The document summarizes key aspects of quantum field theory on de Sitter spacetime, including solutions to the Dirac, scalar, electromagnetic, and other field equations. It presents: 1) Fundamental solutions for the Dirac equation and orthonormalization relations for Dirac spinor modes. 2) Solutions to the Klein-Gordon equation for a scalar field and corresponding orthonormalization relations. 3) Quantization of electromagnetic vector potentials in the Coulomb gauge and orthonormalization relations for photon modes.

1. PERTURBATIVE Q.E.D
ON DE SITTER UNIVERSE
WEST UNIVERSITY OF TIMISOARA
FACULTY OF PHYSICS
author: Crucean Cosmin
1

2. THE THEORY OF FREE FIELDS ON DE
SITTER UNIVERSE
Dirac field
• De Sitter metric:
1
ds2 = dt2 − e2ωtdx 2 = (dt2 − dx 2), ω > 0, ωtc = −e−ωt (1)
(ωtc)2 c
ˆ
gµν = ηαβ eαeβ
ˆ
ˆ
ˆ µ ν (2)
• In the Cartesian gauge the nonvanishing tetradic components are:
ˆ 1 ˆ ˆ 1
e0 = −ωtc, eˆ = −δˆωtc, e0 = − , ei = −δj .
ˆ
0
i
j
i
j 0 j
i
(3)
ωtc ωtc
2

3. • Action for Dirac field
√i ¯ α
S [e, ψ] = d x −g [ψγ ˆ Dαψ − (Dαψ)γ αψ]
4
ˆ ˆ
ˆ
2
¯
−mψψ , (4)
where
Dα = eµ Dµ = ∂α + Γα
ˆ α
ˆ ˆ ˆ (5)
• In this gauge the Dirac operator reads:
3iω 0
ED = iγ 0∂t + ie−ωtγ i∂i + γ
2
0 i 3iω 0
= −iωtc(γ ∂tc + γ ∂i) + γ (6)
2
• the Dirac equation
(ED − m)ψ = 0 (7)
• The fundamental solutions are:
1 πk/2 (1)
πp/ω e Hν− (qe−ωt)ξλ(p )
Up, λ(t, x ) = i 2
(1) eip·x−2ωt
(2π)3/2 λe−πk/2Hν+ (qe−ωt)ξλ(p )
(2)
πp/ω −λe−πk/2Hν− (qe−ωt)ηλ(p )
Vp, λ(t, x ) = i 1 πk/2 (2)
e−ip·x−2ωt , (8)
(2π)3/2 2 e Hν+ (qe−ωt)ηλ(p )
3

4. • Charge conjugation
¯
Up, λ(x) → Vp, λ(x) = iγ 2γ 0(Up, λ(x))T . (9)
• The orthonormalization relations :
¯
d3x(−g)1/2Up, λ(x)γ 0Up , λ (x) =
¯
d3x(−g)1/2Vp, λ(x)γ 0Vp , λ (x) = δλλ δ 3(p − p )
¯
d3x(−g)1/2Up,λ(x)γ 0Vp ,λ (x) = 0, (10)
d3p Up, λ(t, x )Up, λ(t, x ) + Vp, λ(t, x )Vp, λ(t, x ) = e−3ωtδ 3(x − x ).
+ +
(11)
λ
• The field operator
ψ(t, x ) = ψ (+)(t, x ) + ψ (−)(t, x )
= d3p Up, λ(t, x )a(p, λ) + Vp, λ(t, x )b+(p, λ) . (12)
λ
• Canonic cuantization
{a(p, λ), a+(p , λ )} = {b(p, λ), b+(p , λ )} = δλλ δ 3(p − p ) , (13)
4

