HPO formalism
Encyclopedia
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
proposition
s asserted at different points in time.
. 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 
.
A physical proposition
about the system at a fixed time can be represented by a projection operator 
on 
(See quantum logic). 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 
specified at different times 
. These times are called the temporal support of the history. We shall denote the proposition 
as 
and read it as
"
at time 
is true and then 
at time 
is true and then 
and then 
at time 
is true"
OR 
for two homogeneous histories 
.
For a homogeneous history
we can use the tensor product to define a projector
where
is the projection operator on 
that represents the proposition 
at time 
.
This
is a projection operator on the tensor product "history Hilbert space" 
Not all projection operators on
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.
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.
If two homogeneous histories
and 
don't share the same temporal support they can be modified so that they do. If 
is in the temporal support of 
but not 
(for example) then a new homogeneous history proposition which differs from 
by including the "always true" proposition at each time 
can be formed. In this way the temporal supports of 
can always be joined together. What shall therefore assume that all homogeneous histories share the same temporal support.
We now present the logical operations for homogeneous history propositions
and 
such that 
and 
are two homogeneous histories then the history proposition "
and 
" is also a homogeneous history. It is represented by the projection operator
and 
are two homogeneous histories then the history proposition "
or 
" is in general not a homogeneous history. It is represented by the projection operator
to
where
is the identity operator on the Hilbert space. Thus the projector used to represent the proposition 
(i.e. ``not 
) is
where
is the identity operator on the history Hilbert space.
. The projector to represent the proposition 
is
The terms which appear in this expression:
can each be interpreted as follows:
These three homogeneous histories, joined together with the OR operation, include all the possibilities for how the proposition "
and then 
" can be false. We therefore see that the definition of 
agrees with what the proposition 
should mean.
Temporal logic
In logic, the term temporal logic is used to describe any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time. In a temporal logic we can then express statements like "I am always hungry", "I will eventually be hungry", or "I will be hungry...
quantum logic
Quantum logic
In quantum mechanics, quantum logic is a set of rules for reasoning about propositions which takes the principles of quantum theory into account...
developed by Chris Isham
Christopher Isham
Christopher Isham is a theoretical physicist at Imperial College London. His main research interests are quantum gravity and foundational studies in quantum theory. He was the inventor of an approach to temporal quantum logic called the HPO formalism, and has worked on loop quantum gravity and...
. It deals with the logical structure of quantum mechanical
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
s asserted at different points in time.
Introduction
In standard quantum mechanics a physical system is associated with a Hilbert spaceHilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
. 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 .A physical proposition
about the system at a fixed time can be represented by a projection operator on (See quantum logic). This representation links together the latticeLattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
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.
Homogeneous Histories
A homogeneous history proposition is a sequence of single-time propositions specified at different times . These times are called the temporal support of the history. We shall denote the proposition as and read it as"
at time is true and then at time is true and then and then at time is true"Inhomogeneous Histories
Not all history propositions can be represented by a sequence of single-time propositions are different times. These are called inhomogeneous history propositions. An example is the proposition OR for two homogeneous histories .History Projection Operators
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.For a homogeneous history
we can use the tensor product to define a projectorwhere
is the projection operator on that represents the proposition at time .This
is a projection operator on the tensor product "history Hilbert space" Not all projection operators on
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.Temporal Quantum Logic
Representing history propositions by projectors on the history Hilbert space naturally encodes the logical structure of history propositions. The latticeLattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
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.If two homogeneous histories
and don't share the same temporal support they can be modified so that they do. If is in the temporal support of but not (for example) then a new homogeneous history proposition which differs from by including the "always true" proposition at each time can be formed. In this way the temporal supports of can always be joined together. What shall therefore assume that all homogeneous histories share the same temporal support.We now present the logical operations for homogeneous history propositions
and such that Conjunction (AND)
If and are two homogeneous histories then the history proposition " and " is also a homogeneous history. It is represented by the projection operator Disjunction (OR)
If and are two homogeneous histories then the history proposition " or " is in general not a homogeneous history. It is represented by the projection operatorNegation (NOT)
The negation operation in the lattice of projection operators takes towhere
is the identity operator on the Hilbert space. Thus the projector used to represent the proposition (i.e. ``not ) iswhere
is the identity operator on the history Hilbert space.Example: Two-time history
As an example, consider the negation of the two-time homogeneous history proposition. The projector to represent the proposition isThe terms which appear in this expression:
-
.
can each be interpreted as follows:
-
is false and 
is true
-
is true and 
is false
- both
is false and 
is false
These three homogeneous histories, joined together with the OR operation, include all the possibilities for how the proposition "
and then " can be false. We therefore see that the definition of agrees with what the proposition should mean.