Noncommutative Space-Time of the Relativistic Equations with a Coulomb Potential Using Seiberg-Witten Map
We present an important contribution to the noncommutative approach to the hydrogen atom to deal with Lamb shift corrections. This can be done by studying the Klein{Gordon and Dirac equations in a non-commutative space-time up to first-order of the noncommutativity parameter using the Seiberg-Witten...
Gespeichert in:
| Datum: | 2016 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2016
|
| Schriftenreihe: | Журнал математической физики, анализа, геометрии |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/140560 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Noncommutative Space-Time of the Relativistic Equations with a Coulomb Potential Using Seiberg-Witten Map / S. Zaim // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 4. — С. 359-373. — Бібліогр.: 46 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-140560 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1405602018-07-11T01:23:16Z Noncommutative Space-Time of the Relativistic Equations with a Coulomb Potential Using Seiberg-Witten Map Zaim, S. We present an important contribution to the noncommutative approach to the hydrogen atom to deal with Lamb shift corrections. This can be done by studying the Klein{Gordon and Dirac equations in a non-commutative space-time up to first-order of the noncommutativity parameter using the Seiberg-Witten maps. We thus find the noncommutative modification of the energy levels and by comparing with the current experimental results on the Lamb shift of the 2P level to extract a bound on the parameter of noncommutativity, we show that the fundamental length (√Θ) is compatible with the value of the electroweak length scale (l). Phenomenologically, this effectively confirms the presence of gravity at this level. 2016 Article Noncommutative Space-Time of the Relativistic Equations with a Coulomb Potential Using Seiberg-Witten Map / S. Zaim // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 4. — С. 359-373. — Бібліогр.: 46 назв. — англ. 1812-9471 DOI : doi.org/10.15407/mag12.04.359 Mathematics Subject Classification 2000: 81T80, 37K05, 81Q05 http://dspace.nbuv.gov.ua/handle/123456789/140560 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We present an important contribution to the noncommutative approach to the hydrogen atom to deal with Lamb shift corrections. This can be done by studying the Klein{Gordon and Dirac equations in a non-commutative space-time up to first-order of the noncommutativity parameter using the Seiberg-Witten maps. We thus find the noncommutative modification of the energy levels and by comparing with the current experimental results on the Lamb shift of the 2P level to extract a bound on the parameter of noncommutativity, we show that the fundamental length (√Θ) is compatible with the value of the electroweak length scale (l). Phenomenologically, this effectively confirms the presence of gravity at this level. |
| format |
Article |
| author |
Zaim, S. |
| spellingShingle |
Zaim, S. Noncommutative Space-Time of the Relativistic Equations with a Coulomb Potential Using Seiberg-Witten Map Журнал математической физики, анализа, геометрии |
| author_facet |
Zaim, S. |
| author_sort |
Zaim, S. |
| title |
Noncommutative Space-Time of the Relativistic Equations with a Coulomb Potential Using Seiberg-Witten Map |
| title_short |
Noncommutative Space-Time of the Relativistic Equations with a Coulomb Potential Using Seiberg-Witten Map |
| title_full |
Noncommutative Space-Time of the Relativistic Equations with a Coulomb Potential Using Seiberg-Witten Map |
| title_fullStr |
Noncommutative Space-Time of the Relativistic Equations with a Coulomb Potential Using Seiberg-Witten Map |
| title_full_unstemmed |
Noncommutative Space-Time of the Relativistic Equations with a Coulomb Potential Using Seiberg-Witten Map |
| title_sort |
noncommutative space-time of the relativistic equations with a coulomb potential using seiberg-witten map |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2016 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/140560 |
| citation_txt |
Noncommutative Space-Time of the Relativistic Equations with a Coulomb Potential Using Seiberg-Witten Map / S. Zaim // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 4. — С. 359-373. — Бібліогр.: 46 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT zaims noncommutativespacetimeoftherelativisticequationswithacoulombpotentialusingseibergwittenmap |
| first_indexed |
2025-07-10T10:44:17Z |
| last_indexed |
2025-07-10T10:44:17Z |
| _version_ |
1837256424061665280 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2016, vol. 12, No. 4, pp. 359–373
Noncommutative Space-Time of the Relativistic
Equations with a Coulomb Potential Using
Seiberg–Witten Map
S. Zaim
Département de Physique, Faculté des Sciences de la Matière,
Université Batna1, Algeria
E-mail: zaim69slimane@yahoo.com
Received July 8, 2015
We present an important contribution to the noncommutative approach
to the hydrogen atom to deal with Lamb shift corrections. This can be done
by studying the Klein–Gordon and Dirac equations in a non-commutative
space-time up to first-order of the noncommutativity parameter using the
Seiberg–Witten maps. We thus find the noncommutative modification of
the energy levels and by comparing with the current experimental results
on the Lamb shift of the 2P level to extract a bound on the parameter of
noncommutativity, we show that the fundamental length (
√
Θ) is compatible
with the value of the electroweak length scale (l). Phenomenologically, this
effectively confirms the presence of gravity at this level.
