In his paper “Sur la dynamique de l’électron” from July 1905 [12], Henri Poincaré formulated the “Principle of Relativity”, introduces the concepts of Lorentz transformation and the 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 are simply connected. In describing physics, one usually considers the component connected to the identity, called the proper orthochronous Lorentz group $\mathrm{Lor}(1,3)$.

An important subgroup of $\mathrm{Lor}(1,3)$ that preserves a given four-vector p is Wigner’s little group. For p describing the momentum of a massive particle, the condition ${\mathsf{\Lambda }}_{p}p=p$ for the elements ${\mathsf{\Lambda }}_p$ of the little group can be solved in the rest frame of the particle where the normalised momentum vector is given by $\widehat{p}={(1;0,0,0)}^T$, leading to the block structure where $D{D}^T={\mathbb{1}}_3=D^{T}D$. Therefore, the little group of a massive particle is isomorphic to $\mathrm{SO}\left(3\right)$. However, for a massless particle, the momentum vectors $p=(1;0,0,\epsilon )$ with $\epsilon =\pm 1$ for a movement along the z axis are projective vectors. Therefore, solving the generalised equations $\mathsf{\Lambda }p=\lambda p$ and ${\mathsf{\Lambda }}^T\eta p={\lambda }^{-1}\eta p$ for $p=(1;0,0,\epsilon )$ with a general value of $\epsilon$ and the Minkowskian metric $\eta =\mathrm{diag}(1;-1,-1,-1)$ via the block ansatz leads to $\epsilon B_3=\lambda -A$, $\epsilon C_3=A-{\lambda }^{-1}$, $\epsilon D_{33}=\epsilon \lambda -C_3$, and $\epsilon D_{33}=B_3+\epsilon {\lambda }^{-1}$. 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 $\lambda ={\lambda }^{-1}=1$ ($\lambda >0$ for the proper orthochronous Lorentz group), one ends up again with Wigner’s little group $\mathrm{SO}\left(3\right)$. For massless particles, however, one has ${\epsilon }^2=1$ and, therefore, one can keep $\lambda >0$ arbitrary, ending up with the Borel subgroup explained in the following.

The introduction of an extension of Wigner’s little group needs justification. Wigner introduced 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,

In 1956, Armand Borel generalised the Lie–Kolchin theorem as a fixed-point theorem for algebraic varieties [13] and, therefore, also for a projective space.

As expressed by Equation (3), the projectivity of the fixed-point is broken if ${\epsilon }^2<1$, i.e., if the particle gains mass. In this case, we fall back to Wigner’s little group. The extension can also be understood on the level of Lie algebras, as for massless particles the interchange of the space and time components of the momentum vector is an additional symmetry that is absent for massive particles. Note that in this context $E\left(2\right)$, as the little group for massless particles proposed by Wigner, is also a solvable group, though not maximal.

Even though the main emphasis of this paper is on the 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 ${\widehat{\mathsf{\Lambda }}}_p=B_{r,s,t}{R}_{u,v,w}$ can be written as a 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 $B_p=B_{0,0,\epsilon {\xi }_p}$ parametrised by the momentum vector p, where $c_p=cosh{\xi }_p$ and $s_p=sinh{\xi }_p$ with a rapidity of ${\xi }_p$. 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 $c_p,s_p\to \infty$ in the massless limit, r and s (but not t) have to be renormalised in order to obtain a finite matrix $B_p{B}_{r,s,t}{B}_p^{-1}$. This can be performed by replacing r by $xr$ and s by $xs$ where $x{c}_p=x{s}_p\to 1$ in the massless limit $x\to 0$. Raising the generic element in Equation (15) 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 $B_p{B}_t^{-1}=B_t^{-1}{B}_p$ and obtain Because of the renormalisation, $B_p{B}_{xr}{B}_{xs}{B}_p^{-1}$ is finite in the massless limit and gives ${\mathsf{\Lambda }}_{\epsilon r,\epsilon s}$, which can be seen by comparing the result of the exponentiation with Equation (10).

