# Quantum incompleteness?

#### TrickyDicky

Every so often discussions come up about completeness of quantum theory and I often can't see what their point is so I might be missing something.

Is it not possible for a theory to be incomplete and at the same time give very accurate predictions in its domain of applicability? Newton mechanics comes to mind as an example.

How is the Newtonian case in principle different from the quantum theory case besides the obvious the fact that the theory that would extend the domain of QM ("quantum gravity") hasn't benn found yet while in the Newtonian case we have relativistic mechanics?

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#### dx

Homework Helper
Gold Member
I believe the discussions about the completeness of quantum mechanics (for example Einstein's paper "Can the quantum mechanical description of physical reality be considered complete?") are concerned with the adequacy of the quantum state as a representation of the physical situation. Einstein argues that since it is possible to have a situation where either P or X can be predicted "without disturbing the particle" by making measurements on another system with which the particle has been in interaction, that both P and X have an element of 'physical reality', and therefore he concludes that the quantum mechanical state cannot be a complete description.

#### VantagePoint72

Einstein argues that since it is possible to have a situation where either P or X can be predicted "without disturbing the particle" by making measurements on another system with which the particle has been in interaction, that both P and X have an element of 'physical reality', and therefore he concludes that the quantum mechanical state cannot be a complete description.
And, since any discussion of EPR without mentioning Bell is incomplete, it should be pointed out for OP's benefit that Einstein seems to have been wrong. Bell showed that a theory that respects Einstein's notion of locality (no "spooky action-at-a-distance") and 'physical reality' must disagree with quantum mechanics as we know it in certain experiments. Thus far, such experiments have agreed with quantum mechanics.

#### DrChinese

Gold Member
Adding to what dx says: You could also say either "QM is incomplete" with the caveat "no more complete representation is possible". Some folks will quibble with the terminology. I think if you read the "incompleteness" word as a statement that there is always uncertainty, then it is easier to swallow. On the other hand, EPR intended the term as a criticism.

Obviously, "incomplete" QM is a very successful theory.

#### dx

Homework Helper
Gold Member
Also, Bohr wrote a paper in response to Einstein, soon after Einstein published his, with the same title.

Abstract: "It is shown that a certain "criterion of physical reality" formulated in a recent article contains an essential ambiguity when it is applied to quantum phenomena. In this connection a viewpoint termed 'complementarity' is explained from which quantum mechanical description of physical phenomena would seem to fulfill, within its scope, all rational demands of completeness"

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#### Charles Wilson

Yeah, Bohr pushed "Complementarity" which is as Metaphysical as the EPR assertion.

It appears that Einstein and EPR make an assertion, "Assume an electron here, going through a slit to a screen which catches it." At every presented duration, you can "measure" that there is an electron "there".

THEREFORE, the electron WAS THERE - atomic, isolated - even when we didn't "measure" it being there. This step is denied by QM. That this is so is to be found in the examples of the "2 gloves". "If I open a box that contains a right handed glove and the other glove is in another box, I know it must be the left handed glove".
At this point, Bohr wants to stick his nose in where it doesn't belong and argue that "SUDDENLY!" the situation changed to a complementarian description, which also goes beyond the data and the QM math which describes "All we can know."

If Bohr believed that "The observation created the situation", HE should have been the one to produce the EPR experiment. But both Einstein and Bohr could not see beyond the assumptions. "Is there NO THING that can travel faster than light?" is only a start to the possibilities.

Entanglement is the game changer here and Stapp is correct.
If you accept that there is NO THING that can travel faster than light AND Entanglement exists, then NECESSARILY you are positing a Universe greater than this one, which encompasses this one. Entanglement must map onto this universe somehow or we are left in a Solipsistic Universe where we "seemingly" perceive a "das Noumena" (The "Why" of entanglement.) but can never ever get to it.
QM therefore is complete in this universe (Bell's FAPP) but may be supplemented by the final (Mathematics) of the Universe Above.

