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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
by empath75 7y ago
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.
- MCompeau 7y agoThis is a very eloquent explanation, thank you.
- deleted 7y ago[deleted]
- solinent 7y agoGreat 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.
- leereeves 7y agoI 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.
- solinent 7y agoMy 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.
- magicalhippo 7y agoFound 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. https://mathoverflow.net/questions/44946/why-is-abelian-infrequently-capitalized https://mathoverflow.net/questions/44946/why-is-abelian-infr...
- WhiteSage 7y agoIf 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.
- dlivingston 7y agoI can't say that this is true. Examples of mathematical operators/functions that are always capitalized: * Laplacian operator * Gaussian distribution * Lorentz factor * Taylor series * Bohr radius etc...
- 7y ago
- amelius 7y agoWait, if a Rubik's cube has this property, then why does the title state that the phenomenon was just observed for the first time?
- chc 7y agoThe 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.