- #36
Mehdi_
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Hi Garrett
Can we then say that the quaternion of norm 1 belong to SU(2) group ?
Can we then say that the quaternion of norm 1 belong to SU(2) group ?
Yes.Mehdi_ said:Can we then say that the quaternion of norm 1 belong to SU(2) group?
Heh. Read my last paper. :)I know that spinors are related to quaternions... tomorrow I will try to find the link between them...
garrett said:Great!
The related wiki page is here:
http://deferentialgeometry.org/#[[vector-form algebra]]
Explicitly, every tangent vector gets an arrow over it,
[tex]
\vec{v}=v^i \vec{\partial_i}[/tex]
and every 1-form gets an arrow under it,
[tex]\underrightarrow{f} = f_i \underrightarrow{dx^i}[/tex]
These vectors and forms all anti-commute with one another. And the coordinate vector and form basis elements contract:
[tex]
\vec{\partial_i} \underrightarrow{dx^j} = \delta_i^j
[/tex]
so
[tex]
\vec{v} \underrightarrow{f} = v^i f_i
[/tex]
Mehdi_ said:These vectors and forms all anti-commute with one another should means:
[tex]\vec{v}=v^i \vec{\partial_i}=-\vec{\partial_i}v^i[/tex]
[tex]\underrightarrow{f} = f_i \underrightarrow{dx^i}=-\underrightarrow{dx^i}f_i[/tex]
That means that order is important... it is a non-commutative algebra
I had never realized that!garrett said:Sure Patrick, glad you're liking this thread.
By "the vectors and forms all anticommute with one another" I mean
[tex]
\underrightarrow{dx^i} \underrightarrow{dx^j} = - \underrightarrow{dx^j} \underrightarrow{dx^i}
[/tex]
which is the wedge product of two forms, without the wedge written. And
[tex]
\vec{\partial_i} \vec{\partial_j} = -
\vec{\partial_j} \vec{\partial_i}
[/tex]
which tangent vectors have to do for contraction with 2-forms to be consistent. And
[tex]
\vec{\partial_i} \underrightarrow{dx^j} = -
\underrightarrow{dx^j} \vec{\partial_i} = \delta_i^j
[/tex]
which is an anticommutation rule you can avoid if you always write vectors on the left, but otherwise is necessary for algebraic consistency.
1-form anticommutation is pretty standard, as is vector-form contraction -- often called the vector-form inner product. The vector anticommutation follows from that. And the vector-form anticommutation from that. (Though I haven't seen this done elsewhere.) It makes for a consistant algebra, but it's non-associative for many intermixed vectors and forms, so you need to use parenthesis to enclose the desired contracting elements.
garrett said:By "the vectors and forms all anticommute with one another" I mean
[tex]
\underrightarrow{dx^i} \underrightarrow{dx^j} = - \underrightarrow{dx^j} \underrightarrow{dx^i}
[/tex]
which is the wedge product of two forms, without the wedge written. And
[tex]
\vec{\partial_i} \vec{\partial_j} = -
\vec{\partial_j} \vec{\partial_i}
[/tex]
which tangent vectors have to do for contraction with 2-forms to be consistent. And
[tex]
\vec{\partial_i} \underrightarrow{dx^j} = -
\underrightarrow{dx^j} \vec{\partial_i} = \delta_i^j
[/tex]
which is an anticommutation rule you can avoid if you always write vectors on the left, but otherwise is necessary for algebraic consistency.
garrett said:The algebra of vectors and forms at a manifold point, spanned by the coordinate basis elements [itex]\vec{\partial_i}[/itex] and [itex]\underrightarrow{dx^i}[/itex], are completely independent from the algebra of Clifford elements, spanned by [itex]\gamma_\alpha[/itex], or, if you like, they're independent of all Lie algebra elements. By the algebra being independent, I mean that all elements commute.
garrett said:The expression you calculated,
[tex]
g(x) = e^{x^i T_i} = \cos(r) + x^i T_i \frac{\sin(r)}{r}
[/tex]
is a perfectly valid element of SU(2) for all values of x. Go ahead and multiply it times its Hermitian conjugate and you'll get precisely 1.
