arXiv:2105.14965v3 [hep-th] 6 Sep 2021 icse nteltrtr:tecvratcodntsapp coordinates covariant the literature: the W in decades. discussed last the during investigated extensively was lem background general in D-branes of physics low-energy i of ing theory gauge non-commutative of definition consistent The Introduction 1 * Email: ae oso structures Poisson differnates consider We action. invariant gauge the t from for obtained equations field The identities. Bianchi corresponding nstePisnfil teghwihi oain ne h P standard the the under covariant reproduces is and which strength field Poisson relation the commutator fines The field. matter of derivative variant ftefl o-omttv ag hoywt oriaede coordinate with theory Θ gauge non-commutative full the investiof to designed is model algebra t proposed gauge The gauge Poisson symmetries. Poisson gauge the the having model formulate theoretical we field work cal present the In algebra. o l rpsdcntutos u oe suiu pt inv to up unique is transformations. model coordinate Our constructions. proposed all for ab h oso ag ler sasm-lsia ii fcompl of limit semi-classical a is algebra gauge Poisson The ( [email protected] x ) seilywoewt o-osatrn.W eieteex the derive We rank. non-constant a with whose especially , nvriaeFdrld B,SnoAdé P Brazil SP, André, Santo ABC, do Federal Universidade etod aeáia optçoeCognição e Computação Matemática, de Centro and os tt nvriy os,Russia Tomsk, University, State Tomsk Θ ab oso ag theory gauge Poisson ( x ldsa .Kupriyanov G. Vladislav U ) (1) swl snnlna ns n banepii expression explicit obtain and ones, non-linear as well as , etme ,2021 8, September edsrnt ntecmuaielmt edrv the derive We limit. commutative the in strength field Abstract 1 oc 4 ] h ws prah[6]-[9], approach twist the 5], [4, roach egueadtemte ed are fields matter the and gauge he eto eetemi approaches main the here mention e fsrn hoy[][] hsprob- This [1]-[3]. theory string of s n xmlso ieri coordi- in linear of examples ent o h oain eiaie de- derivatives covariant the for nipratse o understand- for step important an s aetesm-lsia features semi-classical the gate sacrepnigagbaof algebra corresponding a as isnguetransformations gauge oisson ril edrdfiiin and redefinitions field ertible * edn non-commutativity pendent t o-omttv gauge non-commutative ete er hc sadynami- a is which heory rsinfrteco- the for pression s the L∞-bootstrap approach [10] and also recently proposed approach based on the combination of the L∞ algebras and the twist [11]. Nevertheless the structure of the non-commutative field theory still requires the better understanding, especially in case when the non-commutativity Θab(x) is not constant. In the previous work [12] in collaboration with Richard Szabo we for- mulated a novel approach to the construction of algebra of non-commutative gauge symmetries in the semi-classical limit, based on the symplectic embeddings of (almost)-Poisson gauge struc- ture. The aim of the present research is to construct the dynamical sector of the corresponding theory. The associative non-commutativity of space-time is usually introduced in the theory by sub- stituting the standard pointwise multiplication of fields f g on some manifold M with the star · multiplication, i~ f⋆g = f g + f,g + (~2) , (1.1) · 2 { } O defined as an associative deformation of the pointwise product along the given Poisson structure,

xi, xj = α Θij(x) , (1.2) { } where α is a small parameter which we will refere to as the non-commutativity parameter. The non-commutativity field Θij(x) is considered to be an external field. The higher order in ~ contributions to (1.1) are defined from the condition of the associativity, (f⋆g)⋆h = f⋆(g ⋆h), and can be constructed according to the Formality theorem [13], or also the polydifferential approach [14]. NC The non-commutative U(1) gauge transformations δf Aa are defined as transformations satisfying the following two conditions: they should close the algebra,

NC NC NC [δf , δg ]Aa = δ−i[f,g]⋆ Aa , (1.3)

with, [f,g]⋆ = f⋆g g ⋆ f, and should reproduce the standard U(1) gauge transformations 0 − δf Aa = ∂af, in the commutative limit. Reminding that the Poisson bracket is the semi-classical limit of the star commutator, f,g = lim~ [f,g]⋆/i~, we define the Poisson gauge trans- { } →0 formations δf Aa, following [12], as the semi-classical limit of the full non-commutative U(1) gauge transformations. They should close the algebra,

[δf , δg]Aa = δ{f,g}Aa , (1.4) called Poisson gauge algebra, and reproduce the standard U(1) gauge transformations in the commutative limit, limα→0 δf Aa = ∂af. ij If Θ is constant, one may easily see that the expression, δf Aa = ∂af + f, Aa , satisfies { } (1.4). However, for non-constant Θij(x) the standard Leibniz rule with respect to the partial derivative is violated, ∂a f,g = ∂af,g + f, ∂ag , therefore the same expression will not { } 6 { } { } close the algebra (1.4) anymore. To overcome this problem one has to modify the expression for the gauge transformations introducing the corrections proportional to the derivatives of the non- ij commutativity ∂aΘ (x) whichwould compensatetheviolation of the Leibnizrule. The problem

was solved in [15] using the L∞-bootstrap approach to the non-commutative gauge theories

2 [10]. We stress that L∞-formalism is a powerful tool for the construction of perturbative order by order in α expressions for the consistent non-commutative deformations of gauge theories [16]. Though, to get an explicit all order expressions normally one needs to invoke additional considerations. The approach to the solution of this problem proposed in [12] is based on the symplectic embeddings of Poisson manifolds [17, 18] and is especially good for obtaining explicit form of the deformed constructions. The problem with violation of the Leibniz rule for the original Poisson bracket (1.2) can be solved in an extended space. To each coordinate xi we introduce a conjugate variable pi, in such a way that the corresponding Poisson brackets,

i i x ,pj = γ (x, p) , pi,pj =0 , (1.5) { } j { } should satisfy the Jacobi identity. In our constructionwe will need the vanishing bracket between i i α ik 2 p-variables, pi,pj = 0, while the matrix, γj(x, p) = δj 2 ∂jΘ pk + (α ), defining the { i} − ij O Poisson bracket x ,pj will be constructed in Sec. 2. For constant Θ this matrix is constant, i i { } γj(x, p) = δj, so, f(x),pi = ∂if(x), i.e., the Poisson bracket between the function f(x) on { } ij M and the auxiliary variable pi is just a partial derivative of this function. In case if Θ (x) is i not constant the expression for γj(x, p) is more complicated, however the action of the operator ,pi which we will call ‘twisted’ derivative on functions f(x) is similar to that of the partial {· } derivatives ∂i. First of all because the Jacobi identity for the Poisson brackets (1.5) and the fact that pi,pj = 0 imply that these operators comute, f(x),pi ,pj = f(x),pj ,pi . { } {{ } } {{ } } Second, because the ‘twisted’ derivative satisfies the Leibniz rule,

f(x),g(x) ,pi = f(x),pi ,g(x) + f(x), g(x),pi , (1.6) {{ } } {{ } } { { }} which also follows from the Jacobi identity. However, the price to pay is that the expression

f(x),pi depends also on the auxiliary non-physical p-variables. It turns out that the auxiliary { } * variables can be eliminated in the consistent way by introducing the constraints, pa = Aa(x) . In the Sec. 3 we will prove that the gauge transformations defined by,

l δf Aa = γ (A) ∂lf(x)+ Aa(x), f(x) , (1.7) a { } l l where, γa(A) := γa(x, p) pa=Aa(x), close the algebra (1.4) and reproduce the standard U(1) 0 | gauge transformations, δf Aa = ∂af, in the commutative limit. The new results of the present research are related to the consistent definition of the dynam- ical part of field-theoretical model having the Poisson gauge algebra (1.4) as the corresponding algebra of gauge symmetries. Working in the formalism of the symplectic embeddings we intro-

duce the matter field ψ by postulating the corresponding gauge transformation as, δf ψ = ψ, f . { } In the Sec. 4 we construct the gauge covariant derivative a(ψ) satisfying two key requirements: D it transforms covariantly under the gauge transformation, δf a(ψ) = a(ψ), f , and repro- D {D } duces the standard partial derivative in the commutative limit, a(ψ) ∂aψ, as α 0. In the D → → *Note that in [19] the symplectic embeddings were used to construct the consistent Hamiltonian description of the electrically charged particle in the field of magnetic monopole distributions. However in that case the elimination the auxiliary p-variables was not possible.

