All Questions
Tagged with special-relativity spinors
212
questions
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Split Pauli Four-vector as quadratic terms of spinors
If I have the Pauli Four-vector $$x_{\mu}\sigma^{\mu} = \left(\begin{array}{cc}
t+z & x-i y \\
x+i y & t-z
\end{array}\right)$$ with $\sigma^0$ as Identity Matrix. Is there some way to write ...
0
votes
1
answer
44
views
Is the Dirac adjoint in the representation dual to Dirac spinor?
As seen in this Wikipedia page, the Lorentz group is not compact and the Dirac spinor (spin $\frac{1}{2}$) representation is NOT unitary.
Therefore, the complex conjugate representation does NOT ...
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0
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30
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Schwichtenberg Physics from Symmetry p. 83 Eq 3.225
Firstly - an apology. This is my first question to Stack Exchange and also my first attempt at using Latex. I need to show a subscript letter with a dot above it, but can't work out how to do that ...
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answers
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Trying to solve the energy levels of a spin 1/2 particle in a one-dimensional box using Dirac Equation
I was studying the problem I asked above in the title and found the article P Alberto et al 1996 Eur. J. Phys. 17 19.
The wave function inside the walls is:
$$
\psi(z)=B\ exp(ikz) \left[\begin{array}{...
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0
answers
98
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How to motivate spinors from the Dirac equation? [closed]
I am trying to motivate spinors by making sure the Dirac equation is relativistically invariant (and it suffices to discuss just the Dirac operator).
Let $\{ e_i \}$ be an orthonormal frame and $x^i$ ...
- 498
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0
answers
55
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What is the connection between Lorentz transforms on spinors and vectors?
When deriving the (1/2,0) and (0,1/2) representations of the Lorentz group one usually starts by describing how points in Minkowski space transform while preserving the speed of light (or the metric).
...
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votes
1
answer
357
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Can we make a Bloch sphere for Weyl spinors?
If spinors are the "square root" of 3-vectors [$\mathrm{SU}(2)$ double cover of $\mathrm{SO}(3)$], Weyl spinors can be thought of as the "square root" of 4-vectors [$\mathrm{SL}(2,\...
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Interpretation of "spin-1/2" in classical Dirac field
I emphasize that the proceeding is purely classical physics. Consider the Grassmann-valued field (where $\mathcal{N}$ is a Grassmann number), which is a solution to the Dirac equation, given by
$$\psi(...
- 2,676
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0
answers
54
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Weyl spinors under the Lorentz transformation
I am reading An Modern Introduction to Quantum Field Theory by Maggiore. On page 28, it says
Using the property of the Pauli matrices $\sigma^2 \sigma^i \sigma^2 = -\sigma^{i*}$ and the explicit form ...
- 41
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1
answer
94
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Why is the derivative necessary to connect left and right-hand spinors?
I am studying Weyl and Dirac spinors. Suppose we have two Weyl fermions $\eta, \chi$ transforming under $(1/2,0)$ representation of the Lorentz group. I learned that to construct Lorentz invariant ...
- 1,783
2
votes
1
answer
120
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Columns, rows, dotted, undotted, $SL(2, \mathbb{C})$ reps, and building Dirac spinors from Weyl spinors
I'm looking through Introduction to Supersymmetry by Muller-Kirsten and Wiedemann, along with any other resource I can find. I'm specifically trying to understand the concepts and notations for ...
- 213
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vote
1
answer
73
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Transformation of spinor reps and why the Dirac rep is its own conjugate
In Polchinski's String Theory volume 2, appendix B, on page 433 (in the section on Spinors and SUSY in various dimensions, specifically the subsection on Majorana spinors) he says:
"It follows ...
- 213
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0
answers
186
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Represent the Pauli 4-vector $\sigma^\mu$ as hermitian matrix of matrices due to the $SL(2,C)$ universal double cover of $SO^+(3,1)$
It's known that it's possible to map a 4-vector $x^\mu=(t,x,y,z)$, here i use $c=1$, into a 2x2 hermitian matrix as linear combination of Pauli matrices, thus the mapping $x^\mu \leftrightarrow X$. ...
- 310
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1
answer
213
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Building 4-vectors out of Weyl spinors: Combining 2 independent Weyl spinors and a sigma matrix to get a 4-vector
i'm struggling with this problem
In Exercise 2.3 of A Modern Introduction to Quantum Field Theory of Michele Maggiore I am asked to show that, if $\xi_R$ and $\psi_R$ are right-handed spinors, then
$$...
- 310
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Representation theoretic constraints in SUSY algebra
Let's try to build from scratch the SUSY commutator $[Q_\alpha^I, P_\mu]$. We know that the result of this commutator must be a fermonic generator belonging to $(1/2, 0)\otimes(1/2,1/2) \simeq (1, 1/2)...
- 81
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1
answer
266
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Understanding spinors, double cover and professor's expanation
I'm following an introductory course in QFT, and we are facing the spin group part. I think that most of the details are left apart because it would take too much time to be developd, and my profesor ...
