Noncommutative integration of the Dirac equation in homogeneous spaces

We develop a noncommutative integration method for the Dirac equation in homogeneous spaces. The Dirac equation with an invariant metric is shown to be equivalent to a system of equations on a Lie group of transformations of a homogeneous space. This allows us to effectively apply the noncommutative integration method of linear partial differential equations on Lie groups. This method differs from the well-known method of separation of variables and to some extent can often supplement it. The general structure of the method developed is illustrated with an example of a homogeneous space which does not admit separation of variables in the Dirac equation. However, the basis of exact solutions to the Dirac equation is constructed explicitly by the noncommutative integration method. Also, we construct a complete set of new exact solutions to the Dirac equation in the three-dimensional de Sitter space-time $\mathrm{AdS_{3}}$ using the method developed. The solutions obtained are found in terms of elementary functions, which is characteristic of the noncommutative integration method.


I. INTRODUCTION
Exact solutions of the relativistic wave equations in strong gravitational and electromagnetic fields are the basis for studying quantum effects in the framework of quantum field theory in curved space-time (see, e.g. [1][2][3][4][5][6]). A construction of the complete set of exact solutions to these equations in many cases is associated with the presence of integrals of motion. For example, to separate the variables in a wave equation, it is necessary to have dim M − 1 commuting integrals, where M is the space of independent variables. In this paper, by integrability of the wave equation we mean an explicit possibility of reducing the original equation to a system of ordinary differential equations, the solution of which provides a complete set of solutions to the original wave equation.
The best-known technique for such a reduction is based on the method of separation of variables (SoV) (various aspects of the SoV method can be found, e.g., in [7][8][9]). There is a broad scope of research dealing with separation of variables in relativistic quantum wave equations, mainly for the Klein-Gordon and Dirac equations, and with classification of external fields admitting SoV in these equations (see, e.g., [10] and references therein). This motivates the development of methods for the exact integration of wave equations other than SoV that can give some new possibilities in relativistic quantum theory.
In this regard, we focus on homogeneous spaces as geometric objects with high symmetry.
We also note that most of the physically interesting problems and effects are associated with gravitational fields possessing symmetries. Mathematically, these symmetries indicate the presence of various groups of transformations that leave invariant the gravitational field.
Representing the space-time as a homogeneous space with a group-invariant metric, we can consider a large class of gravitational fields and cosmological models [11,12] with rich symmetries, and the corresponding relativistic equations in these fields have integrals of motion.
We note that the relativistic wave equations on a homogeneous space may not allow separation of variables. The matter is that in accordance with the theorem of Refs. [13,14], for the separation of variables in the wave equation in an appropriate coordinate system the equation should admit a complete set of mutually commuting symmetry operators (integrals of motion, details can be found in [13,14], see also [15]). Therefore, the problem arises of constructing exact solutions to the wave equation in the case when it has symmetry oper-ators, but they do not form a complete set and separation of variables can not be carried out. We consider the noncommutative integration method (NCIM) based on noncommutative algebras of symmetry operators admitted by the equation [16][17][18][19][20]. This method can be thought as a generalization of the method of SoV. A reduction of the wave equation to a system of ODEs according to the NCIM (we use the term noncommutative reduction) can be carried out in a way that is substantially different from the method of separation of variables.
We note that the method of noncommutative integration has shown its effectiveness in constructing bases of exact solutions to the Klein-Gordon and Dirac equations in some spaces with invariance groups.
For instance, the NCIM was applied to the Klein-Gordon equation in homogeneous spaces with an invariant metric in [19,20]. The polarization vacuum effect of a scalar field in a homogeneous space was studied using NCIM in [19][20][21].
The noncommutative reduction of the Dirac equation to a system of ordinary differential equations in the Riemannian and pseudo-Riemannian spaces with a nontrivial group of motions was considered in [22][23][24][25][26][27]. In Refs. [28,29] the NCIM was applied to the Dirac equation in the four-dimensional flat space and in the de Sitter space. The Dirac equation on Lie groups that can be a special case of homogeneous spaces with a trivial isotropy subgroup, was explored in terms of the NCIM in Refs. [30,31]. In the present work we consider non-commutative symmetries of the Dirac equation in homogeneous spaces. We also develop the method of noncommutative integration of the Dirac equation in homogeneous spaces. Using the group-theoretic approach, we reduce the Dirac equation on the homogeneous space to such a system of equations on the transformation group that lets us to apply the noncommutative reduction and construct exact solutions of the Dirac equation. In this paper, for the first time, we explicitly take into account the identities for generators of the transformation group in the problem of noncommutative reduction for the Dirac equation.
