The Hilbert space costratification for the orbit type strata of \(\mathrm{SU}(2)\)-lattice gauge theory. (English) Zbl 1398.81157
Different from perturbation theory approaches, in lattice gauge theories the full theory (like for instance QCD) is becoming implemented on a lattice. By doing so, theorists hope to observe effects which are out of the reach of perturbative approaches. Taking the gauge group \(\mathrm{SU}(2)\) as an example, the authors deal with mathematical aspects of this implementation. \(\mathrm{SU}(2)\) is the Lie group of the theory of angular momentum. For single particles, the combination of two angular momenta is conducted by the Clebsch-Gordan coefficients. On the lattice, the combination of angular momenta needs an iterative approach. Two angular momenta are combined to an intermediate momentum which can again be combined with the next angular momentum. This iteration, symbolically expressed in terms of binary trees, leads to an iterative construction of the costratifiaction of the quantum Hilbert space according to Huebschmann. In the following, the authors show how two such trees can assembled to a combined tree for the lattice. For a practicioner like the referee, the publication is difficult to understand, and it takes a long time before the essence is found. However, for readers with the necessary mathematical background the authors present a lot of details for a rigorous treatment of the subject.
Reviewer: Stefan Groote (Tartu)
MSC:
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
| 81T25 | Quantum field theory on lattices |
| 57N80 | Stratifications in topological manifolds |
| 51A10 | Homomorphism, automorphism and dualities in linear incidence geometry |
| 20E36 | Automorphisms of infinite groups |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
Keywords:
costratification of the Hilbert space; angular momentum algebra; binary trees; Clebsch-Gordan coefficients; Wigner 3nj symbolsCitations:
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