Pith. sign in

REVIEW 2 major objections 4 minor 1 cited by

Dependence of scalar matter vacuum energy, induced by a magnetic topological defect, on the coupling to space-time curvature

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper shows that in flat spacetime the total vacuum energy induced by an impenetrable magnetic flux tube depends on the curvature-coupling parameter ξ for Robin boundary conditions with −π/2<θ<0, while Dirichlet and Neumann boundary…

desk verdict Modest but genuine new result on xi-dependence of total vacuum energy for Robin tubes; the numerical derivative in Eq. (37) needs to be made reproducible before I'd bet on it. read the letter →

arxiv 2412.06814 v2 pith:WARM3TGO submitted 2024-12-01 hep-th math-phmath.MPquant-ph

V.M. Gorkavenko , A.O. Zaporozhchenko , M.S. Tsarenkova This is my paper · ORCID
classification hep-thmath-phmath.MPquant-ph MSC 81T2081T55
keywords vacuumpolarizationtopologicaldefectAharonov-BohmeffectCasimirRobinboundaryconditionscurvaturecouplingmagneticfluxtube
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks whether the vacuum energy induced by an impenetrable magnetic flux tube depends on the coupling ξ of a charged scalar field to spacetime curvature, even in flat spacetime. It answers yes: for Robin boundary conditions on the tube surface with parameter θ in (−π/2,0), the total induced vacuum energy contains a positive ξ-proportional term E_ξ, while for the special Dirichlet (θ=0) and Neumann (θ=−π/2) cases that term vanishes. The result is obtained in (2+1) dimensions for half-integer magnetic flux, with numerical evaluation of the mode sums, and is carried to higher dimensions via a known reduction argument. This matters because it identifies a flat-space observable—the Casimir energy of a flux tube—that can depend on a curvature-coupling parameter, and because it isolates which boundary conditions preserve conformal coupling independence.

What carries the argument

The load-bearing object is the ξ-dependent part of the vacuum energy density, expressed through the function α_−(θ,x,x0,F), whose transverse Laplacian gives the term proportional to (1/4−ξ). The total integral of this term is reduced by integration by parts to a surface value at the tube radius x=x0, namely E_ξ = −2πm [x (α_−/x)']_{x=x0}, so the entire question reduces to whether the radial derivative of α_−/x at the boundary is nonzero. The mode functions are built from Bessel functions J_ρ and Y_ρ with a Robin-phase combination Ω_ρ(θ,u,v)=sin μ_ρ J_ρ(u)−cos μ_ρ Y_ρ(u), and the parameter θ encodes the boundary condition (θ=0 Dirichlet, θ=−π/2 Neumann). Numerical computation of α_− for finite tube thickness and half-integer flux supplies the derivative, giving the positive curve E_ξ(θ) in Fig. 5.

What would settle it

Solve the radial Fock–Klein–Gordon equation with the Robin boundary condition for a few values of θ in (−π/2,0) and search for normalizable eigenfunctions with energy below the continuum threshold; the existence of even one such bound state would require adding its contribution to the vacuum energy and would invalidate the computed E_ξ for that θ. Alternatively, recompute E_ξ from (37) with a finer radial grid or an analytic expression for α_− near x=x0; a change in sign or magnitude beyond numerical error would expose the interpolation-dependent derivative.

Watch

Extended reading notes

Core claim

The central claim is that in flat spacetime, the total induced vacuum energy of a quantized charged scalar field outside an impenetrable finite-thickness magnetic tube is independent of the curvature-coupling parameter ξ only for the Dirichlet and Neumann boundary conditions. For generalized Robin conditions with −π/2<θ<0, the ξ-dependent part of the total energy, E_ξ = −2πm [x ∂/∂x (α_−(θ,x,x0,F)/x)] at x=x0, is positive and vanishes only at the endpoints θ=−π/2 and θ=0. This is demonstrated numerically for the (2+1)-dimensional case with half-integer flux F=1/2: the α_− function is computed by truncated mode sums and interpolation, and the boundary term is extracted. Positive values of θ are set aside because bound-state solutions are expected to contribute there, as in the related induced-magnetic-flux problem. The authors argue from an earlier dimensional-reduction result that the same ξ dependence persists in higher dimensions.

Load-bearing premise

The central result rests on the assumption that for −π/2<θ<0 the field has no bound states, so the vacuum energy comes entirely from continuum modes; if a bound state exists in that interval, the mode sum and therefore E_ξ would change.

