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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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