Groenewold-Moyal Product, α^\star-Cohomology, and Classification of Translation-Invariant Non-Commutative Structures
aa r X i v : . [ m a t h - ph ] F e b [math-ph] Groenewold-Moyal Product, α ∗ -Cohomology, and Classification ofTranslation-Invariant Non-Commutative Structures Amir Abbass Varshovi ∗ School of Mathematics,Institute for Research in Fundamental Sciences (IPM).School of Physics,Institute for Research in Fundamental Sciences (IPM).Tehran-IRAN
Abstract:
The theory of α ∗ -cohomology is studied thoroughly and it is shown that ineach cohomology class there exists a unique 2-cocycle, the harmonic form, which generatesa particular Groenewold-Moyal star product. This leads to an algebraic classification oftranslation-invariant non-commutative structures and shows that any general translation-invariant non-commutative quantum field theory is physically equivalent to a Groenewold-Moyal non-commutative quantum field theory. I. INTRODUCTION
Translation-invariant star products as the most natural generalization of non-commutativeGroenewold-Moyal ⋆ product have been introduced [1, 2] and discussed partially [3–7] in the frame-work of non-commutative quantum field theories by generally considering the pathological behaviorof UV/IR mixing through the renormalization programs. In fact the most prominent motivationof translation-invariant products was essentially inspired by considering the Wick-Voros formalism[8] in deformation quantization approach [1–4, 9–11] as an alternative technique of quantizationfor Groenewold-Moyal star product. But on the other hand, it was then shown [1–3, 9–11] thatWick-Voros non-commutative field theories are physically exactly equivalent to Groenewold-Moyalones at all quantum levels. Particularly Wick-Voros and Groenewold-Moyal star products leadprecisely to the same Green functions and consequently by LSZ theorem result exactly in the samescattering matrix for respectively the Wick-Voros and the Groenewold-Moyal non-commutativeversions of any given renormalizable quantum field theory [1, 2, 5, 7].The equivalence of non-commutative Wick-Voros and Groenewold-Moyal quantum field theorieswas then understood to be intimately correlated to a cohomology theory, so called α ∗ -cohomology [1–3, 5, 7], as an algebraic theory for classifying translation-invariant star products thoroughly inspiredby the Hochschild theory of cohomology. In fact, by definition two translation-invariant complex star products ⋆ and ⋆ on C ∞ ( R m ) are α ∗ -cohomologous if and only if there exists a fixed smooth ∗ Electronic address: [email protected] The star product ⋆ is said to be complex if; ( f ⋆ g ) ∗ = g ∗ ⋆ f ∗ for any f, g ∈ C ∞ ( R m ). unction over R m , say β , such that for any n ≥
1, and any set of { f i } n ⊂ C ∞ ( R m ) one finds that; Z R m f ′ ⋆ ... ⋆ f ′ n = Z R m f ⋆ ... ⋆ f n , (I.1)for: f ′ ( x ) = Z d m p (2 π ) m e ip.x ˜ f ( p ) e β ( p ) , (I.2) f ∈ C ∞ ( R m ), and for ˜ f the Fourier transform of f [7].It is seen that (I.1) leads to the most general classification of ⋆ products from the viewpoints ofquantum physics, so called the quantum equivalence [7]. By definition, two star products ⋆ and ⋆ are quantum equivalent if and only if there exists a fixed β ∈ C ∞ ( R m ), with β (0) = 0, such that forany n ≥
