We present a picture of the world suggested by quantum mechanics in a manner accessible to nonspecialists. We construct a model of subatomic reality and use the model to present a geometrical proof of the Kochen–Specker–Bell no-go theorem, which demonstrates that the properties of a microscopic system are not completely defined before we look at it.

1.
More precisely, the projection of the spin along the direction of the magnetic field can have only two values in this case.
2.
In general, the state of a system is a vector (a ray, properly speaking) of a Hilbert space defined over the field of complex numbers. The necessity of complex numbers is already present in the experiments discussed here: For a magnetic field pointing in the direction, the beam that deflects right would be in the state (1/√)(ẑ−iŷ), and the one that deflects left would be in the state (1/√)(ẑ+iŷ).
3.
To use a Stern–Gerlach apparatus to answer the question corresponding to the projector Px means that we recombine the m=1 and m=−1 beams, so that only two alternatives remain: m=0 and m≠0. The technical difficulties will not concern us here. The mere possibility of realization suffices.
4.
That “If the system is in the -state, the probability that Pn will be answered affirmatively is equal to the square of the projection of onto x̂.” is considered one of the basic axioms of quantum mechanics, although Gleason, in his remarkable theorem published in 1957 (Ref. 12), proved that a probability measure μ defined over a spherical surface [or a linear space of dimension 3 if for any vector “a” and any number “λ,” μ(a)=μ (λ a) (a and λa are the same physical state)] has to be of the form: prob(Pθ=1/Pz=1)=cos2 θ. (Note: Pθ=1 is a convenient way of saying that Pθ takes on the value 1. Other authors write v(Pθ)=1instead). Gleason showed the equivalent result for any dimension of Hilbert space.
5.
We are leaving aside the randomness due to chaotic systems.
6.
A. Pais, Subtle is the Lord (Oxford U.P., New York, 1982), p. 455.
7.
As we have said the projectors take on only two values, 0 or 1, with a certain probability. When we say that the projectors are defined, we mean that they take on the value 1 (or 0) with probability 1. Therefore the expectation value of any projector in that case is 〈Pθ〉=prob(Pθ=1)=0 or 1. In brief “definite values” means 〈Pθ〉=0 or 1.
8.
We observe that the assumption 〈Pθ〉=0 or 1 is by itself contradictory to quantum mechanics because in the latter 〈Pθ〉=prob(Pθ=1/Pz=1)=cos2 θ. We do not have to wait for the conclusion of the theorem. It is a matter of principle: a theory with definite values clashes with quantum mechanics. So for that matter we do not need the KSB theorem. The merit of the theorem is to show that any theory with definite values and satisfying prob(ΣPi)=Σprob(Pi)=1(Pi mutually orthogonal) is contradictory by itself. We should mention that the hypothesis 〈Pθ〉=0 or 1 is not as explicit as it should in many papers about the KSB theorem. Note that the hypothesis 〈Pθ〉=0 or 1 is the only one that distinguishes the asumptions of the KSB theorem from those of Gleason’s theorem. In the original work by KS and in many other papers the initial asumption is that the existence of hidden variables (noncontextual) would cause that, if for certain mutually commuting observables A,B,C,…, is true that f(A,B,C,…)=0, the value of these observables would satisfy the same algebraic relation: f[v(A),v(B),v(C),…]=0. But it is straightforward to derive, from that assertion, that 〈Pθ〉=0 or 1.
9.
J. von Neumann, Mathematical Foundation of Quantum Mechanics (Princeton U.P., Princeton, 1955), Chap. 4, Secs. 1 and 2.
10.
Here we are considering only noncontextual hidden variables. (The KSB theorem only deals with noncontextual hidden variables.) We do not discuss the hidden variables that “conspire” with the measuring apparatus—such as that suggested by Bell—and depend on the observable being measured. Perhaps that kind does not deserve the name of hidden variables. We note that the well-known Bohm model, which reproduces the statistical results of quantum mechanics, is a contextual hidden variable theory because the variable position (of a particle) depends on the measuring device.
11.
J.
