Gerhard Mack

All unitary ray representations of the conformal group SU(2,2) with positive energy

AbstractWe find all those unitary irreducible representations of the ∞-sheeted covering group

Proof of confinement of static quarks in 3-dimensionalU(1) lattice gauge theory for all values of the coupling constant

\tilde G

Convergence of Operator Product Expansions on the Vacuum in Conformal Invariant Quantum Field Theory

of the conformal group SU(2,2)/ℤ4 which have positive energyP0≧0. They are all finite component field representations and are labelled by dimensiond and a finite dimensional irreducible representation (j1,j2) of the Lorentz group SL(2ℂ). They all decompose into a finite number of unitary irreducible representations of the Poincaré subgroup with dilations.

Duality in quantum field theory

In quantum theory, internal symmetries more general than groups are possible. We show that quasi-triangular quasi Hopf algebras G∗ (“quasi quantum groups”) as introduced by Drinfeld [1] permit a consistent formulation of a transformation law of states in the physical Hilbert space H, of invariance of the ground state, and of a transformation law of field operators which is consistent with local braid relations of field operators which generalise those proposed by Frohlich [2]. All this remains true when Drinfelds axioms are suitably weakened in order to build in truncated tensor products. Conversely, all the axioms of a weak quasi-triangular quasi Hopf algebra are motivated from what physics demands of a symmetry. Unitarity requires in addition that G∗ admits a ∗-operation with certain properties. Invariance properties of Green functions follow from invariance of the ground state and covariance of field operators as usual. Covariant adjoints and covariant products of field operators can be defined. The R-matrix elements in the local braid relations are in general operators in H. They are determined by the symmetry up to a phase factor. Quantum group algebras like Uq(sl2) with |q| = 1 are examples of symmetries with special properties. We show that a weak quasi-triangular quasi Hopf algebra G∗ is canonically associated with Uq(sl2) if qp = 1. We argue that these weak quasi Hopf algebras are the true symmetries of minimal conformal models. Their dual algebras G (“functions on the group”) are neither commutative nor associative.

Monopoles, Vortices, and Confinement

AbstractWe study the 3-dimensional pureU(1) lattice gauge theory with Villain action which is related to the 3-dimensional ℤ-ferromagnet by an exact duality transformation (and also to a Coulomb system). We show that its string tension α is nonzero for all values of the coupling constantg2, and obeys a bound α≧const·mDβ−1 for smallag2, with β=4π2/g2 and

Colour screening and quark confinement

m_D^2 = (2{\beta \mathord{\left/ {\vphantom {\beta a}} \right. \kern-\nulldelimiterspace} a}^3 )e^{ - \beta \upsilon _{Cb} {{(0)} \mathord{\left/ {\vphantom {{(0)} 2}} \right. \kern-\nulldelimiterspace} 2}} (a = lattice spacing)