  Feynman's sum-over-histories formulation of quantum mechanics has been
considered a useful calculational tool in which virtual Feynman histories
entering into a coherent quantum superposition cannot be individually measured.
Here we show that sequential weak values, inferred by consecutive weak
measurements of projectors, allow direct experimental probing of individual
virtual Feynman histories thereby revealing the exact nature of quantum
interference of coherently superposed histories. Because the total sum of
sequential weak values of multi-time projection operators for a complete set of
orthogonal quantum histories is unity, complete sets of weak values could be
interpreted in agreement with the standard quantum mechanical picture. We also
elucidate the relationship between sequential weak values of quantum histories
with different coarse-graining in time and establish the incompatibility of
weak values for non-orthogonal quantum histories in history Hilbert space.
Bridging theory and experiment, the presented results may enhance our
understanding of both weak values and quantum histories.
