Classical and Quantum Mechanics via Lie algebras

A. Neumaier

Most of my book is independent of any interpretation, in the same way as an ordinary QM textbook. Therefore one can use it in a shut-up-and-calculate fashion. But it is also written in a way to make the analogy between the quantum world an the classical world as apparent as possible, resulting naturally in the thermal representation.

When the latter is applied to quantum fields, it yields the results discussed in this and related threads. But actually the current version of the book contains almost no field theory, as I haven't had the time yet to present the latter coherently.

gpran

I came to know about this concept of ‘Thermal Interpretation’ from the thread ‘Quantum Interpretation Poll (2011)’. I am writing this to get clarification about some of the basic concepts.

1) Please see the slide show: http://www.mat.univie.ac.at/~neum/ms/optslides.pdf. It mentions that the intensity of the beam is S0 = ψ*ψ. Does it mean that ψ*ψ gives classical intensity of the beam and not probability? I believe that probability is of statistical nature whereas intensity is real. May be, it is suggested that probability of finding a particle is more if intensity of beam is greater in a particular location. This is acceptable where we have large number of particles but what about a single particle?
2) The Schrödinger equation is obtained in the paper through a mathematical exercise. Can we say that the equation has been derived and not presented as a postulate? Is it because we are assuming a classical beam of particles for the derivation?
3) What is exact picture of a particle? If you suggest that a particle is like a beam or wavepacket then it is equally confusing or abstract. If a charged particle electron is like a beam then does it mean that the mass and charge are spread throughout the space? If there are two particles then the two beams may mix with each other leading to a bigger particle. For neutral particles like photons this is acceptable but for charged particles like electron this may not be acceptable. In widely accepted Q.M. interpretation, ψ is not real and therefore addition does not lead to a bigger particle.
4) I presume that there is no problem of wavefunction collapse in this approach. Is it because the theory assumes a classical beam of particles/photons?

I may be asking these basic questions because I have not really understood what is said in the slides. My problem is that I am trying to compare every statement made in the slides with the traditional interpretations taught in the text books. I feel that a short note/chart about the concept giving the differences with the presently accepted interpretations may help. I request help from any body who is working on this theory.

A. Neumaier

Everything in Section 1 is classical physics. Neither particles nor probabilities are involved, only the electromagnetic field.

You may read as background Chapters 2 and 6 of the book by Mandel & Wolf. (It has quantum optics in its title but the first 8 chapters are purely classical.)

Section 1 demonstrates that a simple quantum system, which is usually described in terms of particles and probabilities (and associated interpretation problems), can as well be described by a classical field (and was in fact so described, almost 50 years before Planck first suggested quantization), without losing anything in predictive value.

The remainder of the paper extends this equivalence to everything that can be done with a single photon.

However, entangled multiphoton states cannot be described by the classical electromagnetic field. But the thermal interpretation can be extended - though this is yet to be written up.
The derivation shows that with the assumptions and approximations made, the Schroedinger equation holds in the classical setting. Therefore it is derived, not assumed.

Assumed was only classical physics.
There is no exact picture of a particle, just as there is no exact picture of a cloud.

A particle is a localized field concentration that consistently behaves like a classical point at the length scales probed. Its boundary is a bit fuzzy but the fuzziness doesn't matter since it is below the scale of resolution of the description.
Confusing is to think particles are well-defined points. Real particles, no matter of which size, are extended objects with fuzzy boundaries. Point particles are unreal abstractions of real particles, obtained by deliberately ignoring detail in order to gain simplicity of the description.

In celestial mechanics, where the particle picture originated, stars and planets are particles. Where does the star or planet begin and end? One cannot tell - the atmosphere just gets thinner and thiner as one goes outward, and at some point its density is so small that one doesn't care anymore. Thus stars and planets are ill-defined as exact objects, but they are well-defined as a point for most practical purposes. Except for the planet Earth, which is too close to us observers to treat it as a point particle. Therefore we use a field description of the earth: At each point we know the composition and density of the materials.

In the quantum realm things are fully analogous. As long as we don't consider length scales comparable to its size, an atom or elementary particle behaves like a point - it is a particle. But once shorter scales become relevant (going through a narrow slit, say), the particle description becomes inappropriate and one needs more detail - provided by the field description,.
Yes. Just as the mass of the particle Earth considered in celestial mechanics is spread out throughout the space.
Not usually. They will pass each other, and occasionally, particles in the beams will scatter. It is uncommon that particles from different beams stick together.
No. The thermal interpretation is an interpretation of quantum systems, described by the usual shut-up-and-calculate attitute, but giving intuitive words so that one can open one's mouth without talking nonsense.

