THE DIRAC EQUATION IN ROBERTSON-WALKER

THE DIRAC EQUATION IN ROBERTSON-WALKER SPACES: A CLASS OF SOLUTIONS

´gare ´ M. Kovalyov and M. Le Department of Mathematics University of Alberta Edmonton, Alberta Canada T6G 2G1

ABSTRACT Exact solutions of the Dirac equation in open and closed Robertson-Walker spaces are presented. A set of massive solutions is given for static metrics. In the case of non-trivial and arbitrary expansion factors, massless solutions are obtained via a conformal transformation. The set of massless solutions in open RobertsonWalker spaces is shown to be complete.

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1. INTRODUCTION: Dirac spinor fields in the background of a gravitational field have been the subject of many investigations. In the last two decades, a number of such studies have been devoted to the determination of the renormalized vacuum expectation value of the energy-momentum tensor and the problem of creation of particles in expanding universes [1-9]. As a starting point, a complete set of solutions to the generalized Dirac equation is desirable. For the flat and closed Robertson-Walker (R.-W.) spaces, sets of exact solutions to the massless and massive Dirac equations have already been found [1,2,4,9-11]. In the following, we present a complete set of massless solutions in open R.-W. spaces using the Poincar´e (or upper half-space) model of the three dimensional hyperbolic space (Hk3 ). Even if massless solutions in R.-W. spaces generate a conformally trivial case, preventing any creation of particles with expansion, these solutions can still be used to evaluate back reaction effects of quantum spinor fields on the gravitational field. The sketch of the paper is as follows. In Section 2, we briefly introduce the generally covariant formulation of the Dirac equation and we present its explicit form for the three classes of R.-W. metrics (open, flat, and closed). Section 3 is devoted to the description of the upper half-space model of the three dimensional hyperbolic space (Hk3 ) and to a discussion of its isometry group. In Section 4, the Dirac equation in the space IR × Hk3

with static metric is solved under specific requirements for the general

massive case. Then a set of solutions is generated by the action of isometries of Hk3 and massless solutions to the Dirac equation in open R.-W. spaces are obtained by application of a conformal transformation to the massless spinor solutions determined for the static metric. In Section 5, the spinor solutions found in IR × Hk3 are rewritten in terms of spherical coordinates. As a byproduct, a transformation effected on the curvature parameter of these solutions will give rise to solutions in closed R.-W. spaces. We show that the massless solutions in open R.-W. spaces form a complete set in Section 6. Finally, a summary of the results and possible

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future developments are given in the last section.

2. THE DIRAC EQUATION IN ROBERTSON-WALKER SPACES: The covariant formulation of the Dirac equation in curved spaces is presented in Lichnerowicz [13] and Choquet-Bruhat et al. [14]. First, we summarize some of the definitions and notations. Let M

of signature (+ − −−) . A Dirac spinor field ψ

with a hyperbolic metric g on

M

is a

(C ∞ )

be a four-dimensional manifold endowed

section of the vector bundle associated to the spin bundle 1

1

corresponding to (M, g) via the D( 2 ,0) ⊕ D(0, 2 ) representation of SL(2, C). For each class of Robertson-Walker spaces, a spin bundle exists and thus the spinor fields are globally well defined. Moreover, the Levi-Civita connection associated to g (with coefficients denoted by Γµνλ ) on the frame bundle over M determines a connection on the spin bundle (spin connection), which then defines a covariant derivative of the spinor field ψ :

∇µ ψ = (∂µ + Σµ )ψ,

where

Σµ , µ = 0, 1, 2, 3, 1

(2.1)

stand for the spin connection coefficients. They take 1

values in a D( 2 ,0) ⊕ D(0, 2 ) representation of the Lie algebra of SL(2, C) and satisfy the following equation: ∂˜ γµ + Γµνλ γ˜ λ (x) + [Σν , γ˜ µ (x)] = 0. ν ∂x

(2.2)

The γ˜ ’s appearing in equation (2.2) are constrained by:

γ˜ µ γ˜ ν + γ˜ ν γ˜ µ = 2g µν 1 4 .

(2.3)

With a choice of orthonormal frames {eα | α = 0, 1, 2, 3} on M :

eα = eµα (x)

∂ , ∂xµ

(2.4)

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where α = 0, 1, 2, 3, the standard Dirac matrices are retrieved:

γα = eµα (x)˜ γµ (x).

If ηαβ

(2.5)

denotes the Minkowski metric, it follows from (2.5) that the γ’s obey

the relation defining a Clifford algebra on the Minkowski space:

γα γβ + γβ γα = 2ηαβ 1 4 .

