the square of the time-derivative plus the square of the gradients, and sum of squares is always positive). About this quantity for the Dirac field it is my original question.

Then for the reference you gave me, many thanks, but it is a little long: could you pin-point more precisely? That reference has a structure that is very similar to your book "Double Space-Time and more": the generalization you talked about can also be found in this book? If yes, can you tell me the page, since I have it at home? Many thanks.

Varying L with respect to g supposes that the L is written as a function of g. This is not the case with the Dirac equation or with my non-linear wave equation which has the Dirac equation as linear approximation. An answer to your question supposes that it is possible to write the Dirac equation in any frame, not only in the inertial frame implicitly used by the Dirac wave. It is not proved that this is possible, because the representations of the SL(2,C) group used for the spin 1/2 are not isomorphisms into the Lorentz group.. And the energy-momentum tensor of the Dirac theory, which comes from the invariance under translations via the Noether's therorem, is not symmetric, contrary to the Ricci tensor of general relativity.

Therefore my answer is not directly an answer to your question. I have suppressed, in the Dirac Lagrangien, the cos(beta) term in factor of m ρ (beta is the Yvon-Takabayasi angle). And this m ρ  term is general and always positive. Both for the lepton or for the quark wave of the first generation  ρ is the norm of the J current which is the probability current. But with the general mass term the temporal component of the momentum-energy tensor is always positive.

http://dx.doi.org/10.4236/jmp.2016.712143

http://dx.doi.org/10.4236/jmp.2016.716207

Right, for the Dirac case you have to vary L with respect to the tetrads: this gives you a generalized form of Einstein equations that are not symmetric, but on which we can split in symmetric and antisymmetric parts: the antisymmetric part is the divergence of the spin-torsion field equation, and then the symmetric part is the usual Einstein equation with spin contributions.

For the Takabayashi angle I agree absolutely that it should be included, but I do not see how that could render the T_00 always positive: any reference?

It is in fact possible to write the Dirac equation in terms of the metric without a tetrad if one uses a nonlinear group realization---roughly, the spinor has a metric-dependent (and quite nonlinear) coordinate transformation rule.  Infinitesimally, this was done by Ogievetsky and Polubarinov in 1965.  Ranat Bilyalov did the finite case in the 1990s.  The works were all translated into English.  References can be found in my paper https://arxiv.org/abs/1111.4586.  Thus there is no difficulty in principle in using the Hilbert metric stress-energy definition.  A quick and dirty way to understand this formalism (roughly) is to impose the symmetric tetrad gauge condition (as many authors do):  in each coordinate system, perform a local Lorentz boost-rotation to make the tetrad symmetric.  Bilyalov and I address what happens far from the identity. 

Always is not appropriate, "for all stationary states" is explained in "The standard model of quantume physics in Clifford Algebra", C. Daviau and J. Bertrand, World Scientific, page 139 that I join.

There is this symmetric tensor: 

https://en.wikipedia.org/wiki/Belinfante%E2%80%93Rosenfeld_stress%E2%80%93energy_tensor

You say that this was done by Ogievetsky and Polubarinov, but I think that all the difficulties were put in the part of the "integration by parts" that was never precisely described, and nobody proved that the cancellation of the neglected part has a physical meaning. My opinion comes from :

-the necessity to account for the irreversibility of the time,

-the necessity to account for the orientation of the physical space (separately),

-the non existence of a meta-physical reason for a universal Lagrangian formalism.

-the fact that the exponential function is not able to reach, from its Lie algebra, all elements in SL(2,C).

So I search a way to get the previous notions with calculations actually, not only potentially feasible.

Claude, ok, now we are on the same page, always for stationary states, I agree, which doesn't mean always in general.

For classical fields with half integer spin, in particular for the Dirac field, the charge density and thus the total charge, is positive definite. It was first wrongly interpreted as a probability density. On the other hand the energy density and the total energy are indefinite. There is an intimate relation between these fact and the possible mode of quantization -- "spin-statistic connection" --, one of the most important insights of quantum field theory. (For a "modern" treatment, see for instance the famous book of Streater and Wightman.) Classical spin-1/2 fields are on physical grounds not acceptable. It makes therefore also no sense to use them in GR, which is a classical field theory. I never understood why there are in the GR literature so many articles which include classical spinor fields. This is rubbish. They will only have room as quantum fields in a future quantum gravity theory. (Of course, in the sense of a mean field approximation, we can use them in GR when we derive, for instance, equations of state for neutron star matter; in a volume of matter, that is tiny in comparison to typical distances over which the curvature varies sizeably, there are so many particles that we can use there SR, including quantum field theory.)

... I see... however, classical spinors can be seen as quantum spinors at tree-level: quantum spinors are defined, tree-level is also an acceptable approximation, so putting the two things together should lead to acceptable results although approximated: if you say that this is rubbish, then I would argue that the tree-level approximation must fail at some point. Could you show or give some references that tree-level approximations cannon in principle be possible?

And if yes, is it possible to understand also why in so many other contexts tree-level approximations are so successful?

Many thanks.

What is rubbish ? The fact that the electron is a particle with spin 1/2 and is a magnet ?  I agree totally with the opinion that spin 1/2 fields are on physical grounds not acceptable in quantum field theory, and therefore quantum field theory must be corrected !

Dear Luca,

A quick answer: Given the tree level approximation (for pair production, say), unitarity gives rise to loop contributions (we use that even in practical calculations, for instance in computing g-2 for the electron), and this can not be described in the framework of a classical field theory. A derivation of Feynman rules requires quantum field theory.

I know, but it is always possible to make an approximation for no radiative corrections and I cannot see how that approximation fails: according to what you said the classical theory is not only limited, it is really rubbish, that is it shouldn't even exist, and therefore this means that the no-loop approximation should not only approximate, but also annihilate the classical theory, and I can see no way in which this approximation can fail so much as to render non-sensical the limit. It may be as you say, I just do not see how.

Can you elucidate this form me?

Antiparticles are mathematically described by particles with momentum vector pointing backwards in time. The momentum 2-tensor is unaffected, as can be seen in the simplest cases: a classical fluid, a Klein-Gordon scalar field.

This is particularly clear in a polar representation of the Klein-Gordon momentum tensor. (notation: m=mass, n= particle density, r = sqrt(n), p[mu]=momentum, '^' = upper index) 

In this case, the momentum tensor is: P[mu,nu] = n*p[mu]*p[nu]/m

+ hbar^2/(4m) (grad(n).grad(n)/n - delsquared(n)*g[mu,nu]).

