# “Classical Physics Is Wrong” Fallacy

One of the common questions or comments we get on PF is the claim that classical physics or classical mechanics (i.e. Newton’s laws, etc.) is wrong because it has been superseded by Special Relativity (SR) and General Relativity (GR), and/or Quantum Mechanics (QM). Such claims are typically made by either a student who barely learned anything about physics, or by someone who have not had a formal education in physics. There is somehow a notion that SR, GR, and QM have shown that classical physics is wrong, and so, it shouldn’t be used.

There is a need to debunk that idea, and it needs to be done in the clearest possible manner. This is because the misunderstanding that results in such an erroneous conclusion is not just simply due to lack of knowledge of physics, but rather due to something more inherent in the difference between science and our everyday world/practices. It is rooted in how people accept certain things and not being able to see how certain idea can merge into something else under different circumstances.

Before we deal with specific examples, let’s get one **FACT** straighten out:

Classical physics is used in an overwhelming majority of situations in our lives. Your houses, buildings, bridges, airplanes, and physical structures were built using the classical laws. The heat engines, motors, etc. were designed based on classical thermodynamics laws. And your radio reception, antennae, TV transmitters, wi-fi signals, etc. are all based on classical electromagnetic description.

These are all FACTS, not a matter of opinion. You are welcome to check for yourself and see how many of these were done using SR, GR, or QM. Most, if not all, of these would endanger your life and the lives of your loved ones if they were not designed or described accurately. So how can one claim that classical physics is wrong, or incorrect, if they work, and work so well in such situations?

What actually is true is that we discovered a more accurate, and more general description of our world. In this description, it turns out that classical physics appears as a “simplification” or “approximation” whereby it becomes more and more valid as various parameters approach the common, everyday, terrestrial values. And this is an extremely important point to remember, because since classical physics works under our ordinary situation, **any new theory or description must somehow converge and look like the classical physics description under such ordinary conditions**. Otherwise, this new theory must show that it produces the same set of results as classical physics for **all** of our known phenomena that classical physics can already accurately described.

So in this part of the article, I will show two specific examples where the more general theory of SR and QM merge smoothly into the classical physics form when one adopts the appropriate approximation. This means that at some limit, both SR and QM description will be the same as the classical description.

### Example 1: Special Relativity Velocity Addition

The first example is from SR and deals with the velocity addition in different inertial reference frames. This is illustrated in the figure below:

The reference frame S’ is moving with velocity v with respect to reference frame S. A vessel is moving with velocity v’ with respect to S’ frame. What is u, the velocity of that vessel with respect to frame S?

The “normal” way to find this velocity is using what is known as Galilean transformation. Here, since both v and v’ and in the same direction, the velocity of the vessel with respect to S frame is a simple addition, i.e.

u = v + v’ (1)

Keep that result in mind.

Now let’s look at how we do this in SR, which is the more general description of such kinematics. Here, we use what is known as the Lorentz transformation. Using the figure from above as before, the velocity u of the vessel in S frame is:

(2)

where c is the speed of light in vacuum.

Now this looks different than the Galilean transformation that we are used to in Eq. 1. This works for reference frame S’ at any velocity v, even if it approaches c. At that value, Eq. 1 does not work, and the velocity addition that we are used to fails miserably.

However, what happens when v«c, i.e. when reference frame S’ moves much slower than the speed of light? This is what we normally encounter, i.e. someone moving in a vehicle or an airplane. For v«c, Eq. 2 simplifies quite a bit. Without having to do any kind of Taylor series expansion on the denominator of Eq. 2, we can already see that the ratio vv’/c^2 « 1, i.e. it is a very small fraction less than one (v’ cannot be greater than c). This means that, to a good approximation, the denominator of Eq. 2 is essentially just 1.

When that happens, Eq. 2 then simplifies to u = v + v’, which is **exactly** Eq. 1! We got back our familiar result when we apply the more general equation (Eq. 2) to our normal, terrestrial condition! This means that all of the velocity addition equations and concepts that we already know using Galilean transformation are derivable from the more general Lorentz transformation equations. The Lorentz transformation is the more accurate, more encompassing description of velocity addition, while the Galilean transformation, which is what we know and are familiar with, is simply a special case for when the other reference frame is moving much slower than the speed of light. Eq. 1 isn’t wrong. It has a limited range of situation when it is valid or accurate enough.

