Questions tagged [spinors]

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Physical interpretation of spinor parameter $\epsilon$ in supersymmetry

Studying supersymmetry, I came across the introduction of the idea of SUSY field variations involving spinor parameters $\epsilon_{\alpha}$ under which actions must be invariant. This spinor parameter ...
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Finding neutrino mass eigenstates from Lagrangian?

I was given this homework assignment. We have a Lagrangian density for left- and right-handed spinors: $$\mathcal{L} = i\nu_L^\dagger\bar{\sigma}^\mu\partial_\mu\nu_L + i\nu_R^\dagger\sigma^\mu\...
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Why $(\vec{\sigma}\cdot\vec{p}) (\vec{\sigma}\cdot\vec{p}) = (\vec{\sigma}\cdot\vec{p}) p^0$ for Pauli matrices?

I am trying to verify that the following equation is true: $$(\vec{\sigma}\cdot\vec{p}) (\vec{\sigma}\cdot\vec{p}) = (\vec{\sigma}\cdot\vec{p}) p^0$$ where $p^\mu=(p^0,\vec{p})$ is the four momentum ...
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$qg \rightarrow qg$: Spinor Helicity Formalism

I am calculating the cross-section for quark gluon to quark gluon scattering in the spinor helicity formalism. This process has contributions from the Mandelstam channels $s$, $t$ and $u$. Using the ...
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1answer
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Spinor matrix representation of a spacetime vector in 2+1D

Let $a^\mu$ be a spacetime vector in 2+1D: $(t,x,y)$. What would be the spinor-matrix representation ${A^\alpha}_\beta$ of the spacetime vector $a^\mu$? I have not been able to even find examples or ...
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970 views

Calculation of Parity in Quantum Field Theory

In the book "Relativistic Quantum Mechanics An Introduction To Relativistic Quantum Fields" by Luciano Maiani Omar Benhar, page 174, the picture of that page is provided below. I don't ...
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1answer
27 views

Charge conjugation of fields

This page on Wikipedia says, "In the language of quantum field theory, charge conjugation transforms as - $\psi \Rightarrow -i\big(\bar{\psi} \gamma ^0 \gamma ^2 \big)^T $ $\bar{\psi} \...
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Interchanging positions of Gell-Mann matrices with Dirac matrices, Pauli matrices

The anti-commutation relations for Gamma matrices $\big\{\gamma ^\mu , \gamma ^\nu \big\} = 2g ^{\mu \nu} $ can be used for interchanging positions of the respective matrices in a given expression, ...
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Conformal generators in spinor-helicity variables

The question in particular pertains to Section D.1 of https://arxiv.org/abs/2005.04234. In this section, they have written the conformal generators of 3d in terms of the spinor-helicity variables. ...
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1answer
53 views

Equation 11.27 in M.D. Schwartz book

I am having trouble understanding the steps taken in (11.27) equation in Quantum Field Theory and the Standard Model by M.D.Schwartz. I don't understand how to get the middle diagonal matrix in the ...
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1answer
48 views

Are the solutions to the Dirac equation orthogonal?

The massless Dirac Hamiltonian is given by $H = -i \gamma^0 \gamma^i \partial_i \equiv -i \alpha^i \partial_i $. If I define an inner product of spinors as $$ ( \psi , \phi ) = \int d^n x \psi^\dagger ...
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Are super-symmetrical complements of physically realised theories (scalar, tensor, Yang-Mills etc.) compellingly first-order differential equations?

I made the observation that all super-symmetrical extensions of "standard" theories (SM and GR) are given by first order differential equations. In particular if in case of EFEs the super-...
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Spinor covariant derivative conventions

The covariant derivative of a spinor $\psi$ is given by $$ \nabla_\mu \psi = \partial_\mu \psi + \Omega_\mu \psi $$ where $\Omega_\mu$ is the spin connection. In equation (7.227) of Geometry, Topology ...
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When a Hilbert space's state vector becomes a spinor?

Given a Hilbert space $\mathcal{H}$, we pick up some state vector $| \psi\rangle$ which lives in the Hilbert space $\mathcal{H}$. The $| \psi\rangle$ is a vector of the Hilbert space satisfies the ...
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420 views

Rotation of a Spinor

I have a question about an intuitive approach on spinors as certain mathematical objects which have certain properties that make them similar to vectors but on the other hand there is a property which ...
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Counting supersymmetries in $d=4$ vs in $d=1+1$

Having studied supersymmetry in $d=4$, my understanding is that we count supersymmetries by the number of pair of complex supercharges $$ Q_\alpha^I = \begin{pmatrix} Q_1^I \\ Q_2^I \end{pmatrix}~,~ \...
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Dirac Bilinear Transformation Laws (vector)

