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3
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0answers
82 views

Transformations of gamma-matrices through Pauli matrices transformations

I have the transformation law of the Lorentz group for Pauli matrices: $$ \tag 0 (\sigma^{\mu})_{a \dot {a}}{'} = \Lambda^{\mu}_{\quad \nu} N_{a}^{\quad c}(\sigma^{\nu})_{c \dot {c}}(N^{-1})^{\dot ...
0
votes
0answers
60 views

Angular momentum operator for 2 dimensions?

Recently I get the task to build (2 + 1)-Dirac theory. I wrote corresponding Dirac equation in a form $$ (i\sigma_{0}\partial_{0} + i\sigma_{1}\partial_{1} + i\sigma_{2}\partial_{2} - m)\Psi = 0, $$ ...
2
votes
0answers
77 views

One more relation with spherical spinors

Let's have the spherical spinors: $$ \mathbf {Y}_{j, m, l = j \pm \frac{1}{2}} = \frac{1}{\sqrt{2l + 1}}\begin{pmatrix} \pm \sqrt{l \pm m +\frac{1}{2}}Y_{l, m - \frac{1}{2}} \\ \sqrt{l \mp m ...
0
votes
0answers
37 views

How to find an action of $(\hat {\sigma} \cdot \hat {\mathbf L} )$ on spherical spinors?

Let's have the spherical spinors $\psi_{j, m, l = j \pm \frac{1}{2}}$, $$ Y_{j, m, l = j \pm \frac{1}{2}} = \frac{1}{\sqrt{2l + 1}}\begin{pmatrix} \pm \sqrt{l \pm m +\frac{1}{2}}Y_{l, m - \frac{1}{2}} ...
1
vote
0answers
65 views

Two pairs of projection operators of the Dirac equation

The Dirac equation may be interpreted as the action of projection operator $\frac{1 - \Delta}{2}\Psi = 0$, where $$ \Delta = \begin{pmatrix} 0 & \Delta_{b \dot {a}} \\ \Delta^{\dot {b}a} & 0 ...
2
votes
1answer
85 views

A simple question about matrix product with spinor indices

I have a big problem with dotted and undotted spinor indices. For example, suppose we have two convolutions: $$ \sigma^{\dot {a} a}F_{ab}, \quad \sigma^{\dot {a} a}F_{\dot {a} \dot {b}}, \quad F_{ab} ...
3
votes
0answers
120 views

Parity violating Dirac particle

We normally write down the Dirac Lagrangian as \begin{equation} {\cal L} _D = \bar{\psi} ( i \partial _\mu \gamma ^\mu - m ) \psi \end{equation} but are the Lagrangian's, \begin{equation} ...
2
votes
0answers
111 views

Pauli matrices product identity

How to prove the identity $$ \tilde {\sigma}_{\alpha}\sigma_{\beta}\tilde {\sigma}_{\gamma} = g_{\alpha \beta}\tilde {\sigma}_{\gamma} + g_{\alpha \gamma}\tilde {\sigma}_{\beta} - g_{\beta \gamma} ...
6
votes
1answer
231 views

Helicity and Chirality

Does the concept of both helicity and and chirality make sense for a massive Dirac spinor? A massive electron in chiral basis is written as a column made up of $\psi_L$ and $\psi_R$. What are the ...
1
vote
1answer
130 views

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 ...
7
votes
1answer
236 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
votes
1answer
273 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 ...
3
votes
0answers
93 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 ...
2
votes
0answers
60 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} ...
4
votes
1answer
272 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 ...
4
votes
3answers
1k 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 ...
1
vote
1answer
96 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 ...
5
votes
1answer
301 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
votes
2answers
177 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 ...
2
votes
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 ...
3
votes
2answers
205 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 ...
2
votes
0answers
100 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 ...
2
votes
0answers
91 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
votes
1answer
283 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 ...
0
votes
1answer
81 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 ...
1
vote
1answer
351 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 ...
2
votes
0answers
251 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 ...
3
votes
2answers
589 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, ...
6
votes
1answer
320 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} = ...
0
votes
0answers
115 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 ...
8
votes
2answers
488 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 ...
1
vote
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 ...
1
vote
1answer
211 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 ...
1
vote
1answer
173 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 ...
0
votes
1answer
112 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 & ...
1
vote
1answer
108 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
votes
1answer
188 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 ...
11
votes
2answers
526 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
votes
1answer
242 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 ...
1
vote
0answers
51 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 ...
4
votes
2answers
143 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 ...
1
vote
1answer
165 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 ...
3
votes
1answer
216 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?
4
votes
1answer
150 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 ...
1
vote
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 ...
2
votes
1answer
63 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 ...
11
votes
1answer
695 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 ...
0
votes
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
votes
1answer
361 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 ...
3
votes
0answers
44 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, ...