All Questions
23
questions
2
votes
0
answers
43
views
Fundamental invariants of a Lorentz tensor
As answered in this question, an antisymmetric tensor on 4D Minkowski space has two Lorentz-invariant degrees of freedom. These are the two scalar combinations of the electromagnetic tensor (as proven ...
- 21
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
2
votes
2
answers
132
views
Which finite-dimensional representations of the Lorentz group do $p$-forms correspond to?
On the Wikipedia article about the representation theory of the Lorentz group, the finite-dimensional representations $(1,0)$ and $(0,1)$ are referred to as "$2$-form" representations. On ...
- 737
1
vote
0
answers
66
views
Doubt on transformation laws of tensors and spinors using standard tensor calculus and group theory
1) Introduction
From standard tensor calculus, here restricted to Minkowski spacetime, we learned that:
A scalar field is a object that transforms as:
$$\phi'(x^{\mu'}) = \phi(x^{\mu})\tag{1}$$
A ...
- 3,085
1
vote
1
answer
219
views
How to contract spinor indices?
In normal vector representation, vectors can be contracted as follows:
$$v^\mu v_\mu$$
with one covariant and one contravariant index. But in spinor representation, there are 4 possible type of ...
- 1,324
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
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
0
answers
178
views
Representing a four-vector by a rank 2 tensor
When representing a 3-vector $(x,y,z)$ as a skew-symmetric matrix like this:
$X=\begin{pmatrix}
0 & -z & y\\
z & 0 & -x\\
-y & x & 0
\end{pmatrix}$
$X$ transforms a a rank 2 ...
- 586
4
votes
1
answer
396
views
Decomposition of the torsion tensor
See Update below.
Consider the torsion tensor $T_{\mu\nu\rho} = -T_{\mu\rho\nu}$. In a local Lorentz frame, as determined by a vierbein $e^{a}{}_{\mu}$, it may equivalently be given as $T_{abc} = e^{\...
- 403
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
0
votes
1
answer
151
views
$\mathcal N=2$ SUSY representation
I want to understand why in $\mathcal N=2$ SUSY representation (from Wess & Bagger book on SUSY and SUGRA, the second table on page 14):
$$Q_\alpha^A Q^B_\beta |1\rangle=(1)^4 \oplus 0 \oplus 2,\...
- 143
3
votes
1
answer
347
views
General definition of an $n$-rank spinor
I have been looking around for a formal (and easy to comprehend) definition of a general $n$-rank spinor. I have had no luck trying to find such a definition, or any definition for that matter.
So ...
5
votes
1
answer
256
views
Function form of Lorentz invariant functions
In QFT, Green function of gauge field is Lorentz invariant(i.e. $\forall \Lambda \in SO(3,1), f(\Lambda p)=\Lambda f(p)$).And according to the textbook I'm reading, The form of such functions is ...
- 59