Why Quantum loop or string theory

[wolram]

I was wondering why Quantum loop theory is still being followed FAIK it does not predict any thing yet, the same for string theory which seems to have lost its predictability, if these theories fail being tested where will we be
It seems to me like super symmetry is the theory that every one seems to be the one that every one is hanging there hat on, yet even this is un testable yet .
so were are we if these three theories collapse.
I may not be up to date with these theoris so if i am wrong please correct me.

 

[jedishrfu]

I think we just don't know enough yet about these theories to either keep them or discard them. Also, until a new theory with greater promise comes along, we are stuck with them.

It may well be that the EP=EPR conjecture will inspire a new more comprehensive theory.

https://en.wikipedia.org/wiki/ER=EPR

and

https://www.quantamagazine.org/20150424-wormholes-entanglement-firewalls-er-epr/

 

[Chalnoth]

I was wondering why Quantum loop theory is still being followed FAIK it does not predict any thing yet, the same for string theory which seems to have lost its predictability, if these theories fail being tested where will we be
It seems to me like super symmetry is the theory that every one seems to be the one that every one is hanging there hat on, yet even this is un testable yet .
so were are we if these three theories collapse.
I may not be up to date with these theoris so if i am wrong please correct me.

Because General Relativity and Quantum Mechanics are fundamentally incompatible.

Some overarching theory which combines GR and QM must exist, but we don't yet know what that theory is. String Theory and Loop Quantum Gravity are two attempts to resolve the discrepancy.

String Theory comes more from the quantum side: it's an alternative formulation of quantum mechanics which is exciting because it predicts a quantum theory of gravity. It also predicts a large number of spatial dimensions, however. Some people also worry about the excessively large number of possible low-energy laws that could be realized in the theory.

Loop Quantum Gravity comes at the problem from the gravity side of the equation, attempting to produce a theory of gravity which acts like General Relativity in the appropriate limit, but is fundamentally quantum in nature. As far as I know, so far we don't know whether or not Loop Quantum Gravity actually does behave like General Relativity in the appropriate limit.

 

[andrew s 1905]

Some overarching theory which combines GR and QM must exist

Why is this the case? I mean do we have a theoretical or observational justification for believing it exists.
Regards Andrew

 

[Grinkle]

Why is this the case? I mean do we have a theoretical or observational justification for believing it exists.

We exist and we observe that at least locally we live in a continuous universe with no boundary points where one set of physical laws takes over for another set.

So there must be some way of describing our universe that does not stop at two theories that are not compatible with each other.

 

[Chalnoth]

Why is this the case? I mean do we have a theoretical or observational justification for believing it exists.
Regards Andrew

If it weren't the case, then you could use a gravitational measurement to get around the uncertainty principle of quantum mechanics. Furthermore, gravity just assumes a definite energy and position of the particles that make up the matter, while with quantum mechanics each particle will always be in a superposition of energy and/or position states.

 

[andrew s 1905]

We exist and we observe that at least locally we live in a continuous universe with no boundary points where one set of physical laws takes over for another set.

But locally we are not in a region that takes either theory to an extreme so we don't know.

If it weren't the case, then you could use a gravitational measurement to get around the uncertainty principle of quantum mechanics.

Can you explain how?

Thanks for your post but they just seem to restate the assertion that such a theory exists rather than demonstrate it must.

Regards Andrew

 

[Grinkle]

But locally we are not in a region that takes either theory to an extreme so we don't know.

A theory that contained a boundary and an explanation for how the boundary acts to transition from GR to QM and in that manner integrates GR with QM is an example of an overarching theory combining GR and QM.

So if you are speculating that maybe there is, in fact, such a boundary, you are hypothesizing the kind of theory you are asking for a reason to think might exist.

 

[Chalnoth]

Can you explain how?

