Physics Letters B 638 (2006) 292–295

www.elsevier.com/locate/physletb

Schwinger model in noncommutating space–time

Anirban Saha a, Anisur Rahaman b, Pradip Mukherjee a,∗,1

a Department of Physics, Presidency College, 86/1 College Street, Kolkata 700 073, Indiab Department of Physics, Durgapur Government College, Durgapur 713 214, Burdwan, West Bengal, India

Received 8 March 2006; received in revised form 15 May 2006; accepted 16 May 2006

Available online 2 June 2006

Editor: T. Yanagida

Abstract

The (1 + 1)-dimensional bosonized Schwinger model has been studied in a noncommutative scenario. The theory in the reduced phase spaceexhibits a massive boson interacting with a background. The emergence of this background interaction is a novel feature due to noncommutativity.The structure of the theory ensures unitarity and causality.© 2006 Elsevier B.V. All rights reserved.

PACS: 11.15.-q; 11.10.Nx

Keywords: Schwinger model; Hamiltonian analysis; Noncommutativity

The idea of fuzzy space–time where the coordinates xμ sat-isfy the noncommutative (NC) algebra

(1)[xμ, xν

] = iθμν

with constant anti-symmetric θμν , was mooted long ago [1].This idea has been revived in the recent past and field theoriesdefined over this NC space has been studied extensively [2].A contentious issue in this scenario is the aspect of noncommu-tativity in the time–space sector. It was argued that introductionof space–time noncommutativity spoils unitarity [3,4] or evencausality [5]. However, much attention has been devoted in re-cent times to circumvent these difficulties in formulating theo-ries with θ0i �= 0 [6–8]. The (1 + 1)-dimensional field theoreticmodels are particularly important in this context because anynoncommutative extension of such models essentially containsfuzziness in the time–space sector. The two-dimensional fieldtheories have long been recognized as the laboratory where im-

* Corresponding author.E-mail addresses: ani_saha09@dataone.in (A. Saha),

anisur.rahman@saha.ac.in (A. Rahaman), pradip@bose.res.in (P. Mukherjee).1 Also Visiting Associate, S.N. Bose National Centre for Basic Sciences, JD

Block, Sector III, Salt Lake City, Calcutta 700 098, India and IUCAA, Post Bag4, Pune University Campus, Ganeshkhind, Pune 411 007, India.

0370-2693/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2006.05.049

portant ideas can be tested in a simple setting. However, notmuch analysis of the corresponding NC theories is available inthe literature. In the present Letter we will consider the (1 + 1)-dimensional Schwinger model [9] on a NC setting. Apart fromthe specific NC aspect such studies are also motivated by theinherent interest of the model as (1 + 1)-dimensional electro-dynamics.

Historically, the first two-dimensional model was proposedby Thirring [10] describing a pure fermionic current–currentinteraction. The interest increased a huge when Schwinger wasable to obtain an exact solution of two-dimensional electrody-namics with massless spinor [9]. The Schwinger model i.e., thetheory of mass less fermion interacting with an Abelian gaugefield in (1 + 1)-dimensional space–time is an exactly solvablefield theoretical model. It has been extensively studied over theyears [11–26] mainly due to the emergence of phenomena suchas mass generation and confinement of fermions (quarks). Thegauge field acquires a mass via a kind of dynamical symmetrybreaking and the fermions disappear from the physical spectra.

In (1 + 1) dimensions an exact mapping can be establishedbetween the bosonic and fermionic theories. The singularitiesof the Schwinger model can be accommodated by regularizingthe fermionic current. An equivalent approach is to obtain theeffective action by integrating out the fermions. Commonly it is

A. Saha et al. / Physics Letters B 638 (2006) 292–295 293

known as bosonization of the Schwinger model. For extensionto NC scenario the bosonised version is more economical. Sowe will follow this approach in the present Letter.

