# The double copy: from optics to quantum gravity

Chris D. White  1  1Centre for Research in String Theory, School of Physics
and Astronomy, Queen Mary University of London, 327 Mile End Road, London E1
4NS, UK<EMAIL_ADDRESS>
###### Abstract

Recently, an intriguing relationship (the double copy) has been discovered
between theories like electromagnetism, and gravity. This potentially gives us
a new way to think about gravity, and there are also practical applications
involving the efficient calculation of gravitational observables, and also how
to simulate gravity using optical systems. In this tutorial, we will review
what is known about the double copy, and argue that now is the perfect time
for researchers in optics and / or condensed matter to become interested in
this fascinating correspondence.

††journal: osajournal††articletype: Tutorial

QMUL-PH-21-25

##  1 Introduction

Our current understanding of the ultimate building blocks of nature is in
terms of matter, acted on by forces. To the best of our knowledge, there are
only four fundamental forces: electromagnetism, the weak and strong nuclear
forces, and gravity. The first three of these are described by the Standard
Model of Particle Physics, which is a quantum field theory and thus
incorporates the effects of both quantum mechanics and relativity. Gravity, on
the other hand, is best described by General Relativity, which has so far
stubbornly resisted humankind’s attempts to make it quantum. This is
unfortunate, given that quantum gravitational effects are believed to be
important in extreme regions of the universe where GR breaks down e.g. the
centre of black holes, or at the Big Bang itself. However, even classical GR
is at the cutting edge of fundamental physics research: the recent discovery
of gravitational waves by the LIGO experiment [1] provides an entirely new
tool with which to view our universe. Typical signals arise from the the
inspiral and merger of heavy objects such as black holes and neutron stars,
and the precision with which such signals can be understood is limited by our
ability to approximate these physical situations in GR calculations.

The above discussion suggests that new ways of thinking about gravity are
useful for two main reasons: (i) they could generate new conceptual insights
into what a complete (quantum) theory of gravity might look like; (ii) they
might provide clever ways of streamlining difficult calculations, even in the
incomplete theory of gravity (GR) that we know about. The aim of this article
is to review a particularly intriguing way to think about gravity, namely that
it can be seen – in a well-defined sense that we will make precise – as two
copies of a gauge theory (the type of theory describing the other forces in
nature). This is known as the double copy, and was first discovered in the so-
called scattering amplitudes that are related to probabilities for particles
to interact in collider experiments [2, 3, 4]. It has since been extended to
other types of solution, and there are even cases where certain quantities in
GR can be obtained by "copying" counterparts in conventional electromagnetism.
Whilst analogies between electromagnetic and gravitational physics have been
made many times before, the double copy constitutes a very precise
relationship, that opens up the possibility to simulate and explore gravity
using optical or condensed matter systems, in a very systematic way.
Furthermore, the double copy between gauge and gravity theories turns out to
be only one example of a whole host of such copy relationships, that have been
shown to relate quantities in an increasingly complex web of both physical,
and non-physical theories (see e.g. ref. [5] for a wide-ranging review). This
constitutes mounting evidence that our traditional way of thinking about field
theories obscures a profound underlying structure, such that the rules of
quantum field theory that we take for granted may have to be rewritten. We
also do not know if all field theories can be obtained by "copying" other
ones, or whether there is a special set of "copiable" theories, that is picked
out for some sort of special reason.

As the scope and range of applications of the double copy have widened, so has
the network of scientists who are interested in its consequences. The field of
interested people is an interdisciplinary mix of high energy physicists
(trained in either particle physics or string theory); astrophysicists and
cosmologists; and both pure and applied mathematicians. There is even an
entire annual conference devoted to the topic (QCD Meets Gravity), whose most
recent incarnation had over 250 participants from 23 countries! As is common
in academia, most of the barriers for knowledge transfer arise from the lack
of a common language for what is very often the same physics. Indeed, finding
such a language may itself shed light on the origin of the double copy itself.
Furthermore, there is clearly scope for researchers from other branches of
physics – such as optics and condensed matter – to become involved, and to
suggest theoretical methods, or experimental set-ups, that can be used to
further our knowledge of the double copy. Given that this is a young subject
involving unfamiliar territory, there is a high probability of generating
novel and impactful research insights by bringing in well-established ideas
and techniques from other areas. To define the double copy more precisely, we
must first review the various theories that it relates.

##  2 Gauge Theories

All of the forces and matter particles in the Standard Model are described by
fields, whose equations of motion have wave-like solutions. In the quantum
theory, these waves arrive in discrete quanta or lumps, giving rise to the
observed matter and force-carrying particles. For example, the force particles
associated with the electromagnetic and strong nuclear forces are the photon
and gluon respectively. The former is described by the quantum version of
Maxwell’s equations, which in many applications are written in a non-
relativistic form involving separate electric fields $\vec{E}$ and $\vec{B}$.
In high-energy physics applications, however, it is convenient to use a
different language that makes the compatibility of electromagnetism with the
theory of Special Relativity manifest (for a textbook review of the following,
see e.g. ref. [6]). In SR, space and time mix with each other under the
Lorentz transformations that take us between different inertial frames. It
thus makes no sense to separate space and time coordinates, and instead we may
combine them to form so-called 4-vectors:

$x^{\mu}=(t,\vec{x}),$ (1)

where we use natural units such that the speed of light $c=1$ (also Planck’s
constant $\hbar=1$), and the index $\mu\in\\{0,1,2,3\\}$ labels which
component of the 4-vector we are talking about. The 4-vector in eq. (1)
specifies the position of an event in spacetime, and the vector itself
transforms straightforwardly under Lorentz transformations, regarded as
$4\times 4$ matrices. Given eq. (1), it is also convenient to define a vector
with the index downstairs:

$x_{\mu}=(t,-\vec{x})=\eta_{\mu\nu}x^{\nu}.$ (2)

That is, the downstairs 4-vector consists of flipping the spatial components
relative to the upstairs one, and the right-hand side shows that this
operation can be formalised by multiplying the components of the original
4-vector with the metric tensor, expressed in matrix form as

