General solutions of one class of field equations
aa r X i v : . [ m a t h - ph ] N ov GENERAL SOLUTIONS OF ONE CLASS OF FIELD EQUATIONS
N. G. Marchuk ∗ Steklov Mathematical Institute,Russian Academy of Sciences, Gubkina St. 8, 119991 Moscow, Russiae-mail: [email protected]. S. ShirokovNational Research University Higher School of Economics,Myasnitskaya str. 20, 101000, Moscow, Russiae-mail: [email protected] Institute for Information Transmission Problems,Russian Academy of Sciences, Bolshoy Karetny per. 19, 127051, Moscow, Russiae-mail: [email protected]
We find general solutions of some field equations (systems of equations) inpseudo-Euclidian spaces (so-called primitive field equations). These equations areused in the study of the Dirac equation and Yang-Mills equations. These equationsare invariant under orthogonal O( p, q ) coordinate transformations and invariant un-der gauge transformations, which depend on some Lie groups. In this paper we usesome new geometric objects - Clifford field vector and an algebra of h-forms which isa generalization of the algebra of differential forms and the Atiyah-K¨ahler algebra.
Keywords:
Clifford algebra; gauge symmetry; primitive field equation; Dirac equation;Dirac gamma matrices; Yang-Mills equations.
1. Introduction
In physics field equations describe physical fields and (using quantization) elementaryparticles. The following equations are fundamental relativistic field equations: Maxwell’sequations (1862), the Klein-Gordon-Fock equation (1926), the Dirac equation (1928),Yang-Mills equations (1954). These equations are considered in Minkowski space R , ,they are invariant under Lorentz coordinate transformations. They are also invariantunder certain unitary gauge transformations.In this paper we consider a class of so-called primitive field equations (systems ofequations) (27). These equations are considered in pseudo-Euclidian spaces R p,q andhave different Lie groups of gauge symmetry. We find general solutions of primitive ∗ This work was supported by Russian Science Foundation (project RSF 14-50-00005, Steklov Math-ematical Institute). [1]field equations corresponding to a wide class of gauge Lie groups. A partial case ofsuch equations was considered in 1930-1939 in the theory of Dirac equation on curvedpseudo-Riemannian manifolds of signature (1,3). Namely, it is a condition of generalizedcovariant constancy of γ -matrices, which is gauge invariant w.r.t. the spinor Lie groupSpin(1 ,
3) (see, for example, [12] formula (4.5.13)).In the theory of model field equations developed by authors in a series of papersand in monograph [7] there arise necessity to solve primitive field equations in pseudo-Euclidean space R , with various gauge Lie groups (see Section 2). Also, primitive fieldequations are used in the theory of Yang-Mills equations. Namely, we present a newclass of gauge-invariant solutions of Yang-Mills equations, which correspond to solutionsof primitive field equations (see [8]).In Section 2 of this paper we consider the Dirac equation and make some importantnotes about the Lie group SU(2 ,
2) in connection with the Dirac equation. As a resultof these notes we get new system of equations (8), (9). Equation (8) is a special case ofprimitive field equations and we study these equations in the next sections of the paperin pseudo-Euclidian spaces R p,q .In Section 3 we discuss some known facts about Clifford algebras. We actively usetensor fields with values in Clifford algebra. Also we discuss some Lie algebras in Cliffordalgebra, especially Lie algebras w( C ℓ ( p, q ))). Results about Lie subalgebras of the Liealgebras w( C ℓ ( p, q ))) of pseudo-Unitary Lie group are our original results published in[7, 11].In Section 4 we present original results about projection operators and contractionsin Clifford algebras. We use these results in Section 7.In Section 5 of this paper we present some new geometric objects - Clifford fieldvector and an algebra of h-forms which is a generalization of the algebra of differentialforms and the Atiyah-K¨ahler algebra [1, 5] . These objects are helpful for considerationof some problems related to field theory equations.In Sections 6 and 7 we consider a primitive field equation and present original resultsabout its gauge symmetry and about general solutions of this equation.Note that all considerations of this paper are valid for general pseudo-euclidian metricof signature ( p, q ) and, in particular, for the Lorentzian metric (+ , − , − , − ). Results ofthe paper can be understood either on the base of Dirac gamma matrices or on the baseof Clifford algebras.
