[0805.3428] Microscopic and Macroscopic Behaviors of Palatini Modified Gravity Theories
Authors:  Baojiu Li, David F. Mota, Douglas J. Shaw 
Abstract:  We show that, within modified gravity, the nonlinear nature of the field equations implies that the usual naive averaging procedure (replacing the microscopic energymomentum by its cosmological average) is invalid. We discuss then how the averaging should be performed correctly and show that, as a consequence, at classical level the physical masses and geodesics of particles, cosmology and astrophysics in Palatini modified gravity theories are all indistinguishable from the results of general relativity plus a cosmological constant. Palatini gravity is however a different theory from general relativity and predicts different internal structures of particles from the latter. On the other hand, and in contrast to classical particles, the electromagnetic field permeates in the space, hence a different averaging procedure should be applied here. We show that in general Palatini gravity theories would then affect the propagation of photons, thus changing the behaviour of a Universe dominated by radiation. Finally, Palatini theories also predict alterations to particle physics laws. For example, it can lead to sensitive corrections to the hydrogen energy levels, the measurements of which could be used to place very strong constraints on the properties of viable Palatini gravity theories 
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[0805.3428] Microscopic and Macroscopic Behaviors of Palat
I found this paper intriguing. The authors argue that in the Palatini formulation of higherorder curvature theories with the matter treated as discrete clumps on small scales, the cosmology is actually no different from [tex]\Lambda[/tex]CDM, regardless of the gravity action. (Within the [tex]f(R, R^{\alpha\beta} R_{\alpha\beta})[/tex] class of theories.)
The auithors also argue that the situation is different for the electromagnetic field, since its energymomentum tensor is continuous. However, from a quantum mechanical point of view, I don't see a difference between photons and, say, the dark matter particles. Both are described by the excitations of a continuous field. (And even in GR, light can be interpreted as localised clumps in the geometric optics limit.)
This would mean that the large distance physics is completely different depending on whether the matter microscopically consists of infinitesimally small discrete particles stacked close together, or a real continuum, which seems rather odd.
This is a strong claim, it would be interesting to hear what people who do Palatini theories think about it.
The auithors also argue that the situation is different for the electromagnetic field, since its energymomentum tensor is continuous. However, from a quantum mechanical point of view, I don't see a difference between photons and, say, the dark matter particles. Both are described by the excitations of a continuous field. (And even in GR, light can be interpreted as localised clumps in the geometric optics limit.)
This would mean that the large distance physics is completely different depending on whether the matter microscopically consists of infinitesimally small discrete particles stacked close together, or a real continuum, which seems rather odd.
This is a strong claim, it would be interesting to hear what people who do Palatini theories think about it.

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[0805.3428] Microscopic and Macroscopic Behaviors of Palati
Hi Syksy,
Thanks for your interest in our paper. I agree that it is difficult to see what why dark matter and light should be seen to behave differently.
I think the first thing to clarify is that we don't require the particles to be infinitesimally small, just much smaller than the scale you are interested in. Secondly, we're not claiming that the Palatini modification would have no effect on the internal structure of microscopic clumps of dark or normal matter (in fact with respect to the hydrogen atom we show that the opposite is true).
What we are claiming is that classically a massive lump of energy / matter (which we call a particle), when treated as a whole moves in the same way as a massive classical lump of matter would in GR. The situation with a lump of photons is slightly different. This is because in GR the lump would be massless, whereas in the Palatini it would in general (but not in the special case [tex]f=f(R)[/tex]) develop an effective mass, and so it would be observed to move like a massive `particle' rather than a massless one.
Put differently, say a particle as a mass, [tex]m_{p}[/tex], appearing its Lagrangian. We found that in Palatini theories the physical mass of the particle (i.e. its inertial and gravitational mass) is not [tex]m_{p}[/tex], but something different, let's call it [tex]\hat{m}_{p}[/tex]. Now in every classical sense the mass of the particle is [tex]\hat{m}_{p}[/tex]. Thus without examine the inertial structure of the particle, one can't measure [tex]m_{p}[/tex], and hence cannot tell that [tex]\hat{m}_{p} \neq m_{p}[/tex].
Similarly, the Palatini modification can be seen as introducing an effective mass to a lump of photon field. In this case however you know that [tex]m_{p} = 0[/tex], so if you see it moving with [tex]\hat{m}_{p} \neq 0[/tex] you know something is up, and this time one might therefore be able to detect the Palatini modification to gravity.
