LOG#249. Basic twistor theory.

 

This my last normal post. Welcome to those who read me. TSOR is ending and from its ashes will arise another project. That is inevitable. I want to use new TeX packages, and that is not easy here, to simplify things, and to write better things I wish to tell the world.

1. Twistor: the introduction

Roger Penrose formulated twistor theory in the hope of making complex geometry and not the real geometry the fundamental arena of geometric theoretical physics, and a better way to understand quantum mechanics. Quantum Mechanics(QM) is baed on complex structure of Hilbert space of physical states (both of finite or infinite dimensions!). The probability amplitudes are the complex numbers. That is, complex numbers describe oscillations interpreted as probability waves! By the other hand, relativity implies that spacetime points is a real four dimensional (in general D-dimensional) vectors. But this is again a restricted option determined by experiments. That coordinates of spacetime are real numbers is just an hypothesis of our mathematical models, despite the fact it is well supported by experiments! The main difficulty is a consistent formulation of special relativistic quantum theory. Even when possible, that it is in the form of quantum field theory, many questions arise: entanglement, the measurement problem, the collapse of wave functions/state vector reduction and the quantum gravity issue.

Twistor theory was created with the idea of treating real coordinates of spacetime points as composed quantities of more general complex objects called twistors. Therefore, in twistor theory, the most fundamental object are twistors instead of spacetime points. Twistor theory is pointless (in the real sense) geometry.

Mathematically, a first approximation to what a twistor is comes from the conformal O(4,2) spinors. Complex 4-vectors in the fundamental representation of a covering conformal group called SU(2,2) is isomorphic to \overline{O(4,2)}. A correspondence between twistors and the spacetime points is given by the so-called incidence equation or Penrose relation.

The twistor formalism originally introduced by Penrose for 4D spacetime can be extended in two or three ways:

  • Extending the Penrose-relation in a supersymmetric way one obtains a correspondence between the supertwistors and the points of D=4 superspaces.

  • Replacing the complex numbers by quaternions (or octonions, Clifford-numbers) in the Penrose relation, one can bring the quaternionic (octonionic, cliffordonic) twistors into connection with the points of the D=6 (D=10, D=2^n) spacetime (superspace, C-(super)space). Even more, one can extend this quaternionic twistor formalism in a SUSY fashion introducting quaternionic fermionic degrees of freedom.

  • Introducing hypertwistors, and likely hypersuperspace or C-hypersuperspace.

2. 4D spacetime as twistor composite

There is a very pedagogical introduction to 4D twistor theory and the fundamental incidence equation and the Penrose relation. It is well known that ANY spacetime point can be described by the 4D (ND) vector X^\mu=(X^0,X^1,X^2,X^3), or X^\mu=(X^0,\ldots,X^{D-1}). This can be connected and linked to a 2\times 2 hermitean matrix, using the Pauli matrics as follows:

(1)   \begin{equation*}X\rightarrow X=\begin{pmatrix} X^0+X^3 & X^1-iX^2\\ X^1+iX^2 & X^0-X^3\end{pmatrix}=X^\mu \sigma_\mu\end{equation*}

This map is one-to-one. We can also consider the complex 4d vector

(2)   \begin{equation*}Z=(Z^0,Z^1,Z^2,Z^3)\end{equation*}

instead of real 4d vectors. The complex 4-vector Z describes a point of the complexified Minkovski spacetime C\mathbb{R}^{4}. A similar relation to the previous equation gives us the correspondence between the points in this complexified Minkovski spacetime and the 2d complex matrices via Z=Z^\mu \sigma_\mu. You can get the real Minkovski spacetime R\mathbb{R}^{4} by putting the reality condition onto the complex matrix Z as Z=Z^+. A point in the twistor construction or model is the use of the isomorphism between complex 2d matrices Z and the Z-plane in 4d complex vector \mathbb{C}: this is the twistor space \mathbb{T}=\mathbb{C}^4. This isomorphism is given by the following correspondence, called Penrose relation:

(3)   \begin{equation*}Z:\mbox{Subspace spanned by columns of $4\times 2$ matrix} \begin{bmatrix}iZ\\ I_2\end{bmatrix}\end{equation*}

More explicitly, the 4\times 2 matrix are identified with a bitwistor, a couple of twistors (T_1,T_2)\in\mathbb{T}:

