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Home , Borel subgroup, Cartan subalgebra, Pauli matrices

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In his paper “Sur la dynamique de l’electron” from July 1905, Henri Poincar´eformulates the Principle of Relativity, introduces the concepts of Lorentz transformation and , and postulates the covariance of laws of Nature under Lorentz transformations [1]. In 1939, Eug´ene Wigner analysed the unitary representations of the inhomogeneous Lorentz group [2, 3]. However, there are also nonunitary representations. As Wigner pointed out, the irreducible representations of the Lorentz group are important because they define the types of particles. The Lorentz is expressed by introducing the commutative diagram for the covariant wave functions ψC ,

C C ψ : E p ψ (p) 1,3 ∋ −→ U(Λ) Λ T (Λ) (1) ↓ ↓ ↓ U(Λ)ψC :Λx Λp T (Λ)ψC(p) ∋ −→ implying (U(Λ)ψC )(Λp) = T (Λ)ψC(p). Here Lor Λ T (Λ) is the finite dimensional 1,3 ∋ → nonunitary representation. Let the vectors p,λ with four-momentum p and independent parameter λ form a | i complete orthogonal of the irreducible representation of the Poincar´egroup. In this basis, U(Λ) is always expressed by [4, 5]

U(Λ) = Q (W (Λ,p)) P (Λ), where Q is diagonal with respect to λ, P is diagonal with respect to p, and W (Λ,p) is the Thomas–Wigner . The transformation law for Wigner’s ψW reads

W −1 W −1 ′ (U(Λ)ψ )(p,λ)= Qλλ′ W (Λ, Λ p)ψ (Λ p,λ ) . (2) λ′ X  Here Q(W (Λ,p)) is a representation of Wigner’s little group as a subrepresentation of some Lorentz group representation T (Λ). The explicit relation between ψC and ψW is given in Refs. [6, 7, 8, 9, 10, 11, 12, 13, 14].

2 The great importance of Wigner’s theory is that the classification of particles according to their Lorentz transformation properties is entirely determined by the representation of

0 the little group as the subrepresentation of the representation of Lor1,3 = SO1,3 [15, 16]. The most important cases are

◦ p = (1, 0, 0, 0), (1, 0, 0, 1) ◦ Lg(p)= SO3 E(2) | {z } | {z } Bor1,3 In the title we use the word “nophysics”. This| word{z } can be taken as an abbreviation, namely nophysics = new old physics

where “old” stands for Pauli’s massless neutrino hypothesis, “new” for the solvability as symmetry applicable to massless particles. Another more detailed explanation is found in the Conclusions of this paper.

2 The little group

◦ ◦ The little group of the four-momentum p or the stabiliser of p is the maximal closed

subgroup of Lor1,3 defined as

◦ ◦ ◦ ◦ ◦ Lg p = Λ Lor : Λp = p . (3) { ∈ 1,3 }

◦

The orbit of p is a subspace in E1,3,

◦ ◦ ◦ Orb p = Λp :Λ Lor = Lor p, { ∈ 1,3} 1,3

◦ ◦ ◦ given as the bijection Orb p = Lor / Lg p. For all p Orb p the Thomas–Wigner rotation 1,3 ∈ is given by

−1 W (Λ,p)= LΛp ΛLp , (4)

◦ where L Lor is the representative of p Orb p. p ∈ 1,3 ∈ 3 ◦ The characteristic feature of the massive case is that the fixed vector p can be chosen ◦ to be p =(m,~0), for which ◦ locally Lg p = SO3 = SU2

(SU2 is the universal of SO3). Since SU2 is compact and simply con- nected, all its finite-dimensional irreducible representations are single-valued, unitary and parametrised by the eigenvalue s of the Casimir which can take on non-negative half-integer values 1 3 s =0, , 1, ,... 2 2 From the local equivalence, the equivalence of the Lie algebras can be derived, i.e.

◦ so3 = ad su2 = lg p.

It is important that the little group SO3 is the maximal compact simple subgroup of Lor1,3 ◦

while su2 (i.e. lg p) is the simplest semisimple , dimÊ su2 = 3. ◦ For every irreducible unitary representation of the little group Lg p = SO3 one can derive a corresponding induced representation of the Poincar´egroup, = ⋊ Lor . P1,3 T1,3 1,3 The irreducible representations of are characterised by the pairs (m, s), where the P1,3 1 mass m is real and positive and the takes on the values s = 0, 2 , 1,... The states within each irreducible representation are labelled by ξ = s, s +1,...,s which means − − that massive particles of spin s have 2s + 1 degrees of freedom. ◦ In the massless case m = 0 the representative vector p may be taken to be

◦ p =(ω , 0, 0,ω ), ω = ~p . (5) p p | |

To construct the little group, one has to solve the defining equation

◦ ◦ ◦ Λp = p. (6)

The result is the [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]

◦ Lg p = E(2) = ISO(2) = ⋊ SO(2) T2 4 for which the double covering group is given by

E(2) = ⋊ U(1) T2 where is the Abelian two-dimensional group of translations. Thus, the little group is T2 solvable and non-compact, and the restrictions of the finite-dimensional representations of

Lor1,3 to E(2) are in general non-unitary. In fact, the only unitary, irreducible representa- tion of E(2) are one-dimensional, i.e. degenerate representations, since the subgroup of T2 translations has to be realised trivially [4, 5],

E(2) E(2)/ = SO(2). (7) → T2

The requirement that the representations are at most double-valued implies that only the representations SO(2) U (j)(SO(2)), j =0, 1 , 1,... are allowed. This one-dimensional → ± 2 ± (internal) freedom of massless particles is usually called the helicity. Since all the unitary representations on the orbits p2 = 0 are induced by the non-faithful one-dimensional repre- sentation of the little group E(2), massless particles are characterised by a discrete helicity

1 λ =0, , 1,... (8) ±2 ±

Notice that if the parity is included, the helicity takes on two values, λ and λ. For − example, the two states λ = 1 are then referred to as left-handed (λ = 1) and right- ± − handed (λ = +1) photons.

2.1 The

◦ Due to the unitarity of representations of the little group Lg p = E(2), zero-mass particles have only a single value for the helicity (if the parity is not taken into account). Suppose, the most general determined, relativistically invariant, first order single particle equation is of the form

µ (β ∂µ + ρ) ψ(x)=0. (9)

5 Then there is a simple criterion by D. Kwoh under which this equation will have zero-mass solutions [17],

Kwoh’s lemma: A necessary condition that Eq. (9) has a zero-mass solution is that

det( iβµp + λρ)=0 (10) − µ

for all real λ and all light-like p, i.e. all p such that p2 = 0. If Eq. (9) is a defining equation for a single massless particle, Kwoh’s lemma states the gauge invariance p λp as a very special property of the theory. Therefore, it seems to → be reasonable (at least mathematically) to include this gauge transformation into the little group, i.e. instead of Eq. (6) take ◦ ◦ ◦ Λp = λp (11)

◦ as defining equation for the little group, where p = (ε, 0, 0, 1) with ε = 1, λ > 0 and ± ◦ ◦ ◦ Λ Lor . If Λ = exp( 1 ωµνe ), Eq. (11) yields ωµ pν = δλpµ or more explicitly ∈ 1,3 − 2 µν ν

ω = δλε, εω ω =0, εω ω =0 03 01 − 13 02 − 23

(cf. Sec. A.2). Notice that in case of E(2) as little group one has δλ = 0. The solution of Eq. (11) reads

1 ( 1 ξ2 + λ + 1 ) εξ~ T rot ω ε ( 1 ξ2 + λ 1 ) 2 λ λ 2 − λ − λ ◦ Λ= B(ε)(ξ~; λ,ω)=  ε ξ~ rot ω 1 ξ~  , (12) λ − λ  ε ( 1 ξ2 + λ 1 ) ξ~T rot ω 1 ( 1 ξ2 + λ + 1 )   2 λ λ 2 λ λ   − −  where ξ1 cos ω sin ω ~ − 2 2 2 ξ = , rot ω = , ξ = ξ1 + ξ2. ξ2 ! sin ω cos ω ! If one now defines 1 (λ + 1 ) ~0 T ε (λ 1 ) 2 λ 2 − λ (ε) (ε)

~ ½ ~ Bλ = B (0; λ, 0) =  0 2 0  ,  ε (λ 1 ) ~0 T 1 (λ + 1 )   2 λ 2 λ   −  6 1 ~0 T 0

(ε) Rω = B (0;1,ω) = ~0 rot ω ~0  ,  0 ~0 T 1    1+ 1 ξ2 εξ~ T ε ξ2 2 − 2 (ε) (ε) ~ ~ ~ T = B (ξ;1, 0) =  εξ ½2 ξ  , (13) ξ −  ε ξ2 ξ~ T 1 1 ξ2   2 2   −  the general transformation (12) can be written as

(ε) ~ (ε) (ε) B (ξ; λ,ω)= Tξ Bλ Rω.

