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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.

Introduction

In standard quantum mechanics a physical system is associated with a Hilbert space $mathcal H$ . 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 $mathcal H$ .

A physical proposition $,P$ about the system at a fixed time can be represented by a projection operator $hatP$ on $mathcal H$ (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.

History Propositions

Homogeneous Histories

A homogeneous history proposition $,alpha$ is a sequence of single-time propositions $alpha _t_i$ specified at different times $t_1<t_2<ldots <t_n$ . These times are called the temporal support of the history. We shall denote the proposition $,alpha$ as $(alpha _1,alpha _2,ldots ,alpha _n)$ and read it as

" $alpha _t_1$ at time $t_1$ is true and then $alpha _t_2$ at time $t_2$ is true and then $ldots$ and then $alpha _t_n$ at time $t_n$ 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 $,alpha$ OR $,beta$ for two homogeneous histories $,alpha ,beta$ .

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 $alpha =(alpha _1,alpha _2,ldots ,alpha _n)$ we can use the tensor product to define a projector

$hat alpha :=hat alpha _t_1otimes hat alpha _t_2otimes ldots otimes hat alpha _t_n$

where $hat alpha _t_i$ is the projection operator on $mathcal H$ that represents the proposition $alpha _t_i$ at time $t_i$ .

This $hatalpha$ is a projection operator on the tensor product "history Hilbert space" $H=mathcal Hotimes mathcal Hotimes ldots otimes mathcal H$

Not all projection operators on $H$ can be written as the sum of tensor products of the form $hatalpha$ . 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 $H$ can be applied to model the lattice of logical operations on history propositions.

If two homogeneous histories $,alpha$ and $,beta$ don't share the same temporal support they can be modified so that they do. If $,t_i$ is in the temporal support of $,alpha$ but not $,beta$ (for example) then a new homogeneous history proposition which differs from $,beta$ by including the "always true" proposition at each time $,t_i$ can be formed. In this way the temporal supports of $,alpha ,beta$ 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 $,alpha$ and $,beta$ such that $hat alpha hat beta =hat beta hat alpha$

Conjunction (AND)

If $alpha$ and $beta$ are two homogeneous histories then the history proposition " $,alpha$ and $,beta$ " is also a homogeneous history. It is represented by the projection operator

$widehat alpha wedge beta :=hat alpha hat beta$ $(=hat beta hat alpha )$

Disjunction (OR)

If $alpha$ and $beta$ are two homogeneous histories then the history proposition " $,alpha$ or $,beta$ " is in general not a homogeneous history. It is represented by the projection operator

$widehat alpha vee beta :=hat alpha +hat beta -hat alpha hat beta$

Negation (NOT)

The negation operation in the lattice of projection operators takes $hat P$ to

$neg hat P:=mathbb I-hat P$

where $mathbb I$ is the identity operator on the Hilbert space. Thus the projector used to represent the proposition $neg alpha$ (i.e. "not $alpha$ ") is

$widehat neg alpha :=mathbb I-hat alpha$

where $mathbb I$ 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 $,alpha =(alpha _1,alpha _2)$ . The projector to represent the proposition $neg alpha$ is

$widehat neg alpha =mathbb Iotimes mathbb I-hat alpha _1otimes hat alpha _2$
$=(mathbb I-hat alpha _1)otimes hat alpha _2+hat alpha _1otimes (mathbb I-hat alpha _2)+(mathbb I-hat alpha _1)otimes (mathbb I-hat alpha _2)$

The terms which appear in this expression:

$(mathbb I-hat alpha _1)otimes hat alpha _2$

$hat alpha _1otimes (mathbb I-hat alpha _2)$

$(mathbb I-hat alpha _1)otimes (mathbb I-hat alpha _2)$ .

can each be interpreted as follows:

$,alpha _1$ is false and $,alpha _2$ is true

$,alpha _1$ is true and $,alpha _2$ is false

both $,alpha _1$ is false and $,alpha _2$ is false

These three homogeneous histories, joined together with the OR operation, include all the possibilities for how the proposition " $,alpha _1$ and then $,alpha _2$ " can be false. We therefore see that the definition of $widehat neg alpha$ agrees with what the proposition $neg alpha$ should mean.

References

C.J. Isham, Quantum Logic and the Histories Approach to Quantum Theory, J.Math.Phys. 35 (1994) 2157-2185, arXiv:gr-qc/9308006v1

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