The Forgotten Anniversary: 100 Years since Emmy Noether's Theorems

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In summary: The article specifically defends the idea that we need a new physics that breaks all symmetries, without mentioning Emmy Noether. It's possible that the author is referring to a paper by a physicist that was published in the past year, but I can't remember the name or source. I'm glad this didn't come out 30 years ago; I would have switched my concentration to under-water basket weaving.
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fresh_42
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I find on a website as ours, we shouldn't have forgotten this:

On January, 25th it has been 100 years, a complete century, since Emmy Noether published her paper "Invarianten bestimmter Differentialausdrücke" (Invariants of Certain Differential Expressions, Göttingen 1918). It is still the basic concept of so many physical models, from classical mechanics to the standard model of particle physics. Funnily enough that I recently came upon an article, in which a physicist defended his opinion, that we need a new physics without symmetries. As I couldn't imagine how this would look like, especially because of Emmy Noether's theorems, I didn't give it much attention and can't remember source and name. Maybe someone else has read it and remembers, so that we can have a discussion about it.

The paper(s):
https://gdz.sub.uni-goettingen.de/id/PPN252457811_1918

The thread which reminded me:
https://www.physicsforums.com/threa...reserves-symmetry-of-pde.939382/#post-5938898
 
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  • #2
Just for the record, for those interested:

Here are the two theorems in their modern streamlined form, complete with their modern streamlined proofs, and put into the full context of modern QFT:
in the PhysicsForums QFT notes.
 
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  • #3
fresh_42 said:
a physicist defended his opinion, that we need a new physics without symmetries.
It seems to me that without symmetries this new physics would have to take conservation laws as a priori phenomenological. I'm glad this didn't come out 30 years ago; I would have switched my concentration to under-water basket weaving.

Peace
Fred
 
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  • #4
I consider it one of the greatest and most amazing accomplishments ever.

I remember when I first came across it I was simply stunned what it implied - things just about nobody outside physics knows.

A mentor here has posted when he teaches it the class just sits there in silence as its import sinks in - it simply, correctly, is so far reaching in its implications.

For further background on how it came about:
https://arstechnica.com/science/201...-the-course-of-physics-but-couldnt-get-a-job/
' When it came to Einstein’s theory, Hilbert and his Göttingen colleagues simply couldn’t wrap their minds around a peculiarity having to do with energy. All other physical theories—including electromagnetism, hydrodynamics, and the classical theory of gravity—obeyed local energy conservation. With Einstein's theory, one of the many paradoxical consequences of this failure of energy conservation was that an object could speed up as it lost energy by emitting gravity waves, whereas clearly it should slow down. Unable to make progress, Hilbert turned to the only person he believed might have the specialized knowledge and insight to help. This would-be-savior wasn’t even allowed to be a student at Göttingen once upon a time, but Hilbert had long become a fan of this mathematician’s highly "abstract" approach (which Hilbert considered similar to his own style). He managed to recruit this soon-to-be partner to Göttingen about the same time Einstein showed up. And that’s when a woman—one Emmy Noether—created what may be the most important single theoretical result in modern physics.'

Yes - Emily Noether was thought by Hilbert, who is amongst the very greatest, that she was the person to do what he could not - she was that good. And the article does not mention Einsteins reaction:
'Yesterday I received a paper from Miss Noether a very interesting paper on invariant forms. I am impressed that one can comprehend these matters from so general a viewpoint. It would do no harm for the old guard in Gottingen (presumably referring to those who would not treat her on ability, but instead the silly sexism of that age) had they picked up a thing or two from her.'

Yes Noether is a heroine of mine.

Thanks
Bill
 
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  • #5
I have been teaching a small class that includes analytical mechanics for several years. After showing Noether's theorem to the students I have always taken a small 30 s break of awe and to let the result sink in. In all my classes, it is the only result I have done that for.
 
