# Vacuum currents in curved tubes

A. A. Saharian
Institute of Physics, Yerevan State University,
1 Alex Manogian Street, 0025 Yerevan, Armenia

###### Abstract

We investigate the combined effects of spatial curvature and topology on the
properties of the vacuum state for a charged scalar field localized on
rotationally symmetric 2D curved tubes. For a general spatial geometry and for
quasiperiodicity condition with a general phase, the representation of the
Hadamard function is provided where the topological contribution is explicitly
extracted. As an important local characteristic of the vacuum state the
expectation value of the current density is studied. The vacuum current is a
periodic function of the magnetic flux enclosed by the tube with the period of
flux quantum. The general formula is specified for constant radius and conical
tubes. As another application, we consider the Hadamard function and the
vacuum current density for a scalar field on the Beltrami pseudosphere.
Several representations are provided for the corresponding expectation value.
For small values of the proper radius of the tube, compared with the curvature
radius, the effect of spatial curvature on the vacuum current is weak and the
leading term in the corresponding expansion coincides with the current density
on a constant radius tube. The effect of curvature is essential for proper
radii of the tube larger than the radius of spatial curvature. In this limit
the fall-off of the current density, as a function of the proper radius,
follows a power-law for both massless and massive fields. This behavior is in
clear contrast to the one for a constant radius tube with exponential decay
for massive fields. We also compare the vacuum currents on the Beltrami
pseudosphere and on locally de Sitter and anti-de Sitter 2D tubes.

Keywords: vacuum currents; nontrivial topology; emergent gravity; Beltrami
pseudosphere

##  1 Introduction

The canonical quantization of fields is based on the expansion of the field
operator in terms of a complete set of mode functions being the solutions of
the classical field equation. The coefficients of the expansion determine the
annihilation and creation operators that are used for the construction of the
Fock space of states started from the vacuum state. The latter is defined as
the state of quantum field nullified by the action of the annihilation
operator. The mode functions are sensitive to both local and global
characteristics of the background space-time and the same is the case for the
properties of the vacuum and particle states. Among the interesting areas of
research in quantum field theory is the dependence of those properties on the
geometry and topology of the background spacetime. The corresponding effects
play an important role in gravity, cosmology, in physical models with extra
compact dimensions, in condensed matter physics and in finite temperature
field theory (see, for example, [1, 2, 3]). As an example of the combined
influence of the spatial curvature and topology on the properties of quantum
vacuum, in the present paper we consider the generation of vacuum currents for
a scalar field localized on 2D curved tubes. These currents have common roots
with persistent currents in mesoscopic rings, widely studied in the literature
for different physical systems (see, e.g., [4, 5, 6, 7] and references
therein). The experimental detection of persistent currents in normal metal
rings has been reported in [8, 9].

The investigation of field theoretical effects in (2+1)-dimensional spacetimes
has attracted a great deal of attention. The interest is motivated by several
reasons. In addition to be simplified toy models of (3+1)-dimensional physics,
the change in the number of spatial dimension leads to new types of phenomena.
They include new mechanisms for symmetry breaking, parity violation, and
fractionalization of quantum numbers. From the fundamental point of view, an
important aspect in 2D gauge theories is the mechanism that generates a
topological mass for gauge bosons without breaking the gauge symmetry [10]. 2D
models also appear as effective field theories describing the physics of
compact subspace in Kaluza-Klein and braneworld type theories with extra
dimensions. The recent advances in synthesis methods and techniques for planar
condensed matter systems (2D materials) have increased significantly the
interest to 2D physics. The long-wavelength degrees of freedom in a number of
those systems is described by 2D relativistic field theory. Well known
examples are the graphene family materials with the low-energy excitations of
the electronic subsystem described by 2D Dirac equation involving the Fermi
velocity instead of light velocity (see, e.g., [11, 12]).

In the physics of 2D materials the topological issues play an important role
both in the space of states and in the coordinate space. In particular,
topological phases of matter and topological defects are currently the subject
of active research. Here we will investigate the influence of spatial topology
and curvature on the ground state properties of the 2D scalar field. As the
representative of those properties the current density is taken. The simplest
example of nontrivial topology is a cylindrical tube with constant radius. The
vacuum currents in that geometry for scalar and Dirac fields have been
investigated in [13, 14] as special cases of more general geometries with an
arbitrary number of spatial dimensions and with toroidally compactified
subspace. Both cases of infinite and finite length tubes were considered. The
corresponding finite temperature charge and current densities have been
discussed in [15, 16]. The vacuum currents induced by magnetic fluxes in
conical geometries, describing idealized cosmic strings, are considered in
[17, 18, 19, 20]. More realistic geometry of a finite thickness cosmic string
with a cylindrically symmetric core is discussed in [21]. The compactification
of cosmic strings along the axial direction may lead to the component of the
current density along that direction [22, 23]. The ground state charge and
current densities for a fermionic field on a two-dimensional circular ring
were studied in [24].

The geometric curvature of 2D materials provides an additional mechanism to
control their physical characteristics and opens new perspectives for
applications. Among the interesting features is the generation of effective
gauge fields [25, 26]. The investigation of the curvature induced effects in
those systems is also important from the point of view of fundamental physics
giving insights on the influence of gravitational field on quantum matter.
Motivated by these technological and fundamental perspectives, the effects of
curvature in 2D physics are in the focus of active research (see, for example,
[27]-[53], covering various aspects of the topic). Related to the topic of the
present paper, the vacuum expectation value of the current density for scalar
and fermionic fields in locally de Sitter (dS) and anti-de Sitter (AdS)
spacetimes with toroidal subspaces is investigated in [54, 55] (for a review
see [7]).

The paper is organized as follows. In the next section we describe the
background geometry and present some special cases. The scalar field modes and
the Hadamard function for a general rotationally symmetric tube are given in
Section 3. By using the Hadamard function, the vacuum expectation value of the
current density is investigated in Section 4. The vacuum currents for some
special cases of the tube geometry are presented in Section 5. As an
application of general formulas, in Section 6 we investigate the expectation
value of the current density on the Beltrami pseudosphere. The asymptotic
analysis and numerical results are presented. The main results of the are
summarized in Section 7. In Appendix A the expressions for the current
densities in (2+1)-dimensional locally dS and AdS spacetimes with a spatial
dimension compactified to a circle are presented.

##  2 Geometry of the problem and special cases

###  2.1 Background geometry and induced metric

We consider a cylindrical surface in a 3-dimensional Euclidean space with
coordinates $X^{i}=(X,Y,Z)$, $i=1,2,3$. The equation of the surface in the
parametric form is expressed as

$X=f(u)\cos\phi,\;Y=f(u)\sin\phi,\;Z=z(u),$ (2.1)

where $\phi$ is the angular coordinate, $0\leq\phi\leq 2\pi$. The radius of
the surface for a fixed $Z$ is given by the function $f(u)=\sqrt{X^{2}+Y^{2}}$
(see Fig. 1). In the corresponding (3+1)-dimensional Minkowski spacetime the
line element has the form $ds_{4}^{2}=\eta_{\mu\nu}dX^{\mu}dX^{\nu}$, where
$\eta_{\mu\nu}$ is the Minkowskian metric tensor. We will denote the spacetime
coordinates on the cylindrical surface by
$x^{i}=(x^{0}=t=X^{0},x^{1}=u,x^{2}=\phi)$. For the line element on the
surface one has $ds^{2}=g_{ik}dx^{i}dx^{k}$, where the components of the
induced metric $g_{ik}=\eta_{\mu\nu}\partial_{i}X^{\mu}\partial_{k}X^{\nu}$
are given by

$g_{00}=1,\;g_{11}=-\left[f^{\prime 2}(u)+z^{\prime
2}(u)\right],\;g_{22}=-f^{2}(u),$ (2.2)

and the off-diagonal components vanish. Here the prime stands for the
derivative of the function with respect to the argument.

Figure 1: Curved tube threaded by a magnetic flux.

The metric tensor can be further simplified introducing a new coordinate $w$
in accordance with

$w=w(u)=\int_{u_{0}}^{u}du\sqrt{f^{\prime 2}(u)+z^{\prime 2}(u)}.$ (2.3)

In terms of this coordinate the line element on the cylinder is written as

$ds^{2}=dt^{2}-dw^{2}-p^{2}(w)d\phi^{2},$ (2.4)

where the function $p(w)$ is defined by the relation $p(w)=f(u(w))$, with
$u=u(w)$ determined from (2.3). By taking into account that
$p^{\prime}(w)=1/\sqrt{1+z^{\prime 2}(u)/f^{\prime 2}(u)}$, for the function
$h(w)=z(u(w))$ one gets

$h(w)=\pm\int_{w_{0}}^{w}dw\sqrt{1-p^{\prime 2}(w)},$ (2.5)

with $w_{0}$ being a constant. In terms of $w$ the equation of the surface in
the parametric form is rewritten as

$X=p(w)\cos\phi,\;Y=p(w)\sin\phi,\;Z=h(w),$ (2.6)

where the function $h(w)$ is defined by (2.5). In coordinates $(t,w,\phi)$,
the nonzero components of the Ricci tensor and the Ricci scalar for a
(2+1)-dimensional spacetime determined by (2.4) have the form

$R_{1}^{1}=R_{2}^{2}=-\frac{p^{\prime\prime}(w)}{p(w)},\;R=-2\frac{p^{\prime\prime}(w)}{p(w)}.$
(2.7)

The radius of the tube for a fixed $w$ is given by $p(w)$ and for the length
of compact dimension we have $2\pi p(w)$.

###  2.2 Special cases

Let us consider some special cases of the general geometry described above.
The simplest one corresponds to a cylinder with constant radius $L$. In this
case one has

$p(w)=L$ (2.8)

and $-\infty<w<+\infty$. The second example is described by a linear function

$p(w)=\alpha w,\;\alpha=\mathrm{const}.$ (2.9)

Assuming $\alpha<1$, for the function $h(w)$ from (2.5) one finds

$h(w)=\sqrt{1-\alpha^{2}}w.$ (2.10)

where we have taken $w_{0}=0$. The line element takes the form

$ds^{2}=dt^{2}-dw^{2}-\alpha^{2}w^{2}d\phi^{2},$ (2.11)

with $0\leq w<\infty$. For $\alpha=1$ this is the line element on a plane in
polar coordinates. For $\alpha<1$ it describes a cone with an opening angle
$2\pi\alpha$. It is obtained from a plane cutting the angle
$2\pi\left(1-\alpha\right)$ and gluing the edges of the remaining angle. For a
given coordinate $Z=h(w)$ in the embedding space the radius of the circle is
given by (2.9). The tip of the cone is located at $(X,Y,Z)=(0,0,0)$. Note that
for $w\neq 0$ one has $R_{ik}=0$ and both the spacetime and space are flat.
The graphitic cones are among the possible condensed matter realizations of 2D
conical geometry. The corresponding values of the parameter $\alpha$ are
determined by the structure of the graphene hexagonal lattice and are given by
$\alpha=1-N_{c}/6$ with $N_{c}=1,2,3,4,5$. Graphitic cones with all those
values of the parameter $\alpha$ have been observed in experiments (see, e.g.,
[56, 57]).

As the next example we consider a constant curvature space with

$\frac{p^{\prime\prime}(w)}{p(w)}=\pm\frac{1}{a^{2}}=\mathrm{const}.$ (2.12)

For the corresponding Gaussian curvature one has $K=R/2=\mp 1/a^{2}$. In the
case of a constant positive curvature space (lower sign in (2.12)) the
solution can be taken in the form

$p(w)=L\sin\left(w/a\right),\;0\leq w/a\leq\pi.$ (2.13)

Introducing the angular coordinate $\theta=w/a$, $0\leq\theta\leq\pi$, the
line element is written as

$ds^{2}=dt^{2}-a^{2}\left(d\theta^{2}+\sin^{2}\theta d\phi^{2}\right).$ (2.14)

This corresponds to a sphere with radius $a$. The condensed matter
realizations of this geometry include fullerenes and topological insulators
with spherical surfaces [27, 28, 32, 36]. In the long-wavelength approximation
the dynamics of the corresponding electronic subsystem is described by the 2D
Dirac equation with a sphere as a background space.

