On the Correspondence between Poincaré Symmetry of Commutative QFT and Twisted Poincaré Symmetry of Noncommutative QFT
aa r X i v : . [ h e p - t h ] M a y arXiv:0709.1010KEK-TH-1177 On the Correspondence betweenPoincar´e Symmetry of Commutative QFT andTwisted Poincar´e Symmetry of Noncommutative QFT
Yasumi Abe ∗ Institute of Particle and Nuclear StudiesHigh Energy Accelerator Research Organization (KEK)Tsukuba 305-0801, Japan
Abstract
The space-time symmetry of noncommutative quantum field theories with a deformedquantization is described by the twisted Poincar´e algebra, while that of standard commuta-tive quantum field theories is described by the Poincar´e algebra. Based on the equivalence ofthe deformed theory with a commutative field theory, the correspondence between the twistedPoincar´e symmetry of the deformed theory and the Poincar´e symmetry of a commutativetheory is established. As a by-product, we obtain the conserved charge associated with thetwisted Poincar´e transformation to make the twisted Poincar´e symmetry evident in the de-formed theory. Our result implies that the equivalence between the commutative theory andthe deformed theory holds in a deeper level, i.e., it holds not only in correlation functions butalso in (different types of) symmetries. ∗ email: [email protected] Introduction
The twisted Poincar´e algebra is a quantum group that is obtained by Drinfel’d twist of theuniversal enveloping algebra U ( P ) of the Poincar´e algebra P . It describes the symmetry of non-commutative space-time whose coordinates obey the commutation relation of a canonical type,[ˆ x µ , ˆ x ν ] = iθ µν . (1.1)The twisted Poincar´e symmetry has been proposed in [1] as a substitute for the Poincar´e symmetryin field theories on the noncommutative space-time. In terms of the twisted Poincar´e symmetry,the Moyal star product f ( x ) ∗ g ( x ) = exp (cid:20) i θ µν ∂ ′ µ ∂ ′′ ν (cid:21) f ( x ′ ) g ( x ′′ ) (cid:12)(cid:12)(cid:12) x ′ ,x ′′ → x , (1.2)which provides the noncommutative product for fields on the noncommutative space-time is ob-tained as a twisted product of a module algebra of the twisted Poincar´e algebra. This fact impliesthe twisted Poincar´e invariance of noncommutative field theories.Recently, some researchers including the author have proposed a quantum field theory (QFT)which possesses the twisted Poincar´e symmetry [2–8]. In this QFT, the star product on differentspace-time points is used as a product for fields, and thus it can be considered as a deformed theoryof a standard commutative QFT. Taking account of the role of the Poincar´e symmetry playedfor the standard commutative QFT, it seems worthwhile to investigate the consequences such adeformation yields thoroughly. Clarification of the property of the theory associated to the twistedPoincar´e symmetry may lead to a fuller understanding of the implication of the noncommutativityfor quantum field theories.The deformation through the star product brings two remarkable properties to the new QFT.One is the twisted Poincar´e invariance as mentioned above. The other is that correlation functionsof the deformed QFT appear to take the same values as those of the corresponding commutativeQFT . In fact, one can construct a map between field operators of the two theories which suggeststhe equivalence of correlation functions [7]. It is noticed, however, that this equivalence has notbeen verified rigorously as we shall explain in section 2. In this paper, we assume this equivalenceand investigate a consequence of it. Once the equivalence is admitted, it implies, in some sense,a discouraging fact that the nontrivial deformation of the theory results in no new dynamics: thedynamics of the new QFT is exactly the same as that of the commutative QFT. On the other hand,it means that any troublesome properties inherent in the ordinary noncommutative QFT, such asUV/IR mixing [12], disappear in the new deformed QFT, and one can obtain a well defined QFTas long as the corresponding commutative QFT is well defined.Now, what does this equivalence imply for symmetries? From the fact that the deformed QFTis twisted Poincar´e covariant while the commutative QFT is Poincar´e covariant, it is expectedthat these two different symmetries correspond with each other through the equivalence of the twotheories, that is, the Poincar´e transformations in commutative QFTs may be represented as thecorresponding twisted Poincar´e transformations in deformed QFTs. The purpose of this paper isto show that