This page is a space I am using to sketch my presentation of disproof of Bell’s argument for ‘quantum non-locality’ (and, by extension, against the reduction of quantum mechanics to a better theory, free of spookiness). There is no such thing as ‘non-locality’, I am sure; but at least we can know that the arguments and phenomena touted as ‘proof’ of non-locality are no such thing.
The theoretical bogosity of Bell’s and similar arguments was first recognized in the 1980s by E T Jaynes, I guess instigated by the then recent experiments of Alain Aspect. [See ‘Clearing up mysteries’.] Since then, others have correctly identified the error and devised completely ‘classical’ systems that display the supposedly ‘spooky’ behavior. My goal is to help people visualize what’s going on, in a way as purely geometric and unprejudiced (by unwarranted physical assumptions) as possible. Using conformal geometric algebra lets me make the motions explicit, and could make animation easier.
THIS PAGE IS UNDER DEVELOPMENT. Partly it is my way of learning some geometric algebra.
Conformal geometric algebra is a generalization of the ordinary geometric algebra of three-dimensional euclidean space, with extra dimensions added respectively for the point at the origin and the point at infinity. These enhancements let one define such objects as ‘tangent vectors’, which are like ordinary vectors except they are located at a specific point. This ability makes conformal GA a succinct and thorough language for our task.
Basis vectors o,e_{1},e_{2},e_{3},∞. Null vector o is the point at the origin. Null vector ∞ is the point at infinity. Unit vectors e_{1},e_{2},e_{3} are unit vectors parallel to the x,y,z axes, respectively. Inner products o⋅∞ = ∞⋅o = -1.
A vector-at-a-point, coming into existence at time t_{0} and then moving at a constant velocity.
q(t) = moving tangent vector.
e_{q} = a vector in the direction of motion of q.
v_{qo} = a vector giving the initial position of q.
r_{q} = a vector giving the orientation and amplitude of q.
T_{q}(t) = translation versor for q.
q(t) = 0, if t < 0
q(t) = T_{q}(t) or_{q}T_{q}(t)^{-1}, if 0 ≤ t
T_{q}(t) = exp(-(v_{qo} + e_{q}t) ∞/2) = exp(-e_{q}∞ t/2) exp(-v_{qo}∞/2)
A device that reads the amplitude of selected components of a moving tangent vector that impacts with it.
c = a tangent vector representing the reader.
r_{c} = a vector giving the orientation and gain of the reader.
t_{c} = a time value specifying the position of the reader.
c = T_{c}or_{c}T_{c}^{-1}
T_{c} = T_{q}(t_{c})
An amplitude reading is a scalar made from inner products, (-∞⋅q) ⋅ (-∞⋅a), where q is some tangent vector at the same location as an amplitude reader and c is the component-selective reader. For instance, the reading of an unaltered moving tangent vector q(t_{c}) equals
C_{q} = (-∞⋅q(t_{c})) ⋅ (-∞⋅c)
= (-∞ ⋅ (T_{c}or_{q}T_{c}^{-1})) ⋅ (-∞ ⋅ (T_{c}or_{c}T_{c}^{-1}))
= (-(T_{c}∞T_{c}^{-1}) ⋅ (T_{c}or_{q}T_{c}^{-1})) ⋅ (-(T_{c}∞T_{c}^{-1}) ⋅ (T_{c}or_{c}T_{c}^{-1}))
= (-T_{c}(∞⋅ (or_{q}))T_{c}^{-1}) ⋅ (-T_{c}(∞⋅(or_{c})T_{c}^{-1})
= T_{c}(-(∞⋅(or_{q}))⋅ (-(∞⋅(or_{c}))T_{c}^{-1}
= T_{c}(-(∞⋅(o∧r_{q}))⋅ (-(∞⋅(o∧r_{c}))T_{c}^{-1}
= T_{c}(-((∞⋅o)r_{q}- (∞⋅r_{q})o))⋅ (-((∞⋅o)r_{c}- (∞⋅r_{c})o))T_{c}^{-1}
= T_{c}(r_{q}⋅ r_{c})T_{c}^{-1}
= r_{q}⋅ r_{c}
As you can see, I have defined the amplitude reading so it is simply the inner product of the orientation vectors, respectively, of the moving tangent vector and the amplitude reader.
A device that lets pass only selected components of a moving tangent vector.
s = a tangent vector representing the selector.
r_{s} = a vector giving the orientation and gain of the selector.
t_{s} = a time value specifying the position of the selector.
s = T_{s}or_{s}T_{s}^{-1}
T_{s} = T_{q}(t_{s})
A component selector s transforms a moving tangent vector q to a new moving tangent vector q_{s} as follows:
q_{s}(t) = ((-∞⋅q(t_{s})) ⋅ (-∞⋅s)) T_{q}(t) or_{s}T_{q}(t)^{-1}
= (r_{q}⋅ r_{s}) T_{q}(t) or_{s}T_{q}(t)^{-1}
for t_{s} ≤ t. (Note that the gain of s is the square of its weight. This convention simplifies the expressions without loss of generality.)
