/sci/ - Science & Math

>>11236860
>U(4)

Intuitively:
U(1) = 2D rotations
SO(3) = 3D rotations
SU(2) = slight modification of 3D rotations where rotating by 360 degrees doesn't get you back to the same place, but rotating by 720 does.

>>11236879
more intuitively

U(1) = circle
SU(2) = mobius strip
SO(3) = sphere

>>11236914
more intuitive and more wrong

>>11236879
>>11236914
U(1) = Higgs
SU(2) = Electroweak
SU(3) = Chromodynamics

>>11236914
Very wrong. SU(2) and SO(3) are both 3 dimensional manifolds (comparing to your U(1)=circle example) not a 2D sphere. SO(3) is not orientable like a mobius strip, not SU(2).

>>11236930
Partially wrong. U(1) has nothing to do with the Higgs per se. The Higgs is not a gauge field. It has U(1) charge (hypercharge) but so do other particles, and it has 'charge' under SU(2) too.

>>11236941
>SU(2) and SO(3) are both 3 dimensional manifolds (comparing to your U(1)=circle example) not a 2D sphere. SO(3) is not orientable like a mobius strip, not SU(2).
are we talking about the space of transformations, or the space of things being transformed?

>>11236947
I was talking about the space of transformations

The sphere is acted on in the exact same way by both SO(3) and SU(2), and the mobius strip is not acted on in a natural way by either, so claiming you were talking about the space of things transformed doesn't make much sense.

>>11236941
>Partially wrong. U(1) has nothing to do with the Higgs per se. The Higgs is not a gauge field. It has U(1) charge (hypercharge) but so do other particles, and it has 'charge' under SU(2) too.

fine then smart anon. you win. i will correct my post thusly:
U(1) = weak hypercharge
SU(2) = weak isospin
SU(3) = (strong) hypercharge

>>11236955
question: how does this relate to things like color charge and weak isospin?

>>11237000
If something is transformed under the SU(2) part of the standard model it has weak isospin. If something is transformed under the SU(3) part of the standard model it has color charge, but I don't like this terminology to be honest. The SU(3) part is also more like isospin than like charge

>>11237000
Oh I misunderstood which post you were replying to. It doesn't have much to do with color charge. It has a lot to do with isospin since the set of all distinct states of something with isospin 1/2 is a sphere (the Bloch sphere).

>>11237025
isospin or weak isospin? Is there a difference?

>>11236860
OP, to put it simply, in physics we have things like electric charge and color charge. focusing on electric charge, there happens to be a symmetry: if you replace all + charges with - charges and at the same time replace all - charges with + charges [this is an O(1) symmetry] then you get the same answers after you make that swap. ben franklin style (he picked the wrong sign for coulombs but it happens to work out the same either way).

other things similar to electric charge behave this way. if you study chromodynamics of red->green->blue charges, then nothing changes if you permeate them for>green->blue->red labels instead. the labels don't matter. and what is interesting about this is that the fact you CAN swap labels like this puts constraints on the theory. for example, minus charge can't be special and red color charge can't be special. they all need to be symmetric.

i hope this gives you an idea for why symmetry principles are important. but if not then go read a fucking book.

actually, go read a fucking book anyway. it will do you good.

>>11237025
>>11237034
I mean in terms of geometry

>>11237034
The only difference is the physical interpretation of what the SU(2) group means.

In ordinary "spin" (not isospin) SU(2) refers to ordinary spatial rotations

In ordinary "isospin" SU(2) refers to an abstract symmetry where you mix up and down quarks. This is a "flavor" symmetry

In "weak isospin" SU(2) is a gauge symmetry that is associated to the weak force

do you concur?

>>11237038
No difference in geometry

>>11237037
>if you replace all + charges with - charges and at the same time replace all - charges with + charges [this is an O(1) symmetry]

You should read a book as well. This isn't how U(1) symmetry works in electromagnetism. U(1) rotates the phase of complex fields. Positively and negatively charged fields rotate in opposite directions

>>11237062
haha anon, you just confused U(1) symmetry with O(1) symmetry. get back to the textbook before being condescending.

>>11236930
Why are physishits shitting up my board with their perversions of groups?

