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![[parent]](https://images.physicslibrary.org/images/uparrow.png) Viewing Message
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| ``Re: Why we can say a spinor be a representation of SU(2)?''
by Tyger on 2007-02-19 02:44:41 |
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| > What this means in math speak is that SO(3) is a subgroup of
> the group 0(3), which is all orthogonal transformations
> (infinite # of 3x3 matrices) in Euclidean 3D space. The
> "elements" of S0(3) which are also orthogonal
> transformations (3x3 matrices) are also restricted to having
> their determinant = +/- 1. This added restriction means it
> is a subgoup of O(3). What I'm getting at is that instead
> of using the word representation maybe we should be talking
> about the elements of S0(3) and these elements must satisfy
> the above group 'rules' along with the determinant
> restriction. Although in the book "Group Theory in Physics"
> when they wright down a 3x3 matrix they say it represents an
> element in the SO(3) group. Is that what you mean?
Yes, it is.
> Here we are 'leaving' the definition of the SO(3) group(
> i.e. if we multiply two matrices within SO(3) we get another
> element of SO(3) this is because the group is closed under
> the '*' operation (see PlanetMath def). So the vector that
> the matrix is operating on is not an element of S0(3) group
> but just an element of Euclidean space ( a vector space
> without its orgin) However these vector elements are also
> an additive Abelian group. So here we have two different
> groups. So what is the terminology when we have a SO(3)
> group operate on a Additive abelian group? I'm used to
> hearing rotation, but what is the math term for this?
I'm sorry there is not such math term in my mind now, a vector space seemed too limited, however a module space is too general for this question. Maybe I should search its answer in a broader field beyond my brain later.
> This is good for us to try to clarify all of this, I know I
> was definately confused. So hopefully it now makes sense in
> Griffiths book when he says, "Thus spin-1/2 particles
> transform under rotations according to the fundamental, 2D
> representation of SU(2)" Later.
Really sincerely hope.
Tyger
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