Propagator

CausalRetardedPropagatorPath

CausalAdvancedPropagatorPath

FeynmanPropagatorPath

Propagator in the context of quantum mechanics and quantum field theory, refers to a function or distribution that describes the probability amplitude for a particle to travel from one point to another in a given time, or for a quantum state to evolve from one time to another. The concept is central to the understanding of particle interactions and the formulation of particle physics.

Definition

In quantum mechanics, the propagator can be thought of as the Green's function of the Schrödinger equation for a free particle. It provides a solution to the equation of motion for a particle moving from one point in space-time to another, taking into account the principles of superposition and wave-particle duality. The propagator is essential in calculating the evolution of quantum states over time and in determining the outcomes of quantum mechanical interactions.

Mathematical Formulation

The mathematical expression for a propagator depends on the specific quantum theory being considered. In non-relativistic quantum mechanics, the propagator \(K(x,t;x',t')\) for a particle moving from point \(x'\) at time \(t'\) to point \(x\) at time \(t\) is given by the path integral formulation, where the paths of all possible trajectories are summed over, each weighted by the exponential of the action \(S\) in units of the reduced Planck constant \(\hbar\):

\[K(x,t;x',t') = \int \exp\left(\frac{iS}{\hbar}\right) \mathcal{D}[path]\]

In quantum field theory, the propagator is often expressed in terms of Feynman diagrams, where it represents the probability amplitude for a particle to propagate from one point to another. The form of the propagator will depend on the type of particle and the interactions it undergoes.

Types of Propagators

There are several types of propagators, each relevant to different physical situations: - **Feynman propagator**: Used in quantum field theory to describe the propagation of particles and antiparticles. - **Retarded propagator**: Describes the propagation of signals that move forward in time, respecting causality. - **Advanced propagator**: Represents signals moving backward in time, less common in physical situations.

Applications

Propagators are used extensively in theoretical physics to predict the outcomes of experiments and to calculate observable quantities such as cross sections and decay rates in particle physics. They are also crucial in the study of quantum effects in condensed matter physics, such as quantum tunneling and superconductivity.

See Also

• Quantum mechanics
• Quantum field theory
• Feynman diagram
• Green's function
• Path integral formulation