Hi Raul,

I'm not sure about the answer, but I would like to discuss it.

The graviton itself comes from the linearization of Einstein's equations: you consider g=\eta + h, where \eta is the Minkowski metric and h is a perturbation; then you find the equations of motions for the perturbation h; eventually you quantize it to find the associated "particle".

The particle associated to h (the graviton) then should be massless (long range interaction) and spin 2 (from the tensorial nature of the field).

The fact is that, in order to define the graviton, you had to linearize around Minkowski, which could be regarded (if there are no singularities or strange behaviours) as "going locally", right?

This reminds me of the fact that, in presence of gravity (or, equivalently, on a generic curved manifold) every quantized field has a problem in the definition of a global concept of particle. You can't simply define a unique vacuum state in a time-dependent metric (as you do in Minkowski for usual QFT), hence you cannot "build" the particle with its properties in a global way.

I don't know if this is useful at all, but I hope someone will clarify.

Best regards,

GA

Hi, Raul, I don't understand well about your question also. But Giovanni's explanation seems close to mine. As a field theory, one need to define background space-time, and particles are perturbations of this background (or more precisely, the vacuum). However, in a generic curved space-time, how to define a global vacuum is difficult since it will evolution as time going. The more difficult thing is to define a particle in a curved space-time since it will decay! So the only thing we can do is doing the perturbaton on some static background, such as, Minkowski space-time. When the background is given, then one can do the usual field quantization, which is, write down the action( for gravity, this is Einstein-Hilbert action), find conjugate variables, define time-equivalent commutation relation, canonical or path integral quantization. The parity means for this action, it preserve this symmetry, that is called "global parity", this symmetry is not a really Global symmetry, but just for the action itself! If one has done the quantization, and renormalization and then get an effective theory, if this "global symmetry" still preserve, then one can say that this symmetry is a real symmetry of this theory. If it is broken(usually, it will), then it is not a local symmetry, that is called anomaly. Since for the quantum gravity theory, Graviton propagates on the space-time, it will bring the "local" breaking down to everywell on this space-time and of course, there should not be a global parity symmetry. However, if there are no local breaking down of the defined global symmetry, it will survived from the quantization and then be a real global symmetry.

However, these global symmetries have nothing done to the spin or fermionic particles. The defination of a spinful particle is an alternating speaking of a spin representation of the isomorphic group of the space-time. Spin symmetry is not up and down symmetry. For example, in Minkowski space-time, the iso group is SO(1,3), but the motion group is Poincare group, which is SO(1,3) half-producting Gallilea group. SO(1,3) actully is a local group of the motion group of the manifold. spinful particles are local excitations, so they are spin representations of SO(1,3), eg. Dirac particles are spin 1/2 representations of SO(1,3), Gravitino (Graviton's super parterner) is spin 3/2 representaitions of SO(1,3) etc.

I do not like that saying that photon has two helicities which related to each other by parity, I prefer to say that photon has two polarization directions, on this polarization plane, electronic and magetic fields excite each other vertically. since these two fields are always vertical to each other, one can define a Z_2 symmetry to transform between them. A massless particle can have D-2 polarization directions, where D is the dimention of spacetime, eg. in 9+1 dimention (as string theory) massless particles are representations of SO(8) which is the little group of SO(1,9). For massive particles, the independent polarization directions are D-1 (there is an additional backward polarization which makes the particle Massive).

First of all thanks a lot for your words Giovanni Acquaviva and jian-feng Wu. Let me say a few words about quantization of the "photon field" A_i first. A_i, being a Lorentz vector has four independent commponents, so naively you should expect, afther quantization, that there are four diferent photons. On the other hand you know that A_i has gauge freedom, so maybe you can "gauge out" somo of these degrees of fredom, and thats exactly what happends, gauge freedom kill's two components of A_i, leaving just the physical degrees of freedom, namely the two helicity states of the photon. But you also know that there are not two diferent photons out there, there's just one, so there must be some symmetry relating this two particles. The symmetry relating those degrees of freedom is PARITY, wich is a GLOBAL SYMMETRY, not a local one.

Ok, now you try the same procedure with the metric g_ij (linearazed or of your favorite choice), in a general four-manifold, g_ij has 10 independent components, right?, then you should expect, again, at least 10 diferent types of gravitons, but you know that there are not 10 gravitons, there's just one (i just know that, but honestly i dont know why), therefore, the other 9 particles asociated with g_ij should be related to each other by a symmetry, THE ARGUMENT GIVEN BY MY TEACHER WAS THAT THE SYMMETRY RELATING THOSE DEGREES OF FREEDOM IS PARITY, something wich i disagree, as i've said, simply because parity IS NOT A GLOBAL SYMMETRY IN A GENERAL SPACE-TIME, here of course i have in mind Wigner's definition of particle, that in few words says that everything that can be related by a global symmetry should be considered as one and the same particle.

Maybe you will say, ok 10 gravitons is a lot, and i agree with that, maybe you can kill some of these degrees of freedom using a difeomorfism of M, wich is the gauge freedom in this case, but certainly i dont think you can kill 9 "with just one shot"

Greeting from México

Raúl

You don't need to bring QFT into this (in fact, that raises it to the level of a research problem, because gravity isn't renormalizeable). It sounds like the original statement is simply that waves in the space-time metric have polarizations, which they do. However, they have different polarizations than photons do, because photons are spin-1 and gravitons are spin-2. There are some nice visualizations of gravity-wave polarizations. (And visualizations are useful even if you already have a mathematical understanding!)

