Index Theory and Adiabatic Limit in QFT

Abstract

The paper has the form of a proposal concerned with the relationship between the three mathematically rigorous approaches to quantum field theory: (1) local algebraic formulation of Haag, (2) Wightman formulation and (3) the perturbative formulation based on the microlocal renormalization method. In this project we investigate the relationship between (1) and (3) and utilize the known relationships between (1) and (2). The main goal of the proposal lies in obtaining obstructions for the existence of the adiabatic limit (confinement problem in the phenomenological standard model approach). We extend the method of deformation of Dütsch and Fredenhagen (in the Bordeman-Waldmann sense) and apply Fedosov construction of the formal index-an analog of the index for deformed symplectic manifolds, generalizing the Atiyah-Singer index. We present some first steps in realization of the proposal.

A Generalization of Mackey's Theory of Induced Representations

“…This is important if we want to achieve any chance for recovering the asymptotic states in order to preserve any contact with the existing interpretation in terms of particle states along the lines of the algebraic QFT (compare the Haag-Ruelle scattering theory within the algebraic QFT). In our previous paper [74] we have noticed that the algebra of space-time coordinates should be construable (with the possible uniform infinite multiplicity) within the operator-algebra-Connestype format in the Fock-Krein space of local free fields as a natural generalization of the so called DHR theory of global gauge groups ( [35], Chap. IV), and as an answer to a problem set up by Staruszkiewicz [70].…”

“…Before we explain the spectral construction of space-time suggested in [74], several remarks on the Connes' spectral format giving a structure of a pseudo-Riemann manifold to the Gelfand spectrum of a commutative pre-C * -algebra A of operators acting in a Hilbert space H are in order. It has been analysed (in the compact case) by Strohmaier [72], where he recalled a result of H. Baum [1] that: 1) the Hilbert space of square integrable sections of the Clifford module naturally associated to the pseudo-Riemann structure on a (not necessary compact) orientable amd time orientable pseudo-Riemann manifold admits a fundamental symmetry J (induced by a space like reflection) which induces in the space of sections of the module the structure of the Krein space; 2) the natural Dirac operator D associated to the module is not self-adjont with respect to any natural Hilbert space associated to the pseudo-Riemann manifold, but it is self-adjoint in the Krein sense whenever the ordinary Riemann metric associated to the space like reflection is complete (which is automatic for compact manifolds).…”

“…Any way we may assume for our needs that we have the preferred unitization à of A at hand with the reconstruction problem reduced to the unital case. Indeed we are in the situation where Spec A has the natural R 4 -manifold structure with the standard Lorentzian pseudo-metric tensor and with A equal to the algebra of complex smooth Schwartz functions, so that we are at the non compact analogue of the (⇒)-half situation of the Theorem 11.4 of [10] and we do not need the full reconstruction theorem in order to realize our hint of [74] for the construction of the undeformed (A, D, H). It should be stressed that the non compact version of the reconstruction theorem is important as only under its validity we have the operator-algebraic characterisation of space-time justified, and on the other hand the operator-algebra format is capable of the deformation indicated to above.…”

“…It should be stressed that the non compact version of the reconstruction theorem is important as only under its validity we have the operator-algebraic characterisation of space-time justified, and on the other hand the operator-algebra format is capable of the deformation indicated to above. But we would like to stress here that proving the reconstruction theorem is a practically independent problem and do not intervene into the operator-algebra construction of (A, D, H) as proposed in [74] (we have to assume at this stage that the reconstruction theorem holds true for the non compact case with the first five conditions of Connes replaced appropriately with the conditions suggested in particular in [28] and [64]).…”

“…Indeed comparing [33] with the abstract symbol calculus of Connes and Moscovici [15] we should expect that the construction of deformation as well as the Fedosov-type conditions should extend for obtaining the necessity and sufficiency conditions for existence of the representation of the deformation of the more general algebras A containing A. Thus in the plan of the proposal for solving the problem (b), as proposed in [74], the construction of undeformed spectral triple (A, D, H) in the Fock space of free fields is of fundamental importance and perhaps is the most difficult task in this proposal.…”