Question

Given a Rn

valued function on space-time, is the transformation rule prior decided from pure differential geometry? Being said, I mean really given f(x)
do we know instantly what is f′(x′),G:x→x′=G(x) without any physical assumption?
Another way to ask is that We do know the transformation rule for tensor field of arbitrary type (r,s) from differential geometry, (Question 1.1: ) so can we simply regard fields as tensor fields? For example scalar field should be (0,0)
type tensor field. If so, by specifying tensor type we automatically get the transformation rule. And it follows the answer to question 1 YES.

MY thinking: Let’s first give the general transformation rule according to change of coordinates in QFT:
ϕa(x)⟶Λϕa′(x′)=Sab(Λ)ϕb(Λ−1x).
Now, for scalar field this seems to be consistent with DG transformation law, so Question 1.1 may be a YES. HOWEVER, by heart I know this is not true since the above transformation law comes really from representation of Poincare group, this is of course a additional piece of information that not prior given for an arbitrary Rn
valued function on space-time. DG certainly knows nothing about representations and this is of course something we need to specify manually. What’s the problem? Shouldn’t coordinate transformation be consistent with DG?

Answer 1

In pure differential geometry, the transformation rule for a function from one space-time coordinate system to another is not determined, but for tensor fields, the tensor type can be used to determine the transformation rule.

Step-by-step explanation:

In pure differential geometry, the transformation rule for a function from one space-time coordinate system to another is not determined.

Differential geometry alone does not provide information about the transformation rule for a function. However, in the context of tensor fields, which are objects that generalize the concept of a function, we can specify the tensor type and automatically obtain the transformation rule from the differential geometry of tensor fields.

For example, a scalar field can be regarded as a (0,0) type tensor field, and its transformation rule can be determined based on the differential geometry of tensor fields.

However, it's important to note that additional information, such as the representation of the Poincare group, is needed to determine the transformation rule for a specific field. This additional information is not prior given by differential geometry alone.