Horava Gravity proven wrong, ruled out, etc.

[MTd2]

http://arxiv.org/abs/0905.2579

Strong coupling in Horava gravity
Authors: Christos Charmousis, Gustavo Niz, Antonio Padilla, Paul M. Saffin
(Submitted on 15 May 2009)

Abstract: By studying perturbations about the vacuum, we show that Horava gravity suffers from two different strong coupling problems, extending all the way into the deep infra-red. The first of these is associated with the principle of detailed balance and explains why solutions to General Relativity are typically not recovered in models that preserve this structure. The second of these occurs even without detailed balance and is associated with the breaking of diffeomorphism invariance, required for anisotropic scaling in the UV. Since there is a reduced symmetry group there are additional degrees of freedom, which need not decouple in the infra-red. Indeed, we use the Stuckelberg trick to show that one of these extra modes become strongly coupled as the parameters approach their desired infrared fixed point. Whilst we can evade the first strong coupling problem by breaking detailed balance, we cannot avoid the second, whatever the form of the potential. Therefore the original Horava model, and its "phenomenologically viable" extensions do not have a perturbative General Relativity limit at any scale. Experiments which confirm the perturbative gravitational wave prediction of General Relativity, such as the cumulative shift of the periastron time of binary pulsars, will presumably rule out the theory.

******

Diffeomorphism group

Petr assumed that the full spacetime diffeomorphism group can be broken down to a subgroup and the correct long-distance limit can be reproduced, anyway. This is, of course, a very bold assumption because the diffeomorphism symmetry plays a very important role in General Relativity.

The diffeomorphisms are closely linked to the equivalence principle and are responsible for the reduction of a large number of excitations down to the two transverse physical polarizations of the graviton. This reduction is not only an aesthetically pleasing sign of the power of the underlying principles and symmetries of Einstein's theory. It is also an experimentally supported proposition.



The 1993 physics Nobel prize was given to two men at Princeton who have found a binary pulsar whose frequency is increasing exactly according to general relativity, a theory predicting that this binary pulsar emits two polarizations of gravitational waves and loses a particular and calculable amount of energy per unit time. So if you mess up with the number of physical polarizatinos of the graviton, you are likely to fail, both aesthetically and empirically.

Massive gravity

There exists a useful, older example what happens if the diffeomorphism symmetry is not taken seriously, namely the Fierz-Pauli (massive) gravity. Use the symbol "h_{ij}" for "g_{ij} - eta_{ij}" where "g_{ij}" is the dynamical metric and "eta_{ij}" is a background profile for the metric (e.g. the flat one). You can write down a quadratic action for this tensor field, "h_{ij}". This action can also have mass terms and interactions. In general, they break the diffeomorphism symmetry which therefore cannot be promoted to a constraint algebra.

You could think that if you send the mass of "h_{ij}" to zero, the theory converges to General Relativity for the right choice of the interactions. However, the Fierz-Pauli theory didn't respect the diffeomorphism symmetry so it has a higher number of degrees of freedom than General Relativity. Massive spin-two particles have five polarizations (a traceless symmetric tensor in 3 spatial dimensions) and the three excessive ones don't disappear. They're still there, they become strongly coupled in the massless limit, and their influence on physics generates lethal effects usually presented as the vDVZ discontinuity.

This discontinuity has both ultraviolet and infrared manifestations. Concerning the ultraviolet ones, we still have ill-behaved propagators for the unphysical modes at high energies, analogous to the Proca field. The infrared problems are more serious and resilient. The corrections to the Newtonian planetary motion (e.g. the Mercury perihelion's precession) disagree with General Relativity.

Stuckelberg trick

There exists a nice way to see where this discrepancy comes from. In 2002, Arkani-Hamed, Georgi, and Schwartz updated the Stuckelberg trick. They re-introduced the general diffeomorphism symmetry to the massive theory, by adding additional auxiliary fields (essentially "fake" dynamical spacetime coordinates). In the massless limit, the massive theory reduces to the General Relativity coupled to these new fields. And they're strongly coupled, indeed. Their effects can be nicely isolated and the predictions of General Relativity get damaged.

