## 2/22/2006

### Quantum logic question

I've been thinking about quantum logic (QL) recently, and in particular about the usual semantics for 'or' in QL. I've become puzzled, and hopefully someone out there in the blogosphere can help me clear up my confusion.

For the uninitiated: In QL, propositions are represented by/ interpreted as subspaces in a Hilbert space -- including one-dimensional subspaces, i.e. rays. There are multiple ways of formulating in colloquial language what these subspaces are to represent (see final paragraph below), but (atomic) sentences are usually taken to have the form:
'The value of observable O is o1'
where 'observable' just means any physical quantity (e.g., position, momentum, energy, spin), and o1 is just a particular value (or range of values) of that observable. (E.g. 'The energy of this system is between 4 and 6 Joules.') Such sentences are true iff the state-vector of the system lies within the subspace.

Now, think of a particle P in a superposition of spin up and spin down along the y-axis. This particle's state is of course represented by a different vector (call it V_s) than particles in the spin-up state (represented by V_up), or particles in the spin-down (V_down) state. However, because the usual QL semantics assigns to 'p or q' the linear span, instead of the union, of the rays associated with p and q, the claim 'P is spin-up or P is spin-down' will be true -- because V_s is in the linear span of the spin-up ray and the spin-down ray. Each of the disjuncts is false, but the whole disjunction is true. (To me, this feature of QL is even more striking than the failure of the so-called distributive law, i.e., [p&(q or r)] iff [(p&q) or (p&r)], which commentators on QL tend to focus on.)

This seems intuitively wrong to me (or at least as 'wrong' as something can be in formal semantics). In 2-D Euclidean space, suppose we have a unit vector V at a 45-degree angle to the x-axis. I don't think anyone would consider the sentence 'V lies along the x-axis or V lies along the y-axis' to be true. V is not a unit vector on the x-axis or on the y-axis, but a distinct third thing. I don't see why we would change policies in the quantum case, which appears analogous to me.

So now I can ask my question: could we change the semantics for 'or' to avoid these apparent problems? In particular, in the usual semantics for quantum logic, why must all propositions be represented as subspaces on a Hilbert space? -- why not also allow subsets (which might not be closed under linear combinations)? For then we could allow 'or' to mean the union of rays, and 'P is spin-up or P is spin-down' will come out false.

One further note: some people (e.g. R.I.G. Hughes, "Quantum Logic and the Interpretation of Quantum Mechanics," PSA 1980) take the atomic QL propositions to have a different correlate in colloquial language. Instead of
'The value of observable O in system S is within o1,'
they take the subsets of Hilbert space to mean
'The result of a measurement operation for observable O in system S is within o1.'
Under this understanding, my above worries disappear -- for the result of a spin-y measurement surely will be either spin-up or spin-down. However, QL then becomes much less interesting, because it is just about measurement outcomes, instead of about these supremely odd things, superpositions.

Labels: ,