The Formalism of Generalized Contexts and Decay Processes

Abstract

The formalism of generalized contexts for quantum histories is used to investigate the possibility to consider the survival probability as the probability of no decay property at a given time conditional to no decay property at an earlier time. A negative result is found for an isolated system. The inclusion of two quantum measurement instruments at two different times makes possible to interpret the survival probability as a conditional probability of the whole system.

Appendix: Generalized Contexts

For the sake of completeness we present in this section a brief summary of our formalism of generalized contexts [4, 9].

Let us represent a quantum property p at time t by the pair (p;t), or equivalently by $(\widehat{\varPi}_{p};t)$, where $\widehat{\varPi}_{p}$ is the projector representing the property p in the Hilbert space $\mathcal{H}$ of the system. The time translation of the property p at time t to time t′ is defined by the pair (p′;t′), or by $(\widehat{\varPi}_{p^{\prime}};t^{\prime})$, where p′ is the quantum property represented by $\widehat{\varPi}_{p^{\prime}}\equiv\widehat{U}(t^{\prime},t)\,\widehat{\varPi}_{p}\,\widehat{U}^{-1}(t^{\prime},t)$. The unitary operator $\widehat{U}(t^{\prime},t)=\exp(-i\widehat{H}(t^{\prime }-t)/\hbar)$ is the time evolution operator generated by the Hamiltonian operator $\widehat{H}$ of the system. The relation between time translated pairs is transitive, reflexive and symmetric and, therefore, it is an equivalence relation. We use [(p;t)] (or $[(\widehat{\varPi}_{p};t)]$) to name the class of pairs equivalent to (p;t) (or to $(\widehat{\varPi}_{p};t)$). It is interesting to note that the Born rule assigns the same probability to all the pairs of the same equivalence class in a given state, i.e.

$$\Pr(p;t)=\mathrm{Tr}(\widehat{\rho}_{t}\widehat{\varPi}_{p})=\mathrm{Tr}( \widehat{\rho}_{t^{\prime }}\widehat{\varPi}_{p^{\prime}})=\Pr \bigl(p^{\prime};t^{\prime}\bigr)=\Pr\bigl[(p;t)\bigr]. $$

By definition, the equivalence class $[(\widehat{\varPi}^{(1)};t_{1})]$ implies the equivalence class $[(\widehat{\varPi}^{(2)};t_{2})]$ if the representative elements of the classes at a common time t ₀ verify the implication of the usual formalism of quantum mechanics, i.e.

$$ \begin{array}{l} \widehat{\varPi}^{(1,0)}\,\mathcal{H\subset\,}\widehat{\varPi}^{(2,0)}\mathcal{H},\\[6pt] \widehat{\varPi}^{(1,0)}\equiv\widehat{U}(t_{0},t_{1}) \widehat{\varPi}^{(1)}\widehat{U}^{-1}(t_{0},t_{1}), \qquad\widehat{\varPi}^{(2,0)}\equiv\,\widehat {U}(t_{0},t_{2}) \widehat{\varPi}^{(2)}\widehat{U}^{-1}(t_{0},t_{2}). \end{array} $$

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This implication is a transitive, reflexive and antisymmetric relation, being therefore an order relation.

The conjunction (disjunction) of two classes $[(\widehat {\varPi};t)]$ and $[(\widehat{\varPi}^{\prime};t^{\prime})]$ can be obtained as the greatest lower (least upper) bound, i.e.

$$\begin{array}{l} \bigl[(\widehat{\varPi};t)\bigr]\wedge\bigl[\bigl(\widehat{\varPi}^{\prime};t^{\prime} \bigr)\bigr]=\mathrm{Inf}\bigl\{\bigl[(\widehat{\varPi};t)\bigr], \bigl[\bigl(\widehat{\varPi}^{\prime}; t^{\prime}\bigr)\bigr]\bigr\},\\[6pt] \bigl[(\widehat{\varPi};t)\bigr]\vee\bigl[\bigl(\widehat{\varPi}^{\prime };t^{\prime} \bigr)\bigr]=\mathrm{Sup}\bigl\{\bigl[(\widehat{\varPi};t)\bigr],\bigl[\bigl(\widehat{ \varPi}^{\prime};t^{\prime }\bigr)\bigr]\bigr\}. \end{array} $$

