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Goldstein pg 151 says "it is clear that an inversion of a right-handed system into a left-handed one cannot be accomplished by any rigid change in the coordinate axis..." I am trying to understand what he means by a rigid change... is he saying that an inversion is a discontinuous jump that is impossible for an object to achieve? why can't it?

I can see clearly that the inversion (improper rotation) will be associated with a sort of jump(discontinuity) upon the mirror reflection... but I'm a little confused on the definition of a "rigid change". maybe the problem isn't the discontinuity of a mirror reflection but has to do with the change of handedness upon reflection??

goldstein also writes: " An inversion never corresponds to a physical displacement of a rigid body."

i'm a little confused as to what is the problem with inverting the z-axis??? how does that change the physics?

also, please do not talk about the quantum tunnelling aspect, I am having a problem understanding this classically and I don't want to get into all that ...

( let's say you take the vector r = (1,0,1) in a right handed cartesian coordinate system, then you rotate it 180 degrees you get the vector r' = (-1,0,1) in the new coordinate system, now if you "invert" the z-axis what is the problem with that in terms of "rigid change". why is that not a rigid change????)

as a further note in the example I am working with I think it's important to keep the transformations passive ( rotate the coordinate system 180 degrees counterclockwise and then do the inversion).

Bozostein
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  • if you look at the vector r = (1,0,0) maybe there is a way to explaon this with a simple example like that, i'm workin' on it right now... – Bozostein Oct 24 '11 at 04:51
  • I'm not sure what quantum tunneling aspect you're talking about; this has nothing to do with quantum mechanics at all. – David Z Oct 24 '11 at 04:59
  • that has to do with certain molecules doing wierd quantum things ( NH3 does this supposedly) but I don't want to get into all that. supposedly NH3 can tunnel into it's left-handed state or something (i don't understand it classically so I'm not too sure) – Bozostein Oct 24 '11 at 05:05
  • an inversion is like a plane-reflection of a rigid body... without a rotation ( thus the handedness get's changed right?) – Bozostein Oct 24 '11 at 05:11
  • Ah, I know what you're talking about. It's completely unrelated. – David Z Oct 24 '11 at 05:11
  • first we know the improper rotation changes the handedness of the coordinate system.

    now, imagine a rigid body rotating about the origin. say there are initially some right handed set of coordinates fixed in the rigid body frame at t=0, and then we start rotating the rigid body about the origin of this coordinate system. There is simply not way you can rotate such that the handedness will change!! the rotation of a rigid body is completely defined by the euler angles and if you imagine any of these 3 rotations, there is no way to actually invert invert the axis by such a rotation.

     – Bozostein Oct 24 '11 at 05:32

2 Answers2

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He just means that an object which has a three-dimensional structure with no symmetries cannot be turned from a left-handed version to a right-handed version using rotations alone. You can't rotate a left-hand glove to be a right-hand glove. A vector can be inverted by rotating it, but this doesn't invert a general rigid body, because a vector only has one axis, not three.

The proof of the statement that rotations cannot invert is by the continuity of the determinant function. The determinant of a rotation is always 1, and of a reflection-rotation is -1. The determinant cannot smoothly go from 1 to -1.

Ron Maimon
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  • yo ron thx for the tip! as a further point I see also you cannot turn a left-handed glove into a right handed glove by a translation and a rotation. – Bozostein Oct 24 '11 at 06:12
  • yo ron... can you point me to a discussion of how 3-d objects ( or higher dimensional objects behave under inversion etc)? – Bozostein Oct 24 '11 at 06:14
  • yo ron... what if you cut out some of the fabric from the glove and then repatch it? what is the mathematical term for that? – Bozostein Oct 24 '11 at 06:16
  • @Bozostein: You can turn the glove inside out. I don't understand the question about how objects behave under inversion--- inversion is a symmetry of macroscopic physics--- they behave the same way, only inverted! – Ron Maimon Oct 24 '11 at 07:15
  • heh ok... you right – Bozostein Oct 24 '11 at 07:16
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By rigid, we mean the particles within a body maintain their distance from on another - it doesn't break apart. So obviously a rigid displacement keeping one point fixed can only be a rotation. Therefore if the coordinates are also rigid wrt the body and the origin is fixed where the point in the body is fixed, then they too can only rotate. A reflection of one coordinate would mean having to break the body apart, whereas relecting two coordinates is OK because the body can be rotated to produce an equivalent displacement.

Larry Harson
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