Key words: non-commutative geometry methods, field theory, Klein–
Gordon and Dirac equations.
Mathematics Subject Classification 2010: 81T80, 37K05, 81Q05.
1. Introduction
The standard concept of space-time as a geometric manifold is based on the
notion of a manifold whose points are locally labelled by a finite number of
real coordinates. However, it is generally believed that this picture of space-
time as a manifold should break down at very short distances of the order of
the Planck length. This implies that the mathematical concepts of high energy
physics has to be changed or, more precisely, our classical geometric concepts may
not be well-suited for the description of physical phenomenon at short distances
[1–3]. The connection between the string theory and the non-commutativity [4–
7] motivated a large amount of work to study and understand many physical
c© S. Zaim, 2016
S. Zaim
phenomena. The study of this geometry has raised new physical consequences
and thus, recently, a noncommutative description of quantum mechanics has
stimulated a large amount of research [8–15]. The non-commutative field theory
is characterized by the commutation relations between the position coordinate
operators themselves, namely,
[x̂µ, x̂ν ] = iΘµν , (1)
and the star Moyal product ∗ is defined between two fields ψ (x) and ϕ (x) by
ψ (x) ∗ ϕ (x) = exp
(
i
2
Θµν ∂
∂xµ
∂
∂yν
)
ψ (x) ϕ (y) |y=x, (2)
where Θµν are the noncommutative constant parameters in the canonical non-
commutative space-time.
The issue of time-space noncommutativity is worth pursuing on its own right
because of its deep connection with such fundamental notions as unitarity and
causality. Much attention has been devoted in recent times to circumvent these
difficulties in formulating theories with Θ0i 6= 0 [1, 2, 16, 17]. There are similar
examples of theories with time-space noncommutativity in the literature [18–20]
where unitarity is preserved by a perturbative approach [21].
The most obvious natural phenomena to search for noncommutative effects
are simple systems of quantum mechanics in the presence of a magnetic field,
such as a hydrogen atom. In the noncommutative time-space one expects the
degeneracy of the spectrum levels to be lifted, and therefore one can say that the
noncommutativity plays the role of the magnetic field. The study of the exact
and approximate solutions of the relativistic hydrogen atom has proved to be
fruitful and many papers have been published [22–25]. In this work we present
an important contribution to the noncommutative approach to the relativistic
description of the hydrogen atom. Our goal is to solve the Klein–Gordan and
Dirac equations for the Coulomb potential in a noncommutative space-time up
to first-order of the noncommutativity parameter using the Seiberg–Witten maps
and the Moyal product. We thus find the noncommutative modification of the
energy levels of the hydrogen atom and we show that the noncommutativity is
the source of a magnetic field resulting in the Lamb shift corrections. We also
note that the effect of noncommutativity confirms the presence of gravity at the
very short distances.
In a previous work [26, 27], by solving the deformed Klein–Gordon and Dirac
equations in a canonical noncommutative space, we showed that the energy is
shifted, where the correction is proportional to the magnetic quantum number,
which behavior is similar to the Zeeman effect as applied to a system without
spin in a magnetic field, thus we explicitly accounted for spin effects in this space.
360 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4
Noncommutative Space-Time of the Relativistic Equations...
The purpose of this paper is to study the extension of the Klein–Gordon
and Dirac fields in canonical noncommutative time-space by applying the result
obtained to a hydrogen atom.
The paper is organized as follows. In Sec. 2, we propose an invariant action of
the noncommutative boson and fermion fields in the presence of an electromag-
netic field. In Sec. 3, using the generalized Euler–Lagrange field equations, we
derive the deformed Klein–Gordon (KG) and Dirac equations for the hydrogen
atom. We solve these deformed equations and obtain the noncommutative mod-
ification of the energy levels. Furthermore, we derive the non-relativistic limit
of the noncommutative KG equation for a hydrogen atom and solve it using the
perturbation theory. Finally, in Sec. 4, we draw our conclusions.
