Lowest Landau Level of Relativistic Field Theories in a Strong
  Background Field
Lowest Landau Level of Relativistic Field Theories in a
Strong Background Field
Xavier Calmet1 a and Martin Kober2
1 Université Libre de Bruxelles, Service de Physique Théorique, CP225, Boulevard du Triomphe (Campus plaine),
B-1050 Brussels, Belgium.
2 Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität, Max-von-Laue-Straße 1, D-60438 Frankfurt
am Main, Germany.
Abstract. We consider gauge theories in a strong external magnetic like field. This situation can
appear either in conventional four-dimensional theories, but also naturally in extra-dimensional the-
ories and especially in brane world models. We show that in the lowest Landau level approximation,
some of the coordinates become non-commutative. We find physical reasons to formal problems
with non-commutative gauge theories such as the issue with SU(N) gauge symmetries. Our con-
struction is applied to a minimal extension of the standard model. It is shown that the Higgs sector
might be non-commutative whereas the remaining sectors of the standard model remain commuta-
tive. Signatures of this model at the LHC are discussed. We then discuss an application to a dark
matter sector coupled to the Higgs sector of the standard model and show that here again, dark
matter could be non-commutative, the standard model fields remaining commutative.
PACS. 11.10.Nx – 12.60.Fr
Gauge theories formulated on non-commutative
spaces have received a lot of attention over the last
decade. The main reason is that they were discov-
ered to appear in a certain limit of string theory[1,2,
3]. Non-commuting coordinates do appear generically
whenever one studies a physical system in an exter-
nal background field in the first Landau level approx-
imation. This phenomenon was discovered by Landau
in 1930 [4]. A textbook example is an electron in a
strong magnetic field. In the framework of string the-
ory, the effective low energy four dimensional theory
describing strings ending on a brane in the presence
of a strong external background field is shown to be
non-commutative[1,2,3].
It is notoriously difficult to construct a
non-commutative version of the standard model. There
are different approaches in the literature [5,6]. The
main difficulty is to obtain the right gauge symme-
tries i.e. SU(N) groups necessary to describe the stan-
dard model. The issue here is that the commutator of
two non-commutative Lie algebra valued gauge trans-
formations is not a gauge transformation unless one
chooses U(N) Yang-Mills symmetries and the funda-
mental, anti-fundamental or adjoint representations.
This no-go theorem can be avoided if one considers
the enveloping algebra. However this may not seem
very natural. This is the motivation for the present
work. We shall study relativistic field theories in a
strong external potential to identify the physical rea-
a Email: xavier.calmet@ulb.ac.be
son for this technical problem. We shall start from clas-
sical gauge theories formulated on a regular space-time
which are perfectly well behaved and renormalizable
theories and consider them in a strong external field,
then we shall consider their first Landau Level. Differ-
ent field theories have been considered in the lowest
Landau level approximation leading to more or less
exotic non-commutative gauge theories, see e.g. [7] or
[8]. However, we wish to consider physical situations
which lead to the kind of non-commutative gauge the-
ories found in [1,2,3] to understand the physical reason
for some of the pathologies of these theories. Further-
more our construction allows us to consider models
where only certain sectors of the theory are noncom-
mutative.
Let us first consider a charged scalar field in a
strong magnetic field. The action is given by
(D̄µφ)
∗(D̄µφ) − V (φ∗φ) (1)
F̄µν F̄
where D̄µ = ∂µ + iqAµ and F̄µν = −i[D̄µ, D̄ν]. This
theory is gauge invariant under U(1) gauge transfor-
mations and is renormalizable. Let us now study this
theory in the limit of a strong external magnetic field.
We consider quantum fluctuations Aµ aroundCµ which
is the background field which corresponds to the con-
stant magnetic field. We then have D̄µ = ∂µ + iqAµ +
http://arxiv.org/abs/0704.1355v2
Alternatives Parallel
iqCµ = Dµ + iqCµ and the action becomes
(Dµφ)
∗(Dµφ)− V (φ∗φ) (2)
−iqφ∗CµDµφ+ iq(Dµφ)∗Cµφ+ q2φ∗CµCµφ
µν − 1
µν − 1
where Cµν = ∂µCν − ∂νCµ. To be more specific we
shall pick Cµ = (0,
, 0) which leads to a con-
stant magnetic field of magnitude B in the z-direction.