### Example 2: Quantum Mechanics Rate of Change of Momentum

In classical Newtonian mechanics, for an object with mass m, we know about Newton’s Second Law that relates the force F and the resulting acceleration a, which is

F = ma

This familiar equation can actually be written in a more general form, which is in terms of the time rate of change of momentum p, i.e.

F = dp/dt (3)

We also know that force F can be related to the potential energy (V) gradient, i.e.

F = – dV/dx (in 1D) (4)

So that is from the classical mechanics side. Let’s look at what it says on the QM side. Here, we use Ehrenfest theorem, which says:

(5)

Here, H is the Hamiltonian, Q is an operator representing any observable, the square bracket represents the commutator, while the angled bracket represents the average value. These are all the standard notations used in QM.

So what if we want to find the time rate of change of the momentum, p? In QM, p is an operator representing the observable momentum. Thus, Eq. 5 becomes

(6)

From here, it requires quite a bit of knowledge on how to perform the commutator and take the average. You may read the full derivation of it here. In the end, what you end up with is:

(7)

This says that the time rate of change of the average value of the momentum p is equal to the average of the gradient of the potential energy V. But this equation is equivalent to Eq. 3 and 4 from classical mechanics! They have identical form!

It says that what we typically measure in our everyday lives are really the “average” values of many, many, many values at the QM level. The QM description has made the connection to the classical description under the condition that the QM observables have been averaged. Again, as in the case of velocity addition, we get back the classical description from a more general starting point, in this case a QM description, upon applying a particular condition to the QM picture. It shows that the classical picture is not wrong. It is the average over a large number of QM observables.

### Moral of the Story

- Classical physics WORKS for our ordinary situation, so it HAS to be valid at some level.
- Classical physics has been shown to be derivable from SR and QM under special conditions that apply to our ordinary situation.
- Any theory MUST have the ability to show that it merges to the classical description when applied to ordinary situation.
- This can only be shown mathematically. It cannot be shown convincingly via hand-waving or qualitative arguments. It is the equivalent mathematical form that shows that one theory can derive the other.

What this implies here is that, if there are more general and more accurate theories beyond QM, SR/GR, then those theories must also show that they can be “simplified” into the mathematical forms of QM and SR/GR. Subsequent, more general theories must show that they can derive the mathematical forms of existing, already-working theories. The inability to do that will be a fatal flaw in any new theory.

I mentioned towards the beginning of this article that the inability to comprehend this concept of a more general idea merging and agreeing with something less general may have something to do with the differences between science and our everyday lives. It is unusual for many people to accept the possibility that a simplistic, less sophisticated, and apparently different idea is actually a subset of a more general principle. The fact that one can actually start with a more general principle, applies certain criteria, and then get a seemingly different concept, is not something a lot of people are familiar with, or would even accept.

It is why for someone not trained in physics, the idea that classical mechanics can actually be derived from seemingly a different animal of QM or SR/GR would not even cross his/her mind. Yet, in science/physics, this is **quite common**. We always show how new ideas and theories will turn into the old, tested, and well-known ideas and theories under the appropriate parameters. It is very seldom that old theories are discarded wholesale.

Zz.

PhD Physics

Accelerator physics, photocathodes, field-enhancement. tunneling spectroscopy, superconductivity

A theory is not its underlying assumptions – which can never be tested, it is its predictions.Well, that's where we don't agree. I guess we are going nowhere here. anyway, thanks for the chat. I will think about the insights you all gave. see ya.

What difference does it make if you get the old theory from an upper limit or any other arbitrary direction?Uhmmm … Are you serious or just trolling? Obviously it matters that you recover the old theory in the limit where the old theory is known to be applicable or if you do so in the opposite limit.