Editorial extensions

If this is right

  • Dirichlet and Neumann boundary conditions remain the only Robin-type cases in which flat-space total induced vacuum energy is exactly ξ-independent; any other Robin condition in (−π/2,0) breaks this.
  • The ξ-dependent contribution E_ξ is positive for −π/2<θ<0, so the total induced energy is larger than the canonical value for ξ<1/4 and smaller for ξ>1/4; at the conformal value ξ=1/4 the term disappears.
  • The induced vacuum energy inherits the Aharonov–Bohm periodicity in the magnetic flux and depends only on the fractional part F; for integer flux the effect vanishes.
  • Because the (2+1)-dimensional computation generalizes to arbitrary dimension, the ξ dependence should appear in the physical d=3 case of an infinitely long tube as well.
  • The case θ>0 is not covered by the main result; bound-state contributions may make the total energy behave differently there.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper assumes without proof that no bound states exist for −π/2<θ<0; if a numerical search of the radial spectrum found one, the mode sum and E_ξ would need revision, so the curve in Fig. 5 is a prediction that can be checked directly.
  • The same integration-by-parts mechanism would apply to fermionic fields with Robin-type boundary conditions, suggesting an analogous ξ dependence in flat-space fermion Casimir energies; the paper does not treat fermions.
  • If impenetrable flux tubes model cosmic strings or vortices, a positive E_ξ for non-conformal couplings implies that the vacuum energy of a network of such defects depends on ξ; comparing cosmological vacuum-energy estimates with flat-space Casimir measurements could constrain the scalar-curvature coupling.
  • The numerical derivative in (37) is taken from interpolated α_− data with no reported error control; refining the grid or using an analytic asymptotic for α_− near the tube would harden the quantitative curve, even though the qualitative positivity is clear.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies the vacuum polarization energy of a charged massive scalar field outside an impenetrable, finite-thickness magnetic flux tube in flat spacetime, with generalized Robin boundary conditions on the tube surface. It separates the induced vacuum energy density into a canonical part and a part proportional to (1/4 - ξ), where ξ is the coupling to spacetime curvature. The central claim is that, in 2+1 dimensions and for half-integer magnetic flux, the total induced vacuum energy is independent of ξ only for the Dirichlet and Neumann cases, while for Robin parameters -π/2 < θ < 0 the ξ-dependent contribution E_ξ is nonzero and positive, vanishing at the endpoints θ = -π/2 and θ = 0. The main quantitative evidence is a numerical evaluation of the boundary term in Eq. (37), shown in Fig. 5.

Significance. If the numerical result is reliable, the paper reports a genuine and interesting effect: the total Casimir-type energy in flat spacetime acquires a dependence on the curvature coupling ξ for generic Robin boundary conditions, breaking the degeneracy between Dirichlet and Neumann cases. The reduction of the ξ-dependent total energy to a boundary term, Eq. (37), is an exact integration by parts and is a useful structural observation. The paper also provides analytic singular-vortex limits and reproduces the known Dirichlet and Neumann results, which are welcome consistency checks. The quantitative prediction in Fig. 5 is concrete and falsifiable. However, the weight of the paper rests on a numerically differentiated interpolated function, and the manuscript as it stands does not provide the evidence needed to certify that quantity.