Charmousis et al. use the same trick to analyze the Hořava theory. The result is completely analogous.

The new fields - the fake dynamical spacetime coordinates - become strongly coupled in the hypothetical long-distance limit, lambda=1 (in fact, only the temporal one does, but that's enough). This strong coupling means that the coefficient of the derivative terms in the equations becomes infinitely higher than the coefficient of the interaction terms (without derivatives). The unwanted new modes change the physical predictions by a finite amount.

Charmousis et al. argue that this problem is almost certainly shared by all models you could imagine that are based on Hořava's general idea of a diff-breaking Lorentz-violating UV starting point.

Detailed balance

Aside from this general, resilient problem, the authors also discuss a smaller problem with theories based on the detailed balance. Hořava's theory is non-relativistic so there are many more terms one can add to the action. To replace the constraining power of the Lorentz symmetry, Petr imposed another principle, the detailed balance conditions.

Detailed balance is a condition that relates a dynamical system in "p+1" dimensions to a static system in "p" dimensions. The "forward and backward" transition rates of the higher-dimensional theory are required to be equal, up to the ratio of equilibrium probabilities of the two states. In practice, it means that the action of the p+1-dimensional theory can be generated by an action/potential of a p-dimensional theory, while the terms with the time derivatives are required to be the most standard "velocity squared" terms (for all fields) that you could imagine.

Well, there exist mathematical similarities between quantum field theory in "p+1" dimensions and statistical physics in "p" dimensions. Both disciplines use some functions, integrals, and they even know the concepts of the Renormalization Group that apply in both cases. Still, I think that this relationship cannot be viewed as a deep physical principle to constrain theories or as a complete identification of two theories that would be analogous to dualities. The resulting condition on the kinetic terms seems arbitrary and contradicts other, more well-established and motivated conditions that normally link the temporal kinetic terms with the spatial ones, especially the Lorentz symmetry. Also, the lower-dimensional theory is not treated as a full-fledged quantum theory.

In the case of the Hořava gravity, the detailed balance was seen to be incompatible with the proper Schwarzschild-like, spherically symmetric solutions by Lu, Mei, and Pope. The right solutions were not obtained and in the promising ones, functions that should have been calculable remained unconstrained. The authors decided that it was necessary to abandon the detailed balanced condition. But as Charmousis et al. found, it is not sufficient.

Once you lose this condition, you are back in the world of a generic non-relativistic model building with lots of adjustable terms. A huge fine-tuning is surely necessary to obtain a Lorentz-invariant infrared limit but it is probably not sufficient.

Summary

I think that this episode is just another manifestation of the crucial role played by the diffeomorphism symmetry and the local Lorentz symmetry in the context of General Relativity and its extensions. There exist good theoretical reasons why these principles should be obeyed exactly. And in fact, there exist empirical reasons, toohttp://motls.blogspot.com/2009/05/can-horava-gravity-flow-to-einstein.html" [Broken]

 

[Demystifier]

This is a very interesting paper, but it seems to be in contradiction with the results of
http://lanl.arxiv.org/abs/0905.0477
The authors of http://arxiv.org/abs/0905.2579 seem to be unaware of the paper above. In fact, they cite this paper in a bulk of citations, but they do not comment the results of it.

Hence, I think that it is premature to claim that Horava gravity is "proven wrong".

 

[Haelfix]

The criticisms in the paper were apparent early on to the experts (the paper echoes what the gravity people in my department pointed out during a theory seminar), but its nice that people took some time to write it up.

I'm pretty sure that there is no disagreement with the other paper. The strongly coupled scalar only appears when you depart from static GR solutions and its unclear that even a ton of finetuning would evade experimental bounds from binary pulsars and other assorted cosmological situations, you could imagine a bunch of ways it would reappear.

 

[MTd2]

This is a very interesting paper, but it seems to be in contradiction with the results of
http://lanl.arxiv.org/abs/0905.0477
The authors of http://arxiv.org/abs/0905.2579 seem to be unaware of the paper above. In fact, they cite this paper in a bulk of citations, but they do not comment the results of it.

