Classical and Quantum Mechanics via Lie algebras

A. Neumaier

On the level of wave functions, there are huge differences, since these have no classical equivalent. On the level of density matrices, or in the Heisenberg picture, the differences are marginal (more precisely of the order of the Planck constant. My book

Arnold Neumaier and Dennis Westra,
Classical and Quantum Mechanics via Lie algebras,
2008, 2011. http://lanl.arxiv.org/abs/0810.1019

shows how similar the classical and the quantum worlds are when consistently and from the start treated without significant reference to wave functions.

Ken G

Then the program has already been carried out, and in considerable formal rigor. That's impressive, I hope I get the opportunity to learn what's in there.

rodsika

Ken and Dr. Neumaier (herein called Arnold for short with no disrespect...
to put Ken and Arnold in equal footing without biased). Let's focus on this
collapse issue as it is the heart and soul of the measurement problem.

In decoherence. Born rule is not applied. The system coupling with environment
just puts it in mixed state. So we shouldn't technically call it collapse.
Collapse only occur when one eigenvalue is chosen.

Now in pure particle ontology as in vintage QM, where particle positions are the
primary issues. It is difficult how to imagine a single particle can interfere
by itself in the double slit. So we use the concept of superpositon and
collapse. But in QFT, there is no position, in QFT wave function amplitudes.
Its square magnitude has the interpretation of the
probability of finding the field with a certain field configuration. Now what
did Arnold do. He removes the idea of pure collapse. That is. In his view.

collapse = restricting to a subensemble
= replacing a probability by a conditional probability.

Is this valid at all? The following is Arnold complete statement about Collapse
and Quantum Measurement. It is just brief so please give it a thought Ken. How
do you think Arnold deal with definite outcome? You argued very strong in the
other thread that definite outcome can only be perceived by conscious being who
can make a record of the definite outcome because it is not in the equations.
What is the equivalent of definite outcome in the following. Or did it just go
away since the quantum field is the ontology and wave function collapse doesn't
even exist (hence nothing to worry about definite outcomes)? 　　

http://www.mat.univie.ac.at/~neum/ph...opics/collapse

Collapse and quantum measurement
--------------------------------
Experiments involving measurements are oftern interpreted in terms of
a collapse of the state of the system. However, they can be interpreted
without any collapse.

In particular, in photon experiments, the collapse interpretation is
never applicable since a measured photon stops existing rather than
collapsing into an eigenstate of the measured operator.

Instead, a collapse is just a change of the description level.
The moment one changes the description level, everything changes
everywhere instantaeously, without making the slightest change to
the underlying reality.

One has the same instantaneous change already on the classical level.
We can calculate the probability that a star is of a certain kind.
This probability depends, however, on what we consider to be the
relevant ensemble. If we change the ensemble by restricting to a
subensemble, the probability may change. And it does so throughout
the universe, instantaneously, just by making our subjective decision
to consider only the subensemble instead of the whole ensemble.
This is nothing special to physics, it is an experience of everyday
life. It is as simple as this:

(*) collapse = restricting to a subensemble
= replacing a probability by a conditional probability.

The mathematical justification of the equality (*) is easy to see
by considering only commuting observables, in which case quantum
mechanics reduces to classical probability theory. Now measure
just one of a complete set of commuting observables, and interpret
the resulting formula classically.

It is up to the subject making a study when she will switch
to the conditional probability, and has nothing to do with her
knowledge. But once the ensemble is replaced by a subensemble
(by conditioning with respect to a partial observation on Ann's
side of the system), the view changes instantly, since it happens
only in the subjects head -- Ann decided to remodel the situation,
and so it changes accordingly.

But as long as one keeps fixed what is the system considered, we
have objective physics to tell us what happens with the system,
as far as it can be told at all.
　
The objective state of a physical system is a state of the total
system considered, and not one of its many partial traces,
which only give the perspectives of local observers.
Of course, the partial trace is observer-dependent. The dependence
comes from the freedom of a subject to choose what it will consider
as the system.

This is the _only_ subjectivist element in physics. It is already
present in classical physics, where changing the (subjective)
coordinate system changes everything. There we are trained to know
that these subjective elements are to be ignored, and that what
counts is just the coordinate-independent part of physics.

We know that ordinary optical perspective is something subjective,
and we correct for that by developing a more general objective
framework of space in which each perspective has its place.
In this objective framework, perspective is seen to reduce space by
one dimension, hence hiding information that objectively exists and
can be modelled but is ignored by the view.

