Construction of field algebras with quantum symmetry from local observables

It has been discussed earlier that (weak quasi-) quantum groups allow for a conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid-) statistics and locality was established. This work addresses the reconstruction of quantum symmetries and algebras of field operators. For every algebraA of observables satisfying certain standard assumptions, an appropriate quantum symmetry is found. Field operators are obtained which act on a positive definite Hilbert space of states and transform covariantly under the quantum symmetry. As a substitute for Bose/Fermi (anti-) commutation relations, these fields are demonstrated to obey a local braid relation.     

Duality and absence of locally generated superselection sectors for CCR-type algebras

We isolate an abstract algebraic property which implies duality in all locally normal, irreducible representations of a quasilocalC*-algebra if it holds together with two more specific conditions. All these conditions holding for the CCR-algebra ind2 space time dimensions duality follows for representations of the two-dimensional CCR-algebra generated by pure Wightman states ofP()₂-theories. We then show that algebras of this kind have no nontrivial locally generated superselection sectors which ford3 yields a first approximation to a quantum analogue of Derrick's theorem.Supported by Science Research Council     

Superselection sectors in low dimensional quantum field theory

The occurence of braid group statistics in low dimensions is reviewed and an extension of the general theory to solitonic sectors is described where the DHR endomorphisms are replaced by homomorphisms between different extensions of the algebra of observables.     

Infinite conformal symmetry in two-dimensional quantum field theory

We present an investigation of the massless, two-dimentional, interacting field theories. Their basic property is their invariance under an infinite-dimensional group of conformal (analytic) transformations. It is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of Virasoro algebra, and that the correlation functions are built up of the “conformal blocks” which are completely determined by the conformal invariance. Exactly solvable conformal theories associated with the degenerate representations are analyzed. In these theories the anomalous dimensions are known exactly and the correlation functions satisfy the systems of linear differential equations.     

On the type of local algebras in quantum field theory

We give a simple sufficient condition for a von Neumann algebra to be Type III and apply it to some classes of algebras in QFT. For dilatation invariant local systems in particular we find that all sufficiently regular local algebras are Type III.     

On the application of conformal symmetry to quantum field theory

To leading order in α_s(Q²) conformal symmetry specifies the eigensolutions of the evolution equation for meson distribution amplitudes, the wavefunctions which control large-momentum-transfer exclusive mesonic processes in QCD. We find that at next to leading order, the eigensolutions in various field theories depend on the regularization scheme, even for zero β-function. This is contrary to the expectations of conformal symmetry.     

Conformal quantum field theory and half-sided modular inclusions of von-Neumann-Algebras

LetN, be von-Neumann-Algebras on a Hilbert space , a comon cyclic and separarting vector. Assume to be cyclic and separating also forN . Denote byJ , J _N the modular conjugations to (, ), and _N the associated modular operators. If and these data define in a canonical way a conformal quantum field theory in a cricle. Conversely, the chiral part of a conformal quantum field theory in two dimensions always yields such data in a natural way.Partly supported by the DFG, SFB 288 Differentiageometrie und Quantenphysik     

Conformal field theory and the symmetries of string field theory

The two-dimensional conformal field theory representation of Witten's bosonic string field theory is discussed. The basic overlap equations, K_n symmetry and BRST invariance are proved directly, without the usual expansion in oscillators. The conformal field theory approach naturally provides local overlap identities which (when integrated over half the string) can be used to verify properties of the cubic action. In particular, a recently proposed diffeomorphism invariance is shown to be free of anomalies. Finally, a new class of symmetries, including generalizations of the K_n symmetries which are local in spacetime, are presented.     

Theory of superselection sectors for generalized Ising models

We apply the theory of superselection sectors in the same way as done by G. Mack and V. Schomerus for the Ising model to generalizations of this model described by J. Fröhlich and T. Kerler. The sector generating morphisms are found.Supported by the DFG, SFB 288 Differentialgeometrie und Quantenphysik.     