The work is organized as follows. In Section II we briefly introduce basic concepts and notations from the theory of homogeneous spaces [32][33][34], to be used later.
A construction of invariant differential operator with matrix coefficients on a homogeneous space is introduced in Section III following Refs. [35,36]. Also in this section, we show the connection between generators of the representation of a Lie group on a homogeneous space and the other representation induced by representation of a subgroup, whose action on a homogeneous space has a stationary point.
In the next Section IV, we introduce a special irreducible representation of the Lie algebra of the Lie group of transformations of a homogeneous space using the Kirillov orbit method [37], that is necessary for noncommutative reduction.
In Section V we present the Dirac equation in a homogeneous space with an invariant metric in terms of an invariant matrix operator of the first order. The spinor connection and symmetry operators of the Dirac equation are shown to define isotropy representation in a spinor space. Generators of the spinor representation are found explicitly.
We also introduce a system of differential equations on the Lie group of transformations of a homogeneous space, which is equivalent to the original Dirac equation in a homogeneous space.
Then, in Section VI, we present a noncommutative reduction of the Dirac equation on a homogeneous space, using the irreducible λ-representation introduced in section IV and functional relations between symmetry operators (identities) for the Dirac equation.
In Section VII, we consider a homogeneous space with an invariant metric that does not admit separation of variables for the Klein-Gordon and Dirac equations. In this case a complete set of exact solutions of the Dirac equation is constructed using the noncommutative reduction (Section VI).
The next Section VIII is devoted to the Dirac equation in the (2 + 1) anti-de Sitter AdS 3 - In Section IX we give our conclusion remarks.

II. INVARIANT METRIC ON A HOMOGENEOUS SPACE
This section introduces some basic concepts and notations of the homogeneous space theory with an invariant metric.
Let G be a simply connected real Lie group with a Lie algebra g, M be a homogeneous space with right action of the group G, For any A transformation group G can be regarded as a principal bundle (G, π, M, H) with a structure group H, a base M, and a canonical projection π : G → M, π(e) = x 0 , where e is the identity element of G. An arbitrary point g ∈ G can be represented uniquely as , h ∈ H, and s : M → G is a local and smooth section of G, π • s = id.
Differential of the canonical projection π * : T g G → T π(g) M is a surjective map that allows any tangent vector τ ∈ T x M on a homogeneous space to be represented as π * ζ, where ζ ∈ T g G is a tangent vector on G.
In turn, a linear space of the Lie algebra g ≃ T e G is decomposed into a direct sum of We introduce an invariant metric on the homogeneous space M. Let ·, · m be a nondegenerate Ad(H) -invariant scalar product on the subspace m, (2.1) By action of a Lie group G with right shifts on the homogeneous space M, we define the inner product throughout the space M as We choose a section s : M → G so that equalities s a (x) = x a and s α (x) = 0 hold over the domain U.
The tangent vectors where {e α } is a basis of the Lie algebra h ≃T e H H, and {e a } is a basis of the linear space Here (R g ) * , (R g ) * are differentials of the right shifts R g (g ′ ) = gg ′ on the Lie group G.
The right-invariant vector fields η A satisfy the commutation relations [η A , η B ] = C C AB η C , while the right-invariant 1-forms σ A satisfy the Maurer-Cartan relations, The invariant metric tensor in local coordinates {x i } is written as [38]: The contravariant components of the metric tensor are In what follows we will need the Christoffel symbols of the Levi-Civita connection with respect to a G-invariant metric g M given by [19,33] Here i, j, k = 1, . . . , dim M, and Γ a bc are determined by G ab of the quadratic form G and the structure constants of the Lie algebra g, Thus, in a homogeneous space with invariant metric, the Levi-Civita connection is defined by algebraic properties of the homogeneous space.

FERENTIAL OPERATOR WITH MATRIX COEFFICIENTS
Consider algebraic conditions for an invariant first-order linear differential operator with matrix coefficients on a homogeneous space M. We follow Ref. [35] where a more general case of invariant linear matrix differential operator of the second-order was studied.
