# Extreme Length Contraction: Classical to Quantum?

• I
• Silviu

#### Silviu

Hello! If we have a 1m stick (as measured in its stationary reference frame, call it S) and we (S') move with a high enough velocity, we can make the length of the stick in our frame as small as we want. So for high enough velocities, the stick will appear so small in our frame, S', that it will reach a quantum domain i.e. we need a wavefunction to describe it. Does this mean that the same object can be described in one frame using classical physics but in the other we need quantum mechanics? Will the 2 descriptions still coincide, as we have a deterministic description vs a probabilistic ones? I am just a bit confused about the actual transition from classical to quantum.

Hello! If we have a 1m stick (as measured in its stationary reference frame, call it S) and we (S') move with a high enough velocity, we can make the length of the stick in our frame as small as we want. So for high enough velocities, the stick will appear so small in our frame, S', that it will reach a quantum domain i.e. we need a wavefunction to describe it. Does this mean that the same object can be described in one frame using classical physics but in the other we need quantum mechanics? Will the 2 descriptions still coincide, as we have a deterministic description vs a probabilistic ones? I am just a bit confused about the actual transition from classical to quantum.
Since the stick doesn't actually change size and we are only talking about an observed size due to length contraction, I don't see why QM should enter the picture (but I could be wrong)

Since the stick doesn't actually change size and we are only talking about an observed size due to length contraction, I don't see why QM should enter the picture (but I could be wrong)
Oh, I am also not sure what would actually happen. But I was thinking that if the size of the stick get smaller than (say) a proton, why would one be allowed to use classical mechanics? Actually, for a small enough length, using classical mechanics would violate the uncertainty principle.

for high enough velocities, the stick will appear so small in our frame, S', that it will reach a quantum domain

No. Special relativity by itself is a classical theory. If you want to use a quantum model, you need to use it in all frames, not just one. And to properly do that, you need quantum field theory, i.e., a quantum model that takes proper account of relativity and how things change between frames.

I am just a bit confused about the actual transition from classical to quantum.

It's not a "transition", it's a choice of models. Which model works best depends on what you are trying to model. For a stick, unless you are trying to predict the precise wavelengths of light that it will absorb or emit, I can't think of a reason why you would need to use a quantum model.

No. Special relativity by itself is a classical theory. If you want to use a quantum model, you need to use it in all frames, not just one. And to properly do that, you need quantum field theory, i.e., a quantum model that takes proper account of relativity and how things change between frames.

It's not a "transition", it's a choice of models. Which model works best depends on what you are trying to model. For a stick, unless you are trying to predict the precise wavelengths of light that it will absorb or emit, I can't think of a reason why you would need to use a quantum model.
But won't S' need quantum mechanics to describe the dynamics of the stick (or QFT)?

won't S' need quantum mechanics to describe the dynamics of the stick (or QFT)?

Not unless S' is trying to model a phenomenon that requires quantum mechanics. If he's just trying to model, say, how the stick will fit through a hole, he can use the classical model just fine.

Not unless S' is trying to model a phenomenon that requires quantum mechanics. If he's just trying to model, say, how the stick will fit through a hole, he can use the classical model just fine.
But for S' the length of the stick can be very small. So if he has to answer a basic question (for S) such as where is the center of mass of the stick, can he do that without using quantum mechanics? Or what is the momentum of the stick?

for S' the length of the stick can be very small. So if he has to answer a basic question (for S) such as where is the center of mass of the stick, can he do that without using quantum mechanics? Or what is the momentum of the stick?

Sure. As long as no quantum phenomena are involved, he can do all this just fine. And if there are quantum phenomena involved, he'll need to use a quantum model in any frame, not just S'.

