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4
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1answer
240 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
1answer
71 views

Fierz identity for Weyl spinors in tensor currents

Using Fierz identity I found that certain four-fermion operator with left $l_i$ and right-chiral $r_i$ Weyl spinors vanish $\bar{l}_1\sigma_{\mu\nu} r_2 \bar{r}_3 \sigma^{\mu\nu} l_4 =$ $ ...
0
votes
1answer
30 views

Spinor normalizations in Breit-frame: electric and magnetic form factors

We usually have the normalization (see e.g. page 110 in Halzen & Martin "Quarks and Leptons") $$u^{(r)\dagger}u^{(s)} = 2E\delta_{rs}$$ which leads to $${\bar{u}^{(s)}}u^{(s)} = 2m. $$ ...
13
votes
0answers
598 views

Could this model have soliton solutions?

We consider a theory described by the Lagrangian, $$\mathcal{L}=i\bar{\Psi}\gamma^\mu\partial_\mu\Psi-m\bar{\Psi}\Psi+\frac{1}{2}g(\bar{\Psi}\Psi)^2$$ The corresponding field equations are, ...
6
votes
0answers
83 views

2 Component Spinor Formalism

In Chapters 34-36 of the Srednicki QFT book, 2 component spinors and their combinations in Dirac and Majorana spinors are carefully constructed. Specifically, in equations 36.14 and 36.15 the ...
5
votes
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 = ...
3
votes
0answers
67 views

Transformation law for spinor functions multiplication

Let's have Dirac spinor $\Psi (x)$, which formally corresponds to $$ \left( 0, \frac{1}{2} \right) \oplus \left( \frac{1}{2}, 0 \right) $$ representation of the Lorentz group. What representation is ...
3
votes
0answers
69 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 ...
3
votes
0answers
78 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 ...
3
votes
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, ...
3
votes
0answers
165 views

Some more questions on conformal spinors of $SO(n,2)$

This is somewhat of a continuation of my previous question. I had stated there that a conformal spinor ($V$) of $SO(n,2)$ can be created by taking a direct sum of two $SO(n-1,1)$ spinors $Q$ and $S$ ...
2
votes
0answers
69 views

What are Killing spinors?

What are Killing spinors? How can they be motivated? Are they directly related to Killing vectors and Killing tensors and is there an overarching motivation for all three objects? Any answer is ...
2
votes
0answers
69 views

Does anybody know of a source that explains Wick rotation for fermions in 3-dimensional spacetime?

I've been looking for a long time and I've not had a lot of luck. I've found sources that use fermions in 3d Euclidean space but I can't find any that explain the Wick rotation from Minkowski space. ...
2
votes
0answers
47 views

Irreducible representation for the massless particle with helicity 2 and the Weyl tensor

As it can be shown, the equations for the irrep with zero mass and helicity 2, -2 respectively can be given in a form $$ \tag 1 \partial^{\dot {b}a}C_{abcd} = 0, \quad ...
2
votes
0answers
104 views

How does the Gordon Decomposition of Dirac Current give rise to spin angular momentum?

How does the Gordon Decomposition of Dirac Current give rise to spin angular momentum? I used the Gordon Decomposition to split the Probability Current of the Dirac Field into its orbital current and ...
2
votes
0answers
64 views

One question about Weinberg's derivation of arbitrary spin fields expressions

In his book "QFT" (vol. 1) Weinberg writes the expression for an arbitrary spin massive field: $$ \hat {\Psi}_{a}(x) = \sum_{\sigma = -s}^{s} \int \frac{d^{3}\mathbf p}{\sqrt{(2 \pi)^{3}2 ...
2
votes
0answers
75 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 ...
2
votes
0answers
113 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
97 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} ...
2
votes
0answers
55 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} ...
2
votes
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 ...
2
votes
0answers
83 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
0answers
229 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 ...
2
votes
0answers
115 views

Spin 1/2 finite-difference field simulator?

Is there a finite-difference field simulator for spin 1/2 fields, something like meep for electromagnetism (spin 1)? Looking for something free (GNU, MIT or other open/free style license) and easy ...
1
vote
0answers
62 views

Spinor representation of $SO(d+1,1)$

I have been looking over the internet for a resource that tells me the number of dimensions of a spin $s-1$ spinor representation of $SO(d+1,1)$, but unfortunately have yet to be able to find it. In ...
1
vote
0answers
57 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 ...
1
vote
0answers
82 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
0answers
48 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 ...
0
votes
0answers
76 views

General definition of vector spinor and spin

I am looking for basic and exact definitions of fundamental physical consepts in graduate level. I reach this following definitions. Could you please help to improve these definitions. Spin: ...
0
votes
0answers
58 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 $$ (i\sigma_{0}\partial_{0} + i\sigma_{1}\partial_{1} + i\sigma_{2}\partial_{2} - m)\Psi = 0, $$ ...
0
votes
0answers
36 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}} ...
0
votes
0answers
151 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 ...
0
votes
0answers
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 ...
0
votes
0answers
87 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} ...