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The path integral representation gives the quantum amplitude to go from point {{mvar|x}} to point {{mvar|y}} as an integral over all paths. For a free-particle action (for simplicity let {{math|''m'' {{=}} 1}}, {{math|''ħ'' {{=}} 1}})
: <math>S = \int \frac{\dot{x}^2}{2}\,
the integral can be evaluated explicitly.
To do this, it is convenient to start without the factor {{mvar|i}} in the exponential, so that large deviations are suppressed by small numbers, not by cancelling oscillatory contributions. The amplitude (or Kernel) reads:
: <math>K(x - y; T) = \int_{x(0) = x}^{x(T) = y} \exp\left(-\int_0^T \frac{\dot{x}^2}{2} \,
Splitting the integral into time slices:
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