Chapter 35

Lagrangian Methods and

the Path Integral

In this chapter we’ll give a rapid survey of a different starting point for devel-
oping quantum mechanics, based on the Lagrangian rather than Hamiltonian
classical formalism. The Lagrangian point of view is the one taken in most mod-
ern physics textbooks, and we will refer to these for more detail, concentrating
here on explaining the relation to the Hamiltonian approach. Lagrangian meth-
ods have quite different strengths and weaknesses than those of the Hamiltonian
formalism, and we’ll try and point these out,
The Lagrangian formalism leads naturally to an apparently very different
notion of quantization, one based upon formulating quantum theory in terms of
infinite dimensional integrals known as path integrals. A serious investigation of
these would require another and very different volume, so we’ll have to restrict
ourselves to a quick outline of how path integrals work, giving references to
standard texts for the details. We will try and provide some indication of both
the advantages of the path integral method, as well as the significant problems
it entails.

35.1 Lagrangian mechanics

In the Lagrangian formalism, instead of a phase spaceR^2 dof positionsqjand
momentapj, one considers just the position (or configuration) spaceRd. Instead
of a Hamiltonian functionh(q,p), one has:

Definition(Lagrangian).The LagrangianLfor a classical mechanical system
with configuration spaceRdis a function

L: (q,v)∈Rd×Rd→L(q,v)∈R

Given differentiable paths in the configuration space defined by functions

γ:t∈[t 1 ,t 2 ]→Rd