| … | … | @@ -1312,11 +1312,573 @@ what is the mechanism? |
| 1312 | 1312 | \end_layout |
| 1313 | 1313 | |
| 1314 | 1314 | \begin_layout Standard |
| 1315 | | Anyway... |
| 1315 | So what happens if I try to axiomatize wave function collapse in the above model? |
| 1316 | Well, |
| 1317 | I have that |
| 1318 | \begin_inset Formula $\left|\psi\right\rangle \in H$ |
| 1319 | \end_inset |
| 1320 | |
| 1321 | where |
| 1322 | \begin_inset Formula $H$ |
| 1323 | \end_inset |
| 1324 | |
| 1325 | is the Hilbert space with all the conventional trimmings: |
| 1326 | bounded, |
| 1327 | uniform convergence in the weak topology, |
| 1328 | square–integrable, |
| 1329 | |
| 1330 | \emph on |
| 1331 | etc. |
| 1332 | |
| 1333 | \emph default |
| 1334 | Note that square–integrable means normalizable, |
| 1335 | and normalizable means its a projective space. |
| 1336 | A projective space is one where all the vector lengths are of length–one; |
| 1337 | the usual examples are real projective space |
| 1338 | \begin_inset Formula $\mathbb{RP}^{n}$ |
| 1339 | \end_inset |
| 1340 | |
| 1341 | and complex projective space |
| 1342 | \begin_inset Formula $\mathbb{CP}^{n}$ |
| 1343 | \end_inset |
| 1344 | |
| 1345 | . |
| 1346 | Note that projective spaces are |
| 1347 | \begin_inset Quotes eld |
| 1348 | \end_inset |
| 1349 | |
| 1350 | kind of like spheres |
| 1351 | \begin_inset Quotes erd |
| 1352 | \end_inset |
| 1353 | |
| 1354 | , |
| 1355 | and that the conventional Hilbert space of QM is |
| 1356 | \begin_inset Quotes eld |
| 1357 | \end_inset |
| 1358 | |
| 1359 | kind of like |
| 1360 | \begin_inset Quotes erd |
| 1361 | \end_inset |
| 1362 | |
| 1363 | an infinite–dimensional sphere, |
| 1364 | in that |
| 1365 | \begin_inset Formula $1=\left|c_{1}\right|+\left|c_{2}\right|+\cdots$ |
| 1366 | \end_inset |
| 1367 | |
| 1368 | for any |
| 1369 | \begin_inset Formula $\psi=c_{1}\psi_{1}+c_{2}\psi_{2}+\cdots$ |
| 1370 | \end_inset |
| 1371 | |
| 1372 | and we know that high–dimensional spheres are bizarre beasts in their own right. |
| 1373 | \end_layout |
| 1374 | |
| 1375 | \begin_layout Standard |
| 1376 | Anyway, |
| 1377 | a pure–state wave–function |
| 1378 | \begin_inset Formula $\psi=\left|\psi\right\rangle \in H$ |
| 1379 | \end_inset |
| 1380 | |
| 1381 | is a |
| 1382 | \begin_inset Quotes eld |
| 1383 | \end_inset |
| 1384 | |
| 1385 | single point |
| 1386 | \begin_inset Quotes erd |
| 1387 | \end_inset |
| 1388 | |
| 1389 | on this |
| 1390 | \begin_inset Quotes eld |
| 1391 | \end_inset |
| 1392 | |
| 1393 | infinite–dimensional sphere |
| 1394 | \begin_inset Quotes erd |
| 1395 | \end_inset |
| 1396 | |
| 1397 | |
| 1398 | \begin_inset Formula $S^{\omega}\simeq H$ |
| 1399 | \end_inset |
| 1400 | |
| 1401 | (which is then |
| 1402 | \begin_inset Formula $H\simeq\mathbb{RP}^{\omega}$ |
| 1403 | \end_inset |
| 1404 | |
| 1405 | or |
| 1406 | \begin_inset Formula $H\simeq\mathbb{CP}^{\omega}$ |
| 1407 | \end_inset |
| 1408 | |
| 1409 | , |
| 1410 | take your pick.) I now want to introduce a new symbol, |
| 1411 | |
| 1412 | \begin_inset Formula $r$ |
| 1413 | \end_inset |
| 1414 | |
| 1415 | , |
| 1416 | standing for the |
| 1417 | \begin_inset Quotes eld |
| 1418 | \end_inset |
| 1419 | |
| 1420 | reduction |
| 1421 | \begin_inset Quotes erd |
| 1422 | \end_inset |
| 1423 | |
| 1424 | of |
| 1425 | \begin_inset Quotes eld |
| 1426 | \end_inset |
| 1427 | |
| 1428 | wave function collapse |
| 1429 | \begin_inset Quotes erd |
