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Massless neutrinos and Chirality

The massless neutrinos can be represented by two component Weyl spinors. Then how does one say that it is an eigenstate of the chirality operator $\gamma^5$, which is a $4\times 4$ matrix and can act ...
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1answer
234 views

Can a spinor be defined as any quantity which transforms linearly under Lorentz transformations?

Recently I’ve come across a few papers from China (e.g. Xiang-Yao Wu et al., arXiv:1212.4028v1 14 Dec 2012) that make the following statement: ...any quantity which transforms linearly under ...
2
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1answer
267 views

Parity transformation for spinors (pinors) in odd spacetime dimensions

What is the transformation law for spinors (pinors) under parity in an odd number of spacetime dimensions? I know how to derive the transformation properties of spinors (pinors) under parity in an ...
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0answers
88 views

What is the point of path integral for boson and fermion?

I am a beginner to study QFT and confused about path integral for boson or fermion. I have read about the path integral for single particle, and finished some problems. But I cannot understand the ...
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0answers
58 views

Why does the object $\epsilon_L Q_L + \epsilon_R Q_R$ correspond to a 16-component conserved supercharge when we have a Dp-brane?

I understand that when a 10-dimensional superstring theory has a Dp-brane (say, extending in the $x_0, ... , x_p$ directions) we have the total conserved supercharge given by: \begin{equation} ...
3
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1answer
227 views

Components of the Weyl spinor field

In the Weyl basis we can separate the spinor field into 2 components: the right-chiral spinor and the left-chiral spinor. Each of these fields has again 2 components which are coupled. What is the ...
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3answers
923 views

Dirac spinor and Weyl spinor

How can it be shown that the Dirac spinor is the direct sum of a right handed Weyl spinor and a left handed Weyl spinor? EDIT:- Let $\psi_L$ and $\psi_R$ be 2 component left-handed and right-handed ...
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1answer
95 views

Spinor inner products

The spinor inner product in particle physics is given by $\overline{\psi} \psi = \psi^{\dagger} \gamma_0 \psi $, where I take the convention that the zeroth gamma matrix is hermitian while the rest ...
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1answer
267 views

Interpretation of rank 2 spinors

While inspecting the $(\frac{1}{2},\frac{1}{2})$ representation of the Lorentz group and defining a right-handed spinor with upper dotted index and a left-handed spinor with lower undotted index and ...
2
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2answers
173 views

Recovering 4-vector Lorentz transformation from spinor formalism

I'm trying to recover the 4-vector transformation laws using spinors. I have defined $$v^{\dot{a}b} = v^{\nu} \sigma_{\nu}^{\dot{a}b}$$ as usual, with $\sigma_0=1$. Now with the rules for dotted ...
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1answer
65 views

Compatibility conditions of spinors and Riemannian Metrics

I came across an interesting article by Montesinos (J. Geom. Phys. 2 (1985), no. 2, 145–153.). In it, he finds that spin structures (as lifts of $SO(4)$) are not compatible with all Riemannian metrics ...
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2answers
202 views

Wavefunction of an electron

Electron is a spin $\frac{1}{2}$ particle, so needs 2-component wave function but the Dirac theory of electron is based on 4-component wave function, are all components of it independent or only two ...
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0answers
95 views

Two spinor tensors and Maxwell's equations

Let's have two symmetric (by the indices) spinor tensors $F_{ab}, F_{\dot {a}\dot {b}}$ and conditions $$ F_{ab}, \partial^{\dot {a} a}F_{ab} = 0, \quad F_{\dot {a}\dot {b}}, \partial^{\dot ...
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0answers
84 views

Questions on the elementary excitations in the resonating-valence-bond(RVB) states?

It is known that the RVB states can support spin-charge separations and its elementary excitations are spinons and holons. But it seems that there are some different possibilities for the nature of ...
2
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1answer
265 views

Non-linear Dirac equation in Einstein Cartan theory

Can someone explain this Wikipedia article, specifically the section on Einstein-Cartan theory? I have no idea how the equation ...
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1answer
80 views

Why normalizing spin state is different? Why no need dxdydz?

