Quantum irreversibility in arbitrary dimension

Abstract

Some recent ideas are generalized from four dimensions to the general dimension n. In quantum field theory, two terms of the trace anomaly in external gravity, the Euler density Gn and □n/2−1R, 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. This notion relates the trace-anomaly coefficients a and a′ of Gn and □n/2−1R in a universal way (a=a′) and gives a formula expressing the total RG flow of a as the invariant area of the graph of the beta function between the fixed points. I illustrate these facts in detail for n=6 and check the prediction to the fourth-loop order in the ϕ3-theory. The formula of quantum irreversibility for general n even can be extended to n odd by dimensional continuation. Although the trace anomaly in external gravity is zero in odd dimensions, I show that the odd-dimensional formula has a predictive content.