# Questions tagged [spinors]

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### QFT - Allowed transitions for a given vertex

I'm given the following Lagrangian : $$L = L_{f1 }(\psi_1, \bar\psi_1) + L_{f2}(\psi_2, \bar\psi_2) + \lambda (\bar\psi_1 \gamma^{\mu} \psi_2) (\bar\psi_2 \gamma_{\mu} \psi_1)$$ where $L_{fi}$ are ...
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### Coleman–Mandula theorem and Ward Identity

I was reading a paper on Coleman–Mandula theorem and Ward Identity [The Coleman-Mandula Theorem by Sascha Leonhardt]1, where I saw it says that- Let a higher spin current $\hat{B}_{\mu\nu}$ is non ...
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### Summing over spins in QED and calculating the square of Feynman amplitudes

I'm trying to compute the differential decay rate given by the following amplitude: $$M = i g \bar{u}(q,s)\gamma^\mu \displaystyle{\not}\epsilon^\mu_r(p)v(\tilde{q},s')$$ which concerns the ...
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### What is the physical meaning of the components of a spin 1/2 spinor in matrix representation?

If the spin operators for spin $1/2$ can be represented in matrix form using the Pauli Matrices, e.g $S_x = \frac{1}{2}\hbar \sigma_x = \frac{1}{2}\hbar \begin{bmatrix}0&1\\1&0\end{bmatrix}$, ...
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### Another form of the solutions of the Dirac equation

Consider the Dirac equation $[i\gamma^\mu\partial_\mu-m]\psi=0$ and let me focus in particular on the positive-energy solutions by the ansatz $$\psi(x)=e^{-ipx}u(\mathbf p,r).$$ Making this ...
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### Dirac spinor in the chiral basis

In the chiral basis, the gamma matrices take the form $$\gamma^0=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}, \quad \gamma^j=\begin{bmatrix}0 & -\sigma^j \\ \sigma^j & 0\end{bmatrix}$$...
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### How to show that $\sigma^2\psi_L^*$ transforms as a right-handed spinor? (Peskin&Schroeder)

In Peskin & Schroeder, it is written that the quantity $\sigma^2\psi_L^*$ transforms as a right-handed spinor. What confuses me is that I only get the correct result when considering the following:...
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### Factoring the Laplace operator $\Delta$ in dimensions $D \geq 3$

Consider the Laplace operator in 2 dimensions \begin{equation} \Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} = \partial^2_x + \partial^2_y \end{equation} By defining the ...
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### Transformation between left-handed spinors and right-handed spinors

I am learning (Weyl) spinor formalism from Müller-Kirsten and Wiedemann's Introduction to Supersymmetry (2nd Ed., WS, 2010, here). I am quite confused about the transformation between left-handed ...
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### Homogeneous (projective) coordinates and spinors

When a complex number is considered as the stereographic projection from a sphere to the Argand plane, and then is represented by two “homogeneous coordinates” (in order to allow for a “point at ...
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### Weyl and Dirac spinors

I know that the Dirac spinor is composed of two Weyl spinors and each of the Weyl spinors also has two components. Can I see its two components as two different wave functions? Can I see four ...
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### Why does the Dirac beta matrix commute with the angular momentum operator?

This is the Dirac Hamiltonian, and Beta is The question says it all, I don't understand why Beta would commute with $\hat L$
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### Chiral Fierz identity

I am having trouble with proving the following: (\bar{\psi}_1 P_R \psi_2) (\bar{\psi}_3 P_R \psi_4)=-1/2 (\bar{\psi}_1 P_R \psi_4)(\bar{\psi}_3 P_R \psi_2)-\dfrac{1}{8}(\bar{\psi}_1\sigma_{\mu\nu} ...
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### Weyl Spinor Representation and Single Particle States

I'm trying to study representation theory for quantum field theory. Let me first summarize my current state of (hopefully correct, please correct me if I'm wrong about something) knowledge: Single ...