2. A new view on the Dirac equation and γ -matrices Consider the Dirac equation for an electron in the Minkowski space R , with coor-dinates x µ , µ = 0 , , , ∂ µ = ∂/∂x µ – partial derivatives) iγ µ ( ∂ µ ψ − ia µ ψ ) − mψ = 0 , (1) We combine the technique of the Dirac gamma matrices and the technique of differential forms, inparticular, the Atiyah-K¨ahler algebra of differential forms. where γ µ are 4 complex square matrices of order 4 satisfying conditions ∂ µ γ ν = 0 , (2) γ µ γ ν + γ ν γ µ = 2 η µν I, (3)where η = k η µν k = diag(1 , − , − , − I is the identity matrix of order 4, a µ = a µ ( x ) isa covector potential of electromagnetic field, ψ = ψ ( x ) is a Dirac spinor (column of fourcomplex functions ψ : R , → C ), i is the imaginary unit, m is a real number (mass ofelectron).In the theory of the Dirac equation it is assumed that we have a fixed set of matrices γ µ that satisfy conditions (2), (3) and the condition for Hermitian conjugated matrices( γ µ ) † = γ γ µ γ . (4)Matrices γ µ satisfying conditions (2), (3), (4) are defined up to a similarity transfor-mation with a unitary matrix U ∈ U(4), i.e. matrices´ γ µ = U − γ µ U, where U − = U † (5)satisfy the same conditions (2), (3), (4).In particular, matrices γ , γ , γ , γ in the Dirac representation satisfy these condi-tions and the matrix γ is diagonal γ = diag(1 , , − , − γ changesunder unitary transformation (5).Denote β = diag(1 , , − , −
1) and consider Lie group SU(2 ,
2) of special pseudo-unitary matrices and its real Lie algebra s u (2 ,
2) (see [3])SU(2 ,
2) = { S ∈ Mat(4 , C ) : S † βS = β, det S = 1 } , s u (2 ,
2) = { s ∈ Mat(4 , C ) : βs † β = − s, tr s = 0 } , where Mat(4 , C ) is the algebra of complex matrices of order 4. Dirac gamma matrices γ µ satisfy (4) and tr γ µ = 0, therefore iγ µ ∈ s u (2 , . (6)We may consider conditions (2), (3) together with condition (6) and allow a similaritytransformation iγ µ → i ´ γ µ = S − iγ µ S (7)with matrix S ∈ SU(2 , γ µ with condition (6),then we can consider transformation (7) as a global symmetry (it does not depend on x ∈ R , ) of this system of equations.Now we change equations (2) and obtain a system of equations with local (gauge)symmetry with respect to the pseudo-unitary group SU(2 , This condition is required when we consider bilinear covariants of the Dirac spinors.
Namely, consider the following system of equations [7]: ∂ µ γ ν − [ C µ , γ ν ] = 0 , (8) γ µ γ ν + γ ν γ µ = 2 η µν I, (9)where iγ µ = iγ µ ( x ) and C µ = C µ ( x ) are smooth functions of x ∈ R , with values in theLie algebra s u (2 , iγ µ → i ´ γ µ = S − iγ µ S, (10) C µ → ´ C µ = S − C µ S − S − ∂ µ S, (11)where the matrix S = S ( x ) is a function of x ∈ R , with values in the Lie group SU(2 , a primitive field equation . Let us analyze this equation in pseudo-Euclidian spaces. We use a formalism of Clifford algebras because, in our opinion, thisformalism is the most convenient for this task.
3. Clifford algebras
Consider real C ℓ R ( p, q ) or complexified C ℓ ( p, q ) = C ⊗C ℓ R ( p, q ) (see [6]) Clifford algebrawith p + q = n , n ≥
1. Note that C ℓ R ( p, q ) ⊂ C ℓ ( p, q ). When our argumentationis applicable to both cases, we write C ℓ F ( p, q ), implying that F = R or F = C . Theconstruction of Clifford algebra is discussed in details in [6, 10, 11].Let e be the identity element and let e a , a = 1 , . . . , n be generators of the Cliffordalgebra C ℓ F ( p, q ), e a e b + e b e a = 2 η ab e, (12)where η = || η ab || = || η ab || is the diagonal matrix with p pieces of +1 and q pieces of − e a ...a k = e a · · · e a k , a < · · · < a k , k = 1 , . . . , n, (13)together with the identity element e form the basis of the Clifford algebra. The numberof basis elements is equal to 2 n .Any element U of the Clifford algebra C ℓ F ( p, q ) can be expanded in the basis: U = ue + u a e a + X a
