Re: Rovelli on EPR

Eugene Stefanovich <eugenev@xxxxxxxxxxxx> writes:

Cl.Massi wrote:

I believe this program was successfully completed a while ago:

G. Birkhoff and J. von Neumann,
The logic of quantum mechanics, Ann. Math. 37 (1936), 823.

G. W. Mackey, The mathematical foundations of
quantum mechanics (W. A. Benjamin, New York, 1963), see
esp. Section 2-2.

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

They aren't, by far, the only people who claim to have "understood" quantum
mechanics. Alas, all these works either muddy the water, claim to "shed
some light" on an anyway unessential part of QM, or merely reformulate or
renomenclaturize it without solving the interpretation problem.

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.

Charles has presented a paper which he claims provides the
explanation required. I have examined this paper but cannot
find a satisfactory explanation of why quantum mechanics,
rather than some other procedure, is the correct procedure
to use for assigning probabilities to the results of
measurements. There are some things in the paper with which
I agree, but two of the principal foundations, namely
relationalism and quantum logic, seem to me to be unlikely
to be fruitful.

I have explained before why I find relationalism to be incoherent.
Either it is a founding principle on which his argument is based,
which I hope it is not, because it would be difficult to base
a coherent argument on an incoherent principle, or the argument
can be presented without any appeal to relationalism (or any
other "ism"), which would greatly improve its clarity.

The use of quantum logic also seems to me to present no improvement
in understanding, since it merely replaces one unexplained procedure
with another. To say that the procedures used in quantum mechanics
consitute logic is at best a poetical metaphor, because quantum
mechanics is literally not logic. When somebody says that statements
have "truth values" which can be complex, I do not know what they
mean. I can consider maps from statements to sets other than
{true,false}, but if somebody wants to say that these are "truth
values", then there are two possible cases.

One possibility is that this is just the introduction of a new term,
"truth value", and that any other term could have been used just
as well, for example "fribble". In that case, one would have to
conclude that quantum logic is not a more appropriate name than
quantum fribbology, and that 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.

(In mathematics, we often speak of a distance between numbers, and
this is an example of perhaps similar poetry, because numbers do
not have locations in space, and the map from (say) R times R to R
could have been given any other name apart from distance. It could
have been called the alpha function, for example, and mathematics
would be just the same. The usage of terms like "distance" is
whimsical and metaphorical.)

The other possible position that an advocate of quantum logic
could adopt is that the choice of the term "truth value" does
indeed carry an important message of some kind, and then
they have the obligation to tell us what that message is. Is
the message that normal logic has to be abandoned and replaced
by this new system? Why, and how could anybody possibly ever
know such a thing?

In Charles' paper, and in my correspondence with him, he has
been keen to insist that his introduction of the entire Hilbert
space formalism, along with the identification of kets as
propositions about measurement results in a formal language,
is little more than a choice of notation. However, the set
of things which can be proven about the results of measurements
if one merely accepts the Hilbert space formalism and the
rules of quantum mechanics for assigning probabilities to
the results of measurements is non-trivial.

For example, if there are 10 mutually exclusive complete
measurements than I can perform on a system, each
of which can give 10 possible results, then a complete
specification of the probabilities of obtaining each
of the 100 results requires 90 parameters to specify.
That's 100 possible results, grouped into sets of 10.
In each set of 10, I assign 9 probabilities, and the
10th is fixed by the fact that the probabilities must
sum to one.

If I suppose that measuring the same thing twice always
gives the same result, then I can prepare a system by
making a measurement. Then 10 probabilities are fixed
(one probability is 1 - if I measure the same thing
again, I will get the same result, nine are zero - if
I measure the same thing again, the chance of me getting
any of the other nine results is zero). That leaves 81
parameters which I need in order to specify the probabilities
of obtaining the various results of the different measurements
I could perform.

Quantum mechanics, however, tells us that we can
specify all of the probabilities with only 18 parameters.
All that we need to do is specify a ray in a ten-dimensional
Hilbert space, so that's 9 complex numbers or 18 parameters.

So if we have accepted the Hilbert space formalism, and if
we have accepted that the square of the inner product
gives us probabilities of finding particular measurement
results given particular preparations, then we have
already agreed to some very non-trivial statements
about the results of experiments and how many parameters
are needed to describe them. A mere choice of notation
can't change an 81-dimensional space into an 18-dimensional
space. It is noteworthy that this is all before any mention
has been made of time evolution, the Hamiltonian, or any dynamics.

The question is: "Why are there 18 and not 81 parameters needed to
specify the probabilities of obtaining the various possible measurement
results?" Right now, the only answer we have is that the symbols
tell us so, and I do not believe that Charles, Mackey, Rovelli,
Birkhoff & Von Neumann, or anybody else has come even remotely
close to addressing this question.

R.

.

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