Ultraviolet finite quantum field theory on quantum spacetime. (English) Zbl 1046.81093

The authors formulate a quantum field theory on quantum space-time satisfying \([q^\mu,q^\nu]=i\lambda_p^2Q^{\mu\nu}\). The generalized Weyl correspondence \(W (g\otimes f)=g(Q)f(q)\) extends to any symbol \(F\in{\mathcal Z}_0(\Sigma\times \mathbb{R}^4)\), and a \(C^*\)-algebra \({\mathcal Z}\) is given. A product of fields at different points, \(\varphi (q_1)\cdot\cdots\cdot\varphi(q_n)\), is defined by interpreting \(q_1,\dots,q_n\) as mutually independent quantum coordinates: tensor products over the center \({\mathcal Z}={\mathcal Z}_0(\Sigma)\). The limit \(q_j-q_k\to 0\) is replaced by a quantum diagonal map (QDM) \(E^{(n)}: {\mathcal Z}\otimes_z\cdots \otimes_z{\mathcal Z}\to{\mathcal Z}_1(\subset {\mathcal Z})\). They obtain the following Results: \((I)\) The explicit form of the QDM on \(f\) is \(E^{(n)}(f(q_1, \dots,q_n))=\int dk_1 \cdots dk_n\check f(k_1,\dots,k_n)r_n(k_1,\dots,k_n)\cdot e^{i (\Sigma_j k_j)q}\); \(\check f={\mathcal F}^{-1}f\), \(r_n=\exp\{-2^{-1} \Sigma_{\mu=0 \sim 3}(\Sigma_{j=1\sim n}k_{j\mu}^2-n^{-1} \Sigma_{j,l=1\sim n}k_{j\mu} k_{l \mu})\}\). \((II)\) For any Schwartz function \(\lambda\) of the form \(\lambda=\lambda'(t)\lambda ''(x)\), \(\lambda'\in S(\mathbb{R})\), \(\lambda''\in S(\mathbb{R}^3)\), the formal series \(S [\lambda]=T\cdot \exp\{-i\int d^4x\lambda(x)\). \(E^{(n)}(: \varphi (q_1)\cdots\varphi(q_n):) |_{q=x}\}= I+\Sigma_{N=1 \sim\infty}(-i)^N S^{(N)}[\lambda]\) is finite at all orders. They show the relation between their approach and the usual formulation of the theory in terms of Feynman diagrams.


81T75 Noncommutative geometry methods in quantum field theory
81R60 Noncommutative geometry in quantum theory
81T20 Quantum field theory on curved space or space-time backgrounds
83C47 Methods of quantum field theory in general relativity and gravitational theory
Full Text: DOI arXiv