2. Action
The canonical noncommutative space-time is characterized by the commuta-
tion relations of coordinate operators satisfying relation (1). In order to preserve
this relation, the infinitesimal gauge transformation is generalized by the follow-
ing relation:
φ̂A (A) + δ̂λ̂φ̂A (A) = φ̂A (A + δλA) , (3)
where φ̂A = (µ, ψ̂) is a non-commutative generic field, µ and ψ̂ are the non-
commutative gauge and matter fields, respectively, λ is the U(1) gauge Lie-valued
infinitesimal transformation parameter, δλ is the ordinary gauge transformation
and δ̂λ̂ is a non-commutative gauge transformation which are defined by:
δ̂λ̂ψ̂ = iλ̂ ∗ ψ̂, δλψ = iλψ, (4)
δ̂λ̂µ = ∂µλ̂ + i
[
λ̂, µ
]
∗
, δλAµ = ∂µλ. (5)
Now using these transformations one can get at second order in the noncom-
mutative parameter Θµν the following Seiberg–Witten maps [4]:
ψ̂ = ψ + ψ1 +O (
Θ2
)
, (6)
λ̂ = λ + λ1 (λ,Aµ) +O (
Θ2
)
, (7)
Âξ = Aξ + A1
ξ (Aξ) +O (
Θ2
)
, (8)
F̂µξ = Fµξ (Aξ) + F 1
µξ (Aξ) +O (
Θ2
)
, (9)
where
ψ1 = − i
2
Θαβ({Aα, ∂βψ}+
1
2
{[ψ,Aα] , Aβ}), (10)
λ1 = Θαβ∂αλAβ, (11)
A1
ξ =
1
2
ΘαβAα
(
∂ξAβ − 2∂βA
ξ
)
, (12)
F 1
µξ = −Θαβ (Aα∂βFµξ + FµαFβξ) , (13)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4 361
S. Zaim
and
Fµν = ∂µAν − ∂νAµ. (14)
To begin, we consider an action for a free boson and fermion fields in the
presence of an electrodynamic gauge field in a noncommutative space-time. We
propose the following action [28]:
S =
∫
d4x
(
LMB + LMF − 1
4
F̂µν ∗ F̂µν
)
, (15)
where LMB and LMF are the boson and fermion matter densities, respectively, in
the non-commutative space-time and are given by
LMB = ηµν
(
D̂µϕ̂
)†
∗ D̂νϕ̂ + m2ϕ̂† ∗ ϕ̂, (16)
and
LMF = ψ̂ ∗
(
iγνD̂ν −m
)
∗ ψ̂, (17)
where the gauge covariant derivative is defined as D̂µ = ∂µ + ieµ.
From the action variational principle the generalized equations of Lagrange
up to O (
Θ2
)
are [29]:
∂L
∂Φ̂
− ∂µ
∂L
∂
(
∂µΦ̂
) + ∂µ∂ν
∂L
∂
(
∂µ∂νΦ̂
) +O (
Θ2
)
= 0, (18)
where
L = LMB + LMF − 1
4
F̂µν ∗ F̂µν . (19)
3. Noncommutative Time-Space KG Equation
Using the modified field equation (18) , with the generic boson field ϕ̂, one
can find in a free non-commutative space-time and in the presence of the external
potential µ the following modified Klein–Gordon equation:
(
ηµν∂µ∂ν −m2
e
)
ϕ̂ +
(
ieηµν∂µÂν − e2ηµνµ ∗ Âν + 2ieηµνµ∂ν
)
ϕ̂ = 0, (20)
where the deformed external potential µ
(−e/r
0
)
in free noncommutative space-
time is [30]:
â0 = −e
r
− e3
r4
Θ0kxk +O (
Θ2
)
, (21)
âi =
e3
4 r4
Θikxk +O (
Θ2
)
, (22)
362 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4
Noncommutative Space-Time of the Relativistic Equations...
