A Brief Introduction to Consistent Histories


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A Brief Introduction to Consistent Histories

Consistent histories theory was born as an attempt to describe closed systems in quantum mechanics, partly in light of a desire to construct quantum theories of cosmology. In fact, the Copenhagen interpretation of quantum mechanics cannot be applied to closed systems since it rests on the notion of probabilities defined in terms of a sequence of repeated measurements by an external observer. Thus it enforces a, cosmologically inappropriate, division between system and observer. The consistent-history formulation avoids this division since it assigns probabilities without making use of the measurements and the associated state vector reductions.

In the standard Copenhagen interpretation of quantum theory, probability assignments to sequences of measurements are computed using the von Neumann reduction postulate which, roughly speaking, determines a measurement-induced change in the density matrix that represents the state. Therefore, to give meaning to probabilities, the notion of measurement-induced, state vector reduction is essential.

The consistent history formalism was developed in order to make sense of probability assignments but without invoking the notion of measurement. This requires introducing the decoherence functional, $ d$ , which is a map from the space of all histories to the complex numbers. Specifically, given two histories (sequences of projection operators) $ \alpha=(\hat\alpha_{t_1},\hat\alpha_{t_2},\cdots,\hat\alpha_{t_n})$ and $ \beta=(\hat\beta_{t_1},\hat\beta_{t_2},\cdots,\hat\beta_{t_n})$ the decoherence functional is defined as

$\displaystyle d_{\rho,\hat H}(\alpha,\beta)=tr(\tilde{C}^{\dagger}_{\alpha}\rho \tilde{C}_{\beta})=tr(\hat{C}^{\dagger}_{\alpha}\rho \hat{C}_{\beta})$ (1)

where $ \rho$ is the initial density matrix, $ \hat H$ is the Hamiltonian, and $ \tilde{C}_{\alpha}$ represents the `class operator' which is defined in terms of the Schrodinger-picture projection operator $ \alpha_{t_i}$ as

$\displaystyle \tilde{C}_{\alpha}:=\hat{U}(t_0,t_1)\alpha_{t_1}\hat{U}(t_1,t_2)\alpha_{t_2}\cdots\hat{U}(t_{n-1},t_n)\alpha_{t_n}\hat{U}(t_{n},t_0)$ (2)

Thus $ \tilde{C}_{\alpha}$ represents the history proposition `` $ \alpha_{t_1}$ is true at time $ t_1$ , and then $ \alpha_{t_2}$ is true at time $ t_2$ , $ \cdots$ , and then $ \alpha_{t_n}$ is true at time $ t_n$ ''. It is worth noting that the class operator can be written as the product of Heisenberg-picture projection operators in the form $ \hat{C}_\alpha=\hat{\alpha}_{t_{n}}(t_n)\hat{\alpha}_{t_{n-1}}(t_{n-1})\cdots \hat{\alpha}_{t_1}(t_1)$ . Generally speaking this is not itself a projection operator.

The physical meaning associated to the quantity $ d(\alpha,\alpha)$ is that it is the probability of the history $ \alpha$ being realized. However, this interpretation can only be ascribed in a non-contradictory way if the history $ \alpha$ belongs to a special set of histories, namely a consistent set which, is a set $ \{\alpha^1, \alpha^2,\ldots, \alpha^n\}$ of histories which do not interfere with each other, i.e. $ d(\alpha_i,\alpha_j)=0$ for all $ i,j=1,\cdots,n$ . Only within a consistent set does the definition of consistent histories have any physical meaning. In fact, it is only within a given consistent set that the probability assignments are consistent. Each decoherence functional defines such a consistent set(s).

For an in-depth analysis of the axioms and definition of consistent-history theory the reader is referred to [12], [22], [27] and references therein. For the present paper only the following definitions are needed.

  1. A homogeneous history is any sequentially-ordered sequence of projection operators $ \hat{\alpha}_1,\hat{\alpha}_2,\cdots\hat{\alpha}_n,$
  2. The definition of the join $ \vee$ is straight forward when the two histories have the same time support and differ in their values only at one point $ t_i$ . In this case $ \alpha\vee\beta:=(\alpha_{t_1},\alpha_{t_2},\cdots, \alpha_{t_i}\vee\beta_{t_i}, \cdots \alpha_{t_n})= (\beta_{t_1},\beta_{t_2},\cdots, \beta_{t_i}\vee\alpha_{t_i}, \cdots \beta_{t_n})$ is a homogeneous history and satisfies the relation $ \hat{C}_{\alpha\vee\beta}=\hat{C}_{\alpha}\vee\hat{C}_{\beta}$ .

