[HN Gopher] Groups underpin modern math
___________________________________________________________________
Groups underpin modern math
Author : nsoonhui
Score : 113 points
Date : 2024-09-06 23:42 UTC (1 days ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| card_zero wrote:
| This starts out as one of those "how they came up with a part of
| mathematics" stories, or explanations. But it rapidly gives up on
| explaining that. I don't think I've ever read a satisfactory
| explanation of what any mathematician was thinking when devising
| any part of mathematics. The subheading (which is probably
| written by an editor) is especially bad:
|
| > What do the integers have in common with the symmetries of a
| triangle? In the 19th century, mathematicians invented groups as
| an answer to this question.
|
| No. There is no way they were just sitting around and somebody
| said, "you know what, I bet all the ways you can flip a triangle
| have something in common with adding the first 6 integers modulo
| 6, let's try inventing some abstract concepts and see if we can
| find one that makes this true!"
|
| The quote from Sarah Hart says much the same thing: "It's not
| like a bunch of mathematicians got together one day and said,
| 'Let's create an abstract structure just for a laugh.'" But what
| _were_ they doing? One group of size six is adding integers,
| which seems like an important thing (even when constrained to the
| first six). The other is ... flipping a triangle around. Who
| cares about _that?_ The only real-world example of symmetries is
| when your mattress gets lumpy and you have a choice of ways to
| turn it. Why speculate, even for a moment, that turning a
| mattress and adding integers might be the same category of thing?
| Why reify flippin ' shapes like that, why think about symmetries
| as a thing at all? I don't get it.
| 082349872349872 wrote:
| https://en.wikipedia.org/wiki/History_of_group_theory
|
| > _flipping a triangle around. Who cares about that?_
|
| Chemists, for one. (the people who brought us Haber-Bosch,
| plastics, etc.) See
| https://en.wikipedia.org/wiki/Spectroscopy#Molecules .
| defrost wrote:
| Also roller manufacturers that want a smooth ride:
| https://en.wikipedia.org/wiki/Reuleaux_triangle
| card_zero wrote:
| Yes, but who cared about it in 1830 (or earlier), and why
| did they imagine even for a moment that it might have some
| equivalence to the number line?
|
| Looks like it was all about the quintics, somehow, but I
| don't know why they made the leap from that problem to
| geometry. I'm thinking maybe the equivalence to triangle-
| flipping is just like an amusing conceptual side-effect
| that happened by accident when working out stuff about
| permutations?
|
| I don't think a Reuleaux roller functions very well if you
| flip it around a different axis, anyway, but I'll let you
| off because they're cute.
| defrost wrote:
| > Yes, but who cared about it in 1830 (or earlier)
|
| Anybody with a Sphinx to move, struggling to make a
| _purrfect_ circle.
|
| > I don't think a Reuleaux roller functions very well if
| you flip it around a different axis
|
| As an extruded 2D shape -> 3D solid it lacks a little in
| the mirror symmetry department, rotation is pretty much
| its limit .. admittedly a lacklustre submission, but cute
| indeed.
| 082349872349872 wrote:
| > _why did they imagine even for a moment that it might
| have some equivalence to the number line?_
|
| A few of the groups which Galois introduced are what we
| now call Abelian (after Abel), which is to say that we
| can forget the order of elements within a product: AB ==
| BA (if you get up to leave the beach and put on flip-
| flops then put on a shirt, you wind up in the same state
| as if you get up to leave the beach and put on a shirt
| then put on flip-flops)
|
| Number theory studies products of primes, and here,
| always, although we generally must write down
| multiplicands in some order, it doesn't matter which: 2*3
| == 3*2.
|
| This connection would be enough for any modern
| undergraduate to consider applying general machinery
| built for quintics to the specific case of the number
| line, but in those days it took Euler (working before
| Galois) and Gauss[0] (working after? check this) to blaze
| the trails along this particular connection.
|
| > _maybe the equivalence to triangle-flipping is just
| like an amusing conceptual side-effect_
|
| Not just conceptual: spectroscopy is exactly why chemists
| are taught a little group theory, and triangle flipping
| is the simplest non-trivial[1] example.
