A non-associative quaternion scalar field theory Sergio Giardino∗ Instituto de Matema´tica, Estat´ıstica e Computac¸˜ao Cient´ıfica, Universidade Estadual de Campinas Rua S´ergio Buarque de Holanda 651, 13083-859, Campinas, SP, Brazil Paulo Teotoˆnio-Sobrinho† Instituto de F´ısica, Universidade de S˜ao Paulo, CP 66318, 05315-970 S˜ao Paulo, SP, Brazil. Anon-associativeGroenewold-Moyalplaneisconstructedusingquaternion-valuedfunctionalgebras. Thesymmetrizedmulti-particlestates,thescalarproduct,theannihilation/creationalgebraandthe formulation in terms of a Hopf algebra are also developed. Non-associative quantum algebras in 3 termsofpositionandmomentumoperatorsaregivenasthesimplestexamplesofaframeworkwhose 1 applications may involvestring theory and non-linear quantumfield theory. 0 2 PACSnumbers: n a J 5 1 I. INTRODUCTION ] h Non-commutative geometry [1] has a wide range of applications in quantum field theory [2, 3], in the construction p of non-commutative physical models. These non-commutative theories are associative. A more general framework - could be conceived whereby, in addition to non-commutativity, the algebra is also non-associative. Our aim is to h find an example where non-commutative and non-associativealgebraappears naturally in the context of field theory. t a Since most field theories are based on associative algebra, our aim is to obtain a deformation parameter θ such that m associativity is recovered when θ goes to zero. [ In the following pages, this goal is achieved by means of construction: we start with a field theory where the base space is comprised of RD and target space is comprised of quaternions H. The second step is to deform RD into 2 non-commutativealgebrasuchthat[xµ,xν]=iθνµ. Itturnsoutthatthe resultingalgebraoffieldsisnon-associative. v 9 As expected, when θµν goes to zero, associativity is recovered. 4 Let us consider a quaternion-valued field theory, and write the field : RD H in a symplectic notation as 0 =f0+jf1, so that fi=0,1 :RD C. In this theory, the sources of non-Fcommuta→tivity are the quaternion complex 5 Funits i, j and k =ij = ji. By d→eforming the commutative multiplication of the complex-valued functions fi=0,1 to . − 1 a non-commutative, we obtain a theory with non-associativity as a byproduct of the superposition of the these two 1 different non-commutativities. 2 Non-associativephenomenaappearinmany places,andfurther informationcanbe foundinreviewsonthe subject 1 [4, 5]. However, while non-associativity is common place in algebra [6], examples of non-associativity in physics are : v a collection of disconnected problems. The most obvious proposals for finding a physical phenomenon that may be i described by non-associativity involve the octonion field [7–11]. Although octonion algebra is a standard example of X non-associativity,it does not have an associative limit. Recently, non-associativestructures have appeared in general r relativity, [13–17], string theory [18–20] and brane theory [21–26]. The model proposed in this article is an attempt a to obtain a very simple example of non-associativity where associativity can be recoveredat a suitable limit. The field theory described in this article has a natural interpretation since its target space may be understood as the tangent space of a hyper-complex manifold. In the same way a complex manifold is locally complex, a hyper- complex manifold is locally quaternionic. In the context of super-symmetric models, there can be various types of complex and hyper-complex manifolds as found in super-symmetric extensions of non-linear sigma models [27–31], string compactification on K3 surfaces [32], generalized hyper-K¨ahler applied to string theory [33, 34], and even speculationsonthenatureoftime[35]. Therefore,themodelpresentedherecanbeunderstoodasalinearizedversion of such non-linear sigma models. Our results are related to quaternion quantum mechanics and quantum field theory [35–41]. However, these latter models do not consider multi-particle states, and consequently in these theories it is impossible to build states with ∗Electronicaddress: [email