major comments (2)
  1. [§5, Eq. (37), Fig. 5] The central claim E_ξ ≠ 0 for -π/2 < θ < 0 rests entirely on the numerical evaluation of -2π m [x d(α_-/x)/dx]_{x=x0}. The manuscript states no error bars, no convergence tables, and no code; the value at x0 = 10^{-2} is obtained by interpolating α_- values that are themselves computed with truncated sums and integrals in §4, with N_max and z_max chosen by an unquantified 'sufficient accuracy' condition. The derivative at the lower endpoint is a delicate quantity, especially because the integrand involves Y-Bessel subtraction terms near the tube edge. I consider this load-bearing: without a controlled estimate of the truncation, quadrature, interpolation, and differentiation errors, the positive curve in Fig. 5 is not established. The authors should provide a convergence study in N_max, z_max, and the interpolation grid, and ideally evaluate the left side of Eq. (37) as an integral of (α_- - x α_-' + x^2 α_-'')/x^2 as an independent cross-check of the boundary expression.
  2. [§5, §6] The computation for -π/2 < θ < 0 treats the vacuum energy as coming entirely from continuum modes and states that bound-state contributions are relevant only for θ > 0, but no proof is given that no normalizable bound state exists for negative θ. This assumption is load-bearing because the spectral decomposition leading to Eq. (37) and the integrated boundary term would change if a bound state existed. The gap is easily closed: for a normalizable radial mode built from K_ρ(κr), one has K_ρ(κr) > 0 and K'_ρ(κr) < 0 for κ > 0, r > 0, so for θ ≤ 0 the Robin combination cosθ K_ρ + sinθ r κ K'_ρ is strictly positive and cannot vanish. The authors should include this argument or a citation, so that the continuum-mode restriction is explicit and justified.
minor comments (4)
  1. [§5] The text 'induced vacuum energy Exi is also zero' contains a typo; it should read 'E_ξ'.
  2. [§4, Eq. (29)] The statement that N_max and z_max are chosen 'from the condition the result of computation does not change with sufficient accuracy' is not quantitative. The authors should state the actual values used and the tolerance that defines sufficiency.
  3. [Fig. 2 caption] The caption lists 'mr = 1/1000, mr = 1/100, mr = 10' as tube thicknesses, but the dimensionless thickness is mr0, not mr; the notation should be corrected.
  4. [§6] The final sentence asserts without further support that the ξ-dependence of the total energy generalizes to higher dimensions. Since the presented evidence is a numerical computation in 2+1 dimensions, this should be phrased as an expectation or conjecture rather than a conclusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Robin E_xi term is computed from mode sums and an exact boundary integration by parts, not from an assumed answer.

full rationale

The central claim is that E_xi, the coefficient of (1/4 - xi) in the total vacuum energy, vanishes for Dirichlet and Neumann boundary conditions but is nonzero for -pi/2 < theta < 0. This E_xi is defined by Eq. (36) and reduced by an exact integration by parts to Eq. (37), a boundary derivative of the numerically computed function alpha_-(theta,x,x0,F). The derivative is obtained from a numerical mode-sum/integral calculation of alpha_-, not fitted to the claimed sign or magnitude; no parameter is adjusted to make E_xi positive in the interval. The computational inputs (mode functions (9), flux subtraction (15), and truncation criteria) come from the paper itself or from independent, checkable prior calculations. The self-citations ([30]-[36]) are used for renormalization conventions, numerical convergence checks, the Dirichlet E_xi=0 result, and the dimensional-reduction argument; none assumes the Robin E_xi != 0 conclusion. The dimensional generalization in the Summary rests on [32], but that is a separate published calculation that does not incorporate the present paper's target result, so it is independent support rather than a circular premise. There is therefore no step in which an output is equivalent by construction to an input, and no fitted parameter is relabeled as a prediction. Concerns about numerical error control in Eq. (37) are correctness risks, not circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard QFT definitions, a renormalization convention, and an unproven absence of bound states for negative theta. No new physical entities are introduced. The numerical result depends on chosen values of F, mr0, and the theta interval.

free parameters (3)
  • theta (Robin boundary condition parameter) = varied in (-pi/2, 0); Fig.5 shows E_xi/m vs theta
    The central claim is that E_xi is nonzero for this range; the result is computed as a function of theta. Theta is a model parameter, not fitted to data, but the numerical scan is the content of the paper.
  • F (fractional magnetic flux) = 1/2
    Chosen because the vacuum polarization effect is maximal for the singular vortex; all numerical results are for this value. The dependence on flux is only argued by periodicity, not computed for the central claim.
  • mr0 (dimensionless tube thickness) = 10^-2
    Single thickness used for the central E_xi(theta) plot; convergence in thickness is shown only for alpha_- (Fig.2), not for E_xi.
assumptions (4)
  • domain assumption The improved energy-momentum tensor expression (6) with the (1/4 - xi) Laplacian term is the correct definition of vacuum energy density.
    Invoked in Section 2, Eq. (6). This is what creates flat-space xi dependence; standard in curved-space QFT but not derived here.
  • domain assumption Renormalization reduces to subtracting the zero-flux contribution.
    Stated in Section 2 with citation [27]; no independent derivation is given for Robin boundary conditions.
  • ad hoc to paper No bound-state modes contribute for -pi/2 < theta < 0.
    The paper restricts the central claim to negative theta and only discusses bound states for positive theta, implicitly assuming their absence in the scanned interval.
  • ad hoc to paper The (2+1)-dimensional result generalizes to higher dimensions.
    Section 6 concludes higher-dimensional xi dependence from [32] without a calculation, extending the computed result beyond the demonstrated case.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Dependence of scalar matter vacuum energy, induced by a magnetic topological defect, on the coupling to space-time curvature." pith.science (2026). https://pith.science/paper/WARM3TGO