$\eta_{\mu\nu}=\left(\begin{array}[]{rrrr}1&0&0&0\\\ 0&-1&0&0\\\ 0&0&-1&0\\\
0&0&0&-1\end{array}\right).$ (3)

We have also used the Einstein summation convention in eq. (2), where repeated
indices are assumed to be summed over. The convenience of the downstairs index
notation is that one may define a dot product of two 4-vectors

$x\cdot y=x^{\mu}y_{\mu}=\eta_{\mu\nu}x^{\mu}y^{\nu},$ (4)

which turns out to be invariant under Lorentz transformations. Although we
have talked about spacetime positions here, any set of four numbers that mixes
appropriately under Lorentz transformations may be combined to make a
4-vector. A further example is the 4-momentum of a particle:

$p^{\mu}=(E,\vec{p}),$ (5)

where $E$ and $\vec{p}$ are the (relativistic) energy and momentum
respectively. By evaluating the dot product of eq. (5) with itself in a
general frame, and in the rest frame of the particle, one obtains the
relativistic energy-momentum relation

$E^{2}-\vec{p}^{2}=m^{2},$ (6)

where $m$ is the particle mass. In electromagnetism, it turns out that one may
combine the electrostatic potential $\phi$ and magnetic vector potential
$\vec{A}$ into a single 4-vector known as the gauge field:

$A_{\mu}=(\phi,\vec{A}),$ (7)

where the electric and magnetic fields are defined in our notation by

$\vec{E}=-\nabla\phi-\frac{\partial\vec{A}}{\partial
t},\quad\vec{B}=\nabla\times\vec{A}.$ (8)

With these definitions, the usual non-relativistic Maxwell equations can be
shown, after some effort, to arise from the relativistic equations
111Actually, eq. (9) generates only two of the four usual non-relativistic
Maxwell equations. The other two arise from the Bianchi identity
$\partial_{[\alpha}F_{\beta\gamma]}=0$, where square brackets denote
antisymmetrisation over all indices. This identity is an automatic consequence
of the definition of the field strength in eq. (10).

$\displaystyle\partial_{\mu}F^{\mu\nu}$ $\displaystyle=j^{\nu},$ (9)

where $\partial_{\mu}\equiv\partial/\partial x^{\mu}$, and we have defined the
field strength tensor

$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},$ (10)

as well as the current density 4-vector

$j^{\mu}=(\rho,\vec{j})$ (11)

where $\rho$ and $\vec{j}$ are the charge and (3-vector) current densities
respectively.

The theory of electromagnetism has a remarkable rich symmetry known as gauge
invariance: if we subject $A_{\mu}$ to the transformation

$A_{\mu}\rightarrow A^{\prime}_{\mu}=A_{\mu}+\partial_{\mu}\chi,$ (12)

where $\chi$ is a (spacetime-dependent) scalar field, the field strength
tensor of eq. (10) does not change, and thus the equations of electromagnetism
remain invariant. Gauge invariance turns out to be crucial in being able to
formulate a sensible quantum theory of electromagnetism, but also means that
the gauge field corresponding to a given physical system is not unique, but
can be written in infinitely many ways. One may make things precise by
imposing additional constraints on $A_{\mu}$, known as fixing a gauge. A
common gauge choice is the Lorenz gauge

$\partial_{\mu}A^{\mu}=0,$ (13)

for which the first equation in eq. (9) reduces to

$\left(\frac{\partial^{2}}{\partial t^{2}}-\nabla^{2}\right)A_{\mu}=j_{\mu}.$
(14)

In the vacuum case ($j_{\mu}=0$), this reduces to the wave equation, whose
solutions include the well-known plane waves

$A_{\mu}=\epsilon_{\mu}e^{ik\cdot x},\quad k_{\mu}\epsilon^{\mu}=0,$ (15)

where $\epsilon_{\mu}$ is the polarisation vector, and $k_{\mu}$ can be
interpreted as the 4-momentum 222In natural units, the frequency $\omega$ and
energy $E$ are interchangeable, as are the 3-momentum $\vec{p}$ and wave
vector $\vec{k}$.. In quantum field theory, particle states (photons)
correspond to quanta of these plane waves.

Above, we saw how gauge transformations act on the photon field $A_{\mu}$. In
quantum electrodynamics, there is also an electron field $\psi(x^{\mu})$
which, being complex, has a phase at each point in spacetime 333The electron
field is a type of mathematical quantity called a spinor, but the details of
this are irrelevant to our main discussion.. A (local) gauge transformation
acting on the electron is defined to be a phase shift, which may be defined
separately at each point:

$\psi\rightarrow e^{i\theta(x^{\mu})}\psi.$ (16)

There is a nice geometric way to think about this: the phase at each spacetime
point can be represented as an arrow on the unit circle (figure 2(a)), such
that performing a local gauge transformation amounts to rotating this arrow
differently at different points in spacetime.

Figure 1: The phase of the electron field at each point in spacetime can be
thought of as an arrow on the unit circle, where a local gauge transformation
may rotate the arrows at different points by different amounts.

One may show that requiring that the Dirac equation for the electron be
invariant under these local gauge transformations requires the presence of the
field $A_{\mu}$ that transforms as in eq. (12). Thus, electromagnetism can be
completely derived as a consequence of local gauge invariance! The question
naturally arises of whether the other forces in nature can be phrased in terms
of symmetries. Indeed they can, and we may take as an example the strong
force, that acts on fundamental particles called quarks. Quarks carry a type
of charge called colour, which can take three different values (conventionally
labelled as $r$, $g$ and $b$, or "red", "green" and "blue"). We can then think
of the quark field as carrying an abstract "colour arrow" at each point in
spacetime, that tells us how much redness, greenness and blueness there is:
see figure 2.

Figure 2: We can imagine the quark field at each point in spacetime as having
an arrow in an internal colour space, telling us how much of each colour
charge it has.

