Scalar and spinor particles in the spacetime of a domain wall in

Scalar and spinor particles in the spacetime of a domain wall in string theory

V. B. Bezerra,1,*L. P. Colatto,2,†M. E. X. Guimara˜es,3,‡

and R. M. Teixeira Filho4,§

1

Departamento de Fı´sica, Universidade Federal da Paraı´ba, Joa˜o Pessoa, Brazil

2_{Instituto de Fı´sica, Universidade de Brası´lia, Brası´lia, Brazil} 3_{Departamento de Matema´tica, Universidade de Brası´lia, Brası´lia, Brazil}

4_{Instituto de Fı´sica, Universidade Federal da Bahia, Salvador, Brazil} 共Received 19 October 2001; published 9 May 2002兲

We consider scalar and spinor particles in the spacetime of a domain wall in the context of low energy effective string theories, such as the generalized scalar-tensor gravity theories. This class of theories allows for an arbitrary coupling of the wall and the共gravitational兲 scalar field. First, we derive the metric of a wall in the weak-field approximation and we show that it depends on the wall’s surface energy density and on two post-Newtonian parameters. Then, we solve the Klein-Gordon and the Dirac equations in this spacetime. We obtain the spectrum of energy eigenvalues and the current density in the scalar and spinor cases, respectively. We show that these quantities, except in the case of the energy spectrum for a massless spinor particle, depend on the parameters that characterize the scalar-tensor domain wall.

DOI: 10.1103/PhysRevD.65.104027 PACS number共s兲: 04.62.⫹v, 11.25.⫺w, 98.80.Cq

I. INTRODUCTION

Topological defects arise whenever a symmetry is sponta-neously broken. They can be of various types according to the topology of the vacuum manifold of the field theory be-ing under consideration. In this work, we will concentrate our attention on domain walls which are defects arising from a breaking of a discrete symmetry by means of a Higgs field 关1–3兴.

Domain walls have been extensively studied in the litera-ture. In particular, it was soon realized that they may lead to a cosmological catastrophe 关2兴, even if they were produced in a late time phase transition 关4兴. From the gravitational point of view, an interesting feature of the wall’s gravita-tional field is that its weak field approximation does not cor-respond to any exact static solution of the Einstein’s equa-tions, hence implying that they are gravitationally unstable 关5兴. In Ref. 关6兴, a time-dependent metric was obtained and it was shown that observers experience a repulsion from the wall. Current-carrying walls and their cosmological conse-quences were also object of investigations. In Ref. 关7兴, the internal structure of a surface current-carrying wall was stud-ied and the internal quantities such as the energy per unit surface and the surface current were calculated numerically. The above-mentioned features of a domain wall were ana-lyzed in the framework of Einstein’s theory of gravity. How-ever, it has been argued that gravity may be described by a scalar-tensorial gravitational field, at least at sufficiently high energy scales. Indeed, a scalar field␾, which from now on we will call generically a dilaton, appears as a necessary partner of the graviton field g_␮␯ in all superstring models 关8,9兴. Topological defects of various types and their gravita-tional effects have already been studied in the framework of

various low energy effective string models关10,11兴. In what concernes the domain wall solutions, these configurations were the object of Ref.关12兴, in which the authors studied the properties of the wall’s gravitational field in Brans-Dicke and in dilatonic gravities. In this class of solutions, the dilaton can couple to the matter potential forming the wall. It is shown that the dilaton’s solution varies with the spatial dis-tance from the wall giving rise to a defect called the ‘‘dila-tonic domain wall.’’

The aim of this paper is twofold. First, we investigate the gravitational field of a domain wall in the context of a gen-eralized scalar-tensor gravity. Second, we analyze how par-ticles are affected by this particular gravitational field. The gravitational interaction on quantum mechanical systems has been studied by many authors 关13兴. For this purpose the Klein-Gordon and Dirac equations in covariant form have been used and solved in curved spacetimes. The search for these solutions is very interesting and may be accounted for by the scheme of unifying quantum mechanics and general relativity. As examples of works concerning this subject we can mention Audretsch and Scha¨fer 关14兴 who presented a detailed analysis of the energy spectrum of the hydrogen atom in Robertson-Walker universes and Parker关15,16兴 who studied a one-electron atom in a curved spacetime.

