Editing Loop quantum gravity (section)

== Quantization of the constraints – the equations of quantum general relativity ==
=== Pre-history and Ashtekar new variables ===
{{main|Frame fields in general relativity| Ashtekar variables| Self-dual Palatini action}}
Many of the technical problems in canonical quantum gravity revolve around the constraints. Canonical general relativity was originally formulated in terms of metric variables, but there seemed to be insurmountable mathematical difficulties in promoting the constraints to [[quantum operator]]s because of their highly non-linear dependence on the canonical variables. The equations were much simplified with the introduction of Ashtekar's new variables. Ashtekar variables describe canonical general relativity in terms of a new pair of canonical variables closer to those of gauge theories. The first step consists of using densitized triads <math>\tilde{E}_i^a</math> (a triad <math>E_i^a</math> is simply three orthogonal vector fields labeled by <math>i = 1,2,3</math> and the densitized triad is defined by <math display="inline">\tilde{E}_i^a = \sqrt{\det (q)} E_i^a</math>) to encode information about the spatial metric,
<math display="block">\det(q) q^{ab} = \tilde{E}_i^a \tilde{E}_j^b \delta^{ij}.</math>
(where <math>\delta^{ij}</math> is the flat space metric, and the above equation expresses that <math>q^{ab}</math>, when written in terms of the basis <math>E_i^a</math>, is locally flat). (Formulating general relativity with triads instead of metrics was not new.) The densitized triads are not unique, and in fact one can perform a local in space [[rotation]] with respect to the internal indices <math>i</math>. The canonically conjugate variable is related to the extrinsic curvature by <math display="inline">K_a^i = K_{ab} \tilde{E}^{ai} / \sqrt{\det (q)}</math>. But problems similar to using the metric formulation arise when one tries to quantize the theory. Ashtekar's new insight was to introduce a new configuration variable,
<math display="block">A_a^i = \Gamma_a^i - i K_a^i</math>
that behaves as a complex <math>\operatorname{SU}(2)</math> connection where <math>\Gamma_a^i</math> is related to the so-called [[spin connection]] via <math>\Gamma_a^i = \Gamma_{ajk} \epsilon^{jki}</math>. Here <math>A_a^i</math> is called the chiral spin connection. It defines a covariant derivative <math>\mathcal{D}_a</math>. It turns out that <math>\tilde{E}^a_i</math> is the conjugate momentum of <math>A_a^i</math>, and together these form Ashtekar's new variables.

The expressions for the constraints in Ashtekar variables; Gauss's theorem, the spatial diffeomorphism constraint and the (densitized) Hamiltonian constraint then read:
<math display="block">G^i = \mathcal{D}_a \tilde{E}_i^a = 0</math>
<math display="block">C_a = \tilde{E}_i^b F^i_{ab} - A_a^i (\mathcal{D}_b \tilde{E}_i^b) = V_a - A_a^i G^i = 0,</math>
<math display="block">\tilde{H} = \epsilon_{ijk} \tilde{E}_i^a \tilde{E}_j^b F^k_{ab} = 0</math>
respectively, where <math>F^i_{ab}</math> is the field strength tensor of the connection <math>A_a^i</math> and where <math>V_a</math> is referred to as the vector constraint. The above-mentioned local in space rotational invariance is the original of the <math>\operatorname{SU}(2)</math> gauge invariance here expressed by Gauss's theorem. Note that these constraints are polynomial in the fundamental variables, unlike the constraints in the metric formulation. This dramatic simplification seemed to open up the way to quantizing the constraints. (See the article [[Self-dual Palatini action]] for a derivation of Ashtekar's formalism).

