Bohmian histories and decoherent histories

The predictions of the Bohmian and the decoherent (or consistent) histories formulations of the quantum mechanics of a closed system are compared for histories—sequences of alternatives at a series of times. For certain kinds of histories, Bohmian mechanics and decoherent histories may both be formulated in the same mathematical framework within which they can be compared. In that framework, Bohmian mechanics and decoherent histories represent a given history by different operators. Their predictions for the probabilities of histories of a closed system therefore generally differ. However, in an idealized model of measurement, the predictions of Bohmian mechanics and decoherent histories coincide for the probabilities of records of measurement outcomes. The formulations are thus difficult to distinguish experimentally. They may differ in their accounts of the past history of the Universe in quantum cosmology.

Figure 1

The motion of wave packets in the BESSW example. Two wave packets initially localized (at $t=0$) within the circles at left have equal and opposite $y$ components of their expected momentum $\vec{p}$. The wave packets evolve through the shaded regions indicated to the positions shown at $t=t_7$. At intermediate times $t_1,\dots ,t_6$, they are centered in one or the other of the squares.

Figure 2

Coarse-grained histories with high probability in the BESSW example according to decoherent histories quantum mechanics. Coarse-grained histories are defined by the grid of square regions at a sequence of times $t_1,\dots ,t_7$ adjusted so that the wave packets of Fig. 1 are centered in one square at each time. The possible coarse-grained histories are the sequence of any seven squares at the series of times. The two sequences of squares which have any significant probability according to decoherent histories quantum mechanics are shown, distinguished by different shadings.

Figure 3

Coarse-grained histories with high probability in the BESSW example according to Bohmian mechanics. The coarse-grained histories are the same as those in Fig. 2. The two sequences of squares with any significant probability are shown, distinguished by different shadings. They are different from those predicted by decoherent histories.