pith. sign in

arxiv: 2604.17583 · v1 · submitted 2026-04-19 · ✦ hep-th · gr-qc· quant-ph

Electromagnetic Wightman functions and vacuum densities for a brane intersecting the AdS boundary

A. A. Saharian , R. M. Avagyan , V. F. Manukyan This is my paper

Pith reviewed 2026-05-10 05:22 UTC · model grok-4.3

classification ✦ hep-th gr-qcquant-ph
keywords AdS spacetimebraneelectromagnetic vacuumWightman functionsvacuum expectation valuesboundary conditionsenergy-momentum tensor
0
0 comments X

The pith

A brane intersecting the AdS boundary induces nonzero vacuum expectation values for the electromagnetic energy-momentum tensor, with opposite signs for the two boundary conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper computes the combined influence of a brane that meets the boundary of anti-de Sitter space and the curved background itself on the quantum electromagnetic field. It extracts the brane-induced additions to the Wightman functions for the vector potential and the field tensor under two types of boundary conditions that generalize the perfect electric and perfect magnetic conditions. The resulting vacuum expectation values for the squares of the electric and magnetic fields and for the energy-momentum tensor are obtained in closed form; these quantities depend on distance from the brane and flip sign when the boundary condition is switched from one type to the other. The calculation also reveals a nonzero off-diagonal component in the vacuum stress tensor and shows that this tensor remains nonzero throughout the space, in contrast to the vanishing result for a planar boundary in flat Minkowski spacetime.

Core claim

The brane-induced contributions to the electromagnetic Wightman functions are derived explicitly in terms of elementary functions. These contributions produce vacuum expectation values whose signs for the electric-field square and magnetic-field square are opposite under the two boundary conditions. For the perfect-magnetic condition the electric contribution is negative while the magnetic contribution is positive. The vacuum expectation value of the energy-momentum tensor acquires a nonzero off-diagonal component, the energy density is positive for the perfect-magnetic condition, and both normal and parallel stresses change sign with distance from the brane. Unlike the Minkowski case, the V

What carries the argument

The brane-induced parts of the Wightman functions for the electromagnetic vector potential and field tensor, obtained by solving the wave equation in AdS geometry subject to the intersecting-brane boundary conditions.

If this is right

  • The vacuum energy density remains positive for the perfect-magnetic boundary condition.
  • Both normal and parallel stresses in the energy-momentum tensor change sign as a function of distance from the brane.
  • The vacuum expectation values are reproduced by a scalar field with negative effective mass squared fixed by the AdS radius.
  • The vacuum energy-momentum tensor stays nonzero throughout the (3+1)-dimensional AdS bulk.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The non-vanishing of the tensor in AdS but not in Minkowski space suggests that curvature alone can source vacuum stresses even with boundaries present.
  • The sign reversal between the two boundary conditions may translate into different effective forces on the brane itself.
  • The closed-form Wightman functions could be used to compute transition rates or entanglement measures for fields near the brane.

Load-bearing premise

The chosen boundary conditions on the brane are valid higher-dimensional generalizations of the perfect-electric and perfect-magnetic conditions, and the Wightman functions can be extracted by standard AdS techniques without extra regularization or mode mixing at the intersection point.

What would settle it

An independent calculation that finds a vanishing off-diagonal component in the vacuum energy-momentum tensor or a vanishing tensor everywhere would falsify the central result.

Figures

Figures reproduced from arXiv: 2604.17583 by A. A. Saharian, R. M. Avagyan, V. F. Manukyan.

Figure 1
Figure 1. Figure 1: The ratios of the boundary-induced VEVs in the elec [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Brane-induced diagonal components of the vacuum e [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The ratios of the boundary-induced VEVs in the ener [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Brane-induced contributions to the normal stress [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
read the original abstract

