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20
votes
2answers
154 views

Kerr Geometry, Separability and Twistors

One of the remarkable properties of the Kerr black hole geometry is that scalar field equations separate and are exactly solvable (reducible to quadrature), even though naively it does not have enough ...
14
votes
2answers
823 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 ...
14
votes
1answer
251 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 ...
12
votes
2answers
728 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 ...
12
votes
1answer
734 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, ...
11
votes
1answer
973 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 ...
11
votes
4answers
4k 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?
11
votes
1answer
454 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 ...
9
votes
2answers
214 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}, ...
9
votes
3answers
661 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 ...
8
votes
3answers
2k 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 ...
8
votes
4answers
2k views

covariant derivative for spinor fields

scalars (spin-0) derivatives is expressed as: $$\nabla_{i} \phi = \frac{\partial \phi}{ \partial x_{i}}$$ vector (spin-1) derivatives are expressed as: $$\nabla_{i} V^{k} = \frac{\partial V^{k}}{ ...
8
votes
1answer
283 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 ...
7
votes
2answers
2k 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 ...
7
votes
3answers
141 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 ...
7
votes
2answers
166 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 ...
7
votes
2answers
438 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
1answer
264 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 ...
6
votes
3answers
446 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 ...
6
votes
1answer
489 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} = ...
6
votes
1answer
87 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 $e$ ...
6
votes
2answers
897 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 ...
6
votes
1answer
354 views

Does the concept of both helicity and chirality make sense for a massive Dirac spinor?

Does the concept of both helicity and chirality make sense for a massive Dirac spinor? A massive electron in the chiral basis is written as a column made up of $\psi_L$ and $\psi_R$. What is the ...
6
votes
2answers
211 views

Is a spinor in some sense connected to space?

Spinors transform under the representation of $SL(2,\mathbb{C})$ which is the double cover of the Lorentz group $SO(1,3)$ - or in the non-relativistic case under $SU(2)$, the double cover of $SO(3)$. ...
6
votes
1answer
188 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 ...
6
votes
1answer
167 views

Invariance of supersymmetric Yang-Mills theory under supersymmetry

I was following Brink, Scherk and Schwartz, "Supersymmetric Yang-Mills theories". The variation of the Lagrangian w.r.t a supersymmetry transformation can be reduced to $$ \delta L = -igf_{a b c} ...
5
votes
1answer
316 views

Do Killing spinors know global information?

The conformal Killing spinor equations on $R\times S^3$ in Minkowski signature are \begin{equation} \nabla_\mu \epsilon=\pm \frac{i}{2}\gamma_\mu\gamma^0\gamma^5\epsilon \end{equation} whose solution ...
5
votes
2answers
467 views

Relation for Dirac-spinors of different helicities

Assume that we have massless spin-1/2 particles. The Dirac-spinor is the solution of the Dirac equation: $$ p^\mu \gamma_\mu u_\pm(p) = 0, \quad p^2 = 0$$ The subscripts $\pm$ denote two different ...
5
votes
1answer
407 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 ...
5
votes
2answers
633 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 ...
5
votes
1answer
65 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 ...
5
votes
1answer
119 views

Is there a review article that discusses the various suggestions for approaches to the Dirac spinor field?

I've come across many approaches to the Dirac spinor field over the years. A few have held more than passing interest but most of them are rather forgettable, so that I'd like to know of any reviews ...
5
votes
1answer
107 views

Counting degrees of freedom in spinor-helicity formalism

Just a couple of quick questions about the spinor-helicity formalism. We start with $p^\mu$ and $\epsilon^\mu$, so we have eight degrees of freedom. Then we have that $p^\mu p_\mu = 0$ and that ...
5
votes
1answer
360 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 ...
5
votes
0answers
93 views

Lorentz transformation of Gamma matrices $\gamma^{\mu}$

From my understanding, gamma matrices transforms under Lorentz transformation $\Lambda$ as \begin{equation} \gamma^{\mu} \rightarrow S[\Lambda]\gamma^{\mu}S[\Lambda]^{-1} = ...
5
votes
0answers
129 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 = ...
4
votes
3answers
886 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.
4
votes
3answers
2k 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 ...
4
votes
2answers
548 views

How do I derive the transformation law of a Weyl spinor under a Lorentz transformation?

Let $\xi$ be a spinor. If $(\theta ,\phi)$ are the parameters of a rotation and pure Lorentz transformation, then how can we prove that the transformation rule for $\xi$ can be written as $$\xi ...
4
votes
2answers
134 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
2answers
152 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 ...
4
votes
3answers
149 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} \\ ...
4
votes
2answers
877 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, ...
4
votes
1answer
101 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 ...
4
votes
1answer
253 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 ...
4
votes
1answer
140 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) ...
4
votes
1answer
440 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
1answer
181 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 ...
4
votes
1answer
178 views

What is the definition of precession (in the context of Spinors)?

What is the definition of "precession"? How is it applicable to abstract objects such as Spinors? I understand the mathematics, but don't understand what one means by "precession angle" etc when it ...
4
votes
1answer
318 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 ...