>>11237069
You confused U(1) or SO(2) symmetry with something you are calling O(1). Now I was being kind enough not to call you out on that the first time, but if you're gonna be a smart ass about it, you deserve to be embarrassed

>>11237041
Not him, but this is actually relevant to my interests. I'm learning about Kaluza-Klein. From what I understand, the "U(1) circle" mentioned before is an actual spacial dimension rather than a mathmatical abstraction, right? And charge is represented as motion around that loop?

If so, how would this correspond to the weak force and strong force? From the sound of it they seem the same?

>>11237076
O(1) is a group, physishit he just has the wrong one.

>>11237076
no anon, swapping + and - charges in classical electrodynamics is an O(1) symmetry. read more carefully before being a douche

>>11237078
>If so, how would this correspond to the weak force and strong force? From the sound of it they seem the same?

No the strong force has to do with the group SU(3) not SU(2). It's a little confusing because quarks have both flavor (up,down,strange, etc) and color. Isospin is an approximate global (not gauge) symmetry having to do with flavor. The strong force itself comes from a gauge theory built on SU(3) not SU(2).

If you want to get SU(2) or SU(3) gauge theories out of Kaluza-Klein models you need to start thinking about things called complex projective spaces, which are like generalizations of the Bloch sphere

>>11237086
Electromagnetism is a U(1) gauge group. That group U(1) is what makes it electromagnetism rather than something else. You are describing charge conjugation, which is something completely different and not specific to electromagnetism.

Physicists never talk about "O(1)" and I doubt real mathematicians do either. If you want to talk about trivial sign flips as a group we say "Z_2" symmetry

>>11237096
no anon, physicists talk about local vs. global symmetries all the time. the fact you can only think in terms of local symmetries shows that your understanding is limited

>>11236914
>SO(3)
>Sphere
Physishits, ladies and gentlemen.

>>11237102
That wasn't me. I was the "physishit" that corrected him and posted half the posts in this thread.

>>11237101
You need to stop posting. You were talking about a discrete global symmetry that has nothing to do with electromagnetism itself. If you read the posts here you can see that, for one thing, I know a lot more than you, and for the other thing I was already talking about the global symmetry of isospin when it was brought up

>>11237111
>You were talking about a discrete global symmetry that has nothing to do with electromagnetism itself.
oh okay anon, O(1) has nothing to do with electromagnetism. good argument. global symmetries BTFO apparently

>>11237116
At least you knew O(1) was a group unlike this faggot >>11237076

>>11237144
i am the same anon as "this faggot"

maybe i am out of my depth here, assuming you guys are mathfags who jerk off to E8xE8, but i think all my posts about O(1) and global symmetries make complete sense, and the people saying "noooo charge conjugation isn't a symmetry" are retarded

>>11237150
wait sorry i was wrong, i'm not that guy. deleting my mistaken post.

>>11237150
>i am the same anon as "this faggot"
That was my post(?)

I never said charge conjugation symmetry isn't a symmetry. I said three things, all of which are true:

U(1) symmetry is not charge conjugation. U(1) is to electromagnetism as SU(3) is to quantum chromodynamics. And no one uses the term "O(1)," it is called Z_2. Seriously, do a quick google search

>>11237167
whatever, i like the group O(1) better than whatever mathfag shit you can say is equivalent because O means "the circle group" and it makes more sense to talk about a circle looping back on the same thing than worrying about modular arithmetic.

i deleted the post you quoted long before you posted your post because i had a temporary lapse. but i corrected it before you could post anything.so let that go.

the point is that group theory is important, important to the level that it goes to charge conjugation in classical electrodynamics. the fact that people like you seem to think it is far more mysterious than simple things like that is sad. group theory should be a common tool even at undergrad level, but it is not because ivory-tower mathfags like you like to make it seem super obscure. but it is not. simple symmetry principles are the underpinning for how group theory is used in physics, and in my opinion, simple symmetry principles like charge conjugation should be how undergrads are taught group theory. but that is not the case. instead they are taught years of crap about rings and fields and tons of useless shit. because mathematicians have this purity fetish. i recommend making math more aligned with applications as opposed to this purity fetish crap that helps noone in the real world

What is Eric Weinstein talking about here'


https://www.youtube.com/watch?v=C8B4RiYcaW4

>>11237091
>No the strong force has to do with the group SU(3) not SU(2)
I understand that much

>If you want to get SU(2) or SU(3) gauge theories out of Kaluza-Klein models you need to start thinking about things called complex projective spaces, which are like generalizations of the Bloch sphere