for a non-commutative time-space, Θ0k 6= 0 and Θki = 0, where i, k = 1, 2, 3. In
this case, we can check that
ηµν∂µ∂ν = −∂2
0 + ∆, (23)
and
2ieηµνµ∂ν = i
2e2
r
∂0 + 2i
e4
r4
Θ0jxj∂0, (24)
and
−e2ηµνµ ∗ Âν =
e4
r2
+ 2
e6
r5
Θ0jxj , (25)
then the Klein-Gordon equation (20) up to O (
Θ2
)
takes the form
[
−∂2
0 + ∆−m2
e +
e4
r2
+ i
2e2
r
∂0 + 2i
e4
r4
Θ0jxj∂0 + 2
e6
r5
Θ0jxj
]
ϕ̂ = 0. (26)
The solution to equation (26) in spherical polar coordinates (r, θ, φ) takes the
separable form
ϕ̂(r, θ, φ, t) =
1
r
R̂(r)Ŷ (θ, φ) exp(−iEt). (27)
Then (26) reduces to the radial equation
[
d2
dr2
− l(l + 1)− e4
r2
+
2Ee2
r
+
+E2 −m2
e + 2E
e4
r4
Θ0jxj + 2
e6
r5
Θ0jxj
]
R̂(r) = 0. (28)
In (28), the Coulomb potential in noncommutative space-time appears within
the perturbation terms [31]:
HΘ
pert = 2E
e4
r4
Θ0jxj + 2
e6
r5
Θ0jxj , (29)
where the first term is the electric dipole–dipole interaction created by the non-
commutativity, the second term is the electric dipole–quadruple interaction. These
interactions show us that the effect of space-time noncommutativity on the inter-
action of the electron and the proton is equivalent to an extension of two nuclei
interactions at a considerable distance. This idea effectively confirms the pres-
ence of gravity at this level. To investigate the modification of the energy levels
by equation (29), we use the first-order perturbation theory. The spectrum of H0
and the corresponding wave functions are well-known and given by
Rnl(r) =
√
a
n + ν + 1
(
n!
Γ (n + 2ν + 2)
)1/2
xν+1e−x/2L2ν+1
n (x) , (30)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4 363
S. Zaim
where the relativistic energy levels are given by
E = En,l =
me
(
n + 1
2 +
√(
l + 1
2
)2 − α2
)
[(
n + 1
2
)2 +
(
l + 1
2
)2 + 2
(
n + 1
2
) √(
l + 1
2
)2 − α2
] 1
2
, (31)
and L2ν+1
n are the associated Laguerre polynomials [32], with the following nota-
tions:
ν = −1
2
+
√(
l +
1
2
)2
− α2, α = e2, a =
√
m2
e −E2. (32)
3.1. Noncommutative corrections of the relativistic energy
Now to obtain the modification to the energy levels as a result of the terms (29)
due to the noncommutativity of space-time, we use the perturbation theory. For
simplicity, first of all, we choose the coordinate system (t, r, θ, ϕ) so that Θ0j =
−Θj0 = Θδ01, such that Θ0jxj = Θr and assume that the other components are
all zero and also the fact that in first-order perturbation theory the expectation
values of 1/r3 and 1/r4 are as follows:
〈nlm | r−3 | nlm′〉 =
∞∫
0
R2
nl(r)r
−3drδmm′
=
4a3n!
(n + ν + 1)Γ (n + 2ν + 2)
∞∫
0
x2ν−1e−x
[
L2ν+1
n (x)
]2
dxδmm′
=
4a3n!
(n + ν + 1)Γ (n + 2ν + 2)
[
Γ (n + 2ν + 2)
Γ (n + 1) Γ (2ν + 2)
]2
×
∞∫
0
x2ν−1e−x [F (−n; 2ν + 2;x)]2 dxδmm′
=
2a3
ν (2ν + 1) (n + ν + 1)
{
1 +
n
(ν + 1)
}
δmm′ = f(3), (33)
〈nlm | r−4 | nlm′〉 =
4a4
(2ν − 1) ν (2ν + 1) (n + ν + 1)
[
1 +
3n
(ν + 1)
+
3n (n− 1)
(ν + 1) (2ν + 3)
]
δmm′ = f(4), (34)
364 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4
Noncommutative Space-Time of the Relativistic Equations...
Now, the correction to the energy to first order in Θ is
EΘ(1) = 〈ψ0
nlm
∣∣∣HΘ(1)
pert
∣∣∣ψ0
nlm〉. (35)
where H
Θ(1)
pert is the noncommutative correction to the first order in Θ of the
perturbation Hamiltonian, which is given in the following relation:
H
Θ(1)
pert = 2E
e4
r3
Θ + 2
e6
r4
Θ. (36)
To calculate EΘ(1), we use the radial function in Eq. (30) to obtain
EΘ(1) = 2Θα2
(
E0
n,lf (3) + αf (4)
)
.
Finally, the energy correction of the hydrogen atom in the framework of the
non-commutative KG equation is
∆ENC =
EΘ(1)
2E
= Θα2
(
f (3) +
α
E0
n,l
f (4)
)
. (37)
This result is important because it reflects the existence of Lamb shift, which
is induced by the noncommutativity of the space. Obviously, when Θ = 0, then
∆ENC = 0, which is exactly the result of the space-space commuting case, where
the energy-levels are not shifted.
