[[Mod. note -- I have rewrapped long lines.  -- jt]]

The Everett interpretation assumes that there is no wavefunction
collapse - instead everything is unitary.

I'm not sure how this works because objects like me, cars, the
world! are usually a mixture of states, better described by a density
matrix. How is it that these mixtures can be thought of as pure
states which evolve unitarily?

El domingo, 6 de mayo de 2012 15:49:10 UTC+2, Argand  escribi:

The Everett interpretation is not consistent

http://www.mat.univie.ac.at/~neum/physfaq/topics/manyworlds

On 5/6/2012 3:49 PM, Argand wrote:
   ..

The wave function, Psi, evolves unitary by:

       i d Psi/dt = H Psi
so
       Psi(t) = exp(-iHt) Psi(0)

and if a collapse would occur there would be a
projection operator P, changing Psi by:

       Psi  -->  P Psi

The density matrix, rho, evolves unitary as:

       i d rho / dt = H rho - rho H
so
       rho(t) = exp(-iHt) rho(0) exp(iHt)

and if a collapse would occur it would mean:

       rho  -->  P rho P

So you see: in both descriptions you are free to
choose whether you believe there will be a collapse.

No-one says they are pure states. But still their
time evolution can be described using exp(-iHt).

(PS: I didn't say there actually *is* a collapse!)

--
Jos

The time evolution during a measurement unitarily entangles the
indication of a measuring device A with the state which is being
measured.

Before measurement the state Psi = sum_i Lambda_i psi_i
of the quantum system and the indication Phi of the device is
uncorrelated,

Psi x Phi .

By the interaction with the measuring device it is unitarily mapped to

U(Psi x Phi) = sum_i Chi_i x Phi_i psi_i

Here I use the following notation: Lambda_i are the eigenstates
of A, which yield the corresponding result a_i with certainty and
which generate  with certainty the state Phi_i of the indicator

U(Lambda_i x Phi) = Chi_i x Phi_i .

Chi_i are some normalized states, which are not necessarily orthogonal
to each other, Phi_i are mutually orthogonal, because the different
readings of device A can be read off with certainty. Psi is a linear
combination of the eigenstates Lambda_i with complex coefficients psi_i.

If you measure the indicator and the quantum system after the
first measurement with an apparatus B x A~, where A~ reads of the
indicator of A, then the probability to find the result k in the
second measurement and  read of result i of the first measurement is

p(k,i) = |<Gamma_k | Chi_i> psi_i |^2  where Gamma_k is the

eigenstate of the second device B.

In particular, the probability for the first measurement to yield
result number i, irrespective what the second measurement gives, is

p_1(i) = sum_k p(k,i) = | psi_i |^2  (which we knew already)

and the _conditional_ probability, to get result k in the second
measurement in case that the first yielded result number i is

p_i(k) = p(k,i) / p_1(i) = |<Gamma_k | Chi_i>|^2

as if the first measurement and its result had collapsed Psi to Chi_i.

Note, that in case of discrete, nondegenerate results, Chi_i contains
no information about Psi

This collapse, however, it not due to a discontinuous time evolution
of the probabilities but to a discontinuous change from considering
the probability and then the conditional probability. The collapse
of the wave function results from not considering the wave function
anymore but the conditional wave function.

Such a discontinuous change of probabilities is common in classical
probability theory. The chances to win in Lotto (6 out of 49) change
discontinuously with the knowledge of each drawn number and lead to the
collapse of the Lotto player when he realizes that all numbers agree
with his bet.

--
Superstition brings bad luck

www.itp.uni-hannover.de/~dragon

*  Jos Bergervoet writes:

This description is untenable, because P Psi in not
normalized.

I do. The collapse it the transition from a state Psi x Phi,
which evolves unitarily in a measuring device into

U(Psi x Phi) = sum_i Chi_i x Phi_i psi_i

to the conditional state Chi_i in case that the measurement
becomes known to be result number i with the indicator state Phi_i.

