Exact solutions of the Klein-Gordon equation in external electromagnetic fields on 3D de Sitter background

One of the most important problems of modern theoretical and mathematical physics is the problem of finding exact solutions of relativistic wave equations in external fields. In quantum electrodynamics, for example, exact solutions of the Dirac and Klein–Gordon equations allow us to construct the so-called Furry picture. In the framework of this picture, the interaction with an external field is described exactly, and the interaction with a quantized photon field is taken into account using perturbation theory [1]. Exact solutions of the Dirac and Klein–Gordon equations are also extremely useful in studying various vacuum quantum effects in strong external fields, where the methods and approaches of the standard perturbation theory are not applicable (see, for instance, [2, 3]).

This study focuses on exact solutions of the Klein–Gordon equation in the presence of an external electromagnetic field on 3D de Sitter background $\mathrm{dS}_{3}$. When writing this paper, we mainly followed two objectives. First of all, we would like to demonstrate the potential of a new powerful method for constructing exact solutions of relativistic wave equations called the method of noncommutative integration. This method was developed in the early 90s of the last century by Shapovalov and Shirokov [4]. In contrast to the well-known method of separation of variables, the application of which is associated with the study of commutative subalgebras of the first- and second-order symmetry operators (see, for instance, [5, 6, 7, 8]), the method of noncommutative integration allows us to construct exact solutions using only the algebra of the first-order symmetry operators, which is noncommutative in general [9, 10, 11]. Secondly, in the paper, we would like to give the exhaustive classification of electromagnetic fields on $\mathrm{dS}_{3}$ that admit the existence of a first-order symmetry algebra for the Klein-Gordon equation and, using this algebra, to construct the corresponding exact solutions in cases where this is possible.

Note that $n$-dimensional de Sitter space $\mathrm{dS}_{n}$ is regularly in the center of the attention of specialists in mathematical and theoretical physics, which is mainly owing to two reasons. On the one side, from the modern viewpoint, it is considered that, in the four-dimensional case, this space describes the very early stages of the expansion of the universe quite accurately. On the other side, the space $\mathrm{dS}_{n}$ is a maximally symmetric Lorentzian manifold satisfying Einstein’s equations with a positive cosmological constant. In particular, the high symmetry of this space allows one to integrate relativistic wave equations and, as a consequence, to investigate in detail of various quantum effects on its background [2, 12, 13, 14]. The presence of an external electromagnetic field, however, can cause the symmetry breaking that makes the problem of classification of electromagnetic fields admitting exact solutions of relativistic wave equations relevant. We note that although the problem of integrating free relativistic equations on de Sitter background has long been solved (see, for instanse, [2, 15, 16]), exact solutions of these equations in external electromagnetic fields are still poorly systematized. At the same time, such exact solutions are extremely important, for instance, in connection with the study of the particle creation effect in an external electromagnetic field on a curved spacetime [17, 18, 19, 20].

The structure of this paper is as follows. In Section 1, we recall the necessary information about the algebra of the first-order symmetry operators for the Klein–Gordon equation. Based on the defining equations for these operators (see [21, 22]), we present an algorithm for constructing this algebra and describe its structure in terms of Lie algebra extensions. In Section 2, we outline some geometric and group properties of 3D de Sitter space as well as the well-known classification of all inequivalent subalgebras of the algebra $\mathfrak{so}(1,3)$, which is the Lie algebra of the isometry group of $\mathrm{dS}_{3}$. Using this classification, in Section 3, we obtain the list of all electromagnetic fields on $\mathrm{dS}_{3}$ that admit nontrivial first-order symmetry algebras of the Klein–Gordon equation. In the same section, we explicitly construct these algebras and describe their structures by writing down the corresponding commutation relations. In Section 4, we examine the integrability problem for the Klein–Gordon equation in external electromagnetic fields on $\mathrm{dS}_{3}$. We emphasize that by integrability we mean that the Klein-Gordon equation can be constructively solved by the process of its reduction to an ordinary differential or algebraic equation (as it is understood in the theory of separation of variables). Using the condition of noncommutative integrability [4], we find all the integrable cases, and, for each of them, we reduce the original Klein–Gordon equation to an auxiliary ordinary differential equation. In addition, we express the general solution of the auxiliary equation in terms of known special functions where possible.

