Scalar and Spinor Two-Point Functions in Einstein Universe

HUPD-9516 July 1995

arXiv:hep-th/9512205v1 28 Dec 1995

Scalar and Spinor Two-Point Functions in Einstein Universe

T. Inagaki ∗ , K. Ishikawa

†

and T. Muta

‡

Department of Physics, Hiroshima University, Higashi-Hiroshima, Hiroshima 739, Japan

Abstract

Two-point functions for scalar and spinor fields are investigated in Einstein universe (R ⊗ S N−1 ). Equations for massive scalar and spinor two-point functions are solved and the explicit expressions for the two-point functions are given. The simpler expressions for massless cases are obtained both for the scalar and spinor cases.

∗

e-mail : [email protected] e-mail : [email protected] ‡ e-mail : [email protected] †

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It is important to investigate quantum field theories in curved space-time in connection with phenomena under strong gravity as in the early universe. The fundamental object in dealing with quantum field theories in curved space-time is the two-point Green function which we shall study in the present communication. There are numerous previous studies which have dealt with the two-point functions in the maximally symmetric space-time S N and/or H N . In the present paper we push forward the investigation and study the Einstein universe. We consider the manifold R ⊗ S N−1 as an Euclidean analog of the N-dimensional Einstein universe.1 The manifold is defined by the metric ds2 = dr 2 + a2 (dθ2 + sin2 θdΩN−2 ) ,

(1)

where dΩN−2 is the metric on a unit sphere S N−2 . The manifold is a constant curvature space with curvature R = (N − 1)(N − 2)

1 . a2

(2)

We start with the argument on scalar two-point functions. On the manifold R ⊗ S N−1 the scalar two-point function is defined by the equation 1 ((∂0 )2 + 2N−1 − ξR − m2 )GF (y(0) , y) = − √ δ N (y(0) , y) , g

(3)

where g is the determinant of the metric tensor gµν , 2N−1 is the Laplacian on S N−1 , δ N (y(0) , y) the Dirac delta function in the manifold R ⊗ S N−1 and ξ the coupling constant between the scalar field and the curvature. In the following discussions we fix y(0) at the origin and write GF (y(0) , y) = GF (y). After performing the Fourier transformation in the time variable,      

We obtain from Eq.(3),

    

GF (y) = N

δ (y) =

Z

dω −iωy0 ˜ e GF (ω, yy), 2π

Z

dω −iωy0 N−1 e δ (yy), 2π

 ˜ F (ω, yy) = − √1 δ N−1 (yy) , 2N−1 − ξR − (m2 + ω 2 ) G g



2

(4)

(5)

where yy denotes the space component of y. Equation (5) has the same form as the one for scalar Green functions on S N−1 . For the maximally symmetric space S N−1 we can easily solve the equation for the Green function following the method developed by Allen ˜ F (ω, yy) is obtained straightforwardly. Because of and Jacobson.2 The Green function G ˜ F (ω, yy) is represented as a function only of σ = aθ which the symmetry on R ⊗ S N−1 , G is the geodesic distance between yy(0) and yy on S N−1 . The Laplacian acting on a function of σ is found to be

2

1 d d (sin θ)2−N (sin θ)N−2 f (σ) 2 a dθ dθ     σ N −2 ∂σ f (σ) . cot = ∂σ 2 + a a

2N−1 f (σ) =

(6)

Using Eq.(6) we rewrite the Eq.(5) to obtain 

∂σ

2

N −2 σ ˜F = 0 , + ∂σ − ξR − (m2 + ω 2) G cot a a  



(7)

where we restrict ourselves to the region σ 6= 0. To solve Eq.(7) we make a change of the   σ variable z = cos2 and find 2a 

z(1 − z)∂z

2

N −1 ˜F = 0 . − (N − 1)z ∂z − (N − 1)(N − 2)ξ − (m2 + ω 2 )a2 G + 2 (8) 





Equation (8) is known as the hypergeometric differential equation. The solution of this equation is given by the linear combination of hypergeometric functions,3 N −2 N −2 N −1 + iα, − iα, ;z 2 2 2   N −2 N −1 N −2 + iα, − iα, ;1− z , +pF 2 2 2

˜ F = qF G





(9)

where we define α by α=

s

(m2 + ω 2 )a2 + (N − 1)(N − 2)ξ −

3

(N − 2)2 . 4

(10)

