# Rationale behind gravitons

Haelfix
Incidentally, you might wonder? Couldn't gravity as a force, remain classical to all orders? Why do we need to quantize it in the first place?

The problem is we know the other 3 forces are in fact quantum, and lo and behold they show up in the stress energy tensor. So perhaps we could just treat Einsteins equation classically and replace this object with say, its expectation value <Tuv> (which now must live in a hilbert state of spaces and so forth).

But you immediately run into an obstruction. Solving for the metric and then using that to find an operator for the time evolution of states yields a catastrophe. The time evolution operator in question is nonlinear!

We don't know how to make sense of quantum mechanics with nonlinear modifications, all such theories that have ever been constructed have been failures. Evidently, we have to go about finding a sensible theory in a different way (insert your list of favorite quantum gravity proposals)

Haelfix
"Rederives with tensor fields? You lost me again: as far as i know, GR was originally formulated as a metric tensor field."

I'll say it in another terminology: Weinberg rederives all of GR in the weak field approximation or with mostly algebraic methods. He constructs the theory by symmetry arguments and the principle of equivalance, rather than positing geometric structure. The two formulations are mathematically isomorphic.

http://en.wikipedia.org/wiki/Linearized_gravity

"I can calculate colliding black holes using regge calculus just fine, no linearizations needed. "

You can, but you dont have too. It depends what you find easier to calculate with.

But you immediately run into an obstruction. Solving for the metric and then using that to find an operator for the time evolution of states yields a catastrophe. The time evolution operator in question is nonlinear!

In my humble opinion, that is an artifact of real analysis.

Evolving a Schrodinger equation over a metric/geometry produced by regge calculus, or some form of discrete differential geometry, works just fine, no complications at all. It is nonlinear as viewed through the wrong lens, but any equation can be made nonlinear by squaring it.

I really do have the feeling that most of the developments in modern physics are driven more by limitations of, and confusion over real analysis, than by any physical considerations.

My understanding of QED in terms of fields is most certainly limited.

By 'metric' i mean some mathematical object specifying a notion of distance.

How you could describe an EM field purely with a metric, by a deformation of spacetime, is completely beyond me. My naive understanding is that any field quantity lives on a manifold, having some (dynamic) metric. Any photonic or matter fields will influence the underlying metric by the presence of their energy, but the metric and the various fields are otherwise independent quantities, in my understanding.

When you say 'metric', do you mean that in some more abstract mathematical way (ie, the metric of some functional), or in a physical way: that which influences measurements of distance?

Either this is a confusion over terminology, or i really do not get QFT at all.

Sorry i confused you. I was saying that the metric was analgous to the EM field. Not that you could describe EM in terms of a metric. Actually to be acurate the metric tensor g_ab(x) is analgous to the potential A_a(x). Where a and b spacetime indices and take values 0,1,2 and 3 and by x i mean a point in spacetime. So these are both essentially fields. But when i do QTF i have to consider superpostion states such that there isn't just one metric or one potential. if i have some matter in the gravity case or a charge in the QED case then they may gain or lose momentum due to an interaction with the quantum superposition state of the metric or potential. Because momentum is conserved this must be an exchange of momentum.

Sorry i confused you. I was saying that the metric was analgous to the EM field. Not that you could describe EM in terms of a metric. Actually to be acurate the metric tensor g_ab(x) is analgous to the potential A_a(x). Where a and b spacetime indices and take values 0,1,2 and 3 and by x i mean a point in spacetime. So these are both essentially fields. But when i do QTF i have to consider superpostion states such that there isn't just one metric or one potential. if i have some matter in the gravity case or a charge in the QED case then they may gain or lose momentum due to an interaction with the quantum superposition state of the metric or potential. Because momentum is conserved this must be an exchange of momentum.

That sounds suspect to me.

How can you say the metric is analogous to other fields? Other fields are crucially dependent on the metric for their evolution. It defines the space in which the other quantities live.

To regard the metric as 'just another tensor field' seems conceptually borked to me. Even if such a unification pans out mathematically, have you physically done anything but confuse yourself?

atyy
That sounds suspect to me.

How can you say the metric is analogous to other fields? Other fields are crucially dependent on the metric for their evolution. It defines the space in which the other quantities live.

To regard the metric as 'just another tensor field' seems conceptually borked to me. Even if such a unification pans out mathematically, have you physically done anything but confuse yourself?

As a matter of fact, the metric needs the other fields to exist physically. The pure vacuum solutions of GR are undetectable - one always needs a test particle or test photon to see it. Test particles are contrary to the diffeomorphism invariance of GR.

That sounds suspect to me.

How can you say the metric is analogous to other fields? Other fields are crucially dependent on the metric for their evolution. It defines the space in which the other quantities live.

To regard the metric as 'just another tensor field' seems conceptually borked to me. Even if such a unification pans out mathematically, have you physically done anything but confuse yourself?

