SU(2) Algebras and the Lorentz Group O(3,3)

Article SU(2) × SU(2) Algebras and the Lorentz Group O(3,3)

Martin Walker

Independent Researcher, 3958 Grandis Place, Victoria, BC V8N 4H6, Canada; [email protected]

Received: 24 April 2020; Accepted: 7 May 2020; Published: 15 May 2020

Abstract: The Lie algebra of the Lorentz group O(3,3) admits two types of SU(2) SU(2) subalgebras: × a standard form based on spatial rotation generators and a second form based on temporal rotation generators. The units of measurement for the conserved quantity due to invariance under temporal rotations are investigated and found to be the same units of measure as the Planck constant. The breaking of time reversal symmetry is considered and found to aﬀect the chiral properties of a temporal SU(2) SU(2) algebra. Finally, the symmetry between algebras is explored and pairs of × algebras are found to be related by SU(2) U(1) symmetry, while a group of three algebras are related × by SO(4) symmetry.

Keywords: Lie algebra; O(3,3); time rotation; Dirac; Noether

1. Introduction Spinors were first introduced by Elli Cartan in 1913. The ideas were later adopted into quantum mechanics to describe the intrinsic spin of a fermion and play a fundamental role in Dirac’s equation [1]. In group theory, spinors transform under the spin 1 representation of an SU(2) SU(2) Lie algebra, 2 × which is also the Lie algebra of the proper Lorentz group O(3,1) [2]. This article investigates some aspects of symmetry in the Lorentz group O(3,3). This Lie group can be associated with a six-dimensional mathematical space containing three space dimensions and three time dimensions [3]. The corresponding Lie algebra is SO(3,3) in which the symmetry of time and the symmetry of space are isomorphic. As a result, there are two types of SU(2) SU(2) subalgebras: × one containing spatial rotation generators and one containing temporal rotation generators. To better understand the temporal SU(2) SU(2) algebras, we investigate the units of measure for × the conserved quantity due to invariance under temporal rotations, for a restricted definition of action, in an O(3,3) space. Using Noether’s theorem, it is found that the conserved quantity has the same units of measure as the Planck constant. We also consider the effects of breaking time reversal symmetry. For a temporal SU(2) SU(2) × algebra, the two chiralities are related by a time reversal transformation. This suggests that breaking time reversal symmetry affects the chiral properties of a temporal SU(2) SU(2) algebra. × Finally, we explore symmetries between different algebras in SO(3,3). We find pairs of algebras related by SU(2) U(1) symmetry, as well as a group of three algebras related by SO(4) symmetry. × In Section2, two types of SU(2) SU(2) algebras are described. In Section3, we investigate the × units of measure for the conserved quantity due to invariance under temporal rotations. In Section4, we consider the implications of breaking time reversal symmetry. In Section5, the symmetry between algebras is explored.

Symmetry 2020, 12, 817; doi:10.3390/sym12050817 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 817 2 of 10

2. SU(2) SU(2) Subalgebras × One form of SU(2) SU(2) Lie algebra is related to the proper Lorentz group O(3,1). This Lie × group can be associated with transformations in a four-dimensional space containing three space dimensions and one time dimension [4]. It has six generators [2],

J1,J2,J3,K1,K2,K3 (1) where the J’s are spatial rotation generators and the K’s are boosts. The commutation relations for this algebra are, [J ,J ] = i Jm [K ,K ] = i Jm [J ,K ] = i Km (2) j k jkm j k − jkm j k jkm where is the Levi-Civita symbol, i is the imaginary unit and the indexes j, k, m can assume any value from 1 to 3. Using a complexiﬁcation and a change of basis the Lie algebra becomes a direct product of two SU(2) algebras [5], 1 1 1 1 1 1 (J + iK ), (J + iK ), (J + iK ), (J iK ), (J iK ), (J iK ) (3) 2 1 1 2 2 2 2 3 3 2 1 − 1 2 2 − 2 2 3 − 3 with commutation relations