For the moment, I'm ignoring the complex numbers and thinking of vibrations on shapes, though I don't know how badly that fucks things up. Electromagnetism is a ring vibrating in the r direction. This much I'm comfortable with. When I try to picture the weak force though, I think of this tile pattern. Each of the tiles are actually the same tile, it's just lining them up like this makes it easier to picture how everything's connected. This space is 2-dimensional, but motion in one dimension is the same as motion in the other, meaning that really, there are only two directions you can go: clockwise and counterclockwise. I see this as similar to how [iso]spin works, in that you've got a 2-vector representing a single axis of rotation

My question is, am I picturing things right here, or is there some other way I need to arrange the tiles, possibly flipping them to make sense of all this?

>>11237249
fuck, forgot the image

>>11237249
this is the alternate, flipped pattern. I don't know which (if either) is right

>>11237249
>>11237254
>>11237269

first one, dumbass

>>11237249
>similar to how [iso]spin works, in that you've got a 2-vector representing a single axis of rotation

The 2-vector represents all directions. The fact that it is complex is necessary, since the representation of SU(2) is in terms of complex matrices. You should really read about the Bloch sphere if you haven't already (it sounds like you haven't).

I must admit I don't understand what you are doing with your image, but the notion that you can only go clockwise and counterclockwise corresponding to spin up and spin down sounds off to me.

>>11237249
>Electromagnetism is a ring vibrating in the r direction

Also this isn't correct. In Kaluza-Klein theories vibrations of the circles in the radial direction correspond to an unobserved scalar field called the dilaton. It is twists of the circle that lead to something like electromagnetism.

>>11237285
i think what you are saying is model-specific. lisa randall posted an article today on the arXiv which implicitly questions the orthodox string theory interpretations. her example is a good offering in thinking outside “the box”

>>11237278
>I must admit I don't understand what you are doing with your image, but the notion that you can only go clockwise and counterclockwise corresponding to spin up and spin down sounds off to me.
I'm going to admit I very likely fucked up the tile patten. It's the best way for me to picture this stuff but even that isn't intuitive. I think it would make more sense if I could depict it like an arcade game with weird wrapping

I have read about the bloch sphere, but the representation doesn't make sense to me. In my mind, vector dimensions and cardinal directions need to correspond. I find easier to imagine the complex phase using waves.

Part of the issue is I don't have the mathematical background I need to communicate the things I'm trying to say. Another part is that I just have a really fucking aberrant thought process. I had a math teacher in high school who once actually told me to stop paying attention for a few minutes in the middle of a lecture so that he could explain the subject to me separately in terms I could understand. It actually worked as a teaching technique

>>11237321
>I had a math teacher in high school who once actually told me to stop paying attention for a few minutes in the middle of a lecture so that he could explain the subject to me separately in terms I could understand

>>11237043
what kind of advanced shitpost is this

bumping

>>11237191
The image is just the Hopf fibration. It certainly is not "the most important object in the universe." He's making up his own pop sci to try to wow Joe Rogan and the audience.

The Hopf fibration is important as an example of a fiber bundle and it does show up in various places in physics. It is essentially the same thing as the map from a two-component complex spinor to the Bloch sphere that we were talking about in this thread in another context.

But I think what Weinstein is getting at is that gauge theories are important in physics, and they can be described in fiber bundle language. The simplest example is electromagnetism. Electromagnetism can be described by a "principle bundle" which is a copy of the U(1) Lie group (a circle) attached to each point in space-time, with perhaps some nontrivial global topology. This is not so different from the Hopf fibration, so if spacetime were just the two dimensional sphere (it's not), the Hopf fibration could be a principle bundle for electromagnetism.

But beyond just the principle bundle itself you need some extra mathematical structure to describe a gauge theory called a connection. Considering this, and the fact that the Hopf fibration is not literally a model of a gauge theory of the real world, I think Weinstein is really not being very honest and just trying to impress people about physics.

>>11238227
explain this "fiber bundle". what is it in layman's terms?

>>11238315
>a copy of the U(1) Lie group (a circle) attached to each point in space-time
You attach a "fiber," in this case a circle, to each point of some other manifold (geometrical space). This makes a new manifold you can call a fiber bundle. In the case of the Hopf fibration you attach a circle to each point of a sphere and you make a new manifold which is equivalent to a 3D sphere. I can't get much more layman than that

>>11238351
is the fiber always 1-dimensional, or are there also sheet bundles?