For a negative curvature space (the upper sign) one gets the equation

$p^{\prime\prime}(w)=p(w)/a^{2}.$ (2.15)

Three separate subcases are realized. For the first one we take the solution
of (2.15) given by

$p(w)=Le^{w/a},\;-\infty<w<+\infty,$ (2.16)

where $L$ is a constant with dimension of length. For the line element on the
tube one gets

$ds^{2}=dt^{2}-dw^{2}-L^{2}e^{2w/a}d\phi^{2}.$ (2.17)

The spatial geometry described by this element corresponds to the Beltrami
pseudosphere. The coordinates of the embedding space are given by (2.6). For
the corresponding part of the surface we have $p^{\prime}(w)\leq 1$. This
shows that though in (2.16) one has $-\infty<w<+\infty$, only the part of the
space corresponding to

$w\leq w_{\mathrm{b}}\equiv-a\ln(L/a),$ (2.18)

can be embedded in a 3-dimensional Euclidean space. For the function $h(w)$ in
the embedding space (with the upper sign in (2.5)) one gets

$h(w)=a\left[\sqrt{1-e^{2\left(w-w_{\mathrm{b}}\right)/a}}-\mathrm{arctanh}\left(\sqrt{1-e^{2\left(w-w_{\mathrm{b}}\right)/a}}\right)\right].$
(2.19)

On the left panel of Fig. 1 we have presented the spatial geometry
corresponding to (2.17) as a surface (2.6) in an Euclidean space with
coordinates $(X,Y,Z)$.

|  |
---|---|---
Figure 2: The Beltrami (left), elliptic (middle) and hyperbolic (right)
pseudospheres embedded in a 3-dimensional Euclidean space.

For the second subcase of the negative curvature space we take

$p(w)=L\sinh(w/a),\;0\leq w<\infty,$ (2.20)

with the line element

$ds^{2}=dt^{2}-dw^{2}-L^{2}\sinh^{2}(w/a)d\phi^{2}.$ (2.21)

The spatial geometry corresponds to the elliptic pseudosphere. For the part
embedded in a 3-dimensional Euclidean space one finds

$0\leq w\leq w_{\mathrm{be}}\equiv a\,\mathrm{arccosh}(a/L),$ (2.22)

with $L/a\leq 1$ and the function $h(w)$ in (2.6) is given by

$h(w)=a\int_{0}^{w/a}dx\sqrt{1-(L/a)^{2}\cosh^{2}x}.$ (2.23)

The integral in the right-hand side is expressed in terms of the elliptic
functions. The middle panel in Fig. 2 presents the part of the elliptic
pseudosphere embedded in an Euclidean space.

The third subcase for the negative curvature space corresponds to the function

$p(w)=L\cosh(w/a),\;-\infty<w<\infty,$ (2.24)

and the line element is given by

$ds^{2}=dt^{2}-dw^{2}-L^{2}\cosh^{2}(w/a)d\phi^{2}.$ (2.25)

This correspond to the hyperbolic pseudosphere. For the piece of the geometry
embedded in a Euclidean space we have

$-w_{\mathrm{bh}}\leq w\leq w_{\mathrm{bh}}\equiv a\,\mathrm{arcsinh}(a/L).$
(2.26)

The function $h(w)$ is determined from

$h(w)=a\int_{0}^{w/a}dx\sqrt{1-(L/a)^{2}\sinh^{2}x}.$ (2.27)

Similar to the previous case this integral is expressed in terms of the
elliptic functions. On the right panel of Fig. 2 we display the part of the
hyperbolic pseudosphere embedded in Euclidean space in accordance with (2.6).

The condensed matter realizations of negative constant curvature surfaces by
2D materials have been discussed by several authors. In particular, those
structures have been used as simplified models for 2D black holes and
wormholes (see, for example, [33, 34, 35, 37, 41, 44, 45, 46, 49, 50, 51] and
references therein). One can get new kinds of structures combining different
geometries discussed above. For example, cylindrical and planar geometries
have been combined to construct simplified models for 2D wormholes. The
combination of spherical and cylindrical geometries is realized in capped
graphene nanotubes. The vacuum expectation values of the field squared and of
the energy-momentum tensor for a massive scalar field in the latter geometry
have been studied in [58].

##  3 Scalar field modes and the Hadamard function

Having specified the geometry we turn to the field. We consider a complex
scalar field $\varphi(x)$ localized on a cylindrical surface. The
corresponding field equation reads

$\left(g^{ik}D_{i}D_{k}+m^{2}+\xi
R\right)\varphi(x)=0,\;D_{k}=\nabla_{k}+ieA_{k},$ (3.1)

where $A_{i}$ is the vector potential for a classical gauge field, $\xi$ is
the curvature coupling parameter and $\nabla_{k}$ is the covariant derivative
operator corresponding to the metric tensor $g^{ik}$ determined by (2.4). In
the special cases of minimally and conformally coupled scalar fields one has
$\xi=0$ and $\xi=1/8$, respectively. The complex scalar field describes
charged excitations in Bose-Einstein condensates, phononic degrees of freedom
and surface plasmons (see [59, 60, 61] for some recent references). We
consider a background with nontrivial topology and the periodicity condition
along the direction of the angular coordinate $\phi$ needs to be specified. We
will impose a quasiperiodicity condition

$\varphi(t,w,\phi+2\pi)=e^{i\tilde{\alpha}_{p}}\varphi(t,w,\phi),$ (3.2)

with a constant phase $\tilde{\alpha}_{p}$. For the gauge field a simple
configuration will be considered with $A_{k}=(0,0,A_{2}=\mathrm{const})$ in
the spacetime coordinates $(t,w,\phi)$. This corresponds to a magnetic flux
$\Phi=-2\pi A_{2}$ in the embedding space, threading the tube. On the tube,
the field tensor for the gauge field is zero and the effect on the properties
of 2D quantum scalar field is of the Aharonov-Bohm type. The gauge field under
consideration is excluded from the field equation by the gauge transformation
$\varphi^{\prime}(x)=e^{ie\varkappa(x)}\varphi(x)$,$\;A_{k}^{\prime}=A_{k}-\partial_{k}\varkappa(x)$
with the function $\varkappa(x)=A_{2}\phi$. In the new gauge
$A_{k}^{\prime}=0$ and the scalar field obeys the periodicity condition

$\varphi^{\prime}(t,w,\phi+2\pi)=e^{i\alpha_{p}}\varphi^{\prime}(t,w,\phi),\;\alpha_{p}=\tilde{\alpha}_{p}+2\pi
eA_{2}.$ (3.3)

The phase shift in this condition is presented as $2\pi
eA_{2}=-2\pi\Phi/\Phi_{0}$, where $\Phi_{0}=2\pi/e$ is the flux quantum. The
discussion below will be presented in terms of the field $\varphi^{\prime}(x)$
omitting the prime.

The properties of the vacuum state $|0\rangle$ for the field $\varphi(x)$ are
contained in two-point functions describing the correlations of the vacuum
fluctuations in different spacetime points. For the evaluation of two-point
functions we need to have the complete set of positive and negative energy
scalar modes $\\{\varphi_{\sigma}^{(+)}(x),\varphi_{\sigma}^{(-)}(x)\\}$
obeying the field equation (3.1) with $A_{k}=0$ and the condition (3.3). Here,
$\sigma$ stands for the set of quantum numbers specifying the modes. For the
geometry described by (2.4) the mode functions are presented in the form

$\varphi_{\sigma}^{(\pm)}(x)=e^{ik_{n}\phi\mp i\omega
t}z_{\sigma}(w),\;k_{n}=n+\frac{\alpha_{p}}{2\pi},\;n=0,\pm 1,\pm 2,\ldots,$
(3.4)

where the eigenvalues of the momentum $k_{n}$ along the azimuthal direction
are determined by the condition (3.3). Substituting into (3.1), the following
equation is obtained for the function $z_{\sigma}(w)$:

$\frac{\left[p(w)z_{\sigma}^{\prime}(w)\right]^{\prime}}{p(w)}+\left[2\xi\frac{p^{\prime\prime}(w)}{p(w)}-\frac{u^{2}}{p^{2}(w)}+\lambda^{2}\right]z_{\sigma}(w)=0,$
(3.5)

where $\lambda^{2}=\omega^{2}-m^{2}$, $u=k_{n}$. We denote by $k_{w}$ the
quantum number related to the $w$-direction. The parameter $\lambda$ and the
energy $\omega=\sqrt{\lambda^{2}+m^{2}}$ depend on that quantum number. For
some geometries it will be convenient to take $\lambda$ in the role of the
quantum number $k_{w}$. The mode functions are specified by the set of quantum
numbers $\sigma=(k_{w},k_{n})$ and $z_{\sigma}(w)=z_{k_{w},k_{n}}(w)$. The
normalization condition for the modes has the form

$\int
dw\int_{0}^{2\pi}d\phi\sqrt{|g|}\varphi_{\sigma}^{(\pm)}(x)\varphi_{\sigma^{\prime}}^{(\pm)\ast}(x)=\frac{\delta_{nn^{\prime}}}{2\omega}\delta_{k_{w}k_{w}^{\prime}}.$
(3.6)

Here, $\delta_{k_{w}k_{w}^{\prime}}$ is understood as the Kronecker delta for
discrete quantum number $k_{w}$ and as the Dirac delta function
$\delta(k_{w}-k_{w}^{\prime})$ for continuous one. Substituting (3.4) in
(3.6), we get the orthonormalization condition for the function
$z_{k_{w},k_{n}}(w)$:

$\int
dw\,p(w)z_{k_{w},k_{n}}(w)z_{k_{w}^{\prime},k_{n}}^{\ast}(w)=\frac{\delta_{k_{w}k_{w}^{\prime}}}{4\pi\omega},$
(3.7)

where the limits of the integration are determined by the range of variation
of the coordinates $w$ and depend on the specific problem (see examples
below).

Having the complete set of mode functions we can evaluate the Hadamard
function, defined as the vacuum expectation value (VEV)

$G(x,x^{\prime})=\langle
0|\varphi(x)\varphi^{\dagger}(x^{\prime})+\varphi^{\dagger}(x^{\prime})\varphi(x)|0\rangle,$
(3.8)

where the dagger stands for the hermitian conjugate. Expanding the field
operator in terms of the complete set
$\\{\varphi_{\sigma}^{(+)},\varphi_{\sigma}^{(-)}\\}$, the following mode sum
formula is obtained:

$G(x,x^{\prime})=\sum_{\sigma}\sum_{s=+,-}\,\varphi_{\sigma}^{(s)}(x)\varphi_{\sigma}^{(s)\ast}(x^{\prime}),$
(3.9)

where $\sum_{\sigma}$ includes the summation over the discrete components of
the collective index $\sigma$ and the integration over the continuous ones.
Substituting the modes (3.4) we get

$G(x,x^{\prime})=2\sum_{n=-\infty}^{+\infty}e^{ik_{n}\Delta\phi}\sum_{k_{w}}\,z_{k_{w},k_{n}}(w)z_{k_{w},k_{n}}^{\ast}(w^{\prime})\cos(\omega\Delta
t),$ (3.10)

with $\Delta\phi=\phi-\phi^{\prime}$, $\Delta t=t-t^{\prime}$,
$\omega=\sqrt{\lambda^{2}+m^{2}}$, and $\lambda=\lambda(k_{w})$. The symbol
$\sum_{k_{w}}$ is understood as summation for discrete quantum number $k_{w}$
and integration over $k_{w}$ for the continuous one.

In order to see the effects of the compactification, it is of interest to
compare the function $G(x,x^{\prime})$ with the corresponding function in the
geometry where the coordinate $\phi$ is not compactified,
$-\infty<\phi<+\infty$. The line element is still given by (2.4). Now the mode
functions have the form

$\varphi_{0\sigma}^{(\pm)}=e^{iu\phi\mp i\omega
t}z_{k_{w},u}^{(0)}(w),\;-\infty<u<+\infty,$ (3.11)

with $\sigma=(k_{w},u)$. The function $z_{k_{w},u}^{(0)}(w)$ obeys the
equation (3.5). From the corresponding normalization condition one gets

$\int
dw\,p(w)z_{k_{w},u}^{(0)}(w)z_{k_{w}^{\prime},u}^{(0)\ast}(w)=\frac{\delta_{k_{w}k_{w}^{\prime}}}{4\pi\omega},$
(3.12)

which is the same as in (3.7). The corresponding Hadamard function is
presented in the form

$G_{0}(x,x^{\prime})=2\int_{-\infty}^{+\infty}du\,e^{iu\Delta\phi}\sum_{k_{w}}\,z_{k_{w},u}^{(0)}(w)z_{k_{w},u}^{(0)\ast}(w^{\prime})\cos(\omega\Delta
t),$ (3.13)

again with $\omega=\sqrt{\lambda^{2}+m^{2}}$.

