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Field strength renormalization for fermions

Following section 7.1 and 7.2 in Peskin and Schroeder (P&S), I've tried to consider what the derivation of the LSZ formula looks like for (spin $1/2$) fermions (in the text, they explicitly ...
User3141's user avatar
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$C$-number ignored in charge conjugation

In Weinberg’s QFT V1, under equation 5.5.58, he says that an anticommutator ($c$-number) can be ignored when we exchange spinors, $\psi$ and $\bar{\psi}$. I cannot fully appreciate why we can ignore ...
Ting-Kai Hsu's user avatar
1 vote
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10 views

Rotations in $C\ell_{16,0}$ [migrated]

Consider the Clifford algebra $C\ell_{8,0}$. If $x\in C\ell_{8,0}$ and $u$ is a spinor in $C\ell_{8,0}$, then the rotations in this algebra can be written as $$ x' = u x u^{-1}. $$ Now consider $C\...
user38680's user avatar
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How to derive Fermion Propagator for Special Kinetic Term?

I am currently working through chapter 75 of the book on QFT by Srednicki. There, he considers the example of a single left-handed Weyl field $\psi$ in a $U(1)$ gauge theory. The Lagrangian, written ...
Niels Slotboom's user avatar
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Verifying transformation law for Dirac field

I am trying to verify the Lorentz covariance of the Dirac field \begin{equation}\label{transformation}U(A)\psi(x)U(A^{-1})=D(A^{-1})\psi(\Lambda(A)x)\end{equation} where $A\in SL(2,\mathbb{C})$, $\...
user609020's user avatar
1 vote
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35 views

Building vector Noether current from twistor using Dirac formalism

There was a problem formulated during our lectures to build a Killing vector $$ \nabla_{(\mu} k_{\nu)} = 0 $$ from equation $$ {\nabla_{(C}}^{\dot{D}} \varkappa_{A)} = 0. $$ For me it seems that $$ k_\...
Iowo's user avatar
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Fierz identity problem

I am trying to figure out whether the following has any meaningful transformation in Fierz identities. Suppose w's are either u or v spinors. Then $$ \overline{w}_1 P_L w_2 \; \overline{w}_3 P_L w_4 \...
Hubert John's user avatar
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29 views

Schwichtenberg Physics from Symmetry p. 83 Eq 3.225

Firstly - an apology. This is my first question to Stack Exchange and also my first attempt at using Latex. I need to show a subscript letter with a dot above it, but can't work out how to do that ...
user404102's user avatar
1 vote
0 answers
23 views

Understanding the Electron-Positron Transformation in Dirac's 2-Spinor Notation - Penrose

Penrose in his book, the road to reality, chapter 25, it says: non-standard way, by re-examining the Dirac equation in terms of the ‘2-spinor notation’, briefly introduced in §22.8. As remarked in §...
Julián Oviedo's user avatar
1 vote
1 answer
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"Fourier Transformation" of angle spinors to twistor variables

This relates to the derivation of equation (5.15) if Elvang and Huang's textbook. The idea is to transform the spinor helicity variables we are using, $(|i\rangle_{\dot{a}},[j|^a)$ to go into twistor ...
MathZilla's user avatar
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Fierz Identity in 2+1d

Wikipedia states an example of Fierz Identity, under the assumption of commuting spinors, the $\mathrm{V} \times \mathrm{V}$ product that can be expanded as, $$ \left(\bar{\chi} \gamma^\mu \psi\right)\...
Everlin Martins's user avatar
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0 answers
42 views

Why are spinors members of minimal ideals?

Why do we require that spinors live in minimal left ideals of Clifford algebras and not just left ideals? I assume that it has something to do with irreps but a Dirac spinor also lives in an minimal ...
Silas's user avatar
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Spinor bundles and ideals

I‘ve read in the book “Geometric Multivector Analysis” that viewing spinors as members of minimal left ideals is problematic when considering spinor bundles. Is there a simple reason for that?
Silas's user avatar
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Scalar QED amplitudes with BCFW Recursion Relation

(This question comes from exercise 3.5 of Elvang's and Huang's "Scattering amplitudes in Gauge Theory and Gravity" book. This is not for a class, this is to learn a new technique; albeit I ...
MathZilla's user avatar
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Trying to solve the energy levels of a spin 1/2 particle in a one-dimensional box using Dirac Equation

I was studying the problem I asked above in the title and found the article P Alberto et al 1996 Eur. J. Phys. 17 19. The wave function inside the walls is: $$ \psi(z)=B\ exp(ikz) \left[\begin{array}{...
Joao Pedro Medeiros's user avatar
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How do reality conditions on complexified Minkowski space induce conjugation on Spinor space

So I am following this script here: https://arxiv.org/abs/1712.02196 I am already stuck at chapter 1.3: I understand for the three cases Lorentzian, Euclidean and Split case that the coordinates need ...
Confuse-ray30's user avatar
2 votes
0 answers
96 views

How to motivate spinors from the Dirac equation? [closed]

I am trying to motivate spinors by making sure the Dirac equation is relativistically invariant (and it suffices to discuss just the Dirac operator). Let $\{ e_i \}$ be an orthonormal frame and $x^i$ ...
Integral fan's user avatar
1 vote
0 answers
55 views

What is the connection between Lorentz transforms on spinors and vectors?

