The history projection operator (HPO) formalism is an approach to temporal quantum logic developed by Chris Isham.
It deals with the logical structure of quantum mechanical propositions asserted at different points in time.
In standard quantum mechanics a physical system is associated with a Hilbert space
States of the system at a fixed time are represented by normalised vectors in the space and physical observables are represented by Hermitian operators on
about the system at a fixed time can be represented by an orthogonal projection operator
This representation links together the lattice operations in the lattice of logical propositions and the lattice of projection operators on a Hilbert space (See quantum logic).
The HPO formalism is a natural extension of these ideas to propositions about the system that are concerned with more than one time.
is a sequence of single-time propositions
These times are called the temporal support of the history.
is true" Not all history propositions can be represented by a sequence of single-time propositions at different times.
These are called inhomogeneous history propositions.
The key observation of the HPO formalism is to represent history propositions by projection operators on a history Hilbert space.
This is where the name "History Projection Operator" (HPO) comes from.
we can use the tensor product to define a projector
is a projection operator on the tensor product "history Hilbert space"
can be written as the sum of tensor products of the form
These other projection operators are used to represent inhomogeneous histories by applying lattice operations to homogeneous histories.
Representing history propositions by projectors on the history Hilbert space naturally encodes the logical structure of history propositions.
The lattice operations on the set of projection operations on the history Hilbert space
can be applied to model the lattice of logical operations on history propositions.
don't share the same temporal support they can be modified so that they do.
(for example) then a new homogeneous history proposition which differs from
by including the "always true" proposition at each time
We shall therefore assume that all homogeneous histories share the same temporal support.
We now present the logical operations for homogeneous history propositions
It is represented by the projection operator
" is in general not a homogeneous history.
is the identity operator on the Hilbert space.
As an example, consider the negation of the two-time homogeneous history proposition
The terms which appear in this expression: can each be interpreted as follows: These three homogeneous histories, joined with the OR operation, include all the possibilities for how the proposition "