| 1430 | \end_inset |
| 1431 | |
| 1432 | . |
| 1433 | How should that symbol be deployed algebraically? |
| 1434 | Well, |
| 1435 | lets start with some wrong answers. |
| 1436 | its not a function, |
| 1437 | so I should not write |
| 1438 | \end_layout |
| 1439 | |
| 1440 | \begin_layout Standard |
| 1441 | \begin_inset Formula |
| 1442 | \[ |
| 1443 | r:H\to I\qquad\mbox{(wrong)} |
| 1444 | \] |
| 1445 | |
| 1446 | \end_inset |
| 1447 | |
| 1448 | where |
| 1449 | \begin_inset Formula $I$ |
| 1450 | \end_inset |
| 1451 | |
| 1452 | is the unit interval from before. |
| 1453 | The above is wrong for several reasons: |
| 1454 | its not |
| 1455 | \begin_inset Quotes eld |
| 1456 | \end_inset |
| 1457 | |
| 1458 | deterministic |
| 1459 | \begin_inset Quotes erd |
| 1460 | \end_inset |
| 1461 | |
| 1462 | , |
| 1463 | in that I can't say |
| 1464 | \begin_inset Quotes eld |
| 1465 | \end_inset |
| 1466 | |
| 1467 | for any and every point |
| 1468 | \begin_inset Formula $\psi\in H$ |
| 1469 | \end_inset |
| 1470 | |
| 1471 | there is a unique point |
| 1472 | \begin_inset Formula $x\in I$ |
| 1473 | \end_inset |
| 1474 | |
| 1475 | that it collapses to |
| 1476 | \begin_inset Quotes erd |
| 1477 | \end_inset |
| 1478 | |
| 1479 | ; |
| 1480 | at most, |
| 1481 | all I can say is |
| 1482 | \begin_inset Quotes eld |
| 1483 | \end_inset |
| 1484 | |
| 1485 | after preparing a state |
| 1486 | \begin_inset Formula $\psi$ |
| 1487 | \end_inset |
| 1488 | |
| 1489 | and selecting a spatial location |
| 1490 | \begin_inset Formula $x$ |
| 1491 | \end_inset |
| 1492 | |
| 1493 | then I will have a random chance/probability of |
| 1494 | \begin_inset Formula $\left|\psi\left(x\right)\right|^{2}$ |
| 1495 | \end_inset |
| 1496 | |
| 1497 | that my detector, |
| 1498 | located at |
| 1499 | \begin_inset Formula $x$ |
| 1500 | \end_inset |
| 1501 | |
| 1502 | will detect the prepared state |
| 1503 | \begin_inset Quotes erd |
| 1504 | \end_inset |
| 1505 | |
| 1506 | . |
| 1507 | Twiddle the above as needed for states that aren't pure, |
| 1508 | and detectors that are finite–sized. |
| 1509 | Crack open standard textbooks for this. |
| 1510 | \end_layout |
| 1511 | |
| 1512 | \begin_layout Standard |
| 1513 | The mission here is to find an algebraic infrastructure for talking about |
| 1514 | \begin_inset Formula $r$ |
| 1515 | \end_inset |
| 1516 | |
| 1517 | . |
| 1518 | Discovering notation for it. |
| 1519 | And ... |
| 1520 | on this note, |
| 1521 | I will end today's diary entry, |
| 1522 | mostly because it took five hours to write the trivialities above, |
| 1523 | and I'm exhausted, |
| 1524 | and I really don't have any tricks or anything clever to say about |
| 1525 | \begin_inset Formula $r$ |
| 1526 | \end_inset |
| 1527 | |
| 1528 | because it was at this point that I rolled out of bed this morning, |
| 1529 | with an urge to write down the above. |
| 1530 | So; |
| 1531 | mission accomplished: |
| 1532 | I wrote down what I set out to write. |
| 1533 | I don't think I wrote down anything at all that a normal student of QM would not already know. |
| 1534 | I did want to sharpen the outlines and relief of what wave–function collapse is. |
| 1535 | The point is that wave functions |
| 1536 | \begin_inset Quotes eld |
| 1537 | \end_inset |
| 1538 | |
| 1539 | live in a space |
| 1540 | \begin_inset Formula $H$ |
| 1541 | \end_inset |
| 1542 | |
| 1543 | the is out there, |
| 1544 | independent of space and time |