One easily obtains the multiplication table

(ε) (ε) (ε) (ε) (ε) Bµ Bλ = Bµλ = Bλ Bµ ,

RφRω = Rφ+ω = RωRφ,

(ε) (ε) (ε) (ε) (ε) Tη Tξ = Tη+ξ = Tξ Tη ,

(ε) (ε) (ε) (ε) Bλ Tξ = Tλξ Bλ ,

(ε) (ε) RωTξ = Trot ωξRω,

B(ε)(ξ~; λ,ω)B(ε)(~η; µ,ϕ) = B(ε)(ξ~ + λ rot ω~η; λµ,ω + ϕ),

1 1 (B(ε)(ξ~; λ,ω))−1 = B(ε)( rot( ω)ξ~; , ω). (14) −λ − λ −

From the multiplication table (14) it follows that the transformations B(ε)(ξ~; λ,ω) form a group Bor(ε) Lor with non-compact parameter space 1,3 ⊂ 1,3 ~ ξ Ê , 0 ω π, λ> 0 . { ∈ 2 ≤ ≤ }

It can easily be shown that the derived series of for Bor(ε) ends in the D 1,3 identity id. In fact, 2(Bor(ε))= id . D 1,3 { } 7 (ε) Actually, Bor1,3 is a maximal, solvable and non-compact subgroup of Lor1,3, i.e. the non- compact Borel subgroup of Lor1,3. Moreover, one obtains the Borel decomposition as the semidirect product (ε) Bor(ε) = (ε) ⋊ Tor(ε) . (15) B ≡ 1,3 T2 1,3 (ε) (ε) ~ (ε) The set = Gen T : ξ Ê of elements of Bor is a closed nilpotent T2 { ξ ∈ 2} 1,3 subgroup of Bor(ε). It contains the subgroup (Bor(ε)) = (Bor(ε), Bor(ε)) generated by the 1,3 D 1,3 1,3 1,3 commutators and is normal in Bor(ε). Tor(ε) = Bor(ε) / (ε) = Gen B(ε), R :Λ > 0; 0 1,3 1,3 1,3 T2 { λ ω ≤ ω π is the in Bor(ε) (and in Lor ) with dimension dim(Bor(ε) / (ε)), ≤ } 1,3 1,3 1,3 T2 (ε) generated by the semisimple elements Bλ and Rω. By the Lie–Kolchin theorem as it is (ε) written down later, Bor1,3 is upper triangular [18, 19, 20, 21]. At this point a remark of S. Weinberg is in order [23]: “For the case of zero mass there are interesting complications. The little group as Wigner pointed out is a non-semi- simple group, and one must make special remarks about its invariant Abelian subalgebra.” Indeed, ◦ Lg p Abelian ⋊ Abelian . ∼ 2 2 is the semidirect product (15) of two two-parametric Abelian groups.

2.2 Jordan factorisation

The Jordan factorisation of M GL (Ê) into a semisimple and an unipotent component ∈ 4 is given by

M = MuMs.

8 Since Bor1,3 is solvable, according to the Lie–Kolchin theorem a basis can be chosen with respect to which B Bor can be put into a triangular form ∈ 1,3 1 λ 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0  0   0 1   0 λ1 0 0  B = ∗ ∗ ∗ = ∗ ∗ = BuBs  0 0   0 0 1   0 0 λ 0       2   ∗ ∗   ∗           0 0 0   0 0 0 1   0 0 0 λ3   ∗      where Bu is unipotent (i.e. all eigenvalues of Bu are 1) and Bs = diag(λ0,λ1,λ2,λ3) is

semisimple. The eigenvalues of B and Bs are identical. In this form, the Jordan decom-

positon is given by

⋊ Ê Ê Bor U (Ê) D ( ) T ( ),

1,3 ⊂ 4 4 ≡ 4 Ê where U4(Ê) is the group of upper triangular unipotent matrices and D4( ) is the group of invertible diagonal matrices. If B is solvable, then [24]

λ λ 0 0 0 −1 0 ∗ ∗ ∗ 0  0 λ1   0 λ1 0 0  B = ∗ ∗ Bu = B  0 0 λ  ⇒  0 0 λ 0   2   2   ∗         0 0 0 λ3   0 0 0 λ3      and det B = 1. From Refs. [18, 19, 20, 21, 22] we take the following set of well-known theorems, lemmas and propositions.

Theorem: Let G be a connected linear . Then G contains a Borel subgroup , and all other Borel subgroups of G are conjugate to . The homogeneous space G/ is B B B a projective variety ([20], p. 524).

Lie–Kolchin theorem: If π : G GL(V ) is a linear representation of a connected → , then π(G) leaves a flag in V invariant, i.e. π(G) can be put in triangular form (see e.g. [19], p. 406).

Borel Fixpoint Theorem: Let S be a connected, solvable group that acts algebraically

9 on a projective variety X. Then there exists a point x X such that ([18], p. 137) ∈

Λx = x for all Λ S. ∈

Therefore, Eq. (11) is reasonable, as the semi-invariant of Bor1,3 in E1,3 is the non-zero

◦ ◦ Ê vector p spanning the -stable line in E (x = p). B 1,3

Lemma: Let V be a C- of dimension n > 0 and S a connected, solvable subgroup of GL(V ). Then there exists a vector v V 0 such that ([19], p. 407) ∈ \{ }

Sv = Cv.

Let denote the set of semisimple elements of and the set of unipotent elements. Bs B Bu

Proposition: If is connected and solvable, then the set is a closed, connected and B Bu nilpotent subgroup of , containing (G) and, therefore, is normal in ([19] p. 407). B D B From this it follows that the set is not a closed subgroup of because if it would Bs B be a subgroup, would be nilpotent. B

Proposition: Let be connected and solvable and let b ( ) be its Lie algebra. Then B ≡L B the set ( ) is the set of nilpotent elements of g (i.e. ad u is nilpotent for u ( ) ([19], L Bu g ∈L Bu p. 409).

It is important that for solvable the set of unipotent elements of is a connected, B Bu B closed, normal and nilpotent subgroup of , / is a torus, and ( ) . B B Bu D B ⊂ Bu

Theorem (Borel): Let be a connected, solvable, . If Tor is a B maximal torus of , then B = ⋊ Tor . B Bu Otherwise, there exists such a torus Tor such that ⊂ B

= Rad ( ) ⋊ Tor, B u B 10 where Rad ( ) is the unipotent radical of , leading to the factorisation (15) (this theorem u B B can be found in the original work Ref. [18], p. 137 as well as in Ref. [19], p. 410). The generators of the Borel subgroup in the representation (13) are defined by

(ε) (ε) ∂B (ωµ) bµ = ◦, (16) ∂ωµ ω

◦ where ω0 = λ, ω1 = ξ1, ω2 = ξ2, ω3 = ω and ω = (1 , 0, 0, 0). This yields the Lie algebra (ε) bor1,3 as basis for the underlying vector space to be generated by

(ε) (ε) ∂Bλ b0 = = εe03, ∂λ λ=1

(ε) ∂Tξ b1 = = εe01 + e31, ∂ξ1 ξ~=~0

(ε) ∂Tξ b2 = = εe02 + e32, ∂ξ2 ξ~=~0

(ε) ∂Rω b3 = = e21. (16) ∂ω ω=0

The relations are (a, b 1, 2 ) [25, 26, 27, 28, 29] ∈{ }

(ε) (ε) (ε) (ε) (ε) [b0 , ba ]= ba , [b0 , b3 ]=0,

[b(ε), b(ε)]= ǫ b(ε), [b(ε), b(ε)]=0. (17) 3 a − 3ab b a b

The algebra bor1,3 is solvable because

[ (bor(ε)), (bor(ε))] = 2(bor(ε))= 0 D 1,3 D 1,3 D 1,3 { }

and maximal in Lor1,3, i.e. the Borel algebra of lor1,3. Moreover,

(ε) (ε) ⋊ (ε) bor1,3 = t2 tor1,3, (18)

t(ε) ~t (ε) (ε) (ε)

where the vector space underlying is = spanÊ b , b and that of tor = car 2 2 { 1 2 } 1,3 1,3 ~ (ε) (ε) (ε)

is tor = spanÊ b , b (car is the of lor ). Therefore, one 1,3 { 0 3 } 1,3 1,3 (ε) can conclude that in the massless case the (enlarged) little algebra bor1,3 is a maximal, non-compact and solvable Lie subalgebra of lor1,3, i.e. the , and is the

11 (ε) (ε) semidirect sum of two abelian algebras t2 and tor1,3. Notice that there exists no Casimir operator. In the general case [18, 19, 20, 21], if is solvable, its Lie algebra b = ( ) is solvable. B L B Since is normal in , its Lie algebra n = ( ) is an ideal of b and n is the set of Bu B L Bu nilpotent elements of b. Moreover, since ( ) one has [b, b] n. D B ⊂ Bu ⊂

Theorem: There exists a Lie subalgebra a of b obeying the conditions ([19], p. 410)

1. a is abelian and all its elements are semisimple, i.e. a b ; ⊂ s

2. as a vector space one has b = n a. ⊕

Theorem: Let b be algebraic and solvable in gl(V ) and let n be the set of nilpotent endomorphisms of V in b. If h is the maximal commutative Lie subalgebra of b consisting of semisimple elements, b is the semidirect product of h with n ([19], p. 455),

b = n ⋊ h.