Denote by C ∞ (M, V ) and C ∞ (G, V ) the two spaces of functions that map a homogeneous space M and a transformation group G, respectively, to a linear space V . The last one can be regarded as a representation space of the algebra gl(V ).
Functions on the homogeneous space M can be considered as defined on a Lie group G, but invariant over the fibers H of the bundle G [33]. In our case, when the functions take values in a vector space V , the space C ∞ (M, V ) is isomorphic to a subspace of the function where U(h) is an exact representation of the isotropy group H in V . For any function ϕ ∈F, Then we can identify ϕ(s(x)) with a function ϕ ∈ C ∞ (M, V ). Equation (3.1) gives an explicit form of the isomorphismF ≃ C ∞ (M, V ). Differentiating relation (3.1) with respect to h α and assuming h = e H , we obtain Here, Λ α are representation operators of the algebra h on the space V . Equation From (3.2) we can see that a linear differential operator R = R(g, ∂ g ) leaves invariant the function spaceF, if Thus, the space L(F ) of linear differential operators R(g, ∂ g ) :F →F consists of linear differential operators on C ∞ (G, V ) provided that Then given relation (3.1), the action of R(g, ∂ g ) ∈ L(F ) on a function ϕ(g) from the spacê F is written as Multiplying equation (3.3) by U −1 (h) and given η α U(h) = −Λ α U(h), we obtain From here it follows that the operator U −1 (h)R(g, ∂ g )U(h) is independent of h and (3.5) can be written as That is, for any operator R(g, ∂ g ) of L(F ) there exists an operator R M on the homogeneous space M acting on functions of the space C ∞ (M, V ). We say that the operator is the projection of the operator R(g, ∂ g ): R M (x, ∂ x ) = π * R(g, ∂ g ). For example, for a first-order linear differential operator the projection acts as follows: On the other hand, any linear differential operator R M defined on C ∞ (M, V ) corresponds to an operator Thus, we have the isomorphism L(F ) ≃ L(C ∞ (M, V )) whose explicit form is given by (3.6).
Let ξ X (g) = (L g ) * X be a left-invariant vector field on the Lie group G, where (L g ) * : Since the left-invariant vector fields commute with right-invariant ones, the condition of projectivity (3.4) is fulfilled. Using (3.7), we find the corresponding operator on the homogeneous space as where X(x) are the generators of the action of the group G on M, note that X(x) act in the space C ∞ (M). It is easy to verify the following commutation relations for operators (3.8): for all X, Y ∈ g. Consequently, the operators X corresponding to the left-invariant vector we have: where h(x, g) ∈ H is the factor of the homogeneous space [37], which is determined from the system of equations In view of the isomorphismF ≃ C ∞ (M, V ), we obtain from (3.10) a representation of the This representation is called the induced representation of the group G on the homogeneous space M. Note that whence immediately follows the expression for the derivative of the factor at the identity It is easy to see that the operators X(x), as described by (3.9) and (3.11), are differentials of the representation T g on the homogeneous space M: Thus, the projection of left-invariant vector fields on the group gives the infinitesimal operators of the representation of T g induced by the representation U(h) of the subgroup H.
It follows that the operator R M (x, ∂ x ) is invariant with respect to the transformation group if and only if the corresponding operator R(g, ∂ g ) ∈ L(F ) commutes with the left-invariant vector fields: ) be a linear differential operator of the first order, invariant with respect to the group action. By (3.12), this operator corresponds to a first-order polynomial of right-invariant vector fields: As a result of the projection, the expression B α η α (h) becomes constant B α Λ α , which can be eliminated in the operator R M (x, ∂ x ) by changing the variable B =B + B α Λ α . Therefore, we can put B α = 0 without loss of generality. If we substitute the operator R (1) (g, ∂ g ) in the projectivity condition (3.4), then we obtain Also we have a system of algebraic equations for the coefficients B a and B: When equations (3.13) -(3.14) are fulfilled, the projection of R (1) (g, ∂ g ) on the homogeneous space results in the desired form of the invariant linear differential operator of the first order: So, any linear differential operator of the first order acting on the functions of C ∞ (M, V ) and being invariant with respect to the action of the transformation group has the form (3.15) where the matrix coefficients B a and B satisfy the algebraic system of equations (3.13) -(3.14). The matrices Λ α are generators of the isotropy subgroup H in a linear space V .

