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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 bloftin on 2007-02-18 12:57:32 |
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| > OK, maybe I'm like a literal paranoia, but what I don't
> understand is absolutely related to what description we use
> in the group representation.
It is good to try to dig as deep as possible, but once we leave the connection of the physical world with the underlying mathematics we need to focus on math definitions. A mathematician would probably scream at our previous posts :)
I don't want to take a giant step backwards but let us spend a moment on the basic definitions of a group, bear with me because this is as much for my education if you already have this under wraps. Using the definition from PlanetMath, http://planetmath.org/encyclopedia/Inverse.html
and the book "Group Theory and Physics" (if you go to amazon.com you can search within this book to get more info)
"A group is a set on which we are given a binary operation which behaves much like ordinary multiplication; that is, we are given a map of G x G -> G sending the pair(p,q) into pq, satisfying the associative law, the existence of an identity element e, and the existence of an inverse. That is we assume that
- (pq)r = p(qr) for any three elements p,q,r in G
- there exists an element, e, in G such that ep = pe = p for all p in G
- for every p in G there is a p^-1 in G such that pp^-1 = (p^-1)p = e
"
So when you say
> when we say "the representation of group SO(3)", the image
> in my mind is a field full of infinite numbers of 3x3
> matrices.
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?
>Every matrix can operate on a vector, these
> matrices of course represent different rotation
> transformations; and here, when we multiply two matrices
> through the normal matrix multiplication rules, the new
> matrix we get is still can operate on a vector, and
> represent a new space rotation.
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?
>So I inclined to say these
> matrices can represent the group SO(3) instead of the
> claiming "group SO(3) is represented by vector".
This now makes sense, I think I see what you where trying to say. Although I might now put it that these matrices represeent the elements of the group SO(3) but this could mean the same thing when you look at the big picture ie. All these matrices represent all the elements of SO(3) so we then just say the matrices can represent the group SO(3).
>Similar
> things happen about the spinor and SU(2), I think it's
> reasonable to call these 2x2 complex matrices as the
> representation of SU(2), not the spinor on which rotation
> matrices operated.
So yes these complex 2x2 represent the elements of the SU(2) group. The spinor is then something different, but we would use the elements of SU(2) to rotate the spinors.
>
> Do I clarify myself? Maybe this is only a trial question
> about nomenclature, but I'm really confused on it.
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.
Ben |
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