1, the equality ˜ G ⋆ conn. ( p , ..., p n ) = e P ni =1 β ( p i ) ˜ G ⋆ conn. ( p , ..., p n ) (I.3)holds for any given renormalizable quantum field theory, where G conn. is any connected n -pointfunction, G ⋆ conn. is its non-commutative version for the star product ⋆ and ˜ G ⋆ conn. ( p , ..., p n ) is itsFourier transform for the modes { p i } ni =1 .Therefore it is seen [7] that the star products ⋆ and ⋆ are α ∗ -cohomologous if and only ifthey are quantum equivalent. Consequently, all the quantum behaviors of two α ∗ -cohomologousnon-commutative translation-invariant versions of a fixed given quantum field theory are exactly thesame. This result can be considered as an algebraic proof for physical equivalence of Wick-Voros andGroenewold-Moyal non-commutative field theories provided these star products are α ∗ -cohomologous.In this article translation-invariant products are studied in the framework of α ∗ -cohomologytheory. It is precisely shown that for any complex translation-invariant product ⋆ there exists aparticular Groenewold-Moyal star product that is α ∗ -cohomologous to ⋆ . This eventually can beconsidered as a general version of the theorem which equalizes the Wick-Voros and the Groenewold-Moyal star products from the viewpoints of quantum physics. It is then strictly concluded thatthe non-commutative structure of space-time given by the commutation relation of coordinatefunctions via the star product, entirely characterizes the structure of abnormal quantum behaviorsof non-commutative quantum field theories such as the structure of UV/IR mixings. While thiscorrelation of the non-commutative structures of space-time and abnormal quantum behaviors wasintuitively conjectured [12] and partially proved [1–3, 9] only for Wick-Voros and Groenewol-Moyalformalisms, but here it is accurately provided a strict algebraic proof for all general cases.In section II, α ∗ -cohomology theory and its Hodge theorem are introduced and discussed referringto [7]. In section III it is shown that the harmonic forms due to the Hodge theorem, lead precisely toGroenewold-Moyal star products. Some algebraic theorems are also worked out in the following.2 I. α ∗ -COHOMOLOGY AND THE HODGE THEOREM As a definition a translation-invariant star product on R m with respect to coordinate system { x i } mi =1 is an associative multiplication over C ∞ ( R m ) which manifestly doesn’t depend on the coordinatefunctions x i , i = 1 , ..., m . Hence, more precisely, from the physical viewpoints a translation-invariantstar product preserves the behavior of any Lagrangian density under translation when it is usedinstead of the ordinary product. Consequently translation-invariant star products lead to the energy-momentum conservation law for any relativistic quantum filed theory. Strictly speaking star product ⋆ on C ∞ ( R m ) is translation-invariant if; T a ( f ) ⋆ T a ( g ) = T a ( f ⋆ g ) , (II.1)for any vector a ∈ R m and for any f, g ∈ C ∞ ( R m ), where T a , is the translating operator; T a ( f )( x ) = f ( x + a ), f ∈ C ∞ ( R m ). Replacing a with ta , t ∈ R , in (II.1) and differentiating with respect to t at t = 0, the most important property of translation-invariant products, the exactness, is inferred; ∂ µ ( f ⋆ g ) = ∂ µ ( f ) ⋆ g + f ⋆ ∂ µ ( g ) , (II.2) µ = 1 , ..., m . The exactness property shows that the star product ⋆ as a function is not given in termsof the coordinate functions of which are intrinsically encoded in the translation operator T .An equivalent simple definition of translation-invariant products over the Cartesian space R m , isgiven by [1]; ( f ⋆ g )( x ) := Z d m p (2 π ) m d m q (2 π ) m ˜ f ( q )˜ g ( p ) e α ( p + q,q ) e i ( p + q ) .x , (II.3)for f, g ∈ C ∞ ( R m ), their Fourier transformations ˜ f , ˜ g ∈ C ∞ ( R m ), and finally for a 2-cocycle α ∈ C ∞ ( R m × R m ) which obeys the following cyclic property; α ( p, q ) + α ( q, r ) = α ( p, r ) + α ( p − r, q − r ) , (II.4)for any p, q, r ∈ R m . It is seen that (II.4) holds if and only if ⋆ is associative, i.e.; ( f ⋆ g ) ⋆ h = f ⋆ ( g ⋆ h )for f, g, h ∈ C ∞ ( R m ).To have a well-defined definition for translation-invariant products, C ∞ ( R m ) is conventionallyreplaced by S c ( R m ), the Schwartz class functions with compactly supported Fourier transforms [7].On the other hand S c ( R m ) can naturally be extended to a unital algebra with; S c, ( R m ) := S c ( R m ) ⊕ C .It is obvious that for any f ∈ S c, ( R m ), 1 ⋆ f = f ⋆ f if and only if; α ( p, p ) = α ( p,