The Schwinger model is defined by the Lagrangian density

(2)LF = ψ(i/∂ − e/A)ψ − 1

4FμνFμν,

where the Lorentz indices run over the two values 0,1 and therest of the notation is standard. Notice that the coupling con-stant e has unit mass dimension in this situation. Bosonizing thefermion field we get the Lagrangian density involving a scalarfield φ instead of the Dirac field ψ :

(3)LB = 1

2∂μφ∂μφ + e

2εμνF

μνφ + ae2AμAμ + α

4FμνF

μν.

The first piece is the kinetic energy term for the scalar fieldand the second one describes the interaction between the matterfield and gauge field. The last two terms involve something new,viz., two undetermined parameters a and α.2 These are falloutsof the regularization process. To be more specific, if the left-handed and the right-handed component of ψ are integrated outone by one the regularization of the determinant contains suchparameters [16,19,22]. Setting a to be zero and α to be −1 weget the bosonized version of the usual gauge invariant vectorSchwinger model. We will use this version for the NC extensionin the sequel.

The NC action which we consider is∫d2xLNCBS =

∫d2x

[1

2(Dμ φ) (Dμ φ)

(4)+ 1

2eεμνφ Fμν − 1

4Fμν F μν

],

where φ is the NC scalar field and Aμ is the -gauge field. Weadopt the Minkowski metric ημν = diag(+,−). The covariantderivative Dμ φ is defined as

(5)Dμ φ = ∂μφ − i[Aμ,φ].Here denotes that the ordinary multiplication is replaced bythe star multiplication defined by

(6)φ(x) ψ(x) = (φ ψ)(x) = ei2 θαβ∂α∂ ′

β φ(x)ψ(x′)∣∣x′=x

.

The action (4) is invariant under the -gauge transformation

(7)δλAμ = Dμ λ, δ

λφ = −i[φ, λ].

The physics behind the NC theory (4) can be explored byseveral approaches which sometimes compliment each other[27]. Thus one can think of the fields as operators carryingthe realization of the basic algebra (1) or a conventional phasespace may be used where the ordinary product is deformed.A particularly interesting scenario appears in case of the gaugetheories where one can use the Seiberg–Witten (SW) type trans-formations [28–30] to construct commutative equivalent mod-els [31–34] of the actual NC theories in a perturbative frame-work. In the present Letter we adopt this approach. Note that

2 Note that the usual kinetic energy term is absorbed within α.

even if the fields and the coordinates in the commutative equiv-alent model are commuting it is not obvious that the usualHamiltonian procedure could produce dynamics with respectto noncommuting time. This issue has been addressed by Dayi[8] where noncommutativity in time–space sector emerges in atheory with spatial noncommutativity due to a duality transfor-mation. Specifically a Hamiltonian formulation was obtainedwith commutating time which was shown to be identical toorder θ for both the original theory (with noncommutativityin the spatial sector only) and its dual containing space–timenoncommutativity [8]. Following this we propose to carry outour analysis to first order in θ and assume the applicability ofthe usual Hamiltonian dynamics for the commutative equiv-alent model. Since our motivation is to investigate what newfeatures emerge from the presence of noncommutativity in theSchwinger model, introduction of minimal noncommutativitywill be sufficient.

To the lowest order in θ the explicit forms of the SW mapsare known as [28–30]

φ = φ − θmjAm∂jφ,

(8)Ai = Ai − 1

2θmjAm(∂jAi + Fji).

Using these expressions and the star product (6) to order θ in(4) we get

SSW map=

∫d2x

[{1 + 1

2Tr(Fθ)

}Lc − (Fθ)μ

β∂βφ∂μφ

(9)− e

2εμν(FθF )μνφ + 1

2(FθF )μνFμν

],

where Lc stands for the commutative Lagrangean

(10)Lc = 1

2∂μφ∂μφ + e

2εμνF

μνφ − 1

4FμνF

μν.

