| … | … | @@ -7514,7 +7514,7 @@ with |
| 7514 | 7514 | \end_inset |
| 7515 | 7515 | |
| 7516 | 7516 | and Claude claims |
| 7517 | | \begin_inset Formula $\left(\vec{\sigma}\cdot\hat{L}\right)^{2}\chi=-\chi$ |
| 7517 | \begin_inset Formula $\left(\vec{\sigma}\cdot\vec{L}\right)^{2}\chi=-\chi$ |
| 7518 | 7518 | \end_inset |
| 7519 | 7519 | |
| 7520 | 7520 | for the hedgehog. |
| … | … | @@ -7535,25 +7535,25 @@ grand spin quantum number |
| 7535 | 7535 | \end_inset |
| 7536 | 7536 | |
| 7537 | 7537 | |
| 7538 | | \begin_inset Formula $K=0$ |
| 7538 | \begin_inset Formula $k=0$ |
| 7539 | 7539 | \end_inset |
| 7540 | 7540 | |
| 7541 | 7541 | which is OK if all one wants is the ground state, |
| 7542 | 7542 | but clearly fails for the vacuum states. |
| 7543 | 7543 | I didn't catch this early enough, |
| 7544 | 7544 | so the below proceeds with |
| 7545 | | \begin_inset Formula $K=0$ |
| 7545 | \begin_inset Formula $k=0$ |
| 7546 | 7546 | \end_inset |
| 7547 | 7547 | |
| 7548 | 7548 | . |
| 7549 | 7549 | That's OK, |
| 7550 | 7550 | for now. |
| 7551 | | |
| 7551 | We redo it further below. |
| 7552 | 7552 | \end_layout |
| 7553 | 7553 | |
| 7554 | 7554 | \begin_layout Standard |
| 7555 | 7555 | Assuming Claude is otherwise correct, |
| 7556 | | the radial equations become |
| 7556 | the radial equations (for grand spin zero) become |
| 7557 | 7557 | \begin_inset Formula |
| 7558 | 7558 | \begin{align*} |
| 7559 | 7559 | 2AEu & =\left(1-r^{2}\right)v^{\prime}-rv\\ |
| … | … | @@ -7567,7 +7567,22 @@ with |
| 7567 | 7567 | \end_inset |
| 7568 | 7568 | |
| 7569 | 7569 | etc. |
| 7570 | | Substituting, |
| 7570 | XXX When I forced Claude to track grand spin |
| 7571 | \begin_inset Formula $k\ne0$ |
| 7572 | \end_inset |
| 7573 | |
| 7574 | below, |
| 7575 | it begins to appear that the above are not correct. |
| 7576 | Maybe. |
| 7577 | I am once again running out of steam, |
| 7578 | trying to track down the error. |
| 7579 | So the derivation below looks pretty but might be wrong. |
| 7580 | It clearly needs more serious work. |
| 7581 | XXX Caveat Emptor. |
| 7582 | \end_layout |
| 7583 | |
| 7584 | \begin_layout Standard |
| 7585 | Substituting, |
| 7571 | 7586 | Claude claims this gives |
| 7572 | 7587 | \begin_inset Formula |
| 7573 | 7588 | \[ |
| … | … | @@ -7684,10 +7699,119 @@ It seems plausible that this might reduce to sphere bessel functions in the |
| 7684 | 7699 | . |
| 7685 | 7700 | I'm not doing this now; |
| 7686 | 7701 | first we have to get the |
| 7687 | | \begin_inset Formula $K\ne0$ |
| 7702 | \begin_inset Formula $k\ne0$ |
| 7703 | \end_inset |
| 7704 | |
| 7705 | . |
| 7706 | XXX Well, |
| 7707 | in fact, |
| 7708 | the above might be wrong, |
| 7709 | because the |
| 7710 | \begin_inset Formula $k\ne0$ |
| 7711 | \end_inset |
| 7712 | |
| 7713 | efforts below explode into inconsistencies and bad algebra. |
| 7714 | So the results above are suspect, |
| 7715 | too. |
| 7716 | XXX |
| 7717 | \end_layout |
| 7718 | |
| 7719 | \begin_layout Standard |
| 7720 | Here we go. |
| 7721 | I had to force Claude to redo this several times before getting this, |
| 7722 | which smells better than the earlier version |
| 7723 | \begin_inset Formula |
| 7724 | \[ |
| 7725 | \psi_{+}=u\Phi_{k}^{+}+\tilde{u}\Phi_{k}^{-}\qquad\psi_{-}=iv\Phi_{k}^{-}+i\tilde{v}\Phi_{k}^{+} |
| 7726 | \] |
| 7727 | |
| 7728 | \end_inset |
| 7729 | |
| 7730 | for the |
| 7731 | \begin_inset Formula $\Phi_{k}^{\pm}$ |
| 7732 | \end_inset |
| 7733 | |
| 7734 | the spin–isospin spherical harmonics with Casimir operator angular momentum |
| 7735 | \begin_inset Formula $K^{2}\Phi_{k}^{\pm}=k\left(k+1\right)\Phi_{k}^{\pm}$ |
| 7736 | \end_inset |
| 7737 | |
| 7738 | . |
| 7739 | Then |
| 7740 | \begin_inset Formula |