Note: I do not say when and how it is that the result of a
measurement becomes certain. That is the one and only miracle
of quantum mechanics.

--
Superstition brings bad luck

www.itp.uni-hannover.de/~dragon

On 5/7/2012 4:08 PM, Norbert Dragon wrote:
    ...
  ..

That would just make the time evolution non-unitary.
OP explicitly asked about this! (Wavefunction collapse,
as opposed to having everything unitary..)

And normalization of the state is merely convenient,
not necessary. Describing a "ray" in Hilbert space is
usually considered sufficient.

...

But then you leave open the possibility that this
happens only at the end of time? Or never at all?!
The statement seems equivalent to: "time evolution
given by the Schroedinger equation is insufficient
but we do not say when and how."

There are actually more things you do not say (things
that are essential to the story). For instance: when
do we *not* call time evolution a "measurement"? We
live in a universe of interacting fields, so what is
described above applies at every point in time! So we
add nothing with the word "measurement". We just say
that every calculation with Schoedinger time evolution
is invalid because we should apply this additional
final step of which you do not know "when and how".

To some it brings a collapse of uncertainty.

--
Jos

*  Jos Bergervoet writes

At times I muse the idea, that results are only approximately
certain just as in soccer the moment is not clear when it is that
a goal is scored (the last game England Germany showed that it is not
sufficient that the ball passes the line). The problem of discrete
results from continuous motion is not characteristic of quantum
mechanics but arizes with all probabilitystatements. If you watch in TV
the drawing of the lotto numbers when is it precisely that they are
drawn and certain? The observed motion is continuous, the result
"drawn" or "not drawn" is discrete.

I do not think that time evolution given by the Schroedinger equation
is insufficient to describe a measument. The Schroedinger equation
describes also processes  which are sufficiently irreversible, e.g. the
decay of a particle or light emitted from the indicator of a measuring
device.

The time evolution is no measurement if you cannot read off the result,
e.g. that a particle passes a double slit is not a measurement of the
position, also motion in a vacuum does not measure where the particle
is.

Measurement entangles the state, which is to be measured, with the
indicator state of a device.

I disagree.

No experimental physicist is in doubt that there are times before the
measurement and times after the measurement -- just as there are
times before and after the draw of the lotto numbers. If the
theoretical concepts have a continuous time evolution then one
employs triggers to define the begin and the end of a measurement.

--
Superstition brings bad luck

www.itp.uni-hannover.de/~dragon

The thing that seems a bit circular about the quantum-mechanical
notion of a "measurement" is that something is only a measurement
if there is an irreversible change (the formation of a memory, or
an image on a photograph). But things are only really irreversible
in the sense of overwhelming probability: An ice cube melting on a
hot sidewalk is the normal behavior, but the reverse--an ice cube
forming from a warm, wet sidewalk--is theoretically possible, just
very, very unlikely.

So that's the circularity: you need probability in order to decide
what counts as a measurement, and in quantum mechanics, measurement
is necessary to give a meaning to probability.

Any interaction between particles can result in an entangled state;
even two electrons interacting through electromagnetic repulsion.
But it's not a measurement just because the parts are entangled.
One of the two objects has to be "macroscopic" and capable of
forming irreversible memories of the interaction for it to count
as a measurement.

*  Daryl McCullough writes:

Quantum mechanics is as it is. Einstein wished something more complete
but the violation of Bell's inequality shows him wrong.

The basic equation that

p(i,A,Psi) = |<Lambda_i|Psi>|^2

is the probability, to obtain result number i (in case of discrete,
non-degenerate results) if one measures Psi with the device A, is
so simple that up to now we have no simpler explanation. The same
applies to the fundamental notion of what a measurement is. It is
something which ascertains a result which was uncertain before.