IV. λ-REPRESENTATION OF A LIE ALGEBRA
In this section we describe a special representation of the Lie algebra g using the orbit method [37]. The direct and inverse Fourier transforms on the Lie group G are introduced, that in what follows are necessary for the noncommutative reduction of the Dirac equation on the homogeneous space M. Here we also use some results of the previous section.
First, we describe an orbit classification for the coadjoint representation of Lie groups following conventions of Refs. [39,40].
A degenerate Poisson-Lie bracket, endows the space g * with a Poisson structure. Here f A are coordinates of a linear functional f = f A e A ∈ g * relative to the dual basis e A . The number ind g of functionally independent Casimir functions K µ (f ) relative to the bracket (4.1) is called the index of the Lie algebra g.
A coadjoint representation on g * , Ad * : G × g * → g * , stratifies g * into orbits of the coadjoint representation (K-orbits). The restriction of the bracket (4.1) on orbits is nondegenerate and coincides with the Poisson bracket generated by the symplectic Kirillov form The orbits of maximal dimension dim O (0) = dim g − ind g are called non-degenerate, and the those of less dimension are singular. We denote by O (s) λ the orbits of dimension dim g − ind g − 2s, s = 0, . . . , (dim g − ind g)/2 passing through the functional λ ∈ g * , and a number s is called the orbit singularity index.
at a point f is the linear span of vector fields so that the orbit dimension is given by the rank of the matrix C AB (f ). The rank takes a constant value on the orbit, dim O (s) The space g * can be decomposed into a sum of disjoint invariant algebraic surfaces M s consisting of orbits of the same dimension dim g − ind g − 2s: where F s (f ) denotes the set of all minors of the matrix C AB (f ) = C C AB f C of size dim g − ind g − 2s + 2; the notation F s (f ) = 0 implies that all the corresponding minors at the point f vanish, and ¬(F s (f ) = 0) means that at the point f , the corresponding minors do not vanish simultaneously. In the general case, the surface M s is disconnected.
The non-constant functions K The number of functionally independent solutions of this system is determined by the dimension of the surface M (s) : Denote by Ω (s) ⊂ R r (s) a set of values of the mapping K (s) : M (s) → R r (s) and introduce a locally invariant subset If the Casimir functions K Consider a quotient space B (s) = M (s) /G, dimB (s) = r (s) , whose points are the orbits of We introduce a local section λ(j) of the bundle M (s) with base B (s) using real parameters j = (j 1 , . . . , j r (s) ) taking their values in a domain J ⊂ R r (s) : λ(j) be a K-orbit of (s)-type passing through a covector λ = λ(j) ∈ g * and belonging to the same class of orbits for all j ∈ J.
Using the Kirillov orbit method [37], we construct a unitary irreducible representation of the Lie group G on a given orbit. This representation can be constructed if and only if for the functional λ there exists a subalgebra p ⊂ g C in the complex extension g C of the Lie algebra g satisfying the conditions: The subalgebra p is called the polarization of the functional λ. In (4.3), it is assumed that the functionals from the space g * are extended to g C by linearity. Moreover, real polarizations always exist for nilpotent and completely solvable Lie algebras, and the complex polarizations always exist for solvable Lie groups [41]. For non-degenerate orbits O λ there always exists, generally speaking, a complex polarization. In this paper, for simplicity, we restrict ourselves to the case when p is the real polarization.
Denote by P a closed subgroup of the Lie group G whose Lie algebra is p. The Lie group acts on the right homogeneous space Q ≃ G/P : q ′ = qg. According to the orbit method, we introduce a unitary one-dimensional irreducible representation of the Lie group P , which, in the neighborhood of the identity element of the group, has the form The representation of the Lie group G corresponding to the orbit O (s) λ is induced using (4.4) as where ∆ G (g) = det −1 Ad g is the module of the Lie group G, ∆ P (p) = detAd p is the module of the subgroup P , p ∈ P , and e P is the identity element of P . A function p(q, g) is the factor of the homogeneous space Q.
