Uncertainty relations for time averaged weak values*

Time averaging of weak values using the quantum transition path time probability distribution leads to the establishment of a general uncertainty principle for the weak values of two not necessarily Hermitian operators. This new principle is a weak value analog of the Heisenberg-Robertson strong value uncertainty principle. It shows that complementarity does not prevent the simultaneous determination
of weak values of non-commuting operators. When the time fluctuations of the weak values of the two operators under consideration are proportional to each other there is no uncertainty limitation on their variances and, in principle, their means can be determined with arbitrary precision. This will be exemplified by considering weak value uncertainty relations for the time-energy, coordinate-momentum and coordinate-kinetic energy pairs. and an analysis of spin operators and the Stern-Gerlach experiment in weak and strong inhomogeneous magnetic fields. We find that anomalous spin values are associated with large variances implying that their measurement demands increased signal averaging. These examples establish the importance of considering the time dependence of weak values in scattering experiments.