>>11238355
It is usually not 1-dimensional. If you have U(1)xSU(2)xSU(3) gauge theory like the standard model, that is a 12 dimensional fiber

>>11238368
so the bundle's dimension is determined by the number of generators/bosons? why then does string theory only need 11? shouldn't it need 16?

>>11236860
groups of operators. Look it up

where can I read how is electromagnetism
formulated in the language of connections on principal bundles ?

>>11238375
>so the bundle's dimension is determined by the number of generators/bosons?
Yes

>why then does string theory only need 11?
String theory isn't a fiber bundle in the same way. The fibers aren't dimensions of space in a gauge theory. They are something abstract. In string theory the extra dimensions are just like the four dimensions of space-time we do observe. This kind of set up with extra spatial dimensions is called a Kaluza-Klein theory, and it can also be considered a fiber bundle, but it is a different kind of thing from the fiber bundles in gauge theories

>>11236860
U(1) = circle
SU(2) = 3-dimensional sphere
SO(3) = 3-dimensional projective space

>>11238401
Nakahara's Geometry, Topology and Physics. It's not a great textbook, but that's where I first learned it

>>11238425
so, in string theory is it bases on charges instead?

i keep getting confused because im thinking in terms of charges and the standard model tends to think in terms of transformations

>>11238437
No, I'm not sure what you mean by based on charges. The difference between the two is related to the idea that the 2D sphere still has three distinct ways you can rotate it, and these rotations form a group that is 3-dimensional when you consider it as a manifold. From the principal bundle point of view you attach this 3d manifold to each point of space, from the Kaluza-Klein point of view you attach the 2d sphere to each point of space

>>11238445
what i mean is, in string theory is the number of spacial dimensions based on the number of charges instead of the number of bosons?

>>11238517
The charge is only really well defined in U(1) gauge theory. If anything the number of charges is equal to the number of gauge bosons in non-Abelian gauge theory. You can define 3 independent charges in SU(2). You can define 8 independent charges in SU(3). The problem is these charges are not gauge invariant.

If you are just asking whether the counting is 1 for U(1), 2 for SU(2), and 3 for SU(3), the answer is also no.

>>11238570
how is charge not defined? isnt it just two weak isospin states and three color charges? am i using the word charge wrong?

>>11238633
>two weak isospin states
still just one charge

>>11238633
>am i using the word charge wrong?
Yes.

"Spin-upness" and "spin-downness" aren't two conserved charges. There are three conserved components of angular momentum and these are what are like charges. The same situation occurs for more abstract kinds of isopin or color.

bumping because actual good threads are rare

>>11239185
Wikipedia says only the third component of weak isospin is conserved.

also, i was saying weak isospin, not spin

What are the group elements in SO(2)? Is a particular rotation by [eqn]\theta[/eqn]rad an element and so there's the continuous spectrum of them?

>>11240914
The group is isomorphic to the set of complex number of the form [math] {\mathrm e}^{i\theta} [/math] where [math] \theta [/math] is real. So "yes". The standard representation is the 2x2 rotation matrices.

if SU(2) has 720° rotational symmetry, isn't a Klein bottle a better representation?

>>11241010
Best geometric picture I've seen is to think of a flag pole rotating in 3D space. The pole can rotate around, but you can also rotate the direction of the flag by rotating around the axis of the pole. This is double covered by SU(2) so then you attach a +/- sign to the flagpole.

https://arxiv.org/pdf/1312.3824.pdf

do fermions have their own symmetry groups?

>>11238375
Spacetime is 10D in ST (M-theory is 11), but the symmetry group of string theory (for type I, HE & HO) is 496 dimensional, so it would look like a 496 dimensional fibre attatched to every point of a 10D spacetime.
This is a lot more than we observe, but one can hide symmetries with the Higgs mechanism (or some stringy version thereof)

>>11241594
i thought the 10d one wasn't even used anymore.

also, what the fuck? symmetry groups and compactified dimensions are separate things? i thought that was the entire point of compactified dimensions

>>11241163
I like that paper.
I'm not sure how practically useful it is, but it's good for intuition.

>>11241195
They transform under the same groups as the bosons, but they transform differently.
For example, quarks transform in the fundamental representation, whereas gluons transform in the adjoint.

