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3
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1answer
49 views

Schrödinger-Pauli Equation Solutions

The Schrödinger-Pauli equation is the non-relativistic limit of the Dirac equation, and therefore describes spin-1/2 particles in an external electromagnetic field. It is given by: ...
5
votes
1answer
65 views

Coupling a spinor field to a preexisting scalar field?

So I'm not a physicist, but I'm thinking about a mathematical problem where I think physical insight might be useful. We're working on a Riemannian manifold $(M,g)$ (positive definite metric) with a ...
1
vote
1answer
109 views

Deriving Pauli Matrices

How does one derive using, say, the operator formula for reflections $$ R(r) = (I - 2nn^*)(r),$$ the reflection representation of a vector $$ R(r) = R(x\hat{i} + y\hat{j} + z\hat{k}) = xR(\hat{i}) ...
0
votes
1answer
29 views

Left-handed Majorana mass term forbidden by $SU(2)$?

I'm trying to figure out why a left-handed Majorana mass term is mathematically forbidden by the $SU(2)_L$ symmetry in the context of the seesaw model. As far as I get it, it is because the left ...
-3
votes
0answers
37 views

Pauli Spin Matrices [closed]

i) Determine the effect of the pauli spin matrices $\sigma_x$ and $\sigma_y$ on the eigenstates of $\sigma_z$. Which physical process is conveyed on the eigenstates of $\sigma_z$ by applying ...
2
votes
1answer
164 views

How do simple two-component Fierz identities follow from a property of the Pauli matrices?

On page 51 Peskin and Schroeder are beginning to derive basic Fierz interchange relations using two-component right-handed spinors. They start by stating the trivial (but tedious) Pauli sigma identity ...
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0answers
24 views

How does a linearly polarized spin 1/2 wave look like?

Spin 1 waves are easy to illustrate and a linearly polarized spin 2 wave looks like this, but what is the counterpart for a spin 1/2 wave?
0
votes
0answers
26 views

Probability density for spinors

I am approaching Relativistic Quantum Mechanics seriously for the first time, going through Bjorken & Drell and doing all the excercises, but I am facing some problems with 3.1. Derive (3.11) ...
6
votes
1answer
133 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 ...
1
vote
0answers
42 views

Fierz identity for chiral fermions [closed]

First of all I define the convention I use. The matrices $\bar{\sigma}^\mu$ I will use are $\{ Id, \sigma^i \}$ where $\sigma^i$ are the Pauli matrices and $Id$ is the 2x2 identity matrix. I will use ...
0
votes
1answer
51 views

The true dimension of Dirac field

In natural units with $\hbar=1$ and $c=1$, as we know, the energy dimension of the Dirac field $\psi(x)$ in QED is $\frac{3}{2}$. But in cgs units, what is the true dimension of the Dirac field ...
5
votes
1answer
85 views

Spontaneous symmetry breaking of a spinor / vector field [duplicate]

Why does SSB deal only with scalar fields and not with fermion or vector fields? My professor told me that it's closely related to the Lorentz invariance of the theory, but I don't understand at all ...
5
votes
1answer
359 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, ...
0
votes
0answers
13 views

Spinor helicity formalism - original reference?

The spinor helicity formalism is a modern technique widely used in scattering amplitude calculations nowadays. However, it is hard to find a reference for who first came up with the formalism. Maybe ...
2
votes
1answer
76 views

Is a Weyl fermion its own antiparticle?

Majorana fermions are their own antiparticles, and Weyl fermions are just Majorana fermions without mass. However, I haven't been able to find any source that says whether a Weyl fermion is its own ...
5
votes
1answer
438 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 = -\frac{3}{2} ...
4
votes
1answer
161 views

Two conflicting definitions of chirality

Consider a Majorana fermion embedded in a Dirac spinor, $$\psi = \begin{pmatrix} \psi_L \\ i \sigma_2 \psi_L^* \end{pmatrix}.$$ The Majorana fermion $\psi_L$ is left-chiral, i.e. it transforms in the ...
3
votes
2answers
141 views

Spinor field normalisation from poles in the propagator

In the theory of free scalar bosons (KG field) it is a basic result that the propagator $\Delta(p)$ has poles at $p^2=m^2$, with residue $1$ (or any other constant, depending on conventions). Thinking ...
0
votes
1answer
53 views

When dealing with spinor indices, how exactly do we obtain the barred Pauli operator?

