lukeburns

Dirac ideal

Spinors in the Dirac ideal are given by $$\begin{aligned} N &= M \frac{1+\gamma_0}{2} \frac{1+\gamma_5\sigma_3}{2} \ &= (M_+ + M_-)\frac{1+\gamma_0}{2} \frac{1+\gamma_5\sigma_3}{2} \ &= (M_+ + M_-\gamma_0)\frac{1+\gamma_0}{2} \frac{1+\gamma_5\sigma_3}{2} \ &= M_+'\frac{1+\gamma_0}{2} \frac{1+\gamma_5\sigma_3}{2} \ &= (\text{Re}M_+' + \text{Im} M_+' I \sigma_3)\frac{1+\gamma_5\sigma_3}{2}\frac{1+\gamma_0}{2} \ &= \psi \frac{1+\gamma_5\sigma_3}{2}\frac{1+\gamma_0}{2} \ \end{aligned} $$where $\psi \in \mathcal Cl_{1,3}^+$.

lukeburns

Let $$ \begin{gathered} \gamma_0 = \begin{bmatrix} 0 & - i I_2 \ i I_2 & 0 \end{bmatrix} = -i \beta \ \gamma_1 = \begin{bmatrix} -i\sigma_3 & 0 \ 0 & i\sigma_3

lukeburns

const { RecursiveResolver, dnssec, wire: { Record, types } } = require('bns')

const port = 5349
const resolver = new RecursiveResolver()
resolver.setStub('127.0.0.1', port, hnsDS())
resolver.on('log', console.log)
resolver.on('error', console.error)

resolver.lookup('com', types.A).then(console.log).catch(console.error)

lukeburns

### A Pluto.jl notebook ###
# v0.16.1

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    quote
        local el = $(esc(element))

lukeburns

# Put this in your espanso config folder and run `julia latex-completions.jl` to add all Julia REPL completions to espanso.

# Note: each expansion includes a space at the end (otherwise e.g. \c will autocomplete before you finish typing \cdot).
# You can try using a tab to trigger: "\\\\$(word[2:end])\t" instead, but this led to some unpredictable behavior for me.

# https://github.com/JunoLab/atom-latex-completions/blob/master/completions/generate.jl

import REPL
open(joinpath(dirname(@__FILE__), "user/latex-completions.yml"), "w") do io
  println(io, "name: latex-completions")

lukeburns

chrome.contextMenus.create({
  title: 'Open Frame in New Tab',
  contexts: ['frame'],
  onclick: ({ frameUrl }) => {
    chrome.tabs.create({
      active: false,
      url: frameUrl
    })
  }
})

lukeburns

macro require(mod)
    try
        Base.require(__module__, Symbol(mod)) # like require("module")
    catch e
        mod = string(mod)
        include(mod) # like require("./module.js")
    end
end

macro assign(ex)

lukeburns

hello world

lukeburns

Rendered $\LaTeX$ from [Github Gists](https://gist.github.com/)

lukeburns

So…this is the expression I’m finding for the “inhomogeneous” equations:

$\nabla_\mu R^{\mu \nu}_{ \rho \sigma} = J^\nu_{\rho \sigma}$ where  $J^\nu_{\rho \sigma} \equiv \nabla_\mu W^{\mu \nu}_{\rho \sigma} + \nabla_{[\rho} t_{\sigma]}^\nu$, $t_{\mu \nu} = \frac{1}{2} T_{\mu \nu} - \frac{1}{3} g_{\mu \nu} T$, and $T$ is the trace of $T_{\mu \nu}$. I think this can be placed entirely in terms of $T_{\mu \nu}$ using self-duality of the Weyl tensor.

And I expect the "electric" and "magnetic" fields to be given by $E_{i \rho \sigma} = R_{0 i \rho \sigma}$ and $B_{i \rho \sigma} = -\frac{1}{2} \epsilon_{ijk} R^{jk}_{\rho \sigma}$. 

I think the only real dent in the analogy is that $J^\nu_{\rho \sigma}$ is not a conserved current in general.