How do fields change when you transform spacetime coordinates? In field theory, I have seen the notion that when you infinitessimaly transform your coordinates in the form $$x'_\mu = x_\mu + \delta x_\mu$$
The fields transform as
$$\phi(x) \rightarrow \phi'(x') =\phi(x) + \delta\phi(x) + \partial_\mu\phi x^\mu$$
Why do the fields transform this way? I am a bit confused about what $\phi'(x')$ really means here. Isn't $\phi'$ already the transformed field after you perform the coordinate transformation? Why isn't it something like $\phi(x') = \phi(x) + \delta\phi(x) + \partial_\mu\phi x^\mu$, where instead on the LHS you have an unprimed field?
 A: We should distinguish between two different things:

*

*We can define a new field by $\phi'(x)\equiv \phi(x+\delta x)$. Then $\phi'(x)$ and $\phi(x)$ are two different configurations of the scalar field, both expresed in the same coordinate system.


*Wwe can write the original field $\phi$ in a new coordinate system: $\phi'(x')\equiv\phi(x)$. Then $\phi'(x')$ and $\phi(x)$ both describe the same configuration of the scalar field, expressed in two different coordinate systems. This is the case shown in the question.
This might be a little more clear if we write $x(p)$ for the coordinates of the point $p$. Then the two cases above look like this:

*

*$\phi'(x(p))= \phi(x(p'))$


*$\phi'(x'(p)) = \phi(x(p))$.
In the first case, $x(p')$ is a new point expressed in the original coordinate system. In the second case, $x'(p)$ is the original point expressed in a new coordinate system. The second case is the one shown in the question. Both cases define a new function $\phi'$ of the list of coordinates (list of real numbers), but only the first case defines a new function of spacetime (function of $p$). In the second case, we're just writing the same function of spacetime (function of $p$) in two different ways.
