# Conformal field theory for particle physicists

Marc Gillioz

(SISSA, via Bonomea 265, 34136 Trieste, Italy111Former affiliation. Current
correspondence address<EMAIL_ADDRESS>
###### Abstract

This is a set of introductory lecture notes on conformal field theory. Unlike
most existing reviews on the subject, CFT is presented here from the
perspective of a unitary quantum field theory in Minkowski space-time. It
begins with a non-perturbative formulation of quantum field theory (Wightman
axioms), and then gradually focuses on the implications of scale and special
conformal symmetry, all the way to the modern conformal bootstrap. This
approach includes topics often left out, such as subtleties of conformal
transformations in Minkowski space-time, the construction of Wightman
functions and time-ordered correlators both in position- and momentum-space,
unitarity bounds derived from the spectral representation, and the appearance
of UV and IR divergences.

_These notes were created for a graduate class on conformal field theory at
the Institute for Theoretical Physics of the University of Bern, taught in the
spring semester of 2022. They are aimed at physicists with a good basic
knowledge of quantum field theory but without prior experience in conformal
field theory._

This is a preprint of the following work: Marc Gillioz, “Conformal Field
Theory for Particle Physicists: From QFT Axioms to the Modern Conformal
Bootstrap”, 2023, SpringerBriefs in Physics, reproduced with permission of
Springer Nature Switzerland AG. The final authenticated version is available
online at: https://doi.org/10.1007/978-3-031-27086-4.

###### Contents

  1. 1 Introduction
  2. 2 Classical conformal transformations
    1. 2.1 Infinitesimal transformations
    2. 2.2 The conformal algebra
    3. 2.3 Finite transformations
    4. 2.4 Compactifications
    5. 2.5 Minkowski space-time
    6. 2.6 Conformal symmetry in classical field theory
  3. 3 Conformal quantum field theory
    1. 3.1 Non-perturbative quantum field theory
    2. 3.2 Wightman functions
    3. 3.3 Spectral representation
    4. 3.4 Scale symmetry
    5. 3.5 Special conformal symmetry
    6. 3.6 UV/IR divergences and anomalies
  4. 4 Conformal correlation functions
    1. 4.1 From Minkowksi space-time to Euclidean space
    2. 4.2 From Euclidean space to embedding space
    3. 4.3 3-point functions
    4. 4.4 4-point functions
  5. 5 State-operator correspondence and OPE
    1. 5.1 The OPE in quantum field theory
    2. 5.2 The state/operator correspondence
    3. 5.3 The conformal OPE
  6. 6 The conformal bootstrap
    1. 6.1 Conformal blocks
    2. 6.2 OPE convergence
    3. 6.3 The crossing equation and simple solutions
    4. 6.4 The numerical bootstrap
    5. 6.5 Example: the Ising model in 3 dimensions
    6. 6.6 Other conformal bootstrap results
  7. 7 Conclusions
  8. Acknowledgments

##  1 Introduction

Conformal field theory is a ubiquitous subject in modern theoretical physics.
Every local quantum field theory approaches a CFT in the large- and small-
distance limits,222Massless free theories are conformal. The “empty”, low-
energy limit of massive theories can also be viewed as a special kind of CFT.
and it even plays a key role in the study of quantum gravity through the
AdS/CFT correspondence. CFT is also one of the rare frameworks in which
quantum field theory can be studied outside the realm of perturbation theory.
There a several excellent modern reviews on the subject [1, 2, 3, 4, 5], and
large parts of these lecture notes are directly inspired by them.

Most introductory courses on CFT treat the conformal group as a whole. This
approach usually requires working in Euclidean space (we will see why),333An
exception is Slava Rychkov’s unpublished lecture notes on _Lorentzian methods
in conformal field theory_ , available at https://courses.ipht.fr/node/226.
and the connection with “traditional” quantum field theory appears very late
in the course, if at all. This can be frustrating for a particle physicist. We
propose to start here instead with the following definition:

CFT $\displaystyle=\text{relativistic QFT}$ (1.1)
$\displaystyle\quad+\text{scale symmetry}$ $\displaystyle\quad+\text{special
conformal symmetry},$

and go through each part one after the other. The first part is quantum field
theory in flat Minkowski space-time. We expect the reader to be familiar with
it, but our attention will not be restricted to theories that have a nice
classical limit and a perturbative definition. We will therefore go through
some of the basics of QFT in section 3, namely the Wightman axioms, but
without unnecessary mathematical rigor.

Scale symmetry will come next. Particle physicists typically also have a good
physical intuition of it from the renormalization group. The main novelty in
CFT is that scale symmetry forbids the existence of particles with a definite
mass; instead, in a scale-invariant theory, there are states of _any_ energy
(unless we are in the very special case of a free theory).