protected] †Electronicaddress: [email protected] 2 particlestatistics,aproblemthathasbeensolvedherebydefiningannihilationandcreationoperatorsofsymmetrized states. Thispaperisorganizedasfollows: inthesecondsectionwepresentthenon-deformedquaternionscalarfieldtheory anditsmulti-particlestatesandstatistics. Inthethirdsectionadeformedalgebraoffunctionsisformulatedaccording to the Groenenwold-Moyalprocedure. We then show that the resulting algebra is non-associative. Examples of non- associative quantum algebra obtained from introducing quaternion unity are presented as well. The last section contains our conclusions and future perspectives. II. THE QUATERNION SCALAR FIELD THEORY The afore mentioned quaternion field theories have only one-particle states. This means that multi-particle states cannot be built according to boson-fermion statistics. In this section this void is filled in the mathematical struc- ture of quaternion field theory following the Hopf algebra formalism of [42], where the Poincar´e group acts on the Groenenwold-Moyalplanewithadeformedcoproduct. Inthissectionthedeformationisthehyper-complexquaternion structure. A second deformation, in the usual multiplication, is introduced in the third section. Poincar´e invariance Ifg isanelementbelongingtothePoincar´ealgebra,theactionofthesymmetryalgebra(⊲)onspace-timefunctions , H must obey F G ∈ g⊲( )=(g⊲ ) (g⊲ ), (1) F ·G F · G where the dot representsordinary multiplication. It is adopted the symplectic notation for quaternionic functions, so that =f0+f1j, with fi=1,2 C functions, and j is the complex element of quaternion algebra, and thus ij = ji. F − − In terms ofHopf algebras,the actionof the elements of analgebraovera productof complex functions is determined by the co-product. By way of example, the translation generator pˆ = i∂ of the Poincar´e group acts on complex x function algebra according to the co-product ∆(pˆ)=1 pˆ+pˆ 1, (2) ⊗ ⊗ which, acting on f, g C with multiplication m, is subject to the consistency constraint m ∆(pˆ)(f g) =pˆ(f g), ∈ ⊗ · where m takes the elements of the tensor product and multiplies them. On the other han(cid:0)d, taking the(cid:1)quaternion functions fj and gj, again with f, g C functions, we obtain m(∆(p)(f g)) = p(f g). This difficulty is solved by − ⊗ 6 · defining a quaternion tensor product, namely (f g) (m n)=(f m) (g n), (3) ⊗ · ⊗ · ⊗ · (f gj) (m n)=(f m¯) (gj n), (4) ⊗ · ⊗ · ⊗ · (f g) (mj n)=(f mj) (g¯ n), (5) ⊗ · ⊗ · ⊗ · (f gj) (mj n)=(f m¯j) (g¯j n). (6) ⊗ · ⊗ · ⊗ · f, g, m and n are complex-valued functions and barred functions are the complex conjugates. This result follows for f, n Haswell. Thiskindofstructureis similartothe Z2 tensorproductfoundinLiesuper-algebras[43]. Adopting ∈ this tensor product the co-product satisfies the identity ∆(pˆqˆ)=∆(pˆ)∆(qˆ), and the first element of a multi-particle quaternionic state is given: a well defined co-product. As the co-product of the translation operator of the Poincar´e algebra has the expression (2), either in the quaternion case or in the complex case, it will have the same behavior when the multiplication operation is deformed according to the Moyal procedure in both cases. Thus the deformed co-product of the rotation operator of the Poincar´e group in the non-commutative complex function algebra [44] is valid for the deformed quaternion spaces discussed in the next section as well. State statistics States endowed with well-defined statistics have a permutation operator which interchanges the positions of the functions describing the particles in a state. As the particles are represented by quaternion functions, for a generic quaternion state =f0 g0+f0 g1j+f1j g0+f1j g1j, (7) F ⊗G ⊗ ⊗ ⊗ ⊗ 3 the following operators are defined: σˆ⊲( )=( ) (8) F ⊗G G⊗F τˆ⊲( )=g0 f0+g1j f¯0+g¯0 f1j+g¯1j f¯1j (9) F ⊗G ⊗ ⊗ ⊗ ⊗ κˆ⊲( )=f0 g0+f¯0 g1j+f1j g¯0+f¯1j g¯1j, (10) F ⊗G ⊗ ⊗ ⊗ ⊗ so that σˆ2 =τˆ2 =κˆ2 =1 and κˆ =σˆτˆ. Symmetrized states and anti-symmetrized states are defined as follows 1 1, 2 ± = (1 τˆ)⊲ 1, 2 , (11) |F F i 2 ± |F F i with 1, 2 = 1 2, the defined states are eigen-states of the permutation operator τˆ according to |F F i