Unsimultaneity and quantum physics

[exponent137]

Distinction between general relativity and quantum mechanics is also, that QM time is absolute, but GR time is relative.

But, special relativity also contains unsimultaneity, but SR is quantized.
How it is solved the problem of unsimultaneity, or it cause some problems?

I know Dirac equation, but I do not find how it treats and solves the problem of unsimultaneity.

 

[my2cts]

Dirac's theory fully complies with special relativity so there is no issue.

 

[exponent137]

Dirac's theory fully complies with special relativity
Yes, i know, but how it can be visualized how relativistic quantum theory deals with unsimultaneity?

I suppose that Klein-Gordon equation is easier for visualization of this than Dirac equation.

 

[bhobba]

Yes, i know, but how it can be visualized how relativistic quantum theory deals with unsimultaneity?

It doesn't - or rather it sidesteps the issue by treating everything as a field. That's why we need QFT where both time and position are just parameters.

There is a fundamental issue with QM and relativity. Time is a parameter, and position an observable - but relativity treats them the same. To get around it you either need to promote time to an operator or demote position to a parameter. Time as operator leads to severe mathematical difficulties, but position as a parameter gives QFT.

Thanks
Bill

 

[my2cts]

It doesn't - or rather it sidesteps the issue by treating everything as a field. That's why we need QFT where both time and position are just parameters.

There is a fundamental issue with QM and relativity. Time is a parameter, and position an observable - but relativity treats them the same. To get around it you either need to promote time to an operator or demote position to a parameter. Time as operator leads to severe mathematical difficulties, but position as a parameter gives QFT.

Thanks
Bill

I am not sure, this may be a separate issue.
The Dirac equation and the Klein-Gordon equations are Lorentz covariant and have no issue with simultaneity.

 

[bhobba]

I am not sure, this may be a separate issue.
The Dirac equation and the Klein-Gordon equations are Lorentz covariant and have no issue with simultaneity.

The Klein Gordon Equation has problems with negative probabilities:
http://www.phy.ohiou.edu/~elster/lectures/advqm_3.pdf

The Dirac Equation had problems with negative energies - the hole theory only partly solved the issues (it created others worse eg what about the charge of these negative energy electrons in the infinite sea):
http://www.phy.ohiou.edu/~elster/lectures/advqm_4.pdf

These are all fixed in QFT.

Thanks
Bill

 

[my2cts]

The Klein Gordon Equation has problems with negative probabilities:
http://www.phy.ohiou.edu/~elster/lectures/advqm_3.pdf

The Dirac Equation with negative energies - the hole theory only partly solved the issues:
http://www.phy.ohiou.edu/~elster/lectures/advqm_4.pdf

These are all fixed in QFT.

Thanks
Bill

I know but how are these issues connected with simultaneity?
By the way, in KG theory there is no probability density.
psi^2 is just charge density which is not of definite sign.

 

[bhobba]

I know but how are these issues connected with simultaneity?
By the way, in KG theory there is no probability density.
psi^2 is just charge density which is not of definite sign.

Lack of simultaneity is a symptom that we need space-time not space and an absolute time. Space and time must be treated on the same footing - no way out. I don't know what more I can really say. If you don't do it you get symptoms like I mentioned - one must second quantisize the field to remove them ie go to QFT.

Cant follow your comment about KG - what I said is well known eg from the paper I linked to:
This means that the Klein-Gordon equation allows negative energies as solution. Formally,
one can see that from the square of Eq. (2.60) the information about the sign is lost.
However, when starting from Eq. (2.71) all solutions have to be considered, and there is
the problem of the physical interpretation of negative energies.

Thanks
Bill

 

[WannabeNewton]

By the way, in KG theory there is no probability density.

Eh? In Relativistic QM, solutions $\varphi(x)$ to the KG equation are particle states and the probability 4-current is given by $j^{\mu} = i[(\partial^{\mu}\varphi )\varphi^{\dagger}- (\partial^{\mu}\varphi^{\dagger})\varphi]$ where $\int i[(\partial^{0}\varphi)\varphi^{\dagger} - (\partial^{0}\varphi^{\dagger})\varphi]d^{3}x = 1$ (in contrast to $\int \varphi^{\dagger}\varphi d^{3}x = 1$ in non-relativistic QM). The current conservation $\partial^{\mu}j_{\mu} = 0$ ensures that $\int j^0 d^3 x = 1$ is a Lorenz scalar.

Reinterpreting $j^0$ as charge density arises in QFT where $\varphi(x)$ are fields instead of particle states.

But I also do not see how relativity of simultaneity is an issue here...

 

[my2cts]

Multiply that with e and you have the charge current. It is the Noether current associated with invariance under global phase transition. Note that charge current is an observable while probability is not.

 

[Bill_K]

The Dirac equation and the Klein-Gordon equations are Lorentz covariant and have no issue with simultaneity.

