Production of Dirac particle in twisted Minkowsky space-time

Abstract

In this paper we study the Dirac equation interacting with external
gravitation field. This curve background, which correspond to the deformation
of Minkowsky space-time is described with the tetrad of the form
$e_b^\mu(x)=\varepsilon(\delta_b^\mu+\omega_{ba}^\mu x^a)$, where
$\varepsilon=1$ for $\mu=0$ and $\varepsilon=i$ for $\mu=1,2,3.$ Using
separation of variables the corresponding Dirac equation is solved. The
probability density of the vacuum-vacuum pair creation is given. In particular
case of vanishing electromagnetic fields, we point out that, this external
gravitation field modify weakly the well know probability of pair production of
the Dirac particle given in ordinary space-time.

Full text

Production of Dirac particle in a deformed Minkowsky space-time
Dine Ousmane Samary,1, 2, 3, ∗Sˆecloka Lazare Guedezounme,2, †and Antonin Kanfon3, ‡
1Perimeter Institute for Theoretical Physics, Waterloo, ON, N2L 2Y5, Canada
2International Chair in Mathematical Physics and Applications 072B.P.50, Cotonou, Benin
3Facult´e des Sciences et Techniques, University of Abomey-Calavi, Benin
(Dated: Wednesday 21st October, 2015)
In this paper we study the Dirac field theory interacting with external gravitation field, described
with tetrad of the form eµ
b(x) = ε(δµ
b+ωµ
baxa), where ε= 1 for µ= 0 and ε=ifor µ= 1,2,3.The
probability density of the vacuum-vacuum pair creation is given. In particular case of vanishing
electromagnetic fields, we point out how this deformation modify the amplitude transition. The
corresponding Dirac equation is solved.
I. INTRODUCTION
The Dirac particle theory arise in the theoret-
ical description of the fermion particles phenom-
ena. They also become central to elementary par-
ticle physics, as the starting point for quantum the-
ory of electromagnetic interaction. In the past few
years this theory have been widely studied in various
curved backgrounds due to its importance in both
astrophysics and cosmology, as well as in the study
of particle creation processes [1]-[6]. Pair produc-
tion is a phenomenon of nature where energy is con-
verted to mass. nevertheless there are only few prob-
lems for which the Dirac equation can be solved ex-
actly. Some of them are give in [7]-[15] and references
therein. The explicit solution is crucial in the parti-
cle creation processes and is the base of the standard
cosmological model [25]-[27]. This amounts to claim
that, the formulation and behaviour of fermion par-
ticles physics including the gravitation field is per-
formed using solution of Dirac equation.
In the present work, the transition amplitude and
the probability density of the pair creation of the
Dirac particle is examined in the twisted Minkowsky
space-time. A solution of the Dirac equation is pro-
posed. The metric is chosen to be the first-order
fluctuation of the flat Minkowsky pseudo-metric ten-
sor ηab. We pointed out how this deformation of flat
metric modify the well know probability density of
creation of Dirac particle, a while ago computed in
[2], [4] and references therein.
The paper is organized as follows. In section (II),
we quickly review Dirac particle theory, interacting
with gravity field. In (III) we compute the probabil-
ity density of pair production. The case of vanishing
electromagnetic (EM) fields is examined. The solu-
∗Electronic address: dsamary@perimeterinstitute.ca
†Electronic address: guesel10@yahoo.fr
‡Electronic address: kanfon@yahoo.fr
tion of the corresponding Dirac equation is proposed
in the section (IV). In the last section (V), we con-
clude our work and make some remarks.
II. DIRAC EQUATION COUPLED WITH A
WEAK GRAVITATION FIELD
In the curve space-time the conventional affine
connection ∇µis replaced by the spin connection
Γµwhich is expressed in terms of the vierbein fields
(eµ
a(x)) (see [16]-[24]). Then any curved space de-
scription of physics can be replaced by an equivalent
and simpler flat space physics, through the vierbein
transformation. So there was an equivalent formu-
lation of general relativity involving the dynamics of
the so-called spin-connection. This approach came
to be known as Einstein-Cartan theory and leads to
consider general relativity as gauge theory approach
of gravity [18].
An arbitrary geometrical object defined on the
Riemann space-time manifold can be locally pro-
jected on the tangent Minkowski space, simply by
contracting its curved indices with the vierbein and
its inverse. For rank ntensor object T, we can write
Ta1a2···an=ea1
µ1(x)ea2
µ2(x)···ean
µn(x)e
Tµ1µ2···µn,(1)
Ta1a2···an=eµ1
a1(x)eµ2
a2(x)·· ·eµn
an(x)e
Tµ1µ2···µn,(2)
where the Latin indices (a, b, c, ·· ·) is used only
for the flat space-time and the Greek indices
(α, β, µ, · ·· ) for the curve space-time. The “tilde
notation” is used only for the curve space variables.
ea
µ(x) represent the inverse of eµ
a(x). The metric ten-
sors gµν and ηab = diag(1,−1,−1,−1) are related by
gµν (x) = eµ
a(x)eν
b(x)ηab,or ηab =ea
µ(x)eb
ν(x)gµν (x).
The connection Γµis
Γµ=: 1
4gαβ ∂ea
ν
∂xµeβ
a−Γβ
νµ eσαν ,(3)
where eσµν =1
2[eγµ(x),eγν(x)].
arXiv:1507.01472v2 [hep-th] 13 Sep 2015

