Chapter

Ultracold Quantum Fields

Part of the series Theoretical and Mathematical Physics pp 85-107

Path Integrals

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Abstract

In this chapter, we discuss the Feynman path-integral formulation of quantum mechanics. It is the generalization of the classical Lagrange and Hamilton formalisms to quantum mechanics. In the path-integral approach, the central object that determines the dynamics of the system is called the action. We start by considering the mathematical properties of an action, which is a functional, where we focus initially on functional differentiation. This allows us to determine the minimum of an action, which, according to the principle of least action, determines the classical motion. Having become familiar with actions, we go through the derivation of the path-integral expression for the quantum-mechanical transition amplitude using the time-slicing procedure. The same procedure returns in Chap. 7, when we derive the functional-integral formalism for quantum field theory. Having obtained the path integral, we discuss various ways of solving it and apply these methods to the free particle and to a particle in a potential. We also derive the path-integral expressions for matrix elements of operators and expectation values, where we see that the path-integral formalism gives rise to time-ordered expectation values. All these important concepts return many times in parts II and III of the book, when we use the generalization of the path-integral formalism to quantum field theory in the treatment of interacting quantum gases.