We showed that the energy-level shift for 1S is
∆ENC
1S = Θα2
(
f1S (3) +
α
E0
1,0
f1S (4)
)
. (38)
In our analysis, we simply identify spin up if the noncommutativity parameter
takes the eigenvalue +Θ and spin down if the noncommutativity parameter takes
the eigenvalue −Θ. Also we can say that the Lamb shift is actually induced by the
space-time noncommutativity which plays the role of a magnetic field and spin
in the same moment (Zemann effect). This represents Lamb shift corrections for
l = 0. The result is very important: as a possible means of introducing electron
spin we replace l → ± (
j + 1
2
)
and n → n − j − 1 − 1
2 , where j is the quantum
number associated to the total angular momentum. Then the l = 0 state has the
same total quantum number j = 1
2 . In this case, the noncommutative value of
the energy levels indicates the splitting of the 1s states.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4 365
S. Zaim
3.2. Non-relativistic limit
The non-relativistic limit of noncommutative K-G Eq. (26) is written as
[33, 34]:
[
d2
dr2
− l(l + 1)
r2
+
2mee
2
r
+ 2meε + 2me
e4
r3
Θ + 2
e6
r4
Θ
]
R̂(r) = 0. (39)
In this non-relativistic limit the charged boson does not represent a single charged
particle, but is a distribution of positive and negative charges which are different
and extended in space linearly in
√
Θ. The absence of a perturbation term
of the form Θ/r2 in the noncommutative Coulomb interaction shows that the
distribution of positive and negative charges is spherically symmetric. This can
be interpreted as the spherically symmetric distribution of charges of the quarks
inside the proton.
Now to obtain the modification of energy levels as a result of the noncommu-
tative terms in (39), we use the first-order perturbation theory. The spectrum of
H0 (Θ = 0) and the corresponding wave functions are well-known and given by
εn = −meα
2
2~2n2
, (40)
and
Rnl(r) =
1
n
(
(n− l − 1)!
a (n + l)!
)1/2
xl+1e−x/2L2l+1
n−l−1(x), x =
2
an
r, (41)
where a = ~2/(meα) is the Bohr radius of the hydrogen atom. The Coulomb
potential in noncommutative space-time appears within the perturbation terms
HΘ
pert = 2Θα2
(me
r3
+
α
r4
)
+O (
Θ2
)
, (42)
where the expectation values of 1/r3 and 1/r4 are as follows:
〈nlm | r−3 | nlm′〉l>0 =
2
a3n3l(l + 1)(2l + 1)
δmm′ , (43)
〈nlm | r−4 | nlm′〉l>0 =
[
4
(
3n2 − l(l + 1)
)
a4n5l(l + 1)(2l − 1)(2l + 1)(2l + 3)
+
35
(
3n2 − l(l + 1)
)
3(l − 1)(l + 2)(2l − 1)(2l + 1)(2l + 3)
]
δmm′ . (44)
Hence the modification to the energy levels is given by
∆ENC = Θα2
[
f (3) +
α
me
f (4)
]
+O (
Θ2
)
. (45)
366 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4
Noncommutative Space-Time of the Relativistic Equations...
We can also compute the correction to the Lamb shift of the 2P level where we
have
∆ENC
2P = 0.243 156 Θ (MeV)3 . (46)
According to [35], the current theoretical result for the Lamb shift is 0.08 kHz.
From the splitting (46), this then gives the following bound on Θ:
Θ ≤ (8.5TeV)−2 . (47)
This corresponds to a lower bound for the energy scale of 8.5 TeV, which is in
the range that was obtained in [36–39], namely 1–10 TeV.
4. Noncommutative Time-Space Dirac Equation
Now, concerning the Dirac equation in the free non-commutative time-space
and in the presence of the vector potential µ and using the modified field Eq.
(18), with the generic field ψ̂, we can find the modified Dirac equation up to
O (
Θ2
)
as
(iγµ∂µ −me) ψ̂ − eγµAµψ̂ − eγµA1
µψ̂ +
ie
2
Θαβγµ∂αAµ∂βψ̂ = 0. (48)
For a noncommutative time-space (Θki = 0, where i, k = 1, 2, 3), in this case we
can write:
iγµ∂µ −me = iγ0∂0 + iγi∂i −me, (49)
−eγµÂµ =
e2
r
γ0 +
e4
r4
γ0Θ0kxk, (50)
ie
2
Θαβγµ∂αAµ∂β = −i
e2
2
γ0 Θ0kxk
r3
∂0. (51)
Then the noncommutative Dirac equation (48) up to O (
Θ2
)
takes the following
form:
[
iγ0∂0 + iγi∂i −me +
e2
r
γ0 +
e4
r4
γ0Θ0kxk − i
e2
2
γ0 Θ0kxk
r3
∂0
]
ψ̂ = 0. (52)
We can write this equation as
Ĥψ̂ (t, r, θ, ϕ) = i∂0ψ̂ (t, r, θ, ϕ) . (53)
Then replacing
ψ̂ (t, r, θ, ϕ) → exp (−iEt) ψ̂ (r, θ, ϕ) (54)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4 367
S. Zaim
gives the stationary noncommutative Dirac equation
Ĥψ̂ (r, θ, ϕ) = Eψ̂ (r, θ, ϕ) ,
where E is the ordinary energy of the electron and Ĥ is the noncommutative
Hamiltonian of the form
Ĥ = Ĥ0 + ĤΘ
pert, (55)
where H0 is the relativistic Hamiltonian for the hydrogen atom
Ĥ0 = ~α.