With Ashtekar's new variables, given the configuration variable <math>A^i_a</math>, it is natural to consider wavefunctions <math>\Psi (A^i_a)</math>. This is the connection representation. It is analogous to ordinary quantum mechanics with configuration variable <math>q</math> and wavefunctions <math>\psi (q)</math>. The configuration variable gets promoted to a quantum operator via:
<math display="block">\hat{A}_a^i \Psi (A) = A_a^i \Psi (A),</math>
(analogous to <math>\hat{q} \psi (q) = q \psi (q)</math>) and the triads are (functional) derivatives,
<math display="block">\hat{\tilde{E_i^a}} \Psi (A) = - i {\delta \Psi (A) \over \delta A_a^i}.</math>
(analogous to <math>\hat{p} \psi (q) = -i \hbar d \psi (q) / dq</math>). In passing over to the quantum theory the constraints become operators on a kinematic Hilbert space (the unconstrained <math>\operatorname{SU}(2)</math> Yang–Mills Hilbert space). Note that different ordering of the <math>A</math>'s and <math>\tilde{E}</math>'s when replacing the <math>\tilde{E}</math>'s with derivatives give rise to different operators – the choice made is called the factor ordering and should be chosen via physical reasoning. Formally they read
<math display="block">\hat{G}_j \vert\psi \rangle = 0</math>
<math display="block">\hat{C}_a \vert\psi \rangle = 0</math>
<math display="block">\hat{\tilde{H}} \vert\psi \rangle = 0.</math>

There are still problems in properly defining all these equations and solving them. For example, the Hamiltonian constraint Ashtekar worked with was the densitized version instead of the original Hamiltonian, that is, he worked with <math display="inline">\tilde{H} = \sqrt{\det (q)} H</math>. There were serious difficulties in promoting this quantity to a quantum operator. Moreover, although Ashtekar variables had the virtue of simplifying the Hamiltonian, they are complex. When one quantizes the theory, it is difficult to ensure that one recovers real general relativity as opposed to complex general relativity.

=== Quantum constraints as the equations of quantum general relativity ===
The classical result of the Poisson bracket of the smeared Gauss' law <math display="inline">G(\lambda) = \int d^3x \lambda^j (D_a E^a)^j</math> with the connections is
<math display="block">\{ G(\lambda), A_a^i \} = \partial_a \lambda^i + g \epsilon^{ijk} A_a^j \lambda^k = (D_a \lambda)^i.</math>

The quantum Gauss' law reads
<math display="block">\hat{G}_j \Psi (A) = - i D_a {\delta \lambda \Psi [A] \over \delta A_a^j} = 0.</math>

If one smears the quantum Gauss' law and study its action on the quantum state one finds that the action of the constraint on the quantum state is equivalent to shifting the argument of <math>\Psi</math> by an infinitesimal (in the sense of the parameter <math>\lambda</math> small) gauge transformation,
<math display="block">\left [ 1 + \int d^3x \lambda^j (x) \hat{G}_j \right] \Psi (A) = \Psi [A + D \lambda] = \Psi [A],</math>
and the last identity comes from the fact that the constraint annihilates the state. So the constraint, as a quantum operator, is imposing the same symmetry that its vanishing imposed classically: it is telling us that the functions <math>\Psi [A]</math> have to be gauge invariant functions of the connection. The same idea is true for the other constraints.

Therefore, the two step process in the classical theory of solving the constraints <math>C_I = 0</math> (equivalent to solving the admissibility conditions for the initial data) and looking for the gauge orbits (solving the 'evolution' equations) is replaced by a one step process in the quantum theory, namely looking for solutions <math>\Psi</math> of the quantum equations <math>\hat{C}_I \Psi = 0</math>. This is because it solves the constraint at the quantum level and it simultaneously looks for states that are gauge invariant because <math>\hat{C}_I</math> is the quantum generator of gauge transformations (gauge invariant functions are constant along the gauge orbits and thus characterize them).{{sfn|Thiemann|2003|pp=41–135}} Recall that, at the classical level, solving the admissibility conditions and evolution equations was equivalent to solving all of Einstein's field equations, this underlines the central role of the quantum constraint equations in canonical quantum gravity.

=== Introduction of the loop representation ===
{{main| Holonomy| Wilson loop| Knot invariant}}
It was in particular the inability to have good control over the space of solutions to Gauss's law and spatial diffeomorphism constraints that led Rovelli and Smolin to consider the [[loop representation in gauge theories and quantum gravity]].{{sfn|Rovelli|Smolin|1988|pp=1155–1958}}

LQG includes the concept of a [[holonomy]]. A holonomy is a measure of how much the initial and final values of a spinor or vector differ after [[parallel transport]] around a closed loop; it is denoted
<math display="block">h_\gamma [A] .</math>

Knowledge of the holonomies is equivalent to knowledge of the connection, up to gauge equivalence. Holonomies can also be associated with an edge; under a Gauss Law these transform as
<math display="block">(h'_e)_{\alpha \beta} = U_{\alpha \gamma}^{-1} (x) (h_e)_{\gamma \sigma} U_{\sigma \beta} (y).</math>