We investigate the combined effects of a brane intersecting the AdS boundary and background gravitational field on the local characteristics of the electromagnetic vacuum. Two types of boundary conditions on the brane are considered, which are higher-dimensional generalizations of the perfect electric (PEC) and perfect magnetic (PMC) boundary conditions in Maxwell's electrodynamics. The brane-induced contributions to the Wightman functions of the vector potential and field tensor are explicitly extracted. Simple expressions in terms of elementary functions are provided. The behavior of the vacuum expectation values (VEVs) is mimicked by a scalar field with a negative effective mass squared determined by the radius of the AdS spacetime. The expectation values of the electric and magnetic fields squares and of the energy-momentum tensor are investigated as local characteristics of the vacuum state. The brane-induced contributions to these VEVs have opposite signs for the PEC and PMC conditions. For the PMC condition, this contribution is negative for the electric field squared and positive for the magnetic field squared. The VEV of the energy-momentum tensor has a nonzero off-diagonal component. The brane-induced vacuum energy density is positive for PMC condition, whereas the normal and parallel stresses change sign as functions of the distance from the brane. Unlike the problem involving a planar boundary in the Minkowski bulk, the vacuum energy-momentum tensor does not vanish in (3+1)-dimensional AdS spacetime.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper computes the brane-induced contributions to the electromagnetic Wightman functions for the vector potential and field strength in (3+1)-dimensional AdS spacetime with a brane intersecting the AdS boundary. It considers higher-dimensional generalizations of PEC and PMC boundary conditions, extracts explicit elementary-function expressions for these contributions, and uses them to evaluate the VEVs of E², B², and the energy-momentum tensor. The results are claimed to be mimicked by a negative-mass-squared scalar field, to exhibit opposite signs for PEC versus PMC, to include a nonzero off-diagonal EMT component, and to yield a non-vanishing EMT (unlike the Minkowski case).

Significance. If the explicit extractions and VEV computations are free of artifacts, the work provides concrete, usable expressions for vacuum polarization effects in a geometrically nontrivial AdS setup with intersecting boundaries. The mimicry by a negative-mass scalar and the curvature-induced non-vanishing of the EMT are potentially useful for holographic applications and for understanding how AdS curvature modifies Casimir-type physics. The elementary-function form is a strength that would facilitate further analytic or numerical studies.

major comments (3)
  1. [Section 3 (Wightman functions)] The central extraction of the brane-induced Wightman functions (the step that isolates the contributions from the intersecting brane while imposing both PEC/PMC conditions and AdS boundary fall-off) must be shown to be free of mode mixing or unsubtracted divergences at the codimension-2 intersection. The skeptic concern is load-bearing because the signs of the PEC/PMC contributions to <E²> and <B²>, the off-diagonal <T_zx>, and the positive brane-induced energy density for PMC all depend on the finite parts of these functions.
  2. [Section 4 (VEVs of field squares)] The statement that the VEVs are mimicked by a scalar field with negative effective mass squared determined by the AdS radius should be accompanied by an explicit identification of that mass value and a direct comparison of the functional forms (e.g., the distance dependence from the brane).
  3. [Section 5 (energy-momentum tensor)] The nonzero off-diagonal component of the energy-momentum tensor and the claimed sign changes of the normal and parallel stresses as functions of distance from the brane must be verified to satisfy the covariant conservation law ∇_μ <T^μ_ν> = 0 in AdS, including any boundary contributions at the intersection.
minor comments (2)
  1. [Section 2 (setup and boundary conditions)] Clarify the precise higher-dimensional definitions of the PEC and PMC conditions on the brane (tangential vs. normal components of F or A) and state whether they reduce exactly to the 3+1 Minkowski versions when the AdS radius is taken to infinity.
  2. [Section 3] The abstract claims 'simple expressions in terms of elementary functions'; ensure that all steps leading to these expressions (including any image-charge or mode-sum techniques adapted to AdS) are written out with sufficient intermediate steps for reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive major comments. We address each point below and will revise the manuscript to strengthen the derivations and verifications as indicated.

read point-by-point responses

  1. Referee: [Section 3 (Wightman functions)] The central extraction of the brane-induced Wightman functions (the step that isolates the contributions from the intersecting brane while imposing both PEC/PMC conditions and AdS boundary fall-off) must be shown to be free of mode mixing or unsubtracted divergences at the codimension-2 intersection. The skeptic concern is load-bearing because the signs of the PEC/PMC contributions to <E²> and <B²>, the off-diagonal <T_zx>, and the positive brane-induced energy density for PMC all depend on the finite parts of these functions.

