Luke Burns 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 $\psi \in \mathcal Cl_{1,3}^+$.

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

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	### 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))

	# 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")

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

	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)

	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.