I Quantum mechanics is not weird, unless presented as such

[A. Neumaier]

I understand how bistable potentials can be similar in some respects, but I don't think that works for distant correlations such as EPR.

The mathematics of projection operators does not distinguish between a tensor product of two qubits very close to each other and two qubits very far apart. It doesn't distinguish between whether a system is described only by diagonal density operators (classical deterministic or stochastic system) or by nondiagonal ones (quantum deterministic or stochastic system).Both together are enough to expect that it will work as well for long-distance entangled states of qubits as for classical multistable states, in both cases reproducing the expectations of the corresponding theories.

The detailed predictions are of course different since the dynamics is different. But the statistical principle underlying both is exactly the same (projection operators - same abstract formulas!) and the resulting qualitative dynamical principles (dissipation leads under the correct conditions to discrete limiting states, and they are achieved in a fashion following an exponential law in time) are also precisely the same. Moreover there are already statistical mechanics investigations (such as the 160 page paper I had referred to) that show that the microscopic and the macroscopic are consistent., roughly in the way I discuss.

Thus I (the professional mathematician who has many years of experience in how to build correct intuition about how to qualitatively relate different instances of a common mathematical scheme) don't have any doubt that the details will work out as well when pursued with the required persistence. It would be mathematically weird if it didn't work out. Of course, this is no proof, and occasionally mathematics produces weird truths. So there is merit in doing a detailed model calculation. But as any new detailed application of statistical mechanics to a not completely toy situation is a research project that can easily take the dimensions of a PhD thesis I haven't done yet such a model calculation, and don't know when I'll find the leisure to do it. (I have a full professor's share of work to do in mathematics, and do all physics in my spare time,)

So yes, I agree that detailed calculations are desirable and would give additional insight in the mechanism. But even without this detailed calculations, the nature of the mathematics is of the kind that leads me to expect that nothing surprising (i.e., deviating from the expected results outlined by me) would come out.

Thus you may view my scenario outlined in that part of this discussion centering around the density matrix as a conjecture well supported by qualitative arguments as well as analogies drawn from detailed studies of related problems. Let us postpone the question of the actual validity of the conjecture until someone with enough time has taken up the challenge and wrote a thesis about it.

 

[Bruno81]

There is and always will be a difference between a qualified matematician and a qualified physicist. This thread is a testament that they are in different leagues.

 

[kith]

The central idea of your thread is that the apparent weirdness lies in the fact that people talk about QM in the wrong way and that we can reduce it by changing the way we talk about QM. In your book, you try to present the mathematics of QM and classical mechanics as close as possible.

What do you think about changing the way we talk about classical mechanics? Because the apparent weirdness of QM would also be reduced if we identified preconceived notions which aren't justified by the mathematics in the way we talk about classical mechanics.

 

[A. Neumaier]

What do you think about changing the way we talk about classical mechanics?

Talk about deterministic classical mechanics needs little change, as it leads to few conceptual problems. One must only avoid the use of the notion of point particles in the context of fields, and realize that particles in classical mechanics are in reality also extended. But in the approximation where particles can be treated as rigid impenetrable spheres and the field they generate can be neglected, one can perform a valid point particle limit and hence has a good justification of the point particle picture. The main use of the latter is the great simplification it brings to theory and computations.

On the other hand, traditional thinking in classical statistical mechanics needs some change. The concept of probability (and the associated ensembles) is philosophically thorny, and the concept of indistinguishable particles flies in the face of true classical thinking, though it is necessary to get the correct statistics. In my book I try to minimize the impact of both by emphasizing expectation rather than probability. The latter then appears as a derived concept in the spirit of Whittle's nice book, rather than as a basic entity.

Did you have any other things in mind?

 

[A. Neumaier]

Since I got no satisfactory amendment to stevendaryl's setting in post #234, I started a new thread with my own abstraction of his setting. I'll discuss on a more technical level interesting variants of the experiments fitting my abstraction in this new thread, and would like to have the current thread reserved for informal discussion of quantum weirdness.

 

[crastinus]

What do you all mean by "weird"? Do you mean counterintuitive? Or inexplicably bizarre? Do you mean "does not fit with how we normally think of the world"?

