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Levy Differential Operators and Gauge Invariant Equations for Dirac and Higgs Fields

We study the Levy infinite-dimensional differential operators (differential operators defined by the analogy with the Levy Laplacian) and their relationship to the Yang-Mills equations. We consider the parallel transport on the space of curves as an infinite-dimensional analogue of chiral fields and show that it is a solution to the system of differential equations if and only if the associated curvature is a solution to the Yang-Mills equations. This system is an analogue of the equation of motion of chiral fields and contains the Levy divergence. The systems of infinite-dimensional equations containing Levy differential operators, that are equivalent to the Yang-Mills-Higgs equations and the Yang-Mills-Dirac equations (the equations of quantum chromodinamics), are obtained. The equivalence of two ways to define the Levy differential operators is shown.


Introduction
The current paper has two goals. We provide a classification of infinitedimensional differential operators defined by the analogy with the Lévy Laplacian and naturally connected to the Yang-Mills equations. And we describe by such differential operators solutions to the Yang-Mills-Higgs equations and the QCD equations (the equations of quantum chromodynamics).
In the current paper we will use the following scheme of the definition of differential operators, and not the form in which they were originally introduced. Let E be a real normed space and S be a linear functional acting on a subspace of L(E, E * ). 1 Then S defines a linear differential operator of the second order D 2,S acting on f ∈ C 2 (E, R) by the formula D 2,S f (x) = S(f ′′ (x)), and the linear differential operator of the first order D 1,S acting on B ∈ C 1 (E, E * ) by the formula For example, if we choose E = R d and S = tr (trace), then D 2,tr is the Laplacian ∆ and D 1,tr is the divergence div. Now let E be a real normed space continuously embedded in a real separable Hilbert space H. Let {e n } be an orthonormal basis in H that consists of elements of E. Then the value of the Lévy trace tr . This operator was introduced by Paul Lévy for the case E = H = L 2 (0, 1) (see Ref. [20]).
Another original definition of the Lévy Laplacian from Ref. [20] (the definition of the Lévy Laplacian by means of a second-order derivative of special form) will be given in terms of the Lévy trace as follows. Let K be a bilinear functional on L 2 (0, 1) such that for all u, v ∈ L 2 (0, 1) the following equality holds where K V ∈ L 2 ([0, 1] × [0, 1]) and K L ∈ L ∞ [0, 1]. The value of the Lévy trace tr L on K is defined by Then the Lévy Laplacian ∆ L is the differential operator D 2,tr L . The orthonormal bases {e n } in L 2 (0, 1), such that tr L is a restriction of tr {en} L , form a special class of weakly uniformly dense bases. Hence for weakly uniformly dense bases ∆ L is a restriction of ∆ {en} L . The article deals with functionals which are generalizations of the traces described above. Using uniformly dense bases it is proved that these functionals coincide under certain conditions. Such functionals will be also called the Lévy traces. We consider differential operators of the first and the second order (Laplacian, d'Alembertian, divergence) generated by these traces arising in the study of gauge fields. Such operators will be called the Lévy differential operators.
The following papers are devoted to the connection between the Lévy differential operators and the Yang-Mills equations. In Ref. [9] by Aref'ieva and Volovich the functional divergence was introduced by the analogy with the Lévy Laplacian defined by means of a second-order derivative of special form. The relationship of this divergence with gauge fields were studied. In the current paper it is shown that the functional divergence is a restriction of the Lévy divergence. In Refs. [2,3] Accardi, Gibilisco and Volovich introduced an analogue of the Lévy Laplacian acting on functions on the space of piecewise smooth functions (see also Ref. [1]). This operator was defined by a more complicated form of the second derivative than in (2). It is also called the Lévy Laplacian. In Refs. [2,3] the following was proved. The connection in the trivial bundle over a Euclidean space is a solution to the Yang-Mills equations if and only if the parallel transport, considered as an operator-valued functional on a space of curves, is a solution to the Laplace equation for such Lévy Laplacian. Moreover in Refs. [2,3] the Lévy d'Alembertian was introduced. The similar theorem about the equivalence of the Yang-Mills equations for a connection on a Minkowski space and the Lévy-d'Alembert equation for the parallel transport holds for this operator. The case of a Riemannian manifold and the stochastic case were considered by Leandre and Volovich in Ref. [19]. In Refs. [25,27] the theorem about the connection between the Lévy Laplacian and the gauge fields was proved in the case of a manifold and the Lévy Laplacian and the Lévy d'Alembertian defined as the Césaro mean of the second directional derivatives (see also Ref. [8] for the definition). Thus, the problem has been solved, set in Ref. [2]. In Refs. [28,29] it was proved that the Lévy Laplacians and Lévy d'Alembertians defined as the Cesàro mean of the second directional derivatives and defined by a special form of the second derivative coincide in the plane case. The instantons were described in terms of the Lévy Laplacian and the parallel transport in Ref. [29]. In Refs. [30,31] the Levy Laplacian defined as the Chesaro mean of the directional derivatives in the stochastic case was studied. 2 It was shown that, unlike the deterministic case, the equivalence of the Yang-Mills equations and the Levy-Laplace equation is not valid for such Laplacian. It means that this Levy Laplacian does not coincide with the Levy Laplacian introduced in Ref. [19].
