| … | … | @@ -6830,6 +6830,170 @@ quantum mechanics |
| 6830 | 6830 | Pop goes the weasel. |
| 6831 | 6831 | \end_layout |
| 6832 | 6832 | |
| 6833 | \begin_layout Subsection* |
| 6834 | 27 January 2026 |
| 6835 | \end_layout |
| 6836 | |
| 6837 | \begin_layout Standard |
| 6838 | Ok, |
| 6839 | So there's stuff I should be doing, |
| 6840 | and then there's a left–over task from last night, |
| 6841 | which has been percolating in my brain for three decades, |
| 6842 | so may as well get around to it and do it. |
| 6843 | Lets try to solve the chiral bad model eqns with the interior of the bag modeled as a hyperbolic space. |
| 6844 | Claude will do the lifting, |
| 6845 | I will ask questions and copy results. |
| 6846 | \end_layout |
| 6847 | |
| 6848 | \begin_layout Standard |
| 6849 | Poincare ball metric with curvature |
| 6850 | \begin_inset Formula $a$ |
| 6851 | \end_inset |
| 6852 | |
| 6853 | , |
| 6854 | conformal factor |
| 6855 | \begin_inset Formula $\lambda=2a/\left(1-r^{2}\right)$ |
| 6856 | \end_inset |
| 6857 | |
| 6858 | |
| 6859 | \begin_inset Formula |
| 6860 | \[ |
| 6861 | ds^{2}=\lambda^{2}\left(dx^{2}+dy^{2}+dz^{2}\right) |
| 6862 | \] |
| 6863 | |
| 6864 | \end_inset |
| 6865 | |
| 6866 | This is conformal to flat space, |
| 6867 | |
| 6868 | \begin_inset Formula |
| 6869 | \[ |
| 6870 | g_{\mu\nu}=\lambda^{2}\delta_{\mu\nu}=e_{\mu}^{a}e_{\nu}^{b}\delta_{ab} |
| 6871 | \] |
| 6872 | |
| 6873 | \end_inset |
| 6874 | |
| 6875 | The dreibein: |
| 6876 | \begin_inset Formula |
| 6877 | \[ |
| 6878 | e_{\mu}^{a}=\lambda\,\delta_{\mu}^{a}\qquad e_{a}^{\mu}=\frac{1}{\lambda}\,\delta_{a}^{\mu} |
| 6879 | \] |
| 6880 | |
| 6881 | \end_inset |
| 6882 | |
| 6883 | Equivalently |
| 6884 | \begin_inset Formula |
| 6885 | \[ |
| 6886 | e^{a}=\lambda dx^{a}\qquad e_{\mu}=\frac{1}{\lambda}\partial_{\mu} |
| 6887 | \] |
| 6888 | |
| 6889 | \end_inset |
| 6890 | |
| 6891 | The spin connection for a conformally flat metric |
| 6892 | \begin_inset Formula |
| 6893 | \[ |
| 6894 | \omega^{ab}=\lambda\left(x^{b}\,dx^{a}-x^{a}\,dx^{b}\right) |
| 6895 | \] |
| 6896 | |
| 6897 | \end_inset |
| 6898 | |
| 6899 | For spinors: |
| 6900 | \begin_inset Formula |
| 6901 | \[ |
| 6902 | \omega_{\mu}=\frac{1}{4}\omega_{\mu}^{ab}\gamma_{a}\gamma_{b} |
| 6903 | \] |
| 6904 | |
| 6905 | \end_inset |
| 6906 | |
| 6907 | |
| 6908 | \end_layout |
| 6909 | |
| 6910 | \begin_layout Standard |
| 6911 | The above dropped the time coordinate, |
| 6912 | but we can just gloss this for now. |
| 6913 | The Dirac eqn (massless) is |
| 6914 | \begin_inset Formula |
| 6915 | \[ |
| 6916 | iD\!\!\!/\,\psi=i\gamma^{a}e_{a}^{\mu}\left(\partial_{\mu}+\omega_{\mu}\right)\psi=0 |
| 6917 | \] |
| 6918 | |
| 6919 | \end_inset |
| 6920 | |
| 6921 | Lets sidetrack to the chiral bag. |
| 6922 | The boundary condition is |
| 6923 | \begin_inset Formula |
| 6924 | \[ |
| 6925 | i\gamma^{\mu}n_{\mu}\psi=\exp\left(i\Theta\gamma_{5}\vec{\tau}\cdot\hat{\pi}\right)\psi |
| 6926 | \] |
| 6927 | |
| 6928 | \end_inset |
| 6929 | |
| 6930 | The hedgehog is |
| 6931 | \begin_inset Formula $\hat{\pi}=\hat{r}$ |
| 6932 | \end_inset |
| 6933 | |
| 6934 | . |
| 6935 | |
| 6936 | \end_layout |
| 6937 | |
| 6938 | \begin_layout Standard |
| 6939 | So we need the Dirac eqn is sphere coords, |
| 6940 | now. |
| 6941 | All the piddly stupid details follow, |
| 6942 | since I can't trust Claude for the final answer. |
| 6943 | And I will make mistakes if I try to solve this in my head. |
| 6944 | So this is a bit remedial, |
| 6945 | but whatever. |
| 6946 | |
| 6947 | \begin_inset Quotes eld |
| 6948 | \end_inset |
| 6949 | |
| 6950 | Good note taking |
| 6951 | \begin_inset Quotes erd |
| 6952 | \end_inset |
| 6953 | |
| 6954 | . |
| 6955 | \end_layout |
| 6956 | |
| 6957 | \begin_layout Standard |
| 6958 | The vierbein |
| 6959 | \begin_inset Formula |
| 6960 | \begin{align*} |
| 6961 | e^{0} & =dt\\ |
| 6962 | e^{1} & =\lambda dr\\ |
| 6963 | e^{2} & =\lambda rd\theta\\ |
| 6964 | e^{3} & =\lambda r\sin\theta d\phi |
| 6965 | \end{align*} |
| 6966 | |
| 6967 | \end_inset |
| 6968 | |
| 6969 | The inverse |
| 6970 | \begin_inset Formula |
| 6971 | \begin{align*} |
| 6972 | e_{0} & =\partial_{t}\\ |
| 6973 | e_{1} & =\frac{1}{\lambda}\partial_{r}\\ |
| 6974 | e_{2} & =\frac{1}{\lambda r}\partial_{\theta}\\ |
| 6975 | e_{3} & =\frac{1}{\lambda r\sin\theta}\partial_{\phi} |
| 6976 | \end{align*} |
| 6977 | |
| 6978 | \end_inset |
| 6979 | |
| 6980 | The spin connection. |
| 6981 | No mixed time–space: |
| 6982 | \begin_inset Formula |
| 6983 | \begin{align*} |
| 6984 | \omega^{0i} & =0\\ |
| 6985 | \omega^{12} & =-\frac{1}{\lambda r}d\theta\\ |
| 6986 | \omega^{13} & =-\frac{1}{\lambda r}\sin\theta\,d\phi\\ |
| 6987 | \omega^{23} & =-\cos\theta\,d\phi |
| 6988 | \end{align*} |
| 6989 | |
| 6990 | \end_inset |
| 6991 | |
| 6992 | I copied these from Claude, |
| 6993 | but they look maybe not right. |
| 6994 | Let me check. |
| 6995 | \end_layout |
| 6996 | |
| 6833 | 6997 | \begin_layout Section* |
| 6834 | 6998 | The End |
| 6835 | 6999 | \end_layout |