  Parks introduced a formulation of time dependent weak values in 2008, which
is the formalism we use in this paper. In this paper we extend notions from
time dependent weak values to show that Hamiltonians associated with weak value
measurements can be shown to exhibit even or odd symmetric properties. They
exhibit PT or anti-PT symmetry, respectively. These symmetries are manifested
during the measurement process as pointer translations, which have vanishing
imaginary or vanishing real parts. The consequence of this that one can
characterize some of the aspects of these symmetries of the time dependent
Hamiltonians that arise from time dependent weak values. This allows one to
generalize some work on non-Hermitian variables in quantum mechanics (QM) due
to Bender related to weak values and weak measurement. We also speculate how
these symmetries might apply to distinguishing between the various two-time
interpretations of QM in the Conclusions of this paper.