In the equation (3.5) for $z_{\sigma}(w)$ the parameter $u=k_{n}$ appears in
the form $k_{n}^{2}$ and it can be assumed that
$z_{\sigma}(w)=z_{k_{w},k_{n}}(w)=z_{k_{w}}(w,|k_{n}|)$. We apply to the sum
over $n$ in (3.10) the summation formula [14]

$\sum_{n=-\infty}^{+\infty}g(k_{n})f(|k_{n}|)=\int_{0}^{\infty}du[g(u)+g(-u)]f(u)+i\int_{0}^{\infty}du[f(iu)-f(-iu)]\sum_{s=\pm
1}\frac{g(isu)}{e^{2\pi u+is\alpha_{p}}-1}.$ (3.14)

with $g(u)=e^{iu\Delta\phi}$ and

$f(u)=z_{k_{w}}(w,u)z_{k_{w}}^{\ast}(w^{\prime},u).$ (3.15)

It is worth mentioning that the complex conjugate in (3.15) is taken for real
values of $u$ and then the result is analytically continued in the complex
plane. We will assume that the function obtained in this way is analytic in
the right half-plane. The part with the first term in the right-hand side
gives the Hadamard function (3.13) and we get

$\displaystyle G(x,x^{\prime})$ $\displaystyle=$ $\displaystyle
G_{0}(x,x^{\prime})+2i\sum_{k_{w}}\cos(\omega\Delta
t)\,\int_{0}^{\infty}du\sum_{s=\pm 1}\frac{e^{-su\Delta\phi}}{e^{2\pi
u+is\alpha_{p}}-1}$ (3.16)
$\displaystyle\times[z_{k_{w}}(w,iu)z_{k_{w}}^{\ast}(w^{\prime},iu)-z_{k_{w}}(w,-iu)z_{k_{w}}^{\ast}(w^{\prime},-iu)].$

Note that

$\sum_{s=\pm 1}\frac{e^{-su\Delta\phi}}{e^{2\pi
u+is\alpha_{p}}-1}=\frac{2e^{2\pi
u}\cosh(2\pi\Delta\phi+i\alpha_{p})-2\cosh(u\Delta\phi)}{e^{4\pi u}-2e^{2\pi
u}\cos\alpha_{p}+1}.$ (3.17)

Again, it should be noted that in (3.16) the functions
$z_{k_{w}}^{\ast}(w^{\prime},\pm iu)$ are understood as the following: we
first consider the function $z_{k_{w}}^{\ast}(w^{\prime},u)$ for real $u$ and
then replace $u$ by $\pm iu$. Examples of the function $p(w)$ for which the
solution $z_{k_{w}}(w,u)$ is expressed in terms of the known functions are
given in Section 5.

The function $z_{\sigma}(w)=z_{k_{w}}(w,u)$ in (3.16) obeys the equation
(3.5). From here it follows that for any two solutions $z_{(1)k_{w}}(w,u)$ and
$z_{(2)k_{w}}(w,u)$ of (3.5) the following Wronskian relation takes place

$z_{(1)k_{w}}(w,u)\partial_{w}z_{(2)k_{w}}(w,u)-z_{(2)k_{w}}(w,u)\partial_{w}z_{(1)k_{w}}(w,u)=\frac{C}{p(w)},$
(3.18)

with a constant $C$ which may depend on $k_{w}$ and $u$. This relation is
valid for general $u$, including the complex values.

The representation (3.16) separates explicitly the contribution in the
Hadamard function induced by the compactification of the coordinate $\phi$.
The local geometry for the compactified and decompactified spaces is the same
and, hence, the corresponding divergences in the coincidence limit
$x^{\prime}\rightarrow x$ are the same as well. From here it follows that in
the evaluation of the VEVs for local physical observables, biliniear in the
field operator, the renormalization is required for the parts coming from the
function $G_{0}(x,x^{\prime})$. The renormalized topological contributions in
the VEVs are directly obtained from the second term in the right-hand side of
(3.16) or from its derivatives taking the coincidence limit. In the next
section this will be illustrated for the current density.

##  4 VEV of the current density

Given the Hadamard function, the VEVs of physical observables bilinear in the
field can be evaluated. As an important characteristic of the vacuum state
here we consider the VEV of the current density $\langle
0|j_{k}(x)|0\rangle\equiv\langle j_{k}(x)\rangle$, where the corresponding
operator is given by

$j_{k}(x)=ie[\varphi^{\dagger}(x)\partial_{k}\varphi(x)-(\partial_{k}\varphi(x))^{\dagger}\varphi(x)].$
(4.1)

The VEV is expressed in terms of the Hadamard function as

$\langle j_{k}(x)\rangle=\frac{i}{2}e\lim_{x^{\prime}\rightarrow
x}(\partial_{k}-\partial_{k}^{\prime})G(x,x^{\prime}).$ (4.2)

With the Hadamard function from (3.16), we see that the charge density
vanishes, $\langle j_{0}\rangle=0$.

For the contravariant component of the current density along the
$w$-direction, $\langle j^{1}\rangle=-\langle j_{1}\rangle$, we get

$\langle j^{1}\rangle=\langle
j^{1}\rangle_{0}+\frac{ie}{p(w)}\sum_{k_{w}}\,\int_{0}^{\infty}du\sum_{s=\pm
1}\frac{C_{k_{w}}(iu)-C_{k_{w}}(-iu)}{e^{2\pi u+is\alpha_{p}}-1},$ (4.3)

where

$\langle
j^{1}\rangle_{0}=\frac{e}{p(w)}\sum_{k_{w}}\int_{-\infty}^{+\infty}du\,C_{k_{w}}(u)$
(4.4)

is the current density in the geometry where the coordinate $\phi$ is not
compactified. Here, the function $C_{k_{w}}(u)$ for real $u$ is defined in
accordance with

$z_{k_{w}}^{\ast}(w,u)\partial_{w}z_{k_{w}}(w,u)-z_{k_{w}}(w,u)\partial_{w}z_{k_{w}}^{\ast}(w,u)=\frac{iC_{k_{w}}(u)}{p(w)}.$
(4.5)

For real $u$ the function $C_{k_{w}}(u)$ is real. As it has been emphasized
above, the renormalization is required only for the part $\langle
j^{1}\rangle_{0}$.

By using (3.16) and (4.2), for the physical component of the azimuthal current
density, $\langle j^{\phi}\rangle=p(w)\langle j^{2}\rangle=-\langle
j_{2}\rangle/p(w)$, one finds

$\langle
j^{\phi}\rangle=-\frac{2e}{p(w)}\sum_{k_{w}}\int_{0}^{\infty}du\,u\sum_{s=\pm
1}s\frac{z_{k_{w}}(w,iu)z_{k_{w}}^{\ast}(w,iu)-z_{k_{w}}(w,-iu)z_{k_{w}}^{\ast}(w,-iu)}{e^{2\pi
u+is\alpha_{p}}-1}.$ (4.6)

The azimuthal current vanishes in the geometry with uncompactified $\phi$ and
the current (4.6) is induced by the compatification. The component $\langle
j^{1}\rangle$ is an even function of $\alpha_{p}$ and the component $\langle
j^{\phi}\rangle$ is an odd function. An alternative expression is obtained
from (4.6) by using the expansion $1/(e^{y}-1)=\sum_{l=1}^{\infty}e^{-ly}$:

$\langle
j^{\phi}\rangle=\frac{4ie}{p(w)}\sum_{l=1}^{\infty}\sin\left(l\alpha_{p}\right)\sum_{k_{w}}\int_{0}^{\infty}du\,ue^{-2\pi
lu}\left[z_{k_{w}}(w,iu)z_{k_{w}}^{\ast}(w,iu)-z_{k_{w}}(w,-iu)z_{k_{w}}^{\ast}(w,-iu)\right].$
(4.7)

This current is an odd periodic function of the magnetic flux, enclosed by the
tube, with the period of flux quantum.

##  5 Vacuum currents in special cases

As applications of the general results given above, in this section we
consider special cases.

###  5.1 Cylinder with a constant radius

For a cylinder with a constant radius $L$ the mode functions have the form

$z_{\sigma}(w)=z_{k_{w}}(w,|k_{n}|)=\frac{e^{ik_{w}w}}{\sqrt{8\pi^{2}L\omega}},\;-\infty<k_{w}<+\infty,$
(5.1)

with $\omega=\sqrt{k_{w}^{2}+k_{n}^{2}/L^{2}+m^{2}}$. Substituting in (3.10)
with $\sum_{k_{w}}\rightarrow\int_{-\infty}^{+\infty}dk_{w}$ and integrating
over $k_{w}$, we get

$G(x,x^{\prime})=\frac{1}{2\pi^{2}L}\sum_{n=-\infty}^{+\infty}e^{ik_{n}\Delta\phi}K_{0}\left(\sqrt{k_{n}^{2}/L^{2}+m^{2}}\sqrt{\left(\Delta
w\right)^{2}-\left(\Delta t\right)^{2}}\right),$ (5.2)

where $K_{\nu}(x)$ is the Macdonald function [62]. For the function
$C_{k_{w}}(u)$ defined by (4.5) one has $C_{k_{w}}(u)=k_{w}/(4\pi^{2}\omega)$.
It is an odd function of $k_{w}$ and the integration over $k_{w}$ gives a zero
current along the $w$-direction, $\langle j^{1}\rangle=0$. For the azimuthal
current density from (4.7) we find

$\langle
j^{\phi}\rangle=\frac{e}{4\pi^{3}L^{2}}\sum_{l=1}^{\infty}\frac{1+2\pi
lLm}{l^{2}e^{2\pi lLm}}\sin\left(l\alpha_{p}\right).$ (5.3)

This result is the special case of the general formula from [15].

###  5.2 Current density on a cone

In the case of a cone with an opening angle $2\pi\alpha$ the function $p(w)$
is given by (2.9). As a quantum number $k_{w}$ we take $k_{w}=\lambda$. The
general solution of the equation (3.5) is a linear combination of the Bessel
and Neumann functions $J_{|k_{n}|/\alpha}(\lambda w)$ and
$Y_{|k_{n}|/\alpha}(\lambda w)$. For $|n|\geq 2$, from the normalizability
condition it follows that the function $J_{|k_{n}|/\alpha}(\lambda w)$ should
be taken. For the modes with $|n|=1$, under the condition
$1-|\alpha_{p}|/2\pi<\alpha$, and for $n=0$, under the condition
$|\alpha_{p}|<2\pi\alpha$, the irregular part with the function
$Y_{|k_{n}|/\alpha}(\lambda w)$ may also be present. In those cases, in order
to uniquely define the mode functions, an additional boundary condition is
required on the cone apex. Here we will consider a special case where the
function

$z_{k_{w}}(w,|k_{n}|)=c_{1}J_{|k_{n}|/\alpha}(\lambda w),\;0\leq
k_{w}=\lambda<\infty$ (5.4)

is taken for all values of $n$. By using the integral

$\int_{0}^{\infty}dw\,wJ_{|k_{n}|/\alpha}(\lambda
w)J_{|k_{n}|/\alpha}(\lambda^{\prime}w)=\frac{1}{\lambda}\delta(\lambda-\lambda^{\prime}),$
(5.5)

from (3.7) for the normalization coefficient one gets
$c_{1}=\sqrt{\lambda/(4\pi\alpha\omega)}$. With this coefficient, the function
$z_{k_{w}}(w,u)$ in (4.5) is real and, hence, $C_{k_{w}}(u)=0$. This shows
that $\langle j^{1}\rangle=\langle j^{1}\rangle_{0}=0$ for the current density
in the axial direction.