Other fields are dependent on the metric but the metric is also dependent on the other fields so it works both ways. This is true of classical field theory ie general relativity aswell. There's no confusion here. The metric tensor is a field because it is a function of spacetime g_ab(x) and it depends on the other fields via the einstein field equations.

As a matter of fact, the metric needs the other fields to exist physically. The pure vacuum solutions of GR are undetectable - one always needs a test particle or test photon to see it. Test particles are contrary to the diffeomorphism invariance of GR.

Yes, i realize the former. The latter statement makes no sense to me.

atyy
Yes, i realize the former. The latter statement makes no sense to me.

A test particle propagates on a fixed background.

Other fields are dependent on the metric but the metric is also dependent on the other fields so it works both ways.
Yes, i realize that, but these dependencies seem conceptually very different to me. Since when are a source term and a metric interchangable concepts?

This is true of classical field theory ie general relativity aswell. There's no confusion here. The metric tensor is a field because it is a function of spacetime g_ab(x) and it depends on the other fields via the einstein field equations.
Yeah, but they are mathematically and physically different dependencies. Space is space and matter is matter.

If gravitons seek to dissolve the distinction between space and matter, thats an ambitious goal, and im surprised i havnt seen it stated like that: ill believe it works when somone does a simulation involving gravitons, that doesnt depend on arguments such as 'yeah it reduces to the einstein field equations because of this general abstract nonsense, so actually, we are solving that instead. The linearized variant, yeah.'

A test particle propagates on a fixed background.

I understand such is customary in real analysis, yeah. That is a limitation of the mathematical tools you are using. Why does everyone insist on confusing that with something physical?

Yes, i realize that, but these dependencies seem conceptually very different to me. Since when are a source term and a metric interchangable concepts?

Yeah, but they are mathematically and physically different dependencies. Space is space and matter is matter.

If gravitons seek to dissolve the distinction between space and matter, thats an ambitious goal, and im surprised i havnt seen it stated like that: ill believe it works when somone does a simulation involving gravitons, that doesnt depend on arguments such as 'yeah it reduces to the einstein field equations because of this general abstract nonsense, so actually, we are solving that instead. The linearized variant, yeah.'

No your still confused. The metric tensor isn't spacetime. Its a function of spacetime. The metric has a value at each point in spacetime. The same goes for the EM potential. These are fields. What exists as absoloute concepts are the fields. We can make a general coordinate transform and change the spacetime coordinates so spacetime isn't an absolte concept.

Look both the EM field(U1 gauge field) and the gravitational field have geometic interpretations. Infact gravity is a gauge theory aswell. Yes gravity is a theory of the metric and therefore defines lengths and yes this leads to many conceptual and mathematical problems. But despite this you have to agree that the gravitational field created by a body A will transfer momentum to a body B. Momentum is consvered and according to general pricplies of QM comes in discrete packets. Therefore we can interprete the exchange of this momentum as a "particle". Buts its just an interpretation. Nobody starts off with the idea of a gravition and produces a quantum theory of gravity. It's just a useful concept when dealing with QM where quantities such as momentum do not take continous values and when also using relativity when means that momentum must travel between two points in spacetime ie there is some notion of propagation.

No your still confused. The metric tensor isn't spacetime. Its a function of spacetime. The metric has a value at each point in spacetime. The same goes for the EM potential. These are fields. What exists as absoloute concepts are the fields. We can make a general coordinate transform and change the spacetime coordinates so spacetime isn't an absolte concept.
You are arguing over real analysis, not over physics. In Regge calculus, id definitely say the metric is spacetime.

Look both the EM field(U1 gauge field) and the gravitational field have geometic interpretations. Infact gravity is a gauge theory aswell. Yes gravity is a theory of the metric and therefore defines lengths and yes this leads to many conceptual and mathematical problems. But despite this you have to agree that the gravitational field created by a body A will transfer momentum to a body B. Momentum is consvered and according to general pricplies of QM comes in discrete packets. Therefore we can interprete the exchange of this momentum as a "particle". Buts its just an interpretation. Nobody starts off with the idea of a gravition and produces a quantum theory of gravity. It's just a useful concept when dealing with QM where quantities such as momentum do not take continous values and when also using relativity when means that momentum must travel between two points in spacetime ie there is some notion of propagation.
I agree, a non-gravitonic spacetime seems hard to reconcile with discrete energy quanta.

That said: why should i care about conservation laws in anything but a time averaged sense, when wavefunction collapse does not either?

You are arguing over real analysis, not over physics. In Regge calculus, id definitely say the metric is spacetime.

I agree, a non-gravitonic spacetime seems hard to reconcile with discrete energy quanta.

That said: why should i care about conservation laws in anything but a time averaged sense, when wavefunction collapse does not either?

What is real analysis? the metric tensor g_ab defines a length ds^2 = dx^a dx^b g_ab(x). So it defined a infintessimal length ds in spacetime. Saying "the metric is spacetime" is totally meaningless.

