From Kontsevich Graphs to Feynman graphs, a Viewpoint from the Star Products of Scalar Fields
aa r X i v : . [ m a t h - ph ] M a y From Kontsevich Graphs to Feynman graphs, a Viewpointfrom the Star Products of Scalar Fields
Zhou Mai ∗ May 6, 2020
Abstract
In the present paper we construct the star products concerning scalar fields in thecovariant case from a new approach. We construct the star products at three levels,which are levels of functions on R d , fields and functionals respectively. We emphasesthat the star product at level of functions is essence and starting point for oursetting. Firstly the star product of functions includes all algebraic and combinatorialinformation of the star products concerning the scalar fields and functionals almost.Secondly, a more interesting point is that the star product of functions concernsonly finite dimensional issue, which is a Moyal-like star product on R d generatedby a bi-vector field with abstract coefficients. Thus the Kontsevich graphs playsome roles naturally. Actually we prove that there is an ono-one correspondencebetween a class of Kontsevich graphs and the Feynman graphs. Additionally theWick theorem, Wick power and the expectation of Wick-monomial are discussed interms of the star product at level of functions. Our construction can be consideredas the generalisation of the star products in perturbative algebraic quantum fieldstheory and twist product introduced in [1],[2]. Contents ∗ address:Colleague of Mathematical Science, Nankai University, Weijin Road, Tianjin City, RepublicChina; email address: [email protected] Wick theorem and Wick power 165 The star products at levels of fields and functionals 23
The deformation quantisation of the fields is a infinite dimensional issue essentially.Up to now, there are a lot of works about the deformation quantisation in the infinitedimensional case (for example, see [5],[6],[7],[8],[9],[10], where the list of references isnot complete). In some sense the star products in the infinite dimensional space wereconstructed as the copies of the classical Moyal product frequently, a typical point is thatthe partial derivatives in the classical Moyal bi-vector field are replaced by variationalderivatives, for example, Frechet derivative or others. If we focus on the deformationquantisation of the fields, for example, the case of scalar fields , we need to pay attentionto the following two facts. The first fact is the commutative relation (or Poisson bracket): { ϕ ( x ) , ϕ ( y ) } = K ( x , y ) , where K ( x , y ) is some propagator and ϕ ( x ) is a scalar field in some physical theory.Above commutative relation suggests the possibility of the Moyal-like product. Anotherone is the variational calculation for a specific functional, for example F ( ϕ ) = Z f ( ϕ ( x ) , · · · , ϕ ( x d )) d x · · · d x d , the variation of F ( ϕ ) can be calculated in terms of the partial derivatives of function f ( y , · · · , y d ). This fact suggests us to work in finite dimensional case probably. Oursetup is motivated by the facts mentioned above.In the present article we discuss a new approach to the deformation quantisationof scalar fields in the covariant case, and then as a main application of our construc-tion about the star product we discuss the connection between the Kontsevich graphs(see [13]) and the Feynman graphs. Other applications are discussed also. Now wemake some explanation about function f ( y , · · · , y d ) at level of terminology. In gen-eral f ( ϕ ( x ) , · · · , ϕ ( x d )) (or more precisely f ( ϕ ( x ) , · · · , ϕ ( x d )) dV where dV is volumeform) is called density from the viewpoint of variational theory in classical fields theory.Here we need to distinguish between the f ( y , · · · , y d ) and f ( ϕ ( x ) , · · · , ϕ ( x d )), so wecall the function f ( y , · · · , y d ) the density function, or function for short.Our approach to the star products is divided into three steps. The first step is toconstruct the star products at level of functions. As the second step, the star productof fields, or densities in the sense mentioned above, can be constructed from the firststep simply. Finally, the star product at level of functionals can be costructed basedon the second step. We show our idea in the following table. Our construction canbe considered as a generalisation of covariant deformation quantisation of the fieldsin perturbative algebraic quantum fields theory (see [7], [8], [9]) and twisted product2 ( x i ) ⋆ g ( y j ) functions ↓ f ( ϕ ) ⋆ g ( ϕ )= f ( x i ) ⋆ g ( y j ) | x i = ϕ ( · ) ,y i = ϕ ( · ) fields or densities ↓ R f ( ϕ ) ⋆ g ( ϕ ) functionalsTable 1: defaultintroduced in [1], [2]. Somehow the main outline of our construction is along the idea inour earlier work (see [15]).The basic starting point of our discussion is the construction of the star product atlevel of functions. The deformation quantisation in the case of the fields is the infinitedimensional issue basically. But in our approach, as a key point, the construction ofthe star product at level of functions involves the finite dimensional issue only. Ourdiscussion below will show that the star product of functions contains all algebraic andcombinatorial information of deformation quantisation of the fields and functionals al-most. Actually, it will be showed that everything can be explicitly calculated