1 1 1 [ 2 (Jj + iKj), 2 (Jk + iKk)] = i jkm 2 (Jm + iKm) 1 1 1 [ (J - iK ), (J iK )] = i (Jm iKm) (4) 2 j j 2 k − k jkm 2 − [ 1 (J + iK ), 1 (J iK )] = 0 2 j j 2 k − k where the indexes j, k, m = 1, 2, 3. This SU(2) SU(2) algebra is associated with the description of × spin angular momentum in quantum mechanics [2,5]. Please note that in the text that follows, an SU(2) SU(2) algebra will often be written in a format like × 1 1 1 { (J iK ), (J iK ), (J iK )} (5) 2 1 ± 1 2 2 ± 2 2 3 ± 3 where the curly brackets are delimiters for a list of generators. This article investigates SU(2) SU(2) algebras in the context of the Lorentz group O(3,3). This Lie × group can be associated with transformations in a six-dimensional space containing three space dimensions and three time dimensions [3,4]. Another label for this group is the special orthogonal Lie group SO(3,3), which has ﬁfteen generators [3,6,7]. The group has three space rotation generators, here labelled Ji (i = 1, 2, 3), it has three time rotation generators, labelled Ti (i = 1, 2, 3), and it has nine boost generators, labelled Kij, where the i index denotes the time dimension (i = 1, 2, 3) and the j index denotes the space dimension (j = 1, 2, 3) (see AppendixA for a matrix representation of the generators). The commutation relations in this notation are,

[Tj,Tk] = i jkm Tm [Jj,Jk] = i jkm Jm [Tj,Jk] = 0 [K ,K ] = i Tm [K ,K ] = i Jm (6) jn kn − jkm nj nk − jkm [Tj,Kkn] = i jkm Kmn [Jj,Knk] = i jkm Knm where the indexes j, k, m, n = 1, 2, 3 The complexiﬁcation of the Lie algebra of SO(3,3) used in this article is one in which all the boost generators are multiplied by the imaginary unit, while the rotation generators are left unchanged. This is the same complexiﬁcation commonly used on the Lie algebra of the Lorentz group O(3,1) [5]. This results in the following commutation relations,

[Tj,Tk] = i jkm Tm [Jj,Jk] = i jkm Jm [Tj,Jk] = 0 [iKjn, iKkn] = i jkm Tm [iKnj, iKnk] = i jkm Jm (7) [Tj, iKkn] = i jkm iKmn [Jj, iKnk] = i jkm iKnm where the indexes j, k, m, n = 1,2,3. Symmetry 2020, 12, 817 3 of 10

Complexified SO(3,3) has three complexified SO(3,1) subspaces which give rise to three SU(2) SU(2) subalgebras containing spatial rotation generators: × e = { 1 (J iK ), 1 (J iK ), 1 (J iK )} 1 2 1 ± 11 2 2 ± 12 2 3 ± 13 e = { 1 (J iK ), 1 (J iK ), 1 (J iK )} (8) 2 2 1 ± 21 2 2 ± 22 2 3 ± 23 e = { 1 (J iK ), 1 (J iK ), 1 (J iK )}. 3 2 1 ± 31 2 2 ± 32 2 3 ± 33 These have the standard form [2], and we are encouraged to think of them as a family, as they differ only by the value of the time index in the boost generators. Complexified SO(3,3) also has three complexified SO(1,3) subspaces which give rise to a family of SU(2) SU(2) subalgebras containing temporal rotation generators: × m = { 1 (T iK ), 1 (T iK ), 1 (T iK )} 1 2 1 ± 11 2 2 ± 21 2 3 ± 31 m = { 1 (T iK ), 1 (T iK ), 1 (T iK )} (9) 2 2 1 ± 12 2 2 ± 22 2 3 ± 32 m = { 1 (T iK ), 1 (T iK ), 1 (T iK )}. 3 2 1 ± 13 2 2 ± 23 2 3 ± 33 These algebras differ only by the value of the space index in the boost generators.

