$T^{0}_{0}$ element of Energy-Momentum tensor derivation from a Lagrangian I'm new here. Given the Lagrangian,
$ L=\frac{1}{2}∂_{μ}\Phi (∂^{μ}\Phi)^{*} − \frac{λ}{4}(\Phi\Phi^{*} - 1)^{2}$
and its energy-momentum tensor
$T^{\mu}_{\nu}=\frac{\partial L}{\partial(\partial^{\mu}\Phi^{*})}\partial_{\nu} \Phi -δ^{μ}_{ν}L $
For the element $T^{0}_{0}$ I get 
$T^{0}_{0}=\frac{1}{2}\dot{\Phi}^{2}+\frac{1}{2}(\nabla{\Phi})^{2}+\frac{λ}{4}(\Phi\Phi^{*} - 1)^{2}$ 
Which seems to be incorrect, could anyone help me find the correct answer?  Thank you for your time!! 
 A: You aren't applying the energy-momentum tensor in the right way. You should sum over both fields. Check again your formula!
Edit: You have to consider the complex field like an other field.
Edit2: The correct formula is given by
$$T^\mu_{\nu} = \frac{\partial L}{\partial (\partial_\mu \phi_a)}\partial_\nu \phi_a - \delta_\nu^\mu L $$
where you have to sum over all fiels $\phi_a = \{\Phi, \Phi^*\}$.
A: The correct canonical normalized Lagrangian should be
$$
L = ( \partial^\mu \Phi)^*( \partial_\mu  \Phi )  - \frac{\lambda}{4} ( \Phi^* \Phi - 1 )^4 \,.
$$
The canonical energy momentum tensor is
$$
T^\mu{}_\nu = \frac{\partial L }{ \partial( \partial_\mu \Phi ) } \partial_\nu \Phi + \frac{\partial L }{ \partial( (\partial_\mu \Phi)^* ) } (\partial_\nu \Phi)^* - \delta^\mu_\nu L .
$$
Applying this to the Lagrangian at hand, we find
$$
T^\mu{}_\nu = (\partial^\mu \Phi)^* \partial_\nu \Phi + \partial^\mu \Phi ( \partial_\nu \Phi )^* - \delta^\mu_\nu  \left[ ( \partial^\alpha \Phi)^*( \partial_\alpha  \Phi )  - \frac{\lambda}{4} ( \Phi^* \Phi - 1 )^4 \right]
$$
