On 13-08-2019 10:21, Philip Thrift wrote:
On Monday, August 12, 2019 at 7:43:51 PM UTC-5, smitra wrote:
Thing is that the interference we can observe at some position x on
the
screen is Re[<A|x><x|B>], which for general x is nonzero despite the
fact that <A|B> = 0.
Saibal
To a probability (or measure) theorist, one making the conversion from
classical summing rules to quantum summing rules, this should be
easier:
https://arxiv.org/pdf/gr-qc/9401003.pdf
Where quantum theory differs from classical mechanics (in this view)
is in its dynamics, which of course is stochastic rather than
deterministic. As such, the theory functions by furnishing
probabilities for sets of histories. More formally, it associates to a
set A of histories a non-negative real number |A|, which I will call
its quantum measure |A|; and it is this measure that enters into the
sum-rules we will be concerned with.
...
NOTIONS SUCH AS STATE-VECTORS AND OBSERVABLES NEVER APPEAR, EXCEPT FOR
THE SAKE OF COMPUTATIONAL CONVENIENCE.
...
In the two-slit experiment, for example, the probability that a
particular detector will register the arrival of the electron is
(proportional to) the measure |C| of the set C of all electron world
lines which in fact pass close enough to that detector to trigger it.
When we contemplate also blocking off one or the other slit, there are
(for a fixed detector) three sets of histories to consider: the set A
of histories which arrive at the detector after traversing the
“first” slit, the corresponding set B for the “second” slit,
and the original set C = A ∐ B, the disjoint † union of A and B.
It is of course characteristic of quantum probability that the
interference term
I(A, B) := |A ∐ B| − |A| − |B|
between the slits is not zero. The surprising thing (once one has
gotten used to the fact of interference itself) is that this violation
of the classical probability sum-rules is in a certain sense so mild,
since the corresponding sum-rule for three alternatives
remains valid.
In any case, the important thing from the standpoint of interpretation
is that the electron follows one and only one path, not somehow two at
once. If probabilities are involved, it is only because the path is
not determined in advance, just as it is
initially undetermined in a classical stochastic process.
Given the failure of the sum rule I(A, B) = 0, it is clear that
quantum probabilities cannot be interpreted in the same manner that
classical ones are wont to be interpreted, in terms of (actual or
fictitious) ensemble frequencies.
Thanks, I'll read that article.
Saibal
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