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Noether's Theorem (in its various incarnations) is an amazing connection between symmetries and conservation laws that applies to a large class of dynamical systems. I personally find it to be exceedingly beautiful, interesting, and surprising.

However, I find myself asking myself the following questions which I can't answer convincingly:

  1. If we did not know Noether's Theorem, would our ability to predict the behavior of any real dynamical system be impaired in practice?
  2. Related to the first question: is there any conservation law known for some system that in all probability we would not have discovered without Noether's Theorem and that has served as an important tool in understanding its dynamics?
  3. Has Noether's Theorem enabled the building of any model(s) for real dynamical systems (even simple ones) that would have been prohibitively difficult to build without knowledge of the theorem?

In short, is Neother's Theorem just a beautiful, shiny thing that's fun to look at and which gives insight into how dynamical systems work, or is it something that has really been necessary in doing practically important stuff related to real systems?

I'd appreciate concrete examples -- the more the better.

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    $\begingroup$ Within field theory, Noether's theorem is such a fundamental tool that it's hard to associate specific things to it. (Compare: "if we did not know the quotient rule, would our ability to predict the behavior of any real dynamical system be impaired in practice?" Well, I can't think of a specific reason I've had to use the quotient rule recently, but if I didn't know it, I wouldn't really understand calculus, and I definitely need calculus to do anything at all.) $\endgroup$
    – knzhou
    Feb 26, 2020 at 2:50
  • $\begingroup$ @knzhou That example has come to mind, but It's not obvious to me that there are e.g. conserved charges/currents in field theories that would have been completely hidden without Noether's Theorem. In other words, are there conserved quantities whose existence were brought to light through Noether's Theorem and that one can reasonably argue would not have been computed without it? $\endgroup$ Feb 26, 2020 at 5:03
  • $\begingroup$ @JonCuster Yes. Without theory, and in particular without mathematical models, I'd claim that no physicist, no matter how practically-minded, could get very far at all in e.g. predicting the dynamics of systems in the real world. I'd argue that this applies to science more generally as well. $\endgroup$ Feb 26, 2020 at 5:05
  • $\begingroup$ I would say that all the discussion of quantum anomalies (with their vast applications to topological phases of matter) is rooted in applications of Noether's theorem. Sometime explicitly, many times implicitly (as @knzhou commented). I don't think that a chiral anomaly, for example, could be understood without the theorem, and we would be left puzzling on the meaning of triangle diagrams etc. $\endgroup$
    – user245141
    Feb 26, 2020 at 10:51
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    $\begingroup$ Si Noether n’existait pas, il faudrait l’inventer $\endgroup$ Apr 27, 2021 at 23:30

2 Answers 2

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Top 10 reasons

This is a great question which I myself have thought long about. The way Noether's theorem is presented in textbooks, it really is just this nice shiny thing that doesn't factor much into anything. However, nothing could be further from the truth. Here I am going to be presenting, in order from least important to most important, why Noether's theorem is so vital and USEFUL to physics!

  1. Aesthetics/Beauty. While I think this is the least important reason, there's just something very pleasing to human beings about saying conservation laws come from symmetries. It's aesthetically pleasing, much like a work of art. Actually, I think that it's far more beautiful to think about it not in the Lagrangian formulation as it is usually presented, but instead in the Hamiltonian formulation. Then you have a whole rich interrelated set of phenomena rather than just a single directional symmetry --> conservation law. But I digress.

  2. Noether’s Theorem gives you an automated way of finding conserved quantities. It’s easy to become an aficionado at reading off all of the conserved quantities from a given Lagrangian. For example, while Isaac Newton knew about the conservation of momentum, he didn’t know about the conservation of energy. It took a long time for people to finally come to realize there was a quantity called energy that was conserved with a lot of convoluted history. In the past it would take a long time for people to find conserved quantities, relying on a mounting basis of empirical observation. Now we can identify conserved quantities very easily, and know where to look to find them.

  3. Conserved quantities make it easy to solve problems. Sometimes if you have enough conserved quantities, your system is integrable. Other times, like in thermodynamics, even though your system behaves in such a complicated manner, certain things like energy will always be conserved. In fact, the conservation of energy is maybe the single most important assumption of thermodynamics, allowing you to define temperature and deduce the Boltzmann distribution and whatnot.

