Anisotropic universe space-time non-commutativity and scalar particle creation in the presence of a constant electric field

0.12 MB·English

by  Slimane Zaim

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0
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Anisotropic universe space-time non-commutativity
n
and scalar particle creation in the presence of a
a
J constant electric field

3

]
h Slimane Zaim
t
-
p
D´epartement des Sciences de la Mati`ere, Facult´e des Sciences,
e
h Universit´e Hadj Lakhdar - Batna, Algeria.
[

1 Abstract
v
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We study the effect of the non-commutativity on the creation of
0
6 scalarparticles fromvacuumin the anisotropicuniversespace-time.
0 We derive the deformed Klein-Gordon equation up to second order
0 inthenon-commutativityparameterusingthegeneralmodifiedfield
.
1 equation. Then the canonical method based on Bogoliubov trans-
0 formation is applied to calculate the probability of particle creation
5 in vacuum and the corresponding number density in the k mode.
1
We deduce that the non-commutative space-time introduces a new
:
v source of particle creation.
i
X
r
a
Keywords: Non-commutative field theory, Bogoliubov transformation, Parti-
cle production.
Pacs numbers: 11.10.Nx, 03.65.Pm,03.70.+k , 25.75.Dw
1 Introduction
Inthe classicaltheoryblackholescanonlyabsorbandnotemitparticles. How-
ever it was shown that quantum mechanical effects cause black holes to create
and emit particles. It is also well known that the most significant prediction
of this theory is the phenomenon of particle creation which leads to the con-
cept of quantum gravity. In this paper we are interested in the issue of particle
production by constructing a simple type of the non-commutative geometry.
Theextensionofquantumfieldtheorytooneinacurvedspaceisthestarting
point towards quantum gravity in curved space-time. However another impor-
tant concept in the context of quantum gravity is non-commutative geometry,
by which the quantization of the space-time leads to quantifying gravity. Thus
the non-commutative space-time is intrinsically connected to quantum gravity
despite the well-known problem of Lorentz-violating symmetry. All other fun-
damentalproblems, suchas the unitarity violation[4], causality[5] and UV/IR
divergences [6], have been discussed in the context of the local Lorentz invari-
ance.
In ref. [7], the authors showed that these problem can arise by inducing a
non-constantmetric into the theory andthey foundthat athighenergygravity
andnon-commutativegeometrymustbecomedependantoneachother. Several
important works were performed in a formally Lorentz-invariant approach (see
for example the reviews [2,8,9]). Various theories of gravity in the context of
non-commutative geometry have amongst others been studied in refs. [10 22]
−
and cosmology on non-commutative space-time has been explored in ref. [23].
Evencertainideasreferringtoquantumgravityhavebeenexploredwithrespect
tonon-commutativegeometry,seerefs. [24 28]forexample. Anotherapproach
−
is based on the twisted Poincar´e algebra constructed for canonically deformed
spacewithaconstantparameterofnon-commutativity,whereinthisformalism
the Lagrangiandensity is invariantandthe gaugeand pure gravitytheories are
consistent.
Inourpreviouswork[29]wehaveattemptedtoconstructanon-commutative
gauge gravity model, where the problem of the unitarity (see for example refs.
[4,,14,15,30,31,32])isovercomebytheconstructionofgeneralizedlocalLorentz
and general coordinate transformations, which preserve the non-commutative
coordinate canonical commutation relations. The phenomenon of scalar parti-
cle creation in anisotropic Bianchi I universe with a constant electric field has
beenanalyzedinref. [33]. Actually,thereisnoelectricchargeintheuniverseto
create an electric field, meaning that the particles can be created from vacuum
by the expansion of the universe itself with no other external field present.
Theaimofthispaperisthestudytheeffectofthenon-commutativityonthe
creation of scalar particles from vacuum in the space-time anisotropic Bianchi
I universe when a constant electric field is present. We compute the number
density of created particles in the cases of strong and weak field. From our
results we clearly deduce that the the non-commutativity plays the role of the
electric field.