(
−i ~∇
)
+ βme − e2
r
, (56)
and HΘ
pert is the leading-order perturbation
ĤΘ
pert =
(
E
2
− e2
r
)
e2
~Θt · ~r
r3
. (57)
The leading long-distance part of HΘ
pert behaves like that of a magnetic dipole
potential where the noncommutativity plays the role of a magnetic moment.
So the noncommutative Coulomb potential is the multipolar contribution and
this means that the distribution is not spherically symmetric. In the above the
matrices ~α and β are given by
β =
(
I 0
0 −I
)
, αi =
(
0 σi
σi 0
)
,
where σi are the Pauli matrices:
σ1 =
(
0 1
1 0
)
, σ2 =
(
0 −i
i 0
)
, σ3 =
(
1 0
0 −1
)
.
To investigate the modification of the energy levels by equation (57), we use
the first-order perturbation theory, where, by restoring the constants c and ~,
the spectrum of Ĥ0 and the corresponding wave functions are well-known and
are given by (see [35, 40–45]):
ψ (r, θ, ϕ) =
(
φ (r, θ, ϕ)
χ (r, θ, ϕ)
)
=
(
f (r)ΩjlM (θ, ϕ)
g (r)ΩjlM (θ, ϕ)
)
, (58)
where the bi-spinors ΩjlM (θ, ϕ) are defined by
ΩjlM (θ, ϕ) =
∓
√
(j+1/2)∓(M−1/2)
2j+(1±1) Yj±1/2,M−1/2 (θ, ϕ)√
(j+1/2)±(M+1/2)
2j+(1±1) Yj±1/2,M+1/2 (θ, ϕ)
, (59)
368 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4
Noncommutative Space-Time of the Relativistic Equations...
with the radial functions f (r) and g (r) given as
(
f (r)
g (r)
)
=
(
a
mc
~
)2 1
ν
√
~c (Eκ −mec2ν) n!
(mec2)2 α (κ − ν) Γ (n + 2ν)
e−
1
2
xxν−1 ×
×
(
f1xL2ν+1
n−1 (x) + f2L
2ν−1
n (x)
g1xL2ν+1
n−1 (x) + g2L
2ν−1
n (x)
)
, (60)
where the ordinary relativistic energy levels are given by
E = En,j =
mec
2 (n + ν)√
α2 + (n + ν)2
, n = 0, 1, 2 · · · (61)
and Lα
n (x) are the associated Laguerre polynomials [32], with the following no-
tations:
a =
1
mec2
√
(mec2)2 − E2, x =
2
~c
√
(mec2)2 − E2 r,
κ = ±
(
j +
1
2
)
, ν =
√
κ2 − α2,
f1 =
aα
E
mec2
κ − ν
, f2 = κ − ν,
g1 =
a (κ − ν)
E
mec2
κ −meν
, g2 =
e2
~c
= α.
In the above, me is the mass of the electron and α is the fine structure constant.
