A solvable algebra for massless fermions

We derive the stabiliser group of the four-vector, also known as Wigner's little group, in case of massless particle states, as the maximal solvable subgroup of the proper orthochronous Lorentz group of dimension four, known as the Borel subgroup. In the absence of mass, particle states are disentangled into left- and right-handed chiral states, governed by the maximal solvable subgroups ${\rm sol}_2^\pm$ of order two. Induced Lorentz transformations are constructed and applied to general representations of particle states. Finally, in our conclusions it is argued how the spin-flip contribution might be closely related to the occurrence of nonphysical spin operators.


Introduction
As neutrino oscillations are observed in experiments, it seems obvious that all fermions carry a mass.Even though the mass spectrum reaches from small fractions of eV for neutrinos up to 175 GeV for the top quark, a hierarchy waiting still for an explanation, the fact that a fermion carries a mass allows to go to the rest frame of the particle and to observe both left-handed and right-handed states.
Therefore, the concept of massless fermions, moving with the speed of light, has to be considered as an approximation.This approximation holds true if some of the masses of fermions interacting in a perturbative calculation can be neglected compared to other, larger fermion masses.However, while assuming a fermion to be massless, one not only obtains an essential simplification of the calculation but also different symmetries which are not given for fermions with small but finite mass.As an example, the breakdown of these symmetries can cause spin-flip effects where the result in the mass-zero limit differs from the result for massless fermions [1,2,3,4,5,6,7,8,9,10,11].This effect can be understood as a discontinuity in freezing the spin of the fermion.However, to the best of our knowledge, a deeper understanding of these effects is still missing.
In this paper, we analyse the structure of Wigner's little group for massless particles by adding a small but essential degree of freedom, given by the fact that the momentum vector of a massless particle defines a projective space.In doing so, we come to the conclusion that the stabiliser subgroup is not given by a semisimple group as for massive particles but by a solvable group.In Sec. 2 we give details on the Borel subgroup as the maximal solvable subgroup describing the stabiliser.In Sec. 3 we deal with the representation space in terms of common eigenvectors which, in a natural way, leads to the split-off of the representation space into a left-and right-handed part, described as Kronecker sum on Sec. 4. The two-dimensional subspaces are governed by the solvable groups sol − 2 and sol + 2 which are expressed in terms of the Chevalley basis in Sec. 5. Finally, in Sec.6 we give our conclusions and present an outlook on how the Weyl equations for these massless states can be combined to a Dirac equation for fermions with mass.

Analysis of Wigner's little group
In his paper "Sur la dynamique de l'électron" from July 1905 [12], Henri Poincaré formulates the "Principle of Relativity", introduces the concepts of Lorentz transformation and Lorentz group, postulating the covariance of the laws of nature under Lorentz transformations.The full Lorentz group is a six-dimensional, noncompact and non-abelian real Lie group which is not connected.The four connected components of this group are related to each other via discrete transformations (parity and time reversal).None of these components is simply connected.In describing physics, one usually considers the component connected to the identity, called the proper orthochronous Lorentz group Lor (1,3).
An important subgroup of Lor (1,3) that preserves a given four-vector p is Wigner's little group.For p describing the momentum of a massive particle, the condition Λ p p = p for the elements Λ p of the little group can be solved in the rest frame of the particle where the normalised momentum vector is given by p = (1; 0, 0, 0) T , leading to the block structure where DD T = ½ 3 = D T D. Therefore, the little group of a massive particle is isomorphic to SO (3).However, for a massless particle, the momentum vectors p = (1; 0, 0, ε) with ε = ±1 for a movement along the z axis are projective vectors.Therefore, in solving the generalized equations Λp = λp and Λ T ηp = λ −1 ηp for p = (1; 0, 0, ε) with a general value of ε and the Minkowskian metric The two last conditions are in agreement if and only if This equation marks the point where two different paths are possible to follow: for λ = λ −1 = 1 (λ > 0 for the proper orthochronous Lorentz group) one ends up again with Wigner's little group SO(3).For massless particles, however, one has ε 2 = 1 and, therefore, one can keep λ > 0 arbitrary, ending up with the Borel subgroup explained in the following.

