# Field Fluctuations and Casimir Energy of 1D-Fermions

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## Abstract

**:**

## 1. Introduction

## 2. Fundamentals of the Approach

- Particles moving from left to right with $E=\omega $ and momentum k, ${\psi}_{+}(x;k)={e}^{ikx}{u}_{+}\left(k\right)$.
- Particles moving from right to left with $E=\omega $ and momentum k, ${\psi}_{+}(x;-k)={e}^{-ikx}{\gamma}^{0}{u}_{+}\left(k\right)$.
- Anti-particles moving from left to right with $E=\omega $ and momentum k, ${\psi}_{-}(x;-k)={e}^{-ikx}{\gamma}^{0}{u}_{-}\left(k\right)$.
- Anti-particles moving from right to left with $E=\omega $ and momentum k, ${\psi}_{-}(x;k)={e}^{ikx}{u}_{-}\left(k\right)$.

## 3. Spectrum of Fluctuations of the Dirac Field Confined in a Finite Filament

#### 3.1. Construction of the Boundary Spinors and the Boundary Condition

- At $x=-L/2$ the components ${\xi}^{\pm}$ are determined by the eigenvalue equation $(-\alpha )\mathbf{v}=\pm \mathbf{v}$. This means that the ${\mathbf{e}}_{\pm}$ component of $\psi (-L/2)$ contributes to ${\xi}^{\mp}$.
- At $x=L/2$ the components ${\xi}^{\pm}$ are determined by the eigenvalue equation $\alpha \mathbf{v}=\pm \mathbf{v}$. This means that the ${\mathbf{e}}_{\pm}$ component of $\psi (L/2)$ contributes to ${\xi}^{\pm}$.

#### 3.2. Normal Modes

#### 3.3. Localised Edge States

## 4. Casimir Energy of the Dirac Field in 1D

#### 4.1. Casimir Energy of Light Fermions

#### 4.2. Casimir Energy of Heavy Fermions

#### A Remark on the Sign of the Energy for Heavy Fermions

**Theorem**

**1.**

## 5. Conclusions and Outlook

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**In light green, it is shown the region of boundary condition parameters for which edge states exist. Here $\lambda =arcsin(tanh1)$, and $mL=1$.

**Figure 2.**Plot of ${\mathcal{E}}_{0}L$ for massless fermions as a function of the self-adjoint extension parameter $\theta $, according to Equation (47).

**Figure 3.**Plot of $2\mathcal{E}/\left(m{e}^{-2mL}\right)$ according to Equation (55) for $mL=20$, as a function of the self-adjoint parameters $\theta $ and $\eta $. The purple line represents the curve $\mathcal{E}=0$. Regions where $1-2sin\theta cos\eta \phantom{\rule{0.222222em}{0ex}}+{cos}^{2}\eta \phantom{\rule{0.222222em}{0ex}}-{cos}^{2}\theta $ is identically zero are excluded.

**Figure 4.**Plot of $2\mathcal{E}/\left(m{e}^{-2mL}\right)$ according to Equation (55) for $mL=20$, as a function of the self-adjoint extension parameter $\theta $, for different values of the parameter $\eta $. In blue, $\eta =-0.5$; in yellow, $\eta =0$; in green, $\eta =0.5$.

**Figure 5.**Plot of $2\mathcal{E}/\left(m{e}^{-2mL}\right)$ according to Equation (55) for $mL=20$. The blue curve corresponds to $\theta =\eta +\pi $ with $\eta \in [-\pi /2,\pi /2]$, to $\theta =-\eta +2\pi $ with $\eta \in [0,\pi /2]$, and to $\theta =-\eta $ with $\eta \in [-\pi /2,0]$. The orange curve corresponds to $\theta =-\eta +\pi $ with $\eta \in [-\pi /2,\pi /2]$, for $\theta =\eta $ with $\eta \in [0,\pi /2]$, and for $\theta =\eta +2\pi $ with $\eta \in [-\pi /2,0]$.

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**MDPI and ACS Style**

Donaire, M.; Muñoz-Castañeda, J.M.; Nieto, L.M.; Tello-Fraile, M. Field Fluctuations and Casimir Energy of 1D-Fermions. *Symmetry* **2019**, *11*, 643.
https://doi.org/10.3390/sym11050643

**AMA Style**

Donaire M, Muñoz-Castañeda JM, Nieto LM, Tello-Fraile M. Field Fluctuations and Casimir Energy of 1D-Fermions. *Symmetry*. 2019; 11(5):643.
https://doi.org/10.3390/sym11050643

**Chicago/Turabian Style**

Donaire, Manuel, José María Muñoz-Castañeda, Luis Miguel Nieto, and Marcos Tello-Fraile. 2019. "Field Fluctuations and Casimir Energy of 1D-Fermions" *Symmetry* 11, no. 5: 643.
https://doi.org/10.3390/sym11050643