#Noether's Theorem

[tojskfarnijler]

[teachersource]

Emmy Noether was born on March 23, 1882. A German mathematician who made many important contributions to abstract algebra. She discovered Noether's Theorem, which is fundamental in mathematical physics. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed some theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws.

[omegaphilosophia]

The Philosophy of Invariance

The philosophy of invariance examines the concept of constancy or unchanging nature within various contexts, including science, mathematics, ethics, and metaphysics. This philosophical exploration seeks to understand what remains constant amidst change and why such constancies are significant for our comprehension of reality, knowledge, and truth.

Key Concepts in the Philosophy of Invariance

Definition of Invariance:

Concept: Invariance refers to properties or principles that remain unchanged under specific transformations or conditions.

Argument: Identifying invariances helps in understanding the fundamental nature of systems and theories, providing a stable foundation for analysis and interpretation.

Invariance in Science and Mathematics:

Physical Laws: Many physical laws, such as the laws of motion and conservation laws, are considered invariant under transformations like time shifts or spatial rotations.

Symmetry: Invariance is closely related to symmetry in physics and mathematics. For example, the invariance of physical laws under certain symmetries leads to conservation laws according to Noether's theorem.

Mathematical Constants: Constants like π (pi) and e (Euler's number) are examples of invariance in mathematics, holding the same value across various contexts.

Invariance in Metaphysics:

Universal Principles: In metaphysics, invariance pertains to principles or truths that remain constant across possible worlds or different contexts.

Identity and Change: Philosophers explore how identity can persist over time despite changes, focusing on the invariant core that defines an entity.

Ethical Invariance:

Moral Principles: The idea that certain ethical principles are invariant, holding true regardless of cultural or situational differences.

Universal Ethics: This approach argues for the existence of universal moral truths that apply to all rational beings.

Theoretical Debates and Implications

Role of Invariance in Scientific Theories:

Concept: Invariance as a criterion for the validity and robustness of scientific theories.

Argument: Scientific theories that exhibit invariance under transformation are often considered more fundamental and reliable.

Philosophical Implications of Mathematical Invariance:

Concept: The philosophical significance of invariant mathematical properties and structures.

Argument: The constancy of mathematical truths supports the notion of an objective mathematical reality, independent of human perception.

Ethical Relativism vs. Invariant Ethics:

Concept: The debate between ethical relativism, which denies invariant moral principles, and invariant ethics, which upholds them.

Argument: While ethical relativism emphasizes cultural and contextual differences, invariant ethics seeks universal moral truths applicable to all.

Metaphysical Invariance and Identity:

Concept: The persistence of identity amidst change and the metaphysical basis for invariance.

Argument: Philosophers debate whether there are essential properties that remain invariant to preserve identity through change.

The philosophy of invariance explores the concept of constancy across different domains, from science and mathematics to ethics and metaphysics. By understanding what remains unchanged amidst transformations, this philosophical inquiry provides insights into the fundamental nature of reality, the stability of scientific theories, and the universality of ethical principles.

[dat-physics-gal]

You’ve fallen for my trap!

So, someone took that option in my poll, giving me an excuse to rant about conserved quantities, and Noether’s Theorem!

A conserved quantity is a mathematical value, which, over the course of a transformation, is, well, conserved.

These transformations range from coordinate transformations to processes, like decay or scattering.

I won’t go into the more complicated ones, like the weak hypercharge and isospin nor color, but i will explain how coordinate transformations and conserved quantities follow from one another. This fact was discovered by Emmie Noether, and it states that for every symmetry transformation of a system, a conserved quantity must exist which describes it.

A couple examples:

If a system’s equations don’t change with a translation in time, energy is conserved. If energy is conserved, the equations describing the systems are invariant under translations in time.

If a system’s equations don’t change with a translation in space, momentum is conserved, and vice versa once more.

An invariance to rotation about an axis corresponds to conserved angular momentum.

This theorem is well known among physicists, but it’s beauty is largely unknown to the wider public, and in many opinions, mine included, it’s actually the most beautiful result in all of physics.

Whenever there’s a conserved quantity, there’s a corresponding symmetry, and whenever there’s a symmetry, there’s a corresponding conserved quantity.

Always.

[ianmiller42]

2022 Nobel Prize in Physics

2022 Nobel Prize in Physics

When I woke up on Wednesday, I heard the Physics prize being announced: it was given for unravelling quantum entanglement, and specifically to Alain Aspect for the first “convincing demonstration of violations of Bell’s Inequalities”. In the unlikely event you recall my posts: https://wordpress.com/post/ianmillerblog.wordpress.com/542  and…

[spectrallysequenced]

Would people be interested in a video course on noncommutative algebra? The prerequisite knowledge would be linear algebra, basic group module theory and in the end some knowledge of Galois theory. Topics I might teach would include: Jacobson Radical Theory Central Simple Algebras Wedderburn-Artin and Wedderburn-Mal'tsev The Double Centraliser Theorem Noether-Skolem Theorem Brauer group Theory Crossed Products and Galois Theory Group Cohomology, briefly, and connection with the Brauer Group Further topics to be decided (Maybe Morita Theory, or more Brauer Theory).

