H op y = Eop
y
Where:
p op
= -i h (¶/ ¶ x) And
Eop = i h (¶ /¶t)
So, the full wave equation in one dimension (x) becomes:
-
h / 2m (¶ 2/ ¶ x2) y +
V(x) y = i h (¶ /¶t) y
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:
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[2]:
"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".
The original proposal[3] by
Rietdjik and Selleri was to show that if a photon is successively transmitted
by 2 polarizers (using appropriate settings or orientations) then the very
first transmission must influence a hidden variable which co-determines the
second one. "Malus law"
was first formulated by Etienne Louis Malus in 1809 and asserts that the
intensity of light transiting an analyzer and polarizer is proportional to cos2(q)
where q
is the angle through which the analyzer is rotated with respect to the
polarizer.
One can proceed by first considering a set-up with 2 orthogonally polarized correlated photons (designated 'g 1' and 'g 2' in the diagram shown below) and these interact with three polarizers denoted A, B and C.
By appropriate computations one is led to a theorem:
For the new (lower panel, Fig. B) setup, the
predictions of quantum mechanics allow us to write:
P1,2 (x –p/2, y) = ½ sin2(x – p /2, y) = ½ cos2(x – y) = p2(x,y)
So long as polarizers A and B of the original setup are orthogonal (e.g. perpendicular to one another) photon g 2 is transmitted by C if 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 φ i,j are identical, viz.:
φ i (x – p /2) = φ j(x)
and
φ*i(x – p /2) = φ*j(x)
Therefore, one can deduce that if the hidden variable φ, (implicit in the lower arrangement) experiences no change from the previous interaction (of g1 and A, via transmission or absorption) then we have:
P1,2(x –p /2, y) = m (φ i(x – p /2) Ç φ j(x))
Where /x\ denotes set intersection
As Rietdjik and Selleri note[4] the preceding expressed “a mutual physical independence of the transmission events at A and B, respectively, in the sense 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.
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[2]:
"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 different analyzer orientation 'sets' were obtained. These might be denoted:
To fix ideas and show differences, in the Aspect experiment four 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 (sum) 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
In the case of David Bohm's hidden variables, and its deterministic model, the above sum came to less than 2.
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 '
In the case of David Bohm's hidden variables, and its deterministic model, the above sum came to less than 2.
The key insight for Bohm was in
terms of the evolution (time change) of the probability density:
dP/dt + Sn (Ñ n) (P Ñn S)/m = 0
dP/dt + Sn (Ñ n) (P Ñn S)/m = 0
From the preceding, and a modified Hamilton
I now examine this in the context of an experimental underpinning. Since Bohm's deterministic theory includes what he calls "hidden variables" - which effectively drive the determinism toward a relative locality- then these must be incorporated. The problem is to invoke an experimental basis which can allow the determinism to be checked.
I now examine this in the context of an experimental underpinning. Since Bohm's deterministic theory includes what he calls "hidden variables" - which effectively drive the determinism toward a relative locality- then these must be incorporated. The problem is to invoke an experimental basis which can allow the determinism to be checked.
One can proceed by first considering a set-up with 2 orthogonally polarized correlated photons (designated 'g 1' and 'g 2' in the diagram shown below) and these interact with three polarizers denoted A, B and C.
If 'E' is the causal event (e.g. photon departure from
location), then by causal determinism one must find:
AE < CE < BE
In other words, the interaction event denoted 'AE' (whereupon the designated photon for g 1 interacts with polarizer A) occurs before CE and CE before BE.
AE < CE < BE
In other words, the interaction event denoted 'AE' (whereupon the designated photon for g 1 interacts with polarizer A) occurs before CE and CE before BE.
By appropriate computations one is led to a theorem:
"A deterministic theory with
hidden variable φ reproduces Malus' law for a photon (g2) 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.
To refine the experiment, one removes the polarizer C from the top panel set up shown and defines p1,2(x - p /2, y) to be the probability that, in the new setup g 1 is transmitted by A, and g2 by B.
To refine the experiment, one removes the polarizer C from the top panel set up shown and defines p1,2(x - p /2, y) to be the probability that, in the new setup g 1 is transmitted by A, and g2 by B.
P1,2 (x –p/2, y) = ½ sin2(x – p /2, y) = ½ cos2(x – y) = p2(x,y)
So long as polarizers A and B of the original setup are orthogonal (e.g. perpendicular to one another) photon g 2 is transmitted by C if 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 φ i,j are identical, viz.:
φ i (x – p /2) = φ j(x)
and
φ*i(x – p /2) = φ*j(x)
Therefore, one can deduce that if the hidden variable φ, (implicit in the lower arrangement) experiences no change from the previous interaction (of g1 and A, via transmission or absorption) then we have:
Where /x\ denotes set intersection
As Rietdjik and Selleri note[4] the preceding expressed “a mutual physical independence of the transmission events at A and B, respectively, in the sense 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.
It is also what paves the way toward validating Bohm's acausal determinism. How so?
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 (- φ i(x); and to transit B, we need φ (- φj(y). Then to transit both:
φ (- φi(x) Ç 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 cos2(x – y)
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 (- φ i(x); and to transit B, we need φ (- φj(y). Then to transit both:
φ (- φi(x) Ç 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 cos2(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 Malus’ law requirement, viz. based on the orientations for axes x, y:
S = ½ [cos2(a – b) + cos2(b – c) + cos2(c – a)]
Let S = p2(a,b) + p2(b,c) + p2(c,a)
And apply the Malus’ law requirement, viz. based on the orientations for axes x, y:
S = ½ [cos2(a – b) + cos2(b – c) + cos2(c – a)]
Take the partial:
¶S/ ¶ a = - ½ [sin 2(a – b) – sin 2(c – a)]
At this point, we briefly 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.
¶S/ ¶ a = - ½ [sin 2(a – b) – sin 2(c – a)]
At this point, we briefly 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. 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 [sin 2(a – b) – sin2(c – a)]
¶S/ ¶a = 0 implies: (a – b) = (c – a)
E.g. let (a – b) = (c – a) = p /2
Then: - ½ [sin2(p/2) – sin2(p/2)] = - ½ [sin(p) – sin(p)] = 0
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
¶S/ ¶a = 0, ¶S /¶b = 0 and ¶S /¶c = 0
We see that since: ¶S/¶a = - ½ [sin 2(a – b) – sin2(c – a)]
¶S/ ¶a = 0 implies: (a – b) = (c – a)
E.g. let (a – b) = (c – a) = p /2
Then: - ½ [sin2(p/2) – sin2(p/2)] = - ½ [sin(p) – sin(p)] = 0
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:
Smin = ½ [cos2(120) + cos2(120) + cos2(120)]
Since cos (120) = ½
Smin = ½ [(½)2 + (½)2 + (½)2] = ¼
Or: Smin = 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.
Smin = ½ [cos2(120) + cos2(120) + cos2(120)]
Since cos (120) = ½
Smin = ½ [(½)2 + (½)2 + (½)2] = ¼
Or: Smin = 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.
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