# On the Majorana representation of the optical Dirac equation

Mark R. Dennis1,2, Teuntje Tijssen2 and Michael A. Morgan3 1 School of Physics
and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK  2
H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue,
Bristol, BS8 1TL, UK  3 Physics Department, Seattle University, 901 12th
Avenue, Seattle, WA 98122, USA<EMAIL_ADDRESS>
###### Abstract

We consider the representations of the optical Dirac equation, especially ones
where the Hamiltonian is purely real-valued. This is equivalent, for Maxwell’s
equations, to the Majorana representation of the massless Dirac (Weyl)
equation. We draw analogies between the Dirac, chiral and Majorana
representations of the Dirac and optical Dirac equations, and derive two new
optical Majorana representations. Just as the Dirac and chiral representations
are related to optical spin and helicity states, these Majorana
representations of the optical Dirac equation are associated with the linear
polarisation of light. This provides a means to compare electron and
electromagnetic wave equations in the context of classical field theory.

Keywords: Majorana equation, wave mechanics, classical fields, optical
polarisation

##  1 Introduction

It is now well understood that structured light beams have many analogies with
structured wavepackets in quantum mechanics. This has led to numerous advances
to our understanding of light, such as accelerating Airy wavepackets, optical
vortices, optical orbital angular momentum, and electromagnetic duality as a
gauge transformation, as well as many other examples [1]. Michael Berry has
contributed to many of these advances over the past five decades (e.g. [2,
3]).

In particular, and perhaps as the best example of this analogy, the nature of
optical spin has been explored by directly comparing the Maxwell equations
with the Schrödinger-Pauli or Dirac equation for the spin-half electron. Thus
a transverse, right or left circularly-polarised electromagnetic plane wave,
as a spin $\pm 1$ state, corresponds to a stationary spin $\pm 1/2$ electron,
or more literally, an ultrarelativistic electron with parallel spin and
momentum directions (helicity states).

While this identification reveals significant properties of the spin angular
momentum of light, it is guilty of a deficit in electromagnetic democracy:
circular polarisation is no more fundamental than elliptical or linear
polarisation. What, therefore, of the analogy with the Dirac equation for
linear optical polarisation? We will address this question by comparing light
beams with solutions of the Dirac equation in the _Majorana representation_
[4]. Such Majorana particle solutions, not appearing in the standard model of
particle physics, are less familiar than electrons, although Majorana
quasiparticles have recently received intense interest in condensed matter
physics [5, 6].

As in many of Michael Berry’s contributions to the study of the overlap
between optics and quantum theory, we work in an entirely wave-mechanical
(“first quantised”) setting. The source-free Maxwell equations
$\partial_{t}\boldsymbol{E}=\nabla\times{\boldsymbol{B}}$,
$\partial_{t}\boldsymbol{B}=-\nabla\times{\boldsymbol{E}}$, for divergence-
free electric $\boldsymbol{E}$ and magnetic $\boldsymbol{B}$ fields, can be
cast as a quantum-mechanical Schrödinger-Pauli/Dirac wave equation
$\rmi\partial_{t}\psi=\widehat{H}\psi$ for an appropriate Hamiltonian operator
$\widehat{H}$ and wavefunction $\psi$. We will directly compare solutions to
these Maxwell equations with solutions to the Dirac equation for massless
“electrons” (i.e., general spin-half fermions), which we claim can be
identified as wavefunctions for a single quantum particle (“photon”,
“electron” or “Majorana particle”). We do not consider interactions between
light and matter, and we use natural units such as Planck’s constant $\hbar$
and the speed of light in vacuum $c$ have the value unity.

In the case of the Dirac equation, the wavefunction $\psi$ is a four-component
Dirac spinor, and the Hamiltonian
$\widehat{H}=\boldsymbol{\alpha}\cdot\widehat{\boldsymbol{p}}+m\alpha_{0}=-\rmi\boldsymbol{\alpha}\cdot\nabla+m\alpha_{0}$
depends on a set of four $4\times 4$ Hermitian _Dirac alpha matrices_
$\alpha_{0},\alpha_{x},\alpha_{y},\alpha_{z}$,
$\boldsymbol{\alpha}=(\alpha_{x},\alpha_{y},\alpha_{z})$ and the mass of the
matter wave $m$. Here we’ve used the fact that the canonical momentum operator
is $\widehat{\boldsymbol{p}}=-\rmi\nabla$. In this paper we deal exclusively
with the case of massless electrons. Then, as we’ll see in the next section,
Maxwell’s equations and the Dirac equation

$\rmi\partial_{t}\psi=-\rmi\boldsymbol{\alpha}\cdot\nabla\psi,$ (1.1)

(sometimes called the Weyl equation) share many structural features as systems
of linear first-order partial differential equations in spacetime, with a
multicomponent vector/spinor solution.

Solutions $\psi$ for different representations of the Dirac matrices can be
identified with different states of massive spin-half particles. In the so-
called _Dirac representation_ , the spin operator $\widehat{S}_{z}$ is
diagonal, so the basis states of $\psi$ are spin up and spin down, stationary
electrons. In the _chiral representation_ (also called the Weyl
representation), high energy electrons traveling almost at light-speed
(effectively massless), have basis states given by the helicity states with
spin parallel or antiparallel to their propagation direction.

Finally, in the Majorana representation, the Dirac equation is completely real
($\boldsymbol{\alpha}$ are real and $\alpha_{0}$ is pure imaginary). This is
suited to describe Majorana solutions, which here we take to be eigenfunctions
of the complex conjugation operator, which can be identified as purely real
(eigenvalue $+1$) or purely imaginary (eigenvalue $-1$) [7, 8].

Direct electromagnetic analogues of the Dirac equation (considered by Iwo
Bialynicki-Birula [9], Stephen Barnett [10], see also [11, 12, 13]) emphasize
that the Maxwell equations can be put into a form analogous to (1.1)

$\rmi\partial_{t}\mathcal{F}=-\rmi\boldsymbol{\beta}\cdot\nabla\mathcal{F}$
(1.2)

where $\mathcal{F}$ is some 6-component _Faraday spinor_ whose entries are
linear combinations of the components of electric and magnetic field vectors,
and $\boldsymbol{\beta}=(\beta_{x},\beta_{y},\beta_{z})$ are an appropriate
set of $6\times 6$ matrices, which in fact form a DKP (Duffin-Petiau-Kemmer)
algebra [14, 15, 16, 17, 18, 19], with matrices which, unlike a Clifford
algebra, may be singular.

