The Hamiltonian formulation of quantum field theory is equivalent to, and independent of, its formulation based on the Feynman path integral and the Lagrangian. There are many advantages of having multiple formulations of the same theory, since for some calculations the Hamiltonian formulation may be more transparent and calculable than the Lagrangian formulation.

The path-integral formulation of the forward interest rates, discussed in some detail Chapter 7, is useful for calculating the expectation values of the quantum fields. To study questions related to the time evolution of quantities of interest, one needs to derive the Hamiltonian for the system from its Lagrangian. This route is the opposite to the one taken in Chapter 5 where the Lagrangian for option pricing was derived starting from the Hamiltonian formulation.

Many of the derivations in Chapter 7 that are feasible for Gaussian field theories cannot be replicated for nonlinear field theories, but which, in some cases, are nevertheless tractable in the Hamiltonian formulation. In particular, the risk-neutral martingale measure for the linear theory of the forward rates was derived by performing a Gaussian path integral, and this derivation is no longer tractable for nonlinear forward rates.

A rather remarkable result for the theory of nonlinear forward rates is that the martingale condition can be solved by generalizing the infinitesimal formulation of the condition for the martingale measure that was discussed for the case of a single security in Section 4.7.

The generator of infinitesimal time evolution of the forward interest rates, namely the Hamiltonian, is obtained for both the linear and nonlinear forward interest rates, as well as for the case of stochastic volatility.