John E. Roberts

The quantum structure of spacetime at the Planck scale and quantum fields

We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenbergs principle and by Einsteins theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations.We outline the definition of free fields and interactions over QST and take the first steps to adapting the usual perturbation theory. The quantum nature of the underlying spacetime replaces a local interaction by a specific nonlocal effective interaction in the ordinary Minkowski space. A detailed study of interacting QFT and of the smoothing of ultraviolet divergences is deferred to a subsequent paper.In the classical limit where the Planck length goes to zero, our Quantum Spacetime reduces to the ordinary Minkowski space times a two component space whose components are homeomorphic to the tangent bundleTS2 of the 2-sphere. The relations with Connes theory of the standard model will be studied elsewhere.

Why there is a field algebra with a compact gauge group describing the superselection structure in particle physics

Given the local observables in the vacuum sector fulfilling a few basic principles of local quantum theory, we show that the superselection structure, intrinsically determined a priori, can always be described by a unique compact global gauge group acting on a field algebra generated by field operators which commute or anticommute at spacelike separations. The field algebra and the gauge group are constructed simultaneously from the local observables. There will be sectors obeying parastatistics, an intrinsic notion derived from the observables, if and only if the gauge group is non-Abelian. Topological charges would manifest themselves in field operators associated with spacelike cones but not localizable in bounded regions of Minkowski space. No assumption on the particle spectrum or even on the covariance of the theory is made. However the notion of superselection sector is tailored to theories without massless particles. When translation or Poincaré covariance of the vacuum sector is assumed, our construction leads to a covariant field algebra describing all covariant sectors.

Fields, observables and gauge transformations I

AbstractStarting from an algebra of fields

A Theory of Dimension

\mathfrak{F}

Local quasiequivalence and adiabatic vacuum states

and a compact gauge group of the first kind ℊ, the observable algebra

Local cohomology and superselection structure

\mathfrak{A}