| 1545 | \begin_inset Quotes erd |
| 1546 | \end_inset |
| 1547 | |
| 1548 | , |
| 1549 | and that the physics happens |
| 1550 | \begin_inset Quotes eld |
| 1551 | \end_inset |
| 1552 | |
| 1553 | in that space |
| 1554 | \begin_inset Quotes erd |
| 1555 | \end_inset |
| 1556 | |
| 1557 | , |
| 1558 | which is just as |
| 1559 | \begin_inset Quotes eld |
| 1560 | \end_inset |
| 1561 | |
| 1562 | real |
| 1563 | \begin_inset Quotes erd |
| 1564 | \end_inset |
| 1565 | |
| 1566 | as the space that I am accustomed to living in. |
| 1567 | \end_layout |
| 1568 | |
| 1569 | \begin_layout Standard |
| 1570 | The mystery question is |
| 1571 | \begin_inset Quotes eld |
| 1572 | \end_inset |
| 1573 | |
| 1574 | why is my space 3D? |
| 1575 | \begin_inset Quotes erd |
| 1576 | \end_inset |
| 1577 | |
| 1578 | and not 5D or something else? |
| 1579 | The above sketch paints a picture for 1D space; |
| 1580 | it does not fix the dimension in any way. |
| 1581 | It also does not explain why the 3D space that I live in is Cartesian, |
| 1582 | with the natural topology, |
| 1583 | and not the product of, |
| 1584 | say, |
| 1585 | three Cantor sets with some other topology, |
| 1586 | e.g. |
| 1587 | |
| 1588 | \begin_inset Formula $p$ |
| 1589 | \end_inset |
| 1590 | |
| 1591 | –adic or whatever. |
| 1592 | Where does the 3D come from, |
| 1593 | and where does the natural topology come from? |
| 1594 | \end_layout |
| 1595 | |
| 1596 | \begin_layout Standard |
| 1597 | Oh, |
| 1598 | I should point out that one of the ingredients to the above construction was the postulation of a differential eqn |
| 1599 | \begin_inset Formula $d^{2}\psi/dx^{2}=E\psi$ |
| 1600 | \end_inset |
| 1601 | |
| 1602 | because that is how first–quantization works. |
| 1603 | If I think of this eqn as a |
| 1604 | \begin_inset Quotes eld |
| 1605 | \end_inset |
| 1606 | |
| 1607 | single point in the space of all differential eqns |
| 1608 | \begin_inset Quotes erd |
| 1609 | \end_inset |
| 1610 | |
| 1611 | , |
| 1612 | then I have to ask |
| 1613 | \begin_inset Quotes eld |
| 1614 | \end_inset |
| 1615 | |
| 1616 | what is the space of all differential eqns? |
| 1617 | \begin_inset Quotes erd |
| 1618 | \end_inset |
| 1619 | |
| 1620 | and |
| 1621 | \begin_inset Quotes eld |
| 1622 | \end_inset |
| 1623 | |
| 1624 | does it have a topology? |
| 1625 | \begin_inset Quotes erd |
| 1626 | \end_inset |
| 1627 | |
| 1628 | and |
| 1629 | \begin_inset Quotes eld |
| 1630 | \end_inset |
| 1631 | |
| 1632 | should I think about non–linear e.g. |
| 1633 | quadratic terms for |
| 1634 | \begin_inset Formula $\psi$ |
| 1635 | \end_inset |
| 1636 | |
| 1637 | or more than two derivatives? |
| 1638 | \begin_inset Quotes erd |
| 1639 | \end_inset |
| 1640 | |
| 1641 | The conventional answer to these questions are |
| 1642 | \begin_inset Quotes eld |
| 1643 | \end_inset |
| 1644 | |
| 1645 | no you must not do that because |
| 1646 | \begin_inset Quotes erd |
| 1647 | \end_inset |
| 1648 | |
| 1649 | (a) the Hamilton–Jacobi eqns are naturally quadratic and are the natural setting for geodesics on Riemanninan manifolds (which then beggars the question |
| 1650 | \begin_inset Quotes eld |
| 1651 | \end_inset |
| 1652 | |
| 1653 | why is it a metric space, |
| 1654 | anyway? |
| 1655 | \begin_inset Quotes erd |
| 1656 | \end_inset |
| 1657 | |
| 1658 | ) or perhaps (b) |
| 1659 | \begin_inset Quotes eld |
| 1660 | \end_inset |
| 1661 | |
