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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?

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  • $\begingroup$ 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? $\endgroup$ – Elio Fabri Apr 25 at 7:51
  • $\begingroup$ @ElioFabri How do I find how $T$ transforms? No idea. $\endgroup$ – mithusengupta123 Apr 25 at 7:54
  • $\begingroup$ @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. $\endgroup$ – mithusengupta123 Apr 25 at 8:05
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This comes down to the definitions of the words fundamental and field.

I would guess that you are taking the word fundamental to have the same meaning as in fundamental particle, in which case it specifically refers to one of the quantum fields in the Standard Model. These fields are necessarily Lorentz covariant because that's the way the Standard Model is constructed.

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.

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  • $\begingroup$ Is it a problem that all field (e.g., density field in my question) can not be quantised? $\endgroup$ – mithusengupta123 Apr 27 at 17:23

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