All local constructions are presented in special local coordinates on $\mathrm{dS}_{3}$, in which the basis vector fields for subalgebras of $\mathfrak{so}(1,3)$ have the simplest form. In Appendix, the original algebraic method for constructing such coordinates is described.

Let $(M,g)$ be a smooth $n$-dimensional Lorentzian manifold of signature $(+,-,\dots,-)$, $x^{1},\dots,x^{n}$ are local coordinates on $M$. We consider the covariant Klein–Gordon equation for a charged massive scalar field in the presence of an external electromagnetic field [2, 3]:

$$\hat{H}\varphi\equiv\left(g^{ab}D_{a}D_{b}+\zeta R+m^{2}\right)\varphi=0.$$

(1)

Here, $\varphi$ denotes the scalar field, $g^{ab}$ are the contravariant components of the metric $g$, $R$ is the Riemannian scalar curvature, $m$ and $e$ are the mass and charge of the field quanta, respectively. The generalized covariant derivatives $D_{a}$ are defined as $D_{a}\equiv\nabla_{a}-ie\mathscr{A}_{a}$, where $\nabla_{a}$ is the covariant derivative with respect to the coordinate vector field $\partial_{x^{a}}\equiv\partial/\partial x^{a}$, and $\mathscr{A}_{a}$ are the components of the electromagnetic potential 1-form $\mathscr{A}$. The dimensionless parameter $\zeta$ can take two values: $\zeta=0$ (the minimally coupled case) and $\zeta=(n-1)/(4n)$ (the conformally coupled case). In what follows we will use the Einstein summation convention unless stated otherwise. Latin indices are raised and lowered by the metric $g$.

An operator $\hat{X}$ is said to be a symmetry operator for Eq. (1) if the following condition is satisfied

$$[\hat{H},\hat{X}]\equiv\hat{H}\hat{X}-\hat{X}\hat{H}=\hat{Q}\hat{H},$$

(2)

where $\hat{Q}$ is an operator which depends on $\hat{X}$ in general. It is known that the symmetry operators map solutions of Eq. (1) to other solutions, and the set of all symmetry operators $\mathfrak{M}$ forms a Lie algebra called the symmetry algebra of the Klein-Gordon equation [5].

The main object of our investigation is a subalgebra of $\mathfrak{M}$ generated by all the differential operators of the form

$$\hat{X}=X^{a}(x)D_{a}+ie\chi(x).$$

(3)

Here, $X^{a}(x)$ and $\chi(x)$ are assumed to be smooth real functions on $M$, possibly not everywhere defined. Obviously, the set of all such operators forms a certain subalgebra in $\mathfrak{M}$, which we denote as $\hat{\mathscr{G}}$.

The conditions under which the operator (3) is a symmetry operator of the Klein–Gordon equation are well known (see, for instance, [21]). They can be obtained by substituting (3) into (2) and equating coefficients of $D_{a}D_{b}$, $D_{a}$, and $1$ on both sides of the resulting relation. As a result, we obtain the system of equations

$$\nabla^{a}X^{b}+\nabla^{b}X^{a}=0,$$

(4)

$$X^{b}\mathscr{F}_{ab}-\nabla_{a}\chi=0,$$

(5)

where $\mathscr{F}_{ab}$ are the components of the electromagnetic tensor defined by

$$\mathscr{F}_{ab}=\frac{\partial\mathscr{A}_{b}}{\partial x^{a}}-\frac{\partial\mathscr{A}_{a}}{\partial x^{b}}.$$

Also, note that, for a massive scalar field, there should be $\hat{Q}=0$.

The system of Eqs. (4) and (5) is a necessary and sufficient condition for $\hat{X}$ to be a symmetry operator for the Klein–Gordon equation. In particular, Eq. (4) means that $X=X^{a}\partial_{x^{a}}$ is a Killing vector of the Lorentzian manifold $(M,g)$. Killing vectors reflect the geometric symmetry of the spacetime and are the infinitesimal generators of its (local) isometry group $\mathrm{Iso}(M,g)$. The set of all Killing vectors of $(M,g)$ forms a Lie algebra (with respect to the commutator of vector fields), which we denote as $\mathfrak{iso}(M,g)$.

