More About Quantum Mechanics
First, some definitions. A statement that we make about the state of a system is called a proposition. One way quantum theorists try to make sense of all the different states of a quantum system is to think of them in terms of points in space. The collection of all possible states of a system is called a Hilbert space. There is also a space representing all possible propositions about the system. In terms of the mathematics of operators and wavefunctions, the operators that represent these wavefunctions are known as projection operators, because, in a sense, they 'project' onto the portions of the Hilbert space in which the proposition is true.
By thinking of quantum mechanics in terms of spaces, it becomes easier to describe the logic, and thus the mathematics, that governs the behaviour of quantum systems.
If we consider a system at one point in time, with a number of possible propositions about the state of the system, there is one Hilbert space, and one space of projection operators onto the Hilbert space. Let's say that  is one projection operator, and Ê is another projection operator. Just to make things easier, let  and Ê also be disjoint, meaning that the projections of both operators onto the Hilbert space do not overlap. We can then use  + Ê to form a new projection operator. In terms of propositions, this is the proposition that either  or Ê occurs, or, in mathematical notation,  ∧ Ê. If  and Ê are not disjoint, however, we can write the proposition that the state of the system is within both propositions  and Ê as  ∨ Ê, or  and Ê. This is not, by the way, a projection operator on the Hilbert space--it is a projection of one operator onto the projection of another operator onto a Hilbert space. Another thing to note about this is that when we can write  ∨ Ê, writing  ∧ Ê becomes impossible. We can also compare both propositions by considering the amount of information each provides about the system--if  contains more information than does Ê, we say it is finer than Ê, and we write this  < Ê.
And then we have histories. These are represented by a series of propositions, which are made about the system at specific times. Each series therefore forms a larger history proposition about the evolution of the state of the system through time. As with the single-time case, we can compare two histories, α and β, by saying that if α contains more information at any point in time than does β, then α is finer than β, or α < β.
We also need a way to describe 'α and β', 'α or β', 'not α and β', and 'not α or β'. 'α and β'is simple enough. It is simply the intersection of every proposition within history α and history β at every point where there is a proposition. 'α or β', however, becomes problematic, even if we simplify things by assuming that the propositions within α and β never overlap.
Naively, we would assume that, like the 'and' operation, 'α or β' simply involves saying, "The proposition from history α or the propostion from history β at time t". While this *does* describe the proposition that history α or history β does take place, the lack of influence between pairs of single-time propositions  and Ê means that each history could exchange a proposition at a particular time with the other history. Effectively, what the above describes is the proposition that any history made up of choices from each of the pairs of propositions at each time takes place.
Likewise, conventionally, we represent ' does not happen' by writing 1 - Â. However, if we describe 'α does not happen' simply by saying that every proposition  in α does not happen, we forget that α also does not happen if every proposition in it except one is fulfilled.
This problem was presented by Isham in Quantum Logic and the Histories Approach to Quantum Theory, part of his investigations in to the mathematical structure of consistent histories quantum theory. This post has been brought to you by my final year project, my need to organise my thoughts before I forget everything I've read, and gratuitous abuse of Unicode.
By thinking of quantum mechanics in terms of spaces, it becomes easier to describe the logic, and thus the mathematics, that governs the behaviour of quantum systems.
If we consider a system at one point in time, with a number of possible propositions about the state of the system, there is one Hilbert space, and one space of projection operators onto the Hilbert space. Let's say that  is one projection operator, and Ê is another projection operator. Just to make things easier, let  and Ê also be disjoint, meaning that the projections of both operators onto the Hilbert space do not overlap. We can then use  + Ê to form a new projection operator. In terms of propositions, this is the proposition that either  or Ê occurs, or, in mathematical notation,  ∧ Ê. If  and Ê are not disjoint, however, we can write the proposition that the state of the system is within both propositions  and Ê as  ∨ Ê, or  and Ê. This is not, by the way, a projection operator on the Hilbert space--it is a projection of one operator onto the projection of another operator onto a Hilbert space. Another thing to note about this is that when we can write  ∨ Ê, writing  ∧ Ê becomes impossible. We can also compare both propositions by considering the amount of information each provides about the system--if  contains more information than does Ê, we say it is finer than Ê, and we write this  < Ê.
And then we have histories. These are represented by a series of propositions, which are made about the system at specific times. Each series therefore forms a larger history proposition about the evolution of the state of the system through time. As with the single-time case, we can compare two histories, α and β, by saying that if α contains more information at any point in time than does β, then α is finer than β, or α < β.
We also need a way to describe 'α and β', 'α or β', 'not α and β', and 'not α or β'. 'α and β'is simple enough. It is simply the intersection of every proposition within history α and history β at every point where there is a proposition. 'α or β', however, becomes problematic, even if we simplify things by assuming that the propositions within α and β never overlap.
Naively, we would assume that, like the 'and' operation, 'α or β' simply involves saying, "The proposition from history α or the propostion from history β at time t". While this *does* describe the proposition that history α or history β does take place, the lack of influence between pairs of single-time propositions  and Ê means that each history could exchange a proposition at a particular time with the other history. Effectively, what the above describes is the proposition that any history made up of choices from each of the pairs of propositions at each time takes place.
Likewise, conventionally, we represent ' does not happen' by writing 1 - Â. However, if we describe 'α does not happen' simply by saying that every proposition  in α does not happen, we forget that α also does not happen if every proposition in it except one is fulfilled.
This problem was presented by Isham in Quantum Logic and the Histories Approach to Quantum Theory, part of his investigations in to the mathematical structure of consistent histories quantum theory. This post has been brought to you by my final year project, my need to organise my thoughts before I forget everything I've read, and gratuitous abuse of Unicode.