Questions tagged [spinors]

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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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47 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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Spinor helicity formalism

The two previous diagrams are the real ones contributing to $H \rightarrow$ gg(g). For the R1 I wrote: $-iM = -Ag^2f^{abc}(\epsilon_1.\epsilon_2)(\epsilon_3.(p_1-p_2)) + \epsilon_2.\epsilon_3)(\...
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33 views

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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59 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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32 views

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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39 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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38 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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49 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
44 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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46 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
104 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
59 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
57 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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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
58 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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1answer
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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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
91 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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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
40 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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16 views

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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23 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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30 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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23 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
119 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
63 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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16 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
91 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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29 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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30 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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49 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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1answer
79 views

Standard model notation on doublets

$\require{cancel}$ I have been introduced to electroweak theory in lectures and I wanted to check I understand the notation for the doublets, triplets etc. Take the first generation lepton left handed ...
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1answer
83 views

Lorentz boost of Dirac spinor

Let $\psi_\vec{0}^+$ be a Dirac wavefunction describing a motionless particle, $$\psi_\vec{0}^+(x) = \sqrt{2m} \begin{pmatrix} \chi \\ 0 \end{pmatrix} e^{ip \cdot x}$$ where $p = (m, \vec{0})$. ...
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1answer
53 views

Charge conjugation on spinors: Am I missing a (-1)? [duplicate]

I'm trying to prove the transformation rules for Dirac Bilinears under charge conjugation as given in "Fundamentals of neutrino physics and astrohysics" by Carlo Giunti et.al. According to ...
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How to describe a time dependent spin state from initial state?

If i have the initial spin state defined like this: Of which has an applied magnetic field along some axis say the $z$-direction. How do you then define $A$ with time dependence and what axis are the ...
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23 views

Weyl spinor being on-mass or off-mass shell

Is there a way to know whether a two component Weyl spinor is on-mass or off-mass shell?
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1answer
68 views

Bhabha scattering in the spinor-helicity formalism

I am trying to calculate the square amplitude for Bhabha scattering $e^-(p_1)e^+(p_2)\rightarrow e^-(p_3)e^+(p_4)$ using the spinor-helicity formalism but one of the interference terms just will not ...
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27 views

Rotation operator from angular momentum or spin operator

My instructor on quantum physics just stated that the total angular momentum operator, $\hat{J}$, can be expressed as $\hat{J}=\hat{L}+\hat{S}$, where $\hat{L}$ is angular momentum operator ...
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1answer
69 views

What are the Feynman rules for the spinor-helicity formalism?

If we do not work with helicity amplitudes, there are Feynman rules for the external legs of a Feynman diagram, i.e. $u_s(k),\overline{v}_s(k),\epsilon_r(k)$ for an incoming fermion, antifermion and ...
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1answer
227 views

Does Wightman's unitary $U(\Lambda)$ really exist for Lorentz boost?

This question is related to another question here. But I am asking a more fundamental question about the existence of Wightman's unitary $U(\Lambda)$ for Lorentz transformation. Let $\psi^\alpha$ be a ...
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What do these matrices represent physically? Are they related to Majorana spinors?

I am studying $\mathfrak{so}(1,3)$ representations and I found this claim that $(m,n)\oplus(n,m)$ representations have a real structure (for which I asked a separate question on math.SE). I tried to ...
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2answers
199 views

Spatial inversion and Time reversal

On the spinor field $\psi^{\mu}(x)$, I found the action of $\psi^{\mu}(x)$ on spatial inversion $P$ by postulating $\psi^{\mu}_{P}(x)=P^{\mu}_{\nu}\psi^{\nu}(P^{-1}x)=P^{\mu}_{\nu}\psi^{\nu}(t,-x)$, ...
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2answers
149 views

Does this argument prove that all fermionic states have zero norm?

The following argument seems to show that all states created by a fermionic field have zero norm. This would surely cause problems in QFT, so I believe there must be an error somewhere, but I can't ...
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2answers
66 views

An essential exercise involving spin composition

Suppose we have two particles with spin $1/2$. They have $S^{tot}=1$ and $S^{tot}_y=0$. How can we write the state of the system in terms of the eigenstates of $S_{1z},S_{2z}$? My attempt: I would ...
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32 views

Fierz Identity calculation

While reading an article, it's said that to simplify the following Dirac structure $$\left(P_Lv_j^d\bar{v}^s_kP_R\right)_{\alpha\beta}\tag{1}\label{1}$$ where $j,k$ are color indices and $d,s$ ...
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1answer
78 views

Spinors and Tensors: what is the form of spin transformation matrix?

The (covariant) vector transformation law is given by: $$V^{'}_{\mu} = t^{\nu}\hspace{0.1mm}_{\mu'}V_{\nu} =\frac{\partial x^{\nu}}{\partial x'^{\mu}}V_{\nu} \tag{1}$$ where the transformation is ...

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