I’m not a physicist, but in general, Abelian means that operations in an algebra commute, while Non Abelian means that they don’t, which is to say that Abelian means that a+b=b+a
To give a non mathematical example of a non-Abelian algebra, think of a Rubik’s cube where moves are the ‘numbers’ and composition of moves is the operation.
If you turn the right face clock wise and then turn the top face counter clockwise, that does not produce the same state as turning the top face counter clockwise and the turning the right face clockwise (try it out if you have one handy). Solving the cube in general requires exploiting this property by using what are called commutators, which are performing a move and then another move and then reversing the second move and reversing the first move. If it were Abelian, then that would leave you in the place where you started — which is to say you reversed time — but the state will actually have changed where the first and second move manipulated the same pieces — which makes it non Abelian and breaks ‘time symmetry’
So in this case, think of some physical operation that changes the state of the object somehow as being the same as a Rubik’s cube move, and assuming the operations have an inverse, that performing a commutator where you do one operation and then another and then invert the second and then the first, that leaves you in a new state, which would not be the case if it were time invariant.
The title says that a phenomenon was observed for the first time, not that this is the first non-Abelian phenomenon we've observed. The article is specifically about a non-Abelian version of the Aharonov-Bohm effect.
Great explanation, small nitpick: in mathematics, if you leave a person's name uncapitalized it's actually a sign of respect. So we usually use the uncapitalized abelian to refer to groups which commute.
edit: This is done consistently, and I think abelian is one of the few examples. I've never seen it capitalized before, which is why I made this post.
I agree that "abelian" is often uncapitalized, but if that's a sign of respect, why is Gaussian (Gaussian elimination, Gaussian distribution, Gaussian process, etc) almost always capitalized? Or Hilbert space, Poisson distribution, Bernoulli trial, Cartesian coordinates, Euclidean distance?
It seems to me that Abelian goes uncapitalized because few people know about Niels Henrik Abel, and it should properly be capitalized.
Found this interesting MathOverflow question with some interesting possible answers. Not sure if it clears it up much, but the difference between English and French capitalization rules seems to me like it might have something to do with some of it.
My professor explained that when it gets used to the extent where people stop capitalizing it it's seen as a sign of respect. You won't see it capitalized anywhere really.
At the very least even with the knowledge that it refers to Abel they will still leave it in noun case.
Maybe it's just a post-rationalization of the phenomenon you're describing.
If it is a name it gets capitalized. If it is an adjective derived from the name (abelian, cartesian, euclidean, ...), it depends on the language, the author and the literature around the word.
No idea, but one difference I noticed with your examples are that the names are adjectives modifying a generic math term, but "abelian" is the whole label. (e.g. "abelian" vs. "Abelian group")
How is it a sign of respect? Are we not respecting Fourier for his transforms? Are we failing to respect Stokes, Green, Riemann, Cauchy, Ramanujan, and Pythagoras? The vast majority of "respected" mathematicians retain capital names on their theorems/equations/conjectures etc.
Right, this doesn't mean that time was reversed, it means a phenomenon was created that would be different if time were reversed (unlike most physical phenomena, which work the same both ways).
Time-reversal symmetry means reversing the order of the inputs results in appearance the same output regardless of input order. The attempt was to break the symmetry in order to provide evidence the intended input sequencing was achieved.
"Time Reversal" would be better written as "Time-reversal" in title.