Note that we can write the action (9) in a form which, modulototal derivative terms, does not contain second- or higher-ordertime derivatives. This happens because we are considering aperturbative calculation to first order in θ . If we would calculateto second order or beyond, higher derivatives of time wouldappear in the Lagrangean from star product expansion whichbrings complication in the Hamiltonian formulation [35,36].

The Eular–Lagrange equations following from the action (9)are

∂ξ

[{1 + 1

2Tr(Fθ)

}∂ξφ − (Fθ + θF )ξμ∂μφ

]

− e

2εμν

[{1 + 1

2Tr(Fθ)

}Fμν + (FθF )μν

]= 0,

∂ξ

[−θξαLc +

{1 + 1

2Tr(Fθ)

}(eεξαφ − Fξα

)

− θαμ∂μφ∂ξφ + θξμ∂μφ∂αφ

− eφ{εξμ(θF )α μ − εαμ(θF )ξ μ

}

(11)+ (FFθ + θFF)ξα + (FθF )ξα

]= 0.

294 A. Saha et al. / Physics Letters B 638 (2006) 292–295

We work out the canonical momenta conjugate to φ and Aα

respectively:

πφ ={

1 + 1

2Tr(Fθ) − (Fθ + θF )00

}φ

− (Fθ + θF )0i∂iφ,

πα = −θ0αLc +{

1 + 1

2Tr(Fθ)

}(eε0αφ − F 0α

)

− θαμ∂μφ∂0φ + θ0μ∂μφ∂αφ

− eφ[ε0μ(θF )α μ − εαμ(θF )0

μ

](12)+ (FFθ + θFF)0μ + (FθF )0μ.

From (12) we get after a few steps

πφ = φ + θF01φ,

π0 = 0,

(13)π1 = F01 + eφ + θ

2

[φ2 − (∂1φ)2 + 3F 2

01

].

The Hamiltonian follows as∫dxHCEV

=∫

dx

[HCS + θ

2

{3π2

φπ1 + 3eφφ′2 − 3π1φ′2

(14)

− 3eφπ2φ + 3eφ

(π1)2 − (

π1)3 − 3e2φ2π1 + e3φ3}],

where HCS is given by

(15)HCS = 1

2

[π2

φ + (π1)2 + φ′2 + e2φ2] + π1A′

0 − eπ1φ.

Note that this θ -independent term is nothing but the Hamil-tonian of the commutative theory.

From the second equation of (12) we get one primary con-straint

(16)π0 ≈ 0.

Conserving this in time a secondary constraint

(17)∂1π1 ≈ 0

emerges. The constraints (16), (17) have vanishing Poissonbrackets with the Hamiltonian as well as between themselves.No new constraint, therefore, is obtained. It is interesting to notethat the constraint structure is identical with the commutativeSchwinger model. This structural similarity is remarkable be-cause the gauge field in our commutative equivalent theory isthe SW map of a NC gauge field belonging to the Groenewold–Moyal deformed C algebra.

To proceed further we require to eliminate the gauge redun-dancy in the equations of motion (11) by invoking appropriategauge fixing conditions. The structure of the constraints (16),(17) suggests the choices: A0 = 0 and A1 = 0. The simplecticstructure in the reduced phase space will be obtained from theDirac brackets which enables to impose the constraints strongly[37]. It is easy to verify that the nontrivial Dirac brackets of the

reduced phase space remains identical with the correspondingcanonical brackets. The Hamiltonian density in the constrainedsubspace can be written down as

(18)

HR = 1

2

[(π2

φ + φ′2 + e2φ2) + eθφ(3φ′2 − 3π2

φ + e2φ2)].

The reduced Hamiltonian (18) along with the Dirac bracketsleads to the following equations of motion.

(19)φ = (1 − 3eθφ)πφ,

πφ = φ′′ − e2φ + 1

2eθ

(3π2

φ − 3φ′2 − 3e2φ2)(20)+ 3eθ

(φφ′′ + φ′2).