| 1662 | mumble spin statistics and general relativity |
| 1663 | \begin_inset Quotes erd |
| 1664 | \end_inset |
| 1665 | |
| 1666 | or perhaps (c) non–renormalizability or perhaps (d) |
| 1667 | \begin_inset Quotes eld |
| 1668 | \end_inset |
| 1669 | |
| 1670 | standard QM is just plain–old linear, |
| 1671 | deal with it |
| 1672 | \begin_inset Quotes erd |
| 1673 | \end_inset |
| 1674 | |
| 1675 | or perhaps (e) the propagator for the Laplacian has the very simple null–light–cone form |
| 1676 | \begin_inset Formula $\delta^{3}\left(x-ct\right)$ |
| 1677 | \end_inset |
| 1678 | |
| 1679 | only in 3+1 D and all other dimensions have derivatives in there, |
| 1680 | or worse, |
| 1681 | e.g. |
| 1682 | water waves in 2+1 D where the propagator has a step function for the retarded wave, |
| 1683 | and likewise for N+1 for any even N. |
| 1684 | But all of these hand–waving replies are, |
| 1685 | well, |
| 1686 | hand–waving, |
| 1687 | and serve to obscure the answer even before the question has been clearly posed. |
| 1688 | The only vaguely reasonable answer (or framework) seems to arise in second quantization, |
| 1689 | where I write the partition function |
| 1690 | \begin_inset Formula |
| 1691 | \[ |
| 1692 | Z=\int\left[d\phi\right]e^{-iA\left(\phi\right)} |
| 1693 | \] |
| 1694 | |
| 1695 | \end_inset |
| 1696 | |
| 1697 | and the action is |
| 1698 | \begin_inset Formula $A\left(\phi\right)=\int d^{n}x\mathcal{L}\left(\phi,\dot{\phi}\right)$ |
| 1699 | \end_inset |
| 1700 | |
| 1701 | for some quadratic Lagrangian |
| 1702 | \begin_inset Formula $\mathcal{L}$ |
| 1703 | \end_inset |
| 1704 | |
| 1705 | . |
| 1706 | Again, |
| 1707 | we are back to |
| 1708 | \begin_inset Formula $\mathcal{L}$ |
| 1709 | \end_inset |
| 1710 | |
| 1711 | being quadratic, |
| 1712 | but this I can somehow live with and accept, |
| 1713 | because now, |
| 1714 | any |
| 1715 | \begin_inset Formula $\phi$ |
| 1716 | \end_inset |
| 1717 | |
| 1718 | that is not classical, |
| 1719 | i.e. |
| 1720 | is not an extremum of the partition function, |
| 1721 | I can |
| 1722 | \begin_inset Quotes eld |
| 1723 | \end_inset |
| 1724 | |
| 1725 | imagine |
| 1726 | \begin_inset Quotes erd |
| 1727 | \end_inset |
| 1728 | |
| 1729 | that |
| 1730 | \begin_inset Formula $\phi$ |
| 1731 | \end_inset |
| 1732 | |
| 1733 | as |
| 1734 | \begin_inset Quotes eld |
| 1735 | \end_inset |
| 1736 | |
| 1737 | solving |
| 1738 | \begin_inset Quotes erd |
| 1739 | \end_inset |
| 1740 | |
| 1741 | some |
| 1742 | \begin_inset Quotes eld |
| 1743 | \end_inset |
| 1744 | |
| 1745 | other |
| 1746 | \begin_inset Quotes erd |
| 1747 | \end_inset |
| 1748 | |
| 1749 | differential equation besides |
| 1750 | \begin_inset Formula $d^{2}\psi/dx^{2}=E\psi$ |
| 1751 | \end_inset |
| 1752 | |
| 1753 | , |
| 1754 | because, |
| 1755 | in second quantization, |
| 1756 | these other paths |
| 1757 | \begin_inset Quotes eld |
| 1758 | \end_inset |
| 1759 | |
| 1760 | exist |
| 1761 | \begin_inset Quotes erd |
| 1762 | \end_inset |
| 1763 | |
| 1764 | , |
| 1765 | and only the |
| 1766 | \begin_inset Quotes eld |
| 1767 | \end_inset |
| 1768 | |
| 1769 | classical |
| 1770 | \begin_inset Quotes erd |
| 1771 | \end_inset |
| 1772 | |
| 1773 | path solves the |
| 1774 | \begin_inset Quotes eld |
| 1775 | \end_inset |
| 1776 | |
| 1777 | first–order–quantized |
| 1778 | \begin_inset Quotes erd |
| 1779 | \end_inset |
| 1780 | |
| 1781 | eqn |
| 1782 | \begin_inset Formula $d^{2}\psi/dx^{2}=E\psi$ |