Collapse exists in a much-used approximation, namely to precisely the extent it is derivable from the standard methods of nonequilibrium statistical mechanics.

The thermal interpretation affects not the collapse but the way one interprets measurements. Measured directly are _not_ eigenvalues of operators, only expectations of macroscopic quantum fields.

But everything one can say about a microscopic system is obtained by inference from the way the microscopic system interacts with the observing macroscopic system according to the standard rules of quantum mechanics and statistical mechanics.
Since different people have very different questions about the thermal interpretation I can prepare such a note only after I have enough feedback from readers about what needs which sort of explanation. This is the main purpose of this discussion thread. (Well, for my whole book, not just for the thermal interpretation, though the latter seems to attract most of the interest here.)

Ultimately I'll write a properly published paper on the subject, giving a reasonably complete view of the thermal interpretation.

At present, simply ask about everything that you don't understand, and I'll do my best to explain.

my_wan

I am busy studying Arnold's work. I have not commented here because Arnold wants feedback from those having difficulty with his interpretation and I am quiet accustomed to thinking about QM in very similar terms. Though Arnold has certainly thought about it in many ways I have not. Perhaps me throwing in a perspective might help, maybe. Otherwise it can be refuted or ignored.

By the way, this thermal interpretation also extends to gravity. Such as outlined by Brustein and Hadad in "The Einstein equations for generalized theories of gravity and the thermodynamic relation δQ = T δS are equivalent", Phys.Rev.Lett.103:101301,2009, and others. Even Erik Verlinde, a string theorist, even created a stir with "On the Origin of Gravity and the Laws of Newton", JHEP 1104:029,2011, describing gravity as an entropic force. I suspect that the connection may run much deeper than mere interpretation can fully justify.

It seems to me that most of the confusion is primarily generated by various levels of conflation between theory and interpretation, which are woefully different beast. The remainder appear to be mostly trying to visualize a mass particle as a group of particles traveling through an otherwise empty space. Even a classical wave cannot be described this way, as there are no distinct set of particles traversing a gas to convey sound. Thinking of a mass particle as a distinct group of parts is equally bogus in this thermodynamic interpretation.

To get the interpretive picture forget the particles and look at the definition of a Hilbert space. Now simply assume this Hilbert space is ontologically real and extends throughout all space like air extends throughout an atmosphere. Now consider the wave function, but instead of defining it as a probability think of it as a variational density change in Hilbert space. Much like sound is a variational density change in a gas. At times the density variations can be highly localized. Much like a classical soliton can. In such cases we can refer to that soliton a distinct entity, just like we refer to a tornado as a distinct entity even though fundamentally it is not, and is not even defined by a distinct set molecules. Likewise for a mass particle in this interpretation. Then when you create a situation with many such particles interacting, density variations (not probabilities), which particle is which becomes an ill defined concept. Like asking which wave is which on a choppy ocean. The difference is the medium in this case is defined by an ontologically real Hilbert space with somewhat different properties than a classical medium. Only it still shares the same basic thermodynamics under the degrees of freedom provided by the Hilbert space. Perhaps, maybe for some, that will give some basic context under which to conceptualize the interpretation. Arnold can take exception to any point he sees fit, and/or consider the general reaction to it.

Arnold, have you looked at the phenomena of "ghost interference"? This fits well into this interpretation and might possibly provide a way to measure the energy associated with the total wavefunction itself. This would allow us to study conservation laws as it applies to the wavefunction as a whole. Of course it also provides an interpretation of virtual particle production and associated momentum fluctuations, interaction free measurements, etc.

I cannot claim this is a perfectly valid interpretation, but nothing I have seen refutes it and that is all that is required so long as it is characterized simply as an interpretation. In fact, given that it is empirically predicated on a standard Hilbert space, it is essentially by definition very difficult to refute.

