Questions tagged [spinors]
The spinors tag has no usage guidance.
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Why can the spin operator be written as a product of fermions?
I was studying the Hubbard model, where we define the spin operator $\vec{S} = \frac{1}{2} c^\dagger \vec{\sigma} c$, where the creation and annihilation operators are both vectors of the form $c^\...
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Four-brackets (Hodges, Momentum Twistors)
I use the reference from Andrew Hodges, available at https://arxiv.org/abs/0905.1473. I am having trouble understanding his use of the four-bracket. I refer to equation 6 and equation 9, where he ...
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My Struggle with Fierz Identity
I am following BUSSTEPP Lectures on Supersymmetry to learn SUSY.
The Lagrangian of a interacting Wess-Zumino model in 4D is given by
$$\mathcal{L}=-\frac{1}{2}(\partial_{\mu}S)(\partial^{\mu}S)-\...
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1answer
140 views
Is there any Classical Mechanics system which needs to be described by a spinor?
We need an ordinary number (scalar) to describe a harmonic oscillator, and a vector to describe, for example, a pendulum.
Is there any similarly simple system which we need to describe using a (two-...
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2answers
84 views
Are pseudospinors valid or useful?
We all know that in addition to scalars and vectors, there are pseudoscalars and pseudovectors, which have an additional sign flip under parity. These are useful and necessary when constructing ...
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A Naive Question about SUSY Variation
I am following BUSSTEPP Lectures on Supersymmetry to learn supersymmetry. My simple question is the following.
My Lagrangian for the Wess-Zumino model in $4D$ is
$$\mathcal{L}=-\frac{1}{2}(\...
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1answer
72 views
A few questions about spinors and gamma matrices
I am following BUSSTEPP Lectures on Supersymmetry and trying to show that the Wess-Zumino action is invariant under SUSY transformations. I encountered the following questions about spinors and gamma ...
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1answer
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Transpose of fermion bilinears
TL;DR
When we take the transpose of two Grassmann-valued spinors (fermions), should we add a minus sign because we end up anticommutating the two spinors?
More details.
I'm studying the behavior of ...
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What does it mean to define a spin-structure on a manifold? [closed]
I'm trying to think about what information I need to add to a manifold that it describes a spin structure?
I know you can have spin-structure on a 2d plane, a 2-sphere.
I also know you can define a ...
2
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1answer
71 views
What is difference between fermions and spins?
A spin model i.e. $H_s = \sum_i^{L-1} S_i^x\cdot S_{i+1}^x$ can be written in matrix form as following
$$H_s = \big(S_1^x \otimes S_2^x \otimes I_3^2 \otimes I_4^2\otimes \cdots\otimes I_{L-1}^2\big)...
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Intuitively understanding complex projective space or twistor space
I'm studying momentum twistor variables, which I understand can be seen to be defined projectively from dual complexified Minkowski space to this complex projective space $\mathbb{C}\mathbb{P}^3$. ...
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Are there supersymmetry algebras with higher spinor representations?
The super-Poincare algebra contains supersymmetry generators $Q^I$ which satisfy fermionic anticommutation relations. By the higher-dimensional analogue of the spin-statistics theorem, they must ...
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Question about Spinors and Probability Densities
So I was toying around attempting to simulate some relativistic wave equations for a recreational project. Now I have never studied spinors in dept and the knowledge I have is from reading online (...
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1answer
57 views
Dirac matrices in 1+1 dimensions
Given $\gamma^\mu$ in 1+3 dimensions with signature $(+,-,-,-)$, how can I obtain Dirac matrices in 1+1 dimensions expressed in terms of the $\gamma^\mu$?
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2answers
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Do I need Gamma matrices in Majorana representation in the Lagrangian of a Majorana fermion?
I understand that the Majorana representation of the Gamma matrices are the real representations of the associated Clifford algebra. A Majorana fermion is defined as a fermion that equals to its ...
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1answer
218 views
How does canonical quantization work with Grassmann variables?
Every quantum field theory textbook I've encountered seems to have the same logical oversight, because of the particular order they cover topics.
First, the books introduce the Dirac Lagrangian,
$$\...
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1answer
40 views
Peskin and Schroeder: derivation of Dirac fields commutator
I'm perplexed by the following non numbered equation at page 54 of Peskin & Schroeder, right between $(3.92)$ and $(3.93)$
$$ [\psi_a(x),\overline{\psi}_b(x)]=\int\frac{d^3p}{(2\pi)^3}\frac{1}{...
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4answers
930 views
What precisely is a *classical* spin-1/2 particle?
