I know that if I am sitting in a spacetime $M$ at point $p$, vectors live in the tangent space $T_pM$, and tensors in the tensor product space etc. If I want to consider general tensor fields, I consider sections of the tensor bundle of $M$. My question generally is how representation theory changes when we switch from considering the tensor product space at a point, to sections of the manifolds tensor bundle.

In particular, it is clear that if at $p$ we have vectors $A$ and $B$, their inner product $s = A^\mu B_\mu$ will be a scalar, and transform trivially under the Lorentz group. That is to say another observer at $p$ with different coordinates related via a Lorentz transform will find vectors $A^\prime, B^\prime$ are related via a Lorentz transform, but they will agree without the need to perform any transforms, that $s^\prime = s$. All inertial observers at $p$ agree on the value of $s$ in their reference frames.

However it would seem something changes when we consider scalar fields. I have seen comments on this site (though I am having trouble relocating them) to the effect of "a scalar $\phi$ may transform in a finite dimensional representation, but $\phi(x)$ must belong to an infinite dimensional representation". This answer only confused me further, as it seems to mention three possible representations for a scalar field, while this answer seems to show two valid methods for showing either it does not transform at all or transforms under an infinite dimensional representation (as well, it relies on analyticity to do so).

It's clear to me that a scalar field $\phi(x)$ in an unprimed coordinate system, should equal $\phi^\prime(x^\prime)$ in the primed coordinate system. This is just a statement that the scalar field is only a function of the spacetime points ($p$) and not of the coordinates. What confuses me is how the discussion of infinite dimensional representations enters at all, and how it relates to generalizing from points to sections or tensors to tensor fields. From a group theoretic point of view, if I have a scalar field on Minkowski spacetime, what do different observers at the same point $p$ need to do in order to be sure they are talking about the same geometric object (the same scalar field)? What makes this different from the trivial case if a single scalar value? What about on more complicated manifolds, under general coordinate transforms?

I'm strictly hoping to understand the classical field theory, so please avoid the nuances of infinite dimensional unitary representations and Wigner's classification for the time being. I do not believe they are relevant to the confusion at hand.



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