Flatland, Dimensionality, and QM Hidden Variables

The animated graphic above shows our 3-D Space plus Time view of the physical world and contrasts it with the very different view of "Flatlanders" who are restricted to 2-D Space plus Time. This posting explores the possibility that insights from consideration of Flatland may be extended to higher dimensionality and shed light on what Feynman called "quantum weirdness" and Einstein called "spooky action at a distance".

In particular, could it be that what we perceive as conflicts between "particles" and "waves" are due to the limits of our perception to 3-D Space plus Time? If we imagine 4-D Space or higher dimensionality, could that help us better understand the "weirdness" and "spooky" nature of Quantum Mechanics (QM)? Could we resolve questions about the Nature of the Universe such as deterministic vs probabilistic, discrete vs continuous, brain vs mind, and so on?

ANIMATED GRAPHIC

Things we recognize as the same appear different to Flatlanders: 3-D Space residents recognize a can of cola as being the exact same object (a cylindrical solid) regardless of whether it is upright or on its side. However, when a 3-D can of cola intrudes upon the 2-D Space of Flatlanders, they see it as several different kinds of figures depending upon its orientation.

At the left edge of the graphic, the can is upright, and, to the Flatlanders, it appears as a CIRCLE of CONSTANT DIAMETER. When an identical can of cola is on its side, the Flatlanders first see a LINE as the lower part of the can intrudes upon their 2-D Space. Then, as the can is lowered, the LINE transforms into a NARROW RECTANGLE. As the can is lowered further, the RECTANGLE WIDENS. So, what is it? A CIRCLE? A LINE? A VARIABLE WIDTH RECTANGLE?

To we 3-D Space persons, the can is one, and only one, 3-D object, a CYLINDRICAL SOLID. To the Flatlanders, it is several different 1-D and 2-D objects.

Things we recognize as different appear the same to Flatlanders: Continuing to view the animated graphic, we see that a ball appears to us to be a 3-D SPHERE. To the Flatlanders, it is first a 0-D POINT, then a 2-D CIRCLE OF VARIABLE DIAMETER.

Furthermore, when the 3-D SPHERE intrudes such that the diameter of the Flatlander's 2-D CIRCLE is the same as the diameter of the upright can, they cannot distinguish between the can and the ball!

Things we recognize as a single object appear as multiple objects to Flatlanders: Continuing to view the graphic, when a moving 3-D hand intrudes into the 2-D Space, the Flatlanders see a wide variety of 0-D, 1-D, and 2-D objects. At first, when only three fingertips intrude, they see three POINTS. Then, as the fingers penetrate further, they see four small CIRCLES plus a POINT representing the thumb. Further penetration of the hand, beyond the wrist, yields an OVAL as viewed by Flatlanders.

APPLYING FLATLANDER INSIGHT TO QUANTUM MECHANICS

In my recent Dialog with Howard Pattee, we speculated on whether the Universe is actually probabilistic, which is the mainstream scientific view, or deterministic, which is definitely the minority view. I speculated that extending the Flatland scenario beyond 2-D and 3-D Space to 4-D and higher-dimensionality, might support an alternative QM interpretation such as that of David Bohm. This posting, which advocates what may be termed "Superdeterminism", is my attempt to support this alternate view.

Classical Physics vs Quantum Physics

Classical physicists accepted the view that the Universe is deterministic and this was the view of  Spinoza, Einstein, Bohm, (and it is also what I -Ira- would like to believe :^).

Quantum physicists generally accept the "Copenhagen Interpretation" of QM which is that the Universe is probabilistic. In our discussion, Howard supported the view that the Universe is probabilistic. While I accept the general consensus that the probabilistic interpretation of QM has stood the test of time and correctly predicted and explained the results of all experiments conducted to date, I nevertheless take the other view.

The Double Slit Experiment - Particles vs Waves

The Double Slit Experiment demonstrates that photons (or electrons) behave like particles when only one slit is open, but like waves when there are two slits. This raises the question: Is matter in general, or sub-atomic matter in particular, really waves or really particles, or something else?

Perhaps residents of a 4-D Space world would see matter as a single type of object and understand why we 3-D Space world residents sometimes see a wave and sometimes a particle? (The can of cola in the graphic above, which we in the 3-D Space world recognize as a single object no matter its orientation, is observed by Flatlanders either a circle with a continuous edge -or- as a line or rectangle with discrete edges.)

The EPR Paradox - Locality vs Realism

Einstein believed that the Universe exhibited both "Locality" (the influence of a distant event cannot be transmitted faster than the speed of light) and "Realism" (the value of a measurement exists before the measurement is made). Experiments conducted in the 1980's appear to prove him wrong and indicate that we must choose between "Locality" and "Realism" - we cannot have both!

I believe Einstein, if he had to choose, would pick "Realism" over "Locality", meaning that a distant event could exert an influence faster than the speed of light. However, perhaps residents of a 4-D Space world would see that the 3-D Space world was curled up within the 4-D Space world such that objects that appear distant in 3-D Space are actually much closer in 4-D Space. In the graphic above, the fingers of the hand appear to Flatlanders as unconnected points or circles, but we, in the 3-D Space world see that they are all parts of a single object.

A cornerstone of the mainstream scientific interpretation of QM is Heisenberg's uncertainty principle (1927), which is that it is impossible to exactly measure both the position and momentum of a sub-atomic particle.

In 1935, Einstein, Podolsky, and Rosen published a paper that proposed a thought experiment ("EPR") designed to show that Heisenberg Uncertainty was not correct and that QM, as understood and interpreted at the time, was not complete. The EPR idea was to have an experimenter produce two electrons (or photons) that were "entangled" such that they would fly apart at the same velocity in opposite directions and with opposite momentum.