0) = 0 , (II.5)for any p ∈ R m . Combining (II.4) and (II.5) leads to; α (0 , p ) = α (0 , − p ) , (II.6)for any p ∈ R m . Using (II.6) it can also be shown that any translation-invariant product admits thetrace property; Z R m f ⋆ ... ⋆ f k − ⋆ f k = Z R m f k ⋆ f ⋆ ... ⋆ f k − , (II.7)3or any k ∈ N and for any set of f , ..., f k − , f k ∈ S c ( R m ).To characterize the translation-invariant star products effectively, 2-cocycles α should be cate-gorized and classified appropriately. Conventionally, the 2-cocycles are studied in the setting of acohomology theory [1, 2, 5, 7] usually referred to as α -cohomology. By definition [7] the α -cohomologygroups are the cohomology groups of the following complex;C ( R m ) ∂ −→ C ( R m ) ∂ −→ ... ∂ n − −→ C n ( R m ) ∂ n −→ ... (II.8)with; • C ( R m ) := { } , • For n = 1; C ( R m ) := { f ∈ C ∞ ( R m ) | f (0) = 0 } , • For n = 2; C ( R m ) := { f ∈ C ∞ ( R m × R m ) | f ( p,
0) = f ( p, p ) = 0; p ∈ R m } , • For n ≥
3; C n ( R m ) ⊆ C ∞ ( R m × ... × R m | {z } n − fold ) consists of smooth functions f with propertiesof f ( p , ..., p n − ,
0) = f ( p , ..., p k , p, p, p k +1 , ..., p n − ) = 0, k ≤ n −
2, for any p, p , ..., p n − ∈ R m ,and for the linear maps ∂ n : C n ( R m ) −→ C n +1 ( R m ) , (II.9)usually denoted by ∂ , defined by; ∂ n f ( p , ..., p n ) := ε n n X i =0 f ( p , ..., p i − , ˆ p i , p i +1 , ..., p n ) + ε n ( − ) n +1 f ( p − p n , ..., p n − − p n ) , (II.10) f ∈ C n ( R m ), with ε n = 1 for odd n and ε n = i for n even. One should note that; ∂ = ∂ n ◦ ∂ n − = 0for any n ∈ N .Conventionally the notation of α ∼ α is used for two α -cohomologous n -cocycles α and α . Alsothe cohomology class of α ∈ Ker∂ n is shown by [ α ]. Therefore, the α -cohomolgy group, H nα ( R m ) := Ker∂ n /Im∂ n − , classifies n -cocycles differing in coboundary terms into the same equivalence classes.Now consider the translation-invariant products given by α ∈ C ∞ ( R m × R m ) due to definition (II.3).According to (II.4), associativity of ⋆ is equivalent to ∂α = 0. Indeed, H α ( R m ) classifies all thetranslation-invariant quantization structures over S c, ( R m ) modulo the coboundary terms. It can beeasily seen from (II.10) that if [ α ] = 0 then α leads to a commutative star product ⋆ , i.e.; f ⋆ g = g ⋆ f , (II.11) f, g ∈ S c, ( R m ). In [7] it has been proven that α leads to a commutative star product if and only if[ α ] = 0. Therefore, α ∼ α if and only if α − α generates a commutative product.4t can be shown that [7] there exists an algebraic version of Hodge theorem for H α ( R m ). Moreprecisely, according to [7] for any given α -cohomology class [ α ] ∈ H α ( R m ), there exists a unique2-cocycle, conventionally referred to as the harmonic form, which obeys the following properties; α ( p, q ) = − α ( p, p − q ) α ( p, q ) = α ( − p, − q ) ,α ( p, q ) = − α ( q, p ) (II.12)for any p, q ∈ R m .Specially for any 2-cocycle α its α -cohomologous harmonic form, α H , is given by; α H ( p, q ) = α ( p + q, q ) − α ( p + q, p )2 , (II.13)for any p, q ∈ R m .The set of all pure imaginary elements of H α ( R m ), denoted by H α ∗ ( R m ), also defines a cohomologytheory as a sub-theory of α -cohomology, called the α ∗ -cohomology, which also admits the Hodgetheorem [7]. Particularly, H α ∗ ( R m ) classifies the complex translation-invariant products modulo thecommutative ones. Consequently, according to the quantum equivalence theorem [7] due to (I.3), H α ∗ ( R m ) classifies all the physically equivalent translation-invariant non-commutative versions ofquantum field theories.For instance the Groenewold-Moyal star product, ⋆ G − M , and the Wick-Voros star product, ⋆ W − V , according to (II.3) are respectively defined with 2-cocycles α G − M ( p, q ) = iq µ θ Aµν p ν and α W − V ( p, q ) = α G − M ( p, q ) + q µ θ Sµν ( p − q ) ν , p, q ∈ R m , for θ A an anti-symmetric and θ S a symmetricreal matrix. Therefore, it is seen that α G − M and α W − V are α ∗ -cohomologous and belong to thesame class of H α ∗ ( R m ) denoted by [ α G − M ]. Consequently, according to the quantum equivalencetheorem, the Groenewold-Moyal and the Wick-Voros non-commutative versions of any quantum fieldtheory lead to the same physics, the fact of which confirms the results of [1, 3, 9]. On the otherhand, according to (II.12) it is easily seen that α G − M is a harmonic form. More precisely, anyGroenewold-Moyal star product defines a particular class of H α ∗ ( R m ). III. HARMONIC FORMS AND GROENEWOLD-MOYAL STAR PRODUCTS
In the last section it was finally shown that any given Groenewold-Moyal star product defines aparticular class of H α ∗ ( R m ). Following this statement in this section it is shown that the converse istrue, i.e. any arbitrary class of H α ∗ ( R m ) is uniquely characterized by a particular Groenewold-Moyalstar product. To see this fact explicitly one initially needs to consider an arbitrary 2-cocycle α whichsatisfies the condition (II.4). Showing the first and the second arguments of 2-cocycle α respectivelyby z and z ′ , and then taking the partial derivative of (II.4) with respect to r i at r = 0, one finds that; ∂α∂z ′ i ( q,
0) = ∂α∂z ′ i ( p, − ∂α∂z i ( p, q ) − ∂α∂z ′ i ( p, q ) , (III.1)5 , q ∈ R m . Taking the partial derivative of (III.1) with respect to q i leads to; ∂ α∂z ′ j ∂z i ( q,
0) = − ∂ α∂z ′ i ∂z j ( p, q ) − ∂ α∂z ′ i ∂z ′ j ( p, q ) , (III.2) p, q ∈ R m . Eventually, one needs to evaluate the partial derivative of (III.2) with respect to p i ;0 = ∂ α∂z i ∂z ′ j ∂z k ( p, q ) + ∂ α∂z i ∂z ′ j ∂z ′ k ( p, q ) , (III.3) p, q ∈ R m . More precisely, it is seen that;( ∂∂z k + ∂∂z ′ k ) ∂ α∂z i ∂z ′ j ( p, q ) = 0 , (III.4) p, q ∈ R m . Using the coordinate transformation ( z + i = ( z i + z ′ i ) / √ ,z − i = ( z i − z ′ i ) / √ , (III.5)one easily finds; ∂∂z + k ∂ α∂z i ∂z ′ j ( p, q ) = 0 , (III.6) p, q ∈ R m . Therefore; ∂ α∂z i ∂z ′ j ( p, q ) = σ ( α ) ij ( p − q ) , (III.7) p, q ∈ R m .According to the last property of (II.12), if α is harmonic it is anti-symmetric under the exchangeof p and q . Thus, for α a harmonic form one finds that; σ ( α ) ij ( p − q ) = − σ ( α ) ji ( q − p ) , (III.8) p, q ∈ R m . Moreover, if α is harmonic then by the second property of (II.12) it is seen that; σ ( α ) ij ( p − q ) = σ ( α ) ij ( q − p ) , (III.9) p, q ∈ R m . Hence, by (III.8) and (III.9); σ ( α ) ij ( p − q ) = − σ ( α ) ji ( p − q ) , (III.10) p, q ∈ R m . Finally it can be seen that if α is harmonic then; σ ( α ) ij ( p − q ) = σ ( α ) ij ( q ) + ∂ α∂z ′ i ∂z ′ j ( p, p − q ) , (III.11) p, q ∈ R m , provided by the first property of (II.12).According to (III.10), σ ( α ) ij is anti-symmetric under the exchange of indices i and j . Therefore by(III.11) one concludes that; ∂ α∂z ′ i ∂z ′ j ( p, q ) = 0 , (III.12)6or any p, q ∈ R m , and consequently (III.11) leads to; σ ( α ) ij = θ ij ∈ C , (III.13)for any harmonic form α . Eventually from (II.12) and (III.13) it is obvious that α is a harmonic formif and only if α ( p, q ) = p i θ ij q j , (III.14)for any p, q ∈ R m , and for θ an anti-symmetric constant matrix. Therefore, by considering the pureimaginary harmonic forms, one concludes that H α ∗ ( R m ) is exactly the collection of Groenewold-Moyalstar products. This lets one to characterize the cohomology groups H α ( R m ) and H α ∗ ( R m ) thoroughlywith anti-symmetric m × m matrices. In fact, according to the Hodge theorem for α -cohomology it isseen that; H α ( R m ) = { θ ∈ M m × m ( C ) | θ is anti-symmetric } . (III.15)Therefore, dim H α ( R m ) = m ( m − H α ∗ ( R m ) is the collection ofpure imaginary elements