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Functional Integral

The functional integral involves integrating over all possible paths, weighted by the exponential of the action's phase, eiS/ħe^{iS/ħ}.
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The statement of the theorem

The functional integral ∫Dx(t)\int \mathcal{D}x(t) represents the summation over all possible paths x(t)x(t) in the configuration space. The weighted amplitude is defined by the exponential of the action S[x(t)]S[x(t)]: \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)]}
Source: Wikipedia