Dear Professor Mould, Feb. 3, 2004
Many thanks for your reply of Jan 23 to my letter ofJan 21.
I believe that your reply fails to answers what I called in my
letter "the main problem". So let me spell out the situation in more
detail.
In your "Born model" proposal, without any observer, you write the state
(after the initial time t_0) as psi(t)D_0 + D_1(t),
where D_0 is the (time independent) ground state of the detector,
which is conceived to be some system with a ground state, and where the
"excited states" D_1 are created by absorption (by the ground state) of the
detectable particle with wave function psi(t).
In a straightforward approach based on the Schr. equation the
evolution in time would take the state at t_0, with D_1(t_0)=0, into the
state at time t_f, after the particle wave packet has past by the detector.
The probability of detection is given by the norm of D_1(t_f), and the
probability of non detection is given by the norm of psi(t_f)D_0.
[By "norm of Psi" I mean |Psi| squared.]
Normally one would say that if after time t_f an observer looks at
the detector, then he will see the state D_0 with probability
|psi(t)D_0| squared or, otherwise, the state D_1.
However, if the packet were very long so that |D_1(t)| changes very
slowly on the scale of conscious awareness, then the question
arises: what would the observer experience during the long time period
between t_0 and t_f.
If the observer plays a purely passive role, and (in this slow case)
has a reasonably well defined flow of consciousness that sees at each
time t only D_0 or D_1 not both, and is "witnessing" what is
"actually happening" then one needs a theory on which the "actual
state" of the device is always either D_0 or D_1, not both.
Of course, consciousness is not instantaneoes, so there is a little
slack.
Your proposal is, essentially, to postulate that the "collapses"
occur on the time scale of consciousness, or faster, so that for a
certain initial period the "collapses" will always be to the state
psi(t)D_0. During that initial period the observer/witness, if added,
will "see" D_0 . Eventually, the outcome D_1 might be selected, in
accordance with the orthodox rules, and the state of the device would
then become D_1, and stay D_1, since there is now no particle.
Thus your proposal can be viewed as an *ontologicalization* of the
orthodox (pragmatic) theory, supplemented by an ontological rule
that specifies that real "collapses" actually occur at the "devices",
and rapidly on the time scale of consciousness, whether an observer is
watching or not.
However, you surely want the predictions for the final probabilities
of D_1 and D_0 to agree with the usual orthodox prediction.
This places constraints on the timings of the automatic collapses.
The intervals between collapses must be large enough so the the scattering
theory rules work. Fermi's golden rule gives transition probabilities that
increase LINEARILY IN TIME for a fixed flux density! This linearity in time,
which arises in a large-time approximation, must be valid
if your collapse rules are to agree with the usual rules. In particular,
if you take the intervals to be short on the time scale of the oscillations
in psi(t) then a quantum Zeno effect will come in, due to the *quadratic*
dependence on time for short times that the Schr. Eqn. entails.
[I am speaking here of the time dependence of the transition probability of
the state psi(t)D_0 into psi(t+delta t)D_0 for small delta t.]
So your proposal is, essentially, to postulate that collapses occur
automatically, at a suitable rate, at the "device".
But one key question is: What determines the times at which these real
collapses occur? Why not a little sooner or later? One needs some further
rule to determine the delta t. The inclusion of the decoherence effect
associated with the interaction with the environment does not by itself do
the job: it merely depresses certain interference effects. It does not
introduce any real collapse of the evolving state vector into *one* of the
possible states.
In the many-minds approach the demand is that no further
process (beyond the Schr. Eqn.) is needed.
But I assume, on the basis of your words, that you are trying to
pursue a one-mind-per-person theory with real collapses. But then you
need to say how the delta t associated with the real collapses is fixed!
This requires some rule that goes beyond what the Schroedinger equation
alone specifies, because the Schr. Eqn. alone gives no such real collapses.
This is a critical point: because this delta t is not determined
by the Schroedinger equation alone, it *could*, rationally, be determined
by the other essential component of orthodox quantum theory, namely our
streams of conscious experiences, which are the only realities the existence
of which we are certain, and which ought to play some role in nature.
In addition to this *timing* rule, some specification is needed to specify
which systems are associated with collapses. You resolve this by giving
special dynamical status to systems called "devices" that have a well
defined Ground State, and Excited States.
The specification that the "device" has a "ground state" D_0 and
"excited states" D_1 allows the "device" to be a single
atom. But you speak of the "device" as a macroscopic system
with strong interactions with the environment. So this (normal)
macroscopic concept of the "device" seems (clearly) to be what you mean.
But then: How "big" does a system have to be to be a "device",
i.e., to be a system that produces real/ontological collapses?
In the Copenhagen interp. the "device" is treated as something that
is described in classical language [i.e., in terms of the everyday
concepts (refined by the concepts of classical physics) that we use to
communicate to our colleagues what we have done and what we have learned.]
