Representation of a gauge field via intrinsic"BRST"operator

We show that there exists a representation of a matrix valued gauge field via intrinsic"BRST"operator assigned to matrix valued generators of a gauge algebra. In this way, we reproduce the standard formulation of the ordinary Yang - Mills theory. In the case of a generating quasigroup/groupoid, we give a natural counterpart to the Yang - Mills action. The latter counterpart does also apply as to the most general case of an involution for matrix-valued gauge generators.


Introduction and summary
All modern models describing the fundamental forces in the Nature are based on the concept of gauge fields [1,2,3,4,5,6,7,8]. It is a well-known fact that the BRST symmetry [9,10] is the most powerful method to represent the invariance properties of a gauge field system [11,12]. Usually, in simple examples, in Hamiltonian formalism, gauge generators have the form of secondary constraints similar to the "Gauss law" represented as covariant divergence of canonical momenta. These generators are in involution that represents a gauge algebra on the phase space of the system [13,14]. By introducing ghost canonical pairs, one is able to define the respective nilpotent BRST operator containing the first class constraints in its lowest terms, linear in ghost coordinates. In the respective Lagrangian formalism, the gauge generators are represented in terms of Lagrangian field variables, as the coefficients linear in the original antifields, entering the minimal master action. In this way, usually, space-like components of relativistic fields are identified with Hamiltonian coordinates , while time -like components are identified with Lagrange multipliers to secondary first-class constraints. In the simplest example, the Yang -Mills theory, Lagrangian matrix-valued gauge field is a linear combination of matrix valued generators of adjoint representation of a generating Lie group. Thus, if one has defined the respective intrinsic "BRST" operator assigned to matrix valued generators of adjoint representation, one can define the matrix valued gauge field as a commutator of intrinsic "BRST" operator with an auxilliary "gauge" Fermion linear in the adjoint component of the Yang -Mills field . Thus, one has arrived at the intrinsic "BRST" representation to the matrix valued gauge field.
In the present article, we study in detail the approach based on the intrinsic "BRST" representation. In the case of a Lie group we have shown that the new approach does reproduce exactly the standard formulation of the Yang -Mills theory. Then, we consider the case of a quasigroup/groupoid [15,16,17,18,19,20,21,22], where the structure coefficients of the intrinsic algebra of matrix valued generators, are matrix valued themselves. In that case, we have found a natural counterpart to the Yang-Mills action. Finally, we consider the most general case of being the matrix valued generators in the general involution among themselves.