3 Sec. 5 it will be shown that the commutator of twocovariant derivatives defines the Poisson field

strength ab which also transforms covariantly, δf ab = ab, f , and reproduces the standard F F {F } U(1) field strength in the commutative limit, limα ab = ∂aAb ∂bAa. We also define the →0 F − corresponding Bianchi identity in the Sec. 5.1. The main relations are resumed by the following ij ij k table, which for simplicity is given for the case of linear Poisson structures, Θ (x)= fk x ,

Object Identity

i l de de a(ψ)= ρ (A) γ (A) ∂lψ + Aa, ψ [ a, b]= ab, + ad Λb bd Λa e D a i { } D D {F ·} F −F D ij l
de
ab = Rab (A)(γ (A) ∂lAj + Ai, Aj ) a ( bc) ad Λb ec + cycl.(abc)=0 F i { } D F −F F where,

cd 1 c d c d de −1 d j m m j −1 e Rab = ρ ρ ρ ρ , Λb = ρ ∂ ρ ∂ ρ ρ , 2 a b − b a j A b − A b m



j i i i with, ∂A = ∂/∂Aj , and the matrix ρa(A) is defined by, ρa(A) := ρa(x, p) pa=Aa(x), where i | ρa(x, p) should satisfy the equation,

i b i f(x), ρ (x, p) + ρ (x, p) ∂ f(x),pb =0 , f(x) . (1.8) { a } a p{ } ∀ For the non-linear Poisson structures Θij(x) there may appear additional contributions in the identities in the right hand side of the table which are given in the Sec. 5. For arbitrary non-commutativity Θij(x) we provide the recurrence relations for the con- l l struction of the matrices, γa(A) and ρa(A) which are the building blocs of our construction. For some particular choices of the non-commutative spaces, like the rotationally invariant NC ij ijk space [20]-[24] described by the su(2)-like Lie Poisson structure, Θε (x)=2αε xk, or the l l κ-Minkowski space [25]-[29] we obtain explicit all-order expressions for γa(A) and ρa(A) in the Sections 2 and 4 correspondingly. In the Sec. 6 we use the gauge covariant objects a(ψ) D and ab to construct the gauge invariant action and derive the field equations for the gauge field F Aa and the matter field ψ. In case of the rotationally invariant non-commutative space [20] the field equations in the pure gauge sector read, ε ab =0. We conclude with the final remarks Da Fε and discussion in the Sec. 7 and provide the useful for
the calculation formulas in the Appendix.

2 Symplectic embeddings of Poisson manifold

In this section we will summarize the necessary ingredients from the symplectic geometry that we will use throughout the paper. All precise mathematical definitions and proofs are given in [12], here our aim is to recollect them and expose on a physics friendly language. The problem of the construction of the symplectic embedding for the given Poisson structure (1.2) formulated i in the introduction consists basically in finding the matrix γj(x, p) which defines the Poisson i bracket x ,pj in such a way that the complete algebra of Poisson brackets (1.2) and (1.5) { } should satisfy the Jacobi identity.

4 The Jacobi identity involving the original coordinates only, xi, xj, xk + cycl. = 0, is { { }} satisfied automatically since Θij(x) is a Poisson bi-vector. The Jacobi identity with two original i j † coordinates and one p-variable, x , x ,pk + cycl. =0 , implies an equation , { { }} l b k k b l lm k km l m lk γ ∂ γ γ ∂ γ + α Θ ∂mγ α Θ ∂mγ αγ ∂mΘ =0 , (2.9) b p a − b p a a − a − a i ij on the function γj(x, p) in terms of given Poisson structure Θ (x). The Jacobi identity with i one x and two p–variables, x , pj,pk + cycl. =0, is also satisfied automatically. It follows { { }} j from the fact that, pi,pj = 0, and antisymmetry of x ,pk . And finally the Jacobi identity { } { } involving the p-variables only, pi, pj,pk + cycl. =0, is trivially satisfied. { { }} i So, the matrix γj(x, p) is defined as a solution of the equation (2.9) with the condition, i i γ (x, p) α = δ , to guarantee that the complete algebra of the Poisson brackets (1.2) and (1.5) j | =0 j is a deformation in α of the canonical Poisson brackets, i.e., forms the symplectic algebra. Up to the second order in α the solution reads,

k k(n) k α kb γ (x, p)= γ = δ ∂aΘ pb (2.10) a a a − 2 Xn=0 2 α cm bk bm kc 3 2Θ ∂a∂mΘ + ∂aΘ ∂mΘ pbpc + (α ) . −12 O
k The recurrence relations for the construction of the matrix γa (x, p) in any order in α are given in [30].

2.1 Arbitrariness

We note that the symplectic embedding is not unique. The arbitrariness is described by the in- vertible transformations of the variables (x, p) which leave the Poisson brackets between the i j coordinates and coordinates x , x , as well as, momenta and momenta pi,pj unchanged, { } i { } while change the brackets between the coordinates and momenta x ,pj . Making the transfor- { } mation which change only the p-variables, φ : p p˜, with φ(p) α = p and leave x-variables → | =0 unchanged we do not change neither the bracket (1.2), nor the pi,pj =0, while for the bracket { } between the coordinates and momenta one finds,

i i i k γ˜ (x, p˜)= x , p˜j(p) = γ (x, p) ∂ p˜j(p) . (2.11) j { }p=p(˜p) k p |p=p(˜p) The new set of Poisson brackets,

xi, xj = α Θij(x) , (2.12) { i } i x , p˜j =γ ˜ (x, p˜) , p˜i, p˜ =0 , { } j { } i i also represents the symplectic embedding of the Poisson structure (1.2) since γ˜ (x, p˜) α = δ . j | =0 j † k k k Note that here we use different notations from [12], the matrix γa here corresponds to δa + αγa used in [12].

5 2.2 Symplectic embeddings of Lie-Poisson structures

The Kirillov-Kostant Poisson bracket, also called sometimes the Lie-Poisson bracket, is defined in [31] as, xi, xj = f ij xk , (2.13) { } k ij where fk are the structure constants of a Lie algebra. The solution for the equation (2.9) in this case is given by [12, 32, 33],

i i 1 ij1 i γ (p)= δ f pj + ( M/2) , (2.14) j j − 2 j 1 Xj − where, i ij1 kj2 Ml = fk fl pj1 pj2 , (2.15) and i ( M/2) is a matrix valued function with, Xl − ∞ n n t t ( 2) B n t (t)= cot 1= − 2 , (2.16) X r2 r2 − (2n)! Xn=1

i with Bn being Bernoulli numbers. We stress that the matrix γj(p) defined in (2.14) does not depend on x-variables. Let us consider an exemple of the su(2)-like Lie-Poisson algebra,

k l kl m x , x =2 αε m x , (2.17) { } physically corresponding to the rotationally invariant non-commutative space. In (2.17) εklm is the Levi-Civita symbol and the factor of 2 is just a matter of convenience. We use the Kronecker delta to raise and lower indices, and summation under the repeated indices is understood. In this

case the matrix Mε appearing as the argument of the third term in the r.h.s. of (2.14) is given by,

i 2 i i 2 [Mε] =4 α p pl δ p , (2.18) l − l
2 m −1 where, p = pmp . This matrix is diagonalizable, it can be written as, Mε = S D S , · 2· 2 where D is the diagonal matrix with the eigenvalues of Mǫ, λ = 0, λ = λ = 4α p , on 1 2 3 − the diagonal and the matrix S is constructed from the corresponding eigenvectors. Therefore following [34] we write,

i −1 i ( Mε/2) = S diag [ ( λ /2) , ( λ /2) , ( λ /2)] S (2.19) Xj − · X − 1 X − 2 X − 3 · j  2 i 2 i 2 2  = α δ p p pj χ(α p ) , j −
where, 1 χ(t)= t−1 (2t)= √t cot √t 1 . (2.20) X t  −  We conclude that,

k 2 2 2 2 k 2 2 2 k kl [γε] (p)= 1+ α p χ α p δ α χ α p pap αεa pl , (2.21) a a − − 

