String in noncommutative background another approach

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String In Noncommutative Backgrund;Another Approach R.Bhattacharyya ?Department of Physics,Dinabandhu Andrews College,Calcutta 700084,India We propose a model action in 1+1?at space-time (compact in spatial dimension)embed-ded in D ?at space-time with a non dynamical space-time dependent two vector.For the above constrained system Dirac brackets of suitably de?ned co-ordinates turn out to be non zero and under speci?c choice of the two vector the model action reduces to a general action of open string in a noncommutative backgrund where we ?nd the natural embedding of the open string action of [6],more generally with variable magnetic ?eld,together with the general form of the action which is assumed in space-time dependent Lagrangian formalism [9].Further the above analysis reveals the signi?cant contribution of constant weight factor for Lagrangian density in space-time noncommutativity θμνalthough it is insigni?cant for the equations of motion.Keywords:String theory.Noncommutative geometry.Constrained system.PACS number(s):11.25.-w 02.40.Gh

The idea of noncommutative space-time is quite old[1,2].The quantum phase space is the?rst example of such structure for the co-ordinates.In the last two decades it has been revived in the study of string theories[3]and in?eld theories[4]specially with the emergence of noncommutative Yang-Mills theory in the context of M-theory compacti?cation in various limits[5]or as low energy limit of open string ending on

a D-brane with constant Nevue-Scwarz B?eld(which is also interpreted as the magnetic?eld on D-brane)

[6].

Noncommutative geometry has also been analysed in some other physical theories resulting from the canonical quantisation of constrained systems with second class constraints[7],speci?cally the models of non commutative relativistic and non-relativistic particles[8]where a Chern-Simons term has been added in a?rst order action.

Recently in the formalism of space-time dependent Lagrangian[9](where the explicit space-time depen-dence of Lagrangian density is assumed)the noncommutative structure of co-ordinates arises naturally on the spatial boundary of the theory both in the analysis of weak-strong duality of1+1Sine-Gordon and massive Thirring models[9b]and in context of electro-magnetic duality in SO(3)Yang-Mills theory with D-brane structure[9c].

In the spirit of space-time dependent Lagrangian formalism here we present a general model action on 1+1?at space-time,compact in its spatial dimension and embedded in D dimensional space-time with a non dynamical space-time dependent two vector.Analysing this constrained system we?nd non zero Dirac brackets of the co-ordinate variables and for a particular choice of the two vector the above model reduces to a general open string action in a noncommutative background which not only embeds the open string action in[6]but with a variable magnetic?eld on D-brane.Further this also recovers the desired assumed form of action in space-time dependent formalism and also couples the weight factor of Lagrangian density(which in usual theories is considered as1and does not contribute to the equations of motion[9])with co-ordinate noncommutativityθμν,where for noncommutative co-ordinates Xμand Xν,

[Xμ,Xν]=iθμν

So speci?cally we propose a model action on a1+1?at Minkowski space-time(compact in spatial dimension σ)embeded in D space-time with the?elds X aμ(σ,τ),V aμ(σ,τ)and a three indexed antisymmetric tensor θaμν(σ,τ)whereθ2=θaμνθaμν.We assume greek indicesμ,νfor D dimensions where roman indices a,b for the1+1embeded dimensions.

S s=?1

2πα′ dτdσΛb?b X aμV aμ?1θ2V bν(1)

Hereα′is a constant which will later be identi?ed with the inverse string tension andΛb is a non-dynamical space-time dependent two vector.It is to be noted that the above action is not of the same form of that in space-time dependent formalism[9]and unlike them it is also Poncare invariant.

Starting from(1)to go over the Hamiltonian formalism the primary consraints for the above model are

A aμ=P aμ+Λ0V aμ

2θ2

=0;P(θ)aμν=0

where P aμ,Πaμand P(θ)aμ,νare the conjugate momenta for X aμ,V aμandθaμν.Assuming usual equal time Poisson bracket

{X aμ(σ,τ),P bν(σ′,τ)}=δabδμνδ(σ?σ′);{V aμ(σ,τ),Πbν(σ′,τ)}=δabδμνδ(σ?σ′) (where all others are zero and here we do not require the Poisson brackets ofθ’s and P(θ)’s),we?nd {A aμ(σ,τ),A bν(σ′,τ)}=0;{B aμ(σ,τ),B bν(σ′,τ)}=?