>>11236874
based

>>11241612
>i thought the 10d one wasn't even used anymore.
No, M-theory provides a non-perturbative description of IIA and HE as an 11D theory, but the other three are still 10D, though they're related to M-theory by dualities, which really shows the difference between 10 and 11D isn't always so clear cut.
The original KK theory used the extra dimension to generate symmetries, but that's not true in ST -- typical string compactifications are Calabi-Yau, so they don't have any continuous symmetries. These stringy symmetries are just like the (non-KK) standard model symmetries, unrelated to extra dimensions: they're just groups that the states of the theory transform under, though they have more in depth explanations than in particle physics -- in type I the symmetry lives at the end points of open strings, in the heterotic theories it's concentrated in the left-moving vibrations of the strings.
Compactification is essential because ST must be 10 (or 11) dimensional, but we see 4.

>>11241634
what about the idea of extended local SUSY in ST bringing things to 4D? i realize this screws up the number of time dimensions but has there been any proposed loopholes to make 3+1 out of this idea?

>>11236860
U(n) = group of n × n unitary matrices.
O(n) = group of n × n orthogonal matrices.
SU(n) = group of n × n unitary matrices of determinant 1.
SO(n) = group of n × n orthogonal matrices of determinant 1.

Simple as.

>>11237043
Imagine having to use actual NUMBERS in your calculations lmao physishits are pathetic

>>11237186
>O means "the circle group"
You're retarded.

>>11236860
>what the fuck is this shit?
Group theory ;^P

>>11237186
>O is the circle group
What the fuck are you trying to say? This isn’t even mathfagging - mathematicians, chemists, and physicists all agree that O(n) or O_n refers to the orthogonal group, and this is insanely important in representation theory when you would start having to learn character tables

>>11241695
That's one way to look at it, albeit a material one. I wouldn't identify SO(3) with its fundamental representation.

>>11241778
What do you mean? Those are literally their definitions.

>>11241778
That’s not “one way to look at it”
That’s their actual definitions as they were made LMAO
The absolute state of physishits trying to loosely analogize actual formal definitions.

>>11241778
>He doesn’t know isometries or basic group theory
Based physics brainlet

>>11240895
You misunderstood Wikipedia. Isospin and spin work exactly the same way.

>>11241832
>>11241886
You misunderstood what he was saying. He was saying he prefers to think of SO(3) as an abstract group rather than a matrix representation of the group.

Protip, if you don't know what "fundamental representation" means you should probably hold off on calling someone a brainlet

>>11241807
> Those are literally their definitions.
No.
Go to
https://en.wikipedia.org/wiki/3D_rotation_group
and look where "matrix" first pops up.

Or
https://en.wikipedia.org/wiki/Group_representation

An SO(2) isn't a matrix with 4 entries. But the group SO(2) has a representation where each element is a 2x2 matrix with sin's and cos's as their elements. SO(2) also has a representation as exp(i*t) as in >>11240949

>>11241807
>Those are literally their definitions.
No really, go to the article and look where "matrix" first pops up. You don't say SO(3) := {matrices such that...}
https://en.wikipedia.org/wiki/3D_rotation_group

>>11241914
No, we’re all aware of what the fundamental representation is - but there’s still a reason why SO stands for special orthogonal

>>11241960
And if you click on SO on the Wikipedia article you get the orthogonal group
Yes, you formulate it as an isometry on the Euclidean plane...but exactly what is that in terms of vector spaces? Orthogonal matrices. One need not mess around with the representation homomorphism to know what O and SO are directly as matrices

>>11241960
It is. Tell me how you plan to explain the unitary (complex) cases as “rotations”. It’s much more straightforward to use the definitions he gave and then say hey, this particular case corresponds to the group of symmetries of X.

>>11241960
Go to this article (which covers the more general case) and see where the word “matrix” pops up:
https://en.wikipedia.org/wiki/Unitary_group

>>11241960
LOL
Dude U, O, SO, and the like are lol subgroups is GL, one of the most well known, basic bitch groups in all of math and science. You can think of them as described by abstract symmetry in the context of group actions, but as pure basic group theoretic objects, they *are* matrices.

>>11242472
*are subgroups of

>>11236879
>SU(2) = slight modification of 3D rotations where rotating by 360 degrees doesn't get you back to the same place, but rotating by 720 does.
Does this have anything to do with spin in physics?

