# Spinor Casimir Densities for a Spherical Shell in the Global Monopole Spacetime

###### Abstract

We investigate the vacuum expectation values of the energy-momentum tensor and the fermionic condensate associated with a massive spinor field obeying the MIT bag boundary condition on a spherical shell in the global monopole spacetime. In order to do that it was used the generalized Abel-Plana summation formula. As we shall see, this procedure allows to extract from the vacuum expectation values the contribution coming from to the unbounded spacetime and explicitly to present the boundary induced parts. As to the boundary induced contribution, two distinct situations are examined: the vacuum average effect inside and outside the spherical shell. The asymptotic behavior of the vacuum densities is investigated near the sphere center and surface, and at large distances from the sphere. In the limit of strong gravitational field corresponding to small values of the parameter describing the solid angle deficit in global monopole geometry, the sphere-induced expectation values are exponentially suppressed. As a special case we discuss the fermionic vacuum densities for the spherical shell on background of the Minkowski spacetime. Previous approaches to this problem within the framework of the QCD bag models have been global and our calculation is a local extension of these contributions.

PACS number(s): 03.70.+k, 04.62.+v, 12.39.Ba

## 1 Introduction

Different types of topological defects [1] may have been formed during the phase transitions in the early universe. Depending on the topology of the vacuum manifold these are domain walls, strings, monopoles and textures corresponding to the homotopy groups , , and , respectively. Physically these topological defects appear as a consequence of spontaneous breakdown of local or global gauge symmetries of the system composed by self-coupling scalar Higgs or Goldstone fields, respectively. Global monopoles are spherically symmetric topological defects created due to phase transition when a global symmetry is spontaneously broken and they have important role in the cosmology and astrophysics.

The simplest theoretical model which provides global monopoles has been proposed a few years ago by Barriola and Vilenkin [2]. This model is composed by a self-coupling iso-scalar Goldstone field triplet , whose original global symmetry is spontaneously broken to . The matter field plays the role of an order parameter which outside the monopole’s core acquires a non-vanishing value. The main part of the monopole’s energy is concentrated into its small core. Coupling this system with the Einstein equations, a spherically symmetric metric tensor is found. Neglecting the small size of the monopole’s core, this tensor can be approximately given by the line element

(1) |

where the parameter , smaller than unity, depends on the
symmetry breaking energy scale and codifies the presence of the
global monopole^{1}^{1}1In fact the parameter , being the energy scale where the global symmetry
is spontaneously broken.. This spacetime corresponds to an
idealized point-like global monopole. It is not flat: the scalar
curvature , and the solid angle of a sphere
of unit radius is , so smaller than the
ordinary one. The energy-momentum tensor associated with this
object has a diagonal form and its non-vanishing components read
.

The quantum effects due to the point-like global monopole spacetime on the matter fields have been considered in Refs. [3] and [4] to massless scalar and fermionic fields, respectively. In order to do that, the scalar and spinor Green functions in this background have been obtained. More recently the effect of the temperature on these polarization effects has been analysed in [5] for scalar and fermionic fields. The calculation of quantum effects on massless scalar field in a higher dimensional global monopole spacetime has also been developed in [6].