(4)   \begin{equation*}\begin{bmatrix}iZ\\ I_2\end{bmatrix}=\begin{bmatrix} iZ^0+iZ^3 & Z^2+iZ^1\\ iZ^1-Z^2 & iZ^0-iZ^3\\ 1 & 0\\ 0 & 1\end{bmatrix}\end{equation*}

From a mathematical viewpoint, this gives us an affine system of coordinates for the Z-plane in the twistor space \mathbb{T}. This subspace is a complex Grassmann manifold G_{2,4}(\mathbb{C}). In other words, the Z-plane is given by the two linearly independent twistors (T_1,T_2), the bitwistor in twistor space! This is also a correspondence between the complexified spacetime point Z\in \mathbb{C}^4 and a complex Z-plane in the twistor space. By the other hand, there is NOT a unique relation between the pair of twistors (T_1,T_2) and the Z-plane generated by this pair. It is clear, that every pair of twistors or bitwistor (T_1',T_2') is related to nonsingular matrices 2\times 2 by (T_1',T_2')=(T_1,T_2)M, and that that gives the same Z-plane in the twistor space \mathbb{T}.

Let the pair (T_1', T_2') has the form of previous matrix, then any equivalent pair of twistors satisfy

(5)   \begin{equation*}\begin{bmatrix}iZ\\ I_2\end{bmatrix}=(T_1,T_2)M=\begin{bmatrix}\Omega & M\\ \Pi & M\end{bmatrix}\leftrightarrow iZ=\Omega M, \;\; I_2=\Pi M\end{equation*}

and where the 2\times 2 complex matrices \Omega, \Pi are constructed of the coordinates of the twistors (T_1,T_2). Thus, we have

    \[\tcboxmath{iZ=\Omega\Pi^{-1}\leftrightarrow \Omega=iZ\Pi}\]

This is the Penrose relation in matrix form! If we write

(6)   \begin{equation*}(T_1,T_2)=\begin{pmatrix}\omega^{\dot{1}1} &\omega^{\dot{1}2}\\ \omega^{\dot{2}1} & \omega^{\dot{2}2}\\ \pi_{11} &\pi_{12}\\ \pi_{21} &\pi_{22}\end{pmatrix}\end{equation*}

now we get

(7)   \begin{align*}\omega^{\alpha\dot{1}}=iZ^{\dot{\alpha}\beta}\pi_{\beta 1}\\ \omega^{\dot{\alpha}2}=iZ^{\dot{\alpha}\beta}\pi_{\beta 2}\end{align*}

and where \alpha, \beta=1,2. In short hand notation, we get

    \[\tcboxmath{\omega^{\dot{\alpha}}=iZ^{\dot{\alpha}\beta}\pi_{\beta},\;\; T=\begin{pmatrix}\omega^{\dot{\alpha}}\\ \pi_\beta\end{pmatrix}}\]

This is the celebrated incidence equation postulated firstly by R. Penrose, also named Penrose relation in his honor. It has a simple physical (geometrical) meaning: the pint Z\in \mathbb{C}^4 corresponds to the twistor

    \[T\leftrightarrow \omega^{\dot{\alpha}}=iZ^{\dot{\alpha}\beta}\pi_\beta\]

It is evident that all the twistors lying on the Z-plane given in the same Penrose relation corresponds to a given Z\in\mathbb{C}^4 point and for a given twistor T satisfying the incidence equation, only one complex spacetime point Z is assigned! If anyone need to describe the real spacetime point X\in \mathbb{R}^4, one should require the matrix Z to be hermitian, i.e., to satisfy

