| … | … | @@ -2917,9 +2917,528 @@ OK, |
| 2917 | 2917 | Fuck me. |
| 2918 | 2918 | I cannot summarize this mess. |
| 2919 | 2919 | Later. |
| 2920 | \end_layout |
| 2921 | |
| 2922 | \begin_layout Subsection* |
| 2923 | 27 Jan 2026, |
| 2924 | well past midnight |
| 2925 | \end_layout |
| 2926 | |
| 2927 | \begin_layout Standard |
| 2928 | If I don't write this down now, |
| 2929 | I'll just feel like shit tomorrow. |
| 2930 | I've been putting this off too long. |
| 2931 | Its about haecceity. |
| 2932 | \end_layout |
| 2933 | |
| 2934 | \begin_layout Standard |
| 2935 | So how do we know space is three–dimensional? |
| 2936 | At the macro level, |
| 2937 | it is obvious. |
| 2938 | At the micro level, |
| 2939 | we have the hydrogen atom, |
| 2940 | and the solution of Schrodinger's eqn. |
| 2941 | which gives the correct spectrum only for 3D and so we know that space is also 3D down to at least the atomic level. |
| 2942 | After that, |
| 2943 | we have the Standard Model, |
| 2944 | which tells us space is 3D down to the nuclear level. |
| 2945 | Certainly, |
| 2946 | textbook–standard QCD is built out of 3D space, |
| 2947 | but still I keep daydreaming that somehow QCD is dual to some hyperbolic space all glued up at the chiral bag boundary. |
| 2948 | Hmm. |
| 2949 | Oddly, |
| 2950 | I never even attempted to write down the equations for that. |
| 2951 | Why? |
| 2952 | I dunno. |
| 2953 | Below that, |
| 2954 | perhaps there is, |
| 2955 | and perhaps there are not some kind of strings or branes in some other dimensions, |
| 2956 | but this seems irrelevant for the present discussion. |
| 2957 | \end_layout |
| 2958 | |
| 2959 | \begin_layout Standard |
| 2960 | (Footnote: |
| 2961 | so how do I write down free fermions on a 3D space with hyperbolic metric? |
| 2962 | Fuck me. |
| 2963 | Do I need to redo my PhD thesis? |
| 2964 | Is this question just another waste of time? |
| 2965 | I mean, |
| 2966 | I suppose someone has already solved this, |
| 2967 | and how much am I really curious about it? |
| 2968 | I will ask Claude tomorrow.) |
| 2969 | \end_layout |
| 2970 | |
| 2971 | \begin_layout Standard |
| 2972 | So what is the haecceity of the hydrogen atom? |
| 2973 | Let me step back: |
| 2974 | does time exist in an empty universe? |
| 2975 | Well, |
| 2976 | if by |
| 2977 | \begin_inset Quotes eld |
| 2978 | \end_inset |
| 2979 | |
| 2980 | empty universe |
| 2981 | \begin_inset Quotes erd |
| 2982 | \end_inset |
| 2983 | |
| 2984 | one means |
| 2985 | \begin_inset Quotes eld |
| 2986 | \end_inset |
| 2987 | |
| 2988 | a solution to the Einstein eqns |
| 2989 | \begin_inset Quotes erd |
| 2990 | \end_inset |
| 2991 | |
| 2992 | , |
| 2993 | then yes, |
| 2994 | there are all kinds of solutions, |
| 2995 | static and expanding, |
| 2996 | even rotating. |
| 2997 | These have parameter directions that we can associate with |
| 2998 | \begin_inset Quotes eld |
| 2999 | \end_inset |
| 3000 | |
| 3001 | the time direction |
| 3002 | \begin_inset Quotes erd |
| 3003 | \end_inset |
| 3004 | |
| 3005 | . |
| 3006 | But the are parameterization invariant: |
| 3007 | there is no |
| 3008 | \begin_inset Quotes eld |
| 3009 | \end_inset |
| 3010 | |
| 3011 | here and now |
| 3012 | \begin_inset Quotes erd |
| 3013 | \end_inset |
| 3014 | |
| 3015 | in any solution to GR. |
| 3016 | Or, |
| 3017 | for that matter, |
| 3018 | QM or QCD or the Standard Model: |
| 3019 | time is reparameterizable. |
| 3020 | and since I am looking for here and now, |
| 3021 | where the past is frozen and inaccessible, |
| 3022 | and the future is unwritten, |