We consider the Dirac equation coupled with both
gravitational and EM fields, given by the following
relation
(ieγµ(x)Dµ−m)ψ(x)=0,(4)
where Dµ=∂µ−Γµ+iAµ.In this expression,
the vectors Γµand Aµare respectively the gravi-
tation and EM gauge vectors. The field ψ, is a four-
components complex functions of space-time coordi-
nates xµ, µ = 0,1,2,3. The Dirac gamma matrices
eγµ(x), acting on the vector fields ψ, satisfy the anti
commutation formula:
{eγµ(x),eγν(x)}= 2gµν such that eγµ(x) = eµ
a(x)γa.(5)
The flat space-time gamma matrices γare expressed
with the Pauli matrices σi, i = 1,2,3 by
γ0=120
0−12, γi=0σi
−σi0, i = 1,2,3.(6)
Remark that the equation (4) provided from the
Euler-Lagrange equation of motion of the action S:
S=Zd4x√−gi¯
ψeγµ(x)Dµψ−m¯
ψψ,(7)
where ¯
ψ=ψ†eγ0(x).
The dynamics described by the relation (4) is in-
variant under an external local transformation of
Lorentz group. In the case of internal local trans-
formation this invariance is satisfy for the dynamics
occurring within the space-time manifold. The sym-
metries of this internal space are chosen to be the
gauge symmetries of some gauge theory, so a unified
theory would contain gravity together with the other
observed fields.
For |ωµ
a|<< 1, we consider the vierbein field eµ
a(x)
as
(eµ
a)(x) = diagh1 + ω0
axa, i(1 + ω1
axa),
i(1 + ω2
axa), i(1 + ω3
axa)i.(8)
We shall use the notation ωµ
axa=: ωµ
baxa. Then, the
metric tensor egµν =: ηµν +fµν (where fµν is the
perturbation tensor), takes the form
(egµν ) = diagh1+2ω0
axa,−1−2ω1
axa,
−1−2ω2
axa,−1−2ω3
axai,(9)
such that the limit where (ω)→0 restore the
Minkowsky pseudo-metric. egµν can be considered
as the first-order fluctuation of the flat Minkowsky
pseudo-metric. Now let us choose the tensor (ω)
such that the metric depend only on the coordinates
(t=x0, x =x1), i.e.
ωµ
2=ωµ
3= 0, ωµ
0=ω, ωµ
1=eω. (10)
The vector Γµis
(Γµ) = 