For a closed loop <math>x = y</math> and assuming <math>\alpha = \beta</math>, yields
<math display="block">(h'_e)_{\alpha \alpha} = U_{\alpha \gamma}^{-1} (x) (h_e)_{\gamma \sigma} U_{\sigma \alpha} (x) = [U_{\sigma \alpha} (x) U_{\alpha \gamma}^{-1} (x)] (h_e)_{\gamma \sigma} = \delta_{\sigma \gamma} (h_e)_{\gamma \sigma} = (h_e)_{\gamma \gamma}</math>
or
<math display="block">\operatorname{Tr} h'_\gamma = \operatorname{Tr} h_\gamma.</math>

The trace of an holonomy around a closed loop is written
<math display="block">W_\gamma [A]</math>
and is called a Wilson loop. Thus Wilson loops are gauge invariant. The explicit form of the Holonomy is
<math display="block">h_\gamma [A] = \mathcal{P} \exp \left \{-\int_{\gamma_0}^{\gamma_1} ds \dot{\gamma}^a A_a^i (\gamma (s)) T_i \right \}</math>
where <math>\gamma</math> is the curve along which the holonomy is evaluated, and <math>s</math> is a parameter along the curve, <math>\mathcal{P}</math> denotes path ordering meaning factors for smaller values of <math>s</math> appear to the left, and <math>T_i</math> are matrices that satisfy the <math>\operatorname{SU}(2)</math> algebra
<math display="block">[T^i ,T^j] = 2i \epsilon^{ijk} T_k.</math>

The [[Pauli matrices]] satisfy the above relation. It turns out that there are infinitely many more examples of sets of matrices that satisfy these relations, where each set comprises <math>(N+1) \times (N+1)</math> matrices with <math>N = 1,2,3,\dots</math>, and where none of these can be thought to 'decompose' into two or more examples of lower dimension. They are called different [[irreducible representations]] of the <math>\operatorname{SU}(2)</math> algebra. The most fundamental representation being the Pauli matrices. The holonomy is labelled by a half integer <math>N/2</math> according to the irreducible representation used.

The use of [[Wilson loop]]s explicitly solves the Gauss gauge constraint. [[Loop representation]] is required to handle the spatial diffeomorphism constraint. With Wilson loops as a basis, any Gauss gauge invariant function expands as,
<math display="block">\Psi [A] = \sum_\gamma \Psi [\gamma] W_\gamma [A].</math>

This is called the loop transform and is analogous to the momentum representation in quantum mechanics (see [[Position and momentum space]]). The QM representation has a basis of states <math>\exp (ikx)</math> labelled by a number <math>k</math> and expands as
<math display="block">\psi [x] = \int dk \psi (k) \exp (ikx). </math>
and works with the coefficients of the expansion <math>\psi (k).</math>

The inverse loop transform is defined by
<math display="block">\Psi [\gamma] = \int [dA] \Psi [A] W_\gamma [A].</math>

This defines the loop representation. Given an operator <math>\hat{O}</math> in the connection representation,
<math display="block">\Phi [A] = \hat{O} \Psi [A] \qquad Eq \; 1,</math>
one should define the corresponding operator <math>\hat{O}'</math> on <math>\Psi [\gamma]</math> in the loop representation via,
<math display="block">\Phi [\gamma] = \hat{O}' \Psi [\gamma] \qquad Eq \; 2,</math>
where <math>\Phi [\gamma]</math> is defined by the usual inverse loop transform,
<math display="block">\Phi [\gamma] = \int [dA] \Phi [A] W_\gamma [A] \qquad Eq \; 3.</math>