Eqs. (19) and (20) after a little algebra reduced to

(21)(� + e2)φ = 3

2eθ

(−φ2 + φ′2 + e2φ2).Eq. (21) is the relevant equation of motion obtained by remov-ing the gauge arbitrariness of the theory. To zero order in θ itgives

(22)(� + e2)φ = 0.

Eq. (21) looks complicated but, thanks to the fact that our theoryis order-θ , in the right-hand side we can substitute φ to 0-order.Again, to this order φ satisfies (22) the solution to which caneasily be expanded in terms of the plane wave solutions

(23)

φ =∫

dp

(2π)

1√2p0

[a(p)e−ip0x0+ipx + a†(p)eip0x0−ipx

],

where xμ ≡ (x0, x), pμ ≡ (p0, p) and p0 = √p2 + e2. Hence

from (21) we get

(24)(� + e2)φ = j (x),

where j (x) is given by

j (x) = 3eθ

2

∫dp dq

(8π2)

ei(p+q)x√p0q0

× [(p0q0 − pq + e2){a(p)a(q)e−i(p0+q0)x0

+ a†(−p)a†(−q)ei(p0+q0)x0}+ (−p0q0 − pq + e2){a(p)a†(−q)e−i(p0−q0)x0

(25)+ a†(−p)a(q)ei(p0−q0)x0}].

We thus have a bosonic field φ interacting with a source j (x). Itis easy to recognize that Eq. (24) represents the Klien–Gordon(KG) theory with a classical source.

A remarkable observation is inherent in (24). To understandthe proper perspective we have to briefly review the results fromthe corresponding commutative theory. There we end up with(22) and interpret that the photon has acquired mass and thefermion has disappeared from the physical spectrum [9,11–15].This means the fermion is confined. The introduction of non-commutativity changes the scenario in a fundamental way. Thegauge boson again acquires mass but this time it is interacting

A. Saha et al. / Physics Letters B 638 (2006) 292–295 295

with a background. The origin of this background interaction isthe fuzziness of space–time. This is a physical effect carryingNC signature. In this context it may be mentioned that gener-ation of interactions by casting noninteracting theories in NCcoordinates have been observed in other contexts also [38–40].

It is easy to formulate the theory guided by (24) as a quantumtheory. Note that the NC parameter θ is a small number and wecan treat the interaction term as a perturbation. The resultingS-matrix can easily be written down as

(26)S ∼ T

{exp

[−i

∫d2x j (x)φ(x)

]}.

Since j (x) is real S is unitary. Again, since the field φ satisfiesa KG equation causality is also ensured. These are good newsin view of the presence of time–space noncommutativity andjustifies our proposition based on [8] that usual Hamiltoniananalysis is applicable in our commutative equivalent model.

In this Letter we have studied the effect of time–space non-commutativity on the bosonized Schwinger model in 1 + 1dimensions. We analyzed the model in the commutative equiv-alent representation [27,31–34] using a perturbative Seiberg–Witten map [28–30]. Following [8] we have assumed the ap-plicability of a usual Hamiltonian analysis of the commutativeequivalent model. The model exhibits emergence of a massiveboson. This is similar to what happens in the usual theory. How-ever, the boson is no longer free as in the commutative counter-part but it interacts with a background. Our analysis thus revealsthe presence of a background interaction term which is manifestonly in the tiny length scale ∼ √

θ . The theory in the reducedphase space can be formulated as a perturbative quantum fieldtheory which is formally similar to the KG theory with a clas-sical source. Consequently, the requirements of unitarity andcausality are satisfied.

Acknowledgements

A.S. wants to thank the Council of Scientific and IndustrialResearch (CSIR), Government of India, for financial supportand the Director, S.N. Bose National Centre for Basic Sciencesfor providing computer facilities. He also likes to acknowledgethe hospitality of IUCAA where part of this work has beendone. Finally the authors thank the referee for his useful com-ments.

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