Justification of the extension
The introduction of an extension of Wigner's little group needs justification.Wigner introduces the little group as a stabiliser group with respect to the momentum vector p.
However, because the four-length of the momentum vector for a massless particle is zero and, therefore, the multiplication of this vector with an arbitrary scale does not change the physics of this particle, the physical situation is better described by a projective space.
The existence of an invariant subspace is guaranteed by the Lie-Kolchin theorem,

Lie-Kolchin theorem
If G is a connected and solvable linear algebraic group defined over an algebraically closed field and ρ : G → GL(V ) is a representation on a nonzero finite-dimensional vector space V, then there exists a one-dimensional linear subspace L of V such that In 1956, Armand Borel generalised the Lie-Kolchin theorem as a fixed-point theorem for algebraic varieties [13] and, therefore, also for the projective space,

Borel fixed-point theorem
If G is a connected, solvable, algebraic group acting regularly on a non-empty, complete algebraic variety V over an algebraically closed field, then there exists a fixed-point of V.
As expressed by Eq. ( 3), the projectivity of the fixed-point is broken if ε 2 < 1, i.e. if the particle gains mass.In this case we are falling back to Wigner's little group.The extension can be understood also on the level of Lie algebras, as for massless particles the interchange of space and time components of the momentum vector is an additional symmetry which is absent for massive particles.Note in this context that also E(2) as the little group for massless particles proposed by Wigner is a solvable group, though not maximal.
2 The Borel subgroup Bor(1, 3; p) From now on we use ε only as the sign of the momentum 3-component.The fact that the momentum vector p ∼ (1; 0, 0, ε) for a massless particle is symmetric (up to the sign ε = ±1) under the interchange of the first and the last component gives an additional element of the algebra which is missing so far in Wigner's little group.In order to see this, one can find solutions for the character problem (summation over repeated indices is implied) Solving this problem for the Lorentz matrix Λ p = (Λ µ ν (p)) with Λ T p ηΛ p = η one obtains1 where we have chosen λ = e t and introduced three additional parameters u, v and w.
The two parts of the second exponential factor commutate with each other.They constitute the maximal torus Tor(1, 1; p) describing the transformations that leave the direction of the momentum vector p invariant: a boost directed along the z axis described by exp(T t), and a rotation about the z axis described by exp(W w).However, these two factors do not commute with the first exponential factor Λ u,v := exp(Uu + V v) which constitutes the physically nontrivial part T (2; p) of the Borel subgroup (translations), Note that due to the solvability, the series expansion breaks at the second order.Together, these two parts of the polar decomposition of Λ p represent the Borel subgroup as a semidirect product, Bor(1, 3; p) = T (2; p) ⋊ Tor(1, 1; p).

A bridge from massive to massless
Even though the main emphasis of this paper is layed on an independent treatment of the little group of massless particles as the maximal noncompact solvable subgroup of the proper orthochronous Lorentz group, there is still a way to find a bridge connecting this part of the Lorentz group to the maximal compact simple subgroup which is quite remarkable.
Starting with a massive particle, in the rest frame of this particle a proper orthochronous Lorentz transformation Λp = B r,s,t R u,v,w can be written as polar decomposition of the Wigner rotation matrix R u,v,w followed by a boost B r,s,t , where The transformation to the laboratory frame where the momentum vector of the particle is given by p is performed with the help of the boost matrix B p = B 0,0,εξp parametrised by the momentum vector p, where c p = cosh ξ p and s p = sinh ξ p with rapidity ξ p .Accordingly, the proper orthochronous Lorentz transformation in the laboratory frame is given by For the generic Lie algebra element generating the boost B r,s,t one obtains Because c p , s p → ∞ in the massless limit, r and s (but not t) have to be renormalized in order to obtain a finite matrix B p B r,s,t B −1 p .This can be done by replacing r by xr and s by xs where xc p = xs p → 1 in the massless limit x → 0. Raising the generic element in Eq. ( 15 Because of the renormalisation, B p B xr B xs B −1 p is finite in the massless limit and gives Λ εr,εs which can be seen by comparing the result of the exponentiation with Eq. ( 10).
Looking at the second main factor in Λ p , a similar consideration can be made for u and v (but not w) have to be renormalized using again x with xc p = xs p → 1. Raising the generic element in Eq. ( 17 this factor can be pulled out, and the remaining product gives Λ u,v in the massless limit.Therefore, in this limit, Λ p will decay into In this product, Λ u,v R w B t constitutes the generic element of the Borel subgroup Bor(1, 3; p) and Λ εr,εs constitutes the rest class Lor(1, 3)/ Bor(1, 3; p).To conclude, the little groups of massive and massless particles are connected by a singular transformation, induced by an infinitesimal boost, interpreted as contraction in the sense of Inonu and Wigner [14].