This humble spinless real-variable representation of the KG momentum makes clear what is going on when you replace p[mu] by -p[mu]. There is no effect on the source of gravitation.

I have never seen a polar decomposition of the Diract momentum tensor. To include spin, we should work in EC with the field equation

modified_torsion[i,j,^mu] = Constant * Spin[i,j,^mu].

Iif the operator that reverses p[mu] preserves the sign of the spin tensor, then it has no effect on the spin-torsion field equation.I very much doubt that these results (for KG and an EC treatment of spin) will change when everything is done with spinors.

Am I missing something?

Dear Luca,

On the basis of the classical theory the world would be completely unstable. The energy of matter would rapidly become more and more negative, and the resulting positive energy would lead to  a firewall, burning everything in no time. It is for this reason that Dirac arrived at his reinterpretation, thereby predicting the existence of antimatter. His main goal was to stabilise the ground state, inventing what we call the Dirac sea. His pictures were soon afterwords translated to our present language of quantum field theory that treats matter and antimatter in a symmetric way. For an account of the early history, I recommend the beautiful article by Gregor Wentzel in the memorial volume to Wolfgang Pauli (edited by M. Fierz and V.F. Weisskopf), with the title: "Quantum Theory of Fields (until 1947)".

To Richard, the idea should be that yes, with that transformation for spinors their energy should change sign. The point of employing torsion is interesting, but my question was more basic than that as it involved only energy and gravity.

With torsion and spin (and for spinors that are not Grassmann valued) C conjugation is no longer a symmetry of the Dirac equation, nor is T inversion.

To Norbert, the fact that instability would occur, and negative energy would be obtained, is precisely the point of my question: are negative energy really there, if gravity is considered? Notice that in no argument of QFT are Einstein field equations employed.

To Norbert, you seem to consider the Dirac equation, and all what was made in the first quantification, as included in the "classical theory". This is unusual, classical physics generally means tensorial physics. Louis de Broglie wrote a whole book in 1934 explaining all advantages of the Dirac equation (L'électron magnétique, Hermann, Paris 1934). Only the Dirac equation gives the true quantum numbers needed in Chemistry, the true number of levels, the Landé factors, the magnetism of the electron which is the source of all our magnets.

I think very astonishing to throw away all these results because QFT got much less results, only the Lamb effect and the calculation of the anomalous magnetic  moment of the electron and the muon. The convergence of the method of calculation is not yet established. The calculation of the anomal magnetic moment of the electron gives both the value of the fine structure constant and the value of the anomaly : two results for one calculation, this is too much. The calculation of the anomalous magnetic moment of the muon has only the precision of the other results (as the energy levels of the Dirac equation for the hydrogen atom). And the calculation of the Lamb effect supposed no movement of the proton, which is a bad non-relativistic approximation.

Actually the entire second quantification is based on the non-relativistic Schrödinger equation used to pass from the usual formulation in space to the Heisenberg formulation. It is then not at all astonishing that the relativistic Dirac equation does not take easily place in QFT.

I also disagree with what you said about the current of probability of the Dirac wave, which suffers no problem. Moreover the Tetrode's tensor coming from the invariance of the Lagrangian density under space-time translations is able to give the Lorentz force (a similar result is obtained only in General Relativity, where the move of a little mass may be deduced from the field equations). So it is perfectly convenient to use this tensor in General Relativity.

The problem of negative energies in the Dirac wave is solved by suppressing the cos(beta) factor in the mass term of the Lagrangian density. We then get an energy-momentum tensor with a positive T00 component, both for plane waves with +E and -E energies. This mass term may be generalized to get the wave equations of all fermions and anti-fermions of the first generation. These wave equations are relativistic invariant, gauge invariant under the gauges groups of the standard model. And these wave equations are  then compatible both with General Relativity and with the gauge symmetries of the standard model.

To Claude, most theoretical physicists abandoned about 80 years ago what you still defend. --

When a theoretical prediction agrees with experiment to 11 digital places you should take that as an enormous success. In the meantime QED has become part of the electroweak theory with a long list of successful predictions; just consult one of many reviews of this. (I will not make comments about de Broglie's book, and suppress nasty remarks by Pauli about it.)

It's not true that "on the basis of the classical theory the world would be completely unstable", there are many examples of classical stable systems in which the so-called "non-radiation condition" holds. 

Moreover A .O. Barut shows how calculate the Lamb shift,  the anomalous magnetic moment, Casimir-Polder forces without second quantization and  E.T. Jaynes shows how to calculate the Einstein A coefficient in a classical context. 

If with "classical theory"  we mean a theory of real fields on space-time, there aren't so many difficulties to explain quantum phenomena. 

To Norbert: at the time of Copernicus most astronomers used the Ptolemaic system. Truth is not an affair of majorities in parliaments.

The 11 digital exact places are not general, they concern only a few results, and they shall be actually proved only when the convergence of the method of approximation shall be proved.  Now these 11 exact digital places are only an affair of confidence in the calculators.

And I cannot get confidence in a method of calculation based on the non-relativistic theory. Even If I should become alone in the world to think so, the fact  that the electron is a spin 1/2 particle and the fact that it is a magnet could not change. The way to get the number of states by Pauli, by adding a factor 2 to the number of states of the Schrödinger equation, in the hydrogen atom case, is bad : there are two kinds of solutions, n(n+1) solutions for one kind and n(n-1) for the second kind. Since 80 years most of theoretical physicists have unhappily forgotten this knowledge. Not Louis de Broglie who discovered the quantum wave.

Even if the electro-weak theory was obtained from the quantized electromagnetic field, these experimental results are able to prove the necessity of the second quantification only if it is impossible to get the same results without second quantification.Moreover the weak interactions have proved the necessity of the use of left and right waves, and the existence of these chiral waves comes from the two kinds of representations of the SL(2,C) group. It is another reason to prefer the Dirac equation to the non-relativistic Hamiltonian theory giving the second quantification.

So before writing as if I was a retarded student, you could begin by reading my books and articles. Before this reading you could also read the works of de Broglie, because obviously you never read neither his dissertation nor his main books.

Claude, I see, and very well, the point of your last comments, I follow your arguments very clearly, and I thank you for being so detailed; on the other hand, I do not think Norbert was considering you a student and even less retarded, I guess he is simply following the fashion. I agree following is not appropriate and we should always have to look for deep understanding, but I really believe he meant that with no evil.

Thanks Luca, my bad reaction comes from the difficulty to publish when you are not exactly in the main stream.

