Is Special Relativity a field theory?

1. Jul 21, 2013

QuantumCurt

Hey everyone, I've got a question. I've been doing a lot of reading about Quantum Field Theories lately, and watching a lot of lectures about field theories in general, and I'm wondering, is Special Relativity a field theory? For instance, in this article and the lecture accompanying it...http://www.symmetrymagazine.org/article/july-2013/real-talk-everything-is-made-of-fields ...Sean Carroll is discussing how essentially all of physics comes down to fields. We're taught from early on in introductory chemistry and physics courses that nature is constructed of point particles, but when we delve deeper into things, it becomes apparent that everything is really constructed of fields, rather than point particles. With Quantum Field Theories, and General Relativity, this is an overriding train of thought, but I haven't really been able to find any information one way or the other for Special Relativity.

Does this apply to Special Relativity? Sorry if this seems like a stupid question, but I'm a lowly undergrad that's very early in my studies, so forgive me.

2. Jul 21, 2013

Fredrik

Staff Emeritus
SR is not a field theory. I would say that it's a mathematical framework in which both classical and quantum theories of particles and/or fields can be defined.

3. Jul 21, 2013

WannabeNewton

Special relativity is a framework upon which field theories can be developed, it is not a field theory in and of itself. It asserts that space-time is given by $(\mathbb{R}^{4},\eta_{ab})$ which we call Minkowski space-time, where $\eta_{ab}$ is the Minkowski metric; on this space-time background we will have fields (classical and quantum) propagating and we write down field equations using the aforementioned geometric structure prescribed by special relativity. There is no dynamical evolution of the Minkowski metric $\eta_{ab}$; as mentioned before it just sits idly in the background and gives us a prior background geometry upon which fields can propagate and evolve under their associated field equations.

EDIT: Fredrik cheated; he has faster internet >.> <.< >.<

Last edited: Jul 21, 2013
4. Jul 21, 2013

QuantumCurt

I see. That makes sense. Thanks for the replies.

So SR basically describes the geometry on which different fields are actually interacting?

5. Jul 21, 2013

Mentz114

There is even a theory of gravity in which the gravitational field is a rank-2 tensor in Minkowski spacetime. Most experts assert that this theory leads to GR and offers no new insights.

See arXiv:gr-qc/9912003, for instance.

6. Jul 21, 2013

atyy

Yes.

Edit: I forgot, in case you are interested in pedantic details, there is an additional condition: the fields or particles which you put on the geometry must interact in a way in which the symmetry of the geometry is preserved. So one could say SR means that theories are constrained to be Poincare invariant, since the Poincare group is the isometry group of the Minkowski spacetime geometry.

Some believe that it leads to new insights. For example, the equivalence principle is argued to be derived, rather than imposed as an additional condition.

http://arxiv.org/abs/1105.3735 : "The real underlying principle of GR has nothing to do with coordinate invariance or equivalence principles or geometry, rather it is the statement: general relativity is the theory of a non-trivially interacting massless helicity 2 particle. The other properties are consequences of this statement, and the implication cannot be reversed."

http://arxiv.org/abs/1007.0435 : "As argued by Weinberg, the equivalence principle can be recovered as the spin-two case of his low energy theorem."

Last edited: Jul 21, 2013
7. Jul 21, 2013

WannabeNewton

Not to mention, field theoretic theories of classical gravity on a Minkowski background also provide a good motivation for why we would even want a classical formulation of gravity as the manifestation of curvature of space-time.

8. Jul 22, 2013

harrylin

Well seen! It is as said a mathematical framework (Fredrik), not a field theory in and of itself (WannabeNewton). It simply relates to observations; that is quite free from interpretation.
However it was inspired by Lorentz's theory of electrons which was a field theory, and directly based on field theory by means of the second postulate. The choice was between ballistic emission models and Maxwell's wave model (from EM fields*) of light propagation. As phrased around the time of inception:

" [..] a definite velocity c which is independent of the state of motion of the emitting body. These two postulates suffice for the attainment of a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell's theory for stationary bodies."
- http://www.fourmilab.ch/etexts/einstein/specrel/www/

So, the "overriding train of thought" that you noticed can also be found in the development of SR.