A local gauge transformation in this case corresponds to rotations of the
colour arrow by different amounts at different spacetime points, and demanding
that the equation of motion of the quark field be invariant under local
rotations of the colour arrow leads to the presence of a gauge field for the
gluon, albeit more complicated than the electromagnetic case above. The group
of all possible colour rotations of the quark field (i.e. rotations in a
3-dimensional complex colour space) is denoted by SU(3), and mathematicians
call such sets of continuous transformations Lie groups. Acting on the quark
field, members of the colour rotation group will be a 3$\times$3 complex
matrix, and it turns out that any such member can be written as

${\bf U}=\exp\left[i\theta^{a}(x^{\mu}){\bf T}^{a}\right],$ (17)

where the matrices ${\bf T}^{a}$ are called generators, and the parameters
$\theta^{a}(x^{\mu})$ label which member of the group we are talking about.
Equation (17) is known as Lie’s theorem, and applies to any continuous group:
note, for example, that the electron phase rotations of eq. (16) have this
form, where there is only one parameter $\theta$, and one generator (the
number 1, which is not explicit!). For SU(3), it turns out that 8 generators
are needed, and thus the index $a$ runs from 1 to 8. The explicit form of the
generator matrices depends on the choice of a suitable basis in colour space,
but the generators for any Lie group can be shown to obey a so-called Lie
algebra

$[{\bf T}^{a},{\bf T}^{b}]=if^{abc}{\bf T}^{c},\quad[{\bf A},{\bf
B}]\equiv{\bf A}{\bf B}-{\bf B}{\bf A}.$ (18)

That is, the commutator of any two generators must itself be equal to a
superposition of generators 444The proof of this result comes from demanding
consistency between eq. (17), and the fact that the product of any two
transformations must itself be a transformation in the group., where the
relevant coefficients $\\{f^{abc}\\}$ are called structure constants, are
unique for any given Lie group, and have the property of being completely
antisymmetric under interchange of any two indices. A familiar example of eq.
(18) from our undergraduate days is for the group SU(2), which arises in the
description of angular momentum. Equation (18) is then the statement that a
commutator of two angular momentum generators is itself a generator of angular
momentum 555The structure constants for SU(2) turn out to be the Levi-Civita
symbol $\epsilon^{abc}$, so that one has e.g. $[J_{x},J_{y}]=iJ_{z}$ for the
angular momentum generators.. Recall that for the phase shifts of
electromagnetism, there was only one generator (the number 1), which clearly
commutes with itself! Thus, the structure constants vanish in
electromagnetism.

In electromagnetism, requiring invariance under phase rotations – which had
one degree of freedom at each point – led to a gauge field $A_{\mu}$. For the
strong force, colour rotation invariance of the quark field ends up leading to
8 separate gauge fields, one for each value of the index $a$ above. We may
then write this as a single field $A_{\mu}^{a}$, where $\mu$ is the spacetime
index, and $a$ the colour index. Despite the abstract nature of the present
discussion – which unavoidably arises from the esoteric nature of the
symmetries of the quark field, and how these must be expressed mathematically
– it is nevertheless possible to understand the extra complications of gluons
with respect to photons in highly physical terms. It turns out that a quark
may change colour by emitting a gluon, such that conservation of colour charge
requires that the gluon itself carry colour charge. The index $a$ in the gluon
field then simply tells us which type of gluon we are talking about.

Armed with the above notation, we can finally state the equations describing
the gluon. We will focus on the case in which the quark fields are absent,
which is known as Yang-Mills theory. One may then define a field strength
tensor

$F_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+gf^{abc}A_{\mu}^{b}A_{\nu}^{c},$
(19)

where $g$ is a number called the coupling constant. It represents the strength
of the strong force, and is the analogue of the electron charge $e$ in
electromagnetism. In terms of this field strength, the vacuum Yang-Mills
equations are 666Similar to the electromagnetic case above, we have ignored a
Bianchi identity that arises from the definition of the field strength tensor.

$\partial^{\mu}F_{\mu\nu}^{a}+gf^{abc}A^{\mu b}F^{c}_{\mu\nu}=0.$ (20)

We see that eqs. (19) and (20) are more complicated than their electromagnetic
counterparts of eqs. (10, 9). First, they involve the structure constants,
which are absent in the electromagnetic case, although their presence here
should not surprise us: the symmetry of the theory (as represented by the
structure constants) is dictating the equations of motion! Secondly, the Yang-
Mills equations are clearly non-linear, in striking contrast to the Maxwell
equations. The physical interpretation of this is that these terms represent
interactions between different gluons, which is entirely to be expected given
that gluons carry colour charge, as noted above. In electromagnetism, the
photon carries no charge, and thus cannot interact directly with itself
777Photons can mutually interact indirectly, by coupling to an intermediate
fermion bubble, but this does not invalidate the above discussion!. We may
further check the consistency of the above equations by showing how
electromagnetism emerges in the appropriate limit. Firstly, we see that all of
the non-linear terms in eqs. (19, 20) involve the structure constants
$f^{abc}$. The latter vanish for electromagnetism, so that the equations
linearise. Secondly, the index $a$ can only take a single value in
electromagnetism, so that eqs. (19, 20) reduce to eqs. (10, 9), as required.

Note that, even in Yang-Mills theory, there are situations where the non-
linear terms vanish or can be ignored, such that $A_{\mu}^{a}$ (for each $a$)
obeys an equation similar to the Maxwell equations. By fixing a gauge, one may
obtain the wave equation of eq. (14). The physical gluon particle then arises
as a quantum of a plane-wave solution.

##  3 Gravity

Having reviewed what a gauge theory is, let us now turn to gravity. This is
described by General Relativity, whose basic idea is that matter and energy
curve the spacetime that they are sitting in. Freely falling test particles
will then follow paths of shortest distance in a curved space, and this
distorted motion then corresponds to the force of gravity. As an analogy,
consider a simple two-dimensional curved space, namely the surface of the
sphere in figure 3. Let us take two people at points $A$ and $B$, and instruct
them both to walk towards the north pole $N$. Each person will feel that they
are walking locally in a straight line with no forces acting upon them.
However, merely the fact that they are walking on a curved surface will lead
to them moving closer together, which looks like an attractive force.

Figure 3: Trajectories of two people walking in a curved space.

