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Feynman Path

Each individual path contributing to the path integral is referred to as a Feynman path, representing a possible trajectory of the quantum particle.
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The statement of the theorem

A Feynman path is any continuous, differentiable trajectory x(t)x(t) connecting the initial point xax_a at tat_a to the final point xbx_b at tbt_b. The path integral sums the contributions of all such paths, weighted by the phase factor eiS[x(t)]/ℏe^{iS[x(t)]/\hbar}: \n⟨xb,tb∣xa,ta⟩=∑all paths x(t)A[x(t)]eiℏS[x(t)]\langle x_b, t_b | x_a, t_a \rangle = \sum_{\text{all paths } x(t)} A[x(t)] e^{\frac{i}{\hbar} S[x(t)]}