(2.6)

The covariant form of the Dirac equation in curved space is then written as:

(i˜ γ µ (x)∇µ − m)ψ = 0.

(2.7)

For instance, let us consider the R.-W. spaces with their metric expressed in terms of spherical coordinates [15]: ( ) g = gµν dxµ ⊗ dxν = R2 (t) dt2 − dr2 − f 2 (r)(dθ2 + sin2 θd2 ϕ) ,

(2.8)

where R(t) is the expansion factor, and    f (r) =

 

r,

with with

0 ≤ r < ∞, 0 ≤ r < ∞,

for the open case with curvature equal to for the flat case

sin kr k ,

with

0≤r≤

for the closed case with curvature equal to k 2 .

sinh kr , k

π k,

If we choose the following set of orthonormal co-frames {θα } :

θ0 = R(t)dt,

θ1 = R(t)dr,

θ2 = R(t)f (r)dθ,

and θ3 = R(t)f (r) sin θdϕ, (2.9)

with: eα (θβ ) = δαβ .

(2.10)

− k2

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Then, the spin connection coefficients are given by: Σ0 = 0 1 R′ 0 1 [γ , γ ] 4 R ) 1 ( R′ Σ2 = − f [γ 0 , γ 2 ] + f ′ [γ 1 , γ 2 ] 4 R ) 1 ( R′ Σ3 = − f sin θ[γ 0 , γ 3 ] − f ′ sin θ[γ 1 , γ 3 ] − cosθ[γ 2 , γ 3 ] , 4 R Σ1 = −

(2.11)

where the apostrophe indicates the differentiation of the function with respect to its argument. Substituting this into the equation (2.7), we obtain the generalized form of the Dirac equation for the R.-W. spaces: (∂ [ 0( ∂ 3R′ ) f ′ ) γ2 ( ∂ cot θ ) γ3 ∂ ] γ + + im R 1 4 + γ 1 + + + + ψ = 0. ∂t 2R ∂r f f ∂θ 2 f sin θ ∂ϕ (2.12) In the massless case, a reduction of the Dirac equations in R.-W. spaces to the Dirac equations in spaces with static (R(t) = 1) metric can be achieved with the conformal mapping [16]: g ′ = R−2 (t)g,

(2.13a)

and ψ ′ = R3/2 (t)ψ.

(2.13b)

As a result, a solution to the massless Dirac equation in IR×Hk3 is also a solution to the massless Dirac equation in open R.-W. spaces up to the above-mentioned conformal factor. Before showing solutions to this reduced equation, we review in the next section some properties of Hk3 which will be useful.

3. THE THREE-DIMENSIONAL HYPERBOLIC SPACE: (Hk3 ).

One of

them, which is more natural, is the geodesic model. In this model, Hk3

can be

We will work with two models of the hyperbolic manifold

viewed as the set of all triplets (r, θ, ϕ) with range 0 ≤ r < ∞,

− π2 ≤ θ ≤

π 2,

and 0 ≤ ϕ < 2π, called spherical coordinates. In these coordinates the metric

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tensor has the form: h = dr2 +

( sinh kr )2 2 (dθ + sin2 θdϕ2 ), k

(3.1)

where k is a positive constant. In the second model, Hk3 is described by the upper half-space of the threedimensional Euclidean space with coordinates

(x1 , x2 , y),

where

y > 0,

en-

dowed with the metric:

h=

(dx1 )2 + (dx2 )2 + dy 2 . k2 y2

(3.2)

Both models are related to each other by the following transformation: sin θ cos ϕ , coth(kr) − cos θ sin θ sin ϕ , x2 = coth(kr) − cos θ 1 and y = , cosh(kr) − sinh(kr) cos θ x1 =

(3.3)

where the origin of the spherical coordinates is mapped to the point (0, 0, 1) of the upper half-space model. The group of isometries of

Hk3

is

P SL(2, C).