6 Note that the explicit form of the function χ(t) was obtained from the general solution (2.14) i diagonalizing the matrix ( Mε/2). Alternatively one may check that the expression (2.21) Xj − satisfies the equation (2.9) only if χ(t) obeys the ODE, 1 2 t χ′ +3 χ + t χ2 +1=0 , χ(0) = , (2.22) −3 whose solution is given by (2.20). The second particular exemple we would like to discuss here is the κ-Minkowski space [25]- [29] yielding the Kirillov-Kostant structure,

xk, xl =2 ai xj aj xi , (2.23) { } −
i k with a being constants. For this Poisson structure the simplest solution for γa (p) reads [35],

k 2 k k [γκ]a(p)= 1+(a p) +(a p) δa a pa . (2.24) hp · · i − 2.3 Change of coordinates

If two different Poisson structures Θij(x) and Θ˜ ij(˜x) are related by the invertible coordinate transformation, υ : x x˜, i.e., → ij i j i kl j Θ˜ (˜x)= x˜ (x), x˜ (x) x x x = ∂kx˜ Θ (x) ∂lx˜ x x x , (2.25) { } = (˜) | = (˜) i and the symplectic embedding γj(x, p) of the first one is known, then the symplectic embedding of the second one can be constructed according to,

i i i k γ˜ (˜x, p)= x˜ (x),pj = ∂kx˜ γ (x, p) . (2.26) j { }x=x(˜x) j |x=x(˜x) Thus formally having in hand a symplectic embedding of a given Poisson structure we may generate symplectic embeddings of the whole family of the related Poisson structures. At the same time we have to be careful with the applications of this construction since the physical observables should not depend on the change of coordinates. Let us consider an example. We start with the symplectic embedding (2.21) of the su(2)- like Lie-Poisson structure (2.17). The corresponding non-commutativity preserves rotational symmetry, see [20]. Let us discuss its generalization, namely the non-linear Poisson structure,

i j ij m 2 x˜ , x˜ =2 αε m x˜ f(˜x ) , (2.27) { } where f(˜x2) is some given function. The non-commutativity (2.27) also preserves rotations, however it does not grows on the infinity provided that f(˜x2) decreases faster then 1/√x˜2, when x˜2 . In this case we have a kind of "local" rotationally invariant non-commutativity. The → ∞ algebra (2.27) can be obtained from (2.17) by the change of variables,

x˜i = g(x2) xi , (2.28)

7 where g(x2) is a function to be determined. Note that, x˜2 = g2(x2) x2. One calculates,

i j ij m 2 2 ij m 2 x˜ , x˜ =2 αε m x g (x )=2 αε m x˜ g(x ) . (2.29) { } Comparing (2.29) to (2.27) and taking into account (2.28) one finds,

f g2(x2) x2 = g(x2) , (2.30)
which is a functional relation on g provided that f is some given function. Taking f as the Gaussian function, f(˜x2) = exp( αx˜2/2), one obtains from (2.30), − W αx2 W αx2 2 − ( ) i − ( ) i g(x )= e 2 , x˜ = e 2 x , (2.31) ⇒ where W (z) is the Lambert W function satisfying the functional relation, z = W (z) eW (z). One may easily verify that in this case, αx˜2 = exp( W (αx2)) αx2 = W (αx2), and thus, − f(˜x2) = exp( αx˜2/2) = exp( W (αx2)/2) = g(x2), i.e., (2.30) holds true. The inverse − − transformation is, 2 i 1 i αx˜ i x = x˜ = e 2 x˜ . (2.32) g(x2) Now from (2.26), (2.28) and (2.32) we derive the symplectic embedding of (2.27),

αx2 i i − ˜ i α x˜k x˜ k γ˜ (˜x, p)= e 2 δ γ (p) , (2.33) j  k − 1+ α x˜2  j

k where the matrix γj (p) was constructed in (2.21) and we remind that we do not change p vari- ables.

3 Poisson gauge transformations

In this section we prove the statement formulated in the introduction regarding the closure con- dition (1.4) of the Poisson gauge transformations (1.7). First we observe that the (1.7) can be written in a more convenient form as,

δf Aa = f(x), Φa , (3.34) { }Φ=0 where,

Φa := pa Aa(x) . (3.35) −

Indeed, the expression f(x),pa a a means that first we calculate the Poisson bracket { }p −A (x)=0 f(x),pa according to (1.5) and then substitute all p with A, so, { } l f(x),pa = γ (A) ∂lf(x) . { }Φ=0 a The main property of the transformation (3.34) if formulated by the following proposition.

Proposition 3.36. The Poisson gauge transformations (3.34) close the algebra (1.4).

8 Proof. We start with some preliminary definitions,

δf F (A) := F (A + δf A) F (A) . (3.37) − In particular,

δg ( Aa, f )= δgAa, f , (3.38) { } { } and if F (A) is a smooth function in A, then,

b δf F (A)= ∂AF δf Ab , (3.39)
b where as before we use the notation, ∂A = ∂/∂Ab. The composition of two gauge transforma- tions can be written,

δ (δ A ) = g(x), δ A + ∂b g(x), Φ δ A (3.40) f g a f a Φ=0 p a Φ=0 f b −{ } { b} = g(x), f(x), Φa + ∂ g
(x), Φa f(x), Φb , −{ { }Φ=0}Φ=0 p{ } Φ=0 { }Φ=0
since, b b ∂ ( g(x), Φa )= ∂ g(x), Φa . (3.41) A { }Φ=0 p{ } Φ=0
And thus we find,

δf (δgAa) δg (δf Aa)= (3.42) − b g(x), f(x), Φa Φ=0 Φ=0 + ∂p g(x), Φa Φ=0 f(x), Φb Φ=0 −{ { } } b{ } { } + f(x), g(x), Φa ∂ f(x), Φa
g(x), Φb . { { }Φ=0}Φ=0 − p{ } Φ=0 { }Φ=0
Using the relation (7.135) from the appendix in the right hand side of (3.42) we represent it as,

g(x), f(x), Φa + f(x), g(x), Φa . (3.43) −{ { }}Φ=0 { { }}Φ=0 Finally applying now the Jacobi identity we end up with,

δf (δgAa) δg (δf Aa)= f(x),g(x) , Φa = δ f,g Aa. (3.44) − {{ } }Φ=0 { } The latter means that the gauge variations (3.34) close the Lie algebra (1.4). 2

