SU( N ) Quantum Yang–Mills theory in two dimensions: A complete solution
Open accessAbstract
A closed expression of the Euclidean Wilson-loop functionals is derived for pure Yang–Mills continuum theories with gauge groups SU(N) and U(1) and space-time topologies R1×R1 and R1×S1. (For the U(1) theory, we also consider the S1×S1 topology.) The treatment is rigorous, manifestly gauge invariant, manifestly invariant under area preserving diffeomorphisms and handles all (piecewise analytic) loops in one stroke. Equivalence between the resulting Euclidean theory and and the Hamiltonian framework is then established. Finally, an extension of the Osterwalder–Schrader axioms for gauge theories is proposed. These axioms are satisfied in the present model.
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For d=2, Yang-Mills theory is represented on this Hilbert space, which can be traced to the theory being invariant under area-preserving diffeomorphisms.
“The fact that for d = 2 also Yang-Mills theory is represented on this Hilbert space [10] can be traced back to the fact that this theory is almost diffeomorphism invariant, it is invariant under area-preserving diffeomorphisms.”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [10]
For the free Maxwell field (scattering processes), this Hilbert space is worthless except in d=2 because it does not coincide with Fock space.
“Of course, from the point of view of, say, the free Maxwell field (scattering processes) this Hilbert space is worthless (except in d = 2 [9, 10]) because it does not coincide with Fock space.”
From Kinematical Hilbert spaces for fermionic and Higgs quantum field theories (1998) · cites this paper as [10]
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References 57 · Cited by 35
Research works printed but not linked: 4
- [2][2] L. Gross, C. King, A. Sengupta, Ann. Phys. 194 (1989), 65; B.K. Driver, Commum. Math. Phys. 123 (1989) 575 -
- [3][3] J. Glimm, A. Jaffe, Quantum Physics , 2nd ed., Springer-Verlag, New York (1987) -
- [10][10] J. Baez, Lett. Math. Phys. 31 (1994), 213-223; Adv. Math. 117 (1996) 253 -
- [26][26] R. Courant, Vorlesungen ¨ uber Differential und Integralrechnung , vol. 1 and 2, Springer-Verlag, Berlin, Heidelberg, New York, (1971) -
- Authors
- Ashtekar, A., Lewandowski, J., Marolf, D., Mourão, J., & Thiemann, T.
- Affiliations
- Pennsylvania State University (US)
- University of Warsaw (PL)
- Syracuse University (US)
- University of Lisbon (PT)
- Max Planck Institute for Gravitational Physics (DE)
- Venue
- Journal of Mathematical Physics
- Publisher
- AIP Publishing
- Published
- 1997-11-01
- Volume
- 38
- Issue
- 11
- Pages
- 5453-5482
- ISSN
- 0022-2488 (ISSN-L)
- 1089-7658
- OpenAlex
- W2059144517
- Cited by
- 35
- References
- 57
- Citations by year
- 2025: 2 · 2022: 1 · 2019: 2 · 2014: 1 · 2012: 1
- FWCI
- 3.34
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Journal of Mathematical Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- Max Planck Institute for Plasma Physics (submittedVersion)
- Max Planck Digital Library (submittedVersion, other-oa)
- MPG.PuRe (Max Planck Society) (submittedVersion, other-oa)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Quantum Chromodynamics and Particle Interactions
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum and Classical Electrodynamics
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
- Keywords
- SU(N) Yang-Mills theory, two-dimensional quantum field theory, analytical solution, U(1) gauge theory, toroidal topology, Osterwalder-Schrader axioms, gauge invariance
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