- #1
liorde
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When constructing the Lie algebra of the Lorentz Transformation, the references usually start with an infinitesimal proper-orthochronous transformation, and then find the infinitesimal generators. Let's call the set of these generators L. after finding L, the references usually compute the commutation relations of L, and then add to L the operators S, corresponding to the spin. The reason is that L+S obey the same commutation relations as L, and therefore L+S is more general then L.
My questions are:
a) Is there any physical motivation to add the S (besides "it agrees with the phenomenon of spin")? I'm looking for a symmetric-inspired motivation.
b) Would the physics be any different if we didn't add S?
c) I thought that the commutation relations of the algebra determine the group structure, so why aren't L and L+S equivalent? why do we need the extra S?
d) could we add another term, let's call it T, to L+S such that L+S+T still obey's the algebra?
e) If L represents a transformation which is originally defined as operating on space-time coordinates, what do S and L+S represent in terms of space-time coordinates? if the answer is that S doesn't care about space-time but operates on a different space, can we say that T above (in question d) operates on a third space?
f) I know that in quantum mechanics, angular momentum has integral eigenvalue while spin has half integral values. The reason is that angular momentum is defined via the position operator X and that leads to integral eigenvalue, while spin is defined to operate on "spin space" which is separate from "space space". What does that have to do with my previous discussion?
Thanks
My questions are:
a) Is there any physical motivation to add the S (besides "it agrees with the phenomenon of spin")? I'm looking for a symmetric-inspired motivation.
b) Would the physics be any different if we didn't add S?
c) I thought that the commutation relations of the algebra determine the group structure, so why aren't L and L+S equivalent? why do we need the extra S?
d) could we add another term, let's call it T, to L+S such that L+S+T still obey's the algebra?
e) If L represents a transformation which is originally defined as operating on space-time coordinates, what do S and L+S represent in terms of space-time coordinates? if the answer is that S doesn't care about space-time but operates on a different space, can we say that T above (in question d) operates on a third space?
f) I know that in quantum mechanics, angular momentum has integral eigenvalue while spin has half integral values. The reason is that angular momentum is defined via the position operator X and that leads to integral eigenvalue, while spin is defined to operate on "spin space" which is separate from "space space". What does that have to do with my previous discussion?
Thanks