Finally, special conformal symmetry will be discussed last. It nearly always
comes along with scale symmetry, but it is vastly more powerful. This will
lead us to the study of conformal correlation functions in section 4, to the
discussion of the operator product expansion (OPE) and its very nice features
in section 5, and finally to the conformal bootstrap in section 6, including
an example of a strongly-coupled theory that has been solved by symmetry
principles only.

Before diving into the _quantum_ theory, however, we begin the course with a
look at _classical_ conformal transformations in section 2. It is convenient
to work in $d$-dimensional space-time and think of $d=4$ as a particular
realization of the more general framework. Lorentz indices $\mu$, $\nu$,
$\ldots$ therefore run between $0$ and $d-1$, with $x^{0}$ being the time
component of the vector $x^{\mu}$, and $p^{0}$ the energy component of the
momentum $p^{\mu}$. Even though we ultimately care about unitary quantum field
theory in Minkowski space-time, we will need to establish a connection with
the same theory defined in flat Euclidean space. For this reason, we
(unfortunately) work with the mostly-plus metric convention $(-,+,+,\dots,+)$,
so that going from Minkowski space-time to Euclidean space is simply achieved
by a rotation of the time coordinate in the complex plane.

##  2 Classical conformal transformations

One of the most fundamental principles of physics is the independence of the
reference frame: observers living at different points might have different
perspectives, but the underlying physical laws are the same. This is true in
space (invariance under translations and rotations), but also in space-time
(invariance under Lorentz boosts).

###  2.1 Infinitesimal transformations

In mathematical language, this means that if we have a coordinate system
$x^{\mu}$, the laws of physics do not change under a transformations

$x^{\mu}\to x^{\prime\mu}.$ (2.1)

This principle applies to all maps that are invertible (isomorphisms) and
differentiable (smooth transformations), hence it is usually called
_diffeomorphism_ invariance. Being differentiable, the transformation (2.1)
can be Taylor-expanded to write

$x^{\mu}\to x^{\prime\mu}=x^{\mu}+\varepsilon^{\mu}(x),$ (2.2)

in terms of an infinitesimal vector $\varepsilon^{\mu}$ (meaning that we will
always ignore terms of order $\varepsilon^{2}$).

In addition to the coordinate system, the description of a physical system
requires a way of measuring distances that is provided by a metric
$g_{\mu\nu}(x)$. Distances are measured integrating an infinitesimal line
element whose square equals

$ds^{2}=g_{\mu\nu}(x)dx^{\mu}dx^{\nu}.$ (2.3)

Since all observers should agree on the measure of distances, we must have

$g^{\prime}_{\mu\nu}(x^{\prime})dx^{\prime\mu}dx^{\prime\nu}=g_{\mu\nu}(x)dx^{\mu}dx^{\nu},$
(2.4)

Here $g_{\mu\nu}$ could be the Euclidean metric $\delta_{\mu\nu}$ or the
Minkowski metric $\eta_{\mu\nu}$; for simplicity we only consider the case in
which $g_{\mu\nu}$ is flat, i.e. $\partial_{\alpha}g_{\mu\nu}=0$. In this
case, we can write

$\displaystyle g^{\prime}_{\mu\nu}$
$\displaystyle=g_{\alpha\beta}\frac{\partial x^{\alpha}}{\partial
x^{\prime\mu}}\frac{\partial x^{\beta}}{\partial x^{\prime\nu}}$ (2.5)
$\displaystyle=g_{\alpha\beta}\left(\delta^{\alpha}_{\mu}-\partial_{\mu}\varepsilon^{\alpha}\right)\left(\delta^{\beta}_{\nu}-\partial_{\nu}\varepsilon^{\beta}\right)$
$\displaystyle=g_{\mu\nu}-\left(\partial_{\mu}\varepsilon_{\nu}+\partial_{\nu}\varepsilon_{\mu}\right).$

If we require the different observers to also agree on the metric, then we
must have $g^{\prime}_{\mu\nu}=g_{\mu\nu}$, which gives a constraint on what
kind of coordinate transformations are possible: they must satisfy

$\partial_{\mu}\varepsilon_{\nu}+\partial_{\nu}\varepsilon_{\mu}=0.$ (2.6)

This condition admits as most general solution

$\varepsilon^{\mu}=a^{\mu}+\omega^{\mu}_{\leavevmode\nobreak\ \nu}x^{\nu},$
(2.7)

where $a^{\mu}$ is a constant vector and
$\omega_{\mu\nu}=g_{\mu\rho}\omega^{\rho}_{\leavevmode\nobreak\ \nu}$ an
antisymmetric tensor, i.e. $\omega_{\nu\mu}=-\omega_{\mu\nu}$. The
transformation

$x^{\mu}\xrightarrow{P}x^{\mu}+a^{\mu}$ (2.8)

is obviously a translation and

$x^{\mu}\xrightarrow{M}\left(\delta^{\mu}_{\nu}+\omega^{\mu}_{\leavevmode\nobreak\
\nu}\right)x^{\nu}$ (2.9)

a rotation/Lorentz transformation around the origin $x=0$: the matrix
$R^{\mu}_{\leavevmode\nobreak\
\nu}=\delta^{\mu}_{\nu}+\omega^{\mu}_{\leavevmode\nobreak\ \nu}$ satisfies
$R^{\mu}_{\leavevmode\nobreak\
\alpha}g^{\alpha\beta}R^{T\nu}_{\beta}=g^{\mu\nu}$. The composition of these
two operations generates the Poincaré group. This is the fundamental symmetry
of space-time underlying all relativistic quantum field theory. It is a
symmetry of nature to a very good approximation, at least up to energy scales
at which quantum gravity becomes important.