This paper is organized as follows. In section 2 we derive the correspond-
1
ing Seiberg-Witten maps up to the first order of θ for the various dynamical
fields and we propose an invariant action of the pure non-commutative gauge
gravity and non-commutative charged scalar field in interaction. In section 3
wederivethe anisotropicuniversenon-commutativityspace-time Klein-Gordon
(KG)equationandobtainits solution. Thenwecomputethe densityofcreated
scalarparticles and discuss the weak and strong field limits. The last section is
devoted to a discussion.
2 Seiberg-Witten maps
One can get at first order in the non-commutative parameter θµν the following
Seiberg–Witten maps [1]:
1
ϕˆ =ϕ θµνA ∂ ϕ+ θ2 ,
ν µ
− 2 O
λˆ =λ + 1θσρ ∂ λ ,ω (cid:0)+(cid:1) θ2 ,
P P σ P ρ
4 { } O
λˆ =λ + 1θσρ ∂ λ ,A + (cid:0) θ2(cid:1) , (1)
G G σ G ρ
4 { } O
Aˆ =A 1θµν A ,∂ A +F (cid:0)+(cid:1) θ2 ,
ξ ξ ν µ ξ µξ
− 4 { } O
F1 = 1θαβ F F 1θαβ A ,(∂ +(cid:0)D(cid:1))F + θ2 ,
µξ 2 { µα ξβ}− 4 { α β β µξ} O
eˆa =ea iθαβ ωac∂ ec + ∂ ωac+Rac ec + θ2(cid:0), (cid:1)
µ µ− 4 α β µ β µ βµ µ O
(cid:0) (cid:0) (cid:1) (cid:1) (cid:0) (cid:1)
where
F =∂ A ∂ A i[A ,A ], (2)
µν µ ν ν µ µ ν
− −
ω =ωabS , (3)
µ µ ab
Aˆ =AˆaTa =Aˆa eˆkTa, (4)
µ µ k∗ µ
ωˆ =ωˆabS =ωˆab eˆkS , (5)
µ µ ab k ∗ µ ab
θµν =eˆµ eˆµ θab, (6)
∗a∗ ∗b
and ωab are the spin connections and eˆµ is the inverse- of the vierbein eˆa
µ ∗a ∗ µ
defined as:
eˆb eˆµ =δb, (7)
µ∗ ∗a a
and
eˆa eˆν =δν. (8)
µ∗ ∗a µ
Tobeginweconsideranon-commutativegaugetheorywithachargedscalar
particleinthepresenceofanelectrodynamicgaugefieldinageneralcurvilinear
system of coordinates. We can write the action as:
1
= d4x ( + ), (9)
S 2κ2 LG LSC
Z
2
where and standforthe puregravityandmatterscalardensitiescorre-
G SC
L L
sponding to the charged scalar particle in the presence of an electric field, and
where
=eˆ Rˆ, (10)
G
L ∗
and
†
=eˆ gˆµν Dˆ ϕˆ Dˆ ϕˆ+m2ϕˆ† ϕˆ . (11)
SC µ ν
L ∗ ∗ ∗ ∗
(cid:18) (cid:16) (cid:17) (cid:19)
The deformed tetrad and scalar curvature are given by:
1
eˆ=det (eˆa) ǫµνρσε eˆa eˆb eˆc eˆd, (12)
∗ µ ≡ 4! abcd µ∗ ν ∗ ρ∗ σ
Rˆ =eˆµ eˆν Rˆab , (13)
∗a∗ ∗b∗ µν
and the gauge covariant derivative is defined as: Dˆ ϕˆ = ∂ ieAˆ ϕˆ.