@misc{pith2026241206814,
  author       = {Pith},
  title        = {Pith review of: Dependence of scalar matter vacuum energy, induced by a magnetic topological defect, on the coupling to space-time curvature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WARM3TGO}},
  note         = {Machine review of arXiv:2412.06814}
}
abstract

We considered the vacuum polarization of a quantized charged scalar matter field in the background of a topological defect modeled by a finite-thickness tube with magnetic flux inside. The tube is impenetrable for quantum matter, and a generalized boundary condition of the Robin type is imposed at its surface. We have shown that in the flat space-time, the total induced vacuum energy does not depend on the coupling $(\xi)$ of the scalar field's interaction with the space-time curvature, only for the partial cases of the Dirichlet and Neumann boundary conditions on the tube's edge. However, for generalized Robin boundary conditions, the total induced energy depends on the coupling $\xi$ in flat space-time, at least for negative values of the boundary condition parameter $-\pi/2<\theta<0$.

Figures

Figures reproduced from arXiv: 2412.06814 by the authors.

Figure 1
Figure 1. α− function for fixed thickness of the impenetrable magnetic tube mr0 = 10−2 , half-integer magnetic flux F = 1/2 and parameter of the boundary condition θ = −π/2, −3π/8, −π/4, −π/8, 0 from down to up line correspondingly. In the case of half-integer magnetic flux F = 1/2, the α+ function can be written as α sing + (mr, F = 1/2) = m3 3π 2 nπ 2 − πmr [K0(2mr)L−1(2mr) + K1(2mr)L0(2mr)] + + K0(2mr) 2mr −  1 − 1 2(mr) … view at source ↗
Figure 2
Figure 2. α− function for the case of a) Dirichlet (θ = 0) and b) Neumann (θ = −π/2) boundary condition for the impenetrable magnetic tube of different thicknesses for mr > mr0 and half-integer magnetic flux F = 1/2. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. ˜α− function for the impenetrable magnetic tube of thickness mr0 = 10−2 , half-integer magnetic flux F = 1/2, and different values of parameter of the boundary condition θ = −π/2, −3π/8, −π/4, −π/8, 0 on its edge. tube of fixed thickness (x0 = mr0) for different values of parameter θ and in Fig.2 for different thickness of the tube for the case of Dirichlet and Neumann boundary conditions on its edge. Following [31]… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The induced vacuum energy density r 3 εren for different values of coupling scalar field to the space-time curvature ξ for the case of a) Dirichlet (θ = 0) and b) Neumann (θ = −π/2) boundary conditions on the edge of the magnetic impenetrable tube of thickness mr0 = 10…
Figure 5
Figure 5. Figure 5: The total induced dimensionless vacuum energy [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Effects of Curvature-Scalar Coupling on Vacuum Energy in Flat (3+1)-Dimensional Space-Time

    hep-th 2026-07 conditional novelty 3.0 of 10

    In flat (3+1)-dimensional spacetime, the vacuum energy induced by an impenetrable magnetic flux tube with Robin boundary conditions depends on the curvature coupling ξ; only Dirichlet and Neumann boundary conditions m...

Reference graph

Works this paper leans on

57 extracted references · 56 canonical work pages · cited by 1 Pith paper

  1. [1]

    H.B.G. Casimir. On the Attraction Between Two Perfectly Conducting Plates. Proc. Kon. Ned. Akad. Wetenschap B51, 793 (1948); Physica 19, 846 (1953)

    work page 1948

  2. [2]

    Elizalde

    E. Elizalde. Ten Physical Applications of Spectral Zeta Functions(Berlin: Springer- Verlag, 1995) [ISBN: 3-540-60230-5]

    work page 1995

  3. [3]

    Mostepanenko, N.N

    V.M. Mostepanenko, N.N. Trunov. The Casimir Effect and Its Applications. Oxford: Clarendon Press, 199 (1997)

    work page 1997

  4. [4]

