
Noether’s Theorem – A Quick Explanation (2019) - panic
https://quantum-friend-theory.tumblr.com/post/172814384897/noethers-theorem-a-quick-explanation
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mikorym
I am not well versed in the Physics counterpart of Noether's work, but the
mathematics side is quick to explain: the isomorphism theorems in group theory
(originally, in ring theory) were one of the main impacts of her work.

A quick skim of Wikipedia tells me that the isomorphism theorems were later
than the theorem in physics [1] [2] by about 5 years. Looking at it quickly,
these are not the same thing, and although in spirit similar, I am not sure
what the exact historical progression was.

I wonder if standard abstract algebra modules could teach this history in a
way that tells more about the story behind the isomorphism theorems. In my
abstract algebra class, Noether was not really mentioned, and usually the
focus is (for example) on the progress from Greek geometry (e.g. squaring the
circle) to modern algebra. I don't think anyone is at fault for this, but I
would personally like to have a more accessible introduction to Noether's
legacy via pure mathematics.

[1]
[https://en.wikipedia.org/wiki/Isomorphism_theorems](https://en.wikipedia.org/wiki/Isomorphism_theorems)
[2]
[https://en.wikipedia.org/wiki/Noether%27s_theorem](https://en.wikipedia.org/wiki/Noether%27s_theorem)

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Chinjut
Ring theory in general was founded in large part via her work. I always find
it funny that everyone only makes such a fuss about one physics conservation
law/symmetry correspondence she observed (for physical theories that happen to
be phrased in terms of Lagrangians/Hamiltonians), while the immense
mathematical achievement of her extensive work on ring theory goes largely
undiscussed in these kinds of conversations.

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thanatropism
Noether's theorem is kind of a grand closure of natural science. It's one of
those theorems that gives the sensation of transcending its axiomatic setting
and reveal deep philosophical truth.

Compare Arrow's impossibility theorem, which is set in the much more familiar
realm of order theory, but still appears to say something about the real
world.

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i4t
I'm amazed I understood all of this thanks to Leonard Susskind.

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mannykannot
The scope of this theorem seems so broad and general, apparently covering any
possible physical symmetry, whether or not conceived of yet, that I wonder if
it could reasonably be called a metaphysical one.

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Filligree
In the same sense as quantum mechanics is, perhaps. It's more math than
physical law, but the laws sit on top of it.

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mannykannot
Yes - the dabates over the rival interpretations are, I think, widely (though
not by everyone) regarded as metaphysical.

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ouid
Just as a thought exercise. Try to imagine a universe in which it was
impossible to define conserved quantities.

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LargoLasskhyfv
Nice site. Permanently bookmarked.

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throwlaplace
now do gauge invariance

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loudmax
Unfortunately, equations like this are hard to read if you aren't a fluent
reader of LaTeX:

$$\left.\frac{\partial L}{\partial q} \right|_{\tilde{q} = \dot{q}} =
\frac{dp}{dt}$$

Perhaps there's a tumblr theme or extension that can convert LaTeX to HTML.

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jordigh
It's running mathjax to convert them. Are you perhaps blocking js?

~~~
loudmax
Indeed I was. The post renders legibly with javascript enabled. Thanks for the
pointer.

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lidHanteyk
Modern physics assumes symmetry and then searches for models which fit
observations under the constraint of symmetry. However, we are starting to
wonder whether the symmetries truly exist. Time, charge, parity, position,
momentum; we _hope_ that they have symmetries.

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wtallis
That doesn't sound like a fair description. Physics doesn't simply assume
symmetries exist. We've observed symmetries that hold up under all
observational and experimental scenarios available to us. From that, it's
quite reasonable to consider the possibility that those symmetries are truly
universal, and to study models that treat those symmetries as universal. Those
models have great predictive power for ordinary scenarios of the sort where we
know from experience that the symmetries _do_ hold up, and those models tend
to be simpler than ones that break symmetry in exotic scenarios. When we
actually find a broken symmetry (eg. space, time and momentum behaving oddly
at speeds close to _c_ ), it's time to update those models, which then usually
reveals a more subtle symmetry and conservation law.

But there's no big trend in physics that casts doubts on _all_ the important
symmetries in physics. Even under as-yet undiscovered theories, the symmetries
we're familiar with will always remain true in the limit as the conditions
approach familiar everyday circumstances. Nobody is worried that concepts like
time, charge, momentum, etc. will be discovered to be a silly, unfounded idea
that the universe casually disregards. We're just not adding extra terms to
the equations until there's a need for them.

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stareatgoats
> Nobody is worried that concepts like time, charge, momentum, etc. will be
> discovered to be a silly, unfounded idea

While not overly worried, I have to confess that I'm at least somewhat
worried. And somewhat convinced that our present conceptions of theses things
will one "day" be turned on their head, in a way that might also seriously
affect our everyday conceptions of them.

I would support adding a couple of properties to most if not all our
established models; notably 1: known unknowns, and 2: unknown unknowns. It
would seem a healthy antidote to the hubris that runs rampant among
scientists, at least before they discover the ubiquity of these two properties
(they usually do around the 50 year mark).

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andygates
This is word salad in a math-heavy field.

