Perturbative Algebraic Quantum Field Theory on Quantum Spacetime: Adiabatic and Ultraviolet Convergence
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Communications in
Mathematical Physics
Perturbative Algebraic Quantum Field Theory on Quantum Spacetime: Adiabatic and Ultraviolet Convergence Sergio Doplicher1 , Gerardo Morsella2 , Nicola Pinamonti3,4 1 Dipartimento di Matematica, Università di Roma “La Sapienza”, Piazzale Aldo Moro, 5, 00185 Rome, Italy.
E-mail: [email protected]
2 Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Rome,
Italy. E-mail: [email protected]
3 Dipartimento di Matematica, Università di Genova, Via Dodecaneso, 35, 16146 Genova, Italy.
E-mail: [email protected]
4 INFN - Sez. di Genova, Via Dodecaneso, 33, 16146 Genova, Italy
Received: 25 July 2019 / Accepted: 3 April 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: The quantum structure of Spacetime at the Planck scale suggests the use, in defining interactions between fields, of the Quantum Wick product. The resulting theory is ultraviolet finite, but subject to an adiabatic cutoff in time which seems difficult to remove. We solve this problem here by another strategy: the fields at a point in the interaction Lagrangian are replaced by the fields at a quantum point, described by an optimally localized state on QST; the resulting Lagrangian density agrees with the previous one after spacetime integration, but gives rise to a different interaction hamiltonian. But now the methods of perturbative Algebraic Quantum Field Theory can be applied, and produce an ultraviolet finite perturbation expansion of the interacting observables. If the obtained theory is tested in an equilibrium state at finite temperature the adiabatic cutoff in time becomes immaterial, namely it has no effect on the correlation function at any order in perturbation theory. Moreover, the interacting vacuum state can be obtained in the vanishing temperature limit. It is nevertheless important to stress that the use of states which are optimally localized for a given observer brakes Lorentz invariance at the very beginning. 1. Introduction Quantum Mechanics and Classical General Relativity are the main achievements of the last century in the theoretical description of physical systems. Quantum Mechanics is used to describe subatomic physics while General Relativity is necessary to accurately describe phenomena at the astrophysical level. Unfortunately, a universally accepted theory which combines both is still lacking. In spite of this, the principles of Quantum Mechanics and of Classical General Relativity meet at least at one crucial point: their concurrence prevents arbitrarily accurate localization of events in Spacetime. For the Heisenberg uncertainty principle would imply that with an observation large amount of energy could be packed in small regions; the corresponding classical backreaction on the curvature could then create trapped regions or even black holes. This limitation on
S. Doplicher, G. Morsella, N. Pinamonti
possible localization manifests itself in Spacetime Uncertainty Relations [15], q0 ·
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