The path 2-form is a function on the space of curves in R d beginning at the origin such that the value of this function on a curve is a matrix-valued 2-form at the endpoint of the curve. In Ref. [17] by Gross the infinitedimensional non-commutative Poincaré lemma was proved. It implies that an infinite-dimensional 1-form (a smooth function from the space of curves to the space of linear matrix-valued operators on the space of curves) is generated by the path 2-form and is closed if and only if it is generated by a parallel transport associated with a connection in the trivial bundle over R d . Using this non-commutative Poincaré lemma in Ref. [17] Gross proved that the differential equations on the path 2-form, that are generalization of Maxwell's equations, are equivalent to the Yang-Mills equations. Also in Ref. [17] the differential equations on the path 2-forms and the infinitedimensional fields, that are equivalent to the QCD equations and the Yang-Mills-Higgs equations, were obtained.
In the current paper we show that an infinite-dimensional 1-form generated by the path 2-form is closed and the value of the Lévy divergence on this 1-form is zero if and only if it is associated with a finite-dimensional connection, that is a solution to the Yang-Mills equations. 3 Thus, we obtain the differential equations that are analogue of the equations of motion of the chiral field and are equivalent to the Yang-Mills equations (the parallel transport can be considered as an analogue of the chiral field). In addition, we study the connection between the Lévy infinite-dimensional differential operators and Yang-Mills-Higgs equations and the QCD equations. For this purpose the equations derived in Ref. [17] are used. But in contrast to our approach the Lévy differential operators were not considered in Ref. [17]. We generalize the Accardi-Gibilisco-Volovich theorem and reformulate the Yang-Mills-Higgs equations and the QCD equations in terms of the parallel transport and the Lévy d'Alembertian. Using the non-commutative Poincaré lemma we reformulate the Yang-Mills-Higgs equations and the QCD equations in terms of the infinite-dimensional 1-form and the Lévy divergence. The relationship of the Lévy d'Alembertian and the QCD equations was considered in Ref. [28].
The paper is organized as follows. The first section provides the definition parallel transport but without using the Lévy differential operators was considered in Refs. [13,10,11]. 3 The converse statement was proved in Ref. [9] for the functional divergence.
of the Lévy traces as the Cesàro mean of diagonal elements and differential operators generated by these traces. The second section provides the definition of the Lévy traces as an integral functional and associated differential operators. Also in the second section the equivalence of two ways to define Lévy differential operators is shown. The connection between the Lévy differential operators and the Yang-Mills equations is discussed in the third section. In the fourth and the fifth sections systems of infinite-dimensional differential equations, containing Lévy differential operators, which are equivalent to the Yang-Mills-Higgs equations and to the QCD equations respectively, are obtained.

Lévy trace and Lévy Differential Operators
Let us recall the general scheme of the definition of homogeneous linear differential operators from the paper Ref. [12] (see also Ref. [23]). Let X, Y, Z be real normed vector spaces. Let C n (X, Y ) be the space of n times Frechét differentiable functions from X to Y . Then for any x ∈ X it is hold that The domain domD n,S of the differential operator D n,S of order n generated by the linear operator Then the value of D n,S on f ∈ domD n,S is defined by the following formula If we choose X = R d , Y = Z = R and S = tr, then D 2,tr is the Laplace operator ∆. If we choose X = Y = R d , Z = R and S = tr, then D 1,tr is the divergence div acting on the space of C 1 -smooth 1-forms. Let E 1 be a real normed space. Let E 1 be continuously embedded in a real separable infinite-dimensional Hilbert space H 1 in such a way that the where σ g = 1 or σ g = i, if g = δ and g = η respectively, and dom tr consists of all T ∈ L(E, E * ) for which the right-hand side of (3) exists.
generated by the orthonormal basis {e n } is the differential operator In the Euclidean case the Lévy divergence div δ L generated by the orthonormal basis {e n } is the operator In the Minkowski case the Lévy divergence div η L generated by the orthonormal basis {e n } is the operator D 1,tr {en},η L . Remark 1. The Lévy trace was introduced for d = 1 in Ref. [5].
