Noether's Theorem Revolutionized Physics
quantamagazine.org
quantamagazine.org
There's some related comments by him here: https://kottke.org/17/06/if-you-cant-explain-something-in-si...
Maybe the issue is that some things are too complicated to be explained without maths and aren't accessible to lay people as they're so counter-intuitive.
I’m also reminded on the scene in “Severance” where the members of MDR are unable to explain their job to another department.
However, when it comes to a "deep" understanding, then maybe numbers aren't enough and we need to be able to intuit what's going on that creates those particular numbers.
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"?
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.
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.
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").
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, Gödel, 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.
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.
You can view wave function collapse as invalidating your priors and it still works without observation.
The special case of quantum superposition being indecomposable function, or one you can't Curry if you prefer.
QM is actually lucky that there are only two exit basins, n≥3 is where you get the stronger form of indeterminism.
Classic Chaos is deterministic with inf precision, but riddled basins require absolute precision which is stronger, and the wada property is still not deterministic with absolute precision.
Example of the Newton fractal and the wada property.
https://gauss.math.yale.edu/fractals/MandelSet/RealNewton/NM...
Binary black holes and the Wada property
https://arxiv.org/abs/1807.10741
Think of it as going north from Mexico, crossing the border and finding yourself in Canada.
You can also approach it from KAM and porous sets etc... if that works better for you.
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.
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.
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 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.
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.
For a simpler example look at Newton's 3 laws. They have time symmetry but still allow for things to move around!
-- https://www.forbes.com/sites/startswithabang/2015/12/19/ask-...
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.
It's such an aha moment.
PBS Space Time: https://www.youtube.com/watch?v=04ERSb06dOg
> 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.
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 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...
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)
This is an old misunderstanding that dates back to the early stages of GR research and has nothing to do with any current crisis.
...and so you have to pick an appropriate underlying vector field you call time to cancel this out and get back the invariant, throwing a wrench into calculations... as a reply to the post you linked points out. At the end of the day, you haven't demonstrated that it preserves the invariant so much as you've changed the question to find another conserved quantity and called that energy instead. This lines up with my broad observation that we're out of runway for the 20th century's symmetry-reliant problem solving and hence have to be increasingly clever with setups to apply generalizations of them.
I definitely learned something new today, though. To boot, these pseudotensors are tamer than I thought they'd be - I expected calculational hacks with no formal analogues explored only in old papers, but sections on jet bundles is something I'd expect in a differential geometry text. Maybe we'll see progress along these lines in the next couple decades.
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.
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...
In general relativity, symmetries that exist in the action for a flat spacetime are violated in curved spacetime. So, in curved spacetime, the corresponding quantities are not conserved. One example is energy. The reason has to do with the fact that integrating a tensor along a closed loop in curved spacetime might yield a nonzero result due to the curvature itself, rather than due to the dynamics of what is being integrated.
I don't think that the article goes into any of this. The introduction about general relativity did seem like a curve ball, even if it's historically accurate.
It is postulated by the linear unitary time-reversible Schrodinger/Dirac equation, and assumed by the Many Worlds Interpretation, but contradicted by the non-linear probabilistic time-irreversible Measurement Axiom of conventional QM.
Some people (esp. Penrose) postulate that gravity can collapse the wavefunction, and hence destroy information (esp. Black Holes). The Black Hole Information Paradox is a hotly contested current topic.
Compared to what's happening now, it's totally frightening.