The problem arises in the context of first quantization (where you try to use ψ as a single particle wavefunction) and include an interaction term. Solutions to the inhomogeneous Dirac and Klein-Gordon equations are nonzero outside the light cone. It's only in QFT, with reinterpretation of the negative energy states as positive energy antiparticles, that this causality violation cancels.

 

[exponent137]

One of the biggest obstacles has been that general relativity and quantum mechanics treat time very differently. In the former theory, time is another dimension alongside space and can bend and stretch, speed up and slow down, in different circumstances. Quantum theories, however, usually assume that time is set apart from space and ticks at a set rate.

one of the key paradigms of quantum and classical mechanics: the paradigm of a state evolving in time,

It is clear that this different time is a problem in merging general relativity and quantum theory. But, why it is not a problem at merging of special relativity and quantum theory? One speciality of special relativity is unsimultaneity? Why it is not a problem at quantum theory? How it is with evolution of wave function is special relativity, where different locations means different time, if a rocket is moving?

 

[WannabeNewton]

Again, in relativistic QM as well as QFT the meta-theory of space-time structure is that of special relativity and not Galilean relativity.

 

[Bill_K]

It is clear that this different time is a problem in merging general relativity and quantum theory. But, why it is not a problem at merging of special relativity and quantum theory? One speciality of special relativity is unsimultaneity? Why it is not a problem at quantum theory?

No, I don't think it is. Instead of regarding the state ψ as a function of hyperplanes t = const, you regard it as a functional on a spacelike hypersurface. In place of ∂/∂t, you use the functional derivative δ/δτ that varies the surface.

There is no essential problem, as you seem to think there is, in doing quantum mechanics on a classical curved background. This is the way Hawking radiation is derived. The problem in merging QM with GR comes when you allow the background to be a further dynamical degree of freedom.

 

[exponent137]

Again, in relativistic QM as well as QFT the meta-theory of space-time structure is that of special relativity and not Galilean relativity.
Yes, but different time in general relativity is problem for quantum theory, but time in special relativity is not a problem for quantum theory. Where is this key difference between special and general relativity? Yes, curved space-time is this difference, but why such impact on quantum theory of this difference?

 

[WannabeNewton]

I think you're sort of confusing two aspects here. We can do QFT on a flat background just fine; this background is Minkowski space-time and the meta-theory is SR. We can also do QFT on a curved background just fine wherein the meta-theory is GR. The "problem" is with quantization of GR itself (very loosely speaking) and this is a whole different ball game.

 

[bhobba]

Note that charge current is an observable while probability is not.

Sure - one can interpret the conserved quantity as a charge density - but then out goes the probability interpretation of QM. Either way indicated a sickness QFT is required to remove.

Thanks
Bill

 

[bhobba]

It is clear that this different time is a problem in merging general relativity and quantum theory.

Not really. Feynman and others have shown spin 2 particles leads to linearized gravity which mathematically is the same as an infinitesimal space-time curvature and leads directly to GR. You will find this approach in Ohanian's textbook on GR:
https://www.amazon.com/dp/0393965015/?tag=pfamazon01-20

The issue with GR and Quantum theory is its not renormalizeable - but still is a perfectly valid theory as an effective theory up to a cutoff about the Plank scale:
http://arxiv.org/abs/1209.3511

But even for theories such as QED that is renormalizeable to all orders we don't expect it to be valid at the Plank scale anyway. Indeed well before that scale the Electroweak takes over and we expect another theory to take over from that until we, hopefully, fingers crossed, have a TOE.

Thanks
Bill

 

[my2cts]

Sure - one can interpret the conserved quantity as a charge density - but then out goes the probability interpretation of QM. Either way indicated a sickness QFT is required to remove.

Thanks
Bill

The probability interpretation is basically non-relativistic. The charge-current interpretation is covariant.

 

[exponent137]

Not really. Feynman and others have shown spin 2 particles leads to linearized gravity which mathematically is the same as an infinitesimal space-time curvature and leads directly to GR. You will find this approach in Ohanian's textbook on GR:
https://www.amazon.com/dp/0393965015/?tag=pfamazon01-20

The issue with GR and Quantum theory is its not renormalizeable - but still is a perfectly valid theory as an effective theory up to a cutoff about the Plank scale:
http://arxiv.org/abs/1209.3511

But even for theories such as QED that is renormalizeable to all orders we don't expect it to be valid at the Plank scale anyway. Indeed well before that scale the Electroweak takes over and we expect another theory to take over from that until we, hopefully, fingers crossed, have a TOE.

Thanks
Bill

I read this fine book (Feynman Lectures on Gravitation), but I did not remember all details. I read some articles of Isham, where he found the problems in all theories of quantum gravity. Do you maybe know where the are the problems in Feynman quantum gravity, except of renormalizability.