Bialynicki-Birula’s “photon wave function” is the analogue of the chiral
representation, involving
$\mathcal{F}=(\bi{E}+\rmi\bi{B},\bi{E}-\rmi\bi{B})^{T}$, demonstrating the
analogy between helicity eigenstates of light and ultra-relativistic
electrons/neutrinos. On the other hand, Barnett’s “six-component spinor” (also
see Darwin [20]) is the analogue of the standard Dirac representation,
involving $\mathcal{F}=(\bi{E},\rmi\bi{B})^{T}$, and allows the comparison
between an electron at rest and static $\bi{E}$ and $\bi{B}$ fields.

Of course, one difference between electrons and light particles (“photons”) is
that the latter are massless, and hence there is no need for $\beta_{0}$ (the
analogue of $\alpha_{0}$) in (1.2). Similarly, the massless Dirac equation (or
“Weyl equation”) requires only the three Pauli matrices
$\boldsymbol{\alpha}=(\sigma_{x},\sigma_{y},\sigma_{z})$ as a sufficient set
of Dirac matrices.

A Majorana particle (massless or not), being an eigenstate of complex
conjugation, is always balanced between its positive and negative frequencies,
just like the true representation of electromagnetic fields.

We will construct a representation of the three $\beta_{i}$ matrices analogous
to Majorana’s own construction for the $\alpha_{i}$ matrices to be purely
imaginary. We anticipate the corresponding spinor $\mathcal{F}$ will somehow
pick out linear polarisations as privileged just as the other representations
pick out circular polarisation. For a better comparison, we should prefer a
spherical basis rather than Cartesian, as this generalises to any spin (and is
necessary for any representation of Dirac spinor). We find that the choice of
Majorana representation is not unique, with different choices not only
revealing different aspects of the relation between linear and circular
polarisation, but hopefully offering insight into the nature of Majorana
solutions.

##  2 On the analogy between the Maxwell and Dirac equations

We begin with a review of the standard Dirac wave equation for free massive
electrons (spin-half fermions) [14, 21, 22, 23, 24],

$\rmi\partial_{t}\psi=-\rmi\boldsymbol{\alpha}\cdot\nabla\psi+m\alpha_{0}\psi=\boldsymbol{\alpha}\cdot\widehat{\boldsymbol{p}}\,\psi+m\alpha_{0}\psi=\widehat{H}\psi.$
(2.1)

For equation (2.1) to be appropriately relativistic (i.e., for it to remain
form-invariant under Lorentz transformations), Dirac showed [24] that the four
Dirac matrices $\alpha_{0},\alpha_{x},\alpha_{y},\alpha_{z}$ are required to
satisfy the anticommutation relations of a Clifford algebra

$\alpha_{i}^{2}={\tt 1},\qquad\alpha_{i}\alpha_{j}+\alpha_{j}\alpha_{i}={\tt
0},\qquad i\neq j,\;i,j=0,x,y,z,$ (2.2)

where ${\tt 1}$ is the identity matrix. The fewest number of dimension for
which four such matrices obey the algebra is four, so the Dirac matrices are
$4\times 4$, and the Dirac spinors $\psi$ have four components. A particular
realization of this algebra (2.2) in terms of complex-valued $4\times 4$
matrices is called a _representation_ of the Dirac algebra. As one often shows
in a course on relativistic quantum mechanics, all representations of (2.2)
are unitarily equivalent.

The physical role of a representation of the Dirac matrices is related to
which operators are diagonalised in that representation. For example, in the
_Dirac representation_ (where $\alpha_{0}$ is diagonal), the $z$-component of
spin is diagonal, whereas in the _chiral representation_ , the helicity
operator—the component of spin in the direction of momentum—is diagonal. In
the _Majorana representation_ , the operator
$(\partial_{t}+\rmi\widehat{H}^{\mathrm{M}})$ is required to be real-valued,
which means the matrix elements of the matrices
$\boldsymbol{\alpha}^{\mathrm{M}}$ are real, and the matrix elements of
$\alpha_{\mathrm{0}}^{\mathrm{M}}$ are purely imaginary. In this paper we
refer to the Majorana representation of a wave equation as a ‘Majorana
equation.’ See Pal [25] for a review of Majorana and Weyl fermions.

For the special case of massless electrons with $m=0$, (2.1) is sometimes
referred to as a _Weyl equation_ , for the spinor $\varphi$

$\quad\rmi\partial_{t}\varphi=(\boldsymbol{\alpha}\cdot\widehat{\boldsymbol{p}})\,\varphi.$
(2.3)

Now that the matrix $\alpha_{0}$ is not required, relativistic covariance of
the Weyl equation only requires three independent, anticommuting $\alpha$
matrices. The standard set of Pauli matrices for $i=x,y,z$

$\sigma_{x}=\left(\matrix{0}&1\\\
10\right),\qquad\sigma_{y}=\left(\matrix{0}&\\!\\!\\!-\rmi\\\ \rmi
0\right),\qquad\sigma_{z}=\left(\matrix{1}&0\\\ 0\\!\\!\\!-1\right),$ (2.4)

obey the Clifford algebra (2.2), so they suffice as a set of $\alpha_{i}$
matrices for the Weyl equation (2.3), and hence the _Weyl spinors_ $\varphi$
have only two components.

The _helicity operator_ for a single quantum particle is defined as the
projection of the spin along the direction of the particle’s momentum or
${\boldsymbol{\Sigma}\cdot\widehat{\boldsymbol{p}}}/{|\boldsymbol{\Sigma}||\boldsymbol{p}|}$
where $\boldsymbol{\Sigma}$ is the spin operator, and
$\widehat{\boldsymbol{p}}$ the momentum operator for the particle. For a spin-
half electron where $\boldsymbol{\Sigma}=\frac{1}{2}\boldsymbol{\sigma}$ the
helicity operator is
${\boldsymbol{\sigma}\cdot\widehat{\boldsymbol{p}}}/{|\boldsymbol{p}|}$. The
choice $\boldsymbol{\alpha}=\boldsymbol{\sigma}$ in (2.3) corresponds to Weyl
spinors $\phi^{+}$ with positive helicity, whereas the choice
$\boldsymbol{\alpha}=-\boldsymbol{\sigma}$, which still obeys (2.2),
corresponds to Weyl spinors $\phi^{-}$ with negative helicity. In other words,
the massless spin-half Hamiltonian is $\pm|\boldsymbol{p}|$ times the helicity
operator.