Looking at the second main factor in ${\mathsf{\Lambda }}_p$, a similar consideration can be made for $B_p{R}_{u,v,w}{B}_p^{-1}{B}_t$. Starting from u and v (but not w) have to be renormalised, again using x with $x{c}_p=x{s}_p\to 1$. Raising the generic element in Equation (17) to the exponent, one obtains $R_{xu,xv,w}=R_{xu}{R}_{xv}{R}_w$ where all three factors again commute with each other. As $R_w$ also commutes with $B_p^{-1}$, this factor can be pulled out, and the remaining product $B_p{R}_{xu}{R}_{xv}{B}_p^{-1}$ gives ${\mathsf{\Lambda }}_{u,v}$ in the massless limit. Therefore, in this limit, ${\mathsf{\Lambda }}_p$ will decay into In this product, ${\mathsf{\Lambda }}_{u,v}{R}_w{B}_t$ constitutes the generic element of the Borel subgroup $\mathrm{Bor}(1,3;p)$ and ${\mathsf{\Lambda }}_{\epsilon r,\epsilon s}$ constitutes the rest class $\mathrm{Lor}(1,3)/\mathrm{Bor}(1,3;p)$. 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].

Even though the four generators have only a single common eigenvector, this is not the case for the generic element ${\mathsf{\Lambda }}_p\in \mathrm{Bor}(1,3;p)$ in Equation (6). Solving the fourth-order equation $det({\mathsf{\Lambda }}_p-\lambda \mathbb{1})=0$ for $\lambda$ leads to $\lambda \in \{e^t,e^{iw},e^{-iw},e^{-t}\}$. 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 one obtains ${\mathsf{\Lambda }}_p=P{K}_{u,v}{K}_{t,w}{P}^{-1}$ with unipotent $K_{u,v}=exp(\widehat{U}u+\widehat{V}v)$ and semisimple $K_{t,w}=exp(\widehat{T}t+\widehat{W}w)$, Because $K_{t,w}$ is a diagonal matrix containing the four eigenvectors, the system of eigenvectors is given by the matrix Q obeying ${\mathsf{\Lambda }}_{p}Q=Q{K}_{t,w}$. Inserting ${\mathsf{\Lambda }}_p=P{K}_{u,v}{K}_{t,w}{P}^{-1}$ into this eigenvalue equation, after some rearrangements, one obtains This equation for the unknown quantity $P^{-1}Q$ can be solved iteratively, starting with $P^{-1}Q=\mathbb{1}$, i.e., $Q=P$. 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 philosophical 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.