(8)   \begin{equation*}  Z=Z^+\leftrightarrow Z=-i\Omega\Pi^{-1}=i(\Pi^{-1})^+\Omega^+\end{equation*}

and then

(9)   \begin{equation*}\Pi^+\Omega+\Omega^+\Pi=0\end{equation*}

Using the notation we introduced above, equivalently

(10)   \begin{align*}\overline{\pi}_{\dot{\alpha}1}\omega^{\dot{\alpha}1}+\overline{\omega}^{\alpha 1}\pi_{\alpha 1}=0\\ \overline{\pi}_{\dot{\alpha}2}\omega^{\dot{\alpha}2}+\overline{\omega}^{\alpha 2}\pi_{\alpha 2}=0\\ \overline{\pi}_{\dot{\alpha}1}\omega^{\dot{\alpha} 2}+\overline{\omega}^{\alpha 2}\pi_{\alpha 1}=0\end{align*}

and where

    \[\overline{\pi}_{\dot{\alpha}\beta}=\overline{\left(\pi_{\alpha\beta}\right)}\]

    \[\overline{\omega}^{\alpha\beta}=\overline{\left(\omega^{\dot{\alpha}\beta}\right)}\]

denotes the complex conjugation. In the twistor framework, these equations say that the twistors (T_1,T_2), the bitwistor, are null-twistors with respect to the U(2,2) norm

    \[\langle T_1,T_2\rangle=\langle T_1,T_1\rangle=\langle T_2,T_2\rangle=0\]

where

(11)   \begin{equation*}\langle T, T\rangle=T^+GT=\begin{pmatrix} \overline{\omega}^\alpha & \overline{\pi}_{\dot{\beta}}\end{pmatrix}\begin{pmatrix}0 & I_2\\ I_2 & 0\end{pmatrix}\begin{pmatrix} \omega^{\dot{\alpha}}\\ \pi_\beta\end{pmatrix}\end{equation*}

Therefore, the reality condition is equivalent to the zero condition for twistors, i.e., to the vanishing of the U(2,2) norm of the bitwistor and couple of twistors. The Z-planes generated by the null twistor (or congruence) are called totally null planes or congruence relation. In this way, we obtain a set of correspondences:

  • Complex planes in twistor space are related one-to-one to points in complexified Minkovski spacetime.

  • Complex planes in twistor space are related  to totally null planes in twistor space, not one-to-one.

  • Totally null planes in twistor space are related one-to-one to points in real Minkovski spacetime.

  • Points of complexified spacetime are related to real spacetime, not one to one, to points of real spacetime.

Remark: from the viewpoint of twistor theory, it is more natural to use twistors (couple of twistors indeed, via a bitwistor), for the description of the complex Minkovski spacetime or the null twistor for the description of the real spacetime!

3. SUSY and Penrose relation

The plan of SUSY is to give a unified mathematical description ob bosonic and fermionic fields. Therefore, one can consider bosons and fermions using the same theoretical background. SUSY or supersymmetry allows us to transorm the descriptions of bosonic fields into fermionic fields and vice versa. In order to have a possible description of bosonic and fermionic fields using the twistor theory, we has to extend it using SUSY. What is SUSY? Surprise…

SUSY replaces the notation of any space-time point X=X^\mu=(X^0,X^1,X^2,X^3) by an appropiate superpoint

(12)   \begin{equation*}Y=(X,\Theta)=(X^0,\ldots,X^3;\theta_1,\ldots,\theta_N)\end{equation*}

Here, the superspace point extends spacetime with a new class of numbers, \theta_i, \theta_i^2=0, \theta_i\theta_j=-\theta_j\theta_i, for all i,j=1,\ldots,N. These numbers are called Grassmann numbers. These numbers allow us to handle fermions, since they anticommute themselves. We can define a supervector representing the D=4 N-extended superspace as follows:

(13)   \begin{equation*} Y=(X,\Theta)\end{equation*}

(14)   \begin{equation*} Y=(X^0,\ldots,X^3;\theta_1,\ldots,\theta_N)=(X^\mu;\Theta_A)\end{equation*}

in such a way

(15)   \begin{equation*}\left[X^\mu,X^\nu\right]=X^\mu X^\nu-X^\nu X^\mu=0\end{equation*}

(16)   \begin{equation*}\{\theta_A,\theta_B\}=\theta_A\theta_B+\theta_B\theta_A=0\end{equation*}

(17)   \begin{equation*}\left[X^\mu,\theta_A\right]=X^\mu\theta_A-\theta_A X^\mu=0\end{equation*}

Commuting coordinates of any supervector are called bosonic coordinates, anticommuting coordinates (c-numbers) are called fermionic coordinatese. In the same spirit, we could generalize twistor theory and the twistor approach introducing N-extended bosonic supertwistors