This shows the significance of the _chiral representation_ of the Dirac
equation (2.1). In the massless case, the 4-component Dirac equation reduces
to two uncoupled Weyl equations, one for each helicity. Note that the Pauli
matrices cannot be used as a set of $\alpha_{i}$ matrices for a massless
Majorana equation, since $\sigma_{y}$ is purely imaginary (see [26] for
exceptions.)

Next, we explain how to put the source-free Maxwell wave equations into a
Hamiltonian form analogous to the Weyl (2.3) and massless Dirac (1.1) wave
equations. In an approach going back to Weber and Silberstein in the first
decade of the twentieth century [27, 28, 29], we begin by rewriting the
classical, source-free electromagnetic field as the complex _Riemann-
Silberstein vector_ (RS vector)

$\boldsymbol{F}^{\pm}\equiv\boldsymbol{E}\pm\rmi\boldsymbol{B},$ (2.5)

for which Maxwell’s equations become simply
$\nabla\cdot\boldsymbol{F}^{\pm}=0$ and

$\rmi\partial_{t}\boldsymbol{F}^{\pm}=\pm\nabla\times\boldsymbol{F}^{\pm},$
(2.6)

called the _Riemann-Silberstein (RS) equation_. The RS vector therefore is
divergenceless, and satisfies a Schrödinger-Pauli-like wave equation whose
Hamiltonian $\widehat{H}=\pm\nabla\times$ is first-order and real-valued. For
this reason, $\boldsymbol{F}^{\pm}$ is sometimes identified as the
‘wavefunction’ for electromagnetism [9]. Furthermore, the continuous duality
symmetry in the electromagnetic fields without sources,
$\boldsymbol{E}\to\boldsymbol{E}\cos\tau+\boldsymbol{B}\sin\tau$,
$\boldsymbol{B}\to-\boldsymbol{E}\sin\tau+\boldsymbol{B}\cos\tau$ for
$0\leq\tau<2\pi$, is a global gauge symmetry for the RS equation, since under
this transformation
$\boldsymbol{F}^{\pm}\to\boldsymbol{F}^{\pm}\rme^{\mp\rmi\tau}$, leaving the
RS equation invariant [9, 30, 31, 32, 33, 34, 35].

We can now express the RS equation (2.6) in the form of a Schrödinger-Pauli
wave equation by finding a Hamiltonian form for the curl operator. The cross
product can be expressed by using a 3-vector of $3\times 3$ spin matrices
$\boldsymbol{S}=(S_{x},S_{y},S_{z})$, where

$\quad S_{x}=\left(\matrix{0}&00\\\ 00-\rmi\\\ 0\rmi 0\right),\quad
S_{y}=\left(\matrix{0}&0\rmi\\\ 000\\\ -\rmi 00\right),\quad
S_{z}=\left(\matrix{0}&-\rmi 0\\\ \rmi 00\\\ 000\right),$ (2.7)

that is the components of $S_{i}$ are $(S_{i})_{jk}=-\rmi\varepsilon_{ijk}$.
So for arbitrary vectors $\boldsymbol{u},\boldsymbol{v}$, we have
$-\rmi(\boldsymbol{S}\cdot\boldsymbol{u})\,\boldsymbol{v}=\boldsymbol{u}\times\boldsymbol{v}$.
These matrices are the generators of rotations for vectors in a Cartesian
basis around the $x,y,z$ axes respectively, satisfying the commutation
relations $S_{x}S_{y}-S_{y}S_{x}=\rmi S_{z}$ and cyclic permutations.

Since in terms of the momentum operator the gradient is
$\nabla=\rmi\,\widehat{\boldsymbol{p}}$, we have that the RS Hamiltonian
$\pm\nabla\times$ is $\pm$ the scalar product of the spin angular momentum
with the momentum, $\pm\boldsymbol{S}\cdot\widehat{\boldsymbol{p}}$. For
vector fields (i.e. spin one electromagnetism), the spin operator is simply
$\boldsymbol{\Sigma}=\boldsymbol{S}$. Thus, the RS Hamiltonian is
$\pm|\boldsymbol{p}|$ times the helicity operator, as was also true for the
massless spin-half case. Hence, the RS equation (2.6) for
$\boldsymbol{F}^{\pm}$ is equivalent to

$\rmi\partial_{t}\boldsymbol{F}^{\pm}=\pm(\boldsymbol{S}\cdot\widehat{\boldsymbol{p}})\boldsymbol{F}^{\pm}.$
(2.8)

which is analogous to the Weyl equations (2.3)
$\rmi\partial_{t}\varphi^{\pm}=\pm(\boldsymbol{\sigma}\cdot\widehat{\boldsymbol{p}})\varphi^{\pm}$
for massless electrons/neutrinos.

We’ve now shown that the 3-component RS vector wave-functions
$\boldsymbol{F^{\pm}}$ for light are analogous to the 2-component Weyl spinors
$\varphi^{\pm}$ for massless electrons. It’s natural then to try to find a
wave equation for light which is the analogue to the massless Dirac equation
(1.1). In fact, it turns out that in the early days of relativistic quantum
mechanics, researchers followed similar reasoning, showing that the most
general first order, Lorentz covariant wave equation
$(\beta_{\mu}\partial^{\mu}+m)\psi=0$ for particles of mass $m$ and integral
spin, requires _beta matrices_ (analogous to the Dirac gamma matrices) that
obey the relation
$\beta_{\mu}\beta_{\nu}\beta_{\lambda}+\beta_{\lambda}\beta_{\nu}\beta_{\mu}=\beta_{\mu}\delta_{\nu\lambda}+\beta_{\lambda}\delta_{\nu\mu}$
for all $\mu,\nu,\lambda=0,1,2,3$ [14, 15, 17, 16, 18]. This defines the
_Duffin-Kemmer-Petiau algebra_ (DKP algebra) of $\beta_{\mu}$ matrices. The
DKP algebra for wave equations of integral spin replaces the Clifford algebra
for the case of a wave equation with spin-half. Note that whereas matrices
obeying the Clifford algebra can not be singular, there is no such requirement
here. As for the Dirac equation, the massless case doesn’t require the matrix
$\beta_{0}$; we only need to work with the spatial components $\beta_{i}$. In
this case the DKP algebra reduces to

