  The widely accepted approach to the foundation of quantum mechanics is that
the Poisson bracket, governing the non-commutative algebra of operators, is
taken as a postulate with no underlying physics. In this manuscript, it is
shown that this postulation is in fact unnecessary and may be replaced by a few
deeper concepts, which ultimately lead to the derivation of Poisson bracket.
One would only need to use Fourier transform pairs and Kramers-Kronig
identities in the complex domain. We present a definition of Hermitian
time-operator and discuss some of its basic properties.