  4. Noether's theorem allows you to identify the expression for the same conserved quantity between different systems! This one is huge. Consider, for instance, Feynman's disk paradox, appearing in section 17-4 of Volume II in his lectures, linked here. Long story short, the paradox asks, "how can this disk rotate if angular momentum is conserved?" The profound answer is that the electromagnetic field itself can have its own angular momentum too! It will exactly cancel out the angular momentum from the disk, allowing the disk to turn! In fact, you can make other thought experiments to identify that the electromagnetic field also can carry momentum, given by the Poynting vector. However, if we could have used Noether's theorem, we wouldn't have had to be so clever, constructing intricate thought experiments to identify what exactly the expressions for angular momentum or momentum are for an EM field. We could just start with the Lagrangian for the coupled disk+EM field system and use Noether's theorem for the rotation symmetry to correctly identify the expression for total angular momentum! (Actually, this might be a sort of bad example, because there's this whole issue of the "improved" stress energy tensor for an EM field which comes from also doing a gauge transformation, but let's put that technicality aside, because its non essential.) So we can see that Noether's theorem allows us to say "THIS quantity is the angular momentum of an EM field, and THIS quantity is the angular momentum for the disk, and their sum is conserved." In other words, Noether's theorem allows us to identify analogous conserved quantities for different systems, it allows us to justifiably figure out what the expression for angular momentum is in totally different systems with different Lagrangians.

  5. Noether's theorem also tells us when stuff isn't conserved (which is when there isn't a symmetry). For instance, in an expanding universe, because time isn't a Killing vector, the energy of say a photon moving across the cosmos won't be conserved but will actually shrink as its wavelength is stretched out.

  6. Conservation Laws are good to know because if part of a quantity that you think ought to be conserved “goes missing,” as it might be carried away by a previously undetected particle. Take, for instance beta decay, where a neutron decays into a proton, electron, and anti neutrino. Before people knew about neutrinos, this presented a bit of an issue. It looked like a neutron was just decaying into a proton and electron, but there is no way for angular momentum (spin) to be conserved in this process. It was Pauli who said that there must be some particle we are not detecting, a neutrino, carrying away this angular momentum. While I doubt Pauli himself was thinking in terms of Noether's theorem, he certainly could have been. As long as rotational symmetry isn't violated, then we know with 100% certainty that angular momentum must be conserved somehow.

  7. Conversely, if you know that you have some conserved quantity, you know what symmetries the Lagrangian must respect. It can not be overstated how crucial this type of logic was for building the Standard Model of particle physics, and how crucial it is for people making models for beyond the Standard Model. Just look at the tables of conserved (or sometimes approximately conserved) quantities in particle physics. Lepton number, baryon number, weak hypercharge, etc. If you are working in a particle collider, and you can see which processes are allowed and which are not, how do you actually convert that into a Lagrangian? If all your interactions obey lepton number conservation, you better make sure that the terms in your Lagrangian obey lepton number symmetry. If you observe that, rarely, lepton number isn't conserved, then you must put a term in your Lagrangian that violates lepton number symmetry! (The nice thing about Noether's theorem in quantum mechanics is that, for compact $U(1)$ symmetries, the only allowed values will be discrete multiples of some minimum value, so you can assign integer (or fractional) values of these quantities to every particle.) In fact, in a standard course on the Standard Model, you almost always talk in terms of the symmetries and conserved quantities, making massive tables of all the values of all the different types of charges for the different particles, and what interactions respect which charge conservation laws, and what symmetries these come from, etc. In other words, the symmetries give a much better way of classifying all these particles than the Lagrangian!

  8. Noether's second theorem is arguably more important than the first. Morally, it says that particles with gauge symmetries can only couple to conserved currents. This is really the same thing as Weinberg's soft photon/graviton theorem. In the graviton case, for instance, diffeomorphism symmetry means that the stress energy tensor of all particles it interacts with must be conserved (and with all the same coupling $G$), giving an explanation for the universality of the gravitational force. (You can read more about this in the context of E&M here.)

  9. New Laws of Physics may not obey old symmetries. Therefore, looking for “approximate symmetries” helps us find new laws which break old symmetries. Here is an example from history. The Hyperon and Kaon are produced copiously when protons are smashed together. However, it was noted that they decayed very slowly. Things that decay slowly, as opposed to quickly, usually do so because the weak force is mediating the decay. Gell-mann then postulated that this was because of a new approximate symmetry called “strangeness,” which was conserved in nuclear interactions but not in weak interactions. Looking back, we can now see what was happening. “Strangeness” referred to the number of strange quarks were present in each particle. The nuclear force conserves “strangeness,” while the weak force can change a quark of one flavor into a different flavor. Now, that is ancient history. What about now? Well, analogous problems, like the non conservation of lepton number, are basically of the same type. One can also ask why is there more matter than anti matter, i.e. the problem of baryogenisis? Another question, do protons decay? The logic behind the "approximate" symmetries is that the terms that break them in the Lagrangian are suppressed severely at low energies, meaning that it's very rare to see them violated. So, stuff like baryon number or lepton number is really nothing more than a "low energy accident," not a truly deep symmetry of nature. Actually, black hole evaporation gives us a huge hint that, in quantum gravity, there are probably no symmetries aside from the ones required by mathematical consistency. This is discussed in a beautiful talk by Witten here. You could, for instance, make a black hole by using mostly baryons, but when it decays it'll radiate off its energy in mostly massless particles like photons and gravitons. This suggests that a final theory of quantum gravity will have none of these extraneous ("global") symmetries like baryon number. However, symmetries required for mathematical consistency, like gauge ("local") symmetries and CPT, aren't violated by black holes evaporation, and therefore quantum gravity. Note that electric charge, which is a global symmetry like baryon number, is actually conserved by black hole evaporation because it is "protected" by the local $U(1)$ gauge symmetry. This is all stuff which is important to keep in mind for people thinking about how to formulate theories of quantum gravity.

