I'm am sure that I must be missing something very simple, so apologies in advance.
Considering the Lorentz transformation Λ of a spinor fields, for the plane-wave solution u(p), I cannot for the life of me agree why
(1) us(Λ−1p′)=Λ12us(p′)
where
p′=Λp
This is in Peskin & Schroeder, pg 59, just above equation (3.110).
I have tried to get this a dozen times, to no avail.
I know that, for a scalar field, under a Lorentz transformation Λ we get, as per Peskin & Schroeder, pg 36, equation (3.2)
ϕ(x)→Λϕ(x)=ϕ′(x)=ϕ(Λ−1x)
This makes sense to me as "the transformed field at the transformed point in spacetime should be the same as the un-transformed field at the untransformed-transformed point in spacetime".
So trying to do that with inverse transformations, now using Λ12 for a spinor plane-wave solution, I get
Λu(p)=u(Λ−1p)
and applying an inverse transformation would give
Λ−1Λu(p)=Λ−1u(Λ−1p)
or
u(Λ−1Λp)=u′(Λ−1p)
so
u(Λ−1p′)=u([Λ−1]−1Λ−1p)
whence
u(Λ−1p′)=u(p)
that is,
u(p)=u(p)
So it's consistent alright, but not of much use!
Can anyone show me what I'm missing to derive equation (1) above. Thank you in advance!
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