# understanding why inversion cannot be accomplished by a rigid change

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).

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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 4:51
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 4:59
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 5:05
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 5:11
Ah, I know what you're talking about. It's completely unrelated. – David Z Oct 24 '11 at 5:11

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.

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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 6:12
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 6:14
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 6:16
@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 7:15
heh ok... you right – Bozostein Oct 24 '11 at 7:16

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.

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