Does GR imply small Lorentz violations in practice?

[TrickyDicky]

QED has infinities

So far so good

after renormalization

You've got this backwards, renormalization is the procedure to get rid of the infinite terms appearing at intermediate stages of every calculation of the bare masses and charges of electrons in QED.
These infinities arise because of the postulated pointlike nature of elementary particles, implying electrons could transmit arbitrary high momenta, integrating over these arbitrary high momenta makes quantities diverge to infinity.
What I'm arguing is that the pointlike particle assumption of QED is founded in Lorentz invariance for particles and seems like GR contradicts it insofar this particles have mass.

even if there is global Lorentz invariance.

not even, but because of, is what I'm pointing out. the global LI comes from the infinite degrees of freedom of the fields (thus its quantum must be pointlike)

I guess what is needed is a QFT in curved space not in flat Minkowskian space, but can that be achieved holding on to Lorentz invariance even if we know it's just an approximation?

 

[TrickyDicky]

Sure, why not? Tons of science is done to first order. If you try to do too high of an order approximation then you wind up just fitting to the noise. It is very important to use the lowest order you can.

You can do it alright. That's not my point.
You can't pretend that what is an approximation is exact and derive of that assumption physical axioms.

 

[Dale]

I don't know what you are talking about. What physical axioms are you referring to?

Science is about getting accurate models. As long as the approximations are buried in the noise you have an accurate model. Approximations are a good thing and a staple of calculus as well as physics. We cannot experimentally distinguish between a low-order model and a high-order model when the high-order terms are undetectable, so we use the low-order model until the high-order terms become detectable.

 

[TrickyDicky]

I don't know what you are talking about. What physical axioms are you referring to?

QFT Lorentz symmetry.

Science is about getting accurate models. As long as the approximations are buried in the noise you have an accurate model. Approximations are a good thing and a staple of calculus as well as physics. We cannot experimentally distinguish between a low-order model and a high-order model when the high-order terms are undetectable, so we use the low-order model until the high-order terms become detectable.

This is right of course, is not what I'm debating though, don't know why you bring it up.

 

[Dale]

QFT Lorentz symmetry.

The Lorentz symmetry of QFT depends only on the mathematical form of the equations and has nothing to do with whether or not a given laboratory experiment is affected by spacetime curvature.

 

[atyy]

You've got this backwards, renormalization is the procedure to get rid of the infinite terms appearing at intermediate stages of every calculation of the bare masses and charges of electrons in QED.
These infinities arise because of the postulated pointlike nature of elementary particles, implying electrons could transmit arbitrary high momenta, integrating over these arbitrary high momenta makes quantities diverge to infinity.
What I'm arguing is that the pointlike particle assumption of QED is founded in Lorentz invariance for particles and seems like GR contradicts it insofar this particles have mass.

No I did not. The infinities remain after renormalization. QED may even break down before Lorentz violation is detected.

not even, but because of, is what I'm pointing out. the global LI comes from the infinite degrees of freedom of the fields (thus its quantum must be pointlike)

I guess what is needed is a QFT in curved space not in flat Minkowskian space, but can that be achieved holding on to Lorentz invariance even if we know it's just an approximation?

QFT in curved spacetime is an approximation, because it treats the gravitational field classically, while the other fields are treated quantum mechanically. QFT in flat spacetime is also an approximation. We have no theory that is not an approximation.

 

[TrickyDicky]

The Lorentz symmetry of QFT depends only on the mathematical form of the equations and has nothing to do with whether or not a given laboratory experiment is affected by spacetime curvature.

That's what I'm saying. I'm centering on the form of those equations, and where they come from.

 

[PAllen]

How is Lorentz invariance handled in GR? I know that there is no global Lorentz invariance in GR, instead it only holds locally, meaning that it is obeyed in the limit at infinity:when r goes to infinity by considering infinite distance or infinitely small point mathematical objects.
But when considering finite distances, does GR imply small Lorentz violations in practice?

Let's go back to the beginning. Yes, there is no global Lorentz invarianced in GR (invariants come from the metric, and the Minkowski metric is not globally valid).

But then, "locally" means limit at infinity?? Nope.

Locally is code word for "limit as region goes to zero size". A sphere is locally Euclidean, but no finite section is exactly Euclidean. It is mathematically precise to say that it is in the limit of zero size. This is not a trivial statement, in that sums of angles of triangles go to pi; ratio of circumference of circle to to radius goes to pi. Thus, such a region, blown up, is still much closer to Euclidean, than a larger section of the orginal sphere.

Similarly, locally lorentz is making an exact limiting statement about GR geometry.

 

[TrickyDicky]

The infinities remain after renormalization. QED may even break down before Lorentz violation is detected.

Please explain, your telegraphic style may lead to misunderstandings.

