To Penrose dodecahedrons (1) in preference permutations.
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Quantum correlations appear not to be in the nature of a usable signal, that can transmit information faster than the speed of light. They have been described by Heinz Pagels as two random sequences of information, that in themselves mean nothing. They only form a pattern when put together.
Might not, then, quantum correlations form, in some sense, an election? In an election, conducted by secret ballot, the voters' choices send no signal as to how they have individually voted. But in the count, patterns or correlations of choice arise.
The following "electoral" interpretation of the quantum rules is only for the smallest scale elections. But the process of interpretation did suggest a possible alternative proof of non-locality, to that given by Roger Penrose. And I amplify this, in the last section, without confusing the proof with "elections" in the sense we are used to them.
Can we have a human election by voters of candidates using Penrose's formulation of the quantum rules of non-locality? Is it possible to give a consistent or thoro-going electoral interpretation of these quantum rules?
The previous page showed that a dodecahedron may represent all 120 permutations of 5 possible choices. Say these are five candidates. The voters have a preference vote, which gives the voters a ranked choice of candidates: 1st, 2nd, 3rd, 4th, 5th. This is evident in the quantum rule, which allows the observers as 'voters' to select any of the twenty dodecahedron vertices and then rank any of its three adjacent vertices in one's chosen order.
The first selected vertex is equivalent to a first preference. The three adjacent vertices are the second to fourth preferences. ( The candidate, going by the label not amongst the first four choices, is the fifth or least prefered candidate. )
But here we have a puzzle. The bell only rings for correlations between the two 'voters'' second, third or fourth preferences, not if they happen to alight on the same select vertex, in the first place. What can this mean?
The select buttons, to which the rules apply, are those in the same or opposite positions on the two aligned but separate dodecahedrons. From the point of view of labeling these dodecahedrons for all the permutations of five choices A, B, C, D, E, similar and opposite positions always represent the same candidate. Each candidate is labeled at four vertices, to allow all permutations of five choices to be represented. ( See previous page's figure 3. ) For instance, candidate A is represented at vertex A and its completely opposite vertex A* and similarly at vertices a and a*.
Therefore, it is always for the same candidate that the select button does
not, in some sense, count for a ring, even tho it would seem to represent the
first and most important choice. There is a kind of election in which this
might conceivably be the case. If all the voters are also candidates, then
their first preferences cannot count towards the election, because the very
fact of them standing as candidates is a vote for themselves.
Then, only the second to fourth preferences can be elective.
But this brings another puzzle. There are two separated observers akin to two voters in a secret ballot. The quantum rules only apply when they choose the same candidate. But the assumption has just been made that all the voters are also candidates voting for themselves in the first instance. So, how can two voters be one candidate?
Well, a political candidate typically has an agent to promote his candidature. An agent has been known to succeed his candidate as an MP. Suppose that two candidates are agents for each other, or partners. Marriage partners may enter the electoral relationship in question. Sometimes, husband and wife take turns to be head of the family. In other words, either partner may vote for the other, as well as themselves.
( One example is a husband's complaint that his wife takes over all the pleasant duties towards the children and leaves him to take care of discipline. )
So, the near and far observers have become five voting candidates here, who have specific partners in five voting candidates there. In line with the quantum rules, it is a random process whether voting candidate, say, A is selected here or far away. He chooses one of four vertices A, A*, a, a*, with their respective three adjacent vertices.
Vertices A and a are adjacent, as are their opposites A* and a*. This
suggests that a given partner has the option to vote for himself first and his
partner second, or vice versa. However, there is no obligation to vote the
partner in second any more than third or fourth. It can only be said that the
partner must not be a fifth or least prefered choice.
This raises questions about the nature of such a 'partnership', which begins to
look like some strange monogamous cum polygamous form of competition, modeled
on a quantum correlation. It might be better to name the partners as
'correlatives'.
