PHYSICAL REVIEW D 69, 105006 ~2004!

Space-time noncommutativity tends to create bound states

Dmitri V. Vassilevich*Institut fur Theoretische Physik, Universita¨t Leipzig, Augustusplatz 10, D-04109 Leipzig, Germany

and V. A. Fock Institute of Physics, St.Petersburg University, Russia

Artyom Yurov†

Department of Theoretical Physics, Kaliningrad State University, 236041, A. Nevskogo 14, Kaliningrad, Russia~Received 26 November 2003; published 7 May 2004!

We study the spectrum of fluctuations about static solutions in (111)-dimensional noncommutative scalarfield models. In the case of soliton solutions noncommutativity leads to the creation of new bound states. In thecase of static singular solutions an infinite tower of bound states is produced whose spectrum has a strikingsimilarity to the spectrum of confined quark states.

DOI: 10.1103/PhysRevD.69.105006 PACS number~s!: 11.10.Nx

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I. INTRODUCTION

Over recent years noncommutative field theory has deoped into a mature discipline~see reviews@1#!. It was argued~cf., e.g., Ref.@2#; more references can be found in Ref.@3#!that because of the presence of an infinite number of tderivatives space-time noncommutative theories cannoquantized properly. However, the situation does not lohopeless. Perturbative unitarity can be successfully mtained@3# if one takes care of the explicit Hermiticity of thLagrangian. Even a canonical formalism can be developethe expense of introducing an additional space-time dimsion @4#. We do not have much to add to this discussioMoreover, our analysis will be essentially classical. Wwould like to mention only that space-time noncommutattheories are not excluded, and that one can expect mnonstandard features from these theories.

In this paper we consider some qualitative featuresspace-time noncommutative theories. Namely, we study fltuations around static classical solutions(111)-dimensional noncommutative models with a real slar field. Note that the solutions themselves look exactly athe commutative models. Therefore, noncommutativity cbe seen only through the fluctuation spectra or throughscattering amplitudes@5#. We find that the frequencydependent potential that appears in the equation for fluction has typically an ‘‘effective width’’ proportional to thefrequency and to the noncommutativity parameter. This pnomenon is somewhat similar to delocalization of statescussed in Ref.@6# in a different context. In our case, thdistortion of the potential leads to creation of new boustates~soliton backgrounds! or even to infinite families ofnew bound states~singular static backgrounds!.

This paper is organized as follows. In the next sectionintroduce our notations and conventions. Section III isvoted to fluctuations about solitonic solutions in thef!

4 andin sine-Gordon models. Singular solutions are discusseSec. IV. Some concluding remarks are given in Sec. V.

*Electronic address: Dmitri.Vassilevich@itp.uni-leipzig.de†Electronic address: yurov@freemail.ru,artyom_yurov@mail.r

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II. NOTATIONS AND CONVENTIONS

Let us consider the noncommutative plane with a coornates5(t,x). The Groenewold-Moyal product is defined bthe equation

f ~s!!g~s!5FexpS i

2umn

]

]sm

]

]s8nD f ~s!g~s8!Gs85s

,

~1!

whereumn52unm is a constant antisymmetric 232 matrix,which can be chosen asumn52uemn with emn52enm ande0151. This product is associative but non-commutativehistorical overview can be found in Ref.@7#. The followingrelations will be useful throughout this paper:

f ~x!!eivt5eivt f ~x1uv!, eivt! f ~x!5eivt f ~x2uv!.~2!

We shall study noncommutative deformations of thetion

S5E d2sF1

2]nf]nf2V~f!G ~3!

for a real one-component fieldf with some potentialV.Noncommutative deformations ofV are constructed~asusual! in the following way. Let

V~f!5 (p>0

cpfp. ~4!

Then a noncommutative counterpart ofV is defined as

V!~f!5 (p>0

cpf!f•••!f, ~5!

where thepth term containspth star-power off. We restrictourselves to polynomial or exponential potential only, so tthere is no problem with the convergence of~4!. Conver-gence of~5! is a more subtle question, but we shall actuawork with noncommutativity only in the perturbative regim

Our primary example will be thef4 model

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D. V. VASSILEVICH AND A. YUROV PHYSICAL REVIEW D 69, 105006 ~2004!