This does not make any sense. If I say that the assumptions are: time is relative and space is Riemaniann and you say: time is absolute and space is euclidian, how can we be talking about the same theory?We are not. We are only talking about the limit of one of the theories. In the classical limit you do recover the very same things as the classical theory. That SR is applicable to a larger set of situations is not the issue here.

No, you are wrong. What I said was that the assumptions are different! If you look at how we treat all the experiments and particles / waves in quantum world, we see that space, time, energy e whatever are all different from what we knew from classical mechanics, which means, we cannot start to say that there exists a derivation from one to another, as they were talking, ever, about the same thing.This is just wrong. As already illustrated in the actual Insight. You can start from different assumptions, but in the end what it boils down to is to make identifications of what concepts in a theory that correspond to the concepts of the previous. A theory is not its underlying assumptions – which can never be tested, it is its predictions.

I'm talking about how do we explain an theory evolution without wanting to find convergencies that were created just to make understanding easier.I am sorry but you are not making any sense here. The point is that the "old" theory typically has made a large number of verified predictions that are well studied enough that we know how experiments behave in a certain range – at least to within experimental uncertainty. Under the same type of conditions, the "new" theory must therefore reproduce exactly the same results up to corrections that are smaller than the experimental uncertainty. This is what it means having the "old" theory as a limiting behaviour. It has absolutely nothing to do with what "basic assumptions" have been made, just about predictions.

I am not saying that Newtonian Mechanics is wrong. this has nothing to do with being right or wrong.Actually,

it does!That's the whole point of my writing the article, that we frequently have members here claiming that Newtonian mechanics iswrong, and questioned why we continue to use it! If you make no such claim, then the article wasn't meant for you.You also read WAY too much philosophical implication to what the article had said. First of all, I never said anything about any "proofs" that one theory equals the other. All I did was show that, in certain limits or situation, one theory can reproduce the mathematical form of the other. None of what I had shown should be a surprise, because we all saw this in undergraduate physics courses!

Furthermore, we do not have to go to wide extreme to see this. Even within classical mechanics itself, the Newtonian mechanics and the Hamiltonian/Lagrangian mechanics already have philosophically different approach and "world view". And yet, they both arrive at the same mathematical form in describing the kinematics of a system.

Being able to show that something can be derived into a familiar form is a powerful and extremely useful argument. This is done in mathematics all the time. The ability to reformulate a differential equation into something that we know the solution of is done often. So it is no different here in physics. It says that the new idea can reproduce all of the results of the old ideas, and also shows why and where the old ideas may fail or no longer accurate enough.

Zz.

I am sorry, but this is just ignorant and the limit is

notarbitrary. Obviously you will not recover classical mechanics in the ultra-relativistic limit, you will recover it in the classical limit where speeds are much smaller than the speed of light.What difference does it make if you get the old theory from an upper limit or any other arbitrary direction? My point was that SR is considering space and time as completly different things comparing to old mechanics. This does not make any sense. If I say that the assumptions are: time is relative and space is Riemaniann and you say: time is absolute and space is euclidian, how can we be talking about the same theory?When it comes to light it is very well explained using Maxwell's theory. When it comes to particles such as electrons, it is not a prediction of the classical theory but your logic is completely flawed. You have taken one of the shortcomings of the classical theory that inspired people to QM and presented it as a counter argument to the quantum theory having to reduce to the classical theory

in the limits where we know that the classical theory works. We already know that the classical theory does not work in this limit and so your argument is empty. Essentially you argument to the statement "A must give the same predictions as B when C is true" is to say "but it does not give the same predictions when C is not true". We come back to your assertion that the limit is arbitrary which, again, is not the case.No, you are wrong. What I said was that the assumptions are different! If you look at how we treat all the experiments and particles / waves in quantum world, we see that space, time, energy e whatever are all different from what we knew from classical mechanics, which means, we cannot start to say that there exists a derivation from one to another, as they were talking, ever, about the same thing. Again, if you start with different assumptions how can you end up with theories converging at a limit that has been chosen to find a connection between them?I am not saying that Newtonian Mechanics is wrong. this has nothing to do with being right or wrong. I'm talking about how do we explain an theory evolution without wanting to find convergencies that were created just to make understanding easier.

