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1answer
53 views

Spinor notation in general relativity

I have a somewhat broad/big question, and I know that there are many references for it available out there. However, so far I couldn't find anything that I can really understand, that's why here is my ...
0
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0answers
44 views

Spinor helicity formalism, exact form of the spinors

I am trying to understand how to perform computations with the spinor helicity formalism, I am studying on this review http://arxiv.org/abs/1308.1697. I have stumbled upon a problem though, in pag. ...
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3answers
60 views

How do I operate on a spin state with a sigma operator?

For any arbitrary spin state $|s\rangle$. How do I operate on it with the Pauli spin matrix, $\hat{\sigma_z}$? Does this have something to do with a Bloch sphere?
3
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1answer
56 views

Evolution of Eigenstates when two spin systems are coupled

I would like to describe the following situation: We have two spin systems: Spin 1 ($S_1$) and Spin 1/2 ($S_2$). Now imagine you somehow change their interaction so that you can finetune the ...
2
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1answer
29 views

Two-component formalism and four-component formalism [closed]

When deriving the Dirac equation for spin-1/2 particles, we realize that the wave function must be four-component. In some works, people use two-component wave function for calculation. So, my ...
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0answers
26 views

dagger operator in spinor representation

I just have trouble understanding how hermitian conjugation is acting like this in the following example (dot represents right-handed Weyl field, undot represents left-handed Weyl field). For ...
0
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1answer
52 views

Hermitian Adjoint of Spinor

Say we have a four component spinor $\psi$: $$ \psi=\begin{pmatrix}\psi_L\\\psi_R\end{pmatrix} $$ Is the Hermitian adjoint of this: $$ \psi^\dagger =\begin{pmatrix}\psi_L^\dagger ...
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1answer
41 views

How to derive the form of the invariant spinor inner product?

So we have gamma matrices that satisfy the spacetime algebra relations, $\{\gamma^\mu, \gamma^\nu\} = 2 \eta^{\mu\nu}$. We know that if we set $\sigma^{\mu\nu} = \frac{1}{4}[\gamma^\mu, \gamma^\nu]$ ...
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0answers
27 views

Spinors and asymmetry of wave function

A spinor is a mathematical entity which changes sign if it is rotated by $2\pi$. Is this connected to the asymmetry of the spinor wave function?
3
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1answer
142 views

What is a spinor?

In a youtube video, sir Michael Atiyah mentioned that even after working during the most of his life on spinors, he doesn't know what a spinor is. Now surely that was part of his humorous introduction ...
0
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1answer
65 views

Connection between two Petrov classification schemes

For the Weyl scalars of all spacetimes, at any point, possess one special structure, the so called principal null directions. Consider a general null tetrad $\{ l_a,n_a,m_a,\overline{m}_a \}$, we ...
1
vote
1answer
345 views

How to find the time evolution for two-component spinor? [closed]

I would like to find the time evolution for the given Hamiltonian, the initial state of the system we choose two spinor wavefunction $\psi_{+}(t=0)$ and $\psi_{-}(t=0)$ as given below: The effective ...
2
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1answer
70 views

Expressing the Schrödinger equation in terms of spinors

I appreciate that the Dirac equation can be thought of in terms of spinors, as it directly implies the presence of spin, in addition to initiating the concept of treating fields as operators. From ...
3
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0answers
61 views

How one can write $\bar{\psi}$ in odd dimension?

I know that the Dirac equation in general dimensions has a form of $$ (i\gamma_{\mu} \nabla_\mu - m ) \psi =0 $$ and the action for that is written as $$ S = \int d^d x \bar{\psi} (i\gamma_{\mu} ...
2
votes
1answer
35 views

Why does dimensional reduction of 10,6,2 to 9,5,1 have an equal number of supersymmetry?

In general, the number of supersymmetry is different for each dimension since from dimensional reduction, the number of supersymmetry is usually increased. $i.e$ 4d N=1 $\rightarrow$ 3d N=2 ...
2
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0answers
26 views

Parker-Taylor formula in the $n=4$ simple case

I am trying to do ex. 2.23 of http://arxiv.org/pdf/1308.1697v2.pdf. I have chosen as reference spinors $q_1,q_2 = p_3$ and $q_3,q_4 = p_1$. Therefore if I compute $A^4[1^- 2^- 3^+ 4^+]$ the ...
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0answers
205 views

writing wave function of a spinor having a spin component [closed]

The effective Hamiltonian for photonic graphene is given by \begin{equation} H=\hbar\nu_{F} {\bf \sigma}\cdot\left(q+\frac{\Delta \vec S^{\prime}}{\hbar\nu_{F}}\right), \end{equation} where ...
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0answers
47 views

How many Killing spinors exist on $S^5$?

