Scheme Independence as an Inherent Redundancy in Quantum Field Theory

The path integral formulation of Quantum Field Theory implies an infinite set of local, Schwinger-Dyson-like relations. Exact renormalization group equations can be cast as a particular instance of these relations. Furthermore, exact scheme independence is turned into a vector field transformation of the kernel of the exact renormalization group equation under field redefinitions. Relations for the Green functions of a quantum field theory are in general traced down to symmetries of the partition function. In particular, the path integral allows for the basic symmetry of field redefinitions. A vast family of Schwinger-Dyson like relations follows by taking the infinitessimal field redefinitioñ ϕ α = ϕ α − θ α [ϕ, t] Dϕ ∂ α θ α e −S = 0 ⇒ ∂ α θ α = θ α ∂ α S , (1) where we have introduced the compact notation ∂ α ≡ δ δϕ α , (2) α standing for momentum and other possible quantum numbers. There is a deep connection between this observation and the exact renormal-ization group. 1 Remarkably, the exact renormalization group equation itself, 2,3 as formulated by Polchinski, 4 can be written as the integrand of a field redefinition transformation. 5,6 To see this, let us first recall the dimensionless form of Polchin-ski's equation ∂ t S + d p ϕ p δS δϕ p + p ϕ p p µ ∂ ∂p µ δS δϕ p = p c ′ (p 2) δS δϕ p δS δϕ −p − δ 2 S δϕ p δϕ −p − 2 p 2 c(p 2) ϕ p δS δϕ p , (3) where t ≡ ln µ Λ , µ being some fixed physical scale and Λ the Wilsonian cutoff which is taken towards 0, S[ϕ, t] is a functional of ϕ and a function of t and c(p 2) is the regulating function defining classes of schemes. We can now check that Polchinski's equation is identical to