>>11238351

what kind of fiber can be attached, does the fiber need to be a group or something?

>>11242508
Yes, a lot to do with it. Integer spin is a representation of both SO(3) and SU(2). Half integer spin is strictly speaking only a representation of SU(2)

>>11242511
For a principal bundle, yes it needs to be a group

>>11242514
Does half-integer spin literally just mean some group associated with the particle is SU(2)?

>>11242519
Sort of. Everything that transforms under SU(2) can be broken up into representations that have a spin m/2, where m is an integer

>>11236990
>strong hypercharge
>not "color"
I'm triggered.

I guess you can't please everyone though.

>>11238633
charges in non-abelian gauge theories aren't well-defined because there's no conserved (Noether) current in those theories. The current is only covariantly conserved, meaning your definition of charge depends on choice of gauge, making it unphysical.

>>11242250
>>11242331
I'm not saying that this would be a good or even the best deifnition, nor do I want to imply to take Wikiepdia as your primary source of definition.

But SO(3) is defined as the isometric transformations, which just as well could be seen as a set of functions which as domain has all the objects in 3D space and as outputs have the rotated versions of that.
The fact that it suffices to specify 3 point coordinates (a choice or orthonormal vectors, and not all points, let alone not extended objects such as cubes) to specifiy a rotation
[math] (1,0,0) \mapsto (R_{11}, R_{12}, R_{13}) [/math]
[math] (1,0,0) \mapsto (R_{21}, R_{22}, R_{23}) [/math]
[math] (1,0,0) \mapsto (R_{31}, R_{32}, R_{33}) [/math]
doesn't have to be assumed to say that SO(3) are all rotations.

You can define G=SO(3) just as the group of transformations and their induced concatenation as their group product, without any reference to matrix multiplication upfront. A map from this group G into the set of all orthogonal matrices can still be defined.

>>11242250
>>11242331
I'm not saying that this would be a good or even the best definition, nor do I want to imply to take Wikipedia as your primary source of definition.

But SO(3) is defined as the isometric transformations, which just as well could be seen as a set of functions [math] G=( \{ f,g,h,\dots \}, \circ) [/math] which as domain have all the points in 3D space (or even objects too) and as outputs have the origin-rotated versions of that, [math] f: {\mathbb R}^3 \to {\mathbb R}^3 [/math], [math] g: {\mathbb R}^3 \to {\mathbb R}^3 [/math]
The fact that it suffices to specify 3 point coordinates (a choice or orthonormal vectors, and not all points, let alone not extended objects such as cubes) to characterize a rotation

[math](1,0,0)(R_{11},R_{12},R_{13})[/math]
[math](0,1,0)(R_{21},R_{22},R_{23})[/math]
[math](0,0,1)(R_{31},R_{32},R_{33})[/math]

doesn't have to be assumed to say that SO(3) are all rotations.

You can define G=SO(3) just as the group of transformations and their induced concatenation as their group product, without any reference to matrix multiplication and bases etc. upfront. A map from this group G into the set of all orthogonal matrices can still be defined.

>>11242250
>>11242331
I'm not saying that this would be a good or even the best definition, nor do I want to imply to take Wikipedia as your primary source of definition.

But SO(3) is defined as the isometric transformations, which just as well could be seen as a set of functions [math] G=( \{ f,g,h,\dots \}, \circ) [/math] which as domain have all the points in 3D space (or even objects too) and as outputs have the origin-rotated versions of that, [math] f: {\mathbb R}^3 \to {\mathbb R}^3 [/math], [math] g: {\mathbb R}^3 \to {\mathbb R}^3 [/math]
The fact that it suffices to specify 3 point coordinates (a choice or orthonormal vectors, and not all points, let alone not extended objects such as cubes) to characterize a rotation

[math](1,0,0) \mapsto (R_{11},R_{12},R_{13})[/math]
[math](0,1,0) \mapsto (R_{21},R_{22},R_{23})[/math]
[math](0,0,1) \mapsto (R_{31},R_{32},R_{33})[/math]

doesn't have to be assumed to say that SO(3) are all rotations.

You can define G=SO(3) just as the group of transformations and their induced concatenation as their group product, without any reference to matrix multiplication and bases etc. upfront. A map from this group G into the set of all orthogonal matrices can still be defined.