The existence of Borel algebras follows from the trianguar decomposition of the g [18, 19, 20, 21, 30, 31, 32],

g = n h n + ⊕ ⊕ − where

n+ = gα, n− = g−α + + αX∈φ αX∈φ and h being the Cartan subalgebra of g. Then b = h n are maximal solvable Lie ± ⊕ ± subalgebras of g, called the Borel subalgebra of g relative to h. Moreover,

n (b )= b and z (b )= 0 g ± ± g ± { } where ng is the normaliser and zg is the centraliser of b± in g.

12 2.3 Generators of the Borel subgroup

It can be easily seen that the exponential operation provides the parametrisation of the generic element B(ε)(ξ~; λ,ω) Bor , ∈ 1,3

(ε) ˜ (ε) ˜ (ε) ˜ (ε) 2 λ˜ B λ = exp(λb ) = ½ + sinh λb + (cosh λ 1)(b ) (λ = e ), e 0 4 0 − 0 (ε) ~~(ε) ~~(ε) 1 ~~(ε) 2 T = exp(ξb ) = ½ + ξb + (ξb ) = ξ 4 2 (ε) (ε) 1 2 (ε) 2 1 2 (ε) 2

= ½ + ξ b + ξ b + ξ (b ) + ξ (b ) , 4 1 1 2 2 2 1 1 2 2 2 (ε) (ε) (ε) 2

R = exp(ωb ) = ½ + sin ωb + (1 cos θ)(b ) , (19) ω 3 4 3 − 3

(ε) The general element of bor1,3 can be written as

0 εy1 εy2 εy0 εy1 0 y3 y1 µ   Y = y bµ = − − . (20)  εy2 y3 0 y2     −   0 1 2   εy y y 0    The ad Y in the basis b , b , b , b of the semidirect sum ⋊ is { 1 2 0 3} calculated to be y0 y3 y1 y2 − −  y3 y0 y2 y1  ad Y (= Reg Y )= − − . (21)  00 0 0         00 0 0    From the secular equation det(ad Y λ)=( λ)2((y0 λ)2 +(y3)2) it follows that − − −

spec(ad Y )= 0, 0,λ = y0 iy3,λ = y0 + iy3 . (22) { 3 − 4 }

The eigenvalue problem yields two eigenfunctions Z3 and Z4,

i ad Y (Z )= λ Z Z = (b(ε) + ib(ε)), 3 3 3 ⇒ 3 2 1 2 i ad Y (Z )= λ Z Z = (b(ε) ib(ε)), (23) 4 4 4 ⇒ 4 −2 1 − 2

13 i.e. Z ,Z t(ε). In the special case Y = y0b(ε) + y3b(ε) one obtains 3 4 ∈ 2 0 3

[y0b + y3b ,Z ] = (y0 iy3)Z , 0 3 3 − 3 [y0b + y3b ,Z ] = (y0 + iy3)Z . (24) 0 3 4 − 4

Two cases for Y are important, namely t(ε) = 1 (b(ε) + ib(ε)) and u(ε) = 1 (b(ε) ib(ε)). 0 2 0 3 0 2 0 − 3 (ε) (ε) Combining in pairs with t+ = Z3 and u+ = Z4,

1 i t(ε) = (b(ε) + ib(ε)), t(ε) = (b(ε) + ib(ε)), 0 2 0 3 + 2 1 2 1 i u(ε) = (b(ε) ib(ε)), u(ε) = (b(ε) ib(ε)), (25) 0 2 0 − 3 + −2 1 − 2

one obtains the commutation relations

(ε) (ε) (ε) (ε) (ε) (ε) (ε) (ε) [t0 , t+ ]= t+ , [u0 ,u+ ]= u+ , [t0,+,u0,+]=0. (26)

The elements

1 1 t(ε) = (b(ε) + ib(ε)) = (εe + ie ) = iJ (ε), (27) 0 2 0 3 2 03 21 − 3 i 1 t(ε) = (b(ε) + ib(ε)) = (iεe εe + ie e ) = J (ε) + 2 1 2 2 01 − 02 31 − 32 +

(ε) generate the minimal solvable algebra sol2 (t) with underlying vector space given by (ε) (ε)

span Ê t , t . Similarly, { 0 + } 1 1 u(ε) = (b(ε) ib(ε)) = (εe ie ) = iK(ε), (28) 0 2 0 − 3 2 03 − 21 3 i 1 u(ε) = (b(ε) ib(ε)) = (iεe + εe + ie + e ) = K(ε) + −2 1 − 2 −2 01 02 31 32 −

(ε) (ε) (ε) generate the algebra sol2 (u), and since [sol2 (t), sol2 (u)] = 0, one obtains the decompo- sition (ε)∗ (ε) ⊞ (ε)

bor1,3 = sol2 (t) sol2 (u), (29) ½ where ⊞ is the Kronecker sum, A ⊞ B = A ½ + B. ⊗ ⊗

14 3 Representations

Every representation of lor1,3 defines a particular representation of the subalgebra bor1,3.

Of course, not all the representations are of that kind but those defined by lor1,3 are of great importance because the classification of particles is determined by their Lorentz transformation properties according to Eqs. (1) and (2). More precisely, the common eigenvectors of the representation space of the solvable algebra bor1,3 are the possible helicity states of the particle.

Theorem: Let g be a solvable algebra and g Γ(g) be a representation on a finite- → dimensional vector space V . Then ([22], p. 200)

1. there exists a vector v V which is a simultaneous eigenvector for all of Γ(g), ∈ 2. there exists a basis of V with respect to which all elements of Γ(g) are represented by upper triangular matrices.

Notice that the common eigenvector is determined by all the elements of Γ(g), i.e. in our

(ε) case g = bor1,3 there is no need to assume Γ(t2 ) = 0. In the complex spaces

ρ ρ 3

span C e : e = η = δ , { (µ) (µ) µ ρµ}0 (ε) the eigenvectors of the solvable algebra sol2 (t) are

(ε) T T ℓ0 = εe(0) + e(3) =(ε, 0, 0, 1) , ℓ1 = e(1) + ie(2) = (0, 1, i, 0) .

Indeed, 1 t(ε)ℓ(ε) = ℓ(ε), t(ε)ℓ(ε) =0, 0 0 2 0 + 0 1 t(ε)ℓ = ℓ , t(ε)ℓ =0. (30i) 0 1 2 1 + 1 Accordingly, the eigenvectors of sol (u) are ℓ(ε) and ℓ = e ie = (0, 1, i, 0)T , where 2 0 2 (1) − (2) − 1 u(ε)ℓ(ε) = ℓ(ε), u(ε)ℓ(ε) =0, 0 0 2 0 + 0 1 u(ε)ℓ = ℓ , u(ε)ℓ =0. (30ii) 0 2 2 2 + 2 15 Since

1 u(ε)ℓ = ℓ , u(ε)ℓ = iℓ(ε), 0 1 −2 1 + 1 − 0 1 t(ε)ℓ = ℓ , t(ε)ℓ = iℓ(ε) 0 2 −2 2 + 2 0 ∗ ⊞ (ε)

and bor = sol (t) sol (u), the subspace spanC ℓ ,ℓ ,ℓ is invariant under the action 1,3 2 2 { 0 1 2} (ε) of bor1,3. However, the vector ℓ0 is already the defining vector for bor1,3 (cf. Eq. (11). (ε) Therefore, there are two helicity states ℓ1 and ℓ2 relative to ℓ0 . More precisely, using the two components D~ and B~ of the Lorentz group defined in Appendix A, the defining Eq. (11) yields the conditions

(ε) (ε) (ε) D3ℓ0 =0, B3ℓ0 =1ℓ0 , (30iii)