It can be realized using

quaternionic notation for the upper half-space coordinates (x1 , x2 , y) [17]: q = x1 · 1 + x2 · i + y · j, q ′ = x′ · 1 + x′ · i + y ′ · j. 1

2

(3.4)

The action of P SL(2, C) on q corresponds to a fractional linear transformation: q ′ = (Aq + B)(Cq + D)−1 ,

(3.5)

[

] A B where the matrix belongs to P SL(2, C). C D Let us restrict ourselves to the subgroup leaving invariant the point (0, 0, 1) in the upper half-space model. One verifies that this subgroup of

P SL(2, C),

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which is SO(3), preserves the origin in the geodesic model. It can be described as the SU (2) subgroup of SL(2, C) quotiented by its center, parametrized by: 

aeiχ



(e1 + ie2 )b

(−e1 + ie2 )b where:

ae

  ∈ SU (2),

(3.6)

iχ

e21 + e22 = 1 and 0 ≤ χ < 2π.

a2 + b2 = 1,

Its action on a point of Hk3 can be written explicitly in terms of the variables x1 , x2 and y : 1 1 [x − 2(e1 x1 + e2 x2 )e1 + ∆ 2 1 x′ = [x2 − 2(e1 x1 + e2 x2 )e2 + ∆ y y′ = , ∆ x′ = 1

a e1 ] − b a e2 ] − b

a e1 , b a e2 , b

(3.7a)

where: ∆ = (bx1 − ae1 )2 + (bx2 − ae2 )2 + b2 y 2 .

(3.7b)

A further geometrical meaning can be attributed to the transformation (3.7) if we look at the Hk3 -analogues of the Euclidean planes the so-called horospheres, which are defined by the equation: (b′ y −

where c > 0,

c 2 c 2 ′ 1 ′ ′ 2 ′ 2 ′ ′ 2 ) + (b x − a e ) + (b x − a e ) = ( ) , 1 2 b′ b′

(3.8)

a′ + b′ = 1, and e′1 + e′2 = 1. 2

2

2

2

If we fix a′ , b′ , e′1 , and e′2 , we obtain a one-parameter family of surfaces ( ′ ) ′ which we call a family of parallel horospheres with direction ab′ e′1 , ab′ e′2 . When b′ ̸= 0, this family is composed of the spheres of radius bc′ tangent to the plane ( a′ ′ a′ ′ ) ′ y = 0 at the point b′ e1 , b′ e2 , 0 . If b = 0, we have the family of parallel horospheres with direction (∞, ∞) which consists of the planes y =

1 2c

and

can be thought of as a family of spheres of infinite radius tangent to the plane y =0

at the point

(∞, ∞, 0).

It can be shown that the transformation (3.7)

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maps the set of parallel horospheres with direction

(a

a b e1 , b e2

)

onto the set of

parallel horospheres with direction (∞, ∞). Moreover, for each set of parallel horospheres with direction ν =

(a

a b e1 , b e2

)

,

we can define a family of geodesics with direction ν by the equations:

and where C

1[ 1 x − 2(e1 x1 + e2 x2 )e1 + ∆ 1[ 2 x − 2(e1 x1 + e2 x2 )e2 + ∆

a ] a e1 − e1 = C, b b a ] a e2 − e2 = D, b b

(3.9)

and D are constants.

The tranformation (3.7) also maps the family of geodesics with direction ν onto the family of geodesics with direction (∞, ∞).

4. SOLUTIONS IN IR × H3k : In this section, we find explicit solutions to the massive Dirac equations in the space IR × H 3 using the upper half-space model of Hk3 . In this model, the nonzero coefficients of the Levi-Civita connection are: Γ113 = Γ223 = Γ333 = − 1/y, (4.1) Γ311 = Γ322 = 1/y. We choose the following set of orthonormal frames and co-frames on IR × Hk3 : dx1 ˜2 dx2 θ˜0 = dt, θ˜1 = , θ = , ky ky

and

dy θ˜3 = ky

(4.2)

with: e˜α (θ˜β ) = δαβ ;

α, β = 0, 1, 2, 3.

(4.3)

From (2.2), we derive the spin connection coefficients in terms of the Dirac matrices {γ α } : Σ0 = Σ3 = 0 1 1 3 γ γ , 2y 1 2 3 Σ2 = − γ γ . 2y

Σ1 = −

(4.4)

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The generalized Dirac equation (2.7) in IR × Hk3 can then be written as: { 0∂ } ∂ ∂ ∂ γ + kyγ 1 1 + kyγ 2 2 + kyγ 3 − kγ 3 + im 1 4 ψ = 0, ∂t ∂x ∂x ∂y

(4.5)

where, for explicit calculations, we will use the representation of the Dirac matrices given below: [

O2 γ = σ2 0

] [ σ2 O2 1 ; γ = O2 −iσ3

with the σi ’s

] [ −iσ3 O2 2 ; γ = O2 12

] [ −112 O2 3 ; γ = O2 iσ1

] iσ1 , O2 (4.6)

(i = 1, 2, 3) standing for the Pauli matrices.