A transformation formula giving the action of the operator <math>\hat{O}'</math> on <math>\Psi [\gamma]</math> in terms of the action of the operator <math>\hat{O}</math> on <math>\Psi [A]</math> is then obtained by equating the R.H.S. of <math>Eq \; 2</math> with the R.H.S. of <math>Eq \; 3</math> with <math>Eq \; 1</math> substituted into <math>Eq \; 3</math>, namely
<math display="block">\hat{O}' \Psi [\gamma] = \int [dA] W_\gamma [A] \hat{O} \Psi [A],</math>
or
<math display="block">\hat{O}' \Psi [\gamma] = \int [dA] (\hat{O}^\dagger W_\gamma [A]) \Psi [A],</math>
where <math>\hat{O}^\dagger</math> means the operator <math>\hat{O}</math> but with the reverse factor ordering (remember from simple quantum mechanics where the product of operators is reversed under conjugation). The action of this operator on the Wilson loop is evaluated as a calculation in the connection representation and the result is rearranged purely as a manipulation in terms of loops (with regard to the action on the Wilson loop, the chosen transformed operator is the one with the opposite factor ordering compared to the one used for its action on wavefunctions <math>\Psi [A]</math>). This gives the physical meaning of the operator <math>\hat{O}'</math>. For example, if <math> \hat{O}^\dagger</math> corresponded to a spatial diffeomorphism, then this can be thought of as keeping the connection field <math>A</math> of <math>W_\gamma [A]</math> where it is while performing a spatial diffeomorphism on <math>\gamma</math> instead. Therefore, the meaning of <math>\hat{O}'</math> is a spatial diffeomorphism on <math>\gamma</math>, the argument of <math>\Psi [\gamma]</math>.

In the loop representation, the spatial diffeomorphism constraint is solved by considering functions of loops <math>\Psi [\gamma]</math> that are invariant under spatial diffeomorphisms of the loop <math>\gamma</math>. That is, [[knot invariant]]s are used. This opens up an unexpected connection between [[knot theory]] and quantum gravity.

Any collection of non-intersecting Wilson loops satisfy Ashtekar's quantum Hamiltonian constraint. Using a particular ordering of terms and replacing <math>\tilde{E}^a_i</math> by a derivative, the action of the quantum Hamiltonian constraint on a Wilson loop is
<math display="block">\hat{\tilde{H}}^\dagger W_\gamma [A] = - \epsilon_{ijk} \hat{F}^k_{ab} \frac{\delta}{\delta A_a^i} \frac{\delta}{\delta A_b^j} W_\gamma [A].</math>

When a derivative is taken it brings down the tangent vector, <math>\dot{\gamma}^a</math>, of the loop, <math>\gamma</math>. So,
<math display="block">\hat{F}^i_{ab} \dot{\gamma}^a \dot{\gamma}^b.</math>

However, as <math>F^i_{ab}</math> is anti-symmetric in the indices <math>a</math> and <math>b</math> this vanishes (this assumes that <math>\gamma</math> is not discontinuous anywhere and so the tangent vector is unique).

With regard to loop representation, the wavefunctions <math>\Psi [\gamma]</math> vanish when the loop has discontinuities and are knot invariants. Such functions solve the Gauss law, the spatial diffeomorphism constraint and (formally) the Hamiltonian constraint. This yields an infinite set of exact (if only formal) solutions to all the equations of quantum general relativity!{{sfn|Rovelli|Smolin|1988|pp=1155–1958}} This generated a lot of interest in the approach and eventually led to LQG.

=== Geometric operators, the need for intersecting Wilson loops and spin network states ===
The easiest geometric quantity is the area. Let us choose coordinates so that the surface <math>\Sigma</math> is characterized by <math>x^3 = 0</math>. The area of small parallelogram of the surface <math>\Sigma</math> is the product of length of each side times <math>\sin \theta</math> where <math>\theta</math> is the angle between the sides. Say one edge is given by the vector <math>\vec{u}</math> and the other by <math>\vec{v}</math> then,
<math display="block">A = \| \vec{u} \| \| \vec{v} \| \sin \theta = \sqrt{\| \vec{u} \|^2 \| \vec{v} \|^2 (1 - \cos^2 \theta)} = \sqrt{\| \vec{u} \|^2 \| \vec{v} \|^2 - (\vec{u} \cdot \vec{v})^2}</math>

