Wednesday, December 8, 2010

Quantum Acausal Determinism (2)

As noted by the late David Bohm, and B. J. Hiley [1]:

"It is well known that Einstein did not accept the fundamental and irreducible indeterminism of the usual interpretation of the quantum theory, e.g. as revealed by his statement: 'God does not play dice with the universe'. However, what is much more important is his rejection of another fundamental and irreducible feature of the quantum theory, i.e. nonlocality"

Let's examine this further, before looking at the E-P-R or Einstein - Podolsky -Rosen thought experiment which sought to show nonlocality didn't float so QM had to be incomplete.

If I have always investigated nature from a reductionist viewpoint, taking some phenomenon 'X' and reducing it into localized sub-assemblies:

X C x1 + x2 + x3 + ...... xN

wherein the total function of X is no greater than its parts, then clearly I will be averse to approaching a natural phenomenon any other way. This is particularly applicable if I have been accustomed to seeing interrelations (in other phenomena) of the form:

[x1 + x2] -> x3
x3 -> [x4 + x5]
[x5 + x6] -> x7

where each of the arrows denotes a 'causal flow' from the component(s) on the left hand side, to the component(s) on the right hand side. Nowhere is this type of 'causal flow' seen more clearly than in examples of Newtonian motion. (For example, tossing a baseball at a wall).

Not coincidentally, this converges where the premise of locality attains near sacramental status. By locality I mean extraneous factors can be ignored in examining the immediate physical situation. If I am studying the dynamics of simple pendulum, I need only pay attention to its mass, length and period, not what the pendulum clock in the next room is doing.

Giving up absolute 'causal flow' via direct interactions (or locality) in models, is not unlike asking a junkie to give up whatever gives him some temporary satisfaction. The equivalent of 'cold turkey' occurred when experiments to test quantum conformity to Bell's Theorem were performed by Alain Aspect and his colleagues at the University of Paris.[2] In these experiments, the detection of the polarizations[3] of photons was the key. These were observed with the photons emanating from a Krypton-Dye laser and arriving at two different analyzers, e.g.

A1 (+ ½ ) <-----------[D]----------->(- ½ )A2

The above scene captures the instant just before each detector intercepts an atomic magnet from the device. The quantum state observed is described by the spin number, which is (+½ ) for A1 and (-½) for A2, corresponding to the spin up and spin down orientations, respectively. It is important to understand that these values can only be known definitely at the instant of observation.

In the orthodox Copenhagen (and most conservative) interpretation of quantum theory, there can be no separation of observed (e.g. spin) state until an observation or measurement is made. Until that instant (of detection) the states are in a superposition, as described above. More importantly, the fact of superposition imposes on all quantum phenomena an inescapable ‘black box’. In other words, no information other than statistical can be extracted before observation.

Meanwhile, in the Bohmian acausal deterministic setting [1]: "it is supposed that each experimental result is determined completely by a set of hidden variables, φ. Thus, the result A of measurement of spin in the direction â depends only on φ, while the result C, of measurement of spin in direction ĉ depends only on φ and ĉ"

This was a brilliant tour-de -force which can be summarized (ibid.):

A = A (â, φ)

C = C(ĉ, φ)

where measuring entanglements such as: A = A(â, ĉ, φ) and C = C(â, ĉ, φ) are specifically excluded.

Thus, as the authors note(ibid.):

"In other words, while nothing is said about the general dynamical laws of the hidden variables, φ, which may be as nonlocally connected as we please, we are requiring that the response of each particular observing instrument to the set φ, depends only on it own state and not the state of any other piece of apparatus that is far away".

This is a critical distinction, because it eliminates the pure objection Einstein had to unlimited nonlocality. Thus, Bohmian quantum physics introduces a measure of locality by tying the action of hidden variables to the particular observing device.