| 3023 | none of these are acceptable at the superficial level. |
| 3024 | That is to say |
| 3025 | \begin_inset Quotes eld |
| 3026 | \end_inset |
| 3027 | |
| 3028 | time does not exist for a free electron in an empty universe |
| 3029 | \begin_inset Quotes erd |
| 3030 | \end_inset |
| 3031 | |
| 3032 | . |
| 3033 | \end_layout |
| 3034 | |
| 3035 | \begin_layout Standard |
| 3036 | Ah, |
| 3037 | but what about a hydrogen atom? |
| 3038 | Something happened, |
| 3039 | there. |
| 3040 | There is a haecceity, |
| 3041 | a |
| 3042 | \begin_inset Quotes eld |
| 3043 | \end_inset |
| 3044 | |
| 3045 | thisness |
| 3046 | \begin_inset Quotes erd |
| 3047 | \end_inset |
| 3048 | |
| 3049 | to the hydrogen atom. |
| 3050 | The electron wave function is not just |
| 3051 | \begin_inset Quotes eld |
| 3052 | \end_inset |
| 3053 | |
| 3054 | somewhere out in the universe, |
| 3055 | some |
| 3056 | \begin_inset Formula $\left\langle \psi\right|$ |
| 3057 | \end_inset |
| 3058 | |
| 3059 | lacking an |
| 3060 | \begin_inset Formula $x,t$ |
| 3061 | \end_inset |
| 3062 | |
| 3063 | coordinate – no, |
| 3064 | it has a very specific ground state in relation to the proton that fixes an origin for a coordinate system. |
| 3065 | Direction is not fixed, |
| 3066 | because we have no fixed stars. |
| 3067 | But if we take the universe to be irrotational (knowing that the universe must be approximated by, |
| 3068 | for example, |
| 3069 | de Sitter space) then we can have some conception of angular momentum, |
| 3070 | even if we don't have fixed stars. |
| 3071 | We can |
| 3072 | \begin_inset Quotes eld |
| 3073 | \end_inset |
| 3074 | |
| 3075 | tell |
| 3076 | \begin_inset Quotes erd |
| 3077 | \end_inset |
| 3078 | |
| 3079 | that the de Sitter space is not rotating, |
| 3080 | because certain rotational tensors vanish. |
| 3081 | So we have an irrotational space, |
| 3082 | and a coordinate system established by the proton, |
| 3083 | and the electron is not very far away. |
| 3084 | Together, |
| 3085 | they are |
| 3086 | \begin_inset Quotes eld |
| 3087 | \end_inset |
| 3088 | |
| 3089 | this atom |
| 3090 | \begin_inset Quotes erd |
| 3091 | \end_inset |
| 3092 | |
| 3093 | . |
| 3094 | \end_layout |
| 3095 | |
| 3096 | \begin_layout Standard |
| 3097 | Translational momentum remains indeterminate. |
| 3098 | Worse, |
| 3099 | the hydrogen atom, |
| 3100 | taken as a whole, |
| 3101 | is still subject to the Heisenberg uncertainty principle, |
| 3102 | so, |
| 3103 | however localized the packet may have started, |
| 3104 | it will spread. |
| 3105 | But what we do know is that, |
| 3106 | henceforth, |
| 3107 | the proton and the electron are traveling together. |
| 3108 | This is the |
| 3109 | \begin_inset Quotes eld |
| 3110 | \end_inset |
| 3111 | |
| 3112 | haecceity |
| 3113 | \begin_inset Quotes erd |
| 3114 | \end_inset |
| 3115 | |
| 3116 | of the hydrogen atom. |
| 3117 | By hand-waving, |
| 3118 | this haecceity extends to the Colorado river flowing downstream, |
| 3119 | all |
| 3120 | \begin_inset Formula $6\times10^{23}$ |
| 3121 | \end_inset |
| 3122 | |
| 3123 | times however many cubic centimeters in question: |
| 3124 | they move together now, |
| 3125 | held by hydroen bonds between the water molecules. |
| 3126 | The molecules move together, |
| 3127 | and the wave function |
| 3128 | \begin_inset Formula $\left\langle \psi\right|$ |
| 3129 | \end_inset |
| 3130 | |
| 3131 | is not spread out over the universe, |
| 3132 | but very highly localized to travel together. |
| 3133 | But that's hand–waving to large numbers. |
| 3134 | Lets return to the single hydrogen atom. |