I think you mean that it seems bizarre and inexplicable that the basic physical processes should be statistical, indeterminate, and with so little analogy to interactions on the classical scale. One can get all of classical mech starting from pushes and pulls and notions like longer than, as B. Hartmann argues at http://arxiv.org/pdf/1307.0499.pdf. As far as I know, this can't be down with QM. That's what I think you mean by "weirdness".

 

[A. Neumaier]

What do you all mean by "weird"?

It is the widespread impression that something is deeply unsatisfactory in the foundations of quantum mechanics. For example,

I find it weird for QM to split things into the three parts: (1) Preparation procedures, (2) Unitary evolution, (3) Measurements. At some level, (1) and (3) are just complicated physical processes, so that should be included in (2).

When people say that the problem in understanding QM is because it is too far removed from human experience and human intuition, I don't agree. To me, what's weird is the parts (1) and (3) above, and what's weird about them is that they seem much too tightly tied to human actions (or to humanly comprehensible actions). Nature does not have preparation procedures and measurements, so it's weird for those to appear in a fundamental theory.

It seems to me that the various ways of explaining away the mystery of QM is akin to trying to prove to somebody that a Mobius strip is actually a cylinder. You point to one section of the strip, and say: "There's no twist in this section." You point to another section of the strip, and say: "There's no twist in this section, either." Then after going over every section, you conclude: "Clearly, there are no twists anywhere. It's a cylinder." The fact that it's twisted is a nonlocal property, you can always remove the twist from anyone section.

 

[Cosmology2015]

First of all congratulations on your work! I recently graduated in electrical engineering and I look forward to studying the two pillars of physics: quantum mechanics and general relativity. It is possible to already start in these disciplines? Regarding quantum mechanics, perhaps I am not the best person to argue about this for not having a desirable knowledge in this area, but it seems to me that the problem with quantum mechanics would not be dealing with a weird approach but with an inconsistent approach. There are two sides of quantum mechanics, the Schrödinger equation and the act of performing a measurement, and these are incompatible. What would be your view on this aspect? I would like to thank you for any response .

 

[ddd123]

There are two sides of quantum mechanics, the Schrödinger equation and the act of performing a measurement, and these are incompatible. What would be your view on this aspect? I would like to thank you for any response .

If "measuring" didn't end up with superluminal influences it would be far less weird and something could be worked out. There are other weird aspects in QM but the EPR physics is the cornerstone of weirdness, without it you could work something out to fix the other aspects.

Maybe an exception is "why the discrete chunks", as in the double slit. If, as Neumaier says, we take fields to be fundamental, why should we get clicks in detectors / why should a detector absorb the whole quantum of energy in one go. Or, if a field quantum takes the form of a spherical wave originating from a source, how can the whole energy distributed in this way end up in one single detector at a certain direction. That's weird too.

 

[A. Neumaier]

the Schrödinger equation and the act of performing a measurement, and these are incompatible.

The latter is an approximation of the former, when one approximates a big system consisting of a small system and a detector by a dynamics for the small system only, combined with conditioning with respect to the result of the experiment.

Quantum mechanics is fully consistent (except for fine points in the construction of relativistic quantum field theory). The inconsistency is in its interpretation only, since the latter is always dominated by imprecise and subjective talk.

To start with quantum mechanics without the usual introduction to a mystery cult, you may try my book mentioned in #2 of this thread. (There the discussion of the mysteries is delayed until Chapter 10, where they are demystified.)

 

[A. Neumaier]

why should we get clicks in detectors / why should a detector absorb the whole quantum of energy in one go

Because of the bistable electrons that make up the detector. An electron cannot fly away to 11.578 percent - either it flies or it doesn't. If it does, it takes away a whole quantum of energy.

if a field quantum takes the form of a spherical wave originating from a source, how can the whole energy distributed in this way end up in one single detector at a certain direction.

That's discussed in a famous paper by Mott.

 

[Feeble Wonk]

Why is material existence absent when there is a mass density? Classically, in classical elasticity theory (which governs the behavior of all solids of our ordinary experience) and hydrodynamics (which governs the behavior of all liquids and gases of our ordinary experience), all you have about material existence is the mass density - unless you go into the microscopic domain where classical descriptions are not applicable.

I've been rolling this around in my head, and I'm still somewhat unclear about a couple of things. I think much of this is due to semantic ambiguity. I'm hoping that you could clarify two points that might help me understand.
1.)Would you mind trying to give me your definition of the term "material"?
2.)Having dismissed the concept of material particles, while viewing the quantum field as being ontologically material, should I interpret that to mean that you view the entire universe as a singular material object?