>>11242855
>>11242331
Btw. the fact that U(n) can best be defined as a subgroup of the general linear (!) group of nxn matrices, and that the rotations can also be obtained this way as a subgroup, doesn't mean you need to organize your understanding of the roation group in 3D along those lines.
The information of an infinite set of functions capturing rotations of all points in 3D together with the information about their concatenation apriori is a bigger burden than just an infinite set of triples of 3-vectors (the matrices) together with a simple multiplication rule. But the former is a group. And which group is it? Well both are isormorphic and both represent the rotations. The latter variant has more structure and thus cuts down the presentation

>>11242855
>>11241960
dude SO(3) = literally the group of real 3x3 orthogonal matrices with determinant 1

yes, you can "look at it" any way you want, I also picture the elements as transformations and not 9-tuples of numbers

for an inner product space V you can define SO(V) = the set of all orthogonal transformations with determinant 1, then choice of a basis gives isomorphism SO(V) -> SO(n), in particular we have natural SO(R^n) -> SO(n) induced by the standard basis blah blah blah blah

but SO(3) is LITERALLY the set of real 3x3 orthogonal matrices with determinant 1. there's no way around that.

>>11242922
>there's no way around that.
We're literally talking just semantics. I think most people think "the rotations in 3D space" when they hear SO(3). It might not be universally the case, especially if you're studying vector spaces more than things in space, but it really doesn't matter.

I personally think the "it's matrices with property X" is the best definition for "SO(3)", let's leave it at that.

>symmetry groups are matrices
No, they're much more general than that.

>>11236914
>group = shape
Groups aren't shapes, they're the symmetries of those shapes. And the fag saying a Moebius was orientable needs to pick up a math book.

>>11243086
You need work on your ability parsing English

>>11243086
>>symmetry groups are matrices
who said that ?

the father
the son
the holy spirit

>>11243670
Why? Because I missed the word "strip"? The rest was completely coherent.

>>11243695
This guy for example:
>>11242942

>>11236860
>mathematicians
>cant even define a few very simple objects

Why are mathematicians so RETARDED? You'd think that spending thousands of hours on your science and you would come up with a definition system for each of your objects.

>>11243927
>>cant even define a few very simple objects
Dude these simple objects have simple definitions. The physishits in this thread can't understand abstract symmetry or vector spaces. They don't see why it's called "unitary," "orthogonal," or "special orthogonal"

>>11243086
>No, they're much more general than that.
Not really.
See: Ado's theorem

>>11241685
I'm not sure what you mean; type II is N=2 (hence the name), it's still 10D.

>>11244023
Have you read what it says? The fact matrices represent something doesn't mean they are the exact same thing.

>>11243919
My post you quoted said that both SO(3) and the mobius strip were not orientable. Maybe I phrased it awkwardly but you misread it as though I was saying the mobius strip was orientable.

I know this is the internet, but sometimes when you think you see someone saying something obviously wrong, maybe it's best to stop and think whether you are the one misinterpreting things

>>11244496
You mean this one?
>>11236941
Then yeah, it totally sounded like you were saying a Moebius strip was orientable. I didn't quote it, though, because I couldn't find it at first.
You shouldn't put the blame on others per default when these things happen. Sometimes written language is ambiguous in its meaning. Perhaps we're simply both at fault here.

>>11236860
it's a stupid mobius strip.

>>11245175
Is there any other kind of möbius strip

>>11243919
>This guy for example
said what?
He says the opposite of "symmetry groups are matrices"

>>11245176
nope.

>>11245184
Are you retarded?
>I personally think the "it's matrices with property X" is the best definition for "SO(3)",

>>11245263
does this imply that the person thinks symmetries = matrices in the most general context ?

also SO(3) literally IS a matrix group.

>>11245266
So that's a "yes", okay then.

>>11245308
I'm not the one you were responding to, and I have legit no idea what's your point. Please explain what's wrong with
>"it's matrices with property X" is the best definition for "SO(3)
hint: nothing

>>11245322
If you could read, you'd know I already said that symmetry groups aren't matrices, and that Ado's theorem specifically doesn't claim so. Feel free to read the article on it.
Your condescension won't help your ignorance.

>>11245322
>>11245328
Sorry, should have included this directly:

See for example versors, rotors or Euler angles. All non-matrices realizing SO(3).

