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Home , Gennadi Sardanashvily, World manifold

arXiv:gr-qc/9709054v1 21 Sep 1997 rte ihrsett basis a to respect with written xsec fDrcfrinfilsipista,i ol manifold world a if that, implies fields fermion Dirac of Existence Introduction 1 by ostsycuiiycniin,i sprleial,ta s h tangent the is, that parallelizable, is it conditions, causility satisfy to h rnia bundle principal the M ia pnr r enda olw 2 ] Let 3]. [2, follows as defined are spinors Dirac hsi smrhct h elCiodalgebra Clifford real the to isomorphic is This . dnie ihadtoa esrfields tensor additional with identified soitdwt erdfield tetrad a with associated rvttoa ed orsodt ieettta o metri (or tetrad different field tetrad to correspond fields gravitational a hn of think can i hr xssteuieslsi structure spin universal the exists a of There solutions with (i) Two associated transformations. covariant structure general spin under the because theory itation rvt eeaie odnmccnetosadfrinfiel fermion and Loguno connections to dynamic come to We generalized law. transformation covariant general the omtoswihke h akrudtta field tetrad background the keep which formations ia emo ed r epnil o pnaeu symmet spontaneous for responsible are fields fermion Dirac NGUEGAIAINTHEORY GRAVITATION GAUGE IN eateto hoeia hsc,Mso tt University, State Moscow Physics, Theoretical of Department h h e n h soitdsi structure spin associated the and sbigeetv erdfils eso htteeeitgauge exist there that show We fields. tetrad effective being as LX AKRUDGEOMETRY BACKGROUND foine rmsin frames oriented of -al [email protected] E-mail: { e end Sardanashvily Gennadi η a h } diag(1 = 124Mso, Moscow, 117234 By . sasbudeo h bundle the of subbundle a is C Abstract q 1 λ , , 3 µ − smattecmlxCiodagbagenerated algebra Clifford complex the meant is eciigdeviations describing 1 1 S X T , → − S M 1 h disagoa eto [1]. section global a admits X , − r xd hl rvttoa ed are fields gravitational while fixed, are eteMnosisaewt h metric the with space Minkowski the be uhta n pnstructure spin any that such 1) R h , 2 n c nteeetv ed by fields effective the on act and , 3 over hspolmcnb suggested. be can problem this )fils i)Abackground A (ii) fields. c) ds. erdfil sntpreserved not is field tetrad ybekn nguegrav- gauge in breaking ry ’ eaiitcTer of Theory Relativistic v’s R S 5 h e X t uagbagenerated subalgebra Its . → a λ bundle = snncmati order in non-compact is X q nti model, this In . λ µ h X T a µ of S stiiland trivial is h h One . trans- → X 5 by M ⊂ R is the real Clifford algebra R1,3. A Dirac spinor space Vs is defined as a minimal left ideal of C1,3 on which this algebra acts on the left. We have the representation

a a a γ : M ⊗ Vs → Vs, e = γ(e )= γ , (1) b of elements of the Minkowski space M ⊂ C1,3 by the Dirac matrices on Vs. The explicit form of this representation depends on the choice of the ideal Vs. Different ideals lead to equivalent representations (1). The spinor space Vs is provided with the spinor metric 1 a(v, v′)= (v+γ0v′ + v′+γ0v). (2) 2 Let us consider morphisms preserving the representation (1). By definition, the Clifford group G1,3 consists of the invertible elemets ls of the real Clifford algebra R1,3 such that the inner automorphisms −1 lsels = l(e), e ∈ M, (3) defined by these elements preserve the Minkowski space M ⊂ R1,3. Since the action (3) of the group G1,3 on M is not effective, one usually considers its spin subgroup

0 Ls = Spin (1, 3) ≃ SL(2, C).

This is the two-fold universal covering group zL : Ls → L of the proper L = SO0(1, 3). The group L acts on M by the generators

c c c Lab d = ηadδb − ηbdδa. (4)

The spin group Ls acts on the spinor space Vs by the generators 1 L = [γ ,γ ]. (5) ab 4 a b

+ 0 0 Since, Labγ = −γ Lab, the group Ls preserves the spinor metric (2). The transformations (4) and (5) preserve the representation (1), that is,

γ(lM ⊗ lsVs)= lsγ(M ⊗ Vs).

