# Why is the Lorentz transformation of fields linear?

I know that the coordinate, $$x^\mu = (t,\vec x)$$ is a 4-vector and it transforms as $$x'^\mu={\Lambda^\mu}_\nu x^\nu.$$ The related (classical or quantized) field, $$\phi_a(x)$$, can be classified into different categories based on how it transforms, such as scalar fields, vector fields, spinor fields and so on. All these transformations are linear.

Is the linearity of the transformation of fields an assumption, or can it be derived from the linearity of coordinate transformations? For example, given a vector field $$A_\mu(x)$$, a new field $$B_\mu(x)$$ can be defined as $$B_\mu(x)=f[A_\mu(x)],$$ where $$f(x)$$ is an arbitrary function. $$B_\mu(x)$$ transforms as $$B'_\mu(x')=f[A'_\mu(x')]=f[{\Lambda_\mu}^\nu A_\nu(x) ].$$

It's not a vector field unless $$f(x)$$ is linear. Is $$B(x)$$ a proper field, or can the above transformation be decomposed into Lorentz transformations?

• Have a look at en.wikipedia.org/wiki/Group_representation. The quantities you mentioned are defined to transform under representations of the Lorenz group. Sep 21, 2023 at 16:08
• @go_science Could you please provide a more detailed explanation? Sep 21, 2023 at 18:19
• Related: Non-linear group representations Sep 22, 2023 at 4:16