based onthe calculations at level of functions almost.Here the star product of functions is a Moyal-like one in R d . The bi-vector field in theMoyal product is replaced by a bi-vector field with abstract coefficients, this bi-vectorfield generates the star product of the functions in our setting. With the help of theMoyal-like product in finite dimensional space our discussion goes into the framework ofKontsevich naturally. We prove that for a special class of the Kontsevich graphs, herewe call that the graphs of Bernoulli type, there is an one-one correspondence betweenthe graphs of Bernoulli type and the Feynman graphs. In this paper we consider onlythe Feynman graphs without self-lines. It is well known that the Moyal product is thesimplest example in the theory of deformation quantisation on the Poisson manifolds,thus the Kontsevich graphs involving the Moyal product should be the simplest case. Oursetting is completely parallel to the Moyal product from the viewpoint of the Kontsevichgraphs. Roughly speaking the set of the graphs of Bernoulli type is generated by aspecial Bernoulli graph (see [11], [12]) which may be the simplest, but non-trivial, grapheven in Bernoulli graphs. We will see that the forms of the graphs of Bernoulli typeunder the structure of product of admissible graphs (see [11], [12]) look like the Feynmanamplitudes very much. This similarity results in the existence of one-one correspondencementioned above.Moreover, as another application of our construction we discuss the various forms ofWick theorem, Wick power and expectation of Wick-monomial in terms of the coordi-nates in R d from the viewpoint of the star product of functions. In the sense of the starproduct the Wick theorem, Wick power and expectation of Wick-monomial for the caseof scalar fields can be obtained from their various forms mentioned above. Observingthe procedure to calculate the star product we find the Feynman amplitudes arise from3he bi-vector field, that explains also why the Kontsevich graphs are relevant to theFeynman graphs.This paper is organised as the following. In section 2 we discuss the star products atlevel of functions. The definitions of star product and Poisson bracket are presented andsome properties are discussed. In section 3 we recall some contents of the Kontsevichgraphs including admissible graphs and their product, Bernoulli graphs, et cetera(see[11], [12], [13]). A combinatorial notation, adjacency matrix, is introduced. In the endof this section we prove the existence of one-one correspondence between the graphs ofBernoulli type and the Feynman graphs. In section 4 we discuss the Wick theorem,Wick power and notion of expectation of Wick-monomial from the viewpoint of starproduct of functions on R d . Here everything is expressed in terms of functions, orspecial, coordinates on R d . In section 5 we discuss the star products of the fields andfunctionals based on the star product of functions on R d . In this section we discuss the star products of functions which plays the role of underlyingstructure about the star products of scalar fields and functionals, moreover, includes allof combinatorial and algebraic information concerning the star products of scalar fieldsalmost.
At first we introduce some notations. Let A be a commutative algebra over R (or C )with finite generators, we consider a free C ∞ ( R d ) module on A denoted by C ∞A , C ∞A = M i ∈A C ∞ ( R d ) . The elements in C ∞A are the linear combinations of the elements in A with coefficientsin C ∞ ( R d ). The partial derivations on C ∞ ( R d ) can be extended to C ∞A where theelements in A are viewed as constants, for example, we have ∂ i ( λf ( x )) = λ∂ i f ( x ) ,λ ∈ A , f ( x ) ∈ C ∞ ( R d ), here we have used the short symbols, x = ( x , · · · , x d ), ∂ i = ∂ x i .In the present article we focus on the situation of real scalar fields, the case of complexones are similar, thus we discuss the problems over real number field R below.Now we consider the derivations of the tenser of functions. Let f i ( x ) ∈ C ∞ ( R d ) , i =1 , · · · , m . We define the partial derivations for tenser of the functions f ( x ) ⊗ · · · ⊗ f m ( x m ) as following: ∂ i, ⊗ m ( m O j =1 f j ( x j )) = ddt ( m O j =1 f j ( · · · , x ( i ) j + t, · · · )) | t =0 . (2.1)Where the variables of f j ( x j ) are denoted by x j = ( x (1) j , · · · , x ( d ) j ). It is easy to check4hat ∂ i, ⊗ m ( m O j =1 f j ( x j )) = X j f ⊗ · · · ⊗ ∂ i f j ⊗ · · · ⊗ f m . (2.2)Furthermore, from the formula (2.2) we can get ∂ i, ⊗ m ( m N j =1 f j ( x j ))= " ∂ i, ⊗ k ( k N j =1 f j ( x j )) ⊗ ( m N j = k +1 f j ( x j ))+ ( k N j =1 f j ( x j )) ⊗ " ∂ i, ⊗ m − k m N j = k +1 f j ( x j ) , (2.3)or, shortly, we have ∂ i, ⊗ m = ∂ i, ⊗ k ⊗ id m − k + id k ⊗ ∂ i, ⊗ m − k . (2.4)Where ∂ i, ⊗ k and id k concern the left k factors of tenser f ⊗ · · · ⊗ f m , ∂ i, ⊗ m − k and id m − k concern the right m − k factors.Combining the natural co-product of the partial derivations ∂ i , △ ∂ i = ∂ i ⊗ ⊗ ∂ i ,we have ∂ i, ⊗ m = (1 ⊗ · · · ⊗ | {z } ⊗△ ) · · · (1 ⊗ △ ) △ ∂ i .m − times (2.5)In this paper the star product at level of functions what we want to construct isMoyal-like one. Let K = ( K ij ) d × d be a matrix with entries in A called propagatormatrix, this matrix determines the star product with tenser form as in the followingdefinition. Definition 2.1.