A Dirac on a X is said to be a pair (Ps, zs) of an Ls- Ps → X and a principal bundle morphism of Ps to the LX with the + structure group GL4 = GL (4, R) [2, 4, 5]. Owing to the group epimorphism zL, every bundle h morphism zs factorizes through a bundle epimorphism of Ps onto a subbundle L X ⊂ LX with

2 the structure group L. Such a subbundle LhX is called a Lorentz structure [6, 7]. It exists since X is parallelizable. By virtue of the well-known theorem [10], there is one-to-one correspondence between the Lorentz subbundles of the frame bundle LX and the global sections h of the quotient bundle

Σ= LX/L → X. (6)

This is the two-fold covering of the bundle of pseudo-Riemannian metrics in TX, and sections h h of Σ are tetrad fields. Let P be the Ls-principal bundle covering L X and

h h S =(P × Vs)/Ls (7)

h the associated spinor bundle. Sections sh of S describe Dirac fermion fields in the presence of the tetrad field h. Indeed, every tetrad field h yields the structure

∗ T X =(LhX × M)/L of the LhX-associated bundle of Minkowski spaces on T ∗X, and defines the representation

∗ h h h h γh : T X ⊗ S =(P × (M ⊗ Vs))/Ls → (P × γ(M ⊗ Vs))/Ls = S , ∗ ∗ λ λ λ a γh : T X ∋ t =x ˙ λdx 7→ x˙ λdx =x ˙ λha(x)γ , (8) b of covectors on X by the Dirac matrices on elements of the spinor bundle Sh. The crucial point is that different tetrad fields h define non-equivalent representations (8) [8, 9]. It follows that every Dirac fermion field must be considered in a pair with a certain tetrad field. Thus, we come to the well-known problem of describing fermion fields in the presence of different gravitational fields and under general covariant transformations. Recall that general covariant transformations are automorphisms of the frame bundle LX which are the canonical lift of diffeomorphisms of a world manifold X. They do not preserve the Lorentz subbundles of LX. The following two solutions of this problem can be suggested. (i) One can describe the total system of the pairs of spinor and tetrad fields if any spinor bundle Sh is represented as a subbundle of some fibre bundle S → X [11-14]. To construct S, let us consider the two-fold universal covering group GL4 of the group GL4 and the corresponding g principal bundle LX covering the frame bundle LX [2,15-17]. Note that the group GL4 admits only infinite-dimensionalg spinor representations [18]. At the same time, LX has theg structure g of the Ls-principal bundle LX → Σ [13, 14]. Then let us consider the associated spinor bundle g

S =(LX × Vs)/Ls → Σ g 3 which is the composite bundle S → Σ → X. We have the representation

∗ γΣ : (Σ × T X) ⊗ S → S, X Σ λ λ a γΣ : dx 7→ σa γ , (9) of covectors on X by the Dirac matrices. One can show that, for any tetrad field h, the restriction of S → Σ to h(X) ⊂ Σ is a subbundle of S → X which is isomorphic to the spinor h bundle S , while the representation γΣ (9) restricted to h(X) is exactly the representation γh (8). The bundle LX inherits general covariant transformations of LX [15]. They, in turn, induce general covariantg transformations of S, which transform the subbundles Sh ⊂ S to each other and preserve the representation (9) [13, 14]. (ii) This work is devoted to a different model. A background tetrad field h and the associ- ated background spin structure Sh are fixed, while gravitational fields are identified with the sections of the group bundle Q → X associated with LX (in the spirit of Logunov’s Relativistic

Gravitation Theory (RGT) [19]). We will show that there exists an automorphism fh of LX over any diffeomorphism f of X which preserves the Lorentz subbundle LhX ⊂ LX. e

2 Gauge transformations

With respect to the tangent holonomic frames {∂µ}, the frame bundle LX is equipped with the λ λ coordinates (x ,p a) such that general covariant transformations of LX over diffeomorphisms f of X take the form