Parity is not an exact symmetry of nature; it's completely violated in weak interactions. However, it is an exact symmetry of general relativity. It's also an exact symmetry of QED. (Interestingly, QCD spontaneously breaks chirality, though that's a much smaller effect than the built-in parity violation of weak interactions.)

Because of the polarization of EM radiation into independent components, you could say that there are 2 different types of photons: vertical ones and horizontal ones, in much the same way as there are 10 independent components in space-time waves (and they mix under Lorentz transformations a lot). No one ever does (largely because of the Lorentz-dependence), but that's just a choice in the way that we speak. They are different words for the same mathematical truth.

As far as i know gravitational waves have four polarizations, a number lower than 10. The question then is what to do with the remaining degrees of freedom?, there, i think you should call quantum mechanics (not ghost busters) via QFT. I would like to read why parity is an exact GLOBAL symmetry of GR, where can i find such result? because as i tried to argue in past posts, i think that PARITY IS AT MOST A LOCAL SYMMETRY IN A GENERAL SPACETIME MANIFOLD.

Also i would like to apologize about a mistake i did in the original question, i write by the end that " the graviton doesn't exist as a spin 1/2 particle in a general spacetime manifold M." when graviton has spin = 2. So sorry!!!

Giovanni, i totally agree with you in the statement " linearization of the metric means goin local", but actually that supports my point about the local nature os parity.

Maybe i should be more explicit in what i mean with global parity. By local parity i' m thinking in changing the orientation of a space-like section in the vecinity of a given point, something that can be done because the "tangent bundle" of a general manifold M is locally trivial, the problem is that you dont know how to paste that construction consistently through whole manifold. There are at least two obstructions, the first one is that you need to be able to identify a time-like direction through the whole space-time (something that as far as i know it's not true in general), the second one has to do with what i've said before, namely, the problem of changing the space-like orientation globaly.

Yes, four polarizations sounds right, rather than ten. The masslessness of the graviton may eat some of the extra degrees of freedom, the way that a vector boson's third polarization (the longitudinal one) is not available to the photon because it is massless.

I don't have any references on hand, I'm speaking from what I remember. A local symmetry is a stronger condition than a global symmetry (local implies global, rather than the other way around). Since it sounds like you're in a GR class now (I was more than 10 years ago), why not apply a parity operator to the GR Lagrangian and see if it's invariant? Actually working it out should provide insight into the reason why.

You know, now that I think about it, what's the difference between a local parity operator and a global parity operator? Parity flips the sign of vectors: there are only two elements in the group. I don't see how something like that could be a function of position, the way that a Lie operator might. It seems to me that parity can only ever be a global symmetry: you can't vary the choice with position, as you can with a continuous group.

The reason why that parity should be a function of position is that, in a given manifold, say M, the infinitesila directions (i.e., the elements of the tangent spaces) are mixed, but the tangent spaces are already parametrized by points of M in a locally trivial fashion. And as you probably know, the relationship between any two "trivializations" of the tangent bundle is a "transition function" (an example of a more general concept called "cocycle"), that in plain words is just the jacobian of the change of coordinates.

Actually the proces i've just described is generic, in the sense that if you want to turn a local construction into a global one, you first have to check that the corresponding pasting functions, or transitions functions, satisfy the so called "cocycle condition". If they don't satisfy that relation, you can't paste the local construction consistently trough M.

The extent by which a given pasting function fails to satisfy the cocycle condition is what i called "obstruction", that are strongly related with the "global topology" of M.

I thought the discussion was about the parity symmetry in general relativity, which is a non-quantum theory. Space-time is continuous (in general relativity), and the words "photon" and "graviton" are sloppy short-hand for "electromagnetic waves" and "gavity waves". "Spin-1" and "Spin-2" are sloppy short-hand for "vector field" and "tensor field" in which the waves have a particular angular momentum (classically, like Pointing vectors). The expansion of electromagnetic waves into quantized energy packets is irrelevant to their parity, and the expansion of gravity waves into quantized energy packets is not a solved problem in physics. Even the fancy gauge symmetries of the Standard Model, a fully renormalizable quantum field theory, can be checked without any reference to their quantization (apply the operator to the Lagrangian). That's how I was using the words; sorry for any confusion.

In Raul's last post, I don't know what he means: almost all of the technical terms are unrecognizable to me. What I was talking about is unrelated to the topology of the manifold.

I do know, however, that gravity is one of the forces that conserves parity, at least in general relativity. It might not in some quantum gravity theories, but I don't know about that.

You must not see the `graviton´ like a classic particle or a corpuscular quantum element, but as waves in space sometimes condensed. You know the wave-corpuscular quantum duality. Is not like a wave or like a particle; is something we don't get to understand from our macroscopic point of view. That `continuous waves´, sometimes condensed, sometimes not (canceled, amplified by probabilities, etc.) are the gravitons. And that vibrations of space-time generates intrinsically the local gravitational interactions between masses.

I find very interesting the book "Before Big Bang", of Martin Bojowald. Quantum Gravitational researcher in Pennsylvania State University and Max Planck Institute.

It may helpful to review the Penrose ideas about twistors. The main motivation to introduce this concept was to try to get rid of the continuum and build up physical theory from discreteness. The physical concept used for Penrose for this goal was the angular momentum. This concept is discrete in a scenario of quantum mechanics, but at the same time can be associated with the structure of space-time at the level of polarization vector of a gravitational waves. Eventually these ideas led to the so called twistor theory. The title of the Penrose work is "ANGULAR MOMENTUM: AN APPROACH TO COMBINATORIAL SPACE-TIME"