The functions ψ λ (q; g) = (T λ g ψ)(q) on the group G satisfy a condition similar to (3.1): The space of all such functions will be denoted by F λ . Restriction of the left-invariant vector fields ξ X (g) to a homogeneous space Q, as follows from results of section III, is correctly defined, and the explicit form of the corresponding operator on the homogeneous space is given by (3.6): Equation (4.6) shows that ℓ X (q, ∂ q , λ) are infinitesimal operators of the induced representation (4.5), Denote by L(Q, h, λ) a space of functions on Q where representation (4.5) is defined.
The representation (4.5) is unitary with respect to a scalar product of the function space The function ρ(q) is determined from the Hermitian condition for the operators −iℓ X (q, ∂ q , λ) with respect to this scalar product (4.8).
The irreducible representation of the Lie algebra g by the linear operators of the first order (4.6) dependent on dim O (s) λ /2 variables is called λrepresentation of the Lie algebra g and it was introduced in Ref. [16].
Next, operators of the λ representation are defined using (4.6) as In other words, finding of these operators is reduced to calculating the left-invariant vector fields on the group G in the trivialization domain of the principal bundle of this group in the fibrations P = exp(p).
As P is the stabilizer of the point q = 0 in the homogeneous space Q, and the group G λ lies in P , we get the equality Restricting the first equality (4.10) to the subgroup G λ and setting q = 0, we find The solution of the system (4.11) up to a constant factor can be represented as The subalgebra g λ is subordinate to the covector λ, and the 1-forms σ λ (h) and σ α (h)β α are closed in G λ . Thus, the integral in (4.12) is well-defined. The local solution (4.12) can be extended to a global one, if the integral on the right-hand side of (4.12) over any closed curve Γ on the subgroup G λ is a multiple of 2πi. Note that since the 1-form σ λ (h) is closed, the value of this integral depends only on the homological class to which the curve Γ belongs. Therefore, for a global solution of the system (4.11), the following condition should be satisfied: 1 2π (4.13) In other words, the 1-form σ λ (h) should belong to an integral cohomology class from Thus, for a simply connected group the coadjoint orbit O A set of generalized functions D λ qq ′ (g) satisfying the system (4.10) was studied in Refs. [16,39] and the hypothesis was proposed that this set of generalized functions has the properties of completeness and orthogonality for a certain choice of the measure dµ(λ) in the parameter space J: g). Here δ(g) is the generalized Dirac delta function with respect to the left Haar measure dµ(g) on the Lie group G.
Note that although there is no rigorous proof of the relations (4.14)-(4.15), in each case it is easy to verify directly their validity.
Consider the space L(G, λ, dµ(g)) of functions of the form Here, a function ψ(q, q ′ , λ) of the two variables q and q ′ belongs to the space L(Q, h, λ). The inverse transform reads where we have used (4.14)-(4.15), and dµ R (g) = dµ(g −1 ) is the right Haar measure on the Lie group G.
The action of the operators ξ X (g) and η X (g) on the function ψ λ (g) from L 2 (G, λ, dµ(g)), according to (4.16) and (4.17), corresponds to action of the operators ℓ † X (q, ∂ q , λ) and ℓ X (q ′ , ∂ q ′ , λ) on the function ψ(q, q ′ , λ) respectively, The functions (4.16) are eigenfunctions for the Casimir operators K (s) Indeed, from the system (4.10) we can obtain It follows that the operators K (s) µ (−i ℓ(q ′ , λ)) are independent of q ′ and Thus, as a result of the generalized Fourier transform (4.16), the left and the right fields become the operators of λ-representations, and the Casimir operators become constants.
This fact is a key point for the method of noncommutative integration of linear partial differential equations on Lie groups, since it allows one to reduce the original differential equation with dim G independent variables to an equation with fewer independent variables equal to dim Q.

V. DIRAC EQUATION IN HOMOGENEOUS SPACE
In this section, we consider the Dirac equation in a n-dimensional homogeneous space M with an invariant metric. We shall assume that in the homogeneous space M an invariant metric g M and the Levi-Civita connection are given. Denote by V Ψ a space of spinor fields on M.
