Noether's theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law.
Suppose our action is of the form S=∫d4xL(ϕ,∂μϕ).
if x→x′ then if S→S′ where
S′=∫d4x′L′(ϕ′,∂μϕ′).
But from calculus we know that S=S′ so does that mean that every change of variable correspond to a conserved quantity? why the quantities conserved under Poincare transformation, for example, is more especial?
Answer
The action shown in the question is a functional of ϕ, not of x. A change of the integration variable x is just a relabeling of the index set. It does not transform the dynamic variables ϕ at all, so no: a change of variable does not correspond to a conserved quantity.
More explicitly, if y(x) is a monotonic smooth function of x, then ∫d4y L(ϕ(y(x)),∂∂yμϕ(y(x)))=∫d4x L(ϕ(x),∂∂xμϕ(x)) identically, for any L whatsoever (as long as it depends on x only via ϕ). This is just a change of variable (a relabeling of the index-set), and there is no associated conserved quantity.
In contrast, suppose that the action has this property: ∫d4x L(ϕ(y(x)),∂∂xμϕ(y(x)))=∫d4x L(ϕ(x),∂∂xμϕ(x)). Unlike equation (1), equation (2) is not identically true for any L and any y(x), though it may be true for some choices of L and y(x). The transformation represented in equation (2) replaces the original function x, namely ϕ(x), with a new function of x, namely ϕ(y(x)). This is the kind of transformation we have in mind when we talk about Poincaré invariance and its associated conserved quantities: it is a change of the function ϕ which we then insert into the original action, not a change of the integration variable.
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