# Should the speed of light be slightly uncertain?

1. Apr 8, 2013

### g.lemaitre

The position and momentum of a photon is uncertain. If that is the case, then shouldn't the speed of light be slight uncertain?

2. Apr 8, 2013

### strangerep

Have you ever tried to construct a good position operator for the photon in the relativistic context? Afaik, no one has done it in a fully satisfactory way.

3. Apr 8, 2013

### Integral

Staff Emeritus
We do not need to measure the location of a photon when finding the speed of light.

By definition the speed of light is an integer, therefore no uncertinatiy

4. Apr 8, 2013

### g.lemaitre

Don't know what you mean by integer. Also didn't understand why you wanted me to read the wiki article on meter.

5. Apr 9, 2013

### Integral

Staff Emeritus

6. Apr 9, 2013

### g.lemaitre

I read it before you advised me to read it of course and I still don't understand what you're getting at. If what you're getting at is the speed of light is relative to the observer, well, I already know that but I still don't see why it's speed cannot be uncertain.

7. Apr 9, 2013

### Integral

Staff Emeritus
If you had read all of it you would know why the speed of light is an integer.

8. Apr 9, 2013

### g.lemaitre

Never mind, I give up

9. Apr 9, 2013

### Staff: Mentor

After 2 hours and 40 minutes. Really?

c is a defined constant, as such its uncertainty is 0. What is uncertain is the length of a meter. That is the point to take from the reading.

10. Apr 9, 2013

### g.lemaitre

11. Apr 9, 2013

### Chopin

My god, you guys, what a bunch of rude and pointless arguing of semantics you're giving out here! Those answers completely ignore the intent of the question. I have no idea what you were trying to accomplish with that.

g.lemaitre: The speed of light is a consequence of relativity, not of quantum mechanics, so the types of uncertainty that pop up in studies of quantum behavior don't really apply to it. Lorentz invariance implies that there is a maximum speed that things asymptotically tend towards as you try to accelerate them, and the principles of Quantum Field Theory tell you that a massless particle can't propagate at any velocity except for that speed, so the speed of light is thus completely fixed.

Last edited: Apr 9, 2013
12. Apr 9, 2013

### Integral

Staff Emeritus
I will spoon feed you some more. From the article I was trying to get you to read.

13. Apr 9, 2013

### Integral

Staff Emeritus
This is a bit backwards, Relativity is a consequence of the constancy of the speed of light. The constancy of the speed of light is due to the nature of the universe. Lorentz invariance was COOKED to model the constancy of the speed of light. So naturally it has a max speed.

14. Apr 9, 2013

### Chopin

Except that that wasn't at all what the question was asking. The question was: since the Heisenberg Uncertainty principle limits our ability to simultaneously know the position and momentum of a particle, is it possible EVEN IN PRINCIPLE to know the speed of light exactly? A satisfactory answer to that question must involve some reference to the physics of relativity and quantum mechanics, not a history lesson.

15. Apr 9, 2013

### Chopin

Right, except for one detail. Lorentz invariance implies there is a maximum speed that anything can go, but it doesn't necessarily tell you that light must go at that speed. In order to show that, you have to use Quantum Field Theory, where you show that massless particles have four-momenta with magnitude zero, meaning that they travel at exactly that top speed. Then Lorentz invariance tells you that they must travel at that speed in all frames of reference.

16. Apr 9, 2013

### Integral

Staff Emeritus
Seems to me in post #10 the OP was happy with the being spoon fed the contends of the article I pointed him at. So maybe he really didn't understand his original question as well as you do.

My entire effort was pointed at getting the OP to think for himself just a bit.

17. Apr 10, 2013

### Crazymechanic

Well the easy way of telling this maybe would be that yes we don't know the exact position of a particle but the speed of light doesn't depend on this one particle to measure , we can just have a beam of light from one point traveling down a certain distance (assume a large one as the speed is very high ) and then take a measurement on a very precise clock to see how many "ticks" or "vibrations" of an cesium atom (in a atomic clock) have passed.
Now the rest is elementary maths so remember that we have measured light speed on the macro scale and to do that doesn't require to know where a particular photon is at a given time as they usually come many not one.

18. Apr 10, 2013

### Naty1

Saying that we now DEFINE light to have a certain 'exact' speed doesn't really address whether it HAS that speed or not, and if it does, can we measure it....'exactly'....

