On Mon, Dec 2, 2019 at 7:19 PM Philip Thrift <cloudver...@gmail.com>
wrote:
On Sunday, December 1, 2019 at 6:24:08 PM UTC-6, Bruce wrote:
On Sat, Nov 30, 2019 at 12:35 PM 'Brent Meeker' via Everything List
<everyth...@googlegroups.com> wrote:
On 11/28/2019 4:17 PM, Bruce Kellett wrote:
Right. The subsystem we are considering (an electron fired at a
screen or through an S-G magnet) is just a subspace of the full
Hilbert space. We can take the tensor product of this subspace with
the rest of the universe to recover the full Hilbert space:
|universe> = |system>{\otimes}|environment>
We can then analyse the system in some basis:
|system> = Sum_i c_i |basis_i>,
where c_i are complex coefficients, and |basis_i> are the basis
vectors for (i = 1, ..,, N), N being the dimension of the subspace.
It is assumed that the normal distributive law of vector algebra
acts over the tensor product, so each basis vector then gets
convoluted with the same 'environment' in each case, we have
|universe> = Sum_i c_i (|basis_i>|environment>).
Each basis vector is a solution of the original Schrodinger
equation, so it carries the full energy, moment, change etc, of the
original state.
?? The basis just defines a coordinate system for the Hilbert
space. It doesn't mean that the wf ray has any component along a
basis vector.
The formalism supposes that the state represented by each basis vector
becomes entangled with the environment to leave a record of the result
of the measurement. Coordinate systems do not become entangled with
anything. So the schematic above must represent the particle or
whatever that is being measured (considered of interest, if you wish
to avoid the "M" word.)
The c_i can be zero; in which case that basis vector doesn't carry
anything. No every Schrodinger equation solution is realized
because initial conditions may make it zero.
Irrelevant to the main point.
The environment is just the rest of the universe minus the quantum
quantities associated with the system of interest. So each term in
this sum has the full energy, charge, and so on of the original
state.
If we take each component of the above sum to represent a
self-contained separate world, then all quantum numbers are
conserved in each world. Whether there is global conservation
depends on how we treat the coefficients c_i. But, on the face of
it, there are N copies of the basis+environment in the above sum,
so everything is copied in each individual world. Exactly how you
treat the weights in this situation is not clear to me -- if they
are treated as probabilities, it seems that you just have a
stochastic single-world model.
Yes, I think that's right. Which is the attraction of the epistemic
interpretation: you treat them as probabilities so you renormalize
after the measurement. And one problem with the ontic
interpretation is saying what probability means. But it seems that
the epistemic interpretation leaves the wf to be a personal belief.
Yes, I find this easier to understand in a single-world situation. In
either case, you have to renormalise the state -- energy, charge and
everything -- for each branch in many-worlds as much as in a
single-world. In fact, as Zurek points out, even in many-worlds you
end up on only one branch (stochastically). So the other branches do
no work, and might as well be discarded. If you are really worried
about the possibility of fully decohered branches recombining, take
out life insurance......
Bruce
"even in many-worlds you end up on only one branch (stochastically)"
Sean Carroll himself has said (in a tweet) that if you let
probabilities (stochasticity) in - like the camel's nose under the
tent - you might as well have a one world - not many worlds - theory.
We do have only one world. Do you know of anyone who lives in more
than one branch of the multiverse?
Bruce