λ λ λ λ µ f :(x ,p a) 7→ (f (x), ∂µf (x)p a). e They induce general covariant transformations

f :(p, v) · GL4 7→ (f(p), v) · GL4 e e of any LX-associated bundle

Y =(LX × V )/GL4,

−1 where the quotient is defined by identification of elements (p, v) and (pg,g v) for all g ∈ GL4. Given a tetrad field h, any general covariant transformation of the frame bundle LX can h be written as the composition f = Φ ◦ fh of its automorphism fh over f which preserves L X and some vertical automorphisme e e

Φ: p 7→ pφ(p), p ∈ LX, (10)

4 where φ is a GL4-valued equivariant function on LX, i.e.,

−1 φ(pg)= g φ(p)g, g ∈ GL4.

h Since X is parallelizable, the automorphism fh exists. Indeed, let z be a global section of LhX. Then, we put e

h h fh : LxX ∋ p = z (x)g 7→ z (f(x))g ∈ Lf(x)X. e h h The automorphism fh restricted to L X induces an automorphism of the principal bundle P e h and the corresponding automorphism fs of the spinor bundle S , which preserve the represen- tation (8). e Turn now to the vertical automorphism Φ. Let us consider the group bundle Q → X associated with LX. Its typical fibre is the group GL4 which acts on itself by the adjoint λ λ representation. Let (x , q µ) be coordinates on Q. There exist the left and right canonical actions of Q on any LX-associated bundle Y :

ρl,r : Q × Y → Y, X ρl : ((p,g) · GL4, (p, v) · GL4) 7→ (p,gv) · GL4, −1 ρr : ((p,g) · GL4, (p, v) · GL4) 7→ (p,g v) · GL4. Let Φ be the vertical automorphism (10) of LX. The corresponding vertical automorphisms of an associated bundle Y and the group bundle Q read

Φ:(p, v) · GL4 7→ (p,φ(p)v) · GL4, −1 Φ:(p,g) · GL4 7→ (p,φ(p)gφ (p)) · GL4. For any Φ (10), there exists the fibre-to-fibre morphism

Φ:(p, q) · GL4 7→ (p,φ(p)q) · GL4 of the group bundle Q such that

ρl(Φ(Q) × Y )=Φ(ρl(Q × Y )), (11)

ρr(Φ(Q) × Φ(Y )) = ρr(Q × Y ). (12) For instance, if Y = T ∗X, the expression (11) takes the coordinate form

λ λ λ λ ρr :(x , q µ, x˙ µ) 7→ (x , x˙ λq µ), λ λ λ λ ν Φ:(x , q µ) 7→ (x ,S νq µ), λ λ ν −1 α λ λ ρr(x ,S ν q µ, x˙ α(S ) λ)=(x , x˙ λq µ).

5 Hence, we obtain the representation

∗ h h γQ :(Q × T X) ⊗(Q × S ) → (Q × S ), Q ∗ λ µ λ µ a γQ = γh ◦ ρr :(q, t ) 7→ x˙ λq µdx =x ˙ λq µha γ , (13) b h on elements of the spinor bundle S . Let q0 be the canonical global section of the group bundle

Q → X whose values are the unit elements of the fibres of Q. Then, the representation γQ (13) restricted to q0(X) comes to the representation γh (8). Sections q(x) of the group bundle Q are dynamic variables of the model under consideration. One can think of them as being tensor gravitational fields of Logunov’s RTG. There is the canonical morphism

ρl : Q × Σ → Σ, X ρl : ((p,g) · GL4, (p, σ) · GL4) 7→ (p,gσ) · GL4, p ∈ LX, λ λ µ λ λ µ ρl :(x , q µ, σa ) 7→ (x , q µσa ).

This morphism restricted to h(X) ⊂ Σ takes the form

ρh : Q → Σ, h ρh : ((p,g) · L, (p, σ0) · L) → (p,gσ0) · L, p ∈ L X, (14) λ λ λ λ µ ρh :(x , q µ) 7→ (x , q µha ), where σ0 is the center of the quotient GL4/L. µ Let Σh, coordinatized by σa , be the quotient of the bundle Q by the kernel Ker hρh of the morphism (14) with respect toe the section h. This is isomorphic to the bundle Σ provided with the Lorentz structure of an LhX-associated bundle. Then the representation (13), which is constant on Ker hρh, reduces to the representation

∗ h h (Σh × T X) ⊗(Σh × S ) → (Σh × S ), Σh ∗ λ a (σ, t ) 7→ x˙ λσa γ . (15) e e Thence, one can think of a section h =6 h of the bundle Σh as being an effective tetrad field, µν µ ν e and can treat g = ha hb ηab as an effective metric. A section h is not a true tetrad field, while e e e a a µ e g is not a true metric. Covectors h = hµdx have the same representation by γ-matrices as e a a µ e e the covectors h = hµdx , while Greek indices go down and go up by means of the background µν µ ν metric g = ha hb ηab.

