When the interaction potential is suitably reordered, the Moyal field theory
admits two types of Galilean symmetries, namely the conventional
mass-parameter-centrally-extended one with commuting boosts, but also the
two-fold centrally extended ``exotic'' Galilean symmetry, where the commutator
of the boosts yields the noncommutative parameter. In the free case, one gets
an ``exotic'' two-parameter central extension of the Schroedinger group. The
conformal symmetry is, however, broken by the interaction.Comment: Corrected version. Further remarks and references added. LaTex, 8
  pages, no figure

Aref'eva

Ballesteros

Brihaye

Duval

Fairlie

Gomis

Grigore

Guralnik

Hagen

Jackiw

L. Martina

Langmann

Lifshitz

Lozano

Lukierski

Lévy-Leblond

Martinez

Nair

Niederer

P.A. Horváthy

P.C. Stichel

Szabo

Physics Letters B

English

arXiv

AbstractWhen the interaction potential is suitably reordered, the Moyal field theory admits two types of Galilean symmetries, namely the conventional mass-parameter-centrally-extended one with commuting boosts, but also the two-fold centrally extended “exotic” Galilean symmetry, where the commutator of the boosts yields the noncommutative parameter. In the free case, one gets an “exotic” two-parameter central extension of the Schrödinger group. The conformal symmetry is, however, broken by the interaction

Horváthy, P.A.

Martina, L.

Stichel, P.C.

Elsevier - Publisher Connector 

Galilean symmetry in noncommutative field theory 

Corrected version. Further remarks and references added. LaTex, 8 pages, no figuresWhen the interaction potential is suitably reordered, the Moyal field theory admits two types of Galilean symmetries, namely the conventional mass-parameter-centrally-extended one with commuting boosts, but also the two-fold centrally extended ``exotic\'\' Galilean symmetry, where the commutator of the boosts yields the noncommutative parameter. In the free case, one gets an ``exotic\'\' two-parameter central extension of the Schroedinger group. The conformal symmetry is, however, broken by the interaction

Horvathy, P. A.

Stichel, P. C.

