Feynman Path Integral
The Feynman path integral is a mathematical formulation in quantum mechanics that describes the evolution of a quantum system. It is based on Richard Feynman's idea of summing over all possible paths of a particle to calculate the probability amplitude associated with a particular process. While the path integral formalism is quite involved and covers a wide range of applications, I can provide you with an overview and key formulas associated with it:
1. Propagator (Green's function):
The propagator, also known as the Green's function, describes the transition amplitude between two points in spacetime for a quantum system. It is given by the Feynman path integral:
K(x'', t''; x', t') = ∫ D[x(t)] exp(iS[x(t)])
where x(t) represents all possible paths of the system, S[x(t)] is the action functional associated with the system, and the integral is taken over all possible paths satisfying the boundary conditions x(t') = x' and x(t'') = x''.
2. Transition amplitude:
The transition amplitude between initial and final states of a quantum system can be expressed using the propagator as:
⟨x'', t''| x', t'⟩ = K(x'', t''; x', t')
where |x, t⟩ represents a state vector associated with a particle at position x and time t.
3. Perturbation theory:
The path integral formalism allows for the development of perturbation theory in quantum field theory. The transition amplitude for a process involving interactions can be expanded as a series of terms, known as Feynman diagrams, which correspond to different orders of perturbation.
4. Path integral for multiple particles:
The path integral can be extended to systems with multiple particles. In such cases, the integral is taken over the paths of all particles involved, and the action functional includes contributions from the individual particles and their interactions.
5. Path integral for fields:
The path integral formalism can be generalized to quantum field theory, where the fields themselves vary over spacetime. In this case, the path integral is taken over all possible field configurations, and the action functional is expressed in terms of the field variables.
It's important to note that the Feynman path integral is a powerful mathematical tool, but its rigorous mathematical foundation requires the framework of functional analysis and measure theory. The formulas provided above serve as a basic introduction to the key concepts and formulas associated with the path integral formalism in quantum mechanics and quantum field theory.