5. {ψ(t, x ), ψ(t, x )} = e−3ωtγ 0δ 3(x − x ) , (14)
SF (t, t , x − x ) = i 0|T [ψ(x)ψ(x )]|0
= θ(t − t )S (+)(t, t , x − x )
−θ(t − t)S (−)(t, t , x − x ) . (15)
[ED (x) − m]SF (t, t , x − x ) = −e−3ωtδ 4(x − x ). (16)
• Conserved operators
Pi =: ψ, P iψ := d3p pi [a+(p, λ)a(p, λ) + b+(p, λ)b(p, λ)] (17)
λ
W =: ψ, W ψ := d3p pλ[a+(p, λ)a(p, λ) + b+(p, λ)b(p, λ)] . (18)
λ
iω 3 i +
↔
+
↔
H= d pp a (p, λ) ∂ pi a(p, λ) + b (p, λ) ∂ pi b(p, λ) . (19)
2
λ
5

6. •
[H, P i] = iωP i. (20)
Scalar field
• Action of the scalar field
∗ √ √
S[φ, φ ] = 4
d x −g L = d x −g ∂ µφ∗∂µφ − m2φ∗φ ,
4
(21)
• The Klein-Gordon equation:
1 √
√ ∂µ −g g µν ∂ν φ + m2φ = 0 . (22)
−g
∂t2 − e−2ωt∆ + 3ω∂t + m2 φ(x) = 0 . (23)
• Solutions of Klein-Gordon equation:
1 π e−3ωt/2 −πk/2 (1) p −ωt ip·x
fp (x) = 3/2
e Hik e e , (24)
2 ω (2π) ω
6

7. 9 m
k= µ2 − , 4 µ = , m > 3ω/2 (25)
ω
• The orthonormalization relations:
↔
3 1/2 ∗
i d x (−g) fp (x) ∂t fp (x) =
↔
3 1/2 ∗
−i d x (−g) fp (x) ∂t fp (x) = δ 3(p − p ) , (26)
↔
3 1/2
i d x (−g) fp (x) ∂t fp (x) = 0 , (27)
↔
∗
i 3
d p fp (t, x ) ∂t fp (t, x ) = e−3ωtδ 3(x − x ) . (28)
• Canonic cuantization
∗
φ(x) = φ(+)(x) + φ(−)(x) = d3p fp (x)a(p ) + fp (x)b+(p ) , (29)
[a(p ), a†(p )] = [b(p ), b†(p )] = δ 3(p − p ) . (30)
•
7

8. [φ(t, x ), ∂tφ†(t, x )] = ie−3ωtδ 3(x − x ) . (31)
• The commutator functions:
D(±)(x, x ) = i[φ(±)(x), φ(±)†(x )], (32)
∗
D(+)(x, x ) = i d3p fp (x)fp (x ),
∗
D(−)(x, x ) = −i d3p fp (x)fp (x ). (33)
• G(x, x ) is a Green function of the Klein-Gordon equation if:
∂t2 − e−2ωt∆x + 3ω∂t + m2 G(x, x ) = e−3ωtδ 4(x − x ) . (34)
DR(t, t , x − x ) = θ(t − t )D(t, t , x − x ) , (35)
DA(t, t , x − x ) = − θ(t − t)D(t, t , x − x ) , (36)
8

9. • The Feynman propagator,
DF (t, t , x − x ) = i 0|T [φ(x)φ†(x )] |0
= θ(t − t )D(+)(t, t , x − x ) − θ(t − t)D(−)(t, t , x − x ) ,
(37)
The electromagnetic field
• The action
√ 1 √
S[A] = d4x −g L = − d4x −g Fµν F µν , (38)
4
Fµν = ∂µAν − ∂ν Aµ
• Field equation √
∂ν ( −g g ναg µβ Fαβ ) = 0 , (39)
• In the Coulomb gauge A0 = 0, (Ai) ; i = 0 ,and chart {tc, x}
¸
(∂t2c − ∆)Ai = 0 (40)
9

10. • The field operator
(+) (−)
Ai(x) = Ai (x) + Ai (x)
= d3k ei(nk , λ)fk (x)a(k, λ) + [ei(nk , λ)fk (x)]∗a∗(k, λ)
λ
(41)
1 1
fk (x) = √ e−iktc+ik·x , (42)
(2π)3/2 2k
• the Hermitian form
↔ ↔
3 ∗
f, g = i d x f (tc, x ) ∂tc g(tc, x ) , f ∂ g = f (∂g) − g(∂f ) (43)
• The orthonormalization relations:
∗ ∗
fk , fk = − fk , fk = δ 3(k − k ) , (44)
∗
fk , fk = 0 , (45)
↔
3 ∗
i d k fk (tc, x) ∂tc fk (tc, x ) = δ 3(x − x ) , (46)
k · e (nk , λ) = 0 , (47)
10

