In what follows we shall consider ``rational rays.'' A ``rational ray'' is the linear span of a non-zero vector of Qn Ì Rn.
Let p be a prime number. A coloring of the rational rays of Rn, n ³ 1, using pn-1+pn-2+¼+1 colors can be constructed in a straightforward manner. We refer to [3,4,5] for the theoretical background of the following construction.
Each rational ray is the linear span of a vector (x1,x2,¼,xn) Î Zn, where x1,x2,...,xn are coprime. Such a vector is unique up to a factor ±1.
Next, let Zp be the field of residue classes modulo p. The vector space Zpn has pn -1 non-zero vectors; each ray through the origin of Zpn has p-1 non-zero vectors. So there are exactly (pn -1)/(p-1) = pn-1+pn-2+¼+1 distinct rays through the origin which can be colored with pn-1+pn-2+¼+1 distinct colors.
Finally, assign to the ray Sp(x1,x2,¼,xn) (``Sp'' denotes linear span) the color of the ray of Zpn which is obtained by taking the modulus of the coprime integers x1,x2,¼,xn modulo p. Observe that x1,x2,¼,xn cannot vanish simultaneously modulo p and that ±(x1,x2,¼,xn) yield the same color. Obviously, all pn-1+pn-2+¼+1 colors are actually used.
In what follows, we consider the case p = 2, n = 3. Here all rational rays Sp(x,y,z) (with x,y,z Î Z coprime) are colored according to the property which ones of the components x,y,z are even (E) and odd (O). There are exactly 7 of such triples OEE, EOE, EEO, OOE, EOO, OEO, OOO which are associated with one of seven different colors #1,#2,#3,#4,#5,#6,#7. Only the EEE triple is excluded. Those seven colors can be identified with the seven points of the projective plane over Z2; cf. Fig. 1.
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Next, we restrict our attention to those rays which meet the rational unit sphere S2ÇQ3. The following statements on a triple (x,y,z) Î Z3 \{(0,0,0)} (not necessarily coprime) are equivalent:
This equivalence can be demonstrated as follows. All points on the rational unit sphere can be written as r = ([a/(a¢)],[b/(b¢)],[c/(c¢)] ) with a,b,c Î Z, a¢,b¢,c¢ Î Z\{0}, and ([a/(a¢)])2+([b/(b¢)])2+([c/(c¢)])2 = 1. Multiplication of r with a¢2b¢2c¢2 results in a vector of Z3 satisfying (ii). Conversely, from x2+y2+z2 = n2, n Î N, we obtain the rational unit vector (x/n,y/n,z/n) Î S2ÇQ3.
Notice that this Pythagorean property is rather restrictive. Not all rational rays intersect the rational unit sphere. For a proof, consider Sp(1,1,0) which intersects the unit sphere at ±(1/Ö2)(1,1,0) Ï S2ÇQ3. Although both the set of rational rays as well as S2ÇQ3 are dense, there are ``many'' rational rays which do not have the Pythagorean property.
If x,y,z are chosen coprime then a necessary condition for x2+y2+z2 being a non-zero square is that precisely one of x, y, and z is odd. This is a direct consequence of the observation that any square is congruent to 0 or 1, modulo 4, and from the fact that at least one of x, y, and z is odd. Hence our coloring of the rational rays induces the following coloring of the rational unit sphere with those three colors that are represented by the standard basis of Z23:
All three colors occur, since the vectors of the standard basis of R3 are colored differently.
Suppose that two points of S2ÇQ3 are on rays Sp(x,y,z) and Sp(x¢,y¢,z¢), each with coprime entries. The inner product xx¢+yy¢+zz¢ is even if and only if the inner product of the corresponding basis vectors of Z23 is zero or, in other words, the points are colored differently. In particular, three points of S2ÇQ3 with mutually orthogonal position vectors are colored differently.
From our considerations above, three colors are sufficient
to obtain a coloring of the rational unit sphere
S2ÇQ3 such that points with orthogonal
position vectors are colored differently, but clearly this
cannot be accomplished with two colors. So the ``chromatic
number'' for the rational unit sphere is three. This result
is due to Godsil and Zaks [1]; they also
showed that the chromatic number of the real unit sphere is
four. However, they obtained their result in a slightly different way.
Following [4] all rational rays are associated with three colors
by making the following identification:
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Kent [6] has shown that there also exist dense sets in higher dimensions which permit a reduced two-coloring. Unpublished results by P. Ovchinnikov, O.G.Okunev and D. Mushtari [7] state that the rational d-dimensional unit sphere is d-colorable if and only if it admits a reduced two-coloring if and only if d < 6.
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Now suppose that the point u = ( a/c,b/c,0) is on the rational unit sphere and that a,c are odd and thus b is even. In the coloring scheme introduced above, u has the same color as (1,0,0) (identify a = c = 1 and b = 0); and so does Fu. This proves that I (the image of all powers of F of the points u) is dense. We shall come back to the physical consequences of this property later.
In the reduced two-color setting, if the two "poles" ±(0,0,1) acquire color #1, then the entire equator acquires color #2. Thus, for example, for the two tripods spanned by {(1,0,0),(0,1,0),(0,0,1)} and {(3,4,0),(-4,3,0),(0,0,1)}, the first two legs have color #2, while (0,0,1) has color #1.
A proof that four colors suffice for the coloring of points of the unit sphere in three dimensions is constructive and rather elementary. Consider first the intersection points of the sphere with the the x-, the y- and the z-axis, colored by green, blue and red, respectively. There are exactly three great circles which pass through two of these three pairs of points. The great circles can be colored with the two colors used on the four points they pass through. The three great circles divide the sphere into eight open octants of equal area. Four octants, say, in the half space z > 0, are colored by the four colors red, white, green and blue. The remaining octants obtain their color from their antipodal octant.