The amplitude reading of such a such a transformed moving tangent vector is
C_{qs}= (-∞⋅q_{s}(t_{c})) ⋅ (-∞⋅c)
= (r_{q}⋅ r_{s}) (r_{s}⋅ r_{c})
Note that if r_{s} = r_{q}/|r_{q}| or r_{s} = r_{c}/|r_{c}| then C_{qs} = C_{q} = r_{q}⋅ r_{c}.
Let’s imagine the simple case where the orientations r_{q}, r_{c}, and r_{s} are parallel to the same plane; for instance, if all three vectors are perpendicular to e_{q}, and thus parallel to its bivector dual e_{q}⋅ (-e_{1}∧e_{2}∧e_{3}). Also assume that all three orientation vectors have weight 1; then the amplitude reading depends entirely on the angles of orientation, φ_{q}, φ_{c}, and φ_{s}. The component-selective amplitude reading is
C_{qs}= (r_{q}⋅ r_{s}) (r_{s}⋅ r_{c}) = cos(φ_{q} - φ_{s}) cos(φ_{s} - φ_{c})
The square of an amplitude reading is
C_{qs}^{2} = cos^{2}(φ_{q} - φ_{s}) cos^{2}(φ_{s} - φ_{c})
The expectation of C_{qs}^{2} over all the possible (coplanar) orientations of r_{c} is
E_{qs}= (1/(2π)) ∫_{0}^{2π} C_{qs}^{2} dφ_{c}
= (1/(2π)) ∫_{0}^{2π} cos^{2}(φ_{q}- φ_{s}) cos^{2}(φ_{s}- φ_{c}) dφ_{c}
= (1/(2π)) cos^{2}(φ_{q}- φ_{s}) ∫_{0}^{2π} cos^{2}(φ_{s}- φ_{c}) dφ_{c}
= (1/(2π)) cos^{2}(φ_{q}- φ_{s}) ∫_{0}^{2π} cos^{2}(φ_{c}- φ_{s}) d(φ_{c}- φ_{s})
= (1/2) cos^{2}(φ_{q}- φ_{s})
If this expectation is the probability (or proportional to the probability) that a trigger device will be activated, then the behavior is the same as that of one arm of an Aspect-style experiment.
Suppose that instead of a single moving tangent vector there are two that move in unison. You can think of them together as a single object. Call the second tangent vector g(t) and define it by
r_{g}= − ∞ ⋅ g(t) = exp(-I_{q}θ/2) r_{q}exp(-I_{q}θ/2)
so that it is like a copy of q(t) rotated around e_{q} by an extra θ radians. However, g(t) is unaffected by the component selector. What it does is set the orientation of the component-selective amplitude reader. None of the analysis above is changed by this addition; however, now there will be a clear reason for coincidences (and it will not be ‘spooky’).
Let us run the set-up a second time, with nothing changed except the orientation of the component selector, which is now r_{s}′, φ_{s}′ instead of r_{s}, φ_{s}. Also the altered moving tangent vector now is q_{s}′, with orientation φ_{s}′. Summarizing, substituting φ_{q}+ θ for both φ_{c} and φ_{c}′:
C_{qs} = cos(φ_{q} - φ_{s}) cos(φ_{q}− φ_{s} + θ)
C_{qs′} = cos(φ_{q} - φ_{s}′) cos(φ_{q}− φ_{s}′ + θ)
E_{qs}= (1/2) cos^{2}(φ_{q}- φ_{s})
E_{qs′}= (1/2) cos^{2}(φ_{q}- φ_{s}′)
Imagine these expectations are the probabilities that a trigger device will be activated, using the symbols A and A′ to refer to activation on the first and second runs, respectively. Thus
P(A) = E_{qs} = (1/2) cos^{2}(φ_{q}- φ_{s})
P(A′) = E_{qs′} = (1/2) cos^{2}(φ_{q}- φ_{s}′)
What is the probability P^{coincidence} of trigger activation on one run given that there was a trigger activation on the other run?
