Measurement Problem in Quantum Mechanics

The measurement problem

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[Eric-uvr]

Very well explained. Cleared many of my confusions.

[QuicksilverSG]

Note that the Born rule is not an integral part of the quantum wave-function. It is a transform used to convert the eigenvalues generated by the quantum wave-function in configuration space into probability densities in 3D physical space. It is this projection from configuration space into physical space that is referred to as the "collapse of the wave-function" in the Copenhagen Interpretation. This transformation does not affect the wave-function itself, which continues to evolve according to the Schrodinger equation in configuration space. It is instead triggered by an act of measurement initiated in 3D physical space.

The problem with this interpretation is that for any such quantum trigger to originate in 3D physical space, it must propagate from its point of origin relativistically, i.e. at the speed of light. This is in contradiction to Bell's Theorem, which recent Nobel Prize-winning experiments have confirmed to show that measurements of distantly-entangled particles are not subject to the relativistic limitations of 3D physical space.

[david_porthouse]

The collapse of the wave function is very likely to be
a nonlinear process, for which computer simulation is
needed. Any simulation needs to make use of a random
number generator. I believe I am stating the obvious.

I will propose in outline how to go about it. This is
likely to be a long posting, I am afraid, but I am
going to offer a tentative solution to the measurement
problem. Our first difficulty is that the Schrödinger
equation, or any equation like it, is extremely
accurate at the ensemble level and we can take it for
granted that the modification of it is forbidden. There
is nothing resembling the viscosity term to be found in
the Navier-Stokes equation. And yet we need to inject
some randomness. There are two ways to do it.

*The first way* is to hypothesise that some nonlocal
degree of freedom is involved, so even if we know
nothing about Bell's Theorem we could have guessed it
anyway. Just playing around with the Minkowski formalism,
we notice that there is more than one way to travel
faster than light. I suggest that the Schrödinger
equation describes an oscillation in one of the ways
which is capable of destructive interference with itself.
We can have an orthogonal tachyonic Wiener process in
the other way, which I will just call tachyonic Brownian
motion (TBM). This comes into action during the nonlinear
interaction between the wave function and the
electromagnetic field, and can then lead to an outcome
which does not have an issue with Schrödinger's cat. No
aetiology is proposed for this TBM and I am guessing that
it is quantified by having the Planck time as its
characteristic time. What else?

Nitrogen tri-iodide has the unique property that it is
so unstable that it can be detonated by an alpha particle
from a substance like polonium-210. Nitrogen trifluoride
is stable by contrast. A computer simulation of tri-iodide
under bombardment needs to have an outcome which is
qualitatively different from the trifluoride. In any
well-written simulation the trifluoride behaviour will be
isentropic, but with the tri-iodide there will be a
destruction of unitarity and a substantial rise in
specific entropy. It is suggested that the missing
ingredient in the simulation is TBM, which being normally
orthogonal is dormant in the trifluoride, but is
sufficient to detonate the tri-iodide once an
electromagnetic field is also present. This is a second
order nonlinear effect and is like a random walk along
the edge of a cliff.

Two molecules of nitrogen tri-iodide are in fact a
detector in the classical sense, and constitute the
smallest detector that I can think of. What is called
the Heisenberg cut comes between one and two molecules
of tri-iodide. Maybe in the future somebody will think
of a smaller detector, but it won't really affect the
argument to be given here. The computer simulation of two
molecules of tri-iodide will need to run in at least
twenty four dimensions of configuration space, just
counting atoms. This is impossible in practice, and gives
us a hint of what we are up against. All detectors are
just too complicated to model by the formal method using
TBM. We really do need a detector to get the collapse of
the wave function in our simulation, but we must adopt
other ideas.

*The second way* to reconcile the immutability of the
Schrödinger equation to the need to use a random
number generator involves a bit of handwaving. We just
throw away the Schrödinger equation and replace it by
a classical system with some ordinary Brownian motion
for any object heavier than the Planck mass. The
Heisenberg Uncertainty Principle is replaced by the
Fürth Uncertainty Principle on the same scale, so hardly
anyone will notice. Yes, it is indeed cheating, but we
have TBM as an aetiology and no known practical
alternative.

Classical BM will be much more disruptive than TBM and
of course we are reinventing decoherence. We shouldn't
have much trouble collapsing wave functions using it
in our simulations. The usual objection to decoherence
is the lack of any means of destroying unitarity in any
closed system, which has been answered by proposing TBM
as the aetiology.

If we have an electron in a potential well, then the
electron is modelled by the Dirac equation plus TBM.
The electromagnetic field is modelled by correlated
TBM so the wave function and the electromagnetic
field working together can act like a nonlocal Vernam
cipher. The potential well is considered to be a dimple
in a heavy object so it is modelled with a bit of
classical BM. I have already written a little computer
simulation of the Dirac wave packet, and I am guessing
that the propensity of the monochromatic wave packet to
be a tachyon is also going to be significant.

Output from any program will be to a cinematic loop
display. Buttons will be provided to replay the loop,
to do a time reversal, and to do a Lorentz boost. A
side effect of pressing any button will be to reseed
the random number generator in use, probably the
Mersenne twister, by reference to the time of pressing
the button. This is the "Protean system" and it ensures
that we can simulate nonlocal phenomena without worrying
about causality. A Lorentz transformation might give the
appearance of being able to swap cause and effect, but
only on a new random event.

I intend to put a series of computer simulations in the
public domain which anyone will be able to modify. If
they wish, they can rip out TBM and install some other
way of doing things. What has been described here is
the projected solution to the measurement problem as
we start our programming. We will just have to see how
we get on.

In summary, adding a random number generator to any
computer simulation of quantum mechanics is likely to
require two different methods in practice. One is pitched
at nonlocal work, the other at the collapse of the wave
function. The single use of a RNG is conceivable in
principle, but we lack the computer power to pursue it.