[HN Gopher] Noether's Theorem Revolutionized Physics
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Noether's Theorem Revolutionized Physics
Author : MindGods
Score : 116 points
Date : 2025-02-07 14:33 UTC (2 days ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| twiceaday wrote:
| 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.
|
| It's such an aha moment.
|
| PBS Space Time: https://www.youtube.com/watch?v=04ERSb06dOg
| gauge_field wrote:
| 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.
| eru wrote:
| 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.
| immibis wrote:
| 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.
| mckirk wrote:
| 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'.
| wholinator2 wrote:
| Well i have heard the term 'conservation of misery' quite a
| few times since starting my PhD
| ruuda wrote:
| John Carlos Baez on that article:
| https://mathstodon.xyz/@johncarlosbaez/113964127171705485
| ndsipa_pomu wrote:
| I wonder if the cut off point of science popularisation is
| related to the point where maths becomes the most useful way to
| explain what's going on?
| jahnu wrote:
| 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.
| cubefox wrote:
| Optimal would be something like 3blue1brown math
| animations, but for physics instead of pure mathematics.
| ndsipa_pomu wrote:
| I'd definitely watch that.
| canjobear wrote:
| This is a booming genre. For example this one popped up
| in my recommendations yesterday
| https://www.youtube.com/watch?v=uVKMY-WTrVo
| mperham wrote:
| Sean Carroll's Biggest Ideas in the Universe YouTube series
| is fantastic. Just enough math to be interesting but
| nothing requiring a math degree.
|
| https://youtu.be/HI09kat_GeI
| mr_toad wrote:
| Feynman might have disagreed.
| practal wrote:
| If you read the baez article, he references a great article by
| Atiyah, I've posted it:
| https://news.ycombinator.com/item?id=42989419
| dustingetz wrote:
| "inverse temperature is imaginary time" is in the final
| sentence of the abstract of the paper he links - cool
|
| https://arxiv.org/pdf/2006.14741
| OskarS wrote:
| 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.
| GoblinSlayer wrote:
| It's for people who know about conservation laws and think
| they are fundamental and somehow unvariable.
| sesm wrote:
| 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?
| MeteorMarc wrote:
| Obligatory:
| https://www.reddit.com/r/physicsmemes/comments/gtzprh/noethe...
| bmacho wrote:
| > 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.
|
| [0] :
| https://www.google.com/search?q=general+relativity+and+conse...
| sigmoid10 wrote:
| 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.
| bmacho wrote:
| That was Einstein's 1905 paper about SR not his 1915 paper
| about gravity.
| talismanick wrote:
| 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)
| sigmoid10 wrote:
| Nope. It's just a bit more complex to define what "energy"
| even is on a dynamical spacetime (remember that our usual
| constant known as _time_ is part of a varying field in GR).
| But there 's nothing stopping you from coming up with an
| equivalent conserved current due to a global symmetry as
| laid out by Noether. This fact is even used e.g. in the
| Hamiltonian formulation of GR. See here for a detailed
| explanation:
| https://physics.stackexchange.com/questions/2597/energy-
| cons...
|
| This is an old misunderstanding that dates back to the
| early stages of GR research and has nothing to do with any
| current crisis.
| drdeca wrote:
| 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).
|
| A new heaven and a new earth.
| immmmmm wrote:
| The beauty and power of the Noether Theorem is what pushed me to
| theoretical physics.
|
| I consider it one of mankind's greatest achievement.
| adastra22 wrote:
| "Noether" nominative determinism strikes again in physics.
| ElDji wrote:
| "Like most Jewish academics in Germany, Emmy Noether was fired
| after the Nazis came to power in 1933. She left later that year
| for Bryn Mawr College in the U.S [...]"
|
| Compared to what's happening now, it's totally frightening.
| seanhunter wrote:
| Also, before that she worked for several years as an unpaid
| faculty member because she joined the faculty because David
| Hilbert recognised the importance of her work but there was
| some sort of Prussian regional bureaucrat who would have had to
| sign off on her getting a paid position or something and they
| and/or the university didn't believe that a woman should teach
| at a university. So one of the giants of abstract algebra who
| made a key discovery in physics got screwed over for being a
| woman _and_ then screwed over again for being Jewish.
| layer8 wrote:
| Incidentally, there was a report on Friday that German research
| institutions are seeing a substantial increase in applications
| from the US, presumably due to the US government cutting
| research funding.
| dist-epoch wrote:
| What about information? We know that it is conserved, so what is
| the corresponding symmetry?
| thrance wrote:
| Time symmetry I think? Any computation has to be reversible in
| theory?
| layer8 wrote:
| See https://physics.stackexchange.com/a/75009.
| Isamu wrote:
| I recommend the book Einstein's Tutor" which came out last year.
