I think he sets quite a high bar here. Like, just going through the basics of Noether’s Theorem is so far beyond what normally passes for popular science (which usually boils down to “the universe is really big you guys” and “quantum stuff is weird and nobody understands it”). Like, just using words like “conservation laws” and “Lagrangian” is risky already.
Personally, I would have liked it to dig deeper (as I already heard the basics of Noether’s theorem, but am not a physicist or have studied it in any great depth), but Quanta is not a scientific journal, it’s a pop-sci magazine. The article is a great intro.
I followed the link, that's what he means by 'cool stuff that's missing':
> In short:
> The key to Noether’s theorem is the requirement that we can freely reinterpret observables as symmetry generators, and vice versa — in a way that’s consistent with the action of symmetry generators on both observables and symmetry generators.
> In classical mechanics this is achieved by a hybrid structure: a Poisson algebra, whose elements are both observables and symmetry generators.
> In an algebraic approach to quantum theory, this requirement singles out complex quantum mechanics. i =√−1 turns observables into symmetry generators, and vice versa.
How would one explain this to the audience of Quanta Magazine?
First have a clear statement of the context. Second, define and explain all the obscure terminology and mathematical notation -- at least give good references.
Gee, in one discussion of Lagrangians, saw "configuration space" and "constraints". Okay, studied topological spaces, vector spaces, inner product spaces, measure spaces, Banach spaces, Hilbert spaces, probability spaces, but never saw a definition or explanation of a "configuration space".
"Constraints"? Kuhn-Tucker optimization theory has a lot on constraints, for one of the issues there was a question, and I solved and published it. Thought I had some background in "constraints", but the Lagrangian explanations didn't make clear what they meant by "constraints".
A "symmetry generator"? This is the first I ever heard of any such thing although ugrad math honors paper was on group representations.
Then there was "phase space": What does that have to do with "phase" in light waves and sound waves? Sounds like one word with two different meanings?
E.g., "Poisson algebra": Okay, there is the Poisson stochastic process, e.g., seems to get assumed in "half-life" calculations, and with more in
Erhan \c Cinlar,
{\it Introduction to Stochastic Processes,\/}
ISBN 0-13-498089-1,
Prentice-Hall,
Englewood Cliffs, NJ,
1975.\ \
and there is abstract algebra, e.g., groups, rings, fields, ... But a "Poisson algebra"?
Yeah I think that’s probably true and why I greatly admire efforts by people like Steven Strogatz and especially Sean Carroll who are leading the way from no-maths pop-sci to high school maths pop-sci where you know you don’t want to actually work with the maths but you can start to get an appreciation for the components of it and what the implications are.
This is probably the best layman’s approach to Noether, her impact, and how she probably didn’t think much about the theorem later because she wasn’t interested in physics and abstract mathematics was her consuming passion.
Here is a cosmological issue from Noether's Theorem. An expanding universe shows time asymmetry, therefore it might not have conservation of energy.
This looks like it actually happens. Photons going through empty space go through cosmological redshift, reducing their energy over time. The energy does not appear to go anywhere - it is just gone.
I have no idea why this example is not more widely discussed.
I think it's widely known amongst physicists that energy conservation doesn't hold at cosmological scales.
I've not heard of redshift being a case of this -- I'd imagine because the scales at which conservation breaks down are, to my recollection, none where you could observe red-shifted photons, or anything at all because these "scales" entail causal isolation. Eg., two areas of the universe which are totally causally isolated from each other, may across them, violate various laws of conservation.
However I do not recall seeing any reason for the latter claim, and it's something I took to be implied about the kinds of conservation violation that GR entails (ie., GR is a locally-conservative theory).
The necessity of time symmetry for energy conservation can be a little overstated. As long as the laws are holonomic, there will be a conserved quantity corresponding to the motion. You can call that quantity the energy. It won't be conserved if you jump forwards in time without allowing the state to change, but that can't ever happen, so is it really an issue?
Under our best current theory (map) General relativity, total energy might not be conserved globally, the divergence of the stress-energy-momentum tensor is zero, meaning that energy is conserved locally within a small region of spacetime.
