# Why should the fields not change under translation but may change under Lorentz transformations?

In order to derive the conservation of four-momentum of a field, $$P^\mu$$, it is assumed that the total change in the field, defined as $$\delta\phi_a(x)\equiv {\phi^{\prime}}_a(x^{\prime})-\phi_a(x)$$ is zero for any field, under a spacetime translation $$x^\mu\to x^{\prime\mu}=x^\mu+\varepsilon^\mu.$$ Here, $$\varepsilon^\mu$$ is an arbitrary constant 4-vector.

Why should the fields not change under translation (but may change under Lorentz transformations)?

• Why should they? Do you have an example for a field on $\mathbb{R}^4$ for which $\delta \phi_a \neq 0$? May 16, 2023 at 9:31
• Is it saying that there are no four vectors that change under translation? Well, I understand that under $x^{\prime\mu}=x^\mu+\varepsilon^\mu$, four-vectors such as four-velocity $U^\mu=dx^\mu/d\tau$, four-acceleration $\alpha^\mu=d^2x^\mu/d\tau^2$ etc are unchanged. But I am not sure why $A^\mu$, the electromagnetic four-potential, must be invariant. May 16, 2023 at 9:49
• ?? look here en.wikipedia.org/wiki/… for the A May 16, 2023 at 10:26
• Could you add the source you are working with? May 16, 2023 at 11:10

A tensor field on $$\mathbb{R}^n$$ changes under a diffeomorphism $$f : \mathbb{R}^n\to\mathbb{R}^n$$ by the Jacobian of $$f$$ (pullback by diffeomorphism), e.g. a vector field has $$A(x)\mapsto Df\cdot A(f(x))$$, where $$Df$$ is the Jacobian matrix. The Jacobian of a translation $$x\mapsto x+a$$ is the identity matrix, while the Jacobian of a Lorentz transformation $$x\mapsto \Lambda x$$ is $$\Lambda$$.
In more common physics notation, this is just the claim that a vector field obeys $$V^{\mu} = \frac{\partial y^\mu}{\partial x^\nu}V^\nu$$ for a coordinate transformation $$x\mapsto y(x)$$, which is more or less the definition of a vector field.
Relativity teaches us that the physics should be independent of our choice of coordinate system, so the (physical) fields don't actually change under (mathematical) coordinate transformations. What the transformations laws for tensor fields are really describing is not how the fields themselves change, but rather how the components of a field expressed in a particular basis change according to how you change the basis you're using to represent the field. In other words, the value of a field at a given point is going to be the same regardless of whether you call that point $$x$$ or $$x'$$, since your still talking about the same point.
An infinitesemal translation can be written as $$\phi_i'(x')=[\exp(-\mbox i a^\mu P_\mu)]_{ij} \phi_j(x),$$ where $$P_\mu$$ is the generator for translation. In representation for scalar fields, $$P_\mu=0$$. So actually $$\phi'(x')=\phi(x)$$. But I don't know what it is for other fields.