    Bordag, U

    M. Bordag, U. Mohideen, V.M. Mostepanenko. New Developments in the Casimir Effect. Phys. Rept.353, 1 (2001)

    work page 2001

  5. [5]

    Klimchitskaya, V.M

    G.L. Klimchitskaya, V.M. Mostepanenko. Testing Gravity and Predictions Be- yond the Standard Model at Short Distances: The Casimir Effect. Pfeifer, C., L¨ammerzahl, C. (eds) Modified and Quantum Gravity. Lecture Notes in Physics, vol 1017 (Springer, Cham, 2023). https://doi.org/10.1007/978-3-031-31520-6 13

    work page doi:10.1007/978-3-031-31520-6 2023

  6. [6]

    Aharonov, D

    Y. Aharonov, D. Bohm. Significance of Electromagnetic Potentials in the Quantum Theory. Phys. Rev.115, 485 (1959)

    work page 1959

  7. [7]

    Sitenko, A.Yu

    Yu.A. Sitenko, A.Yu. Babansky. The Casimir-Aharonov-Bohm effect? Mod. Phys. Lett. A13(5), 379 (1998)

    work page 1998

  8. [8]

    Kibble, Some implications of a cosmological phase transition, Phys

    T.W.B. Kibble, Some implications of a cosmological phase transition, Phys. Rep. 67, 183 (1980)

    work page 1980

Show all 57 references
  1. [9]

    Vilenkin, Cosmic strings, Phys

    A. Vilenkin, Cosmic strings, Phys. Rev. D24, 2082 (1981)

    1981

  2. [10]

    Vilenkin and E.P.S

    A. Vilenkin and E.P.S. Shellard, Cosmic Strings and Other Topological Defects (Cam- bridge Univ. Press, Cambridge UK, 1994)

    1994

  3. [11]

    Hindmarsh and T.W.B

    M.B. Hindmarsh and T.W.B. Kibble, Cosmic strings, Rep. Progr. Phys. 58, 477 (1995)

    1995

  4. [12]

    Abrikosov, On the magnetic properties of superconductors of the second group, Sov

    A.A. Abrikosov, On the magnetic properties of superconductors of the second group, Sov. Phys.-JETP 5, 1174 (1957)

    1957

  5. [13]

    Nielsen and P

    H.B. Nielsen and P. Olesen, Vortex-line models for dual strings, Nucl. Phys. B 61, 45 (1973)

    1973

  6. [14]

    Krishnan, E

    A. Krishnan, E. Dujardin, M.M.J. Treacy, J. Hugdahl, S. Lynum, and T.W. Ebbesen, Graphitic cones and the nucleation of curved carbon surfaces,Nature 388, 451 (1997)

    1997

  7. [15]

    Rieth and W

    H.Heiberg-Andersen, Carbon nanonones in Handbook of Theoretical and Computa- tional Nanotechnology, edited by M. Rieth and W. Schommers (American Scientific Publishers, Valencia, CA, 2006), pp.507–517. 13

    2006

  8. [16]

    Sitenko and N.D

    Yu.A. Sitenko and N.D. Vlasii, Electronic properties of graphene with a topological defect, Nucl. Phys. B787, 241 (2007)

    2007

  9. [17]

    Naess, A

    S.N. Naess, A. Elgsaeetter, G. Helgesen, and K.D. Knudsen, Carbon nanocones: Wall structure and morphology, Sci. Technol. Adv. Mat.10, 065002 (2009)

    2009

  10. [18]

    Sitenko and V.M

    Yu.A. Sitenko and V.M. Gorkavenko, Properties of the ground state of electronic excitations in carbon-like nanocones, Low Temp. Phys. 44, 1261 (2018) [Fiz. Nizk. Temp. 44, 1618 (2018)]

    2018

  11. [19]

    Sitenko, V.M

    Yu.A. Sitenko, V.M. Gorkavenko, Induced vacuum magnetic flux in quantum spinor matter in the background of a topological defect in twodimensional space, Phys. Rev. D 100, 085011 (2019)

    2019

  12. [20]

    Serebrianyi

    E.M. Serebrianyi. Vacuum polarization by magnetic flux: The Aharonov-Bohm effect. Theor. Math. Phys.64, 846 (1985) [ Teor. Mat. Fiz.64, 299 (1985)]