QFT in curved spacetime is an approximation, because it treats the gravitational field classically, while the other fields are treated quantum mechanically. QFT in flat spacetime is also an approximation. We have no theory that is not an approximation.

Of course in a trivial sense all science is just an approximaton, but there are right approximations and wrong ones. If you are alluding to the fact that we don't have a Quantum gravity theory, I'm just suggesting that an obstacle towards that goal might be using equations in a Lorentz invariant form. That's what I'm trying to get across without much success, probably this is not the right forum to debate it.

 

[TrickyDicky]

Let's go back to the beginning. Yes, there is no global Lorentz invarianced in GR (invariants come from the metric, and the Minkowski metric is not globally valid).

But then, "locally" means limit at infinity?? Nope.

Locally is code word for "limit as region goes to zero size". A sphere is locally Euclidean, but no finite section is exactly Euclidean. It is mathematically precise to say that it is in the limit of zero size. This is not a trivial statement, in that sums of angles of triangles go to pi; ratio of circumference of circle to to radius goes to pi. Thus, such a region, blown up, is still much closer to Euclidean, than a larger section of the orginal sphere.

Similarly, locally lorentz is making an exact limiting statement about GR geometry.

I can perfectly agree with this,I have no problem wth GR being locally Lorentz invariant because I understand what that means in geometrical terms,now this is the "geometrical part", I find issue with the physical interpretation of that mathematical statement, let's look at the right hand side, the matter-energy source, an elementary particle is not "local" in the geometrical sense. That's all I'm saying. But hey if I'm wrong, please enlighten me.

 

[Dale]

That's what I'm saying. I'm centering on the form of those equations, and where they come from.

Then I don't understand why you care that a real lab will have some second or higher order tidal gravity effects. If you are just dealing with the math then the form of the equations can be exactly Lorentz invariant.

As far as where the QFT equations come from, you may be better off asking in the QM folder.

 

[atyy]

Please explain, your telegraphic style may lead to misunderstandings.

The fixed point of the renormalization flow in QED is an infrared fixed point. It means that we can do perturbation theory at low energies, which accounts for the empirical success of QED. However, the theory diverges as we go to higher energies. Look up eg. the Landau pole. So even without empirical evidence, we expect the theory to fail at sufficiently high energies.

QCD's renormalization flow is a UV fixed point, meaning that the theory is consistent at arbitrarily high energies. In this case, it is expected to fail not on mathematical grounds, but on empirical grounds, such as not containing gravity.

Of course in a trivial sense all science is just an approximaton, but there are right approximations and wrong ones. If you are alluding to the fact that we don't have a Quantum gravity theory, I'm just suggesting that an obstacle towards that goal might be using equations in a Lorentz invariant form. That's what I'm trying to get across without much success, probably this is not the right forum to debate it.

No, you are not having much success because no one debates it (ie. what you are saying is not controversial). However, there seems to be a coherent theory of quantum gravity that is exactly globally Lorentz invariant at the boundary. The boundary theory completely specifies the theory in the bulk, where one only has local Lorentz invariance. Look up AdS/CFT or gauge/gravity duality. This theory seems not to get the correct cosmology, or the matter content. There are attempts to formulate QG as a Lorentz violating theory, such as Horava-Lifgarbagez gravity.

The major results in QG are:
1) The spacetime metric field treated as a quantum field is not perturbatively renormalizable.
2) The spacetime metric field cannot arise from more fundamental degrees of freedom that are also quantum fields in flat 4D Minkowski spacetime (Weinberg-Witten theorem).

So the possibilities people are trying are:
1) The spacetime metric field is renormalizable, if one looks non perturbatively (asymptotic safety). If this fails, then we must conclude that the spacetime metric field arises from more fundamental degrees of freedom that are either
2) not 4D quantum fields (string theory)
3) or Lorentz violating (Horava Lifgarbagez)
4) you can think of more possibilities yourself ;)

 

[TrickyDicky]

Thanks everyone.
Communiction is hard but I think I can get something out of all this, hopefully someone else can too.