To make the above idea precise, we first need to use the language of 4-vectors
to combine space and time. Next, we need to see mathematically how to describe
a curved space. First, let us consider spacetime without gravity, as in the
previous section. Given the spacetime displacement

$dx^{\mu}=(dt,d\vec{x})$

between two events, one may take its dot product with itself to form an
invariant distance

$ds^{2}=dt^{2}-d\vec{x}\cdot d\vec{x}=\eta_{\mu\nu}dx^{\mu}dx^{\nu},$ (21)

where we have rewritten things to explicitly involve the metric tensor of eq.
(3). We thus see that $\eta_{\mu\nu}$ can be thought of as telling us how to
measure distances in spacetime, and it is for this reason that it is called
the "metric" tensor, which in this particular case is known as the Minkowski
metric.

If we now add gravity, spacetime will be distorted, where this distortion will
potentially be different at different spacetime points. We can easily model
this by replacing $\eta_{\mu\nu}\rightarrow g_{\mu\nu}(x^{\mu})$ in eq. (21),
where the dependence of the new metric tensor $g_{\mu\nu}$ on spacetime
position indicates that we have warped our previously flat space to make a
potentially curved one. Different curved spacetimes will have different metric
tensors $g_{\mu\nu}$, and to complete the theory we need equations that tells
us how to determine the metric in all of spacetime that corresponds to a given
distribution of matter and energy. These are the Einstein field equations, and
are extremely complicated to solve in general. Famous solutions include the
possibility of black holes (regions of spacetime from which not even light can
escape), and universes which expand outwards from a finite time in the past
(the Big Bang).

Note that there are many situations in which we are far away from a given
matter distribution, in which case we can try to find an approximate solution
for the gravitational field. To do this, one may write the following ansatz:

$g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu},$ (22)

where $\kappa=\sqrt{32\pi G_{N}}$ is a conventional constant factor containing
Newton’s constant, and $h_{\mu\nu}$ is the graviton field, which represents
the deviation from flat space. Upon substituting eq. (22) into the Einstein
equations, we can collect terms involving successive powers of the small
number $\kappa$, where keeping each additional power amounts to a better
approximation to the exact metric tensor describing the system. This is
perturbation theory, and a similar procedure may be used to find classical
solutions to the non-linear Yang-Mills theory described in the previous
section, where the coupling constant $g$ is the relevant expansion parameter
in that case.

In gauge theory, we saw that the field $A_{\mu}$ is not unique for a given
physical system, but varies upon making gauge transformations as in eq. (12).
Likewise, one may show that the metric tensor of a given spacetime in GR
transforms in the following way under coordinate transformations
$x^{\mu}\rightarrow y^{\mu}$:

$g_{\alpha\beta}(y^{\mu})=\left(\frac{\partial x^{\mu}}{\partial
y^{\alpha}}\right)\left(\frac{\partial x^{\nu}}{\partial
y^{\beta}}\right)g_{\mu\nu}(x^{\mu}).$ (23)

There are infinitely many different coordinate systems we can choose, and thus
infinitely many ways of writing the metric tensor for the same physical
system! As in gauge theory, we may fix things by imposing additional
constraints on the metric. In perturbation theory, this is usually done at the
level of the graviton field itself, and one example is the transverse-
traceless (TT) gauge, in which the graviton satisfies

$\eta^{\mu\nu}h_{\mu\nu}=0,\quad\partial_{\mu}h^{\mu\nu}=0.$ (24)

The vacuum Einstein equations then become

$\left(\frac{\partial^{2}}{\partial t^{2}}-\nabla^{2}\right)h_{\mu\nu}=0,$
(25)

which can immediately be recognised as the wave equation! The solutions
constitute small ripples in the fabric of spacetime, and are precisely the
gravitational waves discovered by LIGO. A particularly straightforward set of
solutions are the plane waves

$h_{\mu\nu}=\epsilon_{\mu\nu}e^{ik\cdot x},$ (26)

where $\epsilon_{\mu\nu}$ is a polarisation tensor. It turns out that
gravitational waves have two polarisation states, as for photons, and the
similarity between eqs. (15, 26) is our first glimpse of an intimate
relationship between gauge theory and gravity 888The comparison between
electromagnetic and gravitational waves has also been explored in ref. [7], in
a way that makes the common physics exceptionally clear.. Quanta of these
waves are known as gravitons, and are the hypothetical particles of quantum
gravity.

##  4 Scattering amplitudes and the double copy

We have seen in the previous section that the wave equation naturally arises
in both gauge theories and gravity, where the force-carrying particles in each
case arise as quanta of plane wave solutions, generalising the origin of the
photon in electromagnetism. In applications of quantum field theory (QFT), a
commonly occuring situation is a scattering experiment, in which two (beams
of) particles interact, and produce a number of other particles in the final
state. In a quantum theory, we cannot predict exactly what will happen, but
can instead calculate the probability of given final states. Each scattering
process is then associated with a complex number called the scattering
amplitude which, as in non-relativistic quantum mechanics, can be defined in
terms of the overlap $\langle f|i\rangle$ between a given initial state
$|i\rangle$ and final state $|f\rangle$. The amplitude is a complex number,
which makes sense given that the incoming and outgoing particles have a wave-
like character, and thus the amplitude must keep track of relative phase
differences. If there are different possibilities for the final state, we must
add them together to form a total amplitude ${\cal A}$, such that the
probability of a given process depends on $|{\cal A}|^{2}$. This is a real
number as it should be, and the fact that different possibilities are added
before squaring means that quantum interference effects are correctly
included.

New methods for calculating amplitudes have blossomed in recent years (see
e.g. ref. [8] for an up-to-date review), but a more traditional way to
calculate them is using Feynman diagrams. These can be viewed as handy space-
time pictures that describe how particles can interact, and an example is
shown in figure 4(a). This shows two gluons (shown as wavy lines) coming
together to form a single intermediate gluon, which then decays to two gluons
in the final state. More generally, there are different symbols for different
types of particle, and they may be coupled together through interaction
vertices in prescribed ways.

Figure 4: (a) Example Feynman diagram for scattering of two gluons to two
gluons; (b) Momentum conservation in a loop diagram.

