QFT with Twisted Poincar'e Invariance and the Moyal Product

a r X i v :h e p -t h /0703245v 2 7 A p r 2007Preprint typeset in JHEP style -HYPER VERSION

Euihun Joung Laboratoire APC,Universit′e Paris VII,B?a timent Condorcet,75205Paris Cedex 13,France joung@apc.univ-paris7.fr Jihad Mourad Laboratoire APC,Universit′e Paris VII,B?a timent Condorcet,75205Paris Cedex 13,France mourad@apc.univ-paris7.fr Abstract:We study the consequences of twisting the Poincar′e invariance in a quantum ?eld theory.First,we construct a Fock space compatible with the twisting and the corre-sponding creation and annihilation operators.Then,we show that a covariant ?eld linear in creation and annihilation operators does not exist.Relaxing the linearity condition,a covariant ?eld can be determined.We show that it is related to the untwisted ?eld by a unitary transformation and the resulting n-point functions coincide with the untwisted

ones.We also show that invariance under the twisted symmetry can be realized using the covariant ?eld with the usual product or by a non-covariant ?eld with a Moyal product.The resulting S -matrix elements are shown to coincide with the untwisted ones up to a momenta dependent phase.

1.Introduction

Invariance under the Poincar′e group leads to important restrictions on quantum?eld the-ory.In the free linear case,it completely determines the theory[1].It is then of great interest to examine the consequences of deforming this invariance.Many possible defor-mations have been considered in the past as,for instance,replacing the Poincar′e group by (A)dS group or deforming the Lie algebra structure.

Here,we will examine the possibility of keeping unchanged the algebra structure of the Poincar′e universal enveloping algebra(UEA)but deforming its coalgebra structure.The latter is crucial in quantum?eld theory since the coproduct determines the transformation laws of a system of several particles.A simple way to deform the coproduct?can be obtained from the usual coproduct?0and a twist element F[2]as

?=F?0F?1,(1.1) the resulting UEA will be called in the following the twisted Poincar′e UEA.1A consistent example was recently[3,4]considered and is given by

F=e i

1In fact,a theorem by Drinfeld[2]shows that all deformations of the coproduct of a semi-simple Lie algebra are of this form.

It describes the low energy e?ective theory on D-branes with an external B-?eld.By the Weyl-Moyal correspondence,the theory can be reformulated as a?eld theory on commuta-tive space but the usual product between two?elds being replaced by the Moyal?-product, 2

(f?θg)(x)=e i

2We use a simpli?ed notation vθw for vμθμνwν.

in NCFT,does not lead to a covariant result.We also argue that the implementation of the twisted covariance in an interacting theory can be realized by an interaction Hamiltonian which is local in the ?eld Φθor equivalently non-local in the Φnc θ?eld operator.If we insist on having asymptotic states transforming covariantly under the twisted Poincar′e symmetry then,as is shown in Section 3,the S -matrix elements are identical to the untwisted ones up to a momenta dependent phase.The Appendix contains the proof of some technical results used in the text.

2.Twisted Poincar′e UEA and its representation

The twisted coproduct yields transformation laws of multi-particle states under the Poincar′e group which are modi?ed with respect to the usual ones.In particular,the action of the UEA on the tensor product of identical particles does not commute any more with the permutation of particles,leading to a twisted Fock space.In this section,we shall brie?y review the twisted Poincar′e UEA representation and use it for the construction of the Fock space.

Let U be the UEA of the Poincar′e Lie algebra generated by P μand M μν.The one-particle Hilbert space H of a scalar particle corresponds to the scalar unitary irreducible representation (UIR).Let U (X )be the corresponding unitary representation of X ∈U .A basis of H is given by the momentum eigenstates |p .

The tensor product representation U (n )(X )on H ?n ,the tensor product of n copies of H ,is obtained from U (X )and the coassociative coproduct ?as

U (n )(X )=U ?n (?(n )(X ))=U (X (1))?···?U (X (n )),

(2.1)where the n-coproduct ?(n )(X )=X (1)?···?X (n )is de?ned by

?(n +1)(X )=(??id ?(n ?1))(?(n )(X )),?(2)=?.(2.2)

The representation X of X on the tensor algebra T (H )= ∞n =0H

?n is the direct sum of the tensor product representations:

X =1⊕U (X )∞ n =2U (n )(X ),(2.3)

where the ?rst part 1of X is the trivial representation.

