Thursday, 15 March 2018

Lorentz transformations of fields evaluated at a point


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(Λ1p)=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(Λ1x)


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(Λ1p)


and applying an inverse transformation would give


Λ1Λu(p)=Λ1u(Λ1p)


or


u(Λ1Λp)=u(Λ1p)


so


u(Λ1p)=u([Λ1]1Λ1p)


whence


u(Λ1p)=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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