[tanadrin]

Noether’s theorem implies any conservation law arises from a symmetry, e.g., angular momentum from rotational symmetry or energy due to time symmetry. Is the conservation of charge in electromagnetism due to gauge symmetry, or some other symmetry? I feel like I saw something like this mentioned somewhere once but Google is being unhelpful as usual.

[bayesic-bitch]

I am unreasonably annoyed by ada Lovelace being elevated to a household name. She had no actual lasting contributions to the field. For someone talked about as a mathematician, what theorems did she prove? For someone talked about as a computer scientist, what algorithms that we use now are her inventions? I've been studying computer science for a decade now and I have never once heard her name come up in an actual classroom, nor do I know anyone who has (at least, not for her actual contributions to the field, as opposed to the kinda-incorrect "first programmer" thing). She just didn't really *do* anything.

And if you're looking for women in STEM who really did make contributions that are overlooked by the general populace, why not focus on Emmy Noether? My undergrad advisor called Noether's theorem "the most beautiful result in physics" and he was 100% correct. And that's just physics, she's better known as a mathematician.

[minmin-vs-physics]

[18/9/24] I was on my feet pretty much all day today because I was either teaching or in a study hall since 9am, with brief breaks for slipping into a library to work on problem sets. Dear lord, am I tired! I want to be more consistent with posting because don’t get me wrong,,, I’ve been studying! So much so that I don’t have time to post about it.

I’ve really been enjoying QFT and GR. So glad to have the chance to think really deeply about these topics because we’re covering symmetries in both of them, and one thing you have to know about me is that I am in love with symmetries and group theory and Noether’s theorem. Conservation laws, my heart!

Should I do a post about topics I enjoy like this one? I feel like there’s a lot to say about the beautiful mathematical formalism around deep physical topics. There are some concepts in this world that completely take over you once you know they exist. (Another Noether’s Theorem plug)

[spiderfreedom]

What are some of examples of female geniuses you can think of? In any field, not just science.

For me, it's:

Emmy Noether - changed the math and physics game like never before with Noether's theorem and her theory of symmetries

Emily Bronte - NEET who dropped a banger of a book with incredible insight into humanity then died :'(

Andrea Dworkin - visionary unafraid of coming up with insane ideas and moving forward with laserlike precision. sometimes she fucked up (the end of woman hating) but no other feminist writer has had her breadth or depth or creativity

Kate Bush - invented the entire template for dynamic off the wall 'quirky' female artists, then invented another template for boundary-pushing pop music with 'The Dreaming'

Nina Simone - boundary pushing musician melding jazz and classical and social responsibility. she just improvised a fugue once during one of her live performanes, no biggie

[venicepearl]

Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She proved Noether's first and second theorems, which are fundamental in mathematical physics. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed theories of rings, fields, and algebras. In physics, Noether's theorem explains the connection between symmetry and conservation laws.

[macla539ac]

I don't remember ever being dead therefore i must've had lived forever. I think reincarnation or eternal recurrence might be real, minus karmic reincarnation

fun fact I'll forget to associate: under time reversal symmetry the property which is conserved as per noether's theorem is literally called 'kramer's degeneracy' in reference to degenerate energy levels which can be thought of as energy levels which have been corrupted

[lipshits-continuous]

Prime numbers of the ask game let's go!

This is gonna be a long old post haha /pos

2. What math classes did you do best in?:

It's joint between Analysis in Many Variables (literally just Multivariable calculus, I don't know why they gave it a fancy name) and Complex Analysis. Both of which I got 90% in :))

3. What math classes did you like the most?

Out of the ones I've completely finished: complex analysis

Including the ones I'm taking at the moment:

Topology

5. Are there areas of math that you enjoy? What are they?

Yes! They are Topology and Analysis. Analysis was my favourite for a while but topology is even better! (I still like analysis just as much though, topology is just more). I also really like group theory and linear algebra

7. What do you like about math?

The abstractness is really nice. Like I adore how abstract things can be (which is why I really like topology, especially now we're moving onto the algebraic topology stuff). What's better is when the abstract stuff behaves in a satisfying way. Like the definition of homotopy just behaves so nicely with everything (so far) for example.

11. Tell me a funny math story.

A short one but I am not the best at arithmetic at times. During secondary school we had to do these tests every so often that tested out arithmetic and other common maths skills and during one I confidently wrote 8·3=18. I guess it's not all that funny but ¯⁠\⁠_⁠(⁠ツ⁠)⁠_⁠/⁠¯

13. Do you have any stories of Mathematical failure you’d like to share?

I guess the competition I recently took part in counts as a failure? It's supposed to be a similar difficulty to the Putnam and I'm not great at competition maths anyway. I got 1/60 so pretty bad. But it was still interesting to do and I think I'll try it again next year so not wholly a failure I think

17. Are there any great female Mathematicians (living or dead) you would give a shout-out to?

Emmy Noether is an obvious one but I don't you could understate how cool she is. I won't name my lecturers cause I don't want to be doxxed but I have a few who are really cool! One of them gave a cool talk about spectral geometry the other week!