V[4]521

2m2f21

l

4f4. ~6!

We shall also consider the Liouville model1

V[L]5gebf, ~7!

the sine-Gordon model

V[sG]52m4

6lcosSA6l

mf D , ~8!

and the sinh-Gordon model

V[shG]5m2

2cosh~2f!. ~9!

The equation of motion following from the noncommuttive deformation

S!5E d2sF1

2]nf]nf2V!~f!G ~10!

of ~3! reads:

] t2f2]x

2f1@]fV#!50. ~11!

Obviously, the star product of functions depending onx onlycoincides with the ordinary product. Therefore, allstatic so-lutions of a commutative model will also solve the noncomutative equation of motion~11!. Note, that there could beof course non-static localized solutions in space-time ncommutative theories~cf. Ref. @8#!.

III. SOLITONS: NEW BOUND STATES

A. The f!4 model

In this section we consider the spectrum of fluctuatioabout static solitonic solutions~i.e., about localized solutionwith finite energy!. For the f4 model ~6! this is the kinksolution:

F~x!5m

AltanhS mx

A2D . ~12!

As we have already mentioned above, this is also a solitothe noncommutativef!

4.Let us consider small fluctuations about the kink ba

ground,fªF1df. The equation of motion for the fluctuations reads

df2df92m2df1l~df!F21F2!df1F!df!F!50.~13!

We shall look for the solutions in the form

1To avoid confusion we should mention that sometimes the Liville model also includes an interaction with a two-dimensionmetric.

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df5eivth~x!. ~14!

Then, by virtue of~2!, one obtains

2h91l~F12 1F1F21F2

2 !h5~v21m2!h, ~15!

whereF6[F(x6)5F(x6uv).This problem has a natural scalem5m/A2. It is conve-

nient to introduce rescaled dimensionless variables:x5mx,v5v/m, u5m2u. In terms of these variables Eq.~15! reads

2h9~ x!1U~ x!h~ x!5~v212!h, ~16!

where the prime denotes differentiation with respect tox,and

U52@ tanh2~ x1!1tanh~ x1!tanh~ x2!1tanh2~ x2!#.~17!

From now on we omit the carets, which is equivalentsettingm51.

In the commutative case,u50, x65x,

U5U05626

cosh2x. ~18!

There are two bound states for this potential withv150 andv25A3 with the wave functions:

h15A3

2 cosh2x, h25A3

2

sinhx

cosh2x~19!

For uÞ0 we have a complicated problem with a frquency dependent potential. Of course, in the generic casexact solution for the eigenstates is available. For smauand relatively low eigenfrequencies,uv!1, we can use or-dinary perturbation theory to estimate shift of the eigenvues. An example of the potentialU(x) is shown on Fig. 1.

We write to the leading order inuv:

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FIG. 1. The potentialU(x) for ~dimensionless! uv51 ~solidline! as compared toU0 ~dashed line!.

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SPACE-TIME NONCOMMUTATIVITY TENDS TO CREATE . . . PHYSICAL REVIEW D 69, 105006 ~2004!

2h92S 6

cosh2x2U (1)D h5~v224!h, ~20!

where

U (1)52~uv!2

cosh2xS 7

cosh2x26D . ~21!

We assumeh5h01dh, where h0 satisfies~16! with U5U0 given by ~18!. Remember that carets have bedropped. Then one immediately obtains that the bound sfrequencyv150 is not shifted to this order, whilev2 re-ceives a negative correction:

v225v2

2uu501E dxh22U (1)53S 12

8

5u2D ~22!

or

v25~120.8u2!v2uu50 . ~23!

Corrections to this formula are of orderu4. We see that inthis regime the eigenfrequency of one of the bound statebeing shifted. Let us remind that the kink solution itself donot depend onu. Therefore, shift of the eigenfrequenciemay be a measurable manifestation of noncommutativitysmall u and low frequencies.

We have shown thatv decreases due to the noncommtativity. A natural question to ask is whethernew boundstates can appear. The answer is positive. The followanalysis will be made in the large-u limit.