I was recently having a Twitter conversation with a UC Riverside Prof. John Carlos Baez about Geometric Quantization, and he said (about his work) that
"Right. For example, you can get the ...
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1answer
40 views
How to build an antisymmetric selfdual tensor out of two 4-vectors?
In problem C of section 1.4 of Ramon's Field Theory: A Modern Primer, we are asked to build a field bilinear in $\chi_L$ and $\psi_L$, two left-handed weyl spinors, which transforms as the (1,0) ...
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1answer
59 views
Do we need really need tetrads? OR: Does the No-Go theorem for spinors in curved space only apply to a linear connection?
When I first learned General Relativity, the tetrad formalism was introduced with near simultaneity. I was immediately taught that, to utilize spinors in any way, I had to formulate a local ...
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1answer
48 views
Does the external leg contraction of gluon in QCD carry group generator index?
While I am trying to compute the amplitude for the following Feynman diagram
I realized that the external leg contraction of $g$ should carry group generator index $A$ or $B$, is that right? If so, ...
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Motivating the Unintuitive Properties of Spinors
In the usual treatment of (Dirac) spinors, one usually starts with "factoring" the energy-momentum relation, deducing the properties of the $\gamma$ matrices by requiring the cross terms to cancel, ...
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Would Left and Right Weyl spinor components mix to become massive in an expanding space?
Sorry, this might be a dumb question.
I was just reading a very old paper by Schrodinger where he's talking about different frequency modes mixing in an expanding universe.
Basically he says if the ...
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What is the Newman-Penrose (NP) form of Rarita-Schwinger (RS) equation?
Does anyone knows how to write down the Rarita-Schwinger equation in Newman-Penrose formalism?
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Equivalence between Dirac and Majorana action in CFT
In Mussardo's Statistical field theory Chapter 12, section 12.3 about the conformal field theory of a free fermion field he talks about the complex fermion field (Dirac field)
$$
\Psi(z,\bar{z}) = \...
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1answer
55 views
Majorana fermions
If you write the Majorana spinors as
$$\chi = \begin{pmatrix}\psi_L\\ i\sigma_2\psi_L^* \end{pmatrix} \tag1$$
It satisfies the Dirac equation that leads you to the Majorana equation $$i\bar{\sigma}^\...
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1answer
154 views
Commute covariant derivatives of spinors
Consider a spinor field $\psi$ on a general smooth Lorentzian manifold. Let $\Sigma_{ab}$ be a matrix representation of the Lorentz group, and let Greek/Latin letters represent world/Lorentz indices. ...
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1answer
100 views
Could there be a pseudovector kinetic term for fermions?
Could there be a kinetic term of the form $\bar{\Psi} \gamma_5 \gamma^\mu \partial_\mu \Psi $ in addition to the usual one? Or is this forbidden by a symmetry?
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1answer
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Meaning of the subscripts $L,R$ for the two component Weyl spinors $\phi_{L,R}$
For a Dirac spinor $\psi$, its chiral projections are $\psi_{L,R}$ are defined as $$\psi_{R,L}=\frac{1}{2}(1\mp\gamma^5)\psi.\tag{1}$$ Acting with the chirality operator $\gamma^5$, we find $$\gamma^5\...
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1answer
50 views
Calculating the expectation value of a spin operator in a uniform magnetic field
I'm trying
Usually for these types of questions, I'm used to the field being in a specific direction.
For example, if the field was in the z direction, I could find this solution by checking
|< ...
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1answer
84 views
Lorentz invariance from Dirac spinor
I have a really naive question that I didn't manage to explain to myself. If I consider SUSY theory without R-parity conservation there exist an operator that mediates proton decay. This operator is
$...
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Vertex with spinorial structure and scattering amplitude
Consider the Lagrangian $$\mathcal{L}=\bar\psi_1\left(i\partial\!\!\!/-m_1\right)\psi_1 + \bar\psi_2\left(i\partial\!\!\!/-m_2\right)\psi_2 - g\bar\psi_1\gamma_\mu\psi_1\bar\psi_2\gamma^\mu\psi_2.$$
...
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0answers
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Feynman rule for a vertex with spinorial structure
Consider the interaction term $$\mathcal{L}_{\rm{int}}=-g\bar\psi_1\gamma_\mu\psi_1 \bar\psi_2\gamma^\mu\psi_2,$$ where $\psi_i$ are fermions. I would like to calculate the Feynman rule for the vertex....
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3answers
364 views
Is spinor the sum of scalar, vector, bi-vector, pseudo-vector, and pseudo-scalar?
Is spinor $\psi$ actually the sum of scalar, vector, bi-vector, ..., pseudo-scalar?
Before talking about spinors, we have to differentiate two kinds of spacetime, demonstrated with the example of ...