(18)   \begin{equation*} T^{(n)}=\left(\omega^{\dot{\alpha}},\pi_\beta;\xi_1,\cdots,\xi_n\right)\in \mathbb{C}^{4\vert N}\end{equation*}

and the fermionic N-extended supertwistors

(19)   \begin{equation*}\tilde{T}^{(n)}=\left(\eta_1,\ldots,\eta_4;u_1,\ldots,u_N\right)\in \mathbb{C}^{N\vert 4}\end{equation*}

where the \eta_i quantities are fermionic coordinates ant the u_A quantities are the bosonic degrees of freedom. We will discuss the N=1 case, simple supersymmetry, for simplicity. Firstly, two linearly independent supertwistors (T_1^{(1)},T_2^{(1)}) span (2,0)-superplane int he superspace \mathbb{C}^{4\vert 1}. In analogy with the no superspace case, we define and get

(20)   \begin{equation*}\left(T_1^{(1)},T_2^{(1)}\right)=\begin{bmatrix} \omega^{\dot{1}1} & \omega^{\dot{1}2}\\ \omega^{\dot{2}1} &\omega^{\dot{2}2}\\ \xi_1 & \xi_2 \\ \pi_{11} & \pi_{12}\\ \pi_{21} & \pi_{22}\end{bmatrix}=\begin{bmatrix} iZ\\ \theta^1 & \theta^2\\ 1 & 0\\ 0 & 1\end{bmatrix}\Pi\end{equation*}

Here, Z, \Pi are complex matrices 2\times 2 made up of bosonic elements. This can also be expressed using equations

(21)   \begin{align*}\omega^{\alpha\dot{1}}=iZ^{\dot{\alpha}\beta}\pi_{\beta 1}\\ \omega^{\dot{\alpha} 2}=iZ^{\dot{\alpha}\beta}\pi_{\beta 2}\\ \xi_1=\theta^1\pi_{11}+\theta^2\pi_{21}\\ \xi_2=\theta^1\pi_{12}+\theta^2\pi_{22}\end{align*}

Then, the supersymmetric extension of Penrose relation reads off

(22)   \begin{align*}\omega^{\dot{\alpha}}=iZ^{\dot{\alpha}\beta}\pi_{\beta}\\ \xi=\theta^\alpha\pi_\alpha\end{align*}

These equations mean that every T^{i}=(\omega^{\dot{\alpha}},\pi_\beta,\xi) supertwistor corresponds to a Y=(Z,\Theta)=(z^\mu, \theta^\alpha) superspace point or superpoint. However, note that it is not the only option to generalize the Penrose relation!

Apply three linearly independent supertwistors T^{(1)}_1, T^{(1)}_2, \tilde{T}^{(1)}, where the latter is a fermionic twistor, such as \mathbb{C}^{4\vert 1} is our superspace.

(23)   \begin{equation*}\left(T_1^{(1)},T_2^{(1)},\tilde{T}^{(1)}\right)=\begin{bmatrix}\omega^{\dot{1}1} & \omega^{\dot{1}2} & \rho^{\dot{1}}\\ \omega^{\dot{2}1} & \omega^{\dot{2}2} & \rho^{\dot{2}}\\ \pi_{11} & \pi_{12} & \eta_1\\ \pi_{11} & \pi_{12} & \eta_2\\ \xi^1 & \xi^2 & u\end{bmatrix}=\begin{bmatrix}iZ^{\dot{1}1} & iZ^{\dot{1}2} & \theta^1\\ iZ^{\dot{2}1} & iZ^{\dot{2}2} & \theta^2\\ 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}\begin{bmatrix}\pi_{11} & \pi_{12} & \eta_1\\ \pi_{21} & \pi_{22} & \eta_2\\ \xi^1 & \xi^2 & u\end{bmatrix}\end{equation*}

where the fermionic supertwistor includes the four fermionic components

    \[\left(\rho^{\dot{1}},\rho^{\dot{2}},\eta_1,\eta_2\right)\]

and also the bosonic u. The (2;1)-superplane is parametrized by a (Z,\Theta) matrix of 2\times 3 type with elements satisfying the following generalized incidence relations:

(24)   \begin{align*}\omega^{\dot{\alpha}}=iZ^{\dot{\alpha}\beta}\pi_\beta+\theta^{\dot{\alpha}}\xi\\ \rho^{\dot{\alpha}}=iZ^{\dot{\alpha}\beta}\eta_\beta+\theta^{\dot{\alpha}}u\end{align*}

and where the first equation is the bosonic incidence relation, and the second one is the fermionic incidence relation. Thus, this is a different generalization of Penrose relation. For N=1 suspersymmetry, there are 2 possible extensions of Penrose’s relation/incidence equations. In case of the N-extended SUSY, one can generalize those equations in N+1 different ways!