HAL Université de Tours

Galilean symmetry in noncommutative field theory

Crossref

Physics Letters B 564 (2003) 149–154
www.elsevier.com/locate/npe
Galilean symmetry in noncommutative field theory
P.A. Horváthy a, L. Martina b, P.C. Stichel c
a Laboratoire de Mathématiques et de Physique Théorique, Université de Tours, Parc de Grandmont, F-37200 Tours, France
b Dipartimento di Fisica dell’Università and Sezione INFN di Lecce, Via Arnesano, C.P. 193, I-73100 Lecce, Italy
c An der Krebskuhle 21, D-33619 Bielefeld, Germany
Received 6 May 2003; accepted 12 May 2003
Editor: P.V. Landshoff
Abstract
When the interaction potential is suitably reordered, the Moyal field theory admits two types of Galilean symmetries, namely
the conventional mass-parameter-centrally-extended one with commuting boosts, but also the two-fold centrally extended
“exotic” Galilean symmetry, where the commutator of the boosts yields the noncommutative parameter. In the free case, one
gets an “exotic” two-parameter central extension of the Schrödinger group. The conformal symmetry is, however, broken by
the interaction.
 2003 Elsevier Science B.V. Open access under CC BY license.1. Introduction
In a recent paper [1] Bak et al. consider a scalar
field theory on the noncommutative plane, described
by the action S = S0 + Sint =
∫
d2 x dt L,
L= L0 − V 
(1.1)=
(
iψ¯∂tψ + ψ¯ ∆ψ2
)
− λ
2
ψ¯  ψ¯  ψ  ψ,
where the star means the Moyal product associated
with the noncommutative parameter θ . Although this
looks like a nonrelativistic theory, Bak et al. men-
tion (without proof) that both the Galilean and scale
invariance are lost. Our aim here is to point out
that the Galilean symmetry can be restored by suit-
ably reordering the interaction potential. Then we
find that the symmetry can be implemented in two
E-mail address: horvathy@univ-tours.fr (P.A. Horváthy).0370-2693 2003 Elsevier Science B.V.
doi:10.1016/S0370-2693(03)00704-4
Open access under CC BY license.different ways: while the conventional one yields
the usual, one-parameter central extension, another,
“Moyal-type” implementation yields the “exotic” two-
parameter centrally extended Galilean symmetry,
found before in a point particle context [2]. We con-
firm that any nontrivial interaction does indeed break
the scale invariance, but in the free case the symmetry
actually extends to a novel type of “exotic” (i.e., two-
parameter-centrally-extended) conformal (Schrödin-
ger) symmetry.
2. Exotic Galilean symmetry
Let us start with the boosts, whose infinitesimal ac-
tion on nonrelativistic space–time, δb x = bt , δbt = 0,
is conventionally implemented as
(2.1)δ0bψ = i b · xψ − t b · ∇ψ.
150 P.A. Horváthy et al. / Physics Letters B 564 (2003) 149–154This changes the free part of the Lagrange density
in (1.1) by a surface term, δL0 = −t b · ∇L0. In the
commutative case the general potential is V (ρ) with
ρ = ψ¯ψ . δ0ρ = −t b · ∇ρ, and hence the potential
changes in the same way as the free part, δ0V =
V ′δρ =−t b · ∇V. In conclusion,
(2.2)δ0bL=−t b · ∇L
implying the Galilean invariance of the action. In
the noncommutative case, however, the interaction
potential in (1.1) is equivalent rather to
(2.3)
V ∗ = (λ/2)ρrρl, ρr = ψ¯  ψ, ρl =ψ  ψ¯.
Then the relations
f  (xig)= xi(f  g)− iθ2 ij ∂j f  g,
(2.4)(xif )  g = xi(f  g)+ iθ2 ij f  ∂j g
readily inferred from the definition of the Moyal
product allow us to establish
(2.5)δ0bρa =±
θ
2
b× ∇ρa − t b · ∇ρa
with the plus sign for a = r and the minus for a = l.
Hence the potential changes as
(2.6)
δ0bV
∗ = θλb× ( ∇ρr  ρl − ρr  ∇ρl)− t b · ∇V ∗.
Owing to the sign change above, the first term here is
