Why are fields which do not transform in a certain way not fundamental?

When I was first exposed to Math physics textbooks and textbooks on vector calculus, I found:

1. Temperature distribution in a room $$T(x,y,z,t)$$ or the density variation in a fluid $$\rho(x,y,z,t)$$ etc were given as examples scalar fields.

2. The velocity field $$\vec{v}(x,y,z,t)$$ of a fluid stood as an example of a vector field.

But now I know that the examples above do not represent fundamental fields. This is because (I guess) they're only scalars and vectors under rotations but not under Lorentz transformations: neither the temperature field $$T(x,y,z,t)$$ nor the velocity field $$\vec{v}(x,y,z,t)$$ transform like a Lorentz scalar and Lorentz vector respectively. An example of the fundamental scalar field of nature is the Higgs field and that of a fundamental vector field is the $$Z$$-boson field.

But why a field must transform in a particular way under Lorentz transformation for it to be regarded as fundamental field of nature? I know that the nature is relativistic. But does that require the fundamental fields to behave in a certain way? How?

• the temperature field T(x,y,z,t) [doesn't] transform like a Lorentz scalar. I'm not sure. How do you define temperature of a moving body? is there another way than to define it as the temperature measured in its rest frame? – Elio Fabri Apr 25 at 7:51
• @ElioFabri How do I find how $T$ transforms? No idea. – mithusengupta123 Apr 25 at 7:54
• @ElioFabri I can find how the electric field transforms under Gallilen or Lorentz boost by considering the field due to a static point charge in one frame and the field in a frame moving w.r.t it. I think, likewise, I have to consider a particular temperature distribution produced by some source and boost it. – mithusengupta123 Apr 25 at 8:05

But this is a somewhat trivial statement since there are only seventeen such fields in the Standard model and the remaining $$\infty - 17$$ fields possible in the universe are then doomed to be consigned to the non-fundamental waste basket. It would make more sense to ask whether a field is Lorentz covariant or not and avoid the term fundamental. So for example a velocity field is not Lorentz covariant, though of course a four-velocity field is. There is nothing wrong with fields that are not Lorentz covariant as long as you're working in a regime where relativistic effects are negligible.