Background-Independent Quantum Field Theory

The General Boundary Formulation (GBF) of Quantum Field Theory (QFT) (see e.g. [1-3]) has been constructed because of the necessity to overcome the conceptual restrictions imposed by the metric in non-perturbative quantum gravity: In standard quantum theory, one needs a split of the spacetime into spatial slices and a time-direction (also called 3+1-split) which can only be defined from the background metric. If the metric itself is quantized non-perturbatively, there is no background metric from which one could define the spacetime split. 

Instead, the GBF seeks to provide a true generalization of standard quantum field theory that relies neither on a Lorentzian metric nor on a spacetime split. It is an axiomatic framework that allows one to formulate quantum field theory on general spacetime regions with general boundaries. More specifically, the set of axioms assigns algebraic structures to geometrical structures and ensures the consistency of these assignments.

In [4], Frank Hellmann, Ralf Banisch and I established an Unruh-DeWitt detector model in the framework of the GBF and used it to gain insight into the structure of the GBF. In particular, we were able to generalize the notion of initial and final states to a notion of incoming and outgoing states that is general enough to be applied to spacetime regions with timelike hypersurfaces. Using this interpretation for specific examples, we showed that the response of the Unruh-DeWitt detector can be used to fully specify the vacuum state on timelike hypersurfaces.

In [5], Daniele Colosi and I apply the GBF to construct QFT on time-like hypersurfaces in two-dimensional Rindler space, which can be associated with the spacetime region causally connected to an accelerated observer. In [6], we have used the GBF framework to recover the Unruh effect, the prediction that accelerated observers see a thermal bath of particles when moving through vacuum.

[1] R. Oeckl, “A ’General boundary’ formulation for quantum mechanics and quantum gravity,” Phys.Lett., vol. B575, pp. 318–324, 2003. Preprint arXiv:hep-th/0306025. 

[2] D. Colosi, “The general boundary formulation of quantum theory and its relevance for the problem of quantum gravity,” Proceedings of the VIII Mexican School of the Gravitational and Mathematical Physics Division of the Mexican Physical Society, AIP Conf. Proc., vol. 1396, pp. 109–113, 2011 

[3] R. Oeckl, “Observables in the General Boundary Formulation,” in Quantum Field Theory and Gravity (F. Finster, O. M ̈uller, M. Nardmann, J. Tolksdorf, and E. Zeidler, eds.), pp. 137–156, Springer Basel, 2012. Preprint arXiv:1101.0367 

[4] Banisch R., Hellmann F., Rätzel D. "The Unruh-DeWitt detector and the vacuum in the general boundary formalism" Class. Quantum Grav. 30 235026 (2013), doi.org/10.1088/0264-9381/30/23/235026; Preprint arxiv.org/abs/1205.1549

[5] Colosi D., Rätzel D. "Quantum field theory on timelike hypersurfaces in Rindler space" Phys. Rev. D 87, 125001 (2013), doi.org/10.1103/PhysRevD.87.125001; Preprint arxiv.org/abs/1303.5873

[6] Colosi D., Rätzel D. "The Unruh Effect in General Boundary Quantum Field Theory" SIGMA 9 (2013), 019, doi.org/10.3842/SIGMA.2013.019; Preprint: arxiv.org/abs/1204.62