On the Concept of Entropy for Quantum Decaying Systems

Abstract

The classical concept of entropy was successfully extended to quantum mechanics by the introduction of the density operator formalism. However, further extensions to quantum decaying states have been hampered by conceptual difficulties associated to the particular nature of these states. In this work we address this problem, by (i) pointing out the difficulties that appear when one tries a consistent definition for this entropy, and (ii) building up a plausible formalism for it, which is based on the use of coherent complex states in the context of a path integration.

Appendix

In the present Appendix, we shall sketch some mathematical properties related to objects that have been introduced along the present paper.

To begin with, let us define the space $C^{\infty}_{b}({\mathbb{R}})$ as the vector space of all functions $f(x):{\mathbb{R}}\longmapsto{\mathbb{C}}$ having the following properties: i.) They are continuously differentiable at all orders and ii.) they have compact support, i.e., they vanish outside a finite interval of the real line ${\mathbb{R}}$.

This space has the following properties:

Let us consider the Fourier transform $\mathcal{F}$. Then, $\mathcal{F}$ is an one to one onto mapping

$$ \mathcal{F}:C^\infty_b({\mathbb{R}})\longmapsto Z , $$

(75)

where Z are entire analytic functions of exponential type [45]. Let us call $\mathcal{Z}$ the vector space of restrictions of functions of Z on the positive semi-axis ${\mathbb{R}}^{+}\equiv [0,\infty)$. Some properties of the spaces $\mathcal{Z}$ are the following:

(i) Functions in Z are in $L^{2}(\mathbb{R})$.

(ii) $\mathcal{Z}$ is dense in $L^{2}({\mathbb{R}}^{+})$ with respect to the topology of the latter, so that

$$ \mathcal{Z}\subset L^2\bigl({\mathbb{R}}^+\bigr)\subset \mathcal{Z}^\times $$

(76)

are well defined Gelfand triplets or RHS. This can be proven as done for similar triplets of Hardy functions [21].

(iii) For any $t\in\mathbb{R}$ and any $f(E)\in \mathcal{Z}$ we have that

$$ e^{itE}f(E)\in\mathcal{Z} . $$

(77)

Then, take $F\in\mathcal{Z}^{\times}$ and consider the duality formula

$$ \langle e^{itE}f(E)|F \rangle= \langle f(E)|e^{-itE}F \rangle . $$

(78)

This means that for any fixed $t\in\mathbb{R}$ and any $F\in\mathcal{Z}^{\times}$, $e^{-itE}F\in \mathcal{Z}^{\times}$.

(iv) The total Hamiltonian H for the Friedrichs model has non-degenerate absolutely continuous spectrum σ(H)≡[0,∞). Then, there is a unitary operator $U: L^{2}({\mathbb{R}}^{+})\oplus{\mathbb{C}}\longmapsto L^{2}({\mathbb{R}}^{+})$ such that UHU ⁻¹ is the multiplication operator on $L^{2}({\mathbb{R}}^{+})$, i.e., UHU ⁻¹ f(E)=Ef(E), for any $f(E)\in L^{2}({\mathbb{R}}^{+})$ such that $Ef(E)\in L^{2}({\mathbb{R}}^{+})$ [34]. Then, if $\varPhi:=U^{-1}\mathcal{Z}$, we have a new Gelfand triplet [18, 19]:

$$ \varPhi\subset L^2\bigl({\mathbb{R}}^+\bigr)\oplus{ \mathbb{C}} \subset \varPhi^\times . $$

(79)

Any test vector ϕ in Φ can be written as ϕ=U ⁻¹ ϕ(E) for any function $\phi(E)\in\mathcal{Z}$.

The Gamow vectors |ψ ^D〉 and |ψ ^G〉 can be defined as functionals on Φ ^×, as follows:

$$ \langle \phi|\psi^D \rangle= \phi^*(z_R) ;\quad \langle \phi|\psi^G \rangle= \phi^*\bigl(z^*_R\bigr) ; \quad \forall \phi\in\varPhi , $$

(80)

where ϕ ^∗(z _R ) and $\phi^{*}(z^{*}_{R})$ are the values on z _R and $z^{*}_{R}$ respectively of the entire function $\phi(E)\in\mathcal{Z}$ with Uϕ=ϕ(E) and the star denotes complex conjugation. Formula (78) and an analysis as given in [18, 19] shows that

$$ e^{-itH}|\psi^D\rangle =e^{-itE_R} e^{-t\varGamma/2} |\psi^D\rangle ;\quad\quad e^{-itH}| \psi^G\rangle =e^{-itE_R} e^{t\varGamma/2} |\psi^G \rangle , \quad t\in\mathbb{R} . $$

(81)

(v) Let us consider the following Gelfand triplet:

$$ \varPhi\otimes \varPhi\subset {\mathcal{H}}\otimes {\mathcal{H}} \subset \varPhi^\times \otimes {^\times\varPhi} , $$

(82)

where ${\mathcal{H}} =L^{2}({\mathbb{R}}^{+})\oplus{\mathbb{C}}$, Φ ^× (^× Φ) is the space of all continuous antilinear (linear) functionals on Φ. Then, the dyad

$$ |\psi^D\rangle \langle\psi^D|\in \varPhi^\times \otimes {^\times\varPhi} . $$

(83)

satisfies (42) for all values of $t\in{\mathbb{R}}$.

This result is obvious. In fact, if ϕ⊗φ∈Φ⊗Φ, we have for the Liouville operator L=H⊗I−I⊗H:

$$ \langle \phi\otimes\varphi | e^{-itL} |\psi^D \rangle \langle\psi^D| \rangle= e^{-t\varGamma} \langle \phi\otimes\varphi| |\psi^D \rangle \langle \psi^D| \rangle , \quad t\in{\mathbb{R}} . $$

(84)

Since the space spanned by vectors in factorizable form ϕ⊗φ is dense in Φ⊗Φ [46], we have for all values of time:

$$ e^{-itL}|\psi^D\rangle \langle \psi^D|=e^{-t\varGamma} |\psi^D \rangle \langle \psi^D| ,\quad\quad e^{-itL}|\psi^G \rangle \langle\psi^G|=e^{t\varGamma} |\psi^G \rangle \langle\psi^G| . $$

(85)

We see that ρ=|ψ ^D〉〈ψ ^D| decays exponentially to the future and ρ=|ψ ^G〉〈ψ ^G| decays exponentially to the past for all values of time. Note that the latter coincides with the Prigogine’s entropy operator (51). The price that we had to pay for this picture is to give up the formulation based in the Hardy spaces (compare to the results given in [40]) and therefore the standard presentation of the TAQM.