In the present work, we are particularly interested in studying scalar and spinor particles in the spacetime of a scalar-tensorial domain wall. In Sec. II we derived the metric of a wall in the weak field approximation. We show that it depends on the wall’s surface energy density ␴ and on two post-Newtonian parameters, G0 and␣2(␾0). In Sec. III we solve the Klein-Gordon equation, we find the energy eigen-values and we point out the dependence of the current on the parameters that characterize the scalar-tensor domain wall. In Sec. IV we first consider the Dirac equation for a massive spinor field. Then, we solve explicitly the Weyl equations for a massless spinor field and we determine the expression for the energy spectrum and for the current. Finally, in Sec. V, we present our concluding remarks.

*Email address: [email protected]

†_{Email address: [email protected]} ‡_{Email address: [email protected]} §_{Email address: [email protected]}

II. THE METRIC OF A DOMAIN WALL IN THE WEAK-FIELD APPROXIMATION

In this section we will derive the metric of a domain wall in a low energy effective string model, in which the axion field is vanishing. This action is analogous to the class of scalar-tensor theories developed in Refs.关17兴 and, in the case of the scalar sector of the gravitational interaction, is mass-less. For technical purposes, it is better to work in the so-called Einstein 共conformal兲 frame in which the kinematic terms of tensor and scalar fields do not mix. Then, a domain wall solution arises from the action

S⫽ 1 16␲G *

冕

d4_x

冑

_{⫺g关R⫺2g}␮␯_⳵ ␮␾⳵␯␾兴 ⫹

冕

d4x

冑

⫺gA2共␾兲

冋

1 2g ␮␯_⳵ ␮⌽⳵␯⌽⫺V共⌽兲

册

, 共1兲

where g_␮␯is a pure rank-2 metric tensor, R is the curvature scalar associated with it and G

*is some ‘‘bare’’ gravitational coupling constant. The second term in the right-hand side of Eq. 共1兲 is the matter action representing a model of a real Higgs scalar field ⌽ and the symmetry breaking potential

V(⌽) which possesses a discrete set of degenerate minima.

Action共1兲 is obtained from the original action appearing in Refs. 关17兴 by a conformal transformation 共see, for instance, 关18兴兲

g

˜_␮␯⫽A2_共␾兲g

␮␯, 共2兲

where g˜_␮␯is the physical metric which contains both scalar and tensor degrees of freedom, and by a redefinition of the quantities

G

*A

2_共␾兲⫽ 1 ⌽˜ ,

where⌽˜ is the original scalar field, and

␣共␾兲⬅⳵_⳵␾ln A⫽ 1 关2␻共⌽˜_兲⫹3兴1/2,

which can be interpreted as the 共field-dependent兲 coupling strength between matter and the scalar field. We choose to leave A2_{(␾) as an arbitrary function of the dilaton field.}

In the Einstein frame, the field equations are written as follows: R_␮␯⫽2⳵_␮␾⳵_␯␾⫹8␲G *

冉

T␮␯⫺ 1 2g␮␯T

冊

䊐g␾⫽⫺4␲G_*␣共␾兲T 共3兲

where the energy-momentum tensor is obtained as

T_␮␯⬅ 2

冑

⫺g

␦Sm

␦g_␮␯.

In what follows, we will consider the solution of a domain wall in the y z plane in the weak-field approximation. There-fore, we will expand Eqs. 共3兲 to first order in G

*A 2_(␾

0) in such a way that

g_␮␯⫽␩_␮␯⫹h_␮␯

␾⫽␾0⫹␾(1)

A共␾兲⫽A共␾0兲关1⫹␣共␾0兲␾(1)兴

T_␯␮⫽T₍₀₎␮ _␯⫹T₍₁₎␮ _␯. 共4兲 In this approximation, T₍₀₎␮ _␯⫽A2(␾0)T˜(0)␮ ␯ is the energy-momentum tensor of a static domain wall with negligible width and lying in a y z plane. Therefore,

T₍₀₎␮ _␯⫽A2共␾0兲␴␦共x兲diag共1,0,1,1兲 共5兲 in the Cartesian coordinate system (t,x,y ,z). The parameter

␴ is the wall’s surface energy density. In our convention, the metric signature is ⫺2.

