Field transformation in Peskin-Schroeder (chapter 3)

[AnZa85]

Homework Statement

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).

Homework Equations

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:

$\Phi$(x) $\rightarrow$ $\Phi$'(x) = $\Phi$($\Lambda$^-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:

$\Phi$(x(O)) $\rightarrow$ $\Phi$'(x(O')) = $\Phi$($\Lambda$^-1x(O'))

which gives the correct law for scalars:

$\Phi$(x(O)) $\rightarrow$ $\Phi$'(x(O') = $\Phi$(x(O))

Now, in chapter 3.5, I find:

U($\Lambda$)$\Psi$(x)U^-1($\Lambda$) = $\Lambda$_1/2^-1 $\Psi$($\Lambda$x)

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

U($\Lambda$)$\Phi$(x)U^-1($\Lambda$) = $\Phi$($\Lambda$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:

$\Phi$(x(O)) $\rightarrow$ $\Phi$'(x(O)) = $\Phi$($\Lambda$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.

 

[mark726]

Thank you for your question. It seems like you are on the right track with understanding the unitary transformation law for one-particle states. However, there may be an error in the notation you are using. In the equation U(\Lambda)\Psi(x)U^-1(\Lambda) = \Lambda^{\frac{1}{2}} \Psi(\Lambda x), the transformation is actually acting on the coordinates x, not the field \Psi(x). So the correct notation would be \Psi(x) \rightarrow \Psi'(x') = \Lambda^{\frac{1}{2}} \Psi(\Lambda x). This means that the transformed field \Psi'(x') is evaluated at the boosted coordinates x', not the original coordinates x.

Now, going back to your question about the transformation law for the Dirac field, it should be written as \Phi(x(O)) \rightarrow \Phi'(x(O')) = \Phi(\Lambda x(O')). This is consistent with the transformation law for the scalar field, as you have correctly derived. The key difference here is that the transformation is acting on the coordinates x, not the fields \Phi(x).

I hope this clarifies any confusion you may have had. Keep up the good work in understanding the unitary transformation law for one-particle states and its application to the Dirac field!