However, one can also consider the situation in which the two observers use
different systems of units, i.e. they disagree on the overall definition of
scale but agree otherwise on the metric being flat. In this case we must have
$g^{\prime}_{\mu\nu}\propto g_{\mu\nu}$, and therefore the constraint (2.5)
becomes

$\partial_{\mu}\varepsilon_{\nu}+\partial_{\nu}\varepsilon_{\mu}=2\lambda
g_{\mu\nu},$ (2.10)

for some positive real number $\lambda$, with the most general solution

$\varepsilon^{\mu}=a^{\mu}+\omega^{\mu}_{\leavevmode\nobreak\
\nu}x^{\nu}+\lambda x^{\mu}.$ (2.11)

The new infinitesimal transformation is

$x^{\mu}\xrightarrow{D}(1+\lambda)x^{\mu}.$ (2.12)

It is a scale transformation, also known as dilatation. Note that scale
symmetry is not a good symmetry of nature: there is a fundamental energy scale
on which all observers must agree (this can be for instance chosen to be the
mass of the electron). Nevertheless, there are systems in which this is a very
good approximate symmetry, making it worth studying.

If one pushes this logic further, in a scale-invariant world in which
observers have no physical means of agreeing on a fundamental scale, they
might even decide to change their definition of scale as they walk around, or
as time passes. This would correspond to a situation in which the metric
$g^{\prime}_{\mu\nu}$ of one observer can differ from the original metric
$g_{\mu\nu}$ by a function of space-time:

$g^{\prime}_{\mu\nu}(x)=\Omega(x)g_{\mu\nu}.$ (2.13)

Note that we are not saying that $g^{\prime}_{\mu\nu}$ is completely
arbitrary: at every point in space-time, it is related to the flat metric by a
scale transformation. But the scale factor is different at every point. The
condition on $\varepsilon^{\mu}$ becomes in this case

$\partial_{\mu}\varepsilon_{\nu}+\partial_{\nu}\varepsilon_{\mu}=2\sigma
g_{\mu\nu},$ (2.14)

where $\sigma$ is the infinitesimal version of $\Omega$, with the conventional
relation $\Omega(x)=e^{-2\sigma(x)}\approx 1-2\sigma(x)$. To find the most
general solution to this equation, note that contracting the indices with
$g^{\mu\nu}$ gives

$\partial_{\mu}\varepsilon^{\mu}=d\sigma,$ (2.15)

where $d$ is the space(-time) dimension, while acting with $\partial^{\nu}$
gives

$\partial_{\mu}\partial_{\nu}\varepsilon^{\nu}+\partial^{2}\varepsilon_{\mu}=2\partial_{\mu}\sigma,$
(2.16)

so that we get

$\partial^{2}\varepsilon_{\mu}=(2-d)\partial_{\mu}\sigma.$ (2.17)

Acting once again with $\partial^{\mu}$, we arrive at

$(d-1)\partial^{2}\sigma=0,$ (2.18)

while acting with $\partial^{\nu}$ and symmetrizing the indices yields

$(2-d)\partial_{\mu}\partial_{\nu}\sigma=g_{\mu\nu}\partial^{2}\sigma.$ (2.19)

The condition $\partial^{2}\sigma=0$ must therefore be satisfied in all
dimensions ($d>1$), and in $d>2$ the additional condition
$\partial_{\mu}\partial_{\nu}\sigma=0$, which is solved by

$\sigma(x)=\lambda+2b\cdot x.$ (2.20)

The corresponding value of $\varepsilon^{\mu}$ is

$\varepsilon^{\mu}=a^{\mu}+\omega^{\mu}_{\leavevmode\nobreak\
\nu}x^{\nu}+\lambda x^{\mu}+2(b\cdot x)x^{\mu}-x^{2}b^{\mu}.$ (2.21)

Therefore, in addition to the transformations found before, we also have

$x^{\mu}\xrightarrow{K}x^{\mu}+2(b\cdot x)x^{\mu}-x^{2}b^{\mu},$ (2.22)

which is called _special conformal transformation_. If we examine the Jacobian
for this transformation, we find