Substituting the functions (5.4) in (4.6), for the VEV of the azimuthal
current we find

$\langle j^{\phi}\rangle=\frac{2e}{\pi
w}\int_{0}^{\infty}d\lambda\frac{\lambda}{\sqrt{\lambda^{2}+m^{2}}}\int_{0}^{\infty}du\,u\,\mathrm{Im}\left(\frac{1}{e^{2\pi\alpha
u+i\alpha_{p}}-1}\right)\mathrm{Im}\left[J_{iu}^{2}(\lambda w)\right],$ (5.6)

where the integration variable is redefined as $u/\alpha\rightarrow u$. For
the further transformation we employ the integral representation

$\frac{1}{\sqrt{\lambda^{2}+m^{2}}}=\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}d\tau
e^{-\tau^{2}(\lambda^{2}+m^{2})}\ .$ (5.7)

With this representation, the integral over $\lambda$ in (5.6) is evaluated by
using the formula from [63]. This gives

$\int_{0}^{\infty}d\lambda\lambda
e^{-\tau^{2}\lambda^{2}}\mathrm{Im}\left[J_{iu}^{2}(\lambda
w)\right]=-\frac{\sinh\left(\pi u\right)}{2\pi\tau^{2}}e^{-x}K_{iu}(x),$ (5.8)

with $x=w^{2}/(2\tau^{2})$. Passing from $\tau$ to a new integration variable
$x$ one gets

$\langle
j^{\phi}\rangle=\frac{e\sin\alpha_{p}}{\sqrt{2}\pi^{5/2}w^{2}}\int_{0}^{\infty}\frac{dx}{\sqrt{x}}e^{-x-w^{2}m^{2}/2x}\int_{0}^{\infty}du\,\frac{u\sinh\left(\pi
u\right)K_{iu}(x)}{\cosh(2\alpha u)-\cos\alpha_{p}},$ (5.9)

where we have used

$\mathrm{Im}\left(\frac{1}{e^{2\alpha
u+i\alpha_{p}}-1}\right)=-\frac{1}{2}\frac{\sin\alpha_{p}}{\cosh(2\alpha
u)-\cos\alpha_{p}}.$ (5.10)

An alternative expression for the current density in general number of spatial
dimensions $D$ is given in [21]. Specified for the case $D=2$ it has the form

$\displaystyle\langle j^{\phi}\rangle$ $\displaystyle=$
$\displaystyle\frac{ewm^{3}}{\pi}\left[\sum_{n=1}^{[1/2\alpha]}\sin(n\alpha_{p})\sin(2\pi
n\alpha)g(2mw\sin(\pi n\alpha))\right.$ (5.11)
$\displaystyle\left.+\frac{1}{2\pi\alpha}\int_{0}^{\infty}dy\,f(\alpha,\alpha_{p},y)\frac{g(2mw\cosh(y/2))\sinh
y}{\cosh(y/\alpha)-\cos(\pi/\alpha)}\right],$

where $[1/2\alpha]$ stands for the integer part of $1/2\alpha$,
$g(x)=(1+x)e^{-x}/x^{3}$ and

$f(\alpha,\alpha_{p},y)=\sin(\alpha_{p}/2\alpha)\sinh[(1-\alpha_{p}/2\pi)y/\alpha]-\sin[\pi(1-\alpha_{p}/2\pi)/\alpha]\sinh(\alpha_{p}y/2\pi\alpha).$
(5.12)

For $\alpha>1/2$ the first term in the square brackets of (5.11) is absent.

The formula (5.9) is further simplified for a massless field by using the
integration formula [63]

$\int_{0}^{\infty}\frac{dx}{\sqrt{x}}e^{-x}K_{iu}(x)=\sqrt{\frac{\pi}{2}}\frac{\pi}{\cosh\left(\pi
u\right)}.$ (5.13)

This gives

$\langle
j^{\phi}\rangle|_{m=0}=\frac{e\sin\alpha_{p}}{2\pi^{3}w^{2}}\int_{0}^{\infty}du\,\frac{u\tanh
u}{\cosh(2\alpha u)-\cos\alpha_{p}}.$ (5.14)

The right-hand side of (5.14) describes the leading behavior of the current
density for a massive field near the cone apex: $\langle
j^{\phi}\rangle\approx\langle j^{\phi}\rangle|_{m=0}$ for $mw\ll 1$. For
massive fields and at large distances, assuming $mw\gg 1$, the current density
is suppressed by the factor $e^{-2mw\sin(\pi\alpha)}$ for $1/2\leq\alpha<1$
and by the factor $e^{-2mw}$ in the case $\alpha\leq 1/2$.

##  6 Hadamard function and the current density on the Beltrami pseudosphere

In this section we specify the general results described above for the curved
geometry corresponding to the Beltrami pseudosphere.

###  6.1 Mode functions

For the Beltrami pseudosphere the line element has the form (2.17).
Introducing a new coordinate $r$ in accordance with

$r=ae^{-w/a},\;0\leq r<\infty,$ (6.1)

the corresponding expression takes the form

$ds^{2}=dt^{2}-\frac{a^{2}}{r^{2}}\left(dr^{2}+L^{2}d\phi^{2}\right).$ (6.2)

For the part of the manifold that can be embedded in a 3-dimensional Euclidean
space one has $r\geq L$. Note that the geometry (6.2) is conformally related
to the 2D Rinlder spacetime described by the line element
$ds_{\mathrm{R}}^{2}=r^{2}d\tau^{2}-dr^{2}-dy^{2}$ with dimensionless time
coordinate $\tau=t/a$ and compact coordinate $y=L\phi$:

$ds^{2}=\frac{a^{2}}{r^{2}}ds_{\mathrm{R}}^{2}.$ (6.3)

The current densities in Rinlder spacetime with a toroidally compact subspace
in general number of spatial dimensions have been recently investigated in
[64].

For the Beltrami pseudosphere the equation (3.5) is rewritten as

$z_{\sigma}^{\prime\prime}(r)+\left(\frac{2\xi+a^{2}\lambda^{2}}{r^{2}}-\frac{k_{n}^{2}}{L^{2}}\right)z_{\sigma}(r)=0,$
(6.4)

with the general solution

$z_{\sigma}(r)=\sqrt{y}\left[c_{1}I_{i\nu}(y)+c_{2}K_{i\nu}(y)\right],\;y=|k_{n}|r/L\,.$
(6.5)

Here $I_{\mu}(y)$ is the modified Bessel function and
$\nu=\sqrt{a^{2}\lambda^{2}+2\xi-1/4}$. As quantum numbers specifying the mode
functions we can take the set $\sigma=(\nu,n)$. For the energy of a given mode
we get

$\omega=\frac{1}{a}\sqrt{\nu^{2}+\nu_{m}^{2}},$ (6.6)

where

$\nu_{m}^{2}=m^{2}a^{2}+1/4-2\xi.$ (6.7)

For a conformally coupled field one has $\nu_{m}=ma$.

From the normalizability condition for the mode function it follows that
$c_{1}=0$. The normalization condition (3.7), written in terms of the
coordinate $r$, takes the form

$2\pi
La^{2}|c_{2}|^{2}\int_{0}^{\infty}\frac{dy}{y}\,K_{i\nu}(y)K_{i\nu^{\prime}}(y)=\frac{\delta(\nu-\nu^{\prime})}{2\omega}.$
(6.8)

By taking unto account that $K_{-i\nu}(y)=K_{i\nu}(y)$, for the quantum number
$\nu$ one has $0<\nu<\infty$. Note that for the values of the curvature
coupling parameter in the range $2\xi>m^{2}a^{2}+1/4$ the energy becomes
imaginary for the modes with $\nu^{2}<2\xi-m^{2}a^{2}-1/4$. This leads to the
instability of the vacuum state. In the discussion below we will assume that
$2\xi\leq m^{2}a^{2}+1/4$. In particular, this condition is satisfied by the
most important special cases of minimally ($\xi=0$) and conformally
($\xi=1/8$) coupled fields. By using the integration formula [65]

$\int_{0}^{\infty}\frac{dy}{y}\,K_{i\nu}(y)K_{i\nu^{\prime}}(y)=\frac{\pi^{2}\delta\left(\nu-\nu^{\prime}\right)}{2\nu\sinh(\nu\pi)},$
(6.9)

for the normalization coefficient we find

$c_{2}=\sqrt{\frac{\nu\sinh(\nu\pi)}{2\pi^{3}La^{2}\omega}}.$ (6.10)

Hence, the normalized mode functions (3.4) take the form

$\varphi_{\sigma}^{(\pm)}(x)=\sqrt{\frac{\nu\sinh(\nu\pi)y}{2\pi^{3}La^{2}\omega}}K_{i\nu}(y)e^{ik_{n}\phi\mp
i\omega t},\;y=|k_{n}|r/L,$ (6.11)

with $k_{w}=\nu$ and $\omega$ given by (6.6). Note that the corresponding
function

$z_{\sigma}(w)=z_{k_{w}}(w,|k_{n}|)=c_{2}\sqrt{y}K_{i\nu}(y)$ (6.12)

is real.

###  6.2 Hadamard function

Having the mode functions, for the Hadamard function we get

$\displaystyle G(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{rr^{\prime}}}{\pi^{3}La^{2}}\sum_{n=-\infty}^{+\infty}e^{ik_{n}\Delta\phi}\int_{0}^{\infty}d\nu\,\frac{\nu}{\omega}\sinh(\nu\pi)$
(6.13) $\displaystyle\times
K_{i\nu}(|k_{n}|r/L)K_{i\nu}(|k_{n}|r^{\prime}/L)\cos(\omega\Delta t),$

with $\omega$ from (6.6). For the corresponding function in the geometry (6.2)
with uncompactified coordinate $\phi$, $-\infty<\phi<+\infty$, one has

$G_{0}(x,x^{\prime})=\frac{2\sqrt{rr^{\prime}}}{\pi^{3}La^{2}}\int_{0}^{\infty}du\,\cos(u\Delta\phi)\int_{0}^{\infty}d\nu\,\frac{\nu}{\omega}\sinh(\nu\pi)K_{i\nu}(ur/L)K_{i\nu}(ur^{\prime}/L)\cos(\omega\Delta
t).$ (6.14)

The integral over $u$ is evaluated by using the formula [63]

$\int_{0}^{\infty}du\,\cos(u\Delta\phi)K_{i\nu}(ur/L)K_{i\nu}(ur^{\prime}/L)=\frac{L\pi^{2}}{4\sqrt{rr^{\prime}}\cosh(\nu\pi)}P_{i\nu-1/2}\left(\frac{r^{2}+r^{\prime
2}+(L\Delta\phi)^{2}}{2rr^{\prime}}\right),$ (6.15)

where $P_{i\nu-1/2}(x)$ is the Legendre function. This gives

$G_{0}(x,x^{\prime})=\frac{a^{-2}}{2\pi}\int_{0}^{\infty}d\nu\,\frac{\nu}{\omega}\tanh(\nu\pi)\cos(\omega\Delta
t)P_{i\nu-1/2}\left(\frac{r^{2}+r^{\prime
2}+(L\Delta\phi)^{2}}{2rr^{\prime}}\right).$ (6.16)

The Hadamard function is decomposed as (3.16). For the function appearing in
the topological part one has

$z_{k_{w}}(w,iu)z_{k_{w}}^{\ast}(w^{\prime},iu)-z_{k_{w}}(w,-iu)z_{k_{w}}^{\ast}(w^{\prime},-iu)=i\frac{\sqrt{rr^{\prime}}}{2\pi
La^{2}}\frac{\nu}{\omega}\mathrm{Im}[J_{i\nu}(ur/L)J_{i\nu}(ur^{\prime}/L)]\,,$
(6.17)

where we have used the relation

$\mathrm{Im}[K_{i\nu}(iy)K_{i\nu}(iy^{\prime})]=\frac{\pi^{2}}{2}\frac{\mathrm{Im}[J_{i\nu}(y)J_{i\nu}(y^{\prime})]}{\sinh(\nu\pi)}.$
(6.18)

Hence, the Hadamard function is decomposed as

$\displaystyle G(x,x^{\prime})$ $\displaystyle=$ $\displaystyle
G_{0}(x,x^{\prime})-\frac{\sqrt{rr^{\prime}}}{\pi
La^{2}}\int_{0}^{\infty}d\nu\,\frac{\nu}{\omega}\cos(\omega\Delta t)$ (6.19)
$\displaystyle\times\int_{0}^{\infty}du\,\mathrm{Im}[J_{i\nu}(ur/L)J_{i\nu}(ur^{\prime}/L)]\sum_{s=\pm
1}\frac{e^{-su\Delta\phi}}{e^{2\pi u+is\alpha_{p}}-1}.$

We can further transform this formula by using the expansion
$1/(e^{y}-1)=\sum_{l=1}^{\infty}e^{-ly}$. With this expansion, the integral
over $u$ is expressed in terms of the Legendre function $Q_{\mu}(x)$:

$\displaystyle G(x,x^{\prime})$ $\displaystyle=$ $\displaystyle
G_{0}(x,x^{\prime})-\frac{a^{-2}}{\pi^{2}}\sideset{}{{}^{\prime}}{\sum}_{l=-\infty}^{+\infty}e^{-li\alpha_{p}}\int_{0}^{\infty}d\nu\,\frac{\nu}{\omega}\cos(\omega\Delta
t)\,$ (6.20)
$\displaystyle\times\mathrm{Im}\left[Q_{i\nu-1/2}\left(\frac{r^{2}+r^{\prime
2}+L^{2}\left(2\pi l+s\Delta\phi\right)^{2}}{2rr^{\prime}}\right)\right],$

where the prime on the summation sign means that the term $l=0$ should be
omitted. Now, by using the relation between the Legendre functions [62], we
can see that (note that the function $P_{i\nu-1/2}(z)$ is a real function for
real $\nu$ and $z$)

$\mathrm{Im}[Q_{i\nu-1/2}(z)]=-\frac{\pi}{2}\tanh\left(\pi\nu\right)P_{i\nu-1/2}(z).$
(6.21)

By taking into account the expression (6.14) for the function
$G_{0}(x,x^{\prime})$, the final expression is written in the form

$G(x,x^{\prime})=\frac{a^{-2}}{2\pi}\sum_{l=-\infty}^{\infty}e^{-il\alpha_{p}}\int_{0}^{\infty}d\nu\,\frac{\nu}{\omega}\tanh\left(\pi\nu\right)\cos(\omega\Delta
t)\,P_{i\nu-1/2}\left(\frac{r^{2}+r^{\prime 2}+L^{2}\left(\Delta\phi+2\pi
l\right)^{2}}{2rr^{\prime}}\right).$ (6.22)

where the $l=0$ term corresponds to the function $G_{0}(x,x^{\prime})$.

