THE QUANTUM BOX EXPERIMENT:
At this point, you are probably wondering if TSQM is real or merely a mathematical-construct of dubious relevance to reality. Although numerous proofs are proffered in the above-cited papers, the following was presented in the Quantum Paradox class I attended and, to me, was particularly convincing. It also illustrates a phenomena that I will be mentioning in other notes.
The following Quantum Box experiment provides one "proof" (there are many others) that TSQM is “real”. Before I go on to describe the experiment, you may wish to review an early description of the experiment. (See:
http://arxiv.org/abs/quant-ph/0310091v1)
Now, please visualize a set of nine boxes arranged in a three by three matrix with the columns labeled from left to right: Box A, B, and C; and rows labeled from bottom to top: time t, t+1 and t+2. A particle entering the system at the bottom (e.g. at time t) is understood to have a one-third probability of being in Box A, B, or C at all levels, t, t+1 and t+2. I understand that these probabilities were confirmed through ideal (von Neumann) measurements taken at each level. (We will defer the question: “what causes the wave function to “collapse” into one box and not in another” to my discussion of the Anthropic Principle.) In any event, these confirming measurements were not part of the experiment that I am about to describe.
In the experiment that was reported in the lecture I attended, a very large ensemble of particles was introduced into the experiment and, although ideal measurements were taken at time t+2 for Boxes A, B, and C, only the experimental data for those particles found Box A (the post-selection sub-ensemble) were retained for further consideration. The theory behind the experiment is, to my understanding, that the ideal measurement of the sub-ensemble of particles found in Box A at t+2 constitutes a boundary condition, which through the propagation of a time-reversed wave, constrains the potential locations and states of the particle to that subset of positions and states that remain possible given both the t (starting) boundary condition and t+2 (ending) boundary condition. Mathematically, the theory generates for the selected sub-ensemble a probability of “1” that the particle at time t+1 will be found in Box A and also generates a probability of “1” that the particle at time t+1 will be found Box B. This means that if an ideal measurements had been conducted at time t+1 and Box A or Box B were, metaphorically speaking, opened, the particle would always be found inside the selected Box with absolute certainty. While this verification cannot be actually performed using ideal measurements, the prediction can be experimentally confirmed using weak measurements where the selected sub-ensemble includes a large number of particles. (Information on weak measurements may be found in the papers listed above.) The resulting interference pattern that Dr. Tollaksen presented arose from these weak measurements and was proffered as proof that TSQM is not just a mathematical model (with explanatory value) but also reflects an underlying reality (that I will explore in other papers).
Noting that the probability of finding the particle in Box A and Box B at t+1 were both “1”, you may be wondering about Box C. Here, the mathematics predicts something that seemed astounding. Where the subject particles are electrons, TSQM predicts a particle with all of the attributes of a positron – but with a fundamental difference. The particle predicted for Box C must have a negative mass. (Although not discussed by Drs. Aharonov or Tollaksen, it appears that this finding would be necessary under a reasonable extension of the conservation of lepton law.) In any event, this outcome was mathematically demonstrated by Dr. Tollaksen and implicitly confirmed in the Physics Applications class that I later completed where it was shown that the time-reversed evolution of a matter wave was impossible where a positive mass was involved. Additionally, Dr. Tollaksen indicated that experimental verifications of these negative mass particles had been obtained.
Subsequent to my preparation of the above lecture notes, a description of the experiment was published. Here is what Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen had to say:
"What about box 3? There are, altogether, only N particles. But we already know from the pre- and postselections that there are N particles in box 1 and also N particles in box 2. So we are forced to predict that the third box contains −N particles, a negative occupancy! ... The probe that measures the gravitational field of box 3, instead of being attracted to the box, is in fact repelled by it. The paradoxical result is, of course, just a quantum fluctuation, a measurement error, but an error that happens with virtual certainty. And the effect is not restricted to the gravitational field. Any interaction (for example, electric or magnetic) sensitive to the number of particles will be as if there are −N in box 3, so long as the coupling is small enough to be nondisturbing." Source: A time-symmetric formulation of quantum mechanics, Physics Today, November 2010, Pages30-31
philosophyfaculty.ucsd.edu/faculty/wuthrich/philphys/AharonovPopescuTollaksen2010PhysToday_TimeSymQM.pdf