$\\!\\!\\!\beta_{i}^{3}=\beta_{i},\quad\\!\\!\beta_{i}\beta_{j}\beta_{i}={\tt
0},\quad\\!\\!\beta_{i}\beta_{j}^{2}+\beta_{j}^{2}\beta_{i}=\beta_{i},\quad\\!\\!\beta_{i}\beta_{j}\beta_{k}+\beta_{k}\beta_{j}\beta_{i}={\tt
0}\quad\\!\\!$ (2.9)

for distinct spatial indices $i,j,k=1,2,3$. Note that the first and last
equalities in (2.9) are satisfied by the Clifford algebra, but the middle two
are not. In general then, we seek a set of three such matrices,
$(\beta_{x},\beta_{y},\beta_{z})\equiv\boldsymbol{\beta}$ satisfying the
_optical Dirac equation_

$\rmi\partial_{t}\mathcal{F}\quad=\quad-\rmi(\boldsymbol{\beta}\cdot\nabla)\mathcal{F}\quad=\quad(\boldsymbol{\beta}\cdot\widehat{\boldsymbol{p}})\mathcal{F},$
(2.10)

where the Faraday spinor $\mathcal{F}$ is a multicomponent wavefunction
involving linear combinations of the components of the electric and magnetic
field, which generalizes the RS vector wave-function (2.5). The optical Dirac
equation (2.10) is the wave equation for light analogous to the massless Dirac
equation for the electron. Expression (2.8) satisfies this equation where
$\boldsymbol{\beta}=\pm\boldsymbol{S}$ and $\mathcal{F}=\boldsymbol{F}^{\pm}$,
exactly like the Weyl equation (2.3) satisfies the massless Dirac equation
(1.1) with $\boldsymbol{\alpha}=\pm\boldsymbol{\sigma}$ and
$\psi=\varphi^{\pm}$. In general, DKP algebras are analogous to the Dirac
Clifford algebra for integer spin fields, and were historically introduced to
describe spin zero and spin one meson fields, as closely as possible to the
Dirac equation [14, 15, 18, 19, 36].

Both the Dirac and the optical Dirac equations are covariant under proper
Lorentz transformations. The rotation generators are given by the commutators
of the $\alpha$ and $\beta$ matrices

$\Sigma^{\alpha}_{i}=-\frac{\rmi}{4}\varepsilon_{ijk}\alpha_{j}\alpha_{k}\quad\quad\Sigma^{\beta}_{i}=-\rmi\varepsilon_{ijk}\beta_{j}\beta_{k},$
(2.11)

which are also the expressions for the spin angular momentum in the spin-half
and spin-one cases respectively, and will be useful in what follows.

##  3 Representations of the Dirac and optical Dirac equations

In the representations considered here, the Dirac $\alpha$ matrices are $2+2$
block matrices, with the $\sigma_{i}$ matrices appearing in either diagonal or
off-diagonal blocks. Similarly, according to the analogy between the optical
Dirac equation and the Dirac equation discussed in the previous section, the
DKP $\beta$ matrices will be $3+3$ block matrices consisting of the $S_{i}$
matrices. An efficient way of denoting such block matrices is to use the
_direct product_ (or _Kronecker product_) of two matrices, $C=A\times B$,
where $C_{(a,j),(b,k)}\equiv A_{ab}\,B_{jk}$, so the rows and columns of the
product matrix are labeled with pairs of indices corresponding to the original
matrices. In our application here, matrix $A$ is taken from the set of Pauli
matrices $\sigma_{i}$ where $i=0,1,2,3$, and $\sigma_{0}\equiv\tt 1$ is the
$2\times 2$ identity matrix. It turns out that the $4\times 4$ Dirac
$\alpha_{j}$ matrices may be expressed as a direct product
$\sigma_{i}\times\sigma_{j}$, while the $6\times 6$ optical Dirac DKP
$\beta_{j}$ matrices are expressed as $\sigma_{i}\times S_{j}$, the choice of
$i=0,1,2,3$ depending generally on which representation of the $\alpha$ and
$\beta$ matrices we choose, as we’ll show explicitly below. Also, we’ll use
$\sigma_{i}\times\boldsymbol{\sigma}$ for the Cartesian vector of direct
product matrices. The mixed-product rule $(A\times B)(C\times
D)=(AC)\times(BD)$ follows directly from the definition of the direct product,
and is particularly useful when computing products of $\alpha$ matrices or
products of $\beta$ matrices, such as required in equations (2.11) above, and
(3.2) below.

First, we consider the massless Dirac equation, (1.1). The simplest way of
combining the two helicities into the Dirac equation is to work in the _chiral
representation_ ,

$\boldsymbol{\alpha}^{\mathrm{c}}\equiv\sigma_{3}\times\boldsymbol{\sigma}=\left(\matrix{}{\sigma}&{\tt
0}\\\ {\tt 0}-\boldsymbol{\sigma}\right),$ (3.1)

which is evidently the block diagonal (i.e. direct sum) of the positive and
negative helicity operators considered in the previous section. The Clifford
algebra (2.2) is completed by
$\alpha^{\mathrm{c}}_{0}=\sigma_{1}\times\tt{1}=\left(\smallmatrix{\tt 0}&{\tt
1}\\\ {\tt 1}{\tt 0}\right)$. According to the first of equations (2.11), the
choice (3.1) gives the spin operator as $\boldsymbol{\Sigma}^{\mathrm{c}}={\tt
1}\times\frac{1}{2}\boldsymbol{\sigma}=\frac{1}{2}\left(\smallmatrix\boldsymbol{\sigma}&{\tt
0}\\\ {\tt 0}\boldsymbol{\sigma}\right)$, which means that a rotation about
the $z$-axis is diagonal in this representation.