Mehdi_ said:It is related to the condition, [itex]{({x^1})^2 + ({x^2})^2+ ({x^3})^2}=1[/itex]
garrett said:Because I missed that term! You're right, I thought those would all drop out, but they don't -- one of them does survive. ( By the way, becuase of the way I defined <> with a half in it, it's [itex] < T_i T_j T_k > = \epsilon_{ijk} [/itex] ) So, the correct expression for the inverse Killing vector field should be
[tex]
\xi^-_i{}^B = - < \left( (T_i - x^i) \frac{\sin(r)}{r} + x^i x^j T_j ( \frac{\cos(r)}{r^2} - \frac{\sin(r)}{r^3}) \right) \left( \cos(r) - x^k T_k \frac{\sin(r)}{r} \right) T_B >
[/tex]
[tex]
= \delta_{iB} \frac{\sin(r)\cos(r)}{r} + x^i x^B ( \frac{1}{r^2} - \frac{\sin(r)\cos(r)}{r^3} ) + \epsilon_{ikB} x^k \frac{\sin^2(r)}{r^2}
[/tex]
garrett said:[itex]
\underrightarrow{e} = \underrightarrow{e^\alpha} \gamma_\alpha = \underrightarrow{dx^i} \left( e_i \right)^\alpha \gamma_\alpha
[/itex]
Taoy said:I'm talking about the basis elements [itex] \partial_i \equiv \frac{d}{dx^i} [/itex] and their dual one-forms. In your notation you put an over arrow over the top indicating that we are dealing with a complete vector, i.e. [itex] e_i \equiv \vec{\partial_i} [/itex]. You then said that they obey an anti-commutation rule: [itex] e_i e_j = -e_j e_i [/itex].
So, my question was about the kind of product that you are using between these elements. In general the product of two vectors carries a symmetric and an antisymmetric part: [itex] e_i e_j = e_i \cdot e_j + e_i \wedge e_j [/itex], and it is only the antisymmetric part which anti-commutes. However if you are explicitly working in an orthonormal basis they what you say is correct, unless i=j in which case the two commute.
Taoy said:Sure I get that, but the series expansions we use are only valid for small x, for instance substitute [itex]4\pi[/itex] into the series expansion and it doesn't work anymore...
Whilst we're here, where does the condition come from? I thought that [itex] g g^- [/itex] might impose some condition on the x's, but it doesn't. Where does it come from? :)
Taoy said:What kind of object is [itex] e_\alpha ? [/itex], and what kind of object is [itex] \gamma_\alpha ? [/itex]
Are you using upper and lower arrows to purely signify differential geometry objects? Why not arrows on the gamma too; I take it that this is a vector (as apposed to a dual vector)?
selfAdjoint said:The [tex]e_\alpha[/tex] are the "legs" of the vierbien or frame; four orthonormal vectors based at a typical point of the manifold.
I think the [tex]\gamma_\alpha[/tex] are just multipliers (bad choice of notation; they look too d*mn much like Dirac matrices).
Taoy said:What happened to the [itex] x^i x^j x^k \epsilon_{jkB} (\cos(r)/r^2 - \sin^2(r)/r^4) [/itex] term?
p.s. it looks like the right-invariant vectors are just minus the left-invariant ones.
Taoy said:p.s. it looks like the right-invariant vectors are just minus the left-invariant ones.
Originally Posted by Taoy
Whilst we're here, where does the condition [itex]{({x^1})^2 + ({x^2})^2+ ({x^3})^2}=1[/itex] come from?
selfAdjoint said:The [tex]e_\alpha[/tex] are the "legs" of the vierbien or frame; four orthonormal vectors based at a typical point of the manifold. I think the [tex]\gamma_\alpha[/tex] are just multipliers (bad choice of notation; they look too d*mn much like Dirac matrices).
garrett said:It's zero.
[tex] x^j x^k \epsilon_{jkB} = 0 [/tex]
garrett said:[tex]
\gamma_\alpha
[/tex]
is one of the Clifford algebra basis vectors.
Yes, I put arrows over tangent vectors, arrows under forms, and no arrows under or over coefficients or Lie algebra or Clifford algebra elements such as [itex]\gamma_\alpha[/itex] .
Taoy said:I thought that you wanted to keep elements of the vector space and of the dual space separate and distinct? The clifford algebra elements can be geometrically interpretted as a vector basis, and an arbitary vector expanded in them,
[tex] v = v^i \gamma_i = v_i \gamma^i [/tex]
where
[tex] \gamma^i . \gamma_j = \delta^{i}_{j} [/tex]
Are you less worried about preserving the distinction between [itex] \vec \gamma_i [/itex] and [itex] \underrightarrow{\gamma^i} [/itex] because of the presence of an implied metric?
Originally Posted by Mehdi
Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign),
we have a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is { + I, − I}.
What does "whose kernel is { + I, − I}" mean