Just to clarify the basic idea behind the paper a little further. In a general modified gravity theories will introduce extra forces and this is also the case for Palatini gravity. But because of the algebraic nature of the theory, this extra force is of zero range (acting on points). As such whether the matter is continuous microscopically is very relevant, because if two particles are separated by a nonzero distance then the extra forces from them cannot reach each other.
Now if you use quantum mechanics, then inside the particles there are fields governed by QCD or whatsoever, so the similar analysis as for a EM field will be used there, and this is why it is claimed that the internal structure of a particle in Palatini theory would be different from that in GR. But the particle as a whole (regardless of its internal structure) can always be treated as a clump of energy classically; and because in between different particles it is vacuum, so there is no extra forces between these particles.
Also in most astrophysical and cosmological environments, only a very tiny portion of the total volume is occupied by particles, so after volume average the averaged value of a physical quantity is dominated by its vacuum value. This is why macroscopically the theory behaves like LCDM.
In GR, when [tex]GM/R \ll 1[/tex], the gravitational field depends linearly on the local matter density, so the average gravitational field is essentially the same however the matter is distributed.
One should not be surprised that large distance physics depends on whether matter consists of lumps rather than a precise continuum. This behavior is of course even seem in GR where the average cosmological evolution of a perfectly homogeneous Universe is different (at some level) from that of a Universe is only statistically homogeneous but is actually fairly lumpy below some scale. The only difference is the scale that the lumpiness has to be at before it proceeds a noticeable different to the evolution. Palatini theories can be seen as the infinite coupling, infinite mass limit of a BransDicke theory, so one has a very strong force acting on points, and so even a microscopic lumpiness in the distribution of energy can be important.
I realise I've rather rambled on there, so I hope that had been sufficient to clarify what we say is going on in these Palatini theories.
Thanks for your interest in our paper. I agree that it is difficult to see what why dark matter and light should be seen to behave differently.
I think the first thing to clarify is that we don't require the particles to be infinitesimally small, just much smaller than the scale you are interested in. Secondly, we're not claiming that the Palatini modification would have no effect on the internal structure of microscopic clumps of dark or normal matter (in fact with respect to the hydrogen atom we show that the opposite is true).
What we are claiming is that classically a massive lump of energy / matter (which we call a particle), when treated as a whole moves in the same way as a massive classical lump of matter would in GR. The situation with a lump of photons is slightly different. This is because in GR the lump would be massless, whereas in the Palatini it would in general (but not in the special case [tex]f=f(R)[/tex]) develop an effective mass, and so it would be observed to move like a massive `particle' rather than a massless one.
Put differently, say a particle as a mass, [tex]m_{p}[/tex], appearing its Lagrangian. We found that in Palatini theories the physical mass of the particle (i.e. its inertial and gravitational mass) is not [tex]m_{p}[/tex], but something different, let's call it [tex]\hat{m}_{p}[/tex]. Now in every classical sense the mass of the particle is [tex]\hat{m}_{p}[/tex]. Thus without examine the inertial structure of the particle, one can't measure [tex]m_{p}[/tex], and hence cannot tell that [tex]\hat{m}_{p} \neq m_{p}[/tex].
Similarly, the Palatini modification can be seen as introducing an effective mass to a lump of photon field. In this case however you know that [tex]m_{p} = 0[/tex], so if you see it moving with [tex]\hat{m}_{p} \neq 0[/tex] you know something is up, and this time one might therefore be able to detect the Palatini modification to gravity.
Just to clarify the basic idea behind the paper a little further. In a general modified gravity theories will introduce extra forces and this is also the case for Palatini gravity. But because of the algebraic nature of the theory, this extra force is of zero range (acting on points). As such whether the matter is continuous microscopically is very relevant, because if two particles are separated by a nonzero distance then the extra forces from them cannot reach each other.
Now if you use quantum mechanics, then inside the particles there are fields governed by QCD or whatsoever, so the similar analysis as for a EM field will be used there, and this is why it is claimed that the internal structure of a particle in Palatini theory would be different from that in GR. But the particle as a whole (regardless of its internal structure) can always be treated as a clump of energy classically; and because in between different particles it is vacuum, so there is no extra forces between these particles.
Also in most astrophysical and cosmological environments, only a very tiny portion of the total volume is occupied by particles, so after volume average the averaged value of a physical quantity is dominated by its vacuum value. This is why macroscopically the theory behaves like LCDM.