Common (pseudo)eigenvectors
The exponential representation ( 9) is a special case of the representation of the full Lorentz group where (e According to Lie's theorem, a solvable algebra has a single common eigenvector.Solving the equations i ℓ 0 (i = 0, 1, 2, 3), one obtains Not very surprisingly, the common eigenvector is just given by p.In order to specify the defective matrices T ε i of the solvable algebra, the equations are solved step-wise to obtain a system of pseudo-eigenvectors and -eigenvalues.Collecting all these equations in a single one, after some normalisation one obtains where P = (ℓ 0 , ℓ 1 , ℓ 2 , ℓ 3 ) is rearranged in order to be unitary, P −1 = P † .Turning back to the original notation, one obtains T P = P T , UP = P U, V P = P V and W P = P W , where are upper triangular forms of the four generators.

Generating the (pseudo)eigenvectors
Even though the four generators have only a single common eigenvector, this is not the case for the generic element Λ p ∈ Bor(1, 3; p) in Eq. ( 6).Solving the fourth-order equation det(Λ p − λ½) = 0 for λ leads to λ ∈ {e t , e iw , e −iw , e −t }.The corresponding system of eigenvectors can be calculated.However, here we give a more elegant method to calculate this system of eigenvectors.Using the exponential representation ( 9) and Because K t,w is a diagonal matrix containing the four eigenvectors, the system of eigenvectors is given by the matrix Q obeying Λ p Q = QK t,w .Inserting Λ p = P K u,v K t,w P −1 into this eigenvalue equation, after some rearrangements one obtains This equation for the unknown quantity P −1 Q can be solved iteratively, starting with The iterative solution can be shown to converge to Multiplying with P from the left, one finally obtains the system of eigenvectors Expressed in a slightly philosophically manner, one can say that starting from the very sparse boundary of four defective matrices, the Lie algebra (in this case, the Borel subalgebra) knits the sweater Q for the Lie group in a straightforward, iterative way.

Kronecker sum of solvable algebras
Although we were able to analyse the solvable algebra bor(1, 3; p), the representation in terms of the generators T , U, V and W is not the best one to see the structure of this algebra.Therefore, we use a second one, namely obeying The first justification for the sign notations for J ε − and K ε + is given by the commutator relations (31).In terms of the pairs {J ε 3 , J ε − } and {K ε 3 , K ε + } of generators bor(1, 3; p) can be rewritten as a Kronecker sum sol − 2 ⊞ sol + 2 of two two-dimensional solvable algebras, as will be detailed in the following.

Weyl's unitary trick
A deeper look at this change of representation unveils that this change is actually a composition of several steps.In order to illustrate these steps, one can start again with the proper orthochronous Lorentz group Lor(1, 3) ⊂ SO(1, 3), the elements of which are given by the exponential representation (19) where (e . This representation can be written in a different form as using an analogy to the electromagnetic field strength tensor F µν to write Obviously, the algebra lor(1, 3) is a real algebra.It contains a compact subalgebra k related to the B i which is isomorphic to the compact algebra so(3).Actually, lor (1,3) However, [k, p] = 0. Therefore, we used the symbol + instead of the symbol ⊕ for the direct sum.The algebra g can be transformed to a compact form by using Weyl's unitary trick.
The result is an algebra g * = k + ip, where the implications for introducing an imaginary factor will be explained later.In case of lor(1, 3), the involution is given by φ : e µν → η(e µν ) := ηe µν η = −e µνT (36) (matrix indices are suppressed) or φ( B) = B, φ( E) = − E. Therefore, the compact form of lor(1, 3) is given by the generators B i and iE i obeying the commutation relations Considered as a real algebra, this algebra is isomorphic to so(4).However, the generators are antihermitean, B † i = −B i and (iE i ) † = −iE i and, therefore, the group is unitary.In general, Weyl's unitary trick can be seen to lead always to unitary Lie groups.

