Inequalities for Trace Anomalies, Length of the Rg Flow, Distance between the Fixed Points and Irreversibility
نویسنده
چکیده
I argue that in quantum field theory: i) the scheme-invariant area ∆a of the graph of the effective beta function between the fixed points defines the length of the RG flow; ii) the minimum of ∆a in the space of flows connecting the same UV and IR fixed points defines the (oriented) distance between the fixed points; iii) in even dimensions, the distance between the fixed points is equal to ∆a = aUV − aIR. These statements imply, in even dimensions, the inequalities 0 ≤ ∆a ≤ ∆a, the irreversibility of the RG flow and the inequality a ≤ c for free scalars and fermions (but not vectors). I elaborate the axiomatic set-up of irreversibility, based on the notion of oriented distance and the “oriented-triangle inequalities”. I show that these axioms imply the irreversibility of the RG flow without a global a function. I argue that the RG flow is irreversible in odd dimensions (without a global a function) and, in support of this, I check the axioms of irreversibility in a class of d = 3 theories where the RG flow is integrable at each order of the large N expansion.
منابع مشابه
Sum Rules for Trace Anomalies and Irreversibility of the Renormalization-group Flow
The purpose of my talk is to review exact results for the trace anomalies c and a in the critical limits of supersymmetric theories, which provide non-perturbative evidence that the renormalization-group (RG) flow is irreversible, and then derive universal sum rules for c, a and a in even dimensions, formulate a theory of the irreversibility of the RG flow, recapitulate the arguments and the ev...
متن کاملKinematic Sum Rules for Trace Anomalies
I derive a procedure to generate sum rules for the trace anomalies a and a′. Linear combinations of ∆a ≡ aUV − aIR and ∆a′ ≡ aUV − aIR are expressed as multiple flow integrals of the two-, threeand four-point functions of the trace of the stress tensor. Eliminating ∆a′, universal flow invariants are obtained, in particular sum rules for ∆a. The formulas hold in the most general renormalizable q...
متن کاملAnomalies , Unitarity and Quantum Irreversibility
The trace anomaly in external gravity is the sum of three terms at criticality: the square of the Weyl tensor, the Euler density and 2R, with coefficients, properly normalized, called c, a and a ′ , the latter being ambiguously defined by an additive constant. Unitarity and positivity properties of the induced actions allow us to show that the total RG flows of a and a ′ are equal and therefore...
متن کاملQuantum Irreversibility in Arbitrary Dimension
Some recent ideas are generalized from four dimensions to the general dimension n. Two terms of the trace anomaly in external gravity, the Euler density Gn and 2 R, are relevant to the problem of quantum irreversibility. By adding the divergence of a gauge-invariant current, Gn can be extended to a new notion of Euler density G̃n, linear in the conformal factor. We call it pondered Euler density...
متن کامل