Re: Rovelli on EPR

rof@xxxxxxxxxxxx wrote:

On page 71, Mackey introduces "Axiom VII", which says:
The partially ordered set of all questions in quantum mechanics is
isomorphic to the partially ordered set of all closed subspaces
of a separable, infinite dimensional Hilbert space.

That this is an axiom, and not a theorem, is the sense in which
we do not know why quantum mechanics works. Mackey is refreshingly
honest about this point:

This axiom has rather a different character from Axioms I through
VI. These all had some degree of physical naturalness and
plausibility. Axiom VII seems entirely ad hoc. Why do we make it?
Can we justify making it? What else might we assume? We shall
discuss these questions in turn. The first is the easiest to
answer. We make it beacuse it "works," that is, it leads to
a theory which explains physical phenomena and successfully
predicts the results of experiments. It is conceivable that a
quite different assumption would do likewise but this is a
possibility that no one seems to have explored. Indeed, one
would like to have a list of physically plausible assumptions
from which one could deduce Axiom VII. Short of this one
would like a list from which one could deduce a set of
possibilities for the structure of Q, all but one of which
could be shown to be inconsistent with suitably planned
experiments. At the moment such lists are not available
and we are far from being forced to accept axiom VII as
logically inevitable. ...

Mackey's description of the situation is fairly accurate, and he
thankfully does not attempt (as, for example, Gottfried does,
with his "algebra of filters") to dupe the reader into thinking
that we know why the probabilities assigned by quantum mechanics
coincide with experimentally observed frequencies of individual
experimental results. Birkhoff and Von Neumann's argument, which
Mackey refers to, is the closest that anybody has come, as far
as I can determine, to providing some justification for the
use of quantum mechanics, although at best they have shown
that quantum mechanics is one of a number of procedures which
could conceivably be used to assign probabilities.

You are right, the Hilbert space is just postulated in Mackey's axioms.
However, there were quite a few developments in this field since
Mackey's book was published (1963). The most important step is in
Piron's book

C. Piron, Foundations of Quantum Physics,
(W. A. Benjamin, Reading, 1976)

Instead of Mackey's axiom VII and instead of the classical distributive
law of logic, and instead of Birkhoff-von Neumann "modular law",
Piron introduces the "orthomodular law". This law can be formulated in a
number of different ways. One of the most transparent formulations is
"if proposition x implies proposition y, then x and y are compatible".
Then Piron goes on to prove a theorem which says that above axioms can
be realized if logical propositions are identified with closed subspaces
in a Hilbert space over R, C, or quaternions. Quantum theories with
real or quaternionic scalars have been studied, but, as far as I know,
nothing interesting came out of this. So, we are left with the usual
C-number quantum mechanics whose mathematical apparatus directly
follows from Birkhoff-von Neumann-Mackey-Piron axioms via Piron's theorem.

There are quite a few reviews and book where you can find more
details. You can check, for example,

E.G. Beltrametti and G. Casinelli "The logic of quantum
mechanics" (Addison-Wesley, Reading, 1981)

or search the web for "quantum logic", "orthomodular lattice",
etc.

When somebody says that statements
have "truth values" which can be complex, I do not know what they
mean.

In quantum logic the "truth values" are not complex. They are
real numbers from the interval [0,1], i.e., the probabilities that the
result of the "yes-no experiment" is "yes". Complex numbers arise
only after the propositions of quantum logics are mapped into the
set of subspaces of the complex Hilbert space via Piron's theorem.
This mapping identifies the "truth value" as the square of the modulus
of the projection of the state vector on the subspace, i.e., still
a real number from [0,1].

... those who opt to study quantum logic
should be warned that it is not a system used to model inference,
as actual logic is, but that it is a particular mathematical system
to which actual normal true-and-false logic applies, and that calling
it logic is just poetry.

I don't think so. In my opinion, classical logic developed by Aristotle
and Boole refers only to propositions about classical objects.
For quantum objects we need to take into account the statistical nature
of measurements and indeterminism. This requires a change in the rules
of logic. Quantum logic says that all classical axioms are still OK,
except the axiom of distributivity. This axiom wasn't very intuitive
in the classical system anyway. Quantum logic uses the "orthomodular
law" instead. The distributivity axiom is a particular case of the
"orthomodular law".

Eugene.

.

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