I'm trying to understand the definition of the quotient stack $[X/G]$ as defined in Frank Neumann's Algebraic Stacks and Moduli of Vector Bundles.

Explicitly, let $G$ be an affine smooth group $S$-scheme with right action $\rho:X\times G\to X$ on a noetherian $S$-scheme $X$. The quotient stack $[X/G]$ is the pseudofunctor $$ [X/G]:(\mathsf{Sch}/S)^{\operatorname{op}}\to\mathsf{Grpds} $$ defined as follows.

For an $S$-scheme $U$ let $[X/G](U)$ be the category whose objects are diagrams $$ U\xleftarrow{\pi}E\xrightarrow{\alpha}X $$ where $\pi$ is a principal $G$-bundle and $\alpha$ is a $G$-equivariant morphism. The morphisms in $[X/G](U)$ are the isomorphisms of principal $G$-bundles commuting with the $G$-equivariant morphisms.

For a morphism of $S$-schemes $f:U^\prime\to U$ let $$ [X/G](f):[X/G](U)\to[X/G](U^\prime) $$ be the functor induced by pullbacks of principal $G$-bundles.

Now, it is clear to me that $[X/G]$ is isomorphic to the representable functor $\operatorname{Hom}(-,X)$ when $G$ is the trivial group. It is not clear to me how to determine when $[X/G]$ is representable in general. My questions are:

Question 1. Is there a sufficient condition we can impose on $\rho$ to ensure that $[X/G]$ is representable? The Wikipedia article on quotient stacks says something about when the categorical quotient $X/G$ exists the canonical map $[X/G]\to\operatorname{Hom}(-,X/G)$ need not necessarily be an isomorhpism. This is somewhat opaque to me as I'm not even sure how $[X/G]\to\operatorname{Hom}(-,X/G)$ is defined.

Question 2. What is an example where $[X/G]$ is not representable?

If these questions are too broad, I'd be very grateful if someone could point me in the direction of a good reference.

I'm trying to understand the definition of the quotient stack $[X/G]$ as defined in Frank Neumann's Algebraic Stacks and Moduli of Vector Bundles.

Explicitly, let $G$ be an affine smooth group $S$-scheme with right action $\rho:X\times G\to X$ on a noetherian $S$-scheme $X$. The quotient stack $[X/G]$ is the pseudofunctor $$ [X/G]:(\mathsf{Sch}/S)^{\operatorname{op}}\to\mathsf{Grpds} $$ defined as follows.

For an $S$-scheme $U$ let $[X/G](U)$ be the category whose objects are diagrams $$ U\xleftarrow{\pi}E\xrightarrow{\alpha}X $$ where $\pi$ is a principal $G$-bundle and $\alpha$ is a $G$-equivariant morphism. The morphisms in $[X/G](U)$ are the isomorphisms of principal $G$-bundles commuting with the $G$-equivariant morphisms.

For a morphism of $S$-schemes $f:U^\prime\to U$ let $$ [X/G](f):[X/G](U)\to[X/G](U^\prime) $$ be the functor induced by pullbacks of principal $G$-bundles.

It is not clear to me how to determine when $[X/G]$ is representable in general. My questions are:

Question 1. Is there a sufficient condition we can impose on $\rho$ to ensure that $[X/G]$ is representable? The Wikipedia article on quotient stacks says something about when the categorical quotient $X/G$ exists the canonical map $[X/G]\to\operatorname{Hom}(-,X/G)$ need not necessarily be an isomorhpism. This is somewhat opaque to me as I'm not even sure how $[X/G]\to\operatorname{Hom}(-,X/G)$ is defined.

Question 2. What is an example where $[X/G]$ is not representable?

If these questions are too broad, I'd be very grateful if someone could point me in the direction of a good reference.