    1985

  13. [21]

    Gornicki

    P. Gornicki. Aharonov-bohm effect and vacuum polarization. Ann. Phys. (N.Y.)202, 271 (1990)

    1990

  14. [22]

    Flekkoy, J.M

    E.G. Flekkoy, J.M. Leinaas. Vacuum currents around a magnetic flux string. Intern. J. Mod. Phys. A06, 5327 (1991)

    1991

  15. [23]

    Parwani, A.S

    R.R. Parwani, A.S. Goldhaber. Decoupling in (2+1)-dimensional QED?, Nucl. Phys. B 359, 483 (1991)

    1991

  16. [24]

    Yu.A. Sitenko. Self-adjointness of the Dirac hamiltonian and fermion number frac- tionization in the background of a singular magnetic vortex. Phys. Lett. B387, 334 (1996)

    1996

  17. [25]

    Yu.A. Sitenko. Self-Adjointness of the Dirac Hamiltonian and Vacuum Quantum Numbers Induced by a Singular External Field. Phys. Atom. Nucl.60, 2102 (1997) [Yad. Fiz.60, 2285 (1997)]

    1997

  18. [26]

    Sitenko, A.Yu

    Yu.A. Sitenko, A.Yu. Babansky. Effects of boson-vacuum polarization by a singular magnetic vortex. Phys. Atom. Nucl.61, 1594 (1998) [ Yad. Fiz.61, 1706 (1998)]

    1998

  19. [27]

    Babanskii, Ya.A

    A.Yu. Babanskii, Ya.A. Sitenko. Vacuum energy induced by a singular magnetic vortex. Theor. Math. Phys.120, 876 (1999)

    1999

  20. [28]

    Yu. A. Sitenko and V. M. Gorkavenko. On the dependence of the induced vacuum energy-momentum tensor on the coupling to the curvature scalar. Ukr. J. Phys.48, 1286 (2003)

    2003

  21. [29]

    Sitenko, V.M

    Yu.A. Sitenko, V.M. Gorkavenko. Induced vacuum energy-momentum tensor in the background of a (d-2)-brane in (d+1)-dimensional space-time. Phys. Rev. D 67, 085015 (2003)

    2003

  22. [30]

    Gorkavenko, Yu.A

    V.M. Gorkavenko, Yu.A. Sitenko, O.B. Stepanov. Polarization of the vacuum of a quantized scalar field by an impenetrable magnetic vortex of finite thickness.J. Phys. A: Math. Theor.43, 175401 (2010). 14

    2010

  23. [31]

    Gorkavenko, Yu.A

    V.M. Gorkavenko, Yu.A. Sitenko, O.B. Stepanov. Vacuum energy induced by an impenetrable flux tube of finite radius. Int. J. Mod. Phys. A26, 3889 (2011)

    2011

  24. [32]

    Gorkavenko, Yu.A

    V.M. Gorkavenko, Yu.A. Sitenko, O.B. Stepanov. Casimir energy and force induced by an impenetrable flux tube of finite radius. Int. J. Mod. Phys. A 28, 1350161 (2013)

    2013

  25. [33]

    Gorkavenko, T.V

    V.M. Gorkavenko, T.V. Gorkavenko, Yu.A. Sitenko, M.S. Tsarenkova. Induced vac- uum energy density of quantum charged scalar matter in the background of an im- penetrable magnetic tube with the Neumann boundary condition. Ukr. J. Phys.67, 715 (2022)

    2022

  26. [34]

    Gorkavenko, I.V

    V.M. Gorkavenko, I.V. Ivanchenko, Yu.A. Sitenko. Induced vacuum current and mag- netic field in the background of a vortex. Int. J. Mod. Phys. A31, 1650017 (2016)

    2016

  27. [35]

    Gorkavenko, T.V

    V.M. Gorkavenko, T.V. Gorkavenko, Yu.A. Sitenko, M.S. Tsarenkova. Induced vac- uum current and magnetic flux in quantum scalar matter in the background of a vortex defect with the Neumann boundary condition. Ukr. J. Phys.67, 3 (2022)

    2022

  28. [36]

    Sitenko, V.M

    Yu.A. Sitenko, V.M. Gorkavenko, M.S. Tsarenkova. Magnetic flux in the vacuum of quantum bosonic matter in the cosmic string background. Phys. Rev. D106, 105010 (2022)

