# An Algebraic Model for Quantum Unstable States

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## Abstract

**:**

## 1. Introduction

- (i)
- $f\left(I\right)=1$;
- (ii)
- for any $A\in \mathcal{A}$, $f\left({A}^{\u2020}A\right)\ge 0$ (positivity);
- (iii)
- continuity with respect to the topologies on $\mathcal{A}$ and $\mathbb{C}$.

## 2. Rigged Hilbert Spaces: An Overview

**Definition**

**1.**

- (i)
- $\mathcal{H}$ is a separable Hilbert space of infinite dimension.
- (ii)
- Φ is a dense subspace of $\mathcal{H}$ having its own topology under the condition that the canonical injection $i:\mathsf{\Phi}\u27fc\mathcal{H}$, $i\left(\phi \right)=\phi $, $\forall \phantom{\rule{0.166667em}{0ex}}\phi \in \mathsf{\Phi}$, be continuous.
- (iii)
- The space${\mathsf{\Phi}}^{\times}$is the antidual space of Φ, which is the set of all continuous antlinear functionals on Φ.

**Definition**

**2.**

**Theorem**

**1**

**(The Gelfand–Maurin Theorem).**

- (i)
- The space $\mathsf{\Phi}\subset \mathcal{D}\left(A\right)$, $A\mathsf{\Phi}\subset \mathsf{\Phi}$ (which means that for any $\phi \in \mathsf{\Phi}$, $A\phi \in \mathsf{\Phi}$) and A is continuous on Φ with the own topology on Φ.
- (ii)
- For almost all $\lambda \in {\mathbb{R}}^{+}$ (with respect to the Lebesgue measure on ${\mathbb{R}}^{+}$), there exists a functional $|\lambda \rangle \in {\mathsf{\Phi}}^{\times}$ such that $A|\lambda \rangle =\lambda \phantom{\rule{0.166667em}{0ex}}|\lambda \rangle $, where A has been extended to ${\mathsf{\Phi}}^{\times}$ using the duality Formula (3). Functionals in the set ${\left\{\right|\lambda \rangle \}}_{\lambda \in [0,\infty )}$ are often called generalized eigenvectors of A.
- (iii)
- For any $\phi ,\psi \in \mathsf{\Phi}$, one has$$\langle \psi |A\phi \rangle ={\int}_{0}^{\infty}\lambda \phantom{\rule{0.166667em}{0ex}}\langle \psi |\lambda \rangle \langle \lambda |\phi \rangle \phantom{\rule{0.166667em}{0ex}}d\lambda \phantom{\rule{0.166667em}{0ex}},$$
- (iv)
- Each function $\langle \lambda |\phi \rangle $, with $\lambda \in [0,\infty )$, is square integrable, and therefore, belongs to the Hilbert space ${L}^{2}\left({\mathbb{R}}^{+}\right)$.
- (v)
- The mapping $U:\mathcal{H}\u27fc{L}^{2}\left({\mathbb{R}}^{+}\right)$ that assigns to each $\phi \in \mathsf{\Phi}$ a $\langle \lambda |\phi \rangle \in {L}^{2}\left({\mathbb{R}}^{+}\right)$ is unitary, so that the norms of the vector φ in $\mathcal{H}$ and $\langle \lambda |\phi \rangle $ in ${L}^{2}\left({\mathbb{R}}^{+}\right)$ are identical.

**Remark**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

**Remark**

**2.**

## 3. The Model and Gamow States

- Pure statesThe state ${\rho}_{\pm}$ is a pure state if there exists a function $\psi \left(E\right)$ such that ${\rho}_{E}\equiv {\left|\psi \left(E\right)\right|}^{2}$ and ${\rho}_{E{E}^{\prime}}\equiv {\psi}^{*}\left(E\right)\psi \left({E}^{\prime}\right)$. If this were the case, then, ${\rho}_{EE}\equiv {\rho}_{E}$.
- MixturesA state ${\rho}_{\pm}$ is a mixture if it is not a pure state, but yet ${\rho}_{EE}\equiv {\rho}_{E}$. This is the typical situation that arises with quantum mixed states represented as trace class operators.
- Singular diagonal statesA state ${\rho}_{\pm}$ is singular diagonal if and only if ${\rho}_{E}\ne {\rho}_{EE}$. These states are quantum states far from equilibrium [72,73]. It is worthy to mention that the origin of this formalism comes from a paper that intended to accommodate these states within a standard quantum formalism [60].

