Information relationship to Special Relativity

How do we write mathematically that "information" cannot go faster than light? And along a similar line of thought, how do we relate "information" with special relativity. Lastly, what is the relationship between Special Relativity and the fact that the phase velocity of a wave packet can go faster than light (light speed here being the group velocity). Is there a reason we cannot consider the frame of reference of a specific phase in a wave packet?

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Related question: physics.stackexchange.com/q/34653/2451 –  Qmechanic Sep 16 '12 at 23:42

How do we write mathematically that "information" cannot go faster than light? And along a similar line of thought, how do we relate "information" with special relativity.

Since you are looking for an equation (you say "mathematically"), I would undoubtedly choose this: $$\left[\hat O (x),\, \hat O' (y)\right]=0, \, \mbox{if}\; x-y \; \mbox{is spacelike}$$ where $\hat O$ and $\hat O'$ are the (linear self-adjoint) operators corresponding to two physical observables —in particular, both may be the same observable and therefore the same operator ($\hat O=\hat O'$)—, and $x, y$ are two points in space-time. This equation summarizes the fact that information cannot travel faster than light because it says that the results of two experiments separated by a space-like interval cannot be correlated. And this is what "information" means since one codes information with physical effects. Please, see this Definitions: 'locality' vs 'causality' if you are interested in the different usages of the terms "causality" and "locality", which are physically more relevant or why entanglement do not imply faster than light propagation.

The previous formula assumes that the physical laws obey the principles of quantum mechanics and special relativity, and are thus quantum field theories. This is the case for the electromagnetic, the weak and the strong interactions and also likely for the case of the gravitational interaction in the weak field limit and in the sense of an effective field theory; which are the fundamental interactions that we know.

Lastly, what is the relationship between Special Relativity and the fact that the phase velocity of a wave packet can go faster than light (light speed here being the group velocity)

Sometimes defining the speed of a wave is tricky. The signal or information velocity is often the group velocity (which is the velocity of a wave packet), even though in some media (see http://en.wikipedia.org/wiki/Signal_velocity) it is not. But the phase velocity (the rate at which the phase of the wave propagates) cannot carry information (see http://en.wikipedia.org/wiki/Phase_velocity) and may be faster than $c$.

Is there a reason we cannot consider the frame of reference of a specific phase in a wave packet?

You may take any inertial frame provided its speed be lower than $c$. Note that according to special relativity one needs an infinite amount of energy to cross the speed of light $c$ threshold.

Edit: SMeznaric points out —and I agree with him— that space-like separated measurements may give correlated results. What is not possible is to send information one has control over, such as the choice of measurement operators.

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This statement "This equation summarizes the fact that information cannot travel faster than light because it says that the results of two experiments separated by a space-like interval cannot be correlated." is actually not true even classically. For example $\rho = \sum_k p_k |k><k| \otimes |k><k|$. Spacelike separated measurements on this system will give correlated results. What is not possible is to send information one has control over, such as the choice of measurement operators. –  SMeznaric Sep 13 '12 at 13:09
I just checked it out now. There you also say that causality (Einstein locality) means that spacelike separated experiments are not correlated. Several of the other things you mention are very closely related, but seem good to me. –  SMeznaric Sep 13 '12 at 21:13
@SMeznaric I also say that this assumes that there is not previous correlations between the (sub)systems over which one is measuring. –  drake Sep 13 '12 at 21:28
This is what you say: "Results of experiments carried out at a space-like distance are not correlated." –  SMeznaric Sep 13 '12 at 21:48

Information as we know it is either carried by a physical object or a field. They both must obey special relativity and thus are thus c limited.

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Rather than information, it may be clearer to say that causality cannot go faster than the speed of light. That is to say, a particle is only affected by events that happened in its backwards light cone. This is expressed mathematically by the equations of motion for a particle not being dependent on anything outside of its light cone. Of course, if a particle cannot be affected in any way by events outside of its light cone, it also cannot receive information about those events.

As for your second question, regarding a superluminal phase velocity, the answer is that the frame of reference of the phase would not be an inertial frame. That is to say, your focus of attention can of course follow the wave crest but you cannot speak of an observer who is at rest relative to the wave crest. This is because an observer cannot be moving faster than the speed of light. There is no Lorentz transform associated with such a frame.

If neutrinos went faster than the speed of light, then all bets are off. We would discard all the physics that was developed in the last 100 years and start from scratch.

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In order to see how special relativity implies that information cannot go faster than light, one must first consider the physical properties of information. Definition of information usually used in physics is the minimum length of a message needed to encode it. For example, given prior probabilities $p_1, \ldots, p_k$ for an outcome $k$ of some random variable, the minimum length of a message needed to represent the information of which outcome occurred is given by $H({p_k}) = - \sum_k p_k \log_2(p_k)$. You can read in great detail about this as it is a well developed field known as the Shannon information theory, as a start see for example Wikipedia. This is intuitive: given $n$ equally likely outcomes, we would need $\log_2(n)$ objects with two completely distinguishable states each to represent the information on which outcome occurred, as $m$ such objects have $2^m$ possible states. If one has objects with more states, then the base of the logarithm is changed. There is no loss of generality if we use 2, and this is the typical use with the unit of information known as bit.

This brings us to physics of transmitting information. In order to actually impart information on another observer, one needs to prepare $H({p_k})$ physical objects and send them to the receiver. Since special relativity limits how fast such objects can travel to the speed of light, this then implies that the receiver will not be able to learn anything about our part of space until our objects have reached them.

