Algebraic structures in quantum gravity. (English) Zbl 1190.83046
Summary: Starting from a recently introduced algebraic structure on spin foam models, we define a Hopf algebra by dividing with an appropriate quotient. The structure, thus defined, naturally allows for a mirror analysis of spin foam models with quantum field theory, from a combinatorial point of view. A grafting operator is introduced allowing for the equivalent of a Dyson-Schwinger equation to be written. Non-trivial examples are explicitly worked out. Finally, the physical significance of the results is discussed.
MSC:
83C45 | Quantization of the gravitational field |
81V17 | Gravitational interaction in quantum theory |
83C27 | Lattice gravity, Regge calculus and other discrete methods in general relativity and gravitational theory |
16T05 | Hopf algebras and their applications |
81S10 | Geometry and quantization, symplectic methods |
81T20 | Quantum field theory on curved space or space-time backgrounds |
13M10 | Polynomials and finite commutative rings |
57T05 | Hopf algebras (aspects of homology and homotopy of topological groups) |