The Feynman way of explaining Symmetry in Physical laws

[mark2142]

TL;DR Summary

I want to discuss and get a better understanding of the proof about symmetry explained by Feynman in his lecture for undergraduates.

So on this page https://www.feynmanlectures.caltech.edu/I_11.html under heading 11-2 Translations first he tries to proof that there is no origin in space. Joe writes newtons laws after measuring quantities from some origin.
$$m(d^2x/dt^2)=F_x$$
$$m(d^2y/dt^2)=F_y$$
$$m(d^2z/dt^2)=F_z$$

We need to proof-
$$1. m(d^2x'/dt^2)=F_x'$$
$$2. m(d^2y'/dt^2)=F_y'$$
$$3. m(d^2z'/dt^2)=F_z'$$

Now, Moe has his parallel system of coordinates.

The translation eqns are-
$$x'=x-a$$
$$y'=y$$
$$z'=z$$
Forces measured by both are-
$$F_{x'}=F_x$$
$$F_{y'}=F_y$$
$$F_{z'}=F_z$$
Now, $\frac d{dt} {x'}(t)=\frac d{dt}(x-a)=\frac d{dt}{x}(t)-\frac d{dt}{a}(t)$
Then, as we solve we get $$\frac {d^2x'}{dt^2}=\frac {d^2x}{dt^2}$$
Therefore 1. becomes, $$m(d^2x/dt^2)=F_{x'}$$
and since $F_{x'}=F_x$
So, $$m(d^2x/dt^2)=F_{x}$$.
Hence we reproduced the correct eqn from 1. So if the original eqn is true (by Joe) then 1. is also true. Hence we have written the newtons eqn with different coordinates and so there is no one origin in universe.The Newtons Law is same when observed from different points.

(I can't seem to understand the logic used below in bold letters by feynman).

This is also true: if there is a piece of equipment in one place with a certain kind of machinery in it, the same equipment in another place will behave in the same way. Why? Because one machine, when analyzed by Moe, has exactly the same equations as the other one, analyzed by Joe. Since the equations are the same, the phenomena appear the same. So the proof that an apparatus in a new position behaves the same as it did in the old position is the same as the proof that the equations when displaced in space reproduce themselves. Therefore we say that the laws of physics are symmetrical for translational displacements, symmetrical in the sense that the laws do not change when we make a translation of our coordinates. Of course it is quite obvious intuitively that this is true, but it is interesting and entertaining to discuss the mathematics of it.

 

[hutchphd]

Perhaps I do not understand your question. The term "laws of physics" is simply replaced by "measurement of result" then this follows. What specifically do you not flollow in Feynman's argument??

 

[mark2142]

What specifically do you not flollow in Feynman's argument??

This!

if there is a piece of equipment in one place with a certain kind of machinery in it, the same equipment in another place will behave in the same way. Why? Because one machine, when analyzed by Moe, has exactly the same equations as the other one, analyzed by Joe. Since the equations are the same, the phenomena appear the same.

 

[topsquark]

This!

There is a principle of Physics (I can't recall the name at the moment) where we assume that the Laws of Physics are the same at every point in the Universe. Obviously, we can't go to the Andromeda galaxy and test this, though we have plenty of indirect evidence for it. But it's still an assumption.

Now, the whole point of it is that we can't really do Physics otherwise. I mean, if we don't assume this is true then how can we possibly make observations of distant phenomena and try to explain them? For now, we don't really have any choice but to make the assumption and that's what Feynman's statement is about.

Note: Mind you, we've never seen any evidence that the Laws of Physics might be different in remote locations. (We can pretty conclusively say that the principle is true within our Solar System, at the very least.) I only bring up the possibility because we have no direct evidence for it to be so. But I really don't have any doubt that this principle is correct and I doubt that many other Physicists do either, except perhaps as a thought in the back of their minds. The assumption needs to be tested but, as time goes on, it's looking like less of an assumption and more of a fact.

-Dan

 

[Ibix]

There is a principle of Physics (I can't recall the name at the moment) where we assume that the Laws of Physics are the same at every point in the Universe.

Just to add, if the laws of physics do vary and they do so in a systematic way then the problem is that we made a mistake when we said we knew the laws of physics. Or, at least, the laws we wrote were not fundamental. And we can learn the variation and write more general laws.