$\frac{\partial x^{\prime\mu}}{\partial x^{\nu}}=\left(1+2b\cdot
x\right)\delta^{\mu}_{\nu}+2\left(b_{\nu}x^{\mu}-x_{\nu}b^{\mu}\right)\approx\left(1+2b\cdot
x\right)R^{\mu}_{\leavevmode\nobreak\ \nu}(x).$ (2.23)

We have written this as a position-dependent scale factor $(1+2b\cdot x)$,
multiplying an orthogonal matrix

$R^{\mu}_{\leavevmode\nobreak\
\nu}(x)=\delta^{\mu}_{\nu}+2\left(b_{\nu}x^{\mu}-x_{\nu}b^{\mu}\right).$
(2.24)

This shows that special conformal transformations act locally as the
composition of a scale transformation and a rotation (or Lorentz
transformation). This also shows that conformal transformations preserve
angles, which is the origin of their name. Eq. (2.14) is called the conformal
Killing equation and its solutions (2.21) the Killing vectors.

Note that in our derivation the original metric $g_{\mu\nu}$ was flat, but the
new metric $g^{\prime}_{\mu\nu}$ is not. It is however _conformally flat_ : it
is always possible to make a change of coordinate after which it is flat
again. In general, transformations

$g_{\mu\nu}(x)\to\Omega(x)g_{\mu\nu}(x)$ (2.25)

are called Weyl transformations. They change the geometry of space-time. We
found that Weyl transformations that are at most quadratic in $x$ can be
compensated by a change of coordinates to go back to flat space. The
corresponding flat-space transformation is called conformal
transformation.444This implies that the group of conformal transformation is a
subgroup of diffeomorphisms. It is in fact the largest finite-dimensional
subgroup.

In $d=2$ the situation is a bit different: the conditions
$\partial^{2}\sigma=0$ is sufficient to ensure that the Killing equation has a
solution. This is most easily seen in light-cone coordinates,

$x^{+}=\frac{x^{0}+x^{1}}{2},\qquad x^{-}=\frac{x^{0}-x^{1}}{2},$ (2.26)

in terms of which

$\partial^{2}\sigma=\partial_{+}\partial_{-}\sigma.$ (2.27)

This is satisfied by taking for $\sigma$ the sum of an arbitrary function of
the coordinate $x^{+}$ and of another arbitrary function of $x^{-}$. In fact,
if we write $\varepsilon^{\pm}=\varepsilon^{0}\pm\varepsilon^{1}$, we can take
arbitrary functions $\varepsilon^{+}(x^{+})$ and $\varepsilon^{-}(x^{-})$, and
verify that eq. (2.14) is satisfied with
$\sigma=\frac{1}{2}\left(\partial_{+}\varepsilon_{+}+\partial_{-}\varepsilon_{-}\right)$.
In Euclidean space, we define

$z=\frac{x^{1}+ix^{2}}{2},\qquad\bar{z}=\frac{x^{1}-ix^{2}}{2},$ (2.28)

complex-conjugate to each other, and the same logic follows: we can apply
arbitrary holomorphic and anti-holomorphic transformations on $z$ and
$\bar{z}$, and the conformal Killing equation is always satisfied. This shows
that there are (infinitely) many more conformal transformations in $d=2$ than
in $d>2$, and also that there is no significant difference between Euclidean
and Minkowski conformal transformations in $d=2$, as the transformation acts
essentially on the two light-cone/holomorphic coordinates independently.

###  2.2 The conformal algebra

The conformal Killing equation (2.21) determines the most general form of
_infinitesimal_ conformal transformations. _Finite_ conformal transformations
follow from a sequence of infinitesimal transformations. However, one has to
bear in mind that infinitesimal conformal transformations do not commute: for
instance, a translation followed by a rotation is not the same as the
opposite. The conformal transformations form a _group_ : the composition of
conformal transformations is again a conformal transformation.

As we know from quantum field theory, a group is characterized by its
_generators_ and their commutation relations (the _algebra_). A generator $G$
describes an infinitesimal transformation in some direction, and finite
transformations are obtained by exponentiation, $e^{i\theta G}$, with
parameter $\theta$ (the factor of $i$ is a physicist’s convention that makes
the generators Hermitian). A representation of the conformal group can be
obtained from smooth functions of the coordinates, $f(x)$. For instance, under
an infinitesimal translation, we have

$f(x)\xrightarrow{P}f(x^{\prime})=f(x+a)\approx
f(x)+a^{\mu}\partial_{\mu}f(x)$ (2.29)

and we require this to be equal to $e^{-ia_{\mu}P^{\mu}}f(x)$, which means

$P_{\mu}=i\partial_{\mu}.$ (2.30)

Performing the same analysis for the other infinitesimal transformations given
in eq. (2.21), we obtain for the other generators555The sign of these
generators is an arbitrary convention. It defines once and for all the
commutations relations that we will derive next. After that, we will always
refer to the commutation relations as defining the generators for other
representations of the conformal group.

