Classical Interpretation of
EPR- Bell Test Photon Correlation Experiments

Many experiments have been performed in the past in order to answer the question if Quantum Mechanics is complete or if its statistical and indeterministic nature is only an apparent one and some 'hidden variables' exist that would in principle enable a deterministic description.
Typically these experiments use correlated (but spatially separated) photons or particles generated in an atomic decay process. These are each passed through a polarizer and the coincidence count rate in dependence on the relative orientation of the two polarizers is being measured. According to a theory expressed in Bell's Theorem (Bell's Inequality) this should yield different results for the 'Quantum Mechanics' and the 'Hidden Variables' assumption. All experiments are claimed to rule out the existence of Hidden Variables as Bell's Inequality is violated and the coincidence rate (for initially unpolarized radiation) is observed to follow the well known Malus law

(1)       n(θ) = cos2(θ) ,

where θ is the angle between the two polarizers and n(θ) the relative coincidence count rate.
In terms of classical optics this result can not be explained as in general a given direction of polarization has components along any set of two directions even if the latter are perpendicular to each other. For an initially unpolarized signal, one should therefore also expect a coincidence count rate for correlated emissions for θ = 90o, in contradiction to the experiments.
This circumstance has led physicists to the conclusion that in fact the physical property of 'polarization' does not exist objectively and independently but is only created by the polarizers when the light transverses the latter. The requirement that the polarization in both channels should be identical (by virtue of the origin of the radiation) would then imply that for crossed polarizers no coincidence signal is detected. This has then led to the further interpretation that spatially separated events (e.g. as defined by the polarizers) can have some kind of non-local interaction beyond the known forces of interaction, which can cause physical objects to change state dependent on the physical state of another object an arbitrary distance away (entanglement).

However, the flaw with the above argument is that the classical (Hidden Variable) interpretation assumes the light to be infinitely coherent i.e. it completely neglects the stochastic time dependence of the radiation field caused by the finite length of the light pulses and the uncorrelated atomic emissions. As follows from the semi-classical Photoionization Theory for Coherent and Incoherent Light on my site plasmaphysics.org.uk, the time of the photoelectron emission depends on the time dependence of the radiation field i.e. the time dependences should ideally be identical in order to cause coincidences at the two detectors. For the initial correlated atomic emissions, this should in principle be naturally the case (at least if the size of the light source is much smaller than the coherence length of the emissions), but obviously after passing differently orientated polarisers this will in general not hold anymore as the individual atomic emissions are weighted differently:
in general, the time dependence of the electric polarization vector of the wave can be described by

(2)       E(x,y,t) = Ex(t).ex + Ey(t).ey

where ex and ey are the unit vectors in the x and y direction respectively. Ex(t) and Ey(t) are the electric field fluctuations along these axes. These are in general not sinusoidal but, due to the superposition of the emissions of many (uncorrelated) atoms, stochastic functions of time which can only technically be described as a coherent wave on time scales shorter than the coherence length of the atomic emissions (see https://www.plasmaphysics.org.uk/research/coherence.htm for a corresponding numerical simulation); the exact form of the time dependence is however irrelevant as even for separated 'single photons' the 'global' time dependence over the time required to emit the photoelectron will be different for different polarisation directions due to the different amplitudes (the only requirement for this is the condition that the coherence time of the radiation field is shorter than the time required for photoionization, which is easily the case considering typical single detector count rates of about 104/sec)).

Projecting now E(x,y,t) on a given direction of polarization

(3)       P(x,y,θ) = cos(θ).ex + sin(θ).ey

yields

(4)       Ep(t) = E(x,y,t).P(x,y,θ) = Ex(t).cos(θ) + Ey(t).sin(θ) .

If one defines the coordinate system such that the x-axis coincides with the direction of the polarizer of detector 1 (i.e. θ=0) , the projection of the radiation field onto the orbital plane of an atomic electron i (with inclination αi) in detector 1 is therefore

(5)       E1,i(0,t) = Ex(t).cos(αi) .

and the projection onto the orbital plane of an atomic electron j in detector 2 (with the polarizer rotated by a relative angle θ) is

(6)       E2,j(θ,t) = [ Ex(t).cos(θ) + Ey(t).sin(θ) ].cos(αj-θ) ,

which using the addition theorem for trigonometric functions yields

(7)       E2,j(θ,t) = Ex(t). [cos2(θ).cos(αj) + cos(θ).sin(θ).sin(αj)]
                        + Ey(t).[sin2(θ).sin(αj) + cos(θ).sin(θ) .cos(αj)] .

