Path Integrals from Spacetime Quantum Actions

Let us revisit the paper and concentrate on a crucial physical quantity: time correlation functions. These functions are fundamental in quantum field theories, path integrals, and quantum mechanics, as they allow us to recover any physical observable. To illustrate the introduction of a time correlation function within our formalism, we will refer to the expression of the propagator in the particle example. By inserting two operators at different time slices into this expression, we can demonstrate that the results correspond to the time correlation functions. Our manuscript provides a mathematically rigorous proof of this relationship.

For a concrete example, consider six time slices and focus on the two-point correlation functions at times 2 and 4, where time is measured in units of ε. This correlation function can be expressed as the quantum trace of the exponential of the action, incorporating two operators at the specified time slices.

On the right side of the equation, one must manually adjust the individual operators to the correct positions. In contrast, the left side remains static. Time evolution arises from the correlations encoded in the exponential of the quantum action, with the relevant times represented by the geometric positions of the operators. A pictorial representation of this scenario for general two-point correlation functions is available. This approach yields a Hilbert space construction analogous to the path integral formulation, where other correlation functions are defined by the insertion of classical variables at different time points. Evaluating the traces will yield a specific expression.