stevendaryl said:
In my opinion, calling an observable of a C*-algebra an "equivalence class of measuring devices" is more suggestive than rigorous.
More suggestive than rigorous...yes, I suppose so. If we want to do it rigorously, we must start by stating a definition of "theory" that's general enough to include all the classical and all the quantum theories. We can then define terms like "state" and "observable" in a way that's both rigorous and theory-independent (in the sense that the same definition applies to all the classical theories, all the quantum theories, and more).
I spent some time thinking about how to do these things a couple of years ago. I didn't keep at it long enough to work everything out, but I feel very strongly that it can be done. The first step is to provide some motivation for a general definition of "theory". This is of course impossible to do rigorously, but the main ideas are very simple and natural. (Actually, what we want to define here isn't a theory of physics in the sense of my previous posts in this thread. It's just the
purely mathematical part of such a theory, not including any correspondence rules. So maybe we should use some other term for it, but "theory" will have to do in this thread).
The idea that I used as the starting point is that a theory must be able to assign probabilities to statements of the form
"If you use the measuring device \delta on the object \pi, the result will be in the set E".
These statements can be identified by the triples ##(\delta,\pi,E)##. This means that associated with each theory, there are sets ##\Delta,\Pi,\Sigma## and a function
$$P:\Pi\times\Delta\times\Sigma\rightarrow[0,1],$$ such that the maps ##E\mapsto P(\delta,\pi,E)## are probability measures. Note that this implies that ##\Sigma## is a σ-algebra. I call elements of the set ##\Delta## "measuring devices" and elements of the set ##\Pi## "preparations".
After these simple observations and conjectures, we are already very close to being able to write down a definition that we can use as the starting point for rigorous proofs. There are some subtleties that we have to figure out how to deal with before we write down a definition, like what happens if the measured object ##\pi## is too big to fit in the measuring device ##\delta##? I'm not going to try to work out all such issues here, I'm just saying that they look like minor obstacles that are unlikely to prevent us from finding a satisfactory definition.
Now let's jump ahead a bit and suppose that we have already written down a satisfactory definition, and that the sets and functions I've mentioned are a part of it. Then we can use the function P to define equivalence classes and terms like "state" and "observable". This function implicitly defines several others, like the maps E\mapsto P(\pi,\delta,E) already mentioned above. We will be interested in the functions that are suggested by the following notations:
\begin{align}
P(\pi,\delta,E)=P_\pi(\delta,E)=P^\delta(\pi,E)=P_\pi^\delta(E)
\end{align} We use the P_\pi and P^\delta functions to define equivalence relations on \Pi and \Delta: \begin{align*}<br />
&\forall \pi,\rho\in\Pi\qquad &\pi \sim \rho\quad &\text{if}\quad P_\pi=P_\rho\\<br />
&\forall \delta,\epsilon\in\Delta\qquad &\delta \sim \epsilon\quad &\text{if}\quad P^\delta=P^\epsilon<br />
\end{align*}
The sets of equivalence classes are denoted by \mathcal S and \mathcal O respectively. The members of \mathcal S=\Pi/\sim are called
states, and the members of \mathcal O =\Delta/\sim are called
observables. The idea behind these definitions is that if two members of the same set can't be distinguished by experiments, the theory shouldn't distinguish between them either.
stevendaryl said:
I would think that if one really wanted to seriously talk about equivalence classes, then one would have to
- Define what a "measuring device" is.
As you can see, I don't agree with this point. We only have to define the term "theory" in such a way that every theory is associated with a set whose elements we can
call "measuring devices".
If we continue along the lines I've started above, we will not automatically end up with C*-algebras. What I'm describing can (probably) be thought of as a common starting point for both the algebraic approach and the quantum logic approach. So we can proceed in more than one way from here. We can define operations on the set of observables that turn it into a
normed vector space, and then think "wouldn't it be awesome if this is a C*-algebra?", or we can keep messing around with equivalence classes and stuff until we find a
lattice, and then think "wouldn't it be awesome if this is orthocomplemented, orthomodular, and whatever else we need it to be?".
I seems to me that the reason why we don't get the most convenient possibility to appear automatically, is that we started with a definition that's "too" general. It doesn't just include all the classical theories and all the quantum theories, it includes a lot more. So if we want to consider
only classical and quantum theories, we need to impose additional conditions on the structure (a normed vector space or a lattice), that
gets rid of the unwanted theories.
stevendaryl said:
I don't think you can really do that in a noncircular way,
I think the approach I have described doesn't have any circularity problems.
stevendaryl said:
But the C*-algebra approach sure seems to single out measurements (or observables) as being something different. As I said, the fact that some interaction is a measurement of some quantity is not what you start with, it's a conclusion. There's a long chain of deductions involved in reaching that conclusion.
The way I see it, the chain of deductions that lead to this conclusion is based only on the concept of "falsifiability". And the conclusion provides the motivation for a definition of the term "theory of physics".
