Scalar field configuration eigenstate ($\hat{\varphi }(\mathbf{x})|\phi ⟩=\phi (\mathbf{x})|\phi ⟩$)

A scalar field configuration eigenstate assigns eigenvalues of the equal-time field operator at every spatial point. These generalized states play the role of position eigenstates for a system of infinitely many coordinates. Inserting their regulated completeness relations on time slices constructs the scalar field path integral.