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  • #6
fresh_42 said:
I find on a website as ours, we shouldn't have forgotten this:

On January, 25th it has been 100 years, a complete century, since Emmy Noether published her paper "Invarianten bestimmter Differentialausdrücke" (Invariants of Certain Differential Expressions, Göttingen 1918). It is still the basic concept of so many physical models, from classical mechanics to the standard model of particle physics. Funnily enough that I recently came upon an article, in which a physicist defended his opinion, that we need a new physics without symmetries. As I couldn't imagine how this would look like, especially because of Emmy Noether's theorems, I didn't give it much attention and can't remember source and name. Maybe someone else has read it and remembers, so that we can have a discussion about it.

The paper(s):
https://gdz.sub.uni-goettingen.de/id/PPN252457811_1918

The thread which reminded me:
https://www.physicsforums.com/threa...reserves-symmetry-of-pde.939382/#post-5938898

Possibly Smolin: "In fact, when we impose the condition that the universe is spatially compact without boundary, general relativity tells us there are no global spacetime symmetries and no non-zero global conserved charges." https://arxiv.org/abs/hep-th/0507235
 
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  • #7
atyy said:
What are the symmetries of general relativity?

The principle of invariance:
https://www.pitt.edu/~jdnorton/papers/decades.pdf

As you can see covariance is vacuous - invariance however is a statement about physics.

Note many books like MTW use covariance but really mean invariance so things can get a bit confusing.

Technically Anderson's view from the above paper is the more exact: 'According to Anderson, what Einstein really intended with his principle of general covariance is what Anderson calls the ‘principle of general invariance’. This principle requires that the symmetry group of a theory be the general group of transformations or, as Anderson calls them, the ‘manifold mapping group’. This principle rules out the possibility of any non-trivial absolute objects in the theory, that is, those which have more than merely topological properties. In this sense, the principle of general invariance amounts to a noabsolute-objects requirement and offers a precise reading for Einstein’s claim that general covariance has eliminated an absolute from spacetime.'

I however would state it even differently and invoke Lovelocks theorem:
https://en.wikipedia.org/wiki/Lovelock's_theorem

The principle of invariance, covariance, whatever you want to call it is that the Lagrangian describing the dynamics of space-time contains only the metric tensor plus its first and second derivatives. Lovelock takes care of the rest.

Details can be found in Tensors, Differential Forms and Variational Principles by Lovelock and Rund.

This follows from the simple observation the motion of a particle is determined from the principle of maximal time which involves only the metric ie dt = √Guv dxu dxv (t proper time). The metric sort of acts like a field in physics so we expect it has its own Lagrangian. Of course since Guv is a tensor the Lagrangian describing it must also be a tensor equation. - I think that's what all this stuff about covariance and invariance is really saying. Its true - but when looked at this way you have to think - it's so obvious why worry. It may be an example of Feynman's complaint when he went to a gravity conference - they argued mostly about obvious things and asked his wife to remind him to never go to one again. But I also recall someone, I think it was Kip Thorne, said I can't believe some of the stuff I discussed with Feynman was in that category. So maybe it was just selective memory on Feynman's part. It can be shown, strangely, that no Lagrangian can be constructed from just Guv and it's first derivatives - see Lovelock and Rund page 310 - you must go to second order derivatives. And if you do that the previously mentioned theorem, Lovelock's Theorem (page 321 of the previously mentioned text) shows only EFE's can result from any Lagrangian - not just the very simple curvature scalar usually used - although of course in view of the theorem its the obvious one to go for.

GR is a bit of a strange beast for all sorts of reasons.

I am beginning to suspect what Wheeler calls GR, Geometrodynamics, not only is a very apt description but its actual foundation. Covarience, invarience etc more or less follow from that.

The above is the usual presentation, but you can also formulate it as a gauge theory similar to EM:
https://www.physicsforums.com/insights/general-relativity-gauge-theory/

Although as a Yang-Mills theory - I don't think that is possible - just one other reason gravity is different. You can formulate a generalization of gravity as a Yang-Mills theory - but not gravity as usually presented.

Thanks
Bill
 
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  • #8
Fred Wright said:
It seems to me that without symmetries this new physics would have to take conservation laws as a priori phenomenological. I'm glad this didn't come out 30 years ago; I would have switched my concentration to under-water basket weaving.

