The path integral formulation is a mathematical tool used in quantum mechanics to calculate the probability of a particle or system transitioning from one state to another. It involves summing over all possible paths that the particle or system could take between the two states.
The path integral formulation differs from other formulations, such as the Schrödinger equation, in that it does not describe the evolution of a single state vector. Instead, it takes into account all possible paths and assigns a probability to each one.
The path integral formulation has many applications in physics, particularly in quantum field theory and statistical mechanics. It allows for the calculation of transition amplitudes, expectation values, and correlation functions, making it a powerful tool for understanding and predicting the behavior of complex systems.
The path integral formulation was first developed by physicist Richard Feynman in the 1940s. It is derived through the use of functional integrals, which involve summing over all possible values of a function. By applying this concept to quantum mechanical systems, Feynman was able to develop the path integral formulation.
The path integral formulation can be difficult to apply to systems with a large number of particles or complex interactions. Additionally, it does not account for the effects of gravity, so it cannot be used to describe systems on a cosmological scale. However, it remains a valuable tool for understanding and predicting the behavior of many physical systems.