θ0μν

2πα′

δabδμνδ(σ?σ′)

and the inverses

{B aμ(σ,τ),B bν(σ′,τ)}?1=0;{A aμ(σ,τ),A bν(σ′,τ)}?1=?

(2πα′)2θ0μν

{A aμ(σ,τ),B bν(σ′,τ)}?1=?

2πα′

θ2Λ02

δabδ(σ?σ′)

{P aμ(σ,τ),P bν(σ′,τ)}DB=0;{X aμ(σ,τ),P bν(σ′,τ)}DB=δabδμνδ(σ?σ′)

De?ningˉ

X aμ=1

lθ2Λ02

δab

{ˉP aμ,ˉP bν}DB=0;{ˉ

X aμ,ˉP bν}DB=δabδμν

Now before entering into the Hamiltonian description we break the Poincare invariance of above action by choosingΛ1=0andΛ0=Λ.Here it is to be noted that the above Dirac brackets do not respect the choice ofΛ1.So under such choice

H=1

2 dσ?1V bμθ1μν

Λ

+

πα′

Λ

+

πα′

θ2

θ1μν?1V0ν

V1μ=?(?1X0μ)Λ+

πα′

4πα′ dτdσΛ2(?a xμ?a xμ?1θ2?b xν)+ˉS(4)

whereˉS involves termsα′,α′2,cμ,dμand total derivatives.The Dirac brackets reduces to

{ˉxμ,ˉxν}DB=?

(2πα′)2θ0μν

2πα′as the string tension then(4)represents the general action for open string in

noncommutative background where we can identify the embedding of the action of open string in presence

3

of magnetic ?eld on D p -brane.For Λ=1one ?nds S is a Poincare invariant action and having no past history.Thus we can argue that S s is a more general action for the open string in noncommutative back ground where after the breaking of the Poincare invariance by above choice of two vector we get the action in [6]embedded in S which is Poincare invariant.Further S is also general in that sense as it includes the non constant magnetic ?eld which arises naturally in the action due to the θ?eld,as it couples the indices of space-time with that of world sheet.We can identify the magnetic ?eld by

B μν=1

θ2

and under the choice of θ0μν=constant,magnetic ?eld can still be in the action as long as θ1μνis a non constant ?eld of time and thus θ1μνcan produce a smooth change of magnetic ?eld.

Now if we turn to Dirac brackets for the choice of θ0μν

=constant,we get usual noncommutative relations with the signi?cant contributions of constant Λand α′2.So constant Λbecomes non trivial for the space-time noncommutativity although it does not contribute to the equations of motion.

Another interesting thing is the revival of the general form of the action of the space-time dependent Lagrangian formalism.In that formalism for sake of more general Lagrangian density explicit space-time dependence is included in it so also in the action.Precisely,in D space-time dimensions and for any arbitrary ?eld f it is assumed that Lagrangian density L ′(f,?f,x )=ρ(x )L (f,?f )where ρ(

x )is weight factor and the

action corresponding to L ′is S ′= d D xρ(x )L .So in formalism [9],the open string action in [6]would take

the form

S ′=12?a x μ?ab

B μν?b x ν)

but to start with,we do not assume this form for S s but as it reduces to S we just recover that form.The action S s needs further investigation for the following reasons,?rstly whether it contains any other theory for a new choice of two vector and secondly what may be the physical theory (if any such theory exists at all)whose acton is S s .We can also study S as it includes more ineraction terms than that of [6].Lastly for Λ=1we ?nd S ,a Poincare invariant action after breaking that of S s .We can proceed in the reverse direction i.e starting from any Poincare invariant usual physical theory analogous to S for Λ=1,we can analyse whether we can get hold of another Poincare invariant theory analogous of S s and if for several such S ’s,S s is unique then this may be a way for uniting the physical theories.

Acknowledgement

I am greatful to D.Gangopadhyay for useful discussion.