Questions tagged [spinors]
The spinors tag has no usage guidance.
981
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Question on transformation law of spinors and the law $\xi^{\mu '} = D(L)^{\mu '}_{\nu}\xi^{\nu}$ where $D(L)$ is a representation of Lorentz group
In the reference $[1]$, the author presents the tensor quantities via its transformation laws. I'm pretty confortable with Pseudo-Riemannian geometry and tensors. But, when group theory enters the ...
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Lorentz Invariance of kinetic terms for Weyl Spinors
Just to preface things, this exact question has been asked before here, but I don't feel like the answer really clarifies things for me.
The core issue is that we want to construct a 4-vector that we ...
- 145
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How to interpret Pauli spinors?
Recently in my QM course we derived the Pauli equation for an electron in a magnetic field. From what I understand, since we now have a spin-dependent term in our Hamiltonian, the spatial and spin ...
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How is the $SU(2)_L$ conjugation applied?
I'm reading a paper where they introduce the lepton doublets $L$ and "their $SU(2)_L$ conjugations" $\tilde{L}$, which I'm guessing means
$$
\tilde{L} = i\sigma_2L^*.
$$
After $\textit{vev}$,...
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Is the sign of the mass in the Dirac action irrelevant? [duplicate]
In even dimensions all the representations of the gamma matrices are equivalent, in particular $\gamma^\mu$ and $-\gamma^\mu$ are equivalent. Usually the Dirac Lagrangian is
\begin{equation}
\psi^\...
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How does the charge conjugated lepton doublet Lorentz transform?
According to Schwartz, left- and right-handed Weyl spinors transform, infinitesimally, like
$$
\delta\psi_R = \frac{1}{2}(i\theta^j\sigma_j + \beta^j\sigma_j)\psi_R, \quad
\delta\psi_L = \frac{1}{2}(i\...
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What is a fermion doublet exactly?
I am trying to prove that the Weinberg operator is the only dimension-5 operator that can be constructed out of Standard Model fields. To that end, I've tried to write up all the dimension-5 operators ...
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Confusion about Transformation of Spin-Weighted Functions
Short version:
I'm trying to understand spin-weighted spherical functions. Wikipedia defines them as functions of a point $\textbf{x}$ on the sphere and an orthonormal frame $\textbf{a}$, $\textbf{b}$ ...
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How do projective representations act on the QFT vacuum?
Let $U:\mathcal{G}\to \mathcal{U}(\mathcal{H})$ be a unitary projective representation of a symmetry group $\mathcal{G}$ on a Hilbert space $\mathcal{H}$. It satisfies the composition rule:
$$U(g_1)U(...
- 1,203
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Projective representations of the Lorentz group can't occur in QFT! What's wrong with my argument?
In flat-space QFT, consider a spinor operator $\phi_i$, taken to lie at the origin. Given a Lorentz transformation $g$, we have
$$\tag{1} U(g)^\dagger \phi_i U(g) = D_{ij}(g)\phi_j$$
where $D_{ij}$ is ...
- 1,203
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$S$-operator for proper Lorentz transformation
By applying infinitesimal Lorentz transformatios successively (with rotation angle $\omega$ around the $\bf n$ axis) one would get
$$\Psi'(x') = \hat{S}\Psi(x) = e^{-(i/4)\omega\hat{\sigma}_{\mu\nu}(\...
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Bilinear covariants of Dirac field
In the book "Advanced quantum mechanics" by Sakurai there is a section (3.5) about bilinear covariants, however i can't really find a definition of these objects, neither in the book nor ...
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Calculations with tensors give two different results from seemingly equivalent paths:
$\require{cancel}$
I want to calculate some operators in the context of Metric-Affine-Gravity: specifically spinors. I will work in tangent space, where all greek ($\mu,\nu,..$) indices are cast into ...
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In which mathematical space do the spinors act on?
I'm studying QFT and from what I've learnt so far is that a general quantum field $\widehat{\phi}(x)$ can be decomposed (at least for the fermion case) as
$$\widehat{\phi}(x)=∫\frac{d^{3}p}{(2\pi)^{3/...
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Gauge transformation of an adjoint left-handed Weyl spinor in $\rm SU(2)$ fundamental representation
I have a left-handed Weyl spinor field $\Psi_L$ in the fundamental representation of the $\rm SU(2)$ gauge group, which transforms $\Psi_{L,i} \rightarrow \Psi_{L,i} + i\theta^at_{ij}^a\Psi_{L,j}$.
...
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Rotation by 360°, spin-1/2 fermions and quaternions
Rotating a spin-1/2 fermion by 360° multiplies the quantum state by -1.
Representing a continuous 360° rotation as a quaternion is also a multiplication by -1.
Is there a relationship between these ...
