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3
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1answer
63 views

Why complexify in order to construct Dirac representation?

Suppose we have a theory is covariant under the Spin group Spin(2n-1; 1). We consider the real vector space $V = R^{2n-1,1}$, which naturally comes with a Lorentzian inner product. On this vector ...
0
votes
0answers
36 views

Relation between Clifford Algebra and Spinors? [closed]

As short as can be stated, I was wondering what is the exact relation between Clifford Algebra and Spinors? The closest I could get was via Wikipedia: "Spinors are typically defined as elements ...
3
votes
1answer
94 views

Is the Dirac equation equivalent to the Klein-Gordon equation for its left handed component?

The Dirac equation $$(i\gamma^a\partial_a - m)\psi=0\tag{0}$$ is given by a first order operator acting on a Dirac spinor, which is the direct sum of a left handed spinor and a right handed spinor. ...
-3
votes
3answers
201 views

Why do electrons and positrons exhibit opposite helical motion in a magnetic field?

When you throw an electron through a solenoid, it moves helically around the field lines, as per this schoolphysics illustration: © Keith Gibbs 2013 Then if we were to throw a positron through the ...
0
votes
1answer
57 views

Spin angular momentum

Spin angular momentum is defined as $\Sigma^{i}= 1/2 e^{ijk} \sigma_{jk}$ . Thus, I can write $\Sigma^{1}$ as $[\gamma_{2},\gamma_{3}]$ . I want some insights on this definition. Also, Can anybody ...
3
votes
2answers
414 views

Generalization of De Rham cohomology for spinor fields

I am interested in possible generalizations of The De Rham cohomology for spinor fields. I am also interested in applications to physics such as in the construction of topological charges I can see ...
0
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0answers
16 views

Where do certain relations relevant to helicity eigenstates come from in Peskin and Schroeder?

On pages 46-47 of Peskin and Schroeder, equations 3.52 and 3.53 are introduced when we have picked specific values of the numerical two-component spinor $\xi$. We choose a basis of ...
4
votes
1answer
294 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 $\bar{l}_1\sigma_{\mu\nu} r_2 \bar{r}_3 \sigma^{\mu\nu} l_4 =$ $ ...
1
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2answers
49 views

What does it mean to differentiate a spinor-valued field?

Peskin and Schroeder, equation 3.28, states that the Klein-Gordon equation $$(\partial^2+m^2)\psi=0 \tag{3.28}$$ is a valid choice of equation for a Dirac spinor field. Their explanation makes sense ...
12
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1answer
765 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, ...
0
votes
0answers
30 views

Clarification of Type D space-time

I've asked this question on MathOverflow but received no feedback, so I thought I'd try better luck here. I've read the following formulation of the Goldberg-Sachs theorem in Chandrasekhar's ...
6
votes
1answer
182 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} ...
3
votes
1answer
88 views

Hermiticity of Dirac operator in curved spacetime

The Dirac Lagrangian in curved spacetime is usually given by \begin{equation} \mathcal{L} = i\bar{\Psi}\gamma^a e^{\mu}_a(\partial_\mu + \frac{1}{4}\omega_{\mu bc}\gamma^b\gamma^c)\Psi \end{equation} ...
5
votes
1answer
46 views

The chirality of (2+1)D Dirac equation

Are there any definitions about the chirality of (2+1)D Dirac equation? For the (3+1)D Dirac equation, the Dirac field can be written as the sum of left- and right-hand Weyl field. Can this be reduced ...
1
vote
1answer
49 views

Necessity, significance of Spinors

This is an area I am researching at my own pace, general rotations in 3D. I've known about the plate trick for a while as well, and have a very rough understanding of the concept of ...
1
vote
0answers
29 views

Antiparticles, charge conjugation and chirality

(Why/how) are antiparticles and charge-conjugates different things? I am trying to understand the effect of discrete symmetries on spinor fields (neutrinos in particular). In the article, Dirac, ...
1
vote
1answer
68 views