˜ F is regular at the point σ = aπ, we find that p = 0. To determine As Green function G ˜ F in the limit σ → 0 the constant q we consider the singularity 3 of G     N −3 N −1  3−N Γ Γ σ 2 2 ˜ , (11) GF −→ q     N −2 N −2 2a + iα Γ − iα Γ 2 2 and compare it with the singularity of the Green function in flat space-time. This procedure is justified because the singularity on a curved space-time background has the same structure as that in the flat space-time. For σ ∼ 0 the Green function in the flat space-time behaves as ˜ fFlat (σ) ∼ G

N − 3 3−N σ . Γ ( N−1)/2 4π 2 1





(12)

Comparing Eq.(11) with Eq.(12) we obtain the constant q : N −2 N −2 + iα Γ − iα a3−N Γ 2 2 q= .   N −1 (4π)(N−1)/2 Γ 2 From Eqs.(4),(9) and (13) we finally obtain the scalar two-point function GF , 







(13)

N −2 N −2 + iα Γ − iα 3−N dω −iωy a 2 2 e GF (y) =   N −1 (4π)(N−1)/2 2π (14) Γ 2    N −2 N −1 σ N −2 2 . + iα, − iα, ; cos ×F 2 2 2 2a The two-point function (14) develops many singularities at σ = 2πna where n is an Γ 0

Z









arbitrary integer. This property is a direct consequence of the boundedness of the space S N−1 . In other words the geodesic distance σ is bounded in [0, 2πa). Thus the two-point function (14) satisfies the periodic boundary condition GF (y 0 , σ) = GF (y 0 , σ + 2πna). In the two dimensional limit N → 2 the two-point function reduces to the well-known formula a GF (y) → 2 =

dω −iωy0 cosh(α(π − σ/a)) e 2π α sinh(απ) ∞ Z X 1 dω −ipy 1 e , 2 2πa n=−∞ 2π p + m2 Z

4

(15)

where pµ and y µ are given by         

p

µ

y

µ

=



n ω, 2 a



,

0

= (y , θ) =

σ y , a



0



(16) .

In deriving Eq.(15) we employ the following summation formula,3 ∞ 1 X cosh(a(π − x)) cos(nx) = . π n=−∞ n2 + a2 sinh(aπ)

(17)

Equation (15) is nothing but the ordinary two-point function with the periodic boundary condition for the spatial coordinate. N −2 If m = 0 and ξ = , the field equation (3) is invariant under conformal 4(N − 1) transformation. In this case we can perform the Fourier integral in Eq.(14) at y = y(0) , conf ormal

GF

2a2−N N N N −1 Γ (y = y(0) ) = Γ −1 Γ 1− ( N+1)/2 (4π) 2 2 2 











.

(18)

The two-point function at y = y(0) obtained above is known to be useful in calculating the effective potential and studying the vacuum structure of the theory.4 Next we consider the spinor two-point function on R ⊗ S N−1 . The spinor two-point function D is defined by the Dirac equation 1 (∇ / + m)D(y) = − √ δ N (y) . g

(19)

We introduce the bispinor function G defined by (∇ / − m)G(y) = D(y) .

(20)

According to Eq.(19) G(y) satisfies the following equation, 1 (∇ /∇ / − m2 )G(y) = − √ δ N (y) . g

(21)

On R ⊗ S N−1 we rewrite Eq.(21) in the following form 

(∂0 )2 + 2N−1 −

1 R − m2 G(y) = − √ δ N (y) . 4 g 

5

(22)

where 2N−1 is the Laplacian on S N−1 . Performing the Fourier transformation dω −iωy0 ˜ e G(ω, yy) , 2π

(23)

R 1 ˜ − (m2 + ω 2 ) G(ω, yy) = − √ δ N−1 (yy) . 4 g

(24)

G(y) =

Z

we rewrite Eq.(22) in the form 

2N−1 −



Equation (24) is of the same form as the one for the spinor Green function on S N−1 . ˜ According to the method developed by Camporesi,5 we can find the expression of G(ω, yy). ˜ The general form of the Green function G(ω, yy) is written as ′ ˜ G(ω, yy) = U(yy)(gN−1 (ω, σ) + gN−1 (ω, σ)ni γ i ) ,

(25)

′ where U is a matrix in the spinor indices, gN−1 and gN−1 are scalar functions only of ω and

σ, ni is a unit vector tangent to the geodesic ni = ∇i σ, γ i is the Dirac gamma matrices and Roman index i runs over the space components (i = 1, 2, · · · , N − 1). Inserting Eqs.(25) into Eq.(24) we get "