Physics is about producing models that make accurate predictions, it is a map, not the territory itself.
The 'crisis in cosmology' e.g. Hubble tension is most likely a sign that current models of the universe are incomplete.
Energy is conserved in static spacetimes and asymptotically flat spacetimes.
The Friedmann-Robertson-Walker spacetimes that cosmology often uses are not static nor asymptotically flat.
It is widely discussed, but all models are wrong, some are useful.
Noether's theorm is a power tool to find useful models.
> Physics is about producing models that make accurate predictions
Very few models in physics ever make accurate predictions -- only in very limited experimental circumstances, mostly ones inaccessible at the time these models were developed.
The ability to craft these experimental conditions, which enable accurate prediction, is predicted on the models actually describing reality. How else would one control the innumerable number of causes, and construct relevant devices, if these causes did not exist and the devices werent constructed to measure reality?
No no, the hard sciences are not concerned about prediction at all. They are concerned about explanation -- it is engineers who worry about predictions, and they quickly find that vast areas of science -- esp. physics -- is nearly impossible to use for predictive accuracy.
"as the Universe expands, photons lose energy. But that doesn't mean energy isn't conserved; it means that the energy goes into the Universe's expansion itself, in the form of work."
TLDR: Do an experiment, then move 10 meters to the left (or rotate 90 degrees, or wait a few days) and do it again. The results don’t change, because the laws of physics don’t change. This realization alone is enough to produce conservation laws. Translational and rotational symmetries produce conservation of linear and angular momentum, and the time symmetry produces conservation of energy. Each symmetry you find leads to new physics.
Symmetries produce conservation laws if you accept and understand Lagrangian mechanics. That's a big asterisk IMO especially if you've never heard of Lagrangian mechanics and then you try to understand Noether's theorem.
Doesn't getting from Newton to Lagrange already rely on the existence of conservation laws? Apparently if we take Lagrange as fundamental, then it works, and a variation of it works in quantum mechanics, so it does seem to be fundamental, but if you're trying to get from Newton's laws to Noether's theorem, you can't get from here to there without fully grasping Lagrange first.
It also works backwards: for (most) conserved quantities, you can also find a symmetry.
> Each symmetry you find leads to new physics.
There's a few caveats and asterisks for that. Eg Noether's theorem only applies to continuous symmetries. Eg Noether's theorem has nothing to say about mirror symmetry or time reversal symmetry.
Another point to appreciate is how universal this principle of symmetry is. It is used in every branch of physics going from Classical Physics (Lagrangian Formulation) to quantum physics (with Feynman's Path Integral Formulation), from conservation of momentum to conservation of electric charge in (U(1) Symmetry) of fundamental particles. The fact that she was able to do this as a woman 100 years agos is also amazing.
I wonder what the conservation law is connected to the 'analysis invariance', i.e. the fact that no matter how well you've thought through everything beforehand, there will still be some recalcitrant pocket of the experiment that behaves confusingly. Maybe that's the 'conservation of surprise'.
I've never understood why Marie Curie is so celebrated in the popular press, but Noether is largely ignored. Noether's work is much more important, IMHO.
Noether did maths, Marie Curie did physics, physics is more popular than maths.
It is easy to see why, symmetries, conserved properties, boring, nuclear power, atomic bombs, deadly radiation, exciting.
Einstein is a rare instance of a popular theoretical physicist. But all the mathematicians whose work led to the theories of relativity are largely unknown to the public.
Marie Curie won 2 Nobel Prizes, for physics and chemistry. She was the first woman to win a Prize, and the first person to win two. She died of radiation exposure from her experiments. Can't really argue with that.
> In the fall of 1915, the foundations of physics began to crack. Einstein’s new theory of gravity seemed to imply that it should be possible to create and destroy energy, a result that threatened to upend two centuries of thinking in physics.
Not just seem to imply, but they do imply[0]. Does that mean that we can build a machine that generates energy and negentropy forever (e.g. an artificial Sun), thus, we can outlive the heat-death of the rest of the Universe? Yes, absolutely. But there are other existential threats, like the collapse of false-vacuum. In the end, it is not known if we have limited or unlimited time here, but Noether's theorem doesn't answer that.
