- HPO formalism
The History Projection Operator (HPO) formalism is an approach to temporal
quantum logicdeveloped by Chris Isham. It deals with the logical structure of quantum mechanical propositionsasserted 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 observablesare represented by Hermitian operatorson .
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.
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"
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 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.
Temporal Quantum Logic
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.
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
If and are two homogeneous histories then the history proposition " and " is also a homogeneous history. It is represented by the projection operator
If and are two homogeneous histories then the history proposition " or " is in general not a homogeneous history. It is represented by the projection operator
The negation operation in the lattice of projection operators takes to
where is the
identity operatoron the Hilbert space. Thus the the projector used to represent the proposition (i.e. ``not ) is
where 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 is
The 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.
* C.J. Isham, [http://arxiv.org/abs/gr-qc/9308006 Quantum Logic and the Histories Approach to Quantum Theory] , J.Math.Phys. 35 (1994) 2157-2185, arXiv:gr-qc/9308006v1
Wikimedia Foundation. 2010.
Look at other dictionaries:
Consistent histories — Quantum mechanics Uncertainty principle … Wikipedia
Quantum logic — In mathematical physics and quantum mechanics, quantum logic is a set of rules for reasoning about propositions which takes the principles of quantum theory into account. This research area and its name originated in the 1936 paper by Garrett… … Wikipedia
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. It is sometimes also used to refer to tense logic, a particular modal logic… … Wikipedia
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… … Wikipedia