Let f i ( x ) , g j ( x ) ∈ C ∞ ( R d ) , i = 1 , · · · , m, j = 1 , · · · , n , their star prod-uct with tenser form is defined by the following formula: [( f ( x ) ⊗ · · · ⊗ f m ( x m )) ⋆ K ( g ( y ) ⊗ · · · ⊗ g n ( y n ))] ⊗ = exp { ~ K} ( f ( x ) ⊗ · · · ⊗ f m ( x m )) ⊗ ( g ( y ) ⊗ · · · ⊗ g n ( y n )) , (2.6) where K = X ij K ij ∂ i, ⊗ m ⊗ ∂ j, ⊗ n , and ∂ i, ⊗ m acts on f ( x ) ⊗ · · · ⊗ f m ( x m ) , ∂ j, ⊗ n acts on g ( y ) ⊗ · · · ⊗ g n ( y n ) . To simplify the formula (2.6) we introduce short notations in the following way.Let F m ( x ) = N i f i ( x i ), G n ( y ) = N j g j ( y j ), for short, ∂ i, ⊗ m F m ( x ) and ∂ j, ⊗ n G n ( y ) bedenoted by ∂ i, x F m ( x ) and ∂ j, y G n ( y ) respectively. Then the formula (2.6) can be denotedin a short way as following( F m ( x ) ⋆ K G n ( y )) ⊗ = exp { ~ K x , y } ( F m ( x ) ⊗ G n ( y )) , (2.7)5here K x , y = P ij K ij ∂ i, x ⊗ ∂ j, y .With the help of the formula (2.3) or (2.4), by a straightforward computation weknow that the associativity is valid, i.e. we have[( F m ( x ) ⋆ K G n ( y )) ⊗ ⋆ K H p ( z )] ⊗ = [ F m ( x ) ⋆ K ( G n ( y ) ⋆ K H p ( z )) ⊗ ] ⊗ , (2.8)where H p ( z ) = N pk =1 h k ( z k ). Actually, it is easy to check that the both sides of theformula (2.8) are of following formexp { ~ ( K x , y + K x , z + K y , z ) } ( F m ( x ) ⊗ G n ( y ) ⊗ H p ( z )) , (2.9)Explicitly, K x , y = P ij K ij ∂ i, x ⊗ ∂ j, y ⊗ id p , but for simplicity we denote it as K x , y = P ij K ij ∂ i, x ⊗ ∂ j, y without confusion, so are K x , z and K y , z . Remark 2.1. • Let C ∞A , ~ ,n = { P k > ~ k f k | f k ∈ C ∞A ⊗ · · · ⊗ C ∞A } ,n − times then the star product (2.6) can be extended as a map ⋆ K : C ∞A , ~ ,n × C ∞A , ~ ,m → C ∞A , ~ ,n + m . • According to the formula (2.2) the star product (2.6) can be expressed as exp { ~ ( X k,l K kl ) } ( f ( x ) ⊗ · · · ⊗ f m ( x m ) ⊗ g ( y ) ⊗ · · · ⊗ g n ( y n )) , (2.10) where K kl = P ij K ij ∂ i, x k ⊗ ∂ j, y l , ∂ i, x k and ∂ i, y l act on f k and g l respectively asordinary partial derivations. Particularly, in the present paper we are mainly in-terested in the case of ( f ( x ) ⋆ K · · · ⋆ K f m ( x m )) ⊗ . • The star product with tenser form defined in definition 2.1 should be put in Kont-sevich’s framework, but there is some slight modification. • Sometime the tenser form of the star product (2.6) may be more convenient forus, such that we can discuss the star products for some objects even without mul-tiplication, for example, some general fields in physics, or, the case of multiplesymplectic geometry. .2 The ordinary star products Starting from the star product with tenser form in definition 2.1 we can define the starproduct in ordinary sense. We have
Definition 2.2.
Let f i ( x ) , g j ( x ) ∈ C ∞ ( R d ) , i = 1 , · · · , m, j = 1 , · · · , n , F m ( x ) = N i f i ( x i ) , G n ( y ) = N j g j ( y j ) , we define the following star product ( m ◦ F m ( x )) ⋆ K ( m ◦ G n ( y )) = m ◦ ( F m ( x ) ⋆ K G n ( y )) ⊗ . (2.11) Where m denotes the multiplication of point-wise of functions. From the formula (2.11), it is easy for us to check that[( m ◦ F m ( x )) ⋆ K ( m ◦ G n ( y ))] ⋆ K ( m ◦ H p ( z ))= m ◦ (( F m ( x ) ⋆ K G n ( y )) ⊗ ) ⋆ K ( m ◦ H p ( z ))= m ◦ [(( F m ( x ) ⋆ K G n ( y )) ⊗ ⋆ K H p ( z )] ⊗ , where H p ( z ) as mentioned above. Therefore, due to the formula (2.8) the associativityof the star product defined in definition 2.2 is valid. Furthermore we have Definition 2.3.
