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Results tagged with lie-algebra
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user 80687
A vector space $\mathfrak{g}$ over some field $F$ and kitted with a bilinear, antisymmetric and Jacobi-identity-fulfilling product ("Lie Bracket" or "commutator"). In physics, most often arises as the Lie algebra (tangent space to the identity) of a Lie group; in gauge theories, basis vectors of the gauge group's Lie algebra correspond to Noether currents and conserved quantities.
1
vote
Proving gauge transformation of non-abelian field strength
You are making heavy weather of something easy. I'll use slightly different conventions but the result is the same.
Start by defining the gauge-covariant derivative
$$
\nabla_\mu=\partial_\mu+A_\mu.
…
- 56.5k
1
vote
Writing gauge transformation of the gauge fields explicitly in terms of coordinates
Your derivative term should be
$$
(\partial_\mu g) g^{-1}.
$$
This combination is an element of the Lie algebra,and so expressible as a sum of the $t^\alpha$, unlike the group element $g$ itself.
It' …
- 56.5k
2
votes
Accepted
Lie algebra adjoint representation
It's easier if you do this index free.
Define the "ad" action of $A$ on $B$ by ${\rm ad}(A) B= [A,B]$. Then
rewrite the usual form of Jacobi
$$
[A,[B,C]]+ [B,[C,A]]+[C,[A,B]]=0
$$
as
$$
[A,[B,C]]- [ …
- 56.5k
2
votes
Is the Clebsch-Gordan decomposition of Lie algebra or Lie group representations?
The representation $r_j$ are representations of both the group and the algebra, so there is no difference.
The C-G decomposition,however, is a decomposition of representations and is a decomposition …
- 56.5k
2
votes
Accepted
What is the derivative of general 3D rotation with respect to one angular component?
Not straightforward. You can use the CBH formulae
$$
e^{-A} e^{A+\delta A} = 1+ \int_0^1 e^{-sA}(\delta A) e^{sA} ds+ O[(\delta A)^2]
$$
and
$$
e^{-sA}(\delta A) e^{sA}= \delta A- s[A, \delta A]+ \fr …
- 56.5k
3
votes
Accepted
What is the angular momentum operator?
Use the chain rule for partial derivatives to show that with
$$
x=r \cos\theta\\
y= r \sin\theta
$$
you have
$$
-i \hbar (x\partial_y - y\partial_x)= -i \hbar \partial_ \theta.
$$
Your two formulae a …
- 56.5k
1
vote
Accepted
Why doesn't the second Chern form $C_2$ vanish in 4D Euclidean space?
The Chern form does not include a commutator. It has a trace instead:
$$
C_2 = -\frac 1 {8\pi^2}{\rm tr}(t_at_b)F^a\wedge F^b.
$$
Note that ${\rm tr}(t_a t_b)={\rm tr}(t_bt_a)$, so it has no reason to …
- 56.5k
3
votes
Accepted
Evaluating the $A \land A \land A$ in the Chern-Simons action
Does this help:
Consider a lie-lagebra valued form $A=A_\mu^a \lambda_a dx^\mu$ then
$$
{\rm tr}(A^3)= {\rm tr} \{\lambda_a\lambda_b \lambda_c\} A^a_\alpha A^b_\beta A^c_\gamma dx^\alpha \wedge dx^\b …
- 56.5k
3
votes
Accepted
Formalism of Non-abelian Gauge theory?
The first notation hides the Lie algebra generators by writing $A_\mu = A_m^a {\boldsymbol \lambda_a}$ where the implied generators obey
$$
[{\boldsymbol \lambda}_a, {\boldsymbol \lambda}_b]= i {f …
- 56.5k
1
vote
Accepted
What do the different "levels" in a Kac-Moody algebra tell us physically?
The scale of the algebra is always chosen so that the length of the longest root in the underlying finite Lie algebra is 2. This choice simplifies a number of formulae, and in particular ensures that …
- 56.5k
3
votes
Accepted
Derivation of Raising operator of $\rm SU(2)$
I think you mean "Cartan" (after Elie Cartan, the French mathematician) rather than "Cartar"? A semi simple Lie algebra is the direct sum of the Cartan algebra composed of a mutually commuting genera …
- 56.5k
3
votes
Accepted
How can we go from a 4-dimensional representation of $SO(4)$ to the 3-dimensional one of its...
It's not quite what you say. Yes ${\rm SO}(4)$ has an obvious collection of ${\rm SO}(3)$ subgroups where one rotates in 3-dimensional subspaces of of the 4-dimensional space--- but Greiner's ${\rm S …
- 56.5k
2
votes
Accepted
Hermitian operators in the expansion of symmetry operators in Weinberg's QFT
I looked at page 54 and Weinberg does not say that the $t_{ab}$ are Hermitian, only that the $t_a$ are Hermitian. I have the 7th reprinting of the paperback edition. Maybe it was wrong in earlier edit …
- 56.5k
1
vote
Rotation of $su(2)$ generators
I think that you are groping for the equation
$$
U(R) \sigma_i U^{-1}(R)= \sigma_j {R^j}_i,
$$
where ${R^j}_i$ is an ${\rm SO}(3)$ rotation matrix, the $\sigma_i$ are the matrix generators of $\mathf …
- 56.5k
1
vote
Jacobi Identity of structure constants of ${\rm SU}(N)$ group
I've not seen that identity before. The only think that comes to mind is to try taking a trace with an extra generator of the identity
$$
[T_1,\{T_2, T_3\}]+ \{T_2,[T_3, T_1]\}- \{T_3,[T_1, T_2]\}=0
$ …
- 56.5k