| … | … | @@ -6845,14 +6845,18 @@ Ok, |
| 6845 | 6845 | I will ask questions and copy results. |
| 6846 | 6846 | \end_layout |
| 6847 | 6847 | |
| 6848 | \begin_layout Paragraph* |
| 6849 | Metric |
| 6850 | \end_layout |
| 6851 | |
| 6848 | 6852 | \begin_layout Standard |
| 6849 | 6853 | Poincare ball metric with curvature |
| 6850 | | \begin_inset Formula $a$ |
| 6854 | \begin_inset Formula $A$ |
| 6851 | 6855 | \end_inset |
| 6852 | 6856 | |
| 6853 | 6857 | , |
| 6854 | 6858 | conformal factor |
| 6855 | | \begin_inset Formula $\lambda=2a/\left(1-r^{2}\right)$ |
| 6859 | \begin_inset Formula $\lambda=2A/\left(1-r^{2}\right)$ |
| 6856 | 6860 | \end_inset |
| 6857 | 6861 | |
| 6858 | 6862 | |
| … | … | @@ -6875,7 +6879,7 @@ g_{\mu\nu}=\lambda^{2}\delta_{\mu\nu}=e_{\mu}^{a}e_{\nu}^{b}\delta_{ab} |
| 6875 | 6879 | The dreibein: |
| 6876 | 6880 | \begin_inset Formula |
| 6877 | 6881 | \[ |
| 6878 | | e_{\mu}^{a}=\lambda\,\delta_{\mu}^{a}\qquad e_{a}^{\mu}=\frac{1}{\lambda}\,\delta_{a}^{\mu} |
| 6882 | e_{\;\mu}^{a}=\lambda\,\delta_{\mu}^{a}\qquad e_{a}^{\mu}=\frac{1}{\lambda}\,\delta_{a}^{\mu} |
| 6879 | 6883 | \] |
| 6880 | 6884 | |
| 6881 | 6885 | \end_inset |
| … | … | @@ -6888,14 +6892,116 @@ e^{a}=\lambda dx^{a}\qquad e_{\mu}=\frac{1}{\lambda}\partial_{\mu} |
| 6888 | 6892 | |
| 6889 | 6893 | \end_inset |
| 6890 | 6894 | |
| 6891 | | The spin connection for a conformally flat metric |
| 6895 | |
| 6896 | \end_layout |
| 6897 | |
| 6898 | \begin_layout Paragraph* |
| 6899 | Spin connection |
| 6900 | \end_layout |
| 6901 | |
| 6902 | \begin_layout Standard |
| 6903 | The torsion–free spin connection is |
| 6904 | \begin_inset Formula |
| 6905 | \[ |
| 6906 | de^{a}+\omega_{\;b}^{a}\wedge e^{b}=0 |
| 6907 | \] |
| 6908 | |
| 6909 | \end_inset |
| 6910 | |
| 6911 | The anholonomy coefficients aka the Ricci rotation coefficients: |
| 6912 | \begin_inset Formula |
| 6913 | \[ |
| 6914 | \Omega_{ab}^{\;\;c}=e_{a}^{\;\mu}e_{b}^{\;\nu}\left(\partial_{\mu}e_{\;\nu}^{c}-\partial_{\nu}e_{\;\mu}^{c}\right) |
| 6915 | \] |
| 6916 | |
| 6917 | \end_inset |
| 6918 | |
| 6919 | The above formula is vibe–coded but it smells right to me. |
| 6920 | It matches what I remember from textbooks. |
| 6921 | The spin connection: |
| 6922 | \begin_inset Formula |
| 6923 | \[ |
| 6924 | \omega_{abc}=\frac{1}{2}\left(\Omega_{abc}-\Omega_{bca}+\Omega_{cab}\right) |
| 6925 | \] |
| 6926 | |
| 6927 | \end_inset |
| 6928 | |
| 6929 | and |
| 6930 | \begin_inset Formula |
| 6931 | \[ |
| 6932 | \omega_{\;\;\mu}^{ab}=\omega_{\;\;c}^{ab}e_{\;\mu}^{c} |
| 6933 | \] |
| 6934 | |
| 6935 | \end_inset |
| 6936 | |
| 6937 | |
| 6938 | \end_layout |
| 6939 | |
| 6940 | \begin_layout Paragraph* |
| 6941 | Conformally flat spin connection |