In the set of SUSY notes I'm following, the Pauli operator is given as: ${(\sigma^\mu)}_{\alpha\dot{\alpha}} = (I_2, \sigma^1, \sigma^2, \sigma^3)$. The antisymmetric tensor that lowers and raises ...
1
vote
1answer
83 views

Deriving the Spinor Completeness Relation without using a Representation

Reference: DAMTP problem set 3, question 5 but ignore the spinor solutions given. To preface, this has taken up 1 entire day and a further 2 afternoons of work so I will just list the most promising ...
1
vote
1answer
50 views

Behaviour of Dirac Bilinears

Dirac bilinears transform in the Lorentz indices as, $\bar{\psi}\psi$ scalar $\bar{\psi}\gamma^\mu\psi$ vector $\bar{\psi}\sigma^{\mu\nu}\psi$ 2nd rank (antisymmetric) tensor ...
1
vote
1answer
75 views

Different definitions of the parity transformation for the Dirac spinors

There are two definitions of the parity transformation acting on the Dirac spinors: $\Psi_P = \eta \gamma^0 \Psi$ with $\eta = i$ ($P^2=-1$ as in Srednicki) and $\eta=1$ ($P^2=+1$ as in Peskin & ...
3
votes
2answers
77 views

Bilinears in adjoint representation

Below are two statements from my notes and I am trying to verify them explicitly. In both cases the fields are assumed to transform under the fundamental representation of $O(N)$ - --'The kinetic ...
5
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 ...
0
votes
0answers
38 views

Confusion over trying to understand spinor components

I've been reading about the quantisation of the Dirac field $\psi(x)$ and it is stated that the general solution to the Dirac equation $(i\gamma^{\mu}\partial_{\mu}-m)\psi(x)=0$ is given by the ...
9
votes
3answers
2k views

Is there a reason why the spin of particles is integer or half integer instead of, say, 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
votes
1answer
90 views

Gamma matrices relations (Dirac Spinors: QFT) [closed]

The entry question in an exam paper: I think I have made an elementary error in the transpose somewhere invoked by a conceptual misunderstanding of how spinors behave with gamma matrices under a ...
3
votes
1answer
66 views

Confusion with chirality eigenstates

In the Weyl/chiral basis, the four components of the Dirac spinor represent left-chirality spin up, left-chirality spin down, right-chirality spin up, and right-chirality spin down, respectively. When ...
3
votes
0answers
75 views

Schwartz's book: Spinor-helicity formalism

I'm trying to learn the spinor-helicity formalism from Schwartz's QFT book. His equation 27.44 is describes the annihilation of an electron(1)-positron(2) pair to a muon(3)-antimuon(4) pair. He ...
1
vote
1answer
65 views

Spin-1/2 with expectation values other than $\hbar/2$

I came across a problem that is related to the expectation value of spin-1/2. Assuming I want to find a single (or possibly a set of) spin state(s) $$\lvert\psi\rangle$$ that gives me an expectation ...
0
votes
0answers
22 views

Tranformation of a spinor in the self representation and conjugate representation of $SL(2,\mathbb{C})$

The transformation rules for a spinors as per Introduction to Supersymmetry by Wiedamann on Pg.38 is be summarized as: $$\begin{align} \psi_{\alpha} \mapsto \psi'_{\alpha} &= ...
0
votes
1answer
42 views

Ordering of Contravariant and Covariant spinors. Understanding the spinor space

I've been referring to Pg.36-Pg.38 in Introduction to Supersymmetry by Wiedamann. For understanding the precise origin of dotted, undotted indices on Spinors. He starts off my saying that $M$ acts on ...
0
votes
1answer
45 views

Why are half integer and full integer spin properties of elementary particles, not of all points in space?

