Questions tagged [spinors]

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Can we make a Bloch sphere for Weyl spinors?

If spinors are the "square root" of 3-vectors [$\mathrm{SU}(2)$ double cover of $\mathrm{SO}(3)$], Weyl spinors can be thought of as the "square root" of 4-vectors [$\mathrm{SL}(2,\...
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Evaluating expressions involving Dirac spinors

Consider the Dirac equation $(i\gamma^{\mu}\partial_{\mu}-m)\psi=0$. This equation describes the free Dirac field. Consider the plane wave solutions $\psi(x)=u(\vec{p})e^{-ip\cdot x}$ and $\psi(x)=v(\...
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How to derive this form of helicity spinor (massless/high energy limit)

Srednicki 50.7 says that in the massless limit, we can express $$u_-(\textbf{p})\bar{u}_-(\textbf{p}) = \begin{pmatrix} 0&-p_{a\dot{a}}\\0&0\end{pmatrix}$$ This comes from a previously ...
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Understading dimensions in quark bilinears

I have encountered myself with the following definition for $\pi$-fields as quark bilinears: $$ \pi^a = i\bar{q}\tau^a \gamma_5 q \ ,\quad\text{with }\ q = \left(\begin{array}{c}u\\d\end{array}\right) ...
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Different representations of the Yukawa interaction

during studying Yukawa sector of the SM, I got confused with different reps of the Yukawa interaction. First, this is what I am familiar with(let me show only electron mass term): $$y_e \bar{L}_e H ...
김승현's user avatar
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Yukawa Theory with Spinor Helicity Formalism

I am trying to learn spinor helicity formalism, and I am attempting to calculate the all-minus amplitude of 4-fermion scattering at tree level in Yukawa theory, but I have no way to check if my ...
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Question about meaning of "bar"-ing in the context of Dirac fields

Following chapter 38 of Srednicki, "bar"-ing means (based on eq. 38.15) $$\bar{A} = \beta A^\dagger\beta$$ One can show for instance that $$\bar{\gamma^\mu} = \gamma^\mu$$ My question is, ...
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4 answers
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How to rotate an electron mathematically?

Im a mathematics student who just learned about the fact that if you rotate an electron by $2 \pi$ its spin state changes but if you turn it by $4 \pi$ it stays the same. I understand all the ...
Henry T.'s user avatar
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Interpretation of "spin-1/2" in classical Dirac field

I emphasize that the proceeding is purely classical physics. Consider the Grassmann-valued field (where $\mathcal{N}$ is a Grassmann number), which is a solution to the Dirac equation, given by $$\psi(...
Silly Goose's user avatar
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Product of spinors in Dirac field anticommutators

I am reading a "A modern introduction to quantum field theory" by Maggiore and on page 88 it shows the anticommutators of the Dirac field: $$ \{\psi_a(\vec{x},t),\psi_{b}^{\dagger}(\vec{y},t)...
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Prove that spinors satisfying the Killing equation lives on a sphere

Considering that the covariant derivative $D_{\mu}$ acting on spinors be given by $$D_{\mu} \eta = \pm \frac{i}{2} \gamma_{\mu} \eta$$ It is claimed that theses spinors lives on a constant curvature ...
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Quark Combination of Hadrons

I am trying to understand the different combinations of quarks in a hadron. I have seen that the positive pion is written as $\pi^{+}=u\bar{d}$, but I have not seen it written in the opposite order. ...
Anant Badal's user avatar
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Non-relativistic limit of time-dependent Dirac equation

Can someone point me to the derivation of the non-relativistic limit of the time-dependent Dirac equation? I'm presuming that the limit is nothing but the time-dependent Schrodinger equation. I ...
John Doe's user avatar
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Fierz idendity (supersymmetry)

So basically I have two Fierz identities involving spinors: $$\psi^a \psi^b = -\frac{1}{2} \epsilon^{ab} \psi \psi$$ And $$\overline{\psi}^{\dot{a}} \overline{\psi}^{\dot{b}} = \frac{1}{2} \epsilon^{\...
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Physical meaning behind the double rotation of spin 1/2 particles [duplicate]

From the Bloch sphere, it is mathematically clear that a $720°$ rotation is necessary to bring a spin $1/2$ particle back to its initial state, as a full rotation changes the sign of the state. ...
QuantumQuasar's user avatar
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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 ...
user174967's user avatar
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What is the point of arranging (1/2,0) spinor into Majorana spinor?