In GR, when [tex]GM/R \ll 1[/tex], the gravitational field depends linearly on the local matter density, so the average gravitational field is essentially the same however the matter is distributed.
One should not be surprised that large distance physics depends on whether matter consists of lumps rather than a precise continuum. This behavior is of course even seem in GR where the average cosmological evolution of a perfectly homogeneous Universe is different (at some level) from that of a Universe is only statistically homogeneous but is actually fairly lumpy below some scale. The only difference is the scale that the lumpiness has to be at before it proceeds a noticeable different to the evolution. Palatini theories can be seen as the infinite coupling, infinite mass limit of a BransDicke theory, so one has a very strong force acting on points, and so even a microscopic lumpiness in the distribution of energy can be important.
I realise I've rather rambled on there, so I hope that had been sufficient to clarify what we say is going on in these Palatini theories.

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Re: [0805.3428] Microscopic and Macroscopic Behaviors of Pa
Thanks for the clarifications.
In usual GR, this does not matter, because the gravitational field of a clump of zero size and a clump which has infinite extent but is highly peaked in some small region is essentially the same. Here it does not seem obvious that this is the case.
A more appropriate analogy would be to compare two systems with some perturbations, one where the matter is continuous and another where it is discrete. In this case, there is in general a difference. For example, in linear perturbation theory, the modes in the discrete system on average grow slower than in the fluid case. (See astroph/0504213, astroph/0601479.) However, the effect disappears for modes with wavelengths much larger than the lattice spacing, so the large scale evolution is not affected.
OK, this explanation I understand better than the one about continuum vs. discrete particles. (Even in GR, light does not in general move on null geodesics. That is only true in the geometric optics limit, it's a question of how the wavelength compares to the curvature scale.)Douglas Shaw wrote:The situation with a lump of photons is slightly different. This is because in GR the lump would be massless, whereas in the Palatini it would in general (but not in the special case [tex]f=f(R)[/tex]) develop an effective mass, and so it would be observed to move like a massive `particle' rather than a massless one.
The issue of internal structure is distinct from the quantum mechanical nature of particles. In QM, neither photons nor dark matter particles are strictly localised clumps: they have a wavefunction with some spread, so there is strictly speaking no "empty space". (This quite aside from the issue of the quantum field theory vacuum structure  which I guess will also be affected by the new gravitational terms.)Douglas Shaw wrote:Just to clarify the basic idea behind the paper a little further. In a general modified gravity theories will introduce extra forces and this is also the case for Palatini gravity. But because of the algebraic nature of the theory, this extra force is of zero range (acting on points). As such whether the matter is continuous microscopically is very relevant, because if two particles are separated by a nonzero distance then the extra forces from them cannot reach each other.
Now if you use quantum mechanics, then inside the particles there are fields governed by QCD or whatsoever, so the similar analysis as for a EM field will be used there, and this is why it is claimed that the internal structure of a particle in Palatini theory would be different from that in GR. But the particle as a whole (regardless of its internal structure) can always be treated as a clump of energy classically; and because in between different particles it is vacuum, so there is no extra forces between these particles.
In usual GR, this does not matter, because the gravitational field of a clump of zero size and a clump which has infinite extent but is highly peaked in some small region is essentially the same. Here it does not seem obvious that this is the case.
This is not an analogous situation. In this case, you are comparing the evolution of a perfectly smooth medium and a medium which is clumpy. This difference can be expressed in terms of the overall lumpiness of the system (specifically, the variance of the expansion rate and the shear). This is of course determined by the smallscale evolution, but there is a smooth limit between having only a little inhomogeneity and having none at all. (And this is not sensitive to the microscopic details.)Douglas Shaw wrote:One should not be surprised that large distance physics depends on whether matter consists of lumps rather than a precise continuum. This behavior is of course even seen in GR where the average cosmological evolution of a perfectly homogeneous Universe is different (at some level) from that of a Universe is only statistically homogeneous but is actually fairly lumpy below some scale.
A more appropriate analogy would be to compare two systems with some perturbations, one where the matter is continuous and another where it is discrete. In this case, there is in general a difference. For example, in linear perturbation theory, the modes in the discrete system on average grow slower than in the fluid case. (See astroph/0504213, astroph/0601479.) However, the effect disappears for modes with wavelengths much larger than the lattice spacing, so the large scale evolution is not affected.

