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Path Integral Formulation

The path integral formulation, pioneered by Feynman, provides a quantum mechanical description where the probability amplitude for a particle's propagation is calculated as a sum over all possible paths between an initial and final point.
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The statement of the theorem

Let ⟨xb,tb∣xa,ta⟩\langle x_b, t_b | x_a, t_a \rangle be the quantum mechanical propagator (amplitude). The Path Integral Formulation defines this amplitude as the functional integral over all possible paths x(t)x(t) connecting xax_a at tat_a to xbx_b at tbt_b: \n⟨xb,tb∣xa,ta⟩=∫Dx(t)eiℏS[x(t)]\langle x_b, t_b | x_a, t_a \rangle = \int \mathcal{D}x(t) e^{\frac{i}{\hbar} S[x(t)]}