4. Quaternionic extension of Penrose incidence in D=6 spacetime

Taking into account the previous section, there are 2 possible approaches to 6D twistors:

  • Extend Penrose relation from D=4 to D=6 as it has been done in bibligraphy or following these lines.

  • Replace the complex 2\times 2 matrices Z by quaternionic ones. Use quaternionic 2\times 2 matrices describing naturally a 6d real Minkovski spacetime. The previous approach is equivalent to this one if the description of 6d spacetime is careful.

Consider the first case at the moment. 6d twistors are objects

(25)   \begin{equation*} T=\left(\omega^\alpha,\pi_\alpha\right)\in \mathbb{C}^8\end{equation*}

whose structure is determined by the norm of the spinors for 8d complex orthogonal group O(8;\mathbb{C}) given by:

(26)   \begin{equation*}\langle t, t'\rangle=\omega^\alpha\pi'_{\alpha}+\pi_a\omega'^a=0\end{equation*}

Points in 6d complex Minkovski spacetime are represented by a 4\times 4 antisymmetric matrix Z^{\alpha\beta}=-Z^{\beta\alpha}. The Penrose relation becomes

(27)   \begin{equation*}\omega^\alpha= Z^{\alpha\beta}\pi_\beta\end{equation*}

with \alpha,\beta=1,2,3,4. This equation has a nontrivial solution if the twistors T are pure, i.e., if

(28)   \begin{equation*}\langle T,T\rangle=2\omega^\alpha\pi_\alpha\end{equation*}

that is, they have vanishing O(8,\mathbb{C}) norm. The points of the real 6d spacetime are representeed by 4\times 4 complex, antisymmetric matrices Z satisfying a reality condition in the form of

    \[Z=-\overline{Z}\]

    \[\overline{Z}=B^{-1}Z^+B,\;\; B=\begin{bmatrix} 0 & 1& 0 & 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & -1 & 0\end{bmatrix}\]

and Z^+ denotes the hermitian conjugated matrix. This reality condition for Z is equivalent to the following equations

(29)   \begin{align*}\overline{\omega}^\alpha\pi_\alpha+\overline{\pi}_\alpha\omega^\alpha=0\\ \overline{\omega}^\alpha=\overline{\omega}^\beta(B^{-1})_\beta^{\;\;\alpha}\\ \overline{\pi}_\alpha=\overline{\pi}_\beta(B)^\beta_{\;\;\alpha}\end{align*}

The overline means complex conjugation. Indeed, this equation is in fact the condition of vanishing the V(4,4) norm. Thus, 6d twistors describe the points of the real Minkovski spacetime RM^6 if the following two norms are zero:

(30)   \begin{align*}\omega^\alpha\pi_\alpha=0\\ \overline{\omega}^\alpha\pi_\alpha+\overline{\pi}_\alpha\omega^\alpha=0\end{align*}

The first equation above is the O(8,C) norm, and the second equation is the U(4,4) norm. It means that 6d twistors describing points in RM^6 are, indeed, invariant under the quaternionic orthogonal group O(4,H), covering the six dimensional group O(6,2). The chain:

    \[O(4,H)\equiv U_\alpha (4,H)=O(8,C)\cap U(4,4)=\overline{O(6,2)}\]

is true as group isomorphism. Therefore, one can look for the quaternionic extension of 4D twistor formalism which can describe RM^6 Minkovski spacetime. Quaternions are algebraic objects

    \[Q=q_0e_1+q_1e_1+q_2e_2+q_3e_3\]

with

 

    \[e_ie_j=-\delta_{ij}+\epsilon_{ijk}e_k\]

and i,j,k=1,2,3. Real numbers are naturally embedded in quaternions. We can also define quaternionic conjugation

