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We construct the non-standard Lagrangian, called the multiplicative form, ofthe homogeneous scalar field and fermion field through the inverse calculus ofvariations, which the equation of motion still satisfies the Klein-Gordon andDirac equations, respectively. By employing the non-uniqueness of Lagrangian,we show that the Lagrangians can be written between the linear combination ofstandard and non-standard Lagrangian. The stability of the ghost field, anunnatural smallness of cosmological constant, and the chiral condensate arediscussed by applying these new Lagrangians.

The non-standard Lagrangian from non-uniqueness principle of the real scalar field and fermion field