The quantum structure of spacetime of the Planck scale and quantum fields. (English) Zbl 0847.53051

Doplicher, Sergio; Fredenhagen, Klaus; Roberts, John E.

Commun. Math. Phys. 172, No. 1, 187-220 (1995).

It is generally believed that the description of spacetime by a Lorentzian manifold breaks down at very short distances of the order of the Planck length. One possibility to get a generalized theory is the usuage of a “non-commutative manifold” as a model for spacetime, i.e. the commutative algebra \({\mathcal C}_0 (M)\) of complex continuous functions on \(M\) vanishing at infinity is replaced by a non-commutative algebra \({\mathcal E}\) and points of \(M\) by pure states on \({\mathcal E}\). Following these ideas the authors propose in this paper a quantized version of flat Minkowski space. To get the algebra \({\mathcal E}\), the authors introduce uncertainty relations for different coordinates of spacetime events, which are motivated by Heisenberg’s principle and by Einstein’s theory of classical gravity. To implement them they use commutation relations between appropriate unbounded operators \(q_\mu\). The algebra \({\mathcal E}\) is a \(C^*\)-algebra to which the \(q_\mu\) are affiliated. On this “quantum spacetime” the authors develop calculus and they outline the definition of free fields and interacting fields. In particular, some first steps to adapt the usual perturbation theory are made, however, a detailed study of interacting QFT and of the smoothing of ultraviolet divergences is deferred to a subsequent paper. In the classical limit where the Planck length goes to zero, quantum spacetime reduces to the product of ordinary Minkowski space with a two component space whose components are homeomorphic to the tangent bundle of the 2-sphere.

Reviewer: M.Keyl (Berlin)

MSC:

53Z05	Applications of differential geometry to physics
46L60	Applications of selfadjoint operator algebras to physics
81T05	Axiomatic quantum field theory; operator algebras
81T20	Quantum field theory on curved space or space-time backgrounds
81V17	Gravitational interaction in quantum theory

Keywords:

non-commutative manifold; non-commutative algebra; uncertainty relations for coordinates; free fields; interacting fields; classical limit; Planck length

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