Bell Local Theory

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Bell Local Theory

The speed of light is constant and invariant relative to the inertial state of the observer.

Let us consider two spatial positions A and B which are fixed on an inertial reference frame. Let A and B lie on the x axis and be positioned at x = 0 and x = X respectively. On a standard space-time diagram A and B appear as a pair of lines parallel with the time axis.

Now let a beam of light leave A at t = 0. The beam will arrive at B at a time T = X/C. If we suppress the y and z dimensions we now have defined relative to our inertial reference frame two events on positions A and B which are (0, 0) and (X, X/C) respectively.

Now let us determine the “subjective” time experienced by our beam of light whilst travelling from A to B. This we can calculate from Minkowski’s quadratic formulation of the Lorentz transformation. S is invariant and is called the proper interval.

(Delta S)^2 = C^2(Delta t)^2 - (Delta x)^2

(Delta S)^2 = C^2(X/C-0)^2 – (X-0)^2

Whish gives: -

S = 0

The proper time interval experienced by a “photon” passing from A to B has zero magnitude the photon does not experience the passage of time in going from A to B.

It is this result that we will use to show that a world characterised by special relativity is consistent with the violation of Bell’s inequality.

Our commonsense tells us that physical states at A and B are independent of each other and an event at A can only affect events at B if the influence is carried by an intermediary entity; a wave or a particle. It also tells the photon not experiencing the passage of time is down to its inertial condition.

To show the world is Bell local we must abandon our intuitive and commonsense notions of space and time and rely on the simple mathematics of the result.

The idea of super-positioning in wave-mechanics is commonly accepted, but the idea of relativistic super-positioning of the physical states of locations that are separated from each other by distance and time is probably new but can be logically deduced from the above result.

The idea is simple this: -

The proper interval between events (0, 0) and (X, X/C) has zero magnitude regardless of whether or not anything is passing along the path (0, 0) to(X, X/C). In other words the state of the world at (0, 0) and that at (X, X/C) are not separated in space-time. It only takes one more step to infer that the Lorentz Transformation super-positions the physical states of the locations (0, 0) and (X, X/C).

If you accept the validity of Lorentz super-positioning the then the consequences for our understanding of quantum behaviour cannot be over exaggerated, it literally transforms the way we perceive the cosmos. It will be sweet irony if as a result of the violation of Bell’s inequality that Special Relativity becomes a central pillar in our understanding of what is going on in the quantum world.

Some of the consequences we can immediately identify.

Lorentz super-positioned quantum systems can interact directly. There is no requirement for particles to carry the electromagnetic force. By removing the photon we immediately simplify our view of the universe. The light cone becomes a cone of super-positioning, forming an infinity of connectivity between an individual quantum system and the rest of the cosmos.

With a little more deduction we can develop an insight into the relationship between relativity and uncertainty, understand how probability waves functions develop and how light interactions form interference patterns. But our immediate objective is to show that Lorentz super-positioning of quantum states is consistent with the violation of Bell’s in equality.

Note the following argument is specific to correlation experiments involving light, for experiments involving Fermions a somewhat different approach is necessary.

First let’s make it absolutely clear this post recognises that the evidence for the violation of Bell’s inequality is over whelming and considers it a done deal! My argument is not with outcome of correlation experiments but with the interpretation of their meaning for the nature of locality.

The superficial argument is that some kind of super-luminal interaction maintains intimate communication between “the particles” as they fly apart so if one is measured it affects the other. This, it is argued by some, contradicts special relativity and therefore special relativity is in trouble! There are two possibilities; one there is a fundamental problem with special relativity and it is capable of explaining the correlations or Special relativity is correct but requires further development to explain the violation of Bell’s inequality. In other words when Einstein claimed quantum mechanics was incomplete, he’d got the wrong theory! It was Special relativity that was incomplete.

Special relativity was incomplete; in it did not recognise that the existence of the minus sign in the metric of space-time causes quantum entities to be universally super-positioned. That is for any time t on the world line of an object, the moment t of the object is super-positioned with every other object in the world where their world lines intersect the light cone(cone of super-positioning) radiating from t. (Both into the future and into the past.) This extension to special relativity would allow quantum mechanics to be developed within the frame work of relativistic super-positioning. My own view is that the weird and counter-intuitive features of quantum mechanics can be explained in terms of relativistic super-positioning including the explanation for the violation of bell’s inequality.

