Quantum histories

Richard Feynman's sum-over-histories formulation of quantum mechanics has been considered to be just a useful calculational tool in which virtual Feynman histories entering into a coherent quantum superposition cannot be individually measured. Sequential weak values inferred by weak measurements, however, allow direct experimental probing of individual virtual Feynman histories thereby revealing the exact nature of quantum interference of the coherently superposed histories.

In 1988, Yakir Aharonov, David Albert and Lev Vaidman proposed an ingenious way of measuring a quantum system without disturbing it to the point where the wave function $\Psi$ of the system collapses into a definite value of the physical observable being measured. If the coupling between the measuring device and the measured system is weak, it is possible to obtain some information about one observable property (such as position) without disturbing another complementary observable property (momentum) and therefore the future evolution of the system. Although each weak measurement only provides a tiny amount of information, by carrying out multiple measurements and extracting the statistical average, one can assign a weak value to the measured physical observable. Surprisingly, however, the weak values obtained through repeated weak measurement are complex numbers and may lie outside the spectrum of the measured quantum observable calling into question what exactly the weak values measure.

In view of the existing controversy over the meaning and interpretation of weak values, our recent article published in Physical Review A demonstrates that sequential weak values of quantum histories (multi-time projection operators) are not arbitrary, but may reflect true physical properties of the quantum physical system under study. What is measured by sequential weak values is the quantum probability amplitude propagating along a given quantum history. If sequential weak values are interpreted for a complete set of orthogonal quantum histories, the total sum of weak values is unity and the analysis agrees with the standard quantum mechanical picture.