Indeterminism - reach out and superluminally touch some one

This little page originated as a quick response to a friend who teaches undergraduate English Literature.
She announced to her class that QM proved "scientifically" that reality was subject to knowledge. Oh my!

Quick summary: Quantum mechanics (despite the Copenhagen interpretation and popular press to the contrary) has little to say on whether or not the universe is deterministic. Why? Because the temporal evolution of a wave-function is itself deterministic. And because determinism is a philosophic rather than scientific question. But of course, there is much more between the lines - read on.

Central Question:

Is the universe truly random and indeterminate at its core? For most thinkers from Aryabhatta to Newton, the answer to this question was rather obvious – all physical processes are deterministic and therefore non-random. There was a precise logical, certainty behind all events, born in part of the prevailing doxa of determinism inhabiting Christian sentiment in the west, and in part of the legacy of Aristotelian lore. Even when Poincaré formulated his ideas concerning relativity (in his well known 'law about laws') at the end of the 19^th century and Laplace worked on his initial velocity formulations, the postulate of determinism was felt to be at the core of all science and rationality. That is to say, it was believed that a priori states both lead, and are deterministically linked to, a posteriori states.

Little by little however, this central assumption began to be questioned. Both philosophy and science began to propose indeterminacy at the core of causality. For example Whitehead's metaphysical analysis was very close to Heisenberg's views of uncertainty in physics. The former through a non-physical analysis; the later through physical analysis.

Background:

But cat was out of the bag, to badly pun Schrördinger's thought experiment. In the early 20^th century Harry Bateman produced his general theory of relativity. Lorentz modified his Electron Theory to account for the Michelson experiment (spacial curvature), and Pointcaré created his famous equation E=mc² relating energy and matter (Einstein failed to credit Pointcaré as the originator not only of the equation, but of the concept of relativity, although he apparently had discussed Pointcaré findings in depth with friends before he himself published his famous paper (for example, see Keswani). Would we today be dealing with a different paradigm if Pointcaré, Hilbert, or others had been acknowledged by Einstein and so studied in more detail? Perhaps – but as I have pointed out in more formal venues than this, scientific 'paradigms' are as subject to political and moral lapses as are those of the mundane world of office politics.) At any rate, none of these accepted theories violated a core belief in absolute determinism. “God does not play dice with the Universe” as Einstein famously put it. Yes She does said Niels Bohr and Werner Heisenberg, who suggested that the unpredictability to be found in quantum events was due to a deeper inherent probabilistic fuzziness affecting the entirety of any quantum system. Or as systems theorists might put it, any observed system may be considered indivisibly probabilistic. In other words, indeterminate at the core.

EPR Thought Experiment:

Suppose two photons are released from an individual atom. Suppose further that they escape in opposite directions. Like all electromagnetic radiation their movement (in theory at least) propagates at constant and unchanging speed in a vacuum (fortunately readily found in the offices of most university presidents), regardless of reference frames. This propagation through vacuum is at said to be exactly 299792.458 km/sec.

Now according to the so-called Copenhagen interpretation, quantum mechanics stipulates that whenever two quantum systems such as these two photons, interact or have a common origin they can no longer be regarded as separate systems. They are permanently “entangled”. Or said another way, for any experiments (measurements) carried out on these systems, there will always be a relationship in the form of a statistical correlation between them. This is sometimes called Schrördinger's quantum intrication. So the two photons launching off in opposite directions must according to Bohr, always show a statistical correlation. That is, what happens to a hundred million photons must be statistically reflected in what happens to their paired brothers. Entanglement is visible via correlation - it is statistically derived.

Consider this diagram (which loosely represents the thought experiment prosed in 1935 Einstein, Podolsky, and Rosen known as EPR):

detector-B ← polarizer-B ← photon-B ← radiant light source → photon-A → polarizer-A → detector-A

This diagram implies the following: For any pair of photons from the same atom, send each member of the pair to different detectors which can measure spin, momentum, position, etc. Atoms of course can rotate in any direction, so we know that the two photons must have same polarity (angle of rotation) when they start out, as well as some other commonalities. Quantum mechanics and classical physics both say that there is a correlation between photon-A and photon-B in terms of position and momentum. Hence knowing something about position or momentum regarding photon-A should automatically tell us something about photon-B.

There is however, a small problem. According to Heisenberg, as soon as the position of photon-A is measured, its momentum becomes uncertain – it becomes indeterminate. Before measuring position, the photon is said to be in superposition, but its unknown state “collapses” upon measurement. In a sense the experimenter's act of observing the photon changes its state.