Although the deficit solid angle and also curvature associated
with this manifold produce non-vanishing vacuum polarization
effects on matter fields, the influence of boundary conditions
obeyed by the matter fields on the vacuum polarization effects
have been investigated. The Casimir energy associated with massive
scalar field inside a spherical region in the global monopole
background have been analyzed in Refs. [7, 8] using the
zeta function regularization method. More recently the Casimir
densities induced by a single and two concentric spherical shells
have been calculated [9, 10] to higher dimensional
global monopole spacetime by making use of the generalized
Abel-Plana summation formula [11, 12]. This
procedure allows to develop the summation over all discrete mode.
Here we shall calculate the Casimir densities for fermionic fields
obeying MIT bag boundary condition on the spherical shell in the
point-like global monopole spacetime. Specifically we shall
calculate the renormalized vacuum expectation values of the
energy-momentum tensor and the fermionic condensate in the regions
inside and outside the spherical shell. As we shall see using the
generalized Abel-Plana summation formula, all the components of
the vacuum average of the energy-momentum tensor can be separated
in two contributions: boundary dependent and independent ones. The
boundary independent contribution is similar to previous result
obtained in [4] using different approach. It is divergent
and consequently in order to obtain a finite and well defined
expression we must apply some regularization procedure. The
boundary dependent contribution is finite at any strictly interior
or exterior point and does not contain anomalies. Consequently, it
does not require any regularization procedure. Because the
analysis of boundary independent term has been performed before,
in this present analysis we shall concentrate on the boundary
dependent part. Taking , from our results in this paper
as a special case we obtain the fermionic Casimir densities for a
spherical shell on background of the Minkowski spacetime.
Motivated by the MIT bag model in QCD, the corresponding Casimir
effect was considered in a number of papers
[13, 14, 15, 16, 17, 18, 19, 20]
(for reviews and additional references see
[21, 22, 23, 24]). To our knowledge, the most of
the previous studies were focused on global quantities, such as
the total vacuum energy and stress on the surface. The density of
the fermionic vacuum condensate for a massless spinor field inside
the bag was investigated in Ref. [15] (see also
[23]). In the considerations of the Casimir effect it is
of physical interest to calculate not only the total energy but
also the local characteristics of the vacuum, such as the
energy-momentum tensor and vacuum condensates. In addition to
describing the physical structure of the quantum field at a given
point, the energy-momentum tensor acts as the source of gravity in
the Einstein equations ^{2}^{2}2The effects of the back-reaction
corrections on the Einstein equation due to the vacuum
polarization produced by massless scalar field in a global
monopole spacetime has been analyzed in [3].. It therefore
plays an important role in modelling a self-consistent dynamics
involving the gravitational field [25]. For the case of
the Minkowski bulk, our calculation is a local extension of the
previous contributions on the fermionic Casimir effect for a
spherical shell.

This paper is organized as follows: In section 2 we obtain the normalized eigenfuntions for a massive spinor field on the global monopole spacetime inside a spherical shell of finite radius. In section 3, using the generalized Abel-Plana summation formula, we formally obtain the vacuum expectation value of the energy-momentum tensor considering that the fermionic field obeys the MIT bag condition on the spherical shell. Explicit behavior for boundary dependent term is exhibited. The section 4 is devoted to the calculation of the vacuum expectation values for the region outside the shell. In section 5 we present our concluding remarks and leave for the Appendix some relevant calculations.

##
2 Eigenfunctions for a Spinor Field on the Global

Monopole Spacetime

The dynamics of a massive spinor field on a curved spacetime is described by the Dirac differential equation

(2) |

where are the Dirac matrices defined in such curved spacetime, and the spin connection defined as

(3) |

being the standard covariant derivative operator. Notice that, using the usual commutation relations for the Dirac matrices, , we can see that

(4) |

After this brief introduction, let us now specialize in the spacetime associated with the point-like global monopole whose line element is described by (1). In order to develop such procedure we shall adopt the following representation for the Dirac matrices

(5) |

given in terms of the curved space Pauli matrices :

(6) |

These matrices satisfy the relation

(7) |

where are the spatial component of metric tensor and is the corresponding determinant. is the totally anti-symmetric symbol with . Here and below the latin indices run over values . It can be easily checked that with these representations the Dirac matrices satisfy the standard anticommutation relations. Substituting these matrices into formula (4), we can see that

(8) |

Let us write the four-components spinor field in terms of two-components ones as

(9) |

Assuming the time dependence in the form , from (2) one finds the equations for these spinors:

(10a) | |||||

(10b) |

where , and . The angular parts of the spinors are the standard spinor spherical harmonics whose explicit form is given in Ref. [26]:

(11) |

where specifies the value of the total angular momentum, and its projection, , . Using the formula

(12) |

it can be seen that

(13a) | |||||

(13b) |

where we use the notation

(14) |

By taking into account these relations, from (10) we obtain the following set of differential equations for the radial functions

(15a) | |||||

(15b) |

They lead to the second order differential equations for the separate functions:

(16a) | |||||

(16b) |

with the solutions

(17) |

where represents the cylindrical Bessel function of the order . The constants and are related by equations (15):

(18) |

As a result for a given we have two types of eigenfunctions with different parities corresponding to . These functions are specified by the set of quantum numbers and have the form

(19) | |||||

(20) |

where , and ,

(21) |

On the base of formula (19) we define the positive and negative frequency eigenfunctions as

(22) |

These functions are orthonormalized by the condition

(23) |

from which the normalization constant can be determined.

