The tag has no wiki summary.

learn more… | top users | synonyms (1)

1
vote
0answers
41 views

Lorentz transformations on momentum space Weyl spinors

The Weyl spinor in momentum space can be given by the inverse Fourier transform: $$\psi_L(x) = \int \frac {d^4 p} {(2\pi)^4}\psi_L(p)e^{-ip\cdot x}, and we also have$$ $$\psi_L(x)=\int \frac ...
0
votes
0answers
33 views

Interpretation of Dirac Spinor components in Chiral Representation?

I failed to find any book or pdf that explains clearly how we can interpret the different components of a Dirac spinor in the chiral representation and I'm starting to get somewhat desperate. This is ...
2
votes
1answer
61 views

How does interpreting negative energy electrons as positrons solve the negative energy problem?

How does interpreting negative energy electrons as positive energy positrons solve the negative energy problem? How does change of “interpretation” without fixing the mathematics have such a profound ...
7
votes
2answers
139 views

Dirac spinors under Parity transformation or what do the Weyl spinors in a Dirac spinor really stand for?

My problem is understanding the transformation behaviour of a Dirac spinor (in the Weyl basis) under parity transformations. The standard textbook answer is $$\Psi^P = \gamma_0 \Psi = ...
7
votes
3answers
116 views

Are terms with spinors analogous to $ ( \partial_\mu \Phi )(\partial^\mu \Phi)$ forbidden in the Lagrangian?

For scalar particles, the Lagrangian involves terms of the form $ ( \partial_\mu \Phi )(\partial^\mu \Phi)$, which is equivalent through integration by parts to $ ( \partial_\mu \partial^\mu \Phi ...
1
vote
0answers
48 views

Hermitian conjugate of spinors

In any textbook, hermitian conjugate of spinor is defined like $ \psi_{\alpha}^{+}=\bar\psi_{\dot{\alpha}} $ and $(\psi^{\alpha})^{+}=\bar{\psi}^{\dot\alpha}$. We have Pauli matrices ...
3
votes
1answer
39 views

Dirac operator partial integration

When you have an action with bosonic $X$ and fermionic $\psi$ (Majorana) fields and perform a SUSY transformation $\epsilon$ (the constant, infinitesimal parameter of transformation, a real, ...
0
votes
0answers
36 views

Polarization sum rule for Rarita-Schwinger field

There are Rarita-Schwinger equations: $$ \tag 1 (p\!\!\!/ - m)\psi_{\mu} = 0, \quad \gamma_{\mu}\psi^{\mu} = 0, \quad i\partial_{\mu}\psi^{\mu} = 0. $$ So the polarization sum $D_{\mu \nu}(p) = ...
8
votes
3answers
391 views

What do the Pauli matrices mean?

All the introductions I've found to Pauli matrices so far simply state them and then start using them. Accompanying descriptions of their meaning seem frustratingly incomplete; I, at least, can't ...
0
votes
1answer
36 views

Electron can be at rest in Relativistic limit - what? [closed]

Here, I will demonstrate an astonishing fact (and thus an astonishing lack of understanding): It is well known from the uncertainty principle that an electron cannot be at rest. However, consider ...
3
votes
1answer
52 views

How is $\varepsilon_+^\mu(p) = \bar{v}(k) \gamma^\mu u(p)$ derived?

The relation $$\varepsilon_+^\mu(p) = \bar{v}(k) \gamma^\mu u(p)$$ is sometimes used to ease calculations of Feynman amplitudes with external gluons (see for example here at (2.13)). Where does this ...
3
votes
0answers
60 views

Diffeomorphisms and the Dirac action

I have a question concerning fermions in curved space-time. Please read it to the end before suggesting the spin-connection and vierbein-based approach. I heard that there is a special way of ...
6
votes
2answers
446 views

Is there a reason why the spin of particles is integer or half integer instead of even and odd?