So, I know that on $S^n$, a spinor of the form $$ \Sigma^\pm = \frac{1 \pm i\gamma^\alpha z_\alpha}{\sqrt{1+z^2}}\Sigma_0$$ where $\Sigma_0$ is a constant spinor, is a Killing spinor on $S^n$ ...
2
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1answer
38 views

Decomposition of the gravitino into helicity $\pm \frac{3}{2}$ and $\pm \frac{1}{2}$ components

I'm reading this book on string theory. When they decompose two dimensional gravitino (formula 7.16) $$ \chi_\alpha = \frac{1}{2}\rho^\beta \rho_\alpha \chi_\beta + \frac{1}{2}\rho_\alpha \rho^\gamma ...
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0answers
25 views

What are the conditions of wave function continuity when solving for Dirac Spinors as done in “Klein paradox” paper by Novoselov?

In the paper "Chiral tunneling and Klein paradox" paper by Katsnelson, Novoselov, and Geim, they use the wave function for Dirac spinors. What are the conditions for continuity of the wave function ...
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1answer
81 views

Is my Summary of a Spinor Bundle Associated with a String Worldsheet Correct?

I've been having difficulty finding a source that lists all the properties of the spinor bundle of a string worldsheet explicitly, so I've had a go at creating my own description. I'd really ...
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0answers
25 views

Euclidean Continuation of spinor

I would like a clear and useful answer or explanation about these following argument or question that I'll place in logical order for my purpose that deal the Euclidean continuation of spinors in 5 ...
1
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0answers
36 views

How can a spinor represent an “epistemic” state?

I have read a lot of stuff on the seemingly endless debate on ontology/epistemology of the quantum state $\psi$. But I always wonder: how can a spinor be considered epistemic when $\psi$ really ...
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0answers
19 views

Explanation for orientation entanglement

I have to write a summary for "orientation-entanglement": the state of an object/subsystem depends in general not only (locally) on its configuration in space, but also (nonlocally) on its topological ...
1
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0answers
49 views

The Covariant Spinor Derivative in the Locally Supersymmetric Generalisation of the Polyakov Action and Potential Mistakes in the Literature

Questions) I recently came upon the thesis The Landscape of Free Fermionic Gauge Models by D. G. Moore and G.B. Cleaver. On pages 20 and 21 they explain that the locally supersymmetric action, ...
1
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1answer
88 views

What is the role of the spacetime algebra?

For Minkowski space $M^4=\mathbb{R}^{1,3}$ the Clifford algebra $Cl_{1,3}$ (Dirac algebra) with $\{\gamma^\mu, \gamma^\nu \}=2 g^{\mu \nu}$ is sometimes called "spacetime algebra". What is its ...
3
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2answers
246 views

An identity of Pauli matrices

I am studying spin recently, and textbook gives some identities of Pauli matrices, one said that for any two unit vectors $\bf m$ and $\bf n$, $[\bf m \cdot \bf{\sigma},\bf {n \cdot \sigma}]= ...
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0answers
33 views

representation of spinors

I am trying to get from the abstract representation of Spinors, as wave functions $|\Psi \rangle$ in the base of tensors products $| S_z \rangle \otimes | x \rangle$ of eigenvectors of the spin ...
5
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2answers
184 views

Is a spinor in some sense connected to space?

Spinors transform under the representation of $SL(2,\mathbb{C})$ which is the double cover of the Lorentz group $SO(1,3)$ - or in the non-relativistic case under $SU(2)$, the double cover of $SO(3)$. ...
3
votes
1answer
62 views

what does Peskin's square root of a matric mean?

Peskin (Intro to QFT) is using the next symbols when discussing dirac fields - $\sqrt{p\sigma}$ with $\sigma = (1,\sigma^1,\sigma^2,\sigma^3)$ (unit & Pauli). For example he represents the dirac ...
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0answers
17 views

Proof of Pauli-Kofink relation

In an article "Effect of the self-induced torsion of the Dirac sources on gravitational singularities" written by Akira Inomata he proves the Pauli-Fierz relation: $$ \bar{\psi}P\gamma_{\mu}\psi ...
3
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2answers
158 views

Why is the $(\frac{1}{2},\frac{1}{2})$ representation of the Lorentz group realized as the vector space of complex $2\times 2$ matrices?

Why can we write an arbitrary object $v_{a \dot{b} }$ our transformations in this basis act on as $$ v_{a \dot{b} } = v_{\nu} \sigma^{ \nu}_{a \dot{b} } = v^0 \begin{pmatrix} 1&0 \\ 0&1 ...
9
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2answers
210 views

If $v_{a \dot{b}}$ transforms like a four-vector, what does $v_{a}^{\dot{b}}$ describe?

The $( \frac{1}{2}, 0)$ representation of the Lorentz group acts on left-chiral spinors $\chi_a$, the $( 0,\frac{1}{2} )$ representation on right-chiral spinors $\chi^{\dot a}$. The $( \frac{1}{2}, ...
4
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1answer
66 views

Variation of the kinetic quark term of the QCD Lagrangian under gauge transformation

A simple kinetic quark term would look like $$\bar{\psi}(\gamma^{\mu}\partial_{\mu} - m){\psi}.$$ Imposing SU(3) symmetry the Dirac spinor transforms like $$\psi(x) \rightarrow \psi'(x) = e^{ig_s ...
1
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2answers
59 views

Transformation of spinors due to Lorentz group

Assume we have a Dirac spinor $\psi(x)$ which satisfies the Dirac equation: $$(i\gamma^{\mu}\partial_{\mu} - m)\psi(x) = 0.$$ If we boost our spacetime coordinates to a new system with a Lorentz ...
2
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1answer
71 views

How does the Lorentz group act on a 4-vector in the spinor-helicity formalism $p_{\alpha\dot{\alpha}}$?

