Histories
A homogeneous history (here labels different histories) is a sequence of Propositions specified at different moments of time (here labels the times). We write this as:
and read it as "the proposition is true at time and then the proposition is true at time and then ". The times are strictly ordered and called the temporal support of the history.
Inhomogeneous histories are multiple-time propositions which cannot be represented by a homogeneous history. An example is the logical OR of two homogeneous histories: .
These propositions can correspond to any set of questions that include all possibilities. Examples might be the three propositions meaning "the electron went through the left slit", "the electron went through the right slit" and "the electron didn't go through either slit". One of the aims of the theory is to show that classical questions such as, "where are my keys?" are consistent. In this case one might use a large number of propositions each one specifying the location of the keys in some small region of space.
Each single-time proposition can be represented by a projection operator acting on the system's Hilbert space (we use "hats" to denote operators). It is then useful to represent homogeneous histories by the time-ordered tensor product of their single-time projection operators. This is the history projection operator (HPO) formalism developed by Christopher Isham and naturally encodes the logical structure of the history propositions. The homogeneous history is represented by the projection operator
This definition can be extended to define projection operators that represent inhomogeneous histories too.
Read more about this topic: Consistent Histories
Famous quotes containing the word histories:
“... the histories of Blacks and Jews in bondage and out of bondage, have been blood histories pursued through our kindred searchings for self-determination. Let this blood be a stain of honor that we share. Let us not now become enemies to ourselves and to each other.”
—June Jordan (b. 1936)
“The delicious faces of children, the beauty of school-girls, “the sweet seriousness of sixteen,” the lofty air of well-born, well-bred boys, the passionate histories in the looks and manners of youth and early manhood, and the varied power in all that well-known company that escort us through life,—we know how these forms thrill, paralyze, provoke, inspire, and enlarge us.”
—Ralph Waldo Emerson (1803–1882)
“As soon as histories are properly told there is no more need of romances.”
—Walt Whitman (1819–1892)