|
| Like programmers, who spend their days building up data
| structures and picking them apart[2], chemists are
| concerned with synthesis (building up molecules) and
| analysis (picking them apart; in principle this includes
| synthetic steps that make small molecules from bigger
| ones, but in practice this means checking your product at
| the end of synthesis to confirm that you made lots of
| what you were hoping to make[3], and little of what you
| didn't want to make[4]).
|
| In particular, spectroscopy is a useful tool in chemical
| analysis, and very often[5] parts of a molecule will have
| a triangular symmetry, meaning that the peaks in a
| recorded spectrum[6] can be explained via a
| representation of the triangle-flipping group. If you set
| out to make, say, ammonia[7], but don't get any triangle-
| flipping parts in the spectrum when you run your tests
| ("characterise your product"), you know you failed[8].
|
| https://www.smbc-
| comics.com/comics/1725209167-20240901.png
|
| [0] when Laplace was asked who the greatest german
| mathematician was, he replied "Pfaff". when asked why not
| Gauss, he explained "you asked for the greatest _german_
| mathematician; Gauss is the greatest _european_
| mathematician " (compare: the LUB of a set need not be a
| member of that set -- EDIT: I guess the set of working
| mathematicians is always finite, so this comparison
| falls)
|
| [1] in 0-D, a point has only the identity, so it's a
| degenerate group; in 1-D, a line segment does have a
| symmetry group (isomorphic to the booleans which are so
| important to CS) but unfortunately children do not learn
| about digons in elementary school, and must wait until
| they discover computer graphics to learn that edge AB is
| distinct from edge BA; indeed, they're explicitly taught
| to ignore that distinction in high school geometry,
| leaving the first non-trivial pedagogically-suitable
| example to be in 2-D: the triangle
|
| [2] we've made some progress on also building up
| functions with the same aplomb as we handle data, but
| we're still not very comfortable when it comes to taking
| functions apart
|
| [3] just as computer scientists often have a better-than-
| average knowledge of computer cracking, and physicists of
| bomb geometry, chemists tend to have a better-than-
| average knowledge of street syntheses. In particular, I
| have a second hand anecdote of undergraduates, who,
| having been in the process of characterising a synthetic
| product one evening, were interrupted by a grad student
| who, just by looking at the spectral lines, told them he
| hadn't seen anything that night but if they wished to
| continue exploring those particular [synthetic] pathways,
| they had better do so independently of university
| equipment.
|
| [4] just as software engineers (who know what corners to
| cut to produce huge numbers of right answers and an
| acceptable number of wrong ones much more cheaply than
| only right answers) are generally paid better than
| researchers, ChemE's (who know what corners to cut to
| produce huge amounts of wanted product and an acceptable
| amount of unwanted) are generally paid better than their
| purer colleagues.
|
| [5] why? (hint: it's the same mechanism --related to
| Natural primes-- that makes binary taxonomies so popular)
|
| [6] indeed, "spectrum" has been reborrowed back into
| maths to refer to something in algebraic geometry which
| is currently beyond my ken. If you poke around these
| areas long enough, you'll also find that von Neumann (who
| had physical, computational, and mathematical reasons to
| be interested) has had the "von Neumann regular rings"
| named after him, and rings are nothing but a pair of
| groups which interact in a certain manner. (the "regular"
| here being related to the "regular" in regular
| expressions, btw) Exercise: do regular expressions
| contain any rings?
|
| [7] as the last century taught us, being able to make
| ammonia is very powerful, having applications both
| desirable and undesirable.
|
| [8] Exercise: if you do see signs of triangle-flipping,
| is that enough to be sure you just made ammonia?
| card_zero wrote:
| Thank you for the extensive reply. There's a hint in
| there about commutativity inspiring the connection, but
| my mental model is now simply "Euler did it", which
| somewhat like creationism relieves me from having to ask
| further questions.
|
| Something I used to imagine: what if we were radically
| different creatures, like ant colonies, and by habit we
| communicated non-linearly (with thousands of limbs and
| organs swarming in parallel all over our mathematical
| work, perhaps written in 3D)? That could make equations
| mostly trivial, since our symbols wouldn't be constrained
| to any particular arrangement in the first place: and
| that seems kind of advantageous. But then grasping the
| concept of "non-commutative" would be a real strain for
| these poor ant-hills. They'd have to deliberately
| reintroduce linear ordering, maybe with special symbols
| to mark precedence.
