| … | … | @@ -6851,7 +6851,7 @@ Metric |
| 6851 | 6851 | |
| 6852 | 6852 | \begin_layout Standard |
| 6853 | 6853 | Poincare ball metric with curvature |
| 6854 | | \begin_inset Formula $A$ |
| 6854 | \begin_inset Formula $K=-1/A^{2}$ |
| 6855 | 6855 | \end_inset |
| 6856 | 6856 | |
| 6857 | 6857 | , |
| … | … | @@ -7168,7 +7168,14 @@ The spin connection |
| 7168 | 7168 | |
| 7169 | 7169 | \begin_layout Standard |
| 7170 | 7170 | Let assume the earlier expressions for the spin connection were correct. |
| 7171 | | Well, |
| 7171 | (I wrote the diatribe about Claude, |
| 7172 | further below, |
| 7173 | first, |
| 7174 | and then circled back here to do this by hand.) |
| 7175 | \end_layout |
| 7176 | |
| 7177 | \begin_layout Standard |
| 7178 | Well, |
| 7172 | 7179 | lets assume they are not, |
| 7173 | 7180 | and restart from first principles. |
| 7174 | 7181 | So |
| … | … | @@ -7321,6 +7328,203 @@ Finally |
| 7321 | 7328 | \end_inset |
| 7322 | 7329 | |
| 7323 | 7330 | but this is trivially satisfied by the above results. |
| 7331 | I think the above is correct. |
| 7332 | \end_layout |
| 7333 | |
| 7334 | \begin_layout Paragraph* |
| 7335 | Dirac equation |
| 7336 | \end_layout |
| 7337 | |
| 7338 | \begin_layout Standard |
| 7339 | So lets repeat. |
| 7340 | We have to solve |
| 7341 | \begin_inset Formula |
| 7342 | \[ |
| 7343 | \gamma^{a}e_{a}^{\mu}\left(\partial_{\mu}+\Omega_{\mu}\right)\psi=0 |
| 7344 | \] |
| 7345 | |
| 7346 | \end_inset |
| 7347 | |
| 7348 | with |
| 7349 | \begin_inset Formula |
| 7350 | \[ |
| 7351 | \Omega_{\mu}=\frac{1}{4}\omega_{\mu}^{ab}\gamma_{a}\gamma_{b} |
| 7352 | \] |
| 7353 | |
| 7354 | \end_inset |
| 7355 | |
| 7356 | subject to the hedgehog boundary conditions applied at |
| 7357 | \begin_inset Formula $r<1$ |
| 7358 | \end_inset |
| 7359 | |
| 7360 | . |
| 7361 | We comply trickery from here on out. |
| 7362 | I am using Claude to find the tricks. |
| 7363 | I'm not verifying them carefully, |
| 7364 | they smell correct. |
| 7365 | \begin_inset Formula |
| 7366 | \[ |
| 7367 | \left[\gamma_{a},\gamma_{b}\right]=2i\epsilon_{abc}\Sigma^{c} |
| 7368 | \] |
| 7369 | |
| 7370 | \end_inset |
| 7371 | |
| 7372 | with |
| 7373 | \begin_inset Formula |
| 7374 | \[ |
| 7375 | \Sigma^{c}=\left(\begin{array}{cc} |
| 7376 | \sigma^{c} & 0\\ |
| 7377 | 0 & \sigma^{c} |
| 7378 | \end{array}\right) |
| 7379 | \] |
| 7380 | |
| 7381 | \end_inset |
| 7382 | |
| 7383 | so |
| 7384 | \begin_inset Formula |
| 7385 | \[ |
| 7386 | \Omega_{\mu}=\frac{i}{4}\epsilon_{abc}\omega_{\mu}^{ab}\Sigma^{c} |
| 7387 | \] |
| 7388 | |
| 7389 | \end_inset |
| 7390 | |
| 7391 | Well, |
| 7392 | OK, |
| 7393 | baby steps. |
| 7394 | Next, |
| 7395 | we have to hedgehog Ansatz our way through. |
| 7396 | Once again, |