(ε) ℓ1 is called right-handed with respect to ℓ0 , and ℓ2 is called left-handed with respect to (ε) ℓ0 . The value 1 in Eq. (30iii) may be considered as helicity 1 (not spin because there is no rotation SO3). The conditions (30iii) are equivalent to 1 t(ε)ℓ(ε) = u(ε)ℓ(ε) = ℓ(ε), t(ε)ℓ(ε) = u(ε)ℓ(ε) =0. 0 0 0 0 2 0 + 0 + 0

3.1 The Chevalley basis

Depending to the sign of ε = 1, the elements of sol (t) and sol (u) can be represented by ± 2 2 elements of the fundamental sl2 (or Chevalley) basis 1 0 0 1 0 0 h = , e = , f = . 0 1 ! 0 0 ! 1 0 ! − (ε) (ε) According to the commutator relations, for ε = +1 one can represent t0 and u0 by εih/2, t(ε) and u(ε) by ie and if – or vice versa for ε = 1 (cf. Eqs. (A37,A38)). The two

∓ + + − Ê solvable algebras sol (e) = span Ê h, ie and sol (f) = span h, if are representations of 2 { } 2 { } the algebras sol2(t) and sol2(u), respectively, 1+ ε 1 ε 1+ ε 1 ε t(ε) = h ⊞ − h = iJ (ε), t(ε) = ie ⊞ − ie = J (ε), 0 4 2 − 3 + 2 2 +         16 1 ε 1+ ε 1 ε 1+ ε u(ε) = − h ⊞ h = iK(ε), u(ε) = − if ⊞ if = K(ε). 0 − 4 4 3 + 2 2 −         (31)

This can be generalised to arbitrary representations as for instance the subrepresentations of the representation (k,l) of lor1,3. The general representation (k,l) is given by the rule

π(k,l) : bor(ε)∗ = sol(ε)(e) ⊞ sol(ε)(f) 1,3 2 2 → π(k,l)(bor(ε)∗)= π(k)(sol(ε)(e)) ⊞ π(l)(sol(ε)(f)), (32) → 1,3 2 2 where

1+ ε 1 ε 1 ε 1+ ε sol(ε)(e)= sol (e) ⊞ − sol (e), sol(ε)(f)= − sol (f) ⊞ sol (f). (33) 2 2 2 2 2 2 2 2 2 2

By virtue of this construction one obtains

i(1 + ε) i(1 ε) π(k,l)(t(ε)) = π(k)(h) ⊞ − π(l)(h) = 0 − 4 4   i (k) 2 π (h) ½l for ε = +1, = − ⊗ i (l) ( ½ π (h) for ε = 1, − k ⊗ 2 − i(1 + ε) i(1 ε) π(k,l)(t(ε)) = π(k)(e) ⊞ − π(l)(e) = + 2 2 (k) iπ (e) ½l for ε = +1, = ⊗ (l) ( ½ iπ (e) for ε = 1, k ⊗ − i(1 ε) i(1 + ε) π(k,l)(u(ε)) = − π(k)(h) ⊞ π(l)(h) = 0 4 4 i (l) ½k 2 π (h) for ε = +1, = ⊗ i (k) ( π (h) ½ for ε = 1, 2 ⊗ l − i(1 ε) i(1 + ε) π(k,l)(u(ε)) = − π(k)(f) ⊞ π(l)(f) = + 2 2 (l) ½k iπ (f) for ε = +1, = ⊗ (34) (k) ( iπ (f) ½ for ε = 1, ⊗ l − where

π(k)(h) k, m = 2m k, m , | i | i 17 π(k)(e) k, m = ρ(k) k, m +1 , | i (m)| i π(k)(f) k, m = ρ(k) k, m 1 , (35) | i (−m)| − i where ρ(k) = (k m)(k + m + 1). In the representation of π(k,l) by direct products, (m) − p k,l; m , m = k, m l, m : k m k; l m l {| k li | ki⊗| li − ≤ k ≤ − ≤ l ≤ } one obtains common eigenvectors for sol2(t) given by

π(k,l)(t(+)) k,l; k, m = k k,l; k, m , π(k,l)(t(+)) k,l; k, m =0 (36) 0 | li | li + | li with m = l, l +1,...,l, and common eigenvectors for sol (u) given by l − − 2 π(k,l)(u(+)) k,l; m , l = l k,l; m , l , π(k,l)(u(+)) k,l; m , l =0 (37) 0 | k − i | k − i + | k − i with m = k, k +1,...,k. k − −

3.2 Resolution of the solvable group

sol2(e) is a Lie algebra of orientation-conserving affine translations Aff1. The underlying Ê topological space Ê for the corresponding Sol (e) is simply connected and × + 2 2 open in the plane Ê . As a general element of this group can be represented by eα β S(β,α)= 0 1 !

(α, β Ê), the geometrical space on which this group acts is the real line, ∈

1 α 1 Ê Sol (e) S(β,α): Ê x e x + β . 2 ∋ ∋ → ∈ Because of ′ eα+α eαβ′ + β S(β,α)S(β′,α′)= = S(eαβ′ + β,α + α′), 0 1 !

the solvable group is a semidirect product of two abelian groups, ⋊ Ê Sol2(e)= Ê +.

Therefore, Sol2 is

18 1. locally compact

2. simply connected

3. minimal non-abelian

4. non-compact

5. non-semisimple, i.e. solvable,

6. non-unimodular.

1

Via the exponential mapping, these two parts are generated by e and h+ = 2 (½ + h). ⋊ Ê Therefore, one can write sol2(e) = Êe h+ A similar consideration leads to sol2(f) =

1

⋊ Ê ½ Êf h where h = ( h). Moreover, sol (e) and sol (f) are related to the Cartan − − 2 − 2 2 involution. Therefore, we end up with the amusing two-fold decomposition

∗

⋊ Ê ⊞ Ê ⋊ Ê bor1,3 =(Êe h+) ( f h−)

where 1 0 0 1 0 0 0 0 h+ = , e = , f = , h− = . 0 0 ! 0 0 ! 1 0 ! 0 1 ! It is important to note that the Kronecker sum structure is imposed by semisimplicity

because bor1,3 is a of so4(C), while the semidirect product structure is caused by solvability.

3.3 Weinberg’s ansatz

From Steven Weinberg we adopt the following statements which we subsume under the keyword of “Weinberg’s Ansatz” [4, 5]:

1. If a massless particle is equal to it’s antiparticle, it is described by the irreducible representation (k,k) of the proper Lorentz group.

19 2. If a massless particle is not equal to it’s antiparticle, the particle is described by the irreducible representation (k, 0) of the proper Lorentz group, the antiparticle is described by the irreducible representation (0,k) of the proper Lorentz group.

Note that the massless particle is defined via the Borel subgroup by the irreducible repre- sentation of the proper Lorentz group without necessity to introduce parity separately. For the representation (k,k), from Eq. (36) one obtains the helicity states associated

(+) with sol2 (e), k,k; k, k + p , p =0, 1, 2,..., 2k, | − i (+) and from Eq. (37) one obtains the helicity states associated with sol2 (f),

k,k; k p, k , p =0, 1, 2,..., 2k. | − − i

However, the condition (30iii) excludes the state k,k; k, k , since | − i

D(k,k) k,k; k, k = i(k k) k,k; k, k =0, 3 | − i − | − i B(k,k) k,k; k, k = 2k k,k; k, k . 3 | − i | − i

Therefore, for the particle (ε = +1) with zero mass and helicity λ =2k one obtains the 4k helicity states

k,k; k, k + p , k,k; k p, k , p =1, 2,..., 2k (38) | − i | − − i relative to the central state k,k; k, k which is determined by the conditions | − i

t(+) k,k; k, k = u(+) k,k; k, k = k k,k; k, k , 0 | − i 0 | − i | − i t(+) k,k; k, k = u(+) k,k; k, k = 0. (39) + | − i + | − i

On setting ε = 1 in Eq. (34), the helicity states become −

k,k; k + p,k , k, k, k,k p , p =1, 2,..., 2k | − i | − − i

20 The central state k,k; k,k is defined by the conditions | − i

t(−) k,k; k,k = u(−) k,k; k,k = k k,k; k,k , 0 | − i 0 | − i | − i t(−) k,k; k,k = u(−) k,k; k,k = 0. (40) + | − i + | − i

1 1 Notice that for the vector case ( 2 , 2 ) and ε = +1 the helicity states are given by

1 , 1 ; 1 , 1 , 1 , 1 ; 1 , 1 | 2 2 2 2 i | 2 2 − 2 − 2 i relative to the central state 1 , 1 ; 1 , 1 . For ε = 1 the helicity states are the same, | 2 2 2 − 2 i − but now relative to the central state 1 , 1 ; 1 , 1 . Therefore, the two polarisations of the | 2 2 − 2 2 i photon as basic quantities in physics are determined by the proper Lorentz group Lor1,3 without taking refuge to the parity. As a further example, the massless particle of helicity λ = 2 is associated with the representation (1, 1) and has 4 helicity states relative to the state 1, 1;1, 1 : two right- | − i handed states 1, 1;1, 1 , 1, 1;1, 0 and two left-handed states 1, 1;0, 1 , 1, 1; 1, 1 . | i | i | − i | − − i

3.4 Weyl equations

1 1 1 The Weyl equations are of the type ( 2 , 0) and (0, 2 ). The representation ( 2 , 0) is defined by the commutative diagram [33, 34, 35]

µ µ µ ν Lor Λ: E p (Λp) =Λ p 1,3 ∋ 1,3 ∋ −→ ν σ σ ↓ ↓ ↓

µ † À SL (C) A : σ(p)= σ p A σ(p)A 2 ∋ ± Λ 2 ∈ µ −→ Λ Λ where the commutativity of the diagram results in

† µ ν AΛσ(p)AΛ = σ(Λp)= σµΛ νp .

