| … | … | @@ -1081,12 +1081,242 @@ obvious |
| 1081 | 1081 | \end_layout |
| 1082 | 1082 | |
| 1083 | 1083 | \begin_layout Standard |
| 1084 | | which suggests or implies the natrual topol |
| 1084 | Anyway, |
| 1085 | I think I'm done here. |
| 1086 | Before moving to the next step, |
| 1087 | some closing remarks: |
| 1088 | spaces like |
| 1089 | \begin_inset Formula $\prod_{x\in I}\mathbb{R}_{x}$ |
| 1090 | \end_inset |
| 1091 | |
| 1092 | endowed with the box topology (or similar) allow crazy paradoxes like Banach–Tarski, |
| 1093 | discussions of the Axiom of Choice, |
| 1094 | the Borel hierarchy of descriptive set theory, |
| 1095 | and other very unusual, |
| 1096 | arcane and abstract topics that are more–or–less impossible to convey to high–school students or |
| 1097 | \begin_inset Quotes eld |
| 1098 | \end_inset |
| 1099 | |
| 1100 | normal |
| 1101 | \begin_inset Quotes erd |
| 1102 | \end_inset |
| 1103 | |
| 1104 | adults. |
| 1105 | I do not think any of these are relevant to QM, |
| 1106 | but I think it would be wrong to not point these out, |
| 1107 | as they are part of the visible landscape in this particular area. |
| 1085 | 1108 | \end_layout |
| 1086 | 1109 | |
| 1087 | 1110 | \begin_layout Standard |
| 1088 | | fine top box paradoxes e.g. |
| 1111 | A bit more interesting and relevant is the countable product |
| 1112 | \begin_inset Formula |
| 1113 | \[ |
| 1114 | \left(I\cap\mathbb{Q}\right)\times\mathbb{R}\simeq\prod_{p/q\in I\cap\mathbb{Q}}\mathbb{R}_{p/q}=\mathbb{R}\times\mathbb{R}\times\cdots |
| 1115 | \] |
| 1116 | |
| 1117 | \end_inset |
| 1118 | |
| 1119 | where |
| 1120 | \begin_inset Formula $\mathbb{Q}$ |
| 1121 | \end_inset |
| 1122 | |
| 1123 | are the rationals. |
| 1124 | This appears to be interesting when one asks for the function |
| 1125 | \begin_inset Formula $f$ |
| 1126 | \end_inset |
| 1127 | |
| 1128 | to be continuous on the natural topology of |
| 1129 | \begin_inset Formula $\mathbb{R}$ |
| 1130 | \end_inset |
| 1131 | |
| 1132 | . |
| 1133 | Values at the rationals appear to determine values everywhere else. |
| 1134 | This is again in the |
| 1135 | \begin_inset Quotes eld |
| 1136 | \end_inset |
| 1137 | |
| 1138 | duhh, |
| 1139 | obvious |
| 1140 | \begin_inset Quotes erd |
| 1141 | \end_inset |
| 1142 | |
| 1143 | category, |
| 1144 | but I mention this because this is not at all the case when one works with iterated function systems or fractals or other systems having fractal nature. |
| 1145 | Such systems factor through the Cantor set |
| 1146 | \begin_inset Formula $2^{\omega}$ |
| 1147 | \end_inset |
| 1148 | |
| 1149 | ; |
| 1150 | the |
| 1151 | \begin_inset Quotes eld |
| 1152 | \end_inset |
| 1153 | |
| 1154 | conventional |
| 1155 | \begin_inset Quotes erd |
| 1156 | \end_inset |
| 1157 | |
| 1158 | topology for the Cantor set is the infinite binary tree. |
| 1159 | It appears commonly enough in number theory; |
| 1160 | it does not normally appear in physics, |
| 1161 | except peripherally via the fractals as they show in chaotic systems (e.g. |
| 1162 | period doubling in high–dimensional turbulence, |
| 1163 | etc.) |
| 1164 | \end_layout |
| 1165 | |
| 1166 | \begin_layout Standard |
| 1167 | Anyway, |
| 1168 | this is neither here nor there; |
| 1169 | QM is not about continuous functions, |
| 1170 | but about square–integrable ones. |
| 1089 | 1171 | |
| 1172 | \emph on |
| 1173 | i.e. |
| 1174 | |
| 1175 | \emph default |
| 1176 | Hilbert spaces endowed with the topology of uniform convergence. |
| 1177 | These arise in |
| 1178 | \begin_inset Quotes eld |
| 1179 | \end_inset |
| 1180 | |
| 1181 | first quantization |
| 1182 | \begin_inset Quotes erd |
| 1183 | \end_inset |
| 1184 | |
| 1185 | ; |
| 1186 | the textbook example is the guitar string or violin string |