6 Given a general covariant transformation f = Φ◦fh of the frame bundle LX, let us consider the morphism e e

h h ∗ ∗ fQ : Q → Φ ◦ fh(Q), S → fs(S ), T X → f(T X). (16) e e e e

This preserves the representation (13), i.e., γQ ◦ fQ = fs ◦ γQ, and yields the general covariant λ λ µ e e transformation σa 7→ ∂µf σa of the bundle Σh. Thus, we recovere RTGe [19] in the case of a background tetrad field h and dynamic gravita- tional fields q.

3 Gauge theory of RTG

We follow the geometric formulation of field theory where a configuration space of fields, rep- resented by sections of a bundle Y → X, is the finite dimensional jet manifold J 1Y of Y , λ i i 1 coordinatized by (x ,y ,yλ) [14, 21]. Recall that J Y comprises the equivalence classes of sec- tions s of Y → X which are identified by their values and values of their first derivatives at points x ∈ X, i.e.,

i i i i y ◦ s = s (x), yλ ◦ s = ∂λs (x).

A Lagrangian on J 1Y is defined to be a horizontal density

λ i i 1 n L = L(x ,y ,yλ)ω, ω = dx ··· dx , n = dim X.

λ λ The notation πi = ∂i L will be used. A connection Γ on the bundle Y → X is defined as a section of the jet bundle J 1Y → Y , and is given by the tangent-valued form

λ i Γ= dx ⊗ (∂λ +Γλ∂i).

For instance, a linear connection K on the TX reads

∂ K = dxλ ⊗ (∂ + K α x˙ ν ). λ λ ν ∂x˙ α Every world connection K yields the spinor connection 1 K = dxλ ⊗ [∂ + (ηkbha − ηkahb )(∂ hµ − hνK µ )L A yB∂ ] (17) h λ 4 µ µ λ k k λ ν ab B A

7 h λ A on the spinor bundle S with coordinates (x ,y ), where Lab are generators (5) [12, 14, 22].

Using the connection Kh and the representation γQ (13), one can construct the following Dirac operator on the product Q × Sh:

λ µ a DQ = q µha γ Dλ, (18)

A A where Dλ = yλ −Kλ are the covariant derivatives relative to the connection (17). The operator h (18) restricted to q0(X) recovers the familiar Dirac operator on S for fermion fields in the presence of the background tetrad field h and the world connection K. Thus, we obtain the metric-affine generalization of RTG where dynamic variables are tensor gravitational fields q, general linear connections K and Dirac fermion fields in the presence of a background tetrad field h [20]. The configuration space of this model is the jet manifold J 1Y of the product h Y = Q × CK × S , (19) X X 1 where CK = J LX/GL4 is the bundle whose sections are world connections K. The bundle µ µ µ A (19) is coordinatized by (x , q ν,kα ν,y ). A total Lagrangian on this configuration space is the sum

L = LMA + Lq(q,g)+ LD (20) where LMA is a metric-affine Lagrangian, expressed into the curvature

α α α α ε α ε Rλµ β = kλµ β − kµλ β + kµ εkλ β − kλ εkµ β

µν µ ν ab and the effective metric σ = σa σb η , the Lagrangian Lq depends on tensor gravitational fields q and the backgrounde metrice eg, and i 1 L = { σλ[y+(γ0γq)A (yB − (ηkbσ−1a − ηkaσ−1b )(σµ − σνk µ )L B yC ) − D 2 q A B λ 4 µ µ λk k λ ν ab C e1 e e e e (y+ − (ηkbσ−1a − ηkaσ−1b )(σµ − σνk µ )y+L+ C (γ0γq)A yB] − λA 4 µ µ λk k λ ν C ab A B + 0 A e B −e1/2 e e µν myA (γ ) By }| σ | , σ = det(σ ) e e e is the Lagrangian of fermion fields in metric-affine gravitation theory [12-14], where tetrad fields are replaced with the effective tetrad fields σ. If e −1/2 µν −1/2 LAM =(−λ1R + λ2) | σ | , Lq = λ3gµνσ | σ | , LD =0, (21) e e µν α where R = σ R µαν , the familiar Lagrangian of RTG is recovered. e 8 4 Energy-momentum conservation law