Sure. As long as no quantum phenomena are involved, he can do all this just fine. And if there are quantum phenomena involved, he'll need to use a quantum model in any frame, not just S'.
But won't the uncertainty principle be violated? Isn't it like trying to model an electron using classical mechanics? My point is, what is the difference for S' between an electron (in its own frame) and this super contracted stick whose size can become the same as that of an electron (ignoring charge, spin etc. talking only about size). So why you are free to choose between classical and quantum to describe this sub-microscopical object, but you have to use quantum for an electron (in order to obtain the right predictions) when they both have similar sizes?

won't the uncertainty principle be violated? Isn't it like trying to model an electron using classical mechanics?

No, because a stick is not an electron.

My point is, what is the difference for S' between an electron (in its own frame) and this super contracted stick whose size can become the same as that of an electron (ignoring charge, spin etc. talking only about size).

The fact that the stick made of ##10^{23}## or so atoms, each of which contains multiple electrons. That doesn't change just because the length contracted length of the stick in S' happens to be the same as the electron's Compton wavelength, or whatever you think an appropriate "size" of an electron is.

You need to stop thinking of this in terms of "size" and start thinking of it in terms of "what is being modeled".

why you are free to choose between classical and quantum to describe this sub-microscopical object, but you have to use quantum for an electron (in order to obtain the right predictions) when they both have similar sizes?

I didn't say you were "free to choose between classical and quantum" for modeling the stick. I said it depends on what you are modeling. I've already given one example where you would have to use a quantum model for the stick. And if, for example, you were modeling electrons in a cathode ray tube for the purpose of predicting how the tube will work as a television set, you can model the electrons as classical particles emitted by a "gun" inside the tube just fine; you don't need QM, even if you set the tube to emit one electron at a time.

No, because a stick is not an electron.

The fact that the stick made of ##10^{23}## or so atoms, each of which contains multiple electrons. That doesn't change just because the length contracted length of the stick in S' happens to be the same as the electron's Compton wavelength, or whatever you think an appropriate "size" of an electron is.

You need to stop thinking of this in terms of "size" and start thinking of it in terms of "what is being modeled".

I didn't say you were "free to choose between classical and quantum" for modeling the stick. I said it depends on what you are modeling. I've already given one example where you would have to use a quantum model for the stick. And if, for example, you were modeling electrons in a cathode ray tube for the purpose of predicting how the tube will work as a television set, you can model the electrons as classical particles emitted by a "gun" inside the tube just fine; you don't need QM, even if you set the tube to emit one electron at a time.
Ok, so I understand the part with classical vs quantum depends on what you want to model (and the precission you need). What confuses me is that one of the first things we were being told in the QM course is that QM applies to small scales (actually it was more like QM small, SR fast, QFT small and fast, and classical mechanics to big and slow). So all this time, I believed that if you have an object small enough (and yeah, this is also relative, but you can make it as small as you want if you go to high enough velocities), you have to use QM in order to accurately describe it's dynamics (i.e. Schrodinger equation, or Dirac, or KG etc.), you can't apply ##F=ma## (again, assuming that you want high precision). So for example, for a small object, asking where it is located makes no sense. You should ask what is the probability of it being located at a given point. Now this is what I got from my first lesson of QM. In this case, the object is small, so I would be tempted to use for example Schrodinger equation to model it's dynamics (find its wavefunction in position space for example). However you are saying (if I understand it well), that we can use classical mechanics to specify, for example, the position or momentum of the stick. And now I am confused. Was my understanding of "if small use QM" wrong? Just to make things easier, so we know what we want to model, let's say we are interested in the position of the CM of the stick. Can S' use classical mechanics to specify that? And if so, what is wrong with what I understood from the discussion above?

Was my understanding of "if small use QM" wrong?

Yes.

A better heuristic would be that QM is used to model systems with a very small number of microscopic degrees of freedom. A stick has roughly ##10^{23}## of them. A single electron has only a few (one if you only need to model its spin; potentially four if you need to model its spin plus its position in 3 dimensions).

let's say we are interested in the position of the CM of the stick. Can S' use classical mechanics to specify that?