rotations/Lorentz transformations: $\displaystyle M^{\mu\nu}$
$\displaystyle=i\left(x^{\mu}\partial^{\nu}-x^{\nu}\partial^{\mu}\right)$
(2.31) scale transformations: $\displaystyle D$
$\displaystyle=ix^{\mu}\partial_{\mu},$ (2.32) special conformal
transformations: $\displaystyle K^{\mu}$
$\displaystyle=i\left(2x^{\mu}x^{\nu}\partial_{\nu}-x^{2}\partial^{\mu}\right).$
(2.33)

The number of generators matches that of the Killing vectors: there are $d$
translations, $d$ special conformal transformations, $d(d-1)/2$
rotations/Lorentz transformations ($M^{\mu\nu}$ is a $d\times d$ antisymmetric
matrix), and one scale transformation. Therefore the total number of
generators, i.e. the dimension of this group, is $(d+1)(d+2)/2$. In $d=4$
space-time dimensions, the conformal group has 15 generators.

Using the above definition, one can verify that the following commutation
relations are satisfied,

$\displaystyle\left[M^{\mu\nu},M^{\rho\sigma}\right]$
$\displaystyle=-i\left(g^{\mu\rho}M^{\nu\sigma}-g^{\mu\sigma}M^{\nu\rho}-g^{\nu\rho}M^{\mu\sigma}+g^{\nu\sigma}M^{\mu\rho}\right)$
(2.34) $\displaystyle\left[M^{\mu\nu},P^{\rho}\right]$
$\displaystyle=-i\left(g^{\mu\rho}P^{\nu}-g^{\nu\rho}P^{\mu}\right)$
$\displaystyle\left[M^{\mu\nu},K^{\rho}\right]$
$\displaystyle=-i\left(g^{\mu\rho}K^{\nu}-g^{\nu\rho}K^{\mu}\right)$
$\displaystyle\left[D,P^{\mu}\right]$ $\displaystyle=-iP^{\mu}$
$\displaystyle\left[D,K^{\mu}\right]$ $\displaystyle=iK^{\mu}$
$\displaystyle\left[P^{\mu},K^{\nu}\right]$
$\displaystyle=2i\left(g^{\mu\nu}D-M^{\mu\nu}\right)$

while all other commutators vanish:

$\left[M^{\mu\nu},D\right]=\left[P^{\mu},P^{\nu}\right]=\left[K^{\mu},K^{\nu}\right]=0.$
(2.35)

The first two relations in eq. (2.34) are the familiar Poincaré algebra. The
next one states that $K^{\mu}$ transforms like a vector (as $P^{\mu}$ does),
whereas $D$ is obviously a scalar. The next two relations remind us that
$K^{\mu}$ and $P^{\mu}$ have respectively the dimension of length and inverse
length.

Even though it is not immediately obvious, this algebra is isomorphic to that
of the group $\text{SO}(d+1,1)$ if $g^{\mu\nu}$ is the Euclidean metric, or
$\text{SO}(d,2)$ if it is the Minkowski metric. To see that it is the case,
let us introduce a $(d+2)$-dimensional space with coordinates

$X^{\mu},\quad X^{d+1},\quad X^{d+2},$ (2.36)

and a metric defined by the line element

$ds^{2}=g_{\mu\nu}dX^{\mu}dX^{\nu}+dX^{d+1}dX^{d+1}-dX^{d+2}dX^{d+2}\equiv\eta_{MN}dX^{M}dX^{N}.$
(2.37)

$X^{d+1}$ is a new spatial coordinate, and $X^{d+2}$ a new time. Then we can
write all conformal commutation relations as being defined by the Lorentzian
algebra

$\left[J^{MN},J^{RS}\right]=-i\left(\eta^{MR}J^{NS}-\eta^{MS}J^{NR}-\eta^{NR}J^{MS}+\eta^{NS}J^{MR}\right),$
(2.38)

provided that we identify the antisymmetric generators $J^{MN}$ with the
conformal generators as follows:

$\displaystyle M^{\mu\nu}$ $\displaystyle=J^{\mu\nu},$ (2.39) $\displaystyle
P^{\mu}$ $\displaystyle=J^{\mu,d+1}+J^{\mu,d+2},$ $\displaystyle K^{\mu}$
$\displaystyle=J^{\mu,d+1}-J^{\mu,d+2},$ $\displaystyle D$
$\displaystyle=J^{d+1,d+2}.$

###  2.3 Finite transformations

We just saw that the infinitesimal conformal transformations generate a group.
But how can we describe finite conformal transformations? Let us see how each
generator exponentiates into an element of the group; the most general
conformal transformation can then be obtained as a composition of such finite
transformations.