A. Neumaier

Actually I am interested in all sorts of feedback that helps me to give a better exposition of everything I did in this direction.
Yes. It may well turn out that gravitation is a pure thermodynamic effect. But in my book and lectures I am sticking to the most solidly accepted part of quantum mechanics, to avoid any unnecessary friction.
This was Schroedinger's idea, but turned out to be not realizable as the dimensions are vastly different. In the thermal interpretation, the ontological status of beables is given to the field expectations, which are true fields in spacetime rather than objects in a high-dimensional space. This is the improvement upon Schroedinger and the reason why everything works neatly and intuitively.
I never heard of this term. Could you please provide a reference?

my_wan

Yes, well understood and I concur. It is obvious that you are not explicitly trying imply anything outside the standard model, rather just express it from a particular context. I just threw this in as an extension, more or less as an afterthought, to illustrate that the interpretation could potentially run deeper than what you intend convey.

This is where your thinking on the subject appears superior to mine. I was well aware that that once you tried to take the analogies with ontological parts of the field, rather than the field itself, too seriously it runs into very distinct problems. The expectation values in QM are simply NOT the positions and momentums of distinct parts the way they are in classical physics. I was not trying to suggest in the analogy provided held in the particulate sense. Only that in defining a mass particle in terms of a deformable field the apparently localized structure is really no more distinct than a wave is in classical physics. I am still trying to work through the details of precisely how you deal with "field expectations" in an ontological sense. Just because it makes sense in a general way is no garrantee of a lack of incongruencies, but I have nothing to indicate specific incongruencies as yet.


Here is one from Phys. Rev. A 54, 1996, "Direct and ghost interference in double-slit experiments with coincidence measurements".

This phenomena has also been used in "ghost imaging", which allows a camera to take a picture of something the camera cannot see. This is also used in single-pixel detector setups and it is argued by some that this is evidence that it does not depend on non-local quantum correlations.
http://arxiv.org/abs/0812.2633

I am also very curious about this experiment showing interference in uncorrelated separable lights sources, which is given in the context of cross beam experiments mentioned in the paper.
http://arxiv.org/abs/physics/0504166
Though I do not know how much weight to put on these results.

my_wan

Now I think I have a more complete picture of your interpretation. I went back to some of your earlier work. Primarily:
Ensembles and experiments in classical and quantum physics

You place no judgment at all on the noncommutativity of QM, other than as an empirical fact, and conceptually work with the "expectation values" as fluid properties in the classical thermodynamic sense. Thus Hilbert space remains a separate construct in an ontological sense with no specific ontological status assigned directly to it. That certainly does escape many classical issues while still maintaining a direct and unmodified formal transition from one to the other. The difficulty it appears then is making the point when people are so accustomed to assigning distinct empirical properties to distinct points in space.

How do you deal with the conservation issue with wave cancellations? In effect it boils down to, if two quantum waves overlap so as to cancel what happened to the energy associated with those waves? If they simply become non-existent there appears to be a conservation violation. Dirac got around this by simply assuming particles could only self-interact, hence they did not really disappear.

This self-interaction hypothesis is, however, dubious.
Ghenadie N. Mardari, "What is a quantum really like?" AIP Conf. Proc. 810 (2006), 360.

Ken G

I'm only just beginning to be introduced to the thermal approach, but what I've seen so far shows promise. On another thread, I was grappling with the question of how we should regard "what a classical apparatus knows about itself", in a sense. The usual interpretation is that the projection onto the measuring device is a mixed state, and further, that a mixed state is "in a definite state we just don't know which." Then when we look, it merely confirms what was true before we looked. Although this runs into no problems in experience, it does not seem to be strictly true to our own theory-- our own theory tells us that a mixed state is just a mixed state, and not a definite state that we just don't know which. The latter conjures the concept of a probability distribution, but the former seems more inherently "fuzzy" to me. It seems your language offers the possibility of putting that difference on a firmer basis.

If you consider the subtext of what I'm saying, I suggesting that maybe you can take your idea even farther: out of the nebulous quantum realm, where "anything goes" pretty much, and into the well-worn classical realm, where surely there are no new surprises. But the status of a "mixed state" was always a bit nebulous in the classical realm too-- we say that the air in this room, treated classically, is in a definite state "we just don't know which one", but how do we really know that this is what the classical theory asserted? There is no instrument or perceptive agent anywhere that has the power to tell the definite state of the air in the room, so on what basis did we claim there was such a state?

