One needs to define the observer + measurement apparatus. In
conventional QM one associates to a measurement device made ready to
measure some property of a system a Hermitian operator. But if we pursue
the MWI rigorously, then one should associate a set of commuting
operators to a conscious experience of an observer. And this should then
follow from a description of consciousness as the computational state of
some algorithm.
So, at this very moment my experience while I'm typing these words must
be the result of a particular algorithm that is processing a particular
set of data. I cannot in principle see the difference of that exact
algorithm being implemented in different parts of the multiverse. At
least not at that very moment when that particular set of data is
processed.
An algorithm can be described by a time evolution operator U that acts
on an input state and maps it to the state one computational step later.
The preferred basis then arises from selecting the sectors where the
algorithm representing the observer exists. The algorithm will have some
valid input states |input> and corresponding output states |output(in)>
= U|input>, we can then write:
U = sum over all |input>, |output> of |output><output|U|input><input| =
sum over all |input> of |output(input)><input|
Suppose that we seek the observer represented by the operator U who has
some definite experience. Whatever the observer is experiencing will be
some course grained description of the precise inputs the actual data
that U could be processing. So, we then consider limiting the summation
above to only those input states that correspond to the specified coarse
grained description. So, one can say that we need to narrow down U to
some definite experience, but we then still end up summing over a very
large set of states, because a particular conscious experience does not
correspond to a definite physical state.
A simple example is to consider simulating a spin measurement using a
quantum computer. The spin is then represent by a qbuit and the
"observer" measuring the spin would then be the CNOT operator that takes
the qubit representing the spin as the control qubit while the other
qubit that it acts on is initialized to be |0>. So, one can then say
that there exists a "CNOT observer". This definition is then well
defined w.r.t. changing the basis.
Saibal
On 20-12-2021 18:01, Jesse Mazer wrote:
When you say the MWI + Born rule "yields an unambiguous framework for
a fundamental theory" are you assuming the idea of probability being
equal to amplitude squared only applies to "measurements", or that it
would somehow apply at all times in the MWI? If the former there would
seem to be some ambiguity about what a "measurement" is; if the
latter, I believe MWI advocates still don't have an agreed-upon answer
to the "preferred basis problem" discussed at
https://physics.stackexchange.com/questions/65177/is-the-preferred-basis-problem-solved
On Mon, Dec 20, 2021 at 4:03 AM smitra <[email protected]> wrote:
On 20-12-2021 03:05, Bruce Kellett wrote:
On Mon, Dec 20, 2021 at 12:23 PM John Clark <[email protected]>
wrote:
On Sun, Dec 19, 2021 at 7:59 PM Brent Meeker
<[email protected]>
wrote:
On 12/19/2021 5:25 AM, John Clark wrote:
By contrast the Many Worlds Theory only makes one assumption,
Schrodinger's Equation means what it says. So Many Worlds wins.
_> It also makes the assumption that the eigenvalues of a
measurement are realized probabilistically._
What is the eigenvalue of a temperature of 72°F? It doesn't have
one.
A measurement doesn't have an eigenvalue but a matrix does, such
as
the one that describes the Schrodinger Wave. And no quantum
interpretation needs to assume there is a relationship between the
square of the absolute value of that wave and probability because
it
is observed to be true.
The Born Rule cannot be derived from the Schrodinger equation; it
has
to be added as a further independent assumption. So it is not true
that Many Worlds makes only one assumption. It requires just as
many
assumptions as collapse theories.
Bruce
Yes, but with those assumptions it yields an unambiguous framework
for a
fundamental theory. In case of collapse theories, you're stuck with
a
phenomenological theory that cannot be improved, because you are not
allowed to describe observers and observations within the collapse
frameworks. It's a bit like the difference between statistical
mechanics
and thermodynamics, if in the latter case textbooks were to insist
that
you are only allowed to consider certain types of heat engines that
operate in the quasistatic limit.
Saibal
If it were not true Schrodinger's Wave would be completely
useless
and there would be no reason any physicist would bother to
calculate
it.
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