Honoring Epimenides of Crete \(±Δx\): From Quantum Paradoxes, through Weak Measurements, to the Nature of Time
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- Bu səhifədəki naviqasiya:
- 3.5 Temporal Paradoxes Revisited
- 4 Quantum Oblivion: The Underlying Mechanism of several Quantum Feats
- 6 Time: A geometric Parameter or the Real Source of Evolution
- Acknowledgements
Bell's inequality. Therefore, enabling to interpret the future state vector as a hidden variable, which converts spatial nonlocality to backward affect. Weak correlations are also the basis for the Cheshire Cat Paradox [29] according to which a spin-1/2 particle can take one route of an interferometer while its spin takes the other. A series of experiments demonstrating the theory in Secs. 3.3-3.5 and especially in [12] is expected to take place at INRiM during 2014. 3.5 Temporal Paradoxes Revisited The method of weak measurement can validate and shed some light on the above temporal paradoxes. A recent experiment of joint weak and interaction free measurement is suggested in [30,31]. Weak measurements validating the Hardy paradox (Sec. 2.4) and negative weak values of projection operators (Sec. 3.4) are presented in [32]. Furthermore, as we concluded in [10], weak measurements (or more specifically, protective measurements [33]) can be used to track the wavefunction's changes after each partial measurement introduced in Sec. 2.3. We also suggest weak correlations measurements of the entangled pairs in Secs. 2.5-2.7 in the spirit of the ones in [12].
Let us again take a step back to grasp the emerging overall picture of the quantum realm: It is a realm describing the microscopic where time-symmetry is much more common than in the classical, macroscopic realm. Recently, we were able to pinpoint this unique quantum reversibility with a simple gedankenexperiment. Notice first that momentum conservation is one of classical physics' most fundamental laws, which every translational symmetric system must obey. The following quantum interaction, however, seems to defy it.
Let an electron and a positron, with spin states 1 ( ) 2 X X X
and momenta e e P P
, be sent along the y-direction, entering two Stern-Gerlach magnets (drawn for simplicity as beam-splitters) positioned at 0 0
, )
t x y and 0 0 ( , , ) e t x y respectively (Fig. 6). The magnets split the particles’ paths according to their spins in the x-direction: 1 ( 1
2 ) and
2 e e e \
1 ( 3 4 ) 2 e e e \
We shall describe the time evolution of the process with the wave-functions above and with two-state detector: I/II. The total wave function is: (24) ICNFP 2013 00028-p.15 1 [( 1
2 ) 3
( 1 2 ) 4 ] 2
e e e e e I II \
Fig. 6: Possible electron-positron interactions. (a) The setting. (b-c) Annihilations. (d) Interaction- free “collapse.” a b d 4 2 1 3
1 t 2 3 1 3 2 4 2 1
e e e e e e e c (25) EPJ Web of Conferences 00028-p.16
Depending on their positions at t 1 or t 2 , the particles may (not) annihilate and consequently (not) release a pair of photons, which may in turn (not) trigger detectors positioned in the appropriate places. Let these photons, exhibiting the unique superposition emitted/not emitted, be termed “conditional photons” and let the detector’s two corresponding states be denoted by I/II At 0
t t t d d
the superposition does not change: 1 [( 1 2 ) 3
( 1 2 ) 4 ] 2
e e e e e I II \
At
1 2
t t
, if a photon pair is emitted we know that the particles ended up in paths 2 and 3. Otherwise, Fig. 7: The peculiar momentum exchange due to outcome (d) in Fig 6: The positron’s momentum is changed while that of the electron remains intact. e e e e (26)
ICNFP 2013 00028-p.17 1 [( 1
2 ) 4
1 3 ] 3 e e e e e I II \
which is an interesting superposition:one component of it is a definite state while the other is a superposition in itself. At
2 t t ! , if a photon is emitted, we know that the particles ended up in paths 1 and 3. Otherwise, 1 ( 1 2 ) 4
2 e e e I \
which is peculiar. On the other hand the positron is physically affected by the interaction: It is not superposed anymore, hence if we time-reverse its splitting, it may fail to return to its origin. In other words, its momentum has changed. Not so with the electron: It remains superposed, hence its time-reversibility remains intact (Fig. 7). Thus only one party of the interaction "remembers" it by exhibiting change, while the other remains unaffected, apparently violating the momentum conservation law. We have recently shown that this quantum oblivion constitutes the essential ingredient of IFM as well as all its variations presented above. Once again, quantum mechanics turns out to owe its strength to its greater temporal flexibility. 5 Zeno Going Quantum Whereas Zeno formulated his paradox on purely logical grounds, his 20 th -Century followers [4] have shown that Nature herself can make time "stop" under an appropriate form of repeated measurement. The broad philosophical implications of this effect, not yet fully explored, go beyond the scope of this paper. Here we only mention Zeilinger's application of it to enhance the IFM [17]. By an appropriate choice of cyclical measurements, he managed to raise IFM's efficiency close to 100%. Strangely, however, this application was not explored further. This is peculiar because Hardy, in the works reviewed above (Sec. 2), has demonstrated several IFM variants even more striking than the original one. It would therefore be reasonable to expect the quantum Zeno effect to offer similar enhancements in their case too. Consider, e.g., Hardy's paradox [21] where a particle and anti-particle perform mutual IFM on one another. Augmented with the quantum Zeno effect, it can always produce the peculiar result that one member of the pair has visibly changed the other's state while it looks equally obvious that they never came into contact. This line of investigation has been recently taken up by our group, with several such intriguing effects to be reported soon.
