As I mentioned Friday, I’m going to begin a short discussion about this paper on some consequences of special relativity and quantum mechanics on our view of determinism and the Universe. The authors, John H. Conway and Simon Kochen, establish three “axioms” (and a “paradox”) and from these statements establish consequences which have wide ranging implications. All of these measurements and the following discussion regard the behavior of a spin 1 massive particle. Spin 1 massive particles can have three possible measured values of quantum mechanical spin, namely -1, 0, or 1. Part 1 in which the axioms (and the Kochen-Specker paradox) are discussed can be found here.

In this installment of my discussions of this (which will have at least one and perhaps two more parts) I will examine the theorem which is at the heart of this paper. Blog neighbor Jim Anderson, noting my “homework assignment” finds the third paragraph daunting. The statement of the (strong) Free Will theorem is:

The Free Will Theorem. The axioms SPIN, TWIN and MIN imply that the response of a spin 1 particle to a triple experiment is free—that is to say, is not a function of properties of that part of the universe that is earlier than this response with respect to any given inertial frame.

Conway and Kochen prove this theorem by contradiction, that is they assume the theorem is not true and show that leads to a problem, in this case the the contradiction comes in the form of the Kochen-Specker paradox.

The basic form of the proof is to take two TWIN particles subjected to the SPIN measurement and begins to follow the consequences that these particles are “not free”. What is meant by free? This takes a particular meaning. If this measurement is free it means that the result of this measurement is not the consequence (a function of) of anything which has occurred earlier in any reference frame.

So, the authors express this measurement in terms of a collection of parameters denoted as alpha. In brief, the method employed in the proof is to pare down that unconstrained parameters sets (axis or other prior settings) via group arithmetic and MIN (one of the axioms from yesterday) to be able to finally express the measurement as a function which is recognizable as the same function which by the Kocken-Specker paradox cannot exist. Then, since the function cannot exist then the prior constraints on the particles measurement cannot exist either.

The paragraph quoted by Mr Anderson as less than transparent to the worlds most competent reader are placed there largely, I think, are included to these results to bear on a more recent proposal (called GWR and rGWR in the paper) which attempt remove by stochastic arguments the “measurement/collapse” of quantum wave functions which is philosophically speaking, uhm, difficult. I have not read any of the rGWR papers or any discussions of them so I will leave that for another time.

Mr Anderson (and his commenter) remark that this paper perhaps goes too far, offering

From what I can tell, it’s an attempt to demonstrate free will by noting that at least one property of elementary particles is nondeterministic. This still doesn’t prove the philosophical idea of free will, however. It appears only to impute it to an object, with a lot of anthropomorphizing to make it all work.

I don’t think that’s the case at all, however. The notions of free will which they think this offering lacks “intentionality, “responibility” and so on are not being discussed here. In any discussions of free will and compatabilism see for example wiki or the Stanford Encyclopedia, there is indeed a lot of discussions over whether determinism and free will are can co-exist. Yet, the universe in which we live is not deterministic. So the compatibility problem shifts. It is not a question of whether free will and determinism can exist but how free will arises in a fundamentally non-deterministic universe. The usage of the term “free will” for the theorem is to point that the freedom of the elementary particle to choose it’s “101” (squared) spin statistics result is equivalent and indistinguishable from the experimenters free will to determine the axis by which the measurement will be taken. No the axis of measurement (and the particles choice of 101,011, or 110) is not a moral choice obviously. But glancing through the compatiblism articles cited above, little space is seemingly granted to the considering consequences of a non-determinstic universe … or if incompatiblisim may be possible, i.e., “or that free will is true, therefore determinism is not” … and since determinism is not might free will be a possiblity?

The point is much discussion within the philosophical community grounds itself on the notions of whether or not determinism is true, i.e., whether the universe is really or is really not deterministic. Physics insists that there is an answer to that part of the question. The universe is not deterministic. So however you argue about free will that part of the argument should be settled.

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