Neither Einstein nor Bohr win.

CW

#### TrickyDicky

Thanks for the comments, they are all reasonable and dwell on the usual well known quantum mechanical facts, characters and narratives:EPR, Bell, Einstein, complementarity, success of the theory, measurements, etc...
Important as all that is I wanted to take the completeness debate out of there for a moment as that kind of discussion has a tendency to get a bit circular anyway. Thus my comparison with the Newtonian case wich noone chose to address, so I take it there's agreement quantum theory is incomplete in that specific sense and that doesn't detract anything from it as a theory.
But I guess it is is quite difficult to discuss this without at least touching upon all those issues: it is IMO worth noting that most of the Einstein vs Bohr controversy, echoed in different forms countless times refers to problems that arise when considering particles as "real" objects. I thought (please correct me if I'm wrong) that since QFT reigns in physics the dichotomy particle-field had been clearly overcome in favor of fields, and particles as excitations of fields are considered simply a property of the quantum fields, remaining in the language simply as a graphical and practical depiction of those localized properties but not being considered fundamental anymore.

#### audioloop

Thanks for the comments, they are all reasonable and dwell on the usual well known quantum mechanical facts, characters and narratives:EPR, Bell, Einstein, complementarity, success of the theory, measurements, etc...
Important as all that is I wanted to take the completeness debate out of there for a moment as that kind of discussion has a tendency to get a bit circular anyway. Thus my comparison with the Newtonian case wich noone chose to address, so I take it there's agreement quantum theory is incomplete in that specific sense and that doesn't detract anything from it as a theory.
But I guess it is is quite difficult to discuss this without at least touching upon all those issues: it is IMO worth noting that most of the Einstein vs Bohr controversy, echoed in different forms countless times refers to problems that arise when considering particles as "real" objects. I thought (please correct me if I'm wrong) that since QFT reigns in physics the dichotomy particle-field had been clearly overcome in favor of fields, and particles as excitations of fields are considered simply a property of the quantum fields, remaining in the language simply as a graphical and practical depiction of those localized properties but not being considered fundamental anymore.
right and that quantum theory is an approximation to a fuller theory.

#### DrChinese

Gold Member
... quantum theory is an approximation to a fuller theory.
I think that is more or less the opposite of the usual conclusion. There is currently no sign of the existence of a more complete theory than QM.

#### audioloop

you said it "currently"

Gold Member
Every so often discussions come up about completeness of quantum theory and I often can't see what their point is so I might be missing something.

I think inevitably, when you bring the word (in)completeness into a discussion about QM you end up with the famous 20 year long Bohr–Einstein debates. However this seems not to be the issue here, right?

IMHO, these are the facts:

• No scientific theory represents the ultimate truth.
• A scientific theory must be falsifiable.
• A scientific theory is only valid as long as not proven false.
• A new scientific theory must include previous theories and empirical verifications (i.e. Newton's Apple can never go the other way).
• A scientific theory is a working model to help humans understand nature and the universe.
• Gödel’s incompleteness theorems show that there are inherent limitations in all axiomatic systems.
• Quantum mechanics is the most accurate theory we have so far.
Any questions on that?

Now, when it comes to the extremely well working model of quantum mechanics; it all works like a dream – mathematically. But when you start talking about QM and try to provide “natural explanations”, it doesn’t always work that great...

Why??

Because, for one thing; the theory does not say anything about what happens at (macroscopic) measurements, and often it’s right there the ‘weirdness’ starts. Where exact is the border between microscopic QM fields/particles and classical macroscopic objects (if any)? Could two elephants be entangled? No one knows for sure... And furthermore, the “Reality Show”; even if you only use fields – are these fields real? No one knows for sure... And what about the Schrödinger wavefunction, is this ‘thing’ real? And lately Bell's theorem + empirical verifications have shown that QM forces us to give up locality and/or reality – i.e. for real! If we give up locality – what ‘stuff/function’ can transmit casual effects to the other side of the universe instantaneously!? And if we give up reality... eh...