Hence, I think that it is premature to claim that Horava gravity is "proven wrong".

Well, the other paper you mention looks technically acceptable, but the crucial conclusions are wrong. For example, if you look at (2.18) in that paper, you will see

(lambda-1) derivatives (h) = 0.

With sources, there would be a right hand side, and much like in the today's paper, one would get a strongly coupled scalar. Note the subtle inconsistency in their paper - they often need to assume lambda different from 1 (e.g. in eqn. (2.19)) but sometimes lambda=1 where GR could hold (e.g. at the top of page 2). Below eqn (2.23), they also reveal that there is an additional scalar H, besides the two physical transverse polarizations. They just say that it's non-dynamical in empty space, thinking that it means that it can be ignored. But it can't. If you actually put matter, this scalar will be strongly coupled and induce new forces that will create deviations from GR.

The paper you mentioned could actually be fixed to be a correct, useful paper, and there's probably no real discrepancy in the equations between the two papers. But the older paper's authors would first have to understand the background presented e.g. in the today's paper - about strongly coupled scalars, their impact on the discontinuity, etc.

 

[MTd2]

A message to the uncautious. The above messages are not mine, since actualy I copied/paste from Lubos Motl's discussion thread in his blog to here, and from this discussion here back there. Of course, I had to edit out the bad words from him, since I wanted some fair opinions and avoid ofenses.

I'm sorry. You can check his post on the link I provided on the opening post of this thread (the link is on the final point of the post). There are further question from other people there.

 

[yoda jedi]

I wanted some fair opinions there.

http://arxiv.org/PS_cache/arxiv/pdf/0911/0911.1299v4.pdf

 

[marcus]

http://arxiv.org/PS_cache/arxiv/pdf/0911/0911.1299v4.pdf

Hi Yoda, you point to this:
http://arxiv.org/abs/0911.1299
Strong coupling in extended Horava-Lifgarbagez gravity
Antonios Papazoglou (Portsmouth U., ICG), Thomas P. Sotiriou (Cambridge U., DAMTP)
4 pages... version accepted for publication in PLB, addresses issues raised in arXiv:0912.0550
(Submitted on 6 Nov 2009)
"An extension of Horava-Lifgarbagez gravity was recently proposed in order to address the pathological behavior of the scalar mode all previous versions of the theory exhibit. We show that even in this new extension the strong coupling persists, casting doubts on whether such a model can constitute an interesting alternative to general relativity (GR)."

I think there has been a followup, possibly by Ted Jacobson. Let me look.

Yes. http://arxiv.org/abs/1001.4823

==quote from Jacobson's January 2010 paper. page 1==

"Much interest has recently been focused on Horava- Lifgarbagez gravity[1], which proposes the possibility of a renormalizable, non-Lorentz-invariant UV completion of general relativity. There are so-called projectable and non-projectable versions of this proposal, and both have been shown to suffer from various problems (instabilities, over constrained evolution, or strong coupling at low energies) related to a badly behaved scalar mode of gravity brought on by the presence of a non-dynamical spatial foliation in the action[2].

A proposal for evading all of these problems, put forth by Blas, Pujolàs and Sibiryakov (BPS)[3], is an “extension” of Horava gravity. I will call it here BPSH gravity, and below T-theory for reasons to become clear. One can view this extension as promoting the fixed foliation to a dynamical one. This extension could still possesses strong coupling at low energy[4], but it is also possible that higher derivative terms in the action become important below the strong coupling energy scale and prevent this[5].

It was remarked in Ref. [5] that this extended Horava theory is related to a restricted version of Einstein-aether
theory, which is general relativity coupled to a dynamical unit time like vector field (for recent reviewsee[6]) ... "

==endquote==

Jacobson's reference [5] is http://arxiv.org/abs/0912.0550 which was mentioned in the abstract of the paper you cited. The one where BPS propose how to evade the Horava difficulties by an "extension".

