Covariant Canonical Quantization Path to Quantum Field Theory

In this article a formulation of Covariant Canonical Quantization (CCQ) is presented, which works on an extended Hilbert space and reduces to conventional canonical quantization when constraining to the mass-shell a priori. From the formal point of view it may be seen as a formalism between the canonical operator and the functional integral approach. A covariant number operator and two symmetric vacua are constructed, which lead to convergent quantities. It is then discussed how the quantum field theoretical divergences like the vacuum energy arise a posteriori when a spacetime split is performed. This is important for a better understanding of the renormalization limits, which so far lack a satisfying explanation.