Feynman diagrams offer much more than a simple way to visualise particle
interactions: there are highly precise Feynman rules that convert each diagram
into an algebraic contribution to the scattering amplitude! To go into these
rules in detail is clearly beyond the scope of this tutorial, but some of the
main ideas are as follows:

  * •

For a given set of particles in the initial and final states, one must draw
all possible diagrams that connect them, with a given number of vertices.

  * •

Each vertex is associated with a power of the coupling constant of the the
theory ($e$, $g$ or $\kappa$ for electromagnetism, Yang-Mills theory or
gravity respectively), plus additional factors involving the momenta of the
interacting particles. Note, then, that diagrams with increasing numbers of
vertices constitute higher orders in perturbation theory (i.e. an expansion in
powers of the coupling).

  * •

Each line carries a 4-momentum $p^{\mu}=(E,\vec{p})$, such that 4-momentum is
conserved at every vertex. When a diagram involves loops, the external momenta
are not sufficient to define all internal momenta. An example is shown in
figure 4(b), where the lines show the direction of momentum flow. In order to
describe the momentum on every line, we have to define the additional loop
momentum $k$, whose possible values must then be summed over according to the
rule for combining all possibilities to get the total amplitude. Given that
each component of the 4-momentum is a continuous variable, this is a
multidimensional integral

$\int\frac{d^{4}k}{(2\pi)^{4}},$

where the normalisation is conventional.

  * •

Internal lines are associated with a factor 999Note that for intermediate
particles, $E$ and $\vec{p}$ are independent degrees of freedom that are not
related by the energy-momentum relation of eq. (6). Such particles are known
as virtual particles in QFT language, and are not physically measurable
directly.

$\frac{\ldots}{p^{2}}\equiv\frac{\ldots}{E^{2}-\vec{p}^{2}},$ (27)

where the numerator is different for each type of particle.

  * •

External particles are associated with polarisation vectors or tensors, as
appropriate.

Calculating amplitudes in a given theory then amounts to knowing what the
exact Feynman rules are, and applying them. This is easier said than done: the
number of Feynman diagrams increases rapidly with either the number of
external particles or loops. Carrying out the integrals over the loop momenta
is also highly non-trivial, and involves interesting connections with
developing branches of pure mathematics (including special function and number
theory). Furthermore, this way of calculating amplitudes makes Yang-Mills
theory and gravity look completely different. As an example, the vertex factor
for three interacting gluons, written in conventional mathematical notation,
has 6 individual terms in it. The 3-graviton vertex instead has over 170
terms! Remarkably, however, a very special structure emerges if we calculate
amplitudes in both theories, and compare the results.

Consider a gauge theory amplitude ${\cal A}_{m}^{(L)}$ for $m$ external
particles, and with $L$ loops. The above description of the Feynman rules
implies that this will take the general form

${\cal
A}_{m}^{(L)}=g^{m-2+2L}\sum_{i}\left(\prod_{l=1}^{L}\frac{d^{4}p_{l}}{(2\pi)^{4}}\right)\frac{n_{i}c_{i}}{\prod_{\alpha_{i}}p_{\alpha_{i}}^{2}}.$
(28)

Here the sum is over distinct diagrams $i$, where the overall power of the
coupling constant is fixed by the number of external particles and loops.
There are integrals over the $L$ loop momenta $\\{p_{l}\\}$, and each term has
a denominator arising from the internal line factors mentioned above. There
will be a dependence on the colour charges of the gluons for each diagram
(represented by $c_{i}$). Finally, there is a so-called kinematic numerator
$n_{i}$ for each diagram, that collects everything else, and which depends on
particle momenta and polarisation vectors. Although the general form of eq.
(28) is correct, the kinematic numerators $n_{i}$ are not unique. For example,
gauge transformations (and generalisations of them to include other types of
field redefinition) will mix up the numerators associated with different
diagrams, such that ${\cal A}_{m}^{(L)}$ remains invariant. However, it was
noticed in 2008 [2] that there is a particular choice for the numerators
$\\{n_{i}\\}$ for a given amplitude, that means that they obey similar
mathematical identities to the colour factors $c_{i}$, which arise from the
fact that the colour degrees of freedom are described by a Lie group. This
itself implies that there must be some mysterious symmetry underlying the
kinematic degrees of freedom of a scattering amplitude, and is known as BCJ
duality. It has to this day remained mysterious, although the nature of the
symmetry can be glimpsed in certain cases [9]. However, once the numerators
$\\{n_{i}\\}$ have been chosen to have this special BCJ-dual form, it turns
out that the formula

${\cal
M}_{m}^{(L)}=\left(\frac{\kappa}{2}\right)^{m-2+2L}\sum_{i}\left(\prod_{l=1}^{L}\frac{d^{4}p_{l}}{(2\pi)^{4}}\right)\frac{n_{i}\tilde{n}_{i}}{\prod_{\alpha_{i}}p_{\alpha_{i}}^{2}}$
(29)

describes a gravity amplitude, where $\tilde{n}_{i}$ is a second set of
kinematic numerators, and the gauge theory coupling constant $g$ has been
replaced by its gravitational counterpart. Even without knowing what any of
the symbols mean, it is clear that eq. (29) is almost identical to eq. (28)!
The only differences are the replacement of coupling constants, and of colour
information by kinematics. This remarkable relationship is the double copy,
and was first presented in refs. [3, 4].

We have been deliberately imprecise above about which gauge and gravity
theories eqs. (28) and (29) correspond to. There are in fact many different
types of gauge theory obtained from Yang-Mills by e.g. incorporating
additional symmetries such as supersymmetry, that relates bosonic and
fermionic degrees of freedom. One may then take the kinematic numerators
$\\{n_{i}\\}$ and $\tilde{n}_{i}$ from different gauge theories, and generate
amplitudes in various different gravity theories, which are themselves
appropriate generalisations of GR 101010The simplest case of pure Yang-Mills
theory copied with itself does not in fact give GR, but GR plus two additional
matter particles. We will see the reason for this below.. It really is amazing
that all of these theories should have amplitudes related by the simple
replacements above: we stress again that using traditional QFT methods, this
structure is entirely invisible.

