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arXiv:hep-th/9501063 · Reviews in Mathematical Physics

SCALING ALGEBRAS AND RENORMALIZATION GROUP IN ALGEBRAIC QUANTUM FIELD THEORY

Detlev Buchholz A, Rainer Verch

Published Nov 1, 1995• 105 citations
arXivHTMLPDF

Abstract

For any given algebra of local observables in Minkowski space an associated scaling algebra is constructed on which renormalization group (scaling) transformations act in a canonical manner. The method can be carried over to arbitrary space-time manifolds and provides a framework for the systematic analysis of the short distance properties of local quantum field theories. It is shown that every theory has a (possibly non-unique) scaling limit which can be classified according to its classical or quantum nature. Dilation invariant theories are stable under the action of the renormalization group. Within this framework the problem of wedge (Bisognano-Wichmann) duality in the scaling limit is discussed and some of its physical implications are outlined.

DOI
10.1142/s0129055x9500044x

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For any given algebra of local observables in Minkowski space an associated scaling algebra is constructed on which renormalization group (scaling) transformations act in a canonical manner. The method can be carried over to arbitrary space-time manifolds and provides a framework for the systematic analysis of the short distance properties of local quantum field theories. It is shown that every theory has a (possibly non-unique) scaling limit which can be classified according to its classical or quantum nature. Dilation invariant theories are stable under the action of the renormalization group. Within this framework the problem of wedge (Bisognano-Wichmann) duality in the scaling limit is discussed and some of its physical implications are outlined.
For any given algebra of local observables in relativistic quantum field theory there exists an associated scaling algebra which permits one to introduce renormalization group transformations and to construct the scaling (short distance) limit of the theory. On the basis of this result it is discussed how the phase space properties of a theory determine the structure of its scaling limit. Bounds on the number of local degrees of freedom appearing in the scaling limit are given which allow one to distinguish between theories with classical and quantum scaling limits. The results can also be used to establish physically significant algebraic properties of the scaling limit theories, such as the split property.
Short distance scaling limits of a class of integrable models on two-dimensional Minkowski space are considered in the algebraic framework of quantum field theory. Making use of the wedge-local quantum fields generating these models, it is shown that massless scaling limit theories exist, and decompose into (twisted) tensor products of chiral, translation-dilation covariant field theories. On the subspace which is generated from the vacuum by the observables localized in finite light ray intervals, this symmetry can be extended to the M\\"obius group. The structure of the interval-localized algebras in the chiral models is discussed in two explicit examples.
We analyze a class of quantum field theory models illustrating some of the possibilities that have emerged in the general study of the short distance properties of superselection sectors, performed in a previous paper (together with R. Verch). In particular, we show that for each pair (G, N), with G a compact Lie group and N a closed normal subgroup, there is a net of observable algebras which has (a subset of) DHR sectors in 1-1 correspondence with classes of irreducible representations of G, and such that only the sectors corresponding to representations of G/N are preserved in the scaling limit. In the way of achieving this result, we derive sufficient conditions under which the scaling limit of a tensor product theory coincides with the product of the scaling limit theories.
The method of scaling algebras, which has been introduced earlier as a means for analyzing the short-distance behaviour of quantum field theories in the setting of the model-independent, operator-algebraic approach, is extended to the case of fields carrying superselection charges. A criterion for the preservance of superselection charges in the short-distance scaling limit is proposed. Consequences of this preservance of superselection charges are studied. The conjugate charge of a preserved charge is also preserved, and the preservance of all charges of a quantum field theory in the scaling limit leads to equivalence of local and global intertwiners between superselection sectors.
The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].