Energy conservation is always obeyed in physics. Its just a common misconception that QM or the uncertainty principle does't conform to it.

atyy
Isn't Regge calc the motivation behind CDT?

CDT may be a computational version of either Asymptotic safety or Horava-Lifschitz - both of which have gravitons.

Haelfix
Regge calculus is about the earliest form of spacetime discretization that I am aware off that was also solutions of the field equations of GR. So yes, it is a precursor to dynamic triangulations, random triangulations, and so forth. Its heavily used in numerical approximations for hard problems in GR (the aforementioned black hole collisions for instance).

Later people tried to get it to work as a quantum gravity or quantum cosmology programs (not to be confused with the original intent). Like most such work, before CDT arrived, the problem was that all the various primordial simplexes would have a tendency to crumble up in numerical simulations and the classical flat limit was never achieved.

What is real analysis? the metric tensor g_ab defines a length ds^2 = dx^a dx^b g_ab(x). So it defined a infintessimal length ds in spacetime. Saying "the metric is spacetime" is totally meaningless.
Real analysis is most of mathematics, including the calculus of real variables you are talking about here.

Even if you implicitly assume a flat spacetime, you are assuming a metric. When you propose a function of three variables, you are implicltly assuming a metric. There is no spacetime without a metric.

Energy conservation is always obeyed in physics. Its just a common misconception that QM or the uncertainty principle does't conform to it.
Dunno, there are published papers on the subject.

My understanding: The expectation value of energy is conserved. Then your wavefunction collapses, at some arbitrary point, without further particle exchange. Does that state it collapses to not affects its energy?

Regge calc does not have gravitons.

The QG variants thereof might; depending on your interpretation. I dont mind thinking of space in terms of superpositions, and if youd want to call that gravitons, fine. My problem is with propagating the defining property of spacetime, over spacetime. How do you cut that knot? What do you start with? A flat spacetime is no less arbitrary than any other, and the only reason you are picking it, is because otherwise the real analysis gets too complicated.

tom.stoer
How do you formulate energy conservation in gravity theories?
- there is a locally conserverd energy momentum tensor in GR - fine
- if you enlarge your theoretical framework and introduce torsion, this conservation law vanishes
- I do not see how you can define a globally conserved energy (as a volume integral transforming as the zeroth component of a four vector)
- I do not see how you can define energy in QG theories (LQG, CDT, ...)

So we should restrict ourselves to talk about local symmetries; energy conservation may be a concept that works only in certain scenarios with appropriate symmetries, asymptotic conditions etc.

(is there an expert in this forum who can talk about quasi-local mass and things like that?)

Can someone explain the rationale behind gravitons to me?

My background is computational physics, and as such i may be biased towards physics that is actually computable, such as LQG and regge calc. I have some clue what this is all about, but i have some questions:

Is there any reason (beyond aestetics which i disagree with anyway) to favor a particle over a geometric explanation? Any sort of empirical matter gravitons may help explain?
We know that energy stored in a gravitational system can be converted to energy stored in other kinds of systems. all those other kinds of systems require that energy be quantized. If energy in a gravitational system were not quantized, then how could it smoothly flow into another type of system which accepted energy in packets? So this is why gravitational energy must be quantized.

How is something like gravitational lensing explained in a flat spacetime with gravitons? Are there force-carrier-to-force-carrier interactions in such a model? I have a hard time imagining how youd explain bending of light with gravitons. It seems likea pressing question to me, but noone else seems to care, as far as i can tell.

SR and GR, space-time, continuums, manifolds, dimensions, and Newtonian mechanics are descriptions of the large-scale behavior of many individual machines (particle interactions). You are right, there is no reasonable merging with the behavior of individual particle interactions for any of those large-scale theories. Scientists continue to erroneously presume theories developed solely to describe the average behavior of many simple machines will also be the founding theories in describing the behavior of each of those machines. There is no reason to believe that.

We know that energy stored in a gravitational system can be converted to energy stored in other kinds of systems. all those other kinds of systems require that energy be quantized. If energy in a gravitational system were not quantized, then how could it smoothly flow into another type of system which accepted energy in packets? So this is why gravitational energy must be quantized.
Yup, that makes sense to me.

Yet in general, I dont think it is a good thing to get too hung up on things like invariants, or even conservation laws. Yeah, they seem to hold. As far as we can tell, which is only to limited resolution.

If your model can explain all observations, it is good to me. To convince me it can, you need to actually compute stuff with it, and compare it side by side with observations. To me, juggeling mathematical theorems is a means to an end, not a goal in itself.

SR and GR, space-time, continuums, manifolds, dimensions, and Newtonian mechanics are descriptions of the large-scale behavior of many individual machines (particle interactions). You are right, there is no reasonable merging with the behavior of individual particle interactions for any of those large-scale theories. Scientists continue to erroneously presume theories developed solely to describe the average behavior of many simple machines will also be the founding theories in describing the behavior of each of those machines. There is no reason to believe that.

Yeah, we completely agree here.