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).
 A: 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.
A: 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.