3. Invariance under Temporal Rotations We would like to determine the units of measurement for the conserved quantity due to invariance under temporal rotations. The ﬁeld theory treatment of Noether’s theorem that follows is adopted from Schwichtenberg [5] and applied to O(3,3) space. We use the Einstein summation convention in this section. For O(3,3) space, a 6-vector is deﬁned as having the form,

xµ = (x1, x2, x3, x4, x5, x6) (10) where the first three components are space dimensions and the last three components are time dimensions. In the following investigation we will restrict ourselves to the action, S4, with respect to the time variable x4. We define, Z Z 5 S = dx L L = d x = Ψ xµ , ∂µΨ xµ , xµ (11) 4 4 4 4 L4 L4 L4 where Ψ xµ is a scalar field, L is the Lagrangian, and the Lagrangian density, , is a density over an 4 L4 element (δx1, δx2, δx3, δx5, δx6). The equations of motion for this Lagrangian density are then given by the Euler-Lagrange equations:    ∂ 4  ∂  L  4 ∂µ  L = 0. (12)  ∂ ∂µΨ  − ∂Ψ

3.1. Inﬁnitesimal Space-Time Translations for a Scalar Field For an inﬁnitesimal space-time translation we have,

xµ x0 = xµ + δxµ = xµ + aµ (13) → µ where aµ is an arbitrary inﬁnitesimal change. If the transformation does not change the Lagrangian density we get,    ∂ ∂Ψ ν  µ δ = ∂ν L δµ a = 0 (14) L −  ∂(∂νΨ) ∂xµ − L where δ is the Kronecker delta. If aµ is arbitrary then we must have, Symmetry 2020, 12, 817 4 of 10

  ν ν  ∂ ∂Ψ ν  ∂νTµ = 0 where Tµ =  L δµ  (15)  ∂(∂νΨ) ∂xµ − L

ν which gives us one continuity equation for each component µ. The elements Tµ are said to deﬁne components of the energy-momentum tensor. For , there are six continuity equations given by L4 1 + 2 + 3 + 4 + 5 + 6 = ∂1T1 ∂2T1 ∂3T1 ∂4T1 ∂5T1 ∂6T1 0 1 + 2 + 3 + 4 + 5 + 6 = ∂1T2 ∂2T2 ∂3T2 ∂4T2 ∂5T2 ∂6T2 0 1 2 3 4 5 6 ∂1T + ∂2T + ∂3T + ∂4T + ∂5T + ∂6T = 0 3 3 3 3 3 3 (16) 1 + 2 + 3 + 4 + 5 + 6 = ∂1T4 ∂2T4 ∂3T4 ∂4T4 ∂5T4 ∂6T4 0 1 + 2 + 3 + 4 + 5 + 6 = ∂1T5 ∂2T5 ∂3T5 ∂4T5 ∂5T5 ∂6T5 0 1 + 2 + 3 + 4 + 5 + 6 = ∂1T6 ∂2T6 ∂3T6 ∂4T6 ∂5T6 ∂6T6 0

Taking into consideration the fourth equation, we can rearrange it and integrate both sides over an inﬁnite volume,

1 + 2 + 3 + 4 + 5 + 6 = ∂1T4 ∂2T4 ∂3T4 ∂4T4 ∂5T4 ∂6T4 0 4 = 1 + 2 + 3 + 5 + 6 ∂4T4 ∂1T4 ∂2T4 ∂3T4 ∂5T4 ∂6T4 R − R 5 4 = 5 1 + 2 + 3 + 5 + 6 ∂4 V d xT4 V d x ∂1T4 ∂2T4 ∂3T4 ∂5T4 ∂6T4 (17) − R R 5 4 = 5 ∂4 V d xT4 V d x T − R H ∇ ∂ d5xT4 = d4xT − 4 V 4 δV where T = ∂ T1 + ∂ T2 + ∂ T3 + ∂ T5 + ∂ T6, δV is the boundary of volume V and we have used ∇ 1 4 2 4 3 4 5 4 6 4 the divergence theorem in the last step. The surface integral on the right hand side of this equation vanishes because the ﬁeld vanishes at inﬁnity and we are left with, Z 5 4 ∂4 d xT4 = 0 (18) V