Canonical spinors from gauge transformations

In this 2006 paper, http://arxiv.org/abs/hep-th/0610128, there is the concept of gauge transformation and how was it employed that I do not fully understand. Note, what will be talked about below is ...
0
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0answers
29 views

Future causality of Dirac charge current spinor

I am trying to solve following problem: The Dirac equation reads \begin{equation} \nabla^{AA'} \psi_{A} = \mu \chi^{A'}, \quad \nabla_{AA'} \chi^{A'} = -\mu \psi_{A} \end{equation} where $ \mu ...
0
votes
1answer
44 views

Do the eigenstates of the Pauli operators correspond to the six directions of the 3D world?

I understand that the six eigenstates of the three Pauli operators $X, Y, Z$ correspond to the six poles of the Bloch sphere. By fixing an orthonormal basis of our physical word, does "measuring Pauli ...
0
votes
1answer
220 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 ...
1
vote
1answer
43 views

Covariant derivative commutator on spinors [closed]

What is this object $[\nabla_{\mu},\nabla_{\nu}]\epsilon$ in terms of curvature tensor $R_{\mu\nu}$? Where $\nabla_{\mu}$ is the covariant derivative on a four sphere and $\epsilon$ is spinor. PS: I ...
0
votes
1answer
67 views

Regarding the Weyl spinor and its transformation properties

I am trying to prove the Lorentz invariance of the (left-handed) Weyl Lagrangian: $$\mathcal L=i\psi^\dagger\bar\sigma^\mu\partial_\mu\psi$$ A Lorentz transformation is realized as $\psi\to M\psi$, ...
2
votes
1answer
400 views

How can it be derived that particles described by the Dirac equation must have spin 1/2?

I am reading some lecture notes that unfortunately don't seem to be available online, but that are quite close in spirit in their treatment of the Dirac equation to Sakurai's "Advanced Quantum ...
0
votes
1answer
50 views

Question about Majorana spinor's property

I am reading the BBS, Exercise 5.1 This exercise is nothing but showing that two Majorana spinors $\Theta_1$ and $\Theta_2$ \begin{align} \bar{\Theta}_1 \Gamma_{\mu} \Theta_2 = -\bar{\Theta}_2 ...
5
votes
0answers
138 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} = ...
7
votes
1answer
272 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
256 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 ...
1
vote
1answer
233 views

General definition of vector, spinor, and spin

I am looking for basic and exact definitions of fundamental physical concepts in graduate level. I reach this following definitions. Could you please help to improve these definitions. Spin: ...
5
votes
1answer
114 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 ...
9
votes
3answers
691 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
1answer
75 views

Do particles have spin because there exist spinor representations for the Lorentz group?

I am reading Peskin and Schroeder's An introduction to field theory. They first describe the spinor representation of the Lorentz group, and then they mention the fact that different particles have ...
2
votes
1answer
73 views

Why must the supersymmetry generators be spinors?

I have read in a few places (for example, at page 5 here) that the supersymmetry generators must be spinors. Quoting the reference mentioned The generator of the symmetry must relate two types of ...
3
votes
1answer
174 views

Evolution of Eigenstates when two spin systems are coupled

I would like to describe the following situation: We have two spin systems: Spin 1 ($S_1$) and Spin 1/2 ($S_2$). Now imagine you somehow change their interaction so that you can fine-tune the ...
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0answers
41 views

Some questions about Weyl spinor algebra

I am following a review on supersymmetry and I am having trouble with Weyl spinor algebra. These are equations (4.2.1) ...
2
votes
1answer
89 views

Why does the Dirac equation reduce the fermionic degree of freedom by half

We know that in 4D a Dirac spinor has 4 complex components or 8 real components meaning 8 real off shell degrees of freedom (please correct me if I say something wrong here). When we go on-shell i.e ...
1
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0answers
39 views

How can I prove that $\gamma^0$ is the parity operator for Dirac fields? [closed]

How can I prove that the parity operator on a Dirac field is $\gamma^0$? I was trying to prove it through Lorentz transformations but failed shortly.
2
votes
1answer
107 views

Spinors and Möbius strips

I asked this question on Math.SE as I thought the perspective of representation theory might be enlightening. But since the question was provoked by a description of Spinors describing the spin of ...
2
votes
3answers
79 views

How do I operate on a spin state with a sigma operator?

