| … | … | @@ -2513,6 +2513,410 @@ explanation |
| 2513 | 2513 | well, |
| 2514 | 2514 | so far, |
| 2515 | 2515 | I can't offer one. |
| 2516 | \end_layout |
| 2517 | |
| 2518 | \begin_layout Standard |
| 2519 | So what have I learned, |
| 2520 | so far? |
| 2521 | Well, |
| 2522 | the natural topology applies to the past (and not some |
| 2523 | \begin_inset Formula $p$ |
| 2524 | \end_inset |
| 2525 | |
| 2526 | –adic or archimedian metric topology) and that Cartesian space is characterized by the reals (and not the Cantor set) and that the present is a Hilbert space (of very high dimension, |
| 2527 | if not infinite, |
| 2528 | then at least of dimension |
| 2529 | \begin_inset Formula $10^{80}$ |
| 2530 | \end_inset |
| 2531 | |
| 2532 | , |
| 2533 | the number of fermions in the universe) and that there is a map from the present, |
| 2534 | from this Hilbert space, |
| 2535 | to the Cartesian past, |
| 2536 | having the notational algebraic form |
| 2537 | \begin_inset Formula $\left\langle x\right.\left|\psi\right\rangle $ |
| 2538 | \end_inset |
| 2539 | |
| 2540 | . |
| 2541 | |
| 2542 | \end_layout |
| 2543 | |
| 2544 | \begin_layout Standard |
| 2545 | What do I not know or understand? |
| 2546 | How it is that Hilbert–space–occupying quanta in the present appear to get localized in the Cartesian past. |
| 2547 | I look out my window, |
| 2548 | and I see a tree, |
| 2549 | and I know that tree is made of fermions which have become localized in the past, |
| 2550 | not with probability |
| 2551 | \begin_inset Formula $p=\left|\left\langle x\right.\left|\psi\right\rangle \right|^{2}$ |
| 2552 | \end_inset |
| 2553 | |
| 2554 | but with probability |
| 2555 | \begin_inset Formula $p=1$ |
| 2556 | \end_inset |
| 2557 | |
| 2558 | : |
| 2559 | those fermions in my tree are located there, |
| 2560 | and not in some mountain in Colorado. |
| 2561 | It is this localization into Cartesian space from Hilbert space, |
| 2562 | by means of probabilistic calculations, |
| 2563 | that is the source of unease. |
| 2564 | |
| 2565 | \end_layout |
| 2566 | |
| 2567 | \begin_layout Standard |
| 2568 | I keep using the word |
| 2569 | \begin_inset Quotes eld |
| 2570 | \end_inset |
| 2571 | |
| 2572 | fermion |
| 2573 | \begin_inset Quotes erd |
| 2574 | \end_inset |
| 2575 | |
| 2576 | because spin–1/2 is somehow key to this process. |
| 2577 | Light is massless, |
| 2578 | and lives on–shell for the most part, |
| 2579 | and appears to be |
| 2580 | \begin_inset Quotes eld |
| 2581 | \end_inset |
| 2582 | |
| 2583 | part of the present |
| 2584 | \begin_inset Quotes erd |
| 2585 | \end_inset |
| 2586 | |
| 2587 | , |
| 2588 | only. |
| 2589 | I cannot put light |
| 2590 | \begin_inset Quotes eld |
| 2591 | \end_inset |
| 2592 | |
| 2593 | into a box |
| 2594 | \begin_inset Quotes erd |
| 2595 | \end_inset |
| 2596 | |
| 2597 | (say with mirrors and lenses) and publish a paper that says |
| 2598 | \begin_inset Quotes eld |
| 2599 | \end_inset |
| 2600 | |
| 2601 | my box, |
| 2602 | which existed in the past, |
| 2603 | briefly caged light and fixed it to a physical location |
| 2604 | \begin_inset Formula $\left(x,y,z\right)$ |
| 2605 | \end_inset |
| 2606 | |
| 2607 | for an extended period of time. |
| 2608 | \begin_inset Quotes erd |
| 2609 | \end_inset |
| 2610 | |
| 2611 | Because, |
| 2612 | if I made that claim, |
| 2613 | it would be tantamount to making the claim that I was able to manipulate light so that |