| 082349872349872 wrote:
| (a) if you haven't read it, Chiang, _Story of Your Life_
| (1998) might have interesting aliens.
|
| (b) the special symbols is a good point. Reading older
| maths papers is cool because you get to see all sorts of
| things people tried before we settled on what we use now.
| Two works that come immediately to mind: _Principia
| Mathematica_ (1910) uses various numbers of dots instead
| of parentheses to mark precedence, while Peano,
| _Arithmetices principia: nova methodo exposita_ (1889)
| uses very modern-looking notation, including parens as we
| would use them, but its expository text is all in latin!
|
| (c) I don't think your aliens would have any more trouble
| with non-commutative than we have with commutative. Have
| you heard of the Boom Hierarchy? It starts with _trees_ ;
| when we add an associative law, so (AB)C == A(BC), then
| we only have flat _lists_ ; when we add a commutative
| law, so AB == BA, then we only have unordered _bags_ ;
| and finally when we add an idempotent law, so AA == A,
| then we have unduplicated _sets_. It turns out
| (exercise!) that if we have information encoded in any of
| these representations, we always have at least one way to
| represent the same information in all the other
| representations, such that we can "round trip" between
| any two levels of this hierarchy without losing any
| information.
|
| So for programming, where we care about time and space,
| picking ordered or unordered representations can be very
| important, but for maths, where all that matters is the
| existence of invertible functions between all these
| representations, that decision is unimportant. Does that
| make sense?
| kjellsbells wrote:
| I dont want to get too far into a joint and doritos
| speculation here, but I wonder if the fact that humans
| have a very small capacity for holding multiple objects
| in their minds, a small number of physical digits, and
| excellent visual acuity is why we do a lot of math the
| way we do. Group theory comes out of symmetry for
| example. Algorithms come out of linear stepwise problem
| solving. It takes us considerable mental effort to think
| about problems in ways that are not like this.
|
| An example that stayed with me for years is when Adelman
| of RSA fame considered whether DNA could be used as a
| computer. (Spoiler: yes). It basically does all the
| computations at once, and then discards all the non
| optimal solutions.
| User23 wrote:
| > That could make equations mostly trivial, since our
| symbols wouldn't be constrained to any particular
| arrangement in the first place
|
| You might find this little tidbit[1] from C.S. Peirce by
| way of John Sowa interesting then. Existential Graphs
| (EG) are an unordered diagramatic representation of
| mathematical logic. And Peirce is the real deal. His more
| conventional notation was adopted by Peano (who
| substituted the familiar symbols for capital sigma and pi
| (which created confusion when used in the context of
| broader proofs, even though sigma and pi pretty much
| directly correspond to what existence and universality
| mean)) and he is credited as an independent co-discoverer
| of both quantifiers with Frege. For EGs,
| only one axiom is necessary: a blank sheet of assertion,
| from which all the axioms and rules of inference by
| Frege, Whitehead, and Russell can be proved by Peirce's
| rules. As an example, Frege's first axiom, a[?](b[?]a),
| can be proved in five steps by Peirce's rules
|
| Peirce gives us the entire predicate calculus with three
| rules and one axiom. And of course it's all built on
| NAND.
|
| And another teaser: In the Principia
| Mathematica, Whitehead and Russell proved the following
| theorem, which Leibniz called the Praeclarum Theorema
| (Splendid Theorem). It is one of the last and most
| complex theorems in propositional logic in the Principia,
| and the proof required a total of 43 steps ... With
| Peirce's rules, this theorem can be proved in just seven
| steps starting with a blank sheet of paper.
|
| John Sowa himself is also no slouch, having been one of
| the leading lights of the earlier AI push. I expect
| advances in modern AI will come when we stop trying to do
| everything with ngrams and start building on richer
| models of knowledge representation.
|
| [1] https://www.jfsowa.com/pubs/egtut.pdf
| trashtester wrote:
| Galois was the first to realize the connection between
| polynomial roots and geometric symmetries through group
| theory.
|
| Some simple examples can be found in subgroups of U(1).
| For instance in how Z_n is linked to regular polyhedra of
| order n and also n'th roots of complex numbers of the
| Unit Circle.
|
| Kind of like how Z_12 is linked to an analog clock.