For example, you could study free-fall near a small region of Earth's surface and conclude that "everything falls with an acceleration of 9.81m/s". But then you look up at the heavens and find that it isn't generally true. Does that mean that the laws of physics vary across the solar system? Or do we need better laws? Such a better law is Newton's law of gravity, which explains why our earlier idea was plausible and correctly predicts the acceleration where the earlier one doesn't. More detailed study will show that Newton isn't quite right near the Sun. Again, does that mean that the laws of physics vary, or do we just need Einstein's theory of gravity?

On the other hand, if there is variation in the laws and there is no system to it (e.g. planets in Andromeda don't orbit, they dance the conga while the stars play mariachi music) then the scientific method is no use at some point anyway.

 

[gmax137]

There is a principle of Physics (I can't recall the name at the moment) where we assume that the Laws of Physics are the same at every point in the Universe.

I don't think this is what Feynman is talking about. His argument is based on using Newton (F=ma) and showing that regardless of your coordinate system, you get the same "a" from a given "F." He is not addressing the idea that over in galaxy Aristotle, F=mv.

The bit about the machinery, I think he's saying if you have some device that includes, say, a pendulum, and you move it from here on earth, to Andromeda, it will still operate as $T=2 \pi \sqrt{\frac {L} {g}}$

 

[vanhees71]

It's about symmetries, and here it's about symmetries of Newton-Galilei spacetime. By assumption you have time-translation invariance (homogeneity of time), i.e., the physical laws don't change with time. Then there's translational invariance (homogeneity of space, which follows from the assumption that space is a Euclidean affine manifold), spatial rotational invariance (isotropy), and together with the homogeneity symmetry it's rotational invariance around any point of space, and finally there's invariance under "Galilei boosts", i.e., if you go from an inertial frame of reference to another frame of reference which moves with constant velocity against the inertial frame, you have again an inertial frame, and the physical laws must look the same in all inertial frames (also by assumption, i.e., the special principle of relativity).

Now you can use Noether's theorem (assuming that the physical laws can be formulated in terms of Hamilton's action principle) to derive how the fundamental laws have to look to fulfill all the symmetries (for closed systems).

Of course, at Andromeda you won't have the same gravitational acceleration as on Earth, but that's because there's no planet exactly like the Earth. What's however the fundamental law, describing the gravitational interaction is one that has the right form to fulfill all the symmetry constraints of Galilei-Newton spacetime, i.e., it's a central two-body force,
$$\vec{F}_{12}=-G m_1 m_2 \frac{\vec{r}_1-\vec{r}_2}{|\vec{r}_1-\vec{r}_2|^3}=-\vec{\nabla}_1 \left (-\frac{G m_1 m_2}{|\vec{r}_1-\vec{r}_2|} \right)=-\vec{F}_{21}=-\vec{\nabla}_2 \left (-\frac{G m_1 m_2}{|\vec{r}_1-\vec{r}_2|} \right),$$
i.e., also Newton's Lex III follows from the Galilei symmetries (more specifically it follows from spatial translation invariance, related to the conservation of the total momentum of a closed system). From the Galilei symmetry the statement about gravity is that the Newton gravitational constant, $G$, is the same everywhere and at any time.

 

[mark2142]

Thank you guys for your valuable comments. Its very informative. But can somebody please explain me the actual logic used by feynman to prove the symmetry in translational motion?
(In bold letters)

 

[Ibix]

I'm not sure he is proving anything there. I think the maths above it shows that the laws of physics (as we know them) are translation invariant. And the reasoning in bold shows that any measurement you make is also translation invariant because the measuring devices are themselves subject to those translation invariant laws.

There are two threads to that - one is the observation that rulers and clocks and whatever work the same and mesh together the same the world over. The other is that laws we write are translation invariant (even though that wasn't a property anyone explicitly demanded). So we end up with an abstract explanation for why devices work the same wherever they are, which is that they follow natural laws which are translation invariant.

It's perhaps worth contrasting this with electromagnetism in the second half of the 19th century. The laws were translation invariant but not velocity invariant - so me standing by my apparatus and you walking past it couldn't do an analogue of Moe and Joe's analysis, and we would have had very different and conflicting understanding of what was happening. We were clearly missing something, which is why cutting edge physics 1860-1905 was all about hunting for the ether and understanding how that would correct our model of EM. The solution was actually relativity, which allowed us once again to analyse each other's apparatus with the same physical laws and predict the same measurements.