For any arbitrary spin state $|s\rangle$. How do I operate on it with the Pauli spin matrix, $\hat{\sigma_z}$? Does this have something to do with a Bloch sphere?
0
votes
1answer
95 views

Spinor notation in general relativity

I have a somewhat broad/big question, and I know that there are many references for it available out there. However, so far I couldn't find anything that I can really understand, that's why here is my ...
1
vote
0answers
76 views

Spinor helicity formalism, exact form of the spinors

I am trying to understand how to perform computations with the spinor helicity formalism, I am studying on this review http://arxiv.org/abs/1308.1697. I have stumbled upon a problem though, in pag. ...
1
vote
1answer
367 views

How to find the time evolution for two-component spinor? [closed]

I would like to find the time evolution for the given Hamiltonian, the initial state of the system we choose two spinor wavefunction $\psi_{+}(t=0)$ and $\psi_{-}(t=0)$ as given below: The effective ...
3
votes
1answer
965 views

How to construct the charge conjugation matrix for any given dimension?

Generally, Gamma matrices could be constructed based on the Clifford algebra. \begin{equation} \gamma^{i}\gamma^{j}+\gamma^{j}\gamma^{i}=2h^{ij}, \end{equation} My question is how to generally ...
1
vote
1answer
60 views

Two-component formalism and four-component formalism [closed]

When deriving the Dirac equation for spin-1/2 particles, we realize that the wave function must be four-component. In some works, people use two-component wave function for calculation. So, my ...
0
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0answers
34 views

dagger operator in spinor representation

I just have trouble understanding how hermitian conjugation is acting like this in the following example (dot represents right-handed Weyl field, undot represents left-handed Weyl field). For ...
0
votes
1answer
61 views

Hermitian Adjoint of Spinor

Say we have a four component spinor $\psi$: $$ \psi=\begin{pmatrix}\psi_L\\\psi_R\end{pmatrix} $$ Is the Hermitian adjoint of this: $$ \psi^\dagger =\begin{pmatrix}\psi_L^\dagger ...
0
votes
1answer
74 views

Connection between two Petrov classification schemes

For the Weyl scalars of all spacetimes, at any point, possess one special structure, the so called principal null directions. Consider a general null tetrad $\{ l_a,n_a,m_a,\overline{m}_a \}$, we ...
1
vote
1answer
70 views

How to derive the form of the invariant spinor inner product?

So we have gamma matrices that satisfy the spacetime algebra relations, $\{\gamma^\mu, \gamma^\nu\} = 2 \eta^{\mu\nu}$. We know that if we set $\sigma^{\mu\nu} = \frac{1}{4}[\gamma^\mu, \gamma^\nu]$ ...
0
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0answers
32 views

Spinors and asymmetry of wave function

A spinor is a mathematical entity which changes sign if it is rotated by $2\pi$. Is this connected to the asymmetry of the spinor wave function?
3
votes
0answers
70 views

How one can write $\bar{\psi}$ in odd dimension?

I know that the Dirac equation in general dimensions has a form of $$ (i\gamma_{\mu} \nabla_\mu - m ) \psi =0 $$ and the action for that is written as $$ S = \int d^d x \bar{\psi} (i\gamma_{\mu} ...
3
votes
1answer
221 views

What is a spinor?

In a youtube video, sir Michael Atiyah mentioned that even after working during the most of his life on spinors, he doesn't know what a spinor is. Now surely that was part of his humorous introduction ...