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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?

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Some physical intuition from the two principles in special relativity:

All the basic equations of fields are linear, which means the operator acting on fields are linear. These operators are derivates w.s.t. coordinates, so they transform as scalars, vectors and so on. We'd like these equations to be invariant under Lorentz transforations, so the fields themselves must transform in the same way.

Moreover, the Lagrangian (density) is a Lorentz scalar. It's convenient to constuct a scalar from such fields.

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If the fields are scalar, vector or tensorial in nature then by definition they have an underlying vector space structure.

The Lorentz transformations act as a map on these vector spaces. Since Lorentz transformations can be seen as rotation operator, they act as linear transformations on these vector spaces. So in short, the fields under consideration have an underlying vector space structure hence Lorentz transformations act linearly on them.

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