| 1783 | \end_inset |
| 1784 | |
| 1785 | . |
| 1786 | I'm putting everything in quotes, |
| 1787 | here, |
| 1788 | I'm trying to sketch a picture here that fails if the quoted terms are taken at their literal meanings; |
| 1789 | you have to read a bit of poetry into this. |
| 1790 | The correct/intended meaning for this is, |
| 1791 | for example, |
| 1792 | a brownian motion random walk, |
| 1793 | where any path can be taken, |
| 1794 | and the distribution of those paths are clearly Gaussian (i.e. |
| 1795 | quadratic) but if I select any one given path and |
| 1796 | \begin_inset Quotes eld |
| 1797 | \end_inset |
| 1798 | |
| 1799 | smooth it |
| 1800 | \begin_inset Quotes erd |
| 1801 | \end_inset |
| 1802 | |
| 1803 | , |
| 1804 | it seems to |
| 1805 | \begin_inset Quotes eld |
| 1806 | \end_inset |
| 1807 | |
| 1808 | almost solve |
| 1809 | \begin_inset Quotes erd |
| 1810 | \end_inset |
| 1811 | |
| 1812 | something that |
| 1813 | \begin_inset Quotes eld |
| 1814 | \end_inset |
| 1815 | |
| 1816 | looks like |
| 1817 | \begin_inset Quotes erd |
| 1818 | \end_inset |
| 1819 | |
| 1820 | a differential eqn with non–linear terms and higher derivatives, |
| 1821 | even though |
| 1822 | \begin_inset Quotes eld |
| 1823 | \end_inset |
| 1824 | |
| 1825 | the actual diff eq depends on the smoothed path |
| 1826 | \begin_inset Quotes erd |
| 1827 | \end_inset |
| 1828 | |
| 1829 | . |
| 1316 | 1830 | \end_layout |
| 1317 | 1831 | |
| 1318 | 1832 | \begin_layout Standard |
| 1319 | | gaussia |
| 1833 | This sounds insane so let me re–sketch it as a a kind of inverted–physics. |
| 1834 | Start with a single stochastic diff eq, |
| 1835 | say the one for Brownian motion, |
| 1836 | then select a single path (a single random walk) then smooth that path (with some smoothing function with some smoothing parameter(s)) and then ask: |
| 1837 | is there some diffeq that this path just happens to accidentally solve? |
| 1838 | And then perhaps, |
| 1839 | |
| 1840 | \begin_inset Quotes eld |
| 1841 | \end_inset |
| 1842 | |
| 1843 | what other solns does this nasty horrible diffeq solve? |
| 1844 | \begin_inset Quotes erd |
| 1845 | \end_inset |
| 1846 | |
| 1847 | Or perhaps |
| 1848 | \begin_inset Quotes eld |
| 1849 | \end_inset |
| 1850 | |
| 1851 | what is the topology for the space of these kinds of difffeqs that are constructed in this insane kind of way? |
| 1852 | \begin_inset Quotes erd |
| 1853 | \end_inset |
| 1854 | |
| 1855 | (where by |
| 1856 | \begin_inset Quotes eld |
| 1857 | \end_inset |
| 1858 | |
| 1859 | topology |
| 1860 | \begin_inset Quotes erd |
| 1861 | \end_inset |
| 1862 | |
| 1863 | I also mean |
| 1864 | \begin_inset Quotes eld |
| 1865 | \end_inset |
| 1866 | |
| 1867 | measure space |
| 1868 | \begin_inset Quotes erd |
| 1869 | \end_inset |
| 1870 | |
| 1871 | , |
| 1872 | because I want to have some measure on it.) But again, |
| 1873 | this is not only wacky, |
| 1874 | but unworkable at this level of description, |
| 1875 | so I abandon ship here. |
| 1876 | My apologies to leave you disappointed and empty–handed. |
| 1877 | It'll be a few weeks before I get back to this; |
| 1878 | I have other things to do. |
| 1879 | I also had some other ideas about here and now, |
| 1880 | a few months ago, |
| 1881 | and I did not write them down and now I cannot remember them. |
| 1320 | 1882 | \end_layout |
| 1321 | 1883 | |
| 1322 | 1884 | \begin_layout Section* |