On the other hand, if I shuffle a deck of cards, I might struggle with wondering if every card is in a definite micostate of internal particles, but I don't have difficulty asserting that the order of the cards is definite, even before the cards are looked at. This conforms to our tests, because we can objectively determine the order of the cards. So does the concept of "resolution" come up here too, is the status of a deck of cards really something different, from an information theory standpoint, than the microstate of the air in this room? Was there fuzziness in classical physics that we just never noticed?

A. Neumaier

In the thermal interpretation, the classical properties are manifest as the expectations of the field operators. This matches naturally with a hydrodynamical description of classical matter. Thus the problem of identifying the classical properties simply vanishes. Statistical mechnaics guarantes that the fields are measurable in a coarse-grained sense, because of being in a mixed (thermal) state.
One doesn't need a fully precise description, since the obsefvation of a field is itself necessarily coarse-grained. It is enough that the description matches the actually possible resolution. In the thermal interpretation, this is guaranteed by the standard results from statistical mechnaics.

A. Neumaier

In quantum field theory, expectations of the field operators also apply to any point in space, and indeed one gets from the thermal interpretation very naturally the hydrodynamical description of classical matter, with definite properties at every point. Thus there is no such difficulty, once one thinks in terms of quantum fields.
The situation is not really different from that with a classical Maxwell field, where waves can destructively interfere.

Ken G

Yes, that is an attractive feature. Some people are left unsatisfied by that, they want an ontology that is crisp, but then they face questions that cannot be answered satisfactorily. Your approach loosens the ontology, but the questions evaporate. I call that a good trade, but others seem to prefer to keep the questions instead. That way all they have to do is sweep the questions under the rug and they have the best of all possible worlds, though it is something of a mild self-delusion I would say.
And that's the heart of it right there-- when is that enough, and when is it not enough. To me, it raises the same issue that the Copenhagen interpretation raises: which is paramount, our ontologies or our epistemologies? Are we trying to know what is, or are we trying to fit our images of what is to what we can know? It sounds like you are siding with Copenhagen and taking that latter approach, and that is one of the things I like about your approach. You also seem to be more specific about things that Copenhagen is willing to leave uncharacterized.

Ken G

It's interesting that you say this, because on several other threads about weak measurements and Bohm interpretations and so on, I've been trying hard to draw the parallels between the quantum and classical pictures. I'm finding many people are unwilling to consider those kinds of parallels-- I even had one person tell me I was embarrassing myself by trying to point them out! There's a kind of myth that "quantum is quantum and classical is classical and never the twain shall meet." I'm not sure where that thinking comes from, it seems to completely ignore the correspondence principle, but maybe it's because educators have had to stress "quantum weirdness" in order to get students interested in that nether world. If so, they may have succeeded too well!

rodsika

Dear Ken, can't you see anything wrong with the thermal interpretation. Dr.
Neumaier was basically claiming that when a 430 atom molecule in the form of
buckyball was sent to the double slit. The slits literally slit the
buckyball into hundreds or thousands of pieces and spatter them across the
detector. And since a detector is consist of millions of electrons. One of these
get triggered and we erroneously thought this one triggered was the location of
the original buckyball when it was just a part of it. This was possible because
according to him, the buckyball being emitted was not a particle to start with
but a field which is undefined. As a more distinct example in case you haven't
grasped the basic of this interpretation. It's like if you sent a cow to the
double slit. It slits the cow into dozens of pieces. When say the kidney hits one of the existing electrons in the detector. We thought the cow is located in that electron
detector position. Dr. Neumaier reasoning this was possible was because the cow
was a field to start with. Now with all our experimental might. Can't we test
this outrageous claim of Dr. Neumaier, or recalling all your knowledge as full
fledge physicist.. can you think of a way to*scrutinize it.*If you meet your
fellow physicists in the lab. Please ask if they can think of a way to test Dr.
Neumaier conjecture and whether there wasn't already existing test(s) that might
have already refuted it that we might not be aware of.. such as a test that
established the particle nature of matter in an absolute way. Thank you.

Ken G

It might be putting words in his mouth, but if I trust that your rendition is accurate, I would say that I do see that as a potentially valid picture, even if a bit bizarre at first look.