Let us determine spinor solutions propagating along the

y-axis.

If they

stay constant on the horospheres with direction (∞, ∞) (corresponding to the y-axis), then they can be expressed as: ψ = eiwt ϕ(y),

(4.7)

where ϕ(y) satisfies the equation: kyγ 3

∂ϕ + (iwγ 0 − kγ 3 + im 1 4 )ϕ = 0. ∂y

(4.8)

A general solution to (4.8) is:

ϕ(y) =

4 ∑

ca y αa va ,

(4.9)

a=1

where the ca ’s are complex constants, √

α1 =α2 = 1 + m k−w , √ 2 2 α3 =α4 = 1 − m k−w ,   0 √  m2 − w2 − m + iw  v1 = √ 2 , m − w2 − m − iw 0 

2

2

 0 √  − m2 − w2 − m + iw  v3 = √ 2  − m − w2 − m − iw 0

√  m2 − w2 − m − iw   0 , v2 =    0 √ m2 − w2 − m + iw   √ − m2 − w2 − m − iw   0 , and v4 =    0 √ 2 2 − m − w − m + iw

(4.10)

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are respectively the eigenvalues and eigenvectors of the matrix: 14 +

iw 3 0 im 3 γ γ + γ . k k

(4.11)

Thus any solution of (4.5) with direction (∞, ∞) can be put in the form: ψ(t, y) = e

iwt

4 ∑

ca y αa va .

(4.12)

a=1

Let us mention that the plane wave spinor solutions in Minkowski space travelling along the y-axis are recovered in the zero limit of the curvature parameter k with the substitution y +

1 k

for y.

In order to obtain spinor solutions moving in the direction ( ab e1 , ab e2 ), we apply the transformation (3.7) which maps the family of horospheres with direction ( ab e1 , ab e2 )

onto the family of horospheres with direction

(∞, ∞).

Since these

transformations leave the metric invariant, the solutions to (4.5) in the direction ( ab e1 , ab e2 ) are: Ψ(t, x1 , x2 , y) = S † (a, b, e1 , e2 )ψ(t, y ′ ), 1

(4.13)

1

where S represents the D( 2 ,0) ⊕ D(0, 2 ) SL(2, C) representation of the rotation R of the orthonormal frames induced by (3.7), that is: Sγ i S † = (RT )ij γ j

(i, j = 1, 2, 3),

(4.14)

with: R=   [R]ij = 

2 ∗ 2 2 2 |z|2 +y 2 [(Im e z) + e1 y ], −2 2 2 1 2 2 |z|2 +y 2 [e1 e2 ((z ) − (z ) + y )

y J, y′

(4.15a)

1−

+ (e21 − e22 )z 1 z 2 ],

−2yz 1 |z|2 +y 2 ,

(4.15b) −2y −2 1 2 2 2 2 1 2 2 2 2 2 1 2  |z|2 +y 2 [e1 e2 ((z ) − (z ) + y ) + z z (e2 − e1 )], |z|2 +y 2 [(e2 − e1 )z − 2e1 e2 z ] −2y 2 2 2 1  1 − |z|22+y2 [(Im e∗ z)2 + e22 y 2 ], |z|2 +y 2 [(e1 − e2 )z − 2e1 e2 z ]  −2yz 2 2y 2 , 1 − 2 2 |z| +y |z|2 +y 2

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where RT

is the transpose of the rotation matrix R,

the transformation (3.7), z 1 ≡ x1 −

J

is the jacobian of

z 2 ≡ x2 − ab e2 ,

z ≡ z 1 + iz 2 , and

 u | O2 S(a, b, e1 , e2 ) =  − − − −|− − − −  , O2 | u∗

(4.16)

a b

e1 ,

e ≡ e1 + ie2 . Explicitly [18]: 

with: [

1

u= √ |z|2 + y 2

z ∗ e, ye,

Hence, the spinor solutions with direction

−ye∗ ze∗

] ∈ SU (2).

( ab e1 , ab e2 )

(4.17)

to the Dirac equation in

IR × Hk3 have the following form:  † u | O 2 [ y ]αa  − − − −|− − − −  va . Ψ(t, x1 , x2 , y) = eiwt ca ∆ a=1 O2 | uT 4 ∑



(4.18)

From this set of solutions, we obtain massless solutions to the Dirac equation in open R.-W. spaces by letting m = 0 and by applying a conformal transformation (2.13): −3/2 ΨR.-W. (t)Ψm=0 , m=0 = R

(4.19)

and Ψm=0 denote respectively massless solutions in R.-W. space where ΨR.-W. m=0 and the solutions (4.18) with m = 0. We note that the limit of the curvature parameter k to zero in (4.18) leads to a set of plane wave spinor solutions in Minkowski-space. In order to perform this limit, it is more appropriate to go back to the “Ball model” of Hk3 [17].