We note that not every Killing vector $X=X^{a}\partial_{x^{a}}$ from $\mathfrak{iso}(M,g)$ can be extended to a symmetry operator $\hat{X}=X^{a}D_{a}+ie\chi$ of Eq. (1). Carter showed [21] that a scalar function $\chi$ of the required form locally exists if and only if

$$\mathscr{L}_{X}\mathscr{F}=0,$$

(6)

where $\mathscr{L}_{X}$ denotes the Lie derivative along the vector field $X$. Indeed, if we rewrite Eq. (5) in terms of differential forms

$$d\chi=-i_{X}\mathscr{F},$$

(7)

where $i_{X}\mathscr{F}\equiv\mathscr{F}_{ab}X^{a}dx^{b}$ is the interior product of the 2-form $\mathscr{F}=\frac{1}{2}\,\mathscr{F}_{ab}\,dx^{a}\wedge dx^{b}$ and the vector field $X=X^{a}\partial_{x^{a}}$, then, it is easy to see that the local existence of the function $\chi$ is equivalent to the statement that the 1-form $-i_{X}\mathscr{F}$ is closed. This, in turn, is equivalent to Eq. (6), in view of the Cartan formula $\mathscr{L}_{X}=i_{X}d+di_{X}$ and the condition $d\mathscr{F}=0$.

Since the correspondence $X\to\mathscr{L}_{X}$ is a homomorphism of Lie algebras, the set of all Killing vectors satisfying the condition (6) for a given electromagnetic field forms a subalgebra $\mathscr{G}$ of the Lie algebra $\mathfrak{iso}(M,g)$. We will call $\mathscr{G}$ the admissible subalgebra.

Let $X_{A}=X_{A}^{a}(x)\partial_{x^{a}}$ be Killing vectors forming a basis of $\mathscr{G}$, $A=1,\dots,\dim\mathscr{G}$. Since $\mathscr{G}$ is a subalgebra, we have

$$[X_{A},X_{B}]=C_{AB}^{C}X_{C},\quad A,B=1,\dots,\dim\mathscr{G}.$$

(8)

Here, the constants $C_{AB}^{C}$ are the structure constants of $\mathscr{G}$. By the definition of the admissible subalgebra, for each vector field $X_{A}$, there exists a local function $\chi_{A}$ satisfying the condition (7). Note that this function is not uniquely defined; instead of $\chi_{A}$, one can choose the function $\chi_{A}^{\prime}=\chi_{A}+\lambda_{A}$, where $\lambda_{A}$ is an arbitrary constant. The meaning of this ambiguity will be explained below.

By construction, the first-order differential operators

$$\hat{X}_{A}=X_{A}^{a}(x)D_{a}+ie\chi_{A},\quad A=1,\dots,\dim\mathscr{G},$$

(9)

commute with the operator $\hat{H}$, i.e., $\hat{X}_{A}$ are symmetry operators for the Klein–Gordon equation (1). In the general case, however, the linear span of these operators does not form the whole algebra $\hat{\mathscr{G}}$. Indeed, using the commutation relations $[D_{a},D_{b}]=-ie\mathscr{F}_{ab}$ and (8), the commutator of $\hat{X}_{A}$ and $\hat{X}_{B}$ can be written as [11]

$$[\hat{X}_{A},\hat{X}_{B}]=C_{AB}^{C}\hat{X}_{C}+\mathbf{F}_{AB}\hat{X}_{0},$$

(10)

where

$$\mathbf{F}_{AB}\equiv\mathscr{F}(X_{A},X_{B})-C_{AB}^{C}\chi_{C}.$$

(11)

Here, we have introduced the trivial symmetry operator $\hat{X}_{0}\equiv ie$. In Ref. [11], it is shown that the quantities $\mathbf{F}_{AB}$ are constant on $M$ and satisfy the identity

$$C_{AB}^{D}\mathbf{F}_{CD}+C_{BC}^{D}\mathbf{F}_{AD}+C_{CA}^{D}\mathbf{F}_{BD}=0.$$

This allows us to interpret the quantities (11) as components of a cocycle of the admissible subalgebra $\mathscr{G}$ with values in the trivial $\mathscr{G}$-module $\mathbb{R}$ (for a review of basic results in Lie algebra cohomology and their applications in mathematical physics see [23, 24]). Moreover, from the commutation relations (10), it follows that the algebra $\hat{\mathscr{G}}$ is a one-dimensional central extension of the Lie algebra $\mathscr{G}$ corresponding to the cocycle (11).