this is indeed the case with the statement presented as a theorem. To this end, weuse the map between the two theories presented in [7], and thereby obtain generators of Poincar´etransformations in the deformed QFT from those in the commutative QFT. The twisted Poincar´esymmetry of the deformed QFT can then be derived by twisting the Poincare algebra constructedfrom these generators. For definiteness, we will restrict our attention to a real scalar field in d + 1 The equivalence of correlation functions holds depending on the definition of correlation functions in the de-formed theory. In [9], correlation functions are defined without the star product. Constructing the deformed QFTbased on this correlation function, one find the resulting dynamics to be different from that of a commutative QFT.For speculation on the Hopf algebraic symmetry of this theory, see [10, 11]. , − , · · · , − ) whose interaction term is given bypolynomials. Further we assume that the time and space coordinates are commutative with eachother, i.e., θ i = 0, so that the discussion of noncommutative field theories in terms of a canonicalformalism can be presented in a simple form. Presumably, this assumption is not essential toresults presented here [8].This paper is organized as follows. In section 2, we recall the main result of [7] which is neededfor our discussion. Section 3 is devoted to the investigation of the twisted Poincar´e invariance ofthe deformed QFT. We present the standard Poincar´e invariance of commutative QFTs in termsof the Hopf algebraic structure of U ( P ) in section 3.1. The Poincar´e algebra represented in thecommutative QFT is translated to that represented in the deformed QFT by the map betweenthe deformed QFT and the commutative QFT. Then the Poincar´e algebra in the deformed QFTis twisted in order to describe the symmetry of the deformed QFT. With these preparations, weprovide the proof of the equivalence between a twisted Poincar´e transformation in the deformedQFT and a Poincar´e transformation in the commutative QFT in section 4. Our conclusions andremarks are given in section 5. The deformed QFT with twisted Poincar´e symmetry has been investigated in [2–8]. There aresome different approaches to define this theory. In [7], we have taken a star product of fields atdifferent space-time points to define the deformed QFT: f ( x ) ⋆ g ( y ) := exp (cid:20) i ∂ x θ∂ y (cid:21) f ( x ) g ( y ) (2.1)where we introduce the notation ∂ x θ∂ y := ∂ x i θ ij ∂ y j , which will be used generally for a contraction, pθk := p i θ ij k j . By using this star product, we have seen that we can construct a well definedquantum field theory, by starting from the following Lagrangian L θ ( x ) = 12 h ( ∂ µ φ θ ) − m ( φ θ ) i − X n =3 λ n n ! n z }| { φ θ ⋆ · · · ⋆ φ θ (2.2)and quantizing the field through a deformed commutation relation[ φ θ ( t, x ) , π θ ( t, y )] ⋆ = φ θ ( t, x ) ⋆ π θ ( t, y ) − π θ ( t, y ) ⋆ φ θ ( t, x )= iδ ( d ) ( x − y )[ φ θ ( t, x ) , φ θ ( t, y )] ⋆ = [ π θ ( t, x ) , π θ ( t, y )] ⋆ = 0 , (2.3)where π θ = ∂ φ θ . We call the theory given by (2.2) and (2.3) as a deformed noncommutativequantum field theory (dNCQFT) in this paper. Let us define correlation functions of field operatorsbetween arbitrary states in a Hilbert space H θ which carries a representation of φ θ as h α | ⋆ φ θ ( x ) ⋆ · · · ⋆ φ θ ( x n ) ⋆ | β i , | α i , | β i ∈ H θ . (2.4)Then these correlation functions turn out to have the same value as those of a commutativequantum field theory (CQFT) whose Lagrangian is given by L ( x ) = 12 h ( ∂ µ φ ) − m ( φ ) i − X n =3 λ n n ! ( φ ) n , (2.5) The definition of the star product between operators and states will be given in section 3.2.
2n which the field is quantized by the standard canonical commutation relation. That is, there isa correspondence between a state | α i in H θ and a state | α ′ i in H which carries a representationof φ , and we have h α | ⋆ φ θ ( x ) ⋆ · · · ⋆ φ θ ( x n ) ⋆ | β i = h α ′ | φ ( x ) · · · φ ( x n ) | β ′ i . (2.6)This equivalence is found from the following map between φ and φ θ : φ θ ( x ) = exp (cid:20) ∂θP (cid:21) φ ( x )= ∞ X n =0 n n ! θ i j · · · θ i n j n ∂ i · · · ∂ i n φ ( x ) P j · · · P j n ,φ ( x ) = exp (cid:20) − ∂θP θ (cid:21) φ θ ( x )= ∞ X n =0 (cid:18) − (cid:19) n n ! θ i j · · · θ i n j n ∂ i · · · ∂ i n φ θ ( x ) P θj · · · P θj n , (2.7)where P i and P θi are generators of translations in CQFT and dNCQFT respectively . In addition,based on this map, we use the