Relative-Zeta and Casimir Energy for a Semitransparent Hyperplane Selecting Transverse Modes

Claudio Cacciapuoti; Davide Fermi; Andrea Posilicano

doi:10.1007/978-3-319-58904-6_5

Advances in Quantum Mechanics pp 71-97 | Cite as

Relative-Zeta and Casimir Energy for a Semitransparent Hyperplane Selecting Transverse Modes

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Claudio Cacciapuoti
Davide Fermi
Andrea Posilicano

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First Online: 03 August 2017

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Part of the Springer INdAM Series book series (SINDAMS, volume 18)

Abstract

We study the relative zeta function for the couple of operators A₀ and A_α, where A₀ is the free unconstrained Laplacian in L²(R^d) (d ≥ 2) and A_α is the singular perturbation of A₀ associated to the presence of a delta interaction supported by a hyperplane. In our setting the operatorial parameter α, which is related to the strength of the perturbation, is of the kind α = α(−Δ_∥), where −Δ_∥ is the free Laplacian in L²(R^d−1). Thus α may depend on the components of the wave vector parallel to the hyperplane; in this sense A_α describes a semitransparent hyperplane selecting transverse modes.

As an application we give an expression for the associated thermal Casimir energy. Whenever α = χ_I(−Δ_∥), where χ_I is the characteristic function of an interval I, the thermal Casimir energy can be explicitly computed.

Keywords

Casimir effect Delta-interactions Finite temperature quantum fields Relative zeta function Zeta regularization

MSC 2010

81Q10 81T55 81T10

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Notes

Acknowledgements

C.C. and D.F. acknowledge the support of the FIR 2013 project “Condensed Matter in Mathematical Physics”, Ministry of University and Research of Italian Republic (code RBFR13WAET).

Appendix: The Case α = const. in Spatial Dimensions d = 3

In order to make connection with the existing literature, in the present appendix we briefly review the computation of the Casimir energy for the 3-dimensional model described in Eq. (51), that is

$$\displaystyle{\alpha (\rho ) =\alpha _{0}\quad \mbox{ for all}\ \rho \in [0,+\infty )\ \mbox{ and some}\ \alpha _{0}> 0\qquad (d = 3)\;.}$$

As a matter of fact, it can be easily checked that all the arguments described in the present manuscript continue to make sense also in this particular case, even though α does not fulfil the required assumptions since it does not have compact support.

First of all, let us notice that the integral representation (20) for the function r₁(z) continues to make sense for any z ∈ C∖[0, +∞) (the integral in the cited equation is trivially seen to remain finite in the present case). Then, by the same arguments of Sect. 3.1 one obtains an expression like Eq. (26) for the relative spectral measure e₁(v); moreover, the term I_α appearing therein (given by Eq. (27) ) can be evaluated explicitly. This allows to infer for the relative spectral measure the expression

$$\displaystyle{ e_{1}(v) = -{ v \over 2\pi ^{2}}\;\arctan \left ({\tilde{\alpha }_{0} \over v}\right )\,\chi _{(0,+\infty )}(v)\;, }$$

(53)

where $\tilde{\alpha }_{0}$ is defined according to Eq. (49). This shows that the map v ↦ e₁(v) is continuous on (0, +∞) and fulfils, for any N ∈ {0, 1, 2, …},

$$\displaystyle{ e_{1}(v) = \left \{\begin{array}{ll} O(v) &\quad \mbox{ for}\ v \rightarrow 0^{+}\;, \\ \sum _{n=0}^{N}{(-1)^{n+1}\,\tilde{\alpha }_{ 0}^{2n+1} \over 2\pi ^{2}(2n + 1)} \;v^{-2n} + O(v^{-2(N+1)})&\quad \mbox{ for}\ v \rightarrow +\infty \;. \\ \end{array} \right. }$$

Next, let us consider the representation (16) of the relative zeta function ζ₁(s) in terms of e₁(v); in view of the above considerations, it appears that the integral in the cited equation is finite for any complex s inside the strip

$$\displaystyle{ \Big\{s \in \mathbf{C}\;\Big\vert \;{1 \over 2} <\mathop{ \mathrm{Re}}\nolimits \,s <1\Big\}\;. }$$

To proceed, notice that for any such s the integral in Eq. (16) can be evaluated explicitly using the expression (53) for e₁(v); one obtains

$$\displaystyle{ \zeta _{1}(s) = -{ \tilde{\alpha }_{0}^{2-2s} \over 8\pi \,(s - 1)\,\cos (\pi s)}\;, }$$

which determines the analytic continuation of s ↦ ζ₁(s) to a function which is meromorphic on the whole complex plane, with simple pole singularities at s = 1 and s = ±1∕2, ±3∕2, … .