OP:

Why do you think it is 'uncertain' ??

from the current discussion....
first part: Well, as has been discussed in numerous threads in these forums, that is NOT what HUP says.....just briefly from those discussions:

One of the lengthy discussions is here:

Note there are likely conflicting viewpoints along the way....so read along before getting 'all fired up' [lol]

And of course this IS an issue, but has nothing to do with HUP....this is a problem with measurement apparatus.

Another discussion is here:

Do particles have well defined positions at all times

But BEWARE: THE BALLENTINE paper seems to draw some questionable conclusions!!

19. Apr 10, 2013

### Naty1

I forgot to post:

http://en.wikipedia.org/wiki/Speed_of_light#Measurement

......doesn't seem like photons are routinely used to 'measure the speed of light' anyway....

edit: so to answer the original question, it seems like we are limited only via our ingenuity in developing ever more accurate measurement devices.

Last edited: Apr 10, 2013
20. Apr 10, 2013

### phinds

And you did it in a way that I think reflects how good this forum is, that you take the time to figure out what might be helpful to an OP and you nudge him/her towards it. It's hardly your fault that some people resist nudging and thinking.

21. Apr 10, 2013

### BWV

even in a naive model of measuring photons with a hypothetical absolutely precise speed detector the UP is inconsequential

after 1 second light would have traveled 300 million meters & the uncertainty of the speed clocked for the photon would have a standard deviation of 10^-34 meters / second, so even say a 10 sigma error would be immaterial

Last edited: Apr 10, 2013
22. Apr 10, 2013

### Staff: Mentor

For a question of what the HUP has to say about the speed of light, you should read an article on the HUP, not on SR. The wiki on the HUP has a section on this that does indeed confirm that it applies to photons/the speed of light.

23. Apr 10, 2013

### fluidistic

I'd appreciate the section... I'm reading through it (http://en.wikipedia.org/wiki/Uncertainty_principle) but I don't see where they talk about the speed of light.
The question of the OP blows my mind. I don't understand how it's possible to have an uncertainty in the position but not in the velocity, if I consider that the velocity is the derivative of the position with respect to time.

24. Apr 10, 2013

### Staff: Mentor

I'm on a cell phone at the moment and can't easily post quotes: it is the section on critical reactions, several subsections, but the one on Einstein's box in particular. Last sentence says position uncertainty applies to photons.

25. Apr 10, 2013

### Physics Monkey

Here is a little concrete calculation that has some bearing on what I interpreted the OPs question to be.

Consider a relativistic quantum particle of mass $m$. Let its position be $\vec{x}$ and its momentum be $\vec{p}$. We may write a formula for the velocity of the particle which reads
$$\vec{v} = \frac{\vec{p}}{\sqrt{p^2 + m^2}}$$
in units where the "speed of light" (i.e. a unit conversion factor) is one.

Certainly the following things are true for this particle. Unless the particle is in a momentum eigenstate, the velocity of the particle is uncertain. If we prepare the particle in a localized wavepacket, then the momentum uncertainty and hence also the velocity uncertain will be related to the localization in position. However, there is a curious feature here since the velocity is a bounded operator.

To see something interesting, consider a Gaussian wavepacket with $\langle p \rangle \rightarrow \infty$, but with a fixed variance $\delta x^2$. When acting on high momentum states, the "speed squared" operator may be expanded as
$$\vec{v} \cdot \vec{v} = 1 - \frac{m^2}{p^2}+ ...$$
The variance of this speed squared operator is
$$\Delta \equiv \langle (v^2)^2 \rangle - \langle v^2 \rangle^2 .$$
Now I think the following is true (and I have checked it in a simplified wavefunction), the variance $\Delta$ vanishes as $\langle p \rangle \rightarrow \infty$ at fixed $\delta x^2$. Hence we may take a limit by first sending $\langle p \rangle \rightarrow \infty$ to obtain a particle moving at the "speed of light" with no variance in its "speed squared" operator. We may then adjust $\delta x^2$ to localize the particle as we see fit. So here it seems we obtain something like a particle with the usual position-momentum uncertainty but with definite speed, the "speed of light", in a certain limit.

Now I know there are many issues with relativistic position operators and so forth, but perhaps this little calculation will be somewhat helpful.