reference frames - Active versus passive transformations

I am a bit confused by the concepts of active and passive transformations. In all the courses I am doing at the moment we do transformations of the form: $$ \phi(x) \rightarrow\phi'(x') = \phi(x) $$ and $$ \partial_{\mu}\phi(x)\rightarrow\partial' _{\nu}\phi(x') = \frac{\partial x^{\alpha}}{\partial x' ^{\nu}}\partial_{\alpha}\phi(x) $$ This is all perfectly clear to me. However, I am reading Peskin and Schoder at the moment, and they adapt an "active" point of view (their words), such that the above transformations are: $$ \phi(x) \rightarrow\phi'(x) = \phi(\Lambda^{-1}x) $$ and $$ \partial_{\mu}\phi(x)\rightarrow\partial _{\mu}(\phi(\Lambda^{-1}x)) = (\Lambda^{-1})^{\nu}_{\mu}(\partial_{\nu}\phi)(\Lambda^{-1}x). $$ I don't understand how to interpret this and especially how to derive the second equation.

Answer

What you wrote down is the same as what Peskin writes. To see this, notice that if we write the "transformed" position $x'$ as $x' = \Lambda x$, then your first equation can be written as

$\phi'(\Lambda x) = \phi(x)$

but this equivalent to

$\phi'(x) = \phi(\Lambda^{-1} x)$

which is the same as the first Peskin equation you wrote down. Your second equation and Peskin's second equation are equivalent. You can show this by using the definition of $x'$ plus the chain rule for partial differentiation. I can add details if you'd like, but I think it's a good exercise to figure out.

Active v. Passive

The convention in which we define $\phi'(x) = \phi(\Lambda^{-1} x)$ is the active convention because the transformed field value at the transformed point is that same as the non-transformed field value at the non-transformed point, so it's as if we have kept our coordinate system fixed and transformed the field configuration. To get intuition for this, imagine a temperature field $T$ in a 2D laboratory, and imagine keeping the laboratory fixed, but rotating the entire temperature field counterclockwise by a rotation $R$ to obtain a temperature field $T'$. Then (drawing a picture helps) the new temperature field evaluated at a counterclockwise rotated point $R x$ should be the same as the old temperature field evaluated at the non-rotated point $x$, namely $T'(R x) = T(x)$ which is the same as $T'(x) = T(R^{-1} x)$

The passive convention is the one in which we define $\phi'(x) = \phi(\Lambda x)$ and has the interpretation of transforming the coordinates while keeping the field configuration fixed. Try using the temperature analogy to understand this.