Kochen-Specker theorem

Two mathematicians, Kochen and Specker, worked together in 1963 (the year I was born) and 1964 on a result that they proved in 1967; John Bell, a physicist, also proved it in the interval. The point of the result is to show that some of the physical observations we've made (that we explain in terms of quantum mechanics, but this is incidental) imply that the process we describe as measurement isn't quite what it was classically thought to be. The classical view is that things have properties independent of our measurement of them and the process of measurement reveals those properties. The physical observations, that lead folk to develop the theory of quantum mechanics, contardict that model by implying that the properties of the system don't exist except to the extent that they are measured; when a property of a thing isn't measured, the thing doesn't meaningfully have a value for that property.

Spin and angular momentum

One of the physical properties physics conventionally deals with is called angular momentum; when an object is spinning, it will continue spinning unless some torque acts on it to slow down its spin. The angular momentum of a rigid body increases in proportion to how many rotations it completes in a given amount of time, just as the linear momentum of a body of fixed mass increases with that body's speed. The constant of proportionality, known as the moment of inertia, depends on the mass of the body and how it is distributed about the axis of rotation; mass further from the axis contributes more to the moment of inertia than mass closer to the axis; mass exactly on the axis contributes (classically) nothing at all. When the spinning object is not rigid, for example when a spinning dancer moves arms in or out from the axis of spin, the angular momentum stays the same but the moment of inertia changes, causing the rate of rotation to change the other way; pulling arms in reduces the arms' mass's contribution to the moment of inertia, so increases the rate of rotation; extending arms outwards increases their contribution, so slows the rotation.

In electrodynamics, the electromagnetic field can carry a form of angular momentum that isn't associated with massive objects rotating; when electromagnetic effects cause rotating bodies (for example, magnets) to speed up or slow down, there is a transfer between the elecromagnetic field's form of angular momentum and the more familiar mechanical form of it. Likewise, in an atom, when light interacts with electrons orbiting the atom, there can be a transfer between this electromagnetic angular momentum and the mechanical angular momentum associated with the electron's orbital motion about the atom. There can also be an interaction between the electromagnetic angular momentum and an internal property of the electron itself, known as spin by analogy with the mechanical form of angular momentum. The amount of spin that any given fundamental particle can hold is limited to a discrete set of values, all multiples of h/π/4, where h is Planck's constant, and each particle's possible amounts of spin differ from one another by twice this unit; an electron can have ±1 of this unit, for example, but never 0 of it. Particles with odd numbers of this unit of spin are known as fermions, those with an even number of units as bosons. In particular, some bosons can have total spin equal to ±2 or 0 of this unit. Folk usually use twice this unit as the usual unit of spin, Dirac's constant ℏ = h/π/2, which is the size of the steps by which spin can change; in this unit, the electron has spin half and these last-mentioned bosons have spin ±1 or 0; the are thus known as spin-1 particles (the ± and 0 being implicit).

Now, angular momentum is a vector quantity, which is to say it has a magnitude and a direction; its direction is parallel to the axis of rotation. (Formally, it's really a 2-form, an antisymmetric product of vector quantities, just as area is; however, in three dimensions, there is a convenient representation of such quantities by vectors, analogous to representing an area by a surface's vector perpendicular to the surface, with magnitude in proportion to the area.) So we're used to being able to measure its components in each of three directions, to determine its direction and magnitude. However, when we try to do this with the spin of fundamental particles, we get odd experimental results, that can only be sensibly explained by quantum mechanics. The theorem Kochen, Specker and Bell variously proved has to do with that odd behaviour; they show that there can't be an underlying, internal, vector quantity (or two-form equivalent to one), consistently with what we observe in experiments.

There's an experimental set-up (known since about 1926) that lets one measure the component of a particle's spin in one direction at a time. If we take a spin-1 boson, as mentioned above, this experiment will give a ±1 or 0 result (in units of Dirac's constant). We're only going to be interested in distinguishing some from none; so we aren't going to care about the difference between +1 and −1. When we perform this experiment on a particle in each of three perpendicular directions, we consistently find that two of the directions get some and one gets none. The result we're going to prove, following a more recent improvement of Kochen and Specker's proof, is that – assuming this is always true – there can't be an internal quantity whose value determines these results.

If you think about the idea that any three perpendicular directions give one with a zero component and the other two with a component that's the whole of a definite unit (and we never see any other non-zero value than this unit), I'm sure you can see this doesn't match up with our intuitions about the components of a vector quantity; so, in an important sense, what Bell, Specker and Kochen proved is a confirmation of (a strong form of) this intuition.

