Braided Quantum Field Theory

Abstract

We develop a general framework for quantum field theory on noncommutative spaces, i.e., spaces with quantum group symmetry. We use the path integral approach to obtain expressions for n-point functions. Perturbation theory leads us to generalised Feynman diagrams which are braided, i.e., they have non-trivial over-and under-crossings. We demonstrate the power of our approach by applying it to φ 4 -theory on the quantum 2-sphere. We find that the basic divergent diagram of the theory is regularised. *

Twisted statistics of quantum fields on the Moyal plane

“…We will not go here into the full details of the construction of the functional integral which gives the twisted quantum field theory. It has been done by Oeckl [9]. It will suffice here to show that the conventional functional integral does not give a twisted Poincaré invariant theory.…”

Formulation of supersymmetry on a lattice as a representation of a deformed superalgebra

“…Here we just see what this identity implies in our superalgebra, without digging into the detail. Applying the first two terms of (3.38) on Q lat A , we find 8 40) and 41) while the last terms gives…”

“…We now move on to the construction of a field theory which has this Hopf algebraic symmetry. We follow the general scheme formulated by Oeckl [41] as braided quantum field theory (BQFT). To this purpose, we have to specify the complete algebraic nature of the fields which is consistent to the algebraic nature of the Hopf algebra.…”

“…It actually forms a Hopf algebra in a natural way [41,54]: 3.120) where ϕ ∈ X. This Hopf algebraic structure shouldn't be confused with that of the symmetry operators U (A) acting onX.…”

Sums over Graphs and Integration over Discrete Groupoids

“…Here, using the notion of average of v ⊗n , we recast it in a coordinate-free way, which is more suitable for a reinterpretation through the graphical formalism of the previous sections. Recently, Robert Oeckl has proven that a Wick-type lemma holds in the wider context of general braided tensor categories, see [Oec01]. …”