| 2614 | \begin_inset Formula $x^{2}+y^{2}+z^{2}-t^{2}\ne0$ |
| 2615 | \end_inset |
| 2616 | |
| 2617 | for an extended range of time |
| 2618 | \begin_inset Formula $\Delta t$ |
| 2619 | \end_inset |
| 2620 | |
| 2621 | – but this is impossible. |
| 2622 | One cannot grab a hold of light and keep it off–shell, |
| 2623 | the way I can keep a book on my library shelf. |
| 2624 | |
| 2625 | \end_layout |
| 2626 | |
| 2627 | \begin_layout Standard |
| 2628 | Thus, |
| 2629 | I conclude that light is necessarily a part of here and now; |
| 2630 | and more: |
| 2631 | the past null light–cone is a part of here–and–now, |
| 2632 | also. |
| 2633 | Of course, |
| 2634 | one can hypothesize massive Klein–Gordon spin–zero particles, |
| 2635 | but these do not exist in nature. |
| 2636 | Of course, |
| 2637 | we can point at the |
| 2638 | \begin_inset Formula $W^{\pm}$ |
| 2639 | \end_inset |
| 2640 | |
| 2641 | and |
| 2642 | \begin_inset Formula $Z^{0}$ |
| 2643 | \end_inset |
| 2644 | |
| 2645 | bosons, |
| 2646 | and say that they have mass, |
| 2647 | but these are dismissively microscopic; |
| 2648 | the relationship between past and present in that microscopic world might be, |
| 2649 | could be, |
| 2650 | probably is rather different from the relationship between past and present in our macroscopic world. |
| 2651 | So, |
| 2652 | for the present discussion, |
| 2653 | the microscopic theory encoded in the Standard Model has to be dismissed as irrelevant. |
| 2654 | The macroscopic theory of cosmology, |
| 2655 | however, |
| 2656 | has to be a part of here–and–now: |
| 2657 | I have seen the Horse–Head Nebula with mine own bare–naked eyes (admittedly with the help of a light amplifier and a rather large scope, |
| 2658 | some 18 inch or 24 inch Newtonian.) Those photons were very much on–shell, |
| 2659 | and were very much a part of here–and–now for the duration that my eyeball was pressed against the reticle. |
| 2660 | To conclude: |
| 2661 | fermions are the bridge between past and present. |
| 2662 | \end_layout |
| 2663 | |
| 2664 | \begin_layout Standard |
| 2665 | I guess that this forces me into pondering (yet again) |
| 2666 | \begin_inset Quotes eld |
| 2667 | \end_inset |
| 2668 | |
| 2669 | what about spin structures |
| 2670 | \begin_inset Quotes erd |
| 2671 | \end_inset |
| 2672 | |
| 2673 | ? |
| 2674 | If the past is adamantly Riemannian, |
| 2675 | I also know that I can define spin structures on Riemannian manifolds, |
| 2676 | and those spin structures are somehow |
| 2677 | \begin_inset Quotes eld |
| 2678 | \end_inset |
| 2679 | |
| 2680 | more general |
| 2681 | \begin_inset Quotes erd |
| 2682 | \end_inset |
| 2683 | |
| 2684 | or |
| 2685 | \begin_inset Quotes eld |
| 2686 | \end_inset |
| 2687 | |
| 2688 | more flexible |
| 2689 | \begin_inset Quotes erd |
| 2690 | \end_inset |
| 2691 | |
| 2692 | or |
| 2693 | \begin_inset Quotes eld |
| 2694 | \end_inset |
| 2695 | |
| 2696 | encode more structure |
| 2697 | \begin_inset Quotes erd |
| 2698 | \end_inset |
| 2699 | |
| 2700 | than the Riemannian manifold itself. |
| 2701 | What I do not know of is how to Hilbert–space–ize the Clifford algebras that are used to construct the spin structures... |
| 2702 | or rather, |
| 2703 | I don't know who to quantize. |
| 2704 | I know how to define a Dirac operator on any Riemannian manifold, |
| 2705 | and the corresponding spinor fields, |
| 2706 | but these fields are |
| 2707 | \begin_inset Quotes eld |
| 2708 | \end_inset |
| 2709 | |
| 2710 | classical |