same Hilbert space as a representation space of the field operatorfor both CQFT and dNCQFT, that is, we take H θ = H , and | α i = | α ′ i and | β i = | β ′ i in (2.6).In the following, we will denote this Hilbert space by H .The correspondence between correlation functions (2.6) can be seen by noticing the followingequation: O ⋆ ( φ θ ) = X n (cid:18) i (cid:19) n n ! θ i j θ i j · · · θ i n j n [ P i , [ P i , · · · [ P i n , O ( φ )] · · · ]] P j P j · · · P j n , (2.8)where O ( φ ) is an arbitrary operator constructed from φ by the ordinary product, and O ⋆ ( φ θ ) is anoperator replacing all the fields and products between them in O ( φ ) by φ and the star product .For example, let us consider the case of O ( φ ) = φ ( x ) · · · φ ( x n ), in which the correspondingoperator O ⋆ ( φ θ ) is given by φ θ ( x ) ⋆ · · · ⋆ φ θ ( x n ). In this case, (2.8) reads φ θ ( x ) ⋆ · · · ⋆ φ θ ( x n ) = exp " X n ∂ x n θP φ ( x ) · · · φ ( x n ) , (2.9)and this is found by substituting the second equation of (2.7) on the right hand side. Based onthis equation, we can easily verify (2.6).Thus we ”prove” the equivalence of correlation functions of the two theories. However, itshould be noted that this proof is somewhat formal, for we ignore some points which should betreated more carefully. Firstly, in the map between operators of the two theory (2.7) or (2.8), themapped operator is given by a nonlocal form of the original local operator, therefore we have tolook into properties of the map, such as an asymptotic behavior, more carefully. Correspondingly,it is unclear whether we can take H θ = H or not. Even if asymptotic completeness is satisfiedfor both theories, there would be no need for asymptotic states in them to correspond with eachother in the simple way as we have stated above. There might be a representation, and thusasymptotic states peculiar to dNCQFT. This would spoil the equivalence of correlation functions.Though there would be need to examine the validity of this correspondence of the two theoriesmore carefully, we assume it in this paper. In particular, we assume that asymptotic states behavein the same manner in both theories and H θ = H . It is noteworthy that, even without (2.7) and(2.8), once H θ = H is assumed, the equivalence of the correlation functions is proved in all orderof perturbation [7]. In fact, P i = P θi as we will see in the next section. From this relation, we confirm that the two equations in(2.7) are in the relation of the inverse map with each other. Inversely, one may consider (2.8) as a definition of O ⋆ ( φ θ ) which corresponds to O ( φ ). Poincar´e symmetry and twisted Poincar´e symmetry
In this section, we show that dNCQFT has the twisted Poincar´e symmetry. The twistedPoincar´e symmetry of dNCQFT can be understood in terms of the Drinfel’d twist by F = e i θ ij P i ⊗ P j . In dNCQFT, we can construct generators of Poincar´e transformations from the fieldoperator, therefore they form an algebra generated from the field operator on a representationspace of them. By twisting the Poincar´e algebra by F , we obtain the twisted Poincar´e algebra.Correlation functions of dNCQFT turn out to be invariant under a transformation in this twistedalgebra. As a preliminary for introducing the twisted Poincar´e symmetry of dNCQFT, we present thePoincar´e symmetry of CQFTs in terms of the Hopf algebraic structure of U ( P ). Here U ( P ) isequipped with a coproduct ∆ : U ( P ) → U ( P ) ⊗ U ( P ), a counit ε : U ( P ) → C and an antipode S : U ( P ) → U ( P ) in addition to the algebraic structure as an enveloping algebra. These linearmaps are given by standard definitions for an enveloping algebra of a Lie algebra. For precisedefinitions of them, see, for example, [13].The Poincar´e algebra P , for which commutators of generators are given by[ P µ , P ν ] = 0 , [ M µν , M ρσ ] = − i ( g µρ M νσ − g νρ M µσ − g µσ M νρ + g νσ M µρ ) , [ M µν , P ρ ] = − i ( g µρ P ν − g νρ P µ ) , (3.1)is represented in CQFT by P µ = Z d d xT µ ( x ) , M µν = Z d d x [ x µ T ν ( x ) − x ν T µ ( x )] , (3.2)where T µ ( x ) = 12 (cid:0) π ( x ) ∂ µ φ ( x ) + ∂ µ φ ( x ) π ( x ) (cid:1) − g µ L ( x ) , (3.3)and π = ∂ φ is the canonical momentum of φ . Of course, these operators are constant in time: dP µ dt = 1 i [ H , P µ ] = 0 ,dM µν dt = ∂M µν ∂t + 1 i [ H , M µν ] = 0 , (3.4)where H = P . It is trivial to construct the representation of U ( P ) from this representationof P . For the representation (3.2) and (3.3), we can take the following