All Questions
Tagged with group-theory tensor-calculus
65
questions
2
votes
1
answer
132
views
Confusion about tensors in $SU(3)$
I have some confusion regarding the notion of tensors in $SU(3)$ (or some other matrix Lie group, but let's keep the discussion to $SU(3)$).
For concreteness, I will refer to Peskin and Schroeder's ...
- 119
0
votes
1
answer
93
views
Multiplying two $SO(3)$ representations
In Group Theory by Zee in Chapter IV.2, he discusses the multiplication of two $SO(3)$ representations on p. 207. Suppose you have a symmetric traceless tensor $S^{ij}$ which furnishes a $5$-...
- 404
1
vote
1
answer
81
views
Electromagnetic duality and vector/pseudovector transformation properties
One consequence of electromagnetic duality (see e.g., https://doi.org/10.1038/s41467-023-36420-4) is that if we have a system described by permittivity and permeability profile $(\varepsilon, \mu)(\...
- 136
2
votes
1
answer
202
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Youngs' tableau method and its relation with tensors
During my masters my prof. mentioned that the Youngs' tableau method can be used to calculate the number independent components of any arbitrary rank symmetric tensor in arbitrary dimensions. I wasn't ...
- 377
-4
votes
1
answer
66
views
Cannot understand this identity between kronecker and metric tensor [closed]
I'm working on Lorentz generators and I am really not able to understand this relation:
$$\omega_{\rho \sigma} \eta^{\rho\mu} \delta^{\alpha}_{\nu} = \frac{1}{2}\omega_{\rho \sigma} \left(\eta^{\rho\...
- 361
0
votes
0
answers
29
views
Unclear passage in Lorentz generators derivation
It's not clear to me a passage, in the extraction of the generators of Lorentz's group acting on the Minkowksi's space points: we have
\begin{equation*}
\begin{split}
x^{' \alpha} & = \Lambda^{\...
- 361
0
votes
0
answers
55
views
Product of spherical tensors
Consider a spin $ j $ system. A spin $ j $ spherical tensor $ T^k_q(j) $ of rank $ k $ is a $ (2j+1) \times (2j+1) $ matrix.
Given two spherical tensors of spin $ j $, say $ T^{k_1}_{q_1}(j) $ and $ T^...
0
votes
0
answers
54
views
Algebra equation for rank-3 tensor
Suppose I work in $4$ dimensions. I have an algebraic equation in the following form, which contains a rank-3 tensor $X ^{\alpha \lambda \mu }$
\begin{equation}
X ^{\alpha \lambda \mu }\eta ^{\beta \...
- 1
0
votes
1
answer
213
views
How to get $3n-j$ symbols using the young tableau method?
How do one compute the $3n-j$ symbols using the young tableau method,for a $U(n)$ ?
Edit
$3j$ symbols were first introduced by Wiegner which are related to C.G. coefficients, similarly $6j$ symbols ...
- 866
1
vote
1
answer
101
views
Question on the spinor Indices, in non-relativistic quantum mechanics
I've caught by a loop of:
Standard texts of Non-Relativistic Quantum Mechanics $\to$ Representation theory of Lie groups and Lie algebras of $SO(3)$ and $SU(2)$ $\to$ Discussions of infinitesimal ...
- 315
1
vote
1
answer
68
views
Question on transformation law of spinors and the law $\xi^{\mu '} = D(L)^{\mu '}_{\nu}\xi^{\nu}$ where $D(L)$ is a representation of Lorentz group
In the reference $[1]$, the author presents the tensor quantities via its transformation laws. I'm pretty confortable with Pseudo-Riemannian geometry and tensors. But, when group theory enters the ...
- 3,085
2
votes
1
answer
260
views
Why is rank-3 tensor in 3D with two antisymmetric indices equivalent to rank-2 tensor?
I'd like to know how many irreducible representations of $SO(n)$ when it comes to rank 3 tensor. Here $n=3$. Among the rank 3 tensor components, there might be antisymmetric parts and symmetric parts ...
- 169
5
votes
2
answers
580
views
Are the linear Lie groups matrices, tensors, or both?
In some ways, this is a question about notation.
In my experience, I have only seen the classical Lie groups — such as $\operatorname{GL}(n,\mathbb{R})$, $\operatorname{SL}(n,\mathbb{R})$, $\...
- 477
2
votes
1
answer
83
views
From the point of view of physics, why is it useful to know the irreps of rotation group?
In 3D, the rank two tensorial physical quantities, for example, the electric susceptibility, the conductivity, the stress tensor etc, are in general, not irreducible representations i.e. neither ...
- 11.8k
0
votes
1
answer
231
views
Tensor products in Howard Georgi's "Lie Algebras in Particle Physics"
My question is regarding eq.(3.39) in the second edition of Georgi's book (for those who have the book:)). The section deals with tensor product states where the states comprising the product ...
