where Insert Eq. (1) into the time evolution of Eq. (2) at each of the times t$_{i}$ : where The operator ˆ C α defined by Eq. (3) is the chain or class operator for the history α and the state ˆ C α | Φ( t$_{0}$ ) 〉 is the branch of the system state | Φ( t$_{n}$ ) 〉 associated with the history α . We are interested in families of histories such that all branches are mutually orthogonal, that is, These are consistency or decoherence conditions and a family of histories that satisfies them is known as a consistent or decoherent family. p ( α ) then gives the probability for the history α . The left-hand side of Eq. (4) is known as the decoherence functional and is often written in the form where In this form, the decoherence functional can be generalized to density matrices ρ$_{0}$ for non- pure initial states, but this generality will not be needed here. Also, the chain operators as defined by Eq. (3) satisfy It is perhaps more common to define chain operators in the form 5 ˆ K α ≡ ˆ P α$_{n}$ n ( t$_{n}$ ) ˆ P α$_{n}$ − 1 n − 1 ( t$_{n}$$_{−}$$_{1}$ ) · · · ˆ P α 1 1 ( t$_{1}$ ) , ∑ α ˆ C α = T ( t$_{n}$, t$_{0}$ ) . (6) ρ$_{0}$ ≡ | Φ( t$_{0}$ ) 〉〈 Φ( t$_{0}$ ) | . D ( α , α $^{′}$) ≡ 〈 Φ( t$_{0}$ ) | ˆ C α $^{′}$† ˆ C α | Φ( t$_{0}$ ) 〉 = Tr( ˆ C α ρ$_{0}$ ˆ C α ′ $^{†}$) , (5) 〈 Φ( t$_{0}$ ) | ˆ C α $^{′}$† ˆ C α | Φ( t$_{0}$ ) 〉 = δ$_{α}$ ′ $_{α}$p ( α ) . (4) T ( t$_{f}$, t$_{i}$ ) ≡ e − i ∫ t f t i d t ˆ H ( t ) . ˆ C α ≡ ˆ P α$_{n}$ n T ( t$_{n}$, t$_{n}$$_{−}$$_{1}$ ) ˆ P α$_{n}$ − 1 n − 1 T ( t$_{n}$$_{−}$$_{1}$, t$_{n}$$_{−}$$_{2}$ ) · · · ˆ P α 1 1 T ( t$_{1}$, t$_{0}$ ) . (3) | Φ( t$_{n}$ ) 〉 = ∑ α$_{n}$ ˆ P α$_{n}$ n T ( t$_{n}$, t$_{n}$$_{−}$$_{1}$ ) ∑ α$_{n}$ − 1 ˆ P α$_{n}$ − 1 n − 1 T ( t$_{n}$$_{−}$$_{1}$,t$_{n}$$_{−}$$_{2}$ ) · · · ∑ α 1 ˆ P α 1 1 T ( t$_{1}$, t$_{0}$ ) | Φ( t$_{0}$ ) 〉 = ∑ α ˆ C α | Φ( t$_{0}$ ) 〉 ,