Peace
Fred
Hm, but if there are conservation laws, there are symmetries. So if there shouldn't be symmetries there must be no conservation laws. That would be a physics clearly contradicting very well established empirical facts!
 
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  • #9
vanhees71 said:
Hm, but if there are conservation laws, there are symmetries. So if there shouldn't be symmetries there must be no conservation laws. That would be a physics clearly contradicting very well established empirical facts!

But what about this remark by Smolin? "In fact, when we impose the condition that the universe is spatially compact without boundary, general relativity tells us there are no global spacetime symmetries and no non-zero global conserved charges." https://arxiv.org/abs/hep-th/0507235
 
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  • #11
vanhees71 said:
Hm, but if there are conservation laws, there are symmetries. So if there shouldn't be symmetries there must be no conservation laws. That would be a physics clearly contradicting very well established empirical facts!
Wasn't Newtonian mechanics at one time a well established empirical fact?
 
  • #12
Newtonian mechanics still is a well-established empirical fact. Only it's range of applicability is now known ;-)).
 
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  • #13
vanhees71 said:
Newtonian mechanics still is a well-established empirical fact. Only it's range of applicability is now known ;-)).
Yep, one has to cross the Einstein-Rosen Bridge :smile:

(5,4,3, ...)
 
  • #14
vanhees71 said:
Newtonian mechanics still is a well-established empirical fact. Only it's range of applicability is now known ;-)).
Couldn't the same apply to conservation laws?
 
  • #15
haruspex said:
Couldn't the same apply to conservation laws?
But Noether's theorems are pure mathematics, so without symmetries automatically means, without the differential equations which describe those conservation laws. And this basically would question physics since Newton and meant more than just a new point of view.
 
  • #16
fresh_42 said:
But Noether's theorems are pure mathematics, so without symmetries automatically means, without the differential equations which describe those conservation laws. And this basically would question physics since Newton and meant more than just a new point of view.
Sorry, I don't get your point.
Yes, there is an equivalence between certain possible symmetries and certain possible conservation laws. But is it not possible that both a symmetry and its conservation law apply to a very high degree of precision in our universe, but not quite perfectly?
 
  • #17
haruspex said:
Sorry, I don't get your point.
My point is, that physics as we know it, is fundamentally described by differential equations which bring their own symmetries, which we call conservation laws. You cannot even speak about such simple things as velocity without to write a derivative. So since those symmetries are part of the building, a building without them would destroy the entire concept. I just read today about the origin of functional analysis and found thermodynamics and differential calculus as an abstraction to the formerly used calculus of differences. I simply find it more than difficult to imagine a physics written in a different language and simultaneously being as accurate as it is today.
 
  • #18
fresh_42 said:
differential equations which bring their own symmetries,
Yes, those particular differential equations possesses symmetries, but variants might not.
This is getting close to the "why is the universe describable by mathematics we can understand" discussion. My answer has always been the anthropic principle: intelligence is only an evolutionary advantage in a universe that is predictable to a first approximation, and anyway we can only say that our current approximations seem pretty good.
On the other hand, your argument seems to be that the universe is amenable to our mathematics, therefore the symmetries exist, ... and that is somewhat the reverse of the "why is it describable" view.
 
  • #19
haruspex said:
Yes, those particular differential equations possesses symmetries, but variants might not.
And so those variants would not carry the same conserved quantities. This still does not break Noether’s theorem, it just states that there is no symmetry and no conserved quantity.
 
  • #20
Orodruin said:
And so those variants would not carry the same conserved quantities. This still does not break Noether’s theorem, it just states that there is no symmetry and no conserved quantity.
At no time have I suggested any flaw in Noether's theorem. I only question whether her theorems prove symmetry and conservation in the universe we inhabit.
 