    Authors: We agree that an explicit demonstration of regularity at the codimension-2 intersection is necessary. Our derivation employs the method of images in AdS spacetime, subtracting the pure AdS contribution after imposing the PEC/PMC conditions on the brane and the appropriate fall-off at the AdS boundary. The resulting elementary-function expressions for the brane-induced Wightman functions are finite everywhere, including at the intersection, with no residual divergences or mode mixing in the subtracted terms. To address the concern directly, we will add a dedicated paragraph in Section 3 that examines the limiting behavior near the intersection and confirms the absence of artifacts in the finite parts used for the VEVs. revision: yes

  2. Referee: [Section 4 (VEVs of field squares)] The statement that the VEVs are mimicked by a scalar field with negative effective mass squared determined by the AdS radius should be accompanied by an explicit identification of that mass value and a direct comparison of the functional forms (e.g., the distance dependence from the brane).

    Authors: We will revise the relevant paragraph in Section 4 to provide the explicit value of the effective mass squared (fixed by the AdS curvature radius) and include a direct side-by-side comparison of the functional forms. This will explicitly show that the distance dependence of the electromagnetic VEVs from the brane matches the corresponding scalar-field expressions with that mass. revision: yes

  3. Referee: [Section 5 (energy-momentum tensor)] The nonzero off-diagonal component of the energy-momentum tensor and the claimed sign changes of the normal and parallel stresses as functions of distance from the brane must be verified to satisfy the covariant conservation law ∇_μ <T^μ_ν> = 0 in AdS, including any boundary contributions at the intersection.

    Authors: Because the VEVs are constructed from Wightman functions that satisfy the Maxwell equations in AdS, the conservation law is expected to hold by construction. Nevertheless, we will add an explicit verification in Section 5 by direct computation of the covariant divergence of the brane-induced <T^μ_ν>, confirming that it vanishes identically with no additional boundary contributions at the intersection arising from the imposed conditions. revision: yes

Circularity Check

0 steps flagged

No circularity: direct extraction of Wightman functions from AdS geometry and boundary conditions

full rationale

The derivation proceeds by imposing the higher-dimensional PEC/PMC conditions simultaneously with AdS boundary fall-offs, then explicitly constructing the brane-induced parts of the Wightman functions for A_μ and F_μν via standard mode-sum or image-charge techniques in AdS. These yield elementary-function expressions whose VEVs (including opposite signs for PEC/PMC, nonzero off-diagonal <T_zx>, and non-vanishing EMT unlike the Minkowski case) are computed directly as consequences. No parameter is fitted to a subset of data and relabeled a prediction, no self-citation supplies a uniqueness theorem or ansatz that the present work merely renames, and the negative-mass scalar mimicry is an observed outcome rather than an input. The chain is self-contained against external benchmarks and does not reduce to its own definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on abstract; no explicit free parameters, ad-hoc entities, or nonstandard axioms identified beyond standard QFT in curved spacetime.

axioms (2)
  • domain assumption Standard definition and properties of Wightman functions and vacuum expectation values in curved spacetime
    Invoked for extracting brane-induced contributions and computing VEVs of E^2, B^2, and EMT
  • domain assumption Validity of PEC and PMC as higher-dimensional boundary conditions for Maxwell field
    Used to define the two cases whose contributions have opposite signs

pith-pipeline@v0.9.0 · 5567 in / 1356 out tokens · 27907 ms · 2026-05-10T05:22:50.516834+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

76 extracted references · 76 canonical work pages

  1. [1]

    Hawking, G.F.R

    S.W. Hawking, G.F.R. Ellis, The Large Scale Structure of Space-Time, Cambridge, Cambridge University Press, 1973

    work page 1973

  2. [2]

    Griffiths, J

    J.B. Griffiths, J. Podolsky, Exact Space-Times in Einstei n’s General Relativity, Cambridge, Cam- bridge University Press, 2009

    work page 2009

  3. [3]

    Aharony, S.S

    O. Aharony, S.S. Gubser, J. Maldacena, H. Ooguri, Y. Oz, L arge N field theories, string theory and gravity, Phys. Rep. 323 (2000) 183

    work page 2000

  4. [4]

    N˘ astase, Introduction to AdS/CFT Correspondence, C ambridge, Cambridge University Press, 2015

    H. N˘ astase, Introduction to AdS/CFT Correspondence, C ambridge, Cambridge University Press, 2015

    work page 2015

  5. [5]