not a surface term. The invariance is therefore broken,
as stated by Bak et al. [1].
Now we argue that, in the Moyal context, (2.1)
is not the correct way to act for a boost. Remember
that the imaginary factor in front of ψ is in fact a[n
infinitesimal] “compensating gauge transformation”
which, in the present context, acts by the Moyal, rather
then by the ordinary multiplication, ψ → g  ψ , g ∈
U(1)∗ [3]. (2.1) should therefore be modified as
δbψ = i b · x  ψ − t b · ∇ψ
= i b · xψ − θ
2
b× ∇ψ − t b · ∇ψ,
(2.7)
δbψ¯ =−i b · ψ¯  x − t b · ∇ψ¯
=−i b · x ψ¯ − θ
2
b× ∇ψ¯ − t b · ∇ψ¯.The sign change in front of the first term here is
consistent with the formula f  g = g¯  f¯ .
Let us first investigate the free theory. The new
implementation (2.7) changes L0 again by a surface
term
(2.8)δbL0 =−t b · ∇L0 −
θ
2
b× ∇L0,
cf. (2.2), so that the free action S0 is left invariant.
Hence, the free theory admits our new type of Galilean
symmetry.
The new term in (2.7) contributes to the conserved
quantity associated through Noether’s theorem, which
says that if L changes as δL = ∂αKα under an
infinitesimal coordinate change δx, then
(2.9)
∫ (
δL
δ(∂tψ)
δψ + δψ¯ δL
δ(∂t ψ¯)
−Kt
)
d2 x
is a constant of the motion. For a boost, implemented
as in (2.7), we get
(2.10)Gi =−
∫
d2x xi |ψ|2 + tPi + θ2 ijPj ,
where Pi = −i
∫
d2x ψ¯∂iψ is the momentum. Our
clue is that the extra piece proportional to θ changes
the commutator of the boost components
(2.11){Gi ,Gj } = ij k, k ≡ θ
∫
d2x |ψ|2,
where the Poisson bracket is
(2.12){F ,G} = (−i)
∫
d2 x
(
δF
δψ
δG
δψ¯
− δG
δψ
δF
δψ¯
)
.
Adding the energy, the angular momentum, and the
mass,
(2.13)H0 =
∫
d2 x 1
2
| ∇ψ|2,
(2.14)J =−i
∫
d2 x ij xiψ¯∂jψ,
(2.15)M =
∫
d2 x ρr =
∫
d2x |ψ|2
(also derived by Noether’s theorem), we obtain the
“exotic” two-fold centrally extended Galilei alge-
bra [2], whose commutation relations only differ from
those of the usual, singly-extended Galilean algebra
in that the boosts yield the second central charge k
in (2.11). The usual central term is the mass M , as-
sociated with the phase invariance.
P.A. Horváthy et al. / Physics Letters B 564 (2003) 149–154 151It is worth noting that the conventional implemen-
tation (2.1) used by Hagen [4] yields instead (2.10)
without the extra piece, so that the boost components
commute.1
We conclude that, for a free particle, both (2.1) and
(2.7) act as symmetries.
Let us mention that the infinitesimal action associ-
ated with a conserved quantity G is
(2.16)δGψ = {ψ,G}.
Interestingly, choosing the equivalent expression
−iψ∂t ψ¯ in the free action, the phase invariance would
yield
∫
d2 x ρl instead of
∫
d2x ρr ; due to the integral
property
∫
d2 x f  g = ∫ d2 x fg, this is, however, the
same as our previous M . Note also that while the ρa
are not positive definite, their integral, M , is positive.
Both densities satisfy a continuity equation
∂tρa + ∇ · a = 0, r = ∇ψ  ψ¯ −ψ  ∇ψ¯,
(2.17)l = ψ¯  ∇ψ − ∇ψ¯  ψ.
This could be used as the starting point for a non-
commutative hydrodynamics [6]. Note that while the
coordinate-space densities have complicated commu-
tation relations, in momentum-space they satisfy the
trigonometric algebra [7]
(2.18){ρ˜a(q), ρ˜a( p)}=±2 sin(θ2 q × p
)
ρ˜a(q + p)
with the positive/negative sign for a = r and a = l,
respectively.
3. Exotic conformal symmetry
The 2-parameter conformal extension of the Galilei
group (called the Schrödinger group) [8] can now be
considered. The new generators are the dilations and
expansions, implemented infinitesimally according to
δ∆x =∆x, δ∆t = 2∆t,
δ0∆ψ =−∆
[
ψ + x · ∇ψ + 2t∂tψ
]
,