When deriving the (1/2,0) and (0,1/2) representations of the Lorentz group one usually starts by describing how points in Minkowski space transform while preserving the speed of light (or the metric). ...
Alexander Haas's user avatar
2 votes
1 answer
50 views

Helicity operator in spinor-helicity variables

How do I prove that the helicity operator is $$ H = \frac{1}{2} (\tilde{\lambda}_\dot{\alpha} \frac{\partial}{\partial \tilde{\lambda}_\dot{\alpha}} - \lambda_\alpha \frac{\partial}{\partial \lambda_\...
michael pasqui's user avatar
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0 answers
38 views

Proof that commuting Dirac fields violate causality

What is the proof that commuting Dirac fields violate causality? All sources I could find just quote this result, but I couldn't find an explicit derivation anywhere. In particular, the case I am ...
pll04's user avatar
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1 answer
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Does the anticommutator of two spinors affect the transpose of their product?

My lecture notes claim that for an anticommutation relation $$[ \psi_{\mu}(\bf{x},t),{\psi_{{\nu}}^{*}}(\bf{y},t)] = \delta_{\mu \nu} \delta^3(\bf{x}-\bf{y})$$ between two spinors, the transpose of ...
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1 answer
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Fermi tetrad field: Fermi-Walker tetrad formalism?

These days I'm reading Dirac Eq in GR, and I'm confused about "Fermi tetrad field". Is it Fermi-Walker tetrad formalism?
Lou TY's user avatar
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Proof Majorana spinors exists if maximal commutant of Clifford algebra is $\mathbb{R}$

I am searching for a proof of the claim made in this post. It states that Majorana spinors (I refer to both complex pinor and spinor representations which are restricted to the real Spin group and ...
anonymous250's user avatar
5 votes
2 answers
99 views

On the local form of the spin covariant derivative: is this an exterior derivative of a spinor?

I'm reading Hamilton's Mathematical Gauge Theory. I'm currently on section 6.10, about the spin covariant derivative. Letting $S$ be the (Dirac) spinor bundle, a section $\Psi \in \Gamma(S)$ can be ...
Níckolas Alves's user avatar
3 votes
1 answer
74 views

Transformation of Killing spinors under vielbein redefinitions

I presume the explicit form a Killing spinor depends on the chosen vielbein, as it appears explicitly in the Killing spinor equation. Therefore, given a vielbein redefinition $e_\mu{}^a\mapsto R^a{}_b ...
user984949's user avatar
2 votes
1 answer
61 views

Original source "GF92" for two drawings of Dirac belt trick

I am trying to track down the original source/artist of these two drawings of the Dirac belt trick (see link below) to use in my thesis (which is in mathematics, but I believe these pictures likely ...
8 votes
1 answer
351 views

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,\...
Mauricio's user avatar
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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(\...
Anant Badal's user avatar
0 votes
1 answer
96 views

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 ...
JohnA.'s user avatar
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0 answers
30 views

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) ...
SrJaimito's user avatar
  • 601
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1 answer
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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
1 vote
0 answers
33 views

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 ...
MathZilla's user avatar
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0 votes
1 answer
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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, ...
JohnA.'s user avatar
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17 votes
4 answers
3k views

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
  • 492
2 votes
0 answers
70 views

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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2 votes
1 answer
71 views

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)...
Andrea's user avatar
  • 509
1 vote
0 answers
28 views

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 ...
LSS's user avatar
  • 980
1 vote
1 answer
78 views

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
-2 votes
1 answer
65 views

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
1 vote
1 answer
66 views

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^{\...
LSS's user avatar
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0 votes
3 answers
158 views

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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0 answers
53 views

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
-1 votes
1 answer
55 views

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
2 votes
1 answer
87 views

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 ...
Ballanzor's user avatar
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0 answers
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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 ...
MegAmaNeo1's user avatar
0 votes
1 answer
92 views

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 ...
IGY's user avatar
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2 votes
0 answers
60 views

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
2 votes
1 answer
63 views

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
2 votes
1 answer
116 views

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 ...
Gleeson's user avatar
  • 213
0 votes
1 answer
159 views

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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