Using Srednicki's notation: For a massive left-handed spinor $\psi$: $\mathcal{L}=i\psi^{\dagger}\bar{\sigma}^{\mu}\partial_{\mu}\psi-{1\over 2}m\psi\psi-{1\over 2} m\psi^{\dagger}\psi^{\dagger}$ It ...
Bababeluma's user avatar
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Why is the Ramond vacuum a Majorana fermion in type II string theory?

I understand that in order to have a supersymmetric spectrum in string theory, the vacuum has to be a MW (Majorana-Weyl) spinor under $SO(1,9)$. But I don't see where the Majorana condition on the R ...
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Commutation behavior of spinors in Feynman diagrams

I am currently playing around with computing cross sections of several simple interactions in QED like Bhabha and Compton Scattering and I have stumbled upon a question which I havent yet managed to ...
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Why is the derivative necessary to connect left and right-hand spinors?

I am studying Weyl and Dirac spinors. Suppose we have two Weyl fermions $\eta, \chi$ transforming under $(1/2,0)$ representation of the Lorentz group. I learned that to construct Lorentz invariant ...
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What is the difference between a twistor and bispinor?

Reading the book on General Relativity written by R.M. Wald I (tags according to Wald's book) encountered the concept of a twistor $$ Z = (\omega^A, \pi_{A'}) \tag{14.1.9} $$ which looks very much as ...
Frederic Thomas's user avatar
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Spinor Components, Helicity, and Chirality in Dirac Theory

In the Dirac the spinor components are defined by fermion/antifermion (here labeled as $+,−$) and spin component $S_z$ ($↑,↓$): \begin{pmatrix} \psi_-^\uparrow \\ \psi_-^\downarrow \\ \psi_+^\uparrow \...
Julián Oviedo's user avatar
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1 answer
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Columns, rows, dotted, undotted, $SL(2, \mathbb{C})$ reps, and building Dirac spinors from Weyl spinors

I'm looking through Introduction to Supersymmetry by Muller-Kirsten and Wiedemann, along with any other resource I can find. I'm specifically trying to understand the concepts and notations for ...
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Real representation of smallest dimension of Clifford Algebra with $d$ generators

I'm trying to understand the model described in this paper. I have a question about a claim they make. From page 2: To describe the fermionic degrees of freedom let, as a preliminary \begin{align*} ...
Gleeson's user avatar
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Can a single atom spin around its own axis?

I know that molecules can spin around a central axis, but lately I keep wondering if it is possible for a single atom to spin around its own axis (like earth for example). Also does this concept of ...
Luiz Phillyp Sabadini Bazoni's user avatar
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2 answers
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Looking for a Pauli matrix identity [closed]

I know some of the better known Pauli Matrix identities. But I don't know (nor could I not figure out or find online) any that could be used for the following expression: $(\sigma_k)_{\alpha \gamma}(\...
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Why the single copy live on a flat background in type D Weyl double copy?

Showed in this paper, the Weyl double copy relates exact solutions in general relativity to exact solutions in gauge theory, formulated in the spinorial language. The relation can be expressed as $\...
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Transformation of spinor reps and why the Dirac rep is its own conjugate

In Polchinski's String Theory volume 2, appendix B, on page 433 (in the section on Spinors and SUSY in various dimensions, specifically the subsection on Majorana spinors) he says: "It follows ...
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Massless limit of the Dirac theory

What is the physical reason why there is no mixing between left-handed and right-handed Weyl spinors in the massless case of the Dirac theory? Why does the chirality of a massive particle change ...
Michael's user avatar
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1 answer
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General formulation of "X transforms like an X"

It has been discussed several times on this site the phrase "a tensor is something that transforms like a tensor". I'm comfortable with both the mathematical formalism and the physical ...
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Represent the Pauli 4-vector $\sigma^\mu$ as hermitian matrix of matrices due to the $SL(2,C)$ universal double cover of $SO^+(3,1)$

It's known that it's possible to map a 4-vector $x^\mu=(t,x,y,z)$, here i use $c=1$, into a 2x2 hermitian matrix as linear combination of Pauli matrices, thus the mapping $x^\mu \leftrightarrow X$. ...
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What justifies the statement that a Dirac spinor can be written as two Weyl spinors?

I've cross listed this post on math SE in case it is more appropiate there. That post can be found here: https://math.stackexchange.com/q/4833722/. I am approaching this from a Clifford algebra point ...
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Building 4-vectors out of Weyl spinors: Combining 2 independent Weyl spinors and a sigma matrix to get a 4-vector

i'm struggling with this problem In Exercise 2.3 of A Modern Introduction to Quantum Field Theory of Michele Maggiore I am asked to show that, if $\xi_R$ and $\psi_R$ are right-handed spinors, then $$...
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How to define spinors in general relativity?