two vector spaces as arepresentation space.One is the Hilbert space H (or its dual space H ∗ ) on which the field operator φ is represented.Denoting the action of X ∈ U ( P ) to H and H ∗ as X ( | α i ) = X | α i , | α i ∈ H ,X ( h α | ) = h α | S ( X ) , h α | ∈ H ∗ , (3.5)we can see that this action is compatible with the inner product of H . That is, if we write theinner product of H by the pairing map ev : H ∗ ⊗ H → C ,ev( h α | ⊗ | β i ) = h α | β i , (3.6)4hen X (ev( h α | ⊗ | β i )) = ev(∆( X )( h α | ⊗ | β i ))= h α | m(( S ⊗ ◦ ∆( X )) | β i = h α | ε ( X ) | β i , (3.7)where m : U ( P ) ⊗ U ( P ) → U ( P ) is the product map of U ( P ) and we use the standard formula ofa Hopf algebra, m(( S ⊗ ◦ ∆( X )) = ε ( X ) (cid:16) = m((1 ⊗ S ) ◦ ∆( X )) (cid:17) . (3.8)From the explicit value of the counit ε , ε ( c ) = c, c ∈ C ⊂ U ( P ) ,ε ( χ ) = 0 , χ ∈ P ⊂ U ( P ) , (3.9)we see that (3.7) means the invariance of the inner product of H under a Poincar´e transformation,since c ( h α | β i ) = c h α | β i , for c ∈ C ⊂ U ( P ), χ ( h α | β i ) = 0 , for χ ∈ P ⊂ U ( P ). (3.10)It is clear that this implies the invariance of the inner product under an arbitrary transformationin U ( P ).The other representation space of P and U ( P ) is the algebra M ( φ ) generated from the fieldoperator φ . The action of P µ , M µν ∈ P on φ is given by the standard form: P µ ( φ ) := [ P µ , φ ] = − i∂ µ φ ,M µν ( φ ) := [ M µν , φ ] = − i ( x µ ∂ ν − x ν ∂ µ ) φ , (3.11)and the action of an arbitrary element of U ( P ) is obtained through X X ( φ ) := X ( X ( φ )),where X , X ∈ U ( P ). For example, the action of P µ P µ · · · P µ n ∈ U ( P ) on φ is P µ P µ · · · P µ n ( φ ) = [ P µ , [ P µ , · · · [ P µ n , φ ] · · · ]]= ( − i ) n ∂ µ · · · ∂ µ n φ . (3.12)Further, M ( φ ) represents U ( P ) as a module algebra. In fact, denoting the product map of M ( φ )by µ : M ( φ ) ⊗ M ( φ ) → M ( φ ), µ ( O ⊗ O ) = O O , for O , O ∈ M ( φ ), (3.13)the action of X ∈ U ( P ) to the product is written as X ( µ ( O ⊗ O )) = µ (∆( X )( O ⊗ O )) . (3.14)Since M ( φ ) is represented on H and H ∗ , we can consider the compatibility between the actionof M ( φ ) to H and the action of U ( P ) to them. That is, writing the action of M ( φ ) to H and H ∗ by linear maps µ R : M ( φ ) ⊗ H → H and µ L : H ∗ ⊗ M ( φ ) → H ∗ respectively as µ R ( O ⊗ | α i ) = O | α i ,µ L ( h α | ⊗ O ) = h α | O, (3.15)the action of U ( P ) to these states is written as X ( µ R ( O ⊗ | α i )) = µ R (∆( X )( O ⊗ | α i )) ,X ( µ L ( h α | ⊗ O )) = µ L (∆( X )( h α | ⊗ O )) . (3.16)5n addition, we can introduce a linear map e ev : H ∗ ⊗ M ( φ ) ⊗ H → C for matrix elements ofoperators, e ev( h α | ⊗ O ⊗ | β i ) = h α | O | β i . (3.17)By composing ev with µ R or µ L , e ev is rewritten as e ev = ev ◦ (1 ⊗ µ R ) = ev ◦ ( µ L ⊗ . (3.18)Using this expression, we can see the compatibility between e ev and the action of M ( φ ): X e ev( h α | ⊗ O ⊗ | β i ) = e ev((∆ ⊗ ◦ ∆( X )( h α | ⊗ O ⊗ | β i ))= e ev((1 ⊗ ∆) ◦ ∆( X )( h α | ⊗ O ⊗ | β i )) . (3.19)It is easily seen that (3.19) means the invariance of matrix elements of operators under a Poincar´etransformation in the same way as (3.10). By using the relation (3.18) and (3.8), (3.19) is writtenas X ( e ev( h α | ⊗ O ⊗ | β i )) = h α | ε ( X )( O | β i )= ( h α | O ) ε ( X ) | β i . (3.20)Again, from the explicit value of the counit ε , we see that this equation means the invariance ofthe matrix element under a Poincar´e transformation.Finally, we notice that there hold some relations between linear maps introduced here, corre-sponding to the associativity of their action. For example,( O O ) | α i = O ( O | α i ) ⇔ µ R ◦ ( µ ⊗
1) = µ R ◦ (1 ⊗ µ R ) . (3.21) To obtain the twisted Poincar´e algebra U F ( P ) and a twisted module algebra of it, we start fromthe standard Poincar´e algebra and its representation space, and then twist them. In dNCQFT,we can construct the Poincar´e algebra by applying (2.7) to (3.3) and substituting them in (3.2).Then we acquire P µ and M µν in the same form as (3.2) but now T µ in it is given by T µ = ∞ X n =0 (cid:18) − (cid:19) n n ! θ i j · · · θ i n j n ∂ i · · · ∂ i n (cid:20) (cid:0) π θ ⋆ ∂ µ φ θ + ∂ µ φ θ ⋆ π θ (cid:1) − g µ L θ (cid:21) P θj · · · P θj n , (3.22)instead of (3.3). It is obvious that the resulting operators satisfy commutation relations of Poincar´ealgebra (3.1). Since, to derive these operators, we only rewrite field operators in them according to(2.7), their commutation relations do not change. Thus