Noncommutative Rings pp 115-129 | Cite as

# Category Theory and Quantum Field Theory

Conference paper

## Abstract

We describe the recent basic result of Doplicher and Roberts characterizing the category of representations of a compact group. We indicate their motivation, coming from quantum field theory, and then state some important related category-theoretic questions which have arisen recently in low-dimensional quantum field theory. These questions, which are quite open, have interesting connections with quantum groups and knot theory.

## Keywords

Gauge Group Compact Group Category Theory Braid Group Monoidal Category
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## References

- [C]J. Cuntz,
*Simple C*-algebras generated by isometries*, Comm. Math. Phys.**57**(1977), 173–185.CrossRefMATHMathSciNetGoogle Scholar - [CK]J. Cuntz and W. Krieger,
*A class of C*-algebras and topological Markov chains*, Invent. Math.**56**(1980), 251–268.CrossRefMATHMathSciNetGoogle Scholar - [DHR1]S. Doplicher, R. Haag and J. E. Roberts,
*Fields, observables**and gauge transformations I*, Comm. Math. Phys.**13**(1969), 1–23.CrossRefMATHMathSciNetGoogle Scholar - [DHR2]S. Doplicher, R. Haag and J. E. Roberts,
*Fields, observables**and gauge transformations**II*, Comm. Math. Phys.**15**(1969), 173–200.CrossRefMATHMathSciNetGoogle Scholar - [DHR3]S. Doplicher, R. Haag and J. E. Roberts,
*Local observables**and particle statistics**I*, Comm. Math. Phys.**23**(1971), 199–230.CrossRefMathSciNetGoogle Scholar - [DHR4]S. Doplicher, R. Haag and J. E. Roberts,
*Local observables**and particle statistics II*, Comm. Math. Phys.**35**(1974), 49–85.CrossRefMathSciNetGoogle Scholar - [DR1]S. Doplicher and J. E. Roberts,
*Duals of compact Lie groups realized in the Cuntz algebras and their actions on C*-algebras*, J. Funct. Anal.**74**(1987), 96–120.CrossRefMATHMathSciNetGoogle Scholar - [DR2]S. Doplicher and J. E. Roberts,
*Endomorphisms of C*-algebras, cross products and duality for compact groups*, Ann. Math.**130**(1989), 75–119.CrossRefMATHMathSciNetGoogle Scholar - [DR3]S. Doplicher and J. E. Roberts,
*A new duality theory for compact groups*, Invent. Math.**98**(1989), 157–218.CrossRefMATHMathSciNetGoogle Scholar - [DR4]S. Doplicher and J. E. Roberts,
*Why there is a field algebra with a compact gauge group describing the supers election structure in particle physics*, Comm. Math. Phys. (to appear).Google Scholar - [Dr]V. G. Drinfeld,
*Quantum groups*, in “Proceedings of the International Congress of Mathematicians, Berkeley,” Amer. Math. Soc., Providence, R. I., 1987, pp. 798–820.Google Scholar - [FFK]G. Felder, J. Fröhlich, and G. Keller,
*On the structure of unitary conformal field theory II: Representation theoretic approach*, Comm. Math. Phys. (to appear).Google Scholar - [Fr]K. Fredenhagen,
*Structure of superselection sectors in low-dimensional quantum field theory*, in “Proceedings of the XVII International Conference on Differential Geometric Methods in Theoretical Physics: Physics and Geometry, 1989” (to appear).Google Scholar - [FRS]K. Fredenhagen, K. H. Rehren, and B. Schroer,
*Superselection sectors with braid group statistics and exchange algebras, I: General Theory*, Comm. Math. Phys.**125**(1989), 201–226.CrossRefMATHMathSciNetGoogle Scholar - [Fro]J. Fröhlich, Statistics of fields, the Yang-Baxter equation, and the theory of knots and links, in “The 1987 Cargèse lectures. Non-perturbative quantum field theory,” Plenum Press, New York, 1988.Google Scholar
