# Questions tagged [spinors]

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### Question on transformation law of spinors and the law $\xi^{\mu '} = D(L)^{\mu '}_{\nu}\xi^{\nu}$ where $D(L)$ is a representation of Lorentz group

In the reference $$, the author presents the tensor quantities via its transformation laws. I'm pretty confortable with Pseudo-Riemannian geometry and tensors. But, when group theory enters the ...
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### Lorentz Invariance of kinetic terms for Weyl Spinors

Just to preface things, this exact question has been asked before here, but I don't feel like the answer really clarifies things for me. The core issue is that we want to construct a 4-vector that we ...
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### How to interpret Pauli spinors?

Recently in my QM course we derived the Pauli equation for an electron in a magnetic field. From what I understand, since we now have a spin-dependent term in our Hamiltonian, the spatial and spin ...
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### How is the $SU(2)_L$ conjugation applied?

I'm reading a paper where they introduce the lepton doublets $L$ and "their $SU(2)_L$ conjugations" $\tilde{L}$, which I'm guessing means $$\tilde{L} = i\sigma_2L^*.$$ After $\textit{vev}$,...
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1 vote
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### Is the sign of the mass in the Dirac action irrelevant? [duplicate]

In even dimensions all the representations of the gamma matrices are equivalent, in particular $\gamma^\mu$ and $-\gamma^\mu$ are equivalent. Usually the Dirac Lagrangian is \begin{equation} \psi^\...
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### Projective representations of the Lorentz group can't occur in QFT! What's wrong with my argument?

In flat-space QFT, consider a spinor operator $\phi_i$, taken to lie at the origin. Given a Lorentz transformation $g$, we have $$\tag{1} U(g)^\dagger \phi_i U(g) = D_{ij}(g)\phi_j$$ where $D_{ij}$ is ...
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### Gauge transformation of an adjoint left-handed Weyl spinor in $\rm SU(2)$ fundamental representation

I have a left-handed Weyl spinor field $\Psi_L$ in the fundamental representation of the $\rm SU(2)$ gauge group, which transforms $\Psi_{L,i} \rightarrow \Psi_{L,i} + i\theta^at_{ij}^a\Psi_{L,j}$. ...
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1 vote
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### Rotation by 360°, spin-1/2 fermions and quaternions

Rotating a spin-1/2 fermion by 360° multiplies the quantum state by -1. Representing a continuous 360° rotation as a quaternion is also a multiplication by -1. Is there a relationship between these ...
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### Sanity check for a simple calculation involving Dirac spinors and matricies

I've been doing some research related to the Dirac equation and its solutions. To help make sure I understand what's going on, I've done some simple calculations involving planewave solutions. I'm ...
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### What does a dot over a spinor index signify?

My questions should be rather simple. I was trying to get through one of my professor’s papers, and I saw the following notation, first with regards to Dirac and Weyl spinors, but the notation ...
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### Parity invariance of Dirac action

The Dirac action is $$S=\int d^4 x \mathcal{L}(x)$$ where the $\mathcal{L}(x)$ is the Lagrangian density given by $$\mathcal{L}(x)=\bar{\psi}(x)(i\gamma^\mu\partial_\mu-m)\psi(x).$$ In proving the ...
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### How to understand the completeness of the Dirac spinor and why?

I'm searching around to see why $$\sum u^s\bar{u}^s=(\gamma^\mu p_\mu+mc)$$ and $$\sum v^s\bar{v}^s=(\gamma^\mu p_\mu - mc)$$ is called the completeness relation. Also wondering the same question for ...
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### Grassmann numbers and fermionic strings

Is it correct that by introducing Grassmann numbers as new directions of spacetime we can make strings behave like fermions (that is, 1/2-spin objects)? And if so, is it possible to show how that ...
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### Dirac free particle with $x$-momentum

For a free particle with momentum $\mathbf{p}=p\mathbf{x}$, the Dirac Hamiltonian is \begin{equation} H=\alpha_xp+\beta m = \begin{pmatrix} m & 0 & 0 &p\\ 0 & m & p & 0\\ 0 &...
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### On the simplicity of the three-particle amplitude in holomorphic configuration

I am reading Clifford Cheung's 2017 TASI Lectures on Scattering Amplitudes. In section 3, "Bootstrapping Amplitudes", the procedure for bootstrapping the three-particle amplitude for ...
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### Question about fields and state vectors

$\hbar = 1$ and $c=1$ The question is written on section $2)$ $1)$ Introduction So, when you write Klein-Gordon equation, $(\square + m^2)\phi = 0 \hspace{2mm}(1)$ , you know exactly which type of ...
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I read Quantum Field Theory Book written by Peskin & Schroeder, and when a commutator about Dirac field is compile, he compile a general commutator: $$[\psi_a(x),\overline{\psi}_b(x)]$$ having ...