11. e (nk , λ) · e (nk , λ )∗ = δλλ , (48)
∗ k ik j
ei(nk , λ) ej (nk , λ) = δij − 2 . (49)
k
λ
• In local frame
·
Aˆ (x) = e ˆj Aj =
i i·
d3k wˆ k,λ)(x)a(k, λ) + [wˆ k,λ)(x)]∗a∗(k, λ) .
i( i( (50)
λ
• Conformal transformation
gµν (x) = Ω(x)ηµν , (51)
g µν (x) = Ω−1(x)η µν , ds2 = Ω ds2,
c (52)
Aµ = Aµ ; Aµ = Ω−1A µ. (53)
• Quantization
[a(k, λ), a†(k , λ )] = δλλ δ 3(k − k ) . (54)
[Ai(tc, x ), π j (tc, x )] = [Ai(tc, x ), ∂tc Aj (tc, x )] = i δij (x − x ) ,
tr
(55)
11

12. • The momentum density
j √ δL
π = −g = ∂tc Aj (56)
δ(∂tc Aj )
tr 1 q iq j
δij (x ) = d3q δij − 2 eiq·x (57)
(2π)3 q
• Conserved operators
1
Pl = δij : Ai, (P l A)j := d3k k l a†(k, λ)a(k, λ) , (58)
2
λ
1
H= δij : Ai, (HA)j : (59)
2
1
W = δij : Ai, (W A)j : = d3k k λ a†(k, λ)a(k, λ) (60)
2
λ
12

13. [H, P i] = iωP i , [H, W] = iωW , [W, P i] = 0 , (61)
3 i
(Hfk ) (x) = −iω k ∂ki + f (x) (62)
2 k
iω 3 i †
↔
H= d kk a (k, λ) ∂ ki a(k, λ) , (63)
2
• The commutator functions
(±) (±) (±) †
Dij (x − x ) = i[Ai (x), Aj (x )] (64)
(+) 3 ∗ k ik j
Dij (x −x) = i d k fk (x)fk (x ) δij − 2 (65)
k
(+) i d3k k ik j )−ik(tc −tc )
Dij (x −x)= δij − 2 eik·(x−x (66)
(2π)3 2k k
13

14. • at equal times,
(+) 1 tr
∂tc Dij (tc − tc, x − x ) = δij (x − x ) . (67)
tc =tc 2
• The transversal Green functions
tr
∂t2c − ∆x Gij (x − x ) = δ(tc − tc)δij (x − x ) (68)
si ∂iGi· (x) = 0.
¸ ·j
R
Dij (x − x ) = θ(tc − tc)Dij (x − x ) , (69)
A
Dij (x − x ) = −θ(tc − tc)Dij (x − x ) , (70)
• The Feynman propagator ,
F
Dij (x − x ) = i 0|T [Ai(x)Aj (x )] |0
(+) (−)
= θ(tc − tc)Dij (x − x ) − θ(tc − tc)Dij (x − x ) , (71)
14

15. Interaction between Dirac field and
electromagnetic field
• The action for interacting fields:
√
i ¯ α
S [e, ψ, A] = 4
d x −g [ψγ ˆ Dαψ − (Dαψ)γ αψ]
ˆ ˆ
ˆ
2
¯ 1 ¯ ˆ ˆ
−mψψ − Fµν F µν − eψγ αAαψ , (72)
4
• Equations for fields in interaction
iγ αDαψ − mψ = eγ αAαψ,
ˆ
ˆ
ˆ
ˆ
1 ¯ ˆ
∂µ (−g)F µν = eeν ψγ αψ.
α
ˆ (73)
(−g)
15