we can construct Poincar´e algebra P andthe universal enveloping algebra U ( P ) in dNCQFT. Notice that P µ ∈ P are equal to translationgenerators P θµ which are derived from Noether currents in terms of translations in dNCQFT. Infact, the difference between them is only total derivative terms in their integrand: P µ = Z d d xT µ = Z d d x (cid:20) (cid:0) π θ ∂ µ φ θ + ∂ µ φ θ π θ (cid:1) − g µ L θ + (total derivative terms) (cid:21) . (3.23)Since we assume the correspondence of asymptotic behaviors of the two theories, this contributiondoes vanish to give P µ = P θµ . In particular, the Hamiltonian H θ = P θ in dNCQFT is equal to6 = P . Therefore, (3.4) means that operators in P are constant in time also in dNCQFT : dP µ dt = 1 i [ H θ , P µ ] = 0 ,dM µν dt = ∂M µν ∂t + 1 i [ H θ , M µν ] = 0 . (3.24)For the representation space of P and U ( P ) represented by (3.2) and (3.22), we can take theHilbert space H and an algebra M ( φ θ ) generated by products of the field operator φ θ in the sameway as in CQFT. Notice that we can use the same Hilbert space H to represent the field operatorfor both CQFT and dNCQFT by based on the map (2.7), as we mentioned in section 2. For brevity,we use the same symbols for each product maps in H , H ∗ and M ( φ θ ) as those corresponding mapsintroduced in section 3.1. That is, µ : M ( φ θ ) ⊗ M ( φ θ ) → M ( φ θ ) , µ ( O θ ⊗ O θ ) = O θ O θ ,µ R : M ( φ θ ) ⊗ H → H , µ R ( O θ ⊗ | α i ) = O θ | α i ,µ L : H ∗ ⊗ M ( φ θ ) → H ∗ , µ L ( h α | ⊗ O θ ) = h α | O θ , e ev : H ∗ ⊗ M ( φ θ ) ⊗ H → C , e ev( h α | ⊗ O θ ⊗ | β i ) = h α | O θ | β i , (3.25)where O θ , O θ , O θ ∈ M ( φ θ ). The Leibniz rule of the action of U ( P ) on product maps, (3.7),(3.14), (3.16) and (3.19), and relations between product maps such as (3.18) and (3.21) which wehave seen in the previous subsection also hold for these product. In particular, from the relationcorresponding to (3.14) we can see that M ( φ θ ) represents U ( P ) as a module algebra.So far, there seems no difference between the representation of U ( P ) in CQFT and that ofdNCQFT. The difference appears in the action of U ( P ) to M ( φ θ ). For P µ , M µν ∈ P ⊂ U ( P )and φ θ ∈ M ( φ θ ), this action is calculated through the representation (3.2) and (3.22), and thecommutation relation (2.3): P µ ( φ θ ) := [ P µ , φ θ ] = − i∂ µ φ θ ,M µν ( φ θ ) := [ M µν , φ θ ]= − i ( x µ ∂ ν − x ν ∂ µ ) φ θ − i h θ µi ( P i δ αν − P ν δ αi ) − θ ν i ( P i δ αµ − P µ δ αi ) i ∂ α φ θ . (3.26)Notice that the action of generators of a Lorentz transformation M µν to φ θ is different from thestandard one (3.11). This action is exponentiated to give a finite Lorentz transformation Λ µν .This finite Lorentz transformation of φ θ can be written formally as φ θ ( x µ ) −−−−→ Λ φ θ (Λ µν x ν + Λ µν θ νρ P ρ − θ µν Λ νρ P ρ ) . The change of the coordinate induced by the Lorentz transformation has the form similar to thenoncommutative Lorentz transformation in [14]. In fact, it is considered as the field theoreticalexpression of the noncommutative Lorentz transformation in [14]. For the case of free field, thisresult is consistent with [15].Now that the structure of P or U ( P ) represented on dNCQFT is clarified, we twist U ( P ) andits representation spaces. By twisting U ( P ) by the invertible element F = e i θ ij P i ⊗ P j , we obtainthe twisted Poincar´e algebra U F ( P ) which has the following coalgebraic structure:∆ F ( X t ) = F ∆( X ) F − ,ε F ( X t ) = ε ( X ) ,S F ( X t ) = S ( X ) , (3.27) To verify this statement, we must prove that the time evolution of operators in dNCQFT is given by thecommutator with H θ . This can be easily seen by noticing that the time evolution of φ θ and π θ is given by[ H θ , φ θ ] = i ˙ φ θ and [ H θ , π θ ] = i ˙ π θ respectively [7]. X t ∈ U F ( P ) is the same element as X ∈ U ( P ) as an element of the algebra. For P tµ , M tµν ∈P ⊂ U F ( P ), this coproduct gives∆ F ( P tµ ) = P tµ ⊗ ⊗ P tµ , ∆ F ( M tµν ) = M tµν ⊗ ⊗ M tµν − θ ij h ( g iµ P tν − g iν P tµ ) ⊗ P tj + P ti ⊗ ( g jµ P tν − g jν P tµ ) i . (3.28)The procedure of the twist induces the way for deriving a module algebra of U F ( P ) from a modulealgebra of U ( P ). In the case of M ( φ θ ), by twisting the product map µ : M ( φ θ ) ⊗ M ( φ θ ) → M ( φ θ )as µ F ( O θ ⊗ O θ ) := µ ( F − ( O θ ⊗ O θ )) , =: O θ ⋆ O θ , (3.29)we obtain a module algebra M F ( φ θ ) of U F ( P ). That is, the algebra M F ( φ θ ) generated fromproducts of field operators φ θ with the product map µ F