5. SOLUTIONS IN SPHERICAL COORDINATES: For completeness, we present the spinor solutions found for the open R.-W. spaces in terms of the usual spherical coordinates. A simple modification to these solutions will allow us to introduce a set of solutions to the Dirac equation in closed R.-W. spaces. We recall that the spherical coordinates are related to the

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upper half-space coordinates by equation (3.3). However, the transformation of the spinor fields requires the SO(3, 1)-transformation

˜ between the two sets of (Λ)

orthonormal co-frames:

β ˜α θ˜α = Λ βθ ,

(5.1)

˜ given by: with Λ 

1 |  − − − −|− − − − − −  ˜α | cos η [Λ] β =  0 | cos ϕ sin η | sin ϕ sin η

0 −−−−− − sin η cos ϕ cos η sin ϕ cos η

 − − − − −  0   − sin ϕ cos ϕ (5.2)



 1 | 0  − − − −|− − − − − − − − − − − −−  , = −2π   √ (L1 +L2 +L3 ) ϕL ηL 0 | e3 3 e 1e 3

where:

sin η =

sin θ , [cosh kr − sinh kr cos θ] (5.3)

cos η =

cosh kr cos θ − sinh kr , [cosh kr − sinh kr cos θ]

and Li stands for the generator of rotations around the i-th-axis The generators satisfy the commutation rules of the SO(3) algebra: ϵijk Lk

(i = 1, 2, 3). [Li , Lj ] =

(i, j, k = 1, 2, 3), and ϵ123 = 1. 1

1

With respect to our choice of Dirac matrices (4.6), the D( 2 ,0) ⊕D(0, 2 ) SL(2, C) ˜ can be written as: representation of Λ 

(112 + i(σ1 + σ2 + σ3 ))v ˜ Λ) ˜ = 1  − − − − − − − − −− S( 2 O2

 | O2 −|− − − − − − − − − −−  , | (112 − i(σ1 − σ2 + σ3 ))v ∗

(5.4)

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where: ϕ

η

v = e−i 2 σ1 e−i 2 σ3 =

1 (cosh kr − cos θ sinh kr)1/2 [

] kr kr (cos θ2 − ie 2 sin θ2 ) cos ϕ2 , (−i cos θ2 + e 2 sin θ2 ) sin ϕ2 × . kr kr (−i cos θ2 − e− 2 sin θ2 ) sin ϕ2 , (cos θ2 + ie 2 sin θ2 ) cos ϕ2 (5.5) Consequently the spinor field, defined in terms of the upper half-space coordinates, will undergo the following transformation: ˜ Λ) ˜ † Ψ(t, x1 , x2 , y). Ψ(t, r, θ, ϕ) = S(

(5.6)

It follows that the equation (5.6) will give rise to solutions of the Dirac equation in spherical coordinates if Ψ(t, x1 , x2 , y) has the form (4.18). The resulting spinor solutions are:

Ψ(t, r, θ, ϕ) =

4 ]αa eiwt ∑ [ 1 ca 2 a=1 ∆(cosh kr − sinh kr cos θ)



( ) v † (112 − i σ1 − σ2 + σ3 ) u† ×  − − − − − − − − − − −− O2

| −|− |

(5.7) 

O2 − − − − −( − − − − − −)− −  va , v T (112 + i σ1 + σ2 + σ3 ) uT

where ∆ and u are expressed in terms of spherical coordinates via equation (3.3). With the conformal mapping (2.13b) of (5.7), we get the spherical coordinate representation of massless spinor solutions in open R.-W. spaces, that is: −3/2 ΨR.-W. (t)Ψm=0 (t, r, θ, ϕ). m=0 (t, r, θ, ϕ) = R

Let us note that our choice of parameters a, b, e1

and e2

(5.8)

for the solutions in

the upper half-space model does not lead to simple expressions when transcribed in spherical coordinates. However, a more suitable parametrization is derived if, first, the spinor solution in the direction

ν = (∞, ∞)

is mapped to spherical

coordinates by (5.6), and then the spinor is transformed by the SO(3) isometries.