    The problem arises when the time supports are different, in particular when the two histories $ \alpha$ and $ \beta$ are disjoint. The join of such histories would take us outside the class of homogeneous histories. Similarly the negation of a homogeneous history would not itself be a homogeneous history.

  3. An inhomogeneous history arises when two disjoint homogeneous histories are joined using the logical connective ``or''($ \vee$ ) or when taking the negation ($ \neg$ ) of a history proposition. Specifically, given two disjoint homogeneous histories $ \alpha$ and $ \beta$ we can meaningfully talk about the inhomogeneous histories $ \alpha\vee\beta$ and $ \neg \alpha$ . Such histories are generally not a just a sequence of projection operators, but when computing the decoherence functional they are represented by the operator $ \hat{C}_{\alpha\vee\beta}:=\hat{C}_{\alpha}\vee\hat{C}_{\beta}$ and $ \hat{C}_{\neg\alpha}:=\hat{1}-\hat{C}_{\alpha}$

Gel Mann and Hartle, tried to solve the problem of representing inhomogeneous histories using path integrals on the configuration space, $ Q$ , of the system. In this formalism the histories $ \alpha$ and $ \beta$ are seen as subsets of the paths of Q. Then a pair of histories is said to be disjoint if they are disjoint subsets of the path space Q. Seen as path integrals, the additivity property of the decoherence functional is easily satisfied namely

$\displaystyle d(\alpha\vee\beta, \gamma)=d(\alpha,\gamma)+ d(\beta,\gamma)$ (3)

where $ \gamma$ is any subset of the path space Q.

Similarly, the negation of a history proposition $ \neg \alpha$ is represented by the complement of the subset $ \alpha$ of Q. Therefore

$\displaystyle d(\neg\alpha,\gamma)=d(1,\gamma)-d(\alpha,\gamma)$ (4)

where 1 is the unit history4.

The above properties in (3) and 4 are well defined in the context of path integrals. But what happens when defining the decoherence functional on a string of projection operators? Gel'Mann and Hartle solved this problem by postulating the following definitions for the class operators when computing decoherence functionals:

$\displaystyle \tilde{C}_{\alpha\vee\beta}$ $\displaystyle := \tilde{C}_{\alpha}+\tilde{C}_{\beta}$    
$\displaystyle \tilde{C}_{\neg\alpha}$ $\displaystyle :=1-\tilde{C}_{\alpha}$ (5)

if $ \alpha$ and $ \beta$ are disjoint histories. The right hand side of these equations are indeed operators that represent $ \alpha\vee\beta$ and $ \neg \alpha$ when computing the decoherence functional, but as objects in the consistent-history formalism, it is not really clear what $ \alpha\vee\beta$ and $ \neg \alpha$ are.
In fact, as defined above, a homogeneous history is a time ordered sequence of projection operators, but there is no analogue definition for $ \alpha\vee\beta$ or $ \neg \alpha$ . One might try to define the inhomogeneous histories $ \neg \alpha$ and $ \alpha\vee\beta$ component-wise so that, for a simple two-time history $ \alpha=(\hat\alpha_{t_1},\hat\alpha_{t_2})$ , we would have

$\displaystyle \neg\alpha=\neg(\hat{\alpha}_{t_1},\hat{\alpha}_{t_2}):= (\neg\hat{\alpha}_{t_1},\neg\hat{\alpha}_{t_2}).$ (6)

However, this definition of the negation operation is wrong. For $ \alpha$ is the temporal proposition ``$ \alpha_1$ is true at time $ t_1$ , and then $ \alpha_2$ is true at time $ t_2$ '', which we shall write as $ \hat{\alpha}_{t_1}\sqcap\hat{\alpha}_{t_2}$ . It is then intuitively clear that the negation of this proposition should be

$\displaystyle \neg(\hat{\alpha}_{t_1}\sqcap\hat{\alpha}_{t_2})= (\neg\hat{\alpha}_{t_1}\sqcap\hat{\alpha}_{t_2}) \vee(\hat{\alpha}_{t_1}\sqcap\neg\hat{\alpha}_{t_2}) \vee(\neg\hat{\alpha}_{t_1})\sqcap\neg(\hat{\alpha}_{t_2})$ (7)

which is not in any obvious sense the same as (6).