Remark 2. The non-classical Lévy Laplacian is a generalization of the Lévy Laplacian defined in the following way. Let R be a linear mapping from span{e n : n ∈ N} to E 1 . The non-classical Lévy trace tr {en},g LR generated by the orthonormal basis {e n }, the metric g and the linear operator R acts on T ∈ L(E, E * ) by the formula respectively. The nonclassical Lévy divergence div g LR is the differential operator D 1,tr g LR . The nonclassical Lévy trace and the associated non-classical Lévy Laplacian were introduced for d = 1 in Ref. [6]. As we show below, such operators can be useful in the study of gauge fields (see also [31]). Moreover, the exotic Lévy Laplacians (see Refs. [5,7] for the definition), which are studied in the white noise analysis (see Ref. [4] and references therein), can be represented as non-classical Lévy Laplacians (see Refs. [26,28]).

Lévy trace as an integral functional
Let µν is a symmetric tensor: Q L µν = Q L νµ , and Q S µν is an anti-symmetric tensor: Definition 4. The Lévy trace tr g L generated by the metric g is a linear functional on T 2 E defined by the formula Remark 3. The first term in (5) is the Volterra part, the second term is the Lévy part, the third term is a singular part.
In the Minkowski case the Lévy divergence div η L is the differential operator D 1 The definition of the functional divergence by Aref'ieva and Volovich (see Ref. [9]) can be reformulated as follows.
The functional divergence is the differential operator D 1,tr δ v . Thus, the functional divergence is a restriction of the Lévy divergence.
Example 1. The sequence e n (t) = √ 2 sin nπt is a weakly uniformly dense basis in L 2 (0, 1). Theorem 1. Let {e n } be a weakly uniformly dense orthonormal basis in L 2 (0, 1) that consists of functions from E such that e n (1) = 0 for all n ∈ N. Let the sequence {e n } be uniformly bounded. If Q ∈ T 2 E , then Proof. Since e n (0) = e n (1) = 0 for all n ∈ N and Q S µν is anti-symmetric tensor, the following equality holds ) (see Ref. [18]). Since Q L µµ ∈ L 1 (0, 1) and The following theorem is a direct corollary of Theorem 1.
Remark 4. The definitions of the Lévy Laplacian and the Lévy d'Alembertian from section 1 and section 2 can be naturally transferred to the space C 2 (E, M m (C)). Additionally the definitions of the Lévy traces can be transferred to the space L(E, E * 0 ). This fact allows to define the Lévy divergence on the space C 1 (E, L(E 0 , M m (C))). Such divergence will be used in the following sections.
π(n−1) sin(π(n− 1)t) for n > 1. Consider the operator N : span{f n : n ∈ N} → E 1 defined by N f n = (n − 1)f n . Then tr {fn},g LN coincides with tr g L on the space T 2 E . The differential operators generated by the trace tr g L are restrictions of the corresponding differential operators generated by the trace tr

The Yang-Mills equations
The summation over repeated upper and lower indices is assumed. Let G be a closed Lie group realized as a subgroup of U (N ), and let Lie(G) be a Lie algebra realized as a subalgebra of u(N ). A connection in the trivial vector bundle with base R d , fiber C N , structure group G is defined as a Lie(G)-valued C ∞ -smooth 1-form A µ (x)dx µ on R d . For the function ϕ ∈ C 1 (R d , Lie(G)) the covariant derivative along the field ( ∂ ∂µ ) is defined by The tensor of curvature is defined by Lie(G)-valued 2-form The connection A is a solution to the Yang-Mills equations with the current j( For any curve σ ∈ E an operator U A t,s (σ), where 0 ≤ s ≤ t ≤ 1, is defined as a solution to the system of differential equations Here and below the symbol I N denotes the unitary matrix in For a proof of this proposition we refer the reader to Ref. [17] and also to Ref. [14].
The results of Proposition 2, of Theorem 3 and of Theorem 4 for j = 0 belong to Accardi, Gibilisco and Volovich. Proposition 2. If σ ∈ E and u, v ∈ E 0 , the following equality holds where For a proof of this proposition we refer the reader to Ref. [3] and also to Ref. [29].
The following theorem is a direct corollary of Proposition 2.
Theorem 3. The following equality holds If σ ∈ E, everywhere below the symbol σ r , where 0 ≤ r ≤ 1, denotes the curve from E defined by σ r (t) = σ(rt). 2. the equality holds for the parallel transport U A 1,0 .