R 5 4 which implies that d xT4 is conserved. Using a similar method with the other equations gives us six conserved quantities. We know already that the conserved quantities for invariance under time and space translations in O(3,1) are energy and momentum, respectively. We make the following assignments for the conserved quantities, Z Z Z 5 4 5 4 5 4 E1 = d xT4 E2 = d xT5 E3 = d xT6 (19) and Z Z Z 5 4 5 4 5 4 P1 = d xT1 P2 = d xT2 P3 = d xT3 (20) where E1, E2, E3 are energies and P1, P2, P3 are momentums.

3.2. Inﬁnitesimal Space-Time Rotations for a Scalar Field For an inﬁnitesimal space-time rotation we have,

σ xµ x0 = xµ + δxµ = xµ + M xσ (21) → µ µ σ where the Mµ are generators of rotations. Setting the change in the Lagrangian density to zero results in, Symmetry 2020, 12, 817 5 of 10

   ∂ ∂Ψ ν  µσ δ = ∂ν L δµ M xσ = 0 L −  ∂(∂νΨ) ∂xµ − L (22) µν σ σν µ ∂ν(T x T x )Mµσ = 0 → − where there is one continuity equation for each rotation generator Mµσ.The values of µ and σ for the spatial rotation generators, Ji, are obtained from the relation, 1 J = M (23) i 2 ijk jk where is again the Levi-Civita symbol. This gives: 1 1 1 J = M J = M J = M . (24) 1 2 23 2 2 31 3 2 12 For , there are three equations: L4 21 3 31 2 22 3 32 2 23 3 33 2 ∂1 T x T x + ∂2 T x T x + ∂3 T x T x − − − +∂ T24x3 T34x2 + ∂ T25x3 T35x2 + ∂ T26x3 T36x2 = 0 4 − 5 − 6 − 31 1 11 3 32 1 12 3 33 1 13 3 ∂1 T x T x + ∂2 T x T x + ∂3 T x T x − − − (25) +∂ T34x1 T14x3 + ∂ T35x1 T15x3 + ∂ T36x1 T16x3 = 0 4 − 5 − 6 − 11 2 21 1 12 2 22 1 13 2 23 1 ∂1 T x T x + ∂2 T x T x + ∂3 T x T x − − − +∂ T14x2 T24x1 + ∂ T15x2 T25x1 + ∂ T16x2 T26x1 = 0. 4 − 5 − 6 − We can again use the divergence theorem to obtain the three continuity equations corresponding to conserved quantities: R ∂ d5x T24x3 T34x2 = 0 4 − R 5 34 1 14 3 ∂4 d x T x T x = 0 (26) R − ∂ d5x T14x2 T24x1 = 0. 4 − The terms in each integrand are a product of a momentum density (associated with one of P1, P2, P3) and a space variable (one of x1, x2, x3). We conclude that these have units of angular momentum, as required. To determine the conserved quantities related to the temporal rotation generators, Tˇ i, we can get the values of µ and σ using the relation, 1 Tˇ = M (27) i 2 ijk jk This gives: 1 1 1 Tˇ = M Tˇ = M Tˇ = M . (28) 4 2 56 5 2 64 6 2 45 The resulting three continuity equations are,