Hello! Where’s the ‘completeness’ in all that?? :uhh:

As you see, it’s a weird world out there... and I love it! (i.e. if it exists )

P.S:
My very personal guess is that we will be forced to do a ‘slight adjustment’ to QM and/or SR...

#### VantagePoint72

• Gödel’s incompleteness theorems show that there are inherent limitations in all axiomatic systems.
No, they don't. They make very specific claims about specific axiomatic systems—those that describe the properties of the natural numbers. To naively carry this over into axiomatized physics models is to assume, with no justification, that every statement about the natural numbers has an analogous physical statement (i.e. a statement about a physically realizable state in terms of elementary processes). What physical process is a reflection of, e.g., the Goldbach conjecture? Fermat's Last Theorem? If you look at the Gödel statements used in the theorems, the suggestion that they have physical analogues is even sillier. Plus, there's a fundamental difference between choosing axioms for the natural numbers and choosing axioms for physics: the latter have empirical consequences. This wishy-washy pop sci idea that Gödel's incomplete theorems have anything to do with scientific models is just as unfounded (and annoying) as New Agers saying, "Quantum mechanics says anything you can imagine is possible!" Mathematical theorems are precisely worded for a reason: if you start running off with vague assertions based on out-of-context generalizations, you're going to say a lot of very incorrect things.

Now, when it comes to the extremely well working model of quantum mechanics; it all works like a dream – mathematically. But when you start talking about QM and try to provide “natural explanations”, it doesn’t always work that great...
And this is supposed to count against the theory somehow? Our brains evolved in environments completely dominated by classical behaviour. Quantum theory is weird because it's unfamiliar, and likely always will be to some extent. The inability to describe QM with 'natural explanations' just demonstrates the shortcomings of intuition—since a natural explanation is just something that makes sense intuitively to us. That is precisely why we use the mathematical models: they take us where our intuition can't. The difficulty of making intuitive sense of QM has no relevance to whether or not the theory is complete.

Because, for one thing; the theory does not say anything about what happens at (macroscopic) measurements, and often it’s right there the ‘weirdness’ starts. Where exact is the border between microscopic QM fields/particles and classical macroscopic objects (if any)?
This, and nearly everything else in this paragraph, is wrong. QM has no restrictions about length scales. There is no postulate saying, "By the way, all this stuff only applies to microscopic objects (whatever that means)." The emergence of classical behaviour from quantum phenomena is well understood through mechanisms like quantum decoherence, Ehrenfest's theorem (and other specific instances of the correspondence principle), the path integral formulation of QM (in which the ratio of the classical action to $\hbar$ determines how dominant quantum effects are), and many other derived properties of QM. There is no arbitrary cut-off point at which we say, "Well, let's stop using QM here." There is no mystery as to how the equations of classical mechanics emerge from various limiting cases of quantum mechanics; disagreements about the nature of quantum states has absolutely no bearing on that fact.

Hello! Where’s the ‘completeness’ in all that?
Nothing you've said has anything to do with the 'completeness' of QM, by any standard definition of the word. You're apparently confusing 'incomplete' with 'makes me uncomfortable'.

As you see, it’s a weird world out there
And it's plenty weird enough as it is without people pointing to well-understood phenomena while waving their hands about and saying, "Ooh, isn't that mysterious?"

#### Jano L.

Gold Member
This, and nearly everything else in this paragraph, is wrong. QM has no restrictions about length scales. There is no postulate saying, "By the way, all this stuff only applies to microscopic objects (whatever that means)." The emergence of classical behaviour from quantum phenomena is well understood through mechanisms like quantum decoherence, Ehrenfest's theorem (and other specific instances of the correspondence principle), the path integral formulation of QM (in which the ratio of the classical action to ℏ determines how dominant quantum effects are), and many other derived properties of QM. There is no arbitrary cut-off point at which we say, "Well, let's stop using QM here." There is no mystery as to how the equations of classical mechanics emerge from various limiting cases of quantum mechanics; disagreements about the nature of quantum states has absolutely no bearing on that fact.
Let me give you opposite viewpoint.

