A GEOMETRICAL INTERPRETATION OF RENORMALIZATION-GROUP FLOW

The renormalization group (RG) equation in D-dimensional Euclidean spa ce, R(D), is analyzed from a geometrical point of view. A general form of the RG equation is derived which is applicable to composite operat ors as well as tensor operators (on R(D)) which may depend on the Eucl idean metric. It is argued that physical N-point amplitudes should be interpreted as rank N covariant tensors on the space of couplings, G, and that the RG equation can be viewed as an equation for Lie transpor t on G with respect to the vector field generated by the beta function s of the theory. In one sense it is nothing more than the definition o f a Lie derivative. The source of the anomalous dimensions can be inte rpreted as being due to the change of the basis vectors on G under Lie transport. The RG equation acts as & bridge between Euclidean space a nd coupling constant space in that the effect on amplitudes of a diffe omorphism of R(D) (that of dilations) is completely equivalent to a di ffeomorphism of G generated by the beta functions of the theory. A for m of the RG equation for operators is also given. These ideas are deve loped in detail for the example of massive lambdaphi4 theory in four d imensions.