Yet since photon-A and photon-B are correlated then surely measuring photon-A in any manner should allow for a deduction to be made about photon-B. So measuring the momentum of photon-A should yield the momentum of photon-B. At the same time we can also measure the position of photon-B. In this way EPR argued, both the momentum and position of photon-B should be able to be obtained – determined precisely – without violating the Heisenberg uncertainty principle. In other words, we would have determinism of the sort not allowed by that same principle. A problem indeed – either the photons are communicating superluminally thereby breaking a founding assumption of special relativity that lightspeed is the upper limit for information exchange; or the uncertainly principle was incorrect; or Schrördinger's quantum Intrication was wrong; or there were hidden variables at play effecting the results.

Bohr argued that the equations of quantum mechanics could never produce a snapshot image of reality. Things were only probabilistic and therefore not subject to the exactitude of the measurement believed necessary for 'real' world explanation, such as this EPR thought experiment suggested. He also said that EPR did not take intrication into account properly since it treated the two photons as part of separate systems. EPR therefore was not in his view a valid way to view reality. But most of all, of the alternatives listed in the previous paragraph, it was most likely that there were hidden variables at work since all other options were thought to be proven.

Bell's Inequality:

Early on Einstein showed that Brownian motion such as the movement of dust in a stream of sunlight is random and probabilistic on the macro level but fully deterministic at the molecular level. Hence an important part of Bohr's response to EPR that hidden variables were at play in determining photon interaction, seemed a reasonable one and no one thought much more about it for a few decades. Some even suggested that the polarization angles were themselves hidden variables.

In order to clarify matters, some experiments were done:

Suppose that a radiant light source emits photon-A and photon-B, both from the same atom and therefore both with the same characteristics (they are “entangled”). Photon-A at random hits polarizer-A or polarizer-B. (As any photographer knows, polarizers let all, none, or some light through depending upon their angle relative to the light source.) Let us say that photon-A hits polarizer-A. If it is seen by detector-A then we know that it must have a degree of rotation similar to polarizer-A, otherwise it would have been stopped (absorbed) at the polarizer and never reached detector-A. Suppose that we rotate polarizer-B 90⁰ relative to polarizer-A. And that photon-B by chance happens to hit polarizer-B at the same time that photon-A hit polarizer-A. Photon-B remember, has the same polarization as photon-A. So if polarizer-B is rotated 90⁰ relative to polarizer A, what will we see at detector-B? The answer is nothing.

This is because the probability of passing though polarizer-B is proportional to cos²(90⁰) which works out to be equal to zero. This formula - cos²(θ⁰) is a shortened version of Malus's law, the full version of which which states that when a beam of plane-polarized light of intensity X, produced by a polarizer falls on detector, the intensity Y of the transmitted beam varies as the square of the cosine of the angle between the two planes of transmission. Quantum mechanics suggests that the observed yield depends only upon the relative angle between the two main axes of the polarizer, which for ideal (theoretical only) filters gives a normalized angular dependence of cos²(θ⁰).

 In other words, when detector-A is triggered by a photon, we know that since polarizer-A was at 90 degrees photon-A must also have also been at or close to 90 degrees. And since photons A and B are from the same atom and therefore of the same polarization photon-B must also be polarized at 90 degrees. As a result proton-B cannot penetrate polarizer-B. This has been experimentally verified – no photons are detected in the real world at detector-B but are detected at detector-A when polarizer-A is set to 90 degrees relative to polariser-B. (Actually, this is not quite what happens. In the real world one is dealing with many measurements over millions of photons over time and drawing a statistical conclusion, as well as imperfect polarizers, imperfect radiant sources, imperfect dark rooms, some doubts about the applicability of Malus's law, and the imperfect-by-definition graduate students performing the measurements. So some photons do get through to detector-B. But only a small number relative to those at detector-A.)

Let us say that photon-A and photon-B are “same” (S) if they both go through their respective polarizers at the same time; similarly they are “same” (S) if they are both absorbed by their respective polarizers at the same time. And let us say that the two photons are “different” (D) if neither of these scenarios occur. When the polarisers are at 90⁰ relative to one another, we would expect p(S)=1 for each trial run. But if we set polarizer-A to 30 degrees relative to polariser-B we would expect the probability of (S) to be cos²(30⁰)=0.75 for each trial. Similarly rotating the polarizers to differ by 60⁰, we should see (S) only 25% of the time. At 45⁰ we should see (S) 50% of the time. This does not happen.