 

[A. Neumaier]

Would you mind trying to give me your definition of the term "material"?

You had introduced the term, and I used it more or less in the sense that you seemed to use it - physical existence of something you can feel or touch. (Thus excluding the massless and invisible electromagnetic field, which can be said to have physical but not material existence.)

In quantum field theory, the fields exist everywhere, but where they have zero (or small enough) mass or energy density they have no physical effect and are considered absent. For example, the solar system has an appreciable mass density concentrated on a limited number of bodies only (the Sun, the planets, asteroids, comets, and space-crafts and their debris), but additional tiny mass distribution in interplanetary space.

you view the entire universe as a singular material object?

I view the universe as a single physical object (why else does it have a name?) composed of many material and nonmaterial parts. The material parts are called galaxies, stars, planets, houses, bricks, cells, molecules, atoms, quarks, etc., the nonmaterial parts are called light, electric fields, magnetic fields, gravitational fields, etc..

So the universe has a density matrix, and by restriction one can get from it the density matrix of arbitrary parts of it (given a sufficiently well-defined operational definition of which part is meant). For example, one can look at the density matrix of the Sun, the Earth, or the gravitational field in between. Or of a beam of particles, or a detector, or the current Queen of England. (Well, in the last two cases, there will be some ambiguity concerning precisely which part of the universe belongs to the object. But strictly speaking, we have this problem already for objects like the Earth or the Sun, where the atmosphere gets thinner and thinner and one has to make an arbitrary cutoff.)

But one cannot consider the density matrix of Schroedinger's cat, since it is not a well-defined part of the physical universe.

 

[Feeble Wonk]

...physical existence of something you can feel or touch (Thus excluding the massless and invisible electromagnetic field, which can be said to have physical but not material existence.)

I don't intend to be trivially argumentative, but one can certainly feel an electromagnetic field. In fact, even in using the symbolic "particle" concept, the oppositely charged particles typically don't actually touch because of the repulsive force of the field.

I assume that you are actually equating the degree of materiality to the content of mass. Yes?

 

[A. Neumaier]

you are actually equating the degree of materiality to the content of mass. Yes?

Yes. At least for the purpose of this discussion I take this as the definition of the word material. It corresponds fairly well to its meaning in ordinary life.

 

[ProfChuck]

Does quantum mechanics have to be weird?

It sells much better to the general public if it is presented that way, and there is a long history of proceeding that way.

But in fact it is an obstacle for everyone who wants to truly understand quantum mechanics, and to physics students who have to unlearn what they were told as laypersons.

Quantum physics provides a more comprehensive view of reality than does classical physics. Classical physics, which includes both special and general relativity, are very accurate approximations of the behavior of "real" physical systems. However, the determinism of classical physics is an illusion that results from the aggregate behavior of vast numbers of individual quanta. It is all a matter of the statistical behavior of very large numbers of samples. It is recognized that classical physics is incomplete at the microscopic and probably also at the macroscopic scale. This suggests that unification of classical and quantum physics may require a rethinking of what we mean by "classical" phenomena.

 

[A. Neumaier]

unification of classical and quantum physics may require a rethinking of what we mean by "classical" phenomena.

Did you also read post #2 in this thread and look at my book? It achieves unification by instead rethinking of what we mean by "quantum" phenomena.

 

[crastinus]

I looked at your book. Fascinating stuff. It's actually kind of exciting, in my opinion.

Since you wanted this thread reserved for the informal discussion of quantum weirdness, is it possible for you to give a sketch of the unified vision of classical-quantum-statistical mechanics your are proposing? I've seen a few posts that do parts of it (Alice and Bob's ignored detailed have profound effects on measurement, etc.), but I don't have quite an adequate idea of how you actually carry the whole thing out. (Maybe I just missed a post, though.)

 

[ddd123]

From the other thread.

"still exhibiting a remarkable variation in foundational concepts among practicing physicists." That makes weirdness not a personal issue any more, but rather something that can be observed across a community of scientists, which is what I think is good about it.

Something like that. But also reconciling QM with physics' "tradition" of pursuing consistency and generality through falsifiable experiments. This seems to have run into a standstill. You can still do things and advance (a lot!) but there's an underlying feeling of incompleteness.