#### Gamow States

- Positivity: It means that for any ${O}_{+}\in {\mathcal{A}}_{+}$, $\left({\rho}_{D}\right|{O}_{+}^{\u2020}\phantom{\rule{0.166667em}{0ex}}{O}_{+})\ge 0$. This property is indeed satisfied. Taking into account (11) and (14), one has that$$\left({\rho}_{D}\right|{O}_{+}^{\u2020}\phantom{\rule{0.166667em}{0ex}}{O}_{+})=|{O}_{{E}_{R}}{|}^{2}+{\left|{O}_{{z}_{R}\phantom{\rule{0.166667em}{0ex}}{z}_{R}^{*}}\right|}^{2}\ge 0\phantom{\rule{0.166667em}{0ex}}.$$
- Normalization. It means that $\left({\rho}_{D}\right|{I}_{+})=1$. Proving this fact is trivial.
- Continuity. Although this requirement is not essential, we may endowed the algebras ${\mathcal{A}}_{\pm}$ with locally convex topologies, so that this property is satisfied for ${\rho}_{D}$. See Section 4.

## 4. Mathematical Details

- (i)
- They are Schwartz functions with support on $\mathbb{R}$ [77].
- (ii)
- They are entire analytic functions, which implies that their support is the whole $\mathbb{R}$ [78].
- (iii)
- They are Hardy functions on the lower half plane [79] (see Appendix A).

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Hardy Functions on a Half Plane