For the untwisted Poincar′e UEA,the coproduct ?0is given by 3

?0(X )=X (2),?(n )0(X )=X (n ),(2.5)

and the resulting tensor algebra representation X0of X is

X0=1⊕U(X)

∞

n=2U(n)0(X),U(n)0(X)=U?n(?(n)0(X)).(2.6)

In order to construct the Fock space we shall need the?ip operationπij on U?n given by

πij(X1?···?X n)=X1?···?i th

X j?···?

j th

X i?···?X n.(2.7)

The n-coproduct?(n)0is then invariant under the?ip:

πij(?(n)0(X))=?(n)0(X),(2.8) and the tensor product representation U(n)(X)commutes with the analogously de?ned?ip mapsΠij’s on H?n:

[Πij,U(n)

(X)]=0.(2.9) Therefore,H?n is reducible and contains two Fock spaces,the invariant eigenspaces of Πij.Eigenvalue+1(totally symmetric case)corresponds to bosons and?1(totally an-tisymmetric case)corresponds to fermions.In the following,we shall concentrate on the bosonic Fock space F,which is obtained by projecting the tensor algebra T(H)on the +1eigenspace ofΠij:

F0=S0T(H).(2.10) Here the projector S0is the symmetrization map and n-boson states are thus given by

|p1,...,p n 0=S0|p1 ?···?|p n ,(2.11) and their scalar product is given by

0 p1,...,p n|q1,...,q n 0= Pδ(p1?q P1)···δ(p1?q P n),(2.12) where the sum P is over all permutations.

We now turn to the twisted Poincar′e UEA which has the same algebra structure as the untwisted one but is equipped with a twisted coproduct?θ:

?θ(X)=F?0(X)F?1,F=exp G,G=

4see the appendix for a proof.

Consequently,the tensor algebra representation Xθof X is also deformed as

Xθ=1⊕U(X)

∞

n=2U(n)θ(X).(2.16)

On the other hand,from eq.(2.14)we get

U(n)θ(X)=U?n(F n)U(n)

(X)U?n(F n)?1,(2.17)

so if we de?ne F as

F=1⊕1

∞

n=2U?n(F n),(2.18)

we obtain the similarity relation between X0and Xθ:

Xθ=F X0F?1.(2.19)

A similar relation can be in fact established for a general counital2-cocycle F[22].

Contrary to the undeformed case,the actions of the twisted Poincar′e UEA do not commute with the?ipsΠij.From the similarity relation eq.(2.19)we see that they commute with the twisted?ips FΠij F?1which still have eigenvalues±1and associated eigenspaces, a bosonic Fock space with the eigenvalue1and a fermionic Fock space with the eigenvalue ?1.It follows that the bosonic Fock space and a twisted n-bosons state are given by

Fθ=SθT(H),|p1,...,p n θ=Sθ|p1 ?···?|p n ,(2.20)

where the projector Sθis the twisted symmetrization map with respect to the twisted?ip FΠij F?1and is related to the undeformed symmetrization map by Sθ=F S0F?1.

Since F is diagonal when acting on|p1 ?···?|p n ,we obtain a simple relation between the twisted and the untwisted states as

F|p1,...,p n 0=fθp

1···p n

|p1,...,p n θ,(2.21)

where fθp

1···p n is the eigenvalue of F with eigenstate|p1 ?···?|p n .Explicitly,it is given

fθp

1···p n

=e i

by the states|p1,...p n θ.These in turn will determine the creation and annihilation operators.

Notice from eq.(2.21)that the states|p1,...,p n θare not symmetric,the exchange of p i and p j multiplies the state by a phase:

|p1,...,i th p j,...,j th p i,...,p n θ=e ip iθp j|p1,...,p n θ.(2.24) The determination of the creation and annihilation operators will also depend on the scalar product of two states which is also changed by the twisting and is given by

p1,...,p n|q1,...,q n θ= P fθp1...p n f?θq1...q nδ(p1?q P1)···δ(p1?q P n).(2.25)

3.Field operators and the Moyal product

A scalar quantum?eld is an operator acting on the twisted Fock space and can be expressed in terms of the twisted creation and annihilation operators.A covariant?eld satis?es

Φ(Λx)=U(Λ)Φ(x)U?1(Λ),(3.1) whereΛis a Poincar′e transformation.In the untwisted case,the covariance relation(3.1) determines completely a scalar?eld linear in the creation and annihilation operators(see, for instance,Section2of[23]for a short proof).The resulting scalar?eld reads

Φ0(x)=φ(+)

0(x)+φ(?)