The parts of the quantum theory you mention in no way address the problem DevilAvocado's pointing to. There is no derivation of deterministic models of classical physics or at least classical mechanics from the quantum formalism in the sense of direct mathematical limit as there is in the special relativity (excluding perhaps statistical physics). The theory is probabilistic, and there does not seem to be a way to recover deterministic description of a single system approaching that of classical theory. Most practical applications of the quantum theory involve the Born rule. The decoherence studies deal with density matrices, not with variables of individual systems.

There is no mystery as to how the equations of classical mechanics emerge from various limiting cases of quantum mechanics; disagreements about the nature of quantum states has absolutely no bearing on that fact.
Can you explain how the equation of motion of a point-like electron in the Coulomb potential emerges from quantum mechanics?

#### Nugatory

Mentor
This is one of those threads that I'm gonna be so sorry that I ever stepped into....

The theory is probabilistic, and there does not seem to be a way to recover deterministic description of a single system approaching that of classical theory.
Classical thermodynamics is also essentially probabilistic and lacks any way of recovering a deterministic description, yet does amazingly well as a deterministic description of the macroscopic world.

Can you explain how the equation of motion of a point-like electron in the Coulomb potential emerges from quantum mechanics?
We can't, and we don't need to. There is no theory of a point-like electron in the Coulomb potential that comes even remotely close to agreeing with observation, so there is no need to make such a description emerge from quantum mechanics. What we DO need to emerge from QM is a description of how a macroscopic body composed of many electrons in the coulomb potential of many nuclei will act according to Coulomb's law.

And that's clearly possible, even if it involves an unspeakable amount of computational drudgery. I still feel sympathy for a colleague of mine whose PhD thesis (early '70s) was a quantum-mechanical description of a bouncing ball.

#### Jano L.

Gold Member
Classical thermodynamics is also essentially probabilistic and lacks any way of recovering a deterministic description, yet does amazingly well as a deterministic description of the macroscopic world.
I am afraid you are confusing properties of two different theories.

"Classical thermodynamics" usually refers to the theory of heat - work transformation based on macroscopic variables with no involvement of probabilistic description. Indeed it does amazingly well as a deterministic description of the equilibrium processes in macroscopic bodies.

On the other hand, it is classical statistical physics which is essentially probabilistic and indeed it does lack any way of recovering the deterministic description itself alone.

This last property is not a problem, because everybody knows that classical statistical physics is not some basic theory from which we would like to derive deterministic models. Everybody knows it is the other way around: the classical statistical physics is built upon already available deterministic classical mechanics and probability theory.

We can't, and we don't need to. There is no theory of a point-like electron in the Coulomb potential that comes even remotely close to agreeing with observation, so there is no need to make such a description emerge from quantum mechanics.
Plasma theory, Rutherford scattering...

Anyway, the Coulomb potential is not crucial for the problem at hand. What about electron in electrical condenser? in uniform magnetic field? Can you derive the equation of motion ffrom quantum theory? (as you surely know, it is confirmed very well).

#### San K

Every so often discussions come up about completeness of quantum theory and I often can't see what their point is so I might be missing something.

Is it not possible for a theory to be incomplete and at the same time give very accurate predictions in its domain of applicability? Newton mechanics comes to mind as an example.

How is the Newtonian case in principle different from the quantum theory case besides the obvious the fact that the theory that would extend the domain of QM ("quantum gravity") hasn't benn found yet while in the Newtonian case we have relativistic mechanics?
QM is complete in the sense that it answers the questions (related to particular phenomena in QM) correctly mathematically (or mathematically correctly...;)).

QM is incomplete in the sense that we will discover more about the nature of our universe/QM/space-time. QM is incomplete in the sense that there is more to be discovered related to QM itself plus there is, obviously, more to be discovered about non-QM as well.