| andrewflnr wrote:
| I think it starts as a nagging feeling that there's a common
| thread between diverse stuff like transforming triangles,
| transforming more complicated/interesting shapes, numbers, and
| other weirder stuff. It's a natural instinct, at least for math
| people, to try to nail down and name that common thread. And
| then it takes, as the article mentioned, a lot of thinking and
| trying things to figure out the best way to formalize it. At
| least that's how I feel it in a niche way, trying to figure out
| my own formalisms for some things. Maybe you're just not that
| type of person, though.
| bbor wrote:
| This is close, but I think it's more than an instinct: it was
| a philosophical challenge. Math is part of a larger project
| to formalize thought, and "we have a bunch of tools for a
| bunch of different things that work in different ways" is a
| lot less metaphysically/intuitively satisfying than "we have
| a single cohesive system of formalization that can be applied
| to any system of quantities and qualities found in human
| experience, all based on the same solid foundation."
|
| This challenge/goal was best expressed by Bertrand Russell
| and Alfred Whitehead, I think:
| https://plato.stanford.edu/entries/principia-mathematica/
| The present work has two main objects. One of
| these, the proof that all pure mathematics deals exclusively
| with concepts definable in terms of a very small number of
| fundamental concepts, and that all its propositions are
| deducible from a very small number of fundamental logical
| principles... will be established by strict symbolic
| reasoning... The other object of this work... is
| the explanation of the fundamental concepts which mathematics
| accepts as indefinable. This is a purely philosophical task.
| andrewflnr wrote:
| Well yes, but it takes a certain kind of instinct to take
| on that philosophical challenge, and again to guide your
| particular angles of attack on it. I took the question to
| be, within the framework of formalized math, how do people
| arrive at particular abstractions like groups? How do you
| pick the axioms and rules?
| bbor wrote:
| The only real-world example of symmetries is when your mattress
| gets lumpy and you have a choice of ways to turn it.
|
| I appreciate where you're coming from and love the passion, but
| this couldn't be further from the truth. Symmetry is the basis
| of life, not to mention physics nor beauty!
|
| https://people.math.harvard.edu/~knill/teaching/mathe320_201...
|
| > Quantum mechanics represents the state of a physical system
| by a vector in a space of many, actually of infinitely many,
| dimensions. Two states that arise from each other, either by a
| virtual rotation of the system of electrons or by one of their
| permutations, are connected by a linear transformation
| associated with that rotation or that permutation. Hence *the
| profoundest and most systematic part of group theory*, the
| theory of representations of a group by linear transformations,
| comes into play here. I must refrain from giving you a more
| precise account of this difficult subject. But here symmetry
| once more has proved the clue to a field of great variety and
| importance.
|
| > From art, from biology, from crystallography and physics I
| finally turn to _mathematics_ , which I must include all the
| more because the essential concepts, *especially that of a
| group*, were first developed from their applications in
| mathematics.
|
| - Hermann Weyl's _Symmetry_ , p. 135
| j2kun wrote:
| If you're asking for a legitimate explanation for why
| mathematicians came up with groups, it's because they wanted to
| find roots of polynomials (or rather, prove one cannot find a
| general formula for solving large-degree polynomials).
|
| The complex roots of polynomials satisfy symmetry properties.
| The group structure of those symmetries allows one to
| discriminate when one can and cannot solve the polynomial using
| elementary operations (+,-,*,/) and radicals (nth roots). They
| call this "Galois Theory", and group theory grew out of it to
| streamline the ideas about symmetry so they could be applied
| elsewhere, particularly in the study of geometry and non-
| Euclidean geometry.
| jacobolus wrote:
| If someone's interested in a more detailed discussion of
| this, here's a an out-of-copyright paper that turned up in a
| 2 minute literature search:
| https://www.jstor.org/stable/2972411
| Someone wrote:
| Slight nitpick: I don't think anybody ever set out to prove
| the non-existence of a general formula for solving large-
| degree polynomials.
|
| The goal always was to find one but they had to settle for
| second best: proving that there is no such formula, and thus
| that the search was over.