Γ0
Γ1
Γ2
Γ3


=



ieω
2γ0γ1
iω
2γ0γ1
iω
2γ0γ2−
eω
2γ1γ2
iω
2γ0γ3−
eω
2γ1γ3



.(11)
We get the following result:
Proposition 1. The Dirac equation in the curve
background defined with the metric (9)and coupled
with EM fields Aµ= (0,0, Bx, −Et)is given by
hiγ0∂0−γj∂j−iγ2Bx +iγ3Et + 9iωγ0−3eωγ1
−m(1 −ωt −eωx)iψ(xµ)=0, j = 1,2,3.(12)
The solution of the corresponding equation can be
split into
ψ(t, x, y, z) = ψ(t, x) exp i(k2y+k3z),(13)
where ψ(t, x)is a function which depends only on t
and x.
Proof. Using the relation (8), the Christoffel tensors
are:
Γa
aa =−ωa
a,Γb
ab =−ωb
a,
Γb
aa =ηaaηbb ωa
b,Γc
ab = 0, a 6=b6=c, (14)
where the Einstein summation are not taking into
account in the above relations. Also the components
of the Lorentz connection are
Γ0=i
2ω0
1σ01 +ω0
2σ02 +ω0
3σ03,(15)
Γ1=−1
2iω1
0σ10 −ω1
2σ12 −ω1
3σ13,(16)
Γ2=−1
2iω2
0σ20 −ω2
1σ21 −ω2
3σ23,(17)
Γ3=−1
2iω3
0σ30 −ω3
1σ31 −ω3
2σ32,(18)
which are reduced to (11) using (10), and σab =
1
2[γa, γb].Now, by replacing the expressions (15) in
(4), the Dirac equation becomes
ieγµ(x)∂µ−eγµ(x)Aµ+ωaγa−mψ(xµ) = 0,(19)
where ω0=i(ω1
0+ω2
0+ω3
0)=3iω, ω1= (ω0
1−ω2
1−
ω3
1) = −eω, ω2= (ω0
2−ω1
2−ω3
2)=0, ω3= (ω0
3−ω1
3−
ω2
3) = 0.We choose the external electromagnetic
field as E=Eex,B=Bex, where exis the unit
vector in xdirection. One solution of the Maxwell
equation is then Aµ= (0,0, Bx, −Et). Finally, the
relation (12) is well satisfy.
2

III. TRANSITION AMPLITUDE OF THE
MODEL
In this section we study, how this new met-
ric modify the pair creation of fermion particles.
We consider the Hilbert space of coordinates vec-
tors Hxsuch that the space-time coordinates xµ=
(x0, x1, x2, x3) = (t, x, y, z) are eigenvalue of coor-
dinate operators Xµ= (t14, X, Y , Z) acting on Hx,
i.e. for |t, x, y, z >∈ Hx
Xµ|t, x, y, z >=xµ|t, x, y, z > . (20)
The Hilbert space of momentum space vectors Hp
is define as the Fourier transformation of Hx. The
momentum operator Pµ= (P0, P1, P2, P3),(P0=
i∂0,Pj=−i∂j, j = 1,2,3), is defined by
Pµ|k0, k1, k2, k3>=kµ|k0, k1, k2, k3>,
< k0, k1, k2, k3|t, x, y, z >=eikµxµ
(2πN )2, N ∈R.(21)
Also the curve space-time coordinates operators are
given by e
Xµ= (e
t, e
X, e
Y , e
Z) and conjugate momen-
tum operators e
Pµ= ( e
P0,e
P1,e
P2,e
P3) such that e
P0=
ig00∂0and e
Pj=igjj ∂j, j = 1,2,3.
In the path integral point of view, the action (7)
gives the transition amplitude of the model (or the
partition function Z(A, Γ) = NRDψD ¯
ψ eiS ) which
is explicitly written as :
Z(A, Γ) = exp h−Tr ln iγµ∂µ−m+i
Mi,(22)
with M=iγµ(∂µ−Γµ+iAµ)−m+i and
the normalization constant is defined such that
Z(0,0) = 1. Using the followings identities:
CeγaC−1=−eγt
a,, (γµ)t=−eµ
aCeγaC−1=
−Cγ µC−1,Γt
µ=−CΓµC−1,we come to Mt=
iCγ µ(∂µ−Γµ+iAµ)C−1−m+i and then the
conjugate of the functional Z(A, Γ) is given by
Zt(A, Γ) = exp h−Tr ln iC γµC−1∂µ+m+i
Mti,
(23)
where C=ieγ2eγ0.
We now compute the transition amplitude
|Z(A, Γ)|2. For this, let us define the quan-
tities XH(ω, 0) = ωX1
H,XH(ω, 0) = ωX1
H,
YH(ω, 0, E , B) = ωY1
H,YH(0,eω, E, B ) = eωY2
H, such
that
X1
H= 2t|P|2+γ0γ1P1+γ0γ2P2+γ0γ3P3,(24)
X2
H= 2X|P|2+γ0γ1P0+iγ1γ2P2+iγ1γ3P3,(25)
Y1
H= 2t|P|2+ 4γ0γ1P1+ (4γ0γ2+ 4BX t)P2
+ (4γ0γ3−4Et2)P3−6γ0γ3Et + 4γ0γ2BX
+ 2iγ1γ2Bt + 2t(B2X2+E2t2),(26)
Y2
H= 2X|P|2−2γ1γ0P0+ (2iγ1γ2+ 4BX2)P2
+ (2iγ1γ3−4EXt)P3−iγ1γ3Et + 3iγ1γ2BX
−2γ0γ3EX + 2X(B2X2+E2t2).(27)
Then P=: |Z(A, Γ)|2is explicitly written as
P= exp h−Tr ln `(ω, eω)
n(ω, eω, E , B)i
= exp h−Tr Z∞
0
ds
seisn(ω,eω ,E,B)−eis`(ω ,eω)i
(28)
where
`(ω, eω) = XH(0,0) + XH(ω, 0) + XH(0,eω),(29)
n(ω, eω, E , B) = YH(0,0, E, B ) + YH(ω, 0, E, B)
+YH(0,eω, E, B ).(30)
We get the following statement:
Proposition 2. Consider that the EM fields are
vanishing. For very small positif parameter of the
size 1/eω2, the probability of the pair production takes
the form
P= exp (−πMeπm8
1024v h2Nγ−37
12 −1
34 ln b−ln ai),
(31)
where ω=veω,a=3eω
8v(v2+ 1),b=3eω
2v(4v2+ 1),
Nγis the Euler number, M=Rµ(y, z)dy dz, and
µ(y, z)is the test function.
Remark 1. Note that the case where M > 0is not
fulfils. This leads to a infinite probability density.
We choose the test function µ(y, z)such that M < 0,
and then
πM eπm8
1024vh2Nγ−37
12 −1
34 ln b−ln ai>0.
In the figure (1)we give the plot of this probability
density as function of the parameter . This figure
gives asymptotically the values of f() = Pwhen 
tends to zero. Then the limit →0leads to P ≈ 0.
Proof. of the proposition (2): The rest of this section
is devoted to the proof of the proposition (2). The
density Ω =: Ω(ω, eω, E , B) such that P= exp(−Ω)
can be expanded as
Ω=Ω1(ω, eω, E , B)−Ω2(ω, eω, E, B),(32)
where
Ω1=ZdxZ∞
0
ds
shx|eisYH(ω,eω ,E,B)|xi,(33)
3