##
3 Vacuum Expectation Values of the Energy-

Momentum Tensor Inside a Spherical Shell

In this section we shall consider the vacuum expectation values of the energy-momentum tensor inside a spherical shell concentric with the global monopole. The integration in formula (23) goes over the interior region of the sphere and , where is the Bessel function of the first kind. We shall assume that on the sphere surface the field satisfies bag boundary conditions:

(24) |

where is the sphere radius, is the outward-pointing normal to the boundary. For the sphere . In terms of the spinors and this condition is written in the form

(25) |

The imposition of this boundary condition on the eigenfunctions (19) leads to the following equations for the eigenvalues

(26) |

This boundary condition can be written in the form

(27) |

where now and below for a given function we shall use the notation

(28) |

with and . Let us denote by , the roots to equation (27) in the right half plane, arranged in ascending order. By taking into account Eq. (26) and using the standard integral for the Bessel functions, from condition (23) for the normalization coefficient one finds

(29) |

with .

Now we expand the field operator in terms of the complete set of single-particle states :

(30) |

where is the annihilation operator for particles, and is the creation operator for antiparticles. In order to find the vacuum expectation value for the operator of the energy-momentum tensor we substitute the expansion (30) and the analog expansion for the operator into the corresponding expression for the spinor fields,

(31) |

By making use the standard anticommutation relations for the annihilation and creation operators, for the vacuum expectation values one finds the following mode-sum formula

(32) |

where is the amplitude for the corresponding vacuum. Since the spacetime is spherically symmetric and static, the vacuum energy-momentum tensor is diagonal, moreover . So in this case we can write:

(33) |

with the energy density , radial, , and azimuthal, , pressures. As a consequence of the continuity equation , these functions are related by the equation

(34) |

which means that the radial dependence of the radial pressure necessarily leads to the anisotropy in the vacuum stresses.

Substituting eigenfunctions (19) into Eq. (32), the summation over the quantum number can be done by using standard summation formula for the spherical harmonics. For the energy-momentum tensor components one finds

(35) |

where we have introduced the notations

(36) | |||||

(37) | |||||

(38) |

Note that in (35) we have used the relation between the normalization coefficient and the function introduced in Appendix A:

(39) |

The vacuum expectation values (35) are divergent and need some regularization procedure. To make them finite we can introduce a cutoff function , with the cutoff parameter , which decreases with increasing and satisfies the condition , . Now to extract the boundary-free parts we apply to the corresponding sums over the summation formula derived in Appendix A. As a function in this formula we take . As a result the components of the vacuum energy-momentum tensor can be presented in the form

(40) |

where the first term on the right hand side comes from the integral on the left of summation formula (78), and the second term comes from the integral on the right of this formula. Making use the asymptotic formulae for the Bessel modified functions, it can be seen that for the part is finite in the limit and, hence, in this part the cutoff can be removed. As it has been pointed out in Appendix A, the function satisfies relation (81) and, hence, the part of the integral on the right of formula (78) over the interval vanishes after removing the cutoff. Introducing the notation

(41) |

explicitly summing over and transforming from summation over to summation over , one receives

(42) |

where we use the notation

(43) | |||||

(44) | |||||

(45) |

Introducing in the expression for the modified Bessel functions, after some transformations end explicitly summing over , we obtain the formula

(46) |

with

(47) | |||||

(48) | |||||

(49) |

Here and below for given functions and we use the notation

(50) |

As we see, the part in the vacuum expectation value for the energy-momentum tensor do not depend on the radius of the sphere , whereas the contribution of the terms tends to zero as (for large the subintegrand behaves as ). It follows from here that the quantities (42) are the vacuum expectation values for the components of the energy-momentum tensor for the unbounded global monopole space:

