BEYOND PROXIMITY FORCE APPROXIMATION IN THE CASIMIR EFFECT

2009 ◽  
Vol 24 (08n09) ◽  
pp. 1743-1747 ◽  
Author(s):  
M. BORDAG ◽  
V. NIKOLAEV
Keyword(s):  
Boundary Conditions ◽  
Casimir Force ◽  
Dirichlet Boundary ◽  
Casimir Effect ◽  
Neumann Boundary ◽  
Numerical Results ◽  
Dirichlet Boundary Conditions ◽  
Proximity Force Approximation ◽  
Interacting Surfaces ◽  
General Appearance

We compare the analytical and numerical results for the Casimir force for the configuration of a plane and a cylinder in front of a plane. While for Dirichlet boundary conditions on both, plane and sphere or cylinder, agreement is found, for Neumann boundary conditions on either the plane or one of the two, cylinder or sphere, disagreement is found. This holds, for a sphere, also for different boundary conditions on the interacting surfaces. From recent, new numerical results for the cylinder, a general appearance of logarithmic contributions beyond PFA can be predicted.

scholarly journals The Casimir effect in the presence of infrared transparency

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Max Warkentin
Keyword(s):  
Boundary Conditions ◽  
Casimir Force ◽  
Dirichlet Boundary ◽  
Casimir Effect ◽  
Dirichlet Boundary Conditions ◽  
Quantum Field ◽  
Codimension One ◽  
Curvature Term ◽  
Parallel Surfaces ◽  
Arbitrary Dimensions

Abstract We revisit the Casimir effect perceived by two surfaces in the presence of infrared (IR) transparency. To address this problem, we study a model, where such a phenomenon naturally arises: the DGP model with two parallel 3-branes, each endowed with a localized curvature term. In that model, the ultraviolet modes of the 5-dimensional graviton are suppressed on the branes, while the IR modes can penetrate them freely. First, we find that the DGP branes act as “effective” (momentum-dependent) boundary conditions for the gravitational field, so that the (gravitational) Casimir force between them emerges. Second, we discover that the presence of an IR transparency region for the discrete modes modifies the standard Casimir force — as derived for ideal Dirichlet boundary conditions — in two competing ways: i) The exclusion of soft modes from the discrete spectrum leads to an increase of the Casimir force. ii) The non-ideal nature of the boundary conditions gives rise to a “leakage” of hard modes. As a result of i) and ii), the Casimir force becomes weaker. Since the derivation of this result involves only the localized kinetic terms of a quantum field on parallel surfaces (with codimension one), the derived Casimir force is expected to be present in a variety of setups in arbitrary dimensions.

scholarly journals SCALAR CASIMIR EFFECT BETWEEN TWO CONCENTRIC D-DIMENSIONAL SPHERES

2012 ◽  
Vol 27 (18) ◽  
pp. 1250094 ◽  
Author(s):  
MUSTAFA ÖZCAN
Keyword(s):  
Scalar Field ◽  
Boundary Conditions ◽  
Complex Plane ◽  
Casimir Force ◽  
Dirichlet Boundary ◽  
Casimir Effect ◽  
Casimir Energy ◽  
Dirichlet Boundary Conditions ◽  
Massless Scalar ◽  
Massless Scalar Field

The Casimir energy for a massless scalar field between the closely spaced two concentric D-dimensional (for D>3) spheres is calculated by using the mode summation with contour integration in the complex plane of eigenfrequencies and the generalized Abel–Plana formula for evenly spaced eigenfrequency at large argument. The sign of the Casimir energy between closely spaced two concentric D-dimensional spheres for a massless scalar field satisfying the Dirichlet boundary conditions is strictly negative. The Casimir energy between (D-1)-dimensional surfaces, close to each other is regarded as interesting both by itself and as the key to describing of stability of the attractive Casimir force.

scholarly journals The Casimir effect for thick pistons

2016 ◽  
Vol 31 (06) ◽  
pp. 1650012
Author(s):  
Guglielmo Fucci
Keyword(s):  
Boundary Conditions ◽  
Zeta Function ◽  
Regularization Method ◽  
Casimir Force ◽  
Casimir Effect ◽  
Casimir Energy ◽  
Numerical Results ◽  
Spectral Zeta Function

In this work, we analyze the Casimir energy and force for a thick piston configuration. This study is performed by utilizing the spectral zeta function regularization method. The results we obtain for the Casimir energy and force depend explicitly on the parameters that describe the general self-adjoint boundary conditions imposed. Numerical results for the Casimir force are provided for specific types of boundary conditions and are also compared to the corresponding force on an infinitely thin piston.

On the Number of Solutions to Semilinear Boundary Value Problems

2004 ◽  
Vol 4 (3) ◽  
Author(s):  
Markus Kunze ◽  
Rafael Ortega
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Upper Bound ◽  
Dirichlet Boundary ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Number Of Solutions ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems

AbstractWe consider semilinear elliptic problems of the form Δu + g(u) = f(x) with Neumann boundary conditions or Δu+λ1u+g(u) = f(x) with Dirichlet boundary conditions, and we derive conditions on g and f under which an upper bound on the number of solutions can be obtained.

scholarly journals Non-universal Casimir Forces at Approach to Bose–Einstein Condensation of an Ideal Gas: Effect of Dirichlet Boundary Conditions

2020 ◽  
Vol 181 (3) ◽  
pp. 944-951
Author(s):  
M. Napiórkowski ◽  
J. Piasecki ◽  
J. W. Turner
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Neumann Boundary ◽  
Ideal Gas ◽  
Einstein Condensate ◽  
Dirichlet Boundary Conditions ◽  
Einstein Condensation ◽  
Casimir Forces ◽  
Leading Term ◽  
Bose Einstein