| isotypic wrote:
| You might like the book "A History of Abstract Algebra" by
| Israel Kleiner - it goes over specifically the developments
| leading to the invention of the abstract group. The answer to
| your questions is that nobody really sat down and invented the
| group from the ether - its more accurate to say someone sat
| down and said "Hey, all these things we've been studying for
| the past 50 years are all the same thing if we think of it this
| way", and then the mathematical community eventually gets
| around to realizing its a useful abstraction (if it is one) as
| people build on it, or work more without it and eventually
| realize the abstraction would be helpful. For groups, this
| played out in how Cayley defined the abstract group in the
| 1850s, but it only started to gain more widespread usage in the
| 1870s. As for what they were doing, the main areas ways
| appeared around this time were through permutation groups
| (roots of polynomials), abelian groups (various number
| theoretic constructions/statements), and geometry (study
| geometry by studying groups of transformations, like isometries
| for Euclidean geometry).
| mrkandel wrote:
| My professor explained that Galois originally thought about the
| subgroups of S_n, i.e. the set of all the permutations of n
| objects.
|
| Thinking about the permutations of n objects is rather natural,
| especially when thinking about roots of polynomials, as when
| you look at roots of some polynomial you will see that some
| permutations of the roots are legal and some are not. And then
| when you investigate that you start to notice there are sets of
| permutation that can operate independently of the other
| permutation, and those are the subgroups. The concept of a
| "group" as an abstract term in itself was not there. Later on
| it was codified.
| anon291 wrote:
| I mean as someone who was not into this but became into this
| when I started Haskell, it's because as you play with things it
| becomes obvious that things behave similarly.
|
| For example, you learn addition and multiplication with numbers
| for dealing with real world quantities.
|
| Then you mess with Booleans for computers. You'll notice that
| there's a distributive law for those as well as numbers and in
| this that or behaves like addition and and behaves like
| multiplication.
|
| So then one thinks why that's the case (hint: it has to do with
| zero and one)
|
| And as you do this you think... Well what else has things that
| behave this way.
|
| And then you're like... Well what if you don't have
| multiplication, then what? And it goes on and on.
|
| It's the same thing as when you're writing code and you notice
| two classes have similar functionality so you create a base
| class. Same idea . Different terminology
| esperent wrote:
| In the case of group theory you'd probably just find that these
| ideas occured in a flash of mostly subconscious insight while
| the mathematician was actually thinking about how the French
| royalists were conspiring to suppress their previous
| breakthroughs, or how their lover was actually a royalist plant
| put in place to trick them into a fatal pistol duel. Something
| of that ilk, anyway.
| tzs wrote:
| > The only real-world example of symmetries is when your
| mattress gets lumpy and you have a choice of ways to turn it.
|
| I once failed to notice that, and it led to a lot of teasing at
| my expense.
|
| It was my second year as an undergraduate at Caltech. Our rooms
| were rectangles with a door on one of the short sides, a
| windows and radiator opposite that, a bed and a desk along one
| of the long sides, and a closet and sink along the other long
| side.
|
| I decided I wanted to switch where my head and feet were when
| sleeping, so started trying to turn the bed around. The room
| was not wide enough to simply swing it around, but it was tall
| enough that if I lifted one end high enough I'd be able to
| pivot it.
|
| That was difficult as a one man job, not helped by the growing
| crowd of people standing outside my door watching with obvious
| amusement as several times I almost dropped the bed on myself.
|
| I finally managed it and then the spectators then pointed out
| the the bed consisted of a rectangular frame with a rectangular
| mattress which is symmetrical under the rotation operation that
| I had just painfully taken a great deal of time to execute, and
| that the only thing that determines which is the "head" end and
| which is the "feet" end is which end you tuck your top sheet
| in.
|
| I could have simply waited for the weekly linen exchange, which
| I would be removing all the linens for, and then put the new
| linens on with the tuck on the other side and accomplished my
| switch with literally no extra work at all.
|
| It took a long time for people to stop making fun me.
| otoburb wrote:
| >> _which is symmetrical under the rotation operation that I
| had just painfully taken a great deal of time to execute_
|
| This critical symmetry property only holds if the mattress is
| assumed to be uniformly lumpy (including no discernable lumps
| at all), which perhaps in your case it was but you might have
| had the last laugh if you'd challenged them on that point.
| tzs wrote:
| They still would have had the last laugh because even if I
| could justify rotating the mattress I wouldn't have been
| able to justify rotating the frame, which was what made the
| operation difficult.