In some cases the exponentiation is trivial. For instance, with translations
we obtain immediately

$x^{\mu}\xrightarrow{P}x^{\mu}+a^{\mu},$ (2.40)

where $a$ is now any $d$-dimensional vector, not necessarily small. The same
is true of scale transformations,

$x^{\mu}\xrightarrow{D}\lambda x^{\mu}$ (2.41)

with finite $\lambda$. Rotations or Lorentz transformations exponentiate as

$x^{\mu}\xrightarrow{M}\Lambda^{\mu}_{\leavevmode\nobreak\ \nu}x^{\nu}$ (2.42)

where $\Lambda^{\mu}_{\leavevmode\nobreak\ \nu}$ is a $\text{SO}(d)$ or
$\text{SO}(1,d-1)$ matrix, depending whether the metric is Euclidean or
Minkowski. All of this is standard in quantum field theory.

On the contrary, special conformal transformations do not exponentiate
trivially. The easiest way to derive their finite form is to make the
following observation: recall that in infinitesimal form we have

$x^{\prime\mu}=x^{\mu}+2(b\cdot x)x^{\mu}-x^{2}b^{\mu},$ (2.43)

which implies $x^{\prime 2}=\left(1+2b\cdot x\right)x^{2}$, and therefore (as
always neglecting terms of order $b^{2}$)

$\frac{x^{\prime\mu}}{x^{\prime 2}}=\frac{x^{\mu}}{x^{2}}-b^{\mu}.$ (2.44)

The ratio $x^{\mu}/x^{2}$ appearing on both side of the equation is the
_inverse_ of the coordinate $x^{\mu}$, respectively $x^{\prime\mu}$: the
inversion is defined by

$x^{\mu}\xrightarrow{I}\frac{x^{\mu}}{x^{2}}.$ (2.45)

This transformation does not have an infinitesimal form, but otherwise it
shares the essential properties of a conformal transformation: its Jacobian is

$\frac{\partial x^{\prime\mu}}{\partial
x^{\nu}}=\frac{1}{x^{2}}\left[\delta^{\mu}_{\nu}-2\frac{x^{\mu}x_{\nu}}{x^{2}}\right],$
(2.46)

which is the product of a position-dependent scale factor ($x^{-2}$) with an
orthogonal matrix. To understand what this transformation does globally, let
us consider a Euclidean point $\vec{x}=(a,0,\ldots 0)\in\mathbb{R}^{d}$. Then
the matrix in square brackets is diagonal, and equates
$\text{diag}(-1,1,\ldots,1)$. This is an orthogonal matrix with determinant
$-1$, which is part of $\text{O}(d)$ but not $\text{SO}(d)$. This shows that
the inversion is a discrete transformation not connected to the identity. A
conformally-invariant theory might be invariant under inversions, but it needs
not be.

Eq. (2.44) shows that infinitesimal special conformal transformations are
obtained taking an inversion followed by a translation, followed by an
inversion again. Since this process involves the inversion twice, and since
inversion is its own inverse, it does not matter whether inversion is a true
symmetry of the system or not. The advantage of this representation is that it
can easily be exponentiated: the composition of (infinitely) many
infinitesimal special conformal transformation can be written as an inversion
followed by a finite translation, followed by an inversion again. In other
words, eq. (2.44) holds for finite $b^{\mu}$. This can be used to show that

$x^{\mu}\xrightarrow{K}x^{\prime\mu}=\frac{x^{\mu}-x^{2}b^{\mu}}{1-2b\cdot
x+b^{2}x^{2}}.$ (2.47)

Exercise 2.1 Use eq. (2.44) to show (2.47).

What do special transformation do globally? Let us look specifically at
Euclidean space. There are some special points:

  * •

The origin of the coordinate system $x=0$ is mapped onto itself.

  * •

The point $b^{\mu}/b^{2}$ is mapped to $\infty$.

  * •

Conversely, the “point” $x\to\infty$ is mapped to the finite value
$-b^{\mu}/b^{2}$.

These properties can be understood from the fact that special conformal
transformations and translations are related by inversion: special conformal
transformations keep the origin fixed but move every other point, including
$\infty$; translations move every point except $\infty$. The other two
transformations, rotations and scale transformations, keep both $0$ and
$\infty$ fixed.

An essential property of conformal transformations is that they let us map any
3 points $(x_{1},x_{2},x_{3})$ onto another triplet
$(x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3})$. This can be seen as follows:
first, apply a translation to place $x_{1}$ at the origin, followed by a
special conformal transformation that takes $x_{3}$ to $\infty$, after which
the image of the original triplet is $(0,x_{2}^{\prime\prime},\infty)$; then
use rotations and scale transformations to move $x^{\prime\prime}_{2}$ to
another point $x^{\prime\prime\prime}_{2}$, while keeping $0$ and $\infty$
fixed; finally apply again a special conformal transformation that takes
$\infty$ to $x^{\prime}_{3}-x^{\prime}_{1}$, and a translation by
$x^{\prime}_{1}$ to reach the configuration
$(x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3})$. This property has an
immediate physical consequence: in correlation functions involving 2 or 3
local operators (see next sections for a definition), all kinematics is fixed
by conformal symmetry. The only freedom is about the operators themselves, not
about their position in space.