Claude, two final remarks:

1. The calculations of g-2 are fully relativistic, involving thousands of Feynman diagrams and an incredible amount of work. Qualifying them as essentially non-relativistic is obviously wrong to everybody who had a closer look at the calculations. (I gave lectures on this.)

2. I know nobody who expects convergence of the power series in quantum electrodynamics, say, for good reasons (from models in lower dimensions). Divergent series are very common in physics, as we learned especially from Poincaré's work on celestial mechanics. (On my book shelves I have his three volumes "Les Methods Nouvelles De La Mécanique Céleste" by him.) He was, however, not disturbed by this. Asymptotic series are also a subject in pure mathematics. --The sad fact is that we have (in contrast to lower dimensions) no rigorously constructed quantum field theories outside perturbation theory in 4 dimensions. There is the hope that such a construction should be possible for QCD, because of its asymptotic freedom, but this is a very, very difficult problem. People who use QCD outside perturbation theory in lattice simulations are convinced of this.-- Juerg Froehlich could tell you much more on this.

Claude, I am aware of the issue.

Norbert, about your second point, I think it is a common ground between you and Claude: my question is, since you two, like me and many others, clearly know that QFT is mathematically ill-defined (at least in perturbative methods), how is it that people are not disturbed by this, to use your words?

Norbert, I know that everyone made his best hard work. The problem of QFT is: the passing of the wave equation into the path integral method is based on the formalism of the non-relativistic quantum theory (with a reversible unique time), and the mathematical properties of the wave as a function of coordinates into the C field of complex numbers. This complex number has a unique modulus (the square of the probability) and a unique argument (the phase that allows to consider the path integral). When you compute the diagrams, you actually use relativistic mechanics, but you do not use the relativistic quantum mechanics, because the wave of the electron does not follow a Hermitian formalism (nor the photon). The electron has not a unique phase, it has two phases, the usual electric phase and a second one, the Yvon-Takabayasi angle. The right and left currents of the alone electron, even at very low velocity, are isotropic: cutting the "small components" is as much stupid as cutting the magnetic field in the electromagnetic classical theory. This is the very reason of the non-relativistic quality of any calculation in QFT.

Often, and this is the case for plane waves and for the solutions of the H atom, this Yvon-Takabayasi angle is small and negligible and then the method of the path integral is not far from physical reality.Then this method is effective in any non-relativistic framework.

But when you want to integrate the gravitation which is essentially relativistic you have no chance of success if you mix relativistic mechanics to non relativistic quantum theory. So I recall that this discussion came from the problem of the sign of the energy which arises as soon as you use a relativistic quantum wave, because the relation between energy and impulse needs E2. The negative energies are non-physical, experimentally: the creation of any particle or antiparticle necessitates an amount of positive energy. This problem was the origin of the need to quantize every field, the Hermitian formalism becoming a general framework for any quantum field.

The opinion of the Broglie was: the difficulties of the QFT (non convergence, infinities and so on) are typical for a non-linear problem treated by linear approximations. This opinion came from the work of Einstein proving that the law of movement of a singularity of the gravitational field may be deduced from the differential laws of the field itself.

Actually it is very simple to get the awaited non-linear wave equation for the electron. All that you have to change in the Dirac theory is a little simplification of the mass term in the Lagrangian density: instead m rho cos(beta) you suppress the cos(beta) factor, where beta is the Yvon-Takabayasi angle. The wave equation is near the linear Dirac equation in all cases where this angle is small. The Tetrode's energy momentum tensor has then an everywhere positive density of energy. This is why I answered to the question of Luca: I knew the solution.

And this allows to build the rigorously constructed quantum field theory that you hope, because this mass term may be extended to describe all fermions and anti-fermions of one generation (why not the second and the third ones tomorrow?) with wave equations generalizing the relativistic wave equation of the electron.  The wave equations are totally compatible with restricted or general relativity and they are also compatible with the gauge formulation of the standard model (both electro-weak theory and chromodynamics).

Obviously this construction is not ended, it is a work too much difficult for a very little group of physicists like our team. Wave equations are important, but solutions of these wave equations are much more interesting. The second and the third generation, and a fourth neutrino, are only suggested. We get only first results, they comfort the standard model, for instance we understand why electric charges of quarks are +2/3 and -1/3, why leptons do not see strong interactions, we are able to calculate the Weinberg-Salam angle...

I should be very happy to enlarge our team and to go further and faster.

Perturbative renormalizabilty means that the  terms at any given order  of the perturbation series are well-defined, i.e. independent of any procedure used to deal with the infinite number of degrees of freedom that a field implies; it implies nothing about the convergence of the perturbation series. For a perturbatively nonrenormalizable theory the terms at any given order aren't well-defined-they remain dependent on the cutoff procedure  used to define them; so convergence or divergence of the perturbation series is meaningless in this case. 

It is, of course, a theoretical  fact that the perturbation theory of the Standard Model is well-defined because it can be proved that the Standard Model is a perturbatively renormalizable theory and it's an experimental fact that it  does indeed describe quantitatively measured effects-which, in fact, implies the existence of, hitherto unkown, degrees of freedom; since, absent those, the perturbation theory about the Higgs vacuum wouldn't remain stable. 

The problem with negative energy spinors is the same as with the corrresponding states of any  relativistic field-they describe instabilities due to pair production and the stable situations can, also, be described. Scalar fields, also, have this issue-what was known, historically, as the Klein paradox. This is textbook material now as the Introduction to vol I. of S. Weinberg's book shows. 

To Stam, do you think that a positron is unstable in the void ? Do you think that a positron is a negative energy spinor ? I cannot understand what is the physics behind your answer.

No, the positron is stable, due to electric charge conservation-there doesn't exist a lighter particle, with its electric charge.  It's related to the electron by a CPT transformation, that's a symmetry of any system, that's Lorentz invariant and has a Hermitian Hamiltonian. That's all. That implies that a positron has the same rest mass as the electron, but opposite electric charge. Once more, one shouldn't do history, but physics. Antimatter is just a word, it doesn't refer to  anything beyond charge conjugation, in fact, and how this is realized depends on the gauge group-for weak or strong interactions, that aren't abelian, it works as implied by the pproperties of the group. There exist bound states of an electron and a positron. These have a finite lifetime that can be calculated-a standard exercise in QED-and can be measured, also. There is a transition amplitude from the electron-positron initial state to a final state of two or of three photons, whereby energy,momentum, angular momentum and electric charge are conserved. And the minimum energy of an electron-positron pair isn't zero, it's 2mc2 where m is the mass of the electron (or of the positron).