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2answers
243 views
“Dark Matter” vs. “Dark Spin”
I am rephrasing an answer to another PSE question as an new question here.
Dark matter is invoked to explain the 'observed space-time geometry that does not match with predictions'.
According to the ...
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2answers
138 views
Question about the true nature of the Spinor mathematical object [closed]
My question is kind of a silly one,but,I really would like to know what truly is a Spinor. I will explain what is my concept of "truly".
Throught all the question post, consider finite vector spaces ...
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1answer
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How is the complexification of the Lorentz Lie algebra related to the need for Dirac's 4-component spinor in QFT?
There have been several questions with good answers in physics.stackexchange about the motivation of the complexification of the Lorentz Lie algebra, basically as a mathematically nice way to deal ...
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0answers
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Equations of motion for a Weyl spinor in the context of SUSY
I'm learning supergravity from the textbook of Antoine Van Proeyen (this is from page 114).
Suppose I'm given a Lagrangian $$ \mathcal{L} = - \partial^{\mu} \bar{Z} \partial_{\mu} Z - \bar{\chi} \...
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1answer
129 views
How to distinguish a spinor from a 4-vector?
Lets say we are given a four components object. To be explicit lets consider that these components are $ x^\mu = \mu $ with $\mu\in{0,1,2,3}$, i.e.
$$ x^\mu \sim \left[ \begin{matrix}
0\\
1\\
2\\
3
\...
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0answers
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Killing Spinor Equation in 4 Dimensions
From https://en.wikipedia.org/wiki/Killing_spinor we see that a Killing Spinor $\epsilon$ is defined as a solution of the following equation:
$$\nabla_{X}\epsilon = \lambda X * \epsilon \quad , \quad ...
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1answer
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What is a hypercomplex quantity?
An extract from the book "Quantum Mechanics and the Path Integrals" by Richard P. Feynman and A. R. Hibbs:
We can state the correct law for $P(x)$ mathematically by saying that $P(x)$ is the ...
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1answer
54 views
Weyl spinor's spin direction
I am a bit confused about the spin direction for a Weyl spinor. So as far as I understand, a Weyl spinor represents a massless fermion and it is an eigenstate of the helicity operator. Now say we have ...
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Relation between Levi-Civita tensor and the trace of Lorentz transformations
here is this tricky identity to prove in an appendix of W.B. Supersymmetry and Supergravity that's driving me crazy. Some premises first:
This book use the van der Waerden's convention for spinor ...
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Spinors in Classical Mechanics and Geometry?
I'm trying to deepen my understanding of spinors by looking at applications in simple problems, preferably unrelated to quantum mechanics. For this purpose I'd like to refrain from discussing ...
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2answers
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Is entanglement *not* intrinsic to state, but dependent on division into subsystems? (Susskind QM)
I'm working through Susskind's "Quantum Mechanics" book (TTM series), which I quite like.
Background
In Lecture 7 (Chapter 7), he studies a 2-spin system. A single spin has eigenvectors:
$$|u\...
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2answers
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On the proof $\eta \sigma^{\mu\nu} \chi=-\chi \sigma^{\mu\nu} \eta$ (problem with spinor indices)
I am trying to prove that :
$$\eta \sigma^{\mu\nu} \chi=-\chi \sigma^{\mu\nu} \eta$$
or
$$\eta^\alpha (\sigma^{\mu\nu})_\alpha^{\ \ \beta} \chi_\beta=-\chi^\alpha (\sigma^{\mu\nu})_\alpha^{\ \ \...
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1answer
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Algebraic operation on a bilinear expression of Weyl-Spinors
In the book of Srednicki in equation (35.29) a rather peculiar algebraic operation is carried out on spinors that I am not able to understand. It's
$$ [\psi_\dot{a}^{\dagger} \overline{\sigma}^{\mu\...
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Can the existence of antimatter be inferred from Matrix Mechanics?
It is well known that Antimatter was first predicted by interpreting the matrices that show up in the Dirac Equation as indicating its existence.
Dirac factorizes $E^2=p^2+m^2$ ($c=1,\hbar=1$) into $...
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1answer
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Why two different spinors are Grassmann quantities?
In Rydberg Quantum Field Theory page 441 (this edition, unfortunately page 441 is not in the link) it says
If $\xi$ and $\eta$ are Majorana spinors [...] and since $\xi$ and $\eta$ are Grassmann ...
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Metrics and Spinors
this might be better posed in mathematics but I'll ask here anyway.
So the Lagrangian for the spinor field can be viewed as follows. Let $(M,g,\nabla)$ denote a locally Minkowskian spacetime, Where $\...
