3.5.9 Path Integrals
As we have seen in chapter II (section 15), the principle of least action is a general principle of Nature, which had been applied to derive the Newtonian, Lagrangian and Hamiltonian equations of motion, and also in General Relativity, or what is known as the Einstein-Hilbert action, as we have seen in section 3. The same principle was also applied by Dirac in his derivation of the quantum interference of amplitudes, when he derived his wave equation, as we have seen in section 5.2.
Feynman, on his part, generalized the least-action principle, by replacing the classical notion of a single classical trajectory with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude. This formulation is called Path Integral, and it had been proven crucial to the subsequent development of many fields in theoretical physics.
Unlike previous methods, the path integral allows physicists to easily change coordinates between very different canonical descriptions of the same quantum system. Another advantage is that it is in practice easier to guess the correct form of the Lagrangian of a theory, which naturally enters the path integrals, than the Hamiltonian.
The path-integral approach has been proved to be equivalent to the other formalisms of Quantum Mechanics and Quantum Field Theory. Thus, by deriving either approach from the other, problems associated with one or the other approach can be solved.
The path integral also relates quantum and stochastic processes, and this provided the basis for the grand synthesis of the 1970s, which unified Quantum Field Theory with the statistical field theory of a fluctuating field near a second-order phase transition. The Schroedinger equation is a diffusion equation with an imaginary diffusion constant, and the path integral is an analytic continuation of a method for summing up all possible random walks.
In Quantum Mechanics, as in classical mechanics, the Hamiltonian is the generator of time translations. This means that the state at a slightly later time differs from the state at the current time by the result of acting with the Hamiltonian operator, multiplied by the negative imaginary unit:
. For states with a definite energy, this is a statement of the de Broglie relation between frequency and energy, and the general relation is consistent with that plus the superposition principle.
The Hamiltonian in classical mechanics is derived from a Lagrangian, which is a more fundamental quantity relative to special relativity. The Hamiltonian indicates how to march forward in time, but the time is different in different reference frames. The Lagrangian is a Lorentz scalar, while the Hamiltonian is the time component of a four-vector. So the Hamiltonian is different in different frames, and this type of symmetry is not apparent in the original formulation of Quantum Mechanics.
Dirac’s work did not provide a precise prescription to calculate the sum over paths, and he did not show that one could recover the Schroedinger equation or the canonical commutation relations from this rule. This was done by Feynman. That is, the classical path arises naturally in the classical limit.
Feynman showed that Dirac’s quantum action was, for most cases of interest, simply equal to the classical action, appropriately discretized. This means that the classical action is the phase acquired by quantum evolution between two fixed endpoints.
The path integral formulation of Quantum Field Theory represents the transition amplitude, corresponding to the classical correlation function, as a weighted sum of all possible histories of the system from the initial to the final state. A Feynman diagram is a graphical representation of a perturbative contribution to the transition amplitude.