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Sanity check for a simple calculation involving Dirac spinors and matricies
I've been doing some research related to the Dirac equation and its solutions. To help make sure I understand what's going on, I've done some simple calculations involving planewave solutions. I'm ...
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What does a dot over a spinor index signify?
My questions should be rather simple. I was trying to get through one of my professor’s papers, and I saw the following notation, first with regards to Dirac and Weyl spinors, but the notation ...
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Parity invariance of Dirac action
The Dirac action is
$$S=\int d^4 x \mathcal{L}(x)$$
where the $\mathcal{L}(x)$ is the Lagrangian density given by
$$\mathcal{L}(x)=\bar{\psi}(x)(i\gamma^\mu\partial_\mu-m)\psi(x).$$
In proving the ...
- 4,011
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Confusion about Dirac mass term
There are a few threads regarding the mass term arising from the Dirac Lagrangian for a fermion field $\psi$ (see here, or here), but I am still confused. I want to show that $\bar\psi_L\psi_L$ and $\...
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What are Dirac spinors and why did relativistic quantum mechanics need them?
I have a good grasp of the Schrödinger equation and the basics of special relativity But the Dirac equation is alien to me. What are Dirac spinors and why did Dirac use them?
user353451
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What is the relation between Dirac spinors and qubit spin? [closed]
Dirac Spinors is a 4 element vector, and a qubit state vector is two element vector. Two spinors are positive(1 and 4) and negative values (3 and 2), being the first value the spin up and the second ...
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Dirac notation and spin confusion [closed]
In my book I have $$\chi = a \chi_+ + b \chi_- = \begin{bmatrix}a \\ b \end{bmatrix} \tag{1}$$
Also, $$| 1/2, 1/2\rangle = \chi_+$$
The way I see that is that $a \chi_+ =|a/2 , a/2\rangle$, but ...
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About the Hilbert space that carries the representation of $\{\psi (x), \bar{\psi (y) }\}=i\delta (x-y) $?
What is this Hilbert space? Is it the complex Hilbert space of wavefunctionals spanned by using the spinor-field configurations as the basis vectors?
I know that the wavefunctional space carries a ...
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Are spinors intrinsically nonlocal?
I would prefer a purely classical answer since I don't think quantum mechanics (quantum field theory etc.) are necessary to answer this question and such answers will likely complicate matters. If you ...
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Square of invariant matrix element involving the Levi-Civita symbol
Consider the invariant matrix element
$$\mathcal M=2ige\frac{\epsilon^{\mu \nu \rho \sigma}k_{1 \mu}\epsilon_\nu(k_1,s_1)\bar{u}(p_2,r_2)\gamma_\sigma u(p_1,r_1)q_\rho}{q^2}, \quad q=k_2-k_1$$
...
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Are all fields on spacetime spinor-valued?
I'm trying to understand the values that fields can take. For fermions, my understanding is that fields on spacetime take values as Dirac Spinors, which are $\mathbb{C}^4$ vectors. The vector space of ...
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What does it mean an electron needs 720 degrees to return to original state? [duplicate]
I have read that the electron spin is represented by a vector of 2 complex numbers. And a frequently asked question is how can it be that an electron must be rotated 720 degrees to return to its ...
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Check of the Majorana condition in Srednicki's book
I wonder how it is possible to reach at the equation (37.18) also called the Majorana condition:
$$\bar{\Psi} = \Psi^T {\cal{C}}\tag{37.18} $$
of Srednicki's book from (37.16), (37.17) and (37.19).
We ...
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Where does the $i$ come from in the left helicity antimuon spinor?
Context: this appears in $e^{+}e^{-} \rightarrow \mu^{+}\mu^{-}$ scattering.
Page 247 of Larkoski particle physics says $$v_L(p) = \sqrt{2E}(e^{-i\frac{\phi}{2}}\cos(\frac{\theta}{2}), e^{i\frac{\phi}{...
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Creating entangled electron pairs using Stern-Gerlach apparatus?
The following is a drawing of the sequential Stern Gerlach experiment.
As we can detect the fraction of the electrons passing through each magnet, I suppose it is possible to detect how many ...
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Dirac spinor field anti-commutation
I am thinking the anti-commutation property of Dirac field! First, note that the equal time anti-commutation relation (from P&S's QFT):
$$\{ \psi_a(\mathbf{x}),\psi_b^{\dagger}(\mathbf{y}) \}=\...
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An example problem to solve using 100 qubits?
Suppose we have in our possession 100 pairs of electrons.
Each electron A1 - A100 is entangled with its respective twin B1 - B100. Each entangled electron pair has been set up to have opposite spins (...
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Do electron spin retain its up/down direction over time?
When high speed electrons are passed through non-homogeneous magnetic field, 50% of the electron will be deflected up, and the rest 50% will be deflected down (Stern Gerlach experiment).
Suppose the ...