Hilbert space sectors,dynamical charges and spontaneous symmetry breaking in classical field theory

Results concerning existence and properties of Hilbert space sectors in the family of solutions of classical non-linear field equations are presented. Within this framework the existence of conserved dynamical charges and the occurrence of spontaneous symmetry breaking is discussed.     

The symmetry aspects of quantum field theory in general relativity 1

Some proposed models for a quantum field theory in general relativity are briefly analyzed. Their main difficulties are a consequence of the initial choice of the group of symmetries of the (quantum) field equations. The necessity of selecting space-time isometries in a general covariant theory and the unphysical character of the Poincaré translations in a tangent plane theory are discussed. Starting from some basic requirements, a model is proposed in which the groups of symmetries are derived from the proper homogeneous groups of isometries of the minimal isometric local embedding spaces of space-times.This essay received an honorable mention (1977) from the Gravity Research Foundation-Ed.     

Spontaneous breaking of electric-magnetic charge symmetry in local quantum field theory

It is shown that in local quantum field theory with electric and magnetic currents, duality transformations are always spontaneously broken. Possible implications are discussed leading to the screening of magnetic charge and the failure of the cluster property as in two-dimensional quantum electrodynamics.     

Boson realizations of quantum algebras and some physical spaces with quantum algebra symmetry

The generators ofq-boson algebra are expressed in terms of those of boson algebra, and the relations among the representations of a quantum algebra onq-Fock space, on Fock space, and on coherent state space are discussed in a general way. Two examples are also given to present concrete physical spaces with quantum algebra symmetry. Finally, a new homomorphic mapping from a Lie algebra to boson algebra is presented.This work is supported by the National Foundation of Natural Science of China.     

Boolean transfer from coherent quantum logics to quantum logics with continuous superselection rules

Quantum logics with continuous superselection rules are shown to be Booleanvalued coherent quantum logics. Since modern set theory provides a transfer principle from standard mathematics to Boolean-valued mathematics, this makes it possible to transfer automatically well-known results on coherent quantum logics to quantum logics with continuous superselection rules. Many illustrations are given.     

Noncommutative quantum mechanics from noncommutative quantum field theory

We derive noncommutative multiparticle quantum mechanics from noncommutative quantum field theory in the nonrelativistic limit. Particles of opposite charges are found to have opposite noncommutativity. As a result, there is no noncommutative correction to the hydrogen atom spectrum at the tree level. We also comment on the obstacles to take noncommutative phenomenology seriously and propose a way to construct noncommutative SU(5) grand unified theory.     

Kℓ3 decay in quantum field theory with chiral symmetry

In quantum chiral field theory the form factor of K_?3 decay is calculated in the one-loop approximation. The results obtained are in good agreement with the recent experimental data.     

algebras and field theory

We review and develop the general properties of algebras focusing on the gauge structure of the associated field theories. Motivated by the homotopy Lie algebra of closed string field theory and the work of Roytenberg and Weinstein describing the Courant bracket in this language we investigate the structure of general gauge invariant perturbative field theories. We sketch such formulations for non‐abelian gauge theories, Einstein gravity, and for double field theory. We find that there is an algebra for the gauge structure and a larger one for the full interacting field theory. Theories where the gauge structure is a strict Lie algebra often require the full algebra for the interacting theory. The analysis suggests that algebras provide a classification of perturbative gauge invariant classical field theories.     

Spontaneous symmetry breaking in local gauge quantum field theory; the Higgs mechanism

Spontaneous symmetry breakings in indefinite metric quantum field theories are analyzed and a generalization of the Goldstone theorem is proved. The case of local gauge quantum field theories is discussed in detail and a characterization is given of the occurrence of the Higgs mechanism versus the Goldstone mechanism. The Higgs phenomenon is explained on general grounds without the introduction of the so-called Higgs fields. The basic property is the relation between the local internal symmetry group and the local group of gauge transformations of the second kind. Spontaneous symmetry breaking ofc-number gauge transformations of the second kind is shown to always occur if there are charged local fields. The implications about the absence of mass gap in the Wightman functions and the occurrence of massless particles associated with the unbroken generators in the Higgs phenomenon are discussed.