Archive: Give me real chaos over complex indeterminacy

Originally posted 15 Mar 2005 - Phew, I can't believe I got that much to grips with the technical discussion back when I was more 'into it'. As I've mentioned in previous posts (I recommend you read If You Think You Understand This, Then You Don't and Bell's Inequality and Bell Revisited before reading this post), I'm not so exercised now about whether the world is deterministic and local. It seems quite likely that it is at least non-local, which fits with my best intuition at this point anyway.

The Man
Beginning with his 1995 paper, Tim Palmer, from the European Centre for Medium-Range Weather Forecasting, questioned the binding force of Bell's Inequality and demonstrated that wholly deterministic (although non-computably chaotic) non-linear dynamical systems could produce the apparent randomness of quantum state measurement while keeping our understanding of the universe on a local and real footing.  He has refined his thinking and presented it in further papers in 2004 and 2005.  I think that he is onto something real and big.

Through the happy chance of working with someone whose partner works with Tim at ECMWF, I got the opportunity to meet him and talk a bit about his thinking.  Keep in mind that Tim's day job is in meteorological research, so his physics work is in his spare time.  Although I clearly lack schooling in the range of mathematical tools necessary to follow all of the technical details, through reading his papers and talking for that hour or so, I've got a pretty good idea what he's up to.

The core points
There are two common and related themes to his physics work:

1. The quantum state reduction associated with measurement via classical devices is not necessarily stochastic.  It can be regarded as a deterministic non-linear dynamical system.  A central element to this system is the non-computability of certain propositions within it.
2. Bell's Theorem relies on an assumption of counterfactual definiteness, an assumption that is not valid in a deterministic world where counterfactual propositions are essentially non-computable and therefore lack definite truth values.  Therefore, we need not accept the unavoidability of 'spooky action at a distance' in order to save deterministic dynamics.

Riddled Basins
Although the evolution of the state vector through time is a deterministic one, the reduction of the system to an observable state appears to be random.  Conventional QM takes this indeterminacy as given.  Palmer thinks that the apparent randomness hides a chaotic dynamic that is simply too messy to untangle, which makes his approach what is known as a 'hidden variables' one.

Chaos theory uses the concepts of attractors and basins when speaking of how different initial conditions migrate via iterations of some non-linear operation toward some resting place.  A resting place is called an attractor, and the collection of initial states that migrates to that attractor is called its basin.

What I have just said is, of course, a gross over-simplification.  Not all non-linear systems converge to an attractor at all.  Some just explode towards infinity.

Nor does an attractor necessarily constitute a single number at which the system settles forever.  An attractor may be a cyclical one, whose cycle may involve simply flipping regularly between two numbers or may involve cycling through a sequence of numbers so long that it would not repeat in the history of the universe to date.

Also, not all basins are defined by smooth outlines.  An attractor's basin may be very messy indeed, with any point within the basin having other points arbitrarily close to it that DO NOT belong to the basin.  Such basins are said to be riddled.

Now imagine a system with two attractors: whose basins collectively cover the entire possible set of initial conditions; whose basins are of equal area (or volume, if the space is three dimensional) to one another; and whose basins are jointly riddled (that is to say, intertwined) as above.

It is possible to construct such a system that is so riddled that (given truncation errors) it is impossible to compute algorithmically which basin a given set of initial conditions belongs to.  Given the equal size of the basins, there is a 50% chance that any set of initial conditions belongs to either basin.  It is also possible to construct this system in such a way that it is consistent with other aspects of the formalism of QM for the measurement of bivalent properties like spin, and Palmer shows this.

There may be more work to do, but the point is that Palmer has shown that a deterministic system may exist that is consistent with QM.

What about locality?
But isn't such a system bounded by Bell's inequality, which is known to be violated by both QM prediction and experimental evidence? No, says Palmer, because Bell's proof makes an implicit assumption about certain counterfactual propositions having definite (yes or no) truth values.

Where does this notion of counterfactual reasoning enter Bell's proof?  Let's remember the experiment that tests it.  Zero angular momentum electron pairs (Right and Left) are emitted from a special source.  One device measures the spin of each Right electron along some axis in the plane that is orthogonal (perpendicular) to the electrons' path.  Another device measures the spin of each Left electron along one of two axes, each of which constitutes a different rotation (say x for one and z for the other) from the axis of the Right device.  Bell's inequality is then a relationship among the measurements taken at these three (R, Lx, Ly) orientations.

