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Instead of formulating the states of a Quantum Field Theory (QFT) as densitymatrices over a single large Hilbert space, it has been proposed by Kijowski[Kijowski, 1977] to construct them as consistent families of partial densitymatrices, the latter being defined over small 'building block' Hilbert spaces.In this picture, each small Hilbert space can be physically interpreted asextracting from the full theory specific degrees of freedom. This allows toreduce the quantization of a classical field theory to the quantization offinite-dimensional sub-systems, thus sidestepping some of the commonambiguities (specifically, the issues revolving around the choice of a 'vacuumstate'), while obtaining robust and well-controlled quantum states spaces.  The present letter provides a self-contained introduction to this formalism,detailing its motivations as well as its relations to other approaches to QFT(such as conventional Fock-like Hilbert spaces, path-integral quantization, andthe algebraic formulation). At the same time, it can serve as a reading guideto the series of more in-depth articles [arXiv:1411.3589, arXiv:1411.3590,arXiv:1411.3591, arXiv:1510.01926].

Projective Limits of State Spaces: Quantum Field Theory without a Vacuum