51 6 61 5 52 6 62 5 53 6 63 5 ∂1 T x T x + ∂2 T x T x + ∂3 T x T x − − − +∂ T54x6 T64x5 + ∂ T55x6 T65x5 + ∂ T56x6 T66x5 = 0 4 − 5 − 6 − 61 4 41 6 62 4 42 6 63 4 43 6 ∂1 T x T x + ∂2 T x T x + ∂3 T x T x − − − (29) +∂ T64x4 T44x6 + ∂ T65x4 T45x6 + ∂ T66x4 T46x6 = 0 4 − 5 − 6 − 41 5 51 4 42 5 52 4 43 5 53 4 ∂1 T x T x + ∂2 T x T x + ∂3 T x T x − − − +∂ T44x5 T54x4 + ∂ T45x5 T55x4 + ∂ T46x5 T56x4 = 0 4 − 5 − 6 − Symmetry 2020, 12, 817 6 of 10 which simplify to the equations, R ∂ d5x T54x6 T64x5 = 0 4 − R 5 64 4 44 6 ∂4 d x T x T x = 0 (30) R − ∂ d5x T44x5 T54x4 = 0 4 − Here, the terms in each integrand are a product of an energy density (associated with one of 4 5 6 E1, E2, E3) and a time variable (one of x , x , x ). If we consider the ﬁrst equation then the units of measure for the ﬁrst term are,

5 5 54 3 2 6 [d x]MKS = m [T ]MKS = kg m− s− [x ]MKS = s (31) giving h 5 54 6 2 1 (d x (T ) x ]MKS = kg m s− . (32) We conclude that these have the same units of measure as the Planck constant. We note that the units of measure for the conserved quantity due to invariance under spatial rotations are also the same units of measure as the Planck constant and that the conserved quantity, for some non-scalar ﬁelds, has been associated with spin angular momentum [5].

4. Breaking Time Reversal Symmetry The spatial SU(2) SU(2) algebras in complexified SO(3,3) have the basic form × left chirality: { 1 (J + iK ), 1 (J + iK ), 1 (J + iK )} 2 1 a1 2 2 a2 2 3 a3 (33) right chirality: { 1 (J iK ), 1 (J iK ), 1 (J iK )} 2 1 − a1 2 2 − a2 2 3 − a3 where a = 1, 2, 3 and the two chiralities are related by a spatial parity transformation [2]. The temporal SU(2) SU(2) algebras have the basic form × first chirality: { 1 (T + iK ), 1 (T + iK ), 1 (T + iK )} 2 1 1b 2 2 2b 2 3 3b (34) second chirality: { 1 (T iK ), 1 (T iK ), 1 (T iK )} 2 1 − 1b 2 2 − 2b 2 3 − 3b where b = 1, 2, 3 and the two chiralities are related by a time reversal transformation. The two chiral parts of a spatial SU(2) SU(2) algebra are related by a spatial parity transformation × and so appear to be unaffected by breaking time reversal symmetry. The two chiral parts of a temporal SU(2) SU(2) algebra are related by a time reversal transformation. This suggests that breaking time × reversal symmetry affects the chiral properties of a temporal SU(2) SU(2) algebra. × 5. Symmetry between Algebras The special orthogonal Lie group SO(4) can be associated with the group of rotations in a four-dimensional Euclidean space [4]. The group has six generators, here labelled aj,bj (j = 1, 2, 3), and commutation relations: [aj, ak] = i jkm am [bj, bk] = i jkm am (35) [aj, bk] = i jkm bm where the indexes j, k, m = 1, 2, 3. The Lie group SO(3), associated with the group of rotations in three dimensions, has three generators, here labelled wj (j = 1, 2, 3), and commutation relations,

[wj, wk] = i jkm wm (36) Symmetry 2020, 12, 817 7 of 10 where the indexes j, k, m = 1, 2, 3. The direct product SO(3) SO(2) has four generators, here labelled × wj (j = 0, 1, 2, 3), and commutation relations,

[wj, wk] = i jkm wm [w0, wk] = 0 (37) where the indexes j, k, m = 1, 2, 3. We also note that SU(2) and SO(3) have the same Lie algebra, and that U(1) and SO(2) are isomorphic [5].