As we have already noted, the functions $\chi_{A}$ are determined by Eq. (7) up to the addition of arbitrary constants. As can be seen from (9), the transformation $\chi_{A}\to\chi_{A}^{\prime}=\chi_{A}+\lambda_{A}$ implies the transformation $\hat{X}_{A}\to\hat{X}^{\prime}_{A}=\hat{X}_{A}+\lambda_{A}\hat{X}_{0}$, which is a change of basis in the algebra $\hat{\mathscr{G}}$. In the general case, it leads to a change of the commutation relations (10), because the quantities (11) are transformed as follows:

$$\mathbf{F}_{AB}\to\mathbf{F}^{\prime}_{AB}=\mathbf{F}_{AB}-C_{AB}^{C}\lambda_{C}.$$

(12)

The above results allow us to list all electromagnetic fields admitting first-order symmetry operators of Eq. (1) for a given Lorentzian manifold $(M,g)$. It is preliminary convenient to introduce the following equivalence relation. Two electromagnetic fields with closed 2-forms $\mathscr{F}$ and $\mathscr{F}^{\prime}$ are called equivalent if they are connected by an isometry:

$$\mathscr{F}=\tau^{*}\mathscr{F}^{\prime},\quad\tau\in\mathrm{Iso}(M,g).$$

It is easy to show that the admissible subalgebras $\mathscr{G}$ and $\mathscr{G}^{\prime}$ corresponding to the equivalent electromagnetic fields $\mathscr{F}$ and $\mathscr{F}^{\prime}$ are conjugate in $\mathfrak{iso}(M,g)$ by the transformation $\tau$: $\mathscr{G}^{\prime}=\tau\mathscr{G}\tau^{-1}$. Conversely, if the subalgebras $\mathscr{G}$ and $\mathscr{G}^{\prime}$ are $\tau$-conjugate, then the spaces of closed 2-forms on $M$ invariant under the subalgebras $\mathscr{G}$ and $\mathscr{G}^{\prime}$ are connected by the transformation $\tau^{*}$. Thus, in order to describe fully the electromagnetic fields that admit a nontrivial first-order symmetry operator of Eq. (1), one may use the list of subalgebras of the algebra $\mathfrak{iso}(M,g)$ up to conjugations. For each subalgebra $\mathscr{G}=\{X_{A}\}$ from this list, we can find the most general form of a closed 2-form $\mathscr{F}$ such that¹¹1Note that the classes of electromagnetic fields corresponding non-conjugate subalgebras of $\mathfrak{iso}(M,g)$ may have nonzero intersections.

$$\mathscr{L}_{X_{A}}\mathscr{F}=0,\quad A=1,\dots,\dim\mathscr{G},$$

(13)

and then explicitly construct a basis of the Lie algebra $\hat{\mathscr{G}}$ in accordance with the formula (9).

De Sitter space $\mathrm{dS}_{3}$ is the three-dimensional one-sheeted hyperboloid in four-dimensional Minkowski space $\mathbb{R}^{1,3}$, described by the equation [2]

$$x_{0}^{2}-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}=-\alpha^{2}.$$

(14)

Here, $\alpha$ is a nonzero positive constant with units of length called the de Sitter radius. The metric of $\mathrm{dS}_{3}$ induced by the standard Minkowski metric $\eta_{ij}=\mathrm{diag}(1,-1,-1,-1)$ is non-degenerate and has the Lorentzian signature $(+,-,-)$. From the group-theoretic point of view, three-dimensional de Sitter space is the homogeneous space $O(1,3)/O(1,2)$, where $O(1,n)$ denotes the pseudo-orthogonal group of all linear transformations that leave invariant a non-degenerate quadratic form of signature $(1,n)$. From the topological point of view, $\mathrm{dS}_{3}$ is the direct product of the real line and the two-dimensional sphere: $\mathrm{dS}_{3}=\mathbb{R}\times S^{2}$. From now on, we assume that $\alpha=1$. The scalar curvature of $\mathrm{dS}_{3}$, in this case, is equal to $R=6$.