    2022

  29. [37]

    Penrose, in Relativity, Groups and Topology, edited by B.S

    R. Penrose, in Relativity, Groups and Topology, edited by B.S. DeWitt and C. DeWitt (Gordon and Breach, New York, 1964)

    1964

  30. [38]

    Chernikov, E.A

    N.A. Chernikov, E.A. Tagirov. Quantum theory of scalar field in de Sitter space-time. Ann. Inst. Henri Poincare, Sect. A 9, 109 (1968)

    1968

  31. [39]

    Callan, S

    C.G. Callan, S. Coleman, R. Jackiw. A New improved energy-momentum tensor. Annals Phys. 59, 42 (1970)

    1970

  32. [40]

    Birrell, P.C.W

    N.D. Birrell, P.C.W. Davies. Quantum fields in curved space (Cambridge University Press, 1982)

    1982

  33. [41]

    Branchina, E

    V. Branchina, E. Bentivegna, F. Contino, and D. Zappal‘a. Direct Higgs-gravity interaction and stability of our Universe. Phys. Rev. D99, 096029 (2019)

    2019

  34. [42]

    S.A. Fulling. Nonuniqueness of Canonical Field Quantization in Riemannian Space- Time. Phys. Rev.D 7, 2850 (1973)

    1973

  35. [43]

    Grib, S.G

    A.A. Grib, S.G. Mamayev, V.M. Mostepanenko. Vacuum quantum effects in strong fields (St.Petersburg, 1994)

    1994

  36. [44]

    R.M. Wald. Quantum Field Theory in Curved Space-Time and Black Hole Thermo- dynamics. Quantum Field Theory in Curved Spacetime and Black Hole Thermody- namics (University of Chicago Press, 1994)

    1994

  37. [45]

    L.H. Ford. Quantum field theory in curved space-time. arXiv:gr-qc/9707062 (1997)

    1997 arXiv

  38. [46]

    Parker, D

    L.E. Parker, D. Toms. Quantum Field Theory in Curved Spacetime: Quantized Field and Gravity (Cambridge University Press, 2009). 15

    2009

  39. [47]

    Dowker, R

    J.S. Dowker, R. Critchley. Effective Lagrangian and energy-momentum tensor in de Sitter space. Phys. Rev. D.13, 3224 (1976)

    1976

  40. [48]

    S.W. Hawking. Zeta function regularization of path integrals in curved spacetime. Commun. Math. Phys.55, 133 (1977)

    1977

  41. [49]

    Bordag, Vacuum energy in smooth background fields, J

    M. Bordag, Vacuum energy in smooth background fields, J. Phys. A: Math. Gen. 28, 755 (1995)

    1995

  42. [50]

    Bordag and K

    M. Bordag and K. Kirsten, Vacuum energy in a spherically symmetric background field, Phys. Rev. D 53, 5753 (1996)

    1996

  43. [51]

    Cangemi, G

    D. Cangemi, G. Dunne, E. D’Hoker. Effective energy for (2 + 1)-dimensional QED with semilocalized static magnetic fields: A solvable model. Phys. Rev. D.52, 3163 (1995)

    1995

  44. [52]

    M.P. Fry. QED in inhomogeneous magnetic fields. Phys. Rev. D54, 6444 (1996)

    1996

  45. [53]

    Dunne and T.M

    G. Dunne and T.M. Hall. An exact QED 3+1 effective action. Phys. Lett. B419, 322 (1998)

    1998

  46. [54]

    Bordag and K

    M. Bordag and K. Kirsten. The ground state energy of a spinor field in the back- ground of a finite radius flux tube. Phys. Rev. D 60, 105019 (1999)

    1999

  47. [55]

    Scandurra

    M. Scandurra. Vacuum energy in the presence of a magnetic string with a delta function profile. Phys. Rev. D.62, 085024 (2000)

    2000

  48. [56]

    Langfeld, L

    K. Langfeld, L. Moyaerts and H. Gies. Fermion induced quantum action of vortex systems. Nucl. Phys. B646, 158 (2002)

    2002

  49. [57]

    Graham, V

    N. Graham, V. Khemani, M. Quandt, O. Schroeder and H. Weigel. Quantum QED Flux Tubes in 2+1 and 3+1 Dimensions. Nucl. Phys. B707, 233 (2005). 16

    2005

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.