So what? Measurement shares some properties with non-measurement.
Measurement entangles a state with the indicator of a measuring device.
An indicator deserves its name only if its reading can be read with
certainty and if the result can be preserved in a memory.

--
Superstition brings bad luck

www.itp.uni-hannover.de/~dragon

On 5/11/2012 12:43 PM, Norbert Dragon wrote:
   ...

It is no explanation at all. If several probabilities are
nonzero, this equation does nothing to tell us how the
time evolution towards one certain outcome takes place.

And if you keep using Schroedinger, then you only end
up with all possibilities still present, nicely entangled
with corresponding result states of the measurement device,
which in turn will be entangled with the states of the
memory device storing the result, which will be entangled
with the states of mind that you have after inspection.

So, the formula is indeed simple, but it does not
explain why one result is selected. It does not even
describe the selection of one particular result! Your
equation is exactly *the opposite* of an explanation
how a single result is obtained in a measurement (but
yes, it is simple..)

And that is most likely what Einstein wanted to see,
and it is not present in quantum mechanics! In QM
there are no measurements. Nothing is ever decided.
It all remains a summation over all possible outcomes.
Consistently entangled over the whole chain of cause
and effect, but *not* decided, *not* ascertained!

Unless of course you can come with a better proposal.
The equation you give here does nothing.

--
Jos

*  Jos Bergervoet writes:

I only stated that there is no simpler explanation.

The proposed explanation that the universe splits into many worlds is
far more complicated than the simple fact which it should explain. Many
worlds "explains" by making the the question so complicated that one
gives  up to ask questions. That is the technique to blur the water to
catch the trout -- I prefer to listen to Schubert.

The basic statement (1) is simple and uses simple words which
in concrete situations have a definite meaning. _But_ it defies all
attempts to explain it in simpler terms.

Only that I propose to exchange "But" by "therefore" and live with it.

No one can explain (1) by an underlying causal mechanism as the
violation of Bell's inequality shows. The question "How does it come?",
however insists on the impossible, namely to give such causes.

(1) restricts the time evolution because probbilities are subject to
the sum rule

sum_i p(i) = 1

which enforces the Schroedinger equation

i d_t Psi = H Psi  (2)

if Psi contains the complete information about the state and if
the time evolution is linear.

This is not a horrible picture but seems to be true. Only that
the correlations become unmanagable and can safely be neglected
for all practical purposes.

Your dissatisfaction is about nature. The probabilities of
entangled photon polarisations show that measurements do not
read off properties which existed and were certain before, but
ascertain results which were uncertain before.

While at lotto, you can dream about a mechanical explanation of
the drawn balls this is logically excluded by the measured
photon probabilities which agree with the quantum theoretical ones.

My argument that the collapse of the wave function is the
discontinuous change from the entangled state of Psi and the
indicator Phi to the conditional state Chi_i, in case that the
measurement gave result number i, solves some puzzles. In
particular, one can postpone the application of (1) to a later
stage and use the continuous time evolution (2) a long as one wants.

Moreover it shows that invertible time evolution does not deserve the
name measurement. For all practical purposes there are such
irreversible time evolutions, e.g. the ionization path which shows
a charged particle, or the light from a visible particle,
though mathematically (2) is invertible.

For all practical purposes quantum mechanics explains ionization and
photographic films. There is no better theory and no better world even
if Einstein had prefered God not to throw dice and the moon to exist
also in case that no one can see it.

You can read off the indicator whenever you want, now or at doomsday.

The equation (1) leaves only one puzzle: why does it hold?
But the question is already shown to have no answer: polarization
measurements of entangled photons cannot read off properties
which were certain and caused the results.

--
Superstition brings bad luck

www.itp.uni-hannover.de/~dragon

On 11/05/12 08:14, Daryl McCullough wrote:

That's true. Of course, there must be some entanglement between the
measured quantity of the object of interest. Let's take the spin
component of an atom in a given direction, measured by an appropriate
Stern-Gerlach apparatus. The atom runs through the inhomogenous field,
through which its position becomes entangled with the spin component in
the corresponding direction, i.e., the position probability
distributions due to the motion in this field becomes discretely peaked
according to the possible values of the spin component, -s,
-s+1,...,s-1,s (hbar=1).