4.1. Noncommutative Corrections to the Dirac Energy
Now to obtain the modification to the energy levels as a result of the terms
(57) due to the noncommutativity of time-space, we use the perturbation theory
up to the first order. With respect to the selection rule ∆l = 0 and choosing the
coordinate system (t, r, θ, ϕ) so that Θ0k = −Θk0 = Θδ01, we have
∆E
(Θ)
n,j = ∆E
(1)
n,j + ∆E
(2)
n,j , (62)
where
∆E
(1)
n,j =
E
2~c
e2
4π∫
0
ΘdΩ
∞∫
0
dr[ψ†njlM (r, θ, ϕ) ψnj′l′M ′ (r, θ, ϕ)]
=
E
2~c
e2ΘMM ′〈 1
r2
〉, (63)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4 369
S. Zaim
and
∆E
(2)
n,j = − e4
~c
4π∫
0
ΘdΩ
∞∫
0
drr−2[ψ†njlM (r, θ, ϕ) ψnj′l′M ′ (r, θ, ϕ)]
= − e4
~c
ΘMM ′〈 1
r3
〉, (64)
where
〈 1
r2
〉 = 2~c
(mec
~
a
)3
[
κ
(
2Eκ −mec
2
)
(mec2)2 αν (4ν2 − 1)
]
, (65)
〈 1
r3
〉 =
(mc
~
a
)3
[
3Eκ
(
Eκ −mec
2
)
(mec2)2 ν (4ν2 − 1) (ν2 − 1)
−
(
mec
2
)2 (
ν2 − 1
)
(mec2)2 ν (4ν2 − 1) (ν2 − 1)
]
. (66)
From Eq. (62) we obtain the modified energy levels in noncommutative space-
time to the first order of Θ as:
∆E
(Θ)
n,j =
( α
~c
)2 mec
2a3
ν (4ν2 − 1)
×
×
[
Eκ
((
2Eκ −mec
2
)
α2
− 3
(
Eκ −mec
2
)
(ν2 − 1)
)
+
(
mec
2
)2
]
ΘMM ′ . (67)
The selection rules for the transitions between the levels
(
NlMj → NlM
′
j
)
are
∆l = 1 and ∆M = 0,±1, where N = n + |κ| describes the principal quantum
number. The 2S1/2 and 2P1/2 levels correspond respectively to
(N = 1, j = 1/2,κ = ±1,M = ±1/2) . (68)
From (67) and (68) we can write
∆E
(Θ)
2S1/2
= 1.944 64× 10−8Θ (MeV)3 , (69)
∆E
(Θ)
2P1/2
= ±2.160 75× 10−9Θ (MeV)3 . (70)
The non-commutative correction to the transition follows as
∆E
(Θ)
2,1/2
(
2P1/2 → 2S1/2
)
= 2.160715× 10−8Θ (MeV)3 .
370 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4
Noncommutative Space-Time of the Relativistic Equations...
Now again using the current theoretical result on the 2P Lamb shift from
[39], which is about 0.08 kHz, and from the splitting (69), we get the bound
Θ . (0.25TeV)−2 . (71)
Restoring the constants c and ~ in (71), we write the bound on the non-commutativity
parameter as
Θ . 1.7× 10−35m2. (72)
It is interesting that the value of the upper bound on the time-space noncom-
mutativity parameter as derived here is better than the results of [12, 21, 36, 46].
This value is only in the sense of an upper bound and not the value of the pa-
rameter itself for which the fundamental length
√
Θ is compatible with the value
of the electroweak length scale (`ω). This effectively confirms the presence of
gravity at this level.
5. Conclusions
In this work we started from quantum relativistic charged scalar and fermion
particles in a canonical noncommutative space-time to find the action which is
invariant under the infinitesimal gauge transformations. By using the Seiberg–
Witten maps and the Moyal product, we derived the deformed KG and Dirac
equations for noncommutative Coulomb potential up to first order in the non-
commutativity parameter Θ. By solving the deformed KG and Dirac equations,
we found the Θ-correction energy shift. This proves that the non-commutativity
has an effect similar to that of the magnetic field. The corrections induced to the
energy levels by this noncommutative effect and the Lamb shift were induced and
compared with experimental results from high precision hydrogen spectroscopy to
obtain a new bound for the noncommutativity parameter of around (0.25TeV)−2,
for which the fundamental length
√
Θ is compatible with the value of the elec-
troweak length scale (`ω). This effectively confirms the presence of gravity at this
level.
References
[1] S. Doplicher, K. Fredenhagen, and J.E. Roberts, Phys. Lett. B 331 (1994), 39.
[2] S. Doplicher, K. Fredenhagen, and J.E. Roberts, Commun. Math. Phys. 172 (1995),
187.
[3] T. Yoneya, Progr. Theor. Phys. 103 (2000), 1081.
[4] N. Seiberg and E. Witten, JHEP 9909 (1999), 032.
[5] E. Bergshoeff, D.S. Berman, J.P. van der Schaar, and P. Sundell, Nucl. Phys. B
590 (2000), 173.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4 371
S. Zaim
[6] S. Kawamoto and N. Sasakura, JHEP 0007 (2000), 014.
[7] A. Das, J. Maharana, and A. Melikyan, JHEP 0104 (2001), 016.
[8] S, Cai, T. Jing, G. Guo, and R. Zhang, J. Theor. Phys. 49 (2010), 1699.
[9] S. Dulat and K. Li, Chin. Phys. C 32 (2008), 32.
[10] J. Gamboa, F. Mondez, M. Loewe, and J.C. Rojas, Mod. Phys. Lett. A 16 (2001),
2075.