gives a module algebra of U F ( P ). Herewe use the same symbol ⋆ for this product as the extended star product (2.1). It is easily seenthat, for field operators φ θ ( x ) and φ θ ( y ), this product gives the extended star product (2.1): µ F ( φ θ ( x ) ⊗ φ θ ( y )) = φ θ ( x ) ⋆ φ θ ( y ) . (3.30)In addition to µ , we introduce a twisted product for other product maps by the same procedure.First, we twist the map for the inner product of H (3.6),ev F ( h α | ⊗ | β i ) := ev( F − ( h α | ⊗ | β i )) =: h α | ⋆ | β i . (3.31)This seems to provide a new inner product for the Hilbert space H , but in fact, ev F = ev since h α | ⋆ | β i = h α | exp h i P i θ ij P j i | β i = h α | β i . (3.32)We insert ⋆ in the inner product only to make explicit the associativity of products in calculatingmatrix elements, as we shall see below. We also introduce a star products for actions of M F ( φ θ )to H and H ∗ , µ F R : M F ( φ θ ) ⊗ H → H ,µ F R ( O θ ⊗ | α i ) := µ R ( F − ( O θ ⊗ | α i )) =: O θ ⋆ | α i ,µ F L : H ∗ ⊗ M F ( φ θ ) → H ∗ ,µ F L ( h α | ⊗ O θ ) := µ L ( F − ( h α | ⊗ O θ )) =: h α | ⋆ O θ . (3.33)Finally we introduce a linear map e ev F : H ∗ ⊗ M F ( φ θ ) ⊗ H → C for evaluating matrix elements ofoperators in M F ( φ θ ), e ev F ( h α | ⊗ O θ ⊗ | β i ) := ev F ◦ (1 ⊗ µ F R )( h α | ⊗ O θ ⊗ | β i )= ev F ◦ ( µ F L ⊗ h α | ⊗ O θ ⊗ | β i )=: h α | ⋆ O θ ⋆ | β i . (3.34)The second equality of this equation means h α | ⋆ ( O θ ⋆ | β i ) = ( h α | ⋆ O θ ) ⋆ | β i , i.e., associativity ofthe star product. This can be easily proved. In fact, noticing(1 ⊗ ∆)( F − )(1 ⊗ F − ) = e − i θ ij ( P i ⊗ P j ⊗ P i ⊗ ⊗ P j +1 ⊗ P i ⊗ P j ) = (∆ ⊗ F − )( F − ⊗ , (3.35)we findev F ◦ (1 ⊗ µ F R ) = ev ◦ (1 ⊗ µ R ) ◦ (1 ⊗ ∆)( F − )(1 ⊗ F − )= ev ◦ ( µ L ⊗ ◦ (∆ ⊗ F − )( F − ⊗
1) = ev F ◦ ( µ F L ⊗ , (3.36)8here we use (3.18). This proof is essentially the same as the proof of associativity of the ordinaryMoyal star product, which also uses (3.35). Furthermore, we can show associativity for all the starproducts introduced here in the same way. For example, quantities such as O θ ⋆ O θ ⋆ O θ ⋆ | α i , h α | ⋆ O θ ⋆ O θ ⋆ | β i , (3.37)do not depend on an order of taking products in them.Next, we observe a relation between these star products and a twisted Poincar´e transformation.In the first place, since, as we noted above, the algebra M F ( φ θ ) is a module algebra of U F ( P ), atwisted Poincar´e transformation of the star product of M F ( φ θ ) is given by X t ( φ θ ( x ) ⋆ φ θ ( y )) = X t ( µ F ( φ θ ( x ) ⊗ φ θ ( y )))= µ F (∆ F ( X t )( φ θ ( x ) ⊗ φ θ ( y ))) , for X t ∈ U F ( P ). (3.38)For a twisted Poincar´e transformation of other star products, we can verify the twisted Leibnizrule in the same form: X t ( h α | ⋆ | β i ) = X t (ev F ( h α | ⊗ | β i )) = ev F (∆ F ( X t )( h α | ⊗ | β i )) ,X t ( O θ ⋆ | α i ) = X t ( µ F R ( O θ ⊗ | β i )) = µ F R (∆ F ( X t )( O θ ⊗ | β i )) ,X t ( h α | ⋆ O θ ) = X t ( µ F L ( h α | ⊗ O θ )) = µ F L (∆ F ( X t )( h α | ⊗ O θ )) , (3.39)and using these relations and (3.36), we obtain X t ( h α | ⋆ O θ ⋆ | β i ) = X t ( e ev F ( h α | ⊗ O θ ⊗ | β i ))= e ev F ((1 ⊗ ∆ F ) ◦ ∆ F ( X t )( h α | ⊗ O θ ⊗ | β i ))= e ev F ((∆ F ⊗ ◦ ∆ F ( X t )( h α | ⊗ O θ ⊗ | β i )) . (3.40)Finally, we note that the inner product (3.31) is invariant under a twisted Poincar´e transfor-mation, as the inner product in CQFT (3.6) is invariant under a Poincar´e transformation. In fact,using a formula of an antipode of U F ( P ) which corresponds to (3.8),m(( S F ⊗ ◦ ∆ F ( X t )) = ε F ( X t ) (cid:16) = m((1 ⊗ S F ) ◦ ∆ F ( X t )) (cid:17) , (3.41)the action of a twisted Poincar´e transformation to an inner product (i.e., the first equation in(3.39)) is written as X t ( h α | ⋆ | β i ) = h α | ε F ( X t ) | β i . (3.42)Then, from the explicit value of the counit ε F ( X t ) = ε ( X ), (see (3.9)) we find the invariance ofthe inner product h α | ⋆ | β i . Furthermore, from this result and (3.34), we can show that a matrixelement of operators in dNCQFT is also invariant under a twisted Poincar´e transformation. X t ( h α | ⋆ O θ ⋆ | β i ) = h α | ε F ( X t )( O θ ⋆ | β i ) = ( h α | ⋆ O θ ) ε F ( X t ) | β i . (3.43) In section 2, we have seen the correspondence between CQFT and dNCQFT established by(2.7). In this section, we shall prove that this correspondence leads to the correspondence betweenthe Poincar´e symmetry of CQFT and the twisted Poincar´e symmetry of dNCQFT. This statementis precisely expressed in the following theorem: 9 heorem 1.