 

[yoda jedi]

Hi Yoda, you point to this:
http://arxiv.org/abs/0911.1299
Strong coupling in extended Horava-Lifgarbagez gravity
Antonios Papazoglou (Portsmouth U., ICG), Thomas P. Sotiriou (Cambridge U., DAMTP)
4 pages... version accepted for publication in PLB, addresses issues raised in arXiv:0912.0550
(Submitted on 6 Nov 2009)
"An extension of Horava-Lifgarbagez gravity was recently proposed in order to address the pathological behavior of the scalar mode all previous versions of the theory exhibit. We show that even in this new extension the strong coupling persists, casting doubts on whether such a model can constitute an interesting alternative to general relativity (GR)."

I think there has been a followup, possibly by Ted Jacobson. Let me look.

Yes. http://arxiv.org/abs/1001.4823

==quote from Jacobson's January 2010 paper. page 1==

"Much interest has recently been focused on Horava- Lifgarbagez gravity[1], which proposes the possibility of a renormalizable, non-Lorentz-invariant UV completion of general relativity. There are so-called projectable and non-projectable versions of this proposal, and both have been shown to suffer from various problems (instabilities, over constrained evolution, or strong coupling at low energies) related to a badly behaved scalar mode of gravity brought on by the presence of a non-dynamical spatial foliation in the action[2].

A proposal for evading all of these problems, put forth by Blas, Pujolàs and Sibiryakov (BPS)[3], is an “extension” of Horava gravity. I will call it here BPSH gravity, and below T-theory for reasons to become clear. One can view this extension as promoting the fixed foliation to a dynamical one. This extension could still possesses strong coupling at low energy[4], but it is also possible that higher derivative terms in the action become important below the strong coupling energy scale and prevent this[5].

It was remarked in Ref. [5] that this extended Horava theory is related to a restricted version of Einstein-aether
theory, which is general relativity coupled to a dynamical unit time like vector field (for recent reviewsee[6]) ... "

==endquote==

Jacobson's reference [5] is http://arxiv.org/abs/0912.0550 which was mentioned in the abstract of the paper you cited. The one where BPS propose how to evade the Horava difficulties by an "extension".

yes, i read it.
but

http://arxiv.org/PS_cache/arxiv/pdf/0911/0911.1299v4.pdf

In any case, the very existence of such of an effective cutoff would imply that this extended theory cannot be considered a UV completion of GR, which is what it was introduced for. That would be true irrespectively of whether the strong coupling energy scale is low enough to fall within the range in which we can test gravity theories.
Let us summarize. We examined the extended version of Horava–Lifgarbagez gravity
proposed in [14] as a way to address viability issues caused by the anomalous
dynamics of the scalar degree of freedom that is present in the original
version of the theory. Expanding around flat space, it was shown that, even
thought the quadratic action of the theory is “healthy”, there are cubic interactions
that blow up when lambda -> 1. This leaves no space for Lorentz symmetry
to be recovered at low energies. Additionally, even if one is willing to abandon
Lorentz symmetry altogether, experimental constraints imply that lambda has to
be close to 1. The strong coupling scale, which acts as an effective cutoff of
the theory, appears to be lower than the cutoff of GR as an effective theory.
Considering that the motivation of the model is to constitute a renormalizable
theory of gravity, this casts serious doubts on whether it can be an interesting
alternative to GR.

 

[MTd2]

Wow that Antonios and Thoma's article was totally reformulated and re uploaded last Jan 22nd. It seems they kept corresponding with the authors of DBS' article, and at firt site, everything is more clear now. I have to recheck that paper again.

 

[ensabah6]

"
I think that this episode is just another manifestation of the crucial role played by the diffeomorphism symmetry and the local Lorentz symmetry in the context of General Relativity and its extensions. There exist good theoretical reasons why these principles should be obeyed exactly. And in fact, there exist empirical reasons, too."

Does String theory and QFT-gravitons respect
diffeomorphism symmetry and the local Lorentz symmetry ?

Does LQG?

 

[tom.stoer]

Does String theory ... respect diffeomorphism symmetry ... ?
Does LQG?

