Questions tagged [spinors]

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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
38 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}...
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3answers
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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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1answer
61 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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5answers
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Can we do better than “a spinor is something that transforms like a spinor”?

It's common for students to be introduced to tensors as "things that transform like tensors" - that is, their components must transform in a certain way when we change coordinates. However, we can do ...
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1answer
56 views

Klein-gordon, Dirac equation and the spin of the particles [closed]

This question might seem basic, but how does one conclude that the Klein-Gordon equation describes spin zero particles but Dirac equation describes spin half particles. Thanks. EDIT: Adding more ...
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1answer
35 views

Is it right to think of Parity as a change of basis in Dirac's Lagrangian?

I'm trying to understand CPT symmetries in the Dirac Lagrangian but, so far, I've had more questions than answers. My naive view of CPT transformations is the following (please don't doubt to correct ...
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2answers
78 views

How do we know there are only 16 Dirac bilinears?

We know there are 5 types of bilinears in 4 dimensions, all of them add up to contribute with 16 independent DoF (degrees of freedom). Namely, these bilinears are known as: scalar (1DoF), pseudoscalar(...
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1answer
52 views

Spinor transformation matrix derivation

The spinor in the Dirac equation should transform via a 4x4 matrix $S$ that depends on the specific Lorentz boost/rotation: $\psi '(x')=S(\Lambda )\psi(x)\tag1$ Where S satisfies: $S^{-1}\gamma ^{\...
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0answers
48 views

Observables of Dirac equation

So I learned about the Dirac equation which describes a relativistic free particle with spin $\frac{1}{2}$. I get the mathematics but what i can't find nowhere: What are the observables of this ...
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51 views

Lorentz transformation of the spinor fields

I have been reading the Srednicki's QFT textbook (available online at https://web.physics.ucsb.edu/~mark/qft.html) and in Chapter 34 the left and right-handed spinors are discussed. There is a step in ...
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0answers
35 views

Would this correspond to a spinor space?

Take a 2D manifold and thicken it by adding a 3rd dimension of radius $4\pi$. Now we define fields on this $\psi(x,y,z) = \psi(x,y,z+4\pi)$ Then we say the physics is invariant under the following ...
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1answer
37 views

Determine which operator has the following form when expressed in matrix spinor notation

I am new to spinor notation and I came across the following matrix: What operator has the following form when expressed in matrix spinor notation; $[[W]]=\frac{\hbar \xi}{2} \begin{pmatrix}\hat L_z &...
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0answers
137 views

An $SL(2,C)$ representation and Dirac Spinor

In PCT, spin and statistics, and all that book, the following example is given: Let $S(A)$ be a representation of $SL(2,C)$ given as : $$S(A)=\frac{1}{2}\left(a^{0} \mathbf{1}+\mathbf{a} \cdot \...
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PCT Theorem and PCT, spin and statistics, and all that book

I am reading through PCT, spin and statistics, and all that, and trying to understand the construction on page 15 specifically equation (1-26) and the calculations that follows, what I can't see is ...
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Lorentz Binary Group actions in Spin Statistic theorem: $D[-1] = (-1)^{2j}$ in Novozhilov

In Novozhilov's book "Introduction to Elementary Particle Field Theory" there is a reproduction of Weinberg's S-matrix covariant proof of the Spin Statistics Theorem. I've referenced this in other ...
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34 views

Spinor tensor multiplication rules

I’m wondering how to contract a product of gamma matrices like $$(\gamma^5)_b^{~~d}(\gamma^5\gamma^{\mu})_{ad}$$ Is this just $$(\gamma^5)_b^{~~d}(\gamma^5\gamma^{\mu})_{ad} = (\gamma^5\gamma^5\...
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27 views

Plane Wave Spinor solutions in 2+1 dimensions

I have read that if we use 2+1 dimensional space instead of the standard 3+1d space used in quantum mechanics, we can write the Dirac equation as $$i \hbar \frac{\partial}{\partial t} \psi(r, t)=\...
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1answer
42 views

Is the $U(1)_A$ axial vector current even under charge conjugation?

The axial current of a Dirac spinor is given by $j_A^\mu = \bar{\psi} \gamma^5 \gamma^\mu \psi$. In this book, in the paragraph under equation (2.18) it is stated that the current is even under charge ...
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2answers
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Does the $U(1)$ vector current flip under charge conjugation?