Above, we started with gauge theory, and replaced colour information by
kinematics. We could instead have done the opposite, and obtained the formula

$\tilde{\cal
A}_{m}^{(L)}=y^{m-2+2L}\sum_{i}\left(\prod_{l=1}^{L}\frac{d^{4}p_{l}}{(2\pi)^{4}}\right)\frac{c_{i}\tilde{c}_{i}}{\prod_{\alpha_{i}}p_{\alpha_{i}}^{2}},$
(30)

where we have relabelled the coupling constant to $y$, and introduced a second
set of colour factors $\tilde{c}$, that may correspond to a different colour
symmetry group in general. This is called the zeroth copy, and also turns out
to yield a scattering amplitude, in so-called biadjoint scalar theory whose
vacuum field equation is:

$\partial^{2}\Phi^{aa^{\prime}}-yf^{abc}\tilde{f}^{a^{\prime}b^{\prime}c^{\prime}}\Phi^{bb^{\prime}}\Phi^{cc^{\prime}}=0.$
(31)

In this case, the field $\Phi^{aa^{\prime}}$ has no spacetime indices (i.e. it
is a scalar), but has two different types of colour charge, where the
quantities $f^{abc}$ snd $\tilde{f}^{a^{\prime}b^{\prime}c^{\prime}}$ are
structure constants in the two different colour symmetry groups. There are no
indications that biadjoint scalar theory is a physical theory by itself.
However, the above correspondences suggest that at least some of the dynamics
of gauge and gravity theories are inherited from the theory of eq. (31).

The above discussion has been very technical, so let us now summarise it in
more pedestrian terms: scattering amplitudes in a ladder of theories
(biadjoint, gauge and gravity) are related by copying correspondences, such
that colour charge information in one theory gets replaced by kinematic
information (e.g. momenta, polarisations) in another. For the fields
themselves, this involves replacing colour charge indices with spacetime
indices, and these correspondences are depicted in figure 5.

Figure 5: Various theories and the relationships between them.
Electromagnetism sits it in the middle, and General Relativity on the right.

The term "double copy" is sometimes used loosely in the literature to refer to
any of the types of correspondence in the figure. Furthermore, there is
mounting evidence that a whole web of interesting theories is related by
double copies, not just the three types of theory considered here [5]. We hope
it is not too hyperbolic to suggest that the double copy promises to
revolutionise our understanding of field theories, given its indication of a
deep underlying connection between very different areas of physics. However,
for this to be true, we need to know how generally figure 5 should be
interpreted. Is it merely an accident for scattering amplitudes, or does it
apply to the "complete" theories, whatever this means? One way to investigate
this is to think of other types of quantity in each theory, and to see whether
they can be matched up. A natural starting point is to consider exact
classical solutions, and this is the subject of the following section.

##  5 The classical double copy

In thinking about classical solutions, perhaps the simplest case to think
about is that of plane waves, which we have already encountered in gauge and
gravity theories in eqs. (15) and (26). To double copy the electromagnetic
plane wave of eq. (15), we can dress it by an arbitrary constant colour vector
$c^{a}$ to make a Yang-Mills field:

$A_{\mu}^{a}=c^{a}A_{\mu}=c^{a}\epsilon_{\mu}e^{ik\cdot x}.$ (32)

Upon substituting this into the Yang-Mills equations, one may show that they
linearise 111111Each non-linear term involves a contraction of two colour
vectors $c^{a}c^{b}$, and structure constants $f^{abc}$. Such contractions
vanish due to the antisymmetry of the latter., and thus reduce to the Maxwell-
like equations

$\partial^{\mu}(\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a})=c^{a}\partial^{\mu}(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})=0.$
(33)

We may thus happily forget about $c^{a}$ (and hence the Yang-Mills nature of
the field), and regard the gauge field as living in electromagnetism only. The
photon has two polarisation states, which are often chosen to correspond to
the two independent circular polarisations 121212In particle physics language,
these are two helicity states of the photon, corresponding to an eigenvalue
$\pm 1$ respectively for the projection of the spin along the direction of
travel.

$\epsilon^{\pm}_{\mu}=\frac{1}{2}(0,\mp 1,i,0).$ (34)

where the normalisation is such that $\epsilon_{+}\cdot\epsilon_{-}=1$. The
graviton also has two polarisation states, and one may verify that the
appropriate polarisation tensors may be written

$\epsilon^{\pm}_{\mu\nu}=\epsilon^{\pm}_{\mu}\epsilon^{\pm}_{\nu},$ (35)

such that the gravitational plane waves of eq. (26) assume the form (for waves
of definite helicity)

$h^{\pm}_{\mu\nu}=\epsilon^{\pm}_{\mu}\epsilon^{\pm}_{\nu}e^{ik\cdot x}.$ (36)

Comparing with eq. (32), we see that this has the form of a function that does
not change (the phase factor $e^{ik\cdot x}$), times two copies of quantities
taken from a gauge theory. Furthermore, one strips away the colour vector in
the gauge theory, and replaces it with a kinematic quantity (a polarisation
vector) to get the gravity solution. Doing the opposite, one would obtain a
biadjoint field

$\Phi^{aa^{\prime}}=c^{a}\tilde{c}^{a^{\prime}}e^{ik\cdot x},$ (37)

which is indeed a plane wave solution of eq. (31). We thus have a series of
fields

$\Phi^{aa^{\prime}}\quad\leftrightarrow\quad
A_{\mu}^{a}\quad\leftrightarrow\quad h_{\mu\nu},$

which provide a realisation of figure 5 for particular classical solutions.
Indeed, this can be related to the double copy for scattering amplitudes
above, in that the incoming and outgoing states in a scattering process are
themselves obtained from plane waves. There is also a more explicit similarity
with the amplitude story: in going from eq. (30) to eqs. (28) and (29), one
must successively remove the colour factors and replace them with kinematic
numerators, whilst leaving the denominators of the expressions intact.
Likewise, in going from eq. (37) to eqs. (32) and (36), one must strip off the
colour vectors and replace them with polarisation vectors, leaving a certain
function ($e^{ik\cdot x}$) alone. The latter therefore seems to play an
analogous role to the denominator factors in an amplitude, and indeed this
analogy can be made more precise [10].