In a strict sense string theory does NOT. As far as I can see there is no quantization of string theory that does not use a split in a classical spacetime and a quantization on top of it. The classical spacetime is fixed to be M⁴ * CY or something like that. M⁴ has to respect Einstein's equations. To my best knowledge there are additional restrictions, e.g. M⁴ must be static, otherwise the quantization cannot be carried out. The Calaby-You is fixed, too. These limitations are not due to physical but due to technical reasons, as far as I know.
Nevertheless the quantization is consistent, there are no quantization ambiguities or anomalies as far as I can see.

Looking at gauge-gravity duality like AdS/CFT diffeomorphism invariance can be recovered, as 1) the CFT does not contain gravity at all and 2) it is dual to a gravity theory where only some boundary conditions Mⁿ of are fixed. Unfortunately this duality is not understood for all topologically relevant Mⁿ but for a certain class only (prominent example is Anti-deSitter).

In LQG the situation is different again. Here you have three constraints
G|phys> = 0
D|phys> = 0
H|phys> = 0
G is the Gauss law; it acts as generator of small local Lorentz trasformations. Solving this constraint implements local Lorentz symmetry in the physical subspace |phys>. This can be done using the loop representation where the nyme LQG comes from
D is the diffeomorphism constraint. Solving it implements spatial diffeomorphism invariance (spatial as M⁴ is sliced to M³ * R. One uses the so-called cylinder functions which eventually leads to the construction of spin networks.
H is the hamiltonian constraint which I would call "relict" of "timelike diffeomorphism". The kernel of this equation is not known exactly so far; so the physical Hilbert space has not yet been fully constructed. In addition there may be problems with the algebra of constraints G, D and H as the commutators incolving H do not close "off-shell".
So in LQG we are rather close to a solution but some work remains to be done.

 

[Haelfix]

No, that's not quite right. I think the confusion arises because people can't quite make the distinction between a solution of string theory, and the actual construction thereof. There is no split of the spacetime metric as described above. Really, there is no spacetime metric (target space) to begin with a priori. Instead they arise as generic solutions (in fact more than generic, encompassing solutions -- there is always gravity in string theory).

String theory is completely diffeomorphism invariant (see chapter 1-3 of Polchinski). The worldsheet is diff invariant by construction, and the only subletly with the spacetime solutions is that you have to go back and check by hand to see if it persists (eg the symmetry is not necessarily manifest), especially after gauge fixing and worrying about anomalies. Otoh by consistency, they have to be.

This is a subtle point, really requiring a textbook explanation (b/c its wonderfully delicate and roundabout) but it works very roughly like so:

The local world sheet diff invariance removes the timelike and longitudinal normal modes which would otherwise be uncancelled ghost modes. From the spacetime point of view, it is the same thing, the spacetime diffeo symmetry removes these unphysical states.

Einstein's equations show up in s.t in a rather unusual way. They are not imposed by hand, like in all other theories you might have seen before, rather they show up in a backalley way, namely as the condition of Weyl Invariance on the worldsheet. This implies that strings can only propagate consistently on backgrounds that satisfy these field equations and that have an enhanced spacetime symmetry.

 

[tom.stoer]

If I look at the Nambu-Goto action there is a (fixed) target space metric right from the beginning. How shall I get rid of it? What is the correct starting point?

 

[yoda jedi]

"


Does String theory respect
the local Lorentz symmetry ?

yes.
i think the only one.

...A natural consequence appears to be the principle of general covariance (which is what is meant by “general” relativity), namely, that physics should be the same in all reference frames, not merely inertial ones. That is, under a general coordinate transformation xμ→x′μ (xν), where the four-vector xμ=(t,x), the physics should be invariant. However, no physical principle dictates the exact form of the action, beyond general coordinate invariance. For action, Einstein used the integral of R (the scalar measure of the curvature at each point), because it reproduces Newtonian gravity in the appropriate limit (small curvature R, small velocity). But one can certainly add small covariant corrections, such as a power of the curvature scalar. In fact, such terms do appear as quantum corrections in the only known quantum theory that includes gravity, namely, string theory. In that case, however, general covariance remains satisfied.....