| 2711 | \begin_inset Quotes erd |
| 2712 | \end_inset |
| 2713 | |
| 2714 | ; |
| 2715 | they are not quanta. |
| 2716 | They obey equations of motion given by that Dirac operator, |
| 2717 | and they |
| 2718 | \begin_inset Quotes eld |
| 2719 | \end_inset |
| 2720 | |
| 2721 | propagate |
| 2722 | \begin_inset Quotes erd |
| 2723 | \end_inset |
| 2724 | |
| 2725 | on–shell. |
| 2726 | Or rather, |
| 2727 | solving them gives a singular solution, |
| 2728 | the |
| 2729 | \begin_inset Quotes eld |
| 2730 | \end_inset |
| 2731 | |
| 2732 | classical |
| 2733 | \begin_inset Quotes erd |
| 2734 | \end_inset |
| 2735 | |
| 2736 | solution, |
| 2737 | and not a superposition of solutions. |
| 2738 | There is only one manifold, |
| 2739 | there is only one field on that manifold, |
| 2740 | there are no |
| 2741 | \begin_inset Quotes eld |
| 2742 | \end_inset |
| 2743 | |
| 2744 | many worlds |
| 2745 | \begin_inset Quotes erd |
| 2746 | \end_inset |
| 2747 | |
| 2748 | . |
| 2749 | To conclude, |
| 2750 | I know how to define fermions in such a way that they obey classical equations of motion, |
| 2751 | and form a classical, |
| 2752 | non–quantized field. |
| 2753 | \end_layout |
| 2754 | |
| 2755 | \begin_layout Standard |
| 2756 | Do I know how to create a quantized fermion field? |
| 2757 | I dunno. |
| 2758 | This is a trap. |
| 2759 | Answering this question is a trap. |
| 2760 | I can say |
| 2761 | \begin_inset Quotes eld |
| 2762 | \end_inset |
| 2763 | |
| 2764 | yes, |
| 2765 | of course I do! |
| 2766 | \begin_inset Quotes erd |
| 2767 | \end_inset |
| 2768 | |
| 2769 | and jot down |
| 2770 | \begin_inset Formula $Z=\int e^{-iA}$ |
| 2771 | \end_inset |
| 2772 | |
| 2773 | for some action |
| 2774 | \begin_inset Formula $A$ |
| 2775 | \end_inset |
| 2776 | |
| 2777 | that includes fermion fields and so |
| 2778 | \begin_inset Quotes eld |
| 2779 | \end_inset |
| 2780 | |
| 2781 | a hah, |
| 2782 | I have quantized it! |
| 2783 | \begin_inset Quotes erd |
| 2784 | \end_inset |
| 2785 | |
| 2786 | This is true in some very narrow, |
| 2787 | technical sense, |
| 2788 | but false in the general sense, |
| 2789 | because if those fermions have mass, |
| 2790 | I don't know how to write down the back–reaction of those with gravity. |
| 2791 | I don't know how to do |
| 2792 | \begin_inset Quotes eld |
| 2793 | \end_inset |
| 2794 | |
| 2795 | many–worlds with gravity |
| 2796 | \begin_inset Quotes erd |
| 2797 | \end_inset |
| 2798 | |
| 2799 | . |
| 2800 | But lets ignore gravity for now. |
| 2801 | I'm not sure I know how to go from second quantization back to first quantization, |
| 2802 | or how to go from the Lagrangian description back to the Hamiltonian description. |
| 2803 | Gut sense seems to say that the Hamiltonian description is appropriate for the here–and–now, |
| 2804 | because the Lagrangian description requires time derivatives, |
| 2805 | and time is problematic; |
| 2806 | the Hamiltonian description avoids the need to have some explicit time coordinate. |
| 2807 | Now, |
| 2808 | I do know that if I want to toggle between these two interaction pictures, |
| 2809 | the Lagrangian and the Hamiltonian, |
| 2810 | and I want these two to have gauge fields in them, |
| 2811 | then I have to invoke the mechanics of Fadeev–Popov ghosts. |
| 2812 | But |
| 2813 | \begin_inset Quotes eld |
| 2814 | \end_inset |
| 2815 | |
| 2816 | that's just mechanics |
| 2817 | \begin_inset Quotes erd |
| 2818 | \end_inset |