| kelseyfrog wrote:
| Not just groups, magmas, semigroups, monoids, &c.
|
| I acquired a taste for algebraic structures and category theory
| by way of Scala+cats. While I no longer work in that space, the
| concepts are universally applicable.
|
| Just last week, I shared with teammates in our weekly symposium
| how merging counts was an associative operation, and how knowing
| that directly relates to being able to divide and conquer the
| problem into an implicit O(log n) operation. Being able to
| identify the operation as forming a semigroup directly
| contributed toward being effective in that problem space.
|
| The abstractness of the concepts is a double-edged sword. It
| allows them to be broadly applied, but it also requires more
| effort on the part of the observer to form the connection.
| cced wrote:
| Can you expand a bit on what the problem was and what the
| discussion was like and solution, if possible.
|
| Thanks!
| kelseyfrog wrote:
| For sure. The problem was writing an np.unique[1] that could
| handle large datasets. Specifically, the solution involved
| chunking the dataset, mapping np.unique across chunks, and
| then combining chunks. Merging the counts result is an
| associative operation and merging them in a tree-like
| computational graph implies O(log n) merges. The end result
| was being able to perform the calculation in seconds whereas
| the previous duration was days to weeks. Going from O(n) to
| O(log n) is magic.
|
| Specifically this is work related to implementing large
| dataset support for the dedupe library[1]. It's valuable to
| be able to effectively de-duplicate messy datasets. That's
| about as much as I can share.
|
| 1. https://numpy.org/doc/stable/reference/generated/numpy.uni
| qu...
|
| 2. https://github.com/dedupeio/dedupe/blob/main/dedupe/cluste
| ri...
| ReleaseCandidat wrote:
| Actually what you are using is an equivalence relation and
| you are generating the equivalence classes of your set,
| "your" associativity is transitivity. Whenever you are
| comparing something it's almost always better to think of
| it as a relation instead of an algebraic structure. So you
| already know beforehand that your algorithm should be
| reflexive, symmetric and transitive (in theory, of course
| with computers there is numerics involved ;)
|
| The steps to the solution are something like:
|
| - I need all distinct elements of X
|
| - Oh, that's a quotient set (the partition) of X by a/the
| equivalence relation (`==`)
|
| - so my algorithm must be reflexive (yeah, trivially),
| symmetric (not so helpful) and transitive - now this I can
| use (together with the symmetry)
|
| It's generally easier if you know beforehand that it must
| be e.g. transitive.
| dskloet wrote:
| Why do you need to merge counts if you look for unique
| elements? Isn't the count always 1 for each element
| present? Isn't the key to shard your chunks so each element
| can appear in at most one chunk?
| SkiFire13 wrote:
| > merging them in a tree-like computational graph implies
| O(log n) merges.
|
| Intuitively this doesn't make sense to me. You have a tree
| that has N leaves, and you have to perform a merge for each
| parent. There are however N-1 parents, so you'll still
| perform O(N) merges.
| lanstin wrote:
| One of the most impressive early results on the path the the full
| classification (in terms of being able to understand the
| statement of the result, if not the 250 pages of the proof) is
| that all non-Abelian groups of odd order are not simple.
| https://en.wikipedia.org/wiki/Feit%E2%80%93Thompson_theorem
| vlovich123 wrote:
| > Physicists rely on them(opens a new tab) to unify the
| fundamental forces of nature: At high energies, group theory can
| be used to show that electromagnetism and the forces that hold
| atomic nuclei together and cause radioactivity are all
| manifestations of a single underlying force.
|
| Wow. I hadn't read this. Does this force have a name & an
| accessible explanation of how it degrades to 3 separate forces at
| low energies?
| pfortuny wrote:
| Should be this (electroweak interaction):
|
| https://en.m.wikipedia.org/wiki/Electroweak_interaction
| vlovich123 wrote:
| Hmmm mistake in the article then?
|
| > the forces that hold atomic nuclei together
|
| Isn't that the strong nuclear force which isn't unified with
| electroweak according to wikipedia? Ah [1] suggests that
| maybe Wikipedia just hasn't caught up and the math suggests
| that the strong nuclear force and even gravity may all be
| unified into a single such force. That's supremely
| fascinating.