| 7741 | \[ |
| 7742 | \left(\vec{\sigma}\cdot\vec{L}\right)\Phi_{k}^{+}=\lambda\Phi_{k}^{+} |
| 7743 | \] |
| 7744 | |
| 7745 | \end_inset |
| 7746 | |
| 7747 | and the anticommutator identity |
| 7748 | \begin_inset Formula |
| 7749 | \[ |
| 7750 | \left\{ \left(\vec{\sigma}\cdot\hat{r}\right),\left(\vec{\sigma}\cdot\vec{L}\right)\right\} =-2\left(\vec{\sigma}\cdot\hat{r}\right) |
| 7751 | \] |
| 7752 | |
| 7753 | \end_inset |
| 7754 | |
| 7755 | gives |
| 7756 | \begin_inset Formula |
| 7757 | \[ |
| 7758 | \left(\vec{\sigma}\cdot\vec{L}\right)\left[\left(\vec{\sigma}\cdot\hat{r}\right)\Phi_{k}^{+}\right]=-\left(\lambda+2\right)\left(\vec{\sigma}\cdot\hat{r}\right)\Phi_{k}^{+} |
| 7759 | \] |
| 7760 | |
| 7761 | \end_inset |
| 7762 | |
| 7763 | and we seem to be converging on the idea that |
| 7764 | \begin_inset Formula |
| 7765 | \[ |
| 7766 | \Phi_{k}^{-}=\left(\vec{\sigma}\cdot\hat{r}\right)\Phi_{k}^{+} |
| 7767 | \] |
| 7768 | |
| 7769 | \end_inset |
| 7770 | |
| 7771 | which has OK vibes, |
| 7772 | for now. |
| 7773 | \end_layout |
| 7774 | |
| 7775 | \begin_layout Standard |
| 7776 | This means eigenvalues on the lower component are |
| 7777 | \begin_inset Formula $-\lambda-2$ |
| 7778 | \end_inset |
| 7779 | |
| 7780 | . |
| 7781 | For the hedge–hog, |
| 7782 | we get |
| 7783 | \begin_inset Formula $\lambda=k$ |
| 7784 | \end_inset |
| 7785 | |
| 7786 | or |
| 7787 | \begin_inset Formula $\lambda=-\left(k+1\right)$ |
| 7788 | \end_inset |
| 7789 | |
| 7790 | when |
| 7791 | \begin_inset Formula $k\ne0,$ |
| 7792 | \end_inset |
| 7793 | |
| 7794 | while for |
| 7795 | \begin_inset Formula $k=0$ |
| 7796 | \end_inset |
| 7797 | |
| 7798 | the special case is |
| 7799 | \begin_inset Formula $\lambda=-1$ |
| 7688 | 7800 | \end_inset |
| 7689 | 7801 | |
| 7690 | 7802 | . |
| 7803 | Plugging through, |
| 7804 | well, |
| 7805 | at this point, |
| 7806 | Claude runs into serious trouble. |
| 7807 | All sorts of algebra starts fucking up. |
| 7808 | It cannot get the |
| 7809 | \begin_inset Formula $k=0$ |
| 7810 | \end_inset |
| 7811 | |
| 7812 | answer to agree with earlier results. |
| 7813 | I've used up all my allotted time on this task, |
| 7814 | so must stop. |
| 7691 | 7815 | \end_layout |
| 7692 | 7816 | |
| 7693 | 7817 | \begin_layout Standard |
| … | … | @@ -7718,6 +7842,242 @@ Robin boundary condition |
| 7718 | 7842 | |
| 7719 | 7843 | . |
| 7720 | 7844 | Never heard of that but OK. |
| 7845 | The coordinate space version with |
| 7846 | \begin_inset Formula $r$ |
| 7847 | \end_inset |
| 7848 | |
| 7849 | will be simpler to work with. |
| 7850 | \end_layout |
| 7851 | |
| 7852 | \begin_layout Standard |
| 7853 | Lets pause and take stock. |
| 7854 | Lets assume that, |
| 7855 | if/when the algebra above is fixed, |
| 7856 | we get radial functions |
| 7857 | \begin_inset Formula $u\sim{}_{2}F_{1}$ |
| 7858 | \end_inset |
| 7859 | |
| 7860 | . |
| 7861 | These are nice, |
| 7862 | and fairly easy to work with and compute numerically; |
| 7863 | the boundary condition will be solvable in some milliseconds of CPU time, |
| 7864 | so we can do full vacuum sums in dozens of seconds or maybe minutes. |
| 7865 | And then what? |
| 7866 | We have two parameters to play with. |
| 7867 | Naively, |
| 7868 | it seems to be three: |
| 7869 | |
| 7870 | \begin_inset Formula $\Theta$ |
| 7871 | \end_inset |
| 7872 | |
| 7873 | , |
| 7874 | |
| 7875 | \begin_inset Formula $R$ |
| 7876 | \end_inset |
| 7877 | |
| 7878 | and |
| 7879 | \begin_inset Formula $A$ |
| 7880 | \end_inset |
| 7881 | |
| 7882 | but I think we can maybe redo to hold |
| 7883 | \begin_inset Formula $RA$ |
| 7884 | \end_inset |
| 7885 | |
| 7886 | fixed, |
| 7887 | so that it reduces to two paramters. |
