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Feyman's Sum-Over-The-Path Approach

The basic idea behind Feynman's sum-over-paths (or sum-over-histories) approach is as follows: Assume that a particle can travel between two points A and B by a - possibly infinite - number of different paths. Each one of these paths will have a certain probability associated with it. In quantum mechanical terms, these probabilities are encoded in the wavefunction that describes the particle, which assigns to each possible path a different probability amplitude; the square modulus of this amplitude gives the corresponding probability. 

The crucial point is that these different amplitudes have a wavelike nature, and as they spread through space they interfere with each other, their respective wave patterns either reinforcing or canceling each other out at various points. And if you sum over all the amplitudes of all the different paths, i.e. you sum-over-histories, then the different amplitudes will reinforce or cancel each other in such a way that the only path that survives this interference process is the one that the particle actually follows.
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