Questions tagged [spinors]

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How to second-quantize an operator if the field operator is a spinor

In non-relativistic QM, one normally second-quantizes an operator using $$ \hat O=\int d^3r~\hat\psi^\dagger(r)O~\hat\psi(r),\qquad(1)$$ where the field operator $\hat\psi$ is given by $$\hat\psi(r)=\...
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Solution to Dirac equation with external source

The Dirac equation is: \begin{equation} \left[i\gamma^{\mu}(\partial_{\mu}-iA_{\mu})-m\right]\psi=0, \tag{1} \end{equation} where $A_\mu$ is a gauge field. The solution to this equation is:...
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Angular-momentum of the Dirac spinor theory

The standard Dirac action $$ S = \int d^4 x \bar \psi (i \gamma^\mu \partial_\mu - m) \psi $$ is invariant under Lorentz transformation. In David Tong's lecture note, eq (4.96) lists that the ...
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Dirac propagator in Non-Abelian Theory

I am trying to derive equation (16.4) from chapter 16.1 page 506 of Peskin&Schroeder. Here is my derivation My Attempt We start here by considering the dirac spinor part of the Non-Abelian ...
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Chiral Transformation and Dirac Bilinear

I need to compute the following Dirac bilinears: $$\overline{\psi} \psi \quad \text{and} \quad \overline{\psi} \gamma_\mu \psi$$ Under the following Chiral transformation: $$\psi \rightarrow \psi' = \...
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Question about the parity violation of weak interaction Lagrangian

In the textbook of A. Zee, Quantum Field Theory in a Nutshell, the author states that the following Lagrangian: $$ \mathcal{L} = G (\overline{\psi}_{1L} \gamma^\mu \psi_{2L})(\overline{\psi}_{3L} \...
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Normalization of solution of Dirac equation

I know that the solution to the dirac equation are of the form: $\psi(x)=u(\vec{p})e^{ip\cdot x}$ and the spinor can be normalized as $u^\dagger u =E$. I was reading "Lectures on Quantum Field ...
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$A(\phi\phi^*\gamma \gamma)$ with Spinor Helicity Formalism

I am calculating $A_4(\phi\phi^*\gamma\gamma)$ with the spinor helicity formalism (Exercise 2.16). I am following the conventions defined in Elvang's notes Such computation requires three diagrams, ...
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Soldering Spinors in cylindrical coordinate

For my computation, I need to have Solderings in cylindrical coordinate(I think). I know what Solderings are in cartesian coordinate: $\sigma^{\mu}_{A\dot{A}}=(\sigma^{0}_{A\dot{A}},\sigma^{1}_{A\dot{...
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The mathematics of different particle rotations

So, in general (if I understand this correctly): Force particles behave differently than matter particles under rotation The matter particles need a 720° rotation to put them back into their initial ...
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Problem with derivation of the Dirac Hamiltonian

I'm having son trouble when obtaining the Dirac equation. I am working in (1+1)-dimensional Minkowski spacetime with signature $(-, +)$ in coordinates $(t, x)\equiv(1, 2)$. I can think of two ways to ...
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What is the correct way of looking at the Dirac field?

All quantum fields are operators in QFT. However, the Dirac field operator $\hat{\psi}$ has the following difference with the scalar field operator $\hat{\phi}$: For the $\hat{\psi}$, it makes sense ...
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What is the idea behind 2-spinor calculus?

In the book by Penrose & Rindler of "Spinors and Space-Time", the preface says that there is an alternative to differential geometry and tensor calculus techniques known as 2-spinor ...
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Conventions Consistency For the Dirac Field

I am writting a text and I try to be consistent with my definitions. I have expanded my fields in modes in the following way $$\psi(x)=\int\frac{d^3\vec{p}}{(2\pi)^3\sqrt{2p^0}}\sum_s\bigg(u_s(\vec{p})...
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What does it mean by spin 1/2 or spin 2 field?