The question is to verify that the following Dirac Bilinears obey the following transformation law: $$\bar\psi'(x') \gamma^\mu \psi'(x') = \Lambda^\mu_\nu \bar\psi(x) \gamma^\nu \psi(x)$$ What I know ...
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Mass dimension of Dirac spinors (QFT)

From a dimensional analysis (with $\hbar=1=c $) of the Dirac Lagrangian $\mathcal{L}= \bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi $ one obtains that the mass dimension of $\psi$ is $3/2$ (i.e. $[\...
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Evaluating the modulus squared of a spinor chain with different number of spinor and anti-spinors

I want to evaluate the interference between diagrams in a BSM model whose relevant part of the contributions are \begin{equation} \begin{split} A&=[\bar{u}_e(k_2)v_e(k_3)] [\bar{u}_e(k_1)u_\mu(p_1)...
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Do the spinor transformation matrices form a matrix representation of the corresponding Lorentz group?

Suppose $\Psi$ is a Dirac spinor, then let the transformation matrix $S$ be defined as usual: $\Psi'=S(\Lambda)\Psi$, where $\Lambda$ is the Lorentz transformation matrix. Then the questions is: for ...
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3answers
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How do wave particles have spin?

I am not a student nor a scientist. If a photon is a wave until it is measured somehow, how can it have a spin? A wave is a wave. Is spin simply a mathematical tag that we give to particles? Or do ...
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1answer
102 views

What's wrong with using a vielbein to define Wick rotation?

Wick rotation is supposed to be a relationship between field theories with spacetime metrics of Lorentzian and Euclidean signature. I thought the definition of Wick rotation was settled, until I came ...
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1answer
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Triplet states, Dicke states and symmetric spin-1 states

When adding two spin-1/2 particles, it is well known that the basis of the composite system can be written in terms of the spin singlet and triplet states. These sates have a well defined symmetry ...
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Spin frame bundle and Orthogonal frame bundle

I was trying to understand the ACTIVE transformation (rotation) of spin states in terms of group action on orthogonal frame bundle. As we know, there exist a lie group homomorphism $ \rho : {\rm Spin}...
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Both formalisms of a chiral multiplet in Supersymmetry

For the description of a chiral multiplet in Supersymmetry there are 2 formalisms, the one I am used to presented for instance in the Supersymmetry Primer of SP Martin which is based on 2-component (...
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53 views

Fermion Determinant

When we calculate fermion determinant for either Majorana or Weyl spinors, why do we get an extra factor of half as the coefficient of the determinant?
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What kind of tensor is $\psi^\dagger\psi$?

I'm trying to find how does this quantity $$\psi^\dagger\psi$$ transforms under a Lorentz transformation. Where $\psi$ is a Dirac spinor. What I've tried so far: It is known that a Dirac spinor ...
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1answer
53 views

Showing that the Wess-Zumino Lagrangian is invariant under a SUSY transformation

I want to show that the free Wess-Zumino Lagrangian is invariant under a SUSY transformation, e.g. following this reference (section 3.1). However, I have a hard time understanding the daggers and ...
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1answer
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Suppose I give you $2^N$ functions that are eigenvectors of a fermionic $H$. How do I determine which function describes which spin configuration?

Consider the hamiltonian $$ H = - \frac{1}{2} \nabla^2 + V. $$ The potential $V : (\mathbb{R^3})^N \to \mathbb{R}$ is symmetric, so for each eigenvalue, there is an antisymmetric eigenvector. There is ...
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Existences of Majorana spinors in $\rm Spin(4)$ and $\rm Spin(1,3)$

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$ We know that $$ \Spin(1,3)=\SL(2,\mathbb C) $$ and $$ \Spin(4)=\SU(2) \times \SU(2). $$ It is also said $\Spin(1,...
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Elvang and Huang sign error? Polarization Vectors in the Spinor Helicity Formalism

I am reading through Elvang and Huang's treatment of polarization vectors for all outgoing spin-1 massless particles (metric signature $(-,+,+,+)$). It is given in Eq. (2.50) in the PDF (but Eq. (2.51)...
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1answer
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Spinor indices in $\not p = m_P$ (mass renormalization) [duplicate]

I'm reading Schwartz QFT, Chapter 18 (mass renormalization) and I'm confused about the equations about on-shell subtraction/pole mass. He writes: The renormalized propagator should have a single pole ...
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107 views

Spinor Helicity Formalism: Reference Spinors $q$ in Compton Scattering

My question is rather straight forward, but the setup in order to pose the question is a little lengthy; please bear with me! I am trying to calculate the average over initial states and sum over ...
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1answer
82 views

PT transformation of a spinor

By demanding the the Dirac equation be invariant under general Lorentz transformations, we get an equation for the transformation matrix of a Dirac spinor, $$ S^{-1}(\Lambda) \gamma^\mu S(\Lambda) = {\...
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1answer
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How precisely can we set up a spin measurement of an atom about a particular axis?

