Timeline for Maxwell equations as Euler-Lagrange equation without electromagnetic potential
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| Dec 31, 2020 at 11:04 | comment | added | asv | I agree with that. But then why YM-Lagrangian works so well? It is the same as to say "it works because it works". Sometimes this explanation can be accepted. But in this case I do not feel satisfied with it. I think YM works so well because it is similar to electrodynamics Lagrangian, and not the other way around. Moreover I think electrodynamics is more than just a special case of gauge theories. To appreciate that one can go back to 19th century. Why not? | |
| Dec 31, 2020 at 10:58 | comment | added | asv | Not that I confuse historical order with physical causation, but often I find historical explanations more intuitive. Newton's laws without Kepler's laws would look unmotivated to me unless they are verified in some other similar situations. I would not deduce them from Einstein gravity equations. The same notions can be understood in different levels. I asked why the electrodynamics Lagrangian looks as it is. You basically say: because it is a special case of YM-Lagrangian which explains remarkably well many physical phenomena and have non-trivial differential geometric properties. | |
| Dec 30, 2020 at 17:32 | comment | added | Terry Tao | To continue the gravitational analogy, your original question is analogous to the question "Can Kepler's laws be expressed as an Euler-Lagrange equation without the gravitational potential?". Perhaps this is mathematically possible by some artificial trick, such as the sum-of-squares trick in my other answer, but it is hard to see the point of asking such a question in view of the completely satisfying success of Newtonian gravitation in explaining Kepler's laws. Similarly with Maxwell's equations and U(1) Yang-Mills theory. | |
| Dec 30, 2020 at 17:18 | comment | added | Terry Tao | Furthermore, Yang-Mills greatly clarifies electromagnetism by introducing the principle of U(1) gauge covariance, which greatly restricts how the electromagnetic field can interact with the other forces of nature, and is a principle that has no substantial counterpart in the classical Maxwellian theory. It is hard to conceive how one could have discovered or even stated the electroweak interaction without the Yang-Mills interpretation of EM, let alone the standard model. (U(1) gauge invariance also explains many other aspects of EM, such as charge conservation.) | |
| Dec 30, 2020 at 17:14 | comment | added | Terry Tao | You are confusing historical order of discovery of theories with physical causation. Kepler's laws of planetary motion were deep experimental facts that directly led to Newton's laws of universal gravitation, but it is the latter that explains the former, not the other way around. Now, previously unexplained empirical facts such as Kepler's second law are mathematical consequences of deeper facts, in this case rotational covariance (and Noether's theorem). This is completely analogous to how the Gauss-Faraday law is now a mathematical consequence of the deeper fact of the U(1) nature of EM. | |
| Dec 30, 2020 at 13:36 | comment | added | asv | There is one key difference between YM and electrodynamics: in the former potential is a part of definition of the theory, while in the latter it is consequence of a deep experimental fact -Gauss-Faraday law. Let me speculate that without this law potentials were not known to physicists, and no YM-theory would be discovered. Thus not YM explains electrodynamics, but vice versa, to certain degree of course IMHO. | |
| Dec 30, 2020 at 13:31 | comment | added | asv | With all respect due, I do not see how the modern Yang-Mills theory clarifies elctrodynamics, except emphasizing the differential geometric meaning of the potential to be a connection. In fact YM-theories were motivated by the electrodynamics. The YM Lagrangian imitates and generalizes the electrodynamics Lagrangian, but tells nothing new on it. | |
| Dec 29, 2020 at 18:53 | comment | added | Terry Tao | From the modern perspective, the deep and nontrivial fact is that electromagnetism is a U(1) gauge theory. This fact, together with the Bianchi identity, completely explains the Gauss-Faraday law. You are of course entitled to revert back to the nineteenth century perspective of viewing the Gauss-Faraday law as an ineffable empirical mystery and ignore a century of gauge theory if you insist, but I don't see the advantage in doing so. | |
| Dec 29, 2020 at 17:45 | comment | added | asv | I agree that defining EM-field to be a vector potential has an advantage that only in this form the theory has Yang-Mills generalizations, and thus electrodynamics can be considered as a special case of them. But still classical electrodynamics is in a sense deeper than non-abelian Yang-Mills theories: it has the Gauss-Faraday law which is not just Bianchi identity. This is a very non-trivial step to replace $(\vec E,\vec B)$ with vector potential; this step is missing in more recent theories. This step I am trying to understand. | |