FIG. 1: Plot of P=f() with v= 1, M=−2π,m= 1,
ω=√.
Ω2=ZdxZ∞
0
ds
shx|eisXH(ω,eω)|xi.(34)
We consider the mean values:
x1
H=<x|X1
H|x>, x2
H=<x|X2
H|x>,
y1
H=<x|Y1
H|x>, y2
H=<x|Y2
H|x> . (35)
Now we choose ω=veωand vt +x > 0 and we are
focussing on the computation of Ω1(ω, eω, E, B). We
get
Ω1=Z∞
0
ds
shx|eisYH(0,0,E,B)|xi
×Zdxdk1 + isωy1
H+iseωy2
H(36)
with
hx|eisYH|xi=−iEB coth(Es) cot(Bs)
4π2e−ism2.
(37)
A simple routine checking shows that
Zdxdkexp hiseωvy1
H+y2
Hi
=Mπ2eπ
4s2Zdxdt
(vt +x)2n
exp hiseω−P(x, t)
vt +x+Q(x, t)io (38)
with
P(x, t)=2v2(3 + 2Bxtγ0γ2+B2x2t2
−2γ0γ3Et2+E2t4)+2v(2Bx2γ0γ2
+iγ1γ2Bxt + 2B2x3t−2γ0γ3Ext
−iγ1γ3Et2+E2xt3) + 3
2+ 2iγ1γ2Bx2
+ 2B2x4−2iγ1γ3Ext + 2E2x2t2,(39)
and
Q(x, t) = v4γ0γ2Bx −6γ0γ3Et + 2iγ1γ2Bt
+ 2t(B2x2+E2t2)−iγ1γ3Et
+ 3iγ1γ2Bx −2γ0γ3Ex
+ 2x(B2x2+E2t2).(40)
In the same manner Ω2takes the form
Ω2=Z∞
0
ds
sZdxdkhx|eisXH(0,0)|xi
×1 + isωx1
H+iseωx2
H(41)
where
hx|eisXH|xi=−i
16π2s2e−ism2.(42)
We can now show that
Zdxdk1 + isveωx1
H+iseωx2
H
=Zdxπ2eπ
4s2(vt +x)2exp h−3iseω(v2+ 1)
8(vt +x)i.
(43)
However
Ω(eω, E , B) = iMeπ
64Z∞
0
ds e−ism2
s3h1
s2I(t0, x0)
−4EB coth(Es) cot(Bs)J(t0, x0)i(44)
where Mis chosen to be M=Rµ(x, y)dy dz < ∞,
I(t0, x0) = Z∞
x0Z∞
t0
dxdt π2eπ
4s2(vt +x)2n
exp h−3iseω(v2+ 1)
8(vt +x)io (45)
and
J(t0, x0) = Z∞
x0Z∞
t0
dxdt 1
(vt +x)2
×exp hiseω−P(x, t)
(vt +x)+Q(x, t)i.(46)
The integral (44) exhibit the divergence at point x=
t= 0. This shall be regularized by using the Cauchy
principal value. For E=B= 0 we get
I(0,0) = 1
α1Z∞
0
dt hsin(α1/vt)+2isin2(α1/2vt)i
=iπ
2v+1
vh1−Nγ−ln α1
vi,(47)
J(0,0) = 1
α2Z∞
0
dt hsin(α2/vt)+2isin2(α2/2vt)i
=iπ
2v+1
vh1−Nγ−ln α2
vi,(48)
with α1=3seω
8(v2+ 1), α2=seω
2(12v2+ 3) and Nγis
the Euler number given by Nγ= 0.577215664. Also,
for a, b ∈Rthe integral
Q=Z∞
0
ds e−ism2
s54 ln(bs)−ln(as)
4