Let O ( φ ) ∈ M ( φ ) and O ⋆ ( φ θ ) ∈ M F ( φ θ ) be operators related with each otherby (2.7) and (2.8), and | α i be an arbitrary state in the Hilbert space H on which φ and φ θ arerepresented. Then we have O ⋆ ( φ θ ) ⋆ | α i = O ( φ ) | α i . (4.1) Further, this equality holds when one transforms the left hand side by X t ∈ U F ( P ) , and right handside by X ∈ U ( P ) where X t is the same element as X as an element of the algebra: X t ( O ⋆ ( φ θ ) ⋆ | α i ) = X ( O ( φ ) | α i ) , (4.2) or equivalently X t (cid:16) µ F R ( O ⋆ ( φ θ ) ⊗ | α i ) (cid:17) = X (cid:16) µ R ( O ( φ ) ⊗ | α i ) (cid:17) ⇔ µ F R (cid:16) ∆ F ( X t )( O ⋆ ( φ θ ) ⊗ | α i ) (cid:17) = µ R (cid:16) ∆( X )( O ( φ ) ⊗ | α i ) (cid:17) . (4.3) Proof.
It is trivial to prove the first part of this theorem, i.e., (4.1): substituting (2.8) into theleft hand side of (4.1), we immediately obtain the right hand side. To prove the second part, weintroduce the following notation for the twisting element F : F = X i f ′ i ⊗ f ′′ i ∈ U ( P ) ⊗ U ( P ) . (4.4)Using this notation, correspondences between fields (2.7) and between operators (2.8) are rewrittenas φ θ = X i f ′ i ( φ ) f ′′ i , O ⋆ ( φ θ ) = X i f ′ i ( O ( φ )) f ′′ i , (4.5)where f ′′ i is considered as an element not in U F ( P ) but in M F ( φ θ ). In this notation, the inverse F − is given by F − = X i f ′′ i ⊗ f ′ i , (4.6)and thus F · F − = F − · F = 1 ⊗ X i,j f ′ i f ′′ j ⊗ f ′′ i f ′ j = X i,j f ′′ j f ′ i ⊗ f ′ j f ′′ i = 1 ⊗ . (4.7)Since f ′ i and f ′′ i are given by the form of a polynomial of P i and commutative each other, we seefurther X i,j f ′ i f ′′ j ⊗ f ′ j f ′′ i = X i,j f ′′ j f ′ i ⊗ f ′′ i f ′ j = 1 ⊗ . (4.8)To prove (4.3), we first show the following relation: F − (cid:16) X i f ′ i ( O ( φ )) f ′′ i ⊗ | α i (cid:17) = X i,j ( µ ⊗ f ′′ j f ′ i ( O ( φ )) ⊗ f ′′ i ⊗ f ′ j | α i ) . (4.9)For this purpose, we write F − in the equation explicitly by P i :L.H.S of (4.9) = X i,n (cid:18) i (cid:19) n n ! θ i j · · · θ i n j n P i · · · P i n (cid:16) f ′ i ( O ( φ )) f ′′ i (cid:17) ⊗ P j · · · P j n | α i . (4.10)10ince f ′′ i is given by a form of a polynomial of P i and therefore commutes with P i , P i · · · P i n (cid:16) f ′ i ( O ( φ )) f ′′ i (cid:17) = (cid:2) P i , (cid:2) P i , · · · (cid:2) P i n , f ′ i ( O ( φ )) f ′′ i (cid:3) · · · (cid:3)(cid:3) = (cid:2) P i , (cid:2) P i , · · · (cid:2) P i n , f ′ i ( O ( φ )) (cid:3) · · · (cid:3)(cid:3) f ′′ i = (cid:16) P i · · · P i n (cid:0) f ′ i ( O ( φ )) (cid:1)(cid:17) f ′′ i = (cid:0) P i · · · P i n f ′ i ( O ( φ )) (cid:1) f ′′ i . (4.11)Then (4.10) reads(4.10) = X i,n (cid:18) i (cid:19) n n ! θ i j · · · θ i n j n (cid:0) P i · · · P i n f ′ i ( O ( φ )) (cid:1) f ′′ i ⊗ P j · · · P j n | α i = X i,j (cid:0) f ′′ j f ′ i ( O ( φ )) (cid:1) f ′′ i ⊗ f ′ j | α i = X i,j ( µ ⊗ (cid:0) f ′′ j f ′ i ( O ( φ )) ⊗ f ′′ i ⊗ f ′ j | α i (cid:1) (4.12)and this is just the right hand side of (4.9).Using (4.5) and (4.9), the left hand side of (4.3) reads µ F R (cid:16) ∆ F ( X t )( O ⋆ ( φ θ ) ⊗ | α i ) (cid:17) = X i,j µ R (cid:16) ∆( X )( µ ⊗ f ′′ j f ′ i ( O ( φ )) ⊗ f ′′ i ⊗ f ′ j | α i ) (cid:17) = X i,j µ R ◦ ( µ ⊗ (cid:16) (∆ ⊗ ◦ ∆( X )( f ′′ j f ′ i ( O ( φ )) ⊗ f ′′ i ⊗ f ′ j | α i ) (cid:17) = X i,j µ R ◦ (1 ⊗ µ R ) (cid:16) (1 ⊗ ∆) ◦ ∆( X )( f ′′ j f ′ i ( O ( φ )) ⊗ f ′′ i ⊗ f ′ j | α i ) (cid:17) = X i,j X (cid:16) µ R ◦ (1 ⊗ µ R )( f ′′ j f ′ i ( O ( φ )) ⊗ f ′′ i ⊗ f ′ j | α i ) (cid:17) = X i,j X (cid:16) µ R ( f ′′ j f ′ i ( O ( φ )) ⊗ f ′′ i f ′ j | α i ) (cid:17) . (4.13)To show the third equality, we use coassociativity of the coproduct ∆ and associativity of products µ R and µ (3.21). Finally, by using (4.8), we obtain the right hand side of (4.3). That is,R.H.S of (4.13) = X i,j X (cid:16) µ R (( f ′′ j f ′ i ⊗ f ′′ i f ′ j )( O ( φ ) ⊗ | α i )) (cid:17) = X (cid:16) µ R ( O ( φ ) ⊗ | α i ) (cid:17) . (cid:3) (4.14)For completeness, we prove the correspondence between a Poincar´e transformation of the innerproduct and matrix elements in CQFT and a twisted Poincar´e transformation of them in dNCQFT. Theorem 2.