In the space spanned by <math>x^1</math> and <math>x^2</math> there is an infinitesimal parallelogram described by <math>\vec{u} = \vec{e}_1 dx^1</math> and <math>\vec{v} = \vec{e}_2 dx^2</math>. Using <math>q_{AB}^{(2)} = \vec{e}_A \cdot \vec{e}_B</math> (where the indices <math>A</math> and <math>B</math> run from 1 to 2), yields the area of the surface <math>\Sigma</math> given by
<math display="block">A_\Sigma = \int_\Sigma dx^1 dx^2 \sqrt{\det \left(q^{(2)}\right)}</math>
where <math>\det (q^{(2)}) = q_{11} q_{22} - q_{12}^2</math> and is the determinant of the metric induced on <math>\Sigma</math>. The latter can be rewritten <math>\det (q^{(2)}) = \epsilon^{AB} \epsilon^{CD} q_{AC} q_{BD} / 2</math> where the indices <math>A \dots D</math> go from 1 to 2. This can be further rewritten as
<math display="block">\det (q^{(2)}) = {\epsilon^{3ab} \epsilon^{3cd} q_{ac} q_{bc} \over 2}.</math>

The standard formula for an inverse matrix is
<math display="block">q^{ab} = {\epsilon^{bcd} \epsilon^{aef} q_{ce} q_{df} \over 2!\det (q)}.</math>

There is a similarity between this and the expression for <math>\det(q^{(2)})</math>. But in Ashtekar variables, <math>\tilde{E}^a_i\tilde{E}^{bi} = \det (q) q^{ab}</math>. Therefore,
<math display="block">A_\Sigma = \int_\Sigma dx^1 dx^2 \sqrt{\tilde{E}^3_i \tilde{E}^{3i}}.</math>

According to the rules of canonical quantization the triads <math>\tilde{E}^3_i</math> should be promoted to quantum operators,
<math display="block">\hat{\tilde{E}}^3_i \sim {\delta \over \delta A_3^i}.</math>

The area <math>A_\Sigma</math> can be promoted to a well defined quantum operator despite the fact that it contains a product of two functional derivatives and a square-root.{{sfn|Gambini|Pullin|2011|loc=Section 8.2}} Putting <math>N = 2J</math> (<math>J</math>-th representation),
<math display="block">\sum_i T^i T^i = J (J+1) 1.</math>

This quantity is important in the final formula for the area spectrum. The result is
<math display="block">\hat{A}_\Sigma W_\gamma [A] = 8 \pi \ell_{\text{Planck}}^2 \beta \sum_I \sqrt{j_I (j_I + 1)} W_\gamma [A]</math>
where the sum is over all edges <math>I</math> of the Wilson loop that pierce the surface <math>\Sigma</math>.

The formula for the volume of a region <math>R</math> is given by
<math display="block">V = \int_R d^3 x \sqrt{\det (q)} = \int_R dx^3 \sqrt{\frac{1}{3!} \epsilon_{abc} \epsilon^{ijk} \tilde{E}^a_i \tilde{E}^b_j \tilde{E}^c_k}.</math>

The quantization of the volume proceeds the same way as with the area. Each time the derivative is taken, it brings down the tangent vector <math>\dot{\gamma}^a</math>, and when the volume operator acts on non-intersecting Wilson loops the result vanishes. Quantum states with non-zero volume must therefore involve intersections. Given that the anti-symmetric summation is taken over in the formula for the volume, it needs intersections with at least three non-[[coplanar]] lines. At least four-valent vertices are needed for the volume operator to be non-vanishing.

Assuming the real representation where the gauge group is <math>\operatorname{SU}(2)</math>, Wilson loops are an over complete basis as there are identities relating different Wilson loops. These occur because Wilson loops are based on matrices (the holonomy) and these matrices satisfy identities. Given any two <math>\operatorname{SU}(2)</math> matrices <math>\mathbb{A}</math> and <math>\mathbb{B}</math>,
<math display="block">\operatorname{Tr}(\mathbb{A}) \operatorname{Tr}(\mathbb{B}) = \operatorname{Tr}(\mathbb{A}\mathbb{B}) + \operatorname{Tr}(\mathbb{A}\mathbb{B}^{-1}).</math>