Although Godsil and Zaks' [1] paper is not entirely specific, it is easy to write down an explicit coloring scheme according to the above prescription. Consider spherical coordinates: let q be the angle between the z-axis and the line connecting the origin and the point, and j be the angle between the x-axis and the projection of the line connecting the origin and the point onto the x-y-plane. In terms of these coordinates, an arbitrary point on the unit sphere is given by (q, j, r = 1) º (q,j).
The colors of the cartesian coordinate axes (p/2,0), (p/2,p/2), (0,0) are green, blue and red, respectively.
The color of the octant {(q, j) | 0 < q £ p/2, 0 £ j < p/2} is green.
The color of the octant {(q, j) | 0 £ q < p/2, p/2 £ j £ p} is red.
The color of the octant {(q, j) | 0 < q < p/2, p < j < 3p/2} is white.
The color of the octant {(q, j) | 0 < q £ p/2, -p/2 £ j < 0} is blue.
The colors of the points in the half space z < 0 are inherited from their antipodes. This completes the coloring of the sphere.
The fact that three colors are not sufficient is not so obvious. Here we shall not review Godsil and Zaks' proof based on a paper by A. W. Hales and E. G. Straus [4], but refer to a result of S. Kochen and E. Specker [2], which is of great importance in the present debate on hidden parameters in quantum mechanics. They have proven that there does not exist a reduced two-coloring, also termed valuation, on the one dimensional subspaces of real Hilbert space in three dimensions.
Recall that a reduced two-coloring of the one dimensional linear subspaces with two colors could immediately be obtained from any possible appropriate coloring of the sphere with three colors by just identifying two of the three colors. Thus, the impossibility of a reduced two-coloring implies that three colors are not sufficient for an appropriate coloring of the threedimensional real unit sphere. (``Appropriate'' here means: ``points at spherical distance p/2 get different colors.'')
In the same article [2], Kochen and Specker gave an explicit example (their G3) of a finite point set of the sphere with weaker properties which suffice just as well for this purpose: the structure still allows for a reduced two-coloring, yet it cannot be colored by three colors. (The authors did not mention nor discuss this particular feature [9].)
The impossibility of a reduced two-coloring also rules out another attempt to ``nullify'' the Kochen-Specker theorem by identifying pairs of colors of an appropriate four-coloring of the real unit sphere. Any such identification would result in tripods colored by #1-#2-#2, as well as for instance #1-#1-#2, which is not allowed for reduced coloring schemes, which requires colorings of the type #1-#2-#2.
Let us re-state the physical interpretation of the coloring schemes discussed above.
Any linear subspace Sp r of a vector r
can be identified with the associated projection operator Er
and with the quantum mechanical proposition ``the physical system is in
a pure state Er'' [12].
The coloring of the associated point on
the unit sphere (if it exists) is equivalent with a valuation or
two-valued probability measure
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Since, as has been argued before, the rational unit sphere has chromatic number three, two colors suffice for a reduced coloring generated under the assumption that the colors of two rays in any orthogonal tripod are identical. This effectively generates consistent valuations associated with the dense subset of physical properties corresponding to the rational unit sphere [10].
Kent [6] has shown that there also exist dense sets in higher dimensions which permit a reduced two-coloring. Unpublished results by P. Ovchinnikov, O.G.Okunev and D. Mushtari [7] state that the rational unit sphere of the d-dimensional real Hilbert space is d-colorable if and only if it admits a reduced two-coloring if and only if d < 6.
The rational unit sphere is not closed under certain geometrical operations such
as taking an orthogonal ray of the subspace spanned by two
non-collinear rays (the cross product of the associated vectors).
This can be easily demonstrated by considering the two vectors
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Indeed, if instead of S2ÇQ3 one would start with three non-orthogonal, non-collinear rational rays and generate new ones by the cross product, one would end up with all rational rays [15].
In the Birkhoff-von Neumann approach to quantum logics [16], this non-closedness under elementary operations such as the nor-operation might be considered a serious deficiency which rules out the above model as an alternative candidate for Hilbert space quantum mechanics. (However, his argument does not apply to the Kochen-Specker partial algebra approach to quantum logic, since there operations among propositions are only allowed if the propositions are comeasurable.)
Informally speaking, the relative (with respect to other sets such as the rational rays) ``thinness'' guarantees colorability. In such a case, the formation of finite cycles such as the ones introduced by Kochen and Specker [2] are impossible.
To put it differently: given any nonzero measurement uncertainty e and any non-colorable Kochen-Specker graph G(0) [2,17], there exists another graph (in fact, a denumerable infinity thereof) G(d) which lies inside the range of measurement uncertainty d £ e [and thus cannot be discriminated from the non-colorable G(0)] which can be colored. Such a graph, however, might not be connected in the sense that the associated subspaces can be cyclically rotated into itself by local transformation along single axes. The set G(d) might thus correspond to a collection of tripods such that none of the axes coincides with any other, although all of those non-identical single axes are located within d apart from each other. Indeed, this appears to be precisely how the Clifton-Kent construction works [11].
The reason why a Kochen-Specker type contradiction does not occur in such a scenario is the impossibility to ``close'' the argument; to complete the cycle: the necessary propositions are simply not available in the rational sphere model. For the same reason, an equilateral triangle does not exist in Q2 [9]. Yet, while these findings seem to contradict the conclusions of Clifton and Kent [11], we would like to emphasize that this does not relate to their formal arguments but rather is a matter of interpretation and a question of how much should be sacrificed for value definiteness.
Thus, although the colorings of rational spheres offer a rather unexpected possibility to define consistent classical models, a closer examination shows that any such colorings should be excluded for physical reasons.