Differentiating (12) with respect to r, we obtain Choose an appropriate σ ∈ E ∩ C 1 ([0, 1], R d ), we see that the connection A is a solution to the Yang-Mills equations with current (8). The other side of the theorem is obvious.
Consider the function B A : E → L(E 0 , Lie(G)) defined by Proposition 3. The following equalities hold Then, using (9), we obtain that for all σ ∈ E and u, v ∈ E 0 the following holds . Thus, we obtain (13).
The following result belongs to Gross (see Ref. [17]). It is an analogue of the Poincaré lemma.
where σ ∈ E, u ∈ E 0 , h(σ) is a Lie(G)-valued anti-symmetric tensor for any σ ∈ E, the function E ∋ σ → h(σ) is continuously Frechét differentiable. 5 Let 1-form B be closed in the sense that for all σ ∈ E and u, v ∈ E 0 the following equality holds Then there is a connection A such that B = B A .
In the Gross's theorem the space P of piecewise C ∞ -smooth functions with the norm σ = 1 0 |σ(t)|dt instead of E was considered. The proof can be transferred unchanged to the case of the spaces W 1,1 0 ([0, 1], R d ) and Remark 6. It is known (see Ref. [17]) that if an infinite-dimensional 1-form B can be represented as (16) then B ∈ C ∞ (E, L(E 0 , Lie(G)). It was also proved in Ref. [17] that for two connection A and A ′ such that Let B ∈ C ∞ (E, L(E 0 , Lie(G)) can be represented as (16). Then B is a solution to the system if and only if there is a connection A which is a solution to the Yang-Mills equations Proof. Let B be a solution to system (17). The existence of the C ∞ -smooth connection A such that B = B A follows from Theorem 5. Then, due to Proposition 3, the equality D 2 tr g L U A 1,0 (σ) = 0 follows from the equality div g L B A (σ) = 0. This implies A is a solution to (18). The other side of the theorem is trivial.
Remark 7. The parallel transport can be considered as an infinite-dimensional analogue of the general chiral field. Recall that if G is a matrix Lie group, the field b : R d → G is a general chiral field. Its Dirichet integral (see Ref. [15]) has a form where The equations of motion of the chiral field have the form divZ = 0 where Z = (Z 1 , . . . , Z d ). System (17) is an infinite-dimensional analogue of system (20). Conservation laws for the 3-dimensional Yang-Mills theory were considered in Ref. [9] (see also Refs. [21,22]). In particular, it was shown that B A is the first non-trivial conserved current if the connection A is a solution to the Yang-Mills equations. The conservation of the current was understood in the sense that the value of the functional divergence on this current is zero. Thus, Theorem 6 is a generalization of this result.

The Yang-Mills-Higgs equations
Let V be a finite-dimensional vector space. For any h ∈ R d consider a Vvalued linear functional S h defined in the following way. The domain domS h of S h consists of all T ∈ L(E, V ) having the following form In terms of the orthonormal basis {f n } of W 1,2 0 ([0, 1], R) (see Remark 5) the operator S h can be defined in the following way. For any h ∈ R d consider a linear operator S where the domain domS In this and the next sections we assume that the Greek indices run from 0 to 3. We assume the basis {p 0 , p 1 , p 2 , p 3 } is given in the Minkowski space R 1,3 and the Minkowski metric η is given by diag{1, −1, −1, −1} in this basis. We will raise and lower indices using the Minkowski metric. We assume that the connection is defined as a su(N )-valued The Yang-Mills-Higgs equations are the following equations on the C ∞smooth su(N )-valued 1-form A and C ∞ -smooth su(N )-valued field φ: where m, l ≥ 0.
Remark 9. System of the differential equations (22) is the Euler-Lagrange equations for the Lagrangian with the Lagrangian density: In terms of the parallel transport and the Lévy d'Alembertian solutions to the Yang-Mills-Higgs equations can be described as follows.
Let the function Φ A,φ : E → su(N ) be defined by Theorem 7. A pair (A, φ) is a solution to system (22) if and only if for the parallel transport U A 1,0 and the function Φ A,φ the following relations hold Proof. Since Since Relations (24) and (25) and (1)))φ(σ(1))U A 1,0 (σ) imply that the pair (A, φ) satisfies the first equation of system (22) if and only if the pair (U 1,0 , Φ) satisfies the first equation of system (23). Since the pair (A, φ) satisfies the second equation of system (22) if and only if the pair (U A 1,0 , Φ A,φ ) satisfies the second equation of system (23).
Solutions to the Yang-Mills-Higgs equations can be described in terms of the Lévy divergence as follows.
