All Questions
52
questions
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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).
...
8
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1
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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,\...
- 5,568
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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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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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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
$$...
- 310
4
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1
answer
266
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Understanding spinors, double cover and professor's expanation
I'm following an introductory course in QFT, and we are facing the spin group part. I think that most of the details are left apart because it would take too much time to be developd, and my profesor ...
- 361
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47
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Question about spinor inner products
Let a 2D spinor be given by
$$\chi_2(p)=\pmatrix{\xi^1\\\xi^2}+i\pmatrix{\xi^3\\\xi^4}$$
with the $\xi^i$'s being real for $i=\{1,2,3,4\}$.
Assume, now, that I want to represent this spinor by a real-...
- 3,992
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112
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Spin and Representation Theory
So for integer spin, the way I understand it mathematically (in a classical limit), is that under a Lorentz transformation (i.e. change of coordinates), spin $n$ particles transform like rank $n$ ...
- 269
1
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101
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Question on the spinor Indices, in non-relativistic quantum mechanics
I've caught by a loop of:
Standard texts of Non-Relativistic Quantum Mechanics $\to$ Representation theory of Lie groups and Lie algebras of $SO(3)$ and $SU(2)$ $\to$ Discussions of infinitesimal ...
- 315
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149
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Lorentz Invariance of kinetic terms for Weyl Spinors
Just to preface things, this exact question has been asked before here, but I don't feel like the answer really clarifies things for me.
The core issue is that we want to construct a 4-vector that we ...
- 328
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75
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Is there an analog of $\Lambda^T\eta\Lambda = \eta$ in any representation of the restricted Lorentz group?
The Lorentz group $O(1,3)$ is defined by
$$\Lambda^T \eta \Lambda = \eta \quad(1)$$
which we call the defining representation.
Given an irreducible representation of the restricted Lorentz group $SO^+...
user346886
1
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87
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How does the two index spinor $(v_{a\dot{b}})$ transforms?
Using the Van der Waerden Notation, we define the four-vector as:
$$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
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38
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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
3
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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
2
votes
1
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276
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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