18 - Conclusions

Summary

All the chapters of the book have examined and studied various aspects of path integrals and Hamiltonians, which in turn exemplified different aspects of quantum mathematics. The principles of quantum mathematics, stated in Chapter 2, can be summarized as follows:

1. • The fundamental degrees of freedom F form the bedrock of the quantum system.

2. • A linear vector state space V based on the degrees of freedom F provides an exhaustive description of the quantum system.

3. • Operators O, which includes the Hamiltonian H, represent the physical properties of the degree of freedom and act on the state space V. Observable quantities are the matrix elements of the physical operators.

4. • A spacetime description of quantum indeterminacy is encoded in the Lagrangian L and the Dirac–Feynman formula relates it to the Hamiltonian.

5. • The path integral provides a representation of all the physical properties of a quantum system. In particular, the path integral yields the correlation functions of the degrees of freedom as well as the probability amplitudes for quantum transitions.

6. • The interconnection of the path integral with the underlying Hamiltonian and state space is a specific feature of quantum mathematics that distinguishes path integration from functional integration in general.