This reduction of a scene by viewing it in a particular perspective
is in complete analogy to the reduction in quantum mechanics, where
the choice of which subsystem to consider affects the resulting view.
　　
　

Ken G

Yes, to me "collapse" is not a physical thing happening to the system, it is a change in our description of the reality there, so I agree with much of Arnold's language there (I've even used the "collapse is a change in someone's head, not a change in a physical system" idea myself), but there is still a key question: which outcome? The "problem of collapse", then, is to me not "what physically happened to the system and how did it physically happen", but rather, "whatever caused our information about the system to change to outcome X instead of outcome Y." I'm not sure that question goes away when we view each observer as holding only a piece of the full objective reality, as we would in relativity, because we have no access to any observers who get outcome Y. In other words, the problem is somewhat the opposite as in relativity-- it relativity, our illusion is that we know more about reality than we can actually know, we think our time coordinate between two events separated from us is physically meaningful, but something less than that is actually physically meaningful. The problem with collapse is that something more than what we perceive appears to be the objective reality, or at least it would help our formal theories if that were true.

But Arnold raises the interesting example of perspective, where each observer sees a 2D window of a 3d objective reality. That's much more like the situation in quantum mechanics-- if we trust our perceptions, we only get part of the objective reality, we have to add something to it, using other observers, to get the full picture. But the disconnect I see is that in the perspective analogy, we have access to those other observers. We can ask them what they see. What is the analog to that in the collapse case? On the surface, it sounds like Arnold is saying that collapse is like the many-worlds interpretation, with the added element that the outcome we perceive is due to some choice we've made about what we regard as the system, or some choice that was thrust upon us if we did not make it consciously. But again, the real sticky part is that we have no access to the other observers who get outcome Y. This makes sense in many-worlds, since they are incoherent, but it's a stretch.

Let's imagine a many-worlds situation where somehow we can communicate with those other worlds. We ask those other observers what they see, and use their testimony to "flesh out" the full unitary state, the non-unitary "collapse" being just our own perspective on the situation. Now that would resolve everything, the interpretation of measurement would then be perfectly obvious. But why is that kind of communication impossible? What is that fact telling us that isn't there in the "perspective" analogy? That's what I see as the "collapse problem", and it doesn't seem to have gone away, though I'm not sure I'm understanding everything Arnold is saying here.

(ETA: cut out accidental inclusion.)

rodsika

　Arnold Thermal Interpretation has prediction that differs from the Standard
Interpretation. For example. Since the electrons existing already in the
detector are triggered by the impinging field and get its energy from the field.
One can design the source to send field with energy magnitude enough to trigger
more than than one electron in the detector. Can anyone propose this? First can
we create a buckyball (or using other molecules or objects) experiment such that
we know exactly how many are sent out and detected? Because if it is not equal
and let's say 5 sources sent out equals to 20 hits in the detector. Then Arnold
is right. But if 5 source equal 5 hits even though the energy of one source
field is enough to trigger a number of electrons, then Arnold is wrong. Let's
call this test Neumaier's Inequality (counterpart of Bell's Inequality.. lol).
If Neumaier's Inequality was violated. Then Neumaier is right and he gets a
ticket to Stockholm. The person who proposed the right experiment also gets
another ticket! Anyone wanna try?

A. Neumaier

No. It is only a new and more rational way to talk about the formal content of quantum mechanics. That's why it is called an interpretation and not a theory. Since all predictions come from the formal machinery, different ways of talking about the latter cannot change the predictive content.

A. Neumaier

Collapse occurs if and only if one chooses to ignore the environment in favor of a simpler description of the problem. Thus it is determined by the simpler, but more approximate framework chosen.
If the observer chooses the no collapse version, he must add the whole environment to the quantum system - exactly (else the neglected part constitutes a new environment). Thus the observer decides upon what to regard as the system to work with, and this choice triggers a (usually partial) collapse, due to everything neglected.
In QFT, there is a concept of position, but as a argument of the field operators, on the same footing as time. But the field has no p[osition - it has values at each position. Particle-like objects are portions of the field in which these field values are nonzero only in a small neighborhood of a particular position.
Yes. And the observation of this is always partial only. Hence the wave function never collapses fully. Thus collapse becomes a marginal phenomenon in QFT.
Yes. By focussing on a subsystem of the full system (which included the environment), the system is conditioned on the state of the environement (which contains the information about measurement).

Thus the probabilites are now conditioned by the observations. This is the same mechanism in classical and in quantum mechanics.

