Invariance of critical exponents for renormalization groups generated by a flow vector

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Abstract
An invariance theorem for exponents of scaling variables is proved for a class of renormalization groups introduced by Wegner. (Renormalization groups generated by a flow vector.) A convenient form for the renormalization group linearized both around the fixed point and around the renormalization trajectory is derived. The latter is used to derive a coordinate system for perturbations around a critical Hamiltonian from that around a fixed point. It is shown that the classification scaling redundant applies also to the vectors of this critical-point coordinate system. The effect of an infinitesimal perturbation of the flow vector on the fixed point and on the renormalization group linearized around the fixed point is investigated. If the unperturbed linearized group has no marginal scaling operator, two cases arise. (i) The perturbed group has no fixed point near that of the unperturbed group. This case may obtain if there is a marginal redundant observable. (ii) The perturbed group has a fixed point which differs from that of the unperturbed group by a redundant perturbation. In the second case the scaling eigenvalues of the perturbed linearized group will be unchanged. Eigenvalues of redundant observables may change. The effect of a perturbation in a flow vector on a renormalized trajectory is considered and shown to suggest the concept of manifolds of equivalent Hamiltonians. Some difficulties involved in this concept are discussed.