 

[PeroK]

Thank you guys for your valuable comments. Its very informative. But can somebody please explain me the actual logic used by feynman to prove the symmetry in translational motion?
(In bold letters)

The actual logic is the following valid syllogism:

Major premise: things governed by the same laws of physics work the same way.

Minor premise: a machine here and a machine there are governed by the same laws of physics.

Conclusion: a machine here and a machine there work the same way.

 

[mark2142]

This is also true: if there is a piece of equipment in one place with a certain kind of machinery in it, the same equipment in another place will behave in the same way. Why? Because one machine, when analyzed by Moe, has exactly the same equations as the other one, analyzed by Joe. Since the equations are the same, the phenomena appear the same. So the proof that an apparatus in a new position behaves the same as it did in the old position is the same as the proof that the equations when displaced in space reproduce themselves. Therefore we say that the laws of physics are symmetrical for translational displacements, symmetrical in the sense that the laws do not change when we make a translation of our coordinates.

The actual logic is the following valid syllogism:

Major premise: things governed by the same laws of physics work the same way.

Minor premise: a machine here and a machine there are governed by the same laws of physics.

Conclusion: a machine here and a machine there work the same way.

Are there two machines or one machine?
Also I am unable to correlate the math that says-

The Newtons Law is same when observed from different points.

with what you are saying.

 

[mark2142]

I think the maths above it shows that the laws of physics (as we know them) are translation invariant.

The math says the law is same from different observing points. The math itself doesn't prove that law is translational invariant.

 

[Ibix]

The math says the law is same from different observing points. The math itself doesn't prove that law is translational invariant.

It proves that the law is independent of your choice of origin. What do you understand by "translation invariance" if that's not it?

 

[mark2142]

It proves that the law is independent of your choice of origin.

Does this mean we can take a origin and shift the law from that origin and it will make no difference to the law?

 

[PeroK]

Also I am unable to correlate the math that says-

with what you are saying.

You explicitly asked for the "logic" and I gave you the logic.

 

[Ibix]

Does this mean we can take a origin and shift the law from that origin and it will make no difference to the law?

Yes. That's what Moe's $x'=x-a$ does.

 

[PeroK]

Does this mean we can take a origin and shift the law from that origin and it will make no difference to the law?

If not, where is the one absolute origin?

 

[mark2142]

It proves that the law is independent of your choice of origin.

If not, where is the one absolute origin?

Why are we focusing on origin? We should be focusing on the law. That there is no absolute law at a point.

 

[Ibix]

Why are we focusing on origin?

Because there being no unique origin is what translation invariance means.

We should be focusing on the law. That there is no absolute law at a point.

I don't know what you mean by this.

 

[vanhees71]

The point is that we start with an assumption of how to describe spacetime. In the case of Newtonian mechanics it's Lex I, i.e., the special principle of relativity, saying that there exist inertial reference frames. Further it's assumed that time is a coordinate of an oriented one-dimensional affine manifold, defining the causal order and independent of any physical influences (Newton's absolute time; while the assumed equivalence class of inertial reference frames defines in some way Newton's absolute space, but it's not really absolute, because there's no preferred inertial frame that can be physically determined, leading to the symmetry under Galilei boosts, i.e., you cannot define in any meaningful physical way an absolute velocity but only relative velocities between different inertial frames). Then, of course, Newton assumes that for any inertial observer space is a Euclidean affine manifold, i.e., it fulfills the mathematical laws of Euclidean 3D geometry.

All this implies "symmetries", i.e., the impossibility to determine any preferred point in time (time-translation invariance; homogeneity of time), the impossibility to determine any preferred point in space (space-translation invariance; homogeneity of space), the impossibility to determine any preferred direction around any point in space (rotation invariance; isotropy of space), and the already above mentioned impossibility to determine an absolute velocity (Galilei-boost invariance).

All this implies that, for a closed system, the physical laws must be invariant under the corresponding transformations between reference frames, and according to a famous theorem by Emmy Noether each symmetry then implies the validity of a conservation law. In the case of the Newtonian space-time symmetries discussed above these are:

homogeneity of time <-> conservation of total energy
homogeneity of space <-> conservation of total momentum
isotropy of space <-> conservation of total angular momentum
Galilei boost invariance <-> the velocity of the center of mass is constant

One can also derive, how the dynamical laws must look like to a large extent from these symmetries. Particularly Newtons Lex II and III follow from the symmetries.

