Path Integrals from Spacetime Quantum Actions

Before we proceed, I would like to discuss some foundational aspects of standard quantum mechanics. We begin by recalling the construction of quantum mechanics for a particle, where we impose a canonical algebra. For clarity, I will include the Planck constant in this discussion.

In our external Hilbert space, this canonical algebra assumes a new form. Notably, we introduce an additional δ in time, indicating that operators at different time slices commute. While this may initially appear unusual, it is mathematically justified as a consequence of the tensor structure in time that we introduced earlier.