3.1 Field redefinition

The form of the Poisson gauge transformation (3.34) relies on the precise symplectic embedding i γa(x, p) of given Poisson structure (1.2). However, as it was discussed in Sec. 2.1 the symplectic embedding of (1.2) is not unique. Different embeddings are related by the invertible change of p-variables, φ : p p˜, with φ(p) α = p. Having the Poisson gauge transformation (3.34) → | =0 which satisfy the relation (1.4), one may construct another gauge transformation corresponding i to the new symplectic embedding γ˜a(x, p˜),

i δ˜f A˜a =γ ˜ (A˜) ∂if + f, A˜a , (3.45) a { }

9 which will close the same gauge algebra (1.4). From the physical perspective one may expect that the invertible field redefinitions defined by, φ : A A˜, and f˜ = f, will leave our construction invariant. We define the new gauge → transformations δ˜f by setting, −1 δ˜f := φ δf φ , (3.46) ◦ ◦ or in the other words,

˜ ˜ ˜ k ˜ δf Aa := δf Aa(A) = ∂AAa(A) δf Ak . (3.47) A(A˜) A(A˜)

Taking into account (2.11) we see that it is exactly the expression (3.45), meaning that the arbi- trariness in our construction corresponds to the invertible field redefinition. Since the field redefinition is invertible the gauge orbits are mapped onto the gauge orbits and the Seiberg-Witten condition,

A˜ (A + δf A)= A˜(A)+ δ˜f A˜(A) , (3.48)

is trivially satisfied up to the linear order in f. Moreover,

˜ ˜ ˜ ˜ ˜ ˜ −1 ˜ δf , δg = δf δg δg δf = φ (δf δg δg δf ) φ = δ{f,g} , (3.49) h i ◦ − ◦ ◦ ◦ − ◦ ◦ meaning that the gauge algebra remains the same, as expected. In [36] it was shown that the

Seiberg-Witten maps correspond to L∞-quasi-isomorphisms which describe the arbitrariness in the definition of the related L∞ algebra in the L∞-bootstrap approach [10].

4 Covariant derivative

We start with the introduction of the matter fields ψ and the definition of the non-commutative NC U(1) gauge transformations δf ψ which should close the same gauge algebra (1.3) as the cor- responding gauge fields Aa. In case pf the associative star products there are three possibilities to define such a gauge transformation. It can be taken to be left or right star multiplication, NC NC NC δ ψ = if ⋆ ψ or δ ψ = iψ ⋆ f, or even as the star commutator, δ ψ = i[ψ, f]⋆. Only the f f f − third possibility is compatible with the semi-classical limit which is the subject of this research. That is why we define the Poisson gauge transformation of the matter field as,

δf ψ = ψ, f . (4.50) { } The Jacobi identity implies that such a determined gauge transformations close the same algebra as (1.4), i.e.,

[δf , δg]ψ = δ{f,g}ψ . (4.51)

We note that in the commutative limit the Poisson gauge variation of the mater field van-

ishes, limα→0 δf ψ =0, meaning that in this limit the matter field ψ is ’electrically neutral’ and does not interact with the gauge field Aa. The interaction between the gauge field and such a

10 defined matter field is caused by the non-commutativity. At the moment we do not see a clear physical meaning of such an interaction. The field ψ appears in our construction more like an auxiliary object needed for the consistent definition of the covariant derivative, which in turn is a central object of this work. As we will see in the next Section, the commutator of the covariant derivatives will produce the gauge covariant field strength for the gauge field. Also this object is essential for the derivation of the corresponding Bianchi identity and the equations of motion.

Our aim now is to construct the covariant derivative a(ψ), the object satisfying two main D requirements.

• Itshouldtransform covariantlyunderthePoissongaugetransformation δf defined in (3.34) and (4.50), i.e.,

δf ( a(ψ)) = a(ψ), f . (4.52) D {D } This property will be essential for the construction of the gauge covariant Lagrangian and the gauge invariant action for the Poisson gauge theory.

• In the commutative limit it should reproduce the standard partial derivative,

lim a(ψ)= ∂aψ , (4.53) α→0 D since the interaction between the matter field and the gauge field disappears when α 0. → The answer is given by the following proposition. Proposition 4.54. The operator,

i a(ψ)= ρ (A) ψ, Φi , (4.55) D a { }Φ=0 i i i satisfies the above two conditions if the matrix, ρa(A) := ρa(x, p)Φ=0, and ρa(x, p) obeys the equation, i b i f(x), ρ (x, p) + ρ (x, p) ∂ f(x),pb =0 , f(x) . (4.56) { a } a p{ } ∀

Proof. Let us calculate,

i δf ρ (A) ψ, Φi = (4.57) a { }Φ=0 b i
i ∂Aρa(A) f, Φb Φ=0 ψ, Φi Φ=0 + ρa(A) ψ, f , Φi Φ=0 + i b{ } { } i {{ } } ρ (A)(∂ ψ, Φi ) f, Φb ρ (A) ψ, f, Φi . a p{ } Φ=0 { }Φ=0 − a { { }Φ=0}Φ=0 Observe that, b i i i ∂ ρ (A) f, Φb = f, ρ (p) ρ (A) , (4.58) A a { }Φ=0 { a − a }Φ=0 and by (7.135) from the appendix,

b ψ, f, Φi = ψ, f, Φi (∂ f, Φi ) ψ, Φb . (4.59) { { }Φ=0}Φ=0 { { }}Φ=0 − p{ } Φ=0 { }Φ=0

11 So the right hand side of (4.57) becomes,

i i f, ρa(p) Φ=0 ψ, Φi Φ=0 + ρa(A), f Φ=0 ψ, Φi Φ=0 + (4.60) {i } { } i { b } { } ρa(A) ψ, f , Φi Φ=0 + ρa(A)(∂p ψ, Φi )Φ=0 f, Φb Φ=0 + i {{b } } { i } { } ρ (A)(∂ f, Φi ) ψ, Φb ρ (A) ψ, f, Φi . a p{ } Φ=0 { }Φ=0 − a { { }}Φ=0 Using the Jacobi identity we rewrite it as,

i b i f, ρa(p) Φ=0 + ρa(A)(∂p f, Φb )Φ=0 ψ, Φi Φ=0 + (4.61) { i } { i } b { } ρ (A), f ψ, Φi + ρ (A) (∂  ψ, Φi ) f, Φb + ϕ, Φi , f . { a }Φ=0 { }Φ=0 a p{ } Φ=0 { }Φ=0 {{ } }Φ=0   Taking into account that by (7.135),

b (∂ ϕ, Φi ) f, Φb + ϕ, Φi , f = ϕ, Φi , f , (4.62) p{ } Φ=0 { }Φ=0 {{ } }Φ=0 {{ }Φ=0 }Φ=0 the second line in (4.61) becomes,

i ρ (A) ϕ, Φi , f = a(ϕ), f . (4.63) { a { }Φ=0 }Φ=0 {D } Therefore the relation (4.52) holds true if the first line of (4.61) vanishes, i.e.,

i b i f(x), ρ (x, p) + ρ (x, p) ∂ f(x), Φb =0 , f(x) . (4.64) { a } a p{ } ∀ Since Ab(x), f(x) does not depend on p one arrives at (4.56). The statement regarding the { } commutative limit, α 0, follows from the fact that in this limit, ψ, Φi ∂iψ, and → { } → ρi (x, p) δi . 2 a → a i In local coordinates the equation (4.56) on the function ρa(x, p) can be written as, j b i b i j jb i γb ∂p ρa + ρa ∂pγb + α Θ ∂bρa =0 , (4.65) jb j where Θ (x) is a given Poisson structure and γb (x, p) represents its symplectic embedding constructed in the Sec. 2. For the arbitrary non-commutativity parameter Θab(x) a solution of the equation (4.65) can be found in form of the perturbative series, ∞ i i(n) i α ib ρ (x, p)= ρ = δ ∂aΘ pb + (4.66) a a a − 2 Xn=0 2 α cm ib bm ic 3 2Θ ∂a∂mΘ ∂aΘ ∂mΘ pbpc + (α ) . 6 − O
For some specific choices of non-commutativity, one may also discuss the convergence of such series and exhibit a closed expressions.