We write the Dirac equation in the space M as an equation in a n-dimensional Lorentz manifold M with the metric ( is the Planck constant) as follows [43]: Here ∇ i is the covariant derivative corresponding to the Levi-Civita connection on M, m is mass of the field ψ ∈ C ∞ (M, V Ψ ), ψ(x) is a column with 2 ⌊n/2⌋ components, γ i (x) are 2 ⌊n/2⌋ × 2 ⌊n/2⌋ gamma matrices, where E denotes the 2 ⌊n/2⌋ × 2 ⌊n/2⌋ identity matrix, Γ i (x) is the spinor connection satisfying the conditions [∇ i , γ j (x)] = 0, Tr Γ i (x) = 0. The spinor connection Γ i (x) can be written as follows [43]: We seek a solution of (5.2) with the decomposition For the Dirac matrices with subscripts using (2.4) we have The spinor connection is given by the following Lemma.
Proof The function Γ(x) = γ i (x)Γ i (x) with Γ i (x) given by (5.3), can be written as where Γ l ki (x) are the Christoffel symbols, and ∂ x i is the partial derivative. Substituting (2.5), (5.4), and (5.6) in (5.8), we obtain Using property (2.3) of the invariant metric, we reduce the expression C d bαγ bγ From the chain of equalities we obtain for the spinor connection the required expression (5.7).
Thus, the Dirac equation in the homogeneous space M with an invariant metric g M and the Dirac matrices of the form (5.4) takes the form A set of matrices Λ α determines a spinor representation of the isotopy subgroup H in the space V Ψ .

Lemma 2
The matrices Λ α are generators of the isotropy subgroup H representation on the space V Ψ .
Proof We prove that the matrices Λ α satisfy the commutation relations The commutator of Λ α and Λ β can be written as Using (2.3), (5.5) and (5.6), we find the commutator of Λ α withγ a : Similarly, for the γ-matrices with lower indices we have Substitution of (5.11)-(5.12) in (5.10) yields The expression inside the parentheses can be written in the form (5.14) By the Jacobi identity for the structure constants, the expression inside the square brackets is equal to zero. Substituting (5.14) in (5.13), we obtain (5.9).
The Dirac operator D M (x) is a differential operator of the first order with matrix coef-  . From (5.11) it follows that the commutator of Λ α andγ a satisfies the first condition in (3.13). In this case the condition (3.14) is reduced to The commutator of Γ and Λ α can be presented in terms of the commutator [Λ α , Γ a ] as Using (5.7) and (5.11)-(5.12), we get  [ In view of the isomorphismF ≃ C ∞ (M, V Ψ ) and Theorem 1, the Dirac equation (5.1) on M is equivalent to the following system of equations on the transformation group G: (η α + Λ α ) ψ(g) = 0.

VI. NONCOMMUTATIVE INTEGRATION
We will look for a solution to the system (5.23) as a set of functions where the function ψ σ (q ′ ) is a spinor, each component of which belongs to the function space L(Q, h, λ) with respect to the variable q ′ , and D λ qq ′ (g −1 ) is introduced by (4.9). Using (4.10) we can then reduce the system (5.23) to the equations We call the operator D ℓ (q ′ , ∂ q ′ , λ) in (6.3) the Dirac operator in the λ-representation. The gives us a solution of the original Dirac equation (5.1) on the homogeneous space M.
It follows from the equations that the solutions ψ σ (x) of the Dirac equation satisfy the system where X is given by (3.8). The algebraic relations between operators of the λ-representation should correspond to the algebraic relations between the generators X(x) for compatibility of the system (6.4). More precisely, the corollary of the system (6.4) is to be fulfilled for any homogeneous function F of X(x): This condition is obviously satisfied for the commutator of two operators (X, Y ∈ g), and for the Casimir functions, we have The homogeneous functions Γ ∈ C ∞ (g * ) provided that Γ(X(x)) ≡ 0, can exist on the dual space g * to the space M. These functions are called identities on the homogeneous space M.