In Schwartz's QFT and the Standard Model, a vector is defined an object that transforms like a vector. For instance, $V$ is a 4-vector if its components transform like this under Lorentz boost, \begin{...
Tommy Tsang's user avatar
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7 answers
3k views

Can spinors be explained or understood without group or representation theory?

Vectors, either as abstract mathematical elements of a vector space (in this case the definition of the vector is divorced from any notion of transformation), or as elements of tangent spaces on ...
Jagerber48's user avatar
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Why the massive spin-1 photon gets more degrees of freedom than massless case; while the massive spin-1/2 electron stays the same as massless case?

Spin 1 field without mass term like photon has 2 real degrees of freedom. The polarization with two states. I think I can denote it as quantum state $|s,s_z> = |1,1>$ and $|1,-1>$. Spin 1 ...
zeta's user avatar
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Physical significance of left- and right-handed spinors

We know that the $(1/2,0)$ and $(0,1/2)$ representations of Lorentz group represent the left- and right-handed spinors respectively. What’s the reason behind this nomenclature? What they represent ...
Sagar K. Biswal's user avatar
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Representation theoretic constraints in SUSY algebra

Let's try to build from scratch the SUSY commutator $[Q_\alpha^I, P_\mu]$. We know that the result of this commutator must be a fermonic generator belonging to $(1/2, 0)\otimes(1/2,1/2) \simeq (1, 1/2)...
Jack Euler's user avatar
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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 ...
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What does it mean to multiply two spinors?

In Peskin and Schroeder they introduce an initial (incorrect, but that's irrelevant) mode expansion of the Dirac field: $$ \psi(x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2 E_p}} e^{-ix \cdot p} \...
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Understanding Wikipedia's definition of a spinor

I originally asked this question on math SE but I'm asking it again here due to the lack of responses. I should note that I come from a mathematical background and not a physics one so I am not ...
CBBAM's user avatar
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Physical motivation for the definition of Spin structure

I'm pretty confused about the motivations behind defining a spin structure on a manifold. Let me explain. In quantum mechanics, particles are represented by irreducible unitary projective ...
eomp's user avatar
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How is a Fock state from QFT related to the wave function from quantum mechanics?

I am currently studying quantum field theory as part of my degree. I'm just lacking intuition or an understanding of some basic concepts. So please don't hesitate to correct me if i got something ...
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Calculation of Killing Spinors on the 2-Sphere

A Killing spinor on a Riemannian spin manifold $\mathcal{M}$ is a spinor field $\psi$ which satisfies \begin{equation} \nabla_{X}\psi = \lambda X\cdot\psi \end{equation} for all tangent vectors $...
Sidhaarth Kumar's user avatar
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Confusion about spin in 1 dimension

If spin should not be defined for particles in 1 dimension since we can't do rotations in 1d, like it happens for the orbital angular momentum, then why can we talk about fermions in a 1 dimensional ...
abc's user avatar
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Mean field and interacting Dirac QFT: channels and spinors

I am dealing with a QFT of Dirac fermions with an interaction term $$L_I=\bar\psi\psi\bar\psi\psi=\psi^\dagger\gamma^0\psi\psi^\dagger\gamma^0\psi,$$ with $\gamma^0$ a Dirac matrix and $\psi$, $\psi^\...
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Proof of normalising the Dirac spinors

I was reading through my particle physics textbook and saw a property of the Dirac spinors that I did not understand. The spinor is defined by $u^s(p)=\sqrt{\frac{E+m}{2m}} \begin{bmatrix}\phi^s \\ \...
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Are representations of (bosonic) Lie groups over Grassmann variables well understood?

When one studies representations of (bosonic) Lie groups in physics, whether dealing with spacetime symmetries or gauge symmetries, it is often left implicit whether the representations are over real ...
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Question about spinor inner products

Let a 2D spinor be given by $$\chi_2(p)=\pmatrix{\xi^1\\\xi^2}+i\pmatrix{\xi^3\\\xi^4}$$ with the $\xi^i$'s being real for $i=\{1,2,3,4\}$. Assume, now, that I want to represent this spinor by a real-...
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Questions on Lorentz generators in Spinor-Helicity formalism

I have read the following PSE posts on Lorentz generators in Spinor-Helicity formalism: Total Angular Momentum Operator in Spinor-Helicity formalism Derivation of conformal generators in spinor ...
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