Equations共3兲 in the linearized regime reduce to ⵜ2_h ␮␯⫽16␲G_*

冉

T(0)␮␯⫺ 1 2␩␮␯T(0)

冊

ⵜ2_␾ (1)⫽4␲G_*␣共␾0兲T(0). 共6兲 Let us begin by solving the equation for the dilaton field

␾(1)in Eq.共6兲: ⵜ2_␾

(1)⫽12␲␴G0␣共␾0兲␦共x兲

␾(1)⫽6␲␴G0␣共␾0兲兩x兩, 共7兲 where G0⬅G_*A2(␾0).

Now, the linearized Einstein’s equation in Eq.共6兲 with a source given by Eq. 共5兲 are just the same as in Vilenkin’s paper关5兴, except that in our case the metric is multiplied by the linearized factor A2(␾). Therefore, we have共to first or-der in G0):

ds2⫽A2共␾0兲关1⫹4␲␴G0兩x兩共3␣2共␾0兲⫺1兲兴 ⫻关dt2_⫺dx2_⫺dy2_⫺dz2_{兴. 共8兲} The factor A2(␾0) appearing in the above expression can be absorbed by a redefinition of the coordinates (t,x,y ,z). We finally, then, obtain

ds2⫽共1⫹4D兩x兩兲关dt2⫺dx2⫺dy2⫺dz2兴, 共9兲

where D⬅␲␴G0(3␣2⫺1).

This is the line element corresponding to a domain wall in the framework of scalar-tensor gravity in the weak-field ap-proximation. The geometry given by Eq.共9兲 is only valid for

III. KLEIN-GORDON EQUATION IN SCALAR-TENSOR DOMAIN WALL

Let us consider a scalar quantum particle embedded in a classical background gravitational field. Its behavior is de-scribed by the covariant Klein-Gordon equation

冋

1

冑

⫺g⳵␮共

冑

⫺gg

␮␯_⳵

␯兲⫹m2

册

␺⫽0, 共10兲

where m is the mass of the particle, g is the determinant of the metric tensor g_␮␯and we are considering a minimal cou-pling.

In the space-time of a scalar-tensor domain wall given by metric 共9兲, Eq. 共10兲 takes the form

冋

1 1⫹4D兩x兩

冉

⳵t 2_⫺ ⳵x 2_⫺ ⳵y 2_⫺ ⳵z 2_⫺ 4D 1⫹4D兩x兩 d兩x兩 dx ⳵x

冊

⫹m 2

册

_␺ ⫽0. 共11兲

Multiplying this equation by 1⫹4D兩x兩 and neglecting terms of order D2 and up 共because we are working in the weak field aproximation兲, we get

冋

⳵t 2_⫺ ⳵x 2_⫺ ⳵y 2_⫺ ⳵z 2_⫺4Dd兩x兩 dx ⳵x⫹m 2_{共1⫹4D兩x兩兲}

册

_␺⫽0. 共12兲 Since Eq. 共12兲 is invariant under the transformation x→ ⫺x, we shall restrict the allowed values of x to the interval x⬎0. Then, we have 关⳵t 2_⫺ ⳵x 2_⫺ ⳵y 2_⫺ ⳵z 2_⫺4D ⳵x⫹m2共1⫹4Dx兲兴␺⫽0. 共13兲

Let us assume that

␺共t,x,y,z兲⫽e⫺i(Et⫺kyy⫺kzz)_X共x兲, 共14兲

where E, ky and kz are constants. If we substitute relation

共14兲 into Eq. 共13兲, we obtain d2X共x兲 dx2 ⫹4D dX共x兲 dx ⫹关E2_⫺k y 2_⫺k z 2_⫺m2_{共1⫹4Dx兲兴X共x兲⫽0, 共15兲} whose general solution is

X共x兲⫽兵C1Airy Ai关⫺共⫺1兲1/32⫺4/3共m2D兲⫺2/3 ⫻共m2_⫺E2_⫹k y 2_⫹k z 2_⫹4m2_Dx⫹4D2_兲兴 ⫹C2Airy Bi关⫺共⫺1兲 1/3_2⫺4/3共m2_D兲⫺2/3 ⫻共m2_⫺E2_⫹k y 2_⫹k z 2_⫹4m2_Dx⫹4D2_兲兴其e⫺2Dx_, 共16兲 with functions Airy Ai(x) and Airy Bi(x) being the Airy functions and C1 and C2are integration constants. Note that the arguments of the Airy functions have only terms of order less than D2_{. Neglecting terms of order⭓D}2_{, we get}