[11] Pei-Ming Ho and Hsien-Chung Kao, Phys. Rev. Lett. 88 (2002), 151602.
[12] B. Muthukumar, Phys. Rev. D 71 (2005), 105007.
[13] K. Li and S. Dulat, Eur. Phys. J. C 46 (2006), 825.
[14] K. Li and N. Chamoun, Chin. Phys. Lett. 24 (2007), 1183.
[15] S. Dulat and K. Li, Mod. Phys. Lett. A 21 (2006), 2971.
[16] O.F. Dayi and B. Yapiskann, JHEP 10 (2002), 022.
[17] O. Bertolami and L. Guisado, JHEP 0312 (2003) 013.
[18] C.S. Chu, J. Lukierski, and W.J. Zakrzewski, Nucl. Phys. B 632 (2002), 219.
[19] D.A. Eliezer and R.P. Woodard, Nucl. Phys. 325 (1989), 389.
[20] T.C. Cheng, P.M. Ho, and M.C. Yeh, Nucl. Phys. B 625 (2002), 151.
[21] A. Saha, Time-Space Non-Commutativity in Gravitational Quantum Well Scenario.
— Eur. Phys. J. C 51 (2007), 199.
[22] P.A. Dirac, The Principles of Quantum Mechanics. Oxford University Press, USA,
1958.
[23] L.D. Landau and E.M. Lifshitz, Quantum Mechanics, Non-Relativistic Theory. Perg-
amon, Canada, 1977.
[24] W. Greiner, Quantum Mechanics—An Introduction. Berlin, Springer, 1989.
[25] A.D. Antia, A.N. Ikot, I.O. Akpan, and O.A. Awoga, Indian J. Phys. 87 (2013),
155.
[26] S. Zaim, L. Khodja, and Y. Delenda, IJMPA 23 (2011), 4133.
[27] L. Khodja and S. Zaim, IJMPA 27 (2012), No. 19, 1250100.
[28] S. Zaim, A. Boudine, N. Mebarki, and M. Moumni, Rom. J. Phys. 53 (2008), Nos.
3–4, p. 445–462.
[29] N. Mebarki, S. Zaim, L Khodja, and H Aissaoui, Phys. Scripta 78 (2008), 045101.
[30] V. Stern, Phys. Rev. Lett. 100 (2008), 061601.
[31] S. Zaim and Y. Delenda, J. Phys. Conf. Ser. 435 (2013), 012020.
[32] A.F. Nikiforov and V.B. Uvarov, Special Functions of Mathematical Physics. Basel,
Birkhauser, 1988.
372 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4
Noncommutative Space-Time of the Relativistic Equations...
[33] M.R. Setareand and O. Hatami, Commun. Theor. Phys. (Beijing) 51 (2009), 1000.
[34] S. Zaim, L. Khodja, and Y. Delenda, Int. J. Mod. Phys. A 26 (2011), 4133.
[35] C. Itzykson and J.-B. Zuber, Quantum Field Theory, Dover Publications, New York,
2005.
[36] M. Chaichian, M.M. Sheikh-Jabbari, and A. Tureanu, Phys. Rev. Lett. 86 (2001),
2716.
[37] J.L. Hewett, F.J. Petriello, and T.G. Rizzo, Phys. Rev. D 64 (2001), 075012.
[38] S.M. Carroll, J.A. Harvey, V.A. Kostelecky, C.D. Lane, and T. Okamoto, Phys.
Rev. Lett. 87 (2001), 141601.
[39] I. Mocioiu, M. Pospelov, and R. Roiban, Phys. Lett. B 489 (2000), 390.
[40] A. Akhiezer and V.B. Berestetskii, Quantum Electrodynamics. Interscience Pub-
lishers, New York, 1965.
[41] V.B. Berestetskii, E.M. Lifshitz, and L.P. Pitaevskii, Relativistic Quantum Theory.
Pergamon Press, Oxford, 1971.
[42] H.A. Bethe and E.E. Salpeter, Quantum Mechanics of One- and Two-Electron
Atoms. Springer–Verlag, Berlin, 1957; reprinted by Dover, New York, 2008.
[43] A. Messiah, Quantum Mechanics, Vol. 2, North-Holland, Amsterdam, 1961;
reprinted by Dover, New York, 1999.
[44] A.F. Nikiforov and V.B. Uvarov, Special Functions of Mathematical Physics,
Birkhäuser, Basel, Boston, 1988.
[45] S.K. Suslov, J. Phys. B 42 (2009), 185003.
[46] E. Akofor, A.P. Balachandran, A. Joseph, L. Pekowshy, B.A. Qureshi, Phys. Rev.
D 79 (2009), 063004.
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 4 373
|