An electron in a spin state: $$X=A\quad\begin{pmatrix} 2i\\2\end{pmatrix}\quad$$ In order to get A, I have to normalise it, my question is how should write? Since I have to normalise why no need ...
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1answer
312 views

Spinors and Probabilities of Electron-Positron Pair

Question: An electron and positron are moving in opposite directions, and are in the spin singlet state. Two Stern-Gerlach machines are orientated in some ...
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0answers
161 views

Parity Transformation Behavior of (Weyl-)spinors

Suppose $u^\pm (p)$ are Weyl-spinors with helicity $h = \pm$, i.e. solutions of the Dirac equation $p_\mu \gamma^\mu u(p) = 0$. How do these spinors behave under parity transformation? In ...
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234 views

Unitary Lorentz transformation on quantized Dirac spinor

I am stuck again on page 59 of Peskin and Schroeder. In particular, I do not know how they get equation (3.110). Let me first give some background in the way that I understand it (but I might be ...
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2answers
512 views

Charge-conjugation of Weyl spinors

I am having trouble reconciling two facts I am aware of: the fact that the charge conjugate of a spinor tranforms in the same representation as the original spinor, and the fact that (in certain, ...
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1answer
283 views

Lorentz transformation of the Spinor Field

I'm reading chapter 3 of Peskin and Schroeder and am stuck on page 43 of P&S. They have defined the Lorentz generators in the spinor representation as: \begin{equation} S^{\mu \nu} = ...
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111 views

Infinitesimal transformations and Poisson bracket for Dirac spinors

I apologize for the cumbersome calculations. Let's have $\Psi$, $i\Psi^{\dagger}$, which are canonical coordinate and impulse in space of solutions of Dirac equation. It can be showed that they have ...
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2answers
476 views

What do the four components of Dirac Spinors represent in the Standard Model?

I've been trying to get my head around the formalisms used in the Standard Model. From what i've gathered Dirac Spinors are 4 component objects designed to be operated on by Lorentz Transformations ...
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0answers
84 views

Direct sum of the spinors and EM field tensor

EM field tensor refer to the direct sum of $(1, 0), (0, 1)$ spinor representation of the Lorentz group. How to show it? Each of these spinor representations corresponds to the symmetrical spinor ...
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1answer
203 views

Spinor formalism in QFT

We can describe fields by two formalisms: vector and spinor. This is the result of possibility of representation of the Lorentz's group irreducible rep as straight cross product of two $SU(2)$ or two ...
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1answer
166 views

Spinor irreducible reps of the Lorentz group and their algebra

Antisymmetric tensor of rank two can be connected with spinor formalism by the formula $$ M_{\mu \nu} = \frac{1}{2}(\sigma_{\mu \nu})^{\alpha \beta}h_{(\alpha \beta )} - \frac{1}{2}(\sigma_{\mu ...
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1answer
105 views

Pauli matrices identities

There are some definitions and properties for Pauli matrices and their combinations: $$ \varepsilon^{\alpha \beta } = \varepsilon^{\dot {\alpha} \dot {\beta} } = \begin{pmatrix} 0 & 1 \\ -1 & ...
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1answer
104 views

Spinor indices and antisymmetric tensor

Excuse me for long prehistory. Maybe it can be useful for someone. I was little confused with spinor indices when getting an expression relating the spinor and antisymmetric tensors. An antisymmetric ...
3
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1answer
181 views

Connection between particles and fields and spinor representation of the Poincare group

Let's have a definition of massive particle as an irreucible representation of the Poincare group. Then, let's have a spinor field $\psi_{\alpha \alpha_{1}...\alpha_{n - 1}\dot {\beta} \dot ...
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2answers
491 views

Introduction to spinors in physics, and their relation to representations

First, I shall say that I am familiar with the intuitive idea that a spinor is like a vector (or tensor) that only transforms "up to a sign" when acted on by the rotation group. I have even rotated a ...
2
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1answer
230 views

Can one prove the full spin-statistics theorem from the spin 0, 1/2 and 1 cases?

Using second quantization for scalar field, spinor field and vector fields, we can get commutation and anticommutation relations for the birth and destruction operators of the fields, which leads us ...
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0answers
50 views

Can we build spinorial eigenstates of Time reversal symmetry?

In the SM, and general theories with spinors, we can build the Parity left/right eigenspinors. Indeed, there are also ELKO fields, eigenstates of Charge operator (non-standard Wigner classes). Can we ...
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2answers
142 views

BRST transformation of adjoint spinor

in Yang-Mills-Theory with matter fields a dirac spinor $\psi$ transforms under BRST as $$\psi \to \delta_\Omega\psi=i\eta\psi $$ with $\eta$ being a ghost field. If I want to get the transformation of ...
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1answer
155 views

Number operator and Dirac field (with anticommutation relations)

Before using anticommutation relatives the energy, momentum, charge and number operators of the Dirac field have following expressions: $$ \hat {H} = \int \epsilon_{\mathbf p}\left( \hat ...
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1answer
207 views

Why do we use spinors for describing fermions?

