Perturbative algebraic quantum field theory and the renormalization groups

Abstract

A new formalism for the perturbative construction of algebraic quantum field theory is developed. The formalism allows the treatment of low-dimensional theories and of non-polynomial interactions. We discuss the connection between the Stückelberg-Petermann renormalization group which describes the freedom in the perturbative construction with e-print archive: http://lanl.arXiv.org/abs/hep-th//0901.2038 R. BRUNETTI ET AL.the Wilsonian idea of theories at different scales. In particular, we relate the approach to renormalization in terms of Polchinski's Flow Equation to the Epstein-Glaser method. We also show that the renormalization group in the sense of Gell-Mann-Low (which characterizes the behaviour of the theory under the change of all scales) is a one-parametric subfamily of the Stückelberg-Petermann group and that this subfamily is in general only a cocycle. Since the algebraic structure of the Stückelberg-Petermann group does not depend on global quantities, this group can be formulated in the (algebraic) adiabatic limit without meeting any infrared divergencies. In particular we derive an algebraic version of the Callan-Symanzik equation and define the β-function in a state independent way.

The Generalised Principle of Perturbative Agreement and the Thermal Mass

“…In this section we briefly recall the functional approach to perturbative algebraic quantum field theories on curved spacetimes [BDF09,FrRe12,FrRe14], which is the setting of our work. For simplicity we will consider only the case of the real Klein-Gordon field.…”

“…In the last twenty years a solid conceptual framework of perturbative algebraic quantum field theory on curved spacetimes has been established [BFK95, BrFr00, HoWa01, HoWa02, BFV03, HoWa05, Ho08, BDF09,FrRe13]. This was possible thanks to the seminal work [Ra96], which introduced the powerful tools of microlocal analysis to algebraic quantum field theory.…”

Quantum Fields in Curved Spacetime

“…It is easy to see that ω(T µν ) con-tributes a positive energy flux to infinity. It follows from conservation of stressenergy together with the approximate stationarity of ω at late times that ω(T µν ) contributes a corresponding flux of negative energy 15 into the black hole. Consequently, if the black hole is isolated (so that there is no other flux of stress-energy into the black hole), the black hole will slowly lose mass as a result of the quantum field effects.…”

“…In particular, in any Hadamard state, the observable φ ( f T ) on Σ 0 in our above discussion will be highly entangled with corresponding field observables inside the black hole. Since φ [F T ] = φ ( f T ) (see (10)), this means that the Hawking radiation flux measured by a distant detector is highly entangled with field observables 15 Negative energy fluxes of stress-energy or negative energy densities can occur in quantum field theory even for fields that classically satisfy the dominant energy condition, see e.g. [32,34].…”

“…We do not see any canonical way of imposing a cutoff in general Lorentzian curved spacetime, and furthermore, any cutoff is in fundamental conflict with the idea of defining the theory in a "local and covariant manner". Further discussion of the relationship between various different formulations of renormalization group flow can be found in [15].…”