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5
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0answers
63 views

$(\frac{1}{2},\frac{1}{2})$ and $(\frac{1}{2},0)\bigoplus (0,\frac{1}{2})$ [duplicate]

I am confused about the notation. What's the differences between $(\frac{1}{2},\frac{1}{2})$ and $(\frac{1}{2},0)\bigoplus (0,\frac{1}{2})$, or maybe $(\frac{1}{2},0)\bigoplus (\frac{1}{2},0)$ ? (...
1
vote
0answers
45 views

How can we use Majorana spinors for charged fermions in MSSM?

According to "Supersymmetry in Particle Physics" by Ian Aitchison (see e.g. p62 of arXiv), in the Minimal Supersymmetric Standard Model (MSSM) we can use Majorana language to build supermultiplets: ...
0
votes
2answers
28 views

Which particles have helicity?

I know electrons have helicity but can particles which are not fundamental i.e.protons have helicity?
0
votes
2answers
48 views

Pseudoscalar current of Majorana fields

Consider a Majorana spinor $$ \Phi=\left(\begin{array}{c}\phi\\\phi^\dagger\end{array}\right) $$ and an pseudoscalar current $\bar\Phi\gamma^5\Phi$. This term is invariant under hermitian conjugation:...
2
votes
1answer
35 views

Helicity amplitudes and muon/electron currents

I have the following helicity spinor: $$ u_R=\sqrt{E} \begin{pmatrix} c \\ se^{i\phi} \\ c \\ se^{i\phi} \end{pmatrix} $$ We take $s=\sin\theta/2$ and $c=\cos\theta/2$. Also, $E$ and $\phi$ are ...
0
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0answers
52 views

Cyclicity of trace with fermionic arguments

I think this is a non-question, but it has me considerably worried. Consider the piece of a Lagrangian density given by, $$\mathcal{L} = \epsilon_{ij}\mbox{Tr}\left(\chi^{i,\alpha}\left[\chi^{j}_\...
2
votes
0answers
69 views

Tensor product of spin states

I just wanted to check that I carried out this problem correctly. I got the correct answer, but I'm not sure if what I did to get it is completely correct. This is from the second part of problem 3.4 ...
1
vote
3answers
95 views

Why is this proof that all $\overline{\psi}\psi\overline{\psi}\psi$ interactions are trivial incorrect?

This is a homework question for my quantum field theory class. I haven't been able to figure out the answer, and neither has anybody I asked. The homework was due two days ago. Consider a spinor ...
2
votes
2answers
67 views

Local Lorentz transformations

If $\gamma^m$ denotes a tangent space gamma matrix, and $\gamma^\mu$ denotes a curved space gamma matrix, then they are related by $$\gamma^\mu(x) = \gamma^m e_{m}^{\mu}(x)$$ where $e_{m}^{\mu}(x)$ ...
15
votes
4answers
519 views

Are fermions intrinsically non-local?

Background: When one studies quantum mechanics of more than one particle, one learns that all fundamental particles can be classified as either bosonic or fermionic. Fermions have a spinor structure, ...
1
vote
0answers
18 views

Conformal compactification and the use of spinors (Twistor theory)

I was reading the book from Huggett and Tod "An introduction to twistor theory" and as the book evolves they reach to the necessity to "found" a Lie derivative of a spinor respect to a conformal ...
0
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0answers
38 views

Twistor theory. Huggett and Tod (problems with one equation)

I'm reading about twistors from the book of Huggett and Tod: $\textit{ An introduction to twistor theory}$. I'm trying to understand everything and reproduce every equation that comes here. So, ...
0
votes
0answers
48 views

A Lie derivative $\mathcal{L}_{\alpha^A}$ with respect to a spinor $\alpha^A$?

Suppose we work with Minkowski flat space $M$ (just to make things easy). If $\textbf X$ is a Killing vector field it is possible to define the Lie derivative of an spinor $\alpha^A$ with respect to $\...
1
vote
2answers
79 views

Complex conjugation of Weyl Spinors

Let $\chi$ be a left-handed Weyl spinor transforming as $$\delta\chi=\frac{1}{2}\omega_{\mu\nu}\sigma^{\mu\nu}\chi.$$ In my lecture notes it is explicitly stated that complex conjugation interchanges ...
3
votes
1answer
60 views

Is there a heuristic explanation why the possible spin values in another direction are not equiprobable for a spin-1 particle?

