What's the meaning of canonical and phantom scalar field? I read in some paper about non-minimal derivative coupling that $\varepsilon = +1$ correspond to canonical scalar field, while $\varepsilon = -1$ corresponds to phantom scalar field. $\varepsilon$ is a minimal coupling constant.
 A: Phantom scalar fields are spooky scalar fields with a negative kinetic term. In other words, the $\epsilon$ constant corresponds to the Lagrangian
$$\mathcal L = \epsilon \frac{1}{2} g^{\mu\nu} \nabla_\mu \phi \nabla_\nu \phi$$
where indeed the canonical one has $\epsilon = 1$, while the phantom field has $\epsilon = -1$. 
This term doesn't change much of the dynamic of the scalar field (it has the same equation of motion, at least when not interacting), but it does change its stress-energy tensor : 
$$T_{\mu\nu} = \epsilon\nabla_\mu \varphi \nabla_\nu \varphi - \epsilon\frac 12 g_{\mu\nu}  g^{\sigma \tau} \nabla_\sigma \varphi \nabla_\tau \varphi$$
As we have that $T = \epsilon T_{\text{Can.}}$, this means that the field violates the null energy condition : for a null vector $k$, we have
$$T(k,k) = \epsilon T_{\text{Can.}}(k,k)$$
Since $T_{\text{Can.}}(k,k) \geq 0$, we have that the phantom field violates the NEC : it is of negative energy, as can be expected from the negative term to the kinetic energy.
