The quantum structure of spacetime at the Planck scale and quantum fields

Abstract

We propose uncertainty relations for the different coordinates of spacetime events, motivated by Heisenberg's principle and by Einstein's theory of classical gravity. A model of Quantum Spacetime is then discussed where the commutation relations exactly implement our uncertainty relations.We outline the definition of free fields and interactions over QST and take the first steps to adapting the usual perturbation theory. The quantum nature of the underlying spacetime replaces a local interaction by a specific nonlocal effective interaction in the ordinary Minkowski space. 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, our Quantum Spacetime reduces to the ordinary Minkowski space times a two component space whose components are homeomorphic to the tangent bundle T S 2 of the 2-sphere. The relations with Connes' theory of the standard model will be studied elsewhere.

Non Local Theories: New Rules for Old Diagrams

“…The non pointwise nature of optimal localization is the source for the breakdown of the covariance of (1.2) under Lorentz boosts. 1 The lagrangian (1.2) is an effective interaction obtained by considering the quantization :φ(q) n : of :φ(x) n :, and was first proposed in [1]. We recall that the DFR quantizationà la Weyl of φ(x) is φ(q) = dkφ(k) ⊗ e ikµq µ , where φ(x) is the ordinary neutral Klein-Gordon quantum field.…”

“…Σ is the orbit of the standard symplectic matrix σ0 = ( 0 −I I 0 ) under the action of the full Lorentz group, while Σ1 ⊂ Σ is the orbit of σ0 under the rotation subgroup. See [1].…”

“…, y n − a) , the interaction hamiltonian H I (t) = −g dx L I (t, x) is invariant under translations. Motivations come from the quest for effective theories on classical spacetime, describing interactions on the quantum spacetime, as it was proposed by Doplicher, Fredenhagen and Roberts (DFR) in their seminal paper [1]; see also [2]. There, localization is described in terms of the covariant DFR quantum coordinates q µ , whose commutators iλ 2 P Q µν = [q µ , q ν ] are constrained so to ensure stability of spacetime under localization (λ P is the Planck length): no black holes should arise as an effect of localization alone.…”

Schwinger model in noncommutating space–time

“…It was argued that introduction of space-time noncommutativity spoils unitarity [3,4] or even causality [5]. However, much attention has been devoted in recent times to circumvent these difficulties in formulating theories with θ 0i = 0 [6,7,8]. The 1 + 1 dimensional field theoretic models are particularly important in this context because any noncommutative extension of such models essentially contains fuzziness in the time-space sector.…”

Noncommutative QFT and Renormalization

“…This line of work was further developed together with Peter Prešnajder and Ctirad Klimčík [4]. At almost the same time, Filk [5] developed his Feynman rules for the canonically deformed four-dimensional field theory, and Doplicher, Fredenhagen and Roberts [6] published their work on deformed spaces. The subject experienced a major boost after one realized that string theory leads to noncommutative field theory under certain conditions [7,8], and the subject developed very rapidly; see e.g.…”