As discussed in the last section, the massless Dirac equation in chiral
representation separates into two uncoupled equations for two Weyl 2-spinors
$\varphi^{\pm}$, with opposite helicity. (For this reason the chiral
representation is often referred to as the Weyl representation.) The sign of
the helicity is an eigenvalue of what here we call the _chirality matrix_ $X$,

$X\equiv-\rmi\,\alpha_{x}\alpha_{y}\alpha_{z}$ (3.2)

which is easily seen to be the so-called fifth gamma matrix
$\gamma^{5}=\rmi\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$
($\gamma^{0}\equiv\alpha_{0},\gamma^{i}\equiv\alpha_{0}\alpha_{i}$) [21, 22,
23], valid in any representation. In the chiral representation,
$X^{\mathrm{c}}=\sigma_{3}\times\tt{1}=\left(\smallmatrix\tt{1}&\tt 0\\\ \tt
0-\tt{1}\right)$, so the 4-spinor solution to the Dirac equation
$\psi^{\mathrm{c}}$ is composed of Weyl spinors of positive and negative
helicities, $\psi^{\mathrm{c}}=(\varphi^{+},\,\varphi^{-})^{T}$. We note here
that many QFT text books have the reverse ordering (upper components with
negative helicity), corresponding to the opposite signs in (3.1) and
$X^{\mathrm{c}}$ [21, 22].

The matrices representing chirality $X^{\mathrm{c}}$ and $z$-component of spin
$\Sigma_{z}^{\mathrm{c}}$ commute, with the four components of
$\psi^{\mathrm{c}}$ corresponding, in turn, to positive helicity, positive
spin; positive helicity, negative spin; negative helicity, positive spin;
negative helicity, negative spin. For a positive-energy plane wave
corresponding to a massless particle with momentum $p=E>0$ in the $+z$
direction, only positive helicity positive spin corresponding to the spinor
$\psi^{\mathrm{c}}\sim\left(1,\,0,\,0,\,0\right)^{T}\rme^{\rmi(pz-Et)}$, and
negative helicity negative spin with spinor
$\psi^{\mathrm{c}}\sim\left(0,\,0,\,0,\,1\right)^{T}\rme^{\rmi(pz-Et)}$, are
possible. Mixing signs of spin and helicity is only possible for negative
energy plane waves.

In the _Dirac representation_ the $\alpha$ matrices are defined in terms of
the direct product with $\sigma_{1}$, namely

$\boldsymbol{\alpha}^{\mathrm{D}}\equiv\sigma_{1}\times\boldsymbol{\sigma}=\left(\matrix{\tt}0&\boldsymbol{\sigma}\\\
\boldsymbol{\sigma}\tt 0\right),$ (3.3)

with
$\alpha_{0}^{\mathrm{D}}=\sigma_{3}\times\tt{1}=\left(\smallmatrix\tt{1}&\tt
0\\\ \tt 0-\tt{1}\right)$, and (3.2) gives the chirality operator
$X^{\mathrm{D}}=\sigma_{1}\times\tt{1}=\left(\smallmatrix\tt{0}&\tt 1\\\ \tt
1\tt{0}\right)$. As in the chiral representation, (2.11) gives
$\boldsymbol{\Sigma}^{\mathrm{D}}=\tt{1}\times\frac{1}{2}\boldsymbol{\sigma}$,
so again $\Sigma_{z}$ is diagonal. This is the original representation used by
Dirac and is frequently used to describe massive electrons, especially in the
non-relativistic limit. Such a positive-energy electron plane wave is an
eigenstate of spin only if it has zero momentum, given by the spin up and spin
down Dirac spinors $(1,\,0,\,0,\,0)^{T}$ and $(0,\,1,\,0,\,0)^{T}$ (the other
two basis spinors correspond to zero momentum negative energy states for the
electron). The two positive energy massless plane waves discussed in the last
paragraph, with positive and negative helicities, correspond in this
representation to the Dirac spinors
$\psi^{\mathrm{D}}\sim(1,\,0,\,1,\,0)^{T}\rme^{\rmi(pz-Et)}$ and
$(0,\,1,\,0,\,-1)^{T}\rme^{\rmi(pz-Et)}$.

We defer discussing the _Majorana representation_ of the Dirac equation until
the next section, where we work in a spherical basis.

We turn now to representations for the optical Dirac equation (2.10). By
direct analogy with the Dirac equation the chiral representation for the
optical Dirac equation is

$\boldsymbol{\beta}^{\mathrm{c}}\equiv\sigma_{3}\times\boldsymbol{S}=\left(\matrix{}{S}&\tt
0\\\ \tt 0-\boldsymbol{S}\right),$ (3.4)

which corresponds to the positive helicity operator in the upper left block,
and the negative helicity operator in the lower right block. The reader can
easily check that these $\beta$ matrices obey the DKP algebra (2.9). The
6-dimensional Faraday spinor in the chiral representation is given in terms of
the RS vectors (2.5) by
$\mathcal{F}^{\mathrm{c}}=(\boldsymbol{F}^{+},\,\boldsymbol{F}^{-})^{T}$
(appropriately normalized). The upper and lower components of the optical
Dirac equation in chiral representation are the $\pm$ RS equations in (2.8).
This is the form of the optical Dirac equation constructed in Bialynicki-
Birula [9] (and also by Majorana [37], also see [38, 39, 40, 41, 42]). The
chirality operator is clearly
$X^{\mathrm{c}}=\sigma_{3}\times\tt{1}=\left(\smallmatrix\tt{1}&\tt 0\\\ \tt
0-\tt{1}\right)$ (where $\tt 1$ is the $3\times 3$ identity matrix) as with
the chiral representation for the Dirac equation, although there is no $\beta$
matrix analogue to (3.2). According to equation (2.11), the spin operator is
$\boldsymbol{\Sigma}^{\mathrm{c}}=\tt 1\times\boldsymbol{S}$. Notice that
since the 3-vectors here are in the Cartesian basis
$\Sigma_{z}^{\mathrm{c}}=\left(\smallmatrix S_{z}&\tt 0\\\ \tt 0S_{z}\right)$
is not diagonal. We consider the spin basis in the next section.

This chiral representation is the natural one for describing circularly
polarized light. The Faraday spinors corresponding to circularly polarized
plane waves with momentum $p=E>0$ traveling in the $+z$ direction can be
easily found by solving the RS equations (2.8). The solution for the case of
positive helicity (R-handed light) is
$\mathcal{F}^{\mathrm{c}}\sim(\boldsymbol{e}_{+},\boldsymbol{0})^{T}\,\rme^{\rmi(pz-
Et)}$, and for the case of negative helicity (L-handed light) is
$\mathcal{F}^{\mathrm{c}}\sim(\boldsymbol{0},\boldsymbol{e}_{-})^{T}\,\rme^{\rmi(pz-
Et)}$, where $\boldsymbol{e}_{h}$ are the eigenvectors of the helicity
operator $S_{z}$ with eigenvalues $h=\pm 1$. Note the correspondence to the
plane wave solutions for the massless Dirac equation in chiral representation
given above. There, instead of the 3-vectors $\boldsymbol{e}_{h}$, we have
spin-up $\left(\smallmatrix 1\\\ 0\right)$ and spin down $\left(\smallmatrix
0\\\ 1\right)$ Weyl 2-spinors for the case of positive and negative helicity,
respectively. A feature of this chiral description is that the duality gauge
symmetry is simply a phase transformation generated by $-\rmi X^{\mathrm{c}}$.