| Dec 29, 2020 at 17:05 | comment | added | Terry Tao | Basically, modern physicists believe so strongly in the second postulate of relativity that we have elevated it to the status of a tautology by arranging our definitions accordingly. Similarly, modern physicists believe so strongly that electromagnetism is a U(1) gauge theory that the EM field is now defined in terms of that theory, in particular elevating the Gauss-Faraday law to the status of a mathematical tautology. (This is a falsifiable viewpoint; the demonstration of existence of magnetic monopoles would definitely cause a rethink of this position. But this has not occured yet.) | |
| Dec 29, 2020 at 16:52 | comment | added | Terry Tao | It's analogous to how with the modern definition of the meter, the constancy of the speed of light $c$ is no longer an empirical law of nature, but is true by definition. On the other hand, the principle of relativity (that physical laws that involve the meter length unit are the same in all reference frames) acquires much more empirical content with this definition. Ultimately the net physical implications of special or general relativity are not affected by this convention on how to define the meter, but the relative weight assigned to the two axioms of relativity are altered. | |
| Dec 29, 2020 at 16:49 | comment | added | Terry Tao | Basically, in the modern interpretation of electromagnetism the epistemology of the electromagnetic field has been redefined. Classically, the EM-field was what one could observe through the Lorentz force law, in which case Gauss-Faraday becomes an empirical law of nature and the Lorentz force law is largely true by definition. In the modern interpretation, the EM-field is defined as the curl of the vector potential; now Gauss-Faraday is a mathematical tautology and it is the Lorentz force law that becomes an empirical law of nature. | |
| Dec 29, 2020 at 16:42 | comment | added | Terry Tao | Similarly, nineteenth century classical electromagnetism is mathematically equivalent to U(1) Yang-Mills theory, differing mainly in their philosophical designations of which fields are primary and which are secondary. You seem to accept that for Yang-Mills it is indeed the vector potential $A_\alpha$ which is the primary field, and the standard model very definitely interprets electromagnetism as a U(1) Yang-Mills theory, so if you accept modern physics then we should all be in agreement. | |
| Dec 29, 2020 at 16:35 | comment | added | Terry Tao | An analogy would be with the vorticity formulation of the Euler equations for incompressible fluids (in which the vorticity $\omega$ is the primary field and the velocity field $u$ is derived by an "empirical" Biot-Savart law) and the velocity formulation (in which the velocity field $u$ is primary and vorticity is, by definition, the curl of the velocity). Mathematically the two formulations are equivalent, but the latter is more physically natural; vorticity is not naturally interpretable as independent field, but rather as a manifestation of a velocity field. | |
| Dec 29, 2020 at 16:26 | comment | added | Terry Tao | It's a question of interpretation. You are using the nineteenth century interpretation of classical electromagnetism in which the electromagnetic field is taken to be primary and the vector potential is secondary. I am using the post-Yang-Mills interpretation in which the vector potential is primary and the electromagnetic field, by definition, is the curl of the vector potential. The subsequent development of the laws of physics (e.g., the electroweak theory linking electromagnetism to the U(2) Yang-Mills theory of the weak force) has shown this is the more natural interpretation. | |
| Dec 29, 2020 at 6:44 | comment | added | asv | This law is not a mathematical identity like Bianchi identity, but a law of nature established experimentally. For this reason vector potential $A_\alpha$ cannot be considered as an a priori given field, it is a consequence of (Maxwell) equations of motion. The fact that $A_\alpha$ is defined up to a gauge is not the main difficulty, I think. Thus one comes back to the same issue: part of equaitons of motion is replaced by something else mathematically equivalent to a constrain. | |
| Dec 29, 2020 at 6:34 | comment | added | asv | Thanks for the interesting answer, Sorry for being boring, but I think that here the same point I do not understand appears in an equivalent form. "For Maxwell the primary field is the vector potential $A_\alpha$..." In Maxwell case existence of the potential is not given in advance, its existence is not assumed (in the contrary to further Yang-Mills generalizations) but it is equivalent to what you called Gauss-Faraday law $div(\vec B)=0, rot(\vec E)=-\frac{1}{c}\dot{\vec B}$. | |
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| Dec 28, 2020 at 18:41 | history | answered | Terry Tao | CC BY-SA 4.0 |