So far everything is described by the unitary time evolution of the
state. To make a clear measurement of the spin component possible, the
peaks in the probability distribution must be well separated in
comparison to the single peaks' width.

Now you can "measure the position of the atom" by, e.g., let it hit a
photo plate, where the spot gets (for all practical purposes
irreversibly) blackened. I would call this the measurement, and the
point is that you have the interaction of your system of interest with a
"macroscopic body" and you are interested only in a very "coarse grained
macroscopic" observable, namely a little blackened crystal on the
surface of your photo plate. The whole process is still described by a
unitary time evolution, but the "projection" to the pretty rough
"pointer state", which involves a drastic averaging over a lot of
microscopic states, which all contribute to the macroscopic observable,
the "pointer state" which is represented by a statistical operator. Of
course also this pointer state is "entangled" with the spin state of the
atom corresponding to the spot.

I'm a follower of the minimal statistical interpretation, and there is
nothing mysterious with this whole process. The atom's original state,
may be a pure or mixed state, it's in any case described by a
statistical operator (a pure state is a projector, i.e., fulfilling
R^2=R, otherwise for a mixed state one has R^2<=R). If it is not a pure
spin state (i.e., if the reduced statistical operator for the
observation of the spin component is not a projector) then one doesn't
know more about the outcome of a measurement of this spin component than
its probability. Within the measurement nothing special happens. It's
simply the interaction of the atom, which I've prepared with help of the
SG apparatus as an state, where the spin component and the position of
the atom are entangled to a sufficiently high degree (in principle one
can make this entanglement a 100% correlation; here one is only limited
by technical means, not from principles of the quantum natural laws).
Also the very procedure of measurement, i.e., the interaction of the
atom with the photo plate to get an irreversible pointer reading of its
position is nothing special, but simply due to the interaction of the
atom with the plate, described by a unitary time evolution, and then
"coarse graining" the microscopic state, which I cannot resolve by any
practical means, to the only relevant information about which spot on
the screen has been blackened. Then I can simply measure the position of
this black spot, and this gives the possibility to check the
distribution of the black spot with the predicted probabilities from
quantum mechanics.

One doesn't need a collapse or other strange ideas about what happens
during a measurement to simply compare the outcome of measurements with
the predictions of quantum mechanics. There is no more mystery in this
than with any classical statistical description of some process in
nature. Take Norbert Dragon's example of the Lotto drawing. There is no
collapse or the splitting of the universe in some number of parallel
universes simply because somebody notices the Lotto numbers.

Of course, this minimal statistical interpretation has important
consequences on our world view. It leaves only two possibilities:

(a) Quantum Theory is a complete description of nature. This means any
system's state can only be determined as completely as possible by
preparing it in a pure state in the sense of quantum theory. Then
necessarily only some observables have a definite value, namely those
for whose representing operators any representing ket of this state
(which is a ray in Hilbert space) is an eigenvector, and the eigenvalue
then is the definite value of the observable, and (given an ideal
measurement device) any outcome of a measurement of this observable
gives with certainty this value. All other observables are not
determined. One only knows the probabilities (or the probability
distribution in the case of continuous observables) for a certain
possible value. In this case, quantum theory tells us that nature is
inherently probabilistic, i.e., non-deterministic. It has been this
consequence of a strict interpretation of Born's probabilistic
interpretation of the quantum mechanical states which has made a lot of
classical physicists, among them Einstein, Planck, Ehrenfest, and
Schroedinger, uneasy since they didn't like to give up a deterministic
world view.