X共x兲⯝兵C1Airy Ai关⫺共⫺1兲1/32⫺4/3共m2D兲⫺2/3 ⫻共m2_⫺E2_⫹k y 2_⫹k z 2_⫹4m2_Dx⫹4D2_兲兴 ⫹C2Airy Bi关⫺共⫺1兲1/32⫺4/3共m2D兲⫺2/3 ⫻共m2_⫺E2_⫹k y 2_⫹k z 2_⫹4m2_Dx⫹4D2_{兲兴其共1⫺2Dx兲,} 共17兲 In order to determine the bound states energies we must re-quire periodicity conditions in directions y and z, with peri-ods Ly and Lz, respectively, supplemented by the boundary

conditions that the solution vanishes at x⫽a and x⫽b, with

a⬍b and such that DaⰆ1 and DbⰆ1. These boundary

con-ditions can be expressed as

␺共t,x,y,z兲⫽␺共t,x,y⫹Ly,z兲

␺共t,x,y,z兲⫽␺共t,x,y,z⫹Lz兲

␺共t,a,y,z兲⫽␺共t,b,y,z兲⫽0. 共18兲 These boundary conditions lead us to the following results:

X共x兲⫽C1

再

Airy Ai关⫺共⫺1兲1/32⫺4/3共m2D兲⫺2/3共m2⫺E2⫹ky 2_⫹k z 2_⫹4m2_Dx⫹4D2_兲兴 ⫺Airy Ai关⫺共⫺1兲 1/3₂⫺4/3_共m2_D兲⫺2/3_共m2_⫺E2_⫹k y 2_⫹k z 2_⫹4m2_Da⫹4D2_兲兴

Airy Bi关⫺共⫺1兲1/32⫺4/3共m2D兲⫺2/3共m2⫺E2⫹k_y2⫹k_z2⫹4m2Da⫹4D2兲兴Airy Bi关⫺共⫺1兲

1/3₂⫺4/3_共m2_D兲⫺2/3 ⫻共m2_⫺E2_⫹k y 2_⫹k z 2_⫹4m2_Dx⫹4D2_兲兴

冎

_{共1⫺2Dx兲, 共19兲}

where ky⫽ 2␲ny Ly , ny⫽0,⫾1,⫾2, . . . 共20兲 kz⫽ 2␲nz Lz , nz⫽0,⫾1,⫾2, . . . , 共21兲 and X共b兲⫽0. 共22兲

The boundary condition given by Eq. 共22兲 determines the energy levels of the particle in the stationary state in the region under consideration. It should be stressed that in order to solve this problem we must impose that 兩m2⫺E2⫹k2_y ⫹kz

2_⫹4m2_{Dx兩Ⰷ兩(m}2_D)2/3_{兩, in which case the absolute} value of the argument of Airy’s functions are much greater than unity, which allows us to use the following assymptotic expansions关19兴: Airy Ai共z兲⬃ 1 2

冑

␲z ⫺1/4_e⫺␰

兺

k⫽0 ⬁ 共⫺1兲k_c k␰⫺k Airy Bi共z兲⬃ 1

冑

␲z⫺1/4e␰k

兺

⫽0 ⬁ c_k␰⫺k 共23兲 where ␰⫽2₃z3/2 c0⫽1, ck⫽ 共2k⫹1兲共2k⫹3兲•••共6k⫺1兲 216k_k! . 共24兲

In this approximation, Eq. 共22兲 can be written as

e⫺(2/3)W3/2(b)

兺

k⫽0 ⬁ 共⫺1兲k_c k

冋

2 3W 3/2_共b兲

册

⫺k ⫺e⫺(4/3)W3/2_(a) ⫻k

兺

⫽0 ⬁ 共⫺1兲k_c k

冋

2 3W 3/2_共a兲

册

⫺k

兺

k⫽0 ⬁ ck

冋

2 3W 3/2_共a兲

册

⫺k e2/3W 3/2_(b) ⫻

兺

k⫽0 ⬁ ck

冋

2 3W 3/2_共b兲

册

⫺k ⫽0, 共25兲 where W(x)⫽⫺(⫺1)1/32⫺4/3(m2D)⫺2/3(m2⫺E2⫹ky 2_⫹k z 2 ⫹4m2_Dx⫹4D2_).