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[0805.3428] Microscopic and Macroscopic Behaviors of Palati
Thanks for your reply.
Just consider what would have to happen for this approximation to be wrong. Cosmologically < proton per every cubic metre. So we know that the proton is not a localized clump of matter but say its energy density has gaussian distribution with width ~ 1/proton mass, and so some exponentially minute faction of the energy density in such a proton is spread over distances > 1m. Now for the relative motions of two protons (seperated by a few metres say) in the Palatini theories to actually be different from GR (in a significant way), one would have to require that the motion of proton one a few metres away from proton two could be affected in a significant way by the Palatini force caused by tiny gradients in that tiny fraction of the energy density of proton two that is spread over that distance, and so the cosmology would look different from GR. Whilst this may happen in particularly pathological theories, I think it was reasonable of us to assume that this is not the case in any theory vaguely connected with reality, and approximate matter as being in classical lumps.
At any rate, I would say that the homogeneous fluid (or even an almost inhomogeneous fluid) approximation for cosmology matter is certainly much further away from reality that the classical lump approximation we use. These lumps of matter are not assumed to have zero extent, but are assumed to have an extent that is to all intents and purposes essentially finite and much smaller than the interparticle separation.
Precisely the same thing is happening with these Palatini theories. Here the way in which Palatini gravity deviates from GR is governed by a algebraic field equation that depends on the local density of matter and not the spatial extent of it. This equation links the R in the f(R) to the trace of the energy momentum tensor. To the extent that this field equation is linear GR and Palatini are the same, so all the deviations are encoded in the nonlinear terms. As a result deviations of Palatini from GR are at all scales determined by a nonlinear algebraic equation which depends on the local energy density. No matter how big your clumps of matter are, if you average the local energy density, you will therefore get the wrong answer. It's because of this nonlinearity one all scales that the distribution of matter on microscopic scales matter.
In GR this doesn't occur as if you go to small enough scales everything linearizes nicely, but on the scales where the nonlinearities become important then averaging breaks down and you have to be more careful.
It was in that sense that I made the analogy.
The key point though is that in all Palatini theories the effective extra force you introduce (which is essentially the modification to gravity) has zero range. So if you have two particles whose energy densities (to a good approximation) do not overlap, the would not feel any extra force between them. As such their relative motions are unaffected by this extra force. As it well known Palatini theories are such GR with a modified source. The modified source means that you may (and do) alter electron energy levels in atoms (since effective electron mass now depends on the local electron density) or modify nuclear binding energies or other such things. But independent of the internal distribution of energy and momentum in a microscopic particle (or clump of energy momentum), in GR all such clumps move in essentially the same way. If no extra long range forces are present then the particle such move as classical particles under gravity, and so cosmologically they look to a good approximation like dust or as a better approximation collisionless matter.
To change this state of affairs one would have to introduce a long range force, or at the very least a force with nonzero range, and in Palatini theories you don't have one. In metric [tex]f(R)[/tex] theories, there is such a force, which is why they are so tightly constrained by fifth force experiments.
It should be pointed out however that one would still get a cosmological evolution that looks like something one would predict from Palatini with a perfectly homogeneous matter distribution, but microscopically the theory would not be a Palatini theory. It could for instance be a strongly (but not too strongly) coupled chameleon theory (which provided the mass is >> H cosmology would behave like a Palatini theory). These theories do however have 5th forces, and so the possible cosmological evolutions are generally constrained by experiments to be pretty damn close to Lambda CDM.
No debating that, but at least cosmologically light from distant objects moves on what may well be approximated as null geodesics. In a Palatini [tex]f(R_{ab}R^{ab})[/tex] theory even this would not be what was observed (although one could always choose the theory so it was approximately the case). That's all we are saying there.(Even in GR, light does not in general move on null geodesics. That is only true in the geometric optics limit, it's a question of how the wavelength compares to the curvature scale.)
It's true that the classical description of particles separated by vacuum is not strictly accurate, and is an idealization of the more realistic situation where one would quasilocalized clumps of energy separated by large regions of almost empty space (ignoring issues with things popping in and out of the quantum vacuum). This said we think that for many situations of interest, not least cosmology after the radiation the classical particle description is pretty good and certainly much closer to the fluid description that is standardly used.The issue of internal structure is distinct from the quantum mechanical nature of particles. In QM, neither photons nor dark matter particles are strictly localized clumps: they have a wavefunction with some spread, so there is strictly speaking no "empty space".