However, as we know nowadays, quantum theoretical probabilities and
probabilities of a local classical deterministic hidden-variable theory
lead to measurable consequences in form of the violation of Bell's
inequality or similar statements. Quantum theory has all empirical
evidence on its side. Of course there is still a little loop hole that
nature may be deterministic but behaves nonlocal. This would mean

(b) Quantum mechanics is an effective probabilistic theory for a yet
unknown deterministic more complete theory of nature.

The latter possibility is not ruled out completely yet. This is true for
any "fundamental" theory of nature: Any theory is always subject to
being falsified by observations, and when this happens, one has made a
big progress in ones understanding of nature. As long as this is not the
case, we have to live with the theories we have, and for quantum
mechanics this is for sure the case: There is not a single reproducible
observation violating its predictions :-)).

--
Hendrik van Hees
Frankfurt Institute of Advanced Studies
D-60438 Frankfurt am Main
http://fias.uni-frankfurt.de/~hees/

On 5/11/2012 8:06 PM, Norbert Dragon wrote:
   ...

There is no smaller positive number than -17 :-)

It also is unrelated to the discussion here: both before
and after the entangling time evolution there is a
summation over a number of possibilities. If someone
wants to see that as many worlds, then they are present
from the start! So "splits" is not appropriate, but "many
worlds" would make some sense. After all, the dimension of
the Hilbert space is considerably larger than 1. The word
"many" doesn't sound to me as a misnomer. :^)

  ...

I never said it is horrible! But it means that in quantum
mechanics there is no measurement by your own definition:
  ".. the fundamental notion of what a measurement is. It is
something which ascertains a result which was uncertain before."

  ...

Unmanageable doesn't guarantee it can be neglected. Shouldn't
we require that they are negligible? I would prefer to have:
"further time evolution commutes with wave function collapse"
or, to be more specific, if unitary time evolution U(t1, t0)
from t0 to t1, describes the measurement which has given us an
entangled state, further evolution U(t2, t1) has to satisfy:
   P U(t2,t1) = U(t2,t1) P
where P projects onto one outcome of the measurement. This
doesn't require collapse to take place, but *if* it would take
place then it would give the same evolution in the remaining
subspace of the Hilbert space that would also occur without a
collapse. Whether it occurs becomes a meaningless question..

   ...

Not mine! (That was another physicist you mentioned in
your previous post.)

  ..

Really? It deserves a chain of explanations and new
questions like in the case of gravity:
   Why do planets move in an ellipse?
   Why is gravity like 1/r^2?
   Why is curvature proportional to energy?
   Why is string theory correct?
At some point the first of the questions will become a
back-of-the-envelope exercise! It just takes a few
centuries..

--
Jos

Okay, so if wave function collapse (and the origin of probabilities
from deterministic evolution) is due to coarse-graining, then it's
subjective, not physical. That's very similar to the derivation of
irreversibility and the second law of thermodynamics from
deterministic microscopic evolution in classical mechanics. There
is a sense in which it is subjective (since it depends on a subjective
choice of a coarse-graining), but in practice, the subjective details
don't make any difference--the conclusions are practically independent
of the subjective choice.

But it seems to me that this interpretation of probability in
quantum mechanics really is the same as the Everett interpretation,
but described using different language. The Everett interpretation
really is about denying that "wave function collapse" ever happens,
as a physical process, and instead to insist on unitary evolution
for all systems, macroscopic or microscopic. The trick, of course,
is to come up with a satisfying explanation of the *subjective*
appearance of collapse without invoking it as a physical process.

--
Daryl McCullough
Ithaca, NY

stevendaryl wrote

http://arxiv.org/abs/1205.0293

This paper attempts to derive the emergence of observed reality
including the born rule from the assumption that the quantum state is
real and evolves unitarily on a global scale. The linearity of the state
space is broken by introducing classes of subjectively equivalent
states. The result is a subjective evolution of the quantum state in
agreement with the measurement po stulate.

Regards,
Andreas