5.1. SO(3) SO(2) symmetry × The e spatial SU(2) SU(2) algebra might be represented in tabular form as, 1 × 1 1 1 1 1 1 (a1 + b1) (a2 + b2) (a3 + b3) (a1 b1) (a2 b2) (a3 b3) 2 2 2 2 − 2 − 2 − (38) 1 (J + iK ) 1 (J + iK ) 1 (J + iK ) 1 (J iK ) 1 (J iK ) 1 (J iK )} 2 1 11 2 2 12 2 3 13 2 1 − 11 2 2 − 12 2 3 − 13 where the a’s and b’s are the generic SO(4) labels given in (35). With a change of basis this becomes:

a a a b b b 1 2 3 1 2 3 (39) J1 J2 J3 iK11 iK12 iK13

This SO(4) contains four SO(3) subalgebras. There is a spatial SO(3) algebra:

w w w 1 2 3 (40) J1 J2 J3

Here, the w’s are the generic SO(3) labels given in (36). There are also three other SO(3) algebras:

w1 w2 w3 J iK iK 2 13 11 (41) J3 iK11 iK12 J1 iK12 iK13.

Additionally, the SO(4) commutes with a rotation generator, T , which will give us three SO(3) 1 × SO(2) algebras, w1 w2 w3 w0 J iK iK T 2 13 11 1 (42) J3 iK11 iK12 T1 J1 iK12 iK13 T1

1 where the w’s are the generic SO(3) SO(2) labels given in (37). Changing the basis to 2 (w1 w2) and 1 × ± 2 (w0 w3) yields ± 1 1 (w1 w2) (w0 w3) 2 ± 2 ± 1 1 (J2 iK13) (T1 iK11) 2 ± 2 ± (43) 1 (J iK ) 1 (T iK ) 2 3 ± 11 2 1 ± 12 1 (J iK ) 1 (T iK ) 2 1 ± 12 2 1 ± 13 If the columns are considered to be six component algebras then in horizontal form we have

1 1 1 1 (w1 w2) = { (J2 iK13), (J3 iK11), (J1 iK12)} 2 ± 2 ± 2 ± 2 ± (44) 1 (w w ) = { 1 (T iK ), 1 (T iK ), 1 (T iK )}. 2 0 ± 3 2 1 ± 11 2 1 ± 12 2 1 ± 13 Symmetry 2020, 12, 817 8 of 10

Rotating 1 (w w ) within the vector space of the SO(4) then gives 2 1 ± 2 1 1 1 1 (w w )0 = { (J iK ), (J iK ), (J K )}. (45) 2 1 ± 2 2 1 ± 11 2 2 ± 12 2 3 ± 13

We conclude that 1 (w w ) and 1 (w w ) are related by SO(3) SO(2) symmetry plus 2 1 ± 2 0 2 0 ± 3 × a rotation. Inspection shows that the 1 (w w ) algebra is the same as e algebra. This suggests that the 2 1 ± 2 0 1 e-family is related to another family of algebras by SO(3) SO(2) symmetry plus a rotation. This is × the n-family: n = { 1 (T iK ), 1 (T iK ), 1 (T iK )} 1 2 1 ± 11 2 1 ± 12 2 1 ± 13 n = { 1 (T iK ), 1 (T iK ), 1 (T iK )} (46) 2 2 2 ± 21 2 2 ± 22 2 2 ± 23 n = { 1 (T iK ), 1 (T iK ), 1 (T iK )}. 3 2 3 ± 31 2 3 ± 32 2 3 ± 33 These algebras are associated with three spatial dimensions, as indicated by the boost generators. The n-family members are not SU(2) SU(2) algebras. × 5.2. SO(4) Symmetry The members of the n-family are related by SO(4) symmetry. This can be illustrated by constructing an array of generators:

n1 n2 n3 1 1 1 m1 (T iK ) (T iK ) (T iK ) 2 1 ± 11 2 2 ± 21 2 3 ± 31 1 1 1 (47) m2 (T iK ) (T iK ) (T iK ) 2 1 ± 12 2 2 ± 22 2 3 ± 32 1 1 1 m3 (T iK ) (T iK ) (T iK ). 2 1 ± 13 2 2 ± 23 2 3 ± 33 Here, the rows are the m-family algebras which have SO(4) = SO(3) SO(3) symmetry, and the × columns are the n-family. We also note that the n1 algebra shares two of its components with each of m1, m2, and m3. This suggests that an n-family algebra might be described as a mixture of m-family components.

