All Questions
Tagged with special-relativity spinors
212
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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 ...
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1
answer
42
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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 ...
- 1,665
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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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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:
$$
\psi(z)=B\ exp(ikz) \left[\begin{array}{...
2
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98
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How to motivate spinors from the Dirac equation? [closed]
I am trying to motivate spinors by making sure the Dirac equation is relativistically invariant (and it suffices to discuss just the Dirac operator).
Let $\{ e_i \}$ be an orthonormal frame and $x^i$ ...
- 498
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55
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What is the connection between Lorentz transforms on spinors and vectors?
When deriving the (1/2,0) and (0,1/2) representations of the Lorentz group one usually starts by describing how points in Minkowski space transform while preserving the speed of light (or the metric).
...
8
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1
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351
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Can we make a Bloch sphere for Weyl spinors?
If spinors are the "square root" of 3-vectors [$\mathrm{SU}(2)$ double cover of $\mathrm{SO}(3)$], Weyl spinors can be thought of as the "square root" of 4-vectors [$\mathrm{SL}(2,\...
- 5,558
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Interpretation of "spin-1/2" in classical Dirac field
I emphasize that the proceeding is purely classical physics. Consider the Grassmann-valued field (where $\mathcal{N}$ is a Grassmann number), which is a solution to the Dirac equation, given by
$$\psi(...
- 2,666
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53
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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
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93
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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
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1
answer
119
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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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1
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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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178
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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
1
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210
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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)...
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