′ ′ ni γ i ) + 2(∇j U)∇j (gN−1 + gN−1 ni γ i ) U2N−1 (gN−1 + gN−1

#

R ′ + (m2 + ω 2 ) U(gN−1 + gN−1 ni γ i ) = 0 , +(2N−1 U)(gN−1 + gN−1 ni γ ) − 4 ′

i





(26)

where we restrict ourselves to the region σ 6= 0. To evaluate Eq.(26) we have to calculate the covariant derivative of U and ni . U is the operator which makes parallel transport of the spinor at point yy(0) along the geodesic to point yy. Thus the operator U must satisfy the following parallel transport equations:6     

Here we set

   

ni ∇i U = 0 ,

(27)

U(yy(0) ) = 11 ,

∇i U ≡ V i U . 6

(28)

From the integrability condition

4

on Vi ,

∇i Vj − ∇j Vi − [Vi , Vj ] =

1 σij , a2

(29)

and the parallel transport equation (27) we easily find that 1 σ Vi = − tan σij nj , a 2a 



(30)

where σµν are the antisymmetric tensors constructed by the Dirac gamma matrices, 1 σµν = [γµ , γν ]. To find Vi we have used the fact that the maximally symmetric bitensors 4 on S N−1 is represented as a sum of products of ni and gij with coefficients which are functions only of σ. After some calculations we get the Laplacian acting on U 2N−1 U = −

σ N −2 U. tan2 2 4a 2a 



(31)

The first and second derivatives of ni are also the maximally symmetric bitensors and found to be

2

1 σ (gij − ni nj ) , cot a a   1 2 σ (N − 2)ni . = − 2 cot a a  

∇i nj = 2N−1 ni

(32)

Therefore Eq.(26) reads N −2 N −2 R σ σ 2 ∂σ − − cot tan − (m2 + ω 2) gN−1 a a 4a2 2a 4       N −2 R σ σ N −2 2 2 2 2 ′ i ∂σ − − − (m + ω ) gN−1 cot cot = 0. +ni γ ∂σ + 2 a a 4a 2a 4 (33)



 

∂σ 2 +







Since the two terms in Eq.(33) are independent, each term in the left-hand side of Eq.(33) has to vanish. We define the functions hN−1 (ω, σ) and h′N−1 (ω, σ) by gN−1 (ω, σ) = 







′ σ hN−1 (ω, σ) and gN−1 (ω, σ) = sin 2a h′N−1 (ω, σ) respectively and make a change   σ of variable by z = cos2 . We then find that Eq.(33) is rewritten in the form of two 2a hypergeometric differential equations :

cos

"

σ 2a

z(1 − z)∂z

2

N +1 (N − 1)2 + − Nz ∂z − − (m2 + ω 2 )a2 hN−1 (ω, z) = 0 . 2 4 

#



7

(34)

"

z(1 − z)∂z

2

(N − 1)2 N −1 − Nz ∂z − − (m2 + ω 2)a2 h′N−1 (ω, z) = 0 . + 2 4 #





(35)

Noting that the Green function is regular at the point σ = aπ we write the solutions of Eqs.(34) and (35) by only one kind of the hypergeometric function,3 hN−1 (ω, z) = cN−1 F ′

′

hN−1 (ω, z) = cN−1 F



N −1 N −1 N +1 + iβ, − iβ, ;z , 2 2 2

(36)



N −1 N −1 N −1 + iβ, − iβ, ;z , 2 2 2

(37)

√ β = a m2 + ω 2 .

(38)





where we define β by

As we remained in the region where σ 6= 0 the normalization constants cN−1 and c′N−1 are yet undetermined. To obtain cN−1 and c′N−1 we consider the singularity

3

˜ in the of G

limit σ → 0, N +1 N −3  3−N Γ Γ σ 2   2 ˜ −→ cN−1  G  N −1 N −1 2a + iβ Γ − iβ Γ 2 2    N −1 2  3−N Γ σ 2 i ′ +ni γ cN−1  ,    N −1 N −1 2a Γ + iβ Γ − iβ 2 2 







(39)

Comparing Eq.(39) with Eq.(12), the over-all factors cN−1 and c′N−1 are obtained :

cN−1 c′N−1

N −1 N −1 + iβ Γ − iβ a3−N Γ 2 2 = ,   N −1 (4π)(N−1)/2 Γ 2 = 0. 