Here's a little bit more detail. It comes down to the fact that source of gravity in General Relativity, the stress-energy tensor, has a well-defined value at every space-time point. The stress-energy tensor has components which include energy, momentum, pressure, and twisting forces. In principle, it's possible to use gravitational measurements to determine what the stress-energy tensor is. And since there's no uncertainty principle at work with the stress-energy tensor, you can, according to General Relativity, obtain this tensor to infinite accuracy given the right experimental apparatus.

But Quantum Mechanics says that's impossible: the stuff that makes up that stress-energy tensor won't have a well-defined position, energy, momentum, etc. This is the essence of the uncertainty principle: you can't measure a quantum system's position to infinite accuracy even with a perfect experimental apparatus, because a quantum system won't have a position which is perfectly-defined. It will be in a superposition of different positions.

So gravity says you can measure the position exactly. Quantum mechanics says you can't.

For even more detail, the standard way of dealing with this problem is to allow the field to be in a superposition as well. This is the way the electromagnetic field is reconciled with quantum mechanics, for example: the EM field is separated into discrete quanta, which we call photons. But when you do this in detail, there's a problem: the calculations result in a number of infinities. Those infinities come from doing integrals where the particles of the field have infinite energy. There's a relatively simple process for fixing this problem: cut off the integrals at a specific energy, replacing the part of the integral above the cutoff with experimental measurements. This is known as "renormalization".

The problem with gravity is that if you follow the same mathematical process, dividing the gravitational field into a bunch of gravitons, and then try to use renormalization to clean up the infinities, there are still infinitely-many parameters that need to be defined. And we can't exactly do an infinite number of experiments (or store their results in any computer system) to define these parameters.

This doesn't mean that finding the quantum theory of gravity is hopeless, just that we can't do it using the same techniques used to make the electromagnetic field into a quantum theory.

 

[andrew s 1905]

A theory that contained a boundary and an explanation for how the boundary acts to transition from GR to QM and in that manner integrates GR with QM is an example of an overarching theory combining GR and QM.

So if you are speculating that maybe there is, in fact, such a boundary, you are hypothesizing the kind of theory you are asking for a reason to think might exist.

I am not speculating on any theory (it is not allowed on this forum). I was pushing at your comment that we observe a continuous universe with no boundary points at least locally.

So gravity says you can measure the position exactly. Quantum mechanics says you can't.

You just seem to be restating that the two theories are not comparable and I agree with this. However, that is not an argument that a theory without this problem exists.

As far as I understand your positions such a theory must exist as observations must transition smoothly as one moves, from say, the every small to the very large and from low to high gravity etc. This, to me, is a conjecture not a proof but let's leave it here to avoid breaking the forum rules. Thanks for your contributions.

Regards Andrew

 

[Chalnoth]

You just seem to be restating that the two theories are not comparable and I agree with this. However, that is not an argument that a theory without this problem exists.

As far as I understand your positions such a theory must exist as observations must transition smoothly as one moves, from say, the every small to the very large and from low to high gravity etc. This, to me, is a conjecture not a proof but let's leave it here to avoid breaking the forum rules. Thanks for your contributions.

It's a bigger problem than that. The fundamental assumption here is that there exists a theory that accurately describes our universe without contradicting itself. A theory that contradicts itself is by nature unable to provide any predictions about regions/events where the contradictions arise. Furthermore, if the contradictions cannot be isolated, then it's impossible for the theory to ever provide any predictions, as any statement could be made to be true (see, for example, arithmetic where division by zero is allowed: if you permit division by zero, you can prove that 1=2, or indeed that any number equals any other number).

General Relativity alone contradicts itself, for example, as it predicts singularities. Such singularities are rather like the division by zero above. As long as those singularities are always hidden behind horizons (as in a black hole), then the theory remains stable outside of the horizon. But inside the horizon all bets are off. This is a strong indication that General Relativity isn't the correct theory of gravity in the strong-field limit.

 

[Grinkle]

I was pushing at your comment that we observe a continuous universe with no boundary points at least locally.