4.1 Lie-Poisson structures and generalizations

First we observe that for the linear Poisson bi-vectors (2.13) there is no explicit coordinate de- i i pendance of the functions γa(p) and ρa(p). These functions depend only on p-variables pa. The equation (4.65) becomes, j l i l i j γl ∂p ρa + ρa ∂pγl =0 , (4.67) j where γl (p) is given by (2.14). In what follows we will discuss the solution of the eq. (4.67) for two particular examples of Lie-Poisson structures.

12 4.1.1 su(2)-like Poisson structure

Let us consider now the particular case of su(2)-like Lie-Poisson structure (2.17). The pertur- bative calculations indicate the ansatz,

i 2 2 i 2 2 2 i 2 2 ik [ρε]a = σ α p δa + α τ α p p pa + αζ α p εa pk , (4.68)


with the initial condition, σ(0) = 1, to match (4.53). Using this ansatz in the equation (4.67) the latter becomes,

′ i j ij α [2 σ + σ χ + ζ] δ p + α [τ + ζ σ χ] δ pa + (4.69) a − 2 2 2 2 ′ j i 2 2 ij α 1+ α p χ τ +2 σ χ + α p χ δap + α 1+ α p χ ζ + σ εa + 3 ′ j ik 3 2 2 ′ i jk α [2 ζ 2 ζ χ]
p εa pk + α 2 ζ χ +
α p χ τ p ε pk +
 − − a 4 ′ ′ i j α [2 τ 2 σ χ τ χ] p p pa=0.
 − − Thustheequation(4.67) resultsinthe systemof seven equationson coefficient functions σ(α2p2), τ(α2p2) and ζ(α2p2):

2 σ′ + σ χ + ζ =0 , (4.70) τ + ζ σ χ =0 , (4.71) − 1+ α2 p2 χ τ +2 σ χ + α2 p2 χ′ =0 , (4.72) 1+ α2 p2 χ
ζ + σ =0 ,
(4.73) 2 ζ′ 2 ζ χ =0
, (4.74) − 2 ζ χ + α2 p2 χ′ τ =0 (4.75) − 2 τ ′ 2 σ χ′ τ
=0 . (4.76) − − However they are not all independent. The eq. (4.72) is just a consequence of (4.73) and (4.75). The eq. (4.71) is satisfied due (4.73), (4.75) and the equation on the function χ (2.22). The eq. holds true as a consequence of (4.73), (4.74) and (2.22). The eq. (4.76) is satisfied due (4.73), (4.74), (4.75) and (2.22). In fact we end up with only three independent equations on σ(A), τ(A) and ζ(A): (4.73), (4.74) and (4.75). Let us first discuss the initial conditions. Taking into account the initial condition, σ(0) = 1, one finds from (4.73) that, ζ(0) = 1. Taking into account that χ(0) = − 1/3, the equation (4.71) then gives the initial condition for the function τ namely, τ(0) = 2/3. − The solution of the equation (4.74) with this initial condition reads, sin2 √t ζ(t)= . (4.77) − t The equation (4.73) gives, sin 2√t σ(t)= . (4.78) 2√t And from the equation (4.75) one finds, sin 2√t τ(t)= t−1 1 . (4.79) −  2√t − 

13 We conclude that the ansatz (4.68) with the functions σ, τ and ζ defined in (4.78), (4.79) and (4.77) correspondingly solve the equation (4.67). i i An explicit form of the matrix, ρa(A)= ρa(x, p)p=A, reads,

i i ik 2 2 2 i 2 i 2 2 [ρε] (A)= δ + αεa Ak ζ α A α δ A A Aa τ(α A ) . (4.80) a a − a −

And thus we have obtained an explicit all orders expression for the gauge covariant derivative defined in (4.55). It satisfies the gauge covariance condition (4.52) and reproduces in the com- mutative limit the standard partial derivative (4.53).

4.1.2 κ-Minkowski

j For the κ-Minkowski Poisson structure (2.23) the matrix [γκ]l (p) is given by (2.24). Following the same strategy as in the previous particular example we write the anzats which follows from the perturbative calculations,

i i i [ρκ]a(p)= δa ξ(z)+ a pa η(z) , (4.81) where, z = a p, and ξ(0) = 1 to guarantee the correct commutative limit. Substituting this · anzats in (4.67) one finds,

′ i j ′ j i ′ ′ i j ((φ z) ξ ξ) δ a +(φ η + ξφ ) δ a + ((φ z) η 2η + φ η) a a pa =0 , (4.82) − − a a − − where, φ(z) = √1+ z2 + z, according to (2.24). Consequently one arrives at the system of three equations on the coefficient functions ξ and η:

(φ z) ξ′ = ξ , (4.83) − φ η + ξφ′ =0 , (φ z) η′ 2η + φ′ η =0 . − − The solution of the first one with the initial condition, ξ(0) = 1, reads,

ξ(z)= √1+ z2 + z . (4.84) Then from the second equation one finds, √1+ z2 + z η(z)= . (4.85) − √1+ z2 The third of the equations (4.83) just becomes an identity.

4.1.3 Change of coordinates

Under the change of coordinates, υ : x x˜, described in Sec. 2.4 the solution of the equation → (4.56) transforms like,

υ : ρi (x, p) ρ˜i (˜x, p)= ρi (x, p) . (4.86) a → a a |x=x(˜x) i i For Lie-Poisson structures it does not change, since ρa(p) does not depend on coordinates x .

14 5 Fieldstrength

Let us first discuss the commutator relation of the covariant derivatives a(ψ) introduced in the D previous section. Proposition 5.87. The commutator relation for the covariant derivatives defined in the propo- sition 4.54 reads, de de e e [ a, b]= ab, + ad Λb bd Λa + ab ba e (5.88) D D {F ·} F −F K −K D where,
i j ab : = ρa(A) ρb(A) Φi, Φj Φ=0 (5.89) F i j { l } l = ρ (A) ρ (A) γ (A) ∂lAj γ (A) ∂lAi + Ai, Aj , a b i − j { } de −1 d j m m j −1 e
Λb (A) = ρ ∂ ρ (A) ∂ ρ (A) ρ , (5.90) j A b − A b m e i
m j
−1
e ab (A) = ρ (A) γ (A) ∂mρ (x, p) ρ . (5.91) K a i b Φ=0 j

