# Unsimultaneity and quantum physics

1. Dec 14, 2013

### 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.

Last edited by a moderator: Dec 14, 2013
2. Dec 14, 2013

### my2cts

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

3. Dec 14, 2013

### 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.

Last edited: Dec 14, 2013
4. Dec 14, 2013

### Staff: Mentor

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

5. Dec 14, 2013

### my2cts

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.

6. Dec 14, 2013

### Staff: Mentor

The Klein Gordon Equation has problems with negative probabilities:

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):

These are all fixed in QFT.

Thanks
Bill

Last edited: Dec 14, 2013
7. Dec 14, 2013

### my2cts

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.

8. Dec 14, 2013

### Staff: Mentor

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.

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

9. Dec 14, 2013

### WannabeNewton

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...

10. Dec 14, 2013

### 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.

11. Dec 14, 2013

### Bill_K

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.

12. Dec 14, 2013

### 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?

Last edited: Dec 14, 2013
13. Dec 14, 2013

### 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.

14. Dec 14, 2013

### Bill_K

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.

15. Dec 14, 2013

### 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?

16. Dec 14, 2013

### 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.

17. Dec 14, 2013

### Staff: Mentor

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

18. Dec 15, 2013

### Staff: Mentor

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/Gravitation-Spacetime-Second-Edition-Ohanian/dp/0393965015

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

Last edited by a moderator: May 6, 2017
19. Dec 15, 2013

### my2cts

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

20. Jan 5, 2014

### exponent137

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.

Last edited by a moderator: May 6, 2017