The formula (6.22) could also be obtained in another way, by using the
integral representation [66]

$K_{i\nu}(br)K_{i\nu}(br^{\prime})=\frac{1}{2}\int_{0}^{\infty}\frac{dy}{y}\,\exp\left(-\frac{y}{2}-b^{2}\frac{r^{2}+r^{\prime
2}}{2y}\right)K_{i\nu}(b^{2}rr^{\prime}/y)$

for the product of the Macdonald functions in (6.13). This gives

$\displaystyle G(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{rr^{\prime}}}{2\pi^{3}La^{2}}\sum_{n=-\infty}^{+\infty}e^{ik_{n}\Delta\phi}\int_{0}^{\infty}d\nu\,\frac{\nu}{\omega}\sinh(\nu\pi)\cos(\omega\Delta
t)$ (6.23)
$\displaystyle\times\int_{0}^{\infty}\frac{du}{u}\exp\left(-\frac{k_{n}^{2}}{2uL^{2}}-u\frac{r^{2}+r^{\prime
2}}{2}\right)K_{i\nu}(rr^{\prime}u),$

where we have introduced a new integration variable $u=k_{n}^{2}/(L^{2}y)$. As
the next step we apply the Poisson resummation formula to the series over $n$.
The following relation is obtained:

$\sum_{n=-\infty}^{+\infty}\exp\left(ik_{n}\Delta\phi-\frac{k_{n}^{2}}{2uL^{2}}\right)=L\sqrt{2\pi
u}\sum_{l=-\infty}^{\infty}\exp\left[-il\alpha_{p}-\frac{u}{2}L^{2}\left(\Delta\phi+2\pi
l\right)^{2}\right].$ (6.24)

Substituting this in (6.23), the integral over $u$ is evaluated with the help
of the formula

$\int_{0}^{\infty}\frac{du}{\sqrt{u}}e^{-pu}K_{i\nu}(cu)=\frac{\pi^{3/2}P_{i\nu-1/2}(p/c)}{\sqrt{2c}\cosh(\pi\nu)},$

and we obtain the representation (6.22).

Another representation for the Hadamard function is obtained from (6.23) by
taking into account the relation (6.24):

$\displaystyle G(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{rr^{\prime}/2}}{\pi^{5/2}a^{2}}\sum_{l=-\infty}^{\infty}e^{-il\alpha_{p}}\int_{0}^{\infty}\frac{du}{\sqrt{u}}\exp\left[-\frac{u}{2}\left(r^{2}+r^{\prime
2}+L^{2}\left(\Delta\phi+2\pi l\right)^{2}\right)\right]$ (6.25)
$\displaystyle\times\int_{0}^{\infty}d\nu\,\frac{\nu}{\omega}\sinh(\nu\pi)\cos(\omega\Delta
t)K_{i\nu}(rr^{\prime}u),$

By using the formula

$\sinh\left(\pi\nu\right)K_{i\nu}(z)=\frac{\pi
i}{2}\left[I_{i\nu}(z)-I_{-i\nu}(z)\right],$ (6.26)

for the Macdonald function, this gives

$\displaystyle G(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{i\sqrt{rr^{\prime}}}{\left(2\pi\right)^{3/2}a^{2}}\sum_{l=-\infty}^{\infty}e^{-il\alpha_{p}}\int_{0}^{\infty}\frac{du}{\sqrt{u}}\exp\left[-\frac{u}{2}\left(r^{2}+r^{\prime
2}+L^{2}\left(\Delta\phi+2\pi l\right)^{2}\right)\right]$ (6.27)
$\displaystyle\times\int_{0}^{\infty}d\nu\,\frac{\nu}{\omega}\sinh(\nu\pi)\cos(\omega\Delta
t)\left[I_{i\nu}(rr^{\prime}u)-I_{-i\nu}(rr^{\prime}u)\right],$

We rotate the integration contour over $\nu$ by the angle $\pi/2$ for the term
with $I_{-i\nu}(xx^{\prime}u)$ and by the angle $-\pi/2$ for the term with
$I_{i\nu}(xx^{\prime}u)$. Assuming that $\nu_{m}^{2}\geq 0$, for the integral
we get

$\int_{0}^{\infty}d\nu\,\frac{\nu}{\omega}\cos(\omega\Delta
t)\left[I_{i\nu}(rr^{\prime}u)-I_{-i\nu}(rr^{\prime}u)\right]=\frac{2}{i}a\int_{\nu_{m}}^{\infty}d\nu\,\frac{\nu\cosh(\sqrt{\nu^{2}-\nu_{m}^{2}}\Delta
t/a)}{\sqrt{\nu^{2}-\nu_{m}^{2}}}I_{\nu}(rr^{\prime}u).$ (6.28)

By using the integration formula [63]

$\int_{0}^{\infty}\frac{du}{\sqrt{u}}e^{-bu/2}I_{\nu}(rr^{\prime}u)=\sqrt{\frac{2}{\pi
rr^{\prime}}}Q_{\nu-1/2}\left(\frac{b}{2rr^{\prime}}\right),$ (6.29)

for the Hadamard function we find the representation

$G(x,x^{\prime})=\frac{1}{\pi^{2}a}\sum_{l=-\infty}^{\infty}e^{il\alpha_{p}}\int_{\nu_{m}}^{\infty}d\nu\,\frac{\nu\cosh(\sqrt{\nu^{2}-\nu_{m}^{2}}\Delta
t/a)}{\sqrt{\nu^{2}-\nu_{m}^{2}}}Q_{\nu-1/2}\left(\frac{r^{2}+r^{\prime
2}+L^{2}\left(\Delta\phi-2\pi l\right)^{2}}{2rr^{\prime}}\right).$ (6.30)

Note that the Legendre function in this formula is expressed in terms of the
hypergeometric function as

$Q_{\nu-1/2}(z)=\frac{\sqrt{\pi}\Gamma\left(\nu+1/2\right)}{\left(2z\right)^{\nu+1/2}\Gamma\left(\nu+2\right)}F\left(\frac{\nu+3/2}{2},\frac{\nu+1/2}{2};\nu+1,\frac{1}{z^{2}}\right).$
(6.31)

As it is seen from (2.17), the geometry of cylindrical tube with constant
radius is obtained from the case of the Beltrami pseudosphere in the limit
$a\rightarrow 0$ for fixed $w$. From (6.1) it follows that the coordinate $r$
is large in that limit and for the argument of the Legendre function in (6.30)
one has

$\frac{r^{2}+r^{\prime 2}+L^{2}\left(\Delta\phi-2\pi
l\right)^{2}}{2rr^{\prime}}\approx 1+\frac{\left(\Delta
x_{l}\right)^{2}}{2a^{2}},\;\Delta x_{l}=\sqrt{\left(\Delta
w\right)^{2}+L^{2}\left(\Delta\phi-2\pi l\right)^{2}}.$ (6.32)

By taking into account that in the limit under consideration $\nu_{m}$ is
large, we use the asymptotic formula for the Legendre function [67]

$Q_{\nu-1/2}(\cosh u)\approx\left(\frac{u}{\sinh u}\right)^{1/2}K_{0}\left(\nu
u\right),\;\nu\gg 1.$ (6.33)

For the function in (6.30) one has $u\approx a\Delta x_{l}$. Substituting
(6.33) in (6.30) and redefining the integration variable as
$a\nu\rightarrow\nu$, in the leading order we get

$G(x,x^{\prime})\approx\frac{1}{\pi^{2}}\sum_{l=-\infty}^{\infty}e^{il\alpha_{p}}\int_{m}^{\infty}d\nu\,\frac{\nu\cosh(\Delta
t\sqrt{\nu^{2}-m^{2}})}{\sqrt{\nu^{2}-m^{2}}}K_{0}\left(\nu\Delta
x_{l}\right).$ (6.34)

The integral is evaluated by using the formula from [63] with the result

$G(x,x^{\prime})\approx\frac{1}{2\pi}\sum_{l=-\infty}^{\infty}\frac{e^{il\alpha_{p}}e^{-m\sqrt{\left(\Delta
x_{l}\right)^{2}-\left(\Delta t\right)^{2}}}}{\sqrt{\left(\Delta
x_{l}\right)^{2}-\left(\Delta t\right)^{2}}}.$ (6.35)

The expression on the right-hand side coincides with the Hadamard function for
a cylindrical tube of constant radius (see, for example, [15]).

###  6.3 Current density

Similar to the case of conical geometry the function $z_{k_{w}}(w,u)$ is real
for real $u$ (see (6.12)) and $C_{k_{w}}(u)=0$. Hence, the current density
along the axial direction vanishes, $\langle j^{1}\rangle=0$. The VEV of the
current density in the compact dimension is obtained from the general formula
(4.6). By using (6.12) we get

$\langle j^{\phi}\rangle=-\frac{er^{2}}{\pi
L^{2}a^{2}}\sin\alpha_{p}\int_{0}^{\infty}d\nu\,\frac{\nu}{\sqrt{\nu^{2}+\nu_{m}^{2}}}\,\int_{0}^{\infty}du\,\frac{u\,\mathrm{Im}[J_{i\nu}^{2}(ur/L)]}{\cosh(2\pi
u)-\cos\alpha_{p}}.$ (6.36)

As seen, the current density depends on $r$ and $L$ in the form of the ratio

$\frac{L}{r}=\frac{L}{a}e^{w/a}.$ (6.37)

For a fixed $w$, the proper length of the compact dimension is given by $2\pi
Le^{w/a}$. The current density is an odd periodic function of the parameter
$\alpha_{p}$ with the period $2\pi$. This corresponds to periodicity with
respect to the magnetic flux with the period of flux quantum. An alternative
representation is obtained by using the formula (6.22) for the Hadamard
function:

$\langle
j^{\phi}\rangle=-\frac{2eL}{ra^{2}}\sum_{l=1}^{\infty}l\sin(l\alpha_{p})\int_{0}^{\infty}d\nu\,\frac{\nu\tanh\left(\pi\nu\right)}{\sqrt{\nu^{2}+\nu_{m}^{2}}}\,P_{i\nu-1/2}^{\prime}\left(1+2\left(\pi
lL/r\right)^{2}\right),$ (6.38)

where the prime means the derivative with respect to the argument of the
function.

Another representation for the current density is obtained from (6.30) by
using (4.2):

$\langle j^{\phi}\rangle=-\frac{4eL}{\pi
ra^{2}}\sum_{l=1}^{\infty}l\sin\left(l\alpha_{p}\right)\int_{\nu_{m}}^{\infty}d\nu\,\nu\frac{Q_{\nu-1/2}^{\prime}\left(1+2\left(\pi
lL/r\right)^{2}\right)}{\sqrt{\nu^{2}-\nu_{m}^{2}}}.$ (6.39)

In the limit $a\rightarrow\infty$, with fixed value of $w$, the parameter
$\nu$ and the coordinate $r$ are large and we use the asymptotic (6.33) for
the Legendre function. It can be seen that in the leading order from (6.39)
the result (5.3) is obtained for the current density in a cylindrical tube
with constant radius.

Let us consider the behavior of the current density in the asymptotic regions
of the ratio $r/L$. For $r/L\ll 1$ the argument of the Legendre function in
(6.39) is large and we use the formula [67]

$Q_{\nu-1/2}\left(x\right)\approx\frac{\sqrt{\pi}\Gamma\left(\nu+1/2\right)}{\Gamma\left(\nu+1\right)\left(2x\right)^{\nu+1/2}},$
(6.40)

for $x\gg 1$. In the leading order this gives

$\langle
j^{\phi}\rangle\approx\frac{e\left(r/L\right)^{2}}{\pi^{7/2}a^{2}}\sum_{l=1}^{\infty}\frac{\sin\left(l\alpha_{p}\right)}{l^{2}}\int_{\nu_{m}}^{\infty}d\nu\,\frac{\Gamma\left(\nu+3/2\right)e^{-2\nu\ln\left(2\pi
l\right)}}{\Gamma\left(\nu\right)\sqrt{\nu^{2}-\nu_{m}^{2}}},$ (6.41)

and the current density tends to zero like $\left(r/L\right)^{2}$. In the
opposite limit $r/L\gg 1$ the dominant contribution to the integral in (6.39)
comes from large $\nu$ and we employ the asymptotic (6.33). The integral is
evaluated by using the formula from [63] and the leading order term is
expressed as

$\langle
j^{\phi}\rangle\approx\frac{er^{2}}{4\pi^{3}L^{2}a^{2}}\sum_{l=1}^{\infty}\frac{\sin\left(l\alpha_{p}\right)}{l^{2}},\;r\gg
L.$ (6.42)

For a given $r$ the radius of the tube is given by $aL/r$ and the limit under
consideration corresponds to the region where the radius of the tube is much
smaller than the curvature radius $a$.

