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2

The first part of OP's construction is directly related to the covariant Hamiltonian formalism for a real scalar field with Lagrangian density $${\cal L} ~=~ \frac{1}{2}\partial_{\alpha} \phi ~\partial^{\alpha} \phi -{\cal V}(\phi), \tag{CW4}$$ see e.g. Ref. [CW] and this Phys.SE post. See also the Wronskian method in this Phys.SE post. [In this answer we ...

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I am going to write a short answer, because no one has answered yet. The left hand side should be $$\eta^{\mu \nu} \partial_\mu \phi \partial_\nu \phi.$$ It is important you only sum raised indicies with lowered indices and you never sum two lowered indices together. Summing two lowered indices gives you a result that is not lorentz invariant. The right ...

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