To see why, suppose the polarisers are randomly shifted between two different angles X, Y. Each time we fire off a couple of entangle photons at opposite polarisers therefore, one of four possible things can happen, as follows:

If the two photons have the same orientation, the polarizers will always read (S). If on photon has X orientation and the other Y, the polarisers will always read (D). Hence p(S) = 2/4=.5

But what if there is something else going on. What if the photons, clever little quanta that they are, decide at random that no matter what angle the polarisers are set to, they will change their orientation before going through the polarizers so that they will always show either (S) (or (D)). In other words, p(S)=1 whenever these quanta decide to change their orientation. They may also decide on (D) of course, in which case p(D)=1 regardless of polarizer orientation. So at random, either the photons change their orientation, or they do not. When they do, they will always show (S) (or D) of 1. When they do not, p(S)=.5. If we look at millions of pairs of photons, we can say that p(S) >= .5, which is known as Bell's inequality. (There are several other versions, such as Harrison's which I particularly like. But all draw the same conclusion – the overall probability of (S) is at least .5. What about doing this with more angles? The argument is the same:

p(S) >= 11/25 = 0.44

But when the experiments, (S) appears less than 50% of the time for two angles, less than 55% of the time for three possible angles, and less than 44% of the time for five possible angles and so on. Of course the actual calculations allow for Malus' law, but the conclusion is still the same - the expected results do not occur.

The best known example of this was work by Alain Aspect and coworkers 1980's papers. They sent two entangled protons (A and B) from the same atom in opposite directions toward two polarizers – polarizer-A and polarizer-B. They then shifted the angle (amount of polarization) of polariser-A while the photons were in mid-flight. The result? (S) did not occur as expected. In fact, it appeared to be the case that photon-B shifted its polarization to match that of photon-A. Photon pairs remained highly correlated regardless of how the polarizers were shifted.

More recently, Olmschenk fired coherent light at a ytterbium ion in a Paul trap (a means of holding one or more ions radially and axially in place). Shining the laser at the ions produces photons. If the photons are entangled, so too are the ions. When the first ion was measured its state (qubit) was shown to exactly mimic by the second ion, even when the two were several meters apart.

There were many others – Clauser, Freedman, Stapp to mention a few of the better known experiments triggered by Bell's work (although there is certainly disagreement with these – see Dickson for an example of this). Overall however, the experimental results have been taken by many to indicate that Schrördinger's quantum intrication is problematic for both special relativity and quantum mechanics.

Now these and literally hundreds of similar experiments draw the conclusion that they are proving/disproving the existence of hidden variables by means of Bell's inequality. Because of course polarization angle - the purported hidden variable - is unknown ahead of time. It is random in this type of experiment. Malus law therefore becomes a random generator for polarization angle. Of course experiments with known polarization angles can be set up as well. Fixed polarization angles rotating so as to average differences, for example. But the argument above still applies. I am ignoring Stern-Gerlach here, as well as effects on electrons using E, V, B fields since this little page is just toe-tip into the water. I am also ignoring string theory and its many knotty problems. As well as Bolm's work which is said to account for the experimental results without recourse to quantum strangeness. Why? Well, consider Bolm's argument concerning that essential element of probability theory and these experiments - randomness.Bolm believed that nothing is random. Rather it is our perceptions that are at fault for we fail to see the higher order functions - like seeing the world from 5 feet versus 30,000 feet. For example,  drop some ink into a glass tank filled with glycerin. Rotate the tank slowly in the horizontal plane. The ink will smear then vanish as it is dissolved into the glycerin. Rotate the tank at the same speed in the opposite direction and the ink will reappear. Bolm believed that the ink is alternating between a low degree of order (when it is clearly visible) and a high degree of order (when it is dissolved and invisible). The former was what he meant by explicate order (i.e. obvious) and the later he termed implicate order (i.e. non-obvious). Implicate order was what he believed is mistakenly termed 'random'. And so the findings of the work above could, Bolm and his followers suggest, be explained in terms of degree of order without recourse to the 'strange' results necessitated by the legacy of the Copenhagen interpretation.... but to continue:

Non-locality:

For the average human, locality extends to the ends of their hands and feet - we can astound our friends with our skill at cat juggling only if 1) we can physically reach the cats and 2) we are wearing face and body armor. But if the cats eschew our overtures and move off beyond our reach, we need something other than the local implements with which most are born. Non-local relationships between objects require something extra. We can easily imagine two types of non-locality - delayed and instant. The former just means that there may be a delay in transmitting a message to or effecting a non-local object from a local one. Instant non-locality means there would be no delay – instant action at a distance. It is the instant type (communication at superluminal speeds) that some believe these experiments are pointing toward.