Tensors and spinors arise mathematically from the representation of the rotation group $SO(3)$ as a ball in 4D with all antipodal points on the surface identified. In this picture it is shown that ...
3
votes
2answers
90 views

Different definitions of spinors

Recently I've read a little about the description of particles with spin in the book Quantum Mechanics by Cohen-Tannoudji. Although I yet didn't fully study the subject, I've read one interesting part ...
4
votes
2answers
240 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 ...
1
vote
0answers
24 views

Spin-Orbit Coupling on Spherical Delta Shell

I'm trying to solve for the energy levels of a rigid rotor in the presence of spin-orbit coupling. In such a case, I'd have the Hamiltonian: $H = \frac{-\hbar^2}{2m}\nabla^2+V_0 ...
2
votes
3answers
155 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?
4
votes
2answers
62 views

Helicities in electron-positron annihilation

Consider the massless limit of a process in which an electron-positron pair annihilates into a virtual photon - the final state doesn't matter. If the electron is massless (or if the energy is high ...
0
votes
1answer
57 views

Gordon Identity confusion

For the Gordon identity $$2m \bar{u}_{s'}(\textbf{p}')\gamma^{\mu}u_{s}(\textbf{p}) = \bar{u}_{s'}(\textbf{p}')[(p'+p)^{\mu} -2iS^{\mu\nu} (p'-p)_{\nu}]u_{s}(\textbf{p}) $$ If I plug in $\mu$=5, ...
0
votes
0answers
43 views

Dirac Spinors as Representation of $SL(2,\mathbb{C})$ over grassmann algebra

Recently, I've learned that the clifford algebra can be regarded as the quantization of grassmann algebra. This is shown from the following two papers by Berezin. 'Classical spin and Grassmann ...
2
votes
1answer
35 views

Weak isospin transformation of $\bar\psi \psi \phi$

In an old exam I found the following question regarding the Higgs potential: Write down the gauge invariant Yukawa interaction term in the Lagrangian that gives rise to the electron mass. The ...
1
vote
1answer
64 views

What are anticommuting spinor parameters $\zeta^\alpha$?

I'm reading Martin F.Sohnius, Introducing supersymmetry, page 82. It is the first time he introduces the anticommuting spinor parameters $\zeta^\alpha$ to calculate the supersymmetry variations of a ...
0
votes
1answer
91 views

Geometry of spacetime and spinor bilinears

In this paper (http://arxiv.org/abs/0704.0247) p.20, the author says in the section titled Geometry of spacetime the following: In order to obtain the spacetime geometry, we consider the spinor ...
6
votes
1answer
237 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
207 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
98 views

Supercovariant Derivative action

My query is with Weinberg Vol3 equation just above 26.7.22 Weinberg follows Majorana Superfield formalism. Where, covariant derivative is defined as, $$D_{R\alpha}=-\epsilon_{\alpha ...
3
votes
0answers
88 views

Angular momentum of the vacuum

I'm studying quantum field theory from "An introduction to Quantum field theory" by Peskin and Schroeder and from "A modern introduction to quantum field theory" by Maggiore. I've read from "An ...
0
votes
2answers
57 views

Density matrices vs Pauli matrices

Studying quantum mechanics, I have suddenly come to the conclusion that Pauli matrices are essentially density matrices for spin systems. Does it make any sense or I have missed something?
2
votes
2answers
90 views

Why can the spin of a relativistic particle not be orthogonal to its momentum?

I have read that the 3-momentum of a relativistic particle cannot be orthogonal to its spin 3-vector. When thinking about how the spin vector transforms when the particle approaches light speed, it ...
3
votes
1answer
484 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 ...