Moving on to the Dirac representation (or Darwin representation) for the
optical Dirac equation and following the same logic as above, we have

$\boldsymbol{\beta}^{\mathrm{D}}\equiv\sigma_{1}\times\boldsymbol{S}=\left(\matrix{0}&\boldsymbol{S}\\\
\boldsymbol{S}0\right),$ (3.5)

with Faraday spinor
$\mathcal{F}^{\mathrm{D}}=(\boldsymbol{E},\,\rmi\boldsymbol{B})^{T}$, which
was discussed at length by Barnett [10] (also see [43]). Unlike the chiral
representation of the optical Dirac equation, the electric and magnetic field
now occur in separate components of the spinor. We recover the RS equations
(2.8) in this representation by adding and subtracting the upper and lower
components of the Faraday tensor.

It is interesting to note that a static, irrotational electric field
$\boldsymbol{E}$ satisfies Maxwell’s equations with $\boldsymbol{B}=0$, which
is a counterpart to the case of a stationary electron being in a spin
eigenstate. For the case of circularly polarized positive energy plane waves
discussed previously, the Faraday spinors are found to be
$\mathcal{F}^{\mathrm{D}}\sim(\boldsymbol{e}_{+},\boldsymbol{e}_{+})^{T}\,\rme^{\rmi(pz-
Et)}$ and
$\mathcal{F}^{\mathrm{D}}\sim(\boldsymbol{e}_{-},-\boldsymbol{e}_{-})^{T}\,\rme^{\rmi(pz-
Et)}$ respectively, for the case of positive helicity and negative helicity.
This again is completely analogous to the electron result in the Dirac
representation, as given above.

Note that in this representation, $X^{\mathrm{D}}=\sigma_{1}\times{\tt 1}$,
and $-\rmi
X^{\mathrm{D}}\mathcal{F}^{\mathrm{D}}=(\boldsymbol{B},\,-\rmi\boldsymbol{E})^{T}$
also satisfies the optical Dirac equation. Just as in the chiral
representation, $-\rmi X^{\mathrm{D}}$ generates the electromagnetic duality
transformation. Additionally, one can show that $\mathcal{F}^{\mathrm{c}}$ and
$\mathcal{F}^{\mathrm{D}}$ behave the same way under spatial rotations, and in
both cases $\boldsymbol{\beta}\cdot\delta\boldsymbol{v}$ generates boosts by a
velocity $\delta\boldsymbol{v}$.

Finally, since we have considered $\beta$ matrices constructed as direct
products $\sigma_{3}\times\boldsymbol{S}$ and
$\sigma_{1}\times\boldsymbol{S}$, it is natural to consider the final case of
$\sigma_{2}\times\boldsymbol{S}$. In this case, the $\beta$ matrices become
purely real, so (2.10) becomes a purely real equation, which can therefore be
considered a Majorana representation in the Cartesian basis. It is convenient
to include a $-$ sign in the definition of the $\beta$ matrices,

$\boldsymbol{\beta}^{\mathrm{M}}\equiv-\sigma_{2}\times\boldsymbol{S}=\left(\matrix{0}&\rmi\boldsymbol{S}\\\
-\rmi\boldsymbol{S}0\right),$ (3.6)

so that the Faraday spinor is
$\mathcal{F}^{\mathrm{M}}=(\boldsymbol{E},\,\boldsymbol{B})^{T}$. This
representation, which is very similar to the Dirac representation above, has
been considered previously in the literature [20, 44, 45], but not explicitly
as a Majorana equation. The Faraday spinor $\mathcal{F}^{\mathrm{M}}$ is
purely real when $\boldsymbol{E}$ and $\boldsymbol{B}$ are, giving classical
electromagnetism with only real numbers; underpinning the fact that
electromagnetism (massless spin one boson) is trivially Majorana. Finally, we
note that duality transformations here are realized by the direct product of
the 2D rotation $\left(\smallmatrix\cos\tau&\sin\tau\\\
\\!\\!\\!-\sin\tau\cos\tau\right)$ matrix with the $3\times 3$ identity
matrix, generated by $-\rmi X^{\mathrm{M}}=\rmi\sigma_{2}\times{\tt 1}$.

##  4 Spin basis representation and the optical Majorana equation

We have so far considered vectors in the Cartesian basis: 3-vectors were
expressed in a basis of real, mutually orthogonal vectors
$\boldsymbol{e}_{x}$, $\boldsymbol{e}_{y}$, $\boldsymbol{e}_{z}$. The
representation $\boldsymbol{\beta}^{\mathrm{M}}$ is not equivalent to
Majorana’s representation of the Dirac equation, however, since
$\boldsymbol{\beta}^{\mathrm{M}}$ gives real matrices in the Cartesian
representation, whereas Majorana was working in the spin basis. In this
section, we consider the complex spin basis (spherical basis)
$\boldsymbol{e}_{+}$, $\boldsymbol{e}_{0}$, $\boldsymbol{e}_{-}$, of
eigenvectors of $S_{z}$ (with eigenvalues $+1$, $0$, $-1$ respectively. The
unitary matrix transforming from the Cartesian basis to the spin basis
$U=\tfrac{1}{\sqrt{2}}\left(\smallmatrix 1&-\rmi 0\\\ 00\sqrt{2}\\\ 1\rmi
0\right)$, such that

$U\left(\boldsymbol{e}_{x},\boldsymbol{e}_{y},\boldsymbol{e}_{z}\right)^{T}=\left(\boldsymbol{e}_{+},\boldsymbol{e}_{0},\boldsymbol{e}_{-}\right)^{T},$
(4.1)

and the inverse $U^{-1}=U^{\dagger}$. It should be noted this matrix slightly
differs from its counterpart in quantum spin physics, as the standard optical
convention does not quite follow the Condon-Shortley convention (which would
define right-handed circular polarisation as $-\boldsymbol{e}_{+}$).