Another interesting property of conformal transformations is that they map
spheres to spheres: this is an obvious property of translations, rotations,
and scale transformations, but it is also true of special conformal
transformations.

Exercise 2.2 Show that under the special conformal transformation (2.47), a
sphere centered at the point $a^{\mu}$ and with radius $R$ gets mapped to a
sphere centered at the point

$a^{\prime\mu}=\frac{a^{\mu}-(a^{2}-R^{2})b^{\mu}}{1-2a\cdot
b+(a^{2}-R^{2})b^{2}}$

and with radius

$R^{\prime}=\frac{R}{\left|1-2a\cdot b+(a^{2}-R^{2})b^{2}\right|}.$

In the special case in which $b^{\mu}/b^{2}$ lies on the surface of the
original sphere, show that the sphere gets mapped to a plane orthogonal to the
vector $a^{\mu}+(R^{2}-a^{2})b^{\mu}$. Note that a plane is a sphere of
infinite radius.

In $d=2$, the additional conformal transformations (infinitely many of them!)
mean that (nearly) any shape can be mapped onto another. This is known as the
Riemann mapping theorem.

###  2.4 Compactifications

We mentioned earlier that conformal symmetry is a symmetry of flat
space(-time). It is true, as we have just seen, provided that we treat the
point $\infty$ as being part of the space. This is quite straightforward in
Euclidean space, but much more subtle in Minkowski space-time, as there are
different, inequivalent ways of reaching $\infty$ there.

To gain a better understanding of this, it is useful to map the flat Euclidean
space $\mathbb{R}^{d}$ or the Minkowski space-time $\mathbb{R}^{1,d-1}$ onto a
curved manifold. In Euclidean space, this is for instance achieved by the
(inverse) stereographic projection that maps $\mathbb{R}^{d}\cup\\{\infty\\}$
to the unit sphere $S^{d}$. Geometrically, the stereographic projection is
constructed as follows (see figure 1): embed $\mathbb{R}^{d}$ as a plane in
$\mathbb{R}^{d+1}$, together with a sphere of unit radius centered at the
origin. Every point on the plane has an image on the sphere obtained by
drawing a segment between the original point and the north pole of the sphere,
and noting where it intersects the sphere. The origin is mapped to the south
pole, $\infty$ to the north pole, and the sphere $S^{d-1}$ of unit radius to
the equator. Algebraically, this is achieved as follows: first write the
Euclidean metric in spherical coordinates,

$ds^{2}=dr^{2}+r^{2}d\Omega_{d-1}^{2},$ (2.48)

where the solid angle is given in $d=2$ by $d\Omega_{1}^{2}=d\phi^{2}$, in
$d=3$ by $d\Omega_{2}^{2}=d\theta^{2}+\sin\theta^{2}d\phi^{2}$, and more
generically by the recursion relation
$d\Omega_{n}^{2}=d\theta^{2}+\sin\theta^{2}d\Omega_{n-1}^{2}$. Let us perform
the change of variable

$r=\frac{\sin\varphi}{1-\cos\varphi},$ (2.49)

and interpret $\varphi\in[0,\pi]$ as the zenith angle on the sphere:
$\varphi=0$ is the north pole, corresponding to $r\to\infty$, and
$\varphi=\pi$ the south pole, corresponding to $r=0$. In these coordinates we
have

$ds^{2}=\frac{1}{(1-\cos\varphi)^{2}}\left(d\varphi^{2}+\sin\varphi^{2}d\Omega_{d-1}^{2}\right)=\frac{1}{(1-\cos\varphi)^{2}}d\Omega_{d}^{2}.$
(2.50)

The new metric is flat up to an overall Weyl factor that depends on $\varphi$.
In these coordinates, conformal transformations are always non-singular. For
this reason, it is often convenient to study _classical_ conformal
transformations on the sphere $S^{d}$ instead of Euclidean space.

Figure 1: Inverse stereographic projection of the Euclidean space
$\mathds{R}^{d}$ (represented here as the horizontal plane) onto the sphere
$S^{d}$, both embedded in $\mathds{R}^{d+1}$. A point $x$ with
$\left|x\right|>1$ gets mapped to the northern hemisphere; a point
$x^{\prime}$ with $\left|x^{\prime}\right|<1$ to the southern hemisphere; the
origin is mapped to the south pole (S), and $\infty$ to the north pole (N).

However, this compactification is not as nice in a _quantum_ theory in which
one wants to foliate the space along some preferred direction: if one chooses
$\varphi$ as the “Euclidean time”, then the “space” direction is a sphere
$S^{d-1}$ whose volume depends on $\varphi$. In other words, the generator of
“time” translations is not a symmetry of the system. This is also true of any
other choice of time direction on the sphere.