  • #21
haruspex said:
This is getting close to the "why is the universe describable by mathematics we can understand" discussion. My answer has always been the anthropic principle: intelligence is only an evolutionary advantage in a universe that is predictable to a first approximation, and anyway we can only say that our current approximations seem pretty good.
This was basically the argumentation of this physicist whose name I've forgotten. He suggested that the symmetries are due to our inclination of pattern recognition and sense of beauty rather than an empiric fact.
On the other hand, your argument seems to be that the universe is amenable to our mathematics, therefore the symmetries exist, ... and that is somewhat the reverse of the "why is it describable" view.
No, my argument is, that either our mathematics describes the universe, and then we have symmetries by construction, or our mathematics isn't appropriate to do so. The latter is in my opinion extremely improbable, as our mathematics is surprisingly accurate in making correct predictions, which I think can't be a simple coincidence. Our mathematics is additionally based on - which I find - a very simple logic and so is the physical description which results from it. I cannot imagine, that all this should be a coincidence, and even less, what should substitute it.
 
  • #22
fresh_42 said:
which I think can't be a simple coincidence
As I posted, no coincidence. For intelligent life to evolve the universe probably needs to follow simple laws to a good approximation. Therefore we cannot use that observation to argue that it likely follows fairly simple laws perfectly. Indeed, such a line of reasoning would have argued that Newton's laws were likely perfect.

Edit: I should add that I share you gut feel that the fundamental laws are symmetric etc., but I claim it is only gut feel, not proven.
 
  • #23
haruspex said:
At no time have I suggested any flaw in Noether's theorem. I only question whether her theorems prove symmetry and conservation in the universe we inhabit.
Obviously not. You never really ”prove” anything in empirical science. You can only prove things based on a given premise and set of logic.

Edit: Noether’s theorem is more of a tool, saying ”if my theory has these symmetries, then those quantities must be conserved” and vice versa. You then go to your lab and measure those quantities. If they are conserved your theory passes that particular test. Or you can use it the other way around to guide your model building - if a measured quantity seemes conserved, try to build a model with the corresponding symmetry.

Edit 2: However, as far as tools go, it is one of the sharpest in the physicist’s toolkit...
 
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  • #24
I'm not a big fan of the anthropological principle. It always occurs to me as a bad excuse, and I see it in the same row as other attempts to explain what isn't understood, or even just believed. In my opinion there is no room for it in science, maybe in philosophy as a science inevitably limited by what happens in our consciousness.
 
  • #25
fresh_42 said:
I'm not a big fan of the anthropological principle. It always occurs to me as a bad excuse, and I see it in the same row as other attempts to explain what isn't understood, or even just believed. In my opinion there is no room for it in science, maybe in philosophy as a science inevitably limited by what happens in our consciousness.
It may be a bad excuse in some contexts, but here it has to be taken into account. The discussion concerns how universes might behave. To ignore that our very existence puts constraints on how our universe can behave is to commit the sin of observation bias.
 
  • #26
Orodruin said:
You never really ”prove” anything in empirical science.
Which is precisely my objection to post #8.
 

What are Emmy Noether's Theorems?

Emmy Noether's Theorems are a set of groundbreaking mathematical theorems developed by German mathematician Emmy Noether in the early 20th century. They revolutionized the fields of abstract algebra and theoretical physics, and are still widely used today in many areas of mathematics and physics.

Why are Emmy Noether's Theorems significant?

Emmy Noether's Theorems are significant because they fundamentally changed our understanding of abstract algebra and the foundations of theoretical physics. They also paved the way for future advancements in these fields, and have been applied to many other areas of mathematics and physics.

What impact did Emmy Noether's Theorems have on the scientific community?

The impact of Emmy Noether's Theorems on the scientific community was immense. They were initially met with skepticism and resistance, but over time they gained widespread recognition and have become one of the most influential contributions to mathematics and physics in the 20th century.

How have Emmy Noether's Theorems been applied in modern science?

Emmy Noether's Theorems have been applied in a wide range of areas in modern science, including quantum mechanics, general relativity, and particle physics. They have also been used in the development of new mathematical concepts and theories, such as Noetherian rings and Noetherian modules.

Why is it important to remember the anniversary of Emmy Noether's Theorems?

It is important to remember the anniversary of Emmy Noether's Theorems because it celebrates the incredible achievements of a woman in the male-dominated fields of mathematics and physics. It also serves as a reminder of the importance of diversity and inclusion in science and the need to recognize and honor the contributions of underrepresented groups.

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