By Paolo Rossi (ed.)
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67) Notice that e4 can be expressed in terms of e2,3 and the tadpole e1 = [K], Hassan and Rosen (2011). The potential interactions can be also in an analog way constructed from the polynomials 4 βn en ( (g μα fαν )). 69) n=0 where the elementary √ symmetric polynomials can also be written in terms of the eigenvalues λi of (g μα fαν ) (Hassan and Rosen 2011) e0 ( (g μα fαν )) = 1 e1 ( (g μα fαν )) = λ1 + λ2 + λ3 + λ4 e2 ( (g μα fαν )) = λ1 λ2 + λ1 λ3 + λ1 λ4 + λ2 λ3 + λ2 λ4 + λ3 λ4 e3 ( (g μα fαν )) = λ1 λ2 λ3 + λ1 λ2 λ4 + λ1 λ3 λ4 + λ2 λ3 λ4 e4 ( (g μα fαν )) = λ1 λ2 λ3 λ4 .
38) This corresponds to a scalar-vector-tensor theory in which the scalar field is kinetically mixed with the tensor whereas the vector field is completely decoupled. In the next Sect. 2 we will try to generalize the interactions of the spin-2 field to the non-linear case and discuss the difficulties one usually encounters. But before doing that let us first introduce the concept of Effective Field Theories since it became an essential tool in particle physics and modern cosmology. 3 Effective Field Theories Nature discloses itself at different energy scales.
Actually this result is not surprising at all. After doing the field redefinition hμν = h˜ μν + πημν the mixing between the scalar and tensor interactions in Eq. 38) vanish at the expense of the coupling of the scalar field to the stress energy tensor πT. It is exactly this coupling that gives rise to the vDVZ discontinuity which survives the m → 0 limit. However, we were working in a regime in which some of the degrees of freedom of the massive graviton were interacting highly non-linearly and therefore the vDVZ discontinuity is just an artifact of the linear approximation.
La Filosofia. III - Le discipline filosofiche. by Paolo Rossi (ed.)