Great article @ZapperZ . I especially liked the part where you talk about the thought processes of the public as compared to physicists (I would use the broader term "technically trained" instead of physicist) .

We usually discourage talk about philosophy here on PF. But in the meaning of the word, "thinking about thought" it goes the heart of your point. I believe that you are correct; that fundamentally different thought processes forever divide the technical and liberal views of our world.

Well obviously Newtonian mechanics is "wrong"Obviously it is not wrong. It has been experimentally verified by a large number of experiments.

People tend to think of experimental falsification as much stronger than experimental verification. But it is not. The fact that a theory later produces incorrect results in some new domain does not change the fact that it does produce correct results in the previous domains.

So “wrong” would be a description for a theory that is never correct in any domain. Other than that theories are “applicable” or not to a given domain, or they are more or less “general” than another theory.

I said that there is no such a thing as a convergence from a new theory to an old one based on some arbitrary limitI am sorry, but this is just ignorant and the limit is

notarbitrary. Obviously you will not recover classical mechanics in the ultra-relativistic limit, you will recover it in the classical limit where speeds are much smaller than the speed of light.In physics and

anyempirical science, a theory is nothing else than the sum of its predictions. This is why discussions on quantum mechanics interpretations tend to degenerate and people either leaving upset or agreeing to disagree.Apart from that:

– Newton considered space as an R³ euclidean and time as absolute.

– Mass and energy were not connected the way we know now.The first of these follows directly in the classical limit of SR. The second also falls out of the theory as separate conservation of mass and kinetic energy at small velocities.

– How would we explain the double slit experiment?When it comes to light it is very well explained using Maxwell's theory. When it comes to particles such as electrons, it is not a prediction of the classical theory but your logic is completely flawed. You have taken one of the shortcomings of the classical theory that inspired people to QM and presented it as a counter argument to the quantum theory having to reduce to the classical theory

in the limits where we know that the classical theory works. We already know that the classical theory does not work in this limit and so your argument is empty. Essentially you argument to the statement "A must give the same predictions as B when C is true" is to say "but it does not give the same predictions when C is not true". We come back to your assertion that the limit is arbitrary which, again, is not the case.Well obviously Newtonian mechanics is "wrong" since it does include relativity or quantum mechanics but that does not imply that it "shouldn't be used". Newtonian mechanics is "wrong" in the sense that it is not the real Universe but that does not imply that it is wrong for you to use it because usually the error intrinsic to Newtonian mechanics is small enough that you can safely ignore it.

@ZapperZ great article! This is why, more than 100 years after Einstein, students still learn Newton. I have had discussions with people who are convinced that because Einstein proved Newton wrong someone will eventually prove Einstein wrong. But future generations of students will still need to learn relativity even if a Lorentz violating theory is eventually verified.

Thank you for an enjoyable, edifying read.

do examples prove that a theory is correct?Obtaining examples where a theory works or doesn’t work is the whole purpose of experiments in the scientific method. We don’t usually use the word “prove”, but examples do verify or falsify theories.

In Helen Quinn's essay in Physics Today, she emphasized the same issue:

We also know that they are not complete. Although they are well tested in some domains, in others uncertainties remain about their detailed application. Indeed, we expect that they will be modified or extended to explain new evidence. But they will not disappear; just as Newton's laws did not become invalid when we understood special relativity, but rather were seen to be a very accurate approximation under well-defined conditions. Theories such as those listed in the previous paragraph are strong enough that we can use them to say we know certain things – we know that protons and neutrons are composed of quarks and gluons, we know the relationship between mass and energy, we know that Earth's surface is not a single rigid structure. These are facts, but not just simple observational facts. They come from the amalgam of observation and theory development and testing that is the essence of scientific knowledge development. It diminishes the status of our understanding greatly to say that scientists "believe" these things. We know them!

