Richard Feynman's interpretation of path integrals in Hamiltonian dynamics is known as the sum-over-paths formulation. This interpretation, also known as the Feynman path integral formulation, is a powerful and alternative way of describing quantum mechanics.
In Feynman's interpretation, the path integral approach is used to calculate the probability amplitude for a particle to travel from one point to another in space and time. It involves summing over all possible paths that the particle could take between the initial and final points. Each path contributes to the probability amplitude with a phase factor, determined by the action of the path.
In the context of Hamiltonian dynamics, the action is defined as the integral of the Lagrangian over time. The Lagrangian is a function that describes the dynamics of a system in terms of its generalized coordinates and their time derivatives. The path integral is then constructed by discretizing the time interval between the initial and final points and summing over all possible configurations of the generalized coordinates at each time step.
The path integral formulation provides a powerful tool for calculating transition amplitudes and expectation values in quantum mechanics. It allows for the incorporation of quantum effects and the treatment of nontrivial systems that may not have simple analytical solutions.
Feynman's interpretation of path integrals in Hamiltonian dynamics provides a deeper understanding of quantum mechanics and has found extensive applications in various areas of theoretical physics, such as quantum field theory and statistical mechanics. It offers an intuitive and pictorial way of visualizing quantum processes by considering all possible paths that particles can take in a given physical situation.