This was initially proposed as an alternative formulation of quantum mechanics by Richard Feynman ^[R. P. Feynman, Rev. of Mod. Phys., 20, 367 ]. Knowing that a given trajectory, $q(\delta t)$ (where $\delta t=t_f-t_i$), for a particle is is probabilistic by [Born's rule](Born%20rule), The path integral formulation postulates that that the [probability amplitude](probability%20amplitude) is the sum of weighted contributions from each possible path, $q(\delta t)$, that can be taken by a given particle (and we square this to get the probability itself). The weight associated with each trajectory is depends on its classical [action](Action.md). The amplitude itself is given as $K(q_F,t_F;q_I,t_I)=\int \mathcal{D}[q(\delta t)]e^{iS[q(\delta t)]/\hbar}$ where here we perform an [[Functional integral]] as indicated by the notation $\mathcal{D}[q(\delta t)]$ where $\mathcal{D}[q(\delta t)]=\lim_{\delta t\rightarrow 0}c(\delta t)\prod_{n=1}^{\infty}da_n$ and $c(\delta t)$ is a normalization factor associated with the particular physical system modeled by the amplitude. # Advantages and when to use it For many well-understood quantum mechanical models, such as the [free particle](Free%20Particle.md), the advantages of this formulation seem unclear and it would appear we're smashing a walnut with a sledgehammer however, the advantages become most apparent in [quantum field theory](Quantum%20Field%20Theory%20(Index)) since it is a coordinate invariant approach. # Derivation ## Argument from the [[Double-slit Experiment]]    #QuantumMechanics/QuantumDynamics/PathIntegrals