of H α ( R m ), thus; H α ∗ ( R m ) = { θ ∈ M m × m ( R ) | θ is anti-symmetric } . (III.16)Consequently; dim H α ∗ ( R m ) = m ( m − / H α ∗ ( R m ). Conversely, (III.14) asserts that any class of H α ∗ ( R m ) is uniquely characterized bya particular Groenewold-Moyal star product. Therefore, one naturally concludes that any complex 2-cocycle α is α ∗ -cohomologous to a particular Groenewold-Moyal 2-cocycle. Particularly (III.14) showsthat for any complex translation-invariant star product ⋆ i , there is a unique Groenewold-Moyal starproduct, say ⋆ i/G − M , such that; ⋆ i ∼ ⋆ i/G − M . (III.17)Using (III.17) and the quantum equivalence theorem due to (I.3) it is seen that for any generaltranslation-invariant non-commutative quantum field theory there is a particular Gronwold-Moyalnon-commutative quantum field theory with exactly the same effects and physical out-comings such as n -point functions and the scattering matrix. Then studying the Groenewold-Moyal non-commutativequantum field theories covers the whole domain of translation-invariant non-commutative quantumfield theories.Translation-invariant star products also can be defined over the polynomials of coordinate func-tions. This leads to non-commutative structures of space-time. More precisely, the non-commutativestructure of space-time due to 2-cocyle α is given by;[ x i , x j ] ⋆ = x i ⋆ x j − x j ⋆ x i = ∂ α∂z j ∂z ′ i (0 , − ∂ α∂z i ∂z ′ j (0 , , (III.18)7 , j = 1 , ..., m , for ⋆ the translation-invariant star product induced by α . It can be seen that if ⋆ iscommutative or equivalently [7] if α = ∂β for 1-cochain β , then;[ x i , x j ] ⋆ = 0 , (III.19) i, j = 1 , ..., m . Equality (III.19) shows that the non-commutative structure of space-time is particularlygiven by the α ∗ -cohomology class of the 2-cocycle. Actually if ⋆ ∼ ⋆ then[ x i , x j ] ⋆ = [ x i , x j ] ⋆ , (III.20) i, j = 1 , ..., m . Consequently the non-commutative structure of space-time due to 2-cocyle α canbe precisely given by its α ∗ -cohomologous harmonic form or more clearly by its α ∗ -cohomologousGroenewold-Moyal star product. In fact, if ⋆ is induced by 2-cocycle α with α ( p, q ) ∼ ip i θ ij q j , p, q ∈ R m , θ ij ∈ R , then; [ x i , x j ] = iθ ij , (III.21) i, j = 1 , ..., m . Equality (III.21) can be considered as the converse proposition for statement (III.20).In fact, (III.21) asserts that if (III.20) holds for complex translation-invariant star products ⋆ and ⋆ , then ⋆ ∼ ⋆ . Consequently by (III.20) and (III.21) one naturally concludes that ⋆ ∼ ⋆ if and only if ⋆ and ⋆ lead to the same non-commutative structure of space-time. Therefore, H α ∗ ( R m ) classifies the non-commutative structures of space-time. One important consequence ofthis achievement with regard to the quantum equivalence theorem is that the non-commutativestructure of space-time thoroughly explains the structure of quantum behaviors of non-commutativequantum field theories. More precisely, the only fundamental data through the quantum physicspoints of view is the non-commutative structure of space-time, but not the analysis of the starproduct. This fact was partly proved for Wick-Voros and Groenewold-Moyal non-commutativestar products [1–3, 9], but here it has been provided a general proof for all cases. In fact, (III.21)can be considered as a modified version of Kontsevich’s theorem [16] which asserts that there is aone to one correspondence between Poisson structures and equivalent star products over a smoothmanifold, noting that any non-commutative structure of space-time is essentially a Poisson structure.But there is a particular difference between Kontsevich’s theorem and equation (III.21): There isconsidered no symmetry in the Kontsevich’s theorem for equivalent star products, while here starproducts are classified with insistence on translation-invariance. Moreover, there