For f ( x ) , g ( x ) ∈ C ∞ ( R d ) , we define f ( x ) ⋆ K g ( x ) = f ( x ) ⋆ K g ( y ) | x = y . (2.12)Precisely we have f ( x ) ⋆ K g ( y ) = exp { ~ K} f ( x ) g ( y ) , where K = P ij K ij ∂ x i ∂ y j . Remark 2.2. • Comparing with the Moyal product f ( x ) ⋆ g ( x ) = f ( x ) g ( x ) + ~ X i,j α ij ∂ i f ( x ) ∂ j g ( x ) + O ( ~ ) , in the present paper the constant coefficients α ij in Moyal star product are replacedby abstract elements K ij in A . However, similar to the case of the Moyal product, K plays the role of bi-vector field with coefficients in A . • It is obvious that the star product defined by the formula (2.12) can be extended tothe case of C ∞A , ~ , where C ∞A , ~ = { X m > ~ m F m | F m ∈ C ∞A , m > } , such that the star product ⋆ K is a map from C ∞A , ~ × C ∞A , ~ to C ∞A , ~ , or from C ∞A , ~ ⊗ C ∞A , ~ into itself. We can extend the star product defined as above to the case where functions de-pending on some parameters. For example, f ( t, x ) ⋆ K g ( t, y ) = exp { ~ K} ( f ( t, x ) g ( t, y )) , where t = ( t , · · · , t k ) . More precisely, we have f ( x ) ⋆ K g ( y ) = f ( x ) g ( y ) + ~ P i,j K ij ∂ i f ( x ) ∂ j g ( y )+ ~ P i ,i ,j ,j K i ,j K i ,j ∂ i ∂ i f ( x ) ∂ j ∂ j g ( y ) + · · · , thus we have f ( x ) ⋆ K g ( y ) − g ( y ) ⋆ K f ( x ) = ~ X i = j ( K ij − K ji ) ∂ i f ( x ) ∂ j g ( y ) + O ( ~ ) . If f ( x ) ⋆ K g ( x ) = g ( x ) ⋆ K f ( x ) we say the star product ⋆ K is commutative. It is obviousthat we have Proposition 2.1.
The star products (2.12) are commutative iff the propagator matrix K is symmetric. For non-commutative case we define the Poisson bracket as following:
Definition 2.4.
Let f ( x ) , g ( x ) ∈ C ∞ ( R d ) , their Poisson bracket is defined to be { f ( x ) , g ( y ) } K = X i = j ( K ij − K ji ) ∂ i f ( x ) ∂ j g ( y ) . (2.13) Specially { f ( x ) , g ( x ) } K = { f ( x ) , g ( y ) } K | x = y . (2.14)The Poisson bracket can be extended to C ∞A also. It is obvious that the Poissonbrackets (2.13) and (2.14) is bi-linear and anti-symmetric, additionally, are derivationsfor both of f and g . The Jacobi identity is valid for the Poisson brackets in abovedefinition. Where the Jacobi identity concerning the Poisson bracket (2.13) is of form {{ f ( x ) , g ( y ) } , h ( z ) } + cycles = 0 . Now we extend the star product to the situation with several factors. We have
Proposition 2.2.
Let f i ( x ) ∈ C ∞ ( R d ) , i = 1 , · · · , m , we have f ( x ) ⋆ K · · · ⋆ K f m ( x m ) = m ◦ ( f ( x ) ⋆ K · · · ⋆ K f m ( x m )) ⊗ (2.15) where ( f ( x ) ⋆ K · · · ⋆ K f m ( x m )) ⊗ = exp { P i For different propagator matrixes K, K ′ we have ( f ( x ) ⋆ K g ( y )) ⊗ = exp { ~ ( K − K ′ ) } ( f ( x ) ⋆ K ′ g ( y )) ⊗ . (2.16) Proof. Actually we have exp { ~ K} = exp { ~ ( K − K ′ ) } exp {K ′ } . Remark 2.3. It is obvious that all formulas in above discussion are valid for elementsin C ∞A , ~ . Thus we will only discuss the issues concerning the star products for smoothfunctions below. In this section we explain how the star products result in the Feynman amplitudes, atsame time, Kontsevich graphs result in the Feynman graphs naturally. At beginning as preparations we recall some contents concerning the Kontsevich graphssimply, for more details we direct readers to [13]. Definition 3.1. An oriented graph Γ is a pair ( V Γ , E Γ ) of two finite sets such that E Γ is a subset of V Γ × V Γ . Elements of V Γ are vertices of Γ , elements of E Γ are edges of Γ .If e = ( v , v ) ∈ E Γ ⊆ V Γ × V Γ is an edge of Γ then we say that e stars at v and endsat v . Definition 3.2. ( Admissible graphs , [13], p.22) Admissible graph G n,m is an orientedgraph with labels such that • The set of vertices V Γ is { , · · · , n } ⊔ { ¯1 , · · · , ¯ m } where n, m ∈ Z > , m + n − > ; vertices from { , · · · , n } are called vertices of the first type, vertices from { ¯1 , · · · , ¯ m } are called vertices of the second type. • Every edge e = ( v , v ) ∈ E Γ stars at a vertex of the first type, v ∈ { , · · · , n } . • There are no loops, i.e. no edges of the type ( v, v ) . • For every vertex k ∈ { , · · · , n } of the first type, the set of edges Star ( k ) = { ( v , v ) ∈ E Γ | v = k } starting from k , is labeled by symbols { e k , · · · , e Star ( k ) k } . 