1. We denote an n -dimensional pseudo-Euclidian space ofsignature ( p, q ) with Cartesian coordinates x µ , µ = 1 , . . . , n by R p,q . Tensor indicescorresponding to the coordinates are denoted by small Greek letters. The metric tensorof pseudo-Euclidian space R p,q is given by the diagonal matrix of order nη = k η µν k = k η µν k = diag(1 , . . . , , − , . . . , − 1) (16)with p copies of 1 and q copies of − R p,q we deal with linear coordinate transformations x µ → ´ x µ = p µν x ν , (17)preserving the metric tensor. So, real numbers p µν satisfy relations p µα p νβ η αβ = η µν , p µα p νβ η µν = η αβ . In matrix formalism we can write P T ηP = η , P ηP T = η , where T isthe matrix transposition and the matrix P = k p µν k is from the pseudo-orthogonal group O ( p, q ) = { P ∈ Mat( n, R ) : P T ηP = η } . We denote the set of ( r, s ) tensor fields (of rank r + s ) of pseudo-Euclidian space R p,q by T rs . Real or complex tensor field u ∈ T rs has components u µ ...µ r ν ...ν s in coordinates x µ .These components are smooth functions R p,q → F , where F is the field of real numbers R There is a difference in notation in literature. We use term “rank” and notation C ℓ k ( p, q ) becausewe take into account a relation with differential forms, see [7]. We use the Einstein summation convention. For example p µν x ν = P nν =1 p µν x ν . or complex numbers C . In all considerations of this work it is sufficient that all functionsof x ∈ R p,q have continuous partial derivatives up to the second order. Functions with values in Clifford algebra. Further we consider functions R p,q →C ℓ ( p, q ) with values in Clifford algebra. We assume that the basis elements (13) do notdepend on the points x ∈ R p,q i.e. ∂ µ e a = 0 , ∀ µ, a = 1 , . . . , n, where ∂ µ = ∂/∂x µ are partial derivatives. The coefficients in the basis expansion of theClifford algebra element u a ...a k = u a ...a k ( x ) may depend on x ∈ R p,q . In the presentpaper we also consider the functions with values in Lie algebras generated by the Cliffordalgebra (see p. 8). Tensor fields with values in Clifford algebra. A tensor at the point x ∈ R p,q withvalues in Clifford algebra is a mathematical object that belongs to the tensor product ofthe tensor algebra and Clifford algebra.If a tensor field of rank ( r, s ) in R p,q has components u µ ...µ r ν ...ν s = u µ ...µ r ν ...ν s ( x ) in Cartesiancoordinates x µ , then these components are considered as functions R p,q → F . Thesefunctions transform by the standard tensor transformation rule.Components U µ ...µ r ν ...ν s of tensor fields with values in Clifford algebra C ℓ F ( p, q ) are con-sidered as functions R p,q → C ℓ F ( p, q ) that transform under changes of coordinates by thestandard tensor transformation rule.We use the following notation for tensor fields with values in Clifford algebra: U µ ...µ r ν ...ν s ∈C ℓ ( p, q )T rs or U ∈ C ℓ ( p, q )T rs . In this notation the letter T means that this object is atensor field. In particular, for scalar functions U : R p,q → C ℓ F ( p, q ) we use the notation U ∈ C ℓ F ( p, q )T.For example, if we consider a tensor field U µν ∈ C ℓ ( p, q )T with values in Cliffordalgebra, then we can write U µν = u µν e + u µνa e a + X a
Let us consider the commutator (Lie bracket) [ U, V ] = U V − V U of Clifford algebraelements U, V ∈ C ℓ ( p, q ). This operation satisfy the Jacobi identity[[ U, V ] , W ] + [[ V, W ] , U ] + [[ W, U ] , V ] = 0 , ∀ U, V, W ∈ C ℓ ( p, q ) . Therefore, Clifford algebra C ℓ ( p, q ) can be considered as a Lie algebra with respect tothe commutator. We can consider vector subspaces L ⊂ C ℓ ( p, q ) of Clifford algebra thatclosed under commutator i.e. with the condition: if U, V ∈ L then [ U, V ] ∈ L . Thesesubspaces are Lie algebras (generated by Clifford algebra), see also [4]. Primarily we areinterested in Lie algebras that are direct sums (as vector spaces) of subspaces of Cliffordalgebra elements of fixed ranks [13].With the help of the operator π : C ℓ F ( p, q ) → C ℓ F ( p, q ) we define operation of Cliffordalgebra trace Tr : C ℓ F → F Tr( U ) = π ( U ) | e → , ∀ U ∈ C ℓ F ( p, q ) . Theorem 1. In Clifford algebra C ℓ F ( p, q ) of arbitrary dimension n = p + q we have Tr([ U, V ]) = 0 for all U, V ∈ C ℓ F ( p, q ) . In Clifford algebra C ℓ F ( p, q ) of odd dimension n = p + q we have π n ([ U, V ]) = 0 for all U, V ∈ C ℓ F ( p, q ) . Proof . Using (12) for 2 arbitrary basis elements we obtain[ e a ...a k , e b ...b l ] = (1 − ( − kl − s ) e a ...a k e b ...b l ∈ C ℓ F k + l − s ( p, q ) , (18)where s is the number of coincident indices in the ordered multi-indices a . . . a k and b . . . b l . If k = l = s , then 1 − ( − kl − s equals 0. If k + l = n , s = 0 and n is odd, thenit equals 0 again. For more details see Theorem 1 in [13].Consider the set of Clifford algebra elements with zero projection onto Clifford algebracenter C ℓ s ( p, q ) = C ℓ ( p, q ) \ Cen( C ℓ ( p, q )) . Theorem 2. The set C ℓ s ( p, q ) is a Lie algebra with respect to the commutator [ A, B ] = AB − BA . Proof . See the previous theorem. Theorem 3.