What makes a field theory relativistic?

[LarryS]

It is my understanding that in both Classical Field Theory and QFT the Lagrangian must be Lorentz invariant in order for the fields to be considered relativistic. Buy what about the field itself (φ or ψ)? As complex-valued functions of space and time do they also have to be Lorentz invariant?

As always, thanks in advance.

 

[DarMM]

Assuming you understand Lorentz covariance (no problem if not, I'll explain).

They must be Lorentz covariant, but not necessarily Lorentz invariant, i.e. they can transform as a vector or a one-form but they don't have to be a scalar.

 

[ftr]

As complex-valued functions of space and time

Do you mean how portability is affected by reference frame.