µ µ µ
− ∗
In the following we consider a symmetric metric gˆµν s(cid:16)uch that: (cid:17)
1
gˆ = (eˆb eˆ +eˆb eˆ ). (14)
µν 2 µ∗ νb ν ∗ µb
As a consequence, the first-order expansion in the non-commutative parameter
θαβ of the scalar curvature Rˆ and metric gˆ vanishes. Thus Rˆ and gˆ can be
µν µν
rewritten as:
Rˆ =R+ θ2 , (15)
O
gˆµν =gµν + (cid:0) θ(cid:1)2 , (16)
O
Next we use the generic field infinitesima(cid:0)l t(cid:1)ransformations (ˆδ ϕˆ = iλˆ ϕˆ),
λˆ ∗
andthestar-producttensorrelations. Wecanprovethattheactionineq. (34)is
actuallyinvariant. Byvaryingthescalardensityunderthegaugetransformation
andfromthegeneralisedfieldequationandtheNoethertheoremweobtain[10]:
∂ ∂ ∂
L ∂ L +∂ ∂ L + θ2 =0. (17)
µ µ ν
∂ϕˆ − ∂(∂ ϕˆ) ∂(∂ ∂ ϕˆ) O
µ µ ν
(cid:0) (cid:1)
3 The solution to the non-commutative Klein-
Gordon equation and particle creation process
Inthissectionweexaminetheparticlecreationphenomenoninducedbyvacuum
instabilities in the context of the non-commutative geometry in presence the
external vector potential A . We shall take the example of the Klein-Gordon
µ
equation in a cosmologicalanisotropic non-commutative Bianchi I universe.
The deformed line element of the Bianchi I universe up to the first-order of
θ takes the following form:
ds2 = dt2+t2 dx2+dy2 +dz2+g(1)dxµdxν + θ2 . (18)
− µν O
(cid:0) (cid:1) (cid:0) (cid:1)
3
We choose for θαβ the following form:
0 0 0 θ
0 0 θ 0
θαβ = , α, β =0,1,2,3. (19)
0 θ 0 0
−
 θ 0 0 0 
 − 
 
Wefollowthesamestepsoutlinedinref. [33]andlookforthenon-commutative
correction of the metric up to the first order in θ.
Choosing the following diagonal tetrads:
e0 = 1, 0, 0, 0 , (20)
µ
e1µ =(cid:0) 0, t, 0, 0 (cid:1), (21)
e2µ =(cid:0) 0, 0, t, 0 (cid:1), (22)
e3µ =(cid:0) 0, 0, 0, 1 (cid:1) , (23)
then the nonzero spin connection(cid:0)s are (cid:1)
ω01 = ω10 =1, (24)
1 − 1
ω02 = ω20 =1. (25)
2 − 2
Usingthe Seiberg-Wittenmap(1)andthechoice(19)wecanobtainthefollow-
ing deformed veirbeins:
eˆ0 = 1, 0, 0, 0 , (26)
µ
eˆ1µ =(cid:0) 0, t, −i4θt, (cid:1)0 , (27)
eˆ2µ =(cid:0) 0, i4θt, t, 0 ,(cid:1) (28)
eˆ3µ =(cid:0) 0, 0, 0, 1 .(cid:1) (29)
As a consequence, the first-orde(cid:0)r expansion in th(cid:1)e non-commutative parameter
θαβ of the Bianchi I metric vanishes. Thus (18) can be rewritten as:
ds2 = dt2+t2 dx2+dy2 +dz2+ θ2 . (30)
− O
Inorderto identify the particl(cid:0)estateswe(cid:1)followthe qu(cid:0)as(cid:1)i-classicalapproach
of ref. [34]. The standard method is to specify the positive and negative fre-
quency modes and solve the classical Hamilton-Jacobi equation looking specif-
ically for the asymptotic limits of the solution t 0 and t . Then one
→ → ∞
solves the Klein Gordon equation by comparing with the quasi-classical limits,
and specifying the positive and negative frequency states. Finally one utilises
Bogouliubovtransformationsandcalculatesthenumberdensityforcreatedpar-
ticles.