The conserved $U(1)$ current of the Dirac Lagrangian is given by $j^\mu = \bar{\psi} \gamma^\mu \psi$, where $\bar{\psi} = \psi^\dagger \gamma^0$. As this is interpreted as electric current I would ...
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1answer
51 views

Spinor expansion [closed]

does anyone know how you get this expansion when expanding terms of 4-momentum to linear order, type of expansion is it? Many thanks.
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1answer
86 views

Spin state rotations and spinors rotations

I've tried to do the calculations to derive the SU(2) matrices that rotates spinors from the rotation of the spin eigenstates. The following is the procedure that I followed but at the end I didn't ...
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1answer
46 views

Rotating $S=1/2$ spinor from $|+x\rangle$ to $|+y\rangle$

This is a simple question on rotating S=1/2 spinors that I can't seem to get. The general rotation matrix for a spinor by an angle $\theta$ about the $\hat{n}$ axis is given by $$ R(\theta,\hat{n}) = ...
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2answers
68 views

Physical interpretation of Lorentz group non-compactness in the case of Weyl spinors

If we consider the generators of Lorentz group $J$ and $K$, it is possible to indroduce the operators $J^{\pm}=\frac{J\pm iK}{2}$ which shows the $SU(2)\times SU(2)$ structure of the Lorentz group. ...
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0answers
17 views

Relation between $y$ and $Z$ coordinates in momentum twistor formalism

I'm trying to understand the basic concepts in momentum twistor formalism (in the context of scattering amplitudes). I'm following the text by Elvang and Huang which can be accesssed here: https://...
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2answers
50 views

Are there two different spinors for the same spin state?

Let's say $ \begin{bmatrix} 1\\ 0\\ \end{bmatrix} $ and $ \begin{bmatrix} 0\\ 1\\ \end{bmatrix} $ are the eigenvector of $\hat S_z$, is the state $ -1\begin{bmatrix} 1\\ 0\\ \end{bmatrix} +0\begin{...
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0answers
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How to get orientation entanglement in network?

Looking at this intuitive description of spinors. Is there a way to add something like this to a network graph. (e.g. the kinds of networks you get in LQG) or any kind of graph? It looks like ...
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25 views

What is the direction of spin in the Fourier transformed space for the following situation?

I have an expression, like, $$\frac{1}{2}\mathbf{\hat{q}}\cdot\xi^{r\dagger}\boldsymbol{\sigma}\xi^s,$$ where, $\mathbf{\hat{q}}$ is the momentum unit vector and $\xi^s$ is a Dirac spinor. The above ...
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1answer
61 views

Spinor method in massless limit

I have this problem where I'm asked to derive an explicit solution for the Dirac equation of massless fermion $p_\mu \gamma^\mu u(p)=0$. I'm instructed to do so by writing $p_\mu \gamma^\mu$ in the ...
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1answer
34 views

What's up with $\mathrm{U}(1)$ regarding the spin homomorphism?

Let $\mathcal H(2)$ be the space of hermitian matrices of size $2\times 2$, and let $\sigma:\mathbb R^{4}\rightarrow\mathcal H(2)$, $$ \sigma(x)=x^\mu\sigma_\mu=\left(\begin{matrix} x^0+x^3 & x^1-...
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Feynman rules of electroweak interactions applied to the muon decay

We know that the muon can decay as follows: $$\mu^{-}(p){\longrightarrow}e^-(p') + \bar{\nu}_e(k')+\nu_{\mu}(k)$$ I want to write down the invariant matrix element for this process at tree level, ...
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25 views

Amplitude of an operator in BSM physics

I am currently working on effective field theory within the standard model (SMEFT). In this formalism, one introduces higher dimensional operators to the standard model Lagrangian, typically operators ...
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37 views

Hamiltonian for the spin-orbit coupling in spinor-notation

After some research I can't find a good source which explains how to write the Hamiltonian of a spin-orbit coupling in spinor notation. The following notation is commonly used $\hat H =\xi \hat L.\...
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30 views

Spinful time reversal and SOC

I'm having some trouble with understanding the distinction between spinful and spinless time-reversal in a real materials context. 1) Does spinful time-reversal (e.g. $\mathcal{T}^2 = -1$) imply spin-...
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1answer
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Internal flavor symmetry of the $N$ left-handed complex Weyl spinors v.s. $N$ real Majorana spinors: ${\rm U}(N)$ vs. ${\rm O}(2N)$ or ${\rm O}(N)$