It's worth noting that, if we measure in two directions that aren't at right angles to one another, and then measure again in the first of those directions, the result tends to change (not always), indicating that such measurements change the spin of the particle; in contrast, it's (at least in principle) possible to measure the three perpendicular directions without change. However, if there were an intrinsic quantity that three perpendicular measurements are all accessing, it would imply that any single choice of three such perpendicular axes must give answers implied by that internal state; in particular, the result of measurement along any axis wouldn't depend on which other axes you chose, provided they're all perpendicular. So, without actually doing any measurement, we can talk about what answers would it give if we measured any particular set of three perpendicular directions; it only makes sense to claim there's an internal state, that implies the results for the directions we actually chose, if that state could imply answers for any set of three directions we were to chose to pick.

The proof

The substance of the proof is to exhibit 33 axes, among which there are several triads of three mutually perpendicular axes, and to reason that there's no pattern of some and none answers that a given internal state can imply for each of these various directions, consistent with the rule that any perpendicular three that we pick must give two some and one none. So no alleged internal state can imply what our measurements shall produce, regardless of which orthogonal triad of axes we measure; the state of the particle has to be a response to our choice of axes to measure along; it effectively has to be making up its answers as it goes along, not revealing a state that existed before we asked the questions.

Take a cube and its three images under turn/8 rotations about its axes, from face-centre to opposite face-centre, of turn/4 symmetry. (The union of the rotated cubes appears atop a pilar in Escher's Waterfall, paired with the union of their dual octohedra.) Mark corners of rotated cubes and edge-centres of all cubes. Each cube has four opposed pairs of corners (marked on three cubes), six opposed pairs of edge-centres (marked on all four); each rotated cube shares one of the latter with each of the other rotated cubes. So we have 3*4 +6 +3*5 = 33 opposed pairs; each of these opposed pairs defines an axis, along which we'll be measuring spin. (Each of these is a symmetry axis of at least one of the rotated cubes.) The points where these axes pierce the original cube are its edge centres, face centres and the corners and edge-centres of a square on each face whose corners are on diagonals of the face, the same distance from the centre as the face's edge-centres. Opposite corners of the inscribed square, on any face, subtend a right angle at the centre; rays from the centre to them have components - parallel to face normals, in units of half the cube's edge-length - with magnitudes 1 and (in opposite directions to one another) [1, 1]/sqrt(2), with inner product 1 -1/2 -1/2 = 0. The third axis perpendicular to the axes through two such points is an edge-centre of the cube, on an edge perpendicular to the inscribed square's face. Start with the original cube's three face-centre axes; two of them are 1, the other 0; call this last the z direction; each axis in the z=0 plane spanned by the other two is then 1. In a face of constant z, parallel to that 1-plane, opposite corners of the inscribed square form an orthogonal triad with an edge-centre in that 1-plane, so each pair of opposite inscribed corners must be a 1 and a 0. We thus get two adjacent 1 corners opposite two adjacent 0 corners on the inscribed square. Call the direction of inscribed edges with the same label (both ends 0 or both ends 1) the x direction; the y direction thus gets us the edges between ends of axes of opposite spin. Consider a 0 to 1 edge of that inscribed square; the plane through it and the cube's centre has a normal that emerges in an edge-centre of the inscribed square of each constant-x face; the edge it's the centre of is parallel to a centre-line of its face, that lies in the initial all-1 plane. Since this point's axis is perpendicular to the plane that lead us to it, which already has one 0 axis in it, we infer that this point is on a 1-axis. Doing this for each of the 0-to-1 edges of a constant-z face's inscribed square puts both y = 0 edge-centres of the constant-x faces on spin-1 axes. As these new 1-axes are in the y = 0 plane, each is perpendicular to the y-axis, which is one of our original spin-1 directions; the axis which complets a triad with them emerges in the centre of one of each constant-z face's inscribed squares from a 1-corner to a 0-corner; the two new spin-1 axes thus, along with the y-axis, imply that both 0-to-1 edges of each constant-z face's inscribed square lies on a spin-0 axis. The plane, through the cube's centre, perpendicular to each of these two new spin-0 axes cuts each constant-x plane in an edge, parallel to the y-axis, of its inscribed square; the two spin-0 axes thus give us spin 1 for the axes through all four corners of the inscribed squares of the of the constant-x faces. Each opposite pair of these corners gives us a pair of perpendicular spin-1 axes, whose triad is completed by an axis through an edge centre of the constant-z faces, in the x = 0 plane; the two pairs of opposite corners of the inscribed square of a constant-x face give us the two such edge-centres available on each constant-z face; but these are centres of opposite edges of that face, so the axes through (the cube's centre and) them are perpendicular; so they cannot both have zero spin. Thus the axes can't all be labelled 0 or 1 consistently with the 101 rule. Result: measurements along directions not parallel or perpendicular to all others previously measured will break consistency with those earlier measurements.

Written by Eddy.