Abstract
We formulate an “action principle” for the operator product expansion (OPE) describing how a given OPE coefficient changes under a deformation induced by a marginal or relevant operator. Our action principle involves no ad-hoc regulator or renormalization and applies to general (Euclidean) quantum field theories. It implies a natural definition of the renormalization group flow for the OPE coefficients and of coupling constants. When applied to the case of conformal theories, the action principle gives a system of coupled dynamical equations for the conformal data. The last result has also recently been derived (without considering tensor structures) independently by Behan (arXiv:1709.03967) using a different argument. Our results were previously announced and outlined at the meetings “In memoriam Rudolf Haag” in September 2016 and the “Wolfhart Zimmermann memorial symposium” in May 2017.
Highlights
One possible viewpoint of quantum field theory (QFT) is that the operator product expansion [41, 42] (OPE) defines a theory, just as the equations of motion define a classical field theory.Informally, the OPE states thatOA1(x1) · · · OAn =CAB1...An(x1, . . . , xn)OB(xn), B (1.1)which is understood in the sense of an insertion into a correlation function, and where {OA} is the set of all composite operators of the theory; for details see sec. 2.2
Such a formula was derived only relatively recently in [19] building on earlier work [22, 25, 29]. It gives an expression for the derivative of an 2-point OPE coefficient w.r.t g in terms of 2- and 3-point OPEs involving the relevant or marginal operator V conjugate to g
These are supplemented by analogous formulas for the derivative of an n-point OPE coefficient w.r.t g in terms of 2- and (n + 1)-point OPEs
Summary
One possible viewpoint of quantum field theory (QFT) is that the operator product expansion [41, 42] (OPE) defines a theory, just as the equations of motion define a classical field theory. The path integral suggests a formal action principle, but such naive formulas require renormalization, while we are looking for an intrinsic formula not requiring such extraneous procedures Such a formula was derived only relatively recently in [19] building on earlier work [22, 25, 29]. The purpose of the present paper is to briefly explain the relationship of the action principle to the renormalisation group, and secondly, to apply it in the case of conformal QFTs (CFTs). In those theories, the n-point OPE coefficients and correlation functions can, in principle, be written in terms of the conformal data, i.e. the dimensions ∆i of the conformal primaries and the structure constants λαijk, where Oi, Oj, . This work is dedicated to the memory of Wolfhart Zimmermann whose work on the OPE [42] has been a major inspiration for us to further study this structure in QFT
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