- 361
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0
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47
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Question about spinor inner products
Let a 2D spinor be given by
$$\chi_2(p)=\pmatrix{\xi^1\\\xi^2}+i\pmatrix{\xi^3\\\xi^4}$$
with the $\xi^i$'s being real for $i=\{1,2,3,4\}$.
Assume, now, that I want to represent this spinor by a real-...
- 3,992
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55
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Questions on Lorentz generators in Spinor-Helicity formalism
I have read the following PSE posts on Lorentz generators in Spinor-Helicity formalism:
Total Angular Momentum Operator in Spinor-Helicity formalism
Derivation of conformal generators in spinor ...
- 3,992
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0
answers
112
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Spin and Representation Theory
So for integer spin, the way I understand it mathematically (in a classical limit), is that under a Lorentz transformation (i.e. change of coordinates), spin $n$ particles transform like rank $n$ ...
- 269
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1
answer
103
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Wavefunction spinor in Dirac equation
Which is the physical interpretation that in Dirac's equation the wavefunction is a spinor?
1
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1
answer
101
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Question on the spinor Indices, in non-relativistic quantum mechanics
I've caught by a loop of:
Standard texts of Non-Relativistic Quantum Mechanics $\to$ Representation theory of Lie groups and Lie algebras of $SO(3)$ and $SU(2)$ $\to$ Discussions of infinitesimal ...
- 315
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149
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Lorentz Invariance of kinetic terms for Weyl Spinors
Just to preface things, this exact question has been asked before here, but I don't feel like the answer really clarifies things for me.
The core issue is that we want to construct a 4-vector that we ...
- 328
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votes
1
answer
156
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How do projective representations act on the QFT vacuum?
Let $U:\mathcal{G}\to \mathcal{U}(\mathcal{H})$ be a unitary projective representation of a symmetry group $\mathcal{G}$ on a Hilbert space $\mathcal{H}$. It satisfies the composition rule:
$$U(g_1)U(...
- 1,455
6
votes
1
answer
236
views
Projective representations of the Lorentz group can't occur in QFT! What's wrong with my argument?
In flat-space QFT, consider a spinor operator $\phi_i$, taken to lie at the origin. Given a Lorentz transformation $g$, we have
$$\tag{1} U(g)^\dagger \phi_i U(g) = D_{ij}(g)\phi_j$$
where $D_{ij}$ is ...
- 1,455
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answers
47
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$S$-operator for proper Lorentz transformation
By applying infinitesimal Lorentz transformatios successively (with rotation angle $\omega$ around the $\bf n$ axis) one would get
$$\Psi'(x') = \hat{S}\Psi(x) = e^{-(i/4)\omega\hat{\sigma}_{\mu\nu}(\...
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0
answers
561
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Bilinear covariants of Dirac field
In the book "Advanced quantum mechanics" by Sakurai there is a section (3.5) about bilinear covariants, however i can't really find a definition of these objects, neither in the book nor ...
- 59
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2
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Rotation by 360°, spin-1/2 fermions and quaternions
Rotating a spin-1/2 fermion by 360° multiplies the quantum state by -1.
Representing a continuous 360° rotation as a quaternion is also a multiplication by -1.
Is there a relationship between these ...
- 190
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votes
3
answers
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What are Dirac spinors and why did relativistic quantum mechanics need them?
I have a good grasp of the Schrödinger equation and the basics of special relativity But the Dirac equation is alien to me. What are Dirac spinors and why did Dirac use them?
user353451
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1
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What does quantization of spin have to do with spinors?
A fermion has half-integer spin. In the context of the theory, this means its wavefunction is made of spinors: geometric objects which, under Lorentz rotations, transform in such a way that they ...
- 2,475
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Is there an analog of $\Lambda^T\eta\Lambda = \eta$ in any representation of the restricted Lorentz group?
The Lorentz group $O(1,3)$ is defined by
$$\Lambda^T \eta \Lambda = \eta \quad(1)$$
which we call the defining representation.
Given an irreducible representation of the restricted Lorentz group $SO^+...
user346886
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0
answers
87
views
How does the two index spinor $(v_{a\dot{b}})$ transforms?
Using the Van der Waerden Notation, we define the four-vector as:
$$v_{a\dot{b}}=v_\nu \sigma^\nu_{a\dot{b}}$$
I'm trying to see how this transforms. Defining:
$$\Lambda \equiv e^{i\vec{\theta}\cdot \...
- 11.9k
1
vote
0
answers
81
views
Derivation of the transformation law for spinors
I'm reading the book Quantum Field Theory: An Integrated Approach by Eduardo Fradkin, and I got stuck where the transformation law for spinors
$$
\psi'(x') = S(\Lambda) \psi(x)
$$
is derived.
In ...
- 601
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votes
1
answer
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What is the relationship between spinors and rotating motion geometrically?
Spinors are famously like spinning tops, but not actually like spinning tops since they are point particles and thus cannot rotate around their axis. It is easy to show algebraically how spinors must ...
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vote
0
answers
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views
Doubt on transformation laws of tensors and spinors using standard tensor calculus and group theory
1) Introduction
From standard tensor calculus, here restricted to Minkowski spacetime, we learned that:
A scalar field is a object that transforms as:
$$\phi'(x^{\mu'}) = \phi(x^{\mu})\tag{1}$$
A ...
