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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61 views

Feynman rules to spinor helicity

The following amplitudes of the 3-gluon vertex are obtained only from momentum conservation (shown in Quantum Field Theory and the Standard Model, Schwartz): $$M^{+++}=C^{abc}\frac{1}{\langle12\rangle\...
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48 views

Adjoint spinors in Feynman diagrams

I am reading through Griffiths textbook "Introduction to Elementary Particles" and I am in the middle of the chapter on Quantum Electrodynamics and Feynman Rules for his diagrams. There is ...
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63 views

How is $j=1/2$ representation, $U(R(\theta,\hat{\bf n}))=e^{i{\sigma}\cdot{\hat {\bf n}}\theta/2}$, is a projective representation of ${\rm SO}(3)$?

A projective unitary representation of ${\rm SO(3)}$ satisfies $$U(R_1)U(R_2)=e^{i\phi(R_1,R_2)}U(R_1R_2)\tag{1}$$ where $R_1,R_2\in {\rm SO(3)}$. How to show that the $j=1/2$ representation, $U(R(\...
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Are Dirac/Weyl/Majorana fermions exclusive?

I think we can be pretty sure that fermions exist. We have several ways to describe them (Dirac, Weyl, Majorana, maybe someone I'm missing?), with different equations and number of components. My ...
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1answer
55 views

Deriving the Generalized Fierz Transformation from Schroeder's Textbook

I am self studying QFT from the textbook An Introduction of Quantum Field Theory and the corresponding solutions from Zhong-Zhi Xianyu. The generalized Fierz Transformation is derived in problem 3.6. ...
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51 views

Are there non-lagrangian field theories for massive Weyl spinors?

It is a well-known fact that a chiral fermion is massless, since the mass term involves both a right-handed and a left-handed field. If you have one chirality only there are no mass terms one can ...
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253 views

Conflicting definitions of a spinor

I'm moving my question from math.stackexchange over here because it got no attention over there, even after 3 weeks and a 50-point bounty, and also because this is a very physics-oriented math ...
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In what meaningful way can we talk about the generators of the super-Poincaré algebra being “spinors” or “4-vectors” because of their indices?

It is often presented without much justification that the generators of the super-Poincaré algebra carry indices that imply they are elements of a representation space of the Poincaré algebra. In ...
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61 views

Levi-Civita symbol in 2-spinor notation

I'm reading An Introduction to Twistor Theory, by Huggett and Tod, and I don’t get the result we're being given page 17: the 2-spinor form of the 4 dimensional Levi-Civita symbol. \begin{equation} \...
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20 views

Flipped sign in solution of Dirac bispinor

I’ve been following David Tong’s lecture notes on QFT, and I came across something that I couldn’t work out for the general solution of the Dirac Spinors. On page 100 in the link here Tong Notes Part ...
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34 views

What is the definition of a Majorana fermion in conformal field theory?

Majorana spinors background According to Eq. (4.84) and (4.85) of these notes, charge conjugation of the spinor $\Psi$ is defined as $$ \Psi^{(c)} = C \Psi^*,$$ where $C$ is the unitary charge ...
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1answer
110 views

Spin $1/2$ motivation for twistor theory in general relativity?

I was watching this Youtube video (I have linked it at the relevant timestamp) and to paraphrase Dr. Woit' s motivation for twistor theory: Within the standard way of thinking about general ...
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Do spinors form a vector space?

In contradiction to a number of other authors (sample ref below), Gerrit Coddens, at France’s prestigious Ecole Poytechnique, asserts that: 2.2 Preliminary caveat: Spinors do not build a vector space ...
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33 views

Majorana fermions in Euclidean and Minkowski signatures - contradiction with Wikipedia Table

In this wonderful lecture note on Clifford Algebra and Spin(N) Representations, http://hitoshi.berkeley.edu/230A/clifford.pdf Somehow I find some inconsistency with his Tables of Euclidean and ...
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33 views

The 'mutual' and the 'self' in terms of the 'conjugacy' of Euclidean and Minkowski Weyl fermions

Euclidean and Minkowski fermions are shown in the Table of Wikipedia. (see the bottom https://en.wikipedia.org/wiki/Spinor#Summary_in_low_dimensions) My question is that what does the conjugacy mean ...
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484 views

Trying to understand spin

My question is fairly simple and straightforward. I'm studying Quantum Mechanics, specifically the spinor formalism. I understand that one can define a generator of rotations, say around axis $z$ by ...
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51 views

Accounts on the solutions of the Dirac equation

Consider the Dirac equation $(i\gamma^{\mu}\partial_{\mu}-m)\psi = 0$. As it is well known, there are different representations for the matrices $\gamma^{\mu}$, $\mu = 0,1,2,3$, the most famous ones ...
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How would the “belt trick” analogy of a pion look like?

Since the (charged) pion has a spin of 0, this by itself would suggest that it has spherical symmetry. However, because if it's non-zero iso-spin property it does not have spherical symmetry (is this ...
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65 views

Why there is a minus sign before $p^0$?