About the Klein paradox: https://arxiv.org/abs/0907.5588

Now I understand why you do not want to consider history : you do not want to consider an irreversible time, you believe that the reversible time of Hermitian Hamiltonians are compulsory for any part of the quantum world (nevertheless, before or after me, you shall die) . I study only physics of the real physical world, where the quantum wave of the electron follows a Lagrangian formalism and this wave has a spin 1/2. CPT is an exact symmetry, not T in the real world. I have no problem wth the CPT symmetry: with the wave equations of any fermion, CP=T so the CPT theorem is trivial.

The CPT theorem of course isn't trivial-if the Hamiltonian isn't Hermitian, or Lorentz invariance isn't exact, it doesn't hold and this is known.  There's no difference in the Lagrangian formalism, either-and waves don't carry spin; the single particle excitations of the field carry spin. For the electron in QED, CP is a good symmetry; for the weak interactions it isn't. But this doesn't have anything to do with any negative energy states. Global Poincaré invariance-that includes the translations-implies that what matters is that the Hamiltonian be bounded from below. In the Lagrangian formalism it's the same, because, otherwise the  correlation functions aren't defined.

And this doesn't have anything to do with spinors, but holds for any quantum fields, i.e. scalars and vectors, too. 

One shouldn't confuse ``formalism'' with ``physics'' either. There are many, equivalent, though mathematically different, ways of describing the same phenomena. That they're equivalent means that it doesn't matter which one uses, there exist results that are the same and those that aren't, don't matter. And the term ``real physical world'' doesn't mean anything, beyond the current, approximate, description of Nature. It turns out that one can formulate a description of known phenomena much more easily, if one starts with conservation laws and looks for how they can be broken, than by trying to set up a formalism where they are broken.

The bottom line is that ``negative energy spinors'' might have been a historically used term-but, now, it's been shown to be meaningless as a description of matter. 

Actually, Stem, you do not seem able to understand that Hamiltonian and Lagrangian physics are not exactly the same thing, and this is precisely the origin of the difficulties about negative energies, the main point of this discussion. I disagree to each point of your response:

1- The CPT is not trivial with the Dirac equation, where P and T symmetries must be added to the relativistic invariance described with the SL(2,C) group. But with the non-linear wave equations of fermions that we have obtained the CPT is trivial because C=PT.

2- Global Poincaré invariance is a symmetry of QED, not of the Nature and the proof is that we live in a world of matter, not a mixture of matter and anti-matter. Our wave equations are not Poincaré invariant, but GL(2,C) invariant, group which includes only the little Lorentz group, not P and T symmetries (the invariance under translations being trivial). You cannot say that this is not correct from physics, only that this is not true in the frame of QED which is only a theoretical model, not the physical reality.

3. The real physical world is simply the one in which I can move and this world exists even if my description of this world is false.

4- The link between wave equations and Lagrangian density is not exactly what you know. The Lagrangian density is only the sum of the real parts (in the Cliffordian sense of what is a real part) of the invariant wave equations. From the Lagrangian density we can deduce the wave equations only for stationary states, and then, but only for stationary states, time means nothing and the Hamiltonian formalism is available. The deduction of the wave equations from the Lagrange equations supposes an integration by parts and the possibility of suppressing one of the parts. This is impossible for propagating waves. So Lagrangian physics is not Hamiltonian physics. The irreversible time also exists in any laboratory, where the annihilation of a pair electron-positron never comes before a pair creation, always after.

5- There is no way for suppressing or adding the spin 1/2 to any fermion. This is only possible from the tale for children that physicists teach to chemistry from the Schrödinger equation and the Pauli up-down tale. This is a tale, even simply when you study magnets.

6- Obviously, if you consider the model of QED as more physical than the universe itself, this discussion is useless.

... guys... guys... the conversation is really nice, but before we warm ourselves up too much, let me just pause a little, in order to go back to my initial question: C maps spinors of positive energy into spinors of negative energy, and this is a mathematical, fact; without gravity, both spinors are solutions, so negative energies would be real and their presence is removed by employing Grassmann variables, but there is no need of Grassmann variables to remove negative energy spinors since these are not solutions of Einstein equations. Dirac equations without gravity may make no sense, but they make perfect sense when the coupling to gravity is taken; and the coupling to gravity must be taken. So there should not be any problem... right?

No, charge conjugation does not change the sign of the energy. That statement is wrong. And it doesn't have anything particular to do with spinors-it acts in a similar way  on charged scalars, for instance. The conjugation depends on the properties of the gauge group. For abelian groups, an example of which is the group of electromagnetic charges, charge conjugation takes a state of definite charge to a state of opposite charge, without affecting spacetime properties, such as energy-the internal symmetry group commutes with the Poincaré group, a statement that's a special case of a proposition known as the Coleman-Mandula theorem. For nonabelian groups charge  conjugation is more complicated, e.g. for QCD. 

The vacuum state of a theory of charged particles is the state that's  annihilated by the annihilation operators of states of all charges. 

The Dirac equation makes perfect sense as the equation of a field that creates particles of spin 1/2-it doesn't make sense as the equation of a single particle, if the motion isn't integrable, so particle number isn't conserved. In the presence of an electric field, this is what leads to pair production. 

All this is standard material in all courses on relativistic quantum field theory.

... oh, no, Stam, now I do not agree with you: for general fields ok, but for spinors C cannot simply be the complex conjugation because the gamma matrices are complex and complex conjugation would change the structure of the spinorial transformation, and hence one has also to multiply on the left by the second gamma matrix, if the charge conjugated spinor is to be a spinor with all correct transformation propoerties; and that transformation does change the sign of the energy and additionally also the spin. Now, one may say that that quantum properties such as Grassmann-valued fields may intervene to C avoid changing the sign of the energy; but we have also to remember that these properties are postulated precisely to solve the problem of negative energies and there would be no need for them if the energy problem is solved otherwise. With no quantum fields, just classical spinors, it is a mathematical fact that C inverts the sign of the energy, there is no doubt about this, I can provide a detailed calculation if needed, although I am sure it won't be needed. The only way in which my question could be invalidated would be if spinors with negative energy and coupled to gravity would still be solutions of the entire system, so again I ask, does anyone know of this?

