# 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

## References

- Asorey, M.; Ibort, A.; Marmo, G. Global theory of quantum boundary conditions and topology change. Int. J. Mod. Phys. A
**2005**, 20, 1001. [Google Scholar] [CrossRef] - Grubb, G. A characterization of the non-local boundary value problems associated with an elliptic operator. Ann. Sc. Norm. Super. Pisa.
**1968**, 22, 425. [Google Scholar] - Asorey, M.; Garcia-Alvarez, D.; Munoz-Castaneda, J.M. Casimir Effect and Global Theory of Boundary Conditions. J. Phys. A Math. Gen.
**2006**, 39, 6127. [Google Scholar] [CrossRef] - Asorey, M.; Garcia-Alvarez, D.; Munoz-Castaneda, J.M. Vacuum Energy and Renormalization on the Edge. J. Phys. A Math. Theor.
**2007**, 40, 6767. [Google Scholar] [CrossRef] - Asorey, M.; Munoz-Castaneda, J.M. Vacuum Boundary Effects. J. Phys. A Math. Theor.
**2008**, 41, 304004. [Google Scholar] [CrossRef] - Asorey, M.; Munoz-Castaneda, J.M. Attractive and Repulsive Casimir Vacuum Energy with General Boundary Conditions. Nucl. Phys. B
**2013**, 874, 852. [Google Scholar] [CrossRef] - Silva, H.O.; Farina, C. A Simple model for the dynamical Casimir effect for a static mirror with time-dependent properties. Phys. Rev. D
**2011**, 84, 045003. [Google Scholar] [CrossRef] - Milton, K.A. The Casimir Effect: Physical Manifestations of Zero-Point Energy; World Sci.: Singapore, 2001. [Google Scholar]
- Milonni, P.W. The Quantum Vacuum; Academic Press: San Diego, CA, USA, 1994. [Google Scholar]
- Ibort, A.; Lledó, F.; Pérez-Pardo, J.M. On Self-Adjoint Extensions and Symmetries in Quantum Mechanics. Ann. Henri Poincaré
**2015**, 1460, 15–54. [Google Scholar] [CrossRef] - Ibort, A. Three lectures on global boundary conditions and the theory of self-adjoint extensions of the covariant Laplace-Beltrami and Dirac operators on Riemannian manifolds with boundary. AIP Conf. Proc.
**2012**, 16, 2367–2397. [Google Scholar] - Pérez-Pardo, J.M.; Barbero-Linán, M.; Ibort, A. Boundary dynamics and topology change in quantum mechanics. Int. J. Geom. Methods Mod. Phys.
**2015**, 12, 1560011. [Google Scholar] [CrossRef] - Facchi, P.; Garnero, G.; Ligabo, L. Quantum cavities with alternating boundary conditions. J. Phys. A Math. Theor.
**2018**, 51, 105301. [Google Scholar] [CrossRef] [Green Version] - Facchi, P.; Garnero, G.; Marmo, G.; Samuel, J.; Sinha, S. Boundaries without boundaries. Ann. Phys.
**2018**, 394, 139. [Google Scholar] [CrossRef] - Asorey, M.; Balachandran, A.P.; Pérez-Pardo, J.M. Edge states at phase boundaries and their stability. Rev. Math. Phys.
**2016**, 28, 1650020. [Google Scholar] [CrossRef] [Green Version] - Chodos, A.; Jaffe, R.L.; Johnson, K.; Thorn, C.B.; Weisskopf, V.F. New Extended Model of Hadrons. Phys. Rev. D
**1974**, 9, 3471. [Google Scholar] [CrossRef] - Johnson, K. The M.I.T. Bag Model. Acta Phys. Polon. B
**1975**, 6, 865. [Google Scholar] - Milton, K.A. Fermionic Casimir Stress on A Spherical Bag. Ann. Phys.
**1983**, 150, 432. [Google Scholar] [CrossRef] - Elizalde, E.; Bordag, M.; Kirsten, K. Casimir energy for a massive fermionic quantum field with a spherical boundary. J. Phys. A
**1998**, 31, 1743. [Google Scholar] [CrossRef] - Asorey, M.; Ibort, A.; Marmo, G. The topology and geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators. Int. J. Geom. Methods Mod. Phys.
**2015**, 12, 1561007. [Google Scholar] [CrossRef] - Balachandran, A.P.; Bimonte, G.; Marmo, G.; Simoni, A. Topology change and quantum physics. Nucl. Phys. B
**1995**, 446, 299. [Google Scholar] [CrossRef] - Mamaev, S.G.; Trunov, N.N. Vacuum expectation values of the energy-momentum tensor of quantized fields on manifolds with different topologies and geometries. III. Sov. Phys.
**1980**, 23, 551. [Google Scholar] [CrossRef] - Vassilevich, D.V. Heat kernel expansion: User’s manual. Phys. Rep.
**2003**, 388, 279–360. [Google Scholar] [CrossRef] - Blau, S.K.; Visser, M.; Wipf, A. Zeta functions and the Casimir energy. Nucl. Phys. B
**1988**, 310, 163–180. [Google Scholar] [CrossRef] [Green Version] - Kirsten, K. Spectral Functions in Mathematics and Physics; Chapman and Hall/CRC: Boca Raton, FL, USA, 2001. [Google Scholar]
- De Paola, R.D.M.; Rodrigues, R.B.; Svaiter, N.F. Casimir Energy of Massless Fermions in the Slab-bag. Mod. Phys. Lett. A
**1999**, 14, 2353. [Google Scholar] [CrossRef] - Elizalde, E.; Santos, F.C.; Tort, A.C. The Casimir Energy of a Massive Fermionic Field Confined in a (d + 1)-dimensional Slab-Bag. Int. J. Mod. Phys. A
**2003**, 18, 1761. [Google Scholar] [CrossRef] - Kenneth, O.; Klich, I. Opposites attract: A Theorem about the Casimir force. Phys. Rev. Lett.
**2006**, 97, 160401. [Google Scholar] [CrossRef] - Mateos-Guilarte, J.; Munoz-Castaneda, J.M.; Pirozhenko, I.; Santamaria-Sanz, L. One-dimensional scattering of fermions on δ-impurities. arXiv
**2019**, arXiv:1903.05568v2. [Google Scholar] - Asorey, M.; Clemente-Gallardo, J.; Munoz-Castaneda, J.M. Path integrals and boundary conditions. J. Phys. Conf. Ser.
**2007**, 87, 012004. [Google Scholar] [CrossRef] - Sundberg, P.; Jaffe, R.L. The Casimir effect for fermions in one-dimension. Ann. Phys. (N. Y.)
**2004**, 309, 442. [Google Scholar] [CrossRef] - Asorey, M.; Ibort, A.; Marmo, G. Path integrals and boundary conditions. In Proceedings of the Meeting on Fundamental Physics ‘A. Galindo’, Madrid, Spain, 26 November 2004. [Google Scholar]

**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