Following the above mentioned argument, coincidences of photoelectron released in the two detectors require time correlated radiation fields i.e. we have to look at their cross correlation.
Generally, the cross correlation between two functions f(t) and g(t) can be defined as

(8)       <f(t).g(t)> := 1/T . 0∫Tdt f(t).g(t) ,

where T is a sufficiently long integration time.
From Eqs.(5) and (7), the cross correlation between E1,i(0,t) and E2,j(θ,t) is therefore

(9)       <E1,i(0,t).E2,j(θ,t)> = cos(αi).[ cos2(θ).cos(αj) + cos(θ).sin(θ).sin(αj)] .<Ex(t). Ex(t)>
                                         + cos(αi).[sin2(θ).sin(αj) + cos(θ) .sin(θ).cos(αj)] .<Ex(t). Ey(t)> .

Now by multiplying Eq.(2) with ex and ey we can write Ex(t) and Ey(t) as

(10)       Ex(t) = E(t).cos(φ(t))
(11)       Ey(t) = E(t).sin(φ(t)) ,

where E(t) is the radial variation of E(x,y,t) and φ(t) is the angle between E(x,y,t) and the x-axis (which may be a stochastic function of time).
Hence we have

(12)       <Ex(t).Ey(t)>= 1/T . 0∫Tdt E2(t).cos(φ(t)).sin(φ(t)) = 0   for T→∞

as the integrand oscillates between positive and negative values (unless the radiation is linearly polarized i.e. φ(t)=const.) and can thus at best grow proportional to √T (if φ(t) is a stochastic function) i.e. the normalized integral will tend to zero for growing T in case of overall unpolarized radiation.

If we now furthermore normalize Eq.(9) to the cross correlation for parallel polarizers

(13)       <E1,i(0,t).E2,j(0,t)> = cos(αi).cos(αj). <Ex(t). Ex(t)> ,

we obtain thus

(14)       nj(θ) := <E1,i(0,t). E2,j(θ,t)>/<E1,i(0,t). E2,j(0,t)> = cos2(θ) + cos(θ).sin(θ).tan(αj) .

If we now perform an ensemble average over all atomic electrons j, the second term vanishes for an isotropic distribution of αj and hence we have

(15)       n(θ) = cos2(θ)

i.e. the relative cross-correlation (and thus the coincidence probability) is distributed according to Malus' law

Note: The assumption of identical functions Ex(t) and Ey(t) at both detectors would strictly speaking imply that both the frequencies and decay times of the two correlated radiative emissions are identical, which is in general not the case. On the other hand, we are of course not actually interested in exact correlations (as implied by Eq.(8)) but only ones that occur within a certain 'coincidence window'. With a correspondingly generalized treatment one should thus expect an identical result to the present one as long as there is some correlation at all, because only the components of the two beams which are polarized along the same direction have maximum correlation as all atomic emissions are reduced by exactly the same amount by both polarizers. If the polarizers point into different directions however, this is not the case anymore and the correlation is lost.

As the Bell test experiments can in this way be explained on a classical basis, they do therefore not introduce any new aspects into physics and in particular do not invalidate the objective physical reality of the polarization of radiation (and consequently neither lead to the conclusion of the existence of a non-local entanglement). Even with the present interpretation, some people may be tempted to again interprete the Malus law as an instantaneous non-local interaction occuring, but it should be obvious from the above treatment that the Malus law is actually only a consequence of the fact that a) the radiation field is a continuous stochastic function of time producing photoionization events in the detectors at certain times and b) that this function is completely uncorrelated for orthogonal directions of polarization. This means that in the latter case no coincidences (apart from accidental ones) are detected. However, this reflects only the nature of the original radiation fields, but does not imply any causal connection between the polarization- or detection processes.


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Thomas Smid (M.Sc. Physics, Ph.D. Astronomy)
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See also my sister site https://www.plasmaphysics.org.uk