9n other articles the vertices of the first type are also called internal vertices andvertices of the second type are called boundary vertices. About the operation of graphswe have Definition 3.3. (see [11], p.7 and [12], p.22)If Γ ∈ G n,m , Γ ∈ G n ′ ,m , we define theproduct Γ Γ ∈ G n + n ′ , m as the graph obtained from disjoint union of two graphs byidentification of the vertices of the first type. It is easy to see that the product of admissible graphs defined above is commutative.For convenience we define the embedding of the admissible graphs G n,m ֒ → G n,m ′ , m ′ > m , or, extend a graph in G n,m to a graph in G n,m ′ . Definition 3.4. Let Γ ∈ G n,m , the all vertices of the second type are labeled by v ¯1 , · · · , v ¯ m ,for a subset of { ¯ i , · · · , ¯ i m } ⊆ { ¯1 , · · · , ¯ m ′ } , ¯ i < · · · < ¯ i m ¯ m ′ , we extend Γ in thefollowing way: • identifying vertex v ¯ j with vertex v ¯ i j , j = 1 , · · · , m .Above procedure define an embedding ι i , ··· ,i m : G n,m ֒ → G n,m ′ , and we denote new graphby Γ i , ··· ,i m . We call ( i , · · · , i m ) the position of ι i , ··· ,i m or Γ i , ··· ,i m . Remark 3.1. Combining the definitions 3.3 and 3.4 we can consider the product ofgeneral admissible graphs. For instance, let Γ ∈ G n,m , Γ ′ ∈ G n ′ ,m ′ , we take m =max { m, m ′ } and choose two positions ( i , · · · , i m ) , ( i ′ , · · · , i ′ m ′ ) , then the product Γ ( i , ··· ,i m ) Γ ′ ( i ′ , ··· ,i ′ m ′ ) ∈ G n + n ′ ,m makes sense. Now we embed the star product defined in definition 2.1 into the Kontsevich’s framework.The Moyal product is one generated by constant Poisson bi-vector field, it may be trivialcase from the viewpoint of Kontsevich theory. However for convention we discuss theproblems in details. Similar to the case of the Moyal product, the basic graph is Bernoulligraph (see [11], [12]) b ∈ G , which endows one vertex of the first type, two vertices ofthe second type named by left and right ones respectively. Now we consider b n = b · · · b | {z } ,n − times where the product of the graphs is in the sense of definition 3.3, thus we know that b n ∈ G n, , or b n ∈ G n more simply (see [13]). b n is simple graph even in G n , actually,there are no edges connecting the different vertices of the first type in b n . The vertices ofthe first type in b n are labeled by { , · · · , n } , and the edges of b n are labeled by symbols e L , e R , · · · , e Ln , e Rn , i.e. E b n = E Ln ∪ E Rn , where e Lk (or e Rk ) denote the edge starting at k -th vertex of the first type and ending at left (or right) vertex of the second type, and10 Ln = { e L , · · · , e Ln } , E Rn = { e R , · · · , e Rn } . We consider the map I : E b n → { , · · · , d } , I L = I | E Ln , I R = I | E Rn . Let I L ( e Lk ) = i k , I R ( e Rk ) = j k , k = 1 , · · · , n , according to theKontsevich’s rule, for star product f ( x ) ⋆ K g ( x ) we have B b n ,K ( f, g ) = X i , ··· ,i n ,j , ··· ,j n K i j · · · K i n j n ∂ i · · · ∂ i n f ∂ j · · · ∂ j n g. In fact due to the bi-vector field with ”constant” coefficients Kontsevich’s rule willwork for the case of star product with tenser form in original way almost, where we makea little modification, i.e. wedge is replaced by tenser product at all. We take poly-vectorfield to be of form K n ⊗ = K ⊗ · · · ⊗ K | {z } .n − times If we use the Kontsevich’s notation U , we have U b n ( K n ⊗ )( f ( x ) ⊗ g ( y ))= P i , ··· ,i n ,j , ··· ,j n K i j · · · K i n j n ∂ i · · · ∂ i n f ( x ) ⊗ ∂ j · · · ∂ j n g ( y ) , or, U b n ( K n ⊗ ) = K n , where K n plays the role of bi-differential operator acting on f ( x ) ⊗ g ( y ). Let b ∈ G , be assigned to the identity, i.e. U b ( id )( f ( x ) ⊗ g ( y )) = f ( x ) ⊗ g ( y ) . Finally we have ( f ( x ) ⋆ K g ( y )) ⊗ = X n > ~ n n ! U b n ( K n ⊗ )( f ( x ) ⊗ g ( y )) . There is a graphical explanation of above discussion. Actually in [12] Kontsevichrule was divided into three steps: • 1. Kontsevich graphs, i.e. admissible graphs (see [13]), • 2. colored graphs, • 3. expressions of coordinates.If we fix the bi-vector field K , under Kontsevich rule the second and third steps men-tioned above can be done in a standard way. Thus Kontsevich graphs include the keyinformation of the star products, for example, associativity of the star products as dis-cussion in [11], [12] and [13]. As what has been done in [12] we consider free R (or C )module with generators { b n } n > o denoted by H b . Because in the present paper we payattention to the star product we restrict us to consider only the special case, where the11olyvector fields are of forms K n ⊗ . Following the idea of [12](Definition 4.7 and Lemma4.1 in [12]), Kontsevich rule can be considered as pairing U : ( graph, poly − vectorf ield ) → poly − dif f erential operator. If we consider only a special class of poly-vector fields, i.e. K n ⊗ , we can get a morphismfrom the set of graphs to poly-differential operators.Now we generalise Kontsevich rule slightly, for a polynomial in H b , P ( b ) = P k c k b n k (where b = b ), we generalise Kontsevich’s morphism in the following way U P ( b ) ( K ) = X k c k U b nk ( K n k ⊗ ) . (3.1)Noting U b n ( K ) = ( U b ( K )) n , we have U P ( b ) ( K ) = X k c k ( U b ( K )) n k . (3.2)Hence we get a morphism from H b to the set of bi-differential operators denoted by U K : U K : X k c k b n k X k c k ( U b ( K )) n k = X k c k K n k . (3.3)Where the bi-differential operators are of froms X ij C ij ∂ α i ⊗ ∂ β j ,C ij ∈ A , α i , β j ∈ N d . Specially we have U K ( b ) = K . It is obvious that U K is injective.Now, formally, we can introduce the following notation which can be thought as agraphical expression of the star product (see [11], [12])exp { ~ b } = X k > ~ k k ! b k . (3.4)Furthermore, we have U K (exp { ~ b } ) = exp { ~ K} . The formula (3.4) will be our starting point of discussion below.For convenience we introduce the notion of adjacency matrix. Definition 3.5. A adjacency matrix is a symmetric matrix with non-negative integerentries. Here we make an additional restriction such that the entries on main diagonalare zeros, i.e.for an adjacency matrix M = ( m ij ) , we have m ij = m ji , m ii = 0 . We call P ij m ij the degree of M denoted by degM . emark 3.2. The notion of adjacency matrix appears in combinatorial theory. In C.Brouder [1] the author emphasised the connection between the Feynman graphs and RSKalgorithm (Robinson-Schensted-Kunth algorithm) and paid attention to the adjacencymatrix. In general the assumption about all zeros along main diagonal of the adjacencymatrixes is not necessary from viewpoint of combinatorial theory. Here we talk aboutthe adjacency matrixes with additional restriction mentioned above due to the followingreasons. On the one hand we will discuss the Feynman graphs from the viewpoint of Wickmonomials, in this case the Feynman graphs do not contain the self-lines (or tadpole).On the other hand, it was noted in [1], there is an one-one correspondence between theFeynman graphs without self-lines and the adjacency matrixes with all zeros on maindiagonal. Actually, for an adjacency matrix M = ( m ij ) k × k , the integer k indicates thenumber of vertices of a Feynman graph and every entry m ij indicates the number of edgesbetween i -th and j -th vertices of the Feynman graph. From the viewpoint of combinatorialtheory, recalling RSK algorithm, we know that there are three one-one correspondencesamong the following three objects: (see [3], [4], [14]) • the set of permutations which are involutions without fixed points, • the set of adjacency matrixes with zeros along the main diagonal, • the set of semi-standard Young tableaus(SSYT) without odd columns. Now we turn to the Feynman amplitudes and Feynman graphs. We need to consider themultiple star product: f ( x ) ⋆ K · · · ⋆ K f m ( x m ) = m ◦ ( f ( x ) ⋆ K · · · ⋆ K f m ( x m )) ⊗ . Recalling proposition 2.2 we know that( f ( x ) ⋆ K · · · ⋆ K f m ( x m )) ⊗ = exp { ~ P i Let M , M be two m × m adjacency matrixes with degM i = 2 k i , i = 1 , , then we have b M + M = b M b M , (3.7) and U K ( b M b M ) = U K ( b M ) U K ( b M ) . (3.8) Proof. Actually, if we consider a graph Q kl =1 b i l j l , where i l < j l , l = 1 , · · · , k , by inductionwe can prove U K ( k Y l =1 b i l j l ) = k Y l =1 U K ( b i i j l ) . On the other hand, b i l j l corresponds to adjacency matrix M ( i l , j l ) with entries m i l j l = m j l i l = 1 , m ij = 0 , for others. For an adjacency matrix M = P i,j m ij M ( i, j ), it is easyto check that b M = Y i Theorem 3.1. Let exp { ~ X i For a given graph of Bernoulli type Q i In summary, up to now we have six ways from different aspects to describe theFeynman graphs showed in table 2. In this section we want to discuss the Wick theorem and Wick power in terms of thestar product at level of functions in C ∞ ( R d ). Therefore we focus on the situation of starproduct with special form as f ( x ) ⋆ K · · · ⋆ K f d ( x d ), where f i ( · ) ∈ C ∞ ( R ) , i = 1 , · · · , d .Firstly we have the following formula obviously,16eynman amplitudes analytic aspectFeynman graphs graphical