Einstein's particular viewpoint in special relativity eschewed this instant kind of non-locality. His theory relied upon the speed of light  being an unchanging constant. As an interesting aside, because of his use of this fundamental Einstein as Miller has pointed out, originally wished to call his theory the Theory of Invariance, rather than of Relativity. At any rate, according to Einstein non-local communication for distant objects should always involve a delay (Hawking's objection to Tipler notwithstanding )

Von Neumann had shown that non-locality at the particle level in quantum mechanics had to imply that the theory forbade instant message transmittal. There would always be a delay. So special relativity and quantum mechanics seemed to be saying something similar - influencing something instantly at a distance or effecting something instantly at a distance was impossible. While indeterminate, random events were certainly acceptable up to a point, non-local instantaneous relationships and/or communication between objects was not acceptable. So the experimental findings stemming from EPR and Bell, particularly the many versions of Aspect's work, were quite troublesome: there appeared to be non-local instant relationships between the photons.

Too many “explanations”:

There have been many theories, experiments, and alternate viewpoints concerning this. Everyone it seems, has an opinion, ranging from pure logic through oddly metaphysical. Here are just a few:

Sometimes there is a comment about the logic of Bell's proof. For example the normal two-valued logic is not necessarily closed (viz. Gödel). Not every theorem is deductible; many are in fact self-referential. There are many systems particularly in n-valued predicate calculus which may be more applicable to interpretation of the statistically observed events in EPR experiments than the deductions used in Bell's theorem.

Sometimes people say that there is no way to know when a particle leaves a polariser unless we detect that particular particle at the polariser. But quantum mechanics suggests one cannot know the location of a particle between detections. And so one also cannot know when the statistics of photon pairs is effected by changing polariser angle.

Some refer to the idea that a signal is not being superluminally transmitted non-locally as some results imply, but rather that the signal is imposed upon the correlation over many trials. Mass-energy as a global properly of any multi-particle system could create multiple energy states which are properties of the system rather than merely locus-specific. And so these experiments would serve as a form of signal detection not of what is transmitted between loci, but rather of system wide traits.

Then there is the question of c. Einstein's initial derivations from Maxwell and of the Lorentz transformations contained velocities of light of c — v, c + v – very different from his later statement that c was independent of the motion of the source. But if his first derivation is correct, then special relativity is incorrect (since c is required to be unchanging). If the second is correct, then one might say that c is a fundamental of Minkowski spacetime geometry (see Maudlin for an interesting take on this, as well as Kastner's work on Cramer's transactional theory). But spacetime's interaction with c may also be considered a logical tautology – space curves to ensure c stays the same; clocks alter their ticks to ensure c stays the same (Michelson and Morley's results therefore may have been about changing geometry only.) Troitskii, amongst others, has suggested a theoretical cosmology in which c is allowed to vary. Milan Pavlovic and many others have explored this, showing it is possible to derive these geometries without requiring 'entanglement'. Pavlovic also demonstrates some straightforward objections to Einstein's work and like Vidmantas Samušis suggests a simple mathematical argument that the dual assumptions of c as a constant and time as being relativistic (i.e. flows at a variable rate) are false.

Others tackle the foundation of matter. For example, John Wheeler discusses a sort of proto-foam below the level of the atom where electrons are not pieces of matter or quantum energy but rather wormholes in the fabric of space time. Hence the apparently superluminal interactions of quanta may be due to connection at a deeper level.

Some have argued that the need for so-called “squeezed cat” (both here and not here) status of photons is due merely to mathematical aberration. Carl von Weizacker for example shows that the axioms of quantum theory give rise to tense logic which describe relationships of all matter in spacetime without the need for Heisenberg uncertainly – i.e. without indeterminate states. David Finkelstein's work in this area is of particular note.

The issue has caused a lot of metaphysical statements to be made as well. Paul Kwiat for example has used quantum interrogation and wave-particle duality to search a region of space without actually entering said space. How? By using a quantum superposition to run and not run a search algorithm in an optical quantum computer to obtain an answer when the state collapses. This and similar work has led Kwiat and his group to the inference that reality and knowledge are linked.

Speaking of collapsing states, some believe (notably David Lindley) that decoherence (when the probability of the photon's superposed state approaches zero, eg. when its probability of being here and not here at the same time approaches zero) has nothing to do with measurement at all. Rather he feels, decoherence is just a process of nature. So in opposition to Kwiat's views the postulate that reality is effected by knowledge becomes completely unnecessary.