Abstract We analyze the Casimir forces for an ideal Bose gas enclosed between two infinite parallel walls separated by the distance D. The walls are characterized by the Dirichlet boundary conditions. We show that if the thermodynamic state with Bose–Einstein condensate present is correctly approached along the path pertinent to the Dirichlet b.c. then the leading term describing the large-distance decay of thermal Casimir force between the walls is $$\sim 1/D^{2}$$ ∼ 1 / D 2 with a non-universal amplitude. The next order correction is $$\sim \ln D/D^3$$ ∼ ln D / D 3 . These observations remain in contrast with the decay law for both the periodic and Neumann boundary conditions for which the leading term is $$\sim 1/D^3$$ ∼ 1 / D 3 with a universal amplitude. We associate this discrepancy with the D-dependent positive value of the one-particle ground state energy in the case of Dirichlet boundary conditions.

scholarly journals Abelian mirror symmetry of $$ \mathcal{N} $$ = (2, 2) boundary conditions

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Tadashi Okazaki
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Mirror Symmetry ◽  
Dirichlet Boundary ◽  
Triangular Matrix ◽  
Neumann Boundary ◽  
Dirichlet Boundary Conditions ◽  
Verma Modules ◽  
Abelian Gauge ◽  
Mirror Theory

Abstract We evaluate half-indices of $$ \mathcal{N} $$ N = (2, 2) half-BPS boundary conditions in 3d $$ \mathcal{N} $$ N = 4 supersymmetric Abelian gauge theories. We confirm that the Neumann boundary condition is dual to the generic Dirichlet boundary condition for its mirror theory as the half-indices perfectly match with each other. We find that a naive mirror symmetry between the exceptional Dirichlet boundary conditions defining the Verma modules of the quantum Coulomb and Higgs branch algebras does not always hold. The triangular matrix obtained from the elliptic stable envelope describes the precise mirror transformation of a collection of half-indices for the exceptional Dirichlet boundary conditions.

scholarly journals On the abundance of supersymmetric strings in AdS 3 x S 3 x S 3 x S 1 describing BPS line operators

2021 ◽  
Author(s):  
Diego H Correa ◽  
Victor I Giraldo-Rivera ◽  
Martín Lagares
Keyword(s):  
Fixed Point ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Neumann Boundary ◽  
Open String ◽  
Neumann Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Open Strings ◽  
Straight Line ◽  
Fermionic Fields

Abstract We study supersymmetric open strings in type IIB $AdS_3 \times S^3 \times S^3 \times S^1$ with mixed R-R and NS-NS fields. We focus on strings ending along a straight line at the boundary of $AdS_3$, which can be interpreted as line operators in a dual CFT$_2$. We study both classical configurations and quadratic fluctuations around them. We find that strings sitting at a fixed point in $S^3 \times S^3 \times S^1$, i.e. satisfying Dirichlet boundary conditions, are 1/2 BPS. We also show that strings sitting at different points of certain submanifolds of $S^3 \times S^3 \times S^1$ can still share some fraction of the supersymmetry. This allows to define supersymmetric smeared configurations by the superposition of them, which range from 1/2 BPS to 1/8 BPS. In addition to the smeared configurations, there are as well 1/4 BPS and 1/8 BPS strings satisfying Neumann boundary conditions. All these supersymmetric strings are shown to be connected by a network of interpolating BPS boundary conditions. Our study reveals the existence of a rich moduli of supersymmetric open string configurations, for which the appearance of massless fermionic fields in the spectrum of quadratic fluctuations is crucial.

scholarly journals Analytical Green’s functions for continuum spectra

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Eugenio Megías ◽  
Mariano Quirós
Keyword(s):  
Boundary Conditions ◽  
Green's Functions ◽  
Dirichlet Boundary ◽  
Green’S Functions ◽  
Neumann Boundary ◽  
Approximate Equation ◽  
New Physics ◽  
Dirichlet Boundary Conditions ◽  
Gauge Bosons ◽  
Linear Dilaton

Abstract Green’s functions with continuum spectra are a way of avoiding the strong bounds on new physics from the absence of new narrow resonances in experimental data. We model such a situation with a five-dimensional model with two branes along the extra dimension z, the ultraviolet (UV) and the infrared (IR) one, such that the metric between the UV and the IR brane is AdS5, thus solving the hierarchy problem, and beyond the IR brane the metric is that of a linear dilaton model, which extends to z → ∞. This simplified metric, which can be considered as an approximation of a more complicated (and smooth) one, leads to analytical Green’s functions (with a mass gap mg ∼ TeV and a continuum for s >$$ {m}_g^2 $$ m g 2 ) which could then be easily incorporated in the experimental codes. The theory contains Standard Model gauge bosons in the bulk with Neumann boundary conditions in the UV brane. To cope with electroweak observables the theory is also endowed with an extra custodial gauge symmetry in the bulk, with gauge bosons with Dirichlet boundary conditions in the UV brane, and without zero (massless) modes. All Green’s functions have analytical expressions and exhibit poles in the second Riemann sheet of the complex plane at s = $$ {M}_n^2 $$ M n 2 − iMnΓn, denoting a discrete (infinite) set of broad resonances with masses (Mn) and widths (Γn). For gauge bosons with Neumann or Dirichlet boundary conditions, the masses and widths of resonances satisfy the (approximate) equation s = −4$$ {m}_g^2{\mathcal{W}}_n^2 $$ m g 2 W n 2 [±(1 + i)/4], where $$ \mathcal{W} $$ W n is the n-th branch of the Lambert function.

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