We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg's principle and by Einstein's theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations.

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

We define and study two-dimensional, chiral conformal field theory by the methods of algebraic field theory. We start by characterizing the vacuum sectors of such theories and show that, under very general hypotheses, their algebras of local observables are isomorphic to the unique hyperfinite type III1 factor. The conformal net determined by the algebras of local observables is proven to satisfy Haag duality. The representation of the Moebius group (and presumably of the entire Virasoro algebra) on the vacuum sector of a conformal field theory is uniquely determined by the Tomita-Takesaki modular operators associated with its vacuum state and its conformal net. We then develop the theory of Moebius covariant representations of a conformal net, using methods of Doplicher, Haag and Roberts. We apply our results to the representation theory of loop groups. Our analysis is motivated by the desire to find a “background-independent” formulation of conformal field theories.

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Operator algebras and conformal field theory

In the book of Haag [Local Quantum Physics (Springer Verlag, Berlin, 1992)] about local quantum field theory the main results are obtained by the older methods of C * - and W * -algebra theory. A great advance, especially in the theory of W * -algebras, is due to Tomita’s discovery of the theory of modular Hilbert algebras [Quasi-standard von Neumann algebras, Preprint (1967)]. Because of the abstract nature of the underlying concepts, this theory became (except for some sporadic results) a technique for quantum field theory only in the beginning of the nineties. In this review the results obtained up to this point will be collected and some problems for the future will be discussed at the end. In the first section the technical tools will be presented. Then in the second section two concepts, the half-sided translations and the half-sided modular inclusions, will be explained. These concepts have revolutionized the handling of quantum field theory. Examples for which the modular groups are explicitly known are presented in the third section. One of the important results of the new theory is the proof of the PCT theorem in the theory of local observables. Questions connected with the proof are discussed in Sec. IV. Section V deals with the structure of local algebras and with questions connected with symmetry groups. In Sec. VI a theory of tensor product decompositions will be presented. In the last section problems that are closely connected with the modular theory and that should be treated in the future will be discussed.

On revolutionizing quantum field theory with Tomita’s modular theory

Quasi Hopf quantum symmetry in quantum theory

Fuzzy Spaces Star Products Scalar Fields on the Fuzzy Sphere Instantons, Monopoles and Projective Modules Fuzzy Nonlinear Sigma Models Fuzzy Gauge Theories The Dirac Operator and Axial Anomaly Fuzzy Supersymmetry SUSY Anomalies on the Fuzzy Supersphere Fuzzy Spaces as Hopf Algebras.

Lectures on Fuzzy and Fuzzy Susy Physics

According to the theory of superselection sectors of Doplicher, Haag, and Roberts, field operators which make transitions between different superselection sectors—i.e. different irreducible representations of the observable algebra—are to be constructed by adjoining localized endomorphisms to the algebra of local observables. We find the relevant endomorphisms of the chiral algebra of observables in the minimal conformal model with central chargec=1/2 (Ising model). We show by explicit and elementary construction how they determine a representation of the braid groupB∞ which is associated with a Temperley-Lieb-Jones algebra. We recover fusion rules, and compute the quantum dimensions of the superselection sectors. We exhibit a field algebra which is quantum group covariant and acts in the Hilbert space of physical states. It obeys local braid relations in an appropriate weak sense.

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Conformal field algebras with quantum symmetry from the theory of superselection sectors

Quantum symmetry for pedestrians

Quantum field planes furnish a noncommutative differential algebra $\Omega$ which substitutes for the commutative algebra of functions and forms on a contractible manifold The data required in their construction come from a quantum field theory The basic idea is to replace the ground field ${\bf C}$ of quantum planes by the noncommutative algebra ${\cal A}$ of observables of the quantum field theory

Models of Quantum Space Time: Quantum Field Planes

Quantum field planes furnish a noncommutative differential algebra $\Omega$ which substitutes for the commutative algebra of functions and forms on a contractible manifold. The data required in their construction come from a quantum field theory. The basic idea is to replace the ground field ${\bf C}$ of quantum planes by the noncommutative algebra ${\cal A}$ of observables of the quantum field theory.

Gerhard Mack

Papers

Conformal field algebras with quantum symmetry from the theory of superselection sectors

Quantum symmetry for pedestrians

Models of Quantum Space Time: Quantum Field Planes

Models of quantum space time: Quantum field planes