I.e., what properties of the spinors gives us a reason for using them for describing of wavefunctions of fermions?
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1answer
143 views

Helicity Representation of Massive Spinor

For massless spinors case we can decompose momentum into Weyl sub-parts as $$p = \lambda_{a}\tilde \lambda_{\dot a}.$$ But for the case of massive fermions can I do something like this? Decompose ...
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3answers
1k views

Energy Spectrum of pair of spin-1/2 particles with general Hamiltonian

I found this problem, and so far I am stumped. I was wondering if anyone wanted to solve it with me, or help me calculate eigenvectors, or just give insight on my questions. Consider a system of ...
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1answer
62 views

Question on spinor brackets

I have been given the Lorentz generator $$J_{ab}=\lambda_a \frac{\partial}{\partial \lambda^b}+a\leftrightarrow b$$ I know $\frac{\partial}{\partial \lambda^b}\lambda_a=\epsilon_{ba}$ and ...
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1answer
654 views

What're the relations and differences between slave-fermion and slave-boson formalism?

As we know, in condensed matter theory, especially in dealing with strongly correlated systems, physicists have constructed various "peculiar" slave-fermion and slave-boson theories. For example, For ...
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0answers
88 views

Finding the coefficients of a spinor

From the Schrödinger equation of a system I'm investigating, where the wave function is a 4-component spinor of coefficients $C_1, C_2, C_3, C_4$, I am able to obtain the expression $\begin{pmatrix} ...
6
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1answer
355 views

Vector and Spinor Representation in Ramond-Neveu-Schwarz Superstring Theory

I am learning Ramnond-Neveu-Schwarz Superstring theory (RNS theory). I often find the following notation, especially in the closed string spectrum etc.: $$\mathbf{8}_s,\mathbf{8}_v $$ And it is ...
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0answers
43 views

Spinors on algebraic plane curves

I'm interested in parameterizing spinors on Riemann surfaces. For my purposes, it's best to represent the Riemann surfaces as immersed in $\mathbb{C}P^2$, i.e. as algebraic plane curves. Apparently, ...
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0answers
125 views

Has hep-th/0312070 forgotten to fix $s_{0} = 1/2$ for the fermionic states in the second table on page 52?

Link to the original paper: The Gauge/String Correspondence Towards Realistic Gauge Theories (arXiv paper) On page 52 we see that, for a theory of Dp-branes placed at an orbifold (orbifold = ...
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2answers
413 views

Two ways to form SU(2) singlets?

I am trying to reconcile the two ways of forming SU(2) singlets out of a pair of doublets. Method (1): If $v=\begin{pmatrix}v^1\\ v^2\end{pmatrix}$ and $w=\begin{pmatrix}w^1\\ w^2\end{pmatrix}$ are ...
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3answers
604 views

Difference between spinor and vector field [duplicate]

How do we distinguish spinors and vector fields? I want to know it in terms of physics with mathematical argument.
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4answers
847 views

Could one argue that h (Planck constant) and $\hbar$/2 (Dirac constant) are in fact independant constants?

My question is very naive and could sound strange but it seems to me natural in so far as the Planck constant is related to the first quantization (of newtonian particle mechanics/galilean relativity) ...
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2answers
513 views

Number of Components of a Spinor

I'm trying to develop my understanding of spinors. In quantum field theory I've learned that a spinor is a 4 component complex vector field on Minkowski space which transforms under the chiral ...
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1answer
402 views

How is the Dirac adjoint generalized?

I am wondering how one can generalize the Dirac adjoint to flat "spacetimes" of arbitrary dimension and signature. To be more specific, a standard situation would be to consider 4 dimensional ...
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1answer
180 views

Inner product of particle-anti-particle spinor components

Suppose I have four-component spinors $\Psi$ and $\bar \Psi$ satisfying the Dirac equation with $$\Psi(\vec x) = \int \frac{\textrm{d}^3 p}{(2\pi)^3} \frac{1}{\sqrt{2 E_{\vec p}}} \sum_{s = \pm ...
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4answers
2k views

What is the difference between a spinor and a vector or a tensor?

Why do we call a 1/2 spin particle satisfying the Dirac equation a spinor, and not a vector or a tensor?