Note that a strong external magnetic like field does not
imply that the quantum fluctuation is strongly cou-
pled to the scalar field. The action (2) has a remaining
gauge invariance: δφ = iαφ, δAµ = ∂µα and the back-
ground field is kept invariant. The classical canonical
momenta of the center of mass of particle φ is given
by πµ = pµ + qAµ + qCµ.
The first quantization of the classical Hamiltonian
implies that the coordinates and the spatial compo-
nents of the canonical momentum do not commute:
[xi, πj ] = ih̄δij . (3)
We now express the canonical momentum in terms of
the kinematical one and find [xi, pj+qAj+qBǫjkxk] =
ih̄δij for i, j ∈ {1, 2}. Let us now consider the limit√
B ≫ m and |Cµ| ≫ |Aµ|. In this limit the terms
involving the kinematical momentum pj and the po-
tential Aj can be neglected:
[xi, xj ] = ih̄
ǫij ≡ iθij . (4)
This means that the scalar field is non-commutative
in the x − y plane. It should be noted that our re-
sult is not a gauge artifact, the very same result would
be obtained if we had chosen e.g. the Landau gauge
Cµ = (0, By, 0, 0). Furthermore, it is easy to see that
since Lorentz covariance is explicitly broken by the
background field new Lorentz violating vertices involv-
ing the gauge boson Aµ will be generated through
its interaction with the background field. In particu-
lar three gauge bosons and four gauge bosons vertices
which are typical of non-commutative gauge theories
are generated. We have just shown that in the limit
B, the coordinates x and y of the scalar field
do not commute, let us rename them x̂ and ŷ. In the
limit m ≪
B, local gauge transformations of the
scalar field involve non-commuting coordinates: δαφ =
iα(t, x̂, ŷ, z)φ(t, x̂, ŷ, z), in order to build a gauge in-
variant action, the gauge boson has to transform ac-
cording to δαAµ(x̂) = ∂µα(x̂)+i[α(x̂), Aµ(x̂)]. The low
energy action is then given by
(Dµφ(x̂))
∗(Dµφ(x̂))− V (φ(x̂)∗φ(x̂))(5)
Fµν (x̂)F
µν(x̂)
with Fµν = −i[Dµ(x̂), Dν(x̂)]. Using the Weyl quanti-
zation procedure, it is easy to replace the
non-commuting coordinates in the argument of the
field φ by commuting ones
(Dµφ)
∗ ⋆ (Dµφ)− V (φ∗ ⋆ φ) (6)
Fµν ⋆ F
where the star product is given by f ⋆ g = fei∂iθ
with θij = h̄
ǫij for i, j ∈ {1, 2} and θµν = 0 in the
time and z-directions.
It should be noted to our derivation that it is not
specific to a scalar field theory since the important
point comes from the equations of motion which are
the Klein-Gordon equations. Since every component of
a spinor field satisfies the Klein-Gordon equations, our
result applies to spinor field as well. Our first result is
that the action (2) is very identical to a U(1) non-
commutative gauge theory with a non-commutativity
in the x − y plane. We find that a non-commutative
gauge theory is very closely related to a commutative
gauge theory in a strong external field in the limit
that the mass of the particle is small compared to the
external background field. It is well known that the ac-
tion we started from is well behaved at the quantum
level and in particular that it is renormalizable. On
the other hand the non-commutative action (6) is not
renormalizable and suffers from UV/IR mixing. This
is a strong hint that the issues with the quantum field
calculations involving the action (6) should disappear
in the limit where more and more Landau levels are
included in the calculations. However, we should point
out that the naive limit B → 0 which would corre-
spond to a vanishing external field implies an infinite
non-commutative parameter. The limits B → 0 and√
B ≫ m do not commute. This is clearly another kind
of UV/IR mixing and probably the origin of UV/IR
mixing in the quantized version of the theory.
