I am reading some literature which is considering translations and boosts in field theory. The reference is Construction of Lagrangians continuum theories, Markus Scholle, 2004, The Royal Society. I am wondering if anyone is able to help me understand applying the transformation. I can see what is done, but I do not understand why.
The translation is defined by:
$\vec{x} \to \vec{x} + \vec{s}$
$t \to t $
$\psi \to \psi$
the boost by:
$\vec{x} \to \vec{x} - \vec{u_0}t$
$t \to t $
$\psi \to \psi- \vec{u_0}.\vec{x}+\frac{1}{2}\vec{u_0^2}t$
(where this, the w.f transformation comes about by ensuring the Schrodinger equation is invariant w.r.t Galilean boosts
[1]the article then will now consider the translation followed by the Galilean and this is done by:
First I get :
1)$\vec{x} \to \vec{x} + \vec{s}, \psi \to \psi$
Then applying the boost to get:
2)$\vec{x} \to \vec{x} + \vec{s} - \vec{u_0}t, \psi \to \psi- \vec{u_0}.\vec{x}+\frac{1}{2}\vec{u_0^2}t$
[2]And now the Galilean boost then the translation:
Boost:
$\vec{x} \to \vec{x} - \vec{u_0}t$
$\psi \to \psi- \vec{u_0}.\vec{x}+\frac{1}{2}\vec{u_0^2}t$
And the translation then gives:
$\vec{x} \to \vec{x} - \vec{u_0}t-\vec{s}$
$\psi \to \psi- \vec{u_0}.(\vec{x}+\vec{s})+\frac{1}{2}\vec{u_0^2}t$
so the argument is that while $\vec{x}$ is the same $\psi$ differs...( that is, in contrast to Newtonian mechanics and the boost and translation commuting, in field theory they do not
So, this is what the article has done. But, I do not really understand this. To me, the $x$ from 1) needs to be plugged into the $\psi$, not treating them separately. I thought it has to be done simultaneously transformation since $\psi$ is a function of $x$ and do not understand how they can be considered separately like this? so that is, i would be getting the same as the result of [2].
