The Hidden Worlds of Quantum Mechanics

1. The Hidden Worlds of Quantum Mechanics Craig Callender Philosophy, UCSD ccallender@ucsd.edu

2. Double Slit Experiment

4. What you would expect is…

5. But what actually happens is… • 1961, Jönsson, Zeitschrift für Physik161 454 • 1974, P. Merli, G. Missiroli and G. Pozzi in Bologna in 1974 • Hitachi (A Tonomura et al). 1989 Demonstration of single-electron buildup of an interference pattern Am. J. Phys. 57

6. The Double Slit Experiment

7. Actual images Bologna 1974 Hitachi 1989

9. Stuff: Newtonian corpuscles State: (xi, pi) Dynamical law: Hamilton’s equations Stuff: ? State: wavefunction or vector |Ψ> Dynamical law: Schrodinger’s equation Structure of Physical Theories

10. Wave function (quantum state) Ψ(x) x

11. Ψ2(x) gives the probability of finding the particle at position x Most probable location of particle Ψ2(x) x

12. where the observable is assumed to have a discrete spectrum of eigenvalues, u are the (normalized) eigenfunctions, and the coefficient cn of the nth term gives the probability of the nth eigenvalue via |cn|2.

13. Schrodinger evolution Deterministic Unitary Linear Schrodinger evolution Ψi(x) Ψf(x)

14. Three Ingredients for Trouble • Linear dynamical evolution • Eigenstate-eigenvalue rule • Determinate outcomes

15. Linearity If the evolution takes |A> →|B>.. And takes |C> → |D>… Then it takes the state |A> + |C> → |B> + |D>

16. Eigenstate-eigenvalue link • A system in the quantum state |Ψ> has the value a for the observable Âif and only if |Ψ> assigns the probability 1 to a and the probability 0 to all other possible values of Â. • Â |Ψ> = a|Ψ> Ψ(x) x

17. Measurement 1 |ready>M|↑>S  |up>M|↑>S 2 |ready>M|↓>S  |down>M|↓>S • |ready>M (a1|↑>S+a2|↓>S)= a1|ready>M|↑>S+a2|ready>M|↓>S  (a1|up>M|↑>S +a2|down>M|↓>S)

18. Schrödinger’s cat 1 |cat ready>|ready>M|↑>S → |cat dead>|up>M|↑>S 2 |cat ready>|ready>M|↓>S → |cat alive>|down>M|↓>S • |cat ready>|ready>M (a1|↑>S +a2|↓>S)= a1|cat ready>|ready>M|↑>S +a2|cat ready>|ready>M|↓>S  (a1|dead>|up>M|↑>S +a2|alive>|down>M|↓>S

19. Measurement Problem • The quantum state is representationally complete, i.e., the eigenstate-eigenvalue link holds • The quantum state always evolves according to a linear laws of evolution, e.g., Schrodinger equation. • Measurements yield definite values Contradiction!

20. Schrodinger evolution Ψf(x) Ψi(x) Final state is a probability distribution; but in the real world something actually happens! non-Schrodinger evolution; miracle; collapse Ψi(x) Ψf(x)

21. MP is Here to Stay • Quantum field theory employs superpositions among distinct macroscopic states • Theories on the horizon do too, e.g., superstring theory, loop theory, etc. • So MP has to be solved

22. The “Standard” Solution • Copenhagen (Bohr, Heisenberg, Dirac, von Neumann, …) Two types of evolution: • Unmeasured evolution • Measurement evolution “Quantum philosophy” of realism about macroscopic entities but anti-realism about microscopic ones

23. Unique? • Rosenfeld: “quantum theory eminently possess this character of uniqueness; every feature of it has been forced upon us as the only way to avoid the ambiguities which would essentially affect any attempt at an analysis in classical terms of typical quantum phenomena”

24. Criticism • “It would seem that the theory is exclusively concerned about "results of measurement", and has nothing to say about anything else. What exactly qualifies some physical systems to play the role of "measurer"? Was the wavefunction of the world waiting to jump for thousands of millions of years until a single-celled living creature appeared? Or did it have to wait a little longer, for some better qualified system ... with a Ph.D.? If the theory is to apply to anything but highly idealized laboratory operations, are we not obliged to admit that more or less "measurement-like" processes are going on more or less all the time, more or less everywhere.”

25. Solution: Deny one of the premises • The quantum state is representationally complete, i.e., the eigenstate-eigenvalue link holds • The quantum state always evolves according to a linear laws of evolution, e.g., Schrodinger equation. • Measurements yield definite values

26. Deny 1 • Bohm-like “hidden variable” theories • De Broglie 1927 • Bohm 1952 • Bohm and Vigier • Nelson’s stochastic mechanics • Bell 1987 • Goldstein, Durr and Zanghi 1991

27. Deny 2 • Physically-specifiable Collapse theories • Pearle 1989 • Pearle and Squires 1994 • Pearle 1996 • Ghiradhi, Rimini and Weber 1986 • Bell 1987 • Penrose

28. Deny 3 • “Many-world” type theories • Everett 1957 • deWitt • “Many minds” • Barbour • Rovelli’s ‘relational’ qm

29. Bohmian Mechanics • De Broglie 1927; David Bohm 1952 • The de Broglie-Bohm “idea seems …so natural and simple, to resolve the wave-particle dilemma in such a clear and ordinary way, that it is a great mystery… that it was so generally ignored.” Bell, 1987.

30. Bohmian Mechanics • Basic idea: Suppose that there are some particles and that their velocities are determined by Ψ… In other words, Ψ is not the whole story; there are also particles.