The standard treatment is that once we measure a spin along the z-direction, it will be aligned with +1 or -1 spin along the z-direction. Then, when we measure it again along the z-direction, it has a ...
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Scattering in QM/NRQM vs QFT

As discussed in this question: How does a wave packet get scattered? And shown in, for example, this video: https://www.youtube.com/watch?v=iq4lGVznr_8 In quantum mechanics particles are scattered due ...
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What does the adjoint of a QFT Dirac plane-wave spinor represent, physically?

So, using the Dirac spinor notation for plane-wave electrons and positrons from HERE, and the definition of the adjoint of $\psi$ = $\psi^{*T} \gamma^0$, and applying that to the "up" ...
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Finding spin up/spin down eigenstates along some arbitrary direction

So let's say I have some particle in some arbitrary state where its ket vector is given as a linear combination of the spin up eigenstate and the spin down eigenstate. The magnitude square of a, which ...
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Is there a convention for left / right-chiral Weyl spinors?

In order to distinguish the representations of the Lorentz group, we can combine the generators for rotations ($J$) and boosts ($K$) into linear combinations, $$ A_i = \tfrac{1}{2}(J_i + \text{i}K_i),\...
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Why is the Lorentz invariant integration measure for a spinor $\frac{d^3 k}{(2\pi)^3}\frac{m}{\omega}$?

I understand for a scalar field theory the integration measure is $\frac{d^3 k}{(2\pi)^3}\frac{1}{2\omega}$ because it has to satisfy the following equation $$\int \frac{d^4 k}{(2\pi)^4}\delta(\omega^...
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Normalization for Clebsch-Gordan Decomposition in Symmetrized Spinor Representation

In various references, including Rovelli's "Quantum Gravity," a method is presented for calculating the Clebsch-Gordan decomposition of two spin states. Suppose we have two spins with $j_{1}$...
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Eigenspinor of helicity of electrons

I am reading the chapter in Griffth's introduction to elementary particle. By solving the momentum space Dirac equation and requiring the solution of the spinor to be the eigenspinor of the helicity ...
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Is the Dirac equation really covariant under Lorentz transfromations or do we just “make” it covariant?

I often read that the Dirac equation is covariant under Lorentz transformations and that this property makes it the right equation and in some sense beautiful. The thing is, the equation $$ \left(i\...
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Why do covariantly constant spinors on the circle require periodic boundary conditions?

I am asking in the context of a paper of Witten on instability of Kaluza-Klein spacetimes (https://www.sciencedirect.com/science/article/pii/0550321382900074). The discussion involves applying Witten'...
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Lorentz transformation of Weyl fields

In the Srednicki's textbook, Chapter 35, the author states (Equation 35.28): $$ U(\Lambda)^{-1}[\psi^\dagger \bar\sigma^\mu \chi ] U(\Lambda) = \Lambda^\mu_{\,\,\nu} [\psi^\dagger \bar\sigma^\nu \chi ]...
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How to find the 3-direction Lorentz boost transformation on Dirac spinor?

I am struggling to work out correct Lorentz transformation for a boost in the 3-direction on a Dirac spinor, $u(p)$. According to Peskin & Schroeder pg. 46, I need to use the equations: $$S^{0i} ...
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1answer
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Confusion about the mass operator in the RNS superstring

In the RNS formulation of superstring theory we have the action: $$S=-\frac{T}{2}\int d^2\sigma(\partial_\alpha X^\mu \partial^\alpha X_\mu + \bar{\psi}^\mu \rho^\alpha \partial_\alpha \psi_\mu)$$ ...
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1answer
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Dirac equation solution - four-component spinors - left-/right-handed in ultrarelativistic limit

I am confused about the solution to the Dirac equation and how it corresponds to left-/right-handed Weyl spinors. In Srednicki, page 242, it is stated that taking the ultrarelativistic limit ($|\bar{p}...
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Why is $\overline{\Lambda_s \psi} = \bar{\psi} \Lambda_s^{-1}$?

In my text Schwartz writes: since $\bar{\psi} \gamma^\mu \psi$ transforms like a 4-vector, we can deduce that $$\Lambda_s^{-1} \gamma^\mu \Lambda_s = (\Lambda_V)^{\mu \nu} \gamma^\nu, \tag{10.78}$$ ...
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81 views

Is this a typo in Peskin's QFT?

In ''An intro to QFT (2018)'' chapter 3, Peskin does the following: Let me introduce some notation first, let $v^s_k=\begin{pmatrix}\;\;\,\sqrt{k\cdot\sigma}\,\xi^{-s}\\-\sqrt{k\cdot\bar{\sigma}}\,\...

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