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Finally, we observe that changing the parameter k by ik and restricting the variable

r

to the interval

[0, πk ]

in the open R.-W. metric, we retrieve

exactly the closed R.-W. class. Applying this transformation to (5.8), we get a set of spinor solutions to the Dirac equation in closed R.-W. spaces. These solutions are well defined everywhere on S 3 , except at the poles r = 0, πk . Let us mention that the massless solutions found in Refs. 2, 10, and 11, which can be expressed as Jacobi polynomials in “ cos θ′′ or “ cos r′′ , differ from the above.

6. COMPLETENESS OF THE SET OF MASSLESS SOLUTIONS: In the following, a sketch of the proof of completeness of the massless solutions in open R.-W. spaces is presented. It is understood that the set of solutions (4.19) with

m = 0

is complete if any solution to the massless equation (4.5) can be

expressed as a linear combination of them. For simplicity, we consider the case when the curvature parameter k = 1. We also ignore the expansion factor R(t), since the conformal map (2.13) preserves the completeness. As a generalization to hyperbolic space of the spatial Fourier transform for massless spinors, we define:



 iw( y )1+iw ˆ β, w) = 1 ψ(t, 16π 3

    H3  

∆

0

∫

y 1−iw −iw( ∆ )

y 1+iw ) iw( ∆

0

     

y 1−iw ) −iw( ∆



 u(x1 , x2 , y; β) | O2 dx1 dx2 dy ×  − − − − − − − − −− −|− − − − − − − − − −−  ψ(t, x1 , x2 , y) , y3 ∗ 1 2 O2 | u (x , x , y; β) (6.1)

15

with the inversion formula: ∫ ∫ ψ(t, x1 , x2 , y) = IR



 u† (x1 , x2 , y; β) | O2  − − − − − − − − −− −|− − − − − − − − − −−  ∂H 3 O2 | uT (x1 , x2 , y; β) 

 −iw( y )1−iw ∆

   ×  

  dβ1 dβ2 dw  ˆ ,  ψ(t, β, w)  (1 + |β|2 )2 (6.2) 

0 y 1+iw iw( ∆ )

y 1−iw −iw( ∆ )

0 y 1+iw −iw( ∆ )

where

β ≡

|x−β|2 +y 2 1+|β|2

a be

∈ ∂H 3

(boundary of

H 3 ), u

is given by (4.17) and

∆ =

(3.7b), with x ≡ x1 + ix2 .

In this representation, we can write the general solution to equation (4.5) with zero mass as: ∫ ∫ ψ(t, x1 , x2 , y) = IR

 u† (x1 , x2 , y; β) | O2 eiwt  − − − − − − − − −− −|− − − − − − − − − −−  3 ∂H O2 | uT (x1 , x2 , y; β) 

 −iw( y )1−iw



∆

   ×  

0 y 1+iw iw( ∆ )

y 1−iw −iw( ∆ )

0

  dβ1 dβ2 dw ˆ ,  ψ(0, β, w)  (1 + |β|2 )2 (6.3) 

y 1+iw iw( ∆ )

ˆ β, w) stand for the “Fourier transform” of the initial data. where ψ(0, The consistence of equations (6.1) and (6.2) follows from the formula: ∫

+∞ −∞

∫

[

′

y 1+iw ( y ′ )1−iw ( ∆ ) 0 u (x , x , y , β) ∆ ′ y 1−iw y ) 0 ( ∆′ )1+iw ( ∆ ∂H 3 †

1′

2′

×u(x1 , x2 , y; β)

′

w2 dwdβ1 dβ2 = 16π 3 y 3 δ(x − x′ , y − y ′ )112 . (1 + |β|2 )2

]

(6.4)

In order to verify this identity, we first split the w-dependent matrix in its

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w-even and w-odd parts, that is: [

′

y 1−iw y 1+iw (∆ (∆ ) 0 ′) y ′ 1+iw y 1−iw 0 ( ∆′ ) (∆ )

Substituting (6.5) in (6.4), the

] =

] y′ y 1 [ y ′ 1−iw y 1+iw ( ′) ( ) + ( ′ )1+iw ( )1−iw 1 2 2 ∆ ∆ ∆ ∆

+

] 1 [ y ′ 1−iw y 1+iw y′ y ( ′) ( ) − ( ′ )1+iw ( )1−iw σ3 2 ∆ ∆ ∆ ∆ (6.5)

w-odd

part vanishes once integrated over

w,

leaving only the w-even contribution. The residual integral can be reduced to the form: ∫

∫

+∞

−∞

[ y ′ 1−iw y 1+iw ] w2 dwdβ1 dβ2 u (x , x , y ; β)u(x , x , y; β) ( ′ ) ( ) ∆ ∆ (1 + |β|2 )2 ∂H 3 †

1′

2′

′

1

2

= 16π 3 δ(x − x′ , y − y ′ )112 .