We follow the standard procedure of constructing Lagrangian conservation laws which is based on the first variational formula [14, 23]. This formula provides the canonical splitting of the Lie derivative of a Lagrangian L along a vector field u on Y → X. We have

λ λ i i i µ λ ∂λu L + [u ∂λ + u ∂i +(dλu − yµ∂λu )∂i ]L = (22) i i µ λ λ µ i i λ (u − yµu )(∂i − dλ∂i )L − dλ[πi (u yµ − u ) − u L].

This identity restricted to the shell

λ i i µ (∂i − dλ∂i )L =0, dλ = ∂λ + yλ∂i + yλµ∂i , comes to the weak equality

λ λ i i i µ λ ∂λu L + [u ∂λ + u ∂i +(dλu − yµ∂λu )∂i ]L ≈ (23) λ µ i i λ −dλ[πi (u yµ − u ) − u L].

If the Lie derivative of L along u vanishes (i.e., L is invariant under the local 1-parameter group of gauge transformations generated by u), we obtain the weak conservation law

λ µ i i λ 0 ≈ −dλ[πi (u yµ − u ) − u L].

For the sake of simplicity, let us replace the bundle Q in the product (19) with the bundle

Σh, and denote

′ h Y = Σh × CK × S . X X There exists the canonical lift on Y ′ of every vector field τ on X:

µ α ν ν α ν α α ∂ µ ν ∂ τ = τ ∂µ +(∂ντ kµ β − ∂βτ kµ ν − ∂µτ kν β + ∂µβτ ) α + ∂ν τ σ a µ + (24) ∂k ∂σa e µ β e 1 ∂ e (ηkbha − ηkahb )(τ λ∂ hµ − hν∂ τ µ)(L A yB∂ − L c σν ), 4 µ µ λ k k ν ab B A ab d d ∂σν e c e c where Lab d are generators (4). This lift is the generator of gauge transformations of the bundle ′ Y induced by morphisms fQ (16). Its part acting on Greek indices is the familiar generator of general covariant transformations,e whereas that acting on the Latin ones is a local generator of vertical Lorentz gauge transformations.

9 Let us examine the weak equality (23) in the case of the Lagrangian (20) and the vector

field (24). In contrast with Lq, the Lagrangians LMA and LD are invariant under the above- mentioned gauge transformations. Then, using the results of [12, 14], we bring (23) into the form ∂L ∂ (τ λL )+(∂ τ µgαν + ∂ τ νgαµ) q ≈ (25) λ q α α ∂gµν e e ∂L e d (2τ µgλα q + τ λL − d U µλ), λ ∂gαµ q µ e e where

∂LAM α α σ U =2 α (∂ν τ − kσ ντ ) ∂Rµλ ν is the generalized Komar superpotential of the energy-momentum of metric-affine gravity [12, 14, 24]. A glance at the expression (25) shows that, if the Lagrangian L (20) containes the Higgs term

Lq, the energy-momentum flow is not reduced to a superpotential, and the familiar covariant conservation law ∂L ∇ tα ≈ 0, tα =2gαµ q , (26) α λ λ ∂gµν f e e takes place. Here, ∇α are covariant derivatives relative to the Levi–Civita connection of the f effective metric g. In the case of the standard Lagrangian Lq (21) of RTG, the equality (26) comes to the well-knowne condition

αµ ∇α(g q| g |) ≈ 0, e e where ∇α are covariant derivatives relative to the Levi-Civita connection of the background metric g. On solutions satisfying this condition, the energy-momentum flow in RTG reduces to the generalized Komar superpotential just as it takes place in General Relativity [25], Palatini formalism [26, 27], metric-affine and gauge gravitation theories [12, 14, 24].

References

[1] R.Geroch, Spinor structure of space-time in general relativity, J. Math. Phys. 9 (1968) 1739.

