All Questions
15
questions
-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
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
5
votes
3
answers
836
views
Decomposition of product of two antisymmetric Lorentz tensors
Suppose I have a tensor $A_{\mu\nu}$ in the $(3,1)\oplus (1,3)$ representation of the Lorentz group where $(a,b) =(2s_a+1,2s_b+1)$. I was wondering on how to decompose explictly in terms of tensors ...
- 2,263
0
votes
2
answers
371
views
Riemannian and Weyl tensors as spinor representation
There is the way of converting vector indices to spinor indices, for example, Maxwell stress tensor $F_{[\mu\nu]}$ can be decomposed to $(1,0) \oplus (0,1)$ irreducible representations of $\mathfrak{...
- 5,707
4
votes
3
answers
1k
views
Showing that 4D rank-2 anti-symmetric tensor always contains a polar and axial vector
In my special relativity course the lecture notes say that in four dimensions a rank-2 anti-symmetric tensor has six independent non-zero elements which can always be written as components of 2 3-...
- 2,604
1
vote
0
answers
58
views
Irrep of stress energy tensor
We have 4-tensor of second rank. For example energy-momentum tensor $T_{\mu\nu}$, which is symmetric and traceless. Then
$T_{\mu\nu}=x_{\mu}x_{\nu}+x_{\nu}x_{\mu}$
where $x_{\mu}$ is 4-vector. Every ...
- 171
0
votes
1
answer
91
views
Which is correct definition between $J^i\equiv \frac{1}{2}\epsilon^i_{~jk}J^{jk}$ and $J^i\equiv \frac{1}{2}\epsilon^{ijk}J^{jk}$?
The generators of the Lorentz group are denoted by $J^{\mu\nu}$ (suppose they are defined in terms of raised indices, as shown). From this, in my opinion, the angular momentum generators $J^i$'s and $...
- 11.8k
1
vote
2
answers
798
views
Symmetry under Lorentz transformation: precise definition
I am studying QFT but I need to fill some gaps in my comprehension of special relativity (I didn't study it very well and I know I still misunderstand things in S.R).
In my book it is written:
" A ...
- 1,450
1
vote
0
answers
84
views
Converting field equation from position to momentum space
Question
I would like to convert the following equation on position space to an algebraic equation on momentum space. $$\partial^{\alpha\dot{\alpha}}\Phi_{\alpha\alpha_1\cdots\alpha_{A-1}\dot{\alpha}...
- 780
3
votes
1
answer
869
views
Lorentz Group Generators: Two Methods
Members of the Lorentz group obey $$\eta=\Lambda^{T}\eta\Lambda$$ where $\eta=\textrm{diag}(1,-1,-1,-1)$ is the Minkowski metric.
First, in matrix form write $$\Lambda=I+T$$ where $T$ is an ...
- 401
4
votes
1
answer
634
views
Srednicki QFT ch34: invariants of the Lorentz group
In chapter 34 of his Quantum Field theory handbook, Srednicki discusses invariants of the Lorentz group and how they appear in the decomposition in irreducible representations of Lorentz tensors. As ...
- 141
9
votes
1
answer
526
views
Is the distinction between covariant and contravariant objects purely for the convenience of mathematical manipulation?
Two kinds of indices, covariant and contravariant, are introduced in special relativity. This, as far as I understand, is solely for mathematical luxury, i.e. write expressions in a concise, self-...
- 26.8k
3
votes
1
answer
1k
views
Why is the Mixed Faraday Tensor a matrix in the algebra so(1,3)?
The mixed Faraday tensor $F^\mu{}_\nu$ explicitly in natural units is:
$$(F^\mu{}_\nu)=\left(\begin{array}{cccc}0&E_x&E_y&E_z\\E_x&0&B_z&-B_y\\E_y&-B_z&0&B_x\\E_z&...
- 88.5k
9
votes
2
answers
478
views
If $v_{a \dot{b}}$ transforms like a four-vector, what does $v_{a}^{\dot{b}}$ describe?
The $( \frac{1}{2}, 0)$ representation of the Lorentz group acts on left-chiral spinors $\chi_a$, the $( 0,\frac{1}{2} )$ representation on right-chiral spinors $\chi^{\dot a}$.
The $( \frac{1}{2}, \...
- 1,872