Bell
, “
On the problem of hidden variables in quantum mechanics
,”
Rev. Mod. Phys.
38
,
447
452
(
1966
).
12.
A.
Gleason
, “
Measures on the closed subspaces of a Hilbert space
,”
J. Math. Mech.
6
(
6
),
885
893
(
1957
).
13.
In fact Gleason showed, as an intermediate step in deriving his famous theorem, that the probability was a continuous function. The hypothesis that the projector takes on definite values clashes with that condition because a function that takes on only two values within an interval cannot be continuous. Thus Gleason’s theorem, besides prescribing the form of the probability in quantum mechanics, rules out dispersion-free states. Bell, after recognizing that the credit of this discovery belongs to Gleason, developed his own procedure to show that it is not posible to assign different values to projectors that are too close. He finds that the angular distance between a one-valued projector and a zero-valued one should be greater than 26.6°. (Other authors find different conditions. One interesting point of the approach to the KSB theorem we follow in this paper is to establish clearly the boundary between 1’s and 0’s).
14.
S.
Kochen
and
E.
Specker
, “
The problem of hidden variables in quantum mechanics
,”
J. Math. Mech.
17
(
1
),
59
87
(
1967
).
15.
R.
Gill
and
M.
Keane
, “
A geometrical proof of the Kochen-Specker no-go theorem
,”
J. Phys. A
29
,
L289
L291
(
1996
).
16.
C. Piron, Foundations of Quantum Physics (Benjamin, New York, 1976), pp. 73–81.
17.
Projected lines and the circumference may appear in color online.
18.
This result is shown in
R.
Cooke
,
M.
Keane
, and
W.
Moran
, “
An elementary proof of Gleason’s theorem
,”
Math. Proc. Cambridge Philos. Soc.
98
,
124
(
1985
).
19.
The result confirms with what was advertised in Ref. 8: the distribution of values we have obtained clashes with quantum mechanics, in which prob(Pθ=1/Pz=1)=cos2 θ. For quantum mechanics, and contrary to the result we have obtained, there is a nonzero probability that a projector within the upper skullcap takes on the value zero.
20.
M. Redhead, Incompleteness, Nonlocality and Realism (Oxford U.P., New York, 1987), p. 123.
21.
A.
Pais
,
Rev. Mod. Phys.
51
,
861
(
1979
).
22.
A.
Peres
, “
Unperformed experiments have no results
,”
Am. J. Phys.
46
(
7
),
745
747
(
1978
).
23.
E.
Specker
, “
The logic of propositions which are not simultaneously decidable
,”
Dialectica
14
,
239
246
(
1960
).
24.
M. Solana, Historia de la Filosofı́a Española (Asociación para el progreso de las ciencias, Madrid, 1941), pp. 360–363.
25.
A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, Dordrecht, 1993), pp. 196–201.
26.
R. Penrose, “On Bell non-locality without probabilities: Some curious geometry,” in Quantum Reflections, edited by J. Ellis and D. Amati (Cambridge U.P., Cambridge, 2000), pp. 1–27.
27.
The 1-valued directions might be drawn in light blue online. The interprenetrating cubes may appear in color online.
28.
We can see again the implicit assumption that projectors have the values 〈Pθ〉=0 or 1. In quantum mechanics we would not be able to admit that; rather we should accept that BS can take on values 0 or 1 with the probabilities sin2(AS,BS)=sin2 45°=1/2 and cos2(AS,BS)=cos2 45°=1/2, respectively.
29.
Quoted by Max Jammer, The Philosophy of Quantum Mechanics (Wiley, New York, 1974), p. 161.
30.
To obtain a deeper and broader knowledge of the meaning of quantum mechanics, we recommend two books, both requiring some technical background in the subject: M. Redhead, Incompleteness, Nonlocality and Realism (Oxford U.P., New York, 1987) and J. Bub, Interpreting the Quantum World (Cambridge U.P., New York, 1997). Redhead’s book is a classic on the foundations of quantum mechanics. Bub’s book is, in a sense, an update to Redhead’s treatise and covers more recent topics.
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