Considering only the first two terms of the summation and neglecting terms of order⭓D2, we find

ei2

冑

E2⫺ky 2 ⫺kz 2 ⫺m2_(b⫺a)

冋

1⫺i 4m 2_D共b2_⫺a2_兲

冑

E2⫺k_y2⫺k_z2⫺m2

册

⫽1. 共26兲

Now, let us take E2⫺k_y2⫺k_z2⫺m2⬎0, then we can rewrite Eq. 共26兲 as a system of equations involving its real and imaginary parts as follows:

1⫹ 4m 2_D共b2_⫺a2_兲

冑

E2⫺k_y2⫺k_z2⫺m2tan关2

冑

E 2_⫺k y 2_⫺k z 2_⫺m2_{兴共b⫺a兲} ⫽sec关2

冑

E2⫺k_y2⫺k_z2⫺m2兴共b⫺a兲, 共27兲 tan关2

冑

E2⫺k_y2⫺k_z2⫺m2兴共b⫺a兲⫽ 4m 2_D共b2_⫺a2_兲

冑

E2⫺k_y2⫺k_z2⫺m2. 共28兲 As the function tan(x) is of order D, we can make the as-sumption that tan(x)⯝x. Then, Eq. 共28兲 results in

2

冑

E2⫺k2_y⫺k_z2⫺m2共b⫺a兲⯝ 4m

2_D共b2_⫺a2_兲

冑

E2⫺k_y2⫺k_z2⫺m2⫹n␲, n⫽0,⫾1,⫾2, . . . . 共29兲

Equation共27兲 is assured if n is even. Then, we have

2

冑

E2⫺k_y2⫺k_z2⫺m2共b⫺a兲⯝ 4m

2_D共b2_⫺a2_兲

冑

E2⫺k_y2⫺k_z2⫺m2⫹2n␲, n⫽0,⫾1,⫾2, . . . . 共30兲

From the previous equation we get, finally, that

E2_⫽m2_⫹k y 2_⫹k z 2_⫹ n 2_␲2 共b⫺a兲2 ⫹

冉

2⫹ n␲ b⫺a

冊

m 2_{D共b⫹a兲, n⫽1,2, . . . . 共31兲} It is worth noting that the presence of the wall increases the energy eigenvalues with parameters1 ␴, ␣(␾0) and G0 and that for D⫽0 共absence of the wall兲 we recover the result corresponding to the Minkowski spacetime as it should be.

In what concerns the current associated with the scalar field given by

J␮⫽i

冑

⫺g

2m 共␺*⳵

␮_␺⫺␺⳵␮_{␺*兲, 共32兲}

1_{Just as a reminder for the reader,␴,␣(␾0) and G}

0are the wall’s

surface energy density, the coupling strength between the wall and the dilaton, and the effective gravitational constant, respectively.

it is clear that the current depends on the parameters that characterize the scalar-tensor domain wall through

冑

⫺g and the solution␺ of the Klein-Gordon equation.

IV. DIRAC EQUATION IN A SCALAR-TENSOR DOMAIN WALL

Now, let us consider a spinor particle embedded in a clas-sical gravitational field. The covariant Dirac equation gov-erning the particle in a curved spacetime for a spinor⌿ may be written as

关i␥␮_共x兲⳵

␮⫹i␥␮共x兲⌫␮⫺m兴⌿共x兲⫽0, 共33兲

where ␥␮(x) are the generalized Dirac matrices and are given in terms of the standard flat space Dirac matrices (␥(a)) as

␥␮_共x兲⫽e

(a)

␮ _共x兲␥(a)_{, 共34兲}

where e_(a)␮ (x) are tetrad components defined by

e_(a)␮ 共x兲e_(b)␯ 共x兲␩(a)(b)⫽g␮␯. 共35兲 The product␥␮(x)⌫_␮ that appears in Dirac equation can be written as

␥␮_共x兲⌫

␮⫽␥(a)关A(a)共x兲⫹i␥(5)B(a)共x兲兴, 共36兲 with␥(5)⫽i␥(0)␥(1)␥(2)␥(3) and A(a) and B(a) are given by

A(a)⫽ 1 2共⳵␮e(a) ␮ _⫹e (a) ␳ _⌫ ␳␮ ␮ _{兲 共37兲} and B(a)⫽ 1 2⑀(a)(b)(c)(d)e (b)␮_e(c)␯_⳵ ␮e␯(d), 共38兲

where ⑀_(a)(b)(c)(d) is the completely antisymmetric fourth-order unit tensor.