We can now push our analysis further and consider
Yang-Mills theories instead of a simple U(1) theory.
To be very concrete let us consider a SU(2) Yang-
Mills theory. In that case there are three gauge po-
tentials B1µ, B
µ and B
µ. We see that the same pro-
cedure as the one outlined in this work leads to two
canonical momenta, one for each of the components
of the doublet φ = (φ1, φ2). To be very precise let us
consider the canonical momentum πi
of the particle
described by the field φ1 and π
which corresponds
to the particle φ2. It is clear that it only depends on
B3µ since the generator T
3 is the only diagonal one.
However T 1 and T 2 are not diagonal and thus B1µ
and B2µ do not contribute to the canonical momenta.
One finds πi
+ gB3i + gD3i, where B3i is the
fluctuation around the strong external field D3i and
−gB3i−gD3i. Let us now assume that the non-
vanishing components of the strong external field D3i
are given by Eǫijx
j , we find [xi, pj+gB
j +gEǫjkx
ih̄δij and [xi, pj − gB3j − gEǫjkxk] = ih̄δij for i, j ∈
{1, 2} let us now consider the limit
E ≫ m and
X. Calmet and M. Kober Lowest Landau Level of Relativistic Field Theories in a Strong Background Field
|D3j | ≫ |B3j | one finds [xi, xj ] = ih̄ 1gE ǫ
ij and simul-
taneously [xi, xj ] = −ih̄ 1
ǫij which is clearly incon-
sistent. In other words, there is no non-commutative
SU(2) theory equivalent to a SU(2) gauge theory re-
stricted to its first Landau Level. However if we had
started from a U(N) gauge group, one of the generators
would be proportional to the identity matrix and we
could have chosen the strong external field in the direc-
tion of the identity matrix and obtained a consistent
non-commutative algebra. In that case the first Lan-
dau level of a gauge theory can be described in terms
of a dual non-commutative gauge theory as long as the
external strong field is chosen in the direction of the
identity matrix. This is the physical origin of the for-
mal problem with SU(N) gauge invariance mentioned
at the beginning of this work.
Furthermore, it should be stressed that the commu-
tative action (2) is not gauge invariant under regular
gauge transformations for U(N) (N>1) gauge groups
unless the background field transforms as well: δAµ =
∂µα + i[α,Aµ] and δCµ = i[α,Cµ]. Note that we are
using a different convention than in the background
quantization technique where the background field trans-
forms as a gauge field whereas the quantum fluctuation
transforms homogeneously [9]. The subtlety only ap-
pears for N>1. This suggests a generalization of non-
commutative gauge transformations to
δAµ = ∂µα+ iα ⋆ Aµ − iAµ ⋆ α (7)
δCµ = iα ⋆ Cµ − iCµ ⋆ α, (8)
where we set g = 1. It is also suggestive that the
background field which is closely related to the non-
commutative parameter through an equation such as
eq. (4) should be introduced in the action:
(Dµφ)
† ⋆ (Dµφ)− V (φ† ⋆ φ) (9)
−iφ† ⋆ Cµ ⋆ Dµφ+ i(Dµφ)† ⋆ Cµ ⋆ φ
+φ† ⋆ Cµ ⋆ C
µφ− 1
Fµν ⋆ F
µν − 1
Cµν ⋆ C
Fµν ⋆ C
µν − 1
Cµν ⋆ F
In the sequel we shall however restrict our consider-
ations to U(1) gauge theories where this subtlety is
irrelevant.