Duplication and complexification
The addition of an imaginary factor i turns the real algebra into a complex algebra, at least for intermediate steps.In general, this process is called complexification and is denoted by a lower index (or additional argument) to the algebra symbol.Given a real Lie algebra L, the duplication of this algebra is given by [15] L + iL := {x + iy | x, y ∈ L}. (38) L + iL is still a real vector space.In defining the multiplication of an element x + iy ∈ L C with a complex number α = a + ib ∈ by (a + ib)(x + iy) := (ax − by) + i(bx + ay), and the commutator of two elements x + iy and x ′ + iy ′ by L + iL constitutes a complex form, denoted by L := ⊗ Ê L. This complex form again is a complex Lie algebra which is called the complexification of L. Applied to the actual case, the complexification turns the real algebra so(1, 3) into the complex algebra so(4, ).
However, it is obvious that the algebra given by the commutator relations (37) is real, not complex.The final algebra, therefore, is a real form of this complex algebra, defined as follows: a subalgebra K of the duplicated algebra L + iL is called real form if the complexification of this subalgebra is the same as the original algebra, K = L.As the duplication is not unique,2 there are also different real forms to a given complex algebra.

Compactified and decompactified real forms
Most important real forms are the normal real form where the duplicates are again taken as separate elements, and the compact real form which exists for all (semi)simple complex Lie algebras.Because we will meet these forms in the lower-dimensional case, we postpone the discussion about the different real forms.In the actual case, one of the compact real forms is so(4).However, another one is given by with the commutation rules Therefore, the algebra decomposes into two separate algebras which are isomorphic to su(2). 3 Turning back to solvable groups, the decomposition into su(2) = span Ê {A i } and su(2) = span Ê { Āi } is not yet conform with the definitions given in Eq. ( 30).Looking at the definitions of T ε i on the one hand and the definitions of E i and B i on the other hand, one obtains and generally As A i and Āi are antihermitean, J ε i and K ε i are hermitean and, therefore, constitute decompactified subgroups generated by exp(ij i J ε i ) and exp(ik i K ε i ).One obtains which is the other justification for the sign notations in J ε − and K ε + .Actually, the algebra looks like sl(2, Ê) with one generator missing (J ε + or K ε − , respectively).In lor (1,3), these missing generators exist.In bor(1, 3; p), however, the generators are found in the resp.other algebra with opposite sign ε, while J ±ε 3 = K ∓ε 3 .Therefore, bor(1, 3; p) splits up into the subalgebras As there is a homomorphism between the two algebras sol + 2 and sol − 2 , both solvable algebras are maximal and, therefore, are Borel subalgebras of the larger algebra sl(2, Ê).For a free choice of ε one can represent the two Borel subalgebras as being generated by a solvable part 3 su( 2) is preferred instead of so(3) because A i and Āi are antihermitean, leading to unitary groups.

In quest of left and right
Searching for eigenvectors of the set {J 3 , J + , J − } one finds that these eigenvectors are disjoint, as known for semisimple algebras.The same holds for the set {K 3 , K + , K − }.However, for each of the solvable subalgebras sol − 2 and sol + 2 one obtains only a single common eigenvector.In order to analyse the eigenvector structure, we return to the eigenvectors of Sec. 3 to which we apply the algebra elements, obtaining For {J 3 , J − } the common eigenvector is given as a linear combination of ℓ 0 and ℓ 2 while for {K 3 , K + } the common eigenvector is given by the linear combination of ℓ 0 and ℓ 1 .On the other hand, the common eigenvector for {J 3 , J + } is a linear combination of ℓ 3 and ℓ 1 while the common eigenvector of {K 3 , K − } is a linear combination of ℓ 3 and ℓ 2 .While ℓ 0 (ℓ 3 ) is proportional to the (space inverted) momentum four-vector p, the interpretation of the eigenvectors ℓ 1 and ℓ 2 deserves more effort.For this one can take refuge to the circular polarisation [16,17].The representation E(z, t) = E 0 Re ( e x + i e y )e ikz−iωt = E 0 ( e x cos(kz − ωt) − e y sin(kz − ωt)) describes the right turn of the electric vector in the (x, y) plane, as can be seen by comparing the solution for z = 0 at t = 0 and after a short time t = ∆t.Therefore, the vector ℓ 1 can be identified with the right turn.However, a turn can be identified with handedness or chirality only in combination with a direction of propagation,4 as in case of the circular polarisation by the argument kz − ωt.This direction is given by ℓ 0 (or ℓ 3 ).Therefore, one can interpret (in case of ε = 1) {J 3 , J − } as forward-propagating left-handed, {K 3 , K + } as forward-propagating right-handed, {J 3 , J + } as backward-propagating left-handed, and