    Ammon, J

    M. Ammon, J. Erdmenger, Gauge/Gravity Duality: Foundat ions and Applications, Cambridge, Cambridge University Press, 2015

    work page 2015

  6. [6]

    Zaanen, Y.-W

    J. Zaanen, Y.-W. Sun, Y. Liu, K. Schalm, Holographic Dual ity in Condensed Matter Physics, Cam- bridge, Cambridge University Press, 2015

    work page 2015

  7. [7]

    P. Brax, C. van de Bruck, A.-C. Davis, Brane world cosmolo gy, Rep. Prog. Phys. 67 (2004) 2183

    work page 2004

  8. [8]

    Maartens, K

    R. Maartens, K. Koyama, Brane-world gravity, Living Rev . Relativity 13 (2010) 5

    work page 2010

  9. [9]

    V. M. Mostepanenko, N.N. Trunov, The Casimir Effect and Its Applications, Oxford, Clarendon, 1997

    work page 1997

  10. [10]

    Milton, The Casimir Effect: Physical Manifestation of Zero-Point Energy, Singapore, World Scientific, 2002

    K.A. Milton, The Casimir Effect: Physical Manifestation of Zero-Point Energy, Singapore, World Scientific, 2002

    work page 2002

  11. [11]

    Bordag, G.L

    M. Bordag, G.L. Klimchitskaya, U. Mohideen, V.M. Moste panenko, Advances in the Casimir Effect, New York, Oxford University Press, 2009

    work page 2009

  12. [12]

    Dalvit, P

    Casimir Physics , edited by D. Dalvit, P. Milonni, D. Roberts, F. da Rosa, Lect ure Notes in Physics Vol. 834, Berlin, Springer-Verlag, 2011

    work page 2011

  13. [13]

    Fabinger, P

    M. Fabinger, P. Horava, Casimir effect between world-bra nes in heterotic M-theory, Nucl. Phys. B 580 (2000) 243

    work page 2000

  14. [14]

    Nojiri, S

    S. Nojiri, S. Odintsov, S. Zerbini, Quantum (in)stabil ity of dilatonic AdS backgrounds and the holographic renormalization group with gravity, Phys. Rev . D 62 (2000) 064006

    work page 2000

  15. [15]

    Toms, Quantised bulk fields in the Randall–Sundrum compactification model, Phys

    D.J. Toms, Quantised bulk fields in the Randall–Sundrum compactification model, Phys. Lett. B 484 (2000) 149

    work page 2000

  16. [16]

    Nojiri, O

    S. Nojiri, O. Obregon, S. Odintsov, (Non)-singular bra ne-world cosmology induced by quantum effects in five-dimensional dilatonic gravity, Phys. Rev. D 62 (2000) 104003

    work page 2000

  17. [17]

    W. D. Goldberger, I. Z. Rothstein, Quantum stabilizati on of compactified AdS 5, Phys. Lett. B 491 (2000) 339

    work page 2000

  18. [18]

    Nojiri, S

    S. Nojiri, S. Odintsov, Brane-world cosmology in highe r derivative gravity or warped compactification in the next-to-leading order of AdS/CFT correspondence, J. High Energy Phys. 07 (2000) 049

    work page 2000

  19. [19]

    Garriga, O

    J. Garriga, O. Pujol´ as, T. Tanaka, Radion effective pote ntial in the brane-world, Nucl. Phys. B 605 (2001) 192. 18

    work page 2001

  20. [20]

    Brevik, K.A

    I.H. Brevik, K.A. Milton, S. Nojiri, S.D. Odintsov, Qua ntum (in)stability of a brane-world AdS 5 universe at nonzero temperature, Nucl. Phys. B 599 (2001) 30 5

    work page 2001

  21. [21]

    Flachi, D.J

    A. Flachi, D.J. Toms, Quantized bulk scalar fields in the Randall-Sundrum brane model, Nucl. Phys. B 610 (2001) 144

    work page 2001

  22. [22]

    Flachi, I.G

    A. Flachi, I.G. Moss, D.J. Toms, Fermion vacuum energie s in brane world models, Phys. Lett. B 518 (2001) 153

    work page 2001

  23. [23]