admits the Cauchy principal value
P v(Q) = m8
1152h415 −300Nγ+ 72N2
γ+ 12π2
+ 96 ln b2−24 ln a2−300 ln(im2)
+ 72 ln(im2)2+ 144Nγln(im2)
+ 4−25 + 12Nγ+ 12 ln(im2)ln a
−16−25 + 12Nγ+ 12 ln(im2)ln bi.
(49)
Remark that the probability density of pair creation
in the limit eω= 0 (see [2]) correspond to
Ω(0, E, B ) = EB
4π2
∞
X
k=1
1
kcoth kπ B
Eexp −kπm2
E.
(50)
Using the Taylor expansion as
Ω(eω, E , B) = Ω(0, E, B ) + eωΩ0(0, E, B) + O(eω2),
(51)
we come to Ω(eω, 0,0) = eωΩ0(0,0,0) + O(eω2) and
Ω(eω, 0,0) = iM eπ
64Z∞
0
ds e−ism2
s5T(0,0),(52)
T(0,0) = I(0,0) −4J(0,0).
We choose the real part of Ω(eω, 0,0) denoted by
<eΩ(eω, 0,0) = [Ω(eω, 0,0) + Ω∗(eω, 0,0)]/2. Using
(47), (48) and (49)
<eΩ(eω, 0,0) = −πMeπm8
1024v h2Nγ−37
12
−1
34 ln b−ln ai.(53)
where a=3eω
8v(v2+ 1), b=3eω
2v(4v2+ 1). Finally it is
straightforward to check the following relation
eωΩ0(0,0,0) = πM eπm8
1024v h2Nγ−37
12
−1
34 ln b−ln ai.(54)
While the probability of the pair production takes
the form
P= exp n−πM eπm8
1024v h2Nγ−37
12
−1
34 ln b−ln aio≈0.(55)
This end the proof of proposition (2).
IV. SOLUTION OF THE DIRAC EQUATION
In this section we give the solution of the Dirac
equation (12). We consider the operators K1and
K2satisfying the commutation relation [K1,K2]=0
and given by
K1=iγ0∂0−iγ2k2−iγ3k3+iγ3Et + 9iωγ0
−3eωγ1−m(1 −ωt),(56)
K2=−γ1∂1−iγ2Bx +meωx. (57)
The equation (12) takes the form (K1+K2)ψ(t, x) =
0 and admit separate variables as ψ(t, x) =
ψ(t)ψ(x). For a constant λ∈C, we get the two
eigenvalue equations
K1ψ(t, x) = λψ(t, x) (58)
K2ψ(t, x) = λψ(t, x).(59)
Consider the equation (58). We write the four
vector ψ(t) as ψ(t) = (ψ1(t), ψ2(t)) and ψj(t) =
(ψja (t), ψjb(t)), j = 1,2, and the equation (58) leads
to
L0L+D2+CC 0ψ1a(t)=0,(60)
LL0+D2+CC 0ψ2a(t)=0,(61)
Cψ1b(t) = L0ψ2a(t)−Dψ1a(t),(62)
Cψ2b(t) = −Lψ1a(t)−Dψ2a(t) (63)
where
L=i∂0+ 9iω −m(1 −ωt) + λ,
L0=−i∂0−9iω +m(1 −ωt) + λ
C=−3eω−k2, C0=−3eω+k2,
D=iEt −ik3.
Let
f(t) = E
2t2+m
E(m−λ)ω−9ω−k3t, (64)
g(t)=(−1)1
4(iE)1
2t(65)
δ1
E=mω
2E3(m−λ)
+1
2Ek2
2−(m−λ)2+imω−1
2(66)
δ2
E=(−1)1
4
(iE)3
2Ek3−(m−λ)mω.(67)
The solution of the equations (60) are a linear com-
bination of Hermite and (1,1)-hypergeometric poly-
nomial given by
ψ1a(t) = c1ef(t)Hhδ1
E, δ2
E+g(t)i
5