Note that particular combinations of polarisation states have been taken in
eq. (36), but that there are two other choices: in combining the two
polarisation states of two photons, we expect four different combinations. For
the missing two, we can take them to be the (anti-)symmetric combinations

$\frac{1}{2}\left[\epsilon^{+}_{\mu}\epsilon^{-}_{\nu}-\epsilon^{-}_{\mu}\epsilon^{+}_{\nu}\right],\quad\frac{1}{2}\left[\epsilon^{+}_{\mu}\epsilon^{-}_{\nu}+\epsilon^{-}_{\mu}\epsilon^{+}_{\nu}\right].$
(38)

Each of these constitutes a single degree of freedom, and the first and second
combinations can be associated with a (pseudo)-scalar field respectively
131313A pseudo-scalar field differs from a scalar in that it picks up a minus
sign under a parity transformation, that reverses the spatial coordinate
axes.. These are known as the axion and the dilaton, and thus the true double
copy of pure Yang-Mills theory is not General Relativity, but gravity coupled
to these two extra fields. There are additional justifications one can give
for this result. For Feynman diagrams with no loops, it turns out that the
double copy can be derived from string theory, where it reproduces known
results [11], and also makes clear that the axion and dilaton should be
present. This argument does not, however, easily generalise to amplitudes with
loops. However, taking a "product" of gauge fields, on general grounds,
generates a field with two indices, which can be decomposed into its symmetric
traceless, antisymmetric, and trace degrees of freedom. These are known to
mathematicians as irreducible representations of the Lorentz group, and
correspond physically to the graviton, axion and dilaton respectively.

The question naturally arises as to whether the above procedure can be
extended, and indeed it can. The double copy of more general exact classical
solutions was first considered in ref. [10], which considered the infinite
family of so-called Kerr-Schild solutions in GR. For each case, the metric can
be written as in eq. (22), with the graviton having the special form

$h_{\mu\nu}=\phi k_{\mu}k_{\nu},$ (39)

where $\phi$ is a scalar field, and the vector field $k_{\mu}$ has to satisfy

$k^{2}=0,\quad k\cdot\partial k^{\mu}=0.$ (40)

Substituting this ansatz into the Einstein equations of GR, they are found
(after a great deal of effort) to linearise. Thus, they become much easier to
solve, and furthermore any solutions are then known to be exact. Looking at
eq. (39), it is tempting to guess how to take its single copy and zeroth
copies. One could simply remove successive factors of the 4-vector $k_{\mu}$,
and replace them with colour vectors to get

$A_{\mu}^{a}=c^{a}\phi
k_{\mu},\quad\Phi^{aa^{\prime}}=c^{a}\tilde{c}^{a^{\prime}}\phi.$ (41)

Reference [10] proved that, for static solutions at least, the fields of eq.
(41) are solutions of the Yang-Mills and biadjoint equations respectively,
where the field equations are also linearised in each case. Some time-
dependent cases are also known, including the plane waves already mentioned
above! Interestingly, the Kerr-Schild double copy involves solutions of pure
GR, without any contamination from the axion and dilaton. To see this, note
that the field of eq. (39) is symmetric in its indices by construction, and
also traceless due to the conditions on $k^{\mu}$ of eq. (40). It thus
corresponds to the graviton only, and no product of gauge fields involving
$k_{\mu}$ would be able to generate the axion (antisymmetric) or dilaton
(trace degree of freedom).

The Kerr-Schild double copy may look abstract, but it in fact includes some of
the most famous gravitational objects known to physics! A first example is the
Schwarzschild black hole, a static, spherically-symmetric solution which can
be sourced by a pointlike mass $M$ at the origin. Its Kerr-Schild scalar field
and vector are found to be

$\phi=\frac{M}{4\pi r},\quad k^{\mu}=(1,\vec{e}_{r}),$ (42)

where $r$ is the spherical radius, and $\vec{e}_{r}$ a unit 3-vector in the
radial direction. Taking the single copy results in the gauge field

$A_{\mu}=\frac{Q}{4\pi r}k^{\mu}\quad\rightarrow\quad\left(\frac{Q}{4\pi
r},0,0,0\right),$ (43)

where we have performed a gauge transformation on the right-hand side, whose
details may be found in ref. [10]. Recalling that the zeroth component of the
gauge field is the electrostatic potential, we recognise eq. (43) as the gauge
field of a point charge. Thus, the double copy relates a simple point charge
at the origin to a point mass in gravity, which is directly analogous to the
replacement of colour information by kinematics in the double copy for
amplitudes!

As a more non-trivial example, one may consider the Kerr black hole, an
axially symmetric system which can be sourced by a rotating disc of mass in
gravity. Similar to above, its single copy turns out to be a rotating disc of
charge, whose profile matches that of its gravitational counterpart. In figure
6, we show the magnetic field of the single copy of the Kerr black hole. As we
zoom out, the field becomes dipole-like, as one expects given that a rotating
disc of charge looks like a nested set of current loops!

Figure 6: The magnetic field of the single copy of the Kerr (rotating) black
hole, corresponding to a rotating disc of charge. The disc is seen sideways
on, and runs from $x=-1$ to $x=1$.

Other examples of the Kerr-Schild double copy involve magnetic monopoles [12],
accelerating particles [13], electromagnetic vortices [14] and novel solutions
in two spatial dimensions [15]. However, the family of solutions that can be
put into a Kerr-Schild form remains very special, so that it would be
desirable to extend the scope of the classical double copy yet further. To
this end, an alternative procedure known as the Weyl double copy has been
defined [16], that relies on the so-called spinorial formalism of field theory
(reviewed both excellently and encyclopaedically in refs. [17, 18]). This can
indeed be made more general than the Kerr-Schild double copy [19, 20, 21], and
has also been explored in the conventional tensor language [22]. Another
deficiency of the Kerr-Schild approach is that it is highly dependent on a
particular gauge (coordinate system) being chosen in gauge theory (gravity),
which itself is reminiscent of the fact that the double copy for amplitudes
only works for a particular generalised gauge such that the kinematic
numerators $\\{n_{i}\\}$ are BCJ-dual. Alternative formalisms have been
developed that can in principle work in any gauge [23, 24, 25, 26, 27, 28],
where the "unphysical" degrees of freedom in each theory (i.e. corresponding
to gauge redundancy in the fields) can themselves be matched up.