Theorem 8. Let B ∈ C ∞ (E, L(E 0 , su(N )) can be represented as (16). Then a pair (B, Φ), where Φ ∈ C ∞ (E, su(N )), is a solution to the system where m, l ≥ 0, if and only if there is a pair (A, φ), where A is a su(N )valued C ∞ -smooth 1-form and φ ∈ C ∞ (R 1,3 , su(N )), which is a solution of the Yang-Mills-Higgs equations (22) such that B = B A and Φ = Φ A,φ .
Proof. Let the pair (B, Φ) be a solution to system (26). Due to Theorem 5 the second equation of the system implies that there is a connection A such that B = B A . Consider the function Φ 1 : E → su(N ) defined by the formula su(N )). Due to the third equation of system (26), if u ∈ E 0 , then This implies that the field φ : holds if and only if the equality holds. Then we can apply Theorem 7. It follows that (A, φ) is a solution to the Yang-Mills-Higgs equations.
If the pair (A, φ) is a solution to the Yang-Mills-Higgs equations, it can be checked by direct calculations that the pair (B A , Φ A,φ ) is a solution to system (26).
The QCD equations (the Yang-Mills-Dirac equations) are the following differential equations on a C ∞ -smooth su(N )-valued 1-form A and a C ∞smooth C N ⊗ C 4 -valued function ψ on R 1,3 : Remark 11. System of differential equations (27) is the Euler-Lagrange equations for the Lagrangian with the Lagrangian density In terms of the parallel transport and the Lévy d'Alembertian the solutions to the Yang-Mills-Dirac equations can be described in the following way.
The function Ψ A,φ : E → C N ⊗ C 4 is defined by the formula Theorem 9. A pair (A, ψ), where ψ ∈ C ∞ (R 1,3 , C N ⊗ C 4 ), is a solution to the Yang-Mills-Dirac equations (27) if and only if for the parallel transport U A 1,0 and the function Ψ A,ψ the following system holds Proof. We obtain by direct calculation This implies the pair (A, ψ) satisfies the first equation of system (27) if and only if the pair (U A 1,0 , Ψ A,ψ ) satisfies the first equation of system (28). Due to Theorem 4 the pair (A, ψ) is a solution to the second equation of system (27) if and only if the following equality holds for all σ ∈ E L U A 1,0 (σ) = 1 0 U A 1,r (σ)(pr su(N ) (iψ(σ(r))γ ν ψ(σ(r))σ ν (r)))U A r,0 (σ)dr.
For any b ∈ SU (N ) and any ϕ ∈ C N the following equality holds (see for example Ref. [17]) Hence, This implies that the pair (A, ψ) satisfies the second equation of system (27) if and only if the pair (U A 1,0 , Ψ A,ψ ) satisfies the second equation of system (28).
In terms of the Lévy divergence the solutions to the Yang-Mills-Dirac equations can be described in the following way.
Theorem 10. Let B ∈ C ∞ (E, L(E 0 , su(N )) can be represented as (16). Then a pair (B, Ψ), where Ψ ∈ C ∞ (E, C N ⊗ C 4 ), is a solution to the system of the equations: if and only if there is a pair (A, ψ) which is a solution to the Yang-Mills-Dirac equations (27) such that B = B A and Ψ = Ψ A,ψ .
Then we can apply Theorem 9 and obtain that (A, ψ) is a solution of the Yang-Mills-Dirac equations. If a pair (A, ψ) is a solution to the Yang-Mills-Dirac equations, it can be checked by direct calculations that the pair (B A , Ψ A,ψ ) is a solution to system (29).
Remark 12. In Ref. [17] systems of infinite-dimensional equations equivalent to the Yang-Mills-Higgs equations and to the Yang-Mills-Dirac equations were obtained. The third and the fourth equation of system (29) and system (26) were derived in that work. Remark 14. Due to Remarks 5 and 8 all differential operators in systems (29) and (26) can be defined by the basis {f n } in W 1,2 0 ([0, 1], R).

Conclusion
We provided the classification of the Levy differential operators and obtained the equations contained these operators that are equivalent to the QCD equations and the Yang-Mills equations. It would be interesting to investigate is it possible to find the stochastic analogue of Gross's noncommutative Poincare lemma. With the help of such analogue it would be possible to develop the results of the paper Ref. [30], where the stochastic parallel transport was considered as a general chiral field. Also it would be interesting to investigate the connection between the stochastic Lévy Laplacians introduced in Ref. [19] and in Ref. [31]. In the next paper we develop some results of the current paper for the case of the Riemannian and the pseudo-Riemannian manifold.