All Questions
20
questions
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54
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Weyl spinors under the Lorentz transformation
I am reading An Modern Introduction to Quantum Field Theory by Maggiore. On page 28, it says
Using the property of the Pauli matrices $\sigma^2 \sigma^i \sigma^2 = -\sigma^{i*}$ and the explicit form ...
- 41
1
vote
1
answer
1k
views
Lorentz boost of Dirac spinor
Let $\psi_\vec{0}^+$ be a Dirac wavefunction describing a motionless particle,
$$\psi_\vec{0}^+(x) = \sqrt{2m} \begin{pmatrix}
\chi \\
0
\end{pmatrix} e^{ip \cdot x}$$
where $p = (m, \vec{0})$. ...
- 653
-2
votes
1
answer
326
views
How to prove these relations for Pauli matrices?
I am reading Schwartz's QFT book and I am trying to verify (10.141) and (10.142). σ means Pauli matrix and $ϵ:=−iσ_2$.
How to prove these relations?
$$\sigma^{\mu}_{\alpha\dot{\alpha}}\sigma^{\nu}_{\...
-1
votes
1
answer
65
views
A question about Lorentz transformations in spinor representation
For
$$\Lambda^{\mu}{}_{\nu}= \frac{1}{2} Tr(\bar{\sigma}^{\mu} S \sigma_{\nu}S^{\dagger}) $$
We need to prove that
$$\Lambda (S)= \Lambda (-S)$$
Am I naive to say that by adding $-S$, $S^{\dagger}...
- 89
0
votes
0
answers
220
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Trying to prove the Wess Zumino invariance under a SUSY transformation
I have the Lagrangian density
$$L=-\partial_\mu \phi^\star \partial^\mu \phi - \bar{\chi}_R \gamma^\mu \partial_\mu \chi_L - \bar{\chi}_L\gamma^\mu\partial_\mu \chi_R.$$
where $\epsilon$ is the ...
- 199
1
vote
1
answer
2k
views
Dirac matrices in 1+1 dimensions
Given $\gamma^\mu$ in 1+3 dimensions with signature $(+,-,-,-)$, how can I obtain Dirac matrices in 1+1 dimensions expressed in terms of the $\gamma^\mu$?
- 311
1
vote
1
answer
215
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How to build an antisymmetric selfdual tensor out of two 4-vectors?
In problem C of section 1.4 of Ramon's Field Theory: A Modern Primer, we are asked to build a field bilinear in $\chi_L$ and $\psi_L$, two left-handed weyl spinors, which transforms as the (1,0) ...
- 3,915
1
vote
0
answers
157
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On the Pauli-Lubansky vector and spin
Lahiri's A First Book on Quantum Field Theory states on problem 4.24 that from the Pauli-Lubansky vector
$$W_\mu=-\frac{1}{2}\epsilon_{\mu\nu\lambda\rho}P^\nu J^{\lambda\rho}$$
one can prove that for ...
- 3,915
0
votes
1
answer
76
views
Is the Heighest weight vector in the Spinor rep of $SO(1,d-1)$ zero?
Consider the highest weight vector of the Spinor rep of $SO(1,d-1)$ where $d=2m+1$. It can be shown that:
$$\gamma_i \gamma_{m+i}v=v\tag{*}$$
I cannot see why this relation does not imply that $v=0$? ...
-1
votes
1
answer
627
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Representation $(1/2,1/2)$ of Lorentz group
I want to show that the Lorentz representation $(1/2,1/2)$ corresponds to the normal vectorial representation $A^\mu$. For this I need to show that the double spinors $A_{ij}=(A_\mu\sigma^\mu\sigma^2)...
- 173
5
votes
1
answer
1k
views
4-vector from a spinor
Currently reading Aitchison's book on SUSY, and on page 35 (section 2.2) he asks the reader to prove that $\bar{\Psi}\gamma^\mu\Psi=\psi^\dagger\sigma^\mu\psi+\chi^\dagger\bar{\sigma}^\mu\chi$ ...
- 1,379
3
votes
1
answer
2k
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How to prove that Weyl spinors equations are Lorentz invariant? [duplicate]
The Dirac equation is given by:
$[iγ^μ ∂_μ − m] ψ(x) = 0$ .
We can prove that it's Lorentz invariant when:
$ψ(x) \to S^{-1} \psi'(x')$ and $\partial_\mu \to \Lambda^\nu_\mu \partial'_\nu$, where
...
- 453
6
votes
1
answer
531
views
Derivation of conformal generators in spinor helicity formalism
I've been trying for some time to find the expressions for conformal generators of Witten's paper in perturbative Yang-Mills.
Given $P_{\alpha \dot{\alpha}} = \lambda_{\alpha} \overline{\lambda}_{\...
- 824
1
vote
0
answers
284
views
The representation of Lorentz boost for two component spinor
It is known that the two components spinor $\chi$ is transformed under the $(\frac{1}{2},0)$ representation of lorentz group. This transformation can be written as $$\chi\rightarrow \exp[-\frac{i}{2}\...
- 33
1
vote
0
answers
625
views
Fierz identity for chiral fermions [closed]
First of all I define the convention I use.
The matrices $\bar{\sigma}^\mu$ I will use are $\{ Id, \sigma^i \}$ where $\sigma^i$ are the Pauli matrices and $Id$ is the 2x2 identity matrix.
I will use ...
- 2,197