It's not true that Dirac equation doesn't make sense as an equation of a single particle. See P. Holland book:

https://books.google.it/books?id=BsEfVBzToRMC&pg=PA504&lpg=PA504&dq=single+particle+interpretation+of+dirac+equation+holland&source=bl&ots=2-xb_fm4G9&sig=HMl1c3Gfx1_QlSZwWPYkN0qqcJY&hl=it&sa=X&sqi=2&ved=0ahUKEwiNzpzqovTRAhWGXRQKHaMRCbQQ6AEIODAD#v=onepage&q=single%20particle%20interpretation%20of%20dirac%20equation%20holland&f=false

Once more: charge conjugation, for abelian gauge groups amounts to flipping the sign of the charge. For non-abelian groups it means something more complicated. 

And, once more, this doesn't affect the energy, because charge conjugation commutes with the generators of the Poincaré group. These are all exercises in any textbook. 

In flat spacetime ``negative'' or ``positive'' energy doesn't make sense, since only energy differences matter and, once more, the vacuum state is annihilated by the annihilation operators for ALL charges. The only thing that matters is that the Hamiltonian be bounded from below. For the Dirac Hamiltonian this is the case, in the absence of an electric field, for electric charges. In the presence of an electric field the vacuum can become unstable and the physical vacuum has a finite density of particles. 

In curved spacetime more care is needed, since time translation isn't a global symmetry, therefore energy isn't globally conserved. 

Therefore, once more, there isn't any problem with anything that could be called a ``negative energy spinor''. 

Stam, with C being defined as

\psi→\gamma^{2}\psi*

then

\overline{\psi}→\overline{\psi}* \gamma^{2}

because \gamma^{2} is anti-Hermitian and anti-commutes with \gamma^{0} as it belongs to a Clifford algebra: because \gamma^{2}\gamma^{a}\gamma^{2}=\gamma^{a}* then

i\overline{\psi}\gamma_{a}D_{b}\psi →

i\overline{\psi}* \gamma^{2}\gamma_{a}D_{b}\gamma^{2}\psi*=

i\overline{\psi}* \gamma_{a}*D_{b}\psi*=

(-i\overline{\psi} \gamma_{a}D_{b}\psi)*

because i*=-i and doing this on both forms of the energy gives

T_{ab}=i/2(\overline{\psi}\gamma_{a}D_{b}\psi-D_{b}\overline{\psi}\gamma_{a}\psi)→

-(i\overline{\psi} \gamma_{a}D_{b}\psi-D_{b}\overline{\psi}\gamma_{a}\psi)*=-T_{ab}*=-T_{ab}

since the energy is real.

Therefore the energy sign is flipped by C for classical spinors.

Now, having settled that there are such things as negative energy spinors, I ask once more, can they be solutions of the Dirac-Einstein system of field equations?

Yes !

1- Only the Dirac equation is compatible with relativity and gives the spin 1/2.

2- The energy-momentum tensor of the Dirac wave is the tensor of Tetrode, linked to the invariance under translations of the Lagrangian density, and the Lorentz force is a consequence of the Dirac wave equation.

3- Solutions with a negative E have a positive T00 energy density if you suppress the cos(beta) term in the Lagrangian density.

Luca is, of course, right. That the Dirac Hamiltonian in the original 1-particle theory is indefinite can be shown by an equally simple calculation (as for instance in my textbook "Relativistische Quantentheorie", Springer-Verlag 2004, p.160.) That this property is physically unacceptable follows, for instance, from the fact that a coupling to external classical time dependent electromagnetic fields induce transitions from positive to negative energy states. (The standard calculation of pair creation in QED for weak time dependent external fields can be reinterpreted in the 1-particle theory as the determination of the transition probability from positive to negative energy states. The mathematics is identical.) This is just an example that the world would be unstable,we would not exist; 

The coupling of a classical Dirac field to the metric of GR was described in a seminal paper by H. Weyl in 1929, that was one of the most important contributions in the development of gauge theories. There are lots of papers in GR which study the (classical) coupled Einstein-Dirac system. On a fundamental level this is meaningless. In the sense of a mean field treatment one can, however, use the quantised Dirac fields (and more generally our standard model, especially QCD), to derive physically meaningful equations of state for neutron star matter, which can then be used for models of neutron stars (by solving  the Tolmann-Oppenheimer-Volkoff-equations). Beyond that we need a quantum theory of gravity.

Indeed-but all the problems are those of interpretation and ambiguous calculations. There's nothing special about spinors, nor about ``negative energy'', that isn't encountered in any other case of relativistic fields and there isn't any particular problem, beyond using these words in ways that are grammatically and syntactically correct, but devoid of meaning. To be able to distinguish ``positive energy'' from ``negative energy'' requires a consistent way of defining what ``zero energy'', i.e. the vacuum state is. In special relativity this is possible and it's the only thing that matters. 

When gravity enters the picture, since time translation  invariance isn't a global symmetry, the vacuum states of matter aren't all equivalent-so observers in curved spacetime don't detect a unique vacuum-which is known, at least since Hawking and  then Unruh and doesn't have anything to do with spinors in particular.  

``Negative energy'' means there's an instability and what matters is what is the corresponding stable vacuum, which can have finite density of particles. Anything else is just meaningless play on words. 

On the classical level there is a very important difference between fields that transform according to a one-valued representation of the Lorentz group on the one hand, and fields belonging belonging to a two-valued representation. This was generally shown for free fields by Markus Fierz in Helv. Phys. Acta 12(1939). (Since Fierz proved in this paper also for the first time the spin-statistic theorem for free fields but arbitrary spin, he obtained for this work the Max Planck Medal), and W. Pauli, Phys. Rev. 58(1940), 716. (At that time Fierz was still assistant of Pauli.) The difference is: For one-valued representations the total energy is positive-definiit, and the total charge is indefinite. However, for fields belonging to a two-valued representation it is just the other way around. These facts determine the different modes of quantization (Bose-Einstein statistics versus Fermi-Dirac statistics). --  It will soon be 80 years since all this was established, but there are obviously still people that do not understand the importance and definiteness of these insights.

@Claude what about the Feynman-Gell Mann equation?

http://adsabs.harvard.edu/abs/2012APS..MARH37010M

But the ambiguities, precisely, are resolved by using the appropriate statistics-so, in the end, there isn't any ambiguity and there isn't any problem with negative energy spinors. Of course mentioning statistics at all means working with fields, i.e. many-particle states and not single-particle states.  

In any event, the statements about issues with gravity that are mentioned in the elaboration of the question are, simply, incorrect. The resolution is that the vacuum state of quantum matter, in a fixed, curved, spacetime, isn't unique, that's all. The explanation is local instead of global time translation invariance and the consequences were explored by Hawking and Unruh. 