(51) |

where is the amplitude for the corresponding vacuum. Note that in expressions (42) for the corresponding components we have not explicitly written the cutoff function. Precisely speaking in this form all terms related with (42) are divergent. The renormalization prescriptions adopted to provide a finite and well defined result to them are the usual ones applied for the curved spacetime without boundary [27, 28, 25]. For the specific system analyzed here the point-splitting renormalization procedure has been applied in previous publication [4]. The part in Eq. (40) is induced by the presence of the spherical shell and can be termed as the boundary part. As we have seen, the application of the generalized Abel-Plana formula allows us to extract from the vacuum expectation value of the energy-momentum tensor the contribution due to the boundary-free monopole spacetime and to present the boundary-induced part in terms of exponentially convergent integrals (for applications of the generalized Abel-Plana formula to a number of Casimir problems with various boundary geometries see Refs. [11, 12, 29, 30, 31, 32, 33]). It can be easily checked that the both terms on the right of formula (40), and , obey the continuity equation (34). In addition, as it is seen from expressions (46)–(49), for a massless spinor field the boundary-induced part of the vacuum energy-momentum tensor is traceless and the trace anomalies are contained only in the purely global monopole part without boundaries.

Having the components of the energy-momentum tensor we can find the corresponding fermionic condensate making use the formula for the trace of the energy-momentum tensor, . It is presented in the form of the sum

(52) |

where the boundary-free part (first summand on the right) and the sphere-induced part (second summand on the right) are determined by formulae similar to Eqs. (42) and (46), respectively, with replacements

(53) | |||||

(54) |

with notation (50). Alternatively one could obtain formulae (52)–(54) applying the summation formula (78) to the corresponding mode-sum for the fermionic condensate.

Note that formulae (40), (42), (46) can be obtained by another equivalent way, applying a certain first-order differential operator on the corresponding Green function and taking the coincidence limit. To construct the Green function we can use the corresponding mode expansion formula with the eigenfunctions (19). This function is a matrix and the angular parts of the corresponding elements are products of the components for the spinor spherical harmonics, , where the upper indices numerate the spinor components. These parts are the same as in the boundary-free case and coincide with the corresponding functions for the Minkowski bulk. The radial parts for the components of the Green function contain the products of the Bessel functions in the forms , , where . To evaluate the sum over we can apply the summation formula (78). The condition (75) is satisfied if . The term with the integral on the left of formula (78) gives the Green function for the boundary-free global monopole spacetime, and the term with the integral on the right will give the boundary-induced part.

In the case the quantities (42) present the vacuum expectation values for the Minkowski spacetime without boundaries. This can be also seen by direct evaluation. For example, in the case of the energy density making use the formula , one finds

(55) | |||||

which is precisely the energy density of the Minkowski vacuum for a spinor field. As for the Monkowski background the renormalized vacuum energy-momentum tensor vanishes, , the vacuum energy-momentum tensor is purely boundary-induced and the corresponding components are given by formulae (46)–(49) with . Note that the previous investigations on the spinor Casimir effect for a spherical boundary (see, for instance, [13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24] and references therein) consider mainly global quantities, such as total vacuum energy. For the case of a massless spinor the density of the fermionic condensate is investigated in [15] (see also [23]). The corresponding formula derived in Ref. [15] is obtained from (46) with replacement (54) in the limit . In Fig. 1 we have presented the dependence of the Casimir densities, on the rescaled radial coordinate for a massless spinor field on the Minkowski bulk. The vacuum energy density and pressures are negative inside the sphere.