For a conformally coupled massless field we have $\nu_{m}=0$ and the
expression for the current density is transformed to

$\langle j^{\phi}\rangle=-\frac{4eL}{\pi
ra^{2}}\sum_{l=1}^{\infty}l\sin\left(l\alpha_{p}\right)\int_{0}^{\infty}d\nu\,Q_{\nu-1/2}^{\prime}\left(1+2\left(\pi
lL/r\right)^{2}\right).$ (6.43)

As it has been mentioned above, the geometry at hand is conformally related to
the 2D Rindler spacetime with a compact dimension (see (6.3)). For a
conformally coupled massless field we expect the relation

$\langle j^{\phi}\rangle=\frac{r^{2}}{a^{2}}\langle
j^{\phi}\rangle_{\mathrm{R}},$ (6.44)

where $\langle j^{\phi}\rangle_{\mathrm{R}}$ is the current density in the
Rindler spacetime. The expression for the latter is obtained from the general
formula in [64] taking $D=2$ for spatial dimension and making the replacement
$L\rightarrow 2\pi L$:

$\langle
j^{\phi}\rangle_{\mathrm{R}}=\frac{eL}{4r^{3}}\sum_{l=1}^{\infty}l\sin\left(l\alpha_{p}\right)\left\\{\frac{1}{\left(\pi
lL/r\right)^{3}}-\int_{0}^{\infty}\,dx\frac{\left[\cosh^{2}x+(\pi
lL/r)^{2}\right]^{-\frac{3}{2}}}{x^{2}+\pi^{2}/4}\right\\}.$ (6.45)

We have checked numerically that the following relation is valid:

$\int_{0}^{\infty}d\nu\,Q_{\nu-1/2}\left(1+2u^{2}\right)=\frac{\pi}{4}\left[\frac{1}{u}-\int_{0}^{\infty}d\nu\frac{\left(\cosh^{2}\nu+u^{2}\right)^{-\frac{1}{2}}}{\nu^{2}+\pi^{2}/4}\right].$
(6.46)

This relation shows that the connection (6.44) indeed takes place between the
current densities on the Beltrami pseudosphere and in 2D Rindler spacetime
with compact dimension.

The left panel of Fig. 3 presents the dependence of the current density on the
mass of the field for $L/r=0.5$ and $\alpha_{p}=2\pi/5$. The right panel
displays the current density versus $\alpha_{p}/2\pi$ for the same value of
$L/r$ and for $ma=0.5$. On both panels the numbers near the curves are the
corresponding values of the curvature coupling parameter.

|
---|---
Figure 3: The current density as a function of the mass (left panel) and of
the phase in the quasiperiodicity condition (right panel) for $L/r=0.5$. For
the left panel we have taken $\alpha_{p}=2\pi/5$ and for the right panel
$ma=0.5$.

In Fig. 4 we have plotted the dependence of the current density on the ratio
$L/r$ for minimally and conformally coupled massless fields. The graphs are
plotted for $\alpha_{p}=\pi/2$. As was expected, the current density tends to
zero in the limit $L/r\rightarrow\infty$.

Figure 4: The current density versus $L/r$ for conformally ($\xi=1/8$) and
minimally ($\xi=0$) coupled massless fields and for fixed $\alpha_{p}=\pi/2$.

It is of interest to compare the vacuum current densities for different
geometries of the 2D tube. The solid curves in Fig. 5 present the ratio of the
current densities on the Beltrami pseudosphere and on the tube with a constant
radius, $\langle j^{\phi}\rangle/\langle j^{\phi}\rangle_{c}$, as a function
of the proper radius $L_{(p)}$ of the tube in units of the curvature radius
$a$. For the Beltrami pseudosphere $L_{(p)}=aL/r$ and for the tube with
constant radius $L_{(p)}=L$. The current density for the tube with a constant
radius is given by (5.3) and the ratio $\langle j^{\phi}\rangle/\langle
j^{\phi}\rangle_{c}$ is evaluated for the same values of the proper radii for
the Beltrami pseudosphere and constant radius tube. The dashed lines in Fig. 5
present the same ratio for the AdS tube instead of the Beltrami pseudosphere.
The proper radius for the AdS tube one has $L_{(p)}=aL/z$ and the
corresponding current density is given by (A.7). The graphs in Fig. 5 are
plotted for $ma=0.5$, $\alpha_{p}=\pi/2$, and the numbers near the curves are
the corresponding values of the curvature coupling parameter $\xi$. As seen
from the graphs, for large values of the proper radius of the tube one has
$\langle j^{\phi}\rangle/\langle j^{\phi}\rangle_{c}\gg 1$ for both the
Beltrami pseudoshpere and AdS tube. This is related to the fact that in that
range the decay of the current densities for the Beltrami pseudoshpere and AdS
tube, as functions of the proper radius, follows a power-law, whereas for a
constant radius tube the decay is exponential for a massive field.

Figure 5: The ratio of the current densities on the Beltrami pseudosphere and
on the tube with constant radius (full curves) versus the proper radius of the
tube in units of the curvature radius $a$. The dashed curves present the same
ratio for the AdS tube and the tube of constant radius. The graphs are plotted
for conformally and minimally coupled fields and for $ma=0.5$,
$\alpha_{p}=\pi/2$.

##  7 Conclusion

We have investigated combined effects of spatial curvature and topology on the
local properties of the ground state for a quantum scalar field in
(2+1)-dimensional spacetime. The corresponding line element has the form (2.4)
and it describes rotationally symmetric 2D curved tube with general dependence
of the radius on the axial coordinate. As special cases, a cylindrical tube
with a constant radius, conical tube and tubes with constant positive and
negative curvature are considered. The local properties of the vacuum state
are encoded in two-point functions. As such, the Hadamard function is taken.
It is evaluated by using the mode-sum formula (3.9) with scalar field modes
given by (3.4) and obeying the quasiperiodicity condition (3.3) with a general
phase $\alpha_{p}$. The dependence of the mode functions on the axial
coordinate is expressed in terms of the function $z_{\sigma}(w)$ being the
solution of the equation (3.5) and normalized by the condition (3.7).

In order to extract the effects induced by the compactification, we have
separated from the Hadamard function the part that corresponds to the same
local geometry with decompactified $\phi$-coordinate. The separation is done
by using the Abel-Plana-type summation formula and the decomposed
representation of the Hadamard function is given by (3.16). The VEVs of local
physical observables, bilinear in the field operator, are obtained from the
Hadamard function and its derivatives in the coincidence limit of the
spacetime arguments. That limit is divergent and a renormalization procedure
is required to extract finite physical values of the VEVs. The advantage of
the presentation (3.16) is that the difference
$G(x,x^{\prime})-G_{0}(x,x^{\prime})$ is finite in the coincidence limit
$x^{\prime}\rightarrow x$ and the last term can be directly used for the
evaluation of renormalized topological contributions in the VEVs.

As an important physical characteristic of the vacuum state we have considered
the expectation value of the current density. The general expressions for the
currents in the axial and angular directions are given by (4.3) and (4.6). In
the special cases of constant radius and conical tubes the axial current
vanishes and the general formulas for the current along compact dimension are
reduced to the results previously investigated in the literature. For a
conical tube, a new representation (5.9) is provided. As another application
of general formulas, in Section 6 the Hadamard function and the current
density on the Beltrami pseudosphere are studied. The corresponding mode
functions are given by (6.12) and the Hadamard function is expressed as
(6.13). An alternative expression is provided by formula (6.30). The component
of the vacuum current along the axial direction vanishes and for the VEV of
the component along the compact dimension we have provided several
representations (see (6.36), (6.38) and (6.39)). The spacetime geometry for
the Beltrami pseudosphere is conformally related to the one for 2D Rindler
spacetime (see (6.3)) and we have shown that for a conformally coupled
massless field the standard relation between the corresponding VEVs in
conformally connected problems take place. For a massive field with general
curvature coupling the dimensionless combination $a^{2}\langle
j^{\phi}\rangle$ is a function of the ratio $L/r$ which gives the proper
radius of compact dimension measured in units of the curvature radius $a$. For
the radius of the tube much smaller than the curvature radius, $L/r\ll 1$, the
effect of the spatial curvature on the current density is weak and, in the
leading order, the VEV coincides with the corresponding current density for a
tube with a constant radius equal to the proper radius $aL/r$ (see (6.42)). In
the opposite limit of large proper radius, $L/r\gg 1$, the effect of curvature
is essential and the asymptotic is described by (6.41). In this limit, the
decay of the current density as a function of the proper radius follows a
power-law, like $\left(r/L\right)^{2}$. This behavior is in clear contrast to
the one for a constant radius tube when the current density for a massive
field is suppressed exponentially.

## Acknowledgments

The work was supported by the grant No. 21AG-1C047 of the Higher Education and
Science Committee of the Ministry of Education, Science, Culture and Sport RA
and by the ANSEF grant 23AN:PS-hepth-2889.

##  Appendix A Vacuum currents in locally dS and AdS tubes

The vacuum currents for a scalar field in locally dS and AdS spacetimes with
toroidal subspaces for general number of spatial dimension have been
considered in Refs. [54] and [55], respectively (for a review and further
discussions see also [7]). Here we specify the corresponding results for the
case of 2D space with a single compact dimension (dS and AdS tubes).

For dS tubes the line element reads

$ds_{\mathrm{dS}}^{2}=dt^{2}-e^{-2t/a}\left(dw^{2}+L^{2}d\phi^{2}\right).$
(A.1)

For the Ricci scalar one has $R=6/a^{2}$ and the nonzero components of the
Ricci tensor are given by (no summation over $i$) $R_{i}^{i}=2/a^{2}$,
$i=0,1,2$. From the general result of [54] for the physical component of the
current density along the compact dimension we get

$\langle
j^{\phi}\rangle_{\mathrm{dS}}=-\frac{2eL}{a^{2}\eta}\sum_{l=1}^{\infty}l\sin(l\alpha_{p})\partial_{u}F_{\nu_{-}}^{\mathrm{(dS)}}(u),\;u=2\left(\pi
l\frac{L}{\eta}\right)^{2}-1,$ (A.2)

where

$\eta=ae^{t/a},\;\nu_{\pm}=\sqrt{1-6\xi\pm m^{2}a^{2}}.$ (A.3)

In (A.2) we have defined the function

$\displaystyle F_{\nu}^{\mathrm{(dS)}}(u)$ $\displaystyle=$
$\displaystyle\frac{\sinh\nu x}{\sin\left(\pi\nu\right)\sinh
x},\;x=\mathrm{arccosh}\,u,\;\text{for }u\geq 1,$ $\displaystyle
F_{\nu}^{\mathrm{(dS)}}(u)$ $\displaystyle=$ $\displaystyle\frac{\sin\nu
x}{\sin\left(\pi\nu\right)\sin x},\;x=\mathrm{arccos}\,u,\;\text{for }u\leq
1.$ (A.4)

In deriving (A.2) it was assumed that the field is prepared in the Bunch-
Davies vacuum state. The parameter $\nu$ can be either positive or purely
imaginary. Note that $\tau=-\eta$, $-\infty<\tau<0$ is the conformal time in
terms of which the line element is written in conformally flat form
$ds_{\mathrm{dS}}^{2}=(a/\tau)^{2}\left(d\tau^{2}-dw^{2}-L^{2}d\phi^{2}\right)$.
In the case of a conformally coupled massless field one has $\nu_{-}=1/2$ and
for the function (A.4) we find
$F_{\nu}^{\mathrm{(dS)}}(u)=2^{3/2}/\sqrt{u+1}$. For the current density this
gives

$\langle
j^{\phi}\rangle_{\mathrm{dS}}=\frac{e\eta^{2}}{4\pi^{3}L^{2}a^{2}}\sum_{l=1}^{\infty}\frac{\sin(l\alpha_{p})}{l^{2}}.$
(A.5)

As expected, in this special case we have the conformal relation $\langle
j^{\phi}\rangle_{\mathrm{dS}}=(\eta/a)^{2}\langle j^{\phi}\rangle$, where
$\langle j^{\phi}\rangle$ is the current density for a massless field on a
cylindrical tube with constant radius (see (5.3) for $m=0$).