The number of functionally independent identities i M is called the index of the homogeneous space M. In Ref. [40] it was shown that any homogeneous function Γ ∈ C ∞ (g * ) satisfying the condition is an identity. In the same Ref. [40] it was shown that the functions F Proof Suppose that a homogeneous function Γ ′ ∈ C ∞ (g * ) is an identity for the generators X(x), i.e., Γ ′ ( X) ≡ 0. Then the symbol of the operator Γ ′ ( X) also equals zero for all (x, p), and where the constants p a are coordinates of the covector f x = p a dx a ∈ T * x M. At a given point (x 0 , p 0 ), we have X a (x 0 , p 0 ) = p 0 a , X α (x 0 , p 0 ) = Λ α , Expanding Γ ′ ( X(x, p)) in terms of the basis B of matrices in the vector space V and putting x = x 0 , p = f x 0 , we get: As a result, for each function Γ σ (f ) we come to equation (6.5). The last one shows that the functions Γ σ (f ) are identities on a homogeneous space, and the identities Γ ′ (f ) for the operators X(x) have the following structure: From this one can see that the number of functionally independent identities between X(x) does not exceed the index i M of the homogeneous space, and the functions Γ σ (f ) depend on identities on the homogeneous space.
For the compatibility of the system (6.4), we have to take into account the identities between the generators X(x), Γ ′ ( X) ≡ 0; namely we impose the following conditions on the operators of the λ-representation: A class of orbits and corresponding parameters j should be restricted by (6.6).
For instance, for the case X(x) = X(x), the condition (6.6) is reduced to The first condition in (6.7) says that the λ-representation has to be constructed by the λ(j) , and the second one imposes a restriction on the parameters j. In Ref. [44], a λ-representation satisfying (6.7) is called a λ-representation corresponding to the homogeneous space M.
Thus, condition (6.7) is stronger than (6.6). One of the important results of our work is the fact that when performing a noncommutative reduction of the Dirac equation, it is necessary to use the weaker condition (6.6) for the correct application of the noncommutative integration method.
The second equation of the system (6.2) can be written as We look for a solution of (6.8) in the form where R σ (q ′ ) is a certain function, and ψ σ (v) is an arbitrary function of the characteristics v = v(q ′ ) of the system (6.2). We carry out a one-to-one change of variables q ′ = q ′ (v, w), where w = w(q ′ ) are some coordinates additional to v. By V and W we denote domains of the variables v and w, respectively. The measure dµ(q ′ ) in the new variables takes the form dµ(q ′ ) = ρ(v, w)dµ(v)dµ(w). Then the solution of the original Dirac equation can be represented as Substituting the solution ψ σ (x) into the Dirac equation (6.2), we obtain a linear first-order The Lie algebra g is a semidirect product of the two-dimensional commutative ideal R 2 = span{e 1 , e 2 } and the three-dimensional simple algebra sl(2) = span{e 3 , e 4 , e 5 }. We also take h = {e 5 } as the one-dimensional subalgebra.
Denote by (x a , h α ) local coordinates on a trivialization domain U of the group G so that g(x, h) = e h 1 e 5 e x 4 e 4 e x 3 e 3 e x 2 e 2 e x 1 e 1 , (7.1) The group G is unimodular and ∆ G (g) = 1. A symmetric non-degenerate matrix defines an invariant metric on the space M, The metric (7.2) has nonzero scalar curvature R = 6c 1 . The group generators in the canonical coordinates (7.1) have the form: The vector fields ξ A (x, e H ), in turn, are determined by the expressions The right-invariant vector fields in the canonical coordinates (7.1) are The gamma matricesγ a can be presented in terms of the standard Dirac gamma matrices γ a as follows:γ The spin connection is independent of local coordinates and has the form The Dirac operator in local coordinates is The first-order symmetry operators are defined by (5.22): The metric (7.2) generally does not admit the Yano vector field and the Yano-Killing tensor field, so the Dirac equation does not admit spin symmetry operators. As a result, the Dirac equation has only two commuting symmetry operators { X 1 (x), X 2 (x)} of the first order. However, the Dirac equation admits a third-order symmetry operator where X · Y = (XY + Y X)/2 is the symmetrized product of the operators X and Y .
As a consequence, the metric (7.2) does not admit separation of variables for the Dirac equation. Note that the Klein-Gordon equation also admits only three commuting symmetry operators {X 1 (x), X 2 (x), K(X)}. One of them, K(X), is the third-order operator, and the Klein-Gordon equation also does not admit separation of variables.
We now carry out a noncommutative reduction of the Dirac equation.