*

Last edited: Jul 22, 2013
9. Jul 22, 2013

Russell E

I think that pre-print is actually a counter-example to your statement. You say "Most experts assert that this theory leads to GR and offers no new insights", and yet the cited pre-print argues just the opposite, i.e., it argues that the "field and geometrical approaches lead to quite different conclusions in strong fields". For example, he argues that black holes do not exist in the field theoretic approach. Also, the pre-print asserts that the field theory approach leads to new insights, and that future astronomical observations will let us decide between the geometrical and field theory approaches.

The paper does, however, acknowledge that many experts in the past have claimed that the two approaches are equivalent. He argues that the proofs of equivalence were applicable only in the weak field limit.

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

Russell E

Right, and Einstein intuited that it was a false choice, and that the answer must actually be a hybrid between the two, as indeed it turned out to be. He already knew in 1905 that Maxwell's equations could not claim unlimited validity, because EM radiation exhibits aspects of an emission model that are inconsistent with Maxwell's equations. Obviously SR needed to be consistent with the empirical successes of Maxwell's equations, so in that generic sense one could say it was "based on" Maxwell's equations, but it also needed to be consistent with the phenomena that are inconsistent with Maxwell's equations, which is why in his first paper on the subject Einstein specifically avoided basing special relativity on Maxwell's equations.

Yes, but it was directly based on emission theory by means of the first postulate - which is only apparently irreconcilable with the second.

Last edited: Jul 22, 2013
11. Jul 22, 2013

Mentz114

Yes, I have read the paper, but I still think most experts disagree with Baryshev, judging from the lack of citations or papers in FTG. Not on tropic for thread though.

12. Jul 23, 2013

harrylin

As you saw, he claimed SR to be based on and in agreement with Maxwell's theory of electrodynamics. What had to be modified was Newton's theory. This is also explained in many textbooks; SR is not the same as SR+QM.

It sounds as if you disagree with Einstein's remark about SR, that 'these two postulates suffice for the attainment of a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell's theory for stationary bodies'.
Both wave theory and emission theory are compatible with the first postulate, which was based on experience and considerations of symmetry. The first postulate was obviously compatible with ballistic emission theory, and only seemingly irreconcilable with wave theory.

Last edited: Jul 23, 2013
13. Jul 23, 2013

Russell E

I don't disagree with Einstein's remark at all, but I disagree with what you think it means. At the time, physicists knew how to apply Maxwell's equations to stationary systems (for classical phenomena), but there were some recognized ambiguities and uncertainties in how to apply Maxwell's equations to moving systems, since it was unclear how motion relative to a putative ether might affect the results. For example, many people were surprised at the results of experiments like Trouton & Noble, etc. Einstein's statement simply asserts that, based on the two foundational postulates, we can apply Maxwell's equations unambiguously to any system in any uniform state of motion, by solving it using "Maxwell's theory for stationary bodies" in terms of coordinates in which the system is at rest, and then (by applying the Lorentz transformation) converting the result to a frame in which the system is in motion. This eliminates the ambiguities and uncertainties in how to apply Maxwell's equations to moving systems. (We have to limit this to low or no acceleration, because accelerating charges imply back-reaction and other features for which classical electrodynamics never gave a fully self-consistent account.) As he put it elsewhere in the same paper:

"All problems in the optics of moving bodies can be solved by the method here employed. What is essential is, that the electric and magnetic force of the light which is influenced by a moving body, be transformed into a system of co-ordinates at rest relatively to the body. By this means all problems in the optics of moving bodies will be reduced to a series of problems in the optics of stationary bodies."