Given a 4-vector $p^\mu$ the Lorentz group acts on it in the vector representation: $$ \tag{1} p^\mu \longrightarrow (J_V[\Lambda])^\mu_{\,\,\nu} p^\nu\equiv \Lambda^\mu_{\,\,\nu} p^\nu. $$ However, I ...
3
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1answer
55 views

In what sense is the chiral decomposition of spinors unique?

We may decompose a spinor field $\psi = \psi_L + \psi_R$ where $\psi_L = \frac12 (1 - \gamma^5) \psi$ and $\psi_R = \frac12 (1 + \gamma^5) \psi$. (I believe this is because the clifford algebra has ...
2
votes
1answer
50 views

Counting d.o.f. and gauge fixing $A_{\mu}$ and $\psi$ in $D$-dimensions

Setup: Let us assume we are in $D$-dimensional Minkowski space-time where $D=d+1$. Consider a free Abelian gauge theory. Then the electromagnetic field will satisfy $$\partial_{\mu}F^{\mu \nu}=0 ...
2
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1answer
114 views

Solutions to Dirac Equation in Weyl Representation

Reading a into QFT I recently came across basically this (Kaku p.94): If $\Psi (x)$ is a solution to the massless Dirac equation in Weyl representation, also $\Phi (x) = \exp(i \Lambda \gamma^5) ...
1
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1answer
39 views

Help with a vector-spinor equation

How can I show that the equation $$\gamma^{abc}\partial_{b}\psi_c=0$$ leads to $$\partial_{b}\psi_{c}-\partial_{c}\psi_{b}=0?$$ I know that $$\gamma^{abc}= \frac{1}{2}\{ \gamma^{a}, \gamma^{bc} \}$$ ...
2
votes
1answer
78 views

Where do the quantum fields encode the spin information?

I know basically the difference between Klein-Gordon and Dirac field is spin. But I am not sure where we need to implement this info. The solutions of both equations are the wave packets which ...
6
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1answer
82 views

Substitution $\partial_\mu \to D_\mu \equiv \partial_\mu + ieA_\mu$ allows the introduction of electromagnetic interactions [duplicate]

I want to show that the substitution $\partial_u \to D_\mu \equiv \partial_\mu + ieA_\mu$, or equivalently $p_\mu \to p_\mu - eA_\mu$ allows the introduction of electromagnetic interactions. Here $e$ ...
3
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0answers
68 views

Substitution $\partial_\mu \to D_\mu \equiv \partial_\mu + ieA_\mu$ allows the introduction of electromagnetic interactions [closed]

I want to show that the substitution $\partial_u \to D_\mu \equiv \partial_\mu + ieA_\mu$, or equivalently $p_\mu \to p_\mu - eA_\mu$ allows the introduction of electromagnetic interactions. Here $e$ ...
2
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2answers
82 views

Higher rank $\gamma$-matrix question

I read that the higher rank $\gamma$ matrices can be written as alternate commutators and anti-commutators. For example, the rank 3 gamma matrix can be written as $$\gamma^{123} = ...
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0answers
65 views

Lorentz transformations of spinors in $SL(2,\mathbb{C})$

I was wondering what the matrix representations of all the coordinate rotations and Lorentz boosts of the $SL(2,\mathbb{C})$ were along with a general method of solving for them. I've been able to do ...
0
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0answers
28 views

Real representations of chiral fields

Why we can´t have real representations of chiral fields, i.e. why does a multiplet of chiral field (Weyl spinors) under a real representaiton of a Lie Group transforms as a "vector". It is easy to see ...
2
votes
1answer
119 views

Treating the spinors as Grassmann numbers or as c-number objects

In the literature on supersymmetry, the following spinor summation convention is often used (eg. Wess & Bagger's book Supersymmetry and Supergravity) $$ \psi\chi = \psi^{\alpha}\chi_{\alpha} = ...
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1answer
67 views

Scalar products in the spinor helicity formalism

In A. Zee's book Quantum Field Theory in a Nutshell (2nd edition), Chapter N.2, page 486, the momentum $p$ is written as a $2\times 2$ matrix: $$ p_{\alpha\dot{\alpha}} = p_{\mu} ...
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0answers
89 views

Killing spinor equation [closed]

The Supersymmetry transformation is: $$\delta \psi_\mu^i=(\partial_\mu +1/4 \gamma^{ab}\omega_{\mu ab})\epsilon^i -1/8\sqrt{2}\kappa \gamma^{ab}F_{ab}\epsilon^{ij} \gamma_\mu \epsilon_j$$ For the ...
4
votes
1answer
196 views

Making sense of the canonical anti-commutation relations for Dirac spinors

When doing scalar QFT one typically imposes the famous 'canonical commutation relations' on the field and canonical momentum: $$[\phi(\vec x),\pi(\vec y)]=i\delta^3 (\vec x-\vec y)$$ at equal times ...