Using the above expression for ζ₁(s) and Eqs. (44)–(45) for the thermal Casimir energy $\mathcal{E}(\beta )$, one easily infers the final result (52), that is

$$\displaystyle{\begin{array}{c} \mathcal{E}(\beta ) ={ \tilde{\alpha }_{0}^{3} \over 12\pi ^{2}}\left ({4 \over 3} -\log (2\ell\tilde{\alpha }_{0})\!\right ) -{ 1 \over 2\pi ^{2}}\int _{0}^{+\infty }dv\;{ v^{2} \over e^{\beta v} - 1}\;\arctan \left ({\tilde{\alpha }_{0} \over v}\right )\,. \\ \end{array} }$$

For completeness, let us mention that the above expression can be easily employed to derive the zero temperature expansion (β → +∞) of the Casimir energy $\mathcal{E}(\beta )$.

References

1.
S. Albeverio, G. Cognola, M. Spreafico, S. Zerbini, Singular perturbations with boundary conditions and the Casimir effect in the half space. J. Math. Phys. 51, 063502 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
2.
S. Albeverio, C. Cacciapuoti, M. Spreafico, Relative partition function of Coulomb plus delta interaction, arXiv:1510.04976 [math-ph], 24pp. To appear in the book J. Dittrich, H. Kovařík, A. Laptev (Eds.): Functional Analysis and Operator Theory for Quantum Physics. A Festschrift in Honor of Pavel Exner. (European Mathematical Society Publishing House, Zurich, 2016).Google Scholar
3.
J.-P. Antoine, F. Gesztesy, J. Shabani, Exactly Solvable Models of Sphere Interactions in Quantum Mechanics. J. Phys. A Math. Gen. 20, 3687–3712 (1987)MathSciNetCrossRefGoogle Scholar
4.
M. Bordag, The Casimir Effect 50 Years Later: Proceedings of the Fourth Workshop on Quantum Field Theory Under the Influence of External Conditions, Leipzig, Germany, 14–18 September 1998 (World Scientific, Singapore, 1999)CrossRefzbMATHGoogle Scholar
5.
M. Bordag, D. Hennig, D. Robaschik, Vacuum energy in quantum field theory with external potentials concentrated on planes. J. Phys. A Math. Gen. 25, 4483–4498 (1992)MathSciNetCrossRefGoogle Scholar
6.
M. Bordag, G.L. Klimchitskaya, U. Mohideen, V.M. Mostepanenko, Advances in the Casimir Effect, vol. 145 (Oxford University Press, Oxford, 2009)CrossRefzbMATHGoogle Scholar
7.
H.B. Casimir, On the attraction between two perfectly conducting plates. Proc. Kongl. Nedel. Akad. Wetensch 51, 793–795 (1948)zbMATHGoogle Scholar
8.
I. Cavero-Pelaez, K.A. Milton, K. Kirsten, Local and global Casimir energies for a semitransparent cylindrical shell. J. Phys. A 40, 3607–3632 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
9.
L. Dabrowski, J. Shabani, Finitely many sphere interactions in quantum mechanics: nonseparated boundary conditions. J. Math. Phys. 29, 2241–2244 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
10.
J.S. Dowker, R. Critchley, Effective Lagrangian and energy-momentum tensor in de Sitter space. Phys. Rev. D 13(12), 3224 (1976)Google Scholar
11.
J. Dowker, G. Kennedy, Finite temperature and boundary effects in static space-times. J. Phys. A Math. Gen. 11(5), 895–920 (1978)MathSciNetCrossRefGoogle Scholar
12.
E. Elizalde, S.D. Odintsov, A. Romeo, A.A. Bytsenko, S. Zerbini, Zeta Regularization Techniques with Applications (World Scientific, Singapore, 1994)CrossRefzbMATHGoogle Scholar
13.
D. Fermi, L. Pizzocchero, Local zeta regularization and the Casimir effect. Prog. Theor. Phys. 126(3), 419–434 (2011)CrossRefzbMATHGoogle Scholar
14.
D. Fermi, L. Pizzocchero, Local zeta regularization and the scalar Casimir effect I. A general approach based on integral kernels, 91pp. (2015), arXiv:1505.00711 [math-ph]Google Scholar
15.
D. Fermi, L. Pizzocchero, Local zeta regularization and the scalar Casimir effect II. Some explicitly solvable cases, 54pp. (2015), arXiv:1505.01044 [math-ph]Google Scholar
16.
D. Fermi, L. Pizzocchero, Local zeta regularization and the scalar Casimir effect III. the case with a background harmonic potential. Int. J. Mod. Phys. A 30(35), 1550213 (2015)Google Scholar
17.