The statement you quoted was obviously just summarizing this, and it is perfectly true, but it does not imply that special relativity was based on Maxwell's equations. To the contrary, the statement you quoted says explicitly that special relativity is based on the two foundational postulates. Then, given special relativity, we can (as Einstein said) solve any problem for a moving system that we can solve for a stationary system. Note that this doesn't pre-suppose that Maxwell's equations are perfectly valid, and in fact Einstein already knew they weren't. (For example, he knew that light has behavior that is inconsistent with the classical wave conception, and that is more consistent with the classical ballistic conception.) It simply says that special relativity suffices to allow us to apply Maxwell's equations to moving systems based on how those equations work for stationary systems. This does not imply that special relativity is based on Maxwell's equations.

Sure, but if SR was based on Maxwell's equations, then SR+QM would be self-contradictory, because Maxwell+QM is self-contradictory. The very fact that SR is compatible with QM (or, more accurately, QFT, since the original QM was not Lorentz invariant) proves that SR was not "based on" Maxwell's equations.

Now, we obviously have to be careful what we mean by "based on". Certainly both Newton's laws (with Galilean relativity) and Maxwell's equations were important guides in the development of special relativity, and the theory obviously needed to be compatible with those things, at least to the extent that those things were empirically successful. For example, the laws of mechanics must reduce to Newton's laws in the low speed limit. So, if you use the term "based on" in this very loose and informal way, then special relativity was "based on" a great many things, including ballistic theories. But your comments suggest that Maxwell's equations and the "wave model" of light are part of the foundations of special relativity. That is clearly false, as proven by the fact that the current best theory of light (QED) is not at all a classical wave theory (in 3+1 dimensional spacetime) and yet it is consistent with special relativity. Yes, Maxwell's equations exhibit Lorentz invariance, but special relativity is based on Lorentz invariance, not on Maxwell's equations. As Einstein said, the key insight of special relativity was that the significance of the Lorentz transformation transcended its connection with Maxwell's equations.

Einstein specifically said that he already knew in 1905 that Maxwell's equations couldn't claim unlimited validity and that this is why he chose not to take Maxwell's equations as part of the foundation of special relativity. The words you quoted don't contradict this.

Sure, and they are both compatible with the second postulate too. That's the point. Special relativity laid the framework for theories in which light has attributes that were classically associated with waves and other attributes that were classically associated with particles. So, the claim that special relativity embodies a decision in favor of the wave model of light completely misses the point of what special relativity accomplished.

Last edited: Jul 24, 2013
14. Jul 24, 2013

harrylin

['these two postulates suffice for the attainment of a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell's theory for stationary bodies']
The statement above says explicitly that special relativity is based on Maxwell's theory for stationary bodies; and I won't argue about it, except if QuantumCurt wants to hear more.

It is well known that SR and QM in their original forms did not match very well; but that's a quite different topic that has been discussed also.
Einstein reduced the wave model to its essential feature for the problem at hand, which is that light propagation is independent of the speed of the source. Or, as Pauli put it in his textbook, Einstein showed that just the following sentence of [Maxwell's] Electrodynamics must be provided: The speed of light is independent of the state of motion of the source.

That is not a logical proof, and it is well known (and I think that it has been explained in many threads on this forum) that wave theory is compatible with the first postulate; in any case that discussion is off-topic so I won't elaborate here. If you start that as a topic, I will participate.

That misses the point of my clarification to the OP. The relationship between Maxwell's theory and SR was also discussed in another thread, where the same point was made by others:

[addendum] Einstein elaborated on that matter in his 1907 follow-up paper which was discussed here (Lorentz's electrodynamics theory was based on Maxwell's theory):

I guess that those threads contain enough elaboration for QuantumCurt. :tongue2:

Last edited: Jul 24, 2013
15. Jul 24, 2013

Russell E

It doesn't. You're completely mis-reading the sentence. It refers to three things:

A. The two postulates of special relativity
B. Maxwell's theory for stationary bodies
C. A theory of electrodynamics for moving bodies

Einstein's quote says that A suffices to attain C based on B. In other words, applying A to B suffices to give us C. Harrylin reads this as saying that A is based on B. That is a complete mis-reading of the sentence. Einstein's statement is correct. Your reading of his statement is completely wrong.