| 6942 | \end_layout |
| 6943 | |
| 6944 | \begin_layout Standard |
| 6945 | For |
| 6946 | \begin_inset Formula $e_{\;\mu}^{a}=\lambda\,\delta_{\mu}^{a}$ |
| 6947 | \end_inset |
| 6948 | |
| 6949 | this works out to |
| 6950 | \begin_inset Formula |
| 6951 | \[ |
| 6952 | \omega_{\;\;\mu}^{ab}=\frac{1}{\lambda}\left(\delta_{\mu}^{a}\partial^{b}\lambda-\delta_{\mu}^{b}\partial^{a}\lambda\right) |
| 6953 | \] |
| 6954 | |
| 6955 | \end_inset |
| 6956 | |
| 6957 | or |
| 6958 | \end_layout |
| 6959 | |
| 6960 | \begin_layout Standard |
| 6961 | \begin_inset Formula |
| 6962 | \[ |
| 6963 | \omega^{ab}=\frac{1}{\lambda}\left(dx^{a}\,\partial^{b}\lambda-dx^{b}\,\partial^{a}\lambda\right) |
| 6964 | \] |
| 6965 | |
| 6966 | \end_inset |
| 6967 | |
| 6968 | so for |
| 6969 | \begin_inset Formula $\lambda=2A/\left(1-r^{2}\right)$ |
| 6970 | \end_inset |
| 6971 | |
| 6972 | we get |
| 6973 | \begin_inset Formula |
| 6974 | \[ |
| 6975 | \partial^{a}\lambda=\frac{4A}{\left(1-r^{2}\right)^{2}}r\partial^{a}r=\frac{\lambda^{2}}{A}r\partial^{a}r=\frac{\lambda^{2}}{A}x^{a} |
| 6976 | \] |
| 6977 | |
| 6978 | \end_inset |
| 6979 | |
| 6980 | Equivalently: |
| 6892 | 6981 | \begin_inset Formula |
| 6893 | 6982 | \[ |
| 6894 | | \omega^{ab}=\lambda\left(x^{b}\,dx^{a}-x^{a}\,dx^{b}\right) |
| 6983 | \partial_{\mu}\lambda=\frac{\lambda^{2}}{A}x_{\mu} |
| 6895 | 6984 | \] |
| 6896 | 6985 | |
| 6897 | 6986 | \end_inset |
| 6898 | 6987 | |
| 6988 | and so as before, |
| 6989 | in Cartesian: |
| 6990 | \begin_inset Formula |
| 6991 | \[ |
| 6992 | \omega^{ab}=\frac{\lambda}{A}\left(x^{b}dx^{a}-x^{a}dx^{b}\right) |
| 6993 | \] |
| 6994 | |
| 6995 | \end_inset |
| 6996 | |
| 6997 | |
| 6998 | \end_layout |
| 6999 | |
| 7000 | \begin_layout Paragraph* |
| 7001 | Spinors |
| 7002 | \end_layout |
| 7003 | |
| 7004 | \begin_layout Standard |
| 6899 | 7005 | For spinors: |
| 6900 | 7006 | \begin_inset Formula |
| 6901 | 7007 | \[ |
| … | … | @@ -6918,6 +7024,14 @@ iD\!\!\!/\,\psi=i\gamma^{a}e_{a}^{\mu}\left(\partial_{\mu}+\omega_{\mu}\right)\p |
| 6918 | 7024 | |
| 6919 | 7025 | \end_inset |
| 6920 | 7026 | |
| 7027 | |
| 7028 | \end_layout |
| 7029 | |
| 7030 | \begin_layout Paragraph* |
| 7031 | Chiral bag |
| 7032 | \end_layout |
| 7033 | |
| 7034 | \begin_layout Standard |
| 6921 | 7035 | Lets sidetrack to the chiral bag. |
| 6922 | 7036 | The boundary condition is |
| 6923 | 7037 | \begin_inset Formula |
| … | … | @@ -6932,7 +7046,10 @@ The hedgehog is |
| 6932 | 7046 | \end_inset |
| 6933 | 7047 | |
| 6934 | 7048 | . |
| 6935 | | |
| 7049 | \end_layout |
| 7050 | |
| 7051 | \begin_layout Paragraph* |
| 7052 | Dirac in sphere coords |
| 6936 | 7053 | \end_layout |
| 6937 | 7054 | |
| 6938 | 7055 | \begin_layout Standard |