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Spinor functional quantization unitarily equivalent and determinant
On P&S's qft page 301 and 302, the book discussed functional quantization of spinor field.
The book define a Grassmann field $\psi(x)$ in terms of any set of orthonormal basis functions:
\begin{...
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Spinor product in QED scattering
Equation 6.36 in Larkoski’s Introduction to Particle Physics says
$v^{\dagger}_{L}\sigma^{\mu}u_R$ =
$E_{cm}(0, -i)(1, \sigma_1, \sigma_2, \sigma_3)\begin{bmatrix}1 & 0\end{bmatrix}$
Which is one ...
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What is meant precisely, when a term in the Lagrangian is "chirally invariant"?
I am reading this paper, where eq. (2):
$$ m_0(\bar{\phi}_{-L}\phi_{+R}-\bar{\phi}_{-R}\phi_{+L}-\bar{\phi}_{+L}\phi_{-R}+\bar{\phi}_{+R}\phi_{-L}) \tag{2} $$
is said to be chirally invariant. Here, ...
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Plane Wave Relations for Dirac Spinors
I am trying to show the following relationships: $\bar{u}_{\pm p\sigma}\gamma^\mu u_{\pm p\sigma'} = 2p^\mu \delta_{\sigma\sigma'}$, $\bar{u}_{\pm p\sigma} u_{\pm p\sigma'} = \pm 2m\delta_{\sigma\...
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How to understand the completeness of the Dirac spinor and why?
I'm searching around to see why $$\sum u^s\bar{u}^s=(\gamma^\mu p_\mu+mc)$$ and $$\sum v^s\bar{v}^s=(\gamma^\mu p_\mu - mc)$$ is called the completeness relation.
Also wondering the same question for ...
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Grassmann numbers and fermionic strings
Is it correct that by introducing Grassmann numbers as new directions of spacetime we can make strings behave like fermions (that is, 1/2-spin objects)?
And if so, is it possible to show how that ...
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Deriving Maxwell equations in Newman-Penrose formalism
I am trying to derive the NP formalism version of Maxwell's, following Chandrasekhar's book and using the xAct package for Mathematica.
First I have the usual Maxwell equation
$$
\nabla_a F^{ab}= J^{...
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How do I show that the fermion creation operator creates a particle with spin 1/2?
Consider the mode expansion of the Dirac spinor:
\begin{equation}
\psi(x) = \int\frac{d^3 k}{(2\pi)^3\omega_k}\sum_\lambda \bigg[b(k, \lambda)u(k, \lambda)\exp(-ikx)+d^\dagger(k, \lambda)v(k, \lambda)\...
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Confusion on inner product between quantum states
I think I have a confusion on some basics of quantum mechanics. To explain my problem I constructed this following simple example.
Let's consider an infinite 1D system made by two sub lattices $A$ and ...
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Verify that $v^s(-p)^{\dagger}\gamma^0u^s(p) = 0$
Given that
$$
\gamma^0 =
\begin{bmatrix}
I & 0\\
0 & -I\\
\end{bmatrix}
$$
Where $I$ is 2 by 2 identity matrix.
and the Dirac spinors
$$
u^s(p) = \sqrt{E_p+m}\begin{bmatrix}
\xi^s \\
\frac{p\...
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Dirac free particle with $x$-momentum
For a free particle with momentum $\mathbf{p}=p\mathbf{x}$, the Dirac Hamiltonian is
\begin{equation}
H=\alpha_xp+\beta m = \begin{pmatrix}
m & 0 & 0 &p\\
0 & m & p & 0\\
0 &...
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Can a Pauli spinor get aligned with the external magnetic field, according the Schrödinger equation?
The Stern-Gerlach experiment shows that spin particles through an inhomogeneous magnetic field are scattered as if their magnetic moment in the direction of the magnetic field could only be $\pm \...
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3-Point Amplitude with Complex Momenta
The following is from Schwartz's Quantum Field Theory and the Standard Model.
Next, consider the MHV amplitude. Again, there are only two possibilities allowed by little-group scaling. Since $\frac{\...
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On the simplicity of the three-particle amplitude in holomorphic configuration
I am reading Clifford Cheung's 2017 TASI Lectures on Scattering Amplitudes. In section 3, "Bootstrapping Amplitudes", the procedure for bootstrapping the three-particle amplitude for ...
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Question about fields and state vectors
$\hbar = 1$ and $c=1$
The question is written on section $2)$
$1)$ Introduction
So, when you write Klein-Gordon equation, $(\square + m^2)\phi = 0 \hspace{2mm}(1)$ , you know exactly which type of ...
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Dirac propagator in Peskin & Schroeder's Book
I read Quantum Field Theory Book written by Peskin & Schroeder, and when a commutator about Dirac field is compile,
he compile a general commutator:
$$ [\psi_a(x),\overline{\psi}_b(x)] $$ having ...
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