| 1187 | \begin_inset Formula |
| 1188 | \[ |
| 1189 | \frac{d^{2}}{dx^{2}}\psi_{n}\left(x\right)=E_{n}\psi_{n}\left(x\right) |
| 1190 | \] |
| 1191 | |
| 1192 | \end_inset |
| 1193 | |
| 1194 | subject to |
| 1195 | \emph on |
| 1196 | e.g. |
| 1197 | |
| 1198 | \emph default |
| 1199 | the boundary condition |
| 1200 | \begin_inset Formula $\psi\left(0\right)=\psi\left(1\right)=0$ |
| 1201 | \end_inset |
| 1202 | |
| 1203 | and (for this example) having the usual sine–wave basis. |
| 1204 | The point here is that this Hilbert space is already non–local. |
| 1205 | If one uses the bra-ket notation |
| 1206 | \begin_inset Formula $\psi\left(x\right)=\left\langle x\left|\psi\right.\right\rangle $ |
| 1207 | \end_inset |
| 1208 | |
| 1209 | and then ask |
| 1210 | \begin_inset Quotes eld |
| 1211 | \end_inset |
| 1212 | |
| 1213 | what is the spatial location of |
| 1214 | \begin_inset Formula $\left|\psi\right\rangle $ |
| 1215 | \end_inset |
| 1216 | |
| 1217 | ? |
| 1218 | \begin_inset Quotes erd |
| 1219 | \end_inset |
| 1220 | |
| 1221 | the answer is |
| 1222 | \begin_inset Quotes eld |
| 1223 | \end_inset |
| 1224 | |
| 1225 | everywhere, |
| 1226 | all at once |
| 1227 | \begin_inset Quotes erd |
| 1228 | \end_inset |
| 1229 | |
| 1230 | , |
| 1231 | because |
| 1232 | \begin_inset Formula $\left|\psi\right\rangle $ |
| 1233 | \end_inset |
| 1234 | |
| 1235 | does not have a |
| 1236 | \begin_inset Quotes eld |
| 1237 | \end_inset |
| 1238 | |
| 1239 | spatial location |
| 1240 | \begin_inset Quotes erd |
| 1241 | \end_inset |
| 1242 | |
| 1243 | ; |
| 1244 | its a vector in Hilbert space. |
| 1245 | Duhh. |
| 1246 | Again, |
| 1247 | this is trivial and trite, |
| 1248 | and taught in college QM courses; |
| 1249 | I'm belaboring the obvious because ... |
| 1250 | well ... |
| 1251 | the next question: |
| 1252 | |
| 1253 | \begin_inset Quotes eld |
| 1254 | \end_inset |
| 1255 | |
| 1256 | what happens when I measure the wave function |
| 1257 | \begin_inset Formula $\psi$ |
| 1258 | \end_inset |
| 1259 | |
| 1260 | at the spatial location |
| 1261 | \begin_inset Formula $x$ |
| 1262 | \end_inset |
| 1263 | |
| 1264 | ? |
| 1265 | \begin_inset Quotes erd |
| 1266 | \end_inset |
| 1267 | |
| 1268 | and |
| 1269 | \begin_inset Quotes eld |
| 1270 | \end_inset |
| 1271 | |
| 1272 | how does that work? |
| 1273 | \begin_inset Quotes erd |
| 1274 | \end_inset |
| 1275 | |
| 1276 | and |
| 1277 | \begin_inset Quotes eld |
| 1278 | \end_inset |
| 1279 | |
| 1280 | what does it mean? |
| 1281 | \begin_inset Quotes erd |
| 1282 | \end_inset |
| 1283 | |
| 1284 | and |
| 1285 | \begin_inset Quotes eld |
| 1286 | \end_inset |
| 1287 | |
| 1288 | what is the mechanism, |
| 1289 | exactly? |
| 1290 | \begin_inset Quotes erd |
| 1291 | \end_inset |
| 1292 | |
| 1293 | This last question, |
| 1294 | |
| 1295 | \begin_inset Quotes eld |
| 1296 | \end_inset |
| 1297 | |
| 1298 | what is the mechanism? |
| 1299 | \begin_inset Quotes erd |
| 1300 | \end_inset |
| 1301 | |
| 1302 | is particularly challenging, |
| 1303 | because the it seems to require some kind of non–linearity or something. |
| 1304 | because linear systems don't collapse, |
| 1305 | can't collapse. |
| 1306 | This last sentence is non–obvious, |
| 1307 | but lies at the core, |
| 1308 | the crux of a lot of hot air and speculation about the mechanisms of QM wave–function collapse. |
| 1309 | At present, |
| 1310 | I can only add to the body of speculation, |
| 1311 | and warm the globe by exhausting my personal hot air on the topic. |
| 1312 | \end_layout |
| 1313 | |
| 1314 | \begin_layout Standard |
| 1315 | Anyway... |
| 1316 | \end_layout |
| 1317 | |
| 1318 | \begin_layout Standard |
| 1319 | gaussia |
| 1090 | 1320 | \end_layout |
| 1091 | 1321 | |
| 1092 | 1322 | \begin_layout Section* |