(6.6)

Since the identity (6.6) is invariant with respect to the action of any isometry of H 3 ity:

′

′

on (x1 , x2 , y ′ ) and (x1 , x2 , y), we can put without loss of general′

′

x1 = x2 = x1 = x2 = 0

H 3 -isometries

y ′ = 1.

and

can be used to map

′

In other words, the group of

′

(x1 , x2 , y ′ )

to

(0, 0, 1)

(x1 , x2 , y)

and

to a point on the positive y-axis, relabelled (0, 0, y). As a consequence, (6.6) is simplified to: ∫

+∞ −∞

∫

u† (0, 0, 1; β)u(0, 0, y; β) ∂H 3

[ y(1 + |β|2 ) ]1+iw w2 dwdβ1 dβ2 y 2 + |β|2 (1 + |β|2 )2

= 16π 3 y 3 δ(x, y − 1)112 .

(6.7)

We obtain from (4.17) that:  u† (0, 0, 1; β)u(0, 0, y; β) = √

1 (|β|2 + 1)(|β|2 + y 2 )



|β|2 + y

−β ∗ (1 − y)

β(1 − y)

|β| + y

 .

2

(6.8)

17

The integration over the parameters β1 and β2 can now be carried out. Inserting (6.8) in (6.7) and passing to polar coordinates:

β = β1 + iβ2 = |β|eiγ , where

0 ≤ γ < 2π, we find that the off-diagonal integrals over the angular variable γ vanish while the diagonal contribution becomes: ∫

∫

+∞

2π

dw w

∞

2

−∞

0

[ y(1 + |β|2 ) ]1+iw d|β| |β| (|β|2 + y) √ 12 (1 + |β|2 )2 (|β|2 + 1)(|β|2 + y 2 ) y 2 + |β|2

= 16π 3 y 3 δ(x, y − 1)112 .

(6.9)

We evaluate the second integral to the left-hand side of (6.9) using the following substitution: √ |β| = where:

1+t , 1−t

(6.10)

−1 ≤ t ≤ 1.

After integration, the equation (6.9) is transformed to: 2πy 3/2 I(y) = 16π 3 y 3 δ(x, y − 1), (1 + y)2 (1 − y)

(6.11)

where: ∫

+∞

dw w2 Re

I(y) = −∞

The change of variables:

{ y − 21 +iw + y 12 −iw } . 1 − iw 2

(6.12)

y = eξ , where ξ ∈ IR, allows us to rewrite (6.11) as:

−1 I(ξ) = 8π 2 y 3 δ(x, y − 1), 4 sinh ξ cosh 2ξ

(6.13)

where: ∫

+∞

I(ξ) = −∞

( eiξ(w+ 2i ) − e−iξ(w+ 2i ) ) 2 w dw. −i(w + 2i )

(6.14)

Differentiating I(ξ), we find that: I ′ (ξ) = 4π cosh

ξ ′′ δ (ξ). 2

(6.16)

18

A symbolic expression for I(ξ) can be derived with help of the following equalities [19]: ξ ′′ δ (ξ) = δ ′′ (ξ) 2 ξ cosh δ ′ (ξ) = δ ′ (ξ). 2

cosh and

(6.17a) (6.17b)

In fact, we get from equations (6.16) and (6.17a): I ′ (ξ) = 4πδ ′′ (ξ).

(6.18)

Using (6.17b), the integration of (6.18) gives rise to:

I(ξ) = 4π cosh

ξ ′ δ (ξ), 2

(6.19)

where the constant of integration vanishes since I(ξ) and δ ′ (ξ) are both odd distributions. Finally, inserting this last result in the identity (6.13), we arrive to: −δ ′ (ξ) = 8πy 3 δ(x, y − 1). sinh ξ

(6.20)

The validity of this identity is shown below. It enables us to conclude that (6.4) is verified, and consequently, that (6.1) and (6.2) are consistent. In order to justify (6.20), we use spherical coordinates. First , we note that |ξ| can be identified with the variable r in (3.3), which is the geodesic distance between (0, 0, 1) and (x1 , x2 , y). The identity (6.20) is then confirmed by verification of the next formula for any test function (f ), expressed in both the upper half-space coordinates (x1 , x2 , y) and the spherical coordinates (r, θ, ϕ) : ∫ −8π

dx1 dx2 dy =2 y δ(x, y−1)f (x , x , y) y3 3

2

∫

2

∞

∫ 2

sinh rdr 0

δ ′ (r) . sinh r (6.21)

sin2 θdθdϕf (r, θ, ϕ)

