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:
“All histories do shew, and wise politicians do hold it necessary that, for the well-governing of every Commonweal, it behoveth man to presuppose that all men are evil, and will declare themselves so to be when occasion is offered.”
—Sir Walter Raleigh (1552–1618)
“I read, with a kind of hopeless envy, histories and legends of people of our craft who “do not write for money.” It must be a pleasant experience to be able to cultivate so delicate a class of motives for the privilege of doing one’s best to express one’s thoughts to people who care for them. Personally, I have yet to breathe the ether of such a transcendent sphere. I am proud to say that I have always been a working woman, and always had to be ...”
—Elizabeth Stuart Phelps (1844–1911)
“The histories of the lives and fortunes of men are full of instances of this nature,—where favorable times and lucky accidents have done for them, what wisdom or skill could not.”
—Laurence Sterne (1713–1768)