Let us now apply this idea to physics beyond the
standard model. If the U(1) external field we are con-
sidering couples only to one specie of particle we would
have a reason to explain why only a certain sector of
the model is non-commutative. It is tempting to iden-
tify the scalar field we have introduced with the Higgs
field of the standard model. However, the Higgs field
of the standard model is charged under SU(2) × U(1)
and this would lead to a SU(2) non-commutative the-
ory which is as explained previously not consistent for
fields which are Lie algebra valued. Furthermore, it is
not possible to gauge the standard model Higgs dou-
blet under a new U(1) without affecting its charge as-
signment under the standard model gauge group. How-
ever, there has been a growing interest [10,11,12,13,
14,15,16] for particles which are not charged under the
gauge group of the standard model or almost decou-
pling from the action of the standard model. Further-
more, scalar singlets are interesting dark matter candi-
dates [17] and could explain why the Higgs boson of the
standard model has not yet been discovered [18]. Let us
consider the coupling of the action (1) to the standard
model and we assume that φ is a SU(3)×SU(2)×U(1)Y
singlet, but that it is charged under a new U(1)E gauge
group under which standard model particles are sin-
glets. We shall call this new particle the e-photon. Let
us assume that the e-photon has a vacuum expecta-
tion which fills the universe which will single out a
preferred direction in space-time. There are different
model building options which will affect the precise
form of the non-commutative tensor θµν . For exam-
ple, the e-photon and φ could for example be living in
extra-dimensions and the standard model confined to
a brane in which case the non-commutativity could be
in three dimensions. If the new degrees of freedom are
confined to live in four dimensions then we would have
non-commutativity in only two-dimension in the plane
perpendicular to the direction of the external strong
field.
Let us consider the scalar sector of the theory. We
(D̄µφ)
∗(D̄µφ)−m2φφ∗φ (10)
−λφ(φ∗φ)2 + (DµH)†(DµH)
−m2HH†H − λH(H†H)2 + λφ∗φH†H
where H is the Higgs doublet of the standard model
and φ is the new scalar singlet charged under the new
U(1)E interaction. The SU(2)× U(1) symmetry of the
standard model has to be spontaneously broken, i.e.
the doublet acquires a vacuum expectation value and
using the unitary gauge, one has H = (0, h+ v) where
v2 = −m2H/(2λH). However, we have two options for
the U(1)E gauge symmetry. Let us first consider the
case where the extra U(1) is not spontaneously bro-
ken, in other words φ does not acquire a vacuum ex-
pectation value. In that case the scalar potential is
given by V [h, φφ∗] = −2m2Hh2 + λH(h + v)4 + λ(h +
v)2φφ∗+m2φφφ
∗+λφφφ
∗φφ∗. It is easy to show that the
e-photon and the usual photon do not mix. Further-
more there is no coupling between the e-photon and
the fermions of the standard model. This new long
range force is thus not in conflict with experiments.
The new charged scalars are protected by the exact
U(1)E symmetry and thus dark matter candidates.
Furthermore, although the carrier of the new force in
the dark matter sector are massless, the bounds on a
fifth force in the dark matter sector [19] do not ap-
ply to our model because the e-photon does not cou-
ple to regular matter. Let us now assume that the e-
photon has some vacuum expectation value such that
it correspond to a strong magnetic-type field in the
z-direction and consider this model in the first Lan-
dau Level approximation. We find that the scalars φ
have non-commuting coordinates in that limit. The
Alternatives Parallel
non-commuting coordinates can be removed at the
expense of introducing a star product V [h, φ ⋆ φ∗] =
−2m2Hh2 +λH(h+ v)4 +λ(h+ v)2φ ⋆ φ∗ +m2φφ ⋆φ∗ +
λφφ⋆φ
∗⋆φ⋆φ∗, i.e. the only non-commutative interac-
tion are those which involve the field φ and obviously
the e-photon. If its mass is low enough, this dark mat-
ter candidate could be produced at the LHC through
the decomposition of a Higgs boson. The decay rate
Γ (Higgs → φφ∗) = 1
is basically
commutative. However, the self-interaction of the φ-
mesons and of the e-photon are non-commutative and
the non-commutative nature of this sector could be
checked by searching for the usual characteristic non-
commutative self-interactions of the e-photon. The cross
section for the dark matter candidate at the Tevatron
and LHC corresponds to the one of a singlet added
to the standard model in the commutative case and
there should thus be a clear signal. However, detecting
the non-commutative nature of the dark matter sector
at a hadron collider will be a difficult task since one
would have to search for the typical self-interactions
of the e-photon. However, one would expect that the
background field will impact the distribution of dark
matter in our universe which would allow to identify
a preferred direction in space-time.
