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A statement of quantum mechanics invented by Richard Feynman. Whereas the Schrodinger Equation and Dirac Equation are analogous to the Hamiltonian method in classical mechanics, the Path Integral is analogous to the Lagrangian method. While cumbersome for nonrelativistic quantum mechanics, the Path Integral formulation is the best available tool for Quantum Field Theory, instances of which are Quantum Electrodynamics and Quantum Chromodynamics. The Path integral formulation can be derived from the wave-mechanical formulation in nonrelativistic and special-relativistic quantum mechanics, though it does not rely on it for validity. The path integral often looks simplest in the Interaction Picture.

Mathematically speaking, the Path Integral provides a Propagator acquired by slicing time up into infinitesimal fragments (indexed by j in the below equation), each of which can be handled with a short time approximation. The short time approximation is essentially every atom in the universe exploding at the speed of light, but with a specific phase relation such that when you add up a bunch of them, the bits that are going the wrong velocity cancel out quite thoroughly.

K(x1, t1, x2, t2) = lim (N → ∞) ∏ ( j = 2 → N-1) ∫ (-∞ → +∞) dxi(ihΔt)(N-1)/2*e2πiS/h

where h is Planck's original constant and S is the classical action,

S = ∫(tj, tj+1) dt L(x(t), ∂x(t)/∂t)

and in turn L is the Lagrangian operator, the Kinetic Energy minus the Potential Energy.

Where classical Lagrangian dynamics dictate that the action is minimized at every moment, period, the Path Integral merely says that paths have a phase shift based on their action. This has the consequence that the magnitude of the wavefunction is maximized where there is minimum interference, which is where the action has a zero derivative. This accomplishes much the same thing in the classical limit.

When you get the propagator in relativistic quantum mechanics, however, your justification is rather different. You still use the Interaction picture. However, instead of looking at individual time-slices, you use a self-consistent method for determining the retarded green function, which can then be integrated to yield the propagator. This derivation is very long, and in order for any of it to be illuminating would require more explanation than can be reasonably crowded into a node.

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