    Flachi, I.G

    A. Flachi, I.G. Moss, D.J. Toms, Quantized bulk fermion s in the Randall-Sundrum brane model, Phys. Rev. D 64 (2001) 105029

    work page 2001

  24. [24]

    Saharian, M.R

    A.A. Saharian, M.R. Setare, The Casimir effect on backgro und of conformally flat brane-world geometries, Phys. Lett. B 552 (2003) 119

    work page 2003

  25. [25]

    Elizalde, S

    E. Elizalde, S. Nojiri, S.D. Odintsov, S. Ogushi, Casim ir effect in de Sitter and anti-de Sitter braneworlds, Phys. Rev. D 67 (2003) 063515

    work page 2003

  26. [26]

    Flachi, J

    A. Flachi, J. Garriga, O. Pujol´ as, T. Tanaka, Moduli st abilization in higher dimensional brane models, J. High Energy Phys. 08 (2003) 053

    work page 2003

  27. [27]

    Flachi, O

    A. Flachi, O. Pujol´ as, Quantum self-consistency of Ad S× Σ brane models Phys. Rev. D 68, 025023 (2003)

    work page 2003

  28. [28]

    Garriga, A

    J. Garriga, A. Pomarol, A stable hierarchy from Casimir forces and the holographic interpretation Phys. Lett. B 560 (2003) 91

    work page 2003

  29. [29]

    Durrer, M

    R. Durrer, M. Ruser, Dynamical Casimir effect in branewor lds, Phys. Rev. Lett. 99 (2007) 071601

    work page 2007

  30. [30]

    Ruser, R

    M. Ruser, R. Durrer, Dynamical Casimir effect for gravito ns in bouncing braneworlds, Phys. Rev. D 76 (2007) 104014

    work page 2007

  31. [31]

    Elizalde, M

    E. Elizalde, M. Minamitsuji, W. Naylor, Casimir effect in rugby-ball type flux compactifications, Phys. Rev. D 75 (2007) 064032

    work page 2007

  32. [32]

    Saharian, A.L

    A.A. Saharian, A.L. Mkhitaryan, Wightman function and vacuum densities for a Z 2-symmetric thick brane in AdS spacetime, J. High Energy Phys. 08 (2007) 063

    work page 2007

  33. [33]

    Frank, I

    M. Frank, I. Turan, L. Ziegler, Casimir force in Randall -Sundrum models, Phys. Rev. D 76 (2007) 015008

    work page 2007

  34. [34]

    Linares, H.A

    R. Linares, H.A. Morales-T´ ecotl, O. Pedraza, Casimir force for a scalar field in warped brane worlds, Phys. Rev. D 77 (2008) 066012

    work page 2008

  35. [35]

    Frank, N

    M. Frank, N. Saad, I. Turan, Casimir force in Randall-Su ndrum models with d+1 dimensions, Phys. Rev. D 78 (2008) 055014

    work page 2008

  36. [36]

    Flachi, T

    A. Flachi, T. Tanaka, Casimir effect on the brane, Phys. Re v. D 80 (2009) 124022

    work page 2009

  37. [37]

    Teo, Casimir effect in spacetime with extra dimensio ns - from Kaluza-Klein to Randall-Sundrum models, Phys

    L.P. Teo, Casimir effect in spacetime with extra dimensio ns - from Kaluza-Klein to Randall-Sundrum models, Phys. Lett. B 682 (2009) 259

    work page 2009

  38. [38]

    Rypestol, I

    M. Rypestol, I. Brevik, Finite-temperature Casimir effe ct in Randall-Sundrum models, New J. Phys. 12 (2010) 013022

    work page 2010

  39. [39]

    Obousy, G

    R. Obousy, G. Cleaver, Casimir energy and brane stabili ty, J. Geom. Phys. 61 (2011) 577. 19

    work page 2011

  40. [40]

    Teo, Finite temperature fermionic Casimir intera ction in anti-de Sitter space-time, Int

    L.P. Teo, Finite temperature fermionic Casimir intera ction in anti-de Sitter space-time, Int. J. Mod. Phys. A 28 (2013) 1350158

    work page 2013

  41. [41]

    Knapman, D.J

    R.A. Knapman, D.J. Toms, Stress-energy tensor for a qua ntized bulk scalar field in the Randall- Sundrum brane model, Phys. Rev. D 69 (2004) 044023

    work page 2004

  42. [42]

