Why symmetry is not important in classical physics?

[wdlang]

but ultra important in quantum physics?

i can see it is important in quantum physics

but i can not see why it is less important in classical physics.

 

[Demystifier]

I don't think that symmetry is less important in classical mechanics than in quantum mechanics. Why do you think it is?

 

[wdlang]

I don't think that symmetry is less important in classical mechanics than in quantum mechanics. Why do you think it is?

i often see this statement in literatures

like 'discrete symmetry is useless in classical physics'

 

[Demystifier]

i often see this statement in literatures

like 'discrete symmetry is useless in classical physics'

I've never seen this statement. And it is certainly wrong; discrete symmetry is not useless in classical physics.

It reminds me on an interesting "paradox" in CLASSICAL physics. Why a mirror exchanges left and right, but does not exchange up and down? (I know the answer, of course.)

 

[MaxwellsDemon]

Symmetry is ultra important in classical physics too. Our most cherished conservation laws are related to symmetries. Without symmetry there would be no conservation of energy or momentum in classical physics, so I'd hardly say that symmetry isn't important. Check out Noether's Theorem.

 

[kote]

Right; I was thinking the same thing. Conservation laws are all about symmetry, and you have "for every action..."

 

[bcrowell]

I think wdlang's question is an intriguing one. It is certainly true that symmetry is prominently featured in most textbook treatments of quantum mechanics, whereas it's typically not even mentioned in most textbook treatments of classical physics. However, this may just be a fact about textbooks. Textbooks tend to be extremely bound by tradition, for economic reasons; it's hard to sell a book that isn't similar to the ones that teachers are familiar with. Noether's theorems are only about a hundred years old, whereas the standard format used for introducing Newton's laws in textbooks is probably much older than that. (Actually if you look at physics textbooks from ca. 1910-1930, they're heavily focused on practical devices like pulleys and butter churns. There's very little abstract theory -- not even much algebra used.) I have taught freshman physics using symmetry and conservation laws as a framework, but that's very uncommon. It is fairly common to see the symmetry features of spacetime discussed in treatments of special relativity. Some books do discuss Galilean invariance in their treatment of Newton's laws (e.g., Penrose's The Road to Reality).

 

[Haelfix]

Any suitably advanced classical mechanics text will have a lengthy discussion of symmetry. For instance, in the Kepler problems. You can derive say the Runge-Lenz vector on symmetry arguments alone, and that's pretty powerful

 

[fermi]

I agree with bcrowell's answer, but I have one other point to add: the dynamics possesses a lot of symmetry for QM and for Classical physics both, but the constraints and the boundary conditions do not necessarily have similar symmetries. Classical Dynamics can often be solved (at least approximately) with very complicated constraints and/or boundary and/or initial conditions. Quantum Dynamics cannot be solved unless the the constraints and the boundary conditions are reasonably simple and symmetric themselves. This is the reason why you will never see a textbook attempting to solve the free (yes I mean free) Schrödinger equation in three dimensions inside a closed surface with a complicated and arbitrary shape with no apparent symmetry. The QM textbooks will never go beyond a simple sphere, or a three dimensional rectangular box.

So the short answer is no, I don't see any difference in the amount of symmetry in classical or Quantum dynamics, but the classical physics is often solvable with complex and non-symmetric boundary conditions. QM is not. Such complicated arbitrary boundary conditions are often relevant for macroscopic objects, which the classical physics deals with. The naturally arising boundary conditions for the microscopic world tend to be much simpler and possesses a lot more symmetry themselves with rare exceptions.

 

[bcrowell]

The naturally arising boundary conditions for the microscopic world tend to be much simpler and possesses a lot more symmetry themselves with rare exceptions.

This is an interesting point. For instance, in low-energy nuclear physics we almost always have axial symmetry, and often we have spherical symmetry.

 

[wdlang]

I've never seen this statement. And it is certainly wrong; discrete symmetry is not useless in classical physics.

It reminds me on an interesting "paradox" in CLASSICAL physics. Why a mirror exchanges left and right, but does not exchange up and down? (I know the answer, of course.)

the paper i am now reading is PRL 67, 158 (1991) by Asher Peres

In his second paragraph, he has the statement above.

 

[Demystifier]

the paper i am now reading is PRL 67, 158 (1991) by Asher Peres

In his second paragraph, he has the statement above.

The sentence in the paper reads:
"... discrete symmetries, which are irrelevant in classical mechanics, may have "surprising" quantum effects."

It does NOT read:
"Discrete symmetries are irrelevant in classical mechanics."

The latter explicit statement is simply wrong, but the former statement can be tollerated as an imprecise expression. I think it was not Peres's intention to imply the latter statement.

 

[turin]

Doesn't E&M count as "classical" physics? For instance, in second semester undergraduate physics, students learn how to use symmetry with Gauss' law in order to determine the fields of some very symmetrical source distributions (e.g. sphere, line, plane). And, further on in the study of E&M (still at the undergrad level), student's find E&M modes in waveguides using very simple geometries (e.g. rectangular and cylindrical).

In other words, I think that fermi's comment applies to wave mechanics rather than quantum mechanics. That is, the wave nature rather than quantization is what makes symmetry essential to finding an exact solution.

 

[peteratcam]

It reminds me on an interesting "paradox" in CLASSICAL physics. Why a mirror exchanges left and right, but does not exchange up and down? (I know the answer, of course.)