We can now write down the $S$ matrices in the spin basis, written
$S^{\mathrm{s}}$, by conjugating with respect to these unitary matrices,
$\boldsymbol{S}^{\mathrm{s}}=U\boldsymbol{S}U^{\dagger}$,

$S^{\mathrm{s}}_{x}=\frac{1}{\sqrt{2}}\left(\matrix{0}&-10\\\ -101\\\
010\right),\quad S^{\mathrm{s}}_{y}=\frac{1}{\sqrt{2}}\left(\matrix{0}&\rmi
0\\\ -\rmi 0-\rmi\\\ 0\rmi 0\right),\quad
S^{\mathrm{s}}_{z}=\left(\matrix{1}&00\\\ 000\\\ 00-1\right).$ (4.2)

With this choice, of course, $S^{\mathrm{s}}_{z}$ becomes diagonal (and real),
with $S^{\mathrm{s}}_{x}$ real and $S^{\mathrm{s}}_{y}$ pure imaginary, just
like the Pauli matrices. These real and imaginary assignments are always
necessary since in the angular momentum algebra formalism for any spin,
raising and lowering operators are given by
$\tfrac{1}{\sqrt{2}}(S^{\mathrm{s}}_{x}\pm\rmi S^{\mathrm{s}}_{y})$, which
must be real-valued matrices.

Considering the Dirac equation, one can follow Majorana’s construction [4] of
a real representation of the $\alpha$ matrices by choosing real $\alpha_{x}$
and $\alpha_{z}$ analogously to an existing representation, such as the Dirac
representation, $\alpha^{\mathrm{M}}_{x}=\sigma_{1}\times\sigma_{x}$,
$\alpha^{\mathrm{M}}_{z}=\sigma_{1}\times\sigma_{z}$. The Clifford algebra
rules (2.2), together with the restriction to real values, give exactly one
choice for $\alpha_{y}^{\mathrm{M}}$, namely
$\alpha_{y}^{\mathrm{M}}=\sigma_{3}\times\tt{1}=\alpha_{0}^{\mathrm{D}}$.
Furthermore, $\alpha_{0}^{\mathrm{M}}$ must be purely imaginary in the
Majorana representation, and is therefore equal to the Dirac representation
$\alpha_{y}^{\mathrm{D}}$. Equation (3.2) gives the chirality operator as
$X^{\mathrm{M}}=\sigma_{3}\times\sigma_{y}$. Clearly, this Majorana
representation is not unique; it would have been just as possible to start
with the analogous chiral representation for
$\alpha_{x}^{\mathrm{M}}=\sigma_{3}\times\sigma_{x}$ and
$\alpha_{z}^{\mathrm{M}}=\sigma_{3}\times\sigma_{z}$, for which then the
unique, real-valued $\alpha_{y}^{\mathrm{M}}$ is
$\alpha_{y}^{\mathrm{M}}=\sigma_{1}\times\tt{1}=\alpha_{0}^{\mathrm{c}}$, and
it follows that $\alpha_{0}^{\mathrm{M}}=\alpha^{\mathrm{c}}_{y}$ and
$X^{\mathrm{M}}=\sigma_{1}\times\sigma_{y}$.

Following the analogous procedure for the $\beta$ matrices, we find the DKP
algebra has a purely real representation

$\beta^{\mathrm{sM1}}_{x}=\sigma_{1}\times
S^{\mathrm{s}}_{x},\qquad\beta^{\mathrm{sM1}}_{z}=\sigma_{1}\times
S^{\mathrm{s}}_{z},\qquad\beta^{\mathrm{sM1}}_{y}=\sigma_{2}\times
S^{\mathrm{s}}_{y},$ (4.3)

where here we’ve assumed the form of the first two matrices, and then used the
DKP algebra (2.9) to find a unique, real-valued $\beta^{\mathrm{sM1}}_{y}$.
The most general special unitary transformations which gives these $\beta$
matrices (choosing to transform from the Cartesian basis Majorana
representation $\boldsymbol{\beta}^{\mathrm{M}}$ defined in Section 3) is a
one-parameter group, parameter $\tau$ results in a Faraday spinor with
components

$\displaystyle\mathcal{F}^{\mathrm{sM}1}$ $\displaystyle=$
$\displaystyle\tfrac{1}{\sqrt{2}}\left\\{\left(E_{x}+E_{y},\,\sqrt{2}E_{z},\,E_{x}-E_{y},\,-B_{x}+B_{y},\,-\sqrt{2}B_{z},\,-B_{x}-B_{y}\right)^{T}\\!\\!\cos\tau\right.$
(4.4)
$\displaystyle+\left.\left(B_{x}+B_{y},\,\sqrt{2}B_{z},\,B_{x}-B_{y},\,E_{x}-E_{y},\,\sqrt{2}E_{z},\,E_{x}+E_{y}\right)^{T}\\!\\!\sin\tau\right\\}.$

Here the freedom of the similarity transformation reproduces the duality gauge
symmetry. Alternatively, with the choice $\tau=0$, we get $-\rmi
X^{\mathrm{M}1}$ acting on the first spinor in (4.4) gives the second spinor.

We now make some observations. Although we started from the spin basis, the
first and third components here are those of the Cartesian components of
$\boldsymbol{E}$ in the basis
$\boldsymbol{e}_{\nearrow}=(\boldsymbol{e}_{x}+\boldsymbol{e}_{y})/\sqrt{2}$,
$\boldsymbol{e}_{0}=\boldsymbol{e}_{z}$ and
$\boldsymbol{e}_{\searrow}=(\boldsymbol{e}_{x}-\boldsymbol{e}_{y})/\sqrt{2}$,
i.e. the standard Cartesian basis rotated $45^{\circ}$ around the $z$-axis,
and similarly for the fourth and sixth components involving the magnetic
field. Strikingly, despite ostensibly working in the spin basis, this is a
real, Cartesian basis, and
$\boldsymbol{e}_{\nearrow},\boldsymbol{e}_{\searrow}$ form a third mutually
unbiased basis, together with $\boldsymbol{e}_{x},\boldsymbol{e}_{y}$ and
$\boldsymbol{e}_{+},\boldsymbol{e}_{-}$ in the transverse plane. Therefore, as
expected, enforcing a Majorana-like condition on the optical Dirac equation in
the spin basis has given us linear polarisation, as anticipated, but in a
different basis from the original Cartesian basis
$\boldsymbol{e}_{x},\boldsymbol{e}_{y}$.