D. Fermi, L. Pizzocchero, Local zeta regularization and the scalar Casimir effect IV: the case of a rectangular box. Int. J. Mod. Phys. A 31, 1650003 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
18.
M. Fierz, On the attraction of conducting planes in vacuum. Helv. Phys. Acta 33, 855 (1960)MathSciNetGoogle Scholar
19.
N. Graham, R.L. Jaffe, V. Khemani, M. Quandt, M. Scandurra, H. Weigel, Calculating vacuum energies in renormalizable quantum field theories: a new approach to the Casimir problem. Nucl. Phys. B 645, 49–84 (2002)CrossRefzbMATHGoogle Scholar
20.
S.W. Hawking, Zeta function regularization of path integrals in curved spacetime. Commun. Math. Phys. 55(2), 133–148 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
21.
R.P. Kanwal, Generalized Functions: Theory and Applications (Springer Science + Business Media, Boston, 2011)Google Scholar
22.
N.R. Khusnutdinov, Zeta-function approach to Casimir energy with singular potentials. Phys. Rev. D 73, 025003 (2006)MathSciNetCrossRefGoogle Scholar
23.
S.G. Mamaev, N.N. Trunov, Vacuum expectation values of the energy-momentum tensor of quantized fields on manifolds of different topology and geometry. IV. Sov. Phys. J. 24(2), 171–174 (1981)CrossRefGoogle Scholar
24.
J. Mehra, Temperature correction to the Casimir effect, Physica 37(1), 145–152 (1967)CrossRefGoogle Scholar
25.
K.A. Milton, The Casimir effect: physical manifestations of zero-point energy (World Scientific, Singapore, 2001)CrossRefzbMATHGoogle Scholar
26.
K.A. Milton, Casimir energies and pressures for δ-function potentials. J. Phys. A 37, 6391–6406 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
27.
S. Minakshisundaram, A generalization of Epstein zeta functions. Can. J. Math. 1, 320–327 (1949)CrossRefzbMATHGoogle Scholar
28.
S. Minakshisundaram, A. Pleijel, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds. Canad. J. Math. 1, 242–256 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
29.
W. Müller, Relative zeta functions, relative determinants and scattering theory. Commun. Math. Phys. 192, 309–347 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
30.
J. Munoz-Castaneda, J.M. Guilarte, A.M. Mosquera, Quantum vacuum energies and Casimir forces between partially transparent δ-function plates. Phys. Rev. D 87, 105020 (2013)CrossRefGoogle Scholar
31.
G. Ortenzi, M. Spreafico, Zeta function regularization for a scalar field in a compact domain. J. Phys. A 37, 11499–11518 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
32.
A. Posilicano, A Krein-like formula for singular perturbations of self-adjoint operators and applications. J. Funct. Anal. 183(1), 109–147 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
33.
D.B. Ray, I.M. Singer, R-torsion and the Laplacian on Riemannian manifolds. Adv. Math. 7(2), 145–210 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
34.
R.T. Seeley, Complex powers of an elliptic operator, in Proceedings of Symposia in Pure Mathematics, vol. 10 (American Mathematical Society, Providence, 1967), pp. 288–307Google Scholar
35.
J. Shabani, Finitely many δ interactions with supports on concentric spheres. J. Math. Phys. 29, 660–664 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
36.
M. Spreafico, S. Zerbini, Finite temperature quantum field theory on noncompact domains and application to delta interactions. Rep. Math. Phys. 63(1), 163–177 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
37.
R.M. Wald, On the Euclidean approach to quantum field theory in curved spacetime. Commun. Math. Phys. 70(3), 221–242 (1979)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

Claudio Cacciapuoti
- 1
Email author
Davide Fermi
- 2
Andrea Posilicano
- 1

1.DiSAT, Sezione di MatematicaUniversità dell’InsubriaComoItaly
2.Dipartimento di MatematicaUniversità di MilanoMilanoItaly

Relative-Zeta and Casimir Energy for a Semitransparent Hyperplane Selecting Transverse Modes

Abstract

Keywords

MSC 2010

Notes

Acknowledgements

References

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