If I understand the perspective, he might say that the buckeyball isn't really a buckeyball in the first place, it is a field that we have labeled a buckeyball because when we have lots of it we have lots of buckeyballs, and when we get just one, we assume there was already one there, but we don't really know what was already there, it's just kind of an assumption on our part. We assert its existence and find no contradiction, but that's not the same as saying we know it existed, if there isn't really anything there called a "real buckeyball" in the first place. To be honest, I'm rather sympathetic to that approach, because I like keeping careful track of what we know versus what we are just assuming we know.
That doesn't sound like a fair characterization, because a kidney is a distinguishably different piece of a cow, whereas the "buckeyball field" he is talking about doesn't have distinguishably different parts like that (we are not actually breaking the bonds in there, after all).

I'm pretty sure Dr. Neumaier's approach is designed to make all the same predictions as more typical modes of thought, so we already know it's not going to be testable. Instead, the question is, does this mindset make certain troubling questions go away, or does it build a picture of "what is" that we find repugnant in some way? I'm afraid questions like that are always going to be matters of personal taste, but I do see a certain plausibility in the idea that there really is no such thing as "a cow." A better ontology might assert the existence of fields of attributes of various systems or combinations of systems that when you put them altogether you end up with something we recognize as a cow by virtue of gross similarities of behavior and appearance to our mental image of what a cow is.

rodsika

No we are not talking about an Ensemble. But a single buckyball at a time double
slit experiment. Dr. Neumaier said that after the buckyball was emitted. The
slits slit the field to various fragments and these hit the detector in all
regions. Since his field is literal with left and middle and right portion
maintained. After it passes thru the slits. The left field would focus on the
left, middle field on middle and right field on the right although diffraction
and interference would also produce constructive and destructive inteference.
Let's analyze just using single buckyball experiment.. let's forget ensemble as
we are scrutinizing what happens in single buckyball emission and detection. If
there is a test that can show a single buckyball still found at the detector.
Then this would refute Dr. Neumaier conjecture. Can you think of a test or other
experiment setup which doesn't use electrons as detection elements? Again Dr.
Neumaier arguments was that a detector is composed of millions of electrons as
detection elements. So a smeared splattered field can trigger just one of them
because after the one was triggered, it would use up all available energies in
the detection circuits with the rest of the electrons in passive mode unable to
fire. So if you can think of a way that we can detect the buckyball or photon
without using electrons. Then his conjecture can be put to experiment test and
be falsifiable. If Dr. Neumaier is right. Then the measurement problem was
solved and we can mention this in all physics textbook from hereon and he become
immortalized in the Physics Hall of Fame in the company of Einstein and Bohr.

Ken G

But that's just it, how do you know you have a single buckeyball in the first place? All you have is a way of emitting a flux of buckeyballs that is emitting them rather rarely, but you have no idea at what point you "actually have a buckeyball." You only know when you detect them. If there never really was a buckeyball in there, just some kind of field we are interpreting as a buckeyball because it produces buckeyball detections, how are you going to say you had a single buckeyball in there? You are starting with assumptions that are not used in the thermal approach, and you cannot show your assumptions are correct, you can only show they work for you-- that doesn't mean one cannot get similar successes without making those assumptions. As I said, I'm pretty sure we are talking about all the same predicted outcomes, so what can we really be talking about here other than the assumptions we make?

Yes, the approach takes the field very literally, granting it a reality in spacetime. That's the part I'm not crazy about, I don't see any particular reason to grant a physical ontology to the fields and not the particles, or to the particles and not the fields. I think they're all mental constructs. But that doesn't make me right and him wrong.

Sure, use buckeyballs. But we already know what you'll see, the experiments have been done. We're just trying to figure out how to talk about what we are seeing.

rodsika

　
We can use electron microscope to view a 430-atom buckyball right, let's say it
is 5 nanometer in diameter? So after we viewed one. We send it out in the
emitter, here we know one buckyball is sent out and let's say we only do this
once.. no more second buckyball. A double slit should just nudge the position of
it after the slit. And we should still find the buckyball in the detector if we
tried to find it. Dr. Neumaier was claiming that the buckyball can no longer be
found.. that is.. the buckyball was no longer the original 5 nanometer size but
it is literally fragmentalized into different components much like a grenade and
these field is splattered all over the detector. And you are saying we can't
even test this out?


Done? Can they modify the setup so the detector elements aren't electrons but
something else?

Don't worry if anything you mention can refute Dr. Neumaier. I don't think he
would excommunicate anyone who has falsified him. He is humble enough that he
may even acknowledge that person. Although it is true other Ph.D.s would be
angry, and shun that person in his circle. Also he is living in Austra so world
away from the United States Academic Circle.