In the case of AdS tubes the line element has the form

$ds_{\mathrm{AdS}}^{2}=e^{2w/a}\left(dt^{2}-L^{2}d\phi^{2}\right)-dw^{2},$
(A.6)

with the curvature scalar and Ricci tensor $R=-6/a^{2}$ and
$R_{i}^{i}=-2/a^{2}$ (no summation over $i$). The expression for the vacuum
current density is obtained from the general formula in Ref. [55]:

$\langle
j^{\phi}\rangle_{\mathrm{AdS}}=-\frac{2eL}{a^{2}z}\sum_{l=1}^{\infty}l\sin(l\alpha_{p})\partial_{u}F_{\nu_{+}}^{\mathrm{(AdS)}}(u),\;u=2\left(\pi
l\frac{L}{z}\right)^{2}+1,$ (A.7)

with $z=ae^{w/a}$, $0<z<\infty$, and

$F_{\nu}^{\mathrm{(AdS)}}(u)=\frac{e^{-\nu x}}{\sinh
x},\;x=\mathrm{arccosh}\,u.$ (A.8)

With the new axial coordinate $z$, the line element is presented in a
conformally flat form
$ds_{\mathrm{dS}}^{2}=(a/z)^{2}\left(d\tau^{2}-dz^{2}-L^{2}d\phi^{2}\right)$.
The circles $z=0$ and $z=\infty$ correspond to the AdS boundary and horizon.

For a conformally coupled massless field $\nu_{+}=1/2$ and

$F_{\nu}^{\mathrm{(AdS)}}(u)=\frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{u-1}}-\frac{1}{\sqrt{u+1}}\right).$
(A.9)

Putting this in (A.7) one gets $\langle
j^{\phi}\rangle_{\mathrm{AdS}}=\left(z/a\right)^{2}\langle
j^{\phi}\rangle_{(1)}$, where

$\langle
j^{\phi}\rangle_{(1)}=\frac{e}{4\pi^{3}L^{2}}\sum_{l=1}^{\infty}\frac{\sin(l\alpha_{p})}{l^{2}}\left\\{1-\left[1+\left(\frac{z}{\pi
lL}\right)^{2}\right]^{-3/2}\right\\}.$ (A.10)

We have a conformal relations between the current densities on AdS tube and on
a cylindrical tube with constant radius having a circular edge at $z=0$. On
the latter the field obeys the Dirichlet boundary condition. The presence of
the edge in the Minkowski counterpart is related to the boundary condition for
the field on the AdS boundary. The edge $z=0$ in the Minkowskian problem is
the conformal image of the AdS boundary. Hence, formula (A.10) provides the
expression of the current density for a massless scalar field with the
boundary condition $\varphi(x)=0$ on the edge $z=0$ of a cylindrical surface
with constant radius. The effects of edges of AdS tubes on the vacuum current
densities for scalar and fermionic fields are discussed in [68, 69, 70, 71].

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# A unified constraint on the Lorentz invariance violation from both short and
long GRBs

Zhe Chang<EMAIL_ADDRESS>Yunguo Jiang<EMAIL_ADDRESS>Hai-Nan Lin
<EMAIL_ADDRESS>Institute of High Energy Physics, Chinese Academy of
Sciences, 100049 Beijing, China  Theoretical Physics Center for Science
Facilities, Chinese Academy of Sciences, 100049 Beijing, China

(August 27, 2024)

###### Abstract

Possible Lorentz invariance violation (LIV) has been investigated for a long
time based on observations of GRBs. These arguments relied on the assumption
that photons with different energy are emitted at the same place and time. In
this work, we try to take account of the intrinsic time delay $\Delta t_{\rm
int}$ between emissions of low and high energy photons by using the magnetic
jet model. The possible LIV effects are discussed in a unified scenario both
for long and short Fermi-detected GRBs. This leads to a unique quantum gravity
energy scale $M_{1}c^{2}\sim 1.0\times 10^{20}$ GeV respecting the linear
dispersion relation.

###### keywords:

gamma-ray burst; Lorentz invariance violation; magnetic jet model

##  1 Introduction

Lorentz invariance is one of the most important cornerstones of modern
physics. Recently, the OPERA collaboration reported that the GeV neutrinos
propagate faster than the speed of light [1]. Although other independent
experiments should be performed to verify the superluminal phenomenon, it is
valuable to ask whether Lorentz invariance violation (LIV) happens in high
energy scale. A favored way to test LIV is to study the most explosive events
in the present universe: gamma-ray bursts (GRBs). The Fermi satellite has
observed several GRBs with photon energy $>100$ MeV in recent years. Fermi
carries two instruments: the Gamma-Ray Burst Monitor (GBM) and the Large Area
Telescope (LAT), which detect the energy band $8{\rm KeV}-40{\rm MeV}$ and
$30{\rm MeV}-300{\rm GeV}$, respectively. An interesting feature of the
observation is that GeV photons arrive several seconds later than MeV photons
[2, 3, 4, 5].

One possible explanation is given by quantum gravity effects. Some quantum
gravity theories predict that high energy photons may interact with the foamy
structure of the space-time, thus photons with different energy propagate with
different velocities [6, 7, 8, 9]. Such effects can be accumulated after
photons travel a cosmological distance. In these theories, high energy photons
are subluminal. But there are still some other theories which show that high
energy photons can be superluminal. For example, LIV can also be induced from
the geometry of the space-time itself, such as the Finsler geometry [10, 11,
12]. One can expect that the velocity of photon also depends on the energy,
and may be superluminal.

A straight forward way to test LIV is studying the individual GRB, several
papers have discussed the upper limits of variations of the light speed [13,
14]. Ellis et al. have proposed a data fitting procedure to test LIV effects
[9, 15, 16, 17]. The linear fitting function is expressed as $\Delta t_{\rm
obs}/(1+z)=a_{\rm LIV}K(z)+b$, where $K(z)$ is a non-linear function of the
redshift measuring the cosmological distance, and $b$ represents the ignorance
of the intrinsic time lag. A statistical ensemble of GRBs was used to fit
values of $a_{\rm LIV}$ and $b$, and no strong evidence of LIV was found. The
explicit form of $K(z)$ depends on the cosmological model. Biesiada and
Piórkowska applied this procedure to various cosmological models [18]. Shao et
al. used this method to discuss four Fermi-detected GRBs [17], which we will
discuss in the present work. Nevertheless, all these investigations
concentrated on the time lag induced by LIV, the intrinsic time lag which
depends on the emission mechanism of GRBs was not considered.

On the other hand, without considering LIV effects, several mainstream GRB
models were proposed to explain the delayed arrival of GeV photons [19, 20,
22, 21]. Mészáros and Rees presented a magnetic dominated jet model to explain
this phenomenon. The MeV photons can escape the plasma when their optical
depth decreases to unity at the photosphere radius. While the GeV photons are
produced by the nuclear collision between protons and neutrons, which happens
at a large radius (compared to the photosphere radius) [20]. Duran and Kumar
considered that photons are emitted by electrons via the synchrotron
radiation, it consumes more time for electrons to be accelerated in order to
radiate GeV photons [21]. Bo$\check{\rm s}$njak and Kumar proposed the
magnetic jet model, the time delay depends linearly on the distance where the
jet is launched [22].

In this work, we argue that the observed time lag for two photons with energy
$E_{\rm high}$ and $E_{\rm low}$ consists of two parts,

$\displaystyle\Delta t_{\rm obs}=\Delta t_{\rm LIV}+\Delta t_{\rm int},$ (1)

where $\Delta t_{\rm int}$ denotes the intrinsic emission time delay, and
$\Delta t_{\rm LIV}$ represents the flying time difference caused by LIV
effects. In section 2, we take use of the magnetic jet model in Ref.[22] to
estimate $\Delta t_{\rm int}$. The LIV induced time lag $\Delta t_{\rm LIV}$
is given, and the quantum gravity energy scale is discussed in Sec. 3.
Finally, the discussion and conclusions are given in Sec. 4.

##  2 Magnetic jet model

In the magnetic jet model, photons with energy less than $10$ MeV can escape
when the jet radius is beyond the Thomson photosphere radius, i.e., the
optical depth for low energy photons is $\tau_{T}\sim 1$. However, GeV photons
will be converted to electron-positron pairs at this radius, and can escape
later when the pair-production optical depth $\tau_{\gamma\gamma}(E)$ drops
below unity.

The bulk Lorentz factor of an expanding spherical fireball increases with the
radius roughly as $\Gamma\propto r$, until reaching a saturate radius $r_{s}$
where the Lorentz factor is saturated [23]. However, for an expanding jet with
small ejecting angle, the effective dynamical dimension is one. The bulk
Lorentz factor increases with the radius roughly as [22]

$\displaystyle\Gamma(r)\approx\begin{cases}(r/r_{0})^{1/3}\quad\ &\ {\rm
for}\quad r_{0}\lesssim r\lesssim r_{s},\\\ \eta\quad\ &\ {\rm for}\quad
r\gtrsim r_{s},\\\ \end{cases}$ (2)

where $r_{0}\approx 10^{7}$ cm is the base of the outflow, which represents
the distance from the central engine where the jet is launched. $\eta$ is the
final bulk Lorentz factor of the jet.

The optical depth for the photon-electron scattering is defined as

$\tau_{T}(r)=\int_{r}^{\infty}\frac{dr^{\prime}}{2\Gamma^{2}}\sigma_{T}n\Gamma,$
(3)

where $\sigma_{T}=e^{4}/(6\pi\varepsilon_{0}^{2}c^{4}m_{e}^{2})\approx
6.65\times 10^{-25}$ cm2 is the Thomson scattering cross-section. The baryon
number density in the observer frame is $n\simeq L/4\pi r^{2}m_{p}\Gamma
c^{3}\sigma_{0}$, where $L$ is the isotropic luminosity, $m_{p}$ is the mass
of protons, $\sigma_{0}\equiv\Gamma(r_{0})[1+\sigma(r_{0})]$, and
$\sigma(r_{0})$ is the initial ratio of the magnetic and baryon energy
densities. When $\tau_{T}(r)=1$, the Thomson photon sphere radius is

$\frac{r_{p}}{r_{0}}\approx 1.36\times
10^{5}L_{52}^{3/5}\sigma_{0,3}^{-3/5}r_{0,7}^{-3/5},$ (4)

where $L_{52}\equiv L/10^{52}\,\,{\rm erg}\cdot{\rm s}^{-1}$,
$\sigma_{0,3}\equiv\sigma_{0}/10^{3}$, and $r_{0,7}\equiv r_{0}/10^{7}$ cm. We
use cgs units for numerical values here and after.

The optical depth for a photon of energy $E_{0}$ to be converted to $e^{\pm}$
while traveling through the jet at a radius $r$, is given by [22]

$\tau_{\pm}(E_{0},r)=\bigg{(}\frac{\beta-2}{\beta-1}\bigg{)}\frac{\sigma_{\gamma\gamma}}{4\pi
r\Gamma^{2}}\frac{L_{>p}}{(1+z)^{3-2\beta}E_{p}c}\bigg{[}\frac{E_{p}E_{0}}{\Gamma^{2}m_{e}^{2}c^{4}}\bigg{]}^{\beta-1},$
(5)

where $\sigma_{\gamma\gamma}=6\times 10^{-26}$ cm2 is the cross-section for
photons producing $e^{\pm}$ just above the threshold energy, $E_{p}$ is the
peak energy of the $\nu F_{\nu}$ spectrum, $L_{>p}$ is the frequency-
integrated luminosity above $E_{p}$, and $\beta\approx 2.2$ is the photon
index of the spectrum above $E_{p}$. Setting $\tau_{\pm}=1$, the pair-
production photonsphere radius is given by

$\frac{r_{\gamma\gamma}(E_{0})}{r_{0}}\approx 4.13\times
10^{6}L_{>p,52}^{0.41}E_{p,-6}^{0.08}E_{0,-4}^{0.49}r_{0,7}^{-0.41}(1+z)^{0.57},$
(6)

where $E_{p,-6}=E_{p}/{\rm MeV}$, and $E_{0,-4}=E_{0}/100{\rm MeV}$.

In the observer frame, the relative time delay between MeV photons and GeV
photons equates to the time for the jet to propagate from $r_{p}$ to
$r_{\gamma\gamma}$ [22]

$\Delta
t=\frac{3r_{0}(1+z)}{2c}\bigg{[}\bigg{(}\frac{r_{\gamma\gamma}(E_{0})}{r_{0}}\bigg{)}^{1/3}-\bigg{(}\frac{r_{p}}{r_{0}}\bigg{)}^{1/3}\bigg{]}.$
(7)

Taking use of Eq.(7) together with Eq.(4) and Eq.(6), one can calculate the
time delay for the arrival of GeV and 100 MeV photons relative to MeV photons.
Thus, the intrinsic time delay is $\Delta t_{\rm int}=\Delta t(E_{\rm
high})-\Delta t(E_{\rm low})$. The observed values of $E_{p}$, $E_{\rm
ios,54}$, T90 and $z$ for four $Fermi$-detected GRBs are taken from [22], and
are listed in Table 1.