I see in common discussions people simply use the terminology spin 1/2 field or spin 2 fields as if it is some common term like hamiltonian. How to think about these fields and understand what it ...
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Griffiths Quantum mechanics example 4.3

The example goes as follows: Imagine a particle of spin $1/2$ at rest in a uniform magnetic field which points in the z-direction: $$\textbf{B} = B_0 \hat{k}$$ The Hamiltonian is $$ H = -\gamma ...
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Properties of rest-frame spinors

The four rest-frame spinors ${\displaystyle u^{(s)}\left({\vec {0}}\right),}$ ${\displaystyle \;v^{(s)}\left({\vec {0}}\right)}$ satisfy $${\displaystyle ({p\!\!\!/}-m)u^{(s)}\left({\vec {p}}\right)=0}...
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The irreducible representation of rank $n$ spinor in 3D

I was reading Ref. 1, where it is asserted that the irreducible resolution of a rank $n$ spinor can be written as totally symmetric spinors. $$\Psi^{n}=\Psi^{\{n\}}+\zeta \Psi^{\{n-2\}}+\zeta \zeta \...
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Relationship between three dimensional spinorial tensor

In a homework assignment, I am asked to show that the components of three dimensional spinorial tensor obey the following relationship $$\Psi_{12}=-\Psi_{1}^{1}=-\Psi^{21}, \Psi_{11}=\Psi_{1}^{2}=\Psi^...
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Does the Schrödinger equation apply to spinors?

I was reading about Larmor precession of the electron in a magnetic field in Griffiths QM when I came across the equation $$ i\hbar \frac{\partial \mathbf \chi}{\partial t} = \mathbf H \mathbf \chi, $$...
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Witten on the hermitian of the Dirac operator

I happened to read Witten SU(2) anomaly paper (1982), and come back to digest again what I said the hermitian of the Dirac operator. According to Prahar https://physics.stackexchange.com/a/701287/...
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Time-reversal transformation acts on the Dirac lagrangian with nonabelian gauge field

Time-reversal transformation acts on the Dirac lagrangian with (non)abelian gauge field Since earlier in Time-reversal transformation acts on the Weyl lagrangian with nonabelian gauge field, we ...
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Implication and proof of conserved charge due to coupling of spin-1(massless) to spin-0 or spin-1/2

I'm following Schwartz's QFT book and problem 11.3 asks to prove that the coupling of massless spin-1 to spin-0 or spin-1/2 implies a conserved charge. It asks to refer to result from section 9.5, ...
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Time-reversal transformation acts on the Weyl lagrangian with nonabelian gauge field

I would like to show time-reversal transformation acts on the Weyl lagrangian in the familiar 4 dimensional space-time. My notation follows the same as Peskin QFT book, such as that of chapter 3. I ...
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Relative velocity of spinors in spinor geometry?

So I'm trying to understand spinor geometry (and not getting anywhere). Is it possible to define relative velocity for spinors? (at a point in the manifold similar to How to calculate relative ...
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Is the Dirac operator $i \not D$ or $\gamma^0 i \not D$ and nonabelian gauge field $A_\mu = A_\mu^\alpha T^\alpha$ hermitian?

In quantum mechanics, it is common to write the momentum operator $$P = i \partial_x.$$ It turns out that $p$ is hermitian although $i^\dagger = -i$ we also have $\partial_x ^ \dagger=-\partial_x$. It ...
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Is $\bar{\psi} \psi$ its own complex (*)? transpose ($T$)? hermitian conjugate ($\dagger$)?

Related to this earlier A common standard model Lagrangian mistake? Here I am treating Dirac equation of Dirac field as QFT. You may want to consider the quantized version or the classical version. ...
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Why is the spin part of the wavefunction contained in a Hilbert space of dimension $2s+1$ for a particle with spin $s$?

The statement in the title of the question is in my lecture notes. I don't understand the reason for this. Furthermore, I'm having trouble understanding what spin is in the first place. I know that ...
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Does the Dirac Spinor live in the complexification of the Lorentz group?

In this question I learned that when working with quaternionic representations of (the double cover of) our relevant orthogonal group, we cannot avoid working in complex vector spaces. However it is ...
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Charge Conjugation of Dirac equation

In contituation of this question In answers of this question people mentioned charged conjugation and formula below $\bar{\psi}\gamma^\mu\psi=u^2-v^2$ With $u$ for particles and $v$ for antiparticles ...
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How to relate mathematicaly rigorous spinor fields to the ones used in physics?

One way to rigorously define spinor fields on metric manifolds is through the language of associated bundles. Namely, we have a principal bundle $P \overset{\pi}{\longrightarrow} M$ over $\mathrm{Spin}...
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Initial value formulation of Dirac equation for spin $>1$ - modern resources?