is no attention toquantum behaviors via classification of star products in Kontsevich’s theorem, while here the criticalproperty of our classification is preserving the quantum effects due to quantum equivalence theorem [7].On the other hand, by (III.14) and (III.21) one simply concludes that there is not any non-commutative translation-invariant star product on S c, ( R m ) which leads to commutative space-time.More precisely, there is no translation-invariant non-commutative star product on S c, ( R m ) whichis commutative at the level of coordinate functions. Therefore, due to path integral formalismwhere the integration is taken over S c, ( R m ), commutative space-time never admits non-commutativetranslation-invariant quantum field theories. 8ne of the other important conclusions of (III.14) and the quantum equivalence theorem due to(I.3) is that the Grosse-Wulkenhaar approach [13, 14] and the method of 1 /p [15] also work properlyfor any given translation-invariant non-commutative version of φ theory. On the other hand, it cansimilarly be concluded that any proposal for renormalizing the Groenewold-Moyal non-commutativegauge theories extends thoroughly to the collection of all translation-invariant non-commutativegauge theories. More precisely, since any given complex 2-cocycle α can be uniquely decomposed to α = α G − M + ∂β due to the Hodge theorem in α ∗ -cohomology [7], the regularization methods forFeynman diagrams in Groenewold-Moyal non-commutative quantum field theories work well for alltranslation-invariant quantum field theories due to canceling out the coboundary terms, such as ∂β ,for internal momenta of loop calculations [7].It has already been shown that the Groenewold-Moyal non-commutative versions of relativisticquantum field theories admit the Drinfeld’s twist of Poincare invariance as a modified concept ofrelativity [2, 9, 17–19]. More precisely, it has been shown that the algebra of S c, ( R m ) ⋆ G − M is also analgebra in the category of U ( P m ) χ G − M -modules [20], where U ( P m ) is the universal enveloping algebraof the Poincare Lie algebra P m for { M µ,ν } m − µ,ν =0 and { P µ } m − µ =0 , respectively the Lorentz and translationLie algebra generators, and U ( P m ) χ G − M is its Drinfeld’s twist due to counital Groenewold-Moyal2-cocycle χ G − M := exp( − i ~ − θ µν P µ N P ν ) [21]. Due to the Hodge decomposition theorem for α ∗ -cohomology which uniquely splits any given 2-cocycle α to a Groenewold-Moyal 2-cocycle, α G − M ,and a coboundary term, say ∂β , it can be easily seen that the algebra of S c, ( R m ) ⋆ is also an algebra inthe category of U ( P m ) χ -modules for χ := exp( β ( ~P N β (1 N ~P )) χ G − M exp( − β ( ~P N N ~P )),where ⋆ is generated by 2-cocycle α G − M + ∂β . Therefore, any translation-invariant non-commutativeversion of a quantum field theory admits the twisted Poincare symmetry (due to the Drinfeld’stwist of Poincare universal enveloping algebra, U ( P m ), for counital 2-cocycle χ ) as a modifiedmeaning of relativistic invariance. It would also be interesting to note that χ = ∂ + γ χ G − M ∂ − γ for counital 1-cochain γ = exp( β ( ~P )), and therefore χ and χ G − M are cohomologous in the thesecond cohomology space of Poincare universal enveloping Hopf algebra H ( U ( P m )) [21], i.e.; theDrinfeld’s twist of Poincare universal enveloping algebra U ( P m ) due to counital 2-cocycles χ G − M and χ lead to isomorphic Hopf algebras. Thus, the cohomology space of H ( U ( P m )) also classifiestranslation-invariant quantum field theories with the same quantum behaviors. Moreover, it canbe easily seen that H ( U ( T m )) ∼ = H α ( R m ) where H ( U ( T m )) is the second cohomology group ofcommutative universal enveloping Hopf algebra U ( T m ) for m -dimensional translation Lie algebra T m generated by { P µ } m − µ =0 . In