Yes, your first response made me realize that continuity is extraneous to my reasoning, I needn't have / shouldn't have included that in my argument. An observation of a discontinuity can as easily lead to a deeper more self-consistent theory as any other new observation.

 

[martinbn]

General Relativity alone contradicts itself, for example, as it predicts singularities. Such singularities are rather like the division by zero above. As long as those singularities are always hidden behind horizons (as in a black hole), then the theory remains stable outside of the horizon. But inside the horizon all bets are off. This is a strong indication that General Relativity isn't the correct theory of gravity in the strong-field limit.

I disagree with this statement. There are no contradictions in general relativity. It may be the case that the consequences do not describe reality, but there are no contradictions. Lornetzian geometry is as good as any other branch of mathematics. The singularities are just a feature, not a contradictions, and they are certainly not like division by zero.

 

[PeterDonis]

General Relativity alone contradicts itself, for example, as it predicts singularities.

This is not a contradiction in the mathematical sense of an inconsistency, as martinbn points out. It is just a mathematical feature of the theory that, in the opinion of most physicists, indicates that it breaks down as far as physical predictions are concerned in the regime close enough to the singularity.

The key point to keep in mind here is that the singularity itself--the "point" (I put that in quotes because it isn't really a point in the mathematical sense) where certain quantities are mathematically undefined (for example, scalar invariants at ##r = 0## in Schwarzschild spacetime)--is not actually part of spacetime. When we say a particular solution of the EFE "has a singularity", what we really mean is that it has certain properties that make physical interpretation problematic--for example, that there are geodesics which can't be extended to arbitrary values of their affine parameter, or that certain invariants increase without bound along such geodesics as some finite value of the affine parameter is approached (but never actually reached). None of these properties make the solution itself mathematically inconsistent or undefined.

 

[George Jones]

General Relativity alone contradicts itself, for example, as it predicts singularities.

It is not general relativity "alone". Perturbative quantum field theory (i.e., Feynman diagrams) also blows up, even after regularization/renormalization. Regularization makes each term in a QFT series finite, but, as first argued by Freeman Dyson more that sixty years ago, the series itself almost certainly diverges. A divergent asymptotic series can still be very predictive; see bottom of post for a mathematical example.

So perturbative quantum field theory blows up; what about non-perturbative quantum.field theory? General non-perturbative quantum field theory is very difficult (much more difficult than GR) and poorly understood. Because of this and other reasons, some respectable folks speculate that QFT is just an approximation to some more general scheme. From two recent bog-standard QFT texts:

The first quote comes "Quantum Field Theory for the Gifted Amateur" (terrible title) by Lancaster and Blundell. The second quote is from "Quantum Field theory and the Standard Model" by Schwartz.

Everything is up fro grabs, not just GR.

I agree that Einstein's equation with the energy-momentum tensor full of quantum sources suggests that quantum theory should apply to gravity, but, as Johnny Lee says, maybe we are "looking for love in all the wrong places."

---------------------------------------------------------------------------------------
Mathematical Example

Define
$$f\left( x \right) = \int_0^\infty \frac{e^{-t}}{1 + \frac{t}{x}} dt$$
This function has an asymptotic expansion series expansion
$$1 - x + \frac{2!}{x^2} - \frac{3!}{x^3} + ...$$
which is a divergent series. If the number of terms is fixed, however, the series does a nice job of approximating ##f\left( x \right)## for large enough ##x##. For example, take ##x = 100##. Then,
$$f\left( x \right) = 0.9901942...$$
The sum of the first four terms of the divergent series is 0.9901940..., which is very close. It takes several hundred terms to see that the series blows up for ##x=100##.

This is what is thought to happen in regularized/renormalized quantum field theory.

 

[Chalnoth]

It is not general relativity "alone". Perturbative quantum field theory (i.e., Feynman diagrams) also blows up, even after regularization/renormalization.