|
| [1] https://www.symmetrymagazine.org/article/what-is-the-
| electro...
| setopt wrote:
| Disclaimer: I'm not a particle physicist, but I am a
| theoretical physicist. My info on this subject might be
| outdated.
|
| My understanding is that the "electroweak unification"
| (electromagnetism + weak force) is considered
| experimentally confirmed, whereas the "grand unification"
| (electroweak + strong force) is something most physicists
| believe to be possible but the theories remain somewhat
| speculative. For instance, if true, such unification likely
| means that the proton should be an unstable particle that
| very slowly decays, and that has never been observed
| experimentally.
|
| Unifying such a "grand unified theory" with gravity
| (general relativity) is however a much harder problem:
| While all the other forces are already understood in terms
| of quantum mechanics, gravity is currently understood in
| terms of _geometry_ , and it's not obvious how to combine
| those in a sane way. That's where string theory, quantum
| loop gravity, etc, come in; but AFAIK, most prospective
| theories are very far from even experimental predictions,
| let alone experimental testing, and not everyone even
| agrees on the premise that gravity "should" be described
| using quantum field theory.
|
| We just know that our current theories can't be complete,
| since thought experiments like "what is gravity during a
| quantum double slit experiment" or "what really happens
| near and inside a black hole" can't be answered in a fully
| satisfactory way using current theories.
| alasr wrote:
| Disclaimer: I'm not a physicist; just a hobbyist interested in
| these topics.
|
| What I understood, while reading this part of the article, was
| the author meant something along the line of supersymmetry[1,2]
| (as Groups are all about symmetry - up to isomorphism).
|
| From CERN Supersymmetry article[1]: If
| supersymmetric particles were included in the Standard Model,
| the interactions of its three forces - electromagnetism and the
| strong and weak nuclear forces - could have the exact same
| strength at very high energies, as in the early universe.
|
| --
|
| [1] - https://home.cern/science/physics/supersymmetry
|
| [2] - https://home.cern/science/physics/unified-forces
| trashtester wrote:
| It's not so much that there aren't 3 different forces, but
| rather that they are linked to each other in a way that cause
| them to "mix".
|
| In particular, electromagnetism (EM) and the weak force (WF)
| are represented mostly by the U(1) group and SU(2) groups,
| respectively.
|
| In pure electromagnetism, U(1)_em is what we're observing. This
| group is linked to a field caused by electric charge.
|
| But if we drill into it, there is also and underlying U(1)_y,
| that is linked to a hypercharge that is a combination of of
| electric charge and the WF interaction strength.
|
| The physics of the combined electroweak force is defined by the
| combined gauge group:
|
| SU(2)_L x U(1)_y
|
| From this fundamental physics we get (as energies get low
| enough) the Higgs mechanism through "spontaneous symmetry
| breaking".
|
| This causes two new independent fields from a linear
| combination of the fields associated with the two above gauge
| symmetries. One of this gives rise to observable photons, and
| one gives rise to observable (kind of) W, Z+ and Z-.
|
| And also quite significantly, this mechanism also gives rise to
| the Higgs field itself, which in turn provides mass (inertia).
| Without the Higgs mechanism, the particles arising from "pure"
| U(1) and SU(2) fields would be massless.
| hiAndrewQuinn wrote:
| Groups are pretty awesome. Understanding group actions in
| particular, which amomg other things is how those guys calculated
| how many different states a Rubik's cube can be in, knitted
| things together and was my "grow up" moment in abstract algebra
| class.
|
| They're also damn hard to explain! You know one when you see one,
| but it takes a lot of practice with different concrete examples
| before what they're _really_ about begins to shine through for
| most of us.
|
| For any undergrads who are tempted to skip AA and go right to
| category theory - don't do it, at least take the group theory
| course. It's a great intermediate intuition pump that makes the
| initial juggling you have to do with the basics of categories
| feel much easier.
| larodi wrote:
| Fascinated how Quanta Magazine can retell university university-
| grade matter in a very entertaining way. Wish all professors read
| it (the magazine) and perhaps be inspired to tell stories this
| way, and maybe more people will stay in math.
| jebarker wrote:
| Seeing the beauty of the correspondence between symmetry groups
| and Platonic Solids for the first time is a standout memory from
| my math undergrad. Groups are awesome.
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