fact, the classification of translation-invariant non-commutative quantumfield theories due to quantum equivalence theorem of α ∗ -cohomology can also be worked out in thesetting of cohomology spaces of quantum groups.Finally according to (III.14) it can be seen that the star product due to 2-cocycle α is naturallyreflected by a modified version of Weyl map [22]. More precisely if α ( p, q ) = ip i θ ij q j + ∂β ( p, q ), p, q ∈ R m , θ i j ∈ R , i, j = 1 , ..., m , for 1-cochain β , then the star product of ⋆ according to (II.3)is particularly reflected by the Weyl-Wigner correspondence [22, 23] due to the following modified9ersion of Weyl map; ˆ f = Z d m p (2 π ) m e i P mi =1 p i ˆ x i ˜ f ( p ) e β ( p ) , (III.22)for f ∈ C ∞ ( R m ), ˜ f its Fourier transform and for ˆ f its corresponding operator with [ˆ x i , ˆ x j ] = iθ ij ,1 ≤ i, j ≤ m . Particularly, (III.14) asserts that any translation-invariant non-commutative starproduct is reflected by a modified version of Moyal-Weyl-Wigner quantization due to (III.22). IV. CONCLUSIONS
In this article α ∗ -cohomology was studied thoroughly and it was shown that in each cohomologyclass there is a unique 2-cocycle, the harmonic form [7], which generates a particular Groenewold-Moyal star product. According to [7] where it was shown that any two α ∗ -cohomologous 2-cocycleslead precisely to two equivalent quantum field theories, i.e. two quantum field theories with ex-actly the same scattering matrix, a one to one correspondence between the collection of Groenewold-Moyal ⋆ products and the set of quantum equivalent translation-invariant non-commutative quantumfield theories was then worked out. More precisely, in this article it was shown that for any gen-eral translation-invariant quantum field theory there is a unique Groenewold-Moyal non-commutativequantum field theory with the same scattering matrix. As a corollary one concludes that studding onlythe Groenewold-Moyal non-commutative quantum field theories covers thoroughly the whole domainof translation-invariant non-commutative quantum field theories. On the other hand, it was explicitlyshown that the non-commutative structure of space-time entirely describes the quantum behaviorsof translation-invariant non-commutative quantum field theories, the conjecture of which was neverprecisely proved before. Consequently, it was particularly proved that for a fixed quantum field theorytwo of its non-commutative versions with complex translation-invariant star products ⋆ and ⋆ ? arequantum equivalent if and only if ⋆ and ⋆ lead to the same non-commutative structure for space-time.Moreover, it was then discussed that the Grosse-Wulkenhaar approach and the method of 1 /p alsowork properly for any given translation-invariant non-commutative version of φ theory. As a conclu-sion, it was illustrated that any proposal for renormalizing the Groenewold-Moyal non-commutativegauge theories extends thoroughly to the collection of all translation-invariant non-commutative gaugetheories. It was also shown that any translation-invariant (non-commutative) quantum field theoryadmits a modified structure of relativistic invariance via the twisted Poincare symmetry due to its starproduct. Finally it was precisely established that any given translation-invariant non-commutativestar product is reflected by a modified version of Moyal-Weyl-Wigner quantization. V. ACKNOWLEDGMENTS
The author should say his special gratitude to M. M. Sheikh-Jabbari for use-full comments and fruit-full discussions. Moreover, my deepest thanks go to A. Shafiei Deh Abad for his kind considerationsand his motivating ideas. Also I should confess that this article owes most of its appearance to S.10iaee, whom my deepest regards goes to for many things. [1] S. Galluccio, F. Lizzi and P. Vitale,
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