| 3135 | \end_layout |
| 3136 | |
| 3137 | \begin_layout Standard |
| 3138 | How did the hydrogen atom come to be? |
| 3139 | Well, |
| 3140 | if we start with the deSitter solution, |
| 3141 | and stick a single proton in it, |
| 3142 | spread uniformly over the entire space, |
| 3143 | and a single electron, |
| 3144 | spread uniformly over the entire space, |
| 3145 | then we say |
| 3146 | \begin_inset Quotes eld |
| 3147 | \end_inset |
| 3148 | |
| 3149 | it is ionized |
| 3150 | \begin_inset Quotes erd |
| 3151 | \end_inset |
| 3152 | |
| 3153 | . |
| 3154 | And if we give both zero momentum, |
| 3155 | then we say that they are at 13.7 eV with respect to each other, |
| 3156 | at the ionization limit: |
| 3157 | motionless with respect to each other, |
| 3158 | and not flying away. |
| 3159 | And since they are motionaless, |
| 3160 | and occupy every possible location in space, |
| 3161 | so that |
| 3162 | \begin_inset Formula $\left\langle \psi\right|\left.x\right\rangle =\mbox{const.}\ne0$ |
| 3163 | \end_inset |
| 3164 | |
| 3165 | for all |
| 3166 | \begin_inset Formula $x$ |
| 3167 | \end_inset |
| 3168 | |
| 3169 | in the universe, |
| 3170 | for both proton and electron. |
| 3171 | well, |
| 3172 | we should be able to apply the Born rule and say that they will emit at least one and perhaps many photons, |
| 3173 | until they turn into a hydrogen atom, |
| 3174 | somewhere in the universe, |
| 3175 | and by extension, |
| 3176 | everywhere in the universe, |
| 3177 | all at once, |
| 3178 | all at the same time. |
| 3179 | And those photons? |
| 3180 | Well, |
| 3181 | they are s-waves, |
| 3182 | (or perhaps p-waves? |
| 3183 | I am so incredibly fuzzy on this...) and again, |
| 3184 | they are everywhere all at once, |
| 3185 | but this times, |
| 3186 | its a bit different, |
| 3187 | because we can now take them to be on–shell, |
| 3188 | and so oscillations of crossed E and B fields, |
| 3189 | this time on a lightcone. |
| 3190 | Or technically, |
| 3191 | on all possible light cones whose origin is located at all possible locations in de Sitter space. |
| 3192 | \end_layout |
| 3193 | |
| 3194 | \begin_layout Standard |
| 3195 | What has changed? |
| 3196 | The emission of the photon(s) appear to have created a specific location in time, |
| 3197 | a mark. |
| 3198 | Now the Born rule doesn't given you a precise delta function, |
| 3199 | its a smear over time. |
| 3200 | But we are talking microseconds here (I have no clue what the timescale is, |
| 3201 | but its short compared to the age of the universe, |
| 3202 | at any rate) so we seem to have a distinct marker in time. |
| 3203 | \end_layout |
| 3204 | |
| 3205 | \begin_layout Standard |
| 3206 | The only problem with this idea is that if the electron and proton |
| 3207 | \begin_inset Quotes eld |
| 3208 | \end_inset |
| 3209 | |
| 3210 | have always been there |
| 3211 | \begin_inset Quotes erd |
| 3212 | \end_inset |
| 3213 | |
| 3214 | , |
| 3215 | since the start of the deSitter universe, |
| 3216 | then we have a problem, |
| 3217 | as they would have deionized on the spot, |
| 3218 | instead of waiting a billion years to |
| 3219 | \begin_inset Quotes eld |
| 3220 | \end_inset |
| 3221 | |
| 3222 | mark the spot |
| 3223 | \begin_inset Quotes erd |
| 3224 | \end_inset |
| 3225 | |
| 3226 | . |
| 3227 | So this is confusing, |
| 3228 | a conundrum. |
| 3229 | \end_layout |
| 3230 | |
| 3231 | \begin_layout Standard |
| 3232 | But at any rate, |
| 3233 | with a hydrogen atom, |
| 3234 | we have a ruler: |
| 3235 | the characteristic size of the atom, |