It is easy to demonstrate that for largeuv the potentialU(x) @cf. ~17!# can be approximated by a square well potetial U ~cf. Fig. 2! such that

U~x!56 for uxu.uv,

U~x!52 for uxu,uv. ~24!

Therefore, we replace~16! by

2h91Uh5~v212!h. ~25!

FIG. 2. The potentialU(x) ~solid line! and the approximating

square well potentialU ~dashed line! for uv510.

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Note, that the characteristic width ofU depends onv.Clearly, bound states can only appear for 0,v,2. By usingthe standard methods which can be found in any quanmechanics text book we obtain that eigenfrequencies ofbound states should satisfy one of the equations

tan~uv2!5A42v2

v, cot~uv2!52

A42v2

v, ~26!

where the first equation gives eigenfrequencies of the stwith a symmetric wave function, while the second equatdescribes the states with antisymmetric wave functions.

In our approximation it is essential thatvu is large.Therefore, we shall consider~26! for v near the upper limitv52. In a small interval nearv254 the functions on theright hand sides of the equations~26! are bounded and continuous, while tan(uv2) and cot(uv2) change from2` to1` when v2 changes fromp(n2 1

2 )/u to p(n1 12 )/u or

from pn/u to p(n21)/u, respectively. This means if thau@p/8 there is always at least one solution for each ofequations~26! nearv52. We can even estimate roughly thnumber of the solutions in the upper half of the allowinterval ~i.e., for v2P@2,4#) to be about 8u/p. We concludethat for a large noncommutativity parameteru there aremany new bound states for the fluctuations about the ksoliton as compared to the commutative case.

B. The sine-Gordon model

To make sure that the phenomenon of creation of nbound states due to the noncommutativity is present not oin the f!

4 model, let us consider the sine-Gordon model~8!.Static solutions in this model in both commutative and nocommutative regimes should satisfy the equation

2f91m3

A6lsinSA6l

mf D 50. ~27!

There is a one-soliton solution of~27! which reads

F~x!54m

A6larctan~emx!. ~28!

To obtain an equation for fluctuations we have to expaa noncommutative exponential. This can be done withhelp of the equation:

e!A1B5e!

A1E0

1

dse!sA!B!e!

(12s)A1O~B2!, ~29!

which has a purely combinatorial origin and is true regaless of the choice of associative product involved~this couldbe the ordinary operator or matrix product, for example,the Groenewold-Moyal star as in our case!. The formulas~29!, ~14!, and~2! yield

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D. V. VASSILEVICH AND A. YUROV PHYSICAL REVIEW D 69, 105006 ~2004!

2h91

m3FsinSA6l

mF1D 2sinSA6l

mF2D G

A6l~F12F2!h5v2h.

~30!

For largeuv the effective potentialU(x) behaves similarlyto that for thef!

4 model ~see Fig. 3!. Namely, the potentiaU(x) can be approximated by a square well potential wthe width 2uv. All arguments of the preceding section appfor this case almost without modification. We conclude thfor large noncommutativity we have new bound states. Tseems to be a generic feature of the fluctuation equationthe background of a static solitonic solution in a twdimensional noncommutative space-time.

IV. SINGULAR SOLUTIONS AND CONFININGPOTENTIALS

A. Masslessf!4 model

Thef!4 model withm50 admits a singular static solutio

which reads

F~x!5A2

xAl. ~31!

Then, by acting exactly as in the previous section we obthe following equation for the fluctuations:

2h912~3x21u2v2!

~x22u2v2!2h5v2h. ~32!

Note, that this equation is scale-invariant, i.e., if we rescx→xm, v→v/m, u→um2 the scaling parameter canceout. As a consequence, we can assume that we are woin dimensionless variable, so thatuv53 in Fig. 4 indeedmakes sense.

Obviously, foru50 there are no bound states. IfuÞ0 thesituation changes drastically~cf. Fig. 4!. In this case we havetwo infinitely high potential barriers located atx56vu. Thephysics between these two barriers can be approximatean infinitely deep well of width 2uv. This approximation is

FIG. 3. The effective potential~30! for the sine-Gordon mode

in dimensionless variablesx5mx, u5m2u, v5v/m at uv510.