19. How did you solve it?

A bit vague? Usually I try messing around with things that might work until one of them does work

23. Will P=NP? Why or why not?

Honestly I'm not really that well versed in this problem but from what I understand I sure hope not.

29. You’re at the club and Grigori Perlman brushes his gorgeous locks of hair to the side and then proves your girl’s conjecture. WYD?

✨polyamory✨

31. Can you share a math pickup line?

Are you a subset of a vector space of the form x+V? Because you're affine plane

37. Have you ever used math in a novel or entertaining way?

Hmm not that I can think of /lh

41. Which is better named? The Chicken McNugget theorem? Or the Hairy Ball theorem?

Hairy Ball Theorem

43. Did you ever fail a math class?

Not so far

47. Just how big is a big number?

At least 3 I'd say

53. Do you collect anything that is math-related?

Textbooks! I probably have between 20 and 30 at the moment! 5 of which are about topology :3

59. Can you reccomend any online resources for math?

The bright side of mathematics is a great YouTube channel! There is a lot of variety in material and the videos aren't too long so are a great way to get exposed to new topics

61. Does 6 really *deserve* to be called a perfect number? What the h*ck did it ever do?

I think it needs to apologise to 7 for mistakingly accusing it of eating 9

67. Do you have any math tatoos?

I don't have any tattoos at all /lh

71. 👀

A monad is a monoid in the category of endofunctors

73. Can you program? What languages do you know?

I used to be decent at using Java but I've not done for years so I'm very rusty. I also know very basic python

Thanks for the ask!!

[wuwojiti]

Math asks 8 9 10 18 19 60 62 (or some subset cause that's a lot) :]

8. least favorite notation you've seen: well there's the low-hanging fruit of logs--i was actually just helping my sibling in high school with those yesterday--but honestly i have pretty strong negative feelings towards using ' to indicate derivatives. it's convenient, sure, but it makes it so much harder to track the independent variable than if you use d/dx explicitly

9. favorite theorems: the curry-howard correspondence is great. honorable mentions to noether's theorem, which sounds cool but i've never actually touched, and the equivalence between bipartite graphs and graphs with no odd cycles

10. least favorite theorems: not a big fan of the master theorem for recursive algorithms, just bc all the classes i've had that used it just take it as "word of god" and apply the formula rather than something that can be proven

18 & 19. can you share a good problem you've solved recently, and how did you solve it? honestly i can't think of any particularly good ones recently :(

60. what's your favorite number? well my favorite integer is 64, but there's so many others that have inexplicably good vibes (i'm looking at you, 196) (side note: 64 is not because of minecraft and 196 is not because of rule 196). then the surreal numbers are really cool so i say * is one of my favorite numbers just bc it's the first thing you learn about among the surreal numbers that suggests that something's not quite right here...

62. are there any non-interesting numbers? well we all know the classic proof by contradiction that there are no non-interesting natural numbers, but it doesn't hold for larger sets like the real numbers. in fact, i would argue that it doesn't even hold for the rationals, because while you *can* define a map from the integers to the rationals, the order it defines in the rationals isn't meaningful (and the density of the rationals means that ordering them by value isn't helpful either). anyway, while the notion of a non-computable number is interesting, i'd consider most actual non-computable numbers non-interesting

[lgbtcorp]

@geniuscorp.

it's a week to the day. she thinks he probably has some quip about that — see, sis? i don't lie to you, but she doesn't listen for it anyway. her keys are strewn across the kitchen island. Iex has spent his time continuing to fuck with her, it seems. tasteful reminders that he was here — a glass on the side of the sink, three fingers of scotch out of the decanter, a book on quantum mechanics splayed open on the coffee table, spine cracked and forced apart on a page on noether's first theorem. he's spelt his name out in the authors in on her bookshelves, moved her couch a quarter inch, and leaves that fucking cologne smell everywhere.

"so, can i assume you're done playing poltergeist?"

she barely looks up at him, but she sees him from her periphery — linen suit, one leg crossed over another, sipping delicately on a scotch by the decanter. make yourself at home, lex. (can you believe this fucking guy?)

[polyphonetic]

hi what's the name of the math concept from your projective plane boyfriend meme so i can do some research on it

Honestly most just the Wikipedia article for Projective Plane as well as Fane Plane! I was reading about it because I want to try to more intuitively understand angular and orbital momentum and spin. This lead to me needing to research Noether's Theorem, Minokowski space, and realizing I need to do more research onto Euclidean space, manifolds, the Real Projective Plane, Fundamental Polygons, etc. Headed to the bottom of trying to intuit projective planes as a whole and that's when I had the idea for the post!

I think it's cool that some of the first instances of the projective plane relate to art in the middle ages! When doing 3D perspective art, parallel lines disappear at the horizon. that's projective space! :)