The irreducible representation
In terms of 4 × 4 matrices the generators J i and K i (i = 3, ±) are, of course, not given in the irreducible representation.However, they can be related to irreducible representations in an easy way.In fact, there is a similarity transformation such that (a deeper understanding of the representation index ⊞ will be given soon), where and S −1 = S † .In detail, one obtains where the outer product is defined by (A ⊗ B) (ik)(jl) := A ij B kl , i.e. the first matrix sets the frame for the second one.The matrices are the usual Pauli matrices, and σ ± = σ 1 ± iσ 2 .The same similarity transformation via S can be applied also to the generators E i and B i of the proper orthochronous Lorentz group Lor(1, 3).One obtains These two results can be rewritten by employing the Kronecker sum Using this notation, one obtains Therefore, the representation index ⊞ indicates that in this representation obtained via the similarity transformation with S the matrix can be written as a Kronecker sum.It is characteristic that contribute only to the first or second component of the Kronecker sum, respectively.Following the argumentation of Sec.4.4 one can conclude that the first component of the Kronecker sum (and thereby J ⊞ i ) is left-handed while the second component of the Kronecker sum (and thereby K ⊞ i ) is right-handed.Finally, we conclude that via the same similarity transformation S the maximal solvable algebra bor(1, 3; p) in the representation of this section can indeed be decomposed into the Kronecker sum sol − 2 ⊞ sol + 2 .

Common eigenvectors
The concept of common eigenvectors introduced in Sec.4.4 pulls through to the very core, i.e. to the irreducible representation.The set of generators {σ 3 , σ + } of sol + 2 have the common eigenvector (1, 0) T and the set {1, 0} of eigenvalues while for {σ 3 , σ − } (i.e.sol − 2 ) the common eigenvector is (0, 1) T with eigenvalues {−1, 0}.Reintroducing the sign ε, the two non-trivial eigenvalue equations can be cast into the form This is the first quantisation step.Indeed, introducing However, this is not the only possible quantisation.Equivalently, one may write ) which is the dual Weyl equation.In using the tilde notation for σ one avoids the breakdown of the covariant notation.Using Weyl's representation of the Dirac matrices, for finite mass m one ends up with the Dirac equation ψ + is the right-handed spinor and ψ − is the left-handed spinor.This is in agreement with the usual definition ψ R = 1 2 (1 + γ 5 )ψ W = (0, ψ + ) T and ψ L = 1 2 (1 − γ 5 )ψ W = (ψ − , 0) T .

Induced Lorentz transformations
The contractions of the momentum four-vector p with σ and σ induces two (proper orthochronous) Lorentz transformations A Λ and ÃΛ which make the diagram commutative.The induced Lorentz transformations are defined by A long but straightforward calculation shows that A Λ and ÃΛ can be written in an exponential form similar to Eq. ( 19), For the exponential coefficients of A Λ one obtains which can be detailed into where b i and e i obey the algebra lor (1,3), For the exponential coefficient of ÃΛ one obtains Formally, the transitions to the induced Lorentz transformations can be considered as mappings π : Λ → A Λ with π(e αβ ) = a αβ and π : Λ → ÃΛ with π(e αβ ) = ãαβ .Under these mappings, the generators J ε i and K ε i of sol ± 2 are mapped onto the Chevalley basis.Under π one obtains while under π one obtains i.e. the same result with ε ↔ −ε and the total sign interchanged.Again, we are faced with the fact that half of the generators are mapped to zero.Taking into account the relations to J ⊞ i and K ⊞ i , one can state that π maps to the first component of the Kronecker sum while π maps to the second component of the Kronecker sum.Due to Sec. 4, π is the mapping to the left-handed sector, π the mapping into the right-handed sector.
Using Eqs.(67) and performing a couple of simple conversions, one obtains the explicit shape for A Λ for Λ of Eq. ( 6) with dependence on ε, ( Ãε Λ = A −ε Λ ), which can be rewritten as where