That some violation of energy-(if-we-don’t-count-energy-from-large-scale-spacetime-shape-stuff) (not counting energy from large-scale spacetime shape stuff may be sensible, because AIUI you can’t really obtain it as just a sum of local quantities, and like, it depends on boundary conditions or something) occurs doesn’t imply it is possible to exploit to obtain more energy.
Does this violation even ever result in more usable energy rather than less?
Like, red-shifting photons reduces their energy…
I suppose if we wanted to do the opposite, it would be making the contraction of space result in photons being blue-shifted, but uh…
Well, that would result in things getting closer together, and unlike expansion, that seems to run into a limit at some point?
I don’t think the laws of physics as they currently are, are sufficient to support an eternity of life (or civilization). For there to be hope of that, it must be hope of something or someone outside of the laws of physics we inhabit (or are well-approximated as inhabiting).
That's wrong, because the quoted part is wrong. Relativity doesn't say you can create or destroy energy. It only says that you can convert mass to energy (and vice-versa) - because in the end they are actually the same thing. And together, they are conserved. That means we still can't have perpetuum mobile stuff unfortunately.
You're talking about E=mc^2, which follows from special relativity. That was revealed in 1905; 1915 marked the advent of general relativity, where energy conservation no longer holds.
The time translation invariance which gives rise to the conservation law is a special case of GR's broader energy-momentum conservation, namely the static one where gravity and such are disregarded altogether as in the Standard Model.
This all ties back to the present crisis of foundations, as string theory and other approaches to reconciling GR with the Standard Model strain at the edges of what Noetherian tools can yield. (see: supersymmetry)
Readit News
Personally, I would have liked it to dig deeper (as I already heard the basics of Noether’s theorem, but am not a physicist or have studied it in any great depth), but Quanta is not a scientific journal, it’s a pop-sci magazine. The article is a great intro.
> In short:
> The key to Noether’s theorem is the requirement that we can freely reinterpret observables as symmetry generators, and vice versa — in a way that’s consistent with the action of symmetry generators on both observables and symmetry generators.
> In classical mechanics this is achieved by a hybrid structure: a Poisson algebra, whose elements are both observables and symmetry generators.
> In an algebraic approach to quantum theory, this requirement singles out complex quantum mechanics. i =√−1 turns observables into symmetry generators, and vice versa.
How would one explain this to the audience of Quanta Magazine?
First have a clear statement of the context. Second, define and explain all the obscure terminology and mathematical notation -- at least give good references.
Gee, in one discussion of Lagrangians, saw "configuration space" and "constraints". Okay, studied topological spaces, vector spaces, inner product spaces, measure spaces, Banach spaces, Hilbert spaces, probability spaces, but never saw a definition or explanation of a "configuration space".
"Constraints"? Kuhn-Tucker optimization theory has a lot on constraints, for one of the issues there was a question, and I solved and published it. Thought I had some background in "constraints", but the Lagrangian explanations didn't make clear what they meant by "constraints".
A "symmetry generator"? This is the first I ever heard of any such thing although ugrad math honors paper was on group representations.
Then there was "phase space": What does that have to do with "phase" in light waves and sound waves? Sounds like one word with two different meanings?
E.g., "Poisson algebra": Okay, there is the Poisson stochastic process, e.g., seems to get assumed in "half-life" calculations, and with more in
Erhan \c Cinlar, {\it Introduction to Stochastic Processes,\/} ISBN 0-13-498089-1, Prentice-Hall, Englewood Cliffs, NJ, 1975.\ \
and there is abstract algebra, e.g., groups, rings, fields, ... But a "Poisson algebra"?
https://arxiv.org/pdf/2006.14741
I consider it one of mankind’s greatest achievement.
https://lee-phillips.org/noether/
This is probably the best layman’s approach to Noether, her impact, and how she probably didn’t think much about the theorem later because she wasn’t interested in physics and abstract mathematics was her consuming passion.
This looks like it actually happens. Photons going through empty space go through cosmological redshift, reducing their energy over time. The energy does not appear to go anywhere - it is just gone.
