| … | … | @@ -7058,7 +7058,7 @@ Spinors |
| 7058 | 7058 | For spinors: |
| 7059 | 7059 | \begin_inset Formula |
| 7060 | 7060 | \[ |
| 7061 | | \omega_{\mu}=\frac{1}{4}\omega_{\mu}^{ab}\gamma_{a}\gamma_{b} |
| 7061 | \Omega_{\mu}=\frac{1}{4}\omega_{\mu}^{ab}\gamma_{a}\gamma_{b} |
| 7062 | 7062 | \] |
| 7063 | 7063 | |
| 7064 | 7064 | \end_inset |
| … | … | @@ -7072,7 +7072,7 @@ The above dropped the time coordinate, |
| 7072 | 7072 | The Dirac eqn (massless) is |
| 7073 | 7073 | \begin_inset Formula |
| 7074 | 7074 | \[ |
| 7075 | | iD\!\!\!/\,\psi=i\gamma^{a}e_{a}^{\mu}\left(\partial_{\mu}+\omega_{\mu}\right)\psi=0 |
| 7075 | iD\!\!\!/\,\psi=i\gamma^{a}e_{a}^{\mu}\left(\partial_{\mu}+\Omega_{\mu}\right)\psi=0 |
| 7076 | 7076 | \] |
| 7077 | 7077 | |
| 7078 | 7078 | \end_inset |
| … | … | @@ -7102,26 +7102,16 @@ The hedgehog is |
| 7102 | 7102 | \end_layout |
| 7103 | 7103 | |
| 7104 | 7104 | \begin_layout Paragraph* |
| 7105 | | Dirac in sphere coords |
| 7105 | Vierbein in sphere coords |
| 7106 | 7106 | \end_layout |
| 7107 | 7107 | |
| 7108 | 7108 | \begin_layout Standard |
| 7109 | | So we need the Dirac eqn is sphere coords, |
| 7110 | | now. |
| 7111 | | All the piddly stupid details follow, |
| 7112 | | since I can't trust Claude for the final answer. |
| 7113 | | And I will make mistakes if I try to solve this in my head. |
| 7114 | | So this is a bit remedial, |
| 7115 | | but whatever. |
| 7109 | So eventually, |
| 7110 | we need the Dirac eqn is sphere coords. |
| 7111 | Since Claude is now consistently lying to me, |
| 7112 | we have to start at square one, |
| 7113 | and just do it the hard way. |
| 7116 | 7114 | |
| 7117 | | \begin_inset Quotes eld |
| 7118 | | \end_inset |
| 7119 | | |
| 7120 | | Good note taking |
| 7121 | | \begin_inset Quotes erd |
| 7122 | | \end_inset |
| 7123 | | |
| 7124 | | . |
| 7125 | 7115 | \end_layout |
| 7126 | 7116 | |
| 7127 | 7117 | \begin_layout Standard |
| … | … | @@ -7147,8 +7137,195 @@ e_{3} & =\frac{1}{\lambda r\sin\theta}\partial_{\phi} |
| 7147 | 7137 | |
| 7148 | 7138 | \end_inset |
| 7149 | 7139 | |
| 7150 | | The spin connection. |
| 7140 | |
| 7141 | \end_layout |
| 7142 | |
| 7143 | \begin_layout Paragraph* |
| 7144 | The spin connection |
| 7145 | \end_layout |
| 7146 | |
| 7147 | \begin_layout Standard |
| 7148 | Lets assume the earlier expressions for the spin connection were correct. |
| 7149 | \end_layout |
| 7150 | |
| 7151 | \begin_layout Standard |
| 7152 | Punt. |
| 7153 | I already wasted a few hours on this. |
| 7154 | It's solvable, |
| 7155 | but turning the crank and double–checking will take a few more hours. |
| 7156 | \end_layout |
| 7157 | |
| 7158 | \begin_layout Standard |
| 7159 | And that's for the simple case, |
| 7160 | (the radial–symmetric case) and not for the case that would really be interesting (which would be to allow local curvature to have some kinematic freedom, |
| 7161 | e.g. |
| 7162 | back–reacting in such as way as to have (vacuum) energy distribute uniformly. |
| 7163 | Or then more broadly to explore how much the chiral bag boundary condition can be made to look like AdS horizons. |
| 7164 | Or, |
| 7165 | in pop–sci–reporter speak, |
| 7166 | that |
| 7167 | \begin_inset Quotes eld |
| 7168 | \end_inset |
| 7169 | |
| 7170 | quark confinement can be modeled by placing the bag boundary infinitely far away, |
| 7171 | from the quarks point of view |
| 7172 | \begin_inset Quotes erd |
| 7173 | \end_inset |
| 7174 | |
| 7175 | . |
| 7176 | Which I think is a cool idea, |
| 7177 | and I'd like to see the answer, |
| 7178 | I'd like to see how the algebra for this works out. |
| 7179 | \end_layout |
| 7180 | |
| 7181 | \begin_layout Standard |
| 7182 | And I could do it. |
| 7183 | How long would it take? |
| 7184 | Well, |
| 7185 | turns out cranking through mathematical algebra is a lot like writing software. |
| 7186 | Once you understand what needs to be done, |
| 7187 | you say |
| 7188 | \begin_inset Quotes eld |
| 7189 | \end_inset |
| 7190 | |
| 7191 | oh a few hours |
| 7192 | \begin_inset Quotes erd |
| 7193 | \end_inset |
| 7194 | |
| 7195 | . |
| 7196 | Plus some time for debugging. |
| 7197 | Software debugging is easy, |
| 7198 | compared to algebra debugging: |
| 7199 | you might have made a mistake somewhere, |
| 7200 | but where? |
| 7201 | Or maybe you didn't, |
| 7202 | so how can you check? |
| 7203 | \end_layout |
| 7204 | |
| 7205 | \begin_layout Standard |
| 7206 | So I could do the spin connection in maybe a few hours. |
| 7207 | Maybe a few hours to double–check that the formulas that Claude quoted are actually correct. |
| 7208 | Then at least a few hours to remember how to do the hedgehog Ansatz, |
| 7209 | in this new setting. |
