HPO formalism

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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[edit]

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 \hat{P} 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[edit]

Homogeneous Histories[edit]

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[edit]

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[edit]

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_1} \otimes \hat{\alpha}_{t_2} \otimes \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 \hat{\alpha} is a projection operator on the tensor product "history Hilbert space" H = \mathcal{H} \otimes \mathcal{H} \otimes \ldots \otimes \mathcal{H}

Not all projection operators on H can be written as the sum of tensor products of the form \hat{\alpha}. These other projection operators are used to represent inhomogeneous histories by applying lattice operations to homogeneous histories.

Temporal Quantum Logic[edit]

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)[edit]

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)[edit]

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)[edit]

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[edit]

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{I} \otimes \mathbb{I} - \hat{\alpha}_1 \otimes \hat{\alpha}_2 =  (\mathbb{I} - \hat{\alpha}_1) \otimes \hat{\alpha}_2 + \hat{\alpha}_1 \otimes (\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}_1 \otimes (\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[edit]