- 3,085
1
vote
1
answer
219
views
How to contract spinor indices?
In normal vector representation, vectors can be contracted as follows:
$$v^\mu v_\mu$$
with one covariant and one contravariant index. But in spinor representation, there are 4 possible type of ...
- 1,324
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votes
1
answer
90
views
Lorentz boost property of gamma matrices
I was watching this video where he boosted the Dirac equation. He reached this equation:
$$S^{-1}(\Lambda)\gamma^\mu S(\Lambda)=\Lambda^\mu{}_\nu \gamma^\nu$$
My question is since $\gamma^\mu$ is a ...
- 1,324
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votes
0
answers
38
views
Can one express the evolution of a particle with a one-parameter group of $SO(3,1)$?
Can one express the evolution of a particles using a sequence of $SO(3,1)$ transformations? If yes, how?
Is it sufficient to apply $SO(3,1)$ transformations to a spinor?
$$
\psi(t) = e^{t\mathfrak{so}(...
- 1,548
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votes
0
answers
146
views
Angular-momentum of the Dirac spinor theory
The standard Dirac action
$$
S = \int d^4 x \bar \psi (i \gamma^\mu \partial_\mu - m) \psi
$$
is invariant under Lorentz transformation.
In David Tong's lecture note, eq (4.96) lists that the ...
- 669
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votes
2
answers
451
views
What is the idea behind 2-spinor calculus?
In the book by Penrose & Rindler of "Spinors and Space-Time", the preface says that there is an alternative to differential geometry and tensor calculus techniques known as 2-spinor ...
- 7,980
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vote
1
answer
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Does the Dirac Spinor live in the complexification of the Lorentz group?
In this question I learned that when working with quaternionic representations of (the double cover of) our relevant orthogonal group, we cannot avoid working in complex vector spaces. However it is ...
- 1,117
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votes
0
answers
147
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Is the real spinor representation of the Lorentz group irreducible?
Specifically the $(\frac{1}{2},0)\oplus(0,\frac{1}{2})$ representation. Given that we label representations by the corresponding representations of the complexified Lie group, the direct sum can be ...
- 1,117
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1
answer
146
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Quantum Field Theory Unitary Transformations
I am currently reading through Itzyskon and Zuber for my quantum field theory class, and I came across this regarding the unitary transformations of the Dirac bispinors in chapter 2. They show that ...
- 329
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1
answer
250
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Linearization of the Klein-Gordon equation and decoupling of ''spinors''! [closed]
We know that the K-G equation is deduced from the Einstein relation:
$E^{2}=m^{2} +\vec{p}^{2} \;\;\;\;$ (with $c=1$)
It is known that :$E^{2}=\frac{m^{2}}{1-\beta^{2}}=\left(\frac{m}{1-\beta}\...
- 1,400
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2
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160
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To construct a Lorentz scalar we use $\psi^{\dagger}\gamma^{0}\psi$. Could we use $\gamma^{5}$ instead of $\gamma^{0}$ seen as both are Hermitian?
Both $\gamma^{0}$ and $\gamma^{5}$ are Hermitian, so could we replace $\gamma^{0}$ with $\gamma^{5}$ to construct a Lorentz scalar with the same properties as $\bar{\psi}\psi$?
- 115
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answers
155
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Unitarity and boost
I wonder if anyone could shed some light on the representation theory of the Lorentz group. In particular, I would like to understand unitary and spinorial representations of boosts better. To my ...
- 9
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0
answers
298
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Lorentz-invariant Lagrangian for spinor field
Schwartz book on QFT (page 167), Zee book on group (page 461) and Maggiore book on QFT (page 55), prove that $\psi_R^{\dagger}\sigma^{\mu}\psi_R$ is a 4-vector, where $\psi_R$ is a right-handed spinor ...
- 613
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votes
1
answer
276
views
How to prove Weyl spinors transform as a representation of Lorentz group?
In my QFT lecture notes, it is written that the Lorentz group elements can be written as
\begin{equation*}
\Lambda = e^{i\vec{\theta}\cdot\vec{J} + i\vec{\eta}\cdot\vec{K}}
\end{equation*}
where $\Big\...
- 1,588
5
votes
1
answer
889
views
Do the Dirac matrices form a proper four-vector?
I'm seeing many conflicting statements about Dirac's gamma matrices. Some say it's a four-vector, some say it isn't, some say it's an invariant four vector. I know the Dirac matrices satisfy the ...
- 187
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vote
2
answers
487
views
Rarita-Schwinger spin projection operators
Chapter 2 of the paper Symmetry of massive Rarita-Schwinger fields by T. Pilling mentions "the usual" spin projection operators. However, to me, they are not usual and I struggle with ...
- 1,312
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votes
2
answers
371
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Riemannian and Weyl tensors as spinor representation
There is the way of converting vector indices to spinor indices, for example, Maxwell stress tensor $F_{[\mu\nu]}$ can be decomposed to $(1,0) \oplus (0,1)$ irreducible representations of $\mathfrak{...
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