Now we turn to the consideration of various limiting cases of the expressions for the sphere-induced vacuum expectation values. In the limit , for the boundary parts (46) the summands with a given behave as , and the leading contributions come from the lowest terms. Making use standard formulae for the Bessel modified functions for small values of the argument, for the sphere-induced parts near the center, , one finds

(56) | |||||

(57) |

where and is the gamma function. Hence, at the sphere center the boundary parts vanish for the global monopole spacetime () and are finite for the Minkowski spacetime (). Note that in the large mass limit, , the integrals in Eqs. (56), (57) are exponentially suppressed by the factor . In the Minkowski background case the vacuum stresses for a massless spinor are isotropic at the sphere center and after the numerical evaluation of the integral one finds

(58) |

The boundary induced parts of the vacuum energy-momentum tensor components diverge at the sphere surface, . These divergences are well-known in quantum field theory with boundaries and are investigated for various types of boundary geometries [34, 35, 36]. In order to find the leading terms of the corresponding asymptotic expansion in powers of the distance from the sphere surface, we note that in the limit the sum over in (46) diverges and, hence, for small the main contribution comes from the large values of . Consequently, rescaling the integration variable and making use the uniform asymptotic expansions for the modified Bessel functions for large values of the order [37], to the leading order one finds

(59) | |||||

(60) |

Notice that the terms in these expansions diverging as the inverse fourth power of the distance have cancelled out. This is a consequence of the conformal invariance of the massless fermionic field and is in agreement with general conclusions of Ref. [35]. Near the sphere surface the energy density is negative for all values of , while the vacuum pressures are negative for and are positive for . It is of interest to note that the leading terms do not depend on the parameter and, hence, are the same for the Minkowski and global monopole bulks. For the latter case due to the divergences, near the sphere surface the total vacuum energy-momentum tensor is dominated by the boundary induced parts . The dependence of these parts on the rescaled radial coordinate is depicted in Fig. 2 for the case of a massless fermionic field on the global monopole background with the solid angle deficit parameter .

Now let us consider the limit for a fixed value . This limit corresponds to strong gravitational fields. In this case from (41) one has , and after introducing in (46) a new integration variable , we can replace the modified Bessel functions by their uniform asymptotic expansions for large values of the order. The integral over can be estimated by making use the Laplace method. The main contribution to the sum over comes from the summands with and the boundary parts of the vacuum energy-momentum tensor components behave as with . Hence, for the boundary-induced vacuum expectation values are exponentially suppressed and the corresponding vacuum stresses are strongly anisotropic. Fig. 3 shows that the nonzero mass can essentially change the behavior of the vacuum energy-momentum tensor components. In this figure we have depicted the dependence of the boundary induced quantities on the parameter for the radial coordinate . The left panel corresponds to the sphere in the Minkowski spacetime () and for the right panel .

## 4 Vacuum Expectation Values Outside a Spherical Shell

Now let us consider the expectation values of the energy–momentum tensor in the region outside a spherical shell, . The corresponding eigenfunctions have the form (19), where now the function is the linear combination of the Bessel functions of the first and second kinds. The coefficient in this linear combination is determined from the boundary condition (25) and one obtains

(61) |

where is the Bessel function of the second kind, and the functions with tilda are defined as (28). Now the spectrum for the quantum number is continuous and the corresponding in Eq. (23) is understood as the Dirac delta function . To find the normalization coefficient from Eq. (23) it is convenient to take for all discrete quantum numbers. As the normalization integral diverges in the limit , the main contribution into the integral over radial coordinate comes from large values of when the Bessel functions can be replaced by their asymptotics for large arguments. The resulting integral is taken elementary and for the normalization coefficient we obtain

(62) |

Substituting the eigenfunctions (19) into the mode-sum formula (32) and taking into account Eqs. (61) and (62), we can see that the vacuum energy-momentum tensor has the form (33). The diagonal components are determined by formulae

(63) |

where the expressions for are obtained from formulae (36)–(38) by replacements

(64) |

To find the parts in the vacuum expectation values of the energy-momentum tensor induced by the presence of the sphere we subtract the corresponding components for the monopole bulk without boundaries, given by Eq. (42). In order to evaluate the corresponding difference we use the relation

(65) |

where , are the Hankel functions. This allows to present the vacuum energy-momentum tensor components in the form (40) with the boundary induced parts

(66) |

On the complex plane we can rotate the integration contour on the right of this formula by the angle for and by the angle for . The integrals over the segments and cancel out and after introducing the Bessel modified functions one obtains

(67) |

Here the expressions for the functions are obtained from formulae (47)–(49) by replacements and