It seems to me that we could change all the current spin values of particles by multiplying them by two. Then we could describe Bosons as even spin particles and Fermions as odd spin particles. Is ...
1
vote
4answers
186 views

Nature of Fields in QFT

I'm not exactly an expert in quantum physics, but this seems to be a simple question, and I can't find an answer anywhere! There are specific types of fields used in physics: scalar fields (i.e. as ...
2
votes
0answers
74 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 ...
1
vote
2answers
79 views

Converting two component product to four component notation

Consider the product of two left Weyl spinors in the notation commonly found in supersymmetry, \begin{equation} \chi ^\alpha\eta_\alpha = \chi ^\alpha \epsilon _{ \alpha \beta } \eta ^\beta ...
1
vote
1answer
44 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 $${\bar{u}^{(s)}}u^{(s)} = 2m. $$ ...
2
votes
2answers
165 views

A whole lot of doubts on Lorentz representation

Can someone tell me in layman's language how the $(1/2,1/2)$ represents a vector field and $(0,1/2)$ or $(1/2,0)$ represents spinors and $(0,0)$ represents scalar field. Please don't be pedantic on ...
0
votes
0answers
99 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: ...
2
votes
0answers
75 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. ...
0
votes
0answers
39 views

What are differences between Spin(3,1), SL(2,C), SO(3,1) and SU(2) representations? Which one is correct exact representation for spinor fields? [duplicate]

I want to understand which group transformations exactly represent spinor fields. That is, do spinor fields transform under the Lorentz group $\mathrm{SO}(3,1)$ or under $\mathrm{Spin}(3,1)$? What ...
0
votes
1answer
63 views

Rotating a complex number

Let us begin in a two-dimensional Euclidean plane. The vector is e.g. $\vec{V}(x,y)$ It is often useful – but in this case, it's just a mathematical trick that doesn't make the complex numbers ...
2
votes
0answers
49 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 ...
3
votes
1answer
161 views

Motivation for spinors

After it was found that the gamma matrices couldn't be Pauli matrices and only had to be larger and even, why was their need to define a new algebraic object (i.e a Dirac spinor)? Why couldn't a ...
3
votes
1answer
131 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 $\bar{l}_1\sigma_{\mu\nu} r_2 \bar{r}_3 \sigma^{\mu\nu} l_4 =$ $ ...
3
votes
2answers
86 views

Fermion as a mixture of particle and antiparticle

The solution to the Dirac equation (in the Dirac basis) are 4 coupled fields. The first 2 of them represent a particle (spin up/down), the other 2 fields are the antiparticle (spin up/down). When the ...
4
votes
1answer
82 views

Why must these Spinors be normalized?

I have just begun studying spin and there are two spinors mentioned: The main spinor $\chi $ and the spin-up spin down spinors (eigenspinors) $\chi_+ ,\chi_- $. I learned that the main spinor is a ...
7
votes
1answer
134 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 ...
3
votes
1answer
125 views

The relationship between spin and spinor curvature

The identity, $$ -\gamma^b{\mathcal{R}}_{ab} = {\mathcal{R}}_{ab}\gamma^b = \frac{1}{2}\gamma^b R_{ab}$$ is presented in the answer to the question Dirac Equation in General Relativity. How does ...
5
votes
1answer
60 views

Spinor reps in $\mathbb{R}^{1,3}\times{}B$ space-times

I am considering spinors in a space-time which is $\mathbb{R}^{1,3}\times{}B$ being $B$ a compact manifold of $D$ dimensions. I know that in ordinary 4 dimensional space-time spinors are ...
3
votes
1answer
98 views

How can a left-handed fermion field create a right-handed antifermion?

My question - which is likely stupid or appears due to some confusion - stems from the following considerations: when quantizing canonically we are told (see any book on QFT) that a Dirac fermion ...
2
votes
0answers
132 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 ...
3
votes
2answers
116 views

Electromagnetic current for interaction with Dirac spinors

The covariant form of the Dirac equation is given by $$(i\gamma^{\mu}\partial_{\mu} - M) \Psi(x) = 0 $$ Einstein's summation is implied here, $x=(x^0,x^1,x^2,x^3)^T$. I am simply looking for the ...
5
votes
3answers
334 views

Is there any relationship between gauge field and spin connection?

