- #1
ComeInSpinor
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Hi, I'm trying to understand spinors better, and I seem to be getting stuck on understanding the reason they're said to transform differently from vectors, and I'd appreciate any help with a justification for that. I'm sure I'm missing something pretty simple, but here goes;
Here's what I've been thinking. Consider the Clifford algebra Cl(2,0)(R), with basis elements 1, e1, e2, e12. Linear combinations of the even-graded elements give the complex numbers, since e12e12=-1. These play the role of spinors, and the linear combinations (ae1+be2) are just vectors. I know that this two-dimensional case is pretty simple, but I think (correct me if I'm wrong) that what I'm going to say next generalizes to higher dimensions. So if we act on a vector with an even-graded element [tex]\gamma[/tex], the vectors conjugate-commute with [tex]\gamma[/tex]. Thus a vector transforms as
v'= [tex]\gamma[/tex] v [tex]\gamma[/tex]*,
and in two dimensions we can get away with rewriting this as [tex]\gamma[/tex][tex]\gamma[/tex] v (by pulling the [tex]\gamma[/tex]* through the v) so the vector transforms through double the angle defined by [tex]\gamma[/tex]. Now a spinor just commutes, rather than conjugate-commuting, so the transformation
s' = [tex]\gamma[/tex] s [tex]\gamma[/tex] *
would just collapse down to
s' = s
Okay, so that's no good. The core of my question then, is why is it "natural" to define the transformation of a spinor as [tex]\gamma[/tex] s, rather than say [tex]\gamma[/tex] s [tex]\gamma[/tex] (which looks more like the transformation of a vector)? If you act on an object with one copy of a transformation, and another object with two copies, it seems obvious that the second object with transform twice as much. So it seems disingenuous to say that a spinor transforms through half the angle that a vector does when acted on by a given transformation, since you're actually acting on the vector with double the transformation that acted upon the spinor. Of course the spinor is going to rotate through half the angle. That doesn't seem to me like a natural, inevitable choice that is forced upon us by the maths. So what am I missing? It would seem more natural if there was a way to associate a (unique?) spinor with any vector, such that the spinor has to turn through [tex]\theta[/tex]/2 when the vector turns through [tex]\theta[/tex]. Is there?
Thanks in advance
Here's what I've been thinking. Consider the Clifford algebra Cl(2,0)(R), with basis elements 1, e1, e2, e12. Linear combinations of the even-graded elements give the complex numbers, since e12e12=-1. These play the role of spinors, and the linear combinations (ae1+be2) are just vectors. I know that this two-dimensional case is pretty simple, but I think (correct me if I'm wrong) that what I'm going to say next generalizes to higher dimensions. So if we act on a vector with an even-graded element [tex]\gamma[/tex], the vectors conjugate-commute with [tex]\gamma[/tex]. Thus a vector transforms as
v'= [tex]\gamma[/tex] v [tex]\gamma[/tex]*,
and in two dimensions we can get away with rewriting this as [tex]\gamma[/tex][tex]\gamma[/tex] v (by pulling the [tex]\gamma[/tex]* through the v) so the vector transforms through double the angle defined by [tex]\gamma[/tex]. Now a spinor just commutes, rather than conjugate-commuting, so the transformation
s' = [tex]\gamma[/tex] s [tex]\gamma[/tex] *
would just collapse down to
s' = s
Okay, so that's no good. The core of my question then, is why is it "natural" to define the transformation of a spinor as [tex]\gamma[/tex] s, rather than say [tex]\gamma[/tex] s [tex]\gamma[/tex] (which looks more like the transformation of a vector)? If you act on an object with one copy of a transformation, and another object with two copies, it seems obvious that the second object with transform twice as much. So it seems disingenuous to say that a spinor transforms through half the angle that a vector does when acted on by a given transformation, since you're actually acting on the vector with double the transformation that acted upon the spinor. Of course the spinor is going to rotate through half the angle. That doesn't seem to me like a natural, inevitable choice that is forced upon us by the maths. So what am I missing? It would seem more natural if there was a way to associate a (unique?) spinor with any vector, such that the spinor has to turn through [tex]\theta[/tex]/2 when the vector turns through [tex]\theta[/tex]. Is there?
Thanks in advance