(28)
(27) EPJ Web of Conferences 00028-p.18
6 Time: A geometric Parameter or the Real Source of Evolution? What, then, is the bearing of these quantum paradoxes on the deeper issues concerning a nature of time? Following are a few speculative reflections, to which we hope later to provide firmer grounding in theory and experiment. Classical physics, to which special and general relativity belong, treats time as a purely geometrical ingredient of the universe, alongside with the three spatial dimensions. Against the perfect logical rigor and experimental support that make relativity so powerful, many physicists find the "block universe" picture emerging from it manifestly awkward. In fact, the very notion of space-time implies that, just as all locations have the same degree of reality in space, so do all past, present, and future somehow exist along the times dimension without any moment being unique as the privileged "now." Against this mainstream view, there are alternative accounts [34] that address a few unresolved physical issues more straightforwardly, even though still lacking empirical support. They suspect that, if we experience time so differently from space, this difference may be objective, no matter how poorly represented in present-day physics. It is well-known that even the founders of the Block Universe, including Einstein, remained highly uncomfortable with it. The minus sign assigned to the zeroth dimension in Minkowski's geometrical formulation of relativity theory is only one hint that time differs from space in a very subtle yet objective sense. The vast unresolved issue of the origins of time asymmetry in the universe [35] is another.
In short, while Einstein and Minkowski are Parmenides' heirs in modern physics' thought, it was left to more intuitive philosophers like Bergson [36] to counter with Heraclitian dynamics. Bergson has ascribed time a genuine "flow" or "passage," characterize by "Becoming". Every event which we perceive as occurring "now" is indeed a novel phenomenon which, prior to that occurrence, did not exist in the most fundamental sense rather than "being already there" in the future direction of time but only inaccessible to observation. No other than de Broglie, one of quantum mechanics' pioneers, paid the following homage to Bergson [37]: [I]f Bergson could have studied the quantum theories in detail, he would have noted certainly with joy that, in the image that they offer us of the evolution of the physical world, they show us nature in all its occasions hesitating between several possibilities, and he would have undoubtedly repeated, as in La Pensée et le Mouvant, that ‘time is that very hesitation or it is nothing at all.’ Far from being mere poetical musings, these comments suggest a possible alternative to the relativistic picture of space-time. Indeed, Cramer, founder of the transactional interpretation of QM which was originally formulated within a strict Block Universe framework, discusses ICNFP 2013 00028-p.19 the "plane of the present" [38]
where the advanced and retarded solutions of relativistic equation of motion perform their "handshake". He states [38]: [W]hile block-universe determinism is consistent with the transactional interpretation, it is not required. A part of the future is emerging into a fixed local existence with each transaction, but the future is not determining the past, and the two are not locked together in a rigid embrace.
Being well-aware of the speculative nature of our model, we briefly sketch its present stage, hopefully to mature into a physical theory. We begin with the following assumption. Just as the speed of light is regarded as a Lorentz scalar, hence, unchanging when transforming between inertial frames, (as opposed to the space and time coordinates that change as a function of v/c), the quantum interaction can be regarded as more fundamental than the space and time coordinates. The space-time interval between two events following an interaction is therefore not a passive, pre-existing background for the interaction between wave-functions. Rather, it is the quantum interaction's very outcome. All the spatial and temporal oddities of quantum measurement, reviewed in the previous sections, would then be natural! Consider, e.g., a quantum position measurement. A particle whose wave-function is widely spread in space is eventually found in one of all the numerous locations along this wave- function. What about all the locations where it has not been found? They are populated by the macroscopic bodies that have performed the interaction-free measurement that seem to indicate that the particle never went on that direction. Yet the wave-function itself, as any interference experiment can prove, did go in all these directions! The Block-Universe account for this case would allow either a "collapse," known for being incompatible with relativity, or some semi-classical hybrid a-la' "hidden variables." Can quantum mechanics offer another alternative? "No" would be the pertinent answer, but the reason for this ineffectiveness is important: It is just what QM lack to this day, namely a full quantum- mechanical description for both microscopic and macroscopic bodies. This is, in other words, the notorious measurement problem. We submit that introducing "Becoming" into physics offers the key for the problematic "macroscopically-superposed state," a direct consequence of the quantum formalism yet never observed. The dead-and-alive cat exists not in a well-defend space-time within the closed box, but rather as a pre-space-time interaction between numerous wave-functions, the completion of which would give either the "dead" or "alive" state with the relevant space- time configurations between all particles involved. "Collapse," then, is the formation of the macroscopic event with its entire space-time configuration. These admittedly speculative hypotheses rely on the insights gained from the TSVF. In contrast to the classical realm, quantum mechanics does not enable a full specification of the initial state at t=0 to predict the results of all measurements performed at later times. However, this fact is responsible also for our ability of defining a final state-vector describing the system (this final state is clearly redundant in classical mechanics). At first EPJ Web of Conferences 00028-p.20
sight, this double boundary condition on the wavefunction seems to resonate with a block- universe approach. The uncertainty principle, however, provides us with freedom to define the present. As Aharonov et al. argue [39] in order to correctly simulate quantum correlations a sequence of moments should be thought of as a chain of pre- and post- selection conditions. A preliminary model based on these assumptions is now in progress. We would feel privileged to present the following stages of its development in future meetings (whether they "already" exist somewhere in a Block Universe or awaiting genuine Becoming) of this highly inspiring forum in the very cradle of science.
It is a pleasure to thank Yakir Aharonov, Boaz Tamir and Shahar Dolev for insightful discussions and fruitful collaborations.
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