Just consider what would have to happen for this approximation to be wrong. Cosmologically < proton per every cubic metre. So we know that the proton is not a localized clump of matter but say its energy density has gaussian distribution with width ~ 1/proton mass, and so some exponentially minute faction of the energy density in such a proton is spread over distances > 1m. Now for the relative motions of two protons (seperated by a few metres say) in the Palatini theories to actually be different from GR (in a significant way), one would have to require that the motion of proton one a few metres away from proton two could be affected in a significant way by the Palatini force caused by tiny gradients in that tiny fraction of the energy density of proton two that is spread over that distance, and so the cosmology would look different from GR. Whilst this may happen in particularly pathological theories, I think it was reasonable of us to assume that this is not the case in any theory vaguely connected with reality, and approximate matter as being in classical lumps.
At any rate, I would say that the homogeneous fluid (or even an almost inhomogeneous fluid) approximation for cosmology matter is certainly much further away from reality that the classical lump approximation we use. These lumps of matter are not assumed to have zero extent, but are assumed to have an extent that is to all intents and purposes essentially finite and much smaller than the interparticle separation.
In making the analogy, I was only trying to say this: In GR the evolution of a system with one scale [tex]L_{1}[/tex] can be sensitive to the distribution of matter on scales [tex]L_{2} \ll L_{1}[/tex] if this distribution is such that the nonlinearity in the GR field equations is important on scales [tex]\sim L_{2}[/tex]. If this is the case then solving the Einstein equations with an averaged energy momentum tensor would give a different answer to solving the metric w.r.t. the microscopic (that is over scales < [tex]L_2[/tex]) energy momentum tensor and then averaging the metric.This is not an analogous situation. In this case, you are comparing the evolution of a perfectly smooth medium and a medium which is clumpy. .... (And this is not sensitive to the microscopic details.)
Precisely the same thing is happening with these Palatini theories. Here the way in which Palatini gravity deviates from GR is governed by a algebraic field equation that depends on the local density of matter and not the spatial extent of it. This equation links the R in the f(R) to the trace of the energy momentum tensor. To the extent that this field equation is linear GR and Palatini are the same, so all the deviations are encoded in the nonlinear terms. As a result deviations of Palatini from GR are at all scales determined by a nonlinear algebraic equation which depends on the local energy density. No matter how big your clumps of matter are, if you average the local energy density, you will therefore get the wrong answer. It's because of this nonlinearity one all scales that the distribution of matter on microscopic scales matter.
In GR this doesn't occur as if you go to small enough scales everything linearizes nicely, but on the scales where the nonlinearities become important then averaging breaks down and you have to be more careful.
It was in that sense that I made the analogy.
The key point though is that in all Palatini theories the effective extra force you introduce (which is essentially the modification to gravity) has zero range. So if you have two particles whose energy densities (to a good approximation) do not overlap, the would not feel any extra force between them. As such their relative motions are unaffected by this extra force. As it well known Palatini theories are such GR with a modified source. The modified source means that you may (and do) alter electron energy levels in atoms (since effective electron mass now depends on the local electron density) or modify nuclear binding energies or other such things. But independent of the internal distribution of energy and momentum in a microscopic particle (or clump of energy momentum), in GR all such clumps move in essentially the same way. If no extra long range forces are present then the particle such move as classical particles under gravity, and so cosmologically they look to a good approximation like dust or as a better approximation collisionless matter.
To change this state of affairs one would have to introduce a long range force, or at the very least a force with nonzero range, and in Palatini theories you don't have one. In metric [tex]f(R)[/tex] theories, there is such a force, which is why they are so tightly constrained by fifth force experiments.
It should be pointed out however that one would still get a cosmological evolution that looks like something one would predict from Palatini with a perfectly homogeneous matter distribution, but microscopically the theory would not be a Palatini theory. It could for instance be a strongly (but not too strongly) coupled chameleon theory (which provided the mass is >> H cosmology would behave like a Palatini theory). These theories do however have 5th forces, and so the possible cosmological evolutions are generally constrained by experiments to be pretty damn close to Lambda CDM.
Re: [0805.3428] Microscopic and Macroscopic Behaviors of Pa
Even in GR, if one zerosize particle lies somewhere in the cloud of another spreading particle, then the force felt by the former is not due to the whole mass of the spreading particle.Syksy Rasanen wrote:
In usual GR, this does not matter, because the gravitational field of a clump of zero size and a clump which has infinite extent but is highly peaked in some small region is essentially the same. Here it does not seem obvious that this is the case.