| 7888 | Or something like that. |
| 7889 | In the flat–space case it was only one parameter, |
| 7890 | |
| 7891 | \begin_inset Formula $\Theta$ |
| 7892 | \end_inset |
| 7893 | |
| 7894 | because the radial coord was always multiplied by energy. |
| 7895 | Here, |
| 7896 | we already see that |
| 7897 | \begin_inset Formula $\epsilon=2AE$ |
| 7898 | \end_inset |
| 7899 | |
| 7900 | and the eigenvalues will be for |
| 7901 | \begin_inset Formula $\epsilon$ |
| 7902 | \end_inset |
| 7903 | |
| 7904 | so... |
| 7905 | whatever. |
| 7906 | Its somehow two parameters. |
| 7907 | \end_layout |
| 7908 | |
| 7909 | \begin_layout Paragraph* |
| 7910 | Why bother? |
| 7911 | \end_layout |
| 7912 | |
| 7913 | \begin_layout Standard |
| 7914 | So, |
| 7915 | now come two fun questions: |
| 7916 | Can the energy be made absolutely independent of the bag radius? |
| 7917 | I assume the answer will be yes. |
| 7918 | Doing so will then fix the curvature as a function of bag radius. |
| 7919 | Presumably, |
| 7920 | the curvature will got to |
| 7921 | \begin_inset Formula $\kappa\to-1$ |
| 7922 | \end_inset |
| 7923 | |
| 7924 | as the bag radius goes to a femtometer. |
| 7925 | This would be the desired model of confinement and asymptotic freedom. |
| 7926 | Next, |
| 7927 | what about the chiral condensate? |
| 7928 | In flat space, |
| 7929 | we had the sigma model |
| 7930 | \begin_inset Formula $\sigma$ |
| 7931 | \end_inset |
| 7932 | |
| 7933 | being exactly zero inside the bag, |
| 7934 | and becoming non–zero only exactly on the boundary. |
| 7935 | So, |
| 7936 | in flat space, |
| 7937 | the bag interior did not look like a sigma model. |
| 7938 | This was a big disappointment. |
| 7939 | However, |
| 7940 | now, |
| 7941 | in curved space, |
| 7942 | could it be otherwise? |
| 7943 | \end_layout |
| 7944 | |
| 7945 | \begin_layout Standard |
| 7946 | Why do it this way? |
| 7947 | The core problem is that geometry is hard. |
| 7948 | If I could do these calculations |
| 7949 | \begin_inset Quotes eld |
| 7950 | \end_inset |
| 7951 | |
| 7952 | in general |
| 7953 | \begin_inset Quotes erd |
| 7954 | \end_inset |
| 7955 | |
| 7956 | , |
| 7957 | then it seems there is almost surely some Atiyah–Singer magic happening. |
| 7958 | But I can't, |
| 7959 | and even if I could I might be disappointed. |
| 7960 | So, |
| 7961 | as a path–finder, |
| 7962 | do the calculations assuming sphere symmetry, |
| 7963 | and explore this simplified model. |
| 7964 | If this simplified model allows the sigma model to emerge from the vacuum, |
| 7965 | well then happy day. |
| 7966 | If not, |
| 7967 | then hopes are dashed. |
| 7968 | But it would have gotten much farther than would otherwise have been possible in the |
| 7969 | \begin_inset Quotes eld |
| 7970 | \end_inset |
| 7971 | |
| 7972 | full general |
| 7973 | \begin_inset Quotes erd |
| 7974 | \end_inset |
| 7975 | |
| 7976 | case. |
| 7977 | \end_layout |
| 7978 | |
| 7979 | \begin_layout Standard |
| 7980 | Suppose the dreams are fulfilled. |
| 7981 | Then the breakthrough is that the AdS boundary condition is re–identified with the bag boundary, |
| 7982 | instead of being some cosmological thing. |
| 7983 | The boundary is also topologically non–trivial, |
| 7984 | so the chiral model now lets us ask about harmonic forms on this combined structure. |
| 7985 | viz a spin manifold having some kind of dynamics inside the bag, |
| 7986 | that spin manifold having some spectrum; |
| 7987 | that spectrum, |
| 7988 | or rather, |
| 7989 | the spectral asymmetry looking like (or I suppose being exactly dual to) a chiral model. |
| 7990 | That would be pretty cool and spiffy, |
| 7991 | if it were true and could be done. |