When we seek to extend and revise our theoretical frameworks, we make hypotheses, build models, and construct untested, alternate, extended theories. These last must incorporate all the well-established elements of prior theories. Experiment not only tests the new hypotheses; any unexplained result both requires and constrains new speculative theory building – new hypotheses. Models, and in the modern world computer simulations too, play an important tole here. They allow us to investigate and formulate the predictions and tests of our theory in complex situations. Our hypotheses are informed guesses, incorporating much that we know. They may or may not pan out, but they are motivated by some aspects or puzzles in the existing data and theory. We actively look for contradictions.There is an under-emphasis of the significance and importance of the claim that "it works!". Nothing is more powerful than having experimental evidence, and having it being verified and used every single day. Sure, as an experimentalist, I am probably very bias towards experimental evidence. However, this is what clearly separates science from non-science, and what makes acceptance in physics extremely difficult due to the requirement of having clear and convincing empirical evidence.

Zz.

I never said that Newtonian mechanics was wrong. never said that. I said that there is no such a thing as a convergence from a new theory to an old one based on some arbitrary limit as they start from different assumptions.So your objection is to the term "convergence", taken as an indication that the new theory must, in the appropriate limit, be precisely identical to the old theory and not merely identical to within current experimental uncertainties.

Yes, that is certainly true. [Though precise agreement in the limit is what we have come to expect]

I think you are rather missing the point. If you are travelling at 30mph down the road and a car passes you going 30mph faster, you can calculate its road speed using Newtonian or Einsteinian relativity. You will get the same answer to the precision you can plausibly measure. You need to take into account the velocity variation from flies crashing into the front of the car long before you need to care about Einstein. So simplifying the maths and using Newton isn't wrong. How can it be? Unless you're saying it's wrong to neglect the flies. And if you believe that then you will get nothing done ever.I never said that Newtonian mechanics was wrong. never said that. I said that there is no such a thing as a convergence from a new theory to an old one based on some arbitrary limit as they start from different assumptions.

It does not matter if you derivade classical physics from modern theories, they have different assumptions! they do not talk about the same thing!I think you are rather missing the point. If you are travelling at 30mph down the road and a car passes you going 30mph faster, you can calculate its road speed using Newtonian or Einsteinian relativity. You will get the same answer to the precision you can plausibly measure. You need to take into account the velocity variation from flies crashing into the front of the car long before you need to care about Einstein. So simplifying the maths and using Newton isn't wrong. How can it be? Unless you're saying it's wrong to neglect the flies. And if you believe that then you will get nothing done ever.

If a new theory does not produce approximately same answers as the classical theory in the realm where the classical theory has been tested experimentally and demonstrated to be accurate then that new theory must not be correct.

Experiment is the final arbiter of correctness.Of course the new theory will have to explain the old stuff, but it does not mean that it will "converge" at some point to the old one. there is no connection between the two things.

Apart from that:

– Newton considered space as an R³ euclidean and time as absolute.

– Mass and energy were not connected the way we know now.

– How would we explain the double slit experiment?

The assumptions can't be ignored.

Another great Insight, thanks Zapperz!

– "any new theory or description must somehow converge and look like the classical physics description under such ordinary conditions.". Who said this "must" be true?If a new theory does not produce approximately same answers as the classical theory in the realm where the classical theory has been tested experimentally and demonstrated to be accurate than that new theory must not be correct.

Experiment is the final arbiter of correctness.

– "Classical physics is used in an overwhelming majority of situations in our lives." so what? do examples prove that a theory is correct? this argument makes no sense at all.

– "any new theory or description must somehow converge and look like the classical physics description under such ordinary conditions.". Who said this "must" be true? The point here is that SR and QM deal with different concepts of reality which are not accounted in classical mechanics. We are looking at the old theories with our "present" eyes. We can't interpret the past this way. The assumptions are totally diferent.

– "They have identical form!". Again: having the identical equation form proves what? that they have the same assumptions? that they work on the same space with the same properties? of course not!

It does not matter if you derivade classical physics from modern theories, they have different assumptions! they do not talk about the same thing!

Definitely worth saying. Asimov wrote an article in a similar vein (tip o' the hat to @Nugatory for introducing me to it): http://chem.tufts.edu/answersinscience/relativityofwrong.htm