...Lorentz symmetry (which says, in the absence of gravity, that physics is the same for all inertial observers moving through (space)......



...The only known way to have a well-defined quantum theory at short distances that preserves general covariance -string theory-....


...since the quantization of gravity, even in the Hořava version, is not very well defined......

 

[Physics Monkey]

One thing which hasn't yet been mentioned is Vainshtein's solution of the vDVZ discontinuity. We should all find it unsettling that one can supposedly claim to know for certain that the graviton is massless via the vDVZ discontinuity. This should not be possible with only limited experimental information. But in fact you can't claim to know that the graviton is truly massless. Vainshtein showed that there is an extra hidden length scale in a theory with massive gravitons. This length scale diverges as the graviton mass is sent to zero, and the predictions of GR are recovered for measurements made at less than this scale.

Now Vainshtein worked with a particular model, but the moral of the story is general: linearization in gravity is an unusually dangerous game. Higher order terms in the low energy effective action can be quite dangerous especially in the presence of background fields and strong coupling. The Arkani-Hamed paper talks about this (massive gravitons can have strong coupled mode), but the Charmousis paper does not as far as I can tell. A quick search revealed nothing, so I wonder if anyone here knows if something like Vainshtein's solution can work in Horava gravity?

 

[jal]

http://arxiv.org/abs/1002.0308
Projectable Horava-Lifgarbagez gravity in a nutshell
Silke Weinfurtner, Thomas P. Sotiriou, Matt Visser
(Submitted on 1 Feb 2010)

To conclude, Hoˇrava–Lifgarbagez gravity constitutes an interesting quantum gravity theory.
It appears challenging to construct a viable model within this framework. In fact there are
numerous issues that have not even been systematically considered yet, such as matter coupling,
renormalization group calculations, etc. Given the fact that sensible renormalizable gravity
theories are not easy to construct, Hoˇrava–Lifgarbagez gravity seems to deserve further study.

It nice to see his position in a paper.
He has not changed his position since the the Emergent Gravity IV Conference in Vancouver.
jal

 

[bcrowell]

This is outside my field, so maybe this question will just show how naive I am. I assume that diffeomorphism invariance is the same as what a relativist would call general covariance or coordinate independence. AFAIK the conventional wisdom these days is that general covariance isn't quite the right notion to capture what makes GR special, since Kretschmann showed in 1917 that any theory can be made generally covariant, and Cartan formulated Newtonian mechanics in a generally covariant way, without making it look ridiculously complex. Is the real issue here general covariance, or is it something more like background independence or the assumption of no prior metric?

[EDIT] Another issue occurs to me -- does the Kretschmann/Cartan argument for the triviality of general covariance only apply to classical theories?

 

[Thomas Larsson]

Re Kretschmann:

Any theory can trivially be made diff invariant by adding a background metric which undergoes a compensating transformation under diffeomorphisms. Analogously, any theory can be made gauge invariant by adding a compensating background gauge field. The point with GR (or gauge theories) is that it is diff (or gauge) invariant without compensating background fields.

Instead, understanding diff invariance is crucial for the following reason. Classically, the physical phase space of GR is the unreduced phase space modulo diffeomorphisms, and quantically the physical Hilbert space is the unreduced Hilbert space modulo diffeomorphisms. The catch is that quantum representations must be of lowest-energy type, as always in quantum theory. Since people do not know how to construct (projective) lowest-energy representations of the diffeomorphism group, the Hilbert space of GR can not be constructed.

 

[Haelfix]

FYI there is some source of dispute in the literature about whether General covariance is physically vacuous. Different people will explain it differently, and it basically boils down to definitions and interpretation.

The way I like it is expressed best by Weinberg in Gravitation. Paraphrasing:
Its worthwhile comparing lorentz invariance to General Covariance.

Take Newtons second law, write it in one coordinate system and then work out what it would look like after a lorentz transformation. Presto, a Lorentz invariant equation. However, you will find a compensating term, which is the velocity of the coordinate frame w.r.t the original reference frame. The principle of Lorentz invariance is the requirement that this term NOT appear in the transformed equation.