Define functions
f_{A}(θ) = C_{qs}^{2}/P(A) = 2 cos^{2}(φ_{q}− φ_{s} + θ)
f_{A′}(θ) = C_{qs′}^{2}/P(A′) = 2 cos^{2}(φ_{q}− φ_{s}′ + θ)
Then (1/(2π)) f_{A′}(θ) dθ represents the proportion of activations in the second run that ‘belong’ to the angle θ, and (1/(2π)) f_{A}(θ) f_{A′}(θ) dθ represents the proportion of those that also involve activation in the first run. So
P^{coincidence} = (1/(2π)) ∫_{0}^{2π} f_{A}(θ) f_{A′}(θ) dθ = (2/π) ∫_{0}^{2π} cos^{2}(φ_{q}− φ_{s} + θ) cos^{2}(φ_{q}− φ_{s}′ + θ) dθ
= cos^{2}(φ_{s} − φ_{s}′)
Summarizing:
P(A)/(P(A)^{max}) = cos^{2}(φ_{q}− φ_{s})
P(A′)/(P(A′)^{max}) = cos^{2}(φ_{q}- φ_{s}′)
P^{coincidence} = cos^{2}(φ_{s} − φ_{s}′)
where P(A)^{max} = P(A′)^{max} = 1/2 are the maximum values of P(A) and P(A′). That’s the same behavior as for individual and coincidence counts in an Aspect-style experiment (see similar expressions in Hofer). Here the coincidences are clearly due to ‘local’ effects, and do not even occur concurrently. All that is necessary is for the moving tangent vectors to have related initial states.
P(A) = (1/2) cos^{2}(φ_{q} − φ_{s}) = (1/2) (r_{q}⋅ r_{s})^{2}
P(A′) = (1/2) cos^{2}(φ_{q} − φ_{s}′) = (1/2) (r_{q}⋅ r_{s}′)^{2}
P(A)^{max} = P(A′)^{max} = 1/2
P(A)/P(A)^{max} = cos^{2}(φ_{q} − φ_{s}) = (r_{q}⋅ r_{s})^{2}
P(A′)/P(A′)^{max} = cos^{2}(φ_{q} − φ_{s}′) = (r_{q}⋅ r_{s}′)^{2}
P^{coincidence} = cos^{2}(φ_{s} − φ_{s}′) = (r_{s}⋅ r_{s}′)^{2}
P^{coincidence} = P(AA′|A + A′) [where + means ‘or’]
P(A + A′) = P(A) + P(A′) − P(AA′) [‘Venn diagram’]
P(AA′) = P(A) + P(A′) − P(A + A′)
P(AA′) + P(A + A′) = P(A) + P(A′)
P(A|A′) = P(AA′)/P(A′) = 1 + P(A)/P(A′) − P(A + A′)/P(A′)
P(A′|A) = P(AA′)/P(A) = 1 + P(A′)/P(A) − P(A + A′)/P(A) = (P(A′)/P(A)) P(A|A′) [Bayes’ theorem]
P^{coincidence} = P(AA′|A + A′) = P(AA′ (A + A′))/P(A + A′) = P(AA′)/P(A + A′) = (P(A) + P(A′))/P(A + A′) − 1 = (P(AA′) + P(A + A′))/P(A + A′) − 1
(P^{coincidence} + 1) P(A + A′) = P(A) + P(A′)
P(A + A′) = (P(A) + P(A′))/(P^{coincidence} + 1) = (1/2) (cos^{2}(φ_{q} − φ_{s}) + cos^{2}(φ_{q} − φ_{s}′))/(cos^{2}(φ_{s} − φ_{s}′) + 1)
= (1/2) ((r_{q}⋅ r_{s})^{2} + (r_{q}⋅ r_{s}′)^{2})/((r_{s}⋅ r_{s}′)^{2} + 1)
P(AA′) = (1/2) ((r_{q}⋅ r_{s})^{2} + (r_{q}⋅ r_{s}′)^{2}) (1 − 1/((r_{s}⋅ r_{s}′)^{2} + 1)) = (1/2) ((r_{q}⋅ r_{s})^{2} + (r_{q}⋅ r_{s}′)^{2}) ((r_{s}⋅ r_{s}′)^{2}/((r_{s}⋅ r_{s}′)^{2} + 1))
Bell erred, and presumably so do many others, by factoring a joint probability as follows:
P(AB) = P(A) P(B)
The correct formula is
P(AB) = P(A) P(B|A) = P(B) P(A|B)
The error seems to have been pointed out first by Jaynes (see ‘Clearing up mysteries’). The error is mathematical, a fallacy of logic. The physical arguments given by Bell in its favor are a striking example of ignoring a common cause, because the laws of probability are specifically devised to help one avoid that error, yet they did not succeed. There are implications for how probability theory is taught and understood. (For more discussion, see Jaynes.)
Screw that. I’m not even going to bother brushing up on these inequalities, because they are derived from an already identified mathematical error. Quoting standard deviations of ‘violation’ of them should have no extra persuasive power.
In my opinion, the notion is so ludicrous that only some psychological pressure to make it true can explain the institutional failure to squelch it through logic. The question then becomes: When and why did fundamental physics become a branch of speculative fiction?
Probably not nearly as clever as it sounds. Maybe there will be spinoff technologies.
Snake oil.
Not only ‘quantum non-locality’ is ‘called into question’, but also the notion that it is impossible to reduce quantum mechanics to something both more fundamental and fully deterministic. (In my opinion, either this is possible or pigs can fly. Everywhere we look, except in the imaginations of human beings, ‘random’ just means unknown or intractable.)