Usingthemodifiedfieldequation(17)withthegenericbosonfieldϕˆ onecan
findinanon-commutativecurvedspace-timeandinthepresenceoftheexternal
potential Aˆ the following modified Klein-Gordon equation:
µ
ηµν∂ ∂ m2 ϕˆ + ieηµν∂ Aˆ e2ηµνAˆ Aˆ +2ieηµνAˆ ∂ ϕˆ =0, (31)
µ ν − e µ ν − µ∗ ν µ ν
(cid:0) (cid:1) (cid:16) (cid:17)
4
withthe deformedexternalpotentialAˆ =(0,0,0,Et)infreenon-commutative
µ
space-time being:
aˆ =a Θµka ∂ a + Θ2 . (32)
3 3 k µ 3
− O
For a non-commutative time-space we have Θ03 =(cid:0)0 a(cid:1)nd Θki =0, where i,k =
6
1,2,3. In this case we can write:
1
ηµν∂ ∂ = ∂2+ ∂2+∂2 +∂2, (33)
µ ν − 0 t2 1 2 3
(cid:0) (cid:1)
and
2ieηµνAˆ ∂ =2ieEt(1+θE)∂ , (34)
µ ν 3
and
e2ηµνAˆ Aˆ =[ieEt(1+θE)]2 . (35)
µ ν
− ∗
TheKlein-Gordonequation(31)(inthepresenceofaconstantexternalfield
A ) up to θ2 then simplifies to:
µ
O
1 (cid:0) (cid:1)
∂2+ ∂2+∂2 +∂2 m2+2ieEt(1+θE)∂ +[ieEt(1+θE)]2 ϕˆ =0.
− 0 t2 1 2 3 − 3
(cid:20) (cid:21)
(cid:0) (cid:1) (36)
In order to keep our results compact and transparent we make use of the ap-
proximation:
1+θg exp(θg), (37)
≈
with g being an arbitrary regular function. Equation (36) commutes with the
operator i−→, and therefore the wave functions ϕˆ can be cast into:
− ∇
ϕˆ =∆˜(t)exp(ik x+ik y+ik z). (38)
x y z
Substituting eq. (38) into eq.(36), one can get:
d2 k2
+ ⊥ +k2+m2+2eE˜tk +e2E˜2t2 ∆˜(t)=0, (39)
dt2 t2 z z
(cid:20) (cid:21)
where
E˜ =Eexp(θE), (40)
and the eigenvalue k is given by:
⊥
k = k2+k2. (41)
⊥ x y
q
We adopt the following change of variable:
ρ=ieE˜t2, (42)
and we deduce that for k =0, equation (39) becomes:
z
d2 1 d k2 1 m2
+ + ⊥ i ∆˜(ρ)=0. (43)
dρ2 2ρdρ 4ρ − 4 − 4eE˜
(cid:20) (cid:21)
5
Following ref. [34], the solution to eq. (43) can be written as a combination
of Whittaker functions M (ρ) and W (ρ):
k˜θ,µ k˜θ,µ
∆˜ (ρ)=ρ−1/4 C M (z)+C W (z) , (44)
1 k˜θ,µ 2 k˜θ,µ
(cid:16) (cid:17)
where k˜ and µ are given by:
θ
m2 i 1
k˜ = i exp( θE), µ= k2 . (45)
θ − 4eE − 2 ⊥− 4
r
Thenthegeneralsolutionof(43)canbeexpressedintermsofthehypergeomet-
ricfunctionsF 1 k˜ +µ,2µ+1,ρ andG 1 k˜ +µ,2µ+1,ρ asfollows:
2 − θ 2 − θ
(cid:16) (cid:17) (cid:16) (cid:17)
1
∆˜ (ρ)=C ρµ+1/4e−ρ/2F k˜ +µ,2µ+1,ρ +
1 θ
2 −
(cid:18) (cid:19)
1
+C ρµ+1/4e−ρ/2G k˜ +µ,2µ+1,ρ , (46)
2 θ
2 −
(cid:18) (cid:19)
where C and C are normalisation constants.