F ⊗F τˆ 1, 2 ± = 1, 2 ±. (12) |F F i ±|F F i After defining symmetrized states and anti-symmetrized states, the particle statistics is guaranteed, and a scalar product is needed, which is presented below. The scalar product By expressing a one-particle state as = f0 + f1j , so that the ortogonality condition f gj = fj g = 0 |Fi | i | i h | i h | i holds, a complex-valued scalar product is obtained as a sum of usual scalar products of complex functions = f0 g0 + f1j g1j (13) hF|Gi h | i h | i = f0 g0 + g1 f1 . h | i h | i In the above scalar product, z = z¯ , where z C and , : R4 H. As a consequence, this fact will h F|Gi 6 hF| Gi ∈ F G → result in the splitting of the creation/annihilation operator algebra, as shown in the next item. On the other hand, when defining the scalar product of two-particle states as , , = , , κˆ , (14) hF G|M Ni |F Gi |M Ni (cid:0) (cid:1) = f0 m0 + f1j m1j , (15) h | ihG|Ni h | ihG|Ni it is observed that if = and = , then F M G N 2 = 2 2. (16) |F ⊗G| |F| |G| The scalar product also obeys the necessary self-adjointness condition τˆ , , , = , , τˆ , . (17) |F Gi |M Ni |F Gi |M Ni (cid:0) (cid:1) (cid:0) (cid:1) Thus, the scalar product constructed above is valid for multi-particles, something which has not been observed in previous quaternion quantum theories. Creation and annihilation operators In principle, the creation a† operator and annihilation operator a of a quaternionic state are F F a† =a† +a† and a =a +a . (18) F f0 f1j F f0 f1j However, as the scalar product constructed above is such that z = z¯ , where z C and , :R4 H, h F|Gi6 hF| Gi ∈ F G → the creation/annihilation algebra will be built in terms of a† , a† , a , and a . These operators create complex f0 f1j f0 f1j fields, and thus satisfy commutation rules with the quaternionic unity j, namely zaf0 =af0z, jaf0 =af¯0j, zaf1j =af1jz¯ and jaf1j =af¯1jj. (19) 4 The operator creates/annihilates either a bosonic or a fermionic state, thus the wave-function must be either sym- metrizedoranti-symmetrized. Inordertoconstructthealgebra,the scalarproductmusthavethesameresultasthat obtained by the creation annihilation operators. The necessary scalar products are f g, m n = f, m g, n f, n g, m) (20) ± ± h ⊗ ⊗ i h ih i±h ih f gj, m nj = f, m gj, nj (21) ± ± h ⊗ ⊗ i h ih i fj gj, mj nj = fj, mj g¯j, n¯j fj, n¯j g¯j, mj , (22) ± ± h ⊗ ⊗ i h ih i±h ih i so that the plus sign corresponds to the symmetric bosonic states and the minus sign corresponds to the fermionic anti-symmetric states. For the bosonic case, the operator algebra reproduces the above results as [a , a ]=[a†, a†]=0 (23) f g f g afagj −agjaf¯=a†fa†gj −a†gja†f¯=0 (24) afjagj −ag¯jaf¯j =a†fja†gj −a†g¯ja†f¯j =0 (25) afja†g−a†g¯afj =0 (26) [a , a†]= f g (27) f g h | i afja†gj −a†g¯jaf¯j =hfj|gji. (28) rememberingthatf andgarecomplex-valuedfunctions,andthat a, zb =z a, b and a, zb =z¯ a, jb areadopted. h i h i h i h i On the other hand, for an anti-symmetric fermionic state, the operator algebra is a , a = a†, a† =0 (29) { f g} { f g} afagj +agjaf¯=a†fa†gj +a†gja†f¯=0 (30) afjagj +ag¯jaf¯j =a†fja†gj +a†g¯ja†f¯j =0 (31) afja†g+a†g¯afj =0 (32) a , a† = f g (33) { f g} h | i afja†gj +a†g¯jaf¯j =hfj|gji. (34) Thus the constructed quaternionic scalar field theory has all the structure necessary: Poincar´e invariant one-particle andmulti-particlestates,symmetrizedandanti-symmetrizedstateswithwell-definedstatistics,ascalarproductanda creation/annihilationoperatoralgebra. This theory canbe deformedaccordingto the Groenenwold-Moyalprocedure generalizing the well-known non-commutative complex field theories, and this is carried out in the next section. III. THE DEFORMED PRODUCT Non-commutativegeometryisobtainedbychangingtheordinarycommutativeproductofcomplex-valuedfunctions f and g into the Groenenwod-Moyal(GM) deformed product ∞ i 1 f(x)⋆g(x)=f(x)g(x)+ θi1j1...θjnjn∂ ...∂ f(x)∂ ...