6. Conclusions This article has considered some of the mathematical properties and relationships associated with SU(2) SU(2) subalgebras in an O(3,3) space. In particular, we ﬁnd the following: × 1. The e-family members are the standard type of SU(2) SU(2) algebra, associated with three space × dimensions and one time dimension. 2. The e algebra is related to the n algebra by SU(2) U(1) symmetry, plus a rotation. 1 1 × 3. We can describe the n1 algebra as being a mixture of components from the three m-family algebras. 4. The m-family members are a second type of SU(2) SU(2) algebra, associated with one space × dimension and three time dimensions. 5. Breaking of time reversal symmetry aﬀects the chiral properties of the m-family algebras. 6. The units of measure of the conserved quantity due to invariance under temporal rotations are the same as those of the Planck constant.

Funding: This research received no external funding. Acknowledgments: I would like to thank Torsten Schoeneberg for his input. I would also like to thank the reviewers for their suggestions. Conﬂicts of Interest: The author declares no conﬂict of interest. Symmetry 2020, 12, 817 9 of 10

Appendix A. SO(3,3) Generators (Referenced in Section2) Time rotation generators:       0 0 0 0 0 0 0 0 i 0 0 0 0 i 0 0 0 0            −   0 0 i 0 0 0   0 0 0 0 0 0   i 0 0 0 0 0   −       0 i 0 0 0 0   i 0 0 0 0 0   0 0 0 0 0 0  =   =   =   T1   T2  −  T3    0 0 0 0 0 0   0 0 0 0 0 0   0 0 0 0 0 0         0 0 0 0 0 0   0 0 0 0 0 0   0 0 0 0 0 0         0 0 0 0 0 0   0 0 0 0 0 0   0 0 0 0 0 0 

Space rotation generators:       0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0              0 0 0 0 0 0   0 0 0 0 0 0   0 0 0 0 0 0         0 0 0 0 0 0   0 0 0 0 0 0   0 0 0 0 0 0  =   =   =   J1   J2   J3    0 0 0 0 0 0   0 0 0 0 0 i   0 0 0 0 i 0       −   0 0 0 0 0 i   0 0 0 0 0 0   0 0 0 i 0 0   −       0 0 0 0 i 0   0 0 0 i 0 0   0 0 0 0 0 0  − Boost generators:       0 0 0 i 0 0 0 0 0 0 i 0 0 0 0 0 0 i              0 0 0 0 0 0   0 0 0 0 0 0   0 0 0 0 0 0         0 0 0 0 0 0   0 0 0 0 0 0   0 0 0 0 0 0  =   =   =   K11   K12   K13    i 0 0 0 0 0   0 0 0 0 0 0   0 0 0 0 0 0         0 0 0 0 0 0   i 0 0 0 0 0   0 0 0 0 0 0         0 0 0 0 0 0   0 0 0 0 0 0   i 0 0 0 0 0 

      0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0              0 0 0 i 0 0   0 0 0 0 i 0   0 0 0 0 0 i         0 0 0 0 0 0   0 0 0 0 0 0   0 0 0 0 0 0  =   =   =   K21   K22   K23    0 i 0 0 0 0   0 0 0 0 0 0   0 0 0 0 0 0         0 0 0 0 0 0   0 i 0 0 0 0   0 0 0 0 0 0         0 0 0 0 0 0   0 0 0 0 0 0   0 i 0 0 0 0        0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0              0 0 0 0 0 0   0 0 0 0 0 0   0 0 0 0 0 0         0 0 0 i 0 0   0 0 0 0 i 0   0 0 0 0 0 i  =   =   =   K31   K32   K33    0 0 i 0 0 0   0 0 0 0 0 0   0 0 0 0 0 0         0 0 0 0 0 0   0 0 i 0 0 0   0 0 0 0 0 0         0 0 0 0 0 0   0 0 0 0 0 0   0 0 i 0 0 0 

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