To fix ideas and show differences, in the Aspect experiment four (not two - as shown) different analyzer orientation 'sets' were obtained. These might be denoted: (A1,A2)I, (A1,A2)II, (A1,A2,)III, and (A1,A2)IV. Each result is expressed as a mathematical (statistical) quantity known as a 'correlation coefficient'. Aspect's final result yielded:

S = (A1,A2)I + (A1,A2)II + (A1,A2,)III + (A1,A2)IV = 2.70 ± 0.05

What is the significance? In a landmark theoretical achievement in 1964, mathematician John S. Bell formulated a thought experiment based on a design similar to that shown. He made the basic assumption of locality (i.e. that no communication could occur between A1 and A2 at any rate faster than light speed). In what is now widely recognized as a seminal work of mathematical physics, he set out to show that a theory which upheld locality could reproduce the predictions of quantum mechanics. His result predicted that the above sum, S, had to be less than or equal to 2 (S less than or equal 2). This is known as the 'Bell Inequality'.

In the case of Bohm's hidden variables, deterministic model, the above sum came to less than 2.

The E-P-R experiment:

This was probably conceived explicitly because Einstein wanted to show quantum theory was incomplete, and hence an unlimited nonlocality could not be accepted - since otherwise anything could occur say at point (x1, y1, z1, t1) 20 million light years from Earth and affect events on Earth, say at point (x2, y2, z2, t2). If that was true, then both locality and determinism went right out the window.

The original form of the experiment invoked the quantum state of a two particle system in which position deifferential (x1 - x2) and momentum sum (p1 + p2)are both determined. Then the wave function is:

U(x1, x2) = f(x1 - x2 -a) = SIGMA_k c_k exp[ik(x1 - x2 -a)]

where SIGMA denotes a summation over the k elements; f(x1 - x2 -a) is a packlet function sharply peaked at (x1 - x2) = a (for an example of such a functional form check out the imposed image in the graphic just below "God" playing dice with Einstein looking on. ) Also c_k is a Fourier coefficient. Thus, in this particular dynamic state, p1 + p2 = 0 and x1 - x2 can be as well-refined as one pleases. (In the normally applied Heisenberg Uncertainty Principle, when nothing is known on the momentum then the position x can be precisely obtained. It's just that both position x and momentum p can't be known to the same exact precision simultaneously)

In the EPR experiment when one measures x1 then one immediately knows: x2 = x1 + a. Alternatively, if p is measured, then one knows immediately that p2 = -p1 since p1 + p2 = 0. In both cases the 1st particle is "disturbed" by measurement and this accounts for the Heisenberg uncertainty relations applied in 1-d. However, the 2nd particle is taken not to interact with the 1st at all- so one can obtain its properties minus the assumption of any disturbance. Still, the Heisenberg principle still applies so: dp2 dx2 >= h-bar.

Where h-bar = h/ 2 pi as before (h is Planck's constant or 6.62 x 10^-34 J-s) , and dp2, dx2 denote the uncertainties in the 2nd particle momentum and position. The key point here is that given the above conditions and parameters Heisenberg's explanation of the uncertainty arising from a "disturbance" can no longer be used. It was exactly this which prompted Einstein, Podolsky and Rosen to argue that since both x2 and p2 were measurable to "arbitrary accuracy" without any disturbance, then they already existed independently (in particle 2) as localized elements of reality with well-defined values before any measurement took place. Hence, they concluded that QM is merely a mathematical abstraction which gives only an incomplete picture of reality - and hence QM itself is incomplete.

Next: Bohr's Reponse to the EPR paradox.

[1]Bohm, D. and Hiley, B.J.:1981, Foundations of Physics, Vol. 11, Nos. 7/8, p. 529.

[2] Aspect, A., Grangier, P. and Roger, G.: 1982, Physical Review Letters, Volume 49, No. 2, p. 91.

[3] Polarization is the orientation in space of the electric field E, associated with light. This can be altered, subject to the imposition of different filters and devices.

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