Otherwise, I also think that the space-time is grained, so this case there is no problems with renormalizability.

 

[atyy]

Yes, the unsimultaneity is a potential problem in special relativity, not just general relativity. How the problem is solved is understood for many cases, but there are still interesting things like

http://arxiv.org/abs/hep-th/0110205
"We show that the nondemolition measurement of a spacelike Wilson loop operator W(C) is impossible in a relativistic non-Abelian gauge theory. In particular, if two spacelike-separated magnetic flux tubes both link with the loop C, then a nondemolition measurement of W(C) would cause electric charge to be transferred from one flux tube to the other, a violation of relativistic causality."

The authors say the problem does not occur for abelian theories (like QED):
"The causality problem arose for the nondemolition measurement of non-Abelian Wilson loops because multiplication of conjugacy classes is ill-defined. Since this problem does not arise if G is Abelian, one might expect that a spacelike Wilson loop operator should be measurable in an Abelian gauge theory (or more generally, if the Wilson loop is evaluated in a one-dimensional irreducible representation of the gauge group). We will see that this is the case."

 

[DevilsAvocado]

Yes, but different time in general relativity is problem for quantum theory, but time in special relativity is not a problem for quantum theory. Where is this key difference between special and general relativity? Yes, curved space-time is this difference, but why such impact on quantum theory of this difference?

The key difference between SR/QM and GR is that in SR/QM we have a fixed spacetime background (Minkowski space/Hilbert space), while in GR the spacetime geometry is dynamic and do not refer to a specific coordinate system.

It might not sound like a big deal, but if you regard the uncertainty principle and the probabilistic nature of QM, involving fluctuations, you have a well-defined background against which to define these fluctuations, while in GR any "fluctuations" will "loopback" on the dynamic background (including space and time) and define itself!

Hence, one could say that GR gives the picture of some sort of "relational loopback" – matter is being located with respect to the gravitational field, and vice versa – and you could maybe imagine what happens when incorporating QM fluctuations into this "loopback brew"...

Tricky huh?

General covariance (or diffeomorphism of smooth manifolds) is not the same as (global) Lorentz covariance, because it also applies to accelerated relative motions.

The image of a rectangular grid on a square under a
diffeomorphism from the square onto itself.

They say QM is weird, but some aspect of GR is at least as bizarre; "Beyond my wildest expectations" to quote Einstein...

Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior geometry" and for a geometric, coordinate-independent formulation of physics. Einstein described both demands by a single phrase, "general covariance." The "no prior geometry" demand actually fathered general relativity, but by doing so anonymously, disguised as "general covariance", it also fathered half a century of confusion.

Here is more on the so-called "problem of time":
http://plato.stanford.edu/entries/quantum-gravity/#5.1

(Of course, QG only applies to extreme situations of energy and/or gravity, like Big Bang or Black Holes)

 

[bhobba]

Do you maybe know where the are the problems in Feynman quantum gravity, except of renormalizability.

To the best of my knowledge there is none.

Without the problems of renormalisability GR is a perfectly good quantum theory as evidenced by the fact we have a EFT valid up to about the Plank scale.

But we think about renoprmalisability differently these days using the EFT paradigm of Wilson. We are pretty sure all our theories break down at the Plank scale so GR isn't really special. Its just that the interesting phenomena with gravity lies beyond that.

Thanks
Bill

 

[bhobba]

The key difference between SR/QM and GR is that in SR/QM we have a fixed spacetime background (Minkowski space/Hilbert space), while in GR the spacetime geometry is dynamic and do not refer to a specific coordinate system.

Actually its not a problem.

From the properties of spin 2 particles one can derive linearized gravity in a flat space-time. Then you show linearized gravity is exactly the same as if spacetime had an infinitesimal curvature. Then from requiring the principle of invarience (a correct varient of the principle of covarience) - lo and behold - you get GR.

You will find a discussion of this in the Ohanian text mentioned previously in this thread, as well as a key error of Einstein with the principle of covarience - it's basically a vacuous statement and he was chided by Kretschmann for it:
http://en.wikipedia.org/wiki/Erich_Kretschmann

But see the book for the detail. And your quote is correct - no-prior geometry is the correct basis of GR - but not the only one.

The key point here is there is really no difference between a field that makes space-time behave as if it as curved and if it actually is curved.

Thanks
Bill

 

[DevilsAvocado]

But we think about renoprmalisability differently these days using the EFT paradigm of Wilson. We are pretty sure all our theories break down at the Plank scale so GR isn't really special. Its just that the interesting phenomena with gravity lies beyond that.