16. Interaction between scalar field and
electromagnetic field
• The action
√ µ † 1
2 †
S[φ, A] = d x −g (∂ φ ∂µφ − m φ φ) − Fµν F µν
4
4
−ie[(∂µφ†)φ − (∂µφ)φ†]Aµ + e2φ†φAµAµ . (74)
• Equations for fields in interaction
1 √ √
√ ∂µ −g g ∂ν φ − ie −gφAµ + m2φ = ie(∂µφ)Aµ + e2φAµAµ,
µν
−g
1
∂µ (−g)F µν = −ie[(∂ ν φ†)φ − (∂ ν φ)φ†]
(−g)
+2e2φ†φAν . (75)
16

17. The in out fields
• The equation of the interaction fields in the Coulomb gauge
¯
(∂t2c − ∆x)A(x) = eψ(x)γψ(x) (76)
• Solution:
ˆ
A(x) = A(x) − e ¯
d3ydtc DG(x − y)ψ(y)γψ(y), (77)
ˆ
A(x) = A(R/A)(x) − e ¯
d3ydtc DR/A(x − y)ψ(y)γψ(y), (78)
ˆ
• the free fields , A(R/A)(x) satisfy:
ˆ
lim (A(x) − A(R/A)(x)) = 0. (79)
t→ ∞
• The fields in/out:
√ ¯
z3Ain/out(x) = A(x) + e d3ydtc DR/A(x − y)ψ(y)γψ(y). (80)
17

18. • The in/out fields, final expressions:
√
z3Ain/out(x) = A(x) + d3ydtc DR/A(x − y)(∂t2c − ∆y )A(y), (81)
↔
3
a(k, λ)in/out = i d xfk (x)e (nk , λ) ∂tc Ain/out(x) (82)
• The case of interaction with scalar field
ˆ + e d4y √−g G(x, y) √ i ∂µ[√−gφ(y)Aµ(y)]
φ(x) = φ(x)
−g
+i[∂µφ(y)]Aµ(y) + eφ(y)Aµ(y)Aµ(y) , (83)
ˆ √ i √
φ(x) = φR/A(x) + e d y −gDR/A(x, y) √ ∂µ[ −gφ(y)Aµ(y)]
4
−g
+i[∂µφ(y)]Aµ(y) + eφ(y)Aµ(y)Aµ(y) . (84)
18

19. • The in/out fields:
√ √
zφin/out(x) = φ(x) − d y −gDR/A(x, y)[EKG(y) + m2]φ(y).
4
(85)
•
↔
3 ∗
a(p )in/out = i d x(−g)1/2fp (x) ∂t φin/out(x)
↔
† 3 1/2
b (p )in/out = i d x(−g) fp (x) ∂t φin/out(x). (86)
19

20. Perturbation theory
• The expression of Green functions
1 ˆ ˆ ˆ
G(y1, y2, ..., yn) = 0|T [φ(y1)φ(y2)...φ(yn), S]|0 , (87)
0|S|0
∞
(x)d4 x (−i)n
S = T e−i LI
=1+ T [LI (x1)...LI (xn)]d4x1...d4xn, (88)
n=1
n!
• The interaction Lagrangian
√
LI (x) = −ie −g : [(∂µφ†)φ − (∂µφ)φ†]Aµ : (89)
(−i)n ˆ ˆ ˆ
0|T [φ(y1)φ(y2)...φ(yn), LI (x1)...LI (xn)]|0 d4x1...d4xn. (90)
n! 0|S|0
20

21. • the amplitude:
3 1 ∗
out, 1(p )|in, 1(p ) = δ (p − p ) − −g(y) −g(z)fp (y)[EKG(y) + m2]
z
× 0|T [φ(y)φ†(z)]|0 [EKG(z) + m2]fp (z)d4yd4z. (91)
• The scattering amplitude in the first order of perturbation theory
↔
∗
Ai→f = −e −g(x) fp (x) ∂µ fp (x) Aµ(x)d4x, (92)
Ai→f = −ie ¯
−g(x)Up ,λ (x)γµ A
µ
ˆ
(x)Up,λ (x)d4x, (93)
21