However, there is another potential way that one could send a message without moving an object. This is to instantaneously (in some frame of reference) change the state of the objects already present at receiver's end so as to match our desired message. Here our understanding of physics implies that objects need to interact through a field (i.e. particles, which are physical objects). Now special relativity requires that fields cannot propagate (translation = particles cannot travel) faster than the speed of light and therefore we are back to not being able to transmit any information.

Of course there is a but coming at the end. The existence of tachyons would throw everything I said above out the window, so everything I have said you should take it in the context of the current theories and experimental evidence. Here by tachyon I mean a particle that can travel faster than the speed of light. Whether nature really works like this or not we do not know.

As an aside, quantum entanglement does not allow one to overcome this requirement. While measurements at the would-be sender's and receiver's sides are correlated, the sender cannot determine the measurement outcome of the receiver through choice of measurements or otherwise. Quantum mechanics is thus fully compliant with the requirements of special relativity.

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I up-voted this answer, even though it does not answer the original OP question: "How do we write mathematically that "information" cannot go faster than light?". Nevertheless, it should be clarified that the definition of tachyion that SMeznaric uses is not the standard field theoretic definition. He means or he should mean a particle which travels faster than light in vacuum. A field-theoretic tachyon (i.e., a field with the opposite sign in the quadratic term of the Hamiltonian) does not imply faster than light propagation. –  drake Sep 13 '12 at 22:33
Thank you for your comment. I have updated the answer to reflect what you say. –  SMeznaric Sep 13 '12 at 22:48

The speed of light can be derived from Maxwell's Equations. So information carried by light cannot go faster than c. But neutrino's were briefly in the news for possibly going faster than light, and the only problem was that there should have been energy radiating from them that was not observed. So it is just a matter of no such energy being observed that we deduce that they do not go measurably faster than light.

The phase velocity can go faster than light for the same reason that you could point a laser at the Moon and observe the reflection spot travelling faster than light. There is no object there. You could use that motion as a frame of reference, but the usual published formulas make the assumption that the speed of the observer is less than c, so those formulas won't work in your example.

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I'm confused what is "information" and what is not. How do mathematically define information vs non-information? Also, I don't understand why if certain phenomena do exhibit faster-than-light motion, we do not expect the standard odd effects to appear in that frame of reference (such as time being negative, etc...)? –  mcFreid Sep 8 '12 at 21:31
The neutrinos never went faster than light, so that doesn't factor into this issue. The only thing that prompted the news was a mechanical timing error. –  David Z Sep 8 '12 at 21:31
mcFried - what information is exactly doesn't really matter for the original question. All that matters is whether some signal can travel faster than light. Whatever form that signal takes could be used to transmit information, such as using AM or FM modulation. And yes, faster than light travel would cause big problems for causality, because it would show that objects could exist in a frame of reference where the consequences of events could be seen at a point in spacetime where it was still possible to prevent the cause. But that already exists in general relativity. –  jcohen79 Sep 8 '12 at 23:57
David - I mentioned neutrinos just to illustrate the process by which it was determined that is was very unlikely that they travel FTL even before the cable problem was found. It would have had a fundamental effect on our understanding of the universe, but we couldn't rule it out completely without other observations such as the lack of Cherenkov radiation. –  jcohen79 Sep 9 '12 at 0:04

[...] how do we relate "information" with special relativity.

The "exchange of information" (a.k.a. "signalling") between participants is arguably a foundational and self-explanatory notion of (special as well as general) relativity:
for instance, participant Alice stated some particular indication $A_i$ (and subsequently remembered having stated it), and later Alice observed and recognized participant Bob having observed Alice's indication $A_i$, and (in coincidence)/or (subsequently) participant Eve having observed Alice's indication $A_i$, and so on;
all without providing any more specific or even quantitative characterization of the "information" thereby exchanged.

How do we write mathematically that "information" cannot go faster than light?

This question seems to consider the exchange of "light" as being distinct from the exchange of "information" in general. However, if so, then necessarily in some more specific "mode" (perhaps "electro-dynamically", or "chromo-dynamically", or "by curvature").

Accordingly, the correct statement would be that "information exchange in some specific mode cannot go faster than information exchange in general";
or in other words, that "light cannot go faster than information".

For suitable setups and specific modes (of "light" exchange) the corresponding ratio of "(general) information speed" to "(specific mode) light speed" can be expressed as the "group speed refractive index, $n_g$"; cmp. http://en.wikipedia.org/wiki/Index_of_refraction#Group_index

Is there a reason we cannot consider the frame of reference of a specific phase in a wave packet?

A "frame of reference" is primarily considered of some specific participant (such as Alice), as a specific geometric relation to other participants (such as "Alice and Bob having been at rest to each other"). In the context of relativity (at least), the underlying though-experimental method of establishing some specific geometric relation between suitable participants is of course based on their mutual exchange of "information", as described above.
However, "a specific phase in a wave packet" may not typically be considered as engaging in the exchange of signals with other participants (and/or with any other "specific phase in a wave packet", for instance).
Therefore, the "frame of reference of a specific phase in a wave packet" may only be identified indirectly, if at all, by suitable proxy, such as the "frame of reference" of a particle moving through a medium of refractive index $n$ at $\beta = 1/n$; cmp. http://en.wikipedia.org/wiki/Cherenkov_radiation

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Please note a follow-up question: physics.stackexchange.com/a/37641/12262 –  user12262 Sep 17 '12 at 19:43