A scalar field is one which is unchanged under rotation. But how do we decide whether a given field (e.g., the temperature in a room) is unchanged under rotation or not? We need to measure the temperature at any point $$P$$ in two coordinates at some time $$t$$. In one $$P$$ has coordinates $$(x,y,z)$$ and in a rotated frame where $$P$$ has coordinates $$(x^\prime,y^\prime,z^\prime)$$. If we find $$T(x,y,z,t)=T^\prime(x^\prime,y^\prime,z^\prime,t)$$ for all points $$P$$, we will call it a scalar field.

To check whether it is unchanged under a Galilean boost or Lorentz boost, do we also need to perform experiments and decide? Can one exclude or establish whether the scalar-like behaviour holds under these transformations (even without measurements)? Stated differently, I mean

$$\bullet$$ Does one expect $$T(x,y,z,t)=T^\prime(x^\prime,y^\prime,z^\prime,t)$$ under Galilean boost where $$\vec{r}^\prime=\vec{r}-\vec{V}t$$ and $$t^\prime=t$$?

$$\bullet$$ Does one expect $$T(x,y,z,t)=T^\prime(x^\prime,y^\prime,z^\prime,t^\prime)$$ where the primed coordinates $${x^\prime}^\mu$$ and unprimed coordinates $$x^\mu$$ are related by Lorentz boost?

Temperature is an annoying example for a scalar field because its definition is $$1/T = \partial_E S(E)$$ but energy $$E$$ is not a scalar in relativity, but rather part of the 4-momentum vector. Depending on what exactly you think the energy apparing there is, and what an observer experiences as "temperature", you can come to the conclusion that it is a covariant, contravariant or scalar quantity, cf. this question and its answers
• "You decide the transformation behaviour of a field based on its definition" what is the definition of the density field $\rho(x,y,z,t)$? How does the Higgs field fit into your answer? – mithusengupta123 May 5 at 12:26