Oh yes, very differently, this guy obviously didn't have a clue:

"The shell game that we play ... is technically called 'renormalization'. But no matter how clever the word, it is still what I would call a dippy process! Having to resort to such hocus-pocus has prevented us from proving that the theory of quantum electrodynamics is mathematically self-consistent. It's surprising that the theory still hasn't been proved self-consistent one way or the other by now; I suspect that renormalization is not mathematically legitimate." -- Richard Feynman (1985)​

Seriously bhobba, I love your – "business as usual" – positivism, but maybe sometimes you push it just a little bit too far? Afaik, an effective field theory should be interpreted as an approximation (reflecting human ignorance), right? It does not (generally) claim to be fundamental or self-consistent, right?

And to make it even more complicated, renormalization fails for GR, and the (only?) way to overcome this problem is to use Steven Weinberg's "Asymptotic Safety", which is promising, but still only a concept, without any rigorous proof of existence.

(Gravitation & Asymptotic Safety starts @53:50)
https://www.youtube.com/watch?v=1mQ0Yu2x9vE
http://www.youtube.com/watch?v=1mQ0Yu2x9vE&t=53m50s

So even if promising, it is still "work in progress", and while Weinberg himself is betting his money on strings, I prefer to keep mine in the wallet until the first physical experiment is about to run...

 

[DevilsAvocado]

Actually its not a problem.

From the properties of spin 2 particles one can derive linearized gravity in a flat space-time. Then you show linearized gravity is exactly the same as if spacetime had an infinitesimal curvature. Then from requiring the principle of invarience (a correct varient of the principle of covarience) - lo and behold - you get GR.

Okay, but then again; linearized gravity is only an approximation, not the whole picture, right?

You will find a discussion of this in the Ohanian text mentioned previously in this thread, as well as a key error of Einstein with the principle of covarience - it's basically a vacuous statement and he was chided by Kretschmann for it:
http://en.wikipedia.org/wiki/Erich_Kretschmann

I don’t get this at all... the discussion between Kretschmann and Einstein was of 'philosophical' nature, but GR is constructed using tensors and covariant derivative, so it would be quite inane to claim there's something wrong about GR without any (experimental) proof...??

But see the book for the detail. And your quote is correct - no-prior geometry is the correct basis of GR - but not the only one.

The key point here is there is really no difference between a field that makes space-time behave as if it as curved and if it actually is curved.

Okay, so what's the name of this successful competing gravitational theory, and will it produce something like this?

Gravitational Lensing in LRG 3-757

(I'm probably wrong, but to me it looks like the main problem is not if spacetime is actually curved or not, but the fundamental GR "feedback loop" between spacetime & matter, which causes severe difficulties when applying QM uncertainty/randomness/fluctuation, which unconditionally will propagate to the dynamic background, creating a "bubbling-random-geometry-brew", in which we're supposed to define everything else, including space & time... gravity "falls down onto itself", so to speak. Einstein also took for granted definite properties of the world, whereas Bell has proven that we cannot have both definite properties and locality, which to me seems very difficult for GR gravity to overcome, in any case... but this of course is just guessing from my side, safest not to pay any serious attention to...)

 

[WannabeNewton]

Okay, but then again; linearized gravity is only an approximation, not the whole picture, right?

Yes. It simply gives you perturbed solutions to Einstein's equation starting from some flat or curved background solution. The perturbed solutions have curvature scales varying to small degrees from the curvature scales of the background metric. They are not exact solutions.

I don’t get this at all... the discussion between Kretschmann and Einstein was of 'philosophical' nature, but GR is constructed using tensors and covariant derivative, so it would be quite inane to claim there's something wrong about GR without any (experimental) proof...??

Actually Kretschmann's comments led to confusions regarding "general covariance" for several decades down the road. Even today laymen tend to fall into the same confusing trap. If you want a beautiful discussion of what covariance vs. invariance really entails I would highly recommend section 3.5 of Straumann "General Relativity". If you don't have access to this text (which is a brilliant text on general relativity by the way) then read this: http://www.pitt.edu/~jdnorton/papers/decades.pdf

 

[DevilsAvocado]

Thanks² WannabeNewton! :thumbs:

 

[atyy]

Actually Kretschmann's comments led to confusions regarding "general covariance" for several decades down the road. Even today laymen tend to fall into the same confusing trap. If you want a beautiful discussion of what covariance vs. invariance really entails I would highly recommend section 3.5 of Straumann "General Relativity". If you don't have access to this text (which is a brilliant text on general relativity by the way) then read this: http://www.pitt.edu/~jdnorton/papers/decades.pdf

Doesn't Norton's paper say that Kretschmann was basically right, and that it was Einstein that was initially confused?

 

[WannabeNewton]

Doesn't Norton's paper say that Kretschmann was basically right, and that it was Einstein that was initially confused?

Yes I wasn't trying to say that Kretschmann himself was confused but rather this his comments led to confusions regarding what "general covariance" was, mainly regarding the difference between invariance and covariance.