*Diagram shows polarizer orientation settings as referred to in the text. Note that the left side-column applies to polarizer C, and the right to polarizer B. (See also Fig. A, of previous blog)*Continued: (

*Note for abbreviation purposes, ‘gamma 1’ = g1, and ‘gamma 2’ = g2*)

For the setup in Fig. B (of previous blog) the predictions of quantum mechanics allow us to write:

P1,2 (x –pi/2, y) = ½ sin^2(x –pi/2, y) = ½ cos^2(x – y) = p2(x,y)

So long as the polarizers and B of Fig. A (previous blog) are orthogonal (e.g. perpendicular to one another) photon g 2 is transmitted by C iff g 1 is transmitted by A. Thus, in the case of A and C orthogonality, then for each micro-condition of the system (g1,g2) for which g1 is transmitted so also is g2. This must meant the relevant sets of hidden variables φi,j are identical:

φi(x – pi/2) = φj(x)

and

φ*i(x – pi/2) = φ*j(x)

Therefore, one can deduce that if the hidden variable φ, implicit in arrangement B (Fig. B, prior blog) experienced no change from the previous interaction (of g1 and A, via transmission or absorption) then we have:

P1,2(x –pi/2, y) = m (φi(x – pi/2) /x\ φj(x))

Where /x\ denotes set intersection

As Rietdjik and Selleri note (op. cit.) the preceding expressed “

*a mutual physical independence of the transmission events at A and B, respectively, in the send that one of the events does not change the hidden variable φ as it is relevant to the other*.” This is precisely what confers a relative locality since the action of φ is constrained.

Here is where some difficulties in reconciling results enter. For this basis we return to the arrangement A (Fig. A) in the previous blog. We focus on the polarizers B and C and arrange three separate settings as follows – keeping in mind the x –orientations refer to C and the y-orientations refer to B (refer to the diagram):

1st setting: x = a and y = b

2nd setting: x = b and y = c

3rd setting: x = c and y = a

One can show from assuming no overlap between hidden variable sets: φi(x) and φ*i(x), then let p2(x,y) be the probability that the

*unpolarized*photon g2 is transmitted by both polarizer C with axis x, and polarizer B with axis y as:

p2(x,y) = m(φi(x) /x\ φ*i(x))

Now either one of two propositions must be valid, supporting or refuting a deterministic effect:

1) Transmission across C generates a change in φ and thereby allows transmission through B, or

2) No change transpires with the crossing at C, so none occurs at B (no influence, so null hypothesis)

Let’s look at each in terms of the hidden variable sets: φi(x), φj(y). Then – in order for g2 to transit C we need: φ (- φi(x); and to transit B, we need φ (- φj(y). Then to transit both:

φ (- φi(x) /x\ φj(y))

Using this one can estimate the probability condition by way of summing all orientations, as applicable to the null hypothesis:

p2(a,b) + p2(b,c) + p2(c,a) >= ½

Which, of course, violates Malus’ law which predicts:

p2(x,y) = ½ cos^2(x – y)

Using partial derivatives (@) we can move further on this.

Let S = p2(a,b) + p2(b,c) + p2(c,a)

And apply the putative Malus’ law requirement, viz. based on the orientations for axes x, y:

S = ½ [cos^2(a – b) + cos^2(b – c) + cos^2(c – a)]

Take the partial:

@S/@a = - ½ [sin2(a – b) – sin2(c – a)]

At this point, we review the concepts of max-min theory to do with partial derivatives of functions of several independent variables (say a, b, c etc.)

Consider a function of x alone such that: F(x) = f(x, a, b, c…)

Which has an extreme value (extremum) at x = a. Then if f has a partial derivative with respect to x at x = a, that partial derivative must be zero by virtue of the theory for max-min functions F(x) of a single independent variable., viz.

@f/@x = 0 at x = a

Similar reasoning allows us to arrive at the necessary conditions for minima say, when one has a function of several independent variables.

The number of simultaneous equations @f/@x = 0 etc. thus obtained is equal to the number of independent variables.

Now, in the case of S, we have three independent variables a, b and c representing the different polarizer positions. For minimization we need:

@S/@a = 0, @S/@b = 0 and @S/@c = 0

We see that since: @S/@a = - ½ [sin2(a – b) – sin2(c – a)]

@S/@a = 0 implies: (a – b) = (c – a)

E.g. let (a – b) = (c – a) = pi/2

Then: - ½ [sin2(pi/2) – sin2(pi/2)] = - ½ [sin(pi) – sin(pi)] = 0

Now, of course, since we’ve three polarizer orientations, a, b and c the minima must hold for all, then also we have:

@S/@b = 0 implies: (a – b) = (b – c)

And:

@S/@c = 0 implies: (b – c) = (c – a)

Further computations disclose that we need:

(a – b) = (b – c) = (c – a) = 120 deg

Then:

S_min = ½ [cos^2(120) + cos^2(120) + cos^2(120)]

Now, since cos (120) = ½

S_min = ½ [(½)^2 + (½)^2 + (½)^2] = ¼

Or: S_min = ½ [(¼) + (¼) + (¼)] = ½ [(3/4)] = 3/8

Since 3/8 is less than ½ this violates the null hypothesis of no influence, and hence proves that a deterministic hidden variables effect is present. This is exactly the theorem noted in the prior blog:

*A deterministic theory with hidden variable φ reproduces Malus' law for a photon (g 2) transmitted by two polarizers C and B, with arbitrarily chosen settings of their axes, only if the hidden variable φ undergoes some change (a redistribution) when g 2 crosses the first polarizer.*

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