>>11245339
>versors, rotors or Euler angles
okay, that's a fair point
for me SO(3) is a group of "universal" rotations, which are yet to be applied to something specific. it's not a group of transformations of some fixed vector space like R^3 (although it almost seems like it). matrices are one way to encode this, I agree that versors and Euler angles are another one. rotors seem more like they're tied to a specific vector space.
I don't know if this makes any sense.

>>11245370
Yeah it does. Now you get what I meant.

>>11237293
No she didn't; her latest paper is from the 13th.

>>11244061
I didn't say they're the same, I said Lie groups aren't MUCH more general than matrix groups, since all (finite dimensional) Lie algebras admit a faithful matrix representation. Note the word faithful -- this isn't just some random representation.

>>11237043
>tfw you can no longer tell if the graphic is made by some crank or Bar-Natan

>>11245884
yukariposter, how do i into CFTs, especially OPEs

>>11244496
>>11245166
>is not orientable like the möbius strip
the more natural interpretation is certainly:
>is not (orientable like the möbius strip)
rather than
>is (not orientable), like the möbius strip
perhaps
>is non-orientable like the möbius strip
would have been the best choice as there is no parsing ambiguity; non-is an affix that must effect only the word it is attached to while not is a free negation particle that can effect single words or entire phrases.

>>11245408
>>11245370
>>11245339
>>11245328
>>11245322, etc
itt: watch people who agree on the reality argue over the semantics for hundreds of replies.

>>11245408
Best definition is pretty much "they are". You probably didn't mean it that way, but I can only reply to words here.

>>11246113
>semantics
No. Try to comprehend what was said.

>>11245370
The vector space definition is from Representation Theory. Just because groups can be expressed using matrices does not mean that is what they are.

>>11245266
No it isn't. I could make an isomorphism from SO(3) -> S0(3) , M ---> Rotate(x radians, y radians, z radians)

>>11237043
>fucking color coded

>>11245339
Those are just *representations* of matrices, which ARE the linear maps.

>>11246649
SO(n) is a matrix group
https://en.wikipedia.org/wiki/Matrix_group

Euler angles suck. They're neither continuous nor bijective (see the gimbal lock problem).
>In formal language, gimbal lock occurs because the map from Euler angles to rotations (topologically, from the 3-torus T3 to the real projective space RP3 which is the same as the space of 3d rotations SO3) is not a local homeomorphism at every point, and thus at some points the rank (degrees of freedom) must drop below 3, at which point gimbal lock occurs. Euler angles provide a means for giving a numerical description of any rotation in three-dimensional space using three numbers, but not only is this description not unique, but there are some points where not every change in the target space (rotations) can be realized by a change in the source space (Euler angles).

>>11246113
Yeah it was an awkward and ambiguous sentence, no doubt.

But the more natural interpretation is
>is (not orientable), like the möbius strip
simply because the möbius strip is used the paradigm example of something which is not orientable. I don't understand how someone could read that sentence, think I meant the möbius strip was orientable, write a post saying what an idiot I was, and at no point have the thought "hey, wait a second, it doesn't quite make sense that he is using words like this and making a mistake so basic, maybe I should read one more time." That says something about discussions on the internet I think.

>>11246684
And? What's your point?

>>11246742
I was talking about the most grammatically natural interpretation without informational context. The "not [adj] like X" structure is extremely commonly used to indicate it is not adj but also provide an example of something that is.
That said, I personally understood your post the first time I read it.
>>11246603
>is
>is represented by
>is defined as

>>11245983
Read Di Francesco

>>11248622

> ~3.4Ghz chips, oscillates with frequency of water, human body is primarily water.

anyone know what else oscillates at 3.4GHz?

>>11248623
thanks, have a lumo

>>11236860
S - special ( determinant = 1 )
U - unitary
O - orthogonal
# - dimension
Referring to transformations

Brainlet here, I know what groups are and most definitions and those are all well and good, and I know what electromagnetism is in the sense that two like charges repel, and so on. What I don‘t get is how is group theory useful to study electromagnetism? Does associating U(1) to electromagnetism tell me something I don‘t already know? If I didn‘t know that like charges repelled, would U(1) tell me that? Why U(1)? Why electromagnetism? I understand the two seperately, but I have no idea why these concepts are related.

So many replies, but OP did not even say the field he is working in. He should have at least told us its characteristic.