Usingσ ˜ =(½; ~σ), one obtains − µν Λ σµσ˜ν µ 1 † AΛ = † , ΛAν = tr(σµAΛσνAΛ). (41) 2tr(AΛ) 2

21 Using the exponential representations Λ = exp 1 ωµνe and A = exp 1 ωµνm , − 2 µν Λ − 2 µν one obtains a relation between the generators,  

i j mkl = ǫkl σj 1 αβ 2 mµν = (eµν ) σασ˜β = 4 1 ( m0j = 2 σj One can obtain the transformation rule of the Weyl by looking at the commutative diagram

ψ : E x ψ (x) R 1,3 ∋ −→ R U(Λ) Λ A ↓ ↓ ↓ Λ

U(Λ)ψ : E Λx (U(Λ)ψ )(Λx) R 1,3 ∋ −→ R (1/2,0) implying (U(Λ)ψR)(Λx)= T ψR(x)= AΛψR(x). The

µ σ˜µ∂ ψR(x)=0

µ ◦ T is given in momentum space byσ ˜µp ψR(p) = 0. For the standard vector p = (ε, 0, 0, 1) this equation reduces to

(ε) (ε) σ3ℓR = εℓR , (42)

having the solutions ℓ(ε) = (1+ ε)aℓ + (1 ε)bℓ with ℓ = (1, 0)T and ℓ = (0, 1)T . The R 1 − 2 1 2 (ε) 1 Borel algebra bor1,3( 2 , 0) can be expressed as ε 1+ ε 1 ε b(ε)( 1 , 0) = h, b(ε)( 1 , 0) = e − f, 0 2 2 1 2 2 − 2 i i(1 + ε) i(1 ε) b(ε)( 1 , 0) = h, b(ε)( 1 , 0) = e − f. (43) 3 2 −2 2 2 − 2 − 2

(ε) (ε) The algebras sol2 (e) and sol2 (f) have the form 1+ ε i(1 + ε) sol(ε)(e) = t(ε)( 1 , 0) = h, t(ε)( 1 , 0) = e , 2 0 2 4 + 2 2   1 ε i(1 ε) sol(ε)(f) = u(ε)( 1 , 0) = − h, u(ε)( 1 , 0) = − f . 2 0 2 − 4 + 2 2   (ε) T The common eigenvector for sol2 (e) is ℓ1 = (1, 0) , 1+ ε t(ε)( 1 , 0)ℓ = ℓ , t(ε)( 1 , 0)ℓ =0, 0 2 1 4 1 + 2 1 22 (ε) T and for sol2 (f) one ontains ℓ2 = (0, 1) , 1 ε u(ε)( 1 , 0)ℓ = − ℓ , u(ε)( 1 , 0)ℓ =0. 0 2 2 − 4 2 + 2 2

(ε) 1 Therefore, the eigenvector of bor1,3( 2 , 0) is exactly equal to the solution of Weyl’s equation,

where ℓ1 is right handed and ℓ2 is left handed. Notice that in case of the irreducible

1 representation ( 2 , 0) of the proper Lorentz group there exists only one single solution, i.e. 1 one helicity state λ = 2 . More generally, the representation space of the Lorentz representation (k, 0) is given by

(+)

V (k, 0) = span C k, 0; m, 0 : m = k, k +1,...,k . {| i − − }

(+) (+) The action of bor1,3 on V (k, 0) can be written as

t(+) k, 0; m, 0 = m k, 0; m, 0 , u(+) k, 0; m, 0 =0, 0 | i | i 0 | i t(+) k, 0; m, 0 = iρ(k) k, 0; m +1, 0 , u(+) k, 0; m, 0 =0. + | i (m)| i + | i

Therefore, there exists only a single eigenvector k, 0; k, 0 V (+)(k, 0) of the Borel algebra | i ∈ (+) bor1,3 (k, 0), i.e. one a single helicity state with

t(+) k, 0; k, 0 = k k, 0; k, 0 , 0 | i | i t(+) k, 0; k, 0 = 0, + | i u(+) k, 0; k, 0 = u(+) k, 0; k, 0 = 0. 0 | i + | i

For ε = 1 the single helicity state can be written as k, 0; k, 0 with − | − i

t(−) k, 0; k, 0 = t(−) k, 0; k, 0 = 0, 0 | − i + | − i u(−) k, 0; k, 0 = k k, 0; k, 0 , 0 | − i | − i u(−) k, 0; k, 0 = 0. + | − i

In the irreducible case (0,k) (and ε = +1) the representation space reads

(+)

V (0,k) = span C 0,k;0, m : m = k, k +1,...,k , {| i − − } 23 (+) and the action of bor1,3 (0,k) is given by

t(+) 0,k;0, m =0, u(+) 0,k;0, m = m 0,k;0, m , 0 | i 0 | i − | i t(+) 0,k;0, m =0, u(+) 0,k;0, m = iρ(k) 0,k;0, m 1 . + | i + | i (−m)| − i

The only eigenvector of bor(+)(0,k) is 0,k;0, k with 1,3 | − i

t(+) 0,k;0, k = t(+) 0,k;0, k = 0, 0 | − i + | − i u(+) 0,k;0, k = k 0,k;0, k , 0 | − i | − i u(+) 0,k;0, k = 0. + | − i

For ε = 1 one obtains the action −

t(−) 0,k;0, m = m 0,k;0, m , u(−) 0,k;0, m =0, 0 | i | i 0 | i t(−) 0,k;0, m = iρ(k) 0,k;0, m +1 , u(−) 0,k;0, m =0 + | i (m)| i + | i and the only eigenvector 0,k;0,k . | i

4 Conclusions

Turning back to the title of our paper, “no” in “nophysics” can also stand for

no semisimple (instead, solvable) solution, •

no abelian (instead, the minimal non-abelian) solution, •

no compact (but locally compact) solution, •

no unimodular (instead, non-unimodular) solution, •

no (because the Cartan–Killing metric • is identically zero on the derived algebra), and

no Casimir invariant found. • 24 If semisimplicity, as we are being used to it, stands for “yes”, solvability represents “no”. However, in our opinion, physics as it should be treated is semisimple as well as solvable, providing a good perspective for our view at the phenomenon of mass. Taking solvability as the internal symmetry of massless particles, the photon of helicity

1 1 1 λ = 1 is represented by ( 2 , 2 ), and Pauli’s neutrino and antineutrino of helicity λ = 2 by 1 1 ( 2 , 0) and (0, 2 ), respectively. Finally, we might ask about the graviton. If the graviton is identical to its own antiparticle, according to Weinberg’s ansatz it is represented by (1, 1) with helicity λ =2. (1, 0) and (0, 1) can stand for an hypothetical massless vector boson and its antiboson, both with helicity λ = 1 but with e.g. opposite charge (e.g. massless charges W bosons). However, since open charges for massless particles have not been seen in any of the accellerator experiments, the validity of Weinberg’s ansatz challenges the graviton as independent particle, i.e. the quantisation of gravity as a whole. Finally, accepting the Borel algebra as symmetry algebra for massless particles, it is reasonable to develop a Yang–Mills theory for solvable groups, treating the elementary al-

gebras su2 and sol2 on the same footing. Indeed, while the algebra su2 generates semisimple algebras via the Cartan , the solvable algebras are constructed gradually as semidi- rect sums of abelian algebras. Moreover, the solvable gauge will more immediately generate the abelian gauge of the field theory of electromagnetism, according to ideas presented by Helfer, Nuyts and others [52, 53, 54].