Consider 4d spacetime, it seems that for massless particles, we can easily change the left-handed complex Weyl spinor basis (2 component in complex $\mathbb{C}$ for Euclidean spacetime Spin(4)) to ...
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Spin non-conservation in Coulomb interaction

In non-relativistic limit, in the lowest order of perturbation, QFT reproduces the classical Coulomb potential. A nice result is that in coulomb interaction the spin of the particles remain conserved ...
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35 views

Manifestation of the spin structure in classical particle physics

Background : Quantum mechanics is formulated with Hilbert spaces, of which the rays (possibly satisfying superselection rules) correspond to physical states. The space of (pure) states is thus a ...
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1answer
62 views

Dirac sea interpretation VS the Feynman-Stueckelberg interpretation for antiparticles

I am trying to understand the difference between the Dirac sea interpretation and the Feynman–Stueckelberg interpretation of the negative energy solutions of the Dirac equation. To do so, I would like ...
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76 views

Is this an alternative Dirac Equation in curved space?

The usual covariant derivative for the Dirac equation in curved space is: $$D_\mu \psi = (\partial_\mu - {i \over 4} {\omega_\mu}^{ab} \sigma_{ab}) \psi$$ However, I think I found another ...
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47 views

Hermitean conjugate of spinor product

Consider spinors $\psi_\alpha$ in the $(0,1/2)$ rep and $\bar{\psi}_{\dot{\alpha}}$ in the $(1/2,0)$ rep. I'm using notation from the following notes. All we will need is the totally antisymmetric 2x2 ...
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37 views

Colour-ordering formula of QCD amplitudes (tree-level)

I have been studying colour-ordered amplitudes and spinor helicity formalism for a while. It is now apparent to me that I do not fully understand the 'master' formula which allows us to relate the ...
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51 views

Covariance of Dirac Equation [closed]

In Bjorken and Drell "Relativistic Quantum Mechanics", problem 3 ch.2: Given a free-particle spinor $u(p)$, construct $u(p+q)$ for $q \rightarrow 0$, with $p\cdot q \rightarrow 0$, in terms of $u(p)$...
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Supersymmetry and Elko spinors

Now, very popular subject is Elko spinors. Is it possible to construct supersymmetric theory with Elko spinors as parameters of susy transformations?
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Kinematics of Spin

By generalizing $n$ component wave function in three dimensions, Under an infinitesimal rotation around z-axis,the wave function is transformed as follows: $$\left[ \begin{array}[c] \psi \psi'_1 \\ \...
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Invariance of superstring action under super-Weyl transformation

Keeping with the notation on page 178 of the text "Basic concepts of string theory", in superstring theory, consider the action \begin{align} S=\frac{1}{8 \pi} \int d^2\sigma e \left(\frac{2}{\alpha} ...
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Fierz Identity problem in P&S

I have a few questions about the Fierz identity. First of all in general form it has terms as $(\bar u_1\Gamma^Au_2)(\bar u_3\Gamma^Bu_4)$ which is a product of Dirac field bilinears. The question is ...
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2answers
114 views

Fermion operators: why $\rm SU(2)$ symmetry and not $\rm U(2)$ symmetry?

Let us consider operators $c_{\uparrow}$ and $c_{\downarrow}$ which destroy a fermion with spin up and a fermion with spin down, respectively. These operators can be found, for example, in the Hubbard ...
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1answer
67 views

Unpolarized $e^+e^-\rightarrow\mu^+\mu^-$ scattering

On Schwartz’s QFT book, page 232, it’s calculating the cross section for $e^+e^-\rightarrow\mu^+\mu^-$ scattering, assuming spin and polarization are not measured, and it writes Let us also assume ...
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1answer
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What is the complex conjugate of two fermionic fields coupled? $(\bar{\psi} \chi)^{\ast} =$ ____?

Suppose $\psi$ and $\chi$ are fermionic fields, and suppose I want to calculate the hermitian conjugate of the operator $\bar{\psi} \chi$ $$ h.c.\ \mathrm{of}\ \bar{\psi} \chi = (\bar{\psi} \chi)^{\...
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1answer
60 views

Spinor and vector representation matrices commutation relation

To show Lorentz invariance of Dirac equation P&S section 3.2 swap $\Lambda$ and $S(\Lambda)$ as both matrices commute. But why is it true? For example taking $${\cal J}^{01}=\left(\begin{matrix} ...

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