This issue with that is that it may just be a feature of the perturbation theory solution of the equations of motion. I don't think it's necessarily true that this represents a fundamental problem in quantum mechanics in the same way as the GR issue with singularities (which are independent of any particular mathematical approach to the equations). However, the fact that the standard model of quantum mechanics doesn't include gravity demonstrates that it can't be correct. The theory also very likely isn't accurate at energies much higher than those we've measured in particle accelerators.

This is not a contradiction in the mathematical sense of an inconsistency, as martinbn points out.

The stability of the theory can be recovered by removing the singularity from the mathematical description. If you include the singularity in the description, then I'm pretty sure you can use the singularity to prove any result from the theory, in a similar way that if you allow division by zero you can prove any integer equals any other integer.

So since General Relativity predicts singularities, but cannot describe a space-time that includes them, the theory is self-contradictory.

 

[PeterDonis]

If you include the singularity in the description, then I'm pretty sure you can use the singularity to prove any result from the theory

I don't think this is the case. If you have particular references that seem to indicate this, I would be very interested in seeing them. My understanding is that the singularities in GR pose no mathematical consistency issues at all, and certainly do not allow you to "prove anything" from the theory. The only issue is physical reasonableness.

since General Relativity predicts singularities, but cannot describe a space-time that includes them

As I posted before, when we say GR "predicts singularities", we do not mean that it predicts the existence of things that are supposed to be part of an actual physical spacetime but that cannot be consistently included in its mathematical description. That is a pop science distortion of what the theory actually says. Any discussion of this issue should be based on specific references that talk about what the theory actually says.

 

[martinbn]

This issue with that is that it may just be a feature of the perturbation theory solution of the equations of motion. I don't think it's necessarily true that this represents a fundamental problem in quantum mechanics in the same way as the GR issue with singularities (which are independent of any particular mathematical approach to the equations).

I think it is the other way around. General relativity is mathematically sound. Incomplete manifolds are a feature of Lorentzian geometry, not an inconsistency. In quantum field theory things are murky. At least as of now there isn't a mathematically consistent theory (as far as I am aware, I may be wrong though).

 

[Chalnoth]

I think it is the other way around. General relativity is mathematically sound. Incomplete manifolds are a feature of Lorentzian geometry, not an inconsistency. In quantum field theory things are murky. At least as of now there isn't a mathematically consistent theory (as far as I am aware, I may be wrong though).

It's sound only if you cut out parts of the space-time and don't try to describe what goes on there.

This means that General Relativity is imminently useful in describing our universe (since it seems to be extremely close to correct), but it still predicts regions it cannot describe.

 

[martinbn]

It's sound only if you cut out parts of the space-time and don't try to describe what goes on there.

No, you don't cut out anything. Those "regions" aren't there to begin with.
This means that General Relativity is imminently useful in describing our universe (since it seems to be extremely close to correct), but it still predicts regions it cannot describe.[/QUOTE]
It doesn't predict regions it cannot describe! It predicts that typical space-times will be geodesically incomplete.

 

[PeterDonis]

It's sound only if you cut out parts of the space-time and don't try to describe what goes on there.

Please produce a specific reference--textbook or peer-reviewed paper--that makes this claim.

 

[Chalnoth]

Please produce a specific reference--textbook or peer-reviewed paper--that makes this claim.

I don't get what the confusion is. This is just a description of geodesic incompleteness, which is a pretty generic prediction of GR. The link between geodesic incompleteness and metric incompleteness is also well-established.

Furthermore, a similar avoidance of singularities is what you have to do in math whenever infinities are encountered. For example, if you want to answer the question about what the value of:

[tex]f(x) = {x^2 - 1 \over x -1}[/tex]

...is at [itex]x = 1[/itex], then the answer is that this equation has no value at [itex]x = 1[/itex], but that we can take the limit as x approaches 1 and show that it converges nicely to 2 arbitrarily-close to the point (but not exactly at the point). This process of taking limits can be so ingrained among people that do this kind of calculation that many people behave as if the function whose limit has been taken actually has that value at that point, but there are some rather nasty mathematical consequences if that is taken seriously.