| 3236 | and we have a clock: |
| 3237 | the characteristic oscillation of the photon (say, |
| 3238 | single 13.7 eV photon) and we have a reference frame origin: |
| 3239 | the position of the proton (which is still |
| 3240 | \begin_inset Quotes eld |
| 3241 | \end_inset |
| 3242 | |
| 3243 | everywhere in the universe |
| 3244 | \begin_inset Quotes erd |
| 3245 | \end_inset |
| 3246 | |
| 3247 | ) and a reference for time (when the deionization occurred,) albeit we have a problem if it did not happen at the instant of the birth of the universe. |
| 3248 | Unless we start doing cosmology or something, |
| 3249 | and talk about temperatures. |
| 3250 | And maybe that is the legitimate way of breaking out of the paradox. |
| 3251 | \end_layout |
| 3252 | |
| 3253 | \begin_layout Standard |
| 3254 | Hmm. |
| 3255 | I don't know how to talk of the temperature of a |
| 3256 | \begin_inset Formula $\left\langle \psi\right|$ |
| 3257 | \end_inset |
| 3258 | |
| 3259 | for a single electron that fills all of the universe. |
| 3260 | It seems to be momentum–less.... |
| 3261 | this is confusing. |
| 3262 | A to–do item is to figure out if there's some rational way to talk about this. |
| 3263 | I don't know how, |
| 3264 | right now. |
| 3265 | \end_layout |
| 3266 | |
| 3267 | \begin_layout Standard |
| 3268 | At any rate, |
| 3269 | we seem to have prepared a state involving a structure consisting of a hydrogen atom, |
| 3270 | together with the light–cone extending from this atom. |
| 3271 | Thus, |
| 3272 | even though the hydrogen atom could be anywhere (is everywhere) the light cone is specifically anchored to the hydrogen atom as well. |
| 3273 | So, |
| 3274 | to be precise, |
| 3275 | the |
| 3276 | \begin_inset Quotes eld |
| 3277 | \end_inset |
| 3278 | |
| 3279 | correct |
| 3280 | \begin_inset Quotes erd |
| 3281 | \end_inset |
| 3282 | |
| 3283 | solution to the Schrodinger equation is not just the ground state (or, |
| 3284 | more generally the |
| 3285 | \begin_inset Formula $\psi_{n,l,m}$ |
| 3286 | \end_inset |
| 3287 | |
| 3288 | ) but that, |
| 3289 | together with a photon field, |
| 3290 | and the on–shell waves emitted upon the descent of the electron to the ground state. |
| 3291 | This is the complete structure. |
| 3292 | \end_layout |
| 3293 | |
| 3294 | \begin_layout Standard |
| 3295 | The philosophical question is then: |
| 3296 | is this the simplest example of |
| 3297 | \begin_inset Quotes eld |
| 3298 | \end_inset |
| 3299 | |
| 3300 | here and now |
| 3301 | \begin_inset Quotes erd |
| 3302 | \end_inset |
| 3303 | |
| 3304 | ? |
| 3305 | An |
| 3306 | \begin_inset Quotes eld |
| 3307 | \end_inset |
| 3308 | |
| 3309 | event happened |
| 3310 | \begin_inset Quotes erd |
| 3311 | \end_inset |
| 3312 | |
| 3313 | ; |
| 3314 | the hydrogen atom came to be; |
| 3315 | it has a specific entangles haecceity of a proton, |
| 3316 | an electron, |
| 3317 | and some garbage in the electromagnetic field that seems to propagate only if you can measure time. |
| 3318 | But the only way to measure time is to measure the frequency of the photon. |
| 3319 | And since the metric is on shell, |
| 3320 | |
| 3321 | \begin_inset Formula $p^{\mu}p_{\mu}=0$ |
| 3322 | \end_inset |
| 3323 | |
| 3324 | and so that photon is perpetually |
| 3325 | \begin_inset Quotes eld |
| 3326 | \end_inset |
| 3327 | |
| 3328 | at zero distance |
| 3329 | \begin_inset Quotes erd |
| 3330 | \end_inset |
| 3331 | |
| 3332 | away from the de–ionization event. |
| 3333 | These all taken together is a single individual here–and–now event, |
| 3334 | and then, |