First, we describe orbits of the coadjoint representation of the Lie group G. The Lie algebra g admits the Casimir function 1 (f ) = f 2 4 + 4f 3 f 5 is the Casimir function of s = 1 type. Each non-degenerate orbit from the class O 0 ω passes through the covector λ(j) = (1, 0, 0, 0, j) and K 1 (λ(j)) = j, where j ∈ R.
The covector λ(j) admits a real polarization p = {e 1 , e 2 , e 5 }, and the λ-representation corresponding to the class of orbits O 0 λ(j) is given by The operators −i ℓ X (q, ∂ q , λ) are symmetric with respect to the measure dµ(q) = dq 1 dq 2 , Q = R 2 . Now solving equations (4.10), we get where δ(q, q ′ ) is the generalized Dirac delta function. The completeness and orthogonality conditions (4.15)-(4.14) are satisfied for the measure Orbits from the class O 0 λ(j) satisfy the integral condition (4.13). The homogeneous space M has zero index, i M = 0, and does not have identities that have to be taken into account in the method of noncommutative integration. So, the λrepresentation (7.4) corresponds to the homogeneous space M.
Integrating the equation Substituting (7.5) into the Dirac equation in the λ-representation (6.2), we obtain the ordinary differential equation for the spinor ψ σ , Then we obtain the solution as where C σ is the normalization factor.
Whence the scalar curvature of the space M reads R = 6ε 2 .
The corresponding λ-representation for the class of orbits O 0 λ(j) is represented in Appendix A (see Eqs. (9.1)). The Casimir operators in the λ-representation are The equation (ℓ α (q ′ , ∂ q ′ , λ) + Λ α ) c λ (q ′ ) = 0 provided that j 2 = s/2 has a nonzero solution c λ (q) = (cos(εq 1 ) cos(εq 2 )) − 3 The Dirac equation in the λ-representation, is reduced to the algebraic equation j 1 + m = 0, then we have j 1 = −m and j 2 = s/2. That is, the eigenvalue of the Casimir operator K 1 (i X) is determined by the particle mass m, and the eigenvalue of the second Casimir operator, K 2 (i X), depends on the parameter s: The solution of the original Dirac equation in our case reads ψ σ (x) = e −tℓ 1 (q,∂q,λ) e −xℓ 2 (q,∂q,λ) e −yℓ 3 (q,∂q.λ) c λ (q), σ = (q 1 , q 2 ) (8.2) Here, the exponentials of operators of the λ-representation for the fixed j 1 = −m and Differently to the above early method, the main idea here is the noncommutative reduction of the corresponding system of equations on the Lie group G and the connection between the solutions of this system and the original Dirac equation.
The noncommutative reduction is defined here using a special irreducible λ-representation of the Lie algebra g of the Lie group G, which we introduce using the orbit method [37].
The key point of the method developed is based on the fact that there exist the identities connecting symmetry operators on a homogeneous space. For the Dirac equation, as follows from the lemma 3, the number of identities is either less than for the Klein-Gordon equation or they are completely absent. For the Klein-Gordon equation, the number of identities is determined by the index of the homogeneous space [19]. The parameters q of solutions (7.7) and (8.2) obtained by the NCIM are in general not eigenvalues of an operator, a fact that crucially distinguishes them from solutions obtained by separation of variables. Nevertheless, the NCIM-solutions can be effectively applied in order to study quantum effects in homogeneous spaces (see, e.g., [19,20]).
The NCIM-solutions of the Dirac equation may have a wide range of applications in the theory of fermion fields [45,46], quantum cosmology [47,48] and other problems of field theory. The NCIM can be applied also to the Dirac-type equation for theoretical models in the condensed matter (graphene, topological insulators, etc.) [49,50]. Note that the technique proposed in the article can be easily generalized to the case of spaces having new spatial dimensions much larger than the weak scale, as large as a millimeter for the case of two extra dimensions [51].
Finally, we note that the NCIM reveals new aspects, both related to the symmetry of the Dirac equation and its integrability, and to study the properties of new solutions constructed. One of the problems is to find out the meaning of the parameters q entering into the NCIM-solutions which, in the general case, do not have to be eigenvalues of operators representing observables. One can notice some similarity of the NCIM-solutions with wellstudied coherent states [53]. In particular, the action of the group on the set of Q data of quantum numbers is defined, that can be found in the theory of coherent states [52][53][54][55].
However, the analysis of the parameters is the subject of special research.