# On quantum-mechanical equations of motion in representation dependent of external sources

## TL;DR

This paper investigates how Heisenberg's equations of motion depend on the choice of external sources and normal symbols, providing a closed-form solution and analyzing the implications for path integral actions.

## Contribution

It offers a detailed analysis of representation dependence in quantum equations of motion and derives a closed solution for variation-derivative equations in a normal form.

## Key findings

- The action in the path integral depends on the choice of normal symbol.
- Boundary conditions for virtual trajectories are affected by the representation.
- The boundary terms in the action are explicitly determined.

## Abstract

In the present paper, we consider in detail the aspects of the Heisenberg's equations of motion, related to their transformation to the representation dependent of external sources. We provide with a closed solution as to the variation-derivative motion equations in the general case of a normal form (symbol) chosen. We show that the action in the path integral does depend actually on a particular choice of a normal symbol. We have determined both the aspects of the latter dependence: the specific boundary conditions for virtual trajectories, and the specific boundary terms in the action.