A. Neumaier

No. In the thermal interpretation, there is only one world. This one world is enough to explain everything that happens.

Collapse is the effect of ignoring the details of the interaction with the environment, keeping only an approximate summary in the form of a history of measurements and a POVM for the average influence of the environment upon measuring.

Even in the analogy with the 2D perspecticve, one doesn't necessarily need other observers, since one notices soon that the different views the single observer gets at different times can be coherently realted only by assuming a third dimension. Once this is realized, moving around is enough to gather rthe information needed to complete the 3D picture.

Therefore, in the thermal interpretation, there is no place anymore for mystery.

rodsika

But your Interpretation differs from the Standard in its prediction.
In your Interpretation. There is no Collapse. And the behavior in the double
slits can vary. For example. In Standard QM. When a Buckyball is emitted, always
one hit would be detected. But in your case, since it doesn't collapse and it is
alway a field, the energy of the buckyball is enough to trigger 5 or even 10
electrons at the detector. So your one Buckyball emission would result in 10 or
more hits due to the energy of the Buckyball field much more than an electron.
Here your interpretion obviously didn't have the same prediction as QM.

my_wan

Collapse is NOT a prediction of QM. It is an interpretation of QM. In requires you to assume the wavefunction is physically real. Is the configuration space of a dice roll real. Does the fact that it is not real make the dice not real?

Ken G

In a restricted way only. Because there is no deterministic explanation that leads to the actual outcome, that's my point-- you have given a way to understand why we get a single outcome, because of how we choose to regard the system (I've called that a role of conscious perception in that other thread), but not how. You have not explained why we get the outcome we get, since asserting that it is random, in an ontologically true sense, runs afoul of Einstein's celebrated complaint about god and dice. As this is the real "measurement problem", a key aspect of it remains, even as a part of it is resolved by noticing the role of the physicist. One can always say, a la Bohr, that we just don't get to know that part, and indeed that's my own feeling, but I wouldn't exactly call that a resolution, merely an acceptance of certain inherent limitations.

That is true, and is just what we'd have to do if there were no other observers, or if we could not trust their testimony. But there are other observers, and we have found we can (usually) trust them, so we need to build a physical ontology that respects these facts. This is related to the "why that outcome" question-- why does everyone we get to talk to agree on that outcome? It's the crucial symmetry principle of relativity, we either need an ontology for that, or we need to recognize we are going to view it as unknowable.

There's always a place for mystery-- I like to say that science is not about removing mystery, it is about replacing superficial mysteries with much more interesting and profound ones.

A. Neumaier

I had already mentioned earlier that what precisely happens when the buckyball field meets the screen must be calculated using statistical mechanics. The general principles suffice to say what happens for a single electron arriving but not for a single buckyball arriving.

And there is just as much collapse in my interpretation as statistical mechanics predicts. The literature contains a number of derivations of POVM's from statistical mechanics, and this captures all the observable features of (partial) collapse.

A. Neumaier

Which outcomes one gets is predictable by the standard machinery of nonequilibrium statistical mechanics. This discipline tells how to compute (at least in principle, and for QED also in practice) field expectations, i.e., the measurable outcomes according to the thermal interpretation, in an approximation sufficient for many purposes.
It is because we all observe the same world, hence (if properly trained) we draw the same conclusions about the stuff we observe. This classical explanation holds in the thermal interpretation also for the quantum domain.

A. Neumaier

Actually, in a scientifically defendable sense (though I haven't yet written it up), God does not play dice, but he acts on incredibly fast time scales, which to us appears as randomness.

rodsika

Ken, do you think Arnold explanation above solves how we get definite outcomes when the equations of physics don't allow them?

Ken G

It still leaves questions for me. The big one is, if we start with a simple system of one quantum in a definite spin state along one axis, and we use a macro instrument to measure its spin along an orthogonal axis, is there enough information in that system (perhaps including the instrument that prepared the quantum in the initial state) to determine what outcome we'll get, even though it is not practical to imagine we could ever have access to that information, or is it fundamentally necessary that we cannot have access to that information to get that probability distribution? If the latter, then how can we give meaning to information present that if we had access to it would not be present?

A. Neumaier

Yes. you get no response because the axis is orthogonal. Things get difficult only if the axes have a nontrivial angle.

In that case, since all macro predictions are made by statistical mechanics, I doubt that one can predict more than a statistics for the resulting event.. Whether it is possible in principle is a different matter - but to be ablre to do this would mean one know every quantum detail of the macro instrument, and whether this is even knowable is questionable.