Let's deal with spin $1/2$ systems and fix a value (e.g. $+1/2$) in a given direction (e.g. $z$). Following a measurement along an orthogonal direction (e.g. $x$), we obtain 50% probability for $+1/2$ ...
5
votes
1answer
246 views

Scattering Amplitudes from Feynman Diagrams (Spinor Helicity Formalism)

$\require{cancel}$ I am trying to do an exercise from Scattering Amplitudes By Elvang (Exercise 2.9) which states: Show that $A_5(f^-\bar{f}^-\phi\phi\phi) = g^3\frac{[12][34]^2}{[13][14][23][24]} ...
3
votes
2answers
142 views

Spinor field normalisation from poles in the propagator

In the theory of free scalar bosons (KG field) it is a basic result that the propagator $\Delta(p)$ has poles at $p^2=m^2$, with residue $1$ (or any other constant, depending on conventions). Thinking ...
3
votes
1answer
76 views

Neutrinos and global $U(1)$ symmetry of Weyl fields

My book on QFT says that neutrinos are well described by left-handed Weyl spinor. The classical Lorentz-invariant Lagrangian density for that field is: $$ \mathcal{L} = i\psi^{\dagger}_L\overline\...
0
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0answers
40 views

Simple question about gamma five and re-writing the Dirac lagrangian

I'm working a problem (Zee, p. 100) asking me to rewrite the Dirac lagrangian in terms of the left and right projections, and along the way I run into: $$\overline{\psi} i \gamma_µ \partial^µ \psi - \...
0
votes
0answers
27 views

Index Placement for Spinors in Relativity

This may ultimately be a silly question, but a pedantic mind like mine gets tied into knots over differing notation. (Disclaimer: I'm a mathematician.) Let $\mathbb{W}$ be a complex two-dimensional ...
0
votes
1answer
95 views

Derivation of Dirac equation in curved spacetime

In all the Literature I have read, the covariant Dirac equation in curved spacetime is given as \begin{equation} \left(i\hbar\gamma^{\mu}(x)\left[\frac{\partial}{{\partial}x^{\mu}}-{\Gamma}_{\mu}(x)...
0
votes
0answers
79 views

Four-momentum and Dirac equation in curved spacetime

Norm of four momentum in Minkowski spacetime is proportional to the square of rest mass as \begin{equation}|P|^2= P^\alpha \eta_{\alpha\beta}P^\beta= (E/c)^2 - p^2 = (mc)^2 \end{equation} While in ...
0
votes
0answers
112 views

Spin 1/2 wavefunction transformation under inversion and mirror symmetry

I'm considering group-theory applications to condensed matter physics now. In particular I work with the following paper: http://journals.aps.org/pr/pdf/10.1103/PhysRev.100.580 and try to understand ...
0
votes
1answer
82 views

Pauli matrices identity with no repeating indices

I was just wondering if there is a proof of, or an example utilizing the following relation: $\sigma^i_{\alpha\beta}\sigma^j_{\gamma\delta}+\sigma^j_{\alpha\beta}\sigma^i_{\gamma\delta}+\delta^{ij}(\...
4
votes
1answer
249 views

Is the Dirac equation equivalent to the Klein-Gordon equation for its left handed component?

The Dirac equation $$(i\gamma^a\partial_a - m)\psi=0\tag{0}$$ is given by a first order operator acting on a Dirac spinor, which is the direct sum of a left handed spinor and a right handed spinor. ...
-2
votes
3answers
606 views

Why do electrons and positrons exhibit opposite helical motion in a magnetic field?

When you throw an electron through a solenoid, it moves helically around the field lines, as per this schoolphysics illustration: © Keith Gibbs 2013 Then if we were to throw a positron through the ...
7
votes
1answer
146 views

Why complexify in order to construct Dirac representation?

Suppose we have a theory is covariant under the Spin group Spin(2n-1; 1). We consider the real vector space $V = R^{2n-1,1}$, which naturally comes with a Lorentzian inner product. On this vector ...
2
votes
1answer
167 views

How do simple two-component Fierz identities follow from a property of the Pauli matrices?

On page 51 Peskin and Schroeder are beginning to derive basic Fierz interchange relations using two-component right-handed spinors. They start by stating the trivial (but tedious) Pauli sigma identity ...
0
votes
0answers
21 views

Where do certain relations relevant to helicity eigenstates come from in Peskin and Schroeder?

On pages 46-47 of Peskin and Schroeder, equations 3.52 and 3.53 are introduced when we have picked specific values of the numerical two-component spinor $\xi$. We choose a basis of $\sigma^3$-...
2
votes
2answers
81 views

What does it mean to differentiate a spinor-valued field?

Peskin and Schroeder, equation 3.28, states that the Klein-Gordon equation $$(\partial^2+m^2)\psi=0 \tag{3.28}$$ is a valid choice of equation for a Dirac spinor field. Their explanation makes sense (...
0
votes
1answer
76 views

Spin angular momentum

Spin angular momentum is defined as $\Sigma^{i}= 1/2 e^{ijk} \sigma_{jk}$ . Thus, I can write $\Sigma^{1}$ as $[\gamma_{2},\gamma_{3}]$ . I want some insights on this definition. Also, Can anybody ...
0
votes
0answers
37 views

Clarification of Type D space-time

I've asked this question on MathOverflow but received no feedback, so I thought I'd try better luck here. I've read the following formulation of the Goldberg-Sachs theorem in Chandrasekhar's ...
1
vote
1answer
61 views

Necessity, significance of Spinors

This is an area I am researching at my own pace, general rotations in 3D. I've known about the plate trick for a while as well, and have a very rough understanding of the concept of simple-...
4
votes
1answer
125 views

