Invariance, covariance and symmetry

Though often heard, often read, often felt being overused, I wonder what are the precise definitions of invariance and covariance. Could you please give me an example from quantum field theory? Thanks!!!

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Possible duplicate: physics.stackexchange.com/q/7700/2451 –  Qmechanic Apr 17 '13 at 2:21

In the context of field theory, one can make these notions precise as follows. Consider a theory of fields $\phi$. Let a transformation $T$ $$\phi \to\phi_T$$ on fields be given. Let a functional $F[\phi]$ of the fields be given (consider the action functional for example). The functional is said to be invariant under the transformation $T$ of the fields provided $$F[\phi_T] = F[\phi]$$ for all fields $\phi$. One the other hand, the equations of motion of the theory are said to be covariant with respect to the transformation $T$ provided if the fields $\phi$ satisfy the equations, then so do the fields $\phi_T$; the form of the equations is left the same by $T$.
For example, the action of a single real Klein-Gordon scalar $\phi$ is Lorentz-invariant meaning that it doesn't change under the transformation $$\phi(x)\to\phi_\Lambda(x) = \phi(\Lambda^{-1}x),$$ and the equations of motion of the theory are Lorentz-covariant in the sense that if $\phi$ satisfies the Klein-Gordon equation, then so does $\phi_\Lambda$.
sorry that this might be a stupid question, but why $\phi_{\Lambda}(x)=\phi({\Lambda}^{-1}x)$? –  Timo Dec 24 '14 at 3:11