The negation of an equivalence class $[(\widehat{\varPi};t)]$ is defined by

$$\overline{\bigl[(\widehat{\varPi};t)\bigr]}=\bigl[(\widehat{\overline{ \varPi}};t)\bigr]=\bigl[\bigl((\widehat{I}-\widehat{\varPi});t\bigr) \bigr]. $$

With the implication, disjunction, conjunction and negation previously obtained, the set of equivalence classes has the structure of an orthocomplemented nondistributive lattice.

The usual concept of context is a subset of all possible simultaneous properties which can be organized as a meaningful description of a quantum system at a given time, and can be endowed with a boolean logic with well-defined probabilities. Our formalism supplies a prescription to obtain, from the nondistributive lattice of equivalence classes of pairs, the valid descriptions involving properties at different times, which we called generalized contexts.

Let us consider a context of properties at time t ₁, generated by atomic properties $p_{j}^{(1)}$ represented by projectors $\widehat{\varPi}_{j}^{(1)}$ verifying

$$\widehat{\varPi}_{i}^{(1)}\widehat{\varPi}_{j}^{(1)}= \delta_{ij}\,\widehat{\varPi}_{i}^{(1)}, \quad\sum_{j\in\sigma^{(1)}}\widehat{\varPi}_{j}^{(1)}= \widehat {I},\ i,j\in\sigma^{(1)}. $$

Let us also consider a context of properties at time t ₂, generated by atomic properties $p_{\mu}^{(2)}$ represented by projectors $\widehat{\varPi }_{\mu}^{(2)}$ verifying

$$\widehat{\varPi}_{\mu}^{(2)}\widehat{\varPi}_{\nu}^{(2)}= \delta_{\mu\nu} \widehat{\varPi}_{\mu}^{(2)}, \quad\sum_{\mu\in\sigma^{(2)}}\widehat{\varPi}_{\mu}^{(2)}= \widehat{I},\ \mu,\nu\in\sigma^{(2)}. $$

We wish to represent with our formalism a universe of discourse able to incorporate expressions like “the property $p_{j}^{(1)}$ at time t ₁ and the property $p_{\mu}^{(2)}$ at time t ₂”. The conjunction of the classes with representative elements $\widehat{\varPi}_{i}^{(1)}$ at t ₁ and $\widehat{\varPi}_{\mu}^{(2)}$ at t ₂ is also the conjunction of the classes with representative elements $\widehat{\varPi}_{i}^{(1,0)}\equiv\widehat{U}(t_{0},t_{1})\widehat{\varPi}_{i}^{(1)}\widehat{U}^{-1}(t_{0},t_{1})$ and $\widehat{\varPi}_{\mu}^{(2,0)}\equiv\widehat{U}(t_{0},t_{2})\widehat{\varPi}_{\mu}^{(2)}\widehat {U}^{-1}(t_{0},t_{2})$ at the common time t ₀.

In usual quantum theory the conjunction of simultaneous properties represented by non-commuting operators has no meaning. So, it seems natural to consider quantum descriptions of a system, involving the properties generated by the projectors $\widehat{\varPi}_{i}^{(1)}$ at time t ₁ and $\widehat{\varPi}_{\mu }^{(2)}$ at time t ₂, only for the cases in which the projectors $\widehat{\varPi}_{i}^{(1)}$ and $\widehat{\varPi}_{\mu}^{(2)}$ commute when translated to a common time t ₀, i.e.

$$\widehat{\varPi}_{i}^{(1,0)}\widehat{\varPi}_{\mu}^{(2,0)}- \widehat{\varPi}_{\mu }^{(2,0)}\widehat{\varPi}_{i}^{(1,0)}=0. $$