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good for high frequencies. In this case the equation for smfluctuations can be easily solved, yielding

vN.ApN

2u, ~33!

where NPN. The accuracy of this formula increases flargeN and/or largeu.

The spectrum obtained in this simple model has a striksimilarity to the spectrum of hadrons. Indeed, we observeinfinite number of bound states with a linear dependencev2

ªM2 on an integer spectral parameter. Since we dohave something like angular momentum in two dimensiowe cannot push these arguments further.

B. Exponential interactions

Again, we would like to test the observation made for tf!

4 model by considering other models admitting similtypes of the classical solution. We start with the Liouvilmodel ~7!. The classical equation of motion for this modreads

f2f91ae!bf50, ~34!

wherea5bg. In the commutative caseu50 there is a gen-eral solution to this equation:

f51

blogS 2

2G8~p!F8~q!

ab@G~p!1F~q!#2D , ~35!

with p5(t1x)/2, q5(t2x)/2. G andF are arbitrary func-tions. The simplest static solution~which again is commonfor the commutative and noncommutative models! is ob-tained by settingG5p, F52q:

F~x!521

blogS abx2

2 D , ab.0. ~36!

Next we use again the expansion~29!, the ansatz~14!, andthe property~2! to write the equation for fluctuations:

FIG. 4. The potentialU from Eq. ~32! for uv;3.

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SPACE-TIME NONCOMMUTATIVITY TENDS TO CREATE . . . PHYSICAL REVIEW D 69, 105006 ~2004!

2h91aS ebF12ebF2

F12F2Dh5v2h. ~37!

The substitution of~36! in ~37! yields the following effectivepotential:

U58uvx

~x1x2!2log~x1 /x2!2. ~38!

This potential is similar to the one appearing in thef!4 model

@cf. ~32!#. Again, we have two infinitely high potential bariers with a ‘‘confinement’’ region between them. The effetive width of this region is 2uv. Therefore, the spectrum ohigher excited states is again given by~33!.

A very similar behavior can be found also in the sinGordon model~9! near the static solution

F~x!51

2log tanh2S mx

2 D . ~39!

We leave this case for the reader as an exercise.We like to stress that in each case the universal form

~33! appears.

V. CONCLUSIONS

Our main result is that in the presence of the space-tnoncommutativity the effective potential describing fluctutions on a static background becomes delocalized witheffective width;uv. As a consequence, in the case of larnoncommutativityu we have much more bound statessolitonic background then in corresponding commutattheories. On singular static backgrounds the picture is emore interesting. Noncommutativity produces an infintower of bound states with linear dependence ofv2 on aninteger quantum number~for large frequencies!. This behav-

y

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ior is universal, the large frequency spectrum depends ouonly, but not on the details of the models. This result sugests that the space-time noncommutativity may have srelation to the problem of quark confinement. Although whave not presented any general proof, the number ofamples considered seems to justify our conclusion thatation of new bound states is a generic feature of space-noncommutative theories in 111 dimensions.

In principle, the frequency spectrum can be used to cculate quantum corrections to the mass of the solitonsnon-commutative theories. In Ref.@9# it was argued that thez function or other heat kernel based methods may bsuitable instrument~although, it is not clear whether the fluctuation operator involved can indeed be reduces to! Lapla-cians considered in Ref.@9#!. As an intermediate step one hato put the system in a large box so that the spectrum becodiscrete. At the end of the calculation the boundary is movto the infinity and, if necessary, the boundary contributionthe vacuum energy is subtracted from the total energy ofsystem~cf. Ref. @10# where this procedure is applied to thkink!. However, in the present case no fixed boundary canfar away enough since the ‘‘effective width’’ of the potentiis proportional to the frequency. This seems to be anotmanifestation of the mixing between ultraviolet and infrarscales in noncommutative theories@11#. Consequences othis mixing for quantum corrections to~space! noncommuta-tive solitons in 211 dimensions were considered recentlyRef. @12#.

ACKNOWLEDGMENTS

This work was supported in part by the DFG Project B1112/12-1, by the Erwin Schro¨dinger Insitute for Mathemati-cal Physics, and by the Helmholtz program. A.Y. is grateto M. Bordag for his kind hospitality at the University oLeipzig.

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