In the spacetime of a scalar-tensor domain wall given by metric 共9兲, let us choose the following set of tetrads:

e_␮(a)⫽关1⫹4D兩x兩兴1/2␦_␮a, 共39兲 which implies that

␥␮_{⫽关1⫺2D兩x兩兴␥}(␮)_{, 共40兲} in which we have neglected terms of order ⭓D2.

Computing the expressions for A(a) and B(a) and putting these results into Eq. 共36兲 and neglecting terms of order ⭓D2_{, we get}

␥␮_共x兲⌫

␮⫽3D␥(1). 共41兲

Now, using Eqs.共40兲 and 共41兲, the Dirac equation 共33兲 in the spacetime of a scalar-tensor domain wall reads

兵i␥(␮)⳵_␮⫹i3D␥(1)⫺关1⫹2Dx兴m其⌿共x兲⫽0, 共42兲

in which we neglected terms of order ⭓D2 and considered only the interval x⬎0. In order to determine the solutions in the spinoral case, let us choose the following representation of Dirac matrices: ␥(0)_⫽

冉

1 0 0 ⫺1

冊

, ␥(i)_⫽

冉

0 ␴i ⫺␴i 0

冊

, i⫽1,2,3 共43兲

where␴i (i⫽1,2,3) are the usual Pauli matrices.

Since we are interested in the qualitative behavior of the particle with respect to the parameters that define the wall, we will simplify our analysis considering the solution of the Dirac equation corresponding to the massless spinor particle in which case Eq. 共42兲 reduces to

兵i␥(␮)⳵_␮⫹i3D␥(1)其⌿共x兲⫽0, 共44兲

and is supplemented by the helicity condition

共1⫹␥5_{兲⌿共x兲⫽0, 共45兲} where

␥5_⫽

冉

0 1

⫺1 0

冊

.

Equations 共44兲 and 共45兲 are the Weyl equations for a mass-less spin-12 particle. The condition given by Eq.共45兲 implies

that the four-spinor⌿(x) is such that

⌿共x兲⫽

冉

⌿_⌿1共x兲 2共x兲

冊

,

with⌿1(x)⫽⌿2(x).

A suitable set of solutions of Weyl’s equations is of the form ⌿共t,x,y,z兲⫽

冉

u_u共x兲_共x兲

冊

e⫺i(Et⫺kyy⫺kzz)_, 共46兲 where u共x兲⫽

冉

u1共x兲 u2共x兲

冊

.

If we substitute relation 共46兲 into Eq. 共44兲, we obtain

idu1共x兲

dx ⫺共ky⫺3D兲u1共x兲⫹共E⫹kz兲u2共x兲⫽0 共47兲 idu2共x兲

dx ⫹共ky⫹3D兲u2共x兲⫹共E⫺kz兲u1共x兲⫽0.

Neglecting terms of order D2and up, the set of Eqs.共47兲 can be written as

d2u1共x兲 dx2 ⫹6D du1共x兲 dx ⫹共E⫺ky 2_⫺k z 2_兲u 1共x兲⫽0, 共48兲 d2u₂共x兲 dx2 ⫹6D du₂共x兲 dx ⫹共E⫺ky 2_⫺k z 2_兲u 2共x兲⫽0, whose general solution reads

u共x兲⫽C1e(⫺3D⫹i

冑

E 2_⫺k y 2 ⫺kz 2 )x_⫹C 2e(⫺3D⫺i

冑

E 2_⫺k y 2 ⫺kz 2 )x_, 共49兲 where C1and C2are constant bispinors. Neglecting terms of order D2 and up, we get

u共x兲⫽共1⫺3Dx兲共C₁ei

冑

E2⫺k_y2⫺k_z2x ⫹C2e⫺i

冑

E 2_⫺k y 2_⫺k z 2_x 兲. 共50兲

Again, we notice that the solutions depend on␴,␣ and G₀, as expected.