Outline of the construction
Let A µ (x) be a Boson N × N matrix valued vector field as defined by the formula where Q is a nilpotent Fermion operator, and Fermion vector field A µ (x) has ghost number −1, In more detail, Q is an N × N matrix valued operator and, at the same time does depend on m Fermion canonical pairs of ghosts (C a , P a ), a = 1, ..., m, ε(C a ) = ε(P a ) = 1, gh(C a ) = 1, gh(P a ) = −1, We assume these operators to be realized as n × n matrices, so that in fact the Q is defined on tensor product of the original matrix arguments and the ones of ghosts in (2.4 These two formulae do follow from the general representation for ghost canonical pairs in terms of two conjugate sets of n × n gamma matrices, where g ab = g ba is a constant invertible metric, g ab = g ba is its inverse, and the γ matrices do commute as It follows from (2.7) that (2.5), (2.6) do generalize to By inserting the doublet (exact) form, the first in (2.1), into the curvature form we have (2.14) where in (2.14) the quantum antibracket (X, Y ) Q is defined for any two operators X, Y , as [23] 2(X, When deriving (2.14), we have used the general property Notice that the quantum antibrackets do satisfy the Jacobi identity modulo a doublet (exact) form where (X, Y, Z) Q is the so-called quantum 3-antibracket, and so on [24] (see also [25,26,27]). The modified Leibnitz rule for quantum antibracket reads In terms of the curvature (2.13), the General "Yang-Mills" Lagrangian reads Let us consider infinitesimal gauge transformations with an operator valued Fermion "pa- (2.21) It follows from the first in (2.1), and (2.16), that the respective variation in A µ can be chosen in the form Due to (2.14), (2.17), it follows that the respective variation in G µν can be chosen in the form Now, we have, as to the respective variation in (2.20) Here, in the second equality we have used (2.16) backward, and we have moved the last commutator to the leftmost position in the second term in the left-hand side of the last (fourth) equality. Thereby, we have confirmed explicitly that the Lagrangian (2.20) is gauge invariant. Thus, we have constructed a family of gauge-invariant classical theories of the type (2.20), closely related to the "general Yang-Mills theory". Every of those classical theories can certainly be considered as a starting point as to apply the Hamiltonian BFV or Lagrangian BV quantization scheme, although we do not do that in the present article.
In what follows below through the article, we assume the operator Q as represented in CP normal form. In that case, it follows in terms of the quantum antibrackets, with no further assumptions, where we have denoted (2.28) and the quantum 3-antibracket of the ghost momenta reads In turn, by commuting the Q with (2.26), we get If we assume that A a µ and Ξ a are c-numbers, and then we get, due to the first in (2.1) and in (2.28), If , moreover, U c ab are c-numbers, and the metric, is invertible, so that η ab is its inverse, then we have for the field components and, therefore, their gauge transformation presents Yang-Mills theory generated by a compact semisimple Lie group Let t a , a = 1, ..., m, be N × N matrix valued Boson generators of a semisimple Lie group, where U c ab = −U c ba = const are structure constants of the group. They satisfy the relations Due to the first in (3.1) and (3.2), the following operator It follows then from (2.1) In turn, due to the first in (3.1) and (3.2), it follows for the second in (3.5) Then, we have where the Yang -Mills curvature (stress tensor ) has the usual form Ghost -extended generators, similar to the second in (3.5), have been first introduced in string theory [28,29], and then generalized and studied systematically in [30,31], being called as "BRST-invariant constraints".

The quasigroup/groupoid case
Now, let us consider a more general situation of quasigroup/groupoid, where the structure coefficient of the algebra are matrix-valued operators rather then constants. In that case we have where we have denoted X abcd + cyclic perm.(a, b, c, d) = 4!S hgf e abcd X ef gh , (4.4) does involve the ghost momenta P a to serve as new generators with their own semi-Abelian subalgebra, Notice that if one commutes the Q with (4.6), one gets no further consequences. So, only the doubled set of generators, does have a closed algebra. However the general formulation of Section 2 appears capable to operate efficiently even in such a complicated situation, as a part of the most general case to be considered below. Here, we restrict ourselves by rewriting the general Lagrangian (2.20) in the form more explicit in respect of being the operator Q (3.3) only linear in ghost momenta P a , even when generalized as for the matrix valued structure operators U c ab specific to the quasigroup /grouppoid case, where, in the right-hand side, every capital index of the "A" type is split into two small indices of the "a" type of a half-dimension, A = {a; a}, and it is denoted in these sectors, where T A is given in (4.9), and G a µν is defined by G µν =: G a µν P a . (4.12)

The most general case
Now, let us consider the most general involution (4.1), without assuming the conditions (4.2), (4.3). In that case, one should seek for a solution to the operator Q in the form of a ghost power series expansion of the form We do assume the following irreducibility condition for the generators t a to be satisfied: where Z cb = −Z bc are arbitrary, and we have denoted where Y e abd just denotes the left-hand side in (4.2). Due to the irreducibility (5.2), one gets which is exactly the relation that does follow from (2.2) to the (CCCPP) order. In this way, one is able, in principle, to show, order by order, that there formally exist all the structure operators in the series expansion (5.1).
In the case of a Lie group, where the generators t a and T a do satisfy the same algebra, there exists a natural counterpart to (5.4) in terms of T a , that is which extends naturally the irredicibility concept as to the ghost-extended generators T a . Then, we have the following relation to hold (5.8)