| 7397 | I think I'll call it quits here, |
| 7398 | for now. |
| 7399 | I've actually asked Claude to take many more steps from here. |
| 7400 | I don't trust it's results, |
| 7401 | so I am not copying them here, |
| 7402 | but so far, |
| 7403 | we're having some amount of fun. |
| 7404 | The vibes are OK, |
| 7405 | so far. |
| 7406 | \end_layout |
| 7407 | |
| 7408 | \begin_layout Paragraph* |
| 7409 | Claude code |
| 7410 | \end_layout |
| 7411 | |
| 7412 | \begin_layout Standard |
| 7413 | Err, |
| 7414 | well I will sketch what Claude is telling me. |
| 7415 | It smells OK. |
| 7416 | It might have errors. |
| 7417 | But the overall picture is not insane. |
| 7418 | I gather that someone has published these details, |
| 7419 | and Claude trained on them, |
| 7420 | and is remembering them. |
| 7421 | Here goes. |
| 7422 | \end_layout |
| 7423 | |
| 7424 | \begin_layout Standard |
| 7425 | One trick is to absorb the spin connection into an angular momentum term. |
| 7426 | I won't write that down. |
| 7427 | \end_layout |
| 7428 | |
| 7429 | \begin_layout Standard |
| 7430 | Next trick is to write |
| 7431 | \begin_inset Formula |
| 7432 | \[ |
| 7433 | \psi=\left(\begin{array}{c} |
| 7434 | \psi_{+}\\ |
| 7435 | \psi_{-} |
| 7436 | \end{array}\right) |
| 7437 | \] |
| 7438 | |
| 7439 | \end_inset |
| 7440 | |
| 7441 | with |
| 7442 | \begin_inset Formula |
| 7443 | \begin{align*} |
| 7444 | E\psi_{+} & =-i\vec{\sigma}\cdot\vec{D}\psi_{-}+m\psi_{+}\\ |
| 7445 | E\psi_{-} & =-i\vec{\sigma}\cdot\vec{D}\psi_{+}-m\psi_{-} |
| 7446 | \end{align*} |
| 7447 | |
| 7448 | \end_inset |
| 7449 | |
| 7450 | where |
| 7451 | \begin_inset Formula |
| 7452 | \begin{align*} |
| 7453 | \vec{\sigma}\cdot\vec{D} & =\frac{\vec{\sigma}\cdot\hat{r}}{\lambda}\left(\partial_{r}+\frac{1}{1-r^{2}}+\frac{1+r^{2}}{1-r^{2}}\frac{\left(\vec{\sigma}\cdot\hat{r}\right)\left(\vec{\sigma}\cdot\vec{L}\right)}{r}\right)\\ |
| 7454 | & =\frac{\vec{\sigma}\cdot\hat{r}}{\lambda}\left(\partial_{r}-\frac{r}{1-r^{2}}+\frac{1+r^{2}}{1-r^{2}}\left[\frac{1+\left(\vec{\sigma}\cdot\hat{r}\right)\left(\vec{\sigma}\cdot\vec{L}\right)}{r}\right]\right)\\ |
| 7455 | & =\frac{\vec{\sigma}\cdot\hat{r}}{2A}\left(\left(1-r^{2}\right)\partial_{r}+1+\left(1+r^{2}\right)\frac{\left(\vec{\sigma}\cdot\hat{r}\right)\left(\vec{\sigma}\cdot\vec{L}\right)}{r}\right) |
| 7456 | \end{align*} |
| 7457 | |
| 7458 | \end_inset |
| 7459 | |
| 7460 | and the flat–space version would have been |
| 7461 | \begin_inset Formula |
| 7462 | \[ |
| 7463 | \vec{\sigma}\cdot\vec{\nabla}=\vec{\sigma}\cdot\hat{r}\left(\partial_{r}+\frac{1+\left(\vec{\sigma}\cdot\hat{r}\right)\left(\vec{\sigma}\cdot\vec{L}\right)}{r}\right) |
| 7464 | \] |
| 7465 | |
| 7466 | \end_inset |
| 7467 | |
| 7468 | To regain flat space |
