Quantum Dynamics for de Sitter Radiation
  We revisit the Hamiltonian formalism for a massive scalar field and study the
particle production in a de Sitter space. In the invariant-operator picture the
time-dependent annihilation and creation operators are constructed in terms of
a complex solution to the classical equation of motion for the field and the
Gaussian wave function for each Fourier mode is found which is an exact
solution to the Schr\"odinger equation. The in-out formalism is reformulated by
the annihilation and creation operators and the Gaussian wave functions. The de
Sitter radiation from the in-out formalism differs from the Gibbons-Hawking
radiation in the planar coordinates, and we discuss the discrepancy of the
particle production by the two method