19

7. SUMMARY: We were concerned in this paper with Dirac fields in the background of R.-W. metrics. In the case of open R.-W. spaces, a set of explicit massless solutions to the Dirac equation has been found. Each solution propagates along the direction defined by a set of parallel horospheres, the analogues of the planes in flat space. For this purpose, we have used the upper half-space representation of the threedimensional spatial submanifold. In the limit of zero curvature, one can show that spinor plane-wave solutions in Minkowski space are retrieved. We also expressed the open R.-W. spinor solutions in terms of spherical coordinates. It follows that the substitution of the parameter k by its imaginary form and the restriction of the domain of the spatial variable r lead to closed R.-W. spinor solutions. Finally, it has been verified that the set of spinor massless solutions presented for the open class is complete, a property which is certainly very important for quantum field theoretical considerations. Let us recall that only massless solutions have been determined for metrics with nonconstant and arbitrary expansion factor R(t), since a conformal map of the static metric was carried out. It would be interesting in a future investigation to exhibit massive spinor solutions in R.-W. spaces with nontrivial expansion factors, as worked out for the flat case in ref. 12.

ACKNOWLEDGMENT: It is a pleasure to thank Dr. J. So for his assistance. This research was supported by grants from the Natural Science and Engineering Research Council of Canada.

REFERENCES: 1. L. Parker, Phys. Rev. D3, 346 (1971). 2. L.H. Ford, Phys. Rev. D14, 3304 (1976). 3. G. Sch¨ afer and H. Dehnen, Astron. Astrophys. 54, 823 (1977).

20

4. J. Audretsch and G. Sch¨ afer, J. Phys. A: Math. Gen. 11, 1583 (1978). 5. A.A. Grib, S.G. Mamayev, V.M. Mostepanenko, Fortsch. Phys. 28, 173 (1980). 6. M. Castagnino, L. Chimento, D.D. Harari and C.A. N´ un ˜ez, J. Math. Phys. 25, 360 (1984). 7. A.H. Najmi and A.C. Ottewill, Phys. Rev. D30, 2573 (1984). 8. L.P. Chimento and M.S. Mollerach, Phys. Rev. D34, 3689 (1986). 9. C.J. Isham and J.E. Nelson, Phys. Rev. D10, 3226 (1974). 10. E. Schr¨ odinger, Comment. Pontif. Acad. Sci. 2, 321 (1938). 11. E. Schr¨ odinger, Proc. R. Ir. Acad. A46, 25 (1940). 12. A.O. Barut and I.H. Duru, Phys. Rev. D36, 3705 (1987). 13. A. Lichnerowicz, Bull. Soc. Math. France 92, 11 (1964). 14. Y. Choquet-Bruhat, C. De Witt-Morette and M. Dillard-Bleick, Analysis, Manifolds and Physics, North-Holland Publ. Co., Revised edition 1982, N.Y. 15. S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press, Cambridge, 1973. 16. Y. Choquet-Bruhat and D. Christodoulou, Ann. Scient. Ec. Norm. Sup., 4 ´eme s´erie, t.14, 481, (1981). 17. A.F. Beardon, The Geometry of Discrete Groups, Graduate Texts in Mathematics, Vol. 91, Springer-Verlag, N.Y., 1983. 18. The spinor transformation S has been determined from the normal eigenvector (⃗n) corresponding to the eigenvalue 1 and the two other eigenvalues (e±iα ) of R. The canonical homomorphism between SU (2) and SO(3) is then used to arrive to u(= cos α2 − i⃗n · ⃗σ sin α2 ). 19. In the formulas (6.17a,b), the absolute value of the variable ξ will later be interpreted as the spherical coordinate r introduced in Section 3. If we consider test functions f (ξ), (6.17a,b) are proved by showing that the following relations are satisfied respectively:

∫

+∞

−∞

ξ f (ξ) cosh δ ′′ (ξ) sinh2 ξdξ = 2

∫

+∞ −∞

f (ξ)δ ′′ (ξ) sinh2 ξdξ,

21

and ∫

+∞

−∞

( f (ξ)

) δ ′ (ξ) sinh2 ξdξ = ξ sinh ξ cosh 2

∫

+∞

−∞

( δ ′ (ξ) ) f (ξ) sinh2 ξdξ. sinh ξ