Classical and Quantum Mechanics via Lie algebras


by A. Neumaier
Tags: algebras, classical, mechanics, quantum
A. Neumaier
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#127
Jun24-11, 08:25 AM
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Quote Quote by Ken G View Post
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,
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
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#128
Jun24-11, 08:48 AM
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Quote Quote by A. Neumaier View Post
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.
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
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#129
Jun24-11, 09:21 AM
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Quote Quote by rodsika View Post
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.
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
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#130
Jun24-11, 10:18 AM
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Quote Quote by A. Neumaier View Post
No. In the thermal interpretation, there is only one world. This one world is enough to explain everything that happens.
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.

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.
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.

Therefore, in the thermal interpretation, there is no place anymore for mystery.
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
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#131
Jun24-11, 10:59 AM
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Quote Quote by rodsika View Post
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.
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
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#132
Jun24-11, 11:06 AM
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Quote Quote by Ken G View Post
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.
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.
Quote Quote by Ken G View Post
I
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 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
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#133
Jun24-11, 02:40 PM
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Quote Quote by Ken G View Post
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.
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
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#134
Jun26-11, 06:08 PM
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Quote Quote by A. Neumaier View Post
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.
Ken, do you think Arnold explanation above solves how we get definite outcomes when the equations of physics don't allow them?
Ken G
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#135
Jun27-11, 01:53 AM
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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
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#136
Jun28-11, 07:01 AM
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Quote Quote by Ken G View Post
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,
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.
Ken G
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#137
Jun28-11, 07:57 AM
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Right, so my point is that if we say there is some unknowable detail that determines the outcome, then nothing science could ever do could distinguish that from something that is simply not determined. So determinism or indeterminism exits science and becomes a personal philosophical choice, and that is what I would call "the measurement problem." It just plain wouldn't qualify as a measurement if it didn't have that property, and that I think is the fundamental paradox/limitation/weirdness of physics. The thermal interpretation has not made it go away, but amazingly, it has relegated it to a completely classical problem. The measurement problem is no longer in some strange quantum/classical interface (a la Copenhagen) nor is it strictly in the quantum domain (a la deBroglie-Bohm), it is still perfectly classical, and applies just as well when we shuffle a deck and pick a card.


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