(40) (41)

′ Therefore gN−1 (ω, σ) disappears in the final expression of the Green function. From

8

˜ Eqs.(36) and (40) we find the expression of G(ω, yy), ˜ G(ω, yy) = U(yy)gN−1 (ω, σ)     N −1 N −1 + iβ Γ − iβ Γ a3−N 2 2 = U(yy)   N +1 (4π)(N−1)/2 Γ 2      σ N −1 N +1 N −1 σ 2 × cos F . + iβ, − iβ, ; cos 2a 2 2 2 2a

(42)

Thus the Green function G(y) on R ⊗ S N−1 is obtained. The spinor two-point function D(y) is derived from the Green function G(y). Inserting Eqs.(23) and (25) into Eq.(20) we get dω −iωy0 e (−iωγ 0 + γ i ∇i − m)UgN−1 2π      Z N −2 σ dω −iωy0 i 0 γi n U ∂σ − gN−1 − (iωγ + m)UgN−1 . (43) e tan = 2π 2a 2a

D =

Z

Substituting Eq.(42) in Eq.(43) the spinor two-point function D(y) is obtained N −1 N −1 + iβ Γ − iβ a3−N dω −iωy0 Γ 2 2 D(y) = − e   N +1 (4π)(N−1)/2 2π Γ 2       N − 1 N − 1 N +1 σ σ 2 0 F + iβ, − iβ, ; cos × (iωγ + m)U(yy) cos 2a 2 2 2 2a N −1 σ +γin U(yy) F sin 2a 2a 

i





Z









N −1 N −1 N −1 σ + iβ, − iβ, ; cos2 2 2 2 2a 



(44)

.

According to the anticommutation relation of spinor fields the two-pint function (44) satisfies the antiperiodic boundary condition D(y 0 , σ) = −D(y 0, σ + 2πna) where n is an arbitrary integer. In the two dimensional limit N → 2 the two-point function (44) simplifies : #

"

a dω −iωy0 sinh(β(π − σ/a)) cosh(β(π − σ/a)) D(y) → − (iωγ 0 + m) e + aγ 1 2 2π β cosh(βπ) a cosh(βπ) Z ∞ dω −ipy 1 1 X e , (45) = 2πa n=−∞ 2π ip/ − m Z

9

where pµ and y µ are defined by         

p

µ

y

µ

2n − 1 = ω, 2a2   σ . = y 0, a 



,

(46)

To obtain Eq.(45) the following formulae were employed, ∞ sinh(a(π − 2x)) cos((2n − 1)x) 1 X = , 2 2 π n=−∞ (2n − 1) + (2a) 4a cosh(aπ) ∞ 1 X cosh(a(π − 2x)) (2n − 1) sin((2n − 1)x) = . π n=−∞ (2n − 1)2 + (2a)2 2 cosh(aπ)

(47)

We easily see that the Eq.(45) is identical to the well-known two-point function of spinor with antiperiodic boundary condition for the spatial coordinate. It should be noted that trD(y = y(0) ) is required in evaluating the effective potential: N −1 N −1   + iβ Γ − iβ 3 − N Z dω Γ tr11ma3−N 2 2 Γ . (48) trD(y = y(0) ) = − (4π)(N−1)/2 2 2π Γ (1 + iβ) Γ (1 − iβ) 







For the massless case we can perform the Fourier integral in Eq.(48) and get trD(y = y(0) , m = 0) N N −1 2tr11a2−N N Γ Γ 1− −→ − Γ ( N+1)/2 m (4π) 2 2 2 











.

(49)

In the present paper we calculated scalar and spinor two-point functions on R ⊗ S N−1 . In the final expression for the scalar and spinor two-point functions, Eqs.(14) and (44), the Fourier integral is remained. For some cases we can perform the Fourier integral in Eqs.(14) and (44) at y = y(0) . The resulting explicit expressions may be useful to investigate the vacuum structure of the Einstein universe.7 We would like to thank K. Fukazawa, S. Mukaigawa and H. Sato for helpful discussions.

10

References [1] J. S. Dowker, Ann. Phys. 62, 361 (1971); J. S. Dowker and R. Critchley, Phys. Rev. D15, 1484 (1977); J. S. Dowker and B. M. Altaie, Phys. Rev. D17, 417 (1978). [2] B. Allen and T. Jacobson, Commun. Math. Phys. 103, 669 (1986). [3] S. Moriguchi, K. Udagawa and S. Hitotsumatsu, Mathematical Formulae Vol.3 (Iwanami Book Co., 1960, in Japanese); I. S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series and Products, (Academic Press Inc., 1965). [4] P. Candelas and D. J. Raine, Phys. Rev. D12, 965 (1975). [5] R. Camporesi, Commun. Math. Phys. 148, 283 (1992). [6] B. Allen and C. A. L¨ utken, Commun. Math. Phys. 106, 201 (1986). [7] T. Inagaki, K. Ishikawa and T. Muta, Hiroshima U. preprint, HUPD-9517 (1995).

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