| 7210 | Then a day to daydream and scramble about and explore related ideas. |
| 7211 | Then a few days to solve the eigenvalue eqn, |
| 7212 | assuming, |
| 7213 | that is, |
| 7214 | that it is easily solvable; |
| 7215 | it might not be. |
| 7216 | A few days to sum the vacuum energy. |
| 7217 | A few days to rethink how this exerts a kinematic force on the curvature. |
| 7218 | A few days to daydream whether this should be turned into a full 3D dynamical model, |
| 7219 | instead of a 1D radial slice. |
| 7220 | And so now weeks or months have gone by on what was intended to be an afternoon project. |
| 7221 | \end_layout |
| 7222 | |
| 7223 | \begin_layout Standard |
| 7224 | Where was Claude in all of this? |
| 7225 | I was imagining some super–intelligence, |
| 7226 | and I just ask it some questions, |
| 7227 | and get some answers. |
| 7228 | This conception worked stunning great, |
| 7229 | when I was asking about electrical generation and transmission around the globe. |
| 7230 | Claude knew all the basic ideas, |
| 7231 | and fetched data from specific web searches, |
| 7232 | as I interrogated. |
| 7233 | It went well, |
| 7234 | it went happily. |
| 7235 | So I vaguely thought that doing some geometry would go equally swimmingly; |
| 7236 | I'd get some answers that have been bugging me for decades. |
| 7237 | \end_layout |
| 7238 | |
| 7239 | \begin_layout Standard |
| 7240 | Hah. |
| 7241 | Once again, |
| 7242 | Claude is revealed to be stupid as a rock. |
| 7243 | It has no fucking clue what it is talking about. |
| 7244 | Yes, |
| 7245 | it trained on some mathematical texts containing latex. |
| 7246 | Yes, |
| 7247 | it has committed them to memory. |
| 7248 | If you ask it a commonplace question, |
| 7249 | it knows what the answer should be, |
| 7250 | because it has memorized the answer after seeing it in a dozen textbooks. |
| 7251 | Ask it to check its work? |
| 7252 | And endless spew of excuses: |
| 7151 | 7253 | |
| 7254 | \begin_inset Quotes eld |
| 7255 | \end_inset |
| 7256 | |
| 7257 | you are right, |
| 7258 | I forgot to divide by r. |
| 7259 | Let me try again and be more careful. |
| 7260 | \begin_inset Quotes erd |
| 7261 | \end_inset |
| 7262 | |
| 7263 | OMG. |
| 7264 | This is exactly like using Claude to write software. |
| 7265 | But in that case, |
| 7266 | when it writes something that doesn't compile, |
| 7267 | I can ask it to use the compiler, |
| 7268 | and try again till it gets it right. |
| 7269 | Here, |
| 7270 | I'm the compiler: |
| 7271 | I have to double–check the formulas that it is claiming to solve. |
| 7272 | And fix all the errors. |
| 7273 | And OMG is it making lots of errors. |
| 7274 | So, |
| 7275 | in this sense, |
| 7276 | I could have made more progress, |
| 7277 | faster, |
| 7278 | without using Claude. |
| 7279 | What's the effective flow? |
| 7280 | To use it as a hinting tool. |
| 7281 | I do all the calculations, |
| 7282 | all the algebra, |
| 7283 | I get it to reformulate, |
| 7284 | using a different path, |
| 7285 | and see if things line up the way they should. |
| 7286 | So I can use Claude as a kind–of computer algebra system, |
| 7287 | and perhaps some super–tedious algebra does go faster, |
| 7288 | and perhaps I can trust it some of the time for some of the results. |
| 7289 | Maybe. |
| 7290 | Do I have time for this? |
| 7291 | Not right now. |
| 7292 | Maybe someday, |
| 7293 | when I need a vacation, |
| 7294 | I can get back to this. |
| 7295 | \end_layout |
| 7296 | |
| 7297 | \begin_layout Standard |
| 7298 | But for now, |
| 7299 | my questions remain unanswered, |
| 7300 | and Claude is revealed, |
| 7301 | yet again, |
| 7302 | to be an LLM. |
| 7303 | Ask it about something that is well known and widely understood? |
| 7304 | You get a great answer. |
| 7305 | Ask it to write javascript for your web site? |
| 7306 | Instant and perfect. |
| 7307 | Why? |
| 7308 | Because it trained on a million web pages, |
| 7309 | tens of thousands of javascript snippets. |
| 7310 | It can interpolate those in an instantaneous and brilliant fashion. |
| 7311 | Ask it to program in an obscure language (like Atomese) or use some obscure API? |
| 7312 | It stumbles. |
| 7313 | Ask it to do geometry? |
| 7314 | Utter and complete failure. |
| 7315 | Its a wordcel. |
| 7316 | It's not a shape rotator. |
| 7317 | And that's where we are, |
| 7318 | for today. |
| 7319 | The End. |
| 7320 | \end_layout |
| 7321 | |
| 7322 | \begin_layout Subsection* |
| 7323 | 28 January 2026 |
| 7324 | \end_layout |
| 7325 | |
| 7326 | \begin_layout Standard |
| 7327 | OK, |
| 7328 | so lets get back to the main topic.... |
| 7152 | 7329 | \end_layout |
| 7153 | 7330 | |
| 7154 | 7331 | \begin_layout Section* |