The dynamics of the entire system can be described by the Liouville equation of motion, χ χ ˙ ˙ = − − i [ H , χ χ ] . This equation, containing an infinite number of degrees of freedom, is impossible to solve analytically except in very particular cases. What's more, under certain approximations, the bath degrees of freedom need not be considered, and an effective master equation can be derived in terms of the system density matrix, ρ ρ = tr B ⁡ ⁡ χ χ . The problem can be analyzed more easily by moving into the interaction picture, defined by the unitary transformation M ~ ~ = U 0 M U 0 † † , where M is an arbitrary operator, and U 0 = e i ( H S + H B ) t . Also note that U ( t , t 0 ) is the total unitary operator of the entire system. It is straightforward to confirm that the Liouville equation becomes