ABSTRACT
This Chapter introduces the Feynman Path Integral description of quantum transition amplitudes, considering the simplest case of a quantum system described by one quantum variable q and its conjugate momentum p. The transition amplitude from time https://www.w3.org/1998/Math/MathML" display="inline"> t 1 to time https://www.w3.org/1998/Math/MathML" display="inline"> t 2 = t 1 + T is obtained by discretising the time axis in N time steps of amplitude https://www.w3.org/1998/Math/MathML" display="inline"> ϵ , implying T=N https://www.w3.org/1998/Math/MathML" display="inline"> ϵ . For https://www.w3.org/1998/Math/MathML" display="inline"> ϵ → 0 , the Feynman Path Integral formula is derived: https://www.w3.org/1998/Math/MathML" display="inline"> < q N | e - i T H / ℏ | q 0 > = ∫ d [ q ( t ) ] e i S q ( t ) , q ' ( t ) / ℏ , where integration is over all trajectories https://www.w3.org/1998/Math/MathML" display="inline"> q ( t ) which satisfy the boundary conditions at https://www.w3.org/1998/Math/MathML" display="inline"> t 1 and t 2 . https://www.w3.org/1998/Math/MathML" display="inline"> S ( q , q ' ) is the classical Action, a function of https://www.w3.org/1998/Math/MathML" display="inline"> q and of its time derivative. The classical limit, corresponding to https://www.w3.org/1998/Math/MathML" display="inline"> ℏ → 0 , is considered, whereby the solution of the Feynman Path Integral reduces to the classical trajectory, https://www.w3.org/1998/Math/MathML" display="inline"> q c ( t ) , which minimises the classical Action. The lattice approximation, i.e. small, non vanishing https://www.w3.org/1998/Math/MathML" display="inline"> ϵ , with evaluation of the Feynman integral by numerical integration, provides a useful approximation in the case of the fundamental theory of the strong interactions (Quantum Chromodynamics, or QCD), to be considered later; Chapters 17 to19. This chapter is completed by consideration of the extension of gthe formalism to imaginary time, where the Feynman Path Integral becomes the Partition Function of Statistical Mechanics, and by the definition of the Green's functions, ground-state expectation values of products of quantum operators q(t) taken at different times, https://www.w3.org/1998/Math/MathML" display="inline"> t 1 , t 2 … t N .