    Saharian, Wightman function and Casimir densitie s on AdS bulk with application to the Randall-Sundrum braneworld, Nucl

    A.A. Saharian, Wightman function and Casimir densitie s on AdS bulk with application to the Randall-Sundrum braneworld, Nucl. Phys. B 712 (2005) 196

    work page 2005

  43. [43]

    Saharian, Wightman function and vacuum fluctuatio ns in higher dimensional brane models, Phys

    A.A. Saharian, Wightman function and vacuum fluctuatio ns in higher dimensional brane models, Phys. Rev. D 73 (2006) 044012

    work page 2006

  44. [44]

    Saharian, Bulk Casimir densities and vacuum inter action forces in higher dimensional brane models, Phys

    A.A. Saharian, Bulk Casimir densities and vacuum inter action forces in higher dimensional brane models, Phys. Rev. D 73 (2006) 064019

    work page 2006

  45. [45]

    S.-H. Shao, P. Chen, J.-A. Gu, Stress-energy tensor ind uced by a bulk Dirac spinor in the Randall- Sundrum model, Phys. Rev. D 81 (2010) 084036

    work page 2010

  46. [46]

    Elizalde, S.D

    E. Elizalde, S.D. Odintsov, A.A. Saharian, Fermionic C asimir densities in anti-de Sitter spacetime, Phys. Rev. D 87 (2013) 084003

    work page 2013

  47. [47]

    Bellucci, W

    S. Bellucci, W. Oliveira dos Santos, E.R. Bezerra de Mel lo, A.A. Saharian, Cosmic string and brane induced effects on the fermionic vacuum in AdS spacetime, J. Hi gh Energy Phys. 05 (2022) 021

    work page 2022

  48. [48]

    Saharian, Surface Casimir densities and induced c osmological constant on parallel branes in AdS spacetime, Phys

    A.A. Saharian, Surface Casimir densities and induced c osmological constant on parallel branes in AdS spacetime, Phys. Rev. D 70 (2004) 064026

    work page 2004

  49. [49]

    Saharian, Surface Casimir densities and induced c osmological constant in higher dimensional braneworlds, Phys

    A.A. Saharian, Surface Casimir densities and induced c osmological constant in higher dimensional braneworlds, Phys. Rev. D 74 (2006) 124009

    work page 2006

  50. [50]

    Teo, Casimir effect of electromagnetic field in Randa ll-Sundrum spacetime, J

    L.P. Teo, Casimir effect of electromagnetic field in Randa ll-Sundrum spacetime, J. High Energy Phys. 10 (2010) 019

    work page 2010

  51. [51]

    Kotanjyan, A.A

    A.S. Kotanjyan, A.A. Saharian, Electromagnetic quant um effects in anti-de Sitter spacetime, Phys. At. Nucl. 80 (2017) 562

    work page 2017

  52. [52]

    Saharian, A.S

    A.A. Saharian, A.S. Kotanjyan, A.A. Saharyan, H. G. Sar gsyan, Electromagnetic Casimir densities for planar boundaries in AdS spacetime, Int. J. Mod. Phys. A 3 5 (2020) 2040029

    work page 2020

  53. [53]

    Saharian, A.S

    A.A. Saharian, A.S. Kotanjyan, H.G. Sargsyan, Electro magnetic field correlators and the Casimir effect for planar boundaries in AdS spacetime with applicatio n in braneworlds, Phys. Rev. D 102 (2020) 105014

    work page 2020

  54. [54]

    Takayanagi, Holographic dual of BCFT, Phys

    T. Takayanagi, Holographic dual of BCFT, Phys. Rev. Let t. 107 (2011) 101602

    work page 2011

  55. [55]

    Fujita, T

    M. Fujita, T. Takayanagi, E. Tonni, Aspects of AdS/BCFT , J. High Energy Phys. 11 (2011) 043

    work page 2011

  56. [56]

    Karch, L

    A. Karch, L. Randall, Open and closed string interpreta tion of SUSY CFT’s on branes with bound- aries, J. High Energy Phys. 06 (2001) 063

    work page 2001

  57. [57]

    Suzuki, Gauge symmetries and conserved currents in A dS/BCFT, J

    K. Suzuki, Gauge symmetries and conserved currents in A dS/BCFT, J. High Energy Phys. 06 (2024) 137

    work page 2024

  58. [58]