Acknowledgements

This work was supported by the Estonian Research Council under Grant No. IUT2-27. The authors are grateful to Z. Oziewicz for interesting discussions on the Lorentz group.

25 A The Lorentz group

The Lorentz group is usually given by its action on the Minkowski vector space E1,3 with the metric η = diag(1; 1, 1, 1). By definition, the Lorentz group O preserves the − − − 1,3 µ ν T invariant x y = x η y = x ηy, x, y E (cf. e.g. Refs. [36, 37, 38, 39, 40, 41, 42]), i.e. · µν ∈ 1,3

Λx Λy = xT ΛT ηΛy =! xT ηy = x y ΛT ηΛ= η, (A1)

· · ⇒ E where O Λ: E x Λx . In the matrix form one obtains 1,3 ∋ 1,3 ∋ → ∈ 1,3

µ µ ν µ ρ (Λx) =Λ νx and Λ νηµρΛ σ = ηνσ.

3 The group O is topologically homeomorphic to O Ê , and the number of connected 1,3 3 × components is four. Henceforth we will consider mainly the component connected to unity,

0 Lor1,3 = SO1,3, called the proper orthochronous Lorentz group,

T 0 Ê Lor = Λ Å ( ):Λ ηΛ= η, det Λ = 1, Λ 1 . (A2) 1,3 { ∈ 4 0 ≥ }

Lor1,3 is a normal subgroup of O1,3. The Lorentz group of all proper orthochronous Lorentz transformations of coordinates on the Minkowski space is a six-parameter matrix Lie group. The domain of the six parameters is given by

D = η , η , η ,ω ,ω ,ω : η Ê, π<ω π, 0 <ω π, π<ω π , { 1 2 3 1 2 3 i ∈ − 1 ≤ 2 ≤ − 3 ≤ }

where the boundary points are topologically identified. The resulting region is homeomor-

3

È È phic to Ê where is the three-dimensional projective space, covered twice by the × 3 3 simply connected three-dimensional unit . Therefore, Lor1,3 is locally compact and doubly connected, path connected, simple and reductive. The universal covering group is

SL2(C), i.e.

C Z C Lor = SL (C) SL ( )/ = SO ( ) (A3) 1,3 2 × 2 2 3 where realification of complex groups is understood.

26 A.1 Matrix representation

Assuming the Minkowski metric, it is convenient to write the defining representation for Λ blockwise, A B~ T Λ= (A4) CD~ ! where A =Λ0 1, B =Λ0 , C =Λk , and 0 ≥ k k k 0

i

D =(D ) GL(3, Ê). j ∈ For Λ Lor , one has ∈ 1,3 A2 =1+ C~ T C~ =1+ C~ 2 | | T T ~ ~ T Λ ηΛ= η  D D = ½3 + BB (A5.1) ⇒   AB~ DT C~ =0 −   A2 =1+ B~ T B~ =1+ B~ 2 | | T T ~ ~ T ΛηΛ = η  DD = ½3 + CC (A5.2) ⇒   AC~ DB~ =0 −  Using these equations, it is easy to see that A C~ T Λ−1 = − . BD~ T ! − Because of det Λ = 1, one has det D = A 1. One can use the equations to rewrite ≥ 1 1 C~ = DB~ or B~ = DT C~ A A to obtain ~ T 1 ~ T A B A A C D Λ= = . (A6) 1 ~ ~ A DBD ! CD ! As a consequence of this, one can apply a polar decomposition [43, 44, 45] to the elements

′ ′ of the Lorentz group Lor1,3,Λ= QP = P Q where Q is orthogonal and P,P are real symmetric positive definite. Using the ansatz 1 0 A B~ T A C~ T Q = , P = , P ′ = (A7) ~ ~ ′ 0 R ! BDP ! CDP !

27 with R SO and D ,D′ symmetric, one obtains ∈ 3 P P 1 1 1 R = (D + AD−1T ),D = (A + DT D),D′ = (A + DDT ). (A8) 1+ A P 1+ A P 1+ A According to Tolhoek’s theorem [24], P = (ΛT Λ)1/2 and P ′ = (ΛΛT )1/2 describe pure Lorentz transformations or boosts, where 1 1 A = , C~ = A~v. 1 v2/c2 c − Vice versa, Λ = QP (B~ )= P (RB~ )pQ can be written as A B~ T A C~ T R Λ= = , (A9) ~ 1 ~ ~ T ~ 1 ~ ~ T RB R + 1+A RBB ! C R + 1+A CC R ! 0 ~ ~ 3 where A = Λ 1, B, C Ê and R SO . The Principal axis theorem for the group 0 ≥ ∈ ∈ 3 Lor , finally, tells us that every Λ Lor has one of the shapes 1,3 ∈ 1,3 lor t 0 −1 −1 SΛsS = S S or 0 rot ω ! 1 0 −1 −1 SΛuS = S S , (A10) 0 N ! where cosh t sinh t

lor t = , t Ê, sinh t cosh t ! ∈ cos ω sin ω rot ω = − SO2, π<ω π. (A11) sin ω cos ω ! ∈ − ≤ S Lor is a similarity transformation. Since spec Λ = et, e−t, eiω, e−iω ,Λ is semisim- ∈ 1,3 s { } s ple. On the contrary Λu us unipotent. Therefore, the Jordan form of N is 1 1 0

 0 1 1   0 0 1      and spec Λ = 1 . Finally, we note that for all Λ Lor , Λ satisfies the minimal u { } ∈ 1,3 equation [46]

4 3 1 2 2 2

Λ (tr Λ)Λ + (trΛ) trΛ Λ (trΛ)Λ + ½ =0. − 2 − − 4  28 Though it is a pure algebraic reason, one concludes from polar decomposition that the maximal compact subgroup of Lor1,3 is isomorphic to SO3. Indeed, Lor1,3 is isomorphic to

SO3(C), and SO3 is the compact real form of the latter. Moreover, since Lor1,3 is stable under the transposition (i.e. Λ Lor ΛT Lor ), Lor is a linear , ∈ 1,3 ⇒ ∈ 1,3 1,3 for which the maximal compact subgroup K is determined by the Cartan involution

θ : Lor Λ θ(Λ) = Λ−1T Lor 1,3 ∋ → ∈ 1,3

as K = Λ Lor : θ(Λ) = Λ−1T = Λ = SO . In this setting, Lor / SO is a { ∈ 1,3 } 3 1,3 3 . It is important that the maximal simple compact subgroup SO Lor determines 3 ⊂ 1,3 the internal symmetry of massive particles, i.e. the spin. On the other hand, the maxi- mal solvable noncompact subgroup called Borel subgroup Bor Lor determines the 1,3 ⊂ 1,3 helicity of massless particles. Notice that Bor1,3 is a semidirect product of the abelian subgroups and Tor , Bor = ⋊ Tor , where Tor is the maximal Torus of Lor . T2 1,3 1,3 T2 1,3 1,3 1,3

A.2 Generators of Lor1,3

In order to linearise the group Lor1,3, one can simply differentiate it and evaluate the derivative at the identity element of the group. The tangent space at the identity element is the Lie algebra

tX

Ê Ê lor = X Å ( ): e Lor t . (A12) 1,3 { ∈ 4 ∈ 1,3 ∀ ∈ } According to Lie’s theorem, the exponential map exp : lor Lor is surjective. 1,3 → 1,3 Therefore, any element Λ Lor that is close to unity can be written as the exponential ∈ 1,3 of an element X lor . From Eq. (A1) one concludes that the defining equation for an ∈ 1,3 element X lor is given by ∈ 1,3 XT η + ηX =0. (A13)

29 Using the infinitesimal transformation 1 Λµ = ηµ + ωµ ηµ (ω eρσ)µ , (A14) ν ν ν ≡ ν − 2 ρσ ν the defining equation (A13) gives ω = ω , and one obtains six independent parameters µν − νµ ω and generators eµν = eνµ. A generic element Λ Lor is written as µν − ∈ 1,3

1 µν µν ∂ Λ(ω) = exp ω eµν , e = Λ(ω) . (A15) −2 −∂ωµν ω=0  

The six independent generators eµν have the form

(eµν )ρ = ηµρην + ηνρηµ (A16) σ − σ σ and obey the commutation relation

[eµν , eρσ]= ηµρeνσ + ηνσeµρ ηµσeνρ ηνρeµσ. (A17) − − The defining equation (A13) applies to the generators in the form

eµνT η + ηeµν =0. (A18)

The minimal equation for X lor ( so ) is given by ∈ 1,3 ≡ 1,3

4 1 2 2

X (tr X )X + (det X)½ =0, − 2 4 and det X 0, tr X = 0. ≤

A.3 Cartan decomposition

3 1 Following Ref. [47], let ~e(i) (i =1, 2, 3) be an orthogonal triad for Ê defined by