- The set of Hardy functions either on the upper half or on the lower half plane is a vector space over the field of complex numbers. Henceforth, we shall denote these spaces as ${\mathcal{H}}_{+}$ and ${\mathcal{H}}_{-}$, respectively.
- Let ${f}_{+}\left(z\right)\equiv {f}_{+}(x+iy)$ be a Hardy function on the upper half plane. Then, for almost all $x\in \mathbb{R}$, with respect to the Lebesgue measure (a.e.), the limit$${f}_{+}\left(x\right):=\underset{y\to 0}{\mathrm{lim}}{f}_{+}(x+iy)\phantom{\rule{0.166667em}{0ex}},$$$${\int}_{-\infty}^{\infty}{\left|{f}_{+}\left(x\right)\right|}^{2}\phantom{\rule{0.166667em}{0ex}}dx<\infty \phantom{\rule{0.166667em}{0ex}}.$$The same result is valid for ${f}_{-}\left(z\right)\equiv {f}_{-}(x-iy)$. For any ${f}_{\pm}\left(z\right)\in {\mathcal{H}}_{\pm}$, their limit functions are a.e. unique.
- Let ${f}_{\pm}\left(x\right)$ be the boundary value function of the Hardy function ${f}_{\pm}\left(z\right)\in {\mathcal{H}}_{+}$. This boundary value function can be used to obtain the values of ${f}_{\pm}\left(z\right)$, for all $z\in {\mathbb{C}}^{\pm}$, by means of the Titchmarsh formula:$${f}_{\pm}\left(z\right)=\pm \phantom{\rule{0.166667em}{0ex}}\frac{1}{2\pi i}{\int}_{-\infty}^{\infty}\frac{{f}_{\pm}\left(x\right)}{x-z}\phantom{\rule{0.166667em}{0ex}}dx\phantom{\rule{0.166667em}{0ex}}.$$In summary, given ${f}_{\pm}\left(z\right)\in {\mathcal{H}}_{\pm}$, we obtain its boundary value function, which is a complex function defined a.e. on the real line. This function is a.e. unique. Conversely, if we have the boundary value function of a function either in ${\mathcal{H}}_{+}$ or in ${\mathcal{H}}_{-}$, we can recover all values of this function on its half plane. Thus, the relation between a Hardy function and its boundary value function is one to one and onto, so that we may somehow identify the boundary value function with its Hardy function. We shall proceed with this identification in the sequel, unless otherwise stated.
- If ${f}_{\pm}\left(x\right)$ is the boundary value function of a Hardy function ${f}_{\pm}\left(z\right)\in {\mathcal{H}}_{\pm}$, it is square integarble, i.e.,$${\int}_{-\infty}^{\infty}{\left|{f}_{\pm}\left(x\right)\right|}^{2}\phantom{\rule{0.166667em}{0ex}}dx<\infty \phantom{\rule{0.166667em}{0ex}}.$$After the identification of the boundary value function with the original Hardy function, we may say that all Hardy functions on a half plane are square integrable, so that ${\mathcal{H}}_{\pm}\subset {L}^{2}\left(\mathbb{R}\right)$.
- Now, the point is: Being given a square integrable function $f\left(x\right)\in {L}^{2}\left(\mathbb{R}\right)$, how we may determine whether this function is the boundary value function of a Hardy function either on the upper or on the lower half planes? The answer is given by the Paley-Wienner theorem, which states the following:The square integrable function ${f}_{\pm}\left(x\right)$ is in ${\mathcal{H}}_{\pm}$ if and only if its inverse Fourier transform is in ${L}^{2}\left({\mathbb{R}}^{\mp}\right)$, where ${L}^{2}\left({\mathbb{R}}^{+}\right)$ is the Hilbert space of square integrable Lebesgue function on the half line ${\mathbb{R}}^{+}$. Similar definition for ${L}^{2}\left({\mathbb{R}}^{-}\right)$. Moreover, if $\mathcal{F}$ represents the Fourier transform operation, one may conclude that$$\mathcal{F}\left[{L}^{2}\left({\mathbb{R}}^{\mp}\right)\right]\equiv {\mathcal{H}}_{\pm}\phantom{\rule{0.166667em}{0ex}}.$$
- The Fourier transform is a unitary mapping on ${L}^{2}\left(\mathbb{R}\right)$. Since ${L}^{2}\left(\mathbb{R}\right)={L}^{2}\left({\mathbb{R}}^{+}\right)\oplus {L}^{2}\left({\mathbb{R}}^{-}\right)$, then Equation (A7) and the properties of the Fourier transform imply that$${L}^{2}\left(\mathbb{R}\right)={\mathcal{H}}_{+}\oplus {\mathcal{H}}_{-}\phantom{\rule{0.166667em}{0ex}}.$$This means that any Lebesgue square integrable function may be decomposed into an orthogonal sum of a Hardy function on the upper half plane plus a Hardy function on the lower half plane.
- The Fourier transform of a Schwartz function is also a Schwartz function. Therefore, the Fourier transform of a Schwartz function supported on ${\mathbb{R}}^{+}$, which means zero outside ${\mathbb{R}}^{+}$, is a Hardy function on the upper half plane for which its boundary value function is a Schwartz function on the whole real line. Analogously, the Fourier transform of a Schwartz function supported on ${\mathbb{R}}^{-}$ is a Hardy function on the lower half plane for which the boundary value function is a Schwartz function supported on the whole $\mathbb{R}$.
- Another Paley-Wiener theorem [78] states that the Fourier transform of a Schwartz function with compact support is entire analytic. Thus, the Fourier transforms of Schwartz functions supported either on ${\mathbb{R}}^{+}$ or in ${\mathbb{R}}^{-}$ are Hardy functions on the corresponding half plane and entire analytic.
- Let $t>0$, ${f}_{+}\left(z\right)\in {\mathcal{H}}_{+}$ and consider the function ${e}^{itz}\phantom{\rule{0.166667em}{0ex}}{f}_{+}\left(z\right)$. Since ${f}_{+}\left(z\right)$ is analytic on the upper half plane, so is ${e}^{itz}\phantom{\rule{0.166667em}{0ex}}{f}_{+}\left(z\right)$. Let us prove that (A1) holds for $t>0$, so that ${e}^{itz}\phantom{\rule{0.166667em}{0ex}}{f}_{+}\left(z\right)$ is in ${\mathcal{H}}_{+}$, for $t>0$. In fact,$${\int}_{-\infty}^{\infty}|{e}^{itz}\phantom{\rule{0.166667em}{0ex}}{f}_{+}{\left(z\right)|}^{2}\phantom{\rule{0.166667em}{0ex}}dx={\int}_{-\infty}^{\infty}{e}^{-ty}\phantom{\rule{0.166667em}{0ex}}|{f}_{+}{(x+iy)|}^{2}\phantom{\rule{0.166667em}{0ex}}dx\le {\int}_{-\infty}^{\infty}{\left|{f}_{+}(x+iy)\right|}^{2}\phantom{\rule{0.166667em}{0ex}}dx\phantom{\rule{0.166667em}{0ex}},$$

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Fortin, S.; Gadella, M.; Holik, F.; Jorge, J.P.; Losada, M.
An Algebraic Model for Quantum Unstable States. *Mathematics* **2022**, *10*, 4562.
https://doi.org/10.3390/math10234562

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Fortin S, Gadella M, Holik F, Jorge JP, Losada M.
An Algebraic Model for Quantum Unstable States. *Mathematics*. 2022; 10(23):4562.
https://doi.org/10.3390/math10234562

**Chicago/Turabian Style**

Fortin, Sebastian, Manuel Gadella, Federico Holik, Juan Pablo Jorge, and Marcelo Losada.
2022. "An Algebraic Model for Quantum Unstable States" *Mathematics* 10, no. 23: 4562.
https://doi.org/10.3390/math10234562