 

[Ibix]

Just to add to @vanhees71's post, that is very much a modern analysis of laws that were developed from experiment and observation. So his "it all comes from the structure of space and time and their symmetries" is backwards with respect to the way physics developed historically. It's a more integrated view, and turned out to be very powerful, but it's very different from the more mechanistic way the subject was developed and frequently is taught.

 

[vanhees71]

It's of course much more demanding concerning the level of mathematical sophistication.

 

[mark2142]

Because there being no unique origin is what translation invariance means.

If I understand it correctly then we measured a law from say O origin and assume its right. Then we measure the same law from O' origin and we don't get the same law. That means there is a absolute origin and space from which the law is right. So if we displace the law the law changes. That means the physical laws are translational variant. Yes?

 

[Ibix]

That means the physical laws are translational variant. Yes?

That would mean the laws were not translationally invariant if that's what were found, yes.

 

[mark2142]

That would mean the laws were not translationally invariant if that's what were found, yes.

This does not happen. We can write the correct laws using the new coordinates. There is no absolute space and origin from which the laws are right. So its right everywhere or from everywhere?

 

[PeroK]

This does not happen. We can write the correct laws using the new coordinates. There is no absolute space and origin from which the laws are right. So its right everywhere or from everywhere?

Yes.

 

[mark2142]

Yes.

I don’t follow. How it’s right everywhere ?

 

[PeroK]

I don’t follow. How it’s right everywhere ?

The idea is that if you do an experiment in your bathtub, you don't have to look up the laws of physics that apply specifically in your bathtub. You can use the same laws of fluid mechanics that everyone else uses.

Otherwise, we'd have a different set of laws of physics for every bathtub - and every swimming pool etc.

 

[PeroK]

... and, for example, Newton's laws of motion didn't only apply in Newton's study or lab in the 17th Century. They apply everywhere and equally well today as they did 350 years ago.

I'm not sure why that's a troublesome idea to be honest.

What's the alternative?

 

[Ibix]

I don’t follow. How it’s right everywhere ?

There's an important distinction here. The maths proves that our mathematical model of the universe has translation invariant laws. Therefore our mathematical model of an instrument functions the same everywhere.

But is that true in the real world? Formally, we can only say that we have never seen it behave otherwise. Every time we apply this force to that spring it always extends so much, wherever we are. That's why our model is built the way it is, after all. But it's always possible that there's some really tiny effect that we haven't spotted yet that does violate translation invariance. It's hard to imagine how such a thing could exist and be still hidden, but it's possible.

So Feynman is proving that his model of the laws of physics are translation invariant and hence his equipment is predicted to be translation invariant. He does not prove (and you cannot prove) that this is a correct model of reality, but it's been an accurate model every time so far.

 

[gmax137]

There's an important distinction here.

Yes, and unfortunately this thread bounces back and forth between the two ideas.

 

[Ibix]

Yes, and unfortunately this thread bounces back and forth between the two ideas.

Actually, I think the original quote does too: "Since the equations are the same, the phenomena appear the same." (Or I'm missing something...)

 

[hutchphd]

So Feynman is proving that his model of the laws of physics are translation invariant and hence his equipment is predicted to be translation invariant. He does not prove (and you cannot prove) that this is a correct model of reality, but it's been an accurate model every time so far.

But this caveat is, unless specifically stated otherwise , fundamentally part of the physics canon.
Perhaps Feynman should have said "Since the equations are the same, canonically the phenomena must appear the same." but this seems a bit pedantic. The book is after all a physics text.

 

[Ibix]

But this caveat is, unless specifically stated otherwise , fundamentally part of the physics canon.

Indeed. But I certainly managed to get a long way through my physics education without coming across a clear statement of the distinction between the mathematical models (about which you can formally prove things) and reality (where you can only test). I don't think Feynman's being particularly clear on that distinction here, and perhaps you are right that it would be pedantic of him to do so, but I wonder if it's confused the OP.

 

[mark2142]

The idea is that if you do an experiment in your bathtub, you don't have to look up the laws of physics that apply specifically in your bathtub. You can use the same laws of fluid mechanics that everyone else uses.

Otherwise, we'd have a different set of laws of physics for every bathtub - and every swimming pool etc.

That I understand. I meant how it’s right everywhere from the logic I gave?

This does not happen. We can write the correct laws using the new coordinates. There is no absolute space and origin from which the laws are right.