I have no idea why this example is not more widely discussed.
I've not heard of redshift being a case of this -- I'd imagine because the scales at which conservation breaks down are, to my recollection, none where you could observe red-shifted photons, or anything at all because these "scales" entail causal isolation. Eg., two areas of the universe which are totally causally isolated from each other, may across them, violate various laws of conservation.
However I do not recall seeing any reason for the latter claim, and it's something I took to be implied about the kinds of conservation violation that GR entails (ie., GR is a locally-conservative theory).
Under our best current theory (map) General relativity, total energy might not be conserved globally, the divergence of the stress-energy-momentum tensor is zero, meaning that energy is conserved locally within a small region of spacetime.
Physics is about producing models that make accurate predictions, it is a map, not the territory itself.
The 'crisis in cosmology' e.g. Hubble tension is most likely a sign that current models of the universe are incomplete.
Energy is conserved in static spacetimes and asymptotically flat spacetimes.
The Friedmann-Robertson-Walker spacetimes that cosmology often uses are not static nor asymptotically flat.
It is widely discussed, but all models are wrong, some are useful.
Noether's theorm is a power tool to find useful models.
Very few models in physics ever make accurate predictions -- only in very limited experimental circumstances, mostly ones inaccessible at the time these models were developed.
The ability to craft these experimental conditions, which enable accurate prediction, is predicted on the models actually describing reality. How else would one control the innumerable number of causes, and construct relevant devices, if these causes did not exist and the devices werent constructed to measure reality?
No no, the hard sciences are not concerned about prediction at all. They are concerned about explanation -- it is engineers who worry about predictions, and they quickly find that vast areas of science -- esp. physics -- is nearly impossible to use for predictive accuracy.
Also you kinda want time symmetry in a physical system otherwise you have no guarantee that today's laws of physics will be valid tomorrow.
-- https://www.forbes.com/sites/startswithabang/2015/12/19/ask-...
It's such an aha moment.
PBS Space Time: https://www.youtube.com/watch?v=04ERSb06dOg
Doesn't getting from Newton to Lagrange already rely on the existence of conservation laws? Apparently if we take Lagrange as fundamental, then it works, and a variation of it works in quantum mechanics, so it does seem to be fundamental, but if you're trying to get from Newton's laws to Noether's theorem, you can't get from here to there without fully grasping Lagrange first.
> Each symmetry you find leads to new physics.
There's a few caveats and asterisks for that. Eg Noether's theorem only applies to continuous symmetries. Eg Noether's theorem has nothing to say about mirror symmetry or time reversal symmetry.
It is easy to see why, symmetries, conserved properties, boring, nuclear power, atomic bombs, deadly radiation, exciting.
Einstein is a rare instance of a popular theoretical physicist. But all the mathematicians whose work led to the theories of relativity are largely unknown to the public.
Not just seem to imply, but they do imply[0]. Does that mean that we can build a machine that generates energy and negentropy forever (e.g. an artificial Sun), thus, we can outlive the heat-death of the rest of the Universe? Yes, absolutely. But there are other existential threats, like the collapse of false-vacuum. In the end, it is not known if we have limited or unlimited time here, but Noether's theorem doesn't answer that.
[0] : https://www.google.com/search?q=general+relativity+and+conse...
Does this violation even ever result in more usable energy rather than less?
Like, red-shifting photons reduces their energy…
I suppose if we wanted to do the opposite, it would be making the contraction of space result in photons being blue-shifted, but uh…
Well, that would result in things getting closer together, and unlike expansion, that seems to run into a limit at some point?
I don’t think the laws of physics as they currently are, are sufficient to support an eternity of life (or civilization). For there to be hope of that, it must be hope of something or someone outside of the laws of physics we inhabit (or are well-approximated as inhabiting).
A new heaven and a new earth.
The time translation invariance which gives rise to the conservation law is a special case of GR's broader energy-momentum conservation, namely the static one where gravity and such are disregarded altogether as in the Standard Model.
This all ties back to the present crisis of foundations, as string theory and other approaches to reconciling GR with the Standard Model strain at the edges of what Noetherian tools can yield. (see: supersymmetry)