  10. And, finally... the theorem just changes the way we think about everything. Noether's theorem simply rewires your brain in ways that are difficult to explain. Emmy Noether I think captured the impact of her work the best (of which this one theorem was just a small part of).

My methods are really methods of working and thinking; this is why they have crept in everywhere anonymously. - Emmy Noether

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    $\begingroup$ Top 10 by whose standards? (Don't get me wrong: they are good reasons!) $\endgroup$
    – TLDR
    Apr 26, 2021 at 20:00
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    $\begingroup$ Importantly, terms in the Lagrangian violating the symmetry are represented in the violation of the conservation of the respective current. Particle physics simply couldn't work without this feature: think chiral symmetry explicit breaking. $\endgroup$ Apr 27, 2021 at 0:22
  • $\begingroup$ Great answer! "I think that it's far more beautiful to think about it not in the Lagrangian formulation as it is usually presented, but instead in the Hamiltonian formulation." may you provide a link/hint/reference? Thank you! $\endgroup$
    – Quillo
    May 9 at 17:12
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    $\begingroup$ @Quillo scholar.harvard.edu/files/noahmiller/files/… $\endgroup$ May 9 at 17:31
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I think that Noether's theorem is not essential from a purely practical point of view, in the following sense. You never need Noether's theorem to define the equations that a given system obeys. In classical mechanics, you can write down the equations of motion for a system, and evolve them on a computer with brute force. In quantum mechanics or quantum field theory, you can write down the theory in whatever your favorite formulation is, and directly compute scattering amplitudes / evolve the wavefunction / evaluate the path integral / whatever without ever making use of Noether's theorem.

Furthermore, I think it is rare that Noether's theorem is used to discover a qualitatively new feature of a physical system.

  • The "standard" examples of conserved quantities -- energy, momentum, angular momentum -- were all known before Noether's theorem.
  • Many "exotic" conserved quantities were discovered by accident, and only later realized to follow via Noether's theorem from some bizarre hidden symmetry. Examples include the Laplace-Runge-Lenz vector and the dual superconformal symmetry of $\mathcal{N}=4$ super-Yang Mills.
  • Noether's theorem is never necessary to find a conserved quantity. For example, in the standard model, while various $U(1)$ symmetries combined with Noether's theorem make it obvious that, say, baryon number will be conserved, it's hard to imagine people would fail to realize baryon number was conserved without Noether's theorem. (For example one would presumably notice when computing scattering amplitudes that all baryon-number-violating diagrams vanish in the Standard Model).

Having said that, Noether's theorem is still tremendously important.

Noether's theorem is an organizing principle by which we understand the internal structure of our theories. Noether's theorem lets us reason about physical theories at a deep and abstract level, and therefore changes our view of physics and suggests questions to us that would be difficult to even ask otherwise, and also tells us that some questions are not interesting to think about.

For example: the world without Noether's theorem is an apparently ad hoc collection of conservation laws. How can we be confident that we have found them all? Noether's theorem explains the origin of these laws. This cuts off a brute direction of trying to check if various random quantities are conserved, and also suggests when an unexpected conservation law is found (like the Laplace-Runge-Lenz vector) to look for a hidden symmetry explaining the conservation law ($SU(4)$) which in turn can be exploited to solve the original problem in an elegant way (the spectrum of the hydrogen atom can be derived using group theory methods by using the $SU(4)$ symmetry of the Kepler problem).

Additionally, quantum field theory is hard and it's not always possible to start from first principles. There are successful phenomenological approaches such as current algebra which are crucially based on the assumption that Noether's theorem will guarantee the existence of operators with certain properties that can be used to parameterize scattering amplitudes, even though we don't know the actual underlying theory. This method was incredibly valuable in the development of the theory of the strong interactions.

The last argument I want to make in favor of Noether's theorem (not the last possible argument by any means!) is that Noether's theorem suggests ways to build new theories. For example, starting from the theory of a complex spin-1/2 field, we can notice there is a $U(1)$ symmetry and a corresponding conserved current. This conserved current can then by used to couple spin-1/2 field to a $U(1)$ gauge field. For more complicated theories like $SU(N)$ Yang-Mills theories and gravity, the Noether method for building consistent couplings is extremely useful. Furthermore, this Noether method can be used to prove the absence of a consistent theory of (a finite number of) higher-spin fields.

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