An experimenter at location A would measure the exact time of arrival (and thus velocity) of particle A and a second experimenter at location B (the exact distance in the opposite direction) would measure the momentum of particle B. Since the particles have the same velocity and opposite momenta, this experiment would yield the exact position and momentum of the particles.

According to Wikipedia:

In his groundbreaking 1964 paper, "On the Einstein Podolsky Rosen paradox", physicist John Stewart Bell presented an analogy (based on spin measurements on pairs of entangled electrons) to EPR's hypothetical paradox. Using their reasoning, he said, a choice of measurement setting here should not affect the outcome of a measurement there (and vice versa). After providing a mathematical formulation of locality and realism based on this, he showed specific cases where this would be inconsistent with the predictions of QM.

In 1982, Alain Aspect performed an experiment that did not turn out well for Einstein's expectations. Aspect (and others) experimentally showed that QM was correct and that Einstein's expectations for both "locality" and "realism" could not be supported. In short, you either had to choose "locality" or "realism", but not both!

Einstein had passed away by the time the EPR experiments overturned his expectations. I believe, given the choice, he would insist upon "realism" and abandon "locality". In other words, he would accept that the action of an experimenter "Alice" at point A could instantaneously affect the results obtained by "Bob" at distant point B! (Please note that the EPR experiments did NOT show that INFORMATION could be transmitted from point A to point B faster than the speed of light, only that the actions of the distant experimenter could influence the results obtained locally.)

"Quantum Non-Locality":  The mainstream view of QM is founded on what Feynman called "quantum weirdness" and Einstein termed "spooky action at a distance". The technical term is "quantum non-locality" which means that microscopic measurements may reveal that sub-atomic "particles" that happen to be far apart in Space may never-the-less be "entangled" such that measurement of the state of one "particle" superluminally (faster than the speed of  light) affects the state of the other, no matter how far away it might be!

According to the mainstream view, the action of an experimenter "Alice" at point A could instantaneously affect the results obtained by "Bob" at distant point B! Despite the apparent "weirdness", Feynman accepts the mainstream view. However, Einstein clung to what is now termed "local realism", which is the view that the Universe has both "Locality" and "Realism".

"Locality" means that an object is DIRECTLY influenced ONLY by its immediate surroundings. Thus, the influence of a distant event will be delayed by a length of Time that is at least the distance multiplied by the speed of light. "Locality" is NOT a property of the mainstream interpretation of QM.

"Realism" means that all objects have a VALUE for any possible measurement and that this value EXISTS PRIOR to the measurement. According to the Schrödinger's cat thought experiment, the Copenhagen Interpretation of QM seems to entail that a cat in a sealed box may be both dead and alive until an experimenter opens the box and looks into it! According to that view, the "collapse of the wave function" requires the intervention of a CONSCIOUSNESS. In other words, the Moon may not exist if no one is currently observing it. "Realism" is NOT a property of the mainstream interpretation of QM.

Superdeterminism

According to Wikipedia:

John Bell discussed "Superdeterminism" in a BBC interview.  

There is a way to escape the inference of superluminal speeds and spooky action at a distance. But it involves absolute determinism in the universe, the complete absence of free will. Suppose the world is super-deterministic, with not just inanimate nature running on behind-the-scenes clockwork, but with our behavior, including our belief that we are free to choose to do one experiment rather than another, absolutely predetermined, including the "decision" by the experimenter to carry out one set of measurements rather than another, the difficulty disappears.  

There is no need for a faster than light signal to tell particle A what measurement has been carried out on particle B, because the universe, including particle A, already "knows" what that measurement, and its outcome, will be.  

Although he [Bell]  acknowledged the loophole, he also argued that it was implausible. Even if the measurements performed are chosen by deterministic random number generators, the choices can be assumed to be "effectively free for the purpose at hand," because the machine's choice is altered by a large number of very small effects. It is unlikely for the hidden variable to be sensitive to all of the same small influences that the random number generator was. 

Superdeterminism has also been criticized because of perceived implications regarding the validity of science itself. For example, Anton Zeilinger has commented: "[W]e always implicitly assume the freedom of the experimentalist... This fundamental assumption is essential to doing science. If this were not true, then, I suggest, it would make no sense at all to ask nature questions in an experiment, since then nature could determine what our questions are, and that could guide our questions such that we arrive at a false picture of nature."

As I read the above objections to Superdeterminism (or as I usually call it "Absolute or Strict Causality") it seems to me that Bell and Zeilinger are wrong to assume that the experimenter is "effectively free" or "we always assume the freedom of the experimentalist". As Bell acknowledges in the quote above, it is well known that what we commonly call a "random" number generator running in a digital computer is actually bit-for-bit DETERMINISTIC. That is, if we repeatedly start the "random" number generator with a given key number, the computer will repeat the exact same sequence of supposedly "random" numbers AND that sequence will pass statistical tests of "randomness"!

We all agree that a digital computer is a discrete, finite, deterministic machine. According to my view, so is the Universe. Yes, the Universe is much, much, much more complex, but it, and all biological organisms within the Universe, including humans, are machines! 


[UPDATE 29 Dec 2013] I have posted a follow-up to this Topic, with a new animated graphic, that extends the Flatland 2-D Space vs our 3-D Space dichotomy into higher dimensionality to further explore implications for our understanding of QM, see Flatland, Particle-Wave Duality and Super-Luminal Effects.

Ira Glickstein