Of more immediate concern is the obligation to make vertex 'a' the choice for a partner of A. It has been assumed A* and a* were one partner's vertices and A and 'a' were the other partner's vertices, thus giving the two partners two different second to fourth preference options.
An example, of how the 'quantum' voting procedure might work, may resolve the inconsistency. Suppose a voter, here, is randomly given his turn to vote. And he can select either himself or his partner, far away, as another voter. The convention could be that the here voter prefers himself by selecting either vertex A or a, depending on whether the three adjacent vertices, to either A or a, constitute his second to fourth preferences.
Alternatively, the here voter might select for first preference his partner, represented by the A* or a* vertices, again depending on whether the three adjacent vertices, to A* or a*, constitute his second to fourth preferences.
Whether, say, the here voter selects himself or his partner, an
inconsistency arises, when he comes to order his choice of the three adjacent
vertices. Because one of them stands for the same choice as his first
preference selection.
This may be alright, provided we make clear, as a matter of procedure, that
when two adjacent vertices both name the same candidate, for selecting him as
first preference, whichever of the two adjacent vertices not so selected, then
switches to become the name of that candidate's partner, for the purpose of
being the second, third or fourth choice.
One may object that this procedure involves an inconsistent classification. And as far as a fixed labeling of the dodecahedron vertices is concerned, this is certainly true. But the re-labeling, that does take place, is according to an orderly procedure. From a static point of view, the labeling is inconsistent, but dynamically there is a consistent order of events.
Discussed on my first page about 'scientific method of elections', S S Stevens' theory is of the four scales of measurement, classificatory, ordinal, interval and ratio scales, each one more accurate and powerful than the one before. With reference to this, one might argue that the consistency of the ordinal scale over-rules the inconsistency of the classificatory scale.
This argument might, or might not, correspond to an inconsistent use of the ordinal scale being over-ruled by a consistent use of the interval scale (exemplified in a weighted count of preference votes).
And some such over-ruling might occur of the interval scale with regard to the ratio scale. Indeed, the ratio scale might be considered an experimental 'control' of the interval scale, as explained in the above reference.
However that may be, the electoral interpretation of the Penrose dodecahedrons, seen as preference permutations, suggests something of more immediate interest. Roger Penrose's disproof, of a classical explanation of the the rules of quantum correlation, depends on not being able to fix any of the vertices as bell-ringing positions, where the order of choice stops.
But the above electoral interpretation, if useful or correct, goes further than saying you can't pin down a final choice ( to a vertex ). It further says that you can't pin down a first choice ( to a vertex ) which is only conditional on the subsequent choices offered by the three adjacent vertices to any one vertex. Two adjacent vertices have to stand for the same initial selection, to offer the option of differing adjacent vertices with different subsequent choices.
The one, of the adjacent vertices, conditionally representing the same first choice, that is not selected, must become a choice for the first choice's partner or 'correlative', which may be taken up as a second, third or fourth choice, in the ordered pressing of three vertex buttons adjacent to the first choice vertex.
Classical assumption of no-influence failed to pin down a bell-ringing vertex, as final preference. Likewise, such independence might be disproved by not being able to pin down an initial select vertex or first preference. Suppose any vertex on the dodecahedron is pre-assigned as a first preference. By the above reasoning, it has an adjacent vertex that also represents the same first preference, but with a different option in possible second to fourth preferences.
If one of any pair of adjacent first preferences is pre-assigned as the first preference, then the other one could no longer serve as the first preference ( with its particular adjacent three choices ). Therefore, the far voter's choosing the corresponding choice of vertex or anti-podal vertex could not elicit the two quantum rules.
I tried to make a human election mimick Penrose's quantum rules. But I had to suppose that when one of two adjacent vertices, for the same candidate, was made the initial selection, the other would automatically change to represent the partner candidate. I dont know whether this could be made to work in quantum terms.