10 [2] H.Lawson and M-L. Michelson, Spin Geometry (Princeton Univ. Press, Princeton, 1989).

[3] J.Crawford, Clifford algebra: Notes on the spinor metric and Lorentz, Poincar´eand conformal groups, J. Math.Phys. 32 (1991) 576.

[4] S.Avis and C.Isham, Generalized spin structure on four dimensional space-times, Comm. Math. Phys. 72 (1980) 103.

[5] I.Benn and R.Tucker, An Introduction to Spinors and Geometry with Applications in Physics (Adam Hilger, Bristol, 1987).

[6] R.Zulanke and P.Wintgen, Differential geometrie und Faserb¨undel, Hochschulbucher fur Mathematik, Band 75 (VEB Deutscher Verlag der Wissenschaften, Berlin, 1972).

[7] F.Gordejuela and J.Masqu´e, Gauge group and G-structures, J. Phys. A 28 (1995) 497.

[8] G.Sardanashvily, , Int. J. Theor. Phys. 30 (1991) 721.

[9] G.Sardanashvily and O.Zakharov, Gauge gravitation theory (World Scientific, Singapore, 1992).

[10] S.Kobayashi and K.Nomizu, Foundations of Differential Geometry, Vol.1. (John Wiley, N.Y. - Singapore, 1963).

[11] G.Sardanashvily, On the geometry of spontaneous symmetry breaking, J. Math. Phys. 33 (1992) 1546.

[12] G.Sardanashvily, Stress-energy-momentum conservation law in gauge gravitation theory, Class. Quant. Grav. 14 (1997) 1371.

[13] G.Giachetta, L.Mangiarotti and G.Sardanashvily, Universal spin structure in gauge grav- itation theory, E-print: gr-qc/9505078.

[14] G.Giachetta, L.Mangiarotti and G.Sardanashvily, New Hamiltonian and Lagrangian Methods in Theory (World Scientific, Singapore, 1997).

[15] L.Dabrowski and R.Percacci, Spinors and diffeomorphisms, Comm. Math. Phys. 106 (1986) 691.

[16] S.Switt, Natural bundles. II. Spin and the diffeomorphism group, J. Math. Phys. 34 (1993) 3825.

11 [17] R.Fulp, J.Lawson and L.Norris, Geometric prequantization on the spin bundle based on n-symplectic geometry: the Dirac equation, Int. J. Theor. Phys. 33 (1994) 1011.

[18] F. Hehl, J. McCrea, E. Mielke and Y. Ne’eman, Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilaton invariance, Phys. Rep. 258 (1995) 1.

[19] A.Logunov, Yu.Loskutov and M.Mestvirishvili, Relativistic theory of gravity, Int. J. Mod. Phys. A 3 (1988) 2067.

[20] G. Sardanashvily, Relativistic theory of gravity: Gauge approach In vol.: Proceedings of the XIX Workshop on High Energy Physics and Field Theory, Protvino, 1996 (Protvino, IHEP Press, 1997) p.184.

[21] G.Sardanashvily, Generalized Hamiltonian Formalism for Field Theory. Constraint Sys- tems (World Scientific, Singapore, 1995).

[22] A.Aringazin and A.Mikhailov, Matter fields in with vector non-metricity, Class. Quant. Grav. 8 (1991) 1685.

[23] G.Sardanashvily, Stress-energy-momentum tensors in constraint field theories, J. Math. Phys. 38 (1997) 847.

[24] G.Giachetta and G.Sardanashvily, Stress-energy-momentum of affine-metric gravity. Gen- eralized Komar superportential, Class. Quant. Grav. 13 (1996) L67.

[25] J.Novotn´y, On the conservation laws in General Relativity, Geometrical Methods in Physics, Proceeding of the Conference on Differential Geometry and its Applications (Czechoslovakia 1983), Ed. D.Krupka (Brno, University of J.E.Purkynˇe, 1984) p.207.

[26] J.Novotn´y, J.Novotn´y, Energy-momentum complex of gravitational field in the Palatini formalism, Int. J. Theor. Phys. 32 (1993) 1033.

[27] A.Borowiec, M.Ferraris, M.Francaviglia and I.Volovich, Energy-momentum complex for nonlinear gravitational Lagrangians in the first-order formalism, GRG 26 (1994) 637.

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