Quantum Reality (Kochen-Specker)
The goal here is to provide evidence that the values measured in experiments aren't always intrinsic properties of the system being measured. It's as if things don't have certain properties until we measure them. Although many people may be familiar with the Heisenberg uncertainty principle, it is often interpreted as "we can't know two properties like the position and momentum of a particle simultaneously." A more appropriate interpretation seems to be "a particle doesn't have a position and momentum simultaneously."
How can such a claim be made? The argument below shows that the math just doesn't work out—there's a contradiction in assuming something simultaneously has values for various measurable properties.
Consider four properties P1, P2, P3, and P4 of a thing at a time. These are not just any four properties, but are chosen such that if all 4 are measured, exactly one of them gives a result "1" while the other three give the result "0" (if you know quantum mechanics, these are mutually commuting projection operators onto 4 orthogonal states S1, S2, S3, and S4 in a 4D Hilbert Space). There are many ways to pick four properties as described above, and we can even define new properties in terms of others: if P1,P2,P3, and P4 are such properties then so are P1+P2,P1-P2,P3,P4. If we keep picking 4 properties in this way, it's possible to show that at least one property, say P1, doesn't have a particular value 0 or 1. How? We'll use the example from [1] below.
- P4, P3, P1+P2, P1-P2
- P4, P2, P1+P3, P1-P3
- P1-P2+P3-P4, P1-P2-P3+P4, P1+P2, P3+P4
- P1-P2+P3-P4, P1+P2+P3+P4, P1-P3, P2-P4
- P3, P2, P1+P4, P1-P4
- P1-P2-P3+P4, P1+P2+P3+P4, P1−P4, P2−P3
- P1+P2-P3+P4, P1+P2-P3+P4, P1+P2+P3-P4, P1-P2, P3+P4
- P1+P2-P3+P4, -P1+P2+P3+P4, P1+P3, P2-P4
- P1+P2+P3-P4, -P1+P2+P3+P4, P1+P4, P2-P3
There are nine rows in the above list. Exactly one property in each row must have the value 1, hence exactly nine 1s must be in the table. On the other hand, every property in the table appears exactly twice, e.g. P4 appears in row 1 and row 2; P3+P4 appears in row 3 and row 7. If a property has it's value independent of what's being measured (e.g. it has the same value regardless of where it occurs in the table), then there must be an even number of 1s in the table (2 for each property that has value 1). So it's impossible to assign 0s and 1s to the properties and end up with exactly one 1 in each row.
AssumptionsIn addition to the postulates of quantum mechanics, it was assumed above that all the properties in rows 1-9 above are measurable. This is called the strong superposition principle and is used in many proofs in quantum mechanics.
References
Work somewhat along these lines was done by several people some credited and some not. Here are just a few references.
[1] http://arxiv.org/abs/quant-ph/9706009v1
[2] http://en.wikipedia.org/wiki/Kochen%E2%80%93Specker_theorem
[3] http://en.wikipedia.org/wiki/Gleason%27s_theorem
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