31. Bohmian mechanics Schrodinger equation Velocity equation at time t0 If at time t then

32. GRW

33. The world is described by two equations, the Schrodinger equation and the velocity equation. The latter is arguably the simplest first order equation for the positions of particles compatible with the Galilean and time reversal invariance of the former. • Given any probability distribution for the initial configuration, Bohmian mechanics defines a probability distribution for the full trajectory. Notice that the velocity equation is simply v= J/p, where J is the quantum probability current and p is the quantum probability density. It follows from the quantum continuity equation that if the distribution of the configuration Q is given by psi at some time (say the initial time) this will be true at all times. • This deterministic theory of particles completely accounts for all the phenomena of nonrelativistic quantum mechanics, from interference effects to spectral lines. Thus Bohmian mechanics provides us with probabilities for complete configurational histories that are consistent with the quantum mechanical probabilities for configurations, including the positions of measuring devices.

34. Many Bohmian Theories • Deterministic alternatives to the velocity equation that are empirically adequate • Indeterministic versions of velocity equation • Spin as fundamental with position (Bohm, Schiller and Tiomno 1955, Dewdney 1992, Holland and Vigier 1988, Bohm and Hiley 1993) • Bell-Bub-Vink dynamics: discrete, indeterministic • Bub: modal interpretation

35. Ghirardi, Rimini & Weber 1986 • Basic Idea: The Schrodinger equation is not quite right. The wavefunction (r1,r2, ..., t) usually evolves according to the Schrodinger equation, but every now and then, at random,  is multiplied (‘hit’) by a gaussian function (and then normalized). How often the state is likely to be ‘hit’ by a gaussian is proportional to how many particles there are in the system. • The effect of this multiplication is to collapse the state to a more localized one. Thus, systems with large N turn out to be overwhelmingly likely to collapse, and systems with small N turn out to be unlikely to collapse.

36. GRW Ψ(x) x Cat Alive Cat Dead 1/2[(C(r1,r2, ...rN...rM, t)S(r1)) + [(C(r1,r2, ...rN...rM, t)S(r1))]

37. Then the Gaussian hits Ψ(x) j(x) = K exp (-[x - ri]2/2a2) * = j(x, ri)(...,t)/ Ri(x). x Cat Alive a=10-5cm, jump time T=1016s For N = 1023, collapse happens around 10-7s, compared to observation time 10-2s

38. Many GRW’s • Ghirardi emphasizes the importance of specifying what he calls ``the physical reality of what exists out there.'' • Mass density interpretation: for the simple GRW theory described here can be identified with the mass weighted sum, over all particles, of the one-particle densities arising from integrating over the coordinates of all but one of the particles. • ‘Hit’ interpretation: Bell [p 205,] “that the space-time points (x,t) at which the hits are centered (which are determined by the wave function trajectory) should themselves serve as the ``local beables of the theory. These are the mathematical counterparts in the theory to real events at definite places and times in the real world (as distinct from the many purely mathematical constructions that occur in the working out of physical theories, as distinct from things which may be real but not localized, and as distinct from the `observables' of other formulations of quantum mechanics, for which we have no use here.) A piece of matter then is a galaxy of such events.'‘ • SL v CSL

40. “There is something wrong with all of these post-Copenhagen interpretations…they don’t offer new predictions” • But: • Why should when the theory is developed be important? • De Broglie 1927 • GRW will differ from Copenhagen; Bohm would if Copenhagen were clear; plus, one never knows…

41. Underdetermination of Theory by Evidence THEORY1 THEORY2 Observable evidence THEORY3 THEORY4

42. Duhem: “Shall we ever dare to assert that no other hypothesis is imaginable? Light may be a swarm of projectiles, or it may be a vibratory motion whose waves are propagated in a medium; is it forbidden to be anything else at all?” (1914)

43. Experimenta Crucis? • Bohm v Copenhagen • Times of arrival, etc. • Conroversial • Bohm v Everett • “in principle” underdetermination • Laudan and Leplin 1991 • GRW v Bohm (or Copenhagen…) • Localizations  greater KE  heating • Resistance of a superconductor different (Gallis & Flemming; Rae & Rimini) • One mole of H: one atom excited/sec • “hope of reaching a crucial test…extremely dim”

44. Theoria crucis? • Solving the measurement problem seems to involve adding some new physics… • The new physics may or may not be experimentally detectable in the future • But it might be crucial to new theories, e.g., Bohmian quantum gravity, GRW’s energy contribution, and so on.

45. Philosophers: are any ‘real theories’ under-determined? • Our evidence is equally compatible with T (our best physical theory) and T* (we and our apparently T-governed world is a computer simulation) • But T* is just skepticism, not a ‘real’ physical theory

46. Philosophers: underdetermination too local to be interesting? The evidence equally supports T (newtonian mechanics plus gravitational theory plus universe is at rest in absolute space) and T* (same, but universe moving 5mph wrt absolute space) 5mph

47. How should we react in QM case?

48. Lesson of Quantum Mechanics: Bad News • GRW, Bohm, Everett, etc. show that there is underdetermination by genuine scientific theories, contrary to what some philosophers suggest. • Furthermore, it’s hardly too local to care about…it involves the central terms of our most fundamental theory • Real life is stranger than fiction or philosophy

49. Good news • Further testing may narrow down the available possibilities • Further theorizing may narrow down the possibilities • The UT doesn’t seem to be of the kind that would challenge realist interpretations of most science. • First, one might appeal to non-observable facts, e.g., simplicity • Second, the underdetermination is not entirely general • Third, there are still levels that are not under-determined wrt the evidence, e.g., energy nuclear levels (Cordero 2002), scattering, etc. • Nevertheless, we’re stuck with a bewildering amount of underdetermination, like it or not—so it’s best to learn to like it.