| 3335 | nothing more happens. |
| 3336 | Nothing more can happen. |
| 3337 | Well, |
| 3338 | since it is the only atom in an otherwise empty de Sitter universe ... |
| 3339 | Hmm. |
| 3340 | \end_layout |
| 3341 | |
| 3342 | \begin_layout Standard |
| 3343 | Blah blah blah diagonalize the Hamiltonian. |
| 3344 | (Punt – not today.) |
| 3345 | \end_layout |
| 3346 | |
| 3347 | \begin_layout Standard |
| 3348 | This has two characteristics: |
| 3349 | the past, |
| 3350 | before the de-ionization event, |
| 3351 | cannot be changed. |
| 3352 | The event happened, |
| 3353 | and it cannot be undone. |
| 3354 | So it would appear that the past is indeed frozen into place. |
| 3355 | The future was unwritten up until the de–ionization happened: |
| 3356 | we did not know when it might happen. |
| 3357 | Again hand–wave blah blah Born rule, |
| 3358 | but the future was unwritten, |
| 3359 | unknown, |
| 3360 | until it |
| 3361 | \begin_inset Quotes eld |
| 3362 | \end_inset |
| 3363 | |
| 3364 | happened |
| 3365 | \begin_inset Quotes erd |
| 3366 | \end_inset |
| 3367 | |
| 3368 | , |
| 3369 | and the hydrogen atom cam into being, |
| 3370 | and then became part of the unchangeable past. |
| 3371 | \end_layout |
| 3372 | |
| 3373 | \begin_layout Standard |
| 3374 | Troublingly here, |
| 3375 | the future also disappeared: |
| 3376 | that hydrogen atom will be, |
| 3377 | forever more. |
| 3378 | So this future is choiceless. |
| 3379 | So, |
| 3380 | if you want to have choices, |
| 3381 | you have to have a universe with more than two particles in it. |
| 2920 | 3382 | |
| 2921 | 3383 | \end_layout |
| 2922 | 3384 | |
| 3385 | \begin_layout Standard |
| 3386 | Say we have two protons and one electron. |
| 3387 | Does that alter anything in any meaningful way? |
| 3388 | Can we say |
| 3389 | \begin_inset Quotes eld |
| 3390 | \end_inset |
| 3391 | |
| 3392 | which |
| 3393 | \begin_inset Quotes erd |
| 3394 | \end_inset |
| 3395 | |
| 3396 | proton ended up being bound? |
| 3397 | Sort of. |
| 3398 | The Pauli exclusion principle says that there is a sign–change in the wave function if we exchange the positions for these two particles, |
| 3399 | so when the electron binds to one and not the other, |
| 3400 | there is some sign change, |
| 3401 | but is that sign change observable somewhere in some way? |
| 3402 | Does answering this question even matter? |
| 3403 | \end_layout |
| 3404 | |
| 3405 | \begin_layout Standard |
| 3406 | The point here is that somehow, |
| 3407 | haecceity is a specific binding of some collection of particles, |
| 3408 | and it is the binding that makes things specific. |
| 3409 | \end_layout |
| 3410 | |
| 3411 | \begin_layout Standard |
| 3412 | But this is only |
| 3413 | \begin_inset Quotes eld |
| 3414 | \end_inset |
| 3415 | |
| 3416 | thisness |
| 3417 | \begin_inset Quotes erd |
| 3418 | \end_inset |
| 3419 | |
| 3420 | , |
| 3421 | I am concerned about |
| 3422 | \begin_inset Quotes eld |
| 3423 | \end_inset |
| 3424 | |
| 3425 | when-ness |
| 3426 | \begin_inset Quotes erd |
| 3427 | \end_inset |
| 3428 | |
| 3429 | . |
| 3430 | Whatever. |
| 3431 | It took me 75 minutes to write the above diary entry, |
| 3432 | and it is now 3:15 AM and I am going to bed. |
| 3433 | Time keeps on slippin slippin slippin into the future. |
| 3434 | That irritating Steve Miller song. |
| 3435 | And then that other song, |
| 3436 | less irritating, |
| 3437 | from Pink Floyd. |
| 3438 | Time! |
| 3439 | Tick tick tick. |
| 3440 | \end_layout |
| 3441 | |
| 2923 | 3442 | \begin_layout Section* |
| 2924 | 3443 | The End |
| 2925 | 3444 | \end_layout |