... well, spin-statistics is proven on the assumption of Lorentz invariance and causality, and positive norm and energy (see Pauli, Phys.Rev. 58, 716 (1940), also cited in Peskin & Schroeder, An introduction to Quantum Field Theory, page 58 (Perseus, 1995)): so I agree totally with Norbert when he says that with spin-statistic there is no problem with negative energy, but this is precisely because spin-statistic is demonstrated with positive energy. So my point remains.

Stam, about your last statement, does it make any sense when the space-time, beyond curved, is also dynamical?

I like the last answer by Luca (no. 55 in this post), since it meets what I  have posed as a question objecting the very popular term of "age of the Universe". Below, I am repeating its first lines for increasing the chance of getting appropriate explanation from the discussion among RGaters:

ResearchGate. Available from: https://www.researchgate.net/post/Mathematically_What_is_the_definition_of_age_of_the_Universe [accessed Feb 4, 2017]:

Mathematically: What is the definition of age of the Universe . . . ?

. . . if the Universe is modelled by a 3+1 space-time manifold? The problem appears in the general relativity since there is no functional representing a universal time on the manifold. On the other hand popular literature bombs us permanently with the problem what is the age of our world. I expect that a popular explanation of this paradox can be build as well:)

A related question:

What does it mean that the Universe expands (more generally: that it evolves)? .. "    etc. etc

A proposal of an answer to the main question is also put within the cited post.

Regards

It's not known how to describe what the dynamical properties of spacetime are, when quantum properties of matter are relevant. 

... yes Stam, in this thread as well as in the other one on which we both commented, you always consider the second-quantized case, about which I agree with you of course; but still I am not interested in this case, and I keep asking what are the problems in the purely classical situation: any?

Claude, I saw your answer only now, sorry: so you say the purely classical case can be treated, and solved by playing with the cosine of the Takabayashi angle: can you send a reference?

... mmh, Claude, I think I see now what you mean: but if I am right, what you say is true only if you define the antiparticle as the spinor corresponding to the Takabayashi angle that is equal to π and in this case only. Now is this what you define?

Luca, when you suppress the cos(beta) term in the Lagrangian density, you change only there the Dirac theory. The C conjugation remains unchanged, a transformation which associates to each solution of the wave equation a solution of another equation, different from the previous one. The initial equation is made of three terms : the differential term, the gauge term and the mass term. With the linear Dirac equation the charge-conjugate equation changes the sign of the differential term and the sign of the mass term, so it is simpler to say that only the sign of the gauge term changes. With the non-linear Dirac equation the charge-conjugate equation changes only the differential term, you can then say that C=PT. Since all derivatives change sign you can say that all gauges change the sign of charges (not only electric, also weak, or colored in the case of quark waves), this was supposed previously by the Standard Model.

The point that is interesting for the link with General Relativity is: charge conjugation now does not change the positive sign of the T00 density of energy, this T tensor remains the Tetrode's tensor coming by the Noether's theorem from the invariance of the wave equation under all space-time translations.

Since you were interested by a solution of the problem of non-physical negative energies in the frame of tensor physics, I have answered to your question that a simple solution actually exists. The resolution of this problem is the main reason that have explained why I studied the non-linear Dirac equation, first in my dissertation (Nantes 1993). Now I have much more reasons to continue the study of this wave equation. The mass term may be generalized, this gives the wave equation of all particles and antiparticles of one generation, invariant under the Cl3* group generalizing the relativistic invariance, gauge invariant under the gauge group of the  Standard Model of quantum physics. The gauge symmetry is an exact symmetry, and with a mass term, then the wave equation is directly compatible with the Einstein's gravitation.

This is better explained in: Daviau, C. and Bertrand, J. (2016) The Standard Model of Quantum Physics in Clifford Algebra. World Science Publishing, Singapore. You shall find the other references in our last article:  Daviau, C., Bertrand, J. and Girardot, D. (2016) Towards the Unification, Part 2: Simplified Equations,

Covariant Derivative, Photons. Journal of Modern Physics, 7, 2398-2417.

http://dx.doi.org/10.4236/jmp.2016.716207

Thanks, I'll have a look

There aren't any problems with classical matter and classical spacetime-that case is perfectly well described by general relativity-just that spinors don't describe  classical matter and  classical electric charges are not described by spinors.

What do you mean with "classical matter"? 

Stam, thanks

Federico, "classical matter" is "quantum matter" without the condition of quantization, so quantum fields with no radiative corrections.

Luca ok, but this is a meaningless definition,  because classically a "point particle" is not  consistent. 

I know, classically there are neither quantum corrections nor point-like particles, point-particles are assumptions of QFT (and the reason of UV divergences in QFT).

There aren't any conceptual issues with point particles, either classically, or quantum mechanically, as long as gravitational effects can be neglected. However a point particle in general relativity, that's not a test particle, is a naked singularity, if it's a classical object; and, if it's  a quantum object, the description, simply, breaks down.

Point particles don't have anything to do with the ultraviolet divergences of quantum field theory. It's just possible to write consistent quantum field theories of point particles, whose classical limit is, also, relevant.  

A classical point charge has infinite self-interaction energy, so there is of course a problem. 

Well Stam, again, I do not follow you: there are classical fields (or classical treatments of quantum fields) that can be considered in Einstein gravity and they do not form singularities; point-like particles are at least one of the reason of UV divergences (see for instance 't Hooft Nobel lecture for a non-technical assessment); and as Federico said, singular things and divergences are problems, I cannot understand why you say there isn't any issue there... additionally, you already told me that C conjugation leaves the energy unchanged while it clearly does not (as I proved it step-by-step in a previous comment); in another thread you even told me that three-level approximation of a quantum field is not the classical field... There has been a number of times you told me something that was plainly against the common knowledge of the field. I am starting to think that maybe we are not at all talking about the same thing.

(Or maybe you are simply enjoying making a fool of me :D )

No-the singularity theorems imply that classical fields, generically, give rise to singularities, in Einstein gravity. However these are hidden, in all known cases, behind event horizons-the statement that this is always the case is the cosmic censorship conjecture. Therefore, there exist observers that will never ``experience'' their effects.

There do exist observers that might be thought to  experience their effects in finite proper time-but the description of the known physics that allows their definition breaks down much before that and it isn't known what takes over. 

The statements about the non-invariance of Einstein's equations are incorrect. 

So much for classical physics. 

It's not the point-like properties of particles that are responsible for the UV divergences of quantum field theories-the reason is different. While, historically, it was thought to be the case, this isn't the case anymore, since the work, in particular, of Wilson. One shouldn't identify formalism and physics-there are very many, apparently different, ways of describing the same physics.  A case in point is that of extended objects, namely strings and branes, whose quantization involves completely different issues and is incomplete precisely for that reason. 