22. Coulomb scattering for the charged scalar field
• External field
ˆ
0 Ze −ωt
A = e (94)
|x|
• initial and final states are:
φf (x) = fpf (x), φi(x) = fpi (x). (95)
• Scattering amplitude
−αZ pi ∞ (2) (1) (2) (1)
Ai→f = − z Hik (pf z)Hik−1(piz) − Hik (pf z)Hik+1(piz) dz
8π|pf − pi|2 2 0
pf ∞ (2) (1) (2) (1)
+ z Hik−1(pf z)Hik (piz) − Hik+1(pf z)Hik (piz) dz , (96)
2 0
e−ωt
z= . (97)
ω
22

23. • Scattering amplitude in final form
−αZ
Ai→f = B ,
2 k
(98)
8π|pf − pi|
(pf + pi) 1 pf pf
Bk = √ 2iδ(pf − pi) + θ(pi − pf ) h∗k − gk
pf pi pi pi pi
1 ∗ pi pi
+ θ(pf − pi) gk − hk . (99)
pf pf pf
3 1 2 1 3 2
e−πk χik 2 F1 2 , 2 + ik; 1 + ik; χ 2 F1 2 , 2 + ik; 1 + ik; χ
gk (χ) = −
sinh2(πk) 1
B( 2 − ik; 1 + ik) B(− 1 − ik; 1 + ik)
2
3 1 2 1 3 2
eπk χ−ik 2 F1 2 , 2 − ik; 1 − ik; χ 2 F1 2 , 2 − ik; 1 − ik; χ
+ −
sinh2(πk) 1
B( 2 + ik; 1 − ik) B(− 1 + ik; 1 − ik)
2
3 1 2 1 3 2
χik 2 F1 2 , 2 + ik; 1 + ik; χ 2 F1 2 , 2 + ik; 1 + ik; χ
+ −
sinh2(πk) 1
B(− 2 ; 1 + ik) 1
B( 2 ; 1 + ik)
3 1 2 1 3 2
χ−ik 2 F1 2 , 2 − ik; 1 − ik; χ 2 F1 2 , 2 − ik; 1 − ik; χ
+ − ,
sinh2(πk) 1
B(− 2 ; 1 − ik) 1
B( 2 ; 1 − ik)
(100)
23

24. 3 3 1 1
e−πk χ−ik 2 F1 , 2 − ik; 2 − ik; χ2 2 F1 2 , 2 − ik; −ik; χ
2
hk (χ) = 2
· χ2 −
sinh2(πk) B(− 1 + ik; 2 − ik)
2 B(−ik; 1 + ik)
2
3 3 2 1 1 2
eπk χik 2 F1 2 , 2 + ik; 2 + ik; χ 2 F1 2 , 2 + ik; ik; χ
+ · χ2 −
sinh2(πk) 1
B(− 2 − ik; 2 + ik) B( 1 − ik; ik)
2
1 1 2 3 3 2
χik 2 F1 2 , 2 + ik; ik; χ 2 F1 2 , 2 + ik; 2 + ik; χ
+ − · χ2
sinh2(πk) B( 1 ; ik)
2 B(− 1 ; 2 + ik)
2
1 1 2 3 3 2
χ−ik 2 F1 2 , 2 − ik; −ik; χ 2 F1 2 , 2 − ik; 2 − ik; χ
+ − · χ2 ,
sinh2(πk) 1
B( 2 ; −ik) 1
B(− 2 ; 2 − ik)
(101)
pf pi
• the notations are χ = pi or pf .
• The Minkowski limit
(pf + pi)
B∞ = √ 2iδ(pf − pi). (102)
pf pi
• Minkowski case
(Ef + Ei)
BM = 4iδ(Ef − Ei), (103)
Ef Ei
24

25. • Ration of the two amplitudes
δ(pf − pi) 1
= . (104)
2δ(Ef − Ei) 2
• The incident flux
↔
∗
Jµ = −i fp (x) ∂µ fp (x) , (105)
∗
↔ πpie−3ωt (2) pi −ωt (1) pi −ωt
j(t) = −i fpi (x) ∂i fpi (x) = 3
Hik e Hik e . (106)
2ω(2π) ω ω
25