Proof. The proof is straightforward, we start writing, i j i j [ a, b](ψ) = ρa(A) ρb(A), Φi Φ=0 ρb(A) ρa(A), Φi Φ=0 ψ, Φj Φ=0 + (5.92) D D i {j } − { } { } ρ (A) ρ (A)( ψ, Φj , Φi ψ, Φi
, Φj ) . a b {{ }Φ=0 }Φ=0 − {{ }Φ=0 }Φ=0 We use the formula (7.136) from the appendix in the second line of the right hand side of the above expression to rewrite it as, i j i j ρa(A) ρb(A), Φi Φ=0 ρb(A) ρa(A), Φi Φ=0 ψ, Φj Φ=0 + (5.93) i {j } − { } { } ρ (A) ρ (A)( ψ, Φj , Φi ψ, Φi , Φj
) a b {{ } }Φ=0 − {{ } }Φ=0 i j m m ρ (A) ρ (A) ∂ ψ, Φj Φm, Φi ∂ ψ, Φi Φm, Φj . − a b p { } Φ=0 { }Φ=0 − p { } Φ=0 { }Φ=0 h

i Employingthe Jacobi identityin the second line and the equation (4.56) in the third line of (5.93) one represents it as, i j i j ρ (A) ρ (A), Φi ρ (A) ρ (A), Φi ψ, Φj + (5.94) a { b }Φ=0 − b { a }Φ=0 { }Φ=0 i j
ρa(A) ρb(A) Φi, Φj , ψ Φ=0 + i j {{ } }i j ρ (A) ρ (p), ψ ρ (A) ρ (p), ψ Φi, Φj a { b }Φ=0 − b { a }Φ=0 { }Φ=0 Applying one more time the eq. (7.136) in the second line of (5
.94) we reorganize it as, i j i j ρa(A) ρb(A), Φi Φ=0 ρb(A) ρa(A), Φi Φ=0 ψ, Φj Φ=0 + (5.95) i {j } − { i }j { m } ρ (A) ρ (A) Φi, Φj , ψ ρ (A) ρ (A
) ∂ Φi, Φj Φm, ψ + { a b { }Φ=0 }Φ=0 − a b p { } Φ=0 { }Φ=0 i j j i j j
ρ (A) ρ (p) ρ (A), ψ ρ (A) ρ (p) ρ (A), ψ Φi, Φj . a { b − b }Φ=0 − b { a − a }Φ=0 { }Φ=0 We use the equation (4.56) in the second term of the second line and also
(4.58) in the last line rewriting (5.95) as, i j j ρa(A) ρb(A), Φi Φ=0 Ai, ρb(p) Φ=0 ψ, Φj Φ=0 (5.96) i { j } −{ j } { } ρ (A) ρ (A), Φi Ai, ρ (p)
ψ, Φj + − b { a }Φ=0 −{ a }Φ=0 { }Φ=0 i j
ρa(A) ρb(A) Φi, Φj Φ=0, ψ Φ=0 + { i m j { } } i m j ρ (A) ∂ ρ (A) Φm, ψ ρ (A) ∂ ρ (A) Φm, ψ Φi, Φj . a A b { }Φ=0 − b A a { }Φ=0 { }Φ=0
15 Let us calculate separately,

j j ρb(A), Φi Φ=0 Ai, ρb(p) Φ=0 = (5.97) {m j } −{ } j m ∂A ρb(A)( Am,pi Φ=0 Ai,pm Φ=0 Am, Ai )+ ∂mρb(x, p) Φ=0 γi (A)= m j { } −{ j } −{ m } ∂ ρ (A) Φm, Φi + ∂mρ (x, p) γ (A) .
− A b { }Φ=0 b Φ=0 i
After the use the relation (5.97) in the expression (5.96) the latter becomes,

i j ρa(A) ρb(A) Φi, Φj Φ=0, ψ Φ=0 (5.98) { i j m{ } m j } +ρ (A) ∂ ρ (A) ∂ ρ (A) Φi, Φj ψ, Φm a A b − A b { }Φ=0 { }Φ=0 i j m m j
ρb(A) ∂Aρa (A) ∂A ρa(A) Φi, Φj Φ=0 ψ, Φm Φ=0 − i m −j { }i { m } j +ρ (A)γ (A) ∂mρ (A) ψ, Φj
ρ (A) γ (A) ∂mρ (A) ψ, Φj . a i b { }Φ=0 − b i a { }Φ=0 Finally using the definitions (5.89), (5.90) and (5.91) as well as the formula (4.55) we represent (5.98) as the right hand side of (5.88) and thus prove the proposition 5.87. 2 Some comments are in order. First of all we remind that in case of Lie-Poisson structures i which are the main example of the present work, the matrix ρa(p) does not depend on coordi- e nates. Therefore the tensor ab defined in (5.91) vanishes and the commutator relation becomes K just,

de de [ a, b]= ab, + ad Λb bd Λa e (5.99) D D {F ·} F −F D
For the canonical Poisson bracket with constant non-commutativity parameter Θij the matrix, i i de ρa(A) = δa, is also constant according to (4.67). So, the tensor Λb determined in (5.90) vanishes and the relation (5.99) becomes the ’usual’ one, [ a, b]= ab, . D D {F ·} Second, in the commutative limit the Poisson bracket vanishes and

i i i lim ρa(A) = lim γa(A)= δa , α→0 α→0 so the quantity introduced in (5.89) reproduces the U(1) field strength when α 0, →

lim ab = ∂a Ab ∂b Aa . (5.100) α→0 F − Moreover, this object enjoys another important property formulated by the following, Proposition 5.101. The quantity (5.89) introduced in the proposition 5.87 transforms covari- antly under the Poisson gauge transformation (3.34),

δf ab = ab, f . (5.102) F {F } Proof. The proof of this proposition is analogous to the proposition 4.54. 2

Because of the properties (5.100) and (5.102) we will call the quantity ab defined in (5.89) F as the Poisson field strength. In the previous works [37, 35] we have derived the quasi-classical limitof the non-commutative U(1) field strength as the quantity,

cd cd ab = Pab (A) ∂cAd + Rab (A) Ac, Ad , (5.103) F { } 16 cd c ld which satisfies the same properties (5.100) and (5.102). In (5.103), Pab = 2 γl Rab , and the cd coefficient function Rab should satisfy the equation,

k l cd kl cd cl d k ld c k γl ∂ARab + α Θ ∂lRab + Rab ∂Aγl + Rab ∂Aγl =0 . (5.104) Comparing (5.89) and (5.103) we see that,

cd 1 c d c d Rab (A)= ρ (A)ρ (A) ρ (A)ρ (A) . (5.105) 2 a b − b a
Substituting (5.105) in the l.h.s. of (5.104) one finds, 1 γk ∂l ρc + ρl ∂c γk + α Θkl ∂ ρc ρd + (5.106) 2 l A a a A l l a b   1 c k l d l d k kl d ρ γ ∂ ρ + ρ ∂ γ + α Θ ∂lρ (a b) . 2 a l A b b A l b − ↔   Observing that in the square brackets we have the l.h.s. of the eq. (4.65) written in A variables c cd we conclude that if ρa obeys the equation (4.56), the tensor Rab (A) defined by the relation (5.105) satisfies the equation (5.104). c It is easy to see that substituting the expression (4.81) for the matrix [ρκ]a corresponding to the κ-Minkowski Lie-Poisson structure (2.23) in (5.105) one obtains exactly the formula for κ cd Rab from [35]. The calculation in case of the su(2)-like Poisson bi-vector (2.17) is a bit less c straightforward. We substitute [ρε]a from (4.80) in (5.105) and then use the formulas from the ε cd appendix of [16] to simplify it. Thus one recovers the expression for Rab from [37]. And finally we note that if the Poisson structures Θij(x) and Θ˜ ij(˜x) are related by the invertible coordinate transformation, υ : x x˜, described in the Sec. 2.3, then the corresponding Poisson field → strengths are also related by the same transformation, ˜ab = ab x x x , as can be seen from F F | = (˜) (2.25), (2.26), (4.86) and (5.89). c From the practical reasons the eq. (4.56) on ρa is easier to solve than the eq. (5.104) on cd Rab because it has less components. However there is more fundamental significance of the covariant derivative (4.55), since it is essential for the definition of the Bianchi identity, the c action principle and the equations of motion. It is therefore one may think of the matrices ρa(A) l and γk(A) as a building blocs of the Poisson gauge theory.