Yes. The CM position is a macroscopic degree of freedom, not a microscopic one. Heuristically, that is because the CM position is obtained by averaging over all of the positions of the ##10^{23}## or so atoms in the stick, so all of the quantum stuff averages out and you're left with one degree of freedom (more precisely three if you are working in three dimensions) that can be modeled classically. Whereas, if you just have one electron, there's no averaging of anything: you just have its position degree of freedom (or three of them in three dimensions, plus its spin, but we can ignore that if position is all we're interested in), which needs to be modeled quantum mechanically unless you are dealing with a situation where no quantum uncertainty is involved (like the electron being shot from the gun in the cathode ray tube).

Yes.

A better heuristic would be that QM is used to model systems with a very small number of microscopic degrees of freedom. A stick has roughly ##10^{23}## of them. A single electron has only a few (one if you only need to model its spin; potentially four if you need to model its spin plus its position in 3 dimensions).

Yes. The CM position is a macroscopic degree of freedom, not a microscopic one. Heuristically, that is because the CM position is obtained by averaging over all of the positions of the ##10^{23}## or so atoms in the stick, so all of the quantum stuff averages out and you're left with one degree of freedom (more precisely three if you are working in three dimensions) that can be modeled classically. Whereas, if you just have one electron, there's no averaging of anything: you just have its position degree of freedom (or three of them in three dimensions, plus its spin, but we can ignore that if position is all we're interested in), which needs to be modeled quantum mechanically unless you are dealing with a situation where no quantum uncertainty is involved (like the electron being shot from the gun in the cathode ray tube).
Ok, I see where I was wrong with my interpretation. However, let's say we will be able to test higher energy (up to Plank scale) and discover that (this is probably not realistic at all, but I don't think it would be forbidden by any law) the electron is formed of many smaller part. Based on what you said, now the electron would be in the same position of the stick i.e. it will have a very large number of degrees of freedom so we can, for example, specify its center of mass. But we know very well that we have to use QM in order to get accurate predictions and we can't specify its CM. So what am I missing here? Based on this, if we didn't know that the stick was made of something, we could have used QM. The fact that we know it is made of electrons, means we don't have to use QM. So our knowledge of it's composition dictates what we can use? Does this mean that the fact that QM works for electron is a proof that it is not formed of (at least not too many) smaller parts?

• Ibix
Based on what you said, now the electron would be in the same position of the stick

No, we wouldn't, because, as you say, we already know we need QM to model single electrons. So even if we end up discovering some underlying theory at the Planck scale that says an electron is composed of smaller parts, the parts won't "fit together" to form an electron the way ##10^{23}## atoms fit together to form a stick.

Based on this, if we didn't know that the stick was made of something, we could have used QM.

Nope. We already knew, even before we discovered QM, that classical physics worked to model sticks for all the cases where classical physics does work to model sticks. That doesn't change just because we discovered something new about what sticks are made of. The only thing that changes because we discovered sticks are made of atoms is that we now have to develop an understanding of why the classical model does work for sticks in the cases where it works, given that we know sticks are made of atoms. That's what the reasoning I gave about quantum effects canceling out does. But just because that reasoning works in that particular case, doesn't mean it has to work for any case at all where something is "made of" a lot of smaller parts. You have to look at the details of each case.

In other words, you are still failing to think in terms of "what are we trying to model". Classical physics and quantum physics are models. We use them in particular cases because those are the cases in which they work. Knowing that they work in a particular case doesn't change if we later discover that different models of the same things work in different cases.

So for high enough velocities, the stick will appear so small in our frame, S', that it will reach a quantum domain i.e. we need a wavefunction to describe it.

It's the rest length that's used to make that consideration. The stick's rest length is invariant, that is, it does not depend on the speed of S'.

I am just a bit confused about the actual transition from classical to quantum.
The transition from classical to quantum does not occur when the distances are sufficiently small. It occurs when the product ##\Delta x \Delta p## is sufficiently small (of the order of ##\hbar## or less).