But that is within a pragmatic rather than ontological stance. If you
want an ontologically construed theory then the question arises whether you
want this "classical description" to describe a true "classical level of
being" that exists independently of human experimenters and observers, or
whether, as per Copenhagen, this classical description it is merely part
of a pragmatic theoretical structure that allows us to make predictions about
relationships between our conscious experiences.
In any case there are two problems: What determines the delta t associated with
the real collapses that you postulate, and what determines which systems
are "devices", in the sense that they induce the real collapses that you
postulate.
The third problem pertains to the choice of the setting of the device.
You say "the *the choice is internal* to the ontological model."
This suggests that your model "might" be essentially the Copenhagen
Interpretation (CI) interpreted ontologically rather that pragmatically.
Then there would be a quantum world, described by quantum rules, imbedded
in a real classically described world. The boundary is drawn so the "device"
is classically described, and the setting of the device is then controlled
by the classically described observer and his classically described brain.
The whole classical description, which the Copenhagen Interp construes
pragmatically ---i.e., as part of a theoretical structure that allows us
to predict correlations among increments of knowledge in the streams of
conscious events of a community of communicating observing agents ---
would be effectively construed literally and ontologically. This would
allow me to understand your claim that choices made by the agents
(pertaining to which setting of the device is chosen) "is internal
in the ontological model": you would be saying that the classically
described agent does the choosing of the classically described device.
But then one would need to specify exactly what this classical-quantum
reality is and what the classical-quantum dynamics is.
Also, this placement of the ontological boundary conflicts with the
fact that the functioning of the material device depends critically
upon the quantum properties of its component atoms. The von Neumann
approach of placing all atoms (including those in the body and brain
of the observer) inside the quantum-described universe seems far
preferable, and I presume that you follow that approach.
The von Neumann model can be made ontological by postulating that
collapses occur in systems of certain specified kinds, under conditions
of certain kinds, at certain specified times. But to achieve this,
these various specifications must be spelled out.
In any case, your ontological stance likewise seems to require answers
to the three questions : (1) How are the "devices" characterized:
(2) What determines the delta t: (3) What determines how the devices will
be placed and oriented?
In regard to this final point it is useful to consider the
Stern-Gerlach experiment, which is the paradigmatic "measurement" in QM.
In this experiment there is not only the "detector" which is denoted by
D, but also the "apparatus". You introduce A, for apparatus, and speak
of the choice by the observer/experimenter as a choice between, say,
A_1 and A_2. You claim that once the (choice part of the) brain of
the experimenter is brought in "the *choice is internal* to the ontological
model."
But how does this work if the brain itself is regarded as a quantum system?
Let theta represent the direction of the axis of the S-G apparatus.
[The incoming particle moves in the z direction, the "axis" is in
the x-y plane]
Let P(theta) represent the associated projection operator acting
in the spin space of the (spin 1/2) particle. This spin space
is a space of vectors (a,b), with a and b two complex numbers.
If theta specifies the direction in the x-y plane then
a= (exp i theta/2)/root 2 and b= (exp -i theta/2)/root 2.
Of course, this angle theta cannot be exactly specified in
a quantum world if the agent is made of atoms and ions etc.
I allow that the experimenter can specify theta to within some
small fraction of a radian. The smearing over this small
interval will change P(theta) to P'(theta), which is *nearly*
equal to P(theta), but is not a projection operator:
P'P' not= P'; no x-y vector is tranformed to zero by P';
and the range of P'(theta) is the whole space of vectors
in the x_y plane.
The usual precise rules (say of von Neumann) for relating theory to
empirical data depend on the notion that the experimental set-up
determines a set of orthogonal projection operators P_e that
correspond to different empirical outcomes e. So a generalization
of the usual vN rules is needed if one is to go over to an ontological
interp.
Your idea of two different choices by the experimenter, yielding
A_1 or A_2, goes over to two different choices, P'(theta_1) or
P'(theta_2).
But suppose the angle theta can be anywhere between 0 and 2pi,
with probability independent of theta?
In particular, suppose the choice of angle specified by the agent's
brain is given by a density matrix rho(theta, theta') that is nearly
diagonal (due to environmentally induced decoherence), and is independent
of theta + theta'. Then how does the agent's brain determine theta?
The corresponding spin space rho is the diagonal matrix (1/2, 1/2), which
is invariant under rotations around the z axis, and specifies no theta at all.
The point is that if the agent's brain is treated quantum
mechanically, then there can be cases where the input conditions
on the brain (combined with all the uncertainties in the brain
dynamics) make the density matrix associated with the brain's
choice not sharply localized in theta space. But then how does
the brain determine the choice of theta? Some extra rule is needed.
The founder's of QM resolved these problems at a practical level
by bringing the choices made by human agents into their theory,
as extra (i.e., unspecified within their physical theory)
input parameters that we human beings are allowed to specify
in practice. Your ontological model appears to leave these questions
unresolved.
Of course, ad hoc rules can be introduced, but the virtue
of the Copenhagen approach is that it can be applied at the practical
level without introducing such extra rules, which would therefore lack
empirical support.
I hope this longer letter will make my solicited comments on your
article more clear.
Best regards,
Henry