Scaling Algebras and Renormalization Group in Algebraic Quantum Field Theory.

Abstract

The concept of scaling algebra provides a novel framework for the general structural analysis and classification of the short distance properties of algebras of local observables in relativistic quantum field theory. In the present article this method is applied to the simple example of massive free field theory in s = 1, 2 and 3 spatial dimensions. Not quite unexpectedly, one obtains for s = 2, 3 in the scaling (short distance) limit the algebra of local observables in massless free field theory. The case s = 1 offers, however, some surprises. There the algebra of observables acquires in the scaling limit a non-trivial center and describes charged physical states satisfying Gauss' law. The latter result is of relevance for the interpretation of the Schwinger model at short distances and illustrates the conceptual and computational virtues of the method.

Scaling Limit for Subsystems and Doplicher–Roberts Reconstruction

“…Furthermore, we denote by 0,ι;r → F (0) be the isomorphism onto the net of the massless scalar field, whose existence is proven in [9]. We will show that φ ↾ A …”

“…Let B ⊂ F be an inclusion of graded-local nets in the vacuum sector, such that F has convergent scaling limit, and let E : F → B be a conditional expectation of nets. Then equation (8) holds. Furthermore B has convergent scaling limit too.…”

“…Also, we associate to the free scalar field of mass m ≥ 0 the net of C * -algebras O → F (m) (O) obtained by considering the elements of the canonical net of von Neumann algebras of the free scalar field of mass m which are norm-continuous under translations. This net is conveniently isomorphically represented on the Fock space of the massless scalar field by local normality, see [9] for details.…”

“…Also, the mathematical tools that are available in this setting are well suited for providing a detailed analysis of subsystems, an issue that is central in order to obtain an intrinsic description of the observable system one starts with [12,10,11]. In another direction, the recently proposed algebraic approach to the renormalization group [8] (see also section 2) has opened the possibility of studying the short distance limit in the local quantum physics framework, and has started to convey new insight into our understanding of physically relevant issues such as confinement of colour charges and renormalization of pointlike fields [3,14,1].…”

Structure functions of the 2d non-linear sigma models

“…In leading order we get 10) which can be further simplified with the help of (A.20), (A.22), (2.3) and (2.7) leading to 11) where the constants G l are equal to V l and T l for the spin and current cases respectively given in (A.21) and (A.23), and further 12) which can also be written as 13) where the Adler functions A (r) were defined in (2.9). The final results for the complete structure functions are…”