δκ x = κt x, δκ t = κt2,
(3.1)
δ0κψ =−κ
[(
− i
2
x2 + t
)
ψ + t x · ∇ψ + t2∂tψ
]
,
1 This corrects an error in calculating the commutator, commit-
ted in [5].respectively, where ∆ > 0 and κ is real. (∆x means
x multiplied by ∆ and ∆t means t multiplied by ∆.)
More generally, an infinitesimal Schrödinger transfor-
mation is of the form δxi = fi(x, t), δt = g(t) with
fi(x, t)= Fi(t)+xiG(t). When implemented conven-
tionally
δ0ψ(x, t)= ih(x)ψ(x, t)− fi(x, t)∂iψ(x, t)
(3.2)+ [k(t)− g(t)∂t )]ψ(x, t),
where the coefficients are suitable real functions, it
leaves invariant L0 and is therefore a symmetry for a
free field. The associated conserved quantities span,
w.r.t. the Poisson bracket (2.12) the one-parameter-
centrally-extended Schrödinger algebra [8].
With hindsight to the noncommutative case, let us
consider instead2
δψ(x, t)= ih(x)  ψ(x, t)− fi(x, t)∂iψ(x, t)
(3.3)+ [k(t)− g(t)∂t ]ψ(x, t).
The “exotic” dilatations act in the same way as the
conventional ones in (3.1); for the “exotic” expansions
we find
δ∗κψ =−κ
[(
− i
2
x2 + t
)
ψ + t x · ∇ψ + t2∂tψ
]
(3.4)− κ
[
θ
2
x × ∇ψ + θ
2
4
∂tψ
]
.
The new transformation is readily seen to be still a
symmetry that extends the “exotic” Galilei algebra by
adding the two conformal generators3
D =−2tH0 + 12i
∫
d2 x xi
(
ψ¯∂iψ − (∂i ψ¯)ψ
)
,
(3.5)
K= t2H0 + tD− 12
∫
d2 x x2|ψ|2 + θ
2
J − θ
2
4
H0.
Unlike in the commutative case, D,H0 and K do
not close to an o(2,1) [8] but yield
{D,H0} = 2H0 {D,K} = −2K+ θJ − θ2H0,
(3.6){H0,K} =D.
2 According to [9] one should take −(1/2)(fi  ∂iψ + ∂iψ  fi)
for the second term. But when fi is at most linear in x, this reduces
to our expression. For ψ¯ the first term becomes −iψ¯ h(x), cf. (2.7).
3 Note that (3.1) is also consistent with (2.16).
152 P.A. Horváthy et al. / Physics Letters B 564 (2003) 149–154When added to the Galilean generators we get a closed
algebra, though, since the nonvanishing brackets read
{K,Gi} = θijGi , {K,Pi} = Gi ,
(3.7){D,Gi} = −Gi + θijPj .
The relations (2.11), (3.6), (3.7) define a new, two-
parameter central extension of the Schrödinger algebra
which seems to have escaped attention so far. We call
it the “exotic Schrödinger algebra”.
4. Symmetry properties of the potential
We now turn our attention to the potential. Let us
first consider a boost, implemented “exotically” i.e.,
as in (2.7). It follows readily (cf. (2.5)) that the right
and left densities transform differently
δbρr =−t b · ∇ρr ,
(4.1)δbρl =−t b · ∇ρl − θ b× ∇ρl.
Hence, for V ∗ = ρrρl we get
(4.2)δbV  =−t b · ∇V ∗r − θ b× (ρr  ∇ρl).
Here the term proportional to θ is not exact. Therefore,
the Galilean invariance is broken by the potential V 
even for the new implementation (2.7).
Let us remember, however, how the potential comes
about [11]. One starts with the second-quantized
expression for the interaction
(4.3)
∫
d2x d2 x ′ ψ¯(x)ψ¯(x ′)U(x − x ′)ψ(x)ψ(x ′),
where U is a two-body potential. Choosing the contact
interaction U = (λ/2)δ(x − x ′) yields a quartic V =
(λ/2)ρ2.
Promoting the commutative theory into a noncom-
muting one requires to replacing the ordinary prod-
ucts by Moyal products. This requires particular care,
though. For example, putting naively Moyal stars be-
tween the various factors in (4.3) while keeping the
original order would lead to inconsistency: expres-
sions of the form ψ(x)  ψ(x ′) that should be defined
due to associativity, would require us to redefine the
Moyal product. Our clue is that this procedure is am-
biguous, as the order of the factors is irrelevant in thecommutative theory, but not in its Moyal version. Re-
arranging as, e.g.,
(4.4)
