Quantum mechanics can describe propagation between two events by assigning a complex amplitude to every possible path connecting them and adding those amplitudes.

In Feynman's formulation, a path $q(t)$ contributes a phase proportional to its classical action:

$$\mathcal A[q]\propto e^{iS[q]/\hbar}.$$

The total transition amplitude is represented schematically as

$$K=\int \mathcal Dq;e^{iS[q]/\hbar},$$

where $\int\mathcal Dq$ denotes a sum over paths rather than an ordinary finite-dimensional integral.

Paths contribute amplitudes, not probabilities

Different paths can reinforce or cancel because their contributions are complex phases. Observable probabilities are calculated only after the amplitudes have been combined.

Relation to the double slit

The double-slit experiment is the simplest discrete analogy: amplitudes through alternative indistinguishable routes are added before squaring their magnitude. The path integral extends that idea from a few alternatives to a continuum of possible trajectories.

A formulation of quantum dynamics

The path-integral and Schrödinger formulations describe the same nonrelativistic quantum theory when applied consistently. The advantage of the path viewpoint is that it makes the role of the classical action and the classical limit especially transparent.