Constraints and generalized gauge transformations on tree-level gluon and graviton amplitudes

Writing the fully color dressed and graviton amplitudes, respectively, as {A} = < {C} {C A}} } {A} > = {C} {C {M| N }} } {{M| N }} > and {{A}_{gr}} = < {{tilde{N}}} { {{tilde{N}} {M| N >.}}} >. } {{M| N >.}} >, where | A > is a set of Kleiss-Kuijf color ordered basis, | N > , | {tilde{N}} > and | C > are the similarly ordered numerators and color coefficients, we show that the propagator matrix M has ( n - 3)( n - 3)! independent eigenvectors | {λ_j^0} > with zero eigenvalue, for n-particle processes. The resulting equations | {λ_j^0} > = 0 are relations among the color ordered amplitudes. The freedom to shift | N > to | N > + sumnolimits_j {{f_j}| {λ_j^0} > } and similarly for | {tilde{N}} > where f _j are ( n - 3)( n - 3)! arbitrary functions, encodes generalized gauge transformations. They yield both BCJ amplitude and KLT relations, when such freedom is accounted for. Furthermore, f _j can be promoted to the role of effective Lagrangian vertices in the field operator space.