The isometry group of de Sitter space $\mathrm{dS}_{3}$ is the Lorentz group $O(1,3)$. The identity component $SO^{+}(1,3)$ of this group, called the restricted Lorentz group, is generated by the six Killing vectors:

$$J_{ij}=x_{i}\,\frac{\partial}{\partial x^{j}}-x_{j}\frac{\partial}{\partial x^{i}},\quad 0\leq i<j\leq 3.$$

(15)

(Raising and lowering the indices are performed by the metric $\eta_{ij}$). The Lie algebra $\mathfrak{so}(1,3)$ defined by these vector fields has the following commutation relations:

$$[J_{ij},J_{kl}]=\eta_{jk}J_{il}-\eta_{ik}J_{jl}+\eta_{il}J_{jk}-\eta_{jl}J_{ik},\quad i,j=0,1,2,3.$$

We note that the vector fields $J_{12}$, $J_{13}$, and $J_{23}$ correspond to ordinary spatial rotations, while the Killing vectors $J_{01},J_{02}$, and $J_{03}$ are associated with the Lorentz boosts.

As noted in the previous section, the first step in the classification of electromagnetic fields that admit first-order symmetry operators of Eq. (1) on $\mathrm{dS}_{3}$ is to list all inequivalent subalgebras of the algebra $\mathfrak{so}(1,3)$. Every such subalgebra $\mathscr{G}\subset\mathfrak{so}(1,3)$ is fixed by a set of $n$ linearly independent vector fields $X_{A}=\sum_{i<j}C_{A}^{ij}J_{ij}$, which are some linear combinations of the Killing vectors (15).

All inequivalent proper subalgebras of the algebra $\mathfrak{so}(1,3)$ are well known (see, for instance, [26, 27, 28]); their exhaustive list (up to conjugation) is given in Table 1. In this table, we label each inequivalent subalgebra $\mathscr{G}_{n,m}$ by its dimension $n$ and its number $m$ in the list. If there is a one-parametric family of subalgebras, we denote this family as $\mathscr{G}^{a}_{n,m}$, where $a$ is a parameter. The third column of the table shows the nonzero commutation relations between the basis vectors of the subalgebras.

Let us give some explanations for the subalgebras listed in Table 1.

The one-dimensional subalgebras $\mathscr{G}_{1,1}$, $\mathscr{G}_{1,2}$, and $\mathscr{G}_{1,4}$ generate one-parameter subgroups of Lorentz boosts $SO(1,1)$, rotations $SO(2)$, and null rotations $SO(0,1)$, respectively. The family of one-dimensional subalgebras $\mathscr{G}^{a}_{1,3}$ parametrized by the positive parameter $a$ generates the family of one-parameter loxodromic transformations.

The subalgebra $\mathscr{G}_{2,1}$ is associated with a two-dimensional Abelian group of transformations consisting of zero rotations. The subalgebra $\mathscr{G}_{2,2}$ corresponds to a two-dimensional Abelian transformation group consisting of Lorentz boosts (in the $x^{3}$ direction), rotations in the plane $x^{1}x^{2}$, and their combinations. The subalgebra $\mathscr{G}_{2,3}$ generates a non-Abelian two-dimensional group isomorphic to the affine group $A(1)$.

The subalgebra $\mathscr{G}_{3,1}$ is the Lie algebra of Bianchi type V; the corresponding group is isomorphic to the group of Euclidean homotheties $\mathrm{Hom}(2)$. The subalgebra $\mathscr{G}_{3,2}$ is the Lie algebra of Bianchi type VII₀; it is isomorphic to the Lie algebra of the euclidean group $E(2)$. The subalgebra $\mathscr{G}_{3,3}^{a}$, where $a>0$, is of Bianchi type VII_a. The subalgebra $\mathscr{G}_{3,4}$ is of Bianchi type IX; the corresponding transformation group is the rotation group $SO(3)$. Finally, the subalgebra $\mathscr{G}_{3,5}$ of Bianchi type VIII is isomorphic to the Lie algebra of the group $SL(2,\mathbb{R})$.

There is only one (up to conjugation) four-dimensional subalgebra $\mathscr{G}_{4,1}$; it is isomorphic to the Lie algebra of the group of Euclidean similitudes $\mathrm{Sim}(2)$.

There are no five-dimensional subalgebras of the Lie algebra $\mathfrak{so}(1,3)$.