26. Coulomb scattering with fermions
• The scattering amplitude
√ ¯ ˆ
Ai→f = −ie −g Upf ,λf (x)γ0A0(x)Upi,λi (x)d4x. (107)
√ ∞
pi pf
Ai→f = −iαZ ξλf (pf )ξλi (pi) eπk
+ (2) (1)
dzzHν+ (pf z)Hν− (piz)
8π|pf − pi|2 0
∞
+sgn(λf λi)e−πk (2) (1)
dzzHν− (pf z)Hν+ (piz) (108)
0
26

27. • Final form of the amplitude
p2

1 1
−i αZ 1  pf eπk 2F1
−ik
2
,1 − ik; − ik;
2
f
p2
+ i
Ai→f = ξ (pf )ξλi (pi) δ(pf − pi) + θ(pi − pf ) i
4π |pf − pi|2 λf pi pi cosh(πk) B( 1 , 1 − ik)
2 2
p2

3 3 f
e−πk 2F1 2 , 1 + ik; 2 + ik; p2 
1+ik 1−ik
pf i 1 pi
−i + θ(pf − pi) i
pi cosh(πk) B(− 1 , 3 + ik)
2 2
pf pf
p2 p2

3 3 1 1
e−πk 2F1 2 , 1 − ik; 2 − ik; p2 ieπk 2F1 2 , 1 + ik; 2 + ik; p2 
i ik i
f pi f
× 1 3 − 1 1
cosh(πk) B(− 2 , 2 − ik) pf cosh(πk) B( 2 , 2 + ik)
ik
1 pf e−πk
+sgn(λf λi)δ(pf − pi) + sgn(λf λi)θ(pi − pf ) i
pi pi cosh(πk)
p2 p2

1 1 3 3
2 F1 2
,1 + ik; + ik;
2
f
p2 pf
1−ik
eπk 2F1 − ik; − ik;2
,1 2
f
p2
i i
× −i 
B( 2 , 1
1
2
+ ik) pi cosh(πk) B(− 1 , 3 − ik)
2 2
2
 pi
3 3
1  pi
1+ik
e πk 2 F1 2
,1 + ik; 2 + ik; p2
f
+sgn(λf λi)θ(pf − pi) i
pf pf cosh(πk) B(− 1 , 3 + ik)
2 2
p2

1 1
pi
−ik
e−πk 2F1 2
,1 − ik; − ik;
2
i
p2
f
−i  . (109)
pf cosh(πk) B( 1 , 1
2 2
− ik)
27

28. • The incident flux
eωt ¯ πpie−3ωt (2) pi −ωt (1) pi −ωt
j= Upi, λi (x)(piγ)Upi, λi (x) = 3 4ω
Hν− e Hν− e
| pi | (2π) ω ω
(2) pi −ωt (1) pi −ωt
+Hν+ e Hν+ e (110)
ω ω
√ ¯ ˆ
Ai→f = −ie −g Upf ,λf (x)γ0A0(x)Vpi,λi (x)d4x (111)
28

29. References
˘
[1] I.I.Cotaescu, Phys. Rev. D 65, 084008 (2002)
˘
[2] I.I.Cotaescu, R.Racoceanu and C.Crucean, Mod.Phys.Lett A 21, 1313 (2006)
[3] C.Crucean, Mod.Phys.Lett A 22, 2573 (2007)
[4] C.Crucean and R.Racoceanu, Int.J.Mod.Phys. A 23, 1075 (2008)
˘
[5] I.I.Cotaescu and C.Crucean, Int.J.Mod.Phys. A 23, 1351 (2008)
˘
[6] I.I.Cotaescu, C.Crucean, A.Pop, Int.J.Mod.Phys. A 23 (2008)
˘
[7] I. I. Cotaescu and C. Crucean, Int.J.Mod.Phys. A 23 (2008)
[8] C.Crucean, Mod.Phys.Lett A 25, (2010)
˘
[9] I.I.Cotaescu and C.Crucean, qr-qc/0806.2515
[10] C.Crucean, R.Racoceanu and A.Pop, Phys.Lett.B 665, 409 (2008)
[11] C.Crucean, math-ph/0912.3659, (2009)
29