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Field transformation in Peskin-Schroeder (chapter 3)

  1. Jun 27, 2012 #1
    1. The problem statement, all variables and given/known data

    There is something I don't understand about eq. 3.110 (there is no need of the complete equation actually) in Peskin Schroeder.
    What I need to do is to use the unitary transformation law obtained for one-particle states to get the usual transformation law for the Dirac field (under Lorentz transformations).

    2. Relevant equations


    3. The attempt at a solution

    I've been able to obtain the law stated in P.S.
    I also checked the result with the similar law for scalar field transformation and still I don't understand.
    I guess I might be wrong somewhere:

    I started from Peskin's law for scalar fields:

    [itex]\Phi[/itex](x) [itex]\rightarrow[/itex] [itex]\Phi[/itex]'(x) = [itex]\Phi[/itex]([itex]\Lambda[/itex]-1x)

    Here the book reads: the transformed field, evaluated at the boosted point, gives the same value as the original field evaluated at the point before boosting.
    From this I understand that the previous relation - with explicit notation for coordinate systems - becomes:

    [itex]\Phi[/itex](x(O)) [itex]\rightarrow[/itex] [itex]\Phi[/itex]'(x(O')) = [itex]\Phi[/itex]([itex]\Lambda[/itex]-1x(O'))

    which gives the correct law for scalars:

    [itex]\Phi[/itex](x(O)) [itex]\rightarrow[/itex] [itex]\Phi[/itex]'(x(O') = [itex]\Phi[/itex](x(O))

    Now, in chapter 3.5, I find:

    U([itex]\Lambda[/itex])[itex]\Psi[/itex](x)U-1([itex]\Lambda[/itex]) = [itex]\Lambda[/itex]1/2-1 [itex]\Psi[/itex]([itex]\Lambda[/itex]x)

    Or the equivalent for scalar field (which is not in Peskin):

    U([itex]\Lambda[/itex])[itex]\Phi[/itex](x)U-1([itex]\Lambda[/itex]) = [itex]\Phi[/itex]([itex]\Lambda[/itex]x)

    That looks good, provided that I understand the change in the tranformation action due to the fact that we are transforming the ladder operators in Dirac field.
    But here comes my question: In deriving these equations, no change was made on coordinate system, so to me they read:

    [itex]\Phi[/itex](x(O)) [itex]\rightarrow[/itex] [itex]\Phi[/itex]'(x(O)) = [itex]\Phi[/itex]([itex]\Lambda[/itex]x(O))

    Which is not the same - even accounting for the transformation change.

    I apologize for the long post on such an inessential question but I could really use some help on this.
    Thanks.
     
  2. jcsd
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