Perturbative Algebraic Quantum Field Theory on Quantum Spacetime: Adiabatic and Ultraviolet Convergence

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.

Quantum Spacetime and the Universe at the Big Bang, vanishing interactions and fading degrees of freedom

“…Moreover, as discussed in Sec. 3, the same holds, at all orders in perturbation theory, for the interacting observables defined using the adaptation of the perturbative approach to Algebraic Quantum Field Theory (pAQFT) of [6] developed in [9].…”

“…Following [7] and the discussion presented in section 2, this limit correspond to an early time limit in a cosmological scenario. A convenient framework in which this limit can be controlled at all orders in perturbation theory is the one deveolped in [9] adapting to QFT on QST the framework of pAQFT. To make the paper reasonably self-contained, we summarize below the main definitions and results of [9], to which we refer the interested reader for more details and proofs.…”

“…A convenient framework in which this limit can be controlled at all orders in perturbation theory is the one deveolped in [9] adapting to QFT on QST the framework of pAQFT. To make the paper reasonably self-contained, we summarize below the main definitions and results of [9], to which we refer the interested reader for more details and proofs. To simplify the notation, from now on we denote the Planck length just by λ.…”

“…The • T λ product is then used in [9] to define the (infrared cutoff) S matrix corresponding to the interaction (3.2) as…”

“…The key step in [9] is the use of a different effective interaction Lagrangian which is equivalent to the one obtained by means of the delocalization kernel discussed in [3] in the adiabatic limit, namely after integration of the effective lagrangian density over the full spacetime. This new interaction Lagrangian is obtained by replacing in the classical Lagrangian density the field operators at a point x by fields at a quantum point described by states optimally localized around x.…”