Helicity
In order to define a helicity which means that the helicity of this state is h = (m k + m l ).

Conclusions and Outlook
In this paper, we have calculated the stabiliser group of the proper orthochronous Lorentz group, which turns out to be the maximal solvable or Borel subgroup of dimension four.
We have explained the continuous transition between the stabiliser groups of massive and massless particles that describes the massless limit but fails for exactly massless states.We have dealt with the system of eigenvectors of the Borel subgroup and shown that the Borel subgroup can be described by a Kronecker sum of two two-dimensional solvable groups sol ± Even though the foundations for an explanation of the spin-flip effect are prepared by this, an exact formulation is not gained here but is aimed for a future publication.The effect is closely related to the concept of mass which we want to understand in more detail.
In our argumentation we obtained unexpected help from a not yet published seminal work explaining in detail the construction of a spin operator by a linear combination of components of the Pauli-Lubanski pseudovector [19].Not unexpectedly, the authors end up with two spin (tensor) operators and corresponding chirality states that are interchanged under parity transformation.Parity eigenstates can be constructed as particle or antiparticle compound states.Applying the Lorentz transformation to the massive states of Ref. [19], the parity eigenstates are shown to evolve to solutions of the Dirac equation.
In Ref. [19] it is emphasised that the two spin operators are neither axial nor Hermitian, and the same holds for the spin operators in the (1/2, 0)⊕(0, 1/2) representation.However, both properties are restored if applied to particle and antiparticle states.On the other hand, as both properties are essential for physical states, we can conclude that massless left-and right-handed states are physical only in the total absence of mass.This "gap of (un)physicalness" as an explanation for the spin-flip effect has to be investigated in detail.
) to the exponent, one obtains B xr,xs,t = B xr B xs B t , where the exponential factors B xr and B xs factorise and commute with each other and with the remaining factor B t due to the smallness of the renormalised parameters xr and xs.The factor B t describes a boost along the z axis.Compared to the boost B p in the same direction, the former is negligible in the massless limit.Therefore, one can replace B p with B p B −1 t = B −1 t B p and obtain

3
one needs a spin vector s.This vector can be defined by s i = i b i , because then the com-mutation relation [b i , b j ] = ǫ ijk b k for the generators of A ε Λ leads to the usual commutation relation [s i , s j ] = i ǫ ijk s k (86)of an angular momentum algebra.For the three-vector part p = (0, 0, 1) T of the momentum vector p generating the Borel subgroup Bor(1, 3; p), one obtainsH( p) = s 3 = i b 3 = 2 σ common eigenvector (1, 0) T of sol +2 has helicity h = + /2 and the common eigenvector (0, 1) T of sol − 2 has helicity h = − /2, in agreement (for ε = +1) with the previous understanding of left and right.As b i is the two-dimensional representation of B i , the concept of helicity can be generalised to representations (k, l),H (k,l) ( p) = π (k,l) (i B 3 ) = 2 π (k,l) (σ 3 ⊞ σ 3 ) ,(88)Applied to the state |k, l; m k , m l one obtains H (k,l) ( p)|k, l; m k , m l = (m k + m l )|k, l; m k , m l (89)

2
representing right-and left-handed chirality states.Finally, in the Chevalley basis we have derived the Weyl and Dirac equations for massless and massive particles and have defined the helicity of the massless states.Note that without the generator T such a Kronecker sum of chiral states would not emerge.The Borel subgroup as the maximal solvable subgroup of the proper orthochronous Lorentz group provides exactly four eigenvectors describing these two chiral states, of which the left handed state is populated by massless fermions, the right handed by antifermions.This is the physical content of our extension.