5.1 Bianchi identity

Proposition 5.107. The field strength defined in (5.89) satisfies the deformed Bianchi identity,

de e e a ( bc) ad Λb ec Γab ec +Γcb ea + cycl.(abc)=0 . (5.108) D F −F F − F F

Proof. Again the proof is straightforward. Using the definitions (4.55) and (5.89) one calculates,

a ( bc)+ cycl.(abc)= (5.109) Di F j k j k ρ (A) ρ (A), Φi ρ (A)+ ρ (A) ρ (A), Φi Φj, Φk + a { b }Φ=0 c b { c }Φ=0 { }Φ=0 i j ( ρ (A) ρ (A) ρ A) Φj, Φk , Φi + cycl.(abc) . a b c {{ }Φ=0 }Φ=0 17 In the last term on the right we use the formula (7.137) from the appendix and rewrite it as,

i j k ρa(A) ρb(A) ρc (A) Φj, Φk Φ=0, Φi Φ=0 + cycl.(abc)= (5.110) i j k {{ } } i j k m ρ (A) ρ (A) ρ (A) Φj, Φk , Φi ρ (A) ρ (A) ρ (A) ∂ Φj, Φk Φm, Φi a b c {{ } }Φ=0 − a b c p { } Φ=0 { }Φ=0 +cycl.(abc) .

The Jacobi identity implies that the first term on the right in (5.110) vanishes,

i j k ρa(A) ρb(A) ρc (A) Φj, Φk , Φi Φ=0 + cycl.(abc)= (5.111) i j k {{ } } ρ (A) ρ (A) ρ (A) Φj, Φk , Φi + Φi, Φj , Φk + Φk, Φi , Φj =0 . a b c {{ } }Φ=0 {{ } }Φ=0 {{ } }Φ=0
Renaming the contracted indices we represent the second term on the right of (5.110) as,

i j k m cycl. (5.112) ρa(A) ρb(A) ρc (A) ∂p Φj, Φk Φ=0 Φm, Φi Φ=0 + (abc)= − k m i j { } { } ρ (A) ρ (A) ρ (A) ∂ Φm, Φi
Φj, Φk + cycl.(abc) . − a b c p{ } Φ=0 { }Φ=0
Note that using the equation (4.56) one finds,

ρk(A) ρm(A) ρi (A) ∂j Φ , Φ = (5.113) a b c p m i Φ=0 − k m i j { } k m i j ρ (A) ρ (A) ρ (A) ∂ Ai,pm
+ ρ (A) ρ (A) ρ (A) ∂ Am,pi = − a b c p{ } Φ=0 a b c p{ } Φ=0 k j i k
i j
ρ (A) Ai, ρ (p) ρ (A) ρ (A) ρ (A) Ai, ρ (p) . a { b }Φ=0 c − a b { c }Φ=0 Taking into account (5.113) and the cyclic permutations we rewrite the r.h.s. of (5.112) as,

i j k i j k ρ (A) Ai, ρ (p) ρ (A) ρ (A) ρ (A) Ai, ρ (p) Φj, Φk +(5.114) − a { b }Φ=0 c − a b { c }Φ=0 { }Φ=0 cycl.(abc) .

Remind that the expression (5.114) represents the modification of the last term in the r.h.s. of (5.109). Substituting (5.114) back into (5.109) we obtain for the r.h.s.,

i j j k ρa(A) ρb(A), Φi Φ=0 Ai, ρb(p) Φ=0 ρc (A) Φj, Φk Φ=0 + (5.115) i {j k } −{ } k {k } ρ (A) ρ (A) ρ (A), Φi Ai, ρ (p
) ρ (A) Φj, Φk + cycl.(abc) . a b { c }Φ=0 −{ c }Φ=0 c { }Φ=0
Let us calculate separately,

j j ρb(A), Φi Φ=0 Ai, ρb(p) Φ=0 = (5.116) {m j } −{ } j m ∂A ρb(A)( Am,pi Φ=0 Ai,pm Φ=0 Am, Ai )+ ∂mρb(A) γi (A)= m j { } −{ j } −{ m } ∂ ρ (A) Φm, Φi + ∂mρ (x, p) γ (A) . − A b { }Φ=0 b Φ=0 i
Using (5.116) in (5.115) the latter becomes

i m j k i j m k ρa(A) ∂A ρb(A) ρc (A)+ ρa(A) ρb(A) ∂A ρc (A) Φi, Φm Φ=0 Φj, Φk Φ=0 + (5.117) i m j k i j{ m } { k } ρ (A) γ (A) ∂mρ (x, p) ρ (A)+ ρ (A) ρ
(A) γ (A) ∂mρ (x, p) Φj, Φk + a i b Φ=0 c a b i c Φ=0 { }Φ=0 cycl.(abc) .

18 Renaming the contracted indices and using the cyclic permutations we may rewrite the first line of (5.117) as,

i m j j m k ρ (A) ∂ ρ (A) ∂ ρ (A) ρ (A) Φi, Φm Φj, Φk + cycl.(abc) . (5.118) a A b − A b c { }Φ=0 { }Φ=0
Finally using (5.118) as well as the definitions (5.89), (5.90) and (5.91) in (5.117) we rewrite the latter as,

de e e ad Λb (A) ec +Γab (A) ec Γcb (A) ea + cycl.(abc) . (5.119) F F F − F The above expression is the right hand side of (5.109), which proves (5.108). 2

6 Action principle and field equations

Having in hand the covariant derivative (4.55) and the Poisson field strength (5.89) now we are in the position to construct the gauge covariant Lagrangian, δf = , f , and gauge invariant L {L } action, δf S =0. To do so, first one ought to introduce an appropriate integration mesure µ(x), which for any two Schwartz functions f and g should satisfy the condition,

N lk d xµ(x) f,g =0 , ∂l µ(x)Θ (x) =0 . (6.120) Z { } ⇔
For the su(2)-like Poisson structure (2.17) the integration measure can be taken to be a constant µ(x)=1. For κ-Minkowski Lie-Poisson bi-vector (2.23) the measure is not constant. An explicit expression for µ(x) in this case was found in [35]. Let us discuss first the pure gauge sector without matter fields ψ. The gauge invariant action in this case reads [37],

N 1 ab Sg = d xµ(x) g , g = ab . (6.121) Z L L −4 F F To derive the corresponding Euler-Lagrange equations we will need the following relation.

i l Proposition 6.122. Provided that the matrices γl (A) and ρa(A) are defined by the proposi- tions 3.36 and 4.54 correspondingly, and the measure µ(x) satisfies (6.120), then the following relation holds true,

l i l cd ∂i µ(x) ρ (A) γ (A) + µ(x) Al, ρ (A) = µ(x)Λa (A) cd (6.123) a l a − F
 l i + ∂i µ(x) ρa(p) γl (p) Φ=0 , 
 cd where the quantities Λa and cd were defined in the proposition 5.87. F Proof. Using the equation (4.56) we represent the left hand side as,

l i m l ∂i µ(x) ρ (p) γ (p) + ∂ ρ (A) Φl, Φm , a l Φ=0 A a { }Φ=0 
 cd and use the definitions of Λa (A) and cd to complete the proof. 2 F 19 Using the relation (6.123) one finds the Euler-Lagrange equations for (6.121), k ab ab k m l m l k l k m µ ρb a µ ρa ∂A ρb + ρa ∂Aρb + ρa ∂Aρb Φl, Φm Φ=0 (6.124) abD F l − Fk i { } + ∂i µ ρ
(p) ρ (p)γ (p) =0 .
F a b l Φ=0 −1 c
 Multiplying by (ρ )k we end up with Poisson gauge equations, ac µ cb de db ce −1 c ab l k i µ a ( )+ Λb de µ Λb de + ρ ∂i µ ρ ρ γ =0 . (6.125) D F 2 F F − F F k F a b l