You ignored the point. QED is consistent with special relativity (in fact, it was based on the requirement to make a Lorentz invariant quantum theory), and yet QED differs profoundly from the classical electrodynamics of Maxwell, which you claim (or rather, claimed, since you modify your position below) is the basis for special relativity.

Well, that's a far cry from your previous claim that special relativity is based on Maxwell's equations. But you're still missing the point. A classical emission model suggested that light propagates at c relative to the source, and a classical wave model suggested that light propagates at c independent of the source. The first suggestion corresponds to the relativity principle, and the second corresponds to the light-speed principle. Special relativity reconciled these two seemingly incompatible models, which was a necessary step along the path of accounting for both the wave-like and particle-like behavior of all physical entities. This is why you completely miss the point when you say that special relativity is based on the wave model - without adding that it is equally based on the emission model. Those were the two seemingly incompatible things that special relativity reconciled. Remember, QED is not a classical wave model, and in fact it's verified predictions include behavior that contradicts the classical wave model.

Sure, but it is equally well known that ballistic theory is compatible with the second postulate. Of course, these compatibilities depend on our understanding of special relativity. In the late 19th century it was thought that the wave model might be inconsistent with the relativity principle (indeed Lorentz expected it to be eventually found inconsistent, even to the end of his life), and of course it was thought that ballistic theory was incompatible with the second postulate. But both of those turned out to be wrong, as we now know. It is possible for light to propagate at a fixed speed relative to the source AND to propagate at that fixed speed independent of the source.

Last edited: Jul 24, 2013
16. Jul 25, 2013

harrylin

One last clarification about an essential, double misunderstanding:
Not at all: I read it as saying that C is based on B. Indeed, A suffices to attain C based on B.
The "simple and consistent theory of the electrodynamics of moving bodies" is the new theory which Einstein later labeled as "special relativity" - the very title of that paper is "ON THE ELECTRODYNAMICS OF MOVING BODIES". He elaborated as follows:

The theory to be developed is based—like all electrodynamics—on the kinematics of the rigid body, since the assertions of any such theory have to do with the relationships between rigid bodies (systems of co-ordinates), clocks, and electromagnetic processes.

And once more, this was also elaborated in the discussion of his 1907 paper.

Last edited: Jul 25, 2013
17. Jul 25, 2013

Russell E

Ah, I see the source of your confusion now. You're conflating special relativity with a specific (and now obsolete) theory of electrodynamics. What we now call special relativity was developed in Part 1 of the 1905 paper (which Einstein called "The Kinematical Part"). In Part 2 of that paper (the Electrodynamical Part) he showed how special relativity could be applied to develop a (now obsolete) theory of electrodynamics of moving bodies based on Maxwell's (now obsolete) theory for stationary bodies. Part 2 of the paper is an application of special relativity to electrodynamics, which was the ostensible subject of the paper. Of course, it turned out that the conceptual framework established in Part 1 of the paper (which we now call special relativity) transcended in scope and importance the application to electrodynamics. Note that the 1907 paper was not limited to electrodynamics, it covered thermodynamics, etc., as well. Remember, special relativity is not a theory of electrodynamics in particular, it is a meta-theory that places a constraint (Lorentz covariance) on all physical theories, including mechanics, thermodynamics, electrodynamics, quantum theory, etc.

Indeed, C is based on B with the help of A. But that's not what you've been claiming. You've been claiming that special relativity (which is A) is based on Maxwell's equations for stationary bodies (which is B).

Right. This is contrary to your claim that A is based on B. The reason you were confused is because you thought special relaivity is C, whereas it is really A.

