To be used in future
 Chapter 5: Path Integrals Up Chapter 5: Path Integrals Section 5.2: Interpretation 

5.1 Sum of paths

The idea is to start from the link between the evolution operator and the transition amplitude↓. We write which is correct when is time independent.
The object is called the propagator↓.

5.1.1 Free particle

For free particles (in one space dimension) we can use the Hamiltonian We evaluate , and insert the complete set of states just before : We now use the momentum eigenstates normalised so that We get
Gaussian integrals
Almost all basic path integrals depend on a single class of Gaussian integrals. The techniques are simple in one dimension, and only rely on completing squares:

5.1.2 General approach

The technique sketched above, which relies on being able to do Gaussian integrals, only works for a free particle. For a general Hamiltonian we can still apply the technique above to take small steps. As usual, we take many small steps to take one giant leap forward:
We define , with a large integer. Now We insert the complete set of states between each of the exponentials. We thus get For convenience define .
We can evaluate the transition amplitude as where (see example sheet). The fact that we evaluate the energy at average position is not crucial; in fact we could have chosen point in the interval and would have been correct with an error proportional to .
We thus obtain This is the first version of the path integral we shall see, so let us look at this and interpret this. What we see is that we sum over all intermediate points in phase space, but the weight given to each point is linked to its energy.
In most standard cases, the Hamiltonian is actually quadratic in the momenta , so we that part of the integration is a high-dimensional Gaussian integral. It is not very hard to perform the integrals in this; let us look at one term We can thus simplify the path integral to
We now take the limit , but increase at the same time so that the value of remains fixed, we see that We can now recognise that Thus, finally we get the coordinate space path integral Here is a mathematically very dangerous object—it is poorly defined, but we shall use it nonetheless! We have also introduced the classical action↓ This is a functional of the fields. The minimal action principle↓ states ; this gives us the classical Lagrangian equations of motion. Remember the boundary conditions and .
 Chapter 5: Path Integrals Up Chapter 5: Path Integrals Section 5.2: Interpretation