In scattering amplitudes, page 244, I am trying to verify that $$ p_{a\dot{b}} = \left( \begin{matrix} -p_0+p_3 & p_1-ip_2 \\ p_1+ip_2 & -p_0-p_3 \end{matrix} \right), \quad (1) $$ i.e, ...
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Riemannian and Weyl tensors as spinor representation

There is the way of converting vector indices to spinor indices, for example, Maxwell stress tensor $F_{[\mu\nu]}$ can be decomposed to $(1,0) \oplus (0,1)$ irreducible representations of $\mathfrak{...
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33 views

Plane waves solutions for Dirac equation in terms of eigenstates of helicity

Suppose $\sigma_{1},\sigma_{2}$ and $\sigma_{3}$ are the Pauli matrices. Given a momentum ${\bf{p}}$, we define the helicity operator: $$ h = \frac{1}{2}\begin{pmatrix} {\bf{\sigma}}\cdot {\bf{\hat{p}}...
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43 views

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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41 views

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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54 views

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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1answer
45 views

Sign error when deriving Weyl spinor transformation laws (3.37) in Peskin Schroesder

I am having some trouble deriving the transormation laws for the weyl spinors, equation (3.37) in the Peskin Schroesder book on quantum field theory. Beginning with the relation $\psi\to(1-\frac{i}{2}\...
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47 views

Quantum Field theory, coupling term, Electroweak Unification

In Anthony Zee, Quantum Field Theory in a Nutshell(2nd edition), Zee writes on Page 381, when $\phi$ - the Higgs field, acquires the vacuum expectation value $\begin{bmatrix} 0 \\ v \end{bmatrix}$, a ...
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1answer
111 views

Spinors and spin group

It seems to me that spinors (pinors) are loosely defined as representations of the spin (pin) group $Spin(p,q)$ ($Pin(p,q)$), which double covers the spacetime symmetry group $SO(p,q)$ ($O(p,q)$). $\...
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1answer
63 views

A question involving chiral transformations and gamma matrices

I'm looking at a calculation that involves an infinitesimal transformation of a Dirac fermion field: $$\Psi \rightarrow e^{i \beta \gamma^5} \Psi.$$ Then the conjugate field $\bar{\Psi} = \Psi^{\...
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1answer
64 views

Solution to Dirac equation

We take the solution of Dirac equation as 4 component wave function (Dirac Spinor). But how do we know that it can't be a square or rectangular matrix like 4x2 or 4x4 matrix?
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Confused about chirality of antiparticles and QED interaction

In the middle of doing some calculations and I seem to have forgotten some basics from QFT. I would be very grateful if someone could help me out! Taking, for example, an interaction proportional to $\...
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38 views

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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1answer
66 views

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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54 views

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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29 views

Dirac spinors completeness relation for $d>4$

In $d=4$ and in the rest mass frame, the four Dirac spinors $u_s(0)$ and $v_s(0)$ satisfy the completeness relation $$ \sum_{s=1}^{2} u_s(0) \overline{u}_s(0) - \sum_{s=1}^{2} v_s(0) \overline{v}_s(0)...
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1answer
105 views

What is Dirac indices?

In Maggiore A Modern Introduction to Quantum Field Theory Eq. 4.31 $$\{\Psi_a(\vec x,t),\Psi_b(\vec x,t)\}=\delta^{(3)}(\vec x-\vec y)) \delta_{ab}$$ where "$a,b=1,2,3,4$ are the Dirac ...
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28 views

Helicity operator for particles and antiparticles

I´ve come across the fact that the helicity operator for particle solutions u(p) is not the same as the helicity operator for anti-particle solutions v(p). For particle solutions: $\begin{equation} h =...
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Confusion with helicity eigenstates of massless spinors

As presented in Schwartz´s QFT and SM one can solve the free Dirac Equation in the Weyl basis. If $p^{\mu} = (E,0,0,p_z)^T$ the four solutions are: \begin{equation} u_1^p = \begin{bmatrix} \...
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1answer
42 views

Are the spinors one can find in the Feynman rules always solutions of the free Dirac equation?

For a given Feynman diagram one can calculate the matrix Element by translating the diagram into math using Feynman rules. In these calculations one will encounter incoming and outgoing particles (and ...
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What's the spin state in the rest frame of an electron which is in a helicity eigenstate?

Helicity is the spin component in the direction of momentum: $\mathbf{\Sigma \cdot \hat{p}}$. For a free electron, the helicity commutes with the free Dirac Hamiltonian $H = c\boldsymbol{\alpha}\cdot\...
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27 views

Problem with constructing a bispinor in the spinor helicity formalism

The $(\frac{1}{2},\frac{1}{2})$ representation of the Lorentz group is constructed as $(\frac{1}{2},0) \otimes (0,\frac{1}{2})$. To get an element of the vector space this specific representation acts ...
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31 views

Under which representation (and how exactly) transforms the given bispinor? [duplicate]

I am currently reading through Chapter 27 (Spinor helicity formalism) in Schwartz´s "QFT and the SM". In this chapter it says that since 4-momenta transform in the (1/2,1/2) representation ...
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30 views

Parity of a fermion bilinear

I'm assuming that the parity transformation of a 4-vector field is: $$x^\mu = (t,\mathbf{x}) \rightarrow x'^\mu = (t',\mathbf{x'}) = (t,-\mathbf{x})$$ $$V^\mu(t,\mathbf{x}) = (V^0(t,\mathbf{x}), V^i(t,...
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1answer
120 views

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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1answer
68 views

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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18 views

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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1answer
95 views

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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30 views

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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38 views

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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50 views

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 ...

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