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 $\mathrm{Lor}(1,3)\subset \mathrm{SO}(1,3)$, the elements of which are given by the exponential representation (19) where ${\left(e^{\alpha \beta }\right)}^{\mu }{}_{\nu }={\eta }^{\alpha }{}_{\nu }{\eta }^{\beta \mu }-{\eta }^{\alpha \mu }{\eta }^{\beta }{}_{\nu }$, $\eta =\mathrm{diag}(1,-1,-1,-1)$. This representation can be written in a different form as using an analogy to the electromagnetic field strength tensor $F^{\mu \nu }$ to write where $\overrightarrow{\tau }=({\omega }_{01},{\omega }_{02},{\omega }_{03})$, $\overrightarrow{\omega }=({\omega }_{23},{\omega }_{31},{\omega }_{12})$, and $\overrightarrow{E}=-(e^{01},e^{02},e^{03})$, $\overrightarrow{B}=-(e^{23},e^{31},e^{12})$, or $e^{0i}=-E_i$, $e^{ij}=-{ϵ}_{ijk}{B}_k$ with the convention of lower indices for $\overrightarrow{E}$ and $\overrightarrow{B}$ and related three-vectors, ${ϵ}_{123}=1$. The $3+3$ generators of $\mathrm{Lor}(1,3)$ obey the commutation relations Obviously, the algebra $\mathrm{lor}(1,3)$ is a real algebra. It contains a compact subalgebra $\mathfrak{k}$ related to the $B^i$ which is isomorphic to the compact algebra $\mathrm{so}\left(3\right)$. Actually, $\mathrm{lor}(1,3)$ is in the shape of a Cartan decomposition $\mathfrak{g}=\mathfrak{k}\dot{+}\mathfrak{p}$ characterised by the values $\varphi \left(\mathfrak{k}\right)=\mathfrak{k}$, $\varphi \left(\mathfrak{p}\right)=-\mathfrak{p}$ of an involution $\varphi$. As vector spaces, $\mathfrak{k}$ and $\mathfrak{p}$ are orthogonal, because given a scalar product $(\mathfrak{k},\mathfrak{p})$ invariant under this involution, one obtains However, $[\mathfrak{k},\mathfrak{p}]\ne 0$. Therefore, we used the symbol $\dot{+}$ instead of the symbol ⊕ for the direct sum. The algebra $\mathfrak{g}$ can be transformed to a compact form by using Weyl’s unitary trick. The result is an algebra ${\mathfrak{g}}^{*}=\mathfrak{k}\dot{+}\mathfrak{i}\mathfrak{p}$, where the implications for introducing an imaginary factor will be explained later. In case of $\mathrm{lor}(1,3)$, the involution is given by (matrix indices are suppressed) or $\varphi \left(\overrightarrow{B}\right)=\overrightarrow{B}$, $\varphi \left(\overrightarrow{E}\right)=-\overrightarrow{E}$. Therefore, the compact form of $\mathrm{lor}(1,3)$ is given by the generators $B^i$ and $i{E}^i$ obeying the commutation relations Considered as a real algebra, this algebra is isomorphic to $\mathrm{so}\left(4\right)$. However, the generators are antihermitean, $B_i^{†}=-B_i$, and ${\left(i{E}_i\right)}^{†}=-i{E}_i$ and, therefore, the group is unitary. In general, Weyl’s unitary trick can be seen to lead always to unitary Lie groups.

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 $\mathbb{C}$ (or additional argument) to the algebra symbol. Given a real Lie algebra L, the duplication of this algebra is given by [15] $L+iL$ is still a real vector space. In defining the multiplication of an element $x+iy\in L_C$ with a complex number $\alpha =a+ib\in \mathbb{C}$ by $(a+ib)(x+iy):=(ax-by)+i(bx+ay)$ and the commutator of two elements $x+iy$ and $x^{\prime }+i{y}^{\prime }$ by $L+iL$ constitutes a complex form, denoted by $L_{\mathbb{C}}:=\mathbb{C}{\otimes }_{\mathbb{R}}L$. 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 $\mathrm{so}(1,3)$ into the complex algebra $\mathrm{so}(4,\mathbb{C})$.

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, $K_{\mathbb{C}}=L$. As the duplication is not unique (for instance, a part of the basis elements can be duplicated with i, another part with $-i$), there are also different real forms to a given complex algebra.