∫
d2 x d2x ′ ψ¯(x)  ψ(x)  U(x − x ′)  ψ¯(x ′)  ψ(x ′)
(where the various products are well-defined) would
yield, instead of V ∗ in (1.1),
(4.5)V˜ ∗ = λ
2
ψ¯  ψ  ψ¯  ψ
equivalent to (λ/2)ρ2r or to V˜ ∗ = (λ/2)ρ2l (which
could also be obtained by a suitable reordering).
Remarkably, it is this expression that had been used by
Lozano et al. in their noncommutative nonrelativistic
Chern–Simons vortex construction [12], and also by
Langmann et al. [13] in their recent exact scalar field
solution in a background magnetic field.
The important fact for us is that the new interaction
is Galilei invariant, since, by (4.1)
(4.6)δ∗bV˜ ∗ = −t b · ∇V˜ ∗.
Similarly, for V˜ ∗ we get δ∗b V˜ ∗ = −t b · ∇V˜ − θ b ×
∇V˜ ∗ which is again a surface term, so that the
Galilean symmetry is again restored. Clearly, the same
statement holds for any “pure” function of ρr or of ρl
alone. For a mixture, ρ = rρr + lρl where the i are
real coefficients, the cross term would break, however
the symmetry, whenever rl = 0.
We failed to find a natural way to reproduce the
potential V ∗ = ψ¯  ψ¯  ψ  ψ in (1.1), proposed by
Bak et al. [1].4 Another argument in favor of our
choice (4.5) is that it is, unlike that of Bak et al.,
V ∗, renormalizable as well as invariant w.r.t. deformed
U(1) transformations [14, p. 25]. It is, therefore, (4.5)
that we shall adopt in what follows.
Interestingly, the modified potential V˜ ∗ also allows
the conventional symmetry (2.1).5 (2.5) indeed implies
that
(4.7)δ0bρ2a =±
θ
2
b× ∇ρ2a − t b · ∇ρ2a ,
a = r, l. Finally, the conventional interaction in terms
of ρ is also δ∗-invariant, proving the Galilean symme-
try also in this case. In conclusion, any of the “pure”
4 Their potential could be obtained by inserting further δ factors.
5 But this is not a surprise as (2.1) and (2.7) just differ by a space-
translation term.
P.A. Horváthy et al. / Physics Letters B 564 (2003) 149–154 153expressions V ∗(ρa) as well as the standard potential
V = ρ2 provide us with a theory which is Galilei-
invariant in two ways: the conventional implemen-
tation yields the usual one-parameter extension with
commuting boosts, and the “star-implementation”
yields the exotic two-parameter extension with non-
commuting boosts.
Let us the record for completeness the equation of
motion associated with our potential (4.5): either by
variation or using the Hamiltonian structure, we get
the “Moyalized” nonlinear Schrödinger equation
(4.8)i∂tψ =−∆2 ψ + λρl  ψ =−
∆
2
ψ + λψ  ρr .
For comparison, for the choice V ∗ of Bak et al.,
the nonlinear term is (λ/2)(ρr  ψ + ψ  ρl). The
commutative counterpart of (4.8) is known to be
non-integrable; coupling our system to an external
magnetic field, yields exact solutions [13], however.
The “ordinary” conformal symmetry is also con-
sistent with adding a quartic potential V (ψ) = ρ2. In
fact, using (3.1) we find that δ0∆ρ =∆(2ρ − x · ∇ρ −
2t∂t ρ) and δ0κρ = κ(2ρ − x · ∇(tρ)− t2∂tρ), so that
δ0∆ρ
2 =−∇ · (∆xρ2)− ∂t(∆2tρ2) for a dilatation,
(4.9)
δ0κρ
2 =−∇ · (κt xρ2)−∂t (κt2ρ2) for an expansion.
The associated conserved quantities, that still form
a representation of the one-parameter centrally ex-
tended Schrödinger algebra, only differ from the free
expressions in that H0 is replaced by H = H0 +
V [10].
As the “exotic” action of a dilatation is the same as
the conventional one in (3.1) we find, using (2.4),
δ∗∆ρr = δ0∆ρr =∆{2ρr − x · ∇ρr − 2t∂t ρr }
(4.10)−∆iθij ∂iψ¯  ∂jψ
which differs from δ∗∆ρ in the second term behind θ .
Thus, owing precisely to this term, ρ2r cannot change
by a surface term. The scaling symmetry is therefore
