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43 views

Energy levels for Hehl-Datta equation

In an article called "Matter-antimatter asymmetry and dark matter from torsion" (http://arxiv.org/abs/1108.6100) its author derives an effective Dirac equation (eq. (2), the author calls it Hehl-Datta ...
1
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0answers
14 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 ...
2
votes
1answer
98 views

Why does the $(\frac{1}{2},\frac{1}{2})$ representation of the Lorentz group act on hermitian 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
189 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}, ...
1
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0answers
22 views

How to take boost of 4-momentum and spinor? [closed]

I am studying Michele Miggiore and Peskin. I am stuck on the missing steps of finding the solutions of Dirac equation (Miggiore equation 3.103). I know how to directly find the solutions but didn't ...
4
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1answer
61 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
48 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
46 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
41 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
34 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 ...
1
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1answer
37 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
71 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 ...
5
votes
1answer
69 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 $e$ ...
2
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0answers
64 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
74 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
56 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 ...
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0answers
27 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
86 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} = ...
1
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1answer
54 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} ...
2
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0answers
80 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 ...
3
votes
1answer
97 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 ...
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2answers
44 views

Transposition of spinors

Suppose we have two 4-components Dirac spinors, that is two non commuting objects, $\psi$ and $\chi$. We know that: $ \bar{\psi} \chi= - \chi \bar{\psi} $ $ \bar{\psi} = \psi^{+} \gamma_0 $ $+=T*$ ...
2
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1answer
83 views

The relationship between the structure of spacetime and the existence of spinor field?

We all know that the existence of spinor fields implies that spacetime must be time-orientable. Thus that spacetime is time-orientable is a necessary condition for existence of spinor fields. Geroch, ...
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0answers
72 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
68 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
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2answers
190 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
124 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
1answer
114 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
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1answer
45 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, ...
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0answers
45 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
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3answers
608 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
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1answer
41 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
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1answer
57 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
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0answers
65 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
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2answers
537 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 ...
2
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4answers
206 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
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0answers
80 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
86 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
45 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 $${\bar{u}^{(s)}}u^{(s)} = 2m. $$ ...
3
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2answers
186 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
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0answers
108 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
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0answers
85 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
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0answers
40 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
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1answer
66 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
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0answers
52 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
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1answer
174 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
179 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 $\bar{l}_1\sigma_{\mu\nu} r_2 \bar{r}_3 \sigma^{\mu\nu} l_4 =$ $ ...
4
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2answers
101 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
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1answer
86 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
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1answer
168 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 ...