 

[TrickyDicky]

So the possibilities people are trying are:
1) The spacetime metric field is renormalizable, if one looks non perturbatively (asymptotic safety). If this fails, then we must conclude that the spacetime metric field arises from more fundamental degrees of freedom that are either
2) not 4D quantum fields (string theory)
3) or Lorentz violating (Horava Lifgarbagez)
4) you can think of more possibilities yourself ;)

There seems to be more possibilities, this is from the wikipedia article about Lorentz violating theories:
Quote
"Lorentz violation refers to theories which are approximately relativistic when it comes to experiments that have actually been performed (and there are quite a number of such experimental tests) but yet contain tiny or hidden Lorentz violating corrections."End quote

The second model of such theories that describes is close to what I was suggesting here:

Quote
"Similar to the approximate Lorentz symmetry of phonons in a lattice (where the speed of sound plays the role of the critical speed), the Lorentz symmetry of special relativity (with the speed of light as the critical speed in vacuum) is only a low-energy limit of the laws of Physics, which involve new phenomena at some fundamental scale. Bare conventional "elementary" particles are not point-like field-theoretical objects at very small distance scales, and a nonzero fundamental length must be taken into account. Lorentz symmetry violation is governed by an energy-dependent parameter which tends to zero as momentum decreases. Such patterns require the existence of a privileged local inertial frame (the "vacuum rest frame"). They can be tested, at least partially, by ultra-high energy cosmic ray experiments like the Pierre Auger Observatory."

"Models belonging to this class can be consistent with experiment if Lorentz breaking happens at Planck scale or beyond it, and if Lorentz symmetry violation is governed by a suitable energy-dependent parameter. One then has a class of models which deviate from Poincaré symmetry near the Planck scale but still flows towards an exact Poincaré group at very large length scales."
End quote

does anyone know of any references of people working on this approach?
Or experiments related in the Pierre Auger Observatory?

 

[atyy]

People have gotten special relativity as a low energy limit, even Maxwell's equations. Gravity, not yet. There are non-relativistic versions of AdS/CFT, but they are probably not UV complete.

2+1D: http://www.als.lbl.gov/als/science/sci_archive/144dirac_fermions.html
3+1D: http://arxiv.org/abs/hep-th/0507118
Long list of references: http://arxiv.org/abs/1102.0789

Non-relativistic AdS/CFT: http://arxiv.org/abs/0804.4053

 

[A. Neumaier]

How is Lorentz invariance handled in GR? [...] does GR imply small Lorentz violations in practice?

No. Lorentz invariance is independent of general covariance = diffeomorphism invariance. The latter only replaces the translation invariance of flat field theories.

Lorentz invariance is seen explicitly in general relativity coupled to a spinor field, where gravity is represented by a frame field and local Lorentz invariance is imposed on the frames.

 

[PAllen]

No. Lorentz invariance is independent of general covariance = diffeomorphism invariance. The latter only replaces the translation invariance of flat field theories.

Lorentz invariance is seen explicitly in general relativity coupled to a spinor field, where gravity is represented by a frame field and local Lorentz invariance is imposed on the frames.

This is way above my math level (conventional treatments of GR). Physically, the way I see Lorentz invariance being violated in GR is:

1) There is no coordinate system where the Minkowski metric can be used to compute path invariants.

2) If two inertial observers set up overlapping Fermi-normal coordinates, the Lorentz transform will not work to translate observations from one to the other.

 

[TrickyDicky]

People have gotten special relativity as a low energy limit, even Maxwell's equations. Gravity, not yet. There are non-relativistic versions of AdS/CFT, but they are probably not UV complete.

2+1D: http://www.als.lbl.gov/als/science/sci_archive/144dirac_fermions.html
3+1D: http://arxiv.org/abs/hep-th/0507118
Long list of references: http://arxiv.org/abs/1102.0789

Non-relativistic AdS/CFT: http://arxiv.org/abs/0804.4053

Thanks for the refs, looks like interesting stuff.

 

[A. Neumaier]

This is way above my math level (conventional treatments of GR). Physically, the way I see Lorentz invariance being violated in GR is:

1) There is no coordinate system where the Minkowski metric can be used to compute path invariants.

2) If two inertial observers set up overlapping Fermi-normal coordinates, the Lorentz transform will not work to translate observations from one to the other.

Local Lorentz invariance is a property of what one finds in the (flat!) tangent hyperplane at each point, which reflects faithfully a neighborhood of that point in terms of Riemann normal coordinates. You can see it if you consider gravity formulated in terms of local orthonormal frames.

 

[PAllen]

Local Lorentz invariance is a property of what one finds in the (flat!) tangent hyperplane at each point, which reflects faithfully a neighborhood of that point in terms of Riemann normal coordinates. You can see it if you consider gravity formulated in terms of local orthonormal frames.

Then I don't think we're disagreeing. If you consider any region of finite size in GR, you will not find exact Lorentz invariance.

 

[TrickyDicky]

If you consider any region of finite size in GR, you will not find exact Lorentz invariance.

Exactly. At least we seem to agree on that.

 

[TrickyDicky]

So if in GR, fermionic particles can't define an SR inertial frame due its having a finite size in the 4-manifold, we must conclude only the vacuum and massless bosons can be real inertial frames as defined in SR, and wouldn't be exactly exchangeable with the fermion frames.
In other words all fermionic matter undergoes some degree of acceleration-is affected by curvature. Is this logic correct?