And of course there is Tim Palmer's very interesting geometric approach to these questions. The hypothesis is that there is a state space (a set of all possible states) – call it Sa. Within Sa is a smaller fractal subset of state space, Sb. Sb  is dynamically invariant – all states within Sb have always belonged to Sb and will always belong to Sb. Sc on the other hand, is a variant state space with no physical reality. Palmer suggests that Sb is where our physical reality resides. Points not within this invariant subset  are 'unreal'.  For example, if a coin toss had landed heads rather than tails when it was tossed. Heads was real, because that was what occurred, tails was unreal because it did not occur. Hence 'unreal' can be considered as counter-factual events. And so like Bohr's complementarity, 'unreal' sets have no definitive solution. Palmer has proposed an 'invariant set postulate' to describe this set beyond Sb but within Sa, which relates fractal geometry and quantum mechanics saying that is Sa is a complex chaotic system the Sb must be fractal.

The idea here is that no algorithmic extension to quantum mechanics, or the body of the theory itself, is capable of determining for every point in Sa whether or not it lies in Sb as an invariant set.  And so quantum theory can only yield algebraically computed probabilities without identifying an underlying complete sample space. Which makes idea breading ground for the various oddities of quantum theory, such as superposition. But in Palmer's view superposition or more generally quantum coherence is due to the failure of quantum theory to discuss the invariant set – only when Schroedinger's cat lies within a dynamic invariance it is a real (aka 'alive') cat. And so measuring does not collapse a state, but rather describes an aspect of the geometry of the invariant set hitherto unmapped. Similarly, wave-particle duality is explained since particles do not destructively interfere but rather lie instead in Sc, with no physical reality. And so on – Palmer's work nicely seems to explain much of the 'strange' guesses (eg. superposition) of quantum theory in thoroughly logical fashion. Of course much work lies ahead, but the theory is very promising. Exciting times.

Postmodernism enters the fray:

And on and on. Interesting experiments are occurring, as are practical applications from cryptography to material science. But the interpretation of it all seems to me to be little more than a rehashing of the philosophies of uncertainty in postmodernism. Or postmodern relativism. Jaspers and his intellectual descendents come to mind here, as an example of someone using the ideas of quantum mechanics in formulating philosophy. Whitehead's intellectual successors still seeking metaphysical links to, and through, the physical sciences. It seems to me that such fancies are fun, but no more than that. Quantum mechanics has nothing to say about these metaphysical questions at all, and even less about post-modernist view points, or new age phantasmagoria exemplified by 'the Tao of Physics', 'Mystical Physics', and similarly popular titles. It has most certainly nothing whatsoever to say about the intertwining of knowledge and “reality”, although of course there are many who interpret results as saying that the one influences the other.

Quantum effects vanish at the macro level. This due to randomization and probabilistic effects, and to complexity as Penrose has so nicely pointed out. For example, whilst there may be quantum gravitational collapse in cellular microtubules which effects of the contents of synaptic junctions and the gross information in the resultant neural net, this is not a link between ideation and physics. An autopoietic system whether biological or mechanical, whether macro or nano, regardless of its mechanisms does not require anthropic or photon-like homunculae telepathing their dreams to the cosmos. The types of quantum effects touched on above are extremely simple – some photons; some correlations; some simple autocorrelative significance; a pinch of Feynman calculus, Lie algebra, Minkowski space, or Lobchevskian topology; a few incantations; and voila - some experimental results. This is not true of the staggeringly complex interactions in the 'normal' world were everything appears to be correlated, as Mandelbaum has suggested. Systems theory and its subset cybernetic theory, chaos theory, quantum theory, hairy cosmic muffin theory ... are theories. They are descriptions and interpretations only. The need for an implied mystical connection between observer and observed is unecessary. Science is merely a temporary acceptance (viz. Kuhnian 'shifts') of what seems the most rational to a group of tenured peer reviewers at the time – a statistically based confidence interval if you will. It is never based upon certainty. Whether or not it is based upon reality is merely noetic semantics.

Whether beliefs produce reality is a game for philosophers, not the mathematics or statistics of quantum mechanics. And certainly not a question that can be, or should be, resolved by assuming a current theory – which will ultimately be replaced by something better. Any theory is little other than a temporary story and prediction about the results of a handful of observable events.

The mere accumulation of observational evidence is not proof.

So what does quantum mechanics, relativity theory, and the like have to say about philosophical indeterminancy? Nothing. For the mathematics which applies to a selected set of infinitesimal particles may be different from that needed to determine the presence or absence of a cat.