This is an invariance principle, and it places constraints on the form of the original equation.

General covariance is different. There too you can bring any equation into a generally covariant form, and you will get extra Guvs and connection terms to compensate. However the principle does NOT require these terms to drop out. Instead, those terms are identified with gravitational field (and nothing else), and no restriction off the form of the initial equation is enforced. All it really says is that a generally covariant physical equation will be true in a gravitational field, if it is true in the absense thereof.

This is an example of something called a dynamic symmetry (like gauge invariance). They are not invariance principles like lorentz invariance.

 

[bcrowell]

Thanks, Thomas Larsson and Haelfix, for your helpful replies.

Instead, understanding diff invariance is crucial for the following reason. Classically, the physical phase space of GR is the unreduced phase space modulo diffeomorphisms, and quantically the physical Hilbert space is the unreduced Hilbert space modulo diffeomorphisms. The catch is that quantum representations must be of lowest-energy type, as always in quantum theory. Since people do not know how to construct (projective) lowest-energy representations of the diffeomorphism group, the Hilbert space of GR can not be constructed.

Since I'm not a specialist in quantum gravity, I'm trying to translate this into terms that I can understand. What does "unreduced" mean in this context?

When you say that the phase space of GR is the unreduced phase space modulo diffeomorphisms, what I'm imagining is something like the distinction between a real gravitational wave (which has a nonvanishing Riemann tensor, carries energy, etc.) and a coordinate wave. I'm assuming that the unreduced phase space would be some mathematical depiction of the physics in which the correct distinction between these is not being made. If so, then I'm having a hard time imaging what this depiction would be, since the only depiction I know is standard GR, and in standard GR, you don't have to remove the coordinate waves by hand -- they just automatically don't carry any observables like curvature or energy. Am I getting this right at all...?

I would like to understand the part about lowest-energy representations, but I'm not quite there yet. Is this referring to something like the need to have a lower bound on the energy spectrum? Or are there concerns about spurious modes appearing in the low-energy spectrum, where "low-energy" refers to phenomena in the classical limit, as opposed to the Planck scale? Since you're saying that "quantum representations must be of lowest-energy type, as always in quantum theory," could you give an example from a more elementary quantum-mechanical context? Is this something in quantum field theories only, or are you talking about something that would be relevant in nonrelativistic quantum mechanics as well?

In general, is it correct to say that the Kretschmann argument, which is controversial for classical field theories, becomes even more uncertain for quantized theories?

Thanks in advance for bearing with my ignorance :-)

 

[atyy]

FYI there is some source of dispute in the literature about whether General covariance is physically vacuous. Different people will explain it differently, and it basically boils down to definitions and interpretation.

The way I like it is expressed best by Weinberg in Gravitation. Paraphrasing:
Its worthwhile comparing lorentz invariance to General Covariance.

Take Newtons second law, write it in one coordinate system and then work out what it would look like after a lorentz transformation. Presto, a Lorentz invariant equation. However, you will find a compensating term, which is the velocity of the coordinate frame w.r.t the original reference frame. The principle of Lorentz invariance is the requirement that this term NOT appear in the transformed equation.

This is an invariance principle, and it places constraints on the form of the original equation.

General covariance is different. There too you can bring any equation into a generally covariant form, and you will get extra Guvs and connection terms to compensate. However the principle does NOT require these terms to drop out. Instead, those terms are identified with gravitational field (and nothing else), and no restriction off the form of the initial equation is enforced. All it really says is that a generally covariant physical equation will be true in a gravitational field, if it is true in the absense thereof.

This is an example of something called a dynamic symmetry (like gauge invariance). They are not invariance principles like lorentz invariance.

Weinberg actually does give and distinguish both usages. He states that general covariance "by itself is empty of physical content. Any equation can be made generally covariant ..."

What Haelfix has described above, Weinberg calls the "Principle of General Covariance" which is a particular version of the Equivalence Principle.

I guess Weinberg likens it to gauge invariance because both are misnamed assumptions of minimal coupling?