martinbn said:
Can anyone (and would anyone) explain what the main ideas are? I mean in less than 160 pages.
let me help a bit
In section 5 we learn that a new investigation to the relationship between spin and statistics has to be performed and that this has to occur in the context of the Clifford algebra. This leads to the investigation of ordinary second quantization of new types of Dirac equations. This excercise reveals some very surprising conclusions : (a) negative energies cannot be avoided within a traditional Hilbert space representation and the usual spin-statistics connection fails (b) to cure this there are two ways out (i) you go directly to nevanlinna spaces or you try to avoid nevalinna spaces and enlarge the theory by including Grasmann nuymbers. In the first case, there is no spin-statistics theorem anymore, but at least the correct connection is allowed for; in the second case, further computation reveals that either (i) you have to consider Nevanlinna spaces or (ii) a new principle of gauge invariance is required (for spin 1/2 particles!). I did not work out this last option and came to the conclusion that Nevanlinna spaces are unavoidable (there are of course the potential interpretational difficulties, but a constistent scheme worked towards in section 6). There are two further lessons which we learn from which the first one is at first sight dramatic. That is (a) there is no spin statistics theorem- not even in Minkowski (b) the whole math suggest to enlarge QFT with the Clifford numbers. With that last statement, I literally mean that we have to consider Clifford-Nevanlinna modules: which is a natural extension of quaternionic quantum mechanis fo Adler and Finkelstein to no-division algebra's. And a very natural one since all associative division algebra's (R,C,H) are Clifford algebra's. But what about the spin statistics relation? The answer to this question is remarkably simple, but no one has figured it out until now. If one looks into the proof of the spin statistics theorem of Weinberg, then you see that causality, Poincare invariance, positive energies and positive probability imply the spin statistics relation. But what is never said is that Poincare invariance, positive energies and positive probabilities and spin-statistics lead to causality. The idea therefore is to exchange causality for the spin-statistics relation and the proof that this point of view is superior to the original one results from the fact that one can weaken the restriction from Hilbert spaces to Nevanlinna spaces and still obtain the correct implication (while the reverse is false as mentioned before). Section seven contains so far a technically rigorous introduction to a very general kind of distribution spaces allowing for well defined scalar products The structures presented here give each observer his own topology and access to the universal module, the unitary equivalences are very weak and of the unbounded kind (which is typical for Nevanlinna spaces) so this framework is large enough to do rigorous QFT. One of the consequences of this investigation is that it is natural to drop the axiom of associativity which gives the mathematical framework an extremely broad scope; still some very stringent results can be obtained.
Ok, so all these ideas together is what I would call a first extension of free QFT, in total there are at least three non trivial ideas for the quantum theory alone (then I do not speak about gravitation yet). What do we do in section 8, well we construct an axiomatic foundation of a theory of quantum gravity. This means that we work out the kinematics, dynamics and part of the ontology. What are the crucial ideas?
(a) Local particle notions ! What do we mean with this and how do we realize this technically? There are several ways to understand this and let me give you one. We only know one particle notion and that is the one associated to a Fock space in the framework of a free theory on Minkowski: everyone who has understood this construction knows that it is inevitable and canonical (apart from the ''statistics theorem'' which gives you ordinary bose/fermi and which requires a number of assumptions we dismissed in section 7). Right, so if we want to have a fundamental particle theory, it seems like we are condemned to a free theory. Wrong ! The realization of how to solve this is the second non-trivial idea (and you immediately notice that if I can solve this problem, we can give a full nonpertubative formulation of interacting relativistic quantum theory). The idea is actually the same Einstein had when he got from special to general relativity: that is, you put ordinary free Minkowski on tangent space. Hence, we move free QFT to TM instead of M (M is the manifold); hence TM is the theater for reality and not M. This implies automatically that we have to pick the vierbein and not the ordinary metric for a theory of gravitation.
Now the development goes two ways - on one side you have in each point of space-time a preferred reference frame given by the vierbein, which gives you an infinite NOW that is isomorphic to the whole of M and on which lives a free relativistic quantum field theory. On the other side your need a new geometry as well as a new idea how to implement interactions. This requires the following:
(a) on the gravitation side you need a new connection having the appropriate transformation laws
under local (in M) Poincare transformations on TM
(b) you need new ideas for quantum theory.