It is very hard for us to visualize (dimensions) "beyond" space-time. However there is still lot out there and QM will play an important role in (explaining, finding) those discoveries.

QM will improve/evolve on understanding of a bigger/better picture.

#### VantagePoint72

Anyway, the Coulomb potential is not crucial for the problem at hand. What about electron in electrical condenser? in uniform magnetic field? Can you derive the equation of motion ffrom quantum theory? (as you surely know, it is confirmed very well).
Yup. Leading order behaviour of QED.

#### Demystifier

2018 Award
Every so often discussions come up about completeness of quantum theory and I often can't see what their point is so I might be missing something.

Is it not possible for a theory to be incomplete and at the same time give very accurate predictions in its domain of applicability? Newton mechanics comes to mind as an example.

How is the Newtonian case in principle different from the quantum theory case besides the obvious the fact that the theory that would extend the domain of QM ("quantum gravity") hasn't benn found yet while in the Newtonian case we have relativistic mechanics?
There is one important difference between incompleteness of Newtonian mechanics (NM) and incompleteness of quantum mechanics (QM).

We know that NM is incomplete because there are EXPERIMENTS which demonstrate so. But there is nothing in the Newtonian THEORY itself suggesting that it should be incomplete on the basis of internal theoretical inconsistencies.

The situation with QM is exactly the opposite. There no experiments demonstrating incompleteness of QM. But there are serious theoretical arguments suggesting that something important must be missing in the standard version of quantum theory.

#### yossell

No, they don't. They make very specific claims about specific axiomatic systems—those that describe the properties of the natural numbers.
No, they make a specific claim about *all* axiomatic systems which logically include a minimal basis for arithmetic. Any axiomatic system which contains a certain amount of arithmetic is subject to Godel's theorems. The mathematical systems employed by physicists certainly include this minimal base and any axiomatisation of the mathematical systems physicists use will be subject to Godel's theorems.

Having said that, I agree that Godel's incompleteness theorem has zilch to do with alleged quantum incompleteness.

I also agree that it's an interesting to ask: what is the weakest mathematical theory needed to `do' physics -- perhaps the systems that are actually used -- which include arithmetic, include calculus, include theories or real numbers -- go too far. But if we want to at least leave it open that spacetime is continuous, or even just that the potential positions of an object have the structure of the reals, it's going to be hard to avoid systems subject to Godel's theorem.

#### VantagePoint72

No, they make a specific claim about *all* axiomatic systems which logically include a minimal basis for arithmetic.
That is the exact same as what I said. If a system includes a minimal basis for arithmetic then it describes the properties of the natural numbers. You're being unnecessarily argumentative.

In any case, I recall Hawking talking about the Gödel angle once. As I understood it, his position was that since number theoretic statements can be well-formed in the mathematical language of physics, incompleteness is inevitable. That's fine, but—I think—entirely uninteresting. If you're going to include mathematical theorems in the scope of your "Theory of Everything" then I think it's an abuse of what physicists mean by the term. If someone wants to say, "Quantum mechanics will never be complete because physics is phrased in terms of mathematical frameworks which are subject to the incompleteness theorems," then they are (trivially) correct. That is completely different from suggesting that the theorems are relevant for whether or not a single mathematical framework describing the four fundamental forces is possible. That would, as I said, require there to be actual physical analogues of number theoretic statements. That, I think, is nonsense.

#### Jano L.

Gold Member
I do not think so. In quantum electrodynamics the electron is described by the operator field, and only "average values" can be directly calculated. How do you arrive at the equation of motion for one electron

$$\frac{d\mathbf p}{dt} = q\mathbf E(\mathbf r) + q\mathbf v\times\mathbf B(\mathbf r),$$

where $\mathbf E,\mathbf B$ are the external fields (due to condenser, magnet...)?

#### VantagePoint72

Huh? Those are the average values. That is the point of the correspondence principle: the average behaviour of quantum systems corresponds to classical behaviour, and in the limit of large systems the variance falls off to zero.

