Questions tagged [spinors]
The spinors tag has no usage guidance.
685
questions
0
votes
0answers
27 views
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 ...
0
votes
0answers
20 views
Finding neutrino mass eigenstates from Lagrangian?
I was given this homework assignment. We have a Lagrangian density for left- and right-handed spinors:
$$\mathcal{L} = i\nu_L^\dagger\bar{\sigma}^\mu\partial_\mu\nu_L + i\nu_R^\dagger\sigma^\mu\...
-1
votes
1answer
32 views
Why $(\vec{\sigma}\cdot\vec{p}) (\vec{\sigma}\cdot\vec{p}) = (\vec{\sigma}\cdot\vec{p}) p^0$ for Pauli matrices?
I am trying to verify that the following equation is true:
$$(\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 ...
2
votes
0answers
60 views
+50
$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 ...
0
votes
1answer
46 views
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 ...
1
vote
1answer
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 ...
1
vote
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} \...
0
votes
1answer
45 views
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, ...
1
vote
0answers
81 views
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. ...
0
votes
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 ...
1
vote
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 ...
0
votes
0answers
21 views
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-...
2
votes
0answers
33 views
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 ...
0
votes
1answer
69 views
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 ...
2
votes
3answers
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 ...
2
votes
0answers
48 views
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 $$ Q_\alpha^I = \begin{pmatrix} Q_1^I \\ Q_2^I \end{pmatrix}~,~ \...
0
votes
0answers
46 views
Dirac Bilinear Transformation Laws (vector)
The question is to verify that the following Dirac Bilinears obey the following transformation law:
$$\bar\psi'(x') \gamma^\mu \psi'(x') = \Lambda^\mu_\nu \bar\psi(x) \gamma^\nu \psi(x)$$
What I know ...
1
vote
0answers
39 views
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. $[\...
0
votes
1answer
64 views
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)...
5
votes
2answers
256 views
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 ...
2
votes
3answers
63 views
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 ...
2
votes
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 ...
0
votes
1answer
37 views
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 ...
0
votes
0answers
28 views
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}...
2
votes
0answers
26 views
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 (...
0
votes
1answer
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?
3
votes
2answers
127 views
What kind of tensor is $\psi^\dagger\psi$?
I'm trying to find how does this quantity $$\psi^\dagger\psi$$ transforms under a Lorentz transformation. Where $\psi$ is a Dirac spinor.
What I've tried so far:
It is known that a Dirac spinor ...
2
votes
1answer
53 views
Showing that the Wess-Zumino Lagrangian is invariant under a SUSY transformation
I want to show that the free Wess-Zumino Lagrangian is invariant under a SUSY transformation, e.g. following this reference (section 3.1).
However, I have a hard time understanding the daggers and ...
1
vote
1answer
48 views
Suppose I give you $2^N$ functions that are eigenvectors of a fermionic $H$. How do I determine which function describes which spin configuration?
Consider the hamiltonian
$$
H = - \frac{1}{2} \nabla^2 + V.
$$
The potential $V : (\mathbb{R^3})^N \to \mathbb{R}$ is symmetric, so for each eigenvalue, there is an antisymmetric eigenvector. There is ...
2
votes
1answer
62 views
Existences of Majorana spinors in $\rm Spin(4)$ and $\rm Spin(1,3)$
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$
We know that
$$
\Spin(1,3)=\SL(2,\mathbb C)
$$
and
$$
\Spin(4)=\SU(2) \times \SU(2).
$$
It is also said $\Spin(1,...
0
votes
0answers
28 views
Elvang and Huang sign error? Polarization Vectors in the Spinor Helicity Formalism
I am reading through Elvang and Huang's treatment of polarization vectors for all outgoing spin-1 massless particles (metric signature $(-,+,+,+)$).
It is given in Eq. (2.50) in the PDF (but Eq. (2.51)...
3
votes
1answer
64 views
Spinor indices in $\not p = m_P$ (mass renormalization) [duplicate]
I'm reading Schwartz QFT, Chapter 18 (mass renormalization) and I'm confused about the equations about on-shell subtraction/pole mass. He writes:
The renormalized propagator should have a single pole ...
4
votes
0answers
107 views
Spinor Helicity Formalism: Reference Spinors $q$ in Compton Scattering
My question is rather straight forward, but the setup in order to pose the question is a little lengthy; please bear with me!
I am trying to calculate the average over initial states and sum over ...
1
vote
1answer
82 views
PT transformation of a spinor
By demanding the the Dirac equation be invariant under general Lorentz transformations, we get an equation for the transformation matrix of a Dirac spinor,
$$ S^{-1}(\Lambda) \gamma^\mu S(\Lambda) = {\...
2
votes
1answer
30 views
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 ...
1
vote
0answers
43 views
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 ...
1
vote
0answers
55 views
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" ...
0
votes
2answers
66 views
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 ...
0
votes
0answers
19 views
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,
$$ A_i = \tfrac{1}{2}(J_i + \text{i}K_i),\...
0
votes
0answers
99 views
Why is the Lorentz invariant integration measure for a spinor $\frac{d^3 k}{(2\pi)^3}\frac{m}{\omega}$?
I understand for a scalar field theory the integration measure is $\frac{d^3 k}{(2\pi)^3}\frac{1}{2\omega}$ because it has to satisfy the following equation
$$\int \frac{d^4 k}{(2\pi)^4}\delta(\omega^...
0
votes
0answers
33 views
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 $j_{1}$...
0
votes
1answer
48 views
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 ...
1
vote
1answer
97 views
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
$$
\left(i\...
1
vote
1answer
61 views
Why do covariantly constant spinors on the circle require periodic boundary conditions?
I am asking in the context of a paper of Witten on instability of Kaluza-Klein spacetimes (https://www.sciencedirect.com/science/article/pii/0550321382900074).
The discussion involves applying Witten'...
1
vote
1answer
54 views
Lorentz transformation of Weyl fields
In the Srednicki's textbook, Chapter 35, the author states (Equation 35.28):
$$
U(\Lambda)^{-1}[\psi^\dagger \bar\sigma^\mu \chi ] U(\Lambda) = \Lambda^\mu_{\,\,\nu} [\psi^\dagger \bar\sigma^\nu \chi ]...
1
vote
1answer
65 views
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:
$$S^{0i} ...
3
votes
1answer
36 views
Confusion about the mass operator in the RNS superstring
In the RNS formulation of superstring theory we have the action:
$$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)$$
...
1
vote
1answer
50 views
Dirac equation solution - four-component spinors - left-/right-handed in ultrarelativistic limit
I am confused about the solution to the Dirac equation and how it corresponds to left-/right-handed Weyl spinors. In Srednicki, page 242, it is stated that taking the ultrarelativistic limit ($|\bar{p}...
0
votes
3answers
60 views
Why is $\overline{\Lambda_s \psi} = \bar{\psi} \Lambda_s^{-1}$?
In my text Schwartz writes: since $\bar{\psi} \gamma^\mu \psi$ transforms like a 4-vector, we can deduce that
$$\Lambda_s^{-1} \gamma^\mu \Lambda_s = (\Lambda_V)^{\mu \nu} \gamma^\nu, \tag{10.78}$$
...
1
vote
1answer
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}}\,\...
