(Background: I know some but not much differential geometry, hopefully enough to formulate this post.)
I want to ask about what physicists mean when they say scalar, vector, etc. The answer in differential geometry is something like:
1) Scalar fields are smooth functions on spacetime ($\phi \in C^{\infty} (\mathcal{M})$)
2) Vector fields are smooth derivations / smooth sections of the tangent bundle ($\vec{A} \in \Gamma(T\mathcal{M})$)
3) One forms and tensors are defined in the usual way as (multi)linear functions satisfying certain properties.
However when we come to studying physics (maybe in QFT) there seems to be an additional requirement:
4) Suppose the map $\Lambda$ sends spacetime to itself (eg a boost). Then we additionally require:
4a) Scalars transform like:
$$\phi(X) \mapsto \phi'(X') = \phi(\Lambda^{-1}(X)) $$
4b) Vectors transform like:
$$\vec{A}(X) \mapsto \vec{A}'(X') = \Lambda \vec{A}(\Lambda^{-1}(x))$$
... and so on for tensors.
I think the above is not completely correct so my questions are:
Q1) Is this a correct summary of the situation, that a "physical vector field" is a mathematical (differential geometry) vector field plus an invariance condition that is a completely separate condition, or can this be motivated from geometry alone?
Q2) What is the correct formulation of 4? Clearly we don't care about all transformations in $\text{Aut}(\mathcal{M})$ but do we fix a subgroup $G\subset\text{Aut}(\mathcal{M})$ arbitrarily? Usually for QFT etc it would be the Lorentz group I know, but are there situations where different $G$ are considered? Also, my writing of 4(b) doesn't make much sense as it stands unless $\mathcal{M}$ is given a vector space structure which I hope isn't necessary.