(~eT ) =(~e )j = δj, ~eT ~e = δ , (~e ~e )k = ǫ k (i) j (i) i (i) (j) ij (i) × (j) ij (1 = ǫ0123 = ǫ ǫ ). In terms of this triad the generators eµν are given by − 0123 ≡ − 123 0 0 k eij = ǫijkD 0 fij ! ≡

1 T µ The notation (X )µ X is used in the following. ≡ 30 or, vice versa, 0 0 1 jk i Di = ǫ0ijke = = D , (A19.1) −2 0 ǫ ~e ~eT ! − − ijk (j) (k) µ µ T T where (Di) ν = ǫ0j ν, fij = ~e(i)~e(j) = ~e(j)~e(i), and

T 0 ~e(i) e0i = Bi, (A19.2) ~e(i) 0 ! ≡ where (B )µ = η µη + η η µ. A general element X lor has the form i ν − 0 iν 0ν i ∈ 1,3 0 η1 η2 η3

 η1 0 ω3 ω2  X = ~ωD~ ~ηB~ = − . (A20) −  η2 ω3 0 ω1     −   3 2 1   η ω ω 0   −  The corresponding finite Lorentz transformation is given by 1 Λ = exp ωµνe = exp(~ωD~ ~ηB~ ), (A21) −2 µν −   where ωi = 1 ǫijkω , ηi = ω = η . The commutation relations can be expressed as 2 jk 0i − i

[Di,Dj] = ǫijkDk,

[Di, Bj] = ǫijkBk,

[B , B ] = ǫ D . (A22) i j − ijk k The compact generators D are antisymmetric (DT = D ) while the noncompact gen- i i − i T erators Bi are symmetric (Bi = Bi). As a consequence, the Lorentz algebra lor1,3 (if considered as vector space) is a symmetric Lie algebra with symmetric decomposition

lor~ = so~ ~p, (A23) 1,3 3 ⊕ ~p 3 p p

where = span Ê B . Indeed, so is a subalgebra, [so , so ] = so , but [so , ] and { i}1 3 3 3 3 3 ⊂ [p, p] so . Given the structure (A20), the generic element X lor can be split up into ⊂ 3 ∈ 1,3 two parts, 0 X~ T 0 0 0 X~ T X = = + (A24) X~ X(3) ! 0 X(3) ! X~ 0 !

31 where the first part is compact, XT = X(3), and contained in so , the second part is (3) − 3 noncompact and contained in p. Notice that so3 and p are orthogonal with respect to the Killing form,

(so3, p)=0. (A25)

Since lor1,3 is simple, the Cartan–Killing form is nonsingular on so3 and p. The symmetric decomposition is determined by the Cartan involution

θ : lor X θ(X)= XT = ηXη lor (A26) 1,3 ∋ → − ∈ 1,3 which is the only external involtive for lor1,3. Indeed, one obtains θ(Di)= DT = D and, therefore, − i i

3

so = X lor : θ(X)=+X = spanÊ D (A27.1) 3 { ∈ 1,3 } { i}1 is the maximal compact subalgebra of lor1,3. Similarly,

p 3

= X lor : θ(X)= X = spanÊ B (A27.2) { ∈ 1,3 − } { i}1

consists of the noncompact elements of lor1,3. Therefore, one ends up with the Cartan decomposition

lor1,3 = so3 +p. (A28)

The map SO p Lor given by 3 × → 1,3

SO p (R,X) R exp X Lor 3 × ∋ → ∈ 1,3 is a diffeomorphism onto Lor1,3. Therefore, the Cartan decomposition (A28) on the level of the Lie algebra lor1,3 induces the polar decomposition (A7) on the level of the Lie group

Lor1,3. The exponential map is a diffeomorphism from the vector space ~p of symmetric matrices to the set of positive definite matrices,

exp : ~p X = XT exp X = (exp X)T . ∋ → 32 A.4 Weyl’s unitary trick

It is an algebraic fact that the symmetric spaces appear in pairs. If the Cartan involution induces g = k + p, it’s companion is

(W ) C

g = k + ip gC = g. (A29) ≡ ⊗

If g is the Lie algebra of a noncompact connected semisimple Lie group G, g(W ) is the Lie algebra of a second Lie group G(W ) which is compact. In this way the noncompact algebras appearing in the Cartan decomposition can be analytically continued to compact algebras by analytic extension, lor = so +p so +ip lor(W ) . (A30) 1,3 3 → 3 ≡ 1,3 This analytical continuation known as Weyl’s unitary trick can be accomplished by using the matrix i 0 Γ=

0 ½3 ! in the way lor X Γ X(W ) =ΓXΓ 1,3 ∋ → (note that Weyl’s unitary trick is not a similarity transformation). One obtains

0 iX~ T X(W ) =ΓXΓ= =Γ−1( ηX)Γ. (A31) iX~ X(3) ! − and for the basis

(W ) (W ) Di =ΓDiΓ= Di, Bi =ΓBiΓ= iBi. (A32)

Accordingly, the commutation relations (A22) change to their compact form

(W ) (W ) (W ) [Di ,Dj ] = ǫijkDk ,

(W ) (W ) (W ) [Di , Bj ] = ǫijkBk ,

(W ) (W ) (W ) [Bi , Bj ] = ǫijkDk . (A33)

33 T Because of (ηX) = ηX, one recovers the algebra so (Ê). The last algebra in turn is − 4 isomorphic to su ⊞ su where ⊞ denotes the Kronecker sum of algebras, i.e. for a g and 2 2 ∈ b h one has

∈

½ ⊞ a ⊞ b a ½ + b g h. (A34) ≡ ⊗ h g ⊗ ∈ (W ) To conclude, the pair of symmetric algebras lor1,3 and lor1,3 is connected by Weyl’s unitary trick, Weyl lor lor(W ) = S(su ⊞ su )S†. (A35) 1,3 −→ 1,3 2 2 The splitting map 0 1 1 0 − 1  10 0 1  S = − (A36) √2  i 0 0 i     − −     011 0  † −1   with S = S gives the decomposition in the su2 basis,

D(W ) S†D(W )S = m ⊞ m D = m ⊞ m , i → i i i ⇒ i i i

B(W ) S†B(W )S = m ⊞ ( m ) B =( im ) ⊞ (im ), (A37) i → i i − i ⇒ i − i i

i where mj = 2 σj, and the decomposition in the sl2 basis,

i i 1 1 D = (e ⊞ e)+ (f ⊞ f), B = (e ⊞ ( e)) + (f ⊞ ( f)) , 1 2 2 1 2 − 2 − 1 1 i i D = (e ⊞ e) (f ⊞ f), B = (e ⊞ ( e)) + (f ⊞ ( f)) , 2 2 − 2 2 −2 − 2 − i 1 D = (h ⊞ h), B = (h ⊞ ( h)) . (A38) 3 2 3 2 −

The meaning of Weyl’s unitary trick is that the representations of the noncompact group

Lor may be viewed as representations of the compact group SO = SU SU /Z ,

1,3 4 2 × 2 2

½ ½ ½ ½ where Z = ( , ), ( , ) is the discrete subgroup. From the unitary nature of 2 { 2 2 − 2 − 2 } the representations of the compact group SO(4) one concludes the full reducibility of the

finite-dimensional representations of Lor1,3. Note that SO4 is the real compact form of

34

C Z SO (C) = SL ( ) SL (C)/ , and Lor is a real noncompact form. Since the simply 4 2 × 2 2 1,3 connected group SU(2) SU(2) is a universal covering group of the doubly connected × group SO4, their Lie algebras are isomorphic. Moreover, since su2 is a compact real form

of sl2(C), the construction of the representations of the algebra lor1,3 may be realised by

3

C Ê using the representations of sl (C). Since SL ( ) as the topological product SU(2) 2 2 × is simply connected, all its representations are single-valued.

