higher cut-off energy of electronic excitations. The alternative to incoherent particle-hole fluctuations are collective modes of electronic fluctuations, to which we turn next. 

## 4. Exchange of collective fluctuations 

Part of the spectral weight of particle-hole fluctuations appears in collective modes if the effective interactions are strong enough compared to the kinetic energy. In general a particle hole-fluctuation is characterized by an internal momentum k and a center of mass momentum q , besides other indices which specify the channel of the excitations, for example, density or spin or current or spin-current or inter-valley, interband, etc. A useful limit is obtained from the fact that a bound state of a fermion particle and hole (and indeed particle and particle or hole and hole) has Boson commutation relations. In a bound state, there is no dispersion in energy at any q as a function of k , so the latter index may be summed over. To regard the fluctuations as collective rather than incoherent, one must make sure that the dependence on k is unimportant to do sensible calculations. Also important is the integrated spectra of the collective modes and the incoherent particle-hole spectral weight of fermions for any given channel, is generally fixed by sum-rules. It is sinful double-counting to introduce collective variables, for example spin-fluctuations, and give them the total weight of S ( S + 1) per spin S , and do calculations of their interactions with fermions in the same way that one does for phonons. 

## 5. Symmetry of Pairing Interaction from Exchange of Spin-Fluctuations 

As we will see later, the frequency dependence of the interaction (with total spectral weight and coupling constant fixed) is of much more importance in determining T$_{c}$ in finite ℓ than for ℓ = 0. The anisotropy of the interaction as well as the details of the geometry of the fermi-surface(s) are also important in determining the irreducible representation favored for pairing. However, essential aspects of finite ℓ pairing, specifically the distinction between the pairing promoted by ferromagnetic and anti-ferromagnetic fluctuations is revealed in a simple model calculation [19] based on a frequency independent and isotropic interaction in a model of a spherical fermi-surface. Consider effective electron-electron interactions of the 

