content_image/340149.jpg content_image/340150.jpg content_image/340151.jpg content_image/340147.jpg content_image/340146.jpg
ψ α S S (Ψ$_{I}$) = 0, then | s α S (Ψ$_{I}$) 〉 0 can be chosen arbitrarily from the eigenspace of ˆ P α S S .) Then, because of Eq. (27), 

Thus, 

<img src='content_image/340149.jpg'>

<img src='content_image/340150.jpg'>

where | s α S , a α S 〉 0 = ˆ G$_{A}$ | s α S 〉 0 (and the explicit dependence on Ψ$_{I}$ has been dropped for simplicity of notation). Because of Eq. (28), a similar relationship to Eq. (31) holds between E and A . Therefore, using this and Eq. (32) in Eq. (29) gives 

where | s α S , a α S , e α S 〉 0 = ˆ G$_{E}$ | s α S , a α S 〉 $_{0}$. 

<img src='content_image/340151.jpg'>

Equation (33) is essentially the form assumed for a measurement state in [24]. Each | s α S , a α S , e α S 〉 0 is a state of the overall system V with outcome α$_{S}$ for S and correlated outcomes for A and E . And these state are orthogonal, so Eq. (33) certainly represents a consistent family of decoherent histories. Thus, one clearly wants to interpret | ψ α S S | 2 as the probability for | s α S , a α S , e α S 〉 0 according to the Born rule. 

The next section turns, then, to establishing the usual Born-rule probability interpreta- tion. Note, though, that the derivation of Eq. (33) is independent of this interpretation. 

## VI. BORN’S RULE 

Consider now a hypervolume V bounded by times T$_{I}$ and T$_{F}$ and divided into a system S and its environment E , such that 

<img src='content_image/340147.jpg'>

for appropriate projection operators ˆ P α E and ˆ P α $_{S}$. Then, following a similar argument to Sec. V, given an initial state | Ψ$_{I}$ 〉 $_{0}$, 

<img src='content_image/340146.jpg'>