Let O ( φ ) , O ⋆ ( φ θ ) , X t and X be as in Theorem 4.1, and h α | and | β i be arbitraryelements in H ∗ and H respectively. Then we have X t ( h α | ⋆ | β i ) = X ( h α | β i ) , (4.15) and X t ( h α | ⋆ O ⋆ ( φ θ ) ⋆ | β i ) = X ( h α | O ( φ ) | β i . (4.16)11 roof. Since (4.15) is given by the case where O ( φ ) = O ⋆ ( φ θ ) = 1 in (4.16), it is suffice to prove(4.16). This is easily done by using (3.20) and (3.43): X ( h α | O ( φ ) | β i ) = h α | ε ( X ) (cid:0) O ( φ ) | β i (cid:1) = h α | ε F ( X t ) (cid:0) O ⋆ ( φ θ ) ⋆ | β i (cid:1) = X t (cid:0) h α | ⋆ O ⋆ ( φ θ ) ⋆ | β i (cid:1) . (4.17)where we use (4.1) to prove second equality. (cid:3) From the results obtained here, in particular (4.3) and (4.17), one can see that the Poincar´ecovariance of CQFT implies the twisted Poincar´e covariance of dNCQFT. Thus, dNCQFT givesan example of a QFT whose symmetry is described by a quantum group. If the symmetry groupof dNCQFT is restricted to a classical group, it is given by a reduced Poincar´e group, e.g., in thecase of four dimensional space-time, the symmetry group is [ O (1 , × SO (2)] ⋊ T . We have discussed the twisted Poincar´e symmetry of noncommutative QFTs with the deformedquantization (dNCQFT) and their correspondence with the Poincar´e symmetry of standard com-mutative QFTs (CQFT). We have seen that the equivalence in correlation functions betweendNCQFT and CQFT is established by the map (2.7) and have presented the rigorous proof ofthe correspondence between symmetries of the two theories. By use of the map, we can representgenerators of the twisted Poincar´e algebra by operators acting on a Hilbert space on which the fieldoperator of dNCQFT is represented. It is easy to see that a twisted Poincar´e transformation ondNCQFT constructed in this way is translated to a Poincar´e transformation on CQFT by the aidof the map between the two theories. This result is seemingly surprising: the two different typesof symmetries correspond with each other through the QFTs with different types of quantizationschemes. We see that actually, this correspondence is made clear by presenting both symmetriesin terms of a Hopf algebra. From a Hopf algebraic point of view, both the Poincar´e algebra andthe twisted Poincar´e algebra are quantum groups and the only difference is that the former iscocommutative while the latter is noncocommutative.In the process of constructing the twisted Poincar´e algebra, we obtain a conserved chargeassociated to the transformation. This is essential to our analysis, since without such operatorsconstant in time, it would be difficult to construct the Poincar´e algebra in dNCQFT and representthe twisted Poincar´e transformation on the dNCQFT as an operator acting in the Hilbert space.Indeed, it has not been obtained by simple application of the Noether procedure extended to thecase of the twisted Poincar´e algebra [16].In this paper, we have proved the correspondence between symmetries of CQFT and dNCQFTunderlying the equivalence between the two theories. We have mentioned in [7] that the equivalenceof correlation functions may be seen for more general theories. In fact, if we use different non-commutative parameters for the interaction term in (2.2) and for commutator in (2.3), say θ ij and˜ θ ij , respectively, then the resulting dynamics of the theory depends only on the difference betweenthem Θ ij = θ ij − ˜ θ ij . In particular, all the deformed QFTs which have the same value of Θ ij areequivalent to the ordinary noncommutative QFT with the noncommutative parameter Θ ij in theirdynamics. This suggests that all the twisted Poincar´e symmetries in theories sharing the same Θ ij would also correspond each other in their generators and coproduct through a map establishingthe equivalence. However, we cannot apply the method employed here straightforwardly to provethis general correspondence of symmetries, because it is not clear how to construct an operatorassociated to a twisted Poincar´e transformation from the field operator in the noncommutative12FT (or in general dNCQFTs) by the same procedure. This prevent us from representing thetwisted Poincar´e algebra on the Hilbert space carrying the representation of the field operator ofthe theory. In other words, the situation becomes especially simple in the case Θ ij = 0 which wehave considered in this paper. The specialty of the case Θ ij = 0 would be expected from the factthat, at least in classical level, Moyal star products with the same rank but different value of θ ij give rise to Morita equivalent algebras. The difference between the property of Θ ij = 0 theory andthat of Θ ij = 0 theory would reflect this equivalence in classical level. Despite the difficulty in theextension, however, we believe that the result and the method presented in this paper provide aclue to a fuller understanding of the symmetry of the dNCQFT. Acknowledgments
I would like to thank Izumi Tsutsui for many useful discussions and a careful reading of themanuscript.
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