Another situation where we have somewhat similar problems would be in classical electrostatics. Consider the situation of a charged point particle. This point particle is represented by a delta function, which is infinite at the point and cannot be evaluated directly (but it can be integrated). It leads to a stable prediction of the electric field in all of space, except for the specific location of the particle. At that position, the electric field diverges and becomes infinite.

We usually don't take this to be a serious problem because the delta function is just an approximation: it's not intended to actually describe a real physical system, but instead describes an idealized model of one. The GR situation is different, because the point at which the curvature diverges isn't just an idealized model. It's an unambiguous prediction of the theory.

 

[Haelfix]

Singularities in physics can be rather subtle creatures. However the existence of a singularity does NOT necessarily signify a breakdown of the theory.

I find that it makes a lot of sense to to take the operational physical point of view. If I have a cup of water, and i approach a singularity like the one in the classical FRW universe under the assumption of classical general relativity, then I expect my cup of water to start heating up and that its properties will diverge as I try to take the limit. It doesn't matter if the point is technically part of spacetime, physical observables blow up there. So just like in the self energy problem of classical electromagnetism, I expect that there is a physical cutoff procedure (in GR's case, quantum gravity could provide this, in the classical e/m case, positron-electron polarization provides the natural cutoff).

On the other hand, you can have theories that have spacetime singularities, but that nevertheless have finite physical observables. One example is the physics of conifolds in string theory. Even though this is an unremovable singularity of the spacetime description, the subtle quantum mechanics of string theory implies that all physical observables (like scattering cross sections) must remain finite at that point.

 

[PeterDonis]

This is just a description of geodesic incompleteness

Geodesic incompleteness is not a mathematical inconsistency, it doesn't mean you have to "cut out parts of spacetime", and it certainly doesn't mean you can "prove anything" from the theory. Those are the claims I am asking you to back up with a reference.

This process of taking limits can be so ingrained among people that do this kind of calculation that many people behave as if the function whose limit has been taken actually has that value at that point

This certainly does not describe how the math of GR is properly done. When properly done, as I have already pointed out, GR does not say that a singularity is actually part of spacetime or that any function "actually has a value" at a singularity. In fact it claims the opposite.

 

[Chalnoth]

Geodesic incompleteness is not a mathematical inconsistency,

It's a logical inconsistency in the theory. The theory predicts regions which cannot be described by the theory. Often this is worded as, "General Relativity predicts its own demise."

it doesn't mean you have to "cut out parts of spacetime", and it certainly doesn't mean you can "prove anything" from the theory. Those are the claims I am asking you to back up with a reference.

I don't see why this is controversial. It's a standard component of the analysis of infinities.

This certainly does not describe how the math of GR is properly done. When properly done, as I have already pointed out, GR does not say that a singularity is actually part of spacetime or that any function "actually has a value" at a singularity. In fact it claims the opposite.

That's the point. The singularity is cut out of the space-time to avoid these problems. You could certainly try to write down a space-time which contained a singularity (and that singularity was actually a part of the space-time). The problem is that any calculation you did that was impacted by the singularity would become undoable. So the singularity is removed to keep the theory stable.

I'm really not understanding why any of this is at all controversial. I'm pretty sure that this process is described quite precisely in most any GR text, if not in the same words I used then in the same basic content.

 

[julcab12]

I don't see why this is controversial. It's a standard component of the analysis of infinities.

... Actually it is not. If you refer to the "singularities" associated with GR--they are the results of division by zero (non-zero mass, zero volume), which is a mathematical error (undefined operation). Singularity is not a place in spacetime as pointed by PeterD. Unless you invoke a pseudo-solution on EFE like the eternal maximally extended spacetimes and so on.

 

[PeterDonis]

Thread closed for moderation.