If it could work, then might Penrose's two rules exchange their application? This supposes a conservation principle in the dodecahedron, whereby the changing of a vertex representing a candidate to representing the partner, has a reverse effect on its anti-podal vertex. ( That is to suggest that rule one might apply to figure two and rule two apply to figure one, on the previous page. )
If the far voter also randomly given her turn to vote, in this instance, is not the here-voter's partner, then there will be no quantum correlation, that is no bell will ring to the partner's pressing vertices that correspond, as either in the same or opposite positions on their respective dodecahedrons.
If the dodecahedrons have opposite position vertices taken for select first preference, the bell still may not ring for any of the second to fourth choices of their adjacent vertices. It is still possible that none of the partners' second to fourth preferences correspond. Even tho the three choices must be the same, the order the two partners press them may be completely different. Then, no bell rings and indeed neither partner can know whether their partner or some other voter, not a partner, made an unrelated selection of vertex.
'Electoral' interpretation of the Penrose dodecahedrons suggested a possible new proof of non-locality. It might be a good idea to re-consider its assumptions without confusing social or political connotations.
The steps in the hypothetical proof started with a dodecahedron as the geometry of all 120 permutations of five choices. Each of the five choices have four vertices each. They are all in two diametrically opposite adjacent pairs of vertices.
The adjacent pairs offer two alternative sets of second to fourth choices. The assumption is made that adjacent vertices, standing for the same choice, are nevertheless distinct in the context of the different choices offered by their respective three adjacent vertices. I trust this may be true of quantum mechanics, or the ensuing proof does not claim to be valid for that science.
This leaves the problem of the diametrically opposite pairs, which appear to be redundant in the way the adjacent pairs are not. The opposed adjacent pairs offer the same two sets of second to fourth choices, in their respective adjacent three vertices. ( See figure 3 on previous page. )
However, we dont worry about that for the moment. Suppose the classical physics assumption is made that a select vertex ( or first choice ) is pre-assigned ( instead of a bell-ringing vertex, which signals a final choice, as in the Penrose proof ).
The pre-assignment of any vertex, on the dodecahedron, as 'select' automatically means that adjacent vertex, which stands for the same first choice, is not available for a quantum correlation between its three adjacent vertices and the corresponding or anti-podal vertices on the far dodecahedron.
Moreover, once a vertex has been selected, its adjacent vertex, which stands for the same first choice, cannot any longer have that meaning. This involves a second assumption, which may or may not apply to quantum mechanics, in this context. Namely, that this vertex not only ceases to be available as first choice. It also must automatically take on another identity.
It is now that the puzzle, of the seemingly redundant opposite pairs, affords a clue. They are redundant as to the succeeding choices they offer. But their very existence, as distinct positions on the dodecahedron, implies the similarity is not complete. Indeed, the distinct rules of quantum correlation, for corresponding vertices and for anti-podal vertices, tell as much.
Therefore, we would seem to have a ready-made new identity for any vertex
that is the foregone first choice and becomes a possible second, third or
fourth choice. It becomes, in effect, its diametric opposite vertex.
That is to say, if any vertex is pre-assigned as select, its adjacent vertex,
for the same first choice ( but different later ones ) might swop identity with
its diametric opposite vertex, also standing for the same choice, but subject
to a different quantum correlation with the other dodecahedron.
The swop is needed to balance the books. By a conservation principle, all the
properties of the dodecahedron must still be systematically represented.
This prediction, of a switch ( between figures one and two on the previous
page ) of the two quantum correlation rules, betraying a pre-assigned first
choice, may still be useful in its implications, even if it proved to be false,
because it could entail alternative explanations.
My experience of electoral method tells me that a proof involving first
preferences ( or initial selection of vertices ) would probably be more potent
than one involving final preferences ( or bell-ringing vertices ). I suspect
this whether or not my own attempt at such a proof is on the right lines.
Richard Lung.
10 April 2002
(with later minor cuts).