|
| 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.
| leephillips wrote:
| I approve this message.
| jakobschwich wrote:
| Not much meat in the article unfortunately. Far too short to
| contain anything substantial
| pfdietz wrote:
| It's sad she died when she did. Had it been a few years later
| penicillin would likely have saved her.
| btilly wrote:
| 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.
| nyrikki wrote:
| You are confusing the map for the territory.
|
| 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.
| mjburgess wrote:
| > 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.
| adrianN wrote:
| Classical mechanics makes sufficiently accurate predictions
| to enable essentially all the engineering we do and was
| invented after we have been engineering things for a few
| millennia.
| mjburgess wrote:
| If you construct highly controlled experimental
| conditions predicated on classical mechanics being true,
| then in those scenarios, the model predicts.
|
| But in almost all cases it fails to predict, because the
| situation is vastly too complex to model. You are only
| able to construct devices (eg., steam engines, baloons,
| etc.) which are "simple" in the relevant ways, because
| classical mechanics successfully explains real properties
| of objects.
|
| If it didn't, you'd have no idea how to take an ordinary
| situation like, "dropping some objects off a cliff" into
| one where you could actually predict where they will land
| (ie., by waiting for a day with no wind, by shaping the
| objects to limit drag, and so on --- without controlling
| for these accidental features, you'd not be able to
| predict where anything would land other than "down there
| somewhere").
| nyrikki wrote:
| Western reductionism/Laplacian determism was falsified
| several times by counterexamples like quantum
| superposition, Cantor diagonalization etc.
|
| The System Identification Problem has also been shown to be
| equivalent to the halting problem.
|
| https://philarchive.org/rec/DIEEOT-2
|
| The common joke about spherical cows in physics also points
| to the predictive, descriptive nature of the field.
|
| The equivalence of various QM interpretations also points
| to the scientific realist views as being incorrect.
|
| Rice's theorm, Godel, Wada property etc... also demonstrate
| the problems of confusing the map with the territory.
|
| There are further topics like indecomposable continua that
| arise frequently and naturally in nonpathological dynamical
| systems. Especially with time delayed ODEs, Hamilton
| systems etc?
|
| Are you arguing that Hamilton systems aren't 'physics'?
|
| The value of western reductionism is finding 'effective
| procedures' but teaching it as being reality is more about
| didactic convention and convenience.
|
| 'Hard science' is a term for study the universe through
| theories, hypotheses and experiments.
|
| It is still about making predictions that match
| observations.
|
| This it is descriptive vs prescriptive.
| mjburgess wrote:
| You're assuming an idealization is a fictionalization,
| rather than a way of getting at an essential property
| (ie., a stable, real, causal feature) of a system.
|
| Treating a cow as spherical is a means of selecting is
| _real_ property of volume, as it is causally efficacious
| in say, a gravitational field -- whilst discarding is
| accidental-random variations in volume across all cows.
|
| That we can treat cows as spherical, and obtain relevant
| dynamics should show that this early 20th C.
| instrumentalism is false. By idealization one _selects_
| the actual properties of objects for explanatory
| modelling -- one does not invent them or otherwise
| construct a merely instrumental fiction. Cows have
| volume, whose variation is accidental across cows, their
| volume expressed as a sphere selects better for their
| essential volume.
|
| Very few, if any, theories of physics are predictive in
| almost any situation without this idealization -- because
| it is impossible to describe, eg., the volume of any
| actual cow. An actual cow has uneven density, shape, etc.
| and would require a significant amount of data to
| describe -- nearly all of which does not bare on the role
| its mass plays in a gravitational field.
|
| What idealization does is create hypothetical scenarios
| which imagination all irrelevant causes are controlled,
| and all accidental properties are uniform (/ of a known
| distribution) -- so that the model can focus on
| _Explaining_ the target _Essential_ property in question.
|
| These hypotheticals are not inventions, they are means of
| targeting _what is being explained_.
|
| If you look at the predictive accuracy of scientific
| models, as applied in any actual scenario, they are fall
| apart -- almost nothing at all can be predicted, because
| all actual situations comprise innumerable accidental
| features which cannot be modelled.
| whatshisface wrote:
| Determinism is falsified by quantum measurement, not
| superposition (which is deterministic), and even then
| only pragmatically, rather than philosophically.