GRB | Ep | Eios,54 | T90 | z
---|---|---|---|---
| keV | erg | s |
080916c | 424 | 8.8 | 66 | 4.35
090510 | 3900 | 0.11 | 0.6 | 0.90
090902b | 726 | 3.7 | 22 | 1.82
090926 | 259 | 2.2 | 13 | 2.11
Table 1: The observed parameters of four $Fermi$-detected GRBs. $E_{p}$ is the
energy at the peak of $\nu f_{\nu}$ spectrum. $E_{\rm ios,54}$ is the
isotropic equivalent energy in unit of $10^{54}$ ergs. T90 is the GRB duration
which $90$% of the counts are above background. $z$ is the GRB redshift [22].

##  3 Test of LIV effects

As is mentioned above, some quantum gravity theories predict that photons with
different wavelength propagate in different speed [6, 7, 8, 13, 18, 24]. The
non-trivial space-time structure may affect the propagation of photons, so
high energy photons may arrive later than low energy ones. Many works have
studied LIV effects of the high energy photons in this direction [9, 15, 25,
26, 27, 28].

Consider two photons emitted at the same time and place, within the LIV
phenomenology, the arrival time delay between them can be written as [25, 27]

$\displaystyle\Delta t_{\rm LIV}=\frac{1+n}{2c}\big{(}\frac{\Delta
E}{M_{n}c^{2}}\big{)}^{n}D_{n},$ (8)

where $n=1$ or $n=2$ denotes the linear or quadratic correction to the
dispersion relation, and $D_{n}$ is defined to be [25, 27]

$\displaystyle
D_{n}\equiv\frac{c}{H_{0}}\int_{0}^{z}\frac{(1+z^{\prime})^{n}dz^{\prime}}{\sqrt{\Omega_{M}(1+z^{\prime})^{3}+\Omega_{\Lambda}}},$
(9)

where $H_{0}\simeq 72$ km sec-1 Mpc-1 is the Hubble constant, $\Omega_{M}$ and
$\Omega_{\Lambda}$ are the present values of the matter density and
cosmological constant density, respectively. In the standard cosmological
model, ($\Omega_{M},\Omega_{\Lambda}$) are given by observations as
($0.3,0.7$) [27]. For $n=1$, the time delay depends linearly on the variation
of energy, which we will consider in the follow. In this case, the effective
LIV energy scale is

$\displaystyle M_{1}c^{2}=\frac{\Delta ED_{1}}{c\Delta t_{\rm LIV}}.$ (10)

In Ref.[17], Shao et al. took use of 4 GRBs the same as in Table 1 to predict
LIV effects, the observed time delay $\Delta t_{\rm obs}/(1+z)$ vs. $K(z)$ was
plotted, where $K(z)$ is defined as

$\displaystyle K(z)\equiv\frac{\Delta E}{(1+z)}\frac{D_{1}}{c}.$ (11)

In their plot, the three long bursts were found to be near one line. However,
the short burst was not fitted well by the same line. The intercept of the
line was interpreted as $\Delta t_{\rm int}/(1+z)$, and the slope of the line
can be interpreted as $1/M_{1}c^{2}$. $\Delta t_{\rm int}/(1+z)$ was found to
be negative, which means that high energy photons are emitted earlier than low
energy ones. This conflicts with standard GRB models. Taking into account of
the four GRBs, the quantum gravity energy scale is estimated to be
$M_{1}c^{2}\sim 2\times 10^{17}$ GeV for the linear energy dependence. In
their work, the intrinsic time delay is assumed to be the same for three long
bursts. Besides, their result strongly depends on the artificial choices. For
example, if the 33.4 GeV photon is replaced by the 11.16 GeV photon in GRB
090902b, the three long bursts can not be fitted well by one line.

In the follow, we first calculate $\Delta t_{\rm int}$ by using Eq.(7). Then,
combining with the observation data, we give $\Delta t_{\rm LIV}$ by the
relation given in Eq.(1). In Ref.[25], the authors used a statistical method
to determine the observed time difference $\Delta t_{\rm obs}$ between photons
with energy $E_{\rm low}$ and $E_{\rm high}$. In our present work, $E_{\rm
low}$ is taken to be 100 MeV, since the onset of $100$ MeV photons can be read
directly from the data of the LAT monitor. $E_{\rm high}$ is the energy of the
most energetic photon in each GRB. One exception is that the second energetic
photon with $E_{\rm high}=11.16$ GeV in GRB 090902b is chosen. The most
energetic $33.4$ GeV photon arriving at $82$ s is excluded due to its
incoincidence with the main burst. In Ref.[17], Shao et al. selected this
event to estimate LIV effects without considering the central engines and
emission mechanism. In the magnetic jet model, if we believe that the delayed
GeV photons are due to the high optical depth, the arrival time of the $33.4$
GeV photon should be 3 s later than the 11.16 GeV photon. However, the
observed time interval $70$ s is far beyond the model’s prediction. This
photon may be due to the inelastic collision of protons and neutrons [19, 20],
and it is quite possible that this individual event happens when the jet
encounter the interstellar medium.

With Eq.(7), we can estimate $\Delta t_{\rm int}\simeq 0.06$ s for GRB 090510,
if $r_{0}=10^{6}$ cm. Then $\Delta t_{\rm LIV}\simeq 0.14$ s, and
$M_{1}c^{2}\sim 9.73\times 10^{19}$ GeV, which is about 8 times of the Planck
energy $E_{\rm Planck}\sim 10^{19}$ GeV [4]. However, if we increase $r_{0}$
to $10^{7}$ cm, then $\Delta t_{\rm int}\simeq 0.46$ s and $\Delta t_{\rm
LIV}\simeq-0.26$ s. In this case, high energy photons become superluminal,
which is against the argument of quantum gravity theories. It is a reasonable
assumption that $r_{0}\simeq 10^{6}$ cm for GRB 090510, because this is a
short burst and its radius should be small than long bursts.

GRB | $E_{\rm low}$ | $E_{\rm high}$ | $\Delta t_{\rm obs}$ | $\Delta t_{\rm LIV}$ | $K(z)$ | $M_{1}c^{2}$
---|---|---|---|---|---|---
| MeV | GeV | s | s | s$\cdot$GeV | GeV
080916c | 100 | 13.22 | 12.94 | 0.24 | 4.50 $\times 10^{18}$ | 10.02 $\times 10^{19}$
090510 | 100 | 31 | 0.20 | 0.14 | 7.02 $\times 10^{18}$ | 9.73 $\times 10^{19}$
090902b | 100 | 11.16 | 9.5 | 0.10 | 3.38 $\times 10^{18}$ | 9.94 $\times 10^{19}$
090926 | 100 | 19.6 | 21.5 | 0.20 | 6.20 $\times 10^{18}$ | 9.59 $\times 10^{19}$
Table 2: The LIV induced time delay $\Delta t_{\rm LIV}$ and quantum gravity
energy scale $M_{1}c^{2}$ derived from four Fermi-detected GRBs. $\Delta
t_{\rm obs}$ is collected from Ref.[2, 3, 4, 5]. $\Delta t_{\rm LIV}=\Delta
t_{\rm obs}-\Delta t_{\rm int}$, where $\Delta t_{\rm int}$ is calculated by
Eq.(4), Eq.(6) and Eq.(7). The value of $\sigma_{0,3}$ in each GRB is
approximately the bulk Lorentz factor of the jet in unit of $10^{3}$ and is
taken as $\sigma_{0,3}\sim 1$ [22]. $r_{0,7}$ is chosen as 16.7, 0.1, 28.7 and
55.0 for GRB 080916c, GRB 090510, GRB 090902b and GRB 090926, respectively.

With the above criteria, we try to use the line fitting method to predict the
LIV effects. In principle, if the linear dispersion relation holds, the
$\Delta t_{\rm LIV}/(1+z)$ vs. $K(z)$ plot should be a zero-intercept line,
whose slope is the inverse of quantum gravity energy scale, i.e.,
$1/M_{1}c^{2}$. By choosing $r_{0}$ of each burst properly, the limits of LIV
effects for both short and long bursts can be unified respecting the linear
dispersion relation. The four GRB points can be fitted well by one line, if we
choose $r_{0,7}=$ 16.7, 0.1, 28.7 and 55.0 for GRB 080916c, GRB 090510, GRB
090902b and GRB 090926, respectively. In this case, LIV effects are calculated
in Table 2, and the $\Delta t_{\rm LIV}/(1+z)$ vs. $K(z)$ plot is given in
Fig.1. The values of $r_{0}$, which indicate the active scale of central
engines, are reasonable. The inverse of the slope gives $M_{1}c^{2}\sim
1.0\times 10^{20}$ GeV, which is roughly the same result of the GRB 090510.
The energy scale of the modified photon dispersion relation is one order of
magnitude higher than the conventional Planck scale. This may suggest that the
quantum gravity scale may be more subtle than one naively thinks, because this
quantity is model dependent. For instance, the effective quantum energy scale
depends on the density of $D$-particles in the D-foam model [7, 8]. It is an
interesting future work to combine both the quantum gravity model and the GRB
model together to study the LIV effects. Supposing LIV effects are strongly
suppressed, which is the assumption taken by the mainstream GRB models, the
observed time delay can well predict the value of $r_{0}$ by using the
magnetic jet model, since the tuning of $r_{0}$ is sensitive to the fitting.

Figure 1: The plot of $\Delta t_{\rm LIV}/(1+z)$ vs. $K(z)$ for four Fermi-
detected GRBs.

##  4 Discussion and conclusion

From Eq.(7), one can infer that the time lag has approximate linear relation
with the initial radius ($\propto r_{0}^{0.86}$) and the redshift
($\propto(1+z)^{1.19}$). It has a weak dependence on the photon energy
$E_{0}$, the peak energy $E_{p}$, and the peak luminosity $L_{>p}$, which are
$\Delta t_{\rm int}\propto E_{0}^{0.17},E_{p}^{0.05},$ and $L_{>p}^{0.14}$,
respectively. In addition, the time lag is hardly dependent on $\sigma_{0}$.
Another model proposed to explain the delayed GeV photons was given by
Mészáros and Rees. In this model, the time lag also depends linearly on
$r_{0}$ (See Eq.(19) in [20]). These GRB models also predict the spectrum, so
they can be verified by the observed spectrum of the whole energy band.

As mentioned above, in Ellis et al.’s proposal, the linear fitting function is
written as $\Delta t_{\rm obs}/(1+z)=a_{\rm LIV}K(z)+b$. In our case,
$b\sim\Delta t_{\rm int}/(1+z)$ is not a constant, and we can estimate it
approximately as

$\displaystyle b\simeq
0.08\,r_{0.7}^{0.86}L_{>p,52}^{0.14}E_{p,-6}^{0.03}E_{0,-4}^{0.16}(1+z)^{0.19},$
(12)

which roughly agrees with Ellis et al.’s result $b\sim 10^{-2}$ [15]. Since
$b$ weakly depends on the redshift, it can be regarded as a distance-
independent quantity [17]. However, $b$ strongly depends on $r_{0}$, so the
long and the short bursts will lead to quite different values of $b$. If the
long bursts have roughly the same $r_{0}$, $b$ can be considered as a
constant, and the linear relation between $\Delta t_{\rm obs}/(1+z)$ and
$K(z)$ holds.

The analysis above gives us the hint that, if we want to consider source
effects of the GRBs, long bursts with small redshifts are preferred. Due to
the short cosmological distance, quantum gravity effects can be attenuated.
With the purpose of enhancing quantum gravity effects, short GRB bursts with
high redshifts should be selected. Therefore, the future observations on short
GRBs will improve the test of LIV effects. If we use an ensemble of GRBs with
both long and short bursts, the fitting function method is not convincing. The
intrinsic time delay plays an important role. Better knowledge of the
intrinsic property of the source will help us to improve the test of LIV
effects. Inversely, better understanding of quantum gravity can help us to
predict the parameters in GRB models.

In this work, we discussed LIV effects by making use of the magnetic jet
model. GeV photons are emitted later than MeV photons, due to their different
optical depths. This physical ingredient should be included in probe of LIV
effects. The neglect of the photon emission mechanism may lead to misleading
results. The constraints of LIV effects can be unified for both long and short
bursts. The calculation of the linear energy dependence of dispersion relation
gives $M_{1}c^{2}\sim 1.0\times 10^{20}$ GeV. Although the magnetic jet model
itself should be tested by further investigations, the analysis of the
intrinsic time delay is important when we study the photons from the
astrophysical sources to test LIV effects.

## Acknowledgments

We are grateful to M. H. Li, X. Li and S. Wang for useful discussion. This
work has been funded in part by the National Natural Science Fund of China
under Grant No. 10875129 and No. 11075166.

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