1 2
To construct the positive and negative frequency modes we use the asymp-
totic limit of the solution (46) and compare the result with that obtained by
solving the Hamilton-Jacobi relativistic equation at t=0 (ρ=0). Thus it may
be shown that the positive and negative frequency modes are given by:
1
∆˜+ =C+ρµ+1/4e−ρ/2F k˜ +µ,2µ+1,ρ , (47)
0 0 2 − θ
(cid:18) (cid:19)
and
∆˜− = ∆˜+ ∗ =C+( 1)µ+14 ρµ+1/4e−ρ/2F 1 k˜ +µ,2µ+1,ρ , (48)
0 0 0 − 2 − θ
(cid:16) (cid:17) (cid:18) (cid:19)
where C+ is normalisation constant. We note that the hypergeometric func-
0
tionsF 1 k˜ +µ,2µ+1,ρ andG 1 k˜ +µ,2µ+1,ρ havethefollowing
2 − θ 2 − θ
asympto(cid:16)tic limits: (cid:17) (cid:16) (cid:17)
1
F k˜ +µ,2µ+1,ρ 1 for ρ 1, (49)
θ
2 − ∼ | |≪
(cid:18) (cid:19)
G 1 k˜ +µ,2µ+1,ρ ρk˜θ−µ−1/2 for ρ . (50)
θ
2 − ∼ | |→∞
(cid:18) (cid:19)
One can show that the positive and negative frequency modes for ρ ,
| |→∞
by observing the asymptotic limit of G 1 k˜ +µ,2µ+1,ρ , is given by:
2 − θ
(cid:16) (cid:17)
1
∆˜+ =C+ρµ+1/4e−ρ/2G k˜ +µ,2µ+1,ρ , (51)
∞ ∞ 2 − θ
(cid:18) (cid:19)
6
and
1
∆˜− =C− ( ρ)µ+1/4eρ/2G +k˜ +µ,2µ+1, ρ , (52)
∞ ∞ − 2 θ −
(cid:18) (cid:19)
where C+ and C− are normalisationconstants.
∞ ∞
Now we utilise the relation:
1 Γ( 2µ) 1
G k˜ +µ,2µ+1,ρ = − F k˜ +µ,2µ+1,ρ +
θ θ
(cid:18)2 − (cid:19) Γ 12 −k˜θ−µ (cid:18)2 − (cid:19)
Γ(2µ)(cid:16) (cid:17) 1
+ ρ−2µF k˜ µ, 2µ+1,ρ , (53)
θ
Γ 12 −k˜θ+µ (cid:18)2 − − − (cid:19)
(cid:16) (cid:17)
whereΓistheGammafunction,andexploitthefactthatthepositivefrequency
mode ∆˜+ can be written in terms of the positive (∆˜+) and negative (∆˜−)
∞ 0 0
frequency modes through the Bogouliubov transformation [35,36,37]:
∆˜+ =αˆ∆˜++βˆ∆˜−, (54)
∞ 0 0
to find that αˆ and βˆ are:
C+Γ( 2µ) C+Γ(2µ)
αˆ = ∞ − , βˆ = ∞ exp(iπ(µ+1/4)), (55)
C+Γ 1 k˜ µ C+Γ 1 k˜ +µ
0 2 − θ− 0 2 − θ
(cid:16) (cid:17) (cid:16) (cid:17)
with
2
αˆ 2 Γ 12 −k˜θ+µ
| | = exp(2πµ). (56)
βˆ 2 (cid:12)(cid:12)(cid:12)Γ(cid:16)12 −k˜θ−µ(cid:17)(cid:12)(cid:12)(cid:12)
Using the followin(cid:12)(cid:12)g p(cid:12)(cid:12)rope(cid:12)(cid:12)(cid:12)rty(cid:16)of the Gam(cid:17)m(cid:12)(cid:12)(cid:12)a function:
(cid:12) (cid:12)
2
1 π
Γ +iρ = , (57)
2 cosh(πρ)
(cid:12) (cid:18) (cid:19)(cid:12)
(cid:12) (cid:12)
and simplifying leads to: (cid:12) (cid:12)
(cid:12) (cid:12)
αˆ 2 cosh π k˜θ+µ
−
| | = exp(2πµ). (58)
βˆ 2 coshh π(cid:16) k˜θ+µ (cid:17)i
(cid:12) (cid:12) h (cid:16) (cid:17)i
The probability to cr(cid:12)ea(cid:12)te a single particle from vacuum is then:
(cid:12) (cid:12)
−1
αˆ 2 cosh π k˜θ+µ
P = | | = exp( 2πµ). (59)
k  βˆ 2 cosh hπ (cid:16) k˜θ+µ(cid:17)i −
−
 
(cid:12) (cid:12)  h (cid:16) (cid:17)i
(cid:12) (cid:12)
(cid:12) (cid:12)
7
Taking into account the fact that k˜ = m2 θm2, for small θ, we easily to
θ 4eE − 4e
obtain:
P =P (θ =0)+Pθ, (60)
k k k
where P (θ =0) denotes the ordinary probability to create a single particle
k
from vacuum in the presence of an electric field and has the expression:
cosh[π(k+µ)] m2
P (θ =0)= exp( 2πµ), k = , (61)
k
cosh[π( k+µ)] − 4eE
−
and Pθ is the generated non-commutative correction of order θ given by:
k
πm2
Pθ = θP (θ =0)[tanhπ(k+µ)+tanhπ( k+µ)]. (62)
k − 4e k −
Nextwecalculatethenon-commutativedensityofthecreatedparticlesnˆ by
the non-commutative curved space-time and electric field. For this we use eq.