These statements don't have anything to do with gravity, because it's not known how to describe a theory, when quantum effects of matter could affect spacetime, nor how quantum effects of spacetime can be described at all, in general.

One can do history, or one can do physics. While the understanding of certain notions evolved in a certain way, historically, this is completely independent of the logic of the final product, which is the only issue of interest for physics.

Regarding, once more, the topic of this thread, there's no problem with spinors that, apparently, describe states with negative energy. It was considered a problem in the past-it's no longer an issue. In particular, the issues, that were thought to involve spinors, are common to all relativistic fields and the solution is the same-how to describe the stable vacuum state of a theory of relativistic fields. 

The statements about the non-invariance of Einstein's equations may be incorrect, but I gave a mathematical proof of that in one of the previous messages, and you did not tell me where you think that was wrong, so I will retain it as right: I am sorry on this, I trust your opinion when you write something, but I trust mathematics a bit more.

About your last statement, I agree there should be no problem with spinors having negative energy, the message in my post is that this lack of problems comes from a correct coupling to gravity in a classical setting.

Now, it is your opinion and that of Norbert that classical spinors and gravity cannot be coupled, so I gather this is why you look for a different solution, but I did not understand why the spinor energy cannot be taken as source of Einstein equations in your and Norbert's view, so I'll have to stick to Clause's answer that problems may be solved by employing the properties of the Takabayashi angle, or at least this looks to me the clearest mathematical answer that I got.

in any case, thanks to all of you :D

It's the energy-momentum tensor that's relevant as a a-classical-source, to Einstein's equations, not the fields from which it's made up-the metric couples only to it, by general coordinate invariance. So the fact that matter is made up of spinors, scalars or vectors  isn't of any particular relevance for this topic.

Strange answer! What about back reactions?

For general relativity only the classical equations of motion of the matter fields matter, of course.  For spinors it's not at all obvious what this would mean.  The reason is that, in the classical limit, one isn't sensitive to the shot noise of the individual quanta. So one would be using an effective description in terms of a density and a current of a fluid, with an equation of state, as happens with neutron stars, for instance. 

" are very many, apparently different, ways of describing the same physics." 

This is true only if your analysis is not philosophically rigorous.  But in physics every mathematical object, every mathematical operation or mathematical assumption,  must have a physical meaning. If the formalism is different,  also the ontology is different, though they apparently describe the same observed phenomena. 

"One can do history, or one can do physics. "

Benjamin Farrington: “History is the most fundamental science, for there is no human knowledge which cannot lose its scientific character when men forget the conditions under which it originated,  the questions which it answered and the functions it was created to serve". 

Said differently: statements involving spinors refer to quantum properties, due to the fact that the statistics isn't Boltzmann statistics. Classical issues aren't sensitive to Fermi-Dirac (or Bose-Einstein) statistics, but to Boltzmann statistics. General relativity isn't sensitive to Fermi-Dirac or Bose-Einstein statistics, only to Boltzmann statistics. 

Stam, this comment was made before, and I already replied to it: you cannot tell there are no negative energies by using the spin-statistic because the spin-statistic theorem is base on the assumption there are only positive energies.

More in general, to my question of negative energies I never wanted to have answers involving quantum concepts precisely because quantum concepts are postulated to solve the problem of negative energy.

I know very well that after decades of training in QFT it becomes virtually impossible to think without it, but I was asking an effort in this direction precisely to avoid touching quantum protocols.

Luca, I agree.

De Broglie understood, fifty years ago, that the explaining power of quantum mechanics was exhausted.

Lochak understood, forty years ago, that it was useless to try convincing the users of QFT. Only new experimental results, impossible with QFT, could change physics. The discovery of magnetic monopoles with spin 1/2 is the true answer to anyone who yet thinks that the physical universe is identical to the virtual world of QFT.

I agree, but I wish to add something: evidences against QFT in the present formulation do not need to be searched exotically, the anomaly of the muon magnetic moment is enough (and it is now at 3.4 sigma); nor in total honesty we need to look at experiments, as nobody who is really open-minded about it could in good conscience admit that in QFT all is fine, and it is suggestive that the first critics came precisely from the inventor of the mathematics of QFT, J. Schwinger, who complained about the use of something so utterly ill-defined as field operators (see for instance his book, "Particles, Sources, and Fields" Vol. 1 (Addison-Wesley,  1970)). While clear to nearly everyone that QFT must be surpassed, and that there already are a few attempts to recover its two results (Lamb shift and anomalous magnetic moment of the electron), there are several reasons for which we do not do it: a social one is that we have to wait that all people who worked in it for forty years get retired, because it is comprehensible that nobody would abandon a theory that made their career, but the most important reason for me is intrinsic to QFT, and it is the fact that for its structure, all unexpected result can fit under the carpet of "new physics". Think about it, if all good results are kept, and all bad results are dismissed by saying that "they will be explained by new physics", it is quite clear, we do not stand a chance. And this holds for all those who might want to think critically about it, because if on top of this you also add the philosophy of the "Shut up and calculate!" hope is gone.

For all this reasons I think that thinking out the box ot QFT is not only hard, it is impossible.

What matters is that positive energy and negative energy states don't mix and the vacuum is stable. What this means is that no negative energy excitations are created during the evolution.  And this is ensured by assuming that spin 1/2 fields are created or annihilated by operators that anticommute at space-like separations. Cf. any textbook on quantum field theory.  

This property means that it isn't possible to describe them as classical quantities. 

 The vacuum state can become unstable in the presence of external fields and then the stable vacuum state has finite density, that's related in a calculable way to the properties of the external field. This holds for flat spacetime.

For curved spacetime the difference is that the vacuum state isn't unique, in general, because the curvature implies that the creation and annihilation operators can't be globally defined-in a fixed, curved, spacetime. Therefore there will be a density profile. However, it's not possible to compute the reaction of the geometry to this profile, since it isn't a classical source-it represents quantum shot noise. 

The question doesn't make sense outside quantum field theory, and its limiting case, classical field theory, because spinors describe physical objects only in the framework of quantum field theory-and their classical limit isn't a spinor field.  There isn't any such notion as a ``quantum protocol''.  Relativistic quantum field theory is a generalization of non-relativistic quantum mechanics-and is identical to ``relativistic quantum mechanics''. Once more, history of physics is different from physics.