    Fu, K.-Y

    Y. Fu, K.-Y. Kim, Wedge holographic complexity in Karch -Randall braneworld, J. High Energy Phys. 01 (2025) 174. 20

    work page 2025

  59. [59]

    Ahn, S.-E

    B. Ahn, S.-E. Bak, K-Y. Kim, M. Nishida, Renyi reflected e ntropy and entanglement wedge cross section with cosmic branes in AdS/BCFT, Phys. Rev. D 111 (202 5) 126006

    work page

  60. [60]

    P.-X. Hao, N. Ogawa, T. Takayanagi, T. Waki, Flat space s olography via AdS/BCFT, arXiv:2509.00652

    work page arXiv

  61. [61]

    S. Ryu, T. Takayanagi, Holographic derivation of entan glement entropy from the anti-de Sitter space/Conformal field theory correspondence, Phys. Rev. Le tt. 96 (2006) 181602

    work page 2006

  62. [62]

    S. Ryu, T. Takayanagi, Aspects of holographic entangle ment entropy, J. High Energy Phys. 08 (2006) 045

    work page 2006

  63. [63]

    Nishioka, S

    T. Nishioka, S. Ryu, T. Takayanagi, Holographic entang lement entropy: an overview, J. Phys. A 42 (2009) 504008

    work page 2009

  64. [64]

    B. Chen, B. Czech, Zi-Zhi Wang, Quantum information in h olographic duality, Rep. Prog. Phys. 85 (2022) 046001

    work page 2022

  65. [65]

    Bezerra de Mello, A.A

    E.R. Bezerra de Mello, A.A. Saharian, M.R. Setare, Vacu um densities for a brane intersecting the AdS boundary, Phys. Rev. D 92 (2015) 104005

    work page 2015

  66. [66]

    Bellucci, A.A

    S. Bellucci, A.A. Saharian, V.Kh. Kotanjyan, Vacuum de nsities and the Casimir forces for branes orthogonal to the AdS boundary, Phys. Rev. D 106 (2022) 06502 1

    work page 2022

  67. [67]

    Saharian, Surface Casimir densities on branes ort hogonal to the boundary of anti-de Sitter spacetime, Physics 5(4) (2023) 1145

    A.A. Saharian, Surface Casimir densities on branes ort hogonal to the boundary of anti-de Sitter spacetime, Physics 5(4) (2023) 1145

    work page 2023

  68. [68]

    Davies, S.D

    P.C.W. Davies, S.D. Unwin, Quantum vacuum energy and th e masslessness of the photon, Phys. Lett. B 98 (1981) 274

    work page 1981

  69. [69]

    Barton, The Casimir effect with finite mass photons, Ann

    G. Barton, The Casimir effect with finite mass photons, Ann . Phys. 162 (1985) 231

    work page 1985

  70. [70]

    Saharian, H.H

    A.A. Saharian, H.H. Asatryan, Two-point functions and the vacuum densities in the Casimir effect for the Proca field, Eur. Phys. J. Plus 141 (2026) 106

    work page 2026

  71. [71]

    B. Allen,T. Jacobson, Vector two point functions in max imally symmetric spaces, Commun. Math. Phys. 103 (1986) 669

    work page 1986

  72. [72]

    Liu, A.A

    H. Liu, A.A. Tseytlin, On four-point functions in the CF T/AdS correspondence, Phys. Rev. D 59 (1999) 086002

    work page 1999

  73. [73]

    D’Hoker, D

    E. D’Hoker, D. Z. Freedman, Gauge boson exchange in AdS d+1, Nucl. Phys. B 544 (1999) 612

    work page 1999

  74. [74]

    Prudnikov, Yu.A

    A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integr als and Series, New York, Gordon and Breach, 1986, Vol. 2

    work page 1986

  75. [75]

    Birrell, P.C.W

    N.D. Birrell, P.C.W. Davies, Quantum Fields in Curved S pace, Cambridge, Cambridge University Press, 1982

    work page 1982

  76. [76]

    Parker, D

    L. Parker, D. Toms, Quantum Field Theory in Curved Space time, Cambridge, Cambridge University Press, 2009. 21

    work page 2009