Now concerning the classical gravitational theory (we come to the issue of gravitons later on), there are a number of important realizations: (i) it is a pure torsion theory (no Einstein tensor, the corresponding Bianchi identities are not adequate because of torsion) (ii) it is a non local theory, more specific the equations are ultrahyperbolic (how this is possible I explain within a minut) (iii) technically therefore it is much better to pass to a boundary value formulation instead of an initial value formulation. Ok, how can it be that these equations are ultrahyperbolic if we work with an ordinary vielbein as well as a Minkowski metric on TM? The answer is easy, but the interpretation as well as the consequences are far reaching. The simple answer is that the equations live on TM and not on M: you notice immediatly that you have two times (one coordinate time on M and one time in TM associated to the tetrad). Now, this appears to be going into the work of Bars, however it does not. That is, the ''time'' in the ultrahyperbolic equations is no time in the metric sense, it is a (linear) combination of space coordinates in the (linearization) of the field equations. Amazing enough, these equations have a 3+3 structure (two coordinates disappear but are recuperated in other equations - so the system is not underdetermined). Such multiple time idea in the naive sense is very problematic (since you think that three ''real times'' are necessary and one needs to make a compactification of two time directions) but here, no problem occurs since the metric on M is hyperbolic. Ok, we have performed a few consistency checks on the theory and also here, the conclusions are pretty radical (but not in contradiction to observation as far as I know): (a) if one gives up locality in gravitational theories, the 1/r^2 force law has no reason of existence anymore. Indeed, we knew already it had to be like that because of the anomalous galaxy rotation curves (where the MOND scenario offers an adequate way out). There is a kind of ''landscape issue'' here ( I don't call it a problem) in the sense that several different gravitational laws can exist within the same universe. To make a long story short: one can verify that short and long distance corrections to MOND are a possible solution (a short distance correction might be that you shut off gravitation on scales smaller than a millimeter and a long distance correction might exist in weakening gravitation where MOND becomes too strong). All those parameters can of course be chosen freely. There are two new classes of black hole solutions and strange enough, Einsteinian black holes are not solutions to the theory. One may guess that the usual no hair theorems do not apply and therefore, one does not have a simple first law of thermodynamics for black holes as is the case for general local metric theories. The physics of these blak holes is really different too: in general, gravitation is much stronger on the effective ''event horizon'' than it is in Einstein's theory. This offers possibilities for new exotic explanations for gamma ray bursts.
Ok, this is what we understood so far of the classical gravitational theory: a complete analysis of the mathematical structure of these equations will have to postponed to future work. Let me talk now about gravitons before we go over to the quantum theory. The only issue we want to treat here is the Weinberg-Witten theorem. So in my logic, gravitons are living on TM and belong to a free theory. Now, Weinberg-Witten says that no Lorentz covariant, conserved energy momentum exists for massless spin two particles. we agree that for a usual quantum gravity theory which is defined pertubatively around a fixed background, this is a death sentence. However, in this theory, it is a logical necessity. Indeed, gravitons are perturbations on a dynamical metric which is fixed through semi-classical equations wherein the relevant energy mometum tensor and spin tensor depend upon matter. Therefore, by definition, gravitons cannot contribute to the semiclassical equations! The conclusion therefore is that gravitons cannot ''gravitate'' but of course, the may belong tot the generators of the Poincare algebra. Weinberg Witten does not say anything about that ! These last generators are the Noether charges of the energy momentum + a graviton contribution. Obviously, gravtions may scatter and gravitate indirectly through interaction with matter. Therefore gravity has two faces, a non-local dynamical ontological side (geometry) and a local particle side (gravitons) and physics for both is essentially different. It is also clear that gravitons have an exceptional position within the particle spectrum.