Maybe a way to put it is this: the very fact that nature's behavior was so peculiar as to break this game is weird.

Still, to play devil's advocate to that version of "weirdness", I would point out that classical mechanics supports many different interpretations as well-- is it forces, is it a principle of least action, or is it just the macroscopic correspondence of quantum mechanics?

There's also open problems like the Landau pole, the pre-acceleration and runaway solutions... But somehow these don't seem to bother people much. Newtonian vs. least action seem to be treated as mutually inclusive, not exclusive. You can derive one from the other. QM interpretations are strongly mutually exclusive. Either there's one world or many, either there's a principle of relativity or not, etc...

 

[rkastner]

Since the author of the thread has offered a reference to his interpretation in book form, I'll do the same and offer my interpretation in book form:

www.cambridge.org/9780521764155 (has some technical chapters)

http://www.worldscientific.com/worldscibooks/10.1142/p993 (for the general reader, extends some of the concepts presented in the first book)

It provides physical referents for the formal objects appearing in QM, including the Born Rule, which has been an ad hoc recipe for calculating empirical predictions from the theory. The interpretation also removes some of the 'weirdness' by resolving a major aspect of the measurement problem--specifying what constitutes the measurement transition from a pure to mixed state.

 

[mikeyork]

I'm late to this thread and don't have time to read it all. So, for what it is worth, QM is weird only to those who can't separate reality from observability.

 

[A. Neumaier]

offer my interpretation in book form:

It it available without a paywall?

 

[A. Neumaier]

is it possible for you to give a sketch of the unified vision of classical-quantum-statistical mechanics your are proposing?

You can find something in my Thermal interpretation FAQ and in Chapter 10 of my online book mentioned in post #2. I plan to write an Insight article here on PF covering the main aspects, but haven't found yet the time for it.

 

[crastinus]

Great. Close enough to what I was looking for. Thanks!

 

[Ken G]

Newtonian vs. least action seem to be treated as mutually inclusive, not exclusive. You can derive one from the other. QM interpretations are strongly mutually exclusive. Either there's one world or many, either there's a principle of relativity or not, etc...

Yet since we can regard all of classical mechanics as emergent from quantum mechanics in the macroscopic limit, classical mechanics automatically inherits all the same interpretations as quantum mechanics does. I agree with you that this doesn't seem to bother people much, but that may be only because it has been around for so long that everyone has kind of settled on a local realist view, which ironically, is not allowed in quantum mechanics. That is probably closest to taking the deBroglie-Bohm interpretation, in the limit where macroscopic decoherences in the pilot wave allow its locality violations to be eliminated or ignored. But we could equally well extrapolate into the classical realm the instrumentalism of the Copenhagen interpretation, we would just say that all that exists for us is the outcome of our experiments, and everything else is a kind of mental fabrication. Or, we could equally well extrapolate the many worlds view, and say that any time something occurs that is fundamentally unpredictable, all outcomes occur, but we are only privy to one of them. These interpretations work fine in classical physics, they are simply not the ones adopted by the majority. So I agree with you that what sets quantum interpretations apart is the absence of widespread agreement on the best one, but I wonder if classical physics did not go through a similar phase, long forgotten. Perhaps the physics itself does not get more or less weird as our discoveries advance, we just get used to it and settle on a majority opinion.

 

[ddd123]

If QM is still the culprit, it's a circular argument. But maybe you mean something like: we could have chosen to believe in the Lorentz Ether instead. But since we have a consensus on which was better, SR or LET, we chose mostly unanimously because the latter feels ad-hoc. Here, we have no better alternative, everything feels ad-hoc and you're just shifting the problem.

 

[Ken G]

I'm saying that I suspect eventually a single interpretation of quantum mechanics will rise to the fore, and quantum mechanics will always be framed according to that interpretation, as happened eventually to classical mechanics. Once that happens, it won't seem weird any more, even if it's many worlds-- because that's what people will learn to be the case. Once we all accept something to be the case, it never continues to seem weird-- we now think the universe had an origin, and that's not supposed to be weird, we think time is local and able to be itself dynamical, and that's not supposed to be weird, we think that past determines future, and that's not supposed to be weird, but somehow we get all hung up on where the random elements of quantum mechanics come from-- and that is supposed to be weird. Once an interpretation is settled on, it will be like all the other weird things we've just come to terms with.