- [FGM1]J. Fröhlich, F. Gabbiani and P.-A. Marchetti,
*Superselection structure and statistics in three-dimensional local quantum theory*, In “Proceedings 12th John Hopkins Workshop on Current Problems in High Energy Particle Theory Florence 1989” (to appear).Google Scholar - [FGM2]J. Fröhlich, F. Gabbiani and P.-A. Marchetti,
*Braid statistics in three-dimensional local quantum theory*, preprint.Google Scholar - [FK]J. Frölich and C. King,
*Two-dimensional conformal field theory and three-dimensional topology*, J. Mod. Phys. A (to appear).Google Scholar - [GJ]J. Glimm and A. Jaffe, “Quantum Physics, 2nd ed.,” Springer-Verlag, New York Berlin Heidelberg, 1987.CrossRefGoogle Scholar
- [HK]R. Haag and D. Kastler,
*An algebraic approach to quantum field theory*, Comm Math. Phys.**5**(1964), 848–861.MATHMathSciNetGoogle Scholar - [Ji]M. Jimbo,
*Introduction to the Yang-Baxter equation*, preprint.Google Scholar - [Jo]V. K. F. Jones,
*Hecke algebra representations of braid groups and link polynomials*, Ann. Math.**126**(1987), 335–388.CrossRefMATHGoogle Scholar - [LI]R. Longo,
*Index of subfactors and statistics of quantum fields*, Comm. Math. Phys. (to appear).Google Scholar - [L2]R. Longo,
*Index of subfactors and statistics of quantum fields II*.*Correspondences, braid group statistics and Jones polynomial*, preprint.Google Scholar - [ML]S. Mac Lane, “Categories for the Working Mathematician,” Springer-Verlag, Berlin, Heidelberg, New York, 1971.MATHGoogle Scholar
- [Mal]S. Majid,
*Quasitriangular Hopf algebras and Yang-Baxter equations*, Int. J. Mod. Phys. A (to appear).Google Scholar - [Ma2]S. Majid,
*Quantum group duality in vertex models and other results in the theory of quaistriangular Hopf algebras*, in “Proc. Int. Conf. Geom. Phys., Tahoe City, 1989” (to appear).Google Scholar - [MS]G. Moore and N. Seiberg,
*Classical and quantum conformal field theory*, Comm. Math. Phys.**123**(1989), 117–254.CrossRefMathSciNetGoogle Scholar - [O]A. Ocneanu,
*Quantized groups, string algebras and Galois theory for algebras*.Google Scholar - [S1]B. Schroer,
*New concepts and results in nonperturbative quantum field theory*, in “Differential Geometric Methods in Theoretical Physics,” Chester, 1988.Google Scholar - [S2]B. Schroer,
*High T*_{C}*superconductivity and a new symmetry concept in low-dimensional quantum field theories*, preprint.Google Scholar - [S3]B. Schroer,
*New kinematics*(*statistics and symmetry*)*in low-dimensional QFT with applications to conformal QFT*_{2}in “Proceedings of the XVII International Conference on Differential Geometric Methods in Theoretical Physics: Physics and Geometry, 1989” (to appear).Google Scholar - [Su]
- [U]K.-H. Ulbrich,
*Tannakian categories for non-commutative Hopf algebras*, preprint.Google Scholar - [Wa]
- [W1]S. L. Woronowicz,
*Twisted SU*(2).*An example of a noncommutative differential calculus*, Publ. Res. Inst. Math. Sci. Kyoto**23**(1987), 117–181.CrossRefMATHMathSciNetGoogle Scholar - [W2]S. L. Woronowicz,
*Compact matrix pseudogroups*, Comm. Math. Phys.**111**(1987), 613–665.CrossRefMATHMathSciNetGoogle Scholar - [W3]S. L. Woronowicz,
*Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU*(*N*)*groups*, Invent, math.**93**(1988), 35–76.CrossRefMATHMathSciNetGoogle Scholar - [W4]S. L. Woronowicz,
*Differential calculus on compact matrix pseudogroups*(*quantum groups*), Comm. Math. Phys.**122**(1989), 125–170.CrossRefMATHMathSciNetGoogle Scholar

## Copyright information

© Springer-Verlag New York, Inc. 1992