These equations transform covariantly under the Poisson gauge transformation (3.34) and re- produce the first pare of the Maxwell equations in the commutative limit α 0, if the measure → µ(x) is constant. As it was already mentioned, for the su(2)-like Poisson structure (2.17) the integration mea- l sure is constant µ(x)=1. Since, for any linear Poisson structures the matrix ρa(p) does not depend on coordinates, the last term on the left in (6.125) disappears. Moreover, one may check that in this case, ab k m l m l k l k m ρ ∂ ρ + ρ ∂ ρ + ρ ∂ ρ Φl, Φm 0 . (6.126) F a A b a A b a A b { }Φ=0 ≡ Therefore the equations of motion of the su(2)-like Poisson
gauge theory become, ε ( ac)=0 , (6.127) Da Fε where the superscript ε in ε and the subscript ε in ac just indicate that these objects cor- Da Fε respond to the su(2)-structure. For the κ-Minkowski case the measure is not constant, so the equations of motion will contain additional terms. Now let us discuss the contribution from the matter fields. First of all let us observe that due proposition 6.122 one has,

N N N cd d x µ ψ a(ψ) = d xµ a(ψ) ψ + d x µ ψ Λa cd ψ (6.128) Z D − Z D Z F N l i + d x ∂i µ(x) ρ (p) γ (p) ψ ψ . Z a l Φ=0 
 The latter means that the operator a is not self-adjoint. Therefore to construct real gauge D invariant action for the Fermionic matter fields ψ we write,

N i a a Sint = d xµ(x) int , = ψ¯ Γ aψ aψ¯ Γ ψ m ψ¯ ψ , (6.129) Z L Lint 2 D − D − 
 where Γa are the gamma matrices. The corresponding Dirac equation reads,

a iµ a cd i a l b iµ Γ a Γ Λa cd Γ ∂b µ ρ γ µ m ψ =0 . (6.130)  D − 2 F − 2 a l − 
To find the contribution of the matter fields to the Poisson Gauge equations one calculates,

δSint k c iµ k l l k c c = i µ ρ ψ¯ Γ , ψ ∂ ρ ∂ ρ ψ,¯ Φl Γ ψ ψ¯ Γ ψ, Φl (6.131) δA c { } − 2 A c − A c { }Φ=0 − { }Φ=0 k
  −1 a Multiplying it by (ρ )k we obtain the right hand side (the current) of the eq. (6.125),

a a iµ ab c c J = iµ ψ¯ Γ , ψ + Λc bψ¯ Γ ψ ψ¯ Γ bψ . (6.132) − { } 2 D − D 
 In the commutative limit the interaction disappears, the field become electrically neutral as it was already discussed in the Sec. 4.

20 7 Conclusions and discussion

Working in the framework of the symplectic embeddings we introduce the Poisson gauge trans- formations (3.34) which close the algebra (1.4) obtained as the semi-classical limit of the full non-commutative gauge algebra (1.3). The proposition 4.54 determines the gauge covariant derivative of the matter field a(ψ). The commutator relation for the covariant derivatives D (5.88) defines the Poisson field strength (5.89) which transforms covariantly under the gauge transformations (3.34) and tends to the standard U(1) field strength in the commutative limit. The corresponding Bianchi identity is set by the proposition 5.107. The gauge invariant action, given by (6.121) for the pure gauge fields and by (6.129) for the interaction part with the matter

fields, is constructed in the Sec. 6 from the gauge covariant objects a(ψ) and ab. From this D F action we derive the field equations (6.125) for the gauge fields without interaction with the mat- ter field, the Dirac equation for the matter field (6.130), and the current term (6.132) describing the interaction between the gauge and the matter fields in the equations of motion. The symplectic embedding of given Poisson structure is not unique, the corresponding arbi- trariness is described in the Sec. 2.1. So, for the same Poisson structure Θab(x) one may obtain different Poisson gauge transformations and other subsequent structures. However the corre- sponding gauge systems are related by the Seiberg-Witten map described in the Sec. 3.1. If two different Poisson structures Θij(x) and Θ˜ ij(˜x) are related by the invertible coordinate transfor- mation, υ : x x˜, then the corresponding Poisson gauge theories constructed following the → proposed here scheme are related by the same coordinate transformation. For arbitrary given Poisson structure Θab(x) all the structures appearing in the paper can be constructed up to arbitrary order in the deformation parameter α. For some specific choices of the non-commutativity, like the rotationally invariant NC space (2.17), κ-Minkowski space (2.23) or the spaces obtained from these two by the coordinate transformations we obtained explicit all-order in α expressions for all structures. It is interesting to work out different explicit examples of NC spaces, like e.g. the angular twist non-commutativity [38]. It would be also interesting to obtain an explicit solution of the equation (4.56) for any Lie-Poisson structure, ab ab c l Θ (x)= fc x , just like it was done in the Sec. 2.2 for the construction of the matrix γa(x, p). In the abstract it is written that the model was designed to investigate the semi-classical features of the complete non-commutative gauge theory. So, it wouldbe interestingto understand what kind of physics may stay behind the Poisson gauge field equations obtained in the Sec. 6. It would be interesting to investigate in particular the corresponding energy-momenta dispersion relations. The most non-trivial and at the same time interesting part of [12] is related to the construc- tion of the almost-Poisson gauge algebra. There are certain configurations in string and M- theory like the non-geometric backgrounds [39]-[42] yielding not only non-commutative but also non-associative deformations of space-time [43]-[45]. In this case the corresponding alge- bra of brackets (1.2) does not satisfy the Jacobi identity. But even so it is possible to construct the Lie algebra of gauge symmetries defined on such non-associative spaces. The key difference with the Poisson gauge algebra (1.4) consists in the fact that the commutator of two gauge trans-

21 formations is again a gauge transformation, however with a field-dependent gauge parameter. It is challenging to generalize the constructions obtained in the present research to the case of almost-Poisson deformations and thus to construct the semi-classical limit of non-associative gauge theory and non-associative gravity.

Acknowledgments

I am grateful to Dima Vassilevich for fruitful discussions. I appreciate the support from the Fundação de Amparo a Pesquisa do Estado de São Paulo (FAPESP, Brazil).

Appendix: Useful formulae

Let us first calculate,

f(x), g(x),pa Φ=0 f(x), g(x),pa Φ=0 Φ=0 = (7.133) { {i }} −{ { i } } f(x),γ (x, p) ∂ig(x) f(x),γ (x, A(x)) ∂ig(x) = { a }Φ=0 −{ a }Φ=0 ∂b g(x),p ( f(x),p f(x), A (x) )= p a Φ=0 b Φ=0 b Φ=0 b{ } { } −{ } ∂ g(x),pa
f(x), Φb . p{ } Φ=0 { }Φ=0
Observe that by (1.2) and (1.5) the Poissonbracket of two functions f(x) and g(x) of coordinates only does not depend on p-variables, so,

f(x),g(x) = f(x),g(x) . (7.134) { }Φ=0 { } The combination of (7.133) and (7.134) implies,

d f(x), g(x), Φa f(x), g(x), Φa = ∂ g(x), Φa f(x), Φd . { { }}Φ=0 −{ { }Φ=0}Φ=0 p { } Φ=0 { }Φ=0
(7.135) One may also check that,

f(x), Φb , Φa Φ=0 f(x), Φb) Φ=0, Φa Φ=0 = (7.136) {{d } } − {{ } } ∂ f(x), Φb) Φd, Φa , p { } Φ=0 { }Φ=0
and,

Φc, Φb , Φa Φ=0 Φc, Φb) Φ=0, Φa Φ=0 = (7.137) {{d } } − {{ } } ∂ Φc, Φb) Φd, Φa . p { } Φ=0 { }Φ=0
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