If this is the case, for the equivalence class of composite properties representing “the property $p_{j}^{(1)}$ at time t ₁ and the property $p_{\mu}^{(2)}$ at time t ₂” we obtain

$$h_{i\mu}=\bigl[\bigl(\widehat{\varPi}_{i}^{(1)};t_{1} \bigr)\bigr]\wedge\bigl[\bigl(\widehat{\varPi}_{\mu }^{(2)};t_{2} \bigr)\bigr]=\Bigl[\Bigl(\lim_{n\rightarrow\infty}\bigl(\widehat{\varPi}_{i}^{(1,0)}\widehat{\varPi}_{\mu}^{(2,0)}\bigr)^{n};t_{0} \Bigr)\Bigr]=\bigl[\bigl(\widehat{\varPi}_{i}^{(1,0)}\widehat{\varPi}_{\mu}^{(2,0)};t_{0}\bigr)\bigr]. $$

As we can see, the conjunction of properties at different times t ₁ and t ₂ is equivalent to a single property, represented by the projector $\widehat{\varPi}_{i\mu}^{(0)}\equiv\widehat{\varPi}_{i}^{(1,0)}\widehat{\varPi}_{\mu }^{(2,0)}$ at a single time t ₀.

If the different contexts at times t ₁ and t ₂ produce commuting projectors $\widehat{\varPi}_{i}^{(1,0)}$ and $\widehat{\varPi}_{\mu}^{(2,0)}$ at the common time t ₀, it is easy to prove that

$$\widehat{\varPi}_{i\mu}^{(0)}\widehat{\varPi}_{j\nu}^{(0)}= \delta_{ij}\delta_{\mu \nu}\widehat{\varPi}_{i\mu}^{(0)}, \quad\sum_{i\mu}\widehat{\varPi}_{i\mu}^{(0)}=\widehat{I}. $$

Therefore, we realize that the composite properties h _iμ , represented at time t ₀ by the complete and exclusive set of projectors $\widehat{\varPi }_{i\mu}^{(0)}$, can be interpreted as the atomic properties generating a usual context in the sense described above. More general properties are obtained from the atomic ones by means of the disjunction operation. For instance, we can represent the property “$p_{j}^{(1)}$ at time t ₁ and $p_{\mu}^{(2)}$ at time t ₂, with j and μ having any value in the subsets Δ⁽¹⁾⊂σ ⁽¹⁾ and Δ⁽²⁾⊂σ ⁽²⁾” as

$$h_{\Delta^{(1)},\Delta^{(2)}}=\biggl[\biggl(\sum_{i\in\Delta^{(1)}}\sum _{\mu\in \Delta^{(2)}}\widehat{\varPi}_{i\mu}^{(0)};t_{0} \biggr)\biggr]. $$

The set of properties obtained in this way is an orthocomplemented and distributive lattice.

If the state of the system at time t ₀ is represented by $\widehat{\rho }_{t_{0}}$, the Born rule gives the following expression for the probability of the class of properties $h_{\Delta^{(1)},\Delta^{(2)}}$,

$$\Pr(h_{\Delta^{(1)},\Delta^{(2)}})=\sum_{i\in\Delta^{(1)}}\sum _{\mu\in \Delta^{(2)}}\mathrm{Tr}\bigl(\widehat{\rho}_{t_{0}}\widehat{ \varPi}_{i\mu}^{(0)}\bigr). $$

As a natural extension of the notion of context, we postulate that a description of a physical system involving properties at two different times t ₁ and t ₂ is valid if these properties are represented by commuting projectors when they are translated to a single time t ₀. We will call each one of those valid descriptions generalized context. On each generalized context, the probabilities given by the Born rule are well-defined (i.e. they are positive, normalized and additive) and, therefore, they may be meaningful in terms of frequencies.

In summary, our formalism is based on the notion of time-translation, allowing to transform the properties at a sequence of different times into properties at a single common time. A usual context of properties is first considered for each time of the sequence. If the projectors representing the atomic properties of each context commute when they are translated to a common time, the contexts at different times can be organized to lead to a generalized context of properties. A generalized context of properties is a distributive and orthocomplemented lattice, a boolean logic with well-defined implication, negation, conjunction and disjunction. This logic can be used to speak and make inferences about the selected properties of the system at different times. Well-defined probabilities on the elements of the lattice of properties are obtained by means of the well-known Born rule.