Here to a good approximation the vast majority of the energy of a particle lies within a Compton wavelengh, which is in most realistic cases much smaller than interparticle distances. Thus even there is some overlap of energy distributions of two particles according to QM, the energy density in this overlapping region must be extremely small. And because the strength of the zerorange extra force in Palatini theory depends on the trace of energy momentum tensor, this force should be negligible for any practical purpose.

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Re: [0805.3428] Microscopic and Macroscopic Behaviors of Pa
Right, I just mentioned the null geodesic thing to illustrate why I can understand the explanation for the different behaviour of light better...Douglas Shaw wrote:No debating that, but at least cosmologically light from distant objects moves on what may well be approximated as null geodesics. In a Palatini [tex]f(R_{ab}R^{ab})[/tex] theory even this would not be what was observed (although one could always choose the theory so it was approximately the case). That's all we are saying there.
I don't disagree. (Though one can note that we don't know what the dark matter is, and how analogous to protons it is  dark matter might for example be an axion condensate.)Douglas Shaw wrote:It's true that the classical description of particles separated by vacuum is not strictly accurate, and is an idealization of the more realistic situation where one would quasilocalized clumps of energy separated by large regions of almost empty space (ignoring issues with things popping in and out of the quantum vacuum). This said we think that for many situations of interest, not least cosmology after the radiation the classical particle description is pretty good and certainly much closer to the fluid description that is standardly used.
Just consider what would have to happen for this approximation to be wrong.
However, in the paper the argument about the point interaction makes it seem (at least to me) that there is a qualitative difference between a continuum and discrete particles. My point was that there is no such qualitative difference. It is a question of scales: if the interparticle separation is of the order of the spread of the wavepacket of individual particles, it should not be possible to make a distinction between a continuum and a set of discrete particles from largescale observables.
As an aside, the effect of clumpiness on the average expansion would persist even if the Einstein equation was linear, as long as averaging and time evolution do not commute.Douglas Shaw wrote:In making the analogy, I was only trying to say this: In GR the evolution of a system with one scale [tex]L_{1}[/tex] can be sensitive to the distribution of matter on scales [tex]L_{2} \ll L_{1}[/tex] if this distribution is such that the nonlinearity in the GR field equations is important on scales [tex]\sim L_{2}[/tex].
What do you mean by averaging the metric? I don't think there is any wellunderstood formalism for averaging the metric, I guess the closest would be the BuchertCarfora use of the Ricci flow (see 0801.0553). Even conceptually, I find the matter very unclear.Douglas Shaw wrote:If this is the case then solving the Einstein equations with an averaged energy momentum tensor would give a different answer to solving the metric w.r.t. the microscopic (that is over scales < [tex]L_2[/tex]) energy momentum tensor and then averaging the metric.

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Re: [0805.3428] Microscopic and Macroscopic Behaviors of Pa
I don't disagree with this. It is then a quantitative question of whether something is treated as a continuum or not, which makes sense physically. (Perhaps this was the idea in the paper as well, and I missed it!)Baojiu Li wrote:Here to a good approximation the vast majority of the energy of a particle lies within a Compton wavelengh, which is in most realistic cases much smaller than interparticle distances. Thus even there is some overlap of energy distributions of two particles according to QM, the energy density in this overlapping region must be extremely small. And because the strength of the zerorange extra force in Palatini theory depends on the trace of energy momentum tensor, this force should be negligible for any practical purpose.
Re: [0805.3428] Microscopic and Macroscopic Behaviors of Pa
I don't disagree with this comment. It might be that the expression is a little vague. However, as we have emphasized several times in the paper, the interparticle distance is actually important because it determines the output the our volume averaging; and if you work in extremely high density environments like a neutron star, then the socalled naive averaging may work well because there is essentially no space between particles.Syksy Rasanen wrote:
However, in the paper the argument about the point interaction makes it seem (at least to me) that there is a qualitative difference between a continuum and discrete particles. My point was that there is no such qualitative difference. It is a question of scales: if the interparticle separation is of the order of the spread of the wavepacket of individual particles, it should not be possible to make a distinction between a continuum and a set of discrete particles from largescale observables.
I agree.Syksy Rasanen wrote: It is then a quantitative question of whether something is treated as a continuum or not, which makes sense physically. (Perhaps this was the idea in the paper as well, and I missed it!)