One electron like you're describing is a quantum system. I don't understand your objection. You're apparently saying the fact that you can't extract the exact classical equations of motion for a quantum-scale system is a problem. That's ridiculous. When I say that the emergence of classical behaviour for classical (i.e. large) systems from quantum behaviour is well-understood, I'm obviously talking about systems for which classical mechanics is known to be a good approximation. Of course quantum mechanics won't yield classical descriptions for systems that classical mechanics is a poor approximation for.

You seem to be bizarrely hung up on the fact that, while QM reproduces the valid descriptions of classical mechanics (à la Nugatory's friend with the bouncing ball), it doesn't reproduce the invalid ones.

#### Jano L.

Gold Member
Huh? Those are the average values. That is the point of the correspondence principle: the average behaviour of quantum systems corresponds to classical behaviour, and in the limit of large systems the variance falls off to zero.
"Average behaviour" is a description of ensemble, or probabilistic description of a single system. You cannot derive deterministic description of a single system from it.

Furthermore, to my knowledge, the above equation cannot be derived from the quantum field theory even in the mean value sense; the quantum field description leads to divergences for point-like electron.

One electron like you're describing is a quantum system. I don't understand your objection. You're apparently saying the fact that you can't extract the exact classical equations of motion for a quantum-scale system is a problem. That's ridiculous.
The electron may be small, but the experiments do not need to be on atomic scale. The equation I wrote above is widely used description used in the classical electron theory (very successful in explaining many optical phenomena) and accelerator design.

#### VantagePoint72

Coulomb's law, with further corrections as needed, comes directly from the QED Lagrangian. You can derive it either as the lowest order perturbation of the free vacuum (as Peskin and Schroeder do in 4.8)—in which it corresponds to the exchange of one virtual photon—or simply by integrating out the gauge freedom of the QED Hamiltonian and observing the (non-perturbative) expression for Coulomb's law that falls out of it. That gives you the first half of your equation. Using the now-justified expression for Coulomb's law, the second half follows immediately from the Lorentz covariance of the electromagnetic field strength tensor (see any derivation of the magnetic force law from Coulomb's law + special relativity). Hence, the total Lorentz force law for a charged particle comes, as I said, as the leading order of QED.

the quantum field description leads to divergences for point-like electron.
So does the classical field theory, which breaks down at a certain length scale do to the electron self-force, self-energy, etc. Once again, you are pointing to the fact that quantum theory doesn't reproduce the bad predictions of classical theory and declaring that it proves the classical theory isn't a limiting case of the quantum theory. I don't think you understand what it means for one theory to reduce to another in some limit.

#### Jano L.

Gold Member
LastOneStanding, the second procedure you mention (separation of the terms with the Coulomb potential from the field Lagrangian) is independent of whether the potential and current are numbers or operators. You can do it in classical theory as well. It thus says nothing about the derivation of classical theory from the quantum theory.

The section in Peskin & Schroeder derives the main part of the scattering amplitude due to the term $1/r$. This quantity refers to probability of the overall result of the process that spans infinite time. It gives no temporal description of the process, and it is probabilistic. You cannot derive classical differential equation of motion from it.

If you do not think that such derivation is necessary then we can discuss that.

The classical equation of motion above is not a prediction, but inference from the experiments done with electrons in vacuum tubes. And it is not bad equation, since CRT tubes and cyclotrons work rather well, don't you think?

So does the classical field theory, which breaks down at a certain length scale do to the electron self-force, self-energy, etc.
Yes, but I did not require derivation of the diverging classical field theory. That is not necessary, as the self-force acting on the electron is entirely hypothetical concept. I am talking about derivation of the equation of motion in external field, which is simpler, free of problems and well confirmed by experiment. (These two are incompatible for point-like particles, by the way.)

I don't think you understand what it means for one theory to reduce to another in some limit.
Then please explain, what does it mean?