Alternatively, we can choose $\beta_{x}$ and $\beta_{z}$ from the chiral
representation to construct a second Majorana-like real representation in the
spin basis,

$\beta^{\mathrm{sM2}}_{x}=\sigma_{3}\times
S^{\mathrm{s}}_{x},\qquad\beta^{\mathrm{sM2}}_{z}=\sigma_{3}\times
S^{\mathrm{s}}_{z},\qquad\beta^{\mathrm{sM2}}_{y}=\sigma_{2}\times
S^{\mathrm{s}}_{y},$ (4.5)

and now the most general Faraday spinor is

$\displaystyle\mathcal{F}^{\mathrm{sM}2}$ $\displaystyle=$
$\displaystyle\tfrac{1}{\sqrt{2}}\left\\{\left(E_{x}+B_{y},\,\sqrt{2}E_{z},\,E_{x}-B_{y},\,-E_{y}-B_{x},\,-\sqrt{2}B_{z},\,E_{y}-B_{x}\right)^{T}\\!\\!\cos\tau\right.$
(4.6)
$\displaystyle\left.\\!+\left(-E_{y}+B_{x},\,\sqrt{2}B_{z},\,E_{y}+B_{x},\,E_{x}-B_{y},\,\sqrt{2}E_{z},\,E_{x}+B_{y}\right)^{T}\\!\\!\sin\tau\right\\}.$

Based on $\sigma_{3}$, this representation involves combinations of electric
and magnetic field components similar to helicity states, but in a new way
with only real coefficients, sharing the real-valuedness of $\boldsymbol{E}$
and $\boldsymbol{B}$. Setting $\tau=0$ initially, we can see that the first
entry combines $E_{x}$ and $B_{y}$ – which are equal for a linearly polarised
electromagnetic wave traveling in $+z$, as are $E_{y}$ and $-B_{x}$ from the
last entry. On the other hand, $E_{x}-B_{y}$ and $E_{y}+B_{x}$ both must
vanish for a linearly polarised plane wave in $+z$, regardless whether the
field is real-valued or complex-valued. The first and last components vanish
for a field propagating in $-z$, again irrespective of whether fields are
real- or complex-valued. Therefore, instead of picking _helicity_ eigenstates
from $z$-propagating plane waves, this second spin basis Majorana
representation detects the _direction of linear momentum_ , based on the
orientation of the electric and magnetic fields!

Elliptic or circular polarisation in the $xy$-plane has the same zero
components in entries 2,3,4,5 of the Faraday spinor, but both the first and
last entries are now complex numbers; similar to helicity, an arbitrary
polarisation of a plane wave with a given direction is a superposition of
orthogonal linear polarisations (as here), or opposite circular polarisations.
The duality angle $\tau$ here rotates the basis choice of these electric-
magnetic components around the $z$-axis, a symmetry always broken on choosing
a direction of linear polarisation.

##  5 Discussion

We have explored various representations of the optical Dirac equation
involving the integral spin DKP algebra, and found a strong analogy to the
Dirac equation for the massless electron. In fact, since electromagnetic
fields (and, upon second quantization, photons) are intrinsically real, they
are automatically their own antiparticles in the formalism. Maxwell’s
equations therefore display a greater similarity to the Majorana
representation of the massless Dirac equation, than to the Dirac equation for
a massive electron.

The main feature was two new Majorana representations, one with Faraday spinor
components being linear polarisation components, but in the mutually unbiased
linearly polarised basis (in the transverse plane); the other roughly
analogous to the helicity representation, but again favouring real
superpositions of electric and magnetic fields which might vanish for certain
linear polarisations.

In the standard optics framework, monochromatic fields lead to time dependence
being suppressed, and fields are often represented by their complex-valued,
positive frequency parts. These naturally resemble complex-valued quantum
matter fields, which are complex in the standard quantum formalism due to the
positive mass and nonzero charge of the electron, although photons are
massless and charge-free. Thus, in the quantum formalism, classical
electromagnetic fields are real-valued, for which it is easier to represent
linear polarisation.

We have avoided giving a physical interpretation to negative-frequency waves.
Of course, any observable effects properly manifest through the energy density
of the field, which is provided by the $00$ component of the fields’
relativistic energy-momentum tensor. The expression of this tensor is
fundamentally different depending on whether the field is integer or half-
integer [46], although there are analogues in the separation of spin and
orbital parts [3].

The main historical reason for the development of the DKP algebra used here
was to describe massive spin one fields (i.e. satisfying the Proca equation),
and a natural extension of this work would be to build an understanding of the
effect of including mass in electromagnetic-like phenomena. However, including
an extra $\beta_{0}$ matrix to the DKP algebra significantly changes the
representation, and the Dirac equation-like analogue to the Proca equation,
whose field quantity is the analogue of the 4-potential rather than
$\boldsymbol{E}$ and $\boldsymbol{B}$, requires $10\times 10$ DKP matrices,
significantly larger than anything discussed here. It would be nevertheless
interesting to explore Weyl, Dirac, and Majorana-like representations of this
more complicated DKP algebra (and the limit, in this algebra, $m\to 0$ [36]).

Much of the current interest in Majorana particles is focused towards
quasiparticle excitations in condensed matter physics (e.g. [47, 48, 49, 50,
51], especially associated with topologically protected edge state modes.
Photonic analogues of various of these have been proposed (e.g. [52, 53, 54,
55]). It would potentially be interesting to take the case of the optical
Majorana equation into materials and interfaces, especially as the natural $s$
and $p$ polarisations associated with reflection and refraction at the edges
of dielectric materials are linear polarisations, i.e. Majorana-like states.
The true exotic nature of Majorana particles—their Majorana mass—may yet arise
in an interpretation of the behavior of light in a suitable medium.

For free-space electromagnetism, however, our analysis merely emphasizes the
fact that we can choose the basis in which we describe electromagnetic
phenomena. The circular or helical bases are analogous to the chiral and Dirac
representations of electrons, and so emphasize spin and angular momentum
aspects. We’ve shown here that the basis states of linear polarization are
analogous to the Majorana representation, which we hope may strengthen the
power of analogies between different areas of physics.

This research was supported by the EPSRC and the Leverhulme Trust Research
Programme RP2013-K-009, SPOCK: Scientific Properties of Complex Knots.

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