I'll give you a prime example of what I mean, and it also comes from quantum mechanics. We often conceptualize white dwarf stars as containing a sea of some 10⁵⁷ identical electrons, and if we like, we can imagine they each occupy a different momentum state (or a pair of electrons, to keep track of spin). Because of the Pauli exclusion principle, this means that an electron undergoing an interaction at one place in that giant star cannot be put into a momentum state that another electron is already in-- even though we don't know where the electron that is already in that state is, we only know it is within several thousand kilometers! There's the grandaddy of all entanglement phenomena right there, a phenomenon that totally shatters the concept of local realism, yet no one even talks about it as strange at all because we've come to terms with it-- we view a white dwarf like a huge molecule, and we know the electrons occupy orbitals, and we're all just fine with it. The weirdness just goes away when there is a common interpretation.

 

[ddd123]

But the winning interpretation emerges for a reason, not arbitrarily, and it's clearly not going to be MWI.

 

[A. Neumaier]

Once an interpretation is settled on, it will be like all the other weird things we've just come to terms with.

The problem with your view is that even after 90 years of quantum mechanics, none of the conventional interpretations looks convincing enough to ''be settled on'' by an overwhelming majority. Why should this ever change?

 

[Ken G]

The problem with your view is that even after 90 years of quantum mechanics, none of the conventional interpretations looks convincing enough to ''be settled on'' by an overwhelming majority. Why should this ever change?

I think you'll find that the proper interpretation of Newtonian mechanics was still very much a matter of discussion 90 years after Newton-- especially in regard to the determinism of that theory. Even today, people don't quite know what to make of chaotic orbits-- in formal mathematical terms, those are still deterministic, but no science experiment could ever establish that they are indeed determined by the initial conditions. Hence, even if quantum mechanics were never necessary, we would still need to debate whether or not classical physics is telling us that the universe is deterministic. Even today, had classical mechanics been the "last word", I wager that a significant fraction of physicists would still regard it as "weird" if the conditions ten seconds after the Big Bang determine, to the minutest detail including what you will have for breakfast tomorrow, everything that has happened since. Weirdness doesn't go away, we just stop thinking about it after awhile.

 

[ddd123]

But you wouldn't have a plethora of conflicting interpretations.

 

[Ken G]

I'm saying we would have a plethora of conflicting classical-mechanics interpretations (such as, whether or not the universe is deterministic), had classical mechanics not been superceded by quantum mechanics. It is always the most fundamental theory that we try to interpret, because we think that's where the "fundamental interpretation" lives. It's just the nature of the beast, it has always been that way in physics. Action at a distance, or not? Material particles, or fields? Aether, or no? GR or QM? At any point in the history of physics, if you want to find where the debate on interpretations was, just look at whatever was regarded as the most fundamental theory of that age.

 

[ddd123]

Fair enough. It's notable though that the pre-quantum mood was that of "we've mostly figured it out", then came the radical crisis of quantum discoveries. Although it was debatable even then, say, determinism wasn't really menaced because those you mention are only theoretical problems, whereas QM manifests in real experiments.

 

[Ken G]

I agree the loss of local realism seems to have had more in the way of aftershocks that the loss of the aether. People dropped Poincare's aether pretty much overnight, but dropping local realism seems weirder. I don't really know why though, both relativity and quantum mechanics have a very satisfactory elegant structure, and neither seems at all like what we experience day to day.

 

[ddd123]

I don't really know why though

Probably because realism was/is the main tenet of the scientific endeavor. You figure out the objective properties of things. So, relativistic space-times are removed from everyday experience but they consist in a very definite objectivity. In a sense they strengthen the realism of science because science convinces us of something that is so intangible: it means it's so powerful. The loss of realism does the opposite.

 

[Ken G]

So the weirdness simply stems from a certain brand of realism. But I don't agree this is the main tenet of science, the main tenet of science is to make sense of observations. So if a certain way of doing that makes us regard science as weird, then get rid of that way of doing it. That's what we did to the aether, and for the same reason.

 

[ddd123]

So the weirdness simply stems from a certain brand of realism. But I don't agree this is the main tenet of science, the main tenet of science is to make sense of observations. So if a certain way of doing that makes us regard science as weird, then get rid of that way of doing it. That's what we did to the aether, and for the same reason.