aspectKontsevich graphs graphical aspectAdjacency matrixes algebraic aspectPermutations combinatorial or algebraic aspectsSSYT combinatorial aspectTable 2: default Lemma 4.1. f ( x ) ⋆ K · · · ⋆ K f d ( x d ) = exp { ~ X i Proposition 4.1. f ( x ) ⋆ K · · · ⋆ K f d ( x d )= P k > ~ k k ! P degM =2 k (cid:18) km , · · · , m d − ,d (cid:19) f ( α )1 · · · f ( α d ) d Q i It is enough for us to count the number of terms with form ~ k K kii x l − ki . Definition 4.1. The Wick power in the sense of the star product ⋆ K of x i is defined tobe : x li : K = x i ⋆ K · · · ⋆ K x i | {z } ,l − times (4.5) where i d . Remark 4.1. • By definition as above we know that the Wick power belongs to C ∞A . Duo to theformula (4.4) the Wick power is expressed by means of Hermite polynomials. • Actually we can get Wick power in another way. We consider the star product x ⋆ K · · · ⋆ K x l (or, more general, x i ⋆ K · · · ⋆ K x i l , i < · · · < i l ). From the formula(4.2) we have x ⋆ K · · · ⋆ K x l = [ l ] X k =0 ~ k k ! X | I | =2 k x I c X P I K P I , where I ⊂ { , · · · , l } , I c = { , · · · , l } r I , x I c = x p · · · x p l − k for I c = { p , · · · , p l − k } , P I is a partition of I with form P I = { i , j } S · · · S { i k , j k } ( i < j , · · · , i k < j k ) , K P I = K i ,j · · · K i k ,j k . Let x , · · · , x l = x i , meanwhile, K pq = K ii , we get theformula (4.4). f ( x ) ⋆ K · · · ⋆ K f d ( x d ) we have moreprecise formula which is a generalisation of the classical Wick theorem. Theorem 4.1. ( Wick theorem ) For different propagator matrixes K = ( K ij ) d × d and K ′ = ( K ′ ij ) d × d we have f ( x ) ⋆ K · · · ⋆ K f d ( x d )= P k > ~ k k ! P degM =2 k (cid:18) km , · · · , m d − ,d (cid:19) f ( α )1 ⋆ K ′ · · · ⋆ K ′ f ( α d ) d . Q i Similar to the proof of theorem 2.2 we haveexp { ~ X i Remark 4.2. • If we take K ′ = 0 in the formula (4.6), we come bake to the formula (4.2). • If we take K = 0 , we have f ( x ) · · · f d ( x d )= P k > − ~ ) k k ! P degM =2 k (cid:18) km , · · · , m d − ,d (cid:19) f ( α )1 ⋆ K ′ · · · ⋆ K ′ f ( α d ) d Q i Observing the formula (4.7), we find that the form of Feynman amplitudesdose not dependent on the choices of propagator matrixes K (1) and K (2) . Thus, whenwe focus on the issues of the Feynman amplitudes, without loss of generality we canchoose the the star product ⋆ K (1) in Wick-monomial : x n : K ⋆ K (1) · · · ⋆ K (1) : x n d d : K to becommutative always. In the traditional sense the Feynman amplitudes arising from the expectation ofGreen functions. But in the previous discussion we talk about the Feynman amplitudeswhich appear as factors arising from the bi-vector field in the coefficients of the starproduct. Above statement is reasonable really. Actually, along the idea of perturbativealgebraic quantum fields theory we can define the expectation of the Wick-monomialas the coefficient of the term with the highest power of ~ in the star product. Now wedefine the expectation of Wick-monomial : x n : K ⋆ K (1) · · · ⋆ K (1) : x n d d : K , denoted by < : x n : K ⋆ K (1) · · · ⋆ K (1) : x n d d : K >, as following: Definition 4.2. If we write the Wick-monomial as a polynomial of ~ : x n : K ⋆ K (1) · · · ⋆ K (1) : x n d d : K = m X k =1 c k ~ k , we define the expectation of above Wick-monomial as following: • When n + · · · + n d = 2 m , < : x n : K ⋆ K (1) · · · ⋆ K (1) : x n d d : K > = c m , (4.8) where c m = 1 m ! X degM =2 m (cid:18) mm , · · · , m d − ,d (cid:19) Y i When the integer sequence ( n , · · · , n d ) satisfies the following conditions: • There is an adjacency matrix M = ( m ij ) d × d , degM = 2 m , such that m = n + · · · + n d ,n i = P j m ij , i = 1 , · · · , d, (4.10) we call this integer sequence ( n , · · · , n d ) admissible. Combining the definition 4.2 and 4.3 we have the following conclusion immediately. Proposition 4.3. The Wick-monomial : x n : K ⋆ K (1) · · · ⋆ K (1) : x n d d : K endows non-zeroexpectation iff ( n , · · · , n d ) is admissible. We have a simpler description of the admissible integer sequence. Here we assumethe star product is commutative and n i > , i = 1 , · · · , d . Theorem 4.2. A integer sequence ( n , · · · , n d ) is admissible iff n + · · · + n d is an eveninteger and n i n + · · · + n d , i = 1 , · · · , d. (4.11) Proof. If the integer sequence ( n , · · · , n d ) is admissible, i.e. there is an adjacency matrix M = ( m ij ) d × d , such that n i = P j m ij , i = 1 , · · · , d . Then