Hermiticity of Dirac operator in curved spacetime

The Dirac Lagrangian in curved spacetime is usually given by \begin{equation} \mathcal{L} = i\bar{\Psi}\gamma^a e^{\mu}_a(\partial_\mu + \frac{1}{4}\omega_{\mu bc}\gamma^b\gamma^c)\Psi \end{equation} ...
5
votes
1answer
393 views

Antiparticles, charge conjugation and chirality

(Why/how) are antiparticles and charge-conjugates different things? I am trying to understand the effect of discrete symmetries on spinor fields (neutrinos in particular). In the article, Dirac, ...
5
votes
1answer
66 views

The chirality of (2+1)D Dirac equation

Are there any definitions about the chirality of (2+1)D Dirac equation? For the (3+1)D Dirac equation, the Dirac field can be written as the sum of left- and right-hand Weyl field. Can this be reduced ...
0
votes
0answers
55 views

Future causality of Dirac charge current spinor

I am trying to solve following problem: The Dirac equation reads \begin{equation} \nabla^{AA'} \psi_{A} = \mu \chi^{A'}, \quad \nabla_{AA'} \chi^{A'} = -\mu \psi_{A} \end{equation} where $ \mu ...
1
vote
1answer
121 views

Canonical spinors from gauge transformations

In this 2006 paper, http://arxiv.org/abs/hep-th/0610128, there is the concept of gauge transformation and how was it employed that I do not fully understand. Note, what will be talked about below is ...
0
votes
1answer
65 views

Do the eigenstates of the Pauli operators correspond to the six directions of the 3D world?

I understand that the six eigenstates of the three Pauli operators $X, Y, Z$ correspond to the six poles of the Bloch sphere. By fixing an orthonormal basis of our physical word, does "measuring Pauli ...
1
vote
1answer
95 views

Covariant derivative commutator on spinors [closed]

What is this object $[\nabla_{\mu},\nabla_{\nu}]\epsilon$ in terms of curvature tensor $R_{\mu\nu}$? Where $\nabla_{\mu}$ is the covariant derivative on a four sphere and $\epsilon$ is spinor. PS: I ...
1
vote
1answer
116 views

Regarding the Weyl spinor and its transformation properties

I am trying to prove the Lorentz invariance of the (left-handed) Weyl Lagrangian: $$\mathcal L=i\psi^\dagger\bar\sigma^\mu\partial_\mu\psi$$ A Lorentz transformation is realized as $\psi\to M\psi$, ...
3
votes
1answer
500 views

How can it be derived that particles described by the Dirac equation must have spin 1/2?

I am reading some lecture notes that unfortunately don't seem to be available online, but that are quite close in spirit in their treatment of the Dirac equation to Sakurai's "Advanced Quantum ...
0
votes
1answer
76 views

Question about Majorana spinor's property

I am reading the BBS, Exercise 5.1 This exercise is nothing but showing that two Majorana spinors $\Theta_1$ and $\Theta_2$ \begin{align} \bar{\Theta}_1 \Gamma_{\mu} \Theta_2 = -\bar{\Theta}_2 \...
8
votes
2answers
627 views

Lorentz transformation of Gamma matrices $\gamma^{\mu}$

From my understanding, gamma matrices transforms under Lorentz transformation $\Lambda$ as \begin{equation} \gamma^{\mu} \rightarrow S[\Lambda]\gamma^{\mu}S[\Lambda]^{-1} = \Lambda^{\mu}_{\nu}\gamma^{\...
3
votes
1answer
475 views

Generalization of De Rham cohomology for spinor fields

I am interested in possible generalizations of The De Rham cohomology for spinor fields. I am also interested in applications to physics such as in the construction of topological charges I can see ...
6
votes
1answer
237 views

Invariance of supersymmetric Yang-Mills theory under supersymmetry

I was following Brink, Scherk and Schwartz, "Supersymmetric Yang-Mills theories". The variation of the Lagrangian w.r.t a supersymmetry transformation can be reduced to $$ \delta L = -igf_{a b c} \bar{...
1
vote
1answer
282 views

Do particles have spin because there exist spinor representations for the Lorentz group?

I am reading Peskin and Schroeder's An introduction to field theory. They first describe the spinor representation of the Lorentz group, and then they mention the fact that different particles have ...
5
votes
1answer
147 views

Counting degrees of freedom in spinor-helicity formalism

Just a couple of quick questions about the spinor-helicity formalism. We start with $p^\mu$ and $\epsilon^\mu$, so we have eight degrees of freedom. Then we have that $p^\mu p_\mu = 0$ and that $\...
3
votes
1answer
106 views

Why must the supersymmetry generators be spinors?

I have read in a few places (for example, at page 5 here) that the supersymmetry generators must be spinors. Quoting the reference mentioned The generator of the symmetry must relate two types of ...
1
vote
0answers
66 views

Some questions about Weyl spinor algebra

I am following a review on supersymmetry and I am having trouble with Weyl spinor algebra. These are equations (4.2.1) $$Q_{\alpha}=i\frac{\partial}{\partial\theta^{\alpha}}-(\sigma^{\mu}\theta^{\...