+c2ef(t)1F1[δ1
E
2,1
2,δ2
E+g(t)2]
ψ2a(t) = c1ef(t)Hh¯
δ1
E, δ2
E+g(t)i
+c2ef(t)1F1[¯
δ1
E
2,1
2,δ2
E+g(t)2].(68)
the solutions of the equation (62) can be simple ob-
tained using the followings identities:
dH(a, b +ct)
dt = 2acH(−1 + a, b +ct),(69)
d1F1(a, b, ct2+dt +e)
dt
=a(d+ 2ct)
b1F1(1 + a, 1 + b, ct2+dt +e).(70)
Now, consider the equation (59). Using the Dirac
matrices (6), we get
−(σ1∂1+iσ2Bx)ψ2(x)+(meωx −λ)ψ1(x) = 0(71)
(σ1∂1+iσ2Bx)ψ1(x)+(meωx −λ)ψ2(x) = 0(72)
where ψ(x) = (ψ1(x), ψ2(x)). Let us define the
quantities b(B), r(B) and s(B) as
b(B) =: λ2
2B,
r(B) = 2meωλ(−1)1
4
√2(iB)3
2
,
s(B)=(−1)1
4√2(iB)1
2.
For ψj(x) = (ψja (x), ψjb (x)), j = 1,2. The solu-
tions of the equation (71) are
ψ1a(x) = c1Dh−b(B),−r(B) + s(B)xi
+c2Dh−b(B),−ir(B) + is(B)xi(73)
ψ1b(x) = c1Dh−1−b(B),−r(B) + s(B)xi
+c2Dh−1−b(B),−ir(B) + is(B)xi.
(74)
However, the equation (72) can be split into
h∂2
1−B−B2x2+λ2−2mλeωxiψ1a(x) = 0 (75)
h∂2
1+B−B2x2+λ2−2mλeωxiψ1b(x)=0,(76)
and the solutions are well given by the following:
ψ2a(x) = 1
λ−meωx (∂1+Bx)ψ1b,(77)
ψ2b(x) = 1
λ−meωx (∂1−Bx)ψ1a.(78)
where the identities
d
dxD(a, bx +c) = b
2(bx +c)D(a, bx +c)
−bD(1 + a, bx +c),(79)
are usefull.
V. CONCLUSION
In this paper, we have computed the probability
density of pair production of the fermion particles.
The case of vanishing EM fields is scrutinized ex-
plicitly. Hereafter we will shed light on the case of
non-vanishing EM fields, which has not been entirely
considered in this paper. In the other hand, we have
solved the Dirac equation coupled with gravitation
field, using the separation of variables. The solutions
are expressed in terms of hypergeometric functions.
The limit where the deformation parameter eωtends
to zero is given.
Acknowledgements
D. O. S. research is supported in part by the
Perimeter Institute for Theoretical Physics (Water-
loo) and by the Fields Institute for Research in
Mathematical Sciences (Toronto). Research at the
Perimeter Institute is supported by the Government
of Canada through Industry Canada and by the
Province of Ontario through the Ministry of Eco-
nomic Development & Innovation.
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