broken, as stated in [1]. (The same statement holds
also for ρ2l and ρrρl .) More generally, (3.3) readily
implies that
δρr = (2k − g∂t )ρr − fi∂iρr
− [(fi∂iψ¯)  ψ + ψ¯  (fi∂iψ)],δρl = (2k− g∂t )ρl − fi∂iρl
(4.11)− [(fi∂iψ)  ψ¯ +ψ  (fi∂i ψ¯)].
As the second brackets involve here expressions of the
form
xi∂iρ ± iθij ∂i ψ¯∂jψ +O
(
θ2
)
the additional terms are not given by a surface term.
that never vanish unless G(t) = 0. In conclusion, any
potential made of products of ρr and ρl necessarily
breaks the conformal invariance.
5. Discussion
An interesting feature of the model studied here
is the two-way Galilean symmetry, and one can be
puzzled how this can happen. Let us consider the
“Moyalized” counterpart of L0, namely
(5.1)L0 = iψ¯  ∂tψ −
1
2
∇ψ  ∇ψ¯.
Now the conventional implementation (2.1) of the
boosts is natural for L0, as is (2.7) for L0. But the
integral property
∫
f  g = ∫ fg implies that L0 and
L0 are equivalent, so one can use either of them to
describe a free field. This resolves the paradox which
says that noncommutativity does not alter the free
theory.
Acknowledgements
We are indebted to F. Schaposnik for enlightening
discussions and to R.J. Szabo for correspondence. One
of us (P.H.) would like to thank for hospitality the
University of Lecce.
References
[1] D. Bak, S.K. Kim, K.-S. Soh, J.H. Yee, Phys. Rev. Lett. 85
(2000) 3087;
Analogous expressions were considered before, e.g., by
J. Gomis, K. Landsteiner, E. Lopez, Phys. Rev. D 62 (2000)
105006;
G.S. Lozano, E.F. Moreno, F.A. Schaposnik, J. High Energy
Phys. 02 (2001) 036;
V.P. Nair, A. Polychronakos, Phys. Lett. B 505 (2001) 267.
154 P.A. Horváthy et al. / Physics Letters B 564 (2003) 149–154[2] J.-M. Lévy-Leblond, in: Loebl Group Theory and Applica-
tions, Vol. II, Academic Press, New York, 1972, p. 222;
A. Ballesteros, N. Gadella, M. del Olmo, J. Math. Phys. 33
(1992) 3379;
Y. Brihaye, C. Gonera, S. Giller, P. Kosin´ski, hep-th/9503046;
D.R. Grigore, J. Math. Phys. 37 (1996) 240;
D.R. Grigore, J. Math. Phys. 37 (1996) 460;
For new developments, see: J. Lukierski, P.C. Stichel, W.J.
Zakrzewski, Ann. Phys. (N.Y.) 260 (1997) 224;
J. Lukierski, P.C. Stichel, W.J. Zakrzewski, hep-th/0207149,
Ann. Phys. (N.Y.), in press;
C. Duval, P.A. Horváthy, Phys. Lett. B 479 (2000) 284.
[3] R.J. Szabo, Phys. Rep. 378 (2003) 203, hep-th/0109162.
[4] C.R. Hagen, Phys. Lett. B 539 (2002) 168.
[5] C. Duval, P.A. Horváthy, Phys. Lett. B 547 (2002) 306.
[6] C. Duval, Z. Horváth, P.A. Horváthy, Int. J. Mod. Phys. B 15
(2001) 3397;
Z. Guralnik, R. Jackiw, S.Y. Pi, A.P. Polychronakos, Phys. Lett.
B 517 (2001) 450.[7] D.B. Fairlie, P. Fletcher, C.K. Zachos, Phys. Lett. B 218 (1989)
203;
D.B. Fairlie, C.K. Zachos, Phys. Lett. B 224 (1989) 101;
J. Martinez, M. Stone, Int. J. Mod. Phys. B 26 (1993) 4389.
[8] R. Jackiw, Phys. Today 25 (1980) 23;
U. Niederer, Helv. Phys. Acta 45 (1972) 802;
C.R. Hagen, Phys. Rev. D 5 (1972) 377.
[9] R. Jackiw, S.-Y. Pi, Phys. Rev. Lett. 88 (2002) 111603.
[10] R. Jackiw, in: A. Ali, P. Hoodboy (Eds.), Bég Memorial
Volume, World Scientific, Singapore, 1991.
[11] E.M. Lifshitz, L.P. Pitaevskii, Statistical Physics. Part 2,
in: Landau–Lifshitz Course of Theoretical Physics, Vol. 9,
Pergamon, Oxford, 1980, Chapter II, Section 7.
[12] See Lozano, Moreno and Schaposnik in Ref. [1]
[13] E. Langmann, R.J. Szabo, Phys. Lett. B 533 (2002) 168;
E. Langmann, Nucl. Phys. B 654 (2003) 404;
E. Langmann, R.J. Szabo, K. Zarembo, hep-th/0303082.
[14] I.Ya. Aref’eva, D.M. Belov, A.A. Giryavets, A.S. Khoshelev,
P.B. Medvedev, hep-th/0111208.


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