(54) so as to arrive at:
2
nˆ = βˆ . (63)
(cid:12) (cid:12)
Using the normalisation condition [37]:(cid:12) (cid:12)
(cid:12) (cid:12)
2
αˆ 2 βˆ =1, (64)
| | −
(cid:12) (cid:12)
wefinallyarriveattheresultforthenon(cid:12)-c(cid:12)ommutativenumberdensityofcreated
(cid:12) (cid:12)
particles nˆ:
−1 cosh π k˜ +µ
1 P θ
nˆ = − k =exp π k˜ µ . (65)
P θ− sihnh(cid:16)(2πµ) (cid:17)i
(cid:18) k (cid:19) h (cid:16) (cid:17)i
Itis alsoveryimportantto considerthe weakandstrongelectricfield limits
and see the behavior of the number density and derive some of the related
thermodynamical quantities.
3.1 The weak field approximation
In this limit, if we set:
m2
k˜ = (1 θE)
θ
4eE −
m2 m2
= θ , (66)
4eE − 4e
such that:
k˜ , (67)
θ
→∞
it is easy to show that the probability P takes the form:
k
1 m2
P =exp π k2 +θ . (68)
k "− r ⊥− 4 4e !#
8
Then the number density nˆ is written up to the second order of θ as:
1
nˆ = . (69)
exp π k2 1 +θm2 1
⊥− 4 4e −
h (cid:16)q (cid:17)i
Thisdensityisthermalandlookslikeatwo-dimensionalBose-Einsteindistri-
butionwithchemicalpotentialµθ = θπm2. Togetthe totalnon-commutative
− 4e
number of the created particles per a unit volume, we have to integrate the
density nˆ over momentum space. Taking into account the fact that nˆ does not
explicitly depend on k , the total non-commutative number Nˆ reads:
z
2
Nˆ = nˆk dk dk , (70)
(2πT)2 ⊥ ⊥ z
Z
where T is the time for the external interaction and the integration over k
z
is equivalent to the integration of the classical equation of motion: dk =
eEdt = eET. Thus the total non-commutative number Nˆ per a unit volume
R
takes the form:
R
2eET k dk
Nˆ = ⊥ ⊥ . (71)
(2πT)2 "Z eπ(cid:16)√k⊥2−14+θm4e2(cid:17) 1#
−
Nowsinceθissmallwehaveexp θm2 1. ConsequentlythetotalnumberNˆ
4e ≪
ineq.(65),writtenuptotheseco(cid:16)ndord(cid:17)erofθ,isgivenbythefollowingrelation:
eE m2 1 m2
Nˆ = exp π θ 1+ exp π θ + θ2 . (72)
∼ 2π4T − 4e 4 − 4e O
(cid:18) (cid:19)(cid:18) (cid:18) (cid:19)(cid:19)
(cid:0) (cid:1)
Notice that the particle creation mechanism is effectively isotropic in the
presence of a constant electric field of the anisotropic Bianchi I universe of the
non-commutativespace-time. The non-commutative number density of created
particles in eq.(66) takes a similar form in the Boltzmann limit in ordinary
commutative space with a chemical potential µθ = θm2. This result was
− 4e
expected due to the fact that the non-commutativity parameter is the smallest
area in space that can be probed.
3.2 The Strong Field Approximation
In this limit if we set
m2
k˜ = (1 θE)
θ
4eE −
m2
= k θ , (73)
− 4e
such that:
k 0, (74)
→
9