It is not yet known whether a fully general and exact statement of the double
copy can be made, that works for any type of solution, be it classical or
quantum. Attempts to generalise the idea include finding (and trying to copy)
non-linear solutions of the biadjoint theory [29, 30, 31, 32], showing how
exact symmetries in different theories can be related [33, 34, 35], and
looking at how topological properties of solutions can be matched up [36, 37].
If one is willing to forego exact statements and work order-by-order in
perturbation theory, there are by now many different approaches for
calculating classical observables in General Relativity by first calcuating in
a (simpler) gluon theory, and then double copying the results (see e.g. ref.
[5] and references therein). This is continuing to attract intense worldwide
interest due to the applications to gravitational waves. A typical signal
observed by the LIGO experiment arises from the coming together of two heavy
objects such as black holes or neutron stars, in a three stage process: (i) an
inspiral phase, in which the objects gradually orbit closer to each other;
(ii) the merger itself, in which e.g. two black holes combine to make a larger
one; (iii) the ringdown phase, in which the combined heavy object wobbles and
settles down (see ref. [38] for an excellent review of contemporary methods
for describing such processes). The double copy has so far been applied to
step (i), and may potentially be used for step (iii). Step (ii) relies on
complex numerical simulation work, but the expense of this may be reduced if
further improvements can be made to the other two steps.

##  6 What’s in it for optics?

One of the key scientific problems of our age is a tendency towards increasing
specialisation and isolation. Physicists start to significantly diverge from
each other even before they have completed their Masters degrees, and the lack
of a common language between apparently diverse fields such as astrophysics /
cosmology, high energy physics, condensed matter and optics too often impedes
the ability of different types of physicist to work together. In this article,
we have reviewed an intriguing set of correspondences – known collectively as
the double copy – that relate solutions in widely different field theories.
Unlike many of the developments that have preoccupied theoretical high energy
physicists in recent years, the double copy connects actual physical theories
that are directly observed in nature! It is thus only natural to try to widen
the notion of the double copy yet further, and to seek situations that may be
tested in the lab. It is difficult for the present author to definitively
state what these experiments might be, especially given that the aim of this
article is to try to stimulate the kinds of conversations that might lead to
the necessary interdisciplinary work. However, there are some potentially
useful connections to optics / condensed matter that have appeared in recent
years.

Reference [39] showed that classical gravitational waves could in principle be
emulated using quantum-entangled photon pairs. It was shown how to obtain the
appropriate gravitational wave polarisations by combining definite photon
helicity states, but it was also noted that two of the possible combinations
were associated with helicity zero, and thus had to be thrown away. This is in
fact the double copy in all but name, and the two "spurious" photon
combinations are precisely the axion and dilaton we discussed above! Rather
than being discarded, they could be used to simulate the full double copy of a
pure gauge theory. Regarding what to simulate, ref. [39] had in mind the
emulation of gravitational waves moving through a background curved spacetime,
where the latter could be modelled by passing the entangled photons through a
metamaterial. Unknown to the authors was the fact that whether or not the
double copy can be be made exact in a curved spacetime is in fact an open
problem (see e.g. refs. [40, 41, 42, 43, 44] for preliminary investigations),
that in turn may shed light on how the double copy may be applied in
astrophysics and / or cosmology. Further studies have involved scattering
particles on localised wave backgrounds, which could be simulated using e.g.
strong laser pulses [45]. It is highly exciting that the possibility exists of
using table-top experiments to probe one of the most intriguing ideas to
emerge in high energy physics in recent years!

Another connection noticed by the present author is that of the study of
topologically non-trivial states of light. A family of solutions of the
Maxwell equations known as Hopfions have electric or magnetic field lines that
are knotted, and generalisations exist such that the field lines form so-
called torus knots. Gravitational counterparts also exist, and have been
studied in refs. [46, 47, 48, 49, 50, 51], where the latter reference comments
that the EM and gravity solutions can be related by the Weyl double copy
mentioned above. The authors use twistor theory [52, 53, 54] – an elegant set
of mathematical ideas relating algebraic geometry and complex analysis – to
classify knotted radiation solutions in a compact way. This was taken further
in refs. [20, 21], which provided a derivation of the Weyl double copy using
twistor ideas (see also refs. [55, 56] for related ideas). It would be
interesting to know whether experimental efforts exist to study the
electromagnetic knotted solutions in their own right. If so, might they be
combined with the above ideas to simulate gravitational knotted radiation?
What could this teach us?

##  7 Conclusion

Field theories occur in many branches of physics, and notable examples include
the gauge theories underlying three of the four fundamental forces of nature
(including electromagnetism), and General Relativity. Recent years have seen a
flurry of activity regarding a newly discovered correspondence between gauge
and gravity theories, called the double copy. Originating in the study of
scattering amplitudes related to the probability for particles to interact,
the double copy has since been extended to classical solutions, including
those of interest in astrophysics and optics. It has also been shown to apply
to an increasing range of theories, with varying degrees of physical
relevance.

As the remit of the double copy has increased, so have the number of
interested researchers from astrophysics, cosmology, high energy physics and
pure mathematics. Now seems to be the perfect time to try to interest
physicists from optics and / or condensed matter, particularly given recent
indications that the scope and applicability of the double copy may be
amenable to table-top experiments in the lab.

The author humbly hopes that this article will stimulate interested
conversations – and indeed conversations about how to have the conversations –
that are able to surmount the often considerable barriers between different
subfields, which are often merely due to a difference in language. Recent
years have taught us that our traditional approach to quantum field theory is
hiding a deep and profound underlying structure. Will the field of optics
shine the light that guides the way?

## Acknowledgments

I thank Johannes Courtial and Kurt Busch for their encouragement in writing
this article. This work has been supported by the UK Science and Technology
Facilities Council (STFC) Consolidated Grant ST/P000754/1 “String theory,
gauge theory and duality”, and by the European Union Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No.
764850 “SAGEX”.

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