(Magnetic monopoles can be described within quantum field theory-they're non-local objects, when the electrically charged fields are local and vice versa. They can have integer spin or half integer spin, depending on the details of the dynamics. For the moment they haven't been experimentally discovered, but their discovery wouldn't affect the consistency of quantum field theory in general, or of the Standard Model, in particular.)

Metaphysical speculations don't have anything to do with the topic of this thread, so it would be more useful to start a new thread about them.

However the recent activity about the  magnetic moment of the muon doesn't indicate any breakdown of quantum field theory-if the deviations from the available  calculations are confirmed it would indicate the contribution of new particles, nothing more. And the reason is simple: it is possible to describe the magnetic moment of the muon through the contribution of particles-there's a plethora of models that do it. However all such models, inevitably, predict *other* effects that can be checked-and these effects aren't found. That's why these models don't explain the deviation. This is the result of experiment, however-the models are all mathematically consistent. Were they not consistent would indicate a breakdown. This isn't the case.  

However it is, also,  known that the contribution of the known quarks, that do contribute,  cannot be computed to discovery accuracy, in perturbation theory and requires a quite lengthy lattice calculation. And it's not obvious that the experimental uncertainties are under such tight control, either. That's why it's, simply, wrong to state that there is a deviation, since the statement implies that there is a controlled calculation to compare with-when there isn't. 

There isn't any such notion as ``good results'' or ``bad results''-there are controlled calculations and experiments and calculations and experiments whose assumptions aren't clear-the vaguer the assumptions, the less useful the statements.   ``New physics'' isn't particular to the quantum field theory that's the Standard Model, but applies to *any* experimental investigation: the known effects are ``old physics'' and any experiment must measure them to discovery precision, before being in a position to state whether any additional effects are beyond the background-these are ``new physics'' and their details are elucidated by comparing the predictions of models-that can recover the old effects in appropriate limits.

One reason why it's not useful to mix up history of physics with physics is that quantum field theory has evolved considerably since Schwinger's time and it is known, in particular, how to resolve the issues he mentions and a lot more. 

``Thinking critically'' means building  the framework for performing calculations and experiments. Otherwise spinors or energy don't make sense.

Quantum concepts aren't postulated to solve the problem of negative energy. They're postulated to solve other problems and the issue of vacuum stability is just part of the new framework, since it is relevant, too.   The Dirac equation already describes a quantum particle, not a classical one, as does the Klein-Gordon equation. The way the solutions of these equations do describe new phenomena is, just, consistent with the way special relativity deals with the sign of the energy. 

For it is possible to describe a  classical, relativistic, particle, without any issues at all-this is just a consequence of properties of the Lorentz group. And this is the appropriate limit of the quantum treatment. Once more, one shouldn't confuse history with  logic.

... ok, we have entered a loop here: I do not agree about what you said and wrote a comment, you did not read it and repeated your previous comment, so I replied with my comment, which you did not read and re-wrote your comment, and if I now try to re-write my comment, you would simply dismiss it and re-write another version of your same comment. This doesn't help me, and it makes you lose time. It will only fill the thread with comments. So I suggest we stop here.

Unless something new comes about, of course

It's the difference between opinions and calculations-in the absence of the latter, there's no way to define what it means to agree or disagree.  The calculations matter, not the opinions, however. So the only new element would be a calculation, that makes sense. Solving the Dirac equation, in general relativity, however,  doesn't satisfy this condition. The spacetime geometry must be  taken to be fixed, just it need not be flat. And this isn't, in fact special to this field, but to any quantum field. It's just that fields with half-integer spin don't have as classical limit the same objects. In the classical limit there isn't any relation between spin and statistics; the statistics is, always, Boltzmann, which means the particles are distinguishable. Spin is just another label. 

I agree, but still, the times I wrote down explicit calculations, you did not reply to them and kept going on your comments... what else could I do?

and what does it mean that solving the Dirac equation in general relativity doesn't satisfy this condition... what condition... explicit calculations?

Lot of thanks to all participants for their interesting contribution to the important questions on Quantum and Relativity of elementary particles.

But: Wouldn't be better to start the discussion about equations describing two Dirac particles? Are such equations known? If yes, would the particles change the mass sign simultaneously under one operation of a suitable invariance operation (if any were admissible)?

Joachim

The Einstein equations are classical. Their RHS is the energy momentum tensor of matter. This energy-momentum tensor is a classical field, function  of the matter fields, that obey their classical equations of motion. The Dirac equation is not the equation of motion of any classical field, because for the evolution problem in that case to be well-defined, the solution of the Dirac equation must obey Fermi-Dirac statistics, not Boltzmann statistics. That's the condition:  Both sides must obey Boltzmann statistics.

It's known that it isn't possible to describe a fixed number of relativistic particles in interaction (``no-interaction theorem'', Leutwyler, 1965). To describe relativistic particles in interaction requires a field, that describes an indefinite number of particles. This is true, whatever the spin of the particles. 

Once more: the mass of a relativistic particle is a Lorentz invariant-it doesn't change sign under charge conjugation. Positrons and electrons have the same mass-they have opposite electric charge. Charge conjugation, as its name indicates, changes the sign of the-additive-charges of the internal symmetry (abelian) groups.  Mass isn't an additive charge of such groups-it's one of the charges of the Lorentz group. 

Energy isn't a Lorentz invariant-t's the time-like component of the energy-momentum 4-vector of a particle. Since it isn't invariant, it isn't useful to focus on it, but on the invariant quantities-that are constructed from non-invariant quantities. Focusing on non-invariant quantities is the source of the confusion on this topic.

Joachim, interesting point: if you open such thread, make sure I can follow it

Stam, let me put everything in mathematics, so that we will solve once and for all: I can certainly define the metric being dynamic and I can certainly define a classical spinor, it certainly has an energy and a spin that can be put as source of curvature and torsion field equations and from the tetrads I can certainly write the spin connection for the spinor field, and the fullly coupled system of field equations is consistently defined with no doubt: now, if you claim that this system of field equations, nevertheless, doesn't make any sense, the only way you have to prove it is to show that with this construction a contradiction will follow, inevitably; do not add hypotheses about QFT because if you do it may always mean that the problem comes from them, and not from field equations. If you can do that, or point to a reference that has done that, I will admit that there is a problem.

If this isn't done, or if this is done by employing a number of additional hypothesis, nobody can really claim there is a problem; the only thing we can do is claim that we have no idea, and since we do not like to have no idea we prefer to say there is a problem.

I'll be waiting for the mathematical demonstration of contradiction.