Could someone elaborate what it means for there to be an inverse relation between transformations on function space and transformation on coordinates? In page 67 of Ballentine's Quantum mechanics, he is talking about Galilean Transformations. He has
$\Psi'(x)=U(\tau)\Psi(x)$
Where the original funciton is defined around $x_0$ and the transformed function is defined around $x_0'=\tau x_0$
He states that the precise relationship between the two functions is $\Psi'(\tau x)=\Psi(x)$ and I dont understand where this relationship comes from.
 A: The motivation behind writting down the expression $\Psi'(\tau x)=\Psi(x)$ has to do with the fact that under a transformation that transforms spacetime points, i.e. $x\rightarrow x'=\tau x$, various functions are also transformed into $\Psi\rightarrow\Psi'$. The new transformed functions describe some physical quantity such as some sort of field, and hence they take as arguments the transformed spacetime points $x'$. Hence, under the $x\rightarrow x'=\tau x$ transformation, the function $\Psi(x)$ transforms as $\Psi(x)\rightarrow\Psi'(x')=\Psi'(\tau x)$
The statement that $\Psi'(\tau x)=\Psi(x)$ means that the value of the function $\Psi$ (which might represent some physical field) is left unchanged under those transformations... In those cases, we say that $\Psi$ transforms as a scalar under the transformations $x\rightarrow x'$, because scalars tend to remain the same under an arbitrary transformation.
An equivalent way you might see the statement above in the literature is
$$\Psi'(x)=\Psi(\tau^{-1}x)$$
I hope this helps