Right, the quantum theory then, the ''main realization'' consisted in bypassing Haag's theorem as well as Coleman-Mandula. The whole idea is that we completely ''relativize'' quantum theory, therefore, away with foliations, away with global Hamiltonian and bye bye Heisenberg comutation relations (on M, not on TM !). All these ''technicalities'' have been major obstacles in case geometry becomes dynamical. Now well, you will say, what do you get into it's place and how do you retrieve ordinary relativistic quantum theory? It is clear that interactions have to be treated essentially different (due to Haag's theorem) and we are just not allowed to recuperate the ordinary Dyson series expansion. Let me reassure you from the start, yes ordinary free QFT is recuperated in an essentially unique way. The theory is constructed from relational ideas: to be brief, there exists something like a universal Clifford-Nevanlinna module which is constructed like a Fock space (she is time and space less). Of every particle species, there exists an infinite number of copies, to every space-time point and tetrad, one attaches the following quantum notions: *local* creation and annihilation operators (so things which the local observer can *measure*) and annihilation/creation operators for particles which you ''perceive'' in the rest of the universe (note that this requires an extension of representations of the Poincare group as constructed by Wigner, and a part of this analysis is already done in section 10). Moreover, one has a LOCAL vacuum state on which the creation/annihilation operators work as usual. Now, dynamics consists in changing particle notions from one spacetime point/vielbein to another spacetime point/vielbein by a general unitary transformation (what I would call a nonlinear Bogoliubov transformation). In other words, the kinematical object is a unitary transformation U(e_b(x),e_b(y),x,y) which depends upon two points and local reference systems. U(e_b(x),e_b(y),x,y) maps local particle notions to local particle notions, local vacua to local vacua and the ''rest of the universe'' to the ''rest of the universe''. One can consider a cycle of unitary mappings from x to y, y to z and z to x; it is trivial to demand that such cycle gives an identical result (other possibilities are discussed but appear to be unphysical). This allows one to write U(e_b(x),e_b(y),x,y) as U(e_b(x),x)U^{*}(e_b(y),y), in other words one has a unitary potential. However, these assumptions of associativity and the mere fact that the product is well defined only hold locally on a coordinate patch and fail globally; therefore the notion of state of the universe gets localized too to a family of compatible states which are each well defined on a coordinate chart only. This is a complete localization of quantum theory entirely in the spirit of general relativity. Now one has to find constraint equations for the unitary potential which requires a new physical idea: local Lorentz covariance. That is, you demand that the equations are invariant under quantum unitary Lorentz tranformations U(e_b(x),x) --> U^{*}(Lambda)U(e_b(x),x), which requires the introduction of a quantum spin connection (it is here that we kick Coleman-Mandula in the but). Anyhow, for U one has tor write down TWO equations instead of one (to ensure unitarity is preserved). This has to do with the Clifford Algebra and it is well know in Clifford analysis where you have something like a left-right monogenic Dirac equation DX = XD = 0, where D is the Dirac operator and X an element of the clifford algebra. There are no Heisenberg commutation relations on M (the spin-statistics information is on TM) and the whole dynamics is fixed by boundary data on a two sphere at infinity (because you have two first order equations in one variable - I did not perform yet the necessary integrability analysis here). This reminded me immediately at the holographic principle and also here one obtains the result that causality is a derived property determined by these boundary conditions. Now, we have verified that one retrieves from this theory ordinary free QFT on Minkowski uniquely (if one shuts off interactions) if one restricts to clifford scalars (these are the natural boundary conditions). Hence ordinary Minkowski causality on M follows from the dynamics as well as causality on TM.
So again, I do not see your problem, perhaps you need flashlights and buzzwords, but I think the technical language is pretty clear for anybody who has mastered it.
If you know something about quantum gravity technically and not just philosophically, then you must understand that these claims are very unusual. Nobody so far had a good answer to Coleman-Mandula, Weinberg-Witten and Haag theorem... these are the cornerstones of quantum gravity.
Sorry, my use of this language is standard: different universes generally means different terms in a superposition with respect to a pointer basis. In the path integral language, this pointer basis consists of the ''classical'' realities. The role of consciousness has been explained by Von Neumann, Wigner and Pauli long time ago. For example you may wish to ask what it means for the same cat to be in a superpostion when the only thing which you have are states formed by creation operators.