we have2 n i = X j m ij + X j m ji X i,j m ij = 2 X i Let K ( x , y ) ∈ C ∞ ( X × X ) , f ( y , · · · , y d ) , g ( y , · · · , y d ) ∈ C ∞ ( R d ) , ϕ ( x ) ∈ C ∞ ( X ) , we define the star product as following: f ( ϕ ( x ) , · · · , ϕ ( x d )) ⋆ K g ( ϕ ( y ) , · · · , ϕ ( y d ))= m ◦ (cid:2) exp { ~ K} f ( x , · · · , x d ) ⊗ g ( y , · · · , y d ) | x i = ϕ ( x i ) ,y i = ϕ ( y i ) (cid:3) . (5.1) Where K = P i,j K ij ∂ x i ⊗ ∂ y j , K ij = K ( x i , y j ) . · · · · · · r r · · · r r · · · dTable 3: default23hen K ( x , y ) = P ∗ K ( x , y ), where P : X × X −→ X × X ; P ( x , y ) = ( y , x ), is per-mutation map, we know that the star product ⋆ K is commutative. For non-commutativecase we have: Definition 5.2. The Poisson bracket is defined to be { f ( ϕ ( x ) , · · · , ϕ ( x d )) , g ( ϕ ( y ) , · · · , ϕ ( y d )) } K = m ◦ hP i,j ( K ij − K ji ) ∂ i f ( x , · · · , x d ) ⊗ ∂ j g ( y , · · · , y d ) | x i = ϕ ( x i ) ,y i = ϕ ( y i ) i . (5.2) Remark 5.1. • The star product defined in definition 5.1 and Poisson bracket defined in definition5.2 are well defined and rely on the issues at level functions. Actually as a specialcase we have f ( ϕ ( x ) , · · · , ϕ ( x d )) ⋆ K g ( ϕ ( x ) , · · · , ϕ ( x d ))= f ( ϕ ( x ) , · · · , ϕ ( x d )) ⋆ K g ( ϕ ( y ) , · · · , ϕ ( y d )) | x i = y i = f ( x , · · · , x d ) ⋆ K g ( x , · · · , x d ) | x i = ϕ ( x i ) . (5.3) and { f ( ϕ ( x ) , · · · , ϕ ( x d )) , g ( ϕ ( x ) , · · · , ϕ ( x d )) } K = { f ( y , · · · , y d ) , g ( y , · · · , y d ) }| y i = ϕ ( x i ) . (5.4) Where K ij = K ( x i , x j ) . • For all of conclusions in section 2 there are parallel ones in the case of scalar fields. In the case of fields we have Wick theorem and Wick power similarly. Recalling thediscussion in section 4, if we take x i = ϕ ( x i ) , i = 1 , · · · , d , in each formula in section 4,we can get a corresponding formula in the case of fields. Theorem 5.1. Let K ( x , y ) , K ′ ( x , y ) be smooth functions on X × X , f ( · ) , · · · , f d ( · ) ∈ C ∞ ( R ) , ϕ ( x ) ∈ C ∞ ( X ) , we have f ( ϕ ( x )) ⋆ K · · · ⋆ K f d ( ϕ ( x d ))= P k > ~ k k ! P degM =2 k (cid:18) km , · · · , m d − ,d (cid:19) f ( α )1 ( ϕ ( x )) ⋆ K ′ · · · ⋆ K ′ f ( α d ) d ( ϕ ( x d )) · Q i Let K ( x , y ) , K (1) ( x , y ) , K (2) ( x , y ) ∈ C ∞ ( X × X ) , then : ϕ n ( x ) : K ⋆ K (1) · · · ⋆ K (1) : ϕ nd ( x d ) : K = P k > ~ k k ! P degM =2 k (cid:18) km , · · · , m d − ,d (cid:19) (cid:18) n α (cid:19) · · · (cid:18) n d α d (cid:19) · : ϕ n − α ( x ) : K ⋆ K (2) · · · ⋆ K (2) : ϕ nd − αd ( x d ) : K Q i For general distribution K ( x , y ) ∈ D ′ ( X × X ) , the power and restrictionon diagonal of X × X make non-sense generally. In this case only the star productwith form ϕ ( x ) ⋆ K · · · ⋆ K ϕ ( x d ) may be well defined, but some analytic conditions, forexample, concerning wave front set W F ( K ) , may be needed. Now we turn to situation of the functionals. We consider the functionals with form F ( ϕ ) = Z X d f ( x , · · · , x d , ϕ ( x ) , · · · , ϕ ( x d )) dV d , (5.10)where f ∈ C ∞ ( X d × R d ), X d = X × · · · × X | {z } ,d − times and dV d is volume form on X d .In the below discussion we assume the integrals make sense always. We state thedefinitions of star product and Poisson bracket of functionals as following. Definition 5.4. Let F ( ϕ ) , G ( ϕ ) be functionals as in (5.10). • We define their star product to be F ( ϕ ) ⋆ K G ( ϕ ) = Z X d Z X d f ( · ) ⋆ K g ( · ) dV d dV d , (5.11) where f ( x , · · · , x d , ϕ ( x ) , · · · , ϕ ( x d )) ⋆ K g ( y , · · · , y d , ϕ ( y ) , · · · , ϕ ( y d ))= m ◦ (cid:2) exp { ~ K} f ( x , · · · , x , · · · , x d ) g ( y , · · · , y , · · · , y d ) | x i = ϕ ( x i ) ,y i = ϕ ( y i ) (cid:3) . where K = P i,j K ij ∂ x i ⊗ ∂ y j , K ij = K ( x i , y j ) . • For the star product between the functionals and fields we have F ( ϕ ) ⋆ K g ( y , · · · , y d , ϕ ( y ) , · · · , ϕ ( y d )) = Z X d f ( · ) ⋆ K g ( · ) dV d , (5.12) in (5.12) the integral concerns the variables of f ( · ) . • We define the Poisson bracket of F ( ϕ ) and G ( ϕ ) to be { F ( ϕ ) , G ( ϕ ) } K = Z X d Z X d { f ( · ) , g ( · ) } K dV d dV d , (5.13) where { f ( x , · · · , x d , ϕ ( x ) , · · · , ϕ ( x d )) , g ( y , · · · , y d , ϕ ( y ) , · · · , ϕ ( y d )) } K = P i,j ( K ij − K ji ) ∂ x i f ( x , · · · , x , · · · , x d ) ∂ y j g ( y , · · · , y , · · · , y d ) | x i = ϕ ( x i ) ,y i = ϕ ( y i ) . 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