C function representation of the Local Potential Approximation

Within the Local Potential Approximation to Wilsonʹs, or Polchinskiʹs, exact renormalization group, and for general spacetime dimension, we construct a function, c, of the coupling constants; it has the property that (for unitary theories) it decreases monotonically along flows, and is stationary only at fixed points — where it ‘counts degrees of freedom’, i.e. is extensive, counting one for each Gaussian scalar. Furthermore, by choosing restrictions to some sub-manifold of coupling constant space, we arrive at a very promising variational approximation method.