| 2819 | |
| 2820 | , |
| 2821 | right? |
| 2822 | There aren't dynamical equations, |
| 2823 | there, |
| 2824 | for these ghosts. |
| 2825 | And that's OK, |
| 2826 | because if they were dynamical, |
| 2827 | it would be a problem. |
| 2828 | \end_layout |
| 2829 | |
| 2830 | \begin_layout Standard |
| 2831 | So is this a red herring, |
| 2832 | a distraction, |
| 2833 | or is it some necessary ingredient? |
| 2834 | If I want to say that the bridge between past and present is embodied in fermions, |
| 2835 | and if I want to say that light, |
| 2836 | or gauge fields, |
| 2837 | exist only in the present, |
| 2838 | and never in the past, |
| 2839 | and if I want to employ a Hamiltonian picture for the present, |
| 2840 | then I guess I have to also include Fadeev–Popov ghosts as one of the ingredients for the present? |
| 2841 | Or is that some bogus arcane distraction? |
| 2842 | I've already claimed that |
| 2843 | \begin_inset Quotes eld |
| 2844 | \end_inset |
| 2845 | |
| 2846 | gravitation is a bogus arcane distraction |
| 2847 | \begin_inset Quotes erd |
| 2848 | \end_inset |
| 2849 | |
| 2850 | , |
| 2851 | so why not FP? |
| 2852 | Is FP just some accounting tool, |
| 2853 | needed to make the books balance, |
| 2854 | or is it something more? |
| 2855 | The ghosts are not dynamical, |
| 2856 | which is great, |
| 2857 | because, |
| 2858 | in a sense, |
| 2859 | only the dynamical things in the present can enter the past. |
| 2860 | Well, |
| 2861 | photons are dynamical, |
| 2862 | but are not part of the past... |
| 2863 | sometimes, |
| 2864 | as I write this stuff, |
| 2865 | it feels like stupid word games. |
| 2866 | Who is to say what is or is not part of the past? |
| 2867 | We can't go there, |
| 2868 | we are imprisoned in the present, |
| 2869 | and have only memories of the past, |
| 2870 | so its kind of nuts to say |
| 2871 | \begin_inset Quotes eld |
| 2872 | \end_inset |
| 2873 | |
| 2874 | something is in the past |
| 2875 | \begin_inset Quotes erd |
| 2876 | \end_inset |
| 2877 | |
| 2878 | , |
| 2879 | when we can only say |
| 2880 | \begin_inset Quotes eld |
| 2881 | \end_inset |
| 2882 | |
| 2883 | there is a phenomenon called object persistence that is fundamental to perception and memory |
| 2884 | \begin_inset Quotes erd |
| 2885 | \end_inset |
| 2886 | |
| 2887 | . |
| 2888 | The word games identify fermions as having something to do with object persistance... |
| 2889 | but the word games also seem to indicate that |
| 2890 | \begin_inset Quotes eld |
| 2891 | \end_inset |
| 2892 | |
| 2893 | the past does not exist |
| 2894 | \begin_inset Quotes erd |
| 2895 | \end_inset |
| 2896 | |
| 2897 | (except in the platonic sense) and that the past is, |
| 2898 | properly speaking, |
| 2899 | an impression, |
| 2900 | a structure, |
| 2901 | encoded/entangled into the here–and–now of the present. |
| 2902 | \end_layout |
| 2903 | |
| 2904 | \begin_layout Standard |
| 2905 | OK, |
| 2906 | well, |
| 2907 | I'm hitting a brick wall, |
| 2908 | again. |
| 2909 | So perhaps this is a good time to call it quits. |
| 2910 | Is there a summary I can make? |
| 2911 | Well, |
| 2912 | not really... |
| 2913 | I've listed many ingredients for the cake, |
| 2914 | with the newest one, |
| 2915 | FP ghosts being either a distraction, |
| 2916 | or being perhaps somehow important. |
| 2917 | Fuck me. |
| 2918 | I cannot summarize this mess. |
| 2919 | Later. |
| 2516 | 2920 | |
| 2517 | 2921 | \end_layout |
| 2518 | 2922 | |