(x),φ(+)

(x)= dμ(p)e ip·x a?0(p),(3.2)

whereφ(?)is the hermitian conjugate ofφ(+)and dμ(p)=d d?1p(2π)1?d2is the invariant measure.The creation operators are de?ned by their action on the multi-particle states as

a?0(p)|p1,...,p n 0=|p,p1,...,p n 0.(3.3) From the symmetry and the scalar product,we obtain the commutation relations: [a?0(p),a?0(q)]=[a0(p),a0(q)]=0,[a0(p),a?0(q)]=δ(p?q).(3.4) We now de?ne the twisted creation and annihilation operators in the same way by

(p)|p1,...,p n θ=|p,p1,...,p n θ,(3.5) and also from the symmetry and the scalar product of the twisted states we obtain the following deformed version of the commutation relations:

a?θ(p)a?

(q)=e?ipθq a?

(q)a?

(p),aθ(p)aθ(q)=e?ipθq aθ(q)aθ(p),

aθ(p)a?

(q)=e ipθq a?

(q)aθ(p)+δ(p?q).(3.6)

The relation between the twisted and untwisted states(2.21)can be written in terms of the above de?ned creation and annihilation operators:

fθp

1···p n |p1,...,p n θ=fθp

1···p n

(p1)···a?θ(p n)|? = e i2p nθP a?θ(p n) |? =F|p1,...,p n 0= F a?0(p1)F?1 ··· F a?0(p1)F?1 |? .(3.7)

where Pμ=Pμθ=Pμ0is the Fock representation of the momentum generator Pμ.From the above equation,we obtain a simple relation between the twisted and untwisted creation operators as

F a?0(p)F?1=e i

2pθP a?

θ(p) Uθ(Λ)?1=e i

(Λp)θP?i

pθP?i

(θ?ΛθΛ)P).(3.12) In fact,we can see from eq.(3.10)that there is no covariant?eld which is linear in the creation and annihilation operators.If we loosen up the linearity condition,we can get one.In order to see this,let us?rst de?ne?a?θ(p)by

?a?θ(p)=e i2

R dμ(q)pθq a?

(q)aθ(q)a?

(p).(3.13)

From the transformation law(3.10),we have

Uθ(Λ)?a?

θ(p)Uθ(Λ)?1=?a?

(Λp),(3.14)

and from eq.(3.8),it satis?es the usual commutation relations:

[?a?

θ(p),?a?

(q)]=[?aθ(p),?aθ(q)]=0,[?aθ(p),?a?

(q)]=δ(p?q).(3.15)

The states generated by successive applications of?a?θ(p)on|? are F|p1,...,p n 0.We deduce that the?eld operator given by

Φθ(x)=φ(+)

θ(x)+φ(?)

(x),φ(+)

(x)= dμ(p)e ip·x?a?θ(p).(3.16)

is covariant under the Poincar′e transformations.Furthermore,from eq.(3.8)it is related

to the untwisted one by a unitary transformation:

FΦ0(x)F?1=Φθ(x).(3.17) Therefore,since F leaves the vacuum state invariant,the QFT with the covariant?eldΦθ

(3.16)based on twisted Poincar′e UEA leads to the same n-point functions as the untwisted

one.This covariant?eldΦθ,which was not considered in the literature before,allows to

construct local interaction Hamiltonian’s which are manifestly invariant under the twisted

symmetry.For example the interaction action? V(Φθ)with V an arbitrary potential,is manifestly invariant under the Poincar′e transformations.Notice that the Moyal product

is not needed to achieve the invariance under the twisted Poincar′e symmetry when using

the covariant?eldΦθ.

