All Questions
Tagged with special-relativity spinors
49
questions
37
votes
3
answers
12k
views
What's the relationship between $SL(2,\mathbb{C})$, $SU(2)\times SU(2)$ and $SO(1,3)$?
I'm a beginner of QFT. Ref. 1 states that
[...] The Lorentz group $SO(1,3)$ is then essentially $SU(2)\times SU(2)$.
But how is it possible, because $SU(2)\times SU(2)$ is a compact Lie group ...
- 5,971
28
votes
2
answers
3k
views
Wick rotation and spinors
I am quite familiar with use of Wick rotations in QFT, but one thing annoys me: let's say we perform it for treating more conveniently (ie. making converge) a functional integral containing spinors; ...
- 2,926
26
votes
4
answers
1k
views
How does complexifying a Lie algebra $\mathfrak{g}$ to $\mathfrak{g}_\mathbb{C}$ help me discover representations of $\mathfrak{g}$?
I have been studying a course on Lie algebras in particle physics and I could never understand how complexifying helps us understand the original Lie algebra.
For example, consider $\mathfrak{su}(2)$...
- 3,986
4
votes
1
answer
266
views
Understanding spinors, double cover and professor's expanation
I'm following an introductory course in QFT, and we are facing the spin group part. I think that most of the details are left apart because it would take too much time to be developd, and my profesor ...
- 361
26
votes
2
answers
13k
views
Dirac, Weyl and Majorana Spinors
To get to the point - what's the defining differences between them? Alas, my current understanding of a spinor is limited. All I know is that they are used to describe fermions (?), but I'm not sure ...
- 1,306
7
votes
2
answers
2k
views
Regarding the Weyl spinor and its transformation properties
I am trying to prove the Lorentz invariance of the (left-handed) Weyl Lagrangian:
$$\mathcal L=i\psi^\dagger\bar\sigma^\mu\partial_\mu\psi$$
A Lorentz transformation is realized as
$\psi\to M\psi$, ...
5
votes
2
answers
1k
views
Why is the $(\frac{1}{2},\frac{1}{2})$ representation of the Lorentz group realized as the vector space of Hermitian $2\times 2$ matrices?
Why can we write an arbitrary object $v_{a \dot{b} }$ our transformations in this basis act on as
$$ v_{a \dot{b} } = v_{\nu} \sigma^{ \nu}_{a \dot{b} } = v^0 \begin{pmatrix} 1&0 \\ 0&1 \...
- 1,872
13
votes
2
answers
5k
views
Conceptual interpretation of the left- and right-handed spinor representations of the Lorentz group
I understand mathematically that the Lorentz group's Lie algrebra $\mathfrak{so(3,1)}$ (given by eqns. (33.11)-(33.13) in Srednicki's QFT book) is isomorphic to $\mathfrak{su(2) \times su(2)}$ (given ...
- 48.4k
8
votes
2
answers
756
views
Does Wightman's unitary $U(\Lambda)$ really exist for Lorentz boost?
This question is related to another question here. But I am asking a more fundamental question about the existence of Wightman's unitary $U(\Lambda)$ for Lorentz transformation.
Let $\psi^\alpha$ be a ...
- 4,452
4
votes
2
answers
2k
views
What are the actual transformation properties of Dirac spinors $u_\sigma(p)$?
Let $u_\sigma(p)$ be a Dirac spinor. As far as I know, it transforms under changes of reference frame according to
$$
u_\sigma(p)=S(\Lambda)u_\sigma(\Lambda p)\tag{1}
$$
where the $\sigma$ label doesn'...
4
votes
1
answer
2k
views
How do simple two-component Fierz identities follow from a property of the Pauli matrices?
On page 51 Peskin and Schroeder are beginning to derive basic Fierz interchange relations using two-component right-handed spinors. They start by stating the trivial (but tedious) Pauli sigma identity ...
- 566
0
votes
1
answer
666
views
Transformation of the spinor indices of Hermitian $2\times 2$ matrices under the Lorentz group
The left-handed Weyl operator is defined by the $2\times 2$ matrix
$$p_{\mu}\bar{\sigma}_{\dot{\beta}\alpha}^{\mu} = \begin{pmatrix} p^0 +p^3 & p^1 - i p^2\\ p^1 + ip^2 & p^0 - p^3 \end{...
- 3,203
29
votes
3
answers
12k
views
Lorentz transformation of Gamma matrices $\gamma^{\mu}$
From my understanding, gamma matrices transforms under Lorentz transformation $\Lambda$ as
\begin{equation}
\gamma^{\mu} \rightarrow S[\Lambda]\gamma^{\mu}S[\Lambda]^{-1} = \Lambda^{\mu}_{\nu}\gamma^{\...
- 595
21
votes
2
answers
5k
views
Interpretation of Dirac Spinor components in Chiral Representation?
I failed to find any book or pdf that explains clearly how we can interpret the different components of a Dirac spinor in the chiral representation and I'm starting to get somewhat desperate. This is ...
- 10.1k
13
votes
3
answers
10k
views
Dirac spinor and Weyl spinor
How can it be shown that the Dirac spinor is the direct sum of a right-handed Weyl spinor and a left-handed Weyl spinor?
EDIT: - Let $\psi_L$ and $\psi_R$ be 2 component left-handed and right-handed ...
- 26.8k