Right, and it was even further elaborated in his extensive writings on the subject during all the remaining years of his life, as well as in many excellent accounts written by others. As Einstein later explained, by 1905 he had already become convinced (rightly, as it turned out) that neither Newtonian mechanics nor Maxwell's electrodynamics could provide a secure foundation, and that only the discovery of a new principle could provide a solid foundation for assured results. He actually took as his example the science of classical thermodynamics, which was based on a small set of principles, independent of any detailed theories. He clearly explained why he did not take Maxwell's equations as part of the foundation of special relativity, although of course the empirical success of Maxwell's "wave" equations (within their domain of applicability) lent plausibility to the light speed principle, just as the empirical success of Newton's "ballistic" equations (within their domain of applicability) lent plausibility to the relativity principle.

Last edited: Jul 25, 2013
18. Jul 27, 2013

QuantumCurt

I believe my question has been quite thoroughly answered. Lots of good information here, thanks everyone. Much of it went right over my head, but I'm still rather early in my physics education. I look forward to the day that I can keep up with discussions like this.

I just gave them a quick glance, looks like a lot of good information. I'm going to give them a more thorough reading when I'm not on the verge of falling asleep.

Thanks for all of the information!!!

19. Jul 27, 2013

harrylin

You are certainly right that SR is much more than electrodynamics; I had in mind to clarify that, to be more precise, the 1905 paper focussed on the electrodynamics but indicated that necessarily the theory has to apply to all physics. And so, although Einstein did not do that in 1907*, one may split hairs by distinguishing the formal generalisation as in the 1907 paper from the earlier papers that focussed on electrodynamics. However, that is irrelevant for the question that was under discussion here: it does in no way change the basis for the second postulate (indeed not a formal basis, but where it came from). That's why I did not think that detail worthy of mention.

I'll now unsubscribe from this topic, the OP has certainly enough literature to base his opinion on.

You're very welcome!

* In what follows, the attempt is made to summarize into a whole the works hitherto emerged from the unification of H.A. Lorentz's theory and the principle of relativity. - Einstein 1907, as translated by Histspec

Last edited: Jul 27, 2013
20. Jul 27, 2013

Russell E

It isn't that SR is "more than electrodynamics", it's that SR is not electrodynamics at all. SR is applicable to electrodynamics, as it is applicable to every other branch of physics, but SR itself is simply the assertion that all physical phenomena are Lorentz invariant.

This discussion is not irrelevant, it directly addresses what the OP asked. Remember, the OP said he had been reading about quantum field theories, and he asked if special relativity is a field theory. He initially got some good replies, explaining (as noted above) that special relativity is not really a theory per se, it is a constraint (Lorentz invariance) that establishes a framework within which theories are formulated. Harrylin (who has now unsubscribed) initially agreed with this, but then remarked that special relativity was "directly based on field theory by means of the second postulate. The choice was between ballistic emission models and Maxwell's wave model of light propagation." This is what prompted my comment that the real accomplishment of special relativity was to reveal that the "choice" he mentioned was a false choice, and that in fact it is possible to reconcile both the ballistic emission models and the wave models of light propagation, and hence that special relativity was not based on either one of them to the exclusion of the other. This was essential to Einstein in 1905 because he had just finished his paper on light quanta, and it was absolutely crucial to the subsequent development of modern quantum field theories - which is precisely what the OP asked about.

Note that the relevant point was actually made in the sentence immediately preceding the one harrylin quoted:

"Only the conception of a luminiferous ether as the carrier of the electric and magnetic forces does not fit into the theory described here; for electromagnetic forces appear here not as states of some substance, but rather as independently existing things that are similar to ponderable matter and share with it the feature of inertia."

He could hardly have been more explicit in disavowing the idea (attributed to him by harrylin) that special relativity is founded on the classical wave model of light. Indeed he says propagating electromagnetic effects are independently existing things (no surprise to anyone who read his 1905 paper on light quanta), not states of some substance, and even goes so far as to say they are inertial entities similar to ponderable matter.

Last edited: Jul 27, 2013