We now turn to examine the relation with the Moyal product which was at the origin

of this twisted symmetry.To cover the most general case,we shall consider a generalized

version of the Moyal product,which gives the?-product between two?elds at di?erent

spacetime points,with an arbitrary twist parameter?as

(f??g)(x,y)=e i

where ?a (+)θ=?a ?θand ?a

(?)θ=?a θ.Using eq.(3.19)and eq.(3.13),the integrand can be rewritten as e s 1ip 1·x 1??···??e s n ip n ·x n ?a (s 1)θ(p 1)···?a (s n )θ(p n )=e s 1ip 1·x 1···e s n ip n ·x n f ??s 1p 1···s n p n e

i 2s n p n θP a (s n )θ(p n ) =e s 1ip 1·x 1···e s n ip n ·x n f ??+θs 1p 1

···s n p n a (s 1)θ(p 1)···a (s n )θ(p n )e i 2(s 1p 1+···+s n p n )θP .

(3.22)Taking x 1=x 2=···=x n =x in eq.(3.20),using eq.(3.22)and integrating over x gives dx Φθ??···??Φθ(x )= dx Φnc θ???θ···???θΦnc θ(x ).(3.23)From the similarity relation (3.17),we can also cast the above integral in the form F

dx Φ0??···??Φ0(x ) F ?1.(3.24)This shows that the invariant interaction given by a monomial potential dx Φn θ(x )can also be written as dx Φnc θ??θ···??θΦnc θ(x )or as F dx Φn 0(x ) F

?1.Acting with the ?eld product (3.20)on the vacuum state,the last factor in eq.(3.22)drops out and we get

(Φθ??···??Φθ)(x 1,···,x n )|? =(Φnc θ???θ···???θΦnc θ)(x 1,···,x n )|?

=F (Φ0??···??Φ0)(x 1,···,x n )|? ,(3.25)

where we used eq.(3.17)to get the last equality.This shows that the n-point functions with di?erent choices of ?elds and products are related to each other in a simple way.The case ?=θshows that using Φ0with the Moyal product,which de?nes NCFT in the Moyal plane,is equivalent to using the non-covariant ?eld Φnc θwith the usual product and is not invariant under the twisted symmetry.The case ?=0leads to covariant results and shows that using Φθwith the usual product is equivalent to Φnc θwith a twisted product but is also related to the untwisted Φ0with the usual product.This agrees with the conclusions of [20,25,21].

The theory is fully determined by the asymptotic states and the S -matrix.We argued that the asymptotic states which transform covariantly under the twisted symmetry are

|p 1,···,p n θ=f ?θp 1···p n

F |p 1,···,p n 0.On the other hand the S -matrix should commute with the transformations U θ(X ).This can be easily realized with an interaction Hamilto-nian local in Φθ.Since Φθ=F Φ0F ?1,the obtained S -matrix is unitarily equivalent to the corresponding untwisted S -matrix S 0:S =F S 0F ?1.The resulting matrix elements are related by phases as

θ q 1,...,q n |S |p 1,...,p m θ=f θq 1,...,q n f ?θp 1,...,p m 0 q 1,...,q n |S 0|p 1,...,p m 0.(3.26)This is the main consequence of the invariance under the twisted Poincar′e symmetry.

Our conclusions apply to the twist given by eq.(1.2),other twists of the Poincar′e are also possible [26],it would be interesting to explore the consequences of the invariance under these twisted symmetries.

Acknowledgments

We would like to thank John Madore,Karim Noui and Renaud Parentani for many useful discussions.

A.Twisted n-coproduct

In this Appendix we prove eq.(2.14)by induction.First,we have

?(2)θ(X )=?θ(X )=F ?0(X )F ?1=F 2?(2)0(X )F ?12.

(A.1)

Assuming ?(n )θ(X )=F n ?(n )0(X )F ?1n ,the next order can be calculated as ?(n +1)θ(X )=(?θ?id ?(n ?1))(?(n )θ(X ))

(A.2)=e G (n +1)12

(?0?id ?(n ?1))(F n ?(n )0(X )F ?1n )e ?G (n +1)12=e G (n +1)12(?0?id ?(n ?1))(F n )?(n +1)0

(X )(?0?id ?(n ?1))(F ?1n )e ?G (n +1)12.

Since ?0?id ?(n ?1)is linear,we get

(?0?id ?(n ?1))(F n )=(?0?id ?(n ?1)) exp P 1≤i

.(A.3)Finally we obtain e

G (n +1)12(?0?id ?(n ?1))(F n )=exp P 1≤i

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