A.5 Higher dimensional representations

In the standard basis 1 0 0 1 0 0 h = , e = , f = (A39) 0 1 ! 0 0 ! 1 0 ! −

of sl (C) with [h, e]=2e,[h, f] = 2f and [e, f] = h, the (2k + 1)-dimensional, real 2 − representations applied to states k, m are given by | i

π(k)(h) k, m = 2m k, m , | i | i π(k)(e) k, m = ρ(k) k, m +1 , | i (m)| i π(k)(f) k, m = ρ(k) k, m 1 , (A40) | i (−m)| − i where k =0, 1 , 1,..., m = k, k +1,...,k and ρ(k) = (k m + 1)(k + m). 2 − − (m) −

p C Theorem: Let 2k Æ and let (V, π) be a simple representation of sl ( ) of dimension ∈ 2 2k + 1. Then ([19], p. 281)

1. π is equivalent to π(k),

2. the eigenvalues of π(k)(h)/2 are k, k +1,...,k = spec 1 π(k)(h), {− − } 2

3. if 0 = v V verifies π(k)(e)v = 0, then π(k)(h)v =2kv, 6 ∈ i.e. π(k)(h) and π(k)(e) have the common eigenvector k,k , | i

35 4. if 0 = v V verifies π(k)(f)v = 0, then π(k)(h)v = 2kv, 6 ∈ − i.e. π(k)(h) and π(k)(f) have the common eigenvector k, k | − i

Since su(2) is the compact real form of sl2(C) and the generators of su(2) are given by

i 1 i m = (e + f), m = (e f), m = h, (A41) 1 2 2 2 − 3 2

one can accordingly define irreducible representations of su(2) given by

(k) (k) (k) π (m1), π (m2) and π (m3). (A42)

Following the procedure given before, the real Lie algebra lor1,3 may be identified via Weyl’s (W ) unitary trick with the algebra lor1,3 which splits into a Kronecker sum of two algebras su(2), lor(W ) su ⊞ su . If π(k) and π(l) are representations of su on the vector spaces 1,3 ∼ 2 2 2 V (k) and V (l), π(k) π(l) is a representation of the Lie algebra lor(W ) on V (k) V (l), defined ⊗ 1,3 ⊗ by m ⊞ m π(k,l)(m ⊞ m )= π(k)(m ) ⊞ π(l)(m ). (A43) i j → i j i j (k,l) The representation π of the Kronecker sum su2 ⊞ su2 on the basis

k,l; m , m k, m l, m , k m k, l m l {| k li≡| ki⊗| li − ≤ k ≤ − ≤ l ≤ }

is given by

π(k,l)(m ⊞ m ) k,l; m , m = i j | k li k l (k) (l) = π (mi) k,l; m, ml + π (mj) k,l; mk, m . (A44) mmk | i mml | i m=−k m=−l X  X  Using the representations of D , B lor in the su basis, one obtains i i ∈ 1,3 2

(k,l) (k) (l) π (Di) = π (mi) ⊞ π (mi),

π(k,l)(B ) = iπ(k)(m ) ⊞ iπ(l)(m ) (A45) i − i i   (k) with π (mi) given by Eq. (A42).

36 A.6 Splitting algebra

(k,l) Theorem: Any finite-dimensional representation of lor1,3 is isomorphic to π for some

1 k,l = 0, 2 , 1,... and is non-antihermitean. The corresponding representation of Lor1,3 is non-unitary [19].

The between so and so so is easily realised by the choice of the basis 4 3 ⊕ 3 1 1+ ε 1 ε J (ε) = (D + iεB ) = m ⊞ − m , i 2 i i 2 i 2 i 1 1 ε 1+ ε K(ε) = (D iεB ) = − m ⊞ m (A46) i 2 i − i 2 i 2 i with J (ε)† = J (ε), K(ε)† = K(ε) (ε = 1). The commutator relations are given by i − i i − i ±

(ε) (ε) (ε) [Ji ,Jj ] = ǫijkJk ,

(ε) (ε) [Ji ,Kj ] = 0,

(ε) (ε) (ε) [Ki ,Kj ] = ǫijkKj . (A47)

Note that the fact that the Lorentz algebra lor1,3 can be written as a Kronecker sum su(2) ⊞ su(2) of two algebras does not mean that lor is the same as su(2) su(2) or 1,3 ⊕ (W ) lor1,3 . Rather, they are the anti-hermitean complex representations of lor1,3.

A.7 Spinor representations

There are two fundamental spinor representations, from which all other may be obtained by tensor product reduction [48, 49, 50, 51]. The Lorentz covariant description needs two

sets of relativistic Pauli matrices, ½ (σ )=(½ , ~σ) and (˜σ )=( , ~σ). µ 2 µ 2 −

The relation between the real Minkowski space E1,3 and the set of all complex hermitean

2 2 matrices À is given by × 2

µ À E p σ(p)= σ p . (A48) 1,3 ∋ → µ ∈ 2 37 The correspondence is a linear isomorphism,

2 µ det σ(p)= p = p pµ, (A49)

and the characteristic polynomial

(p0 + ~p λ)(p0 ~p λ) for p2 > 0, det(σ(p) λ)= | | − −| | − (A50) − ( λ(2p2 λ) for p2 = 0. − Therefore, if p2 > 0, σ(p) is positive semidefinite.

Theorem: C 1. Let σ(p) Å ( ) be positive definite. ∈ 2

† C If A Å ( ) and det A = 0, then Aσ(p)A is positive definite [19].

∈ n 6

C Å C 2. If σ(p) Å ( ) is not positive definite but positive semidefinite and if A ( ), ∈ 2 ∈ n then Aσ(p)A† is always positive semidefinite and not positive definite [19].

1 The fundamental representation ( 2 , 0) can be expressed as the commutative diagram

Lor Λ: E p Λp 1,3 ∋ 1,3 ∋ −→ σ σ (A51) ↓ ↓ ↓

† À SL (C) A : σ(p) A σ(p)A = σ(Λp) 2 ∋ ± Λ 2 ∋ −→ Λ Λ The continuous homomorphism relates an element Λ Lor to two elements A

∈ 1,3 ± Λ ∈ C

SL2(C). The group SL2( ) constitutes the universal covering group of Lor1,3, i.e. Lor1,3 =

Z C SL2(C)/ (on the right hand side the realification of SL2( ) is understood). Using Pauli’s spin matrices one obtains

† µ AΛσνAΛ = σµΛ ν, 1 Λµ = tr(σ A σ A† ), ν 2 µ Λ ν Λ Λµν σ σ˜ A = µ ν , (A52) Λ 1 αβ γδ 1/2 ( 2 Λ Λ tr(σασ˜βσδσ˜γ ))

38 where 1 4(tr A† )2 = ΛµνΛρσ tr(σ σ˜ σ σ˜ ) = (trΛ)2 trΛ2 +4+ iΛµνΛρσǫ Λ 2 µ ν σ ρ − µνρσ

† µ (note the different order of indices). Suppose that Λ and AΛ are given by AΛσν AΛ = σµΛ ν, one can write

† T µ AΛσν AΛ = σµ(Λ ) ν ,

† −1 µ AΛσ˜ν AΛ =σ ˜µ(Λ ) ν,

† −1T µ AΛσ˜ν AΛ =σ ˜µ(Λ ) ν.

Defining Λ = exp( 1 ωµνe ) and A = exp( 1 ωµνe ), for the representation ( 1 , 0) one − 2 µν Λ − 2 µν 2 obtains 1 1 m = (e )αβσ σ˜ = (σ σ˜ σ σ˜ ), µν 4 µν α β −4 µ ν − ν µ 1 i m = ǫ mjk = σ , i −2 ijk 2 i 1 m = σ = im . (A53) 0j 2 j − j

1 The second fundamental representation (0, 2 ) is defined by the commutative diagram

Lor Λ: E p Λp 1,3 ∋ 1,3 ∋ −→ σ˜ σ˜ ↓ ↓ ↓

† À SL (C) B : σ˜(p) B σ˜(p)B =σ ˜(Λp) 2 ∋ ± Λ 2 ∋ −→ Λ Λ whereσ ˜ = C−1σT C = C−1σ∗C, 0 1 C = iσ2 = − − 1 0 ! andσ ˜(p)=2p σ(p). The properties of the Pauli matrices yield 0 −

† µ BΛσ˜ν BΛ =σ ˜µΛ ν, 1 Λµ = tr(˜σ B σ˜ B† ), ν 2 µ Λ ν Λ Λµνσ˜ σ B = µ ν Λ 1 αβ γδ 1/2 ( 2 Λ Λ tr(˜σασβσ˜δσγ ))

39 and 1 4(tr B† )2 = Λµν Λρσ tr(˜σ σ σ˜ σ ) = (trΛ)2 trΛ2 +4 iΛµν Λρσǫ . Λ 2 µ ν σ ρ − − µνρσ Defining the generators of the representation (0, 1 ) by B = exp 1 ωµνm˜ , one obtains 2 Λ − 2 µν 1 1  m˜ = (e )αβσ˜ σ = (˜σ σ σ˜ σ ), µν 4 µν α β −4 µ ν − ν µ 1 i i m˜ = ǫ m˜ jk = σ˜ = σ , i −2 ijk −2 i 2 i 1 1 m˜ = σ˜ = σ = im˜ . 0j 2 j −2 j j

The two nonequivalent fundamental representations AΛ and BΛ are related by

−1 † −1† BΛ = C AΛC =(AΛ) .

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