Now, let us compute the current j␮, which is defined by

j␮⫽⌿¯␥␮⌿. 共51兲

Using Eq.共40兲, the expression for the current in the approxi-mation considered turns into

j␮⫽共1⫺4Dx兲⌿†␥(0)␥(␮)⌿. 共52兲

Substituting Eqs.共46兲 and 共50兲 into Eq. 共52兲 and considering, for simplicity, the case in which C₂⫽0, we get

j␮⫽共1⫺10Dx兲共C₁†C₁†兲␥(0)␥(␮)

冉

C₁

C₁

冊

. 共53兲

In order to obtain the energy spectrum, let us consider the same boundary conditions given by Eq. 共18兲 for the scalar field. Thus, we have

⌿共t,x,y,z兲⫽⌿共t,x,y⫹Ly,z兲

⌿共t,x,y,z兲⫽⌿共t,x,y,z⫹Lz兲

⌿共t,a,y,z兲⫽⌿共t,b,y,z兲⫽0. 共54兲 Analogously to the case of a scalar field, from these bound-ary conditions we get

u共x兲⫽C₁ei

冑

E2⫺k_y2⫺k_z2x_共1⫺e⫺2i

冑

E2⫺k_y2⫺k_z2(x⫺a)_{兲 共55兲} where ky⫽ 2␲ny Ly , ny⫽0,⫾1,⫾2, . . . 共56兲 kz⫽ 2␲nz L_z , nz⫽0,⫾1,⫾2, . . . , 共57兲 and u共b兲⫽0. 共58兲

The energy levels arrive from Eq.共58兲 and are given by

E2_⫽k y 2_⫹k z 2_⫹ n 2_␲2 共b⫺a兲2. 共59兲 Note that the energy spectrum is the same as in the flat Minkowski spacetime case. This result comes from the fact that the spinor field is massless. The same coincidence oc-curs in the case of a massless scalar field.

From previous results we conclude that the current differs from that in the Minkowski spacetime by the terms contain-ing the parameters␴, ␣ and G0 used to describe the scalar-tensor domain wall and tends to the corresponding result in Minkowski spacetime in the absence of the wall as it should be.

V. CONCLUDING REMARKS

Recently there has been growing interest in domain walls as brane world scenarios and also in scalar-tensor theories of gravity due to its possible role in the understanding of the physics of the early Universe when topological defects like domain walls were formed. At that time the dilaton fields as well as the topological defect, such as a scalar-tensor domain wall, were, certainly, very relevant. These points constitute the main motivation for this work.

A scalar or spinor particle placed in the spacetime of a scalar-tensor domain wall is perturbed by this background due to the geometrical and topological features of the space-time under consideration. In other words, the dynamic of atomic systems is determined by the curvature at the position of the system and also by the topology of the background spacetime.

Summarizing our conclusions we can say that the metric of a scalar-tensor domain wall depends on the wall’s surface energy density ␴ and on two post-Newtonian parameters

␣(␾₀) and G₀. The solutions for the scalar and spinor cases differ from the flat Minkowski spacetime case by the pres-ence of these parameters. The prespres-ence of the wall shift the energy levels and alters the current in the scalar case as com-pared with the flat spacetime. In the massless spinor particle case, there is no shift in the energy spectrum, but the current is altered by the presence of the scalar-tensor domain wall.

Finally, it is worth commenting that the study of a quan-tum system in a gravitational field such as, for example, the one considered in this paper, may shed some light on the problems of combining quantum mechanics and gravity. On the other hand, the investigation of topological defects in the framework of general scalar-tensor theories seems to be im-portant in order to understand the role played by these stru-tures in this general context.

ACKNOWLEDGMENTS

V.B.B. would like to thank CNPq for partial financial sup-port. L.P.C. would like to thank CAPES for financial supsup-port. L.P.C. and M.E.X.G. acknowledge the kind hospitality of the

‘‘Grupo de Fı´sica Teo´rica’’ of the Universidade Cato´lica de

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