(31)   \begin{equation*}\overline{Q}=q_0-q_1e_1-q_2e_2-q_3e_3\end{equation*}

and the norm

(32)   \begin{equation*}N(q)^2=\vert Q\vert^2=\overline{Q}Q=q_0^2+q_1^2+q_2^2+q_3^2\end{equation*}

The quaternion algebra has a natural structure of euclidean 4d spacetime. Complex numbers can be seen too as certain subset of quaternion algebras. Identifying complex numbers is easy from quaternions, if you take the couple

(33)   \begin{equation*}Q=z_1+e_2z_2=(q_0+q_3e_3)+e_2(q_2+q_1e_3)\end{equation*}

In analogy to previous arguments, we can associate a Z-plane in 4D quaternionic space \mathbb{H}^4, in quaternionic twistor space as follows: take Z and associate to it the subspace by columns of 4\times 2 quaternionic matrices

(34)   \begin{equation*}\begin{bmatrix} e_2Z\\ I_2\end{bmatrix}\end{equation*}

By a similar procedure, we get quaternionic Penrose relations

(35)   \begin{equation*}\omega^{\dot{\alpha}}=e_2Z^{\dot{\alpha}\beta}\pi_\beta\end{equation*}

and \alpha,\beta=1,2. The quaternionic twistor will be now t=(\omega^{\dot{\alpha}},\pi_\beta). A real 6d Minkovski spacetime point is desecribed by a 6d vector

    \[X^\mu=(X^0,\ldots,X^5)\in RM^6\]

which can be mapped on a quaternionic hermitian 2\times 2 matrix

    \[\mathbb{X}=\begin{pmatrix}X^0+X^5 & X^4+X^ke_k\\ X_4-X^ke_k & X^0-X^5\end{pmatrix}\]

and k=1,2,3, with e_k the imaginary quaternion units. The reality condition X=X^+, with the plus meaning quaternionic conjugation and transposition, is equivalent to the following condition for quaternionic twistors t:

(36)   \begin{equation*}\langle t,t\rangle=\overline{\omega}^\alpha e_2\pi_\alpha+\overline{\pi}_\alpha e_2\omega^\alpha=0\end{equation*}

Thus, twistors t describe a poitn of RM^6 if their norm, on O(4,H)=U_\alpha(4,H), vanishes. Using the decomposition of quaternionic coordinates of twistor in quaternions one can show that Penrose relations are equivalent to the incidence relations, so descriptions of RM^6  by 6d complex twistors and D=6 quaternions are equivalent.

5. Conclusions

We can summarize some simple possible definitions of twistors related to the written stuff:

  • A twistor is a solution of the twistor equation \nabla_{A'}^{\;\; (A}\omega^{B)}=0, that is called twistor space.
  • A spinor of the conformal group (with two elements!) is a twistor.
Features:
  • Point in twistor space=null-line in Minkovski spacetimes.
  • Point in Minkovski spacetime=line in twistor space.
Twistors are generally available in arbitrary complex dimension for conformal groups, BUT, there is a nice emergent relation with commuting spinors in SL(2,\mathbb{K}), where \mathbb{K} is a division algebra, special relativity in D spacetime, supersymmetry and twistors in dimensions D=3,4,6,10 for the Green-Schwarz action for the superstring, twistors in D=4,5,7,11 for the supermembrane and the Lorentz vector of those dimensions. Superspace version of these arguments are available. There is a match between the number of spacetime dimensions of the (super)p-brane embedding, the number of supersymmetries and the dimension of the p-brane.

 

Twistors are a powerful tool for spacetime geometry in complex manifolds or even real manifolds. We note that 2 approaches seen are equivalent only for real spacetime, though. 6D spacetime is special. 6D spacetime can be xtended in two nonequivalent ways: by complexification or quaternionization methods! The quaternionic formulation of twistor theory leads to serious issues in general, and that is why it is not popular. The main issue is the quantization of twistors because of non-commutative of quaternions. However, the description of 6d spacetime with quaternionic procedures allow us to use the same geometry as in case of the complex description of 4d spacetime! In fact, it is natural to extend this to octonions and 10d spacetime. The problem there is that octonions are generally non-associative and matrix multiplications become nasty due to that: octonionic matrices are hardly associative!

There can only be one!!!!!!

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