A similar problem arises with the ``or'' ($ \vee$ ) operation. For, given two homogenous histories $ (\alpha_1,\alpha_2)$ and $ (\beta_1,\beta_2)$ , the ''or'' operation defined component-wise is

$\displaystyle (\alpha_1,\alpha_2)\vee(\beta_1,\beta_2):=(\alpha_1\vee\beta_1, \alpha_2\vee\beta_2)$ (8)

This history would be true (realized) if both $ (\alpha_1\vee\beta_1)$ and $ (\alpha_2\vee\beta_2)$ are true, which implies that either an element in each of the pairs $ (\alpha_1,\alpha_2)$ and $ (\beta_1,\beta_2)$ is true, or both elements in either of the pairs $ (\alpha_1,\alpha_2)$ and $ (\beta_1,\beta_2)$ are true. But this contradicts with the actual meaning of the proposition $ (\alpha_1,\alpha_2)\vee(\beta_1,\beta_2)$ , which states that either history $ (\alpha_1,\alpha_2)$ is realized or history $ (\beta_1,\beta_2)$ is realized. In fact the `or' in the proposition $ (\alpha_1,\alpha_2)\vee(\beta_1,\beta_2)$ should really be as follows

$\displaystyle ({\alpha}_1\sqcap{\alpha}_2)\vee({\beta}_1\sqcap{\beta}_2)=(\neg( {\alpha}_1\sqcap{\alpha}_2)\wedge({\beta}_1\sqcap{\beta}_2))\vee (({\alpha}_1\sqcap{\alpha}_2)\wedge\neg({\beta}_1\sqcap{\beta}_2))$ (9)

Thus for the proposition $ ({\alpha}_1\sqcap{\alpha}_2)\vee({\beta}_1\sqcap{\beta}_2)$ to be true both elements in either of the pairs $ (\alpha_1\sqcap\alpha_2)$ and $ (\beta_1\sqcap\beta_2)$ have to be true, but not all four elements at the same time. If instead we had the history proposition from equation (16), $ ({\alpha}_1\vee{\beta_{1}})\sqcap({\alpha}_2\vee{\beta}_2)$ , this would be equivalent to

$\displaystyle ({\alpha}_1\vee{\beta_{1}})\sqcap({\alpha}_2\vee{\beta}_2):= ({\alpha}_1\sqcap{\alpha}_2)\vee({\alpha}_1\sqcap{\beta}_2)\vee ({\beta}_1\sqcap{\beta}_2)\vee({\beta}_1\sqcap {\alpha}_2)\geq({\alpha}_1\sqcap{\alpha}_2)\vee({\beta}_1 \sqcap{\beta}_2)$ (10)

This shows that it is not possible to define inhomogeneous histories component-wise. Moreover, the appeal to path integrals when defining $ \tilde{C}_{\alpha\vee\beta}$ is realization-dependent and does not uncover what $ \tilde{C}_{\alpha\vee\beta}$ actually is.

However, the right hand sides of equations (5) have a striking similarity to the single-time propositions in quantum logic. In fact, given two single-time propositions P and Q which are disjoint, the proposition $ P\vee Q$ is simply represented by the projection operator $ \hat{P}+\hat{Q}$ ; similarly, the negation $ \neq P$ is represented by the operator $ \hat{1}-\hat{P}$ .

This similarity of the single-time propositions with the right hand side of the equations (5) suggests that somehow it should be possible to identify history propositions with projection operators. Obviously these projection operators cannot be the class operators since, generally, these are not projection operators. The claim that a logic for consistent histories can be defined such that each history proposition is represented by a projection operator on some Hilbert space is also motivated by the fact that the statement that a certain history is ''realized", is itself a proposition. Therefore, the set of all such histories could possess a lattice structure similar to the lattice of single-time propositions in standard quantum logic.

These considerations led Isham to construct the, so-called, HPO formalism. In this new formalism of consistent histories it is possible to identify the entire set $ \mathcal{UP}$ with the projection lattice of some `new' Hilbert space. In the following Section we will describe this formalism in more detail.

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Next: The HPO formulation of Up: A Topos Formulation of Previous: Introduction
2009-04-19