| 7469 | \begin_inset Formula $K=0$ |
| 7470 | \end_inset |
| 7471 | |
| 7472 | in the limit, |
| 7473 | set |
| 7474 | \begin_inset Formula $K\to0$ |
| 7475 | \end_inset |
| 7476 | |
| 7477 | which means |
| 7478 | \begin_inset Formula $A\to\infty$ |
| 7479 | \end_inset |
| 7480 | |
| 7481 | and rescale |
| 7482 | \begin_inset Formula $r=R/2A$ |
| 7483 | \end_inset |
| 7484 | |
| 7485 | with |
| 7486 | \begin_inset Formula $R$ |
| 7487 | \end_inset |
| 7488 | |
| 7489 | held fixed. |
| 7490 | This means all factors |
| 7491 | \begin_inset Formula $r^{2}\to0$ |
| 7492 | \end_inset |
| 7493 | |
| 7494 | and all the rest works out as expected. |
| 7495 | Good. |
| 7496 | \end_layout |
| 7497 | |
| 7498 | \begin_layout Standard |
| 7499 | The Hedgehog Ansatz is |
| 7500 | \begin_inset Formula |
| 7501 | \[ |
| 7502 | \psi_{+}=u\chi\qquad\psi_{-}=iv\left(\vec{\sigma}\cdot\hat{r}\right)\chi |
| 7503 | \] |
| 7504 | |
| 7505 | \end_inset |
| 7506 | |
| 7507 | with |
| 7508 | \begin_inset Formula $\left(\vec{\sigma}+\vec{\tau}\right)\chi=0$ |
| 7509 | \end_inset |
| 7510 | |
| 7511 | . |
| 7512 | So |
| 7513 | \begin_inset Formula $\left(\vec{\sigma}\cdot\hat{r}\right)^{2}=1$ |
| 7514 | \end_inset |
| 7515 | |
| 7516 | and Claude claims |
| 7517 | \begin_inset Formula $\left(\vec{\sigma}\cdot\hat{L}\right)^{2}\chi=-\chi$ |
| 7518 | \end_inset |
| 7519 | |
| 7520 | for the hedgehog. |
| 7521 | Is this right? |
| 7522 | Well, |
| 7523 | maybe, |
| 7524 | I guess. |
| 7525 | Who knows. |
| 7526 | Assuming it is, |
| 7527 | the radial equations become |
| 7324 | 7528 | \end_layout |
| 7325 | 7529 | |
| 7326 | 7530 | \begin_layout Paragraph* |
| … | … | @@ -7502,6 +7706,30 @@ But for now, |
| 7502 | 7706 | The End. |
| 7503 | 7707 | \end_layout |
| 7504 | 7708 | |
| 7709 | \begin_layout Standard |
| 7710 | Or not. |
| 7711 | It's not that bad. |
| 7712 | I gave in to the vibe, |
| 7713 | and pushing further does not look all that bad. |
| 7714 | The formulas look OK–ish. |
| 7715 | I am copying them into place above, |
| 7716 | without really checking. |
| 7717 | I am guessing that this is previously published material and Claude is regurgitating it correctly. |
| 7718 | I feel like this guy and this girl says |
| 7719 | \begin_inset Quotes eld |
| 7720 | \end_inset |
| 7721 | |
| 7722 | this will feel good |
| 7723 | \begin_inset Quotes erd |
| 7724 | \end_inset |
| 7725 | |
| 7726 | and she goes down on me, |
| 7727 | and what's there to complain about? |
| 7728 | So I'll share some consensual hallucinations with Claude for just a bit longer, |
| 7729 | and see if it's OK. |
| 7730 | So scroll back up for the results. |
| 7731 | \end_layout |
| 7732 | |
| 7505 | 7733 | \begin_layout Subsection* |
| 7506 | 7734 | 28 January 2026 |
| 7507 | 7735 | \end_layout |