What's your answer?
(I know the answer, of course :-) )

On symmetry, I remember this thread and thought I had made this point, but obviously I never posted it.

Eigenstates of quantum hamiltonians must transform under some representation of whatever symmetry groups the hamiltonian has - no such idea exists in classical mechanics.

The groundstate of a quantum hamiltonian is expected to have the full symmetry of the hamiltonian, whereas classical groundstates don't. 'Broken symmetry' is such a big deal precisely because of the expectation that quantum groundstates should have the full symmetry of the Hamiltonian.

 

[Demystifier]

What's your answer?
(I know the answer, of course :-) )

To start with, the mirror does NOT exchange left and right, just as it does not exchange up and down, as far as left, right, up, and down are defined as MY left, right, up, and down.

However, when I see my picture in the mirror, I interpret this not as a picture in the mirror, but as my "twin brother" looking at me. (Otherwise, there is no exchange of left and right to talk about.) In order for him to look at me, he must be ROTATED (for 180 degrees) with respect to me. But rotated with respect to which axis? There are essentially TWO different orthogonal choices: a vertical axis and a horizontal axis parallel with the mirror. If he chooses the vertical axis and looks to me like the picture in the mirror, then, in addition, HIS left and right must be exchanged. If he chooses the horizontal axis and looks to me like the picture in the mirror, then, in addition, HIS up and down must be exchanged.

So far we have a complete symmetry. Yet, the symmetry breaks because, for some reason, I tacitly assume that rotation is done with respect to the vertical axis and not with respect to the horizontal one. To explain this tacit assumption we need psychology rather than physics. I'm not a psychologist, but I think there are two reasons why my brain makes such a tacit assumption:
1. Humans are nearly symmetrical with respect to the left-right exchange, so a human with exchanged left and right still looks very much like a human. This is not so for the up-down exchange.
2. Physically it is much easier for a human to make a rotation with respect to the horizontal axis than with respect to the vertical one.

 

[peteratcam]

I think of it this way:
Get a square piece of paper, and write on it: Left Right Top Bottom, (in the appropriate places obviously!)
Stand directly in front of a mirror, but look at the piece of paper directly, so that it looks correct.
Now turn the paper round and look at it in the mirror.
You have an arbitrary choice of how to turn the paper round so that the writing faces the mirror, you can use what ever axis you like. Whether you flip left/right or top/bottom is entirely your choice. You can even exchange Left with Top and Right with Bottom if you want to by choosing a diagonal axis.

So as you say, the mirror doesn't flip anything, you do!

 

[Hurkyl]

I always thought the answer was "it flips front and back".

 

[Frame Dragger]

To start with, the mirror does NOT exchange left and right, just as it does not exchange up and down, as far as left, right, up, and down are defined as MY left, right, up, and down.

However, when I see my picture in the mirror, I interpret this not as a picture in the mirror, but as my "twin brother" looking at me. (Otherwise, there is no exchange of left and right to talk about.) In order for him to look at me, he must be ROTATED (for 180 degrees) with respect to me. But rotated with respect to which axis? There are essentially TWO different orthogonal choices: a vertical axis and a horizontal axis parallel with the mirror. If he chooses the vertical axis and looks to me like the picture in the mirror, then, in addition, HIS left and right must be exchanged. If he chooses the horizontal axis and looks to me like the picture in the mirror, then, in addition, HIS up and down must be exchanged.

So far we have a complete symmetry. Yet, the symmetry breaks because, for some reason, I tacitly assume that rotation is done with respect to the vertical axis and not with respect to the horizontal one. To explain this tacit assumption we need psychology rather than physics. I'm not a psychologist, but I think there are two reasons why my brain makes such a tacit assumption:
1. Humans are nearly symmetrical with respect to the left-right exchange, so a human with exchanged left and right still looks very much like a human. This is not so for the up-down exchange.
2. Physically it is much easier for a human to make a rotation with respect to the horizontal axis than with respect to the vertical one.

For 1&2...

1.) IS true... but as humans we've evolved to consciously and subconcsiously detect, and detest lateral assymetries. Symmetric features are a major factor in the appearence of beauty for instance. That said, left-right to up-down is a world of difference... except...

2.) This is tricky. We ALREADY perform a mental "flip" for all images which strike the retina inverted by the lens. Maybe there is a barrier to performing that flip again, but then I question why we're so easily able to interpret a world seen if we're suspended upside-down (from our perspective).

It's actually HARDER to make the L/R flip, than H/V! Not to mention that people with aphasias don't warp reality around them just because their geometric recognition is destroyed for instance.

 

[Demystifier]

Peteratcam, your explanation is shorter (and correct of course) but does not explain why many people THINK that mirror exchanges left and right but not up and down. In other words, you do not explain the origin of the illusion of asymmetry.

 

[Demystifier]

I always thought the answer was "it flips front and back".

Of course, but it doesn't explain why people THINK that it exchanges left and right.

 

[Frame Dragger]

Peteratcam, your explanation is shorter (and correct of course) but does not explain why many people THINK that mirror exchanges left and right but not up and down. In other words, you do not explain the origin of the illusion of asymmetry.

I think the problem with the mirror example is that most people are so blatantly clueless about the nature of vision to begin with, that it's not too useful. There has to be a better way to describe the issue of apparent assymetry without resorting to our funky sense of vision.