In semiclassical gravity matter fields are quantized on a classical background manifold. For any chosen time-foliation, regions of space at a given time (this room, now) have thus a dual description. From the point of view of GR they are submanifolds, essentially, subsets, of the entire hypersurface at that given time. For the quantized matter fields they are quantum subsystems, accounting for the degrees of freedom of the matter field theory that are localized in that region.
Localization and the idea of treating
Regions of Space as Quantum Subsystems
The reasons why I started working on this are two:
1) A decomposition into subsystems is much more general than one into subsets. There might be regimes (UV, IR) where our usual description of spacetime in terms of a metric manifold breaks down, but you will still be able to talk of something as general as a quantum subsystem.
2) Spacetime relations are operationally defined by means of rods, clocks, particle detectors etc...: i.e. by using the degrees of freedom belonging to the matter fields. Therefore, any region of space should be described, more fundamentally, by the quantum degrees of freedom of the fields ``living therein".
Localization is the rationale to associate a region of space with its quantum degrees of freedom.
According to the standard localization scheme, any region of space should be associated with the relativistic fields operators and conjugate momenta that have spacial labels x within the region itself. According to such a localization, the vacuum entropy of a region of space is UV- divergent and proportional to the area. But do the correlations that such an entropy is accounting for have any operational meaning? Here Fabio Costa and I argue for a negative answer to this question. (The whole issue is more throughly analyzed in this review, chap. 4)
I considered some implications of an alternative localization scheme, related to the Newton-Wigner position operator. In the usual localization, the calculation of the thermal entropy of a region of space is plagued by infinities proportional to the area of the region that cannot be renormalized by local counterterms. Those are the same infinities that one finds in the vacuum. Except that in the vacuum they are cool, as they remind us of BH entropy. Instead, if one wants to recover simple thermodynamics results in QFT, those infinities should be subtracted.
With Sergio Cacciatori and Fabio Costa we calculated the thermal entropy of a region of space in QFT by using Newton-Wigner localization scheme and showed that it is free of divergences and gives a sound thermodynamic result. The calculations of traces in NW-localization can be usefully rearranged in a very pretty diagrammatics!
Bibliography:
F. Piazza, ``Glimmers of a pre-geometric perspective,'' Found. Phys. in press hep-th/0506124
F. Piazza and F. Costa, ``Volumes of Space as Subsystems,'' PoS(QG-Ph)032, 0711.3048
S. Cacciatori, F. Costa and F. Piazza, ``Renormalized Thermal Entropy in Field Theory,'' Phys. Rev. D79, 025006 (2009) 0803.4087.
F. Costa and F. Piazza, ``Modelling a Particle Detector in Field Theory”, New J. Phys. 11 113006 (2009) 0805.0806
F. Piazza, ``Some new views on gravity at low energy'' 0910.4677
Invited Talks:
``Which space-time actually emerges? The world as seen from inside a spin system'', invited talk, Emergent Gravity, MIT, Boston, August 25-29, 2008
``Particle detector models, thermal entropy and localization'', Department of Physics, UCB, Berkeley, September 9, 2008
``Particle Detectors, the Frog Principle and the Unruh Effect'', Hot topics in Modern Cosmology, Cargese (Corse) May 12-17, 2008
``Localization in semi-classical gravity", Quantum Information in Quantum Gravity, Perimeter Institute, December 8 - 10, 2007
``Particle Detector Model questions the Unruh Effect'', sparkling informal discussion with important guests, Perimeter Institute of Theoretical Physics, Waterloo, April 30, 2008