In the standard theory when a quantum system becomes excited, it returns to its ground state by dumping its energy of excitation, in the form of a photon, into free space. In the RSP version of things there is no requirement for a photon, since all quantum entities are super-positioned, instead the excited quantum system becomes sensitive to the states of other systems on its cone of super-positioning. It will literally search out the light cone until it finds a suitable absorber system. When the donor recognises a system that is in a quantum state that can absorb its energy of excitation; it will, because of their super-positioning, donate the energy directly into the absorber system without the need of a mediating particle.

In this transaction ( See Crammer) let the donor system S1 and absorber system S2 be separated by a distance X relative to an inertial reference frame at rest relative to S1.
For the transaction to take place the quantum states of S1 and S2 must be mutually amenable for energy to pass between them. When the two systems interact the proper interval of time separating them has zero magnitude but the time difference on our reference frame is X/C.

So the ability of S1 to exchange energy with S2 depends on the state of S2 at a time X/C into the future of S1 relative to our reference frame.

Now let us consider a quantum system S1 that cascades and donates two quanta of energy in order to lower its energy of excitation. The donor system therefore needs to find two absorber systems to accept its energies of excitation. Because of universal relativistic super-positioning along its light cone S1 can sense the states of other quantum systems. In order for the cascade to be triggered, at some point on S1’s world-line, say time t1, S1 must find two absorber systems S2 and S3 that are simultaneously capable of absorbing its energy of excitation. The light cone radiating from t1 must intersect the two systems when their quantum states are amenable to the absorption of the two quanta of energy held by S1. Let the interactions be configured such that S2 is positioned a distance X1 from S1 and S3 at a distance X2. The timing of S1 donating its energy of excitation is determined by two events: -

1. S2 becoming amenable to receiving a quanta of energy from S1
2. S3 becoming amenable to receiving a quanta of energy from S1

Relative to a space-time diagram these events occur in S1’s future at t1 +X1/C and t1 + X2/C. The proper time intervals between all three events have zero magnitude.
This is an important result and is critically important in explaining how Bell’s inequality is violated.

Now lets look at what happens during a correlation experiment:




In an Aspect’s type experiment let the left hand switch be positioned a distance X1 from the source, the polarisers at X2 and the detectors at X3. Similarly the right hand side components are placed at X4, X5 and X6, respectively.

Therefore the state of the source at a time T0, will be super-positioned with the components of the experiment at T0 +X1/C, T0 +X2/C, T0+X3/C, T0 +X4/C, T0 +X5/C and T0+X6/C relative to our frame of reference. Thus an excited calcium atom at T0 will be “sensitive” to the experimental configuration and quantum states of the systems at the future times defined above.

Assuming that the donation event as seen by the source occurs at T0 then the switching occurs somewhere between T0 and T0 +X1/C and/or T0 +X4/C. The source at T0 is super-positioned with the switches at T0 and T4. The switching occurs before the super-positioned state at interaction is achieved therefore the earlier configuration can have no influence on the outcome of the counts. Immediately before the interaction the source is super-positioned with the detectors at times T3 and T6 and with the polarisers at T2 and T5. At the instance T0 a calcium atom in the source will have sensed the presence of suitable absorber systems to accept its energies of excitation. The absorbers will either be in the polarisers or the detectors depending on the “orientation “of the donor system relative to the polarisers. Whether or not the absorbers are found in the polarisers or the detectors depends on how the calcium atom at T0 is aligned with the polarisers at T2 and T5 and the relative probabilities of a count on either side of the experiment will depend on the alignment of the polarisers relative to each other. What is important is to recognise is that absorber systems on either side of the experiment immediately before acceptance are super-positioned with the donor atom at T0. The same event and same quantum states with the same system orientation! The correlations will be dependent only on the relative setting of the polarisers at T2 and T5 respectively and not on the states of any fictitious properties of particles supposedly mediating the electromagnetic force. Hence there are no super-luminal spooky forces, changing the condition of the “photons” during flight when we alter the angle of a polariser.

Personally I believe the relativistic super-positioning can explain many of the weird and counter-intuitive aspects of QM such as the principle of uncertainty, the nature of the wave-function and interference.

I suspect sometime in the future quantum mechanics will be seen as the child of special relativity.