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 $\mathrm{so}\left(4\right)$. However, another one is given by with the commutation rules Therefore, the algebra decomposes into two separate algebras which are isomorphic to $\mathrm{su}\left(2\right)$ ($\mathrm{su}\left(2\right)$ is preferred instead of $\mathrm{so}\left(3\right)$ because $A_i$ and ${\overline{A}}_i$ are antihermitean, leading to unitary groups). Turning back to solvable groups, the decomposition into $\mathrm{su}\left(2\right)={\mathrm{span}}_{\mathbb{R}}\left\{A_i\right\}$ and $\mathrm{su}\left(2\right)={\mathrm{span}}_{\mathbb{R}}\left\{{\overline{A}}_i\right\}$ does not yet conform with the definitions given in Equation (30). Looking at the definitions of $T_i^{\epsilon }$, on the one hand, and the definitions of $E_i$ and $B_i$, on the other hand, one obtains and generally, $J_i^{\epsilon }=\frac{1}{2}(-\epsilon E_i+i{B}_i)=i{\overline{A}}_i$, $K_i^{\epsilon }=\frac{1}{2}(\epsilon E_i+i{B}_i)=i{A}_i$. As $A_i$ and ${\overline{A}}_i$ are antihermitean, $J_i^{\epsilon }$ and $K_i^{\epsilon }$ are hermitean and, therefore, constitute decompactified subgroups generated by $exp(i{j}_i{J}_i^{\epsilon }$) and $exp\left(i{k}_i{K}_i^{\epsilon }\right)$. One obtains which is the other justification for the sign notations in $J_{-}^{\epsilon }$ and $K_{+}^{\epsilon }$. Actually, the algebra looks like $\mathrm{sl}(2,\mathbb{R})$ with one generator missing ($J_{+}^{\epsilon }$ or $K_{-}^{\epsilon }$, respectively). In $\mathrm{lor}(1,3)$, these missing generators exist. In $\mathrm{bor}(1,3;p)$, however, the generators are found in the respective algebra with an opposite sign $\epsilon$, while $J_3^{\pm \epsilon }=K_3^{\mp \epsilon }$. Therefore, $\mathrm{bor}(1,3;p)$ splits up into the subalgebras As there is a homomorphism between the two algebras ${\mathrm{sol}}_2^{+}$ and ${\mathrm{sol}}_2^{-}$, both solvable algebras are maximal and, therefore, are Borel subalgebras of the larger algebra $\mathrm{sl}(2,\mathbb{R})$. For a free choice of $\epsilon$ one can represent the two Borel subalgebras as being generated by a solvable part of the set $\{J_3^{\pm \epsilon },J_{+}^{\pm \epsilon },J_{-}^{\pm \epsilon }\}$ of generators of $\mathrm{sl}(2,\mathbb{R})$, thereby skipping the second (redundant) set $\{K_3^{\mp \epsilon },K_{+}^{\mp \epsilon },K_{-}^{\mp \epsilon }\}$. Alternatively, one can use the two sets and skip $\epsilon =+1$. Though the first choice is more intriguing, for this paper we stay with the clearer second one.

Searching for eigenvectors of the set $\{J_3,J_{+},J_{-}\}$ one finds that these eigenvectors are disjoint, as is known for semisimple algebras. The same holds for the set $\{K_3,K_{+},K_{-}\}$. However, for each of the solvable subalgebras ${\mathrm{sol}}_2^{-}$ and ${\mathrm{sol}}_2^{+}$ one obtains only a single common eigenvector. In order to analyse the eigenvector structure, we return to the eigenvectors of Section 3 to which we apply the algebra elements, obtaining For $\{J_3,J_{-}\}$ the common eigenvector is given as a linear combination of ${\ell }_0$ and ${\ell }_2$ while for $\{K_3,K_{+}\}$ the common eigenvector is given by the linear combination of ${\ell }_0$ and ${\ell }_1$. On the other hand, the common eigenvector for $\{J_3,J_{+}\}$ is a linear combination of ${\ell }_3$ and ${\ell }_1$ while the common eigenvector of $\{K_3,K_{-}\}$ is a linear combination of ${\ell }_3$ and ${\ell }_2$. While ${\ell }_0$ (${\ell }_3$) is proportional to the (space-inverted) momentum four-vector p, the interpretation of the eigenvectors ${\ell }_1$ and ${\ell }_2$ deserves more effort. For this, one can return to the circular polarisation [16,17]. The representation 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 $t=\Delta t$. Therefore, the vector ${\ell }_1$ can be identified with a right turn. However, a turn can be identified with handedness or chirality only in combination with a direction of propagation as in case of the circular polarisation by the argument $kz-\omega t$. (In optics, this solution is called left polarised, as looked at from the direction the light comes from (passive direction). In our case, however, we consider the direction of propagation (active direction).) This direction is given by ${\ell }_0$ (or ${\ell }_3$). Therefore, one can interpret (in case of $\epsilon =1$)