| mnky9800n wrote:
| I like mostly agree with you but I kind of think as
| prediction versus explanation as more of a spectrum where
| you can weight both. Like I mostly think about it from a
| machine learning perspective where if you do the matrix
| inversion you can say well this is exactly where these
| coefficients come from but a random forest you might only
| get a shap value and an transformer will never give you the
| exact answer as to how it arrived at the solution since it
| is measuring a latent space. And in physics you desire a
| system of equations that can be used to describe some
| dynamics. And if it is terrible at doing it, then you are
| not going to trust the model much. But like the power of a
| model comes from its predictive ability. Like how Ptolemaic
| model mostly gets the planets right but for the wrong
| reasons and newtons law of gravitation gets it mostly right
| for the right reasons and it didn't need to get regular
| adjustments like Ptolemaic. And so in that example you have
| both the predictive ability and the explainability both
| being important in different ways.
|
| I recommend galit Shmueli paper called "to explain or
| predict ". I also like the "two cultures" paper by Leo
| breiman. These are both machine learning / statistics views
| on this topic.
| mjburgess wrote:
| Techniques (eg., of ML or non-ML) do not decide between
| explanation and prediction. It's common in ML to speak
| like many computer scientists do, completely ignorantly
| of science, and suppose somehow it is the algorithm or
| how we "care about" it which matters -- no.
|
| It is entirely due to the experimental conditions which
| are a causal semantics on the data, not given in the data
| or in the algorithm -- something the experimenter or
| scientist will be aware of, but nothing the computer
| scientist will even have access to.
|
| Regression is explanatory if the data set is causal, has
| been causally controlled, the data represents measures of
| causal properties, these measures are reliable in the
| experimental conditions, the variables under question
| each have causal relationship, and so on. Conditions
| entirely absent in the data and in the algorithm, and in
| anything to do with ML.
|
| In a large number majority of cases where ML is applied,
| the data might as well be a teen survey in cosmo magazine
| and the line drawn an instrumental bit of pseudoscience.
| This is why the field is not part of scientific
| statistics -- because it aims to address "data as number"
| _not_ "data as casual measure". The computer scientist
| thinks that ML can be applied to mathematics, or games
| like chess which is a nonsense scientifically (since
| there are no empirical measures of the causal properties
| of chess).
|
| ML is the algorithms of statistics without any awareness,
| or use of, any scientific conditions on the data
| generating process.
| mjburgess wrote:
| 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).
| whatshisface wrote:
| 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?
| mjburgess wrote:
| The problem with 'motion' as the load-bearing property here
| is that there's an infinite number of derivatives at stake:
| x', x'', x''', etc. -- what is "the motion" ? What is
| supposed to be conserved about that motion?
|
| The intuition here seems to be that there's a continuity of
| some property involved in the "transmission of matter over
| time" which is unbroken, but its not clear what this is
| supposed to imply.
|
| It doesnt seem to imply, for example, that the universe
| operates like a closed system of motion simply because this
| property (whatever it is) is unbroken. There can be "global
| motion" without the need for
| discontinutiy/randomization/discretizatino/etc. in the
| trajectories of matter.
| Iolaum wrote:
| In those scales (cosmology+Field Theory) energy is derived and
| defined as the "thing" that is conserved after you apply time
| symmetry.
|
| 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.
| btilly wrote:
| As much as we want time symmetry, we still live in an
| expanding universe. So we have at least some asymmetry.
| Iolaum wrote:
| Time symmetry applies to the physical laws governing the
| universe, not the distribution of mass in it.
|
| For a simpler example look at Newton's 3 laws. They have
| time symmetry but still allow for things to move around!
| btilly wrote:
| It also doesn't apply to the structure of space-time.
| Which is not a situation that Newton's laws can deal
| with.
| ajkjk wrote:
| It's widely discussed in physics, I don't know why you would
| have the impression it isn't.
| invalidator wrote:
| "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."
|
| --
| https://www.forbes.com/sites/startswithabang/2015/12/19/ask-...
| btilly wrote:
| That only works if radiation density and expansion are
| connected. General Relativity says that they aren't.
| mnky9800n wrote:
| How does general relativity say that?
| btilly wrote:
| The term for the stress-energy tensor has the wrong
| impact on expansion for a simple relationship of any
| kind.
|
| You can rescue it with pseudo-tensors, but then you force
| the existence of a preferred reference frame. Which
| undermines the principles of GR.
|
| Some still argue for that. Others don't.
| layer8 wrote:
| Wave-function collapse also violates conservation of energy.
| (Unless you believe in Many-Worlds, that is.)
| whatshisface wrote:
| A state that can have the expected value of the energy change
| post measurement isn't considered to have an energy, rather
| an expected energy.
| layer8 wrote:
| This is about the energy of the system as described by the
| Schrodinger equation.
| PopePompus wrote:
| 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.
| jll29 wrote:
| If you would like to know more about her:
|
| Her legacy https://www.mathgenealogy.org/id.php?id=6967
|
| About her https://en.wikipedia.org/wiki/Emmy_Noether
|
| Where she lived and studied
| https://thonyc.wordpress.com/2011/05/07/the-house-where-emmy...
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