For example, in quantum field theory, one then has to rely on renormalized perturbation theory and has to check for anomalies. There is no general unique procedure to do this and it might fail altogether. Traditionally in quantum theory, one uses the classical form for the generators of symmetries and attempts finding the corresponding expressions in the quantized theory. In the path integral formulation, the situation looks better but mathematically rigorous results are rare in this formalism. A similar connection also holds in quantum physics, but since the Lagrangian is not directly used in canonical quantization, it is less evident. One of the deepest structural results of classical physics is the intimate relation between symmetries and conservation laws which were uncovered by Emmy Noether about 100 years ago. Moreover, the axiom also characterizes symmetries of the theory. The remarkable stability of the algebra under a large class of time evolutions compares well to a similar property in non-relativistic quantum field theory in the framework of Resolvent Algebras by Buchholz and Grundling. We then prove that the new axiom implies the time slice axiom (primitive causality in ), which states that each observable can be expressed in terms of observables in any neighborhood of a Cauchy surface. Here, it is used to characterize possible anomalies of the unitary Master Ward Identity. It corresponds to the Stückelberg–Petermann renormalization group in formal perturbation theory. The crucial ingredient for our formulation is an intrinsic nonperturbative concept of the renormalization group (Definition 4.2). 5)-including anomalies-and show that, in formal perturbation theory, it is essentially equivalent to the infinitesimal version (see Sect 10 and Appendix C). We postulate in this paper a unitary version of this identity (Axiom “Symmetries” in Sect. Its infinitesimal version appears in a somewhat different form in the quantum master equation of the BV formalism, and was precisely analyzed in renormalized perturbation theory under the name Master Ward Identity (MWI) in. An important relation is obtained by the field redefinition (see e.g. We thereby use inspiration from the path integral formulation and check whether the resulting relations are compatible e.g. We discuss in this paper whether further relations can be added in order to enrich the structure and to bring the formalism nearer to standard quantum field theory. In the case of the free Lagrangian, the canonical commutation relations in form of the Weyl relations are a consequence of the formalism. By definition, these unitaries satisfy a Causality Relation corresponding to the ordering in time when these operations are performed, and a relation which determines dynamics in terms of the classical Lagrangian and which is a unitary version of the Schwinger–Dyson equation known from perturbative quantum field theory. F is interpreted as a local variation of the dynamics, and S( F) is the induced operation on the system (the scattering “ S-matrix,” an interpretation as such is offered at the end of Sect. The algebras are generated by unitaries S( F) labeled by local functionals F of the (classical) scalar field configuration. It was formulated for the case of a scalar field with polynomial self-interaction (for the incorporation of fermions, see ). It is a fully nonperturbative construction. Recently, Detlev Buchholz and one of us (KF) succeeded in finding a framework where the classical Lagrangian determines a net of C*-algebras. As a bon mot, Rudolf Haag used to ask colleagues “What is the Lagrangian?” and no answer could satisfy him. On the other hand, the axioms of algebraic quantum field theory have not yet been equally successful in fixing specific interacting theory models. in Quantum Statistical Mechanics via the KMS conditions, in the reconstruction of symmetries via Wiesbrock’s half-sided modular condition/intersection, in localization properties of field theories with their particle interpretation and its mathematical ramifications, and which is fruitfully used nowadays to discuss entropy and entanglement in quantum field theories. Rehren’s contribution in ), and Tomita–Takesaki modular theory which is instrumental e.g. ) which is at the basis of recent work in conformal field theory (see e.g. As particularly important examples, one may mention the theory of superselection sectors of Doplicher, Haag and Roberts (see e.g. In quantum field theory, the algebraic approach of Araki, Borchers, Haag and Kastler has led to deep insights into the structure of the theory. The C*-algebraic formulation of quantum physics is well known for its rather unique combination of conceptual clarity and mathematical precision.
0 Comments
Leave a Reply. |