∂ g(x), (35) (cid:18)2(cid:19)n i1 in j1 jn nX=1 so that θij is anti-symmetric in its indices. Linear functions generate the commutator between coordinates xi⋆xj xj ⋆xi =θij, (36) − and in the limit where θij 0 the commutative geometry is recovered. Both the commutative product and the → non-commutative product of functions are associative. A more general picture may be obtained by deforming the usual product of quaternion-valued functions according to the GM prescription. The quaternionic-valued functions :RD H over a D dimensionalEuclidean space with coordinates xi are represented by F → − =f1+f2j, so that fa=1,2 :RD C. (37) F → 5 f are defined on a Schwarz space, thus allowing a Fourier transform ˜, where a F ˜ =f˜1+f˜2j, and f˜a(k)= dDxe−ikµxµfa(x). (38) F Z Accordingly, the Weyl symbol of a quaternion function may be introduced as well, so that dDk Wˆ[F]=Wˆ[f1]+Wˆ[f2]j, and Wˆ[fa]=Z (2π)De−ikµxµf˜a. (39) The Weyl symbol allows the GM product to be introduced, thus replacing the usual multiplication, so that the complex-valued functions obey Wˆ[f ⋆f ]=Wˆ[f ]⋆Wˆ[f ], which results in a b a b ⋆ =f1⋆g1+(f1⋆g2)j+j(f¯2⋆g1)+j (f¯2⋆g2)j , (40) F G (cid:0) (cid:1) where the bar means the conjugate of the complex function. This star product of quaternion functions is, of course, non-commutative; nevertheless, it is also non associative, so that ( ⋆ )⋆ ⋆( ⋆ )=0. (41) F G H−F G H 6 This is an interesting byproduct for introducing a non-commutative local structure on a former non-commutative complex structure. This simple theory has a number of possible applications, as cited in the introduction of this paper. Non-associative quaternion quantum algebras The simplest example of a non-associative deformed theory comes from quantum mechanics and its celebrated commutation relation [xˆ,pˆ]=i~, (42) whose ~ 0 limit, or classical limit, turns the operators into a commutative algebra. Introducing the quaternion → complex unity j naturally generates a non-associative structure. As j does not commute with [xˆ,pˆ], it does not associatewiththe productsofthe commutatoranymore. Theassociator(xˆ, j, pˆ)=(xˆj)pˆ xˆ(jpˆ)maybe calculated − inthe specific casewhere the quantumquaternionalgebrais analternativealgebra. Using the Moufangidentities [6], the resulting associator is (xˆ, j, pˆ)=k~. (43) so that k =ij. This example in which quantum mechanics turns to a non-associative theory is somewhat surprising, butitshowsverysimplyhowcombiningnon-commutativestructuresgeneratesanon-associativeone. Inthiscase,the associative limit goes to a commutative complex theory, but this is a classical one. In this sense, the commutativity and associativity are coupled. A non-coupled case comes in the more general framework discussed previously. On the other hand, it is possible to further extend the quantum algebra. Defining the operators 1 1 zˆ† = p+ix and zˆ= p ix (44) √2 √2 − (cid:0) (cid:1) (cid:0) (cid:1) so that [zˆ, zˆ†]=~, and with the use of the associator (43), zˆ, j, zˆ† =(zˆ†, j, zˆ = zˆ, zˆ, j = j, zˆ, zˆ = zˆ†, zˆ†, j = j, zˆ†, zˆ† =0 (45) zˆ†, zˆ,(cid:0)j = (cid:1)j, zˆ†, zˆ =(cid:1) zˆ(cid:0), zˆ†, j (cid:1)= j(cid:0), zˆ, zˆ†(cid:1)= (cid:0) zˆ, j, zˆ(cid:1)= (cid:0)zˆ†, j, zˆ†(cid:1)=j. (46) − − − (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) This is also a non-associativeand non-commutativealgebra,althoughit is not alternativeas that formed by xˆ, pˆand j, but its classical limit is also a quaternion classical quaternion theory as expected. The above examples are the simplest examples of the deformed algebras,whose geometry is to be analyzed in forthcoming studies. 6 IV. CONCLUSION In this article two novel quaternion quantum scalar field theories have been presented. Both of them are non- commutative because of the quaternion nature of their fields. In one of them ordinary commutative multiplication is defined, and in this case a multi-particle quaternion scalar theory has been constructed. The second theory is a deformation of the former one according to the Groenenwold-Moyal procedure. This second theory is a non- commutative and non-associative one. These theories are well-defined, and may be used in a number of physical applications, as the models are quite general. Developments in quaternion scalar fields and non-associativegeometry are the most immediate applications. 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