The other day Patrick mentioned to me on #lesswrong that one of the multiverses Briane Greene talks in his book Hidden Reality… is of the MWI type. So I looked through the book, and here is the relevant passage:
Over the years, a number of researchers including Neill Graham;
Bryce DeWitt; James Hartle; Edward Farhi, Jeffrey Goldstone, and Sam
Gutmann; David Deutsch; Sidney Coleman; David Albert; and others,
including me, have independently come upon a striking mathematical fact that
seems central to understanding the nature of probability in quantum mechanics.
For the mathematically inclined reader, here’s what it says: Let be the
wavefunction for a quantum mechanical system, a vector that’s an element of
the Hilbert space H. The wavefunction for n-identical copies of the system is
thus. Let A be any Hermitian operator with eigenvalues αk, and eigenfunctions.
Let Fk(A) be the “frequency” operator that counts the number of timesappears
in a given state lying in. The mathematical result is that lim. That is, as the
number of identical copies of the system grows without bound, the
wavefunction of the composite system approaches an eigenfunction of the
frequency operator, with eigenvalue. This is a remarkable result.
Now, this makes me update more toward the statement “MWI does not require any extra assumptions beyond the Schroedinger equation”, though not all the way there, because it postulates “infinitely many identical copies of the system”, which is still a separate postulate. There is a further problem with this. Brian Greene again, now for a two-state system:
So from the standpoint of
observers (copies of the experimenter) the vast majority would see spin-ups
and spin-downs in a ratio that does not agree with the quantum mechanical
predictions. Only the very few terms in the expansion of that have 98 percent
spin-ups and 2 percent spin-downs are consistent with the quantum mechanical
expectation; the result above tells us that these states are the only ones with
nonzero Hilbert space norm as n goes to infinity. In some sense, then, the vast
majority of terms in the expansion of(the vast majority of copies of the
experimenter) need to be considered as “non existent.” The challenge lies in
understanding what, if anything, that means.
This is a standard problem in MWI, but still, there are some hints that MWI may be a part of the next step in the quantum theory, when it finally happens.
The other day Patrick mentioned to me on #lesswrong that one of the multiverses Briane Greene talks in his book Hidden Reality… is of the MWI type. So I looked through the book, and here is the relevant passage:
Now, this makes me update more toward the statement “MWI does not require any extra assumptions beyond the Schroedinger equation”, though not all the way there, because it postulates “infinitely many identical copies of the system”, which is still a separate postulate. There is a further problem with this. Brian Greene again, now for a two-state system:
This is a standard problem in MWI, but still, there are some hints that MWI may be a part of the next step in the quantum theory, when it finally happens.
Were some characters dropped in the block quotes? It looks like some variable names went missing.
Sorry, this is digitized from a printed book.