That's why I wrote "was/is. I can agree it is not the main tenet, but was? I think we can say objectivity isn't the main goal only because we've started doubting we can have it. So we reflected upon that and came out with the more lax requirement of "making sense of observations" (it's a little vague but I think it's not necessarily bad).

 

[Ken G]

Ah I see, I didn't notice the was/is! Yes, I think that's the main point here, what we regard as weird depends on our philosophy, but we should always expect our philosophy to need to change as science advances. So we should expect constant weirdness, and I think that is exactly what the history of science has always been. We tend to focus only on the current weirdnesses as if they were somehow special.

 

[ddd123]

So we should expect constant weirdness

You might see it differently: we should expect increasing weirdness. Since this is related with our intuition, the more we explore phenomena that are distant from us, the weirder it gets for our intuition, and with QM it even altered the very nature of our knowledge of things, something which we didn't think possible.

 

[adeborts]

When do you think your book will be published?
(You state that what is available on the net is just a draft.)

 

[adeborts]

No, the more I think about quantum mechanics, the less weird it is. I have written a whole book about it, without any weirdness; see post #2.

Quantum mechanics is weird only in the eyes of those who take the talk about it too serious and neglect the formal grounding which contains the real meaning.

You state that what is available on the net is only a draft.
When will the book reach the final format and be published?

 

[adeborts]

Sorry for the repetition!
My first inquiry was not immediately posted and I thought it got lost...

 

[A. Neumaier]

When do you think your book will be published?

A revised version is scheduled to be published in fall 2017. I'll probably add a much more polished and complete discussion of nonequilibrium thermodynamics (except for its field theoretic aspects) and take out the stuff on general manifolds. Field theory needs a second book, and I haven't yet a schedule for its publication.

 

[Feeble Wonk]

Yes. It is only surprising and looks probabilistic to us, because we do only know a very small part of its state.

Please forgive my ignorance on this matter, but is determinism a fundamental principle of QFT? I assume that there are a range of conceptual variations regarding quantum fields. While the version you subscribe to is deterministic, are there versions of QFT that are not?

 

[ddd123]

Please forgive my ignorance on this matter, but is determinism a fundamental principle of QFT?

Actually the opposite, non-determinism is fundamental to QFT. It's a theory, if a deterministic underpinning is found to QFT it will be something else. Actually, since QFT seems to prefer locality in some sense, its non-determinism is pretty essential for it not to violate Bell's inequalities: http://arxiv.org/abs/hep-th/0205105 .

 

[A. Neumaier]

While the version you subscribe to is deterministic, are there versions of QFT that are not?

Most of QFT is applied only to small systems, in which case it is probabilisitc like any (classical or quantum) model that excludes part of the full dynamics from its set of relevant observables.
Whether the full universe (the only system containing us not coupled to an environment) is or is not deterministic is unknown. I believe that it may be taken as deterministic, while those who subscribe to a statistical interpretation would say a quantum model of the universe is meaningless since one cannot replicate it often enough to make statistics about it.

 

[vanhees71]

Well, QFT is also applied to very large systems, and there it's very successful too in at least making it very plausible, why such macroscopic systems usually behave according to classical physics. Particularly QFT of the thermal-equilibrium state is very well developed and very successful in the wide area from condensed-matter physics to cosmology.

 

[rkastner]

It it available without a paywall?

Sorry, I just saw this. You can find a lot of free material on PTI on my blog:
transactionalinterpretation.org

 

[A. Neumaier]

Particularly QFT of the thermal-equilibrium state is very well developed

But it is determinsitic in the thermodynamic limit, and no trace of probabilities is left.

 

[stevendaryl]

But it is determinisitic in the thermodynamic limit, and no trace of probabilities is left.

I don't find that completely satisfying. If you treat Brownian motion using statistical mechanics, then it's deterministic. If you analyze a dust particle suspended in a liquid, your statistical mechanics will give a probability distribution for the location of the particle as a function of time, and that distribution evolves deterministically. But of course, if you're actually looking at a dust particle under a microscope, you'll see it jerk around nondeterministically.

In classical mechanics, we have a theory explaining the actual observation (the dust particle moves when a molecule of the liquid collides with it), as well as the statistical mechanics description. If you only had the statistical mechanics, I would consider the theory incomplete.