At the end of Wald Section 13.2, he mentions that the natural generalization of Dirac equation to curved spacetime does not have a well-posed initial value formulation for $s>1$ , and refers to ...
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Simple time evolution for quantum relativistic particle

I am considering the time evolution of a relativistic particle in 1D, with the time-evolution governed by the equation: \begin{eqnarray} \left(\begin{matrix} \mathrm{i}\partial_{x} && m_{0}\...
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Self or complex Weyl representation in Polchinski

In p.433 of Polchinski String theory volume 2, he said that: In $d$ spacetime dimension, For $d = 2 \mod 4$, each Weyl representation is its own conjugate. ($B \Gamma B^{-1} =-\Gamma $) For $d = 0 \...
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Swapping spin operators when commuting

I'm working on a problem at the moment and its asking me to use ehrenfast's theorum to show that my previous answers were correct, but I'm out by a minus sign. I assumed that if $[S_z, S_x] = ihS_y$, ...
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Analog of the Pauli vector for $SU(4)$

In quantum mechanics and representation theory, it is well known that the Pauli matrices transform as a vector due to the special relationship between $SU(2)$ and $SO(3)$. For example, suppose we have ...
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Is the real spinor representation of the Lorentz group irreducible?

Specifically the $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$ representation. Given that we label representations by the corresponding representations of the complexified Lie group, the direct sum can be ...
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Deriving the Spin Operator Matrices for Spin > 1/2 Using the Direct Product Method

I am trying to derive the spin matrices $S_x$ and $S_y$ in the z-basis for spin $s>1/2$ using the direct product (Kronecker product) method. For simplicity, let's focus on the case $s=1$. I ...
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Time reversal or complex conjugate of Dirac spinor in Peskin & Schroeder QFT book

I have a naive question on the complex conjugate of Dirac spinor in Peskin & Schroeder QFT book (Introduction to quantum field theory), from the part below Eq.(3.137) of the book, $$ u(\tilde{p}, -...
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Scattering amplitude of $e^-\mu^-\to e^-\mu^-$ in terms of matrix elements of $j_\mu$

Using Feynman rules for QED, we can write the Feynman amplitude of a typical electromagnetic scattering process, for example, $$e^-(k)\mu^-(p)\to e^-(k')\mu^-(p'),$$ at the lowest order, in terms of ...
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Is this how spinors transform and is it the square root of a vector?

I have an expression and the transformation rules, and I wonder if this qualifies as a spinor. Can the following expression written with complex Clifford algebra be seen as a spinor? In any case, it ...
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Dirac spinor definition

is it right to say that the Dirac spinor is a mathematical representation of a wave-function that satisfy the Dirac equation? or are there more requirements to it?
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Doubt on dotted indices notation for Weyl spinors

I'm starting Bailin and Love "Supersymmetric gauge field theory and string theory" and trying to get used to dotted indices. Let's consider a Dirac spinor written in terms of left and right ...
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Quantum Field Theory Unitary Transformations

I am currently reading through Itzyskon and Zuber for my quantum field theory class, and I came across this regarding the unitary transformations of the Dirac bispinors in chapter 2. They show that ...
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Obtaining Dirac spectrum on unorientable manifold ($RP^n$) from orientable manifold

The Dirac spectrum for $S^n$ is well known along with its multiplicities. In Appendix D of https://arxiv.org/abs/1510.05663 author computes dirac spectrum of $RP^4$ from that of $S^4$. The argument ...
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Relation between rank-2 antisymmetric tensor and other bilinear covariants

Given a spinor $\psi$, if one defines the bilinear covariants $J=\bar{\psi} \psi$, $J_{5}=i \bar{\psi} \gamma_{5} \psi$, the current $J_{\mu}=i \bar{\psi} \gamma_{\mu} \psi$, the axial current $J_{5 \...
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Error showing the invariance of Dirac equation

Error showing the invariance of Dirac equation Starting from the following equation: $$a’\Psi’(x’)-\partial' \Psi'(x')=0 \tag{1}$$ Where, $a’=a$ is a constant, $\Psi’=A_{4 \times 4}\Psi$, is a spinor ...
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What anticommutes in the two-component formalism for spinors?

I'm studying the 2-component (alias Pauli) formalism for spinors, and I'm confused with the anticommutativity of the objects it uses and describes. For "objects" i mean the spinors $\chi_\...
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Why there are so many spinor components in higher dimensions if the number of degrees of freedom is only 2?

In the book of Freedman & van Proeyen on Supergravity a table (3.2) can be found which shows for dimensions from 2-11 the number of components of Majorana spinors. For instance in 4 dimensions we ...
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Fermions in Euclidean vs Lorentzian Signature

We know that in Lorentzian signature, fermions are representations of \begin{equation} Spin(3,1)\cong SL(2,\mathbb{C})\cong SU(2)\times SU(2)^* \end{equation} where crucially left/right handed ...
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