| 7992 | \end_layout |
| 7993 | |
| 7994 | \begin_layout Standard |
| 7995 | But now, |
| 7996 | I bop my nose up against some age–old (and old–age) concerns. |
| 7997 | The age–old concern is |
| 7998 | \begin_inset Quotes eld |
| 7999 | \end_inset |
| 8000 | |
| 8001 | what is the most important thing I should work on right now? |
| 8002 | \begin_inset Quotes erd |
| 8003 | \end_inset |
| 8004 | |
| 8005 | and the old–age concern is |
| 8006 | \begin_inset Quotes eld |
| 8007 | \end_inset |
| 8008 | |
| 8009 | shit I'm running out of time, |
| 8010 | so I better get the correct answer to the previous question. |
| 8011 | \begin_inset Quotes erd |
| 8012 | \end_inset |
| 8013 | |
| 8014 | |
| 8015 | \end_layout |
| 8016 | |
| 8017 | \begin_layout Standard |
| 8018 | The problem with doing geometry is my meta–imagination. |
| 8019 | The geometrical, |
| 8020 | algebraic universe appears to be constructable by specifying a collection of axioms, |
| 8021 | and a collection of inference rules, |
| 8022 | and turning the crank, |
| 8023 | and discovering gazzillions of identities. |
| 8024 | This is how math has always been done, |
| 8025 | traditionally: |
| 8026 | the more abstract and general and useful identities are given a name, |
| 8027 | placing the discoverer into the ranks of history. |
| 8028 | While still alive, |
| 8029 | mathematicians derive pleasure from finding identities, |
| 8030 | stating and proving theorems, |
| 8031 | grasping at hypothesis. |
| 8032 | The mathematical physics community is less rigorous, |
| 8033 | but questions of harmonic forms on Riemann surfaces certainly counts as mathematical physics, |
| 8034 | especially when minor modifications give things like Landau superconductivity or Yang–Mills eqns. |
| 8035 | So its eminently physics, |
| 8036 | in that sense. |
| 8037 | And the academic grind–core, |
| 8038 | this is what they do: |
| 8039 | they have fun cranking on this stuff. |
| 8040 | And in the last 2-4 years, |
| 8041 | they have been learning how the get LLM's to do some of the more boring parts of the calculations. |
| 8042 | Just like I have learned above. |
| 8043 | \end_layout |
| 8044 | |
| 8045 | \begin_layout Standard |
| 8046 | The personal problem, |
| 8047 | for me, |
| 8048 | is, |
| 8049 | well, |
| 8050 | suppose I do discover that some whizzy chiral condensate happens to be equivalent to, |
| 8051 | dual to some Riemannian manifold? |
| 8052 | That would be pretty cool. |
| 8053 | Would it matter? |
| 8054 | Well, |
| 8055 | I would have had fun, |
| 8056 | once again, |
| 8057 | in the process, |
| 8058 | but I've had lots of fun over my lifetime, |
| 8059 | even if the results of most of it have been utterly unimportant in some local scheme of things. |
| 8060 | I mean, |
| 8061 | how does this work, |
| 8062 | anyway? |
| 8063 | Steven Weinberg wrote an excellent book on gravitation, |
| 8064 | got the Nobel prize in physics for QFT work, |
| 8065 | scowled at me when I approached him at a conference, |
| 8066 | taught lots of students at UTexas for a good salary, |
| 8067 | and then he died. |
| 8068 | Whoopla! |
| 8069 | Hot patootie, |
| 8070 | bless my soul, |
| 8071 | I really love that rock–n–roll. |
| 8072 | \end_layout |
| 8073 | |
| 8074 | \begin_layout Standard |
| 8075 | Anyway, |
| 8076 | the upshot is that I can get paid for doing AI research; |
| 8077 | I cannot get paid for doing mathematical physics. |
| 8078 | This is my current economic environment. |
| 8079 | (This section above was written after the section below. |
| 8080 | Its placed out of line to conetextualize better with the second attempt to solve the eqns above.) |
| 7721 | 8081 | \end_layout |
| 7722 | 8082 | |
| 7723 | 8083 | \begin_layout Paragraph* |