Counterfactual assumption
The important thing to remember, though, is that for any given pair of electrons, only TWO of these measurements can be taken (R & Lx, or R & Ly).  The theorem makes the assumption that the measurements among many pairs of electrons can be lumped together and then relates correlations within the large set.  So, in fact, the relationship observed for any GIVEN pair of electrons is one of two:

1. The relationship among R, Lx and what Lz would have been, had the z orientation been chosen for the Left member of this pair, or
2. The relationship among R, Lz and what Lx would have been, had the x orientation been chosen for the Left member of this pair.

The elements in italics are the counterfactual ones.  In reality, ONLY x OR z can be chosen as the orientation for the Left member of any given pair.  The assumed measure of what it would have been were the other angle chosen is taken from the statistical behaviour of the pairs whose Left element was measured at the other angle.

Determinism, Free Will, and the observer as part of the system
What is the upshot of all of this?  I want to (try to) go into a bit more of the technical detail in a minute, but it is possible to think about this initially at a philosophical level.  IF the universe is deterministic in the philosophical sense, then everything that happens (everything that has ever happened and will ever happen) happens NECESSARILY.  It COULD NOT have happened any other way.  Palmer shows with his demonstration of a particular chaotic system that determinism is consistent with QM observations.

So, in effect, we're saying that the observer only measured, say, R and Lx for a particular electron pair.  And we're saying that the universe has evolved in such a way that - however free the observer felt himself to be in his choice of the L measurement orientation - he COULD NOT have chosen it to be y.  So introducing a counterfactual proposition about what MIGHT have happened HAD he chosen y is meaningless.  Even though it feels like a small hypothetical change in the context of a large universe, it is simply not within the set of possible states of the world.

As uncomfortable as many feel with determinism, because of its implications for our pure notion of free will, this is hard to get around.  Neither the electron pair nor the observer can be taken outside the universe itself.  And if the evolution of that universe is deterministic (as it is if it can be modeled by a non-linear dynamical system) then not only the spin measurements but also the orientations at which they are made follow necessarily from the initial conditions of the universe and the laws that govern is evolution.

Over our heads
Now, Tim Palmer expresses all of this in a much more disciplined way.  He gives an example of a universe defined by a famous attractor, known as the Lorenz Attractor (named after the father of non-linear dynamics, who discovered it).  This attractor is defined by three differential equations on three variables.  If the initial conditions of the universe sit on the attractor, and if these differential equations govern the universe's evolution, then the smallest of perturbations to one of the variables will move the system off of the attractor (given the attractor's fractal nature), thereby violating the laws of the universe.

But Palmer needs to bridge a gap here.  The wave function of quantum mechanics (defined by Schrodinger's equation) uses complex (i.e., using 'i', the square root of -1) linear dynamics.  Palmer is talking about real (i.e. no square root of -1) NON-linear dynamics.  How can his system do the work of Schrodinger's?

At this point, it gets pretty hairy for us non-mathematicians.  Palmer introduces a new definition of i as an operator on a sequence of real numbers.  Quantum states can be defined by sets of these sequences, and Palmer shows how his i operator performs in a way analogous to the maths of the upward cascade of fluctuations in a turbulent flow (something from his meteorological world). 

The effect of these steps is to present a way of describing the state function in granular (like the quantum world itself) terms rather than in the continuous terms of the Hilbert space that is used in conventional QM.  Applying this to the test of Bell's inequality, this means that we can't pick any angle in a continuum but are instead confined to a finite (but as large as we wish) set of angles. Palmer proves that there is no way that measurements for both the Lx and Ly angular differences from the R orientation can be simultaneously defined.  All of this amounts to the more rigorous and mathematical proof of the point I made philosophically and sloppily in the section above.  The bottom line is that any real physical state must be associated with a computable real number (even if the only way to compute it is to let nature 'integrate' it through a physical experiment!).

Repercussions
Where does this take us?  If we re-interpret the wave function as a set of binary sequences as described above, we can think of the elements of those sequences as 'real' bits of quantum reality, which means that even in the absence of a measurement, we take the quantum state to have definite values rather than a superposition of possible values.

Also, a sequence itself encodes information not just about the system it describes but also about that system's relationship to the whole.  Palmer uses an analogy with the DNA in our bodies' cells.  This hearkens back to the explicate and implicate order in David Bohm's interpretation of quantum theory.  Look for more on Bohm in an upcoming post.