Natural canonical equivalence
As to the nilpotency condition (2.2), one can always subject the operator Q to an arbitrary canonical transformation [32] Q → Q ′ = exp{sG}Q exp{−sG}, (6.1) where s is a Boson parameter, and G is a matrix valued and ghost dependent generator, We have ab P c + ..., (6.5) with all matrix valued structure coefficients. It follows from (6.3) that the all the primed structure coefficients of the primed Q ′ satisfy the same equations as their unprimed counterparts do. In turn, it follows from (6.4) and so on. Here in (6.8), G de ab are structure coefficients as to the order CCPP in G (6.6). These equations do determine the transformation law as to all the structure coefficients in (6.5). In particular, the G 0 does determine the canonical transformation in the original matrix valued sector. In turn, the G b a do determine the actual rotations as to the basis of the original generators. In turn, the latter two transformations, as induced to the next structure coefficient U c ab , are determined by the equation (6.8), and so on in (6.5). Our main conjecture claims that the natural arbitrariness (6.1) is maximal, if the irreducibility (5.2) holds for primed basis of the generators t ′ a , as well. In that case, canonical transformations (6.1) are capable to interpolate between the most general generator and Abelian ones.
If one rewrites the (5.4) in the form with enumerated indices, due to the nilpotency (2.2), it becomes rather obvious that there exists a chain of recursive relations extending (6.9) as where the n-th Π (with n uppercases ) is constructed of the first n + 1 structure coefficients in (5.1). That chain of recursive relations extends naturally the irreducibility concept as to higher structure coefficients. As an example, we demonstrate the case n = 2: where we have also used the relation a 2 , a 3 ). (6.13) One can resolve for theŨ operators, 1 , a 2 , a 3 ) , (6.14) to get the following explicit solution

Note added in proof
Here we claim that the standard Faddeev-Popov measure can also be naturally reformulated in terms of the generators T a (2.28), by using the representation similar to (2.1) as applied to the Nakanishi-Lautrup matrix valued fields Π ( Lagrange multipliers for gauge fixing functions), as well as to the ghost and antighost Faddeev-Popov matrix valued field B,B. Then, in the case of the Lorentz gauge, ∂ µ A µ = 0, the gauge fixing part of the total Lagrangian reads where all fields take their values in the T -algebra, If the T a do satisfy a Lie algebra, the BRST invariance holds in a straightforward way, with all the coefficients in (7.2) (field components) being c-numbers. However, in the quasigroup /groupoid case, one should allow for these coefficients to be matrix valued, in general. Then, we have from (2.1) and the first in (2.28) and (2.31) and similar formulae for all other fields. The form of the second term here is quite similar to the one of the second term in (4.6), that makes unclosed the algebra of the generators T a alone. The doubled generators T A , (4.9), do satisfy the closed involution with the structure coefficients U C AB given explicitly in the relations (4.6), (4.8). Any operator X of the form similar to (7.4), X = X A T A , X A = {X a (−1) ε X ; [Q, X a ]}, (7.6) with *matrix valued* coefficients X A , belongs to the closed doubled T -algebra. The latter makes all the commutators entering (7.1), (7.3) well defined as taking their values within the same closed doubled T -algebra. Notice that the (7.6) rewrites in the natural form maintained under commuting of two operators of the form (7.7), due to the ghost number conservation.
where we have denoted It has also been shown in [24], how these equations enable one to derive the modified Jacobi relations for subsequent higher quantum antibrackets.
Notice, in conclusion, that there exists a nice interpretation of the quantum antibracket algebra via the so-called differential polarization [20]. In particular, being B an arbitrary Boson operator, one can 3-times commute that B with the nilpotency equation ( with parameters α, β, γ of the same Grassmann parities as the ones of the operators X, Y, Z, respectively, are, one applies to (A.11) the differential operator ∂ α ∂ β ∂ γ (−1) (εα+1)(εγ +1)+ε β , (A. 13) to get exactly the relation (2.17).