For subsequent purposes, we need to fix a local coordinate system in the space $\mathrm{dS}_{3}$ and rewrite the basis vector fields $X_{A}$ for each of the subalgebras $\mathscr{G}_{n,m}\subset\mathfrak{so}(1,3)$ in these coordinates. Since our problem is to find a closed 2-form $\mathscr{F}$ satisfying Eq. (13), it is reasonable to use a coordinate system in which the vector fields $X_{A}$ have the simplest form. In this sense, local coordinates that “rectify” the integral submanifolds of the system of vector fields $X_{1},\dots,X_{n}$ are most suitable. Let us recall the rigorous definition of such coordinates in the context of the situation under consideration.

Let $\mathscr{G}_{n,m}$ be a subalgebra from Table 1, $X_{1},\dots,X_{n}$ are its basis vector fields. Denote by $r$ the dimension of the space spanned by the vectors $X_{1}|_{x},\dots,X_{n}|_{x}$ for a point $x\in\mathrm{dS}_{3}$ in general position, i. e.

$$r\equiv\sup\limits_{x\in\mathrm{dS}_{3}}\mathrm{rank}\,\|X_{A}^{a}(x)\|.$$

(16)

If $x_{0}\in\mathrm{dS}_{3}$ is a point in general position, then, according to the well-known Frobenius theorem, there are local coordinates $(q,u)=(q^{1},\dots,q^{r}$, $u^{1},\dots,u^{3-r})$ near of $x_{0}\in\mathrm{dS}_{3}$ such that the integral submanifolds of $\{X_{1},\dots,X_{n}\}$ intersect this coordinate chart along the “slices” $u^{1}=c_{1},\dots,u^{3-r}=c_{3-r}$, where $c_{1},\dots,c_{3-r}$ are arbitrary constants (see Ref. [29]). In this case, $q^{1},\dots,q^{r}$ are regarded as local coordinates on the “slices”, and the vector fields $X_{A}$ being tangent to the “slices” are written as

$$X_{A}=\sum\limits_{a=1}^{r}X_{A}^{a}(q,u)\frac{\partial}{\partial q^{a}},\quad A=1,\dots,n.$$

In turn, the coordinates $u^{1},\dots,u^{3-r}$ are local invariants of the transformation group $G_{n,m}$ generated by the Lie algebra $\mathscr{G}_{n,m}$. We call the coordinates $(q,u)=(q^{1},\dots,q^{r}$, $u^{1},\dots,u^{3-r})$ the rectifying coordinates associated with the subalegbras $\mathscr{G}_{n,m}$.

Let us now consider each of the subalgebras from Table 1. In each case, it is not hard to construct local rectifying coordinates associated with $\mathscr{G}_{n,m}$ and then to find the most general form of the closed 2-form $\mathscr{F}$ satisfying Eq. (13). The results of these calculations are summarized in Table 2, in which for each subalgebra $\mathscr{G}_{n,m}$ we indicate the number $r$ calculated by Eq. (16), the vector fields $X_{A}$, and the corresponding invariant closed 2-form $\mathscr{F}$ in the rectifying coordinates. In this table, $f,f_{1},f_{2}$ denote arbitrary smooth functions of their arguments, and $\mu,\mu_{1},\mu_{2}$ are arbitrary constants. The functions $x^{i}=x^{i}(q^{1},\dots,q^{r},u^{1},\dots,u^{3-r})$ defining the rectifying local coordinates associated with the subalgebras $\mathscr{G}_{n,m}$ as well as a constructive algebraic method for their construction are given in Appendix A.

By construction, the electromagnetic fields listed in Table 2 admit nontrivial first-order symmetry algebras of the Klein–Gordon eqiation (1). To construct bases of these algebras explicitly, we again consider each subalgebra $\mathscr{G}_{n,m}$ separately. For each vector fields $X_{A}\in\mathscr{G}_{n,m}$ from Table 2, we find the function $\chi_{A}$, that is, a solution of Eq. (7), and we then write out the explicit form of the associated symmetry operator $\hat{X}_{A}$ according to Eq. (9). Having constructed the basis $\{\hat{X}_{0}=ie,\hat{X}_{1},\dots,\hat{X}_{n}\}$ of the symmetry algebra $\hat{\mathscr{G}}_{n,m}$, we list the nonzero commutation relations between its basis elements.