Instead, another compactification is often preferred to the sphere: from the
Euclidean metric in spherical coordinates, one can make the change of variable

$\tau=\log(r)\qquad\Leftrightarrow\qquad r=e^{\tau},$ (2.51)

after which

$ds^{2}=e^{2\tau}\left(d\tau^{2}+d\Omega_{d-1}^{2}\right).$ (2.52)

This is again a flat metric, up to a Weyl factor $r^{2}=e^{2\tau}$. In this
case, however, the rescaled metric is independent of $\tau$. This space has
the geometry of a _cylinder_ : $\mathbb{R}\times S^{d-1}$. It is not fully
compact: $\tau$ goes from $-\infty$ to $+\infty$. But it has an important
advantage: translations in $\tau$ are generated by dilatations $D$, which will
be taken to be a symmetry of the quantum theory. Foliating the space into
surfaces of constant $\tau$ will later lead us to radial quantization in
conformal field theory.

Note that in $\text{SO}(d+1,1)$ language, the generator $D=J^{d+1,d+2}$ is
completely equivalent to the other generators
$J^{\mu,d+2}=\frac{1}{2}\left(P^{\mu}-K^{\mu}\right)$, since they are related
to them by $\text{SO}(d+1)$ rotations. So we might as well look for a cylinder
compactification in which the non-compact direction corresponds to
transformation generated by $\frac{1}{2}\left(P^{0}-K^{0}\right)$. This
combination of generators obeys

$\frac{1}{2}\left(P^{0}-K^{0}\right)=i\left(\frac{1-(x^{0})^{2}+\vec{x}^{2}}{2}\partial_{0}-x^{0}\,\vec{x}\cdot\vec{\partial}\right),$
(2.53)

with two fixed points at $x^{0}=\pm 1$ with $\vec{x}=0$. The foliation of
space generated by this linear combination of generators is what is used in
N-S quantization,666The line drawn when evolving a point with this Hamiltonian
looks like a magnetic field line connecting the north (N) and south (S) poles
of a magnet, hence the name N-S quantization. discussed later in section 5.
Figure 2 illustrates the two foliations of Euclidean space by $D$ and
$\frac{1}{2}\left(P^{0}-K^{0}\right)$, and the corresponding cylinder
interpretations are shown in figure 3.

Figure 2: Foliations of Euclidean space (here in $d=2$) in radial (left) and
N-S quantization (right). Circles of identical colors are mapped onto each
other by a conformal transformation. In particular, the plane $x^{0}=0$ in N-S
quantization is mapped to the unit sphere in radial quantization.

Exercise 2.3 Find the change of coordinates that makes the Euclidean metric
Weyl-equivalent to a cylinder in which translations in the non-compact
direction are generated by $\frac{1}{2}\left(P^{0}-K^{0}\right)$.
Hint: Find a special conformal transformation followed by a translation that
takes $(0,\infty)$ to $(-1,1)$, and apply it to the radial coordinates.

The cylinder compactifications of Euclidean space are interesting by
themselves, but they are also extremely convenient to understand the
connection between Euclidean and Minkowski space-times: in this last form,
performing a Wick rotation $\tau\to-it$ defines a cylinder on which the
Lorentzian conformal group $\text{SO}(d,2)$ acts naturally. But before we get
there, let us go back to flat Minkowski space-time and make some general
remarks.

Figure 3: Left: the Euclidean cylinder corresponding the radial coordinates
(2.52), in which the evolution in the non-compact direction is given by the
generator $D$ of scale transformations; the blue circle corresponds to the
unit sphere. Middle: same Euclidean cylinder after a conformal transformation,
so that the evolution is given by $\frac{1}{2}(P^{0}-K^{0})$; the blue circle
is now the plane $x^{0}=0$, which includes the origin and the point at
infinity. Right: the Lorentzian cylinder, obtained after a Wick rotation in
$x^{0}$, so that the surface $x^{0}=0$ is unchanged; the image of this surface
under time translations generated by $P^{0}$ is the Poincaré patch, shown as
the blue diamond on the cylinder’s surface; only special conformal
transformations can move a point from inside the Poincaré patch to the
outside.

###  2.5 Minkowski space-time

Translations and Lorentz transformations in Minkowski space-time are familiar,
and even dilatations are a standard tool in the renormalization group
analysis. But what do conformal transformations do?

To understand this, let us place an observer at the origin of Minkowski space-
time. The presence of this observer breaks translations, but not Lorentz
transformations (let us assume that the observer is point-like), nor
dilatations or special conformal transformations. For the observer, space-time
is split into three regions: a future light cone, a past light cone, and a
space-like region from which they know nothing. Lorentz and scale
transformation preserve this causal structure: the future and past light-cones
are mapped onto themselves. In other words, if a point $x$ is space-like
separated from the observer, it will remain space-like separated no matter the
choice of Lorentz frame, or the definition of length. Without loss of
generality, let us choose this point to be at position $x=(0,\vec{n})$, where
$\vec{n}$ is a unit vector (units can be chosen so that this is the case). Now
apply a special conformal transformation with parameter
$b^{\