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Quantum Path Integral Formulation

A formulation of quantum mechanics that calculates amplitudes by summing over all possible paths a particle can take, relevant to quantum optical calculations.
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The statement of the theorem

Let S\mathcal{S} be the space of all continuous paths x:[ta,tb]→Rn\mathbf{x}: [t_a, t_b] \to \mathbb{R}^n such that x(ta)=xa\mathbf{x}(t_a) = \mathbf{x}_a and x(tb)=xb\mathbf{x}(t_b) = \mathbf{x}_b. Define the classical action functional S[x(t)]S[\mathbf{x}(t)] by the Lagrangian L(x,x˙,t)L(\mathbf{x}, \dot{\mathbf{x}}, t): S[x(t)]=∫tatbL(x(t),x˙(t),t)dtS[\mathbf{x}(t)] = \int_{t_a}^{t_b} L(\mathbf{x}(t), \dot{\mathbf{x}}(t), t) dt The quantum mechanical propagator K(xb,tb;xa,ta)K(\mathbf{x}_b, t_b; \mathbf{x}_a, t_a) is then defined by the Feynman path integral: K(xb,tb;xa,ta)=∫SDx(t)exp⁡(i1ℏS[x(t)])K(\mathbf{x}_b, t_b; \mathbf{x}_a, t_a) = \int_{\mathcal{S}} \mathcal{D}\mathbf{x}(t) \exp\left(i \frac{1}{\hbar} S[\mathbf{x}(t)]\right) where Dx(t)\mathcal{D}\mathbf{x}(t) represents the path integral measure, which is formally defined by the limit of the discretized product of Gaussian integrals over small time steps ϵ=(tb−ta)/N\epsilon = (t_b - t_a)/N: Dx(t)=lim⁡N→∞(1A)N/2∏j=1Ndxj\mathcal{D}\mathbf{x}(t) = \lim_{N \to \infty} \left( \frac{1}{A} \right)^{N/2} \prod_{j=1}^{N} d\mathbf{x}_j and AA is a normalization constant dependent on the system's mass and dimensionality.
Source: Wikipedia