Another option is to assume that the remaining
U(1)E is spontaneously broken by a vacuum expecta-
tion value of the φ-mesons in which case the e-photon
acquires a mass. In that scenario, the φ-mesons are not
dark matter candidates. However, the residual degree
of freedom after U(1)E symmetry breaking, which we
call σ, will mix with the standard model Higgs bo-
son. One finds: hphys = cosα h + sinα σ, σphys =
cosα σ − sinα h where α, the mixing angle, is de-
termined by the scalar potential. When we consider
the model in a strong external potential correspond-
ing to a strong magnetic-like field in the z-direction
and in the lowest Landau level limit, we find that
both scalar fields are non-commuting in the x − y
plane whereas the remaining fields of the standard
model are commutative. In that case the only sector
of the theory which would exhibit a non-commutative
nature is the scalar potential sector. The new non-
commutative operators are vhh ⋆ φ ⋆ φ, vφh ⋆ h ⋆ φ and
h ⋆ h ⋆ φ ⋆ φ. Because of the trace property of the star
product (
d4xf ⋆g =
d4xg⋆f =
d4xfg), the inter-
actions of the two scalar degrees of freedom with the
fermions and gauge bosons of the standard model are
commutative.
Conclusions: We have shown that the phenomenon
discovered by Landau in 1930 appears in relativistic
field theories. We find physical reasons to the formal
problems with non-commutative gauge theories such
as the issue with SU(N) gauge symmetries. We apply
our construction to a minimal extension of the stan-
dard model and show that the Higgs sector might be
non-commutative whereas the remaining sectors of the
standard model remain commutative. We discuss the
signatures of this model at the LHC. We then dis-
cuss an application to a dark matter sector coupled to
the Higgs sector of the standard model and show that
here again, dark matter could be non-commutative,
the standard model fields remaining commutative.
Acknowledgments: This work was supported in part
by the IISN and the Belgian science policy office (IAP
V/27).
References
1. A. Connes, M. R. Douglas and A. S. Schwarz, JHEP
9802, 003 (1998).
2. N. Seiberg and E. Witten, JHEP 9909, 032 (1999).
3. V. Schomerus, JHEP 9906, 030 (1999).
4. L. Landau, Z. Phys. 64, 629 (1930).
5. X. Calmet, B. Jurco, P. Schupp, J. Wess and
M. Wohlgenannt, Eur. Phys. J. C 23, 363 (2002).
6. M. Chaichian, A. Kobakhidze and A. Tureanu, Eur.
Phys. J. C 47, 241 (2006).
7. R. Jackiw, Nucl. Phys. Proc. Suppl. 108, 30 (2002)
[Phys. Part. Nucl. 33, S6 (2002 LNPHA,616,294-
304.2003)].
8. P. A. Horvathy, Annals Phys. 299, 128 (2002).
9. S. Weinberg, “The quantum theory of fields. Vol.
2: Modern applications,” Cambridge, UK: Univ. Pr.
(1996) 489 p.
10. A. Hill and J. J. van der Bij, Phys. Rev. D 36, 3463
(1987).
11. J. J. van der Bij, Phys. Lett. B 636, 56 (2006).
12. X. Calmet, Eur. Phys. J. C 28, 451 (2003).
13. X. Calmet, Eur. Phys. J. C 32, 121 (2003).
14. X. Calmet and J. F. Oliver, Europhys. Lett. 77, 51002
(2007) [arXiv:hep-ph/0606209].
15. B. Patt and F. Wilczek, arXiv:hep-ph/0605188.
16. D. O’Connell, M. J. Ramsey-Musolf and M. B. Wise,
Phys. Rev. D 75, 037701 (2007).
17. J. McDonald, Phys. Rev. D 50, 3637 (1994).
18. J. J. van der Bij and S. Dilcher, Phys. Lett. B 638,
234 (2006).
19. M. Kesden and M. Kamionkowski, Phys. Rev. Lett.
97, 131303 (2006); Phys. Rev. D 74, 083007 (2006).
http://arxiv.org/abs/hep-ph/0606209
http://arxiv.org/abs/hep-ph/0605188