This implies that given two loops <math>\gamma</math> and <math>\eta</math> that intersect,
<math display="block">W_\gamma [A] W_\eta [A] = W_{\gamma \circ \eta} [A] + W_{\gamma \circ \eta^{-1}} [A]</math>
where by <math>\eta^{-1}</math> we mean the loop <math>\eta</math> traversed in the opposite direction and <math>\gamma \circ \eta</math> means the loop obtained by going around the loop <math>\gamma</math> and then along <math>\eta</math>. See figure below. Given that the matrices are unitary one has that <math>W_\gamma [A] = W_{\gamma^{-1}} [A]</math>. Also given the cyclic property of the matrix traces (i.e. <math>\operatorname{Tr} (\mathbb{A} \mathbb{B}) = \operatorname{Tr}(\mathbb{B} \mathbb{A})</math>) one has that <math>W_{\gamma \circ \eta} [A] = W_{\eta \circ \gamma} [A]</math>. These identities can be combined with each other into further identities of increasing complexity adding more loops. These identities are the so-called Mandelstam identities. Spin networks certain are linear combinations of intersecting Wilson loops designed to address the over-completeness introduced by the Mandelstam identities (for trivalent intersections they eliminate the over-completeness entirely) and actually constitute a basis for all gauge invariant functions.

[[File:The Mandelstam identity.jpg|right|thumb|upright=2.2|Graphical representation of the simplest non-trivial Mandelstam identity relating different [[Wilson loops]]]]

As mentioned above the holonomy tells one how to propagate test spin half particles. A spin network state assigns an amplitude to a set of spin half particles tracing out a path in space, merging and splitting. These are described by spin networks <math>\gamma</math>: the edges are labelled by spins together with 'intertwiners' at the vertices which are prescription for how to sum over different ways the spins are rerouted. The sum over rerouting are chosen as such to make the form of the intertwiner invariant under Gauss gauge transformations.

=== Hamiltonian constraint of LQG ===
{{main| Hamiltonian constraint of LQG}}

In the long history of canonical quantum gravity formulating the Hamiltonian constraint as a quantum operator ([[Wheeler–DeWitt equation]]) in a mathematically rigorous manner has been a formidable problem. It was in the loop representation that a mathematically well defined Hamiltonian constraint was finally formulated in 1996.{{sfn|Thiemann|1996|pp=257–264}} We leave more details of its construction to the article [[Hamiltonian constraint of LQG]]. This together with the quantum versions of the Gauss law and spatial diffeomorphism constrains written in the loop representation are the central equations of LQG (modern canonical quantum General relativity).

Finding the states that are annihilated by these constraints (the physical states), and finding the corresponding physical inner product, and observables is the main goal of the technical side of LQG.

An important aspect of the Hamiltonian operator is that it only acts at vertices (a consequence of this is that Thiemann's Hamiltonian operator, like Ashtekar's operator, annihilates non-intersecting loops except now it is not just formal and has rigorous mathematical meaning). More precisely, its action is non-zero on at least vertices of valence three and greater and results in a linear combination of new spin networks where the original graph has been modified by the addition of lines at each vertex together and a change in the labels of the adjacent links of the vertex.{{Citation needed|date=July 2021}}

=== Chiral fermions and the fermion doubling problem ===
A significant challenge in theoretical physics lies in unifying LQG, a theory of quantum spacetime, with the [[Standard Model]] of particle physics, which describes fundamental forces and particles. A major obstacle in this endeavor is the [[fermion doubling problem]], which arises when incorporating [[Chiral fermion|chiral fermions]] into the LQG framework.

Chiral fermions, such as electrons and quarks, are fundamental particles characterized by their "handedness" or chirality. This property dictates that a particle and its mirror image behave differently under weak interactions. This asymmetry is fundamental to the Standard Model's success in explaining numerous physical phenomena.

However, attempts to integrate chiral fermions into LQG often result in the appearance of spurious, mirror-image particles. Instead of a single left-handed fermion, for instance, the theory predicts the existence of both a left-handed and a right-handed version.<ref>{{cite arXiv |last1=Barnett |first1=Jacob |title=Fermion Doubling in Loop Quantum Gravity |date=2015-07-05 |eprint =1507.01232 |last2=Smolin |first2=Lee|class=gr-qc }}</ref> This "doubling" contradicts the observed chirality of the Standard Model and disrupts its predictive power. 

The fermion doubling problem poses a significant hurdle in constructing a consistent theory of quantum gravity. The Standard Model's accuracy in describing the universe at the smallest scales relies heavily on the unique properties of chiral fermions. Without a solution to this problem, incorporating matter and its interactions into a unified framework of quantum gravity remains a significant challenge. 

Therefore, resolving the fermion doubling problem is crucial for advancing our understanding of the universe at its most fundamental level and developing a complete theory that unites gravity with the quantum world.