For a spinor on curved spacetime, $D_\mu$ is the covariant derivative for fermionic fields is $$D_\mu = \partial_\mu - \frac{i}{4} \omega_{\mu}^{ab} \sigma_{ab}$$ where $\omega_\mu^{ab}$ are the spin ...
1
vote
1answer
42 views

Difference between non-collinear systems and paramagnetic ones?

Non-collinear magnetism and paramagnetism, are they the same thing?
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 ...
0
votes
1answer
81 views

Dimension of gamma matrices in higher dimensional Dirac equations

Reading about Dirac's equation in higher dimensional space-times I have read that the gamma matrices are $2^{[D/2]}\times{}2^{[D/2]}$. So, if we have $D=11$, for example, how is this formula supposed ...
4
votes
1answer
133 views

The correspondence between Grassmann number and 4-spinor

In canonical quantization, we view the Dirac field $\psi$ as a $4\times1$ matrix of complex number. While in path integral quantization, we view the Dirac field $\psi$ as a Grassmann number. For two ...
2
votes
1answer
68 views

An identity for spinor helicity formalism

I have a question about the spinor helicity formalism from arXiv:1308.1697 Denote the massless spin-1/2 fermions as Eqs. (2.10)-(2.11) in that paper $$v_+(p)= \begin{pmatrix} |p]_a \\ 0 ...
13
votes
2answers
596 views

What's the relationship between $SL(2,\mathbb{C})$, $SU(2)\times SU(2)$ and $SO(1,3)$?

I'm a beginner of QFT. Ref. 1 states that [...] The Lorentz group $SO(1,3)$ is then essentially $SU(2)\times SU(2)$. But how is it possible, because $SU(2)\times SU(2)$ is a compact Lie group ...
2
votes
0answers
65 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 ...
4
votes
1answer
128 views

Anti-symmetric forms on Dirac spinors

In order to describe invariant forms on Dirac spinors $S$ one can find trivial subrepresentations in $S \otimes S$. If we use $S \cong (1/2, 0) \oplus (0, 1/2)$ then \begin{multline} [(1/2, 0) ...
14
votes
1answer
167 views

Chirality, helicity and the weak interaction

From what I'm understanding about Dirac spinors, using the Weyl basis for the $\gamma$ matrices the first two components behave as a left handed Weyl spinor, while the third and the fourth form a ...
6
votes
1answer
114 views

Little confusion with see-saw mechanism

Neutinos are either Dirac particles or Majorana particles but can’t be both at the same time. Then how can we write a general mass term as the sum of a Dirac mass term and a Majorana mass term? When ...
7
votes
2answers
139 views

Definition of a spinor and applications to GR

I understand the construction of the Clifford algebra $C(r,s)$ and in turn the corresponding $Pin$ and $Spin$ groups. I would like first to clarify that $Spin(r,s)^e$ is the universal covering group ...
4
votes
3answers
145 views

How to show the invariant nature of some value by the group theory representations?

Let's have Dirac spinor $\Psi (x)$. It transforms as $\left( \frac{1}{2}, 0 \right) \oplus \left( 0, \frac{1}{2} \right)$ representation of the Lorentz group: $$ \Psi = \begin{pmatrix} \psi_{a} \\ ...
3
votes
0answers
74 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
1answer
257 views

Why do we assume that Dirac spinor $\Psi$ describe the particle, not the field?

It is a well-known fact that Klein-Gordon scalar $\Psi(x)$, $$ (\partial^{2} + m^2) \Psi (x) = 0 $$ as well as 4-vector $A_{\mu}(x)$, $$ (\partial^{2} + m^{2})A_{\mu} = 0,\quad ...
3
votes
2answers
803 views

Dirac, Weyl and Majorana Spinors

To get to the point - what's the defining differences between them? Alas, my current understanding of a spinor is limited. All I know is that they are used to describe fermions (?), but I'm not sure ...
3
votes
2answers
77 views

Some questions about Dirac spinor transformation law

I have perhaps meaningless question about Dirac spinors, but I'm at a loss. The transformation laws for for left-handed and right-handed 2-spinors are $$ \tag 1 \psi_{a} \to \psi_{a}' = N_{a}^{\quad ...