The Holst Spin Foam Model via Cubulations

Spin Foam Models (SFM) are an attempt at a covariant or path integral formulation of canonical Loop Quantum Gravity (LQG). Traditionally, SFM rely on 1. the Plebanski formulation of GR as a constrained BF Theory. 2. simplicial triangulations as a UV regulator and 3. a sum over all triangulations via group field techniques (GFT) in order to get rid off triangulation dependence. Subtle tasks for current SFM are to establish 1. the correct quantum implementation of Plebanski's constraints. 2. the existence of a semiclassical sector implementing additional Regge constraints arising from simplicial triangulations and 3. the physical inner product of LQG via GFT. We propose a new approach which deals with these issues as follows: 1. The simplicity constraints are correctly implemented by starting directly from the Holst action which is also a proper starting point for canonical LQG. 2. Cubulations are chosen rather than triangulations as a regulator. 3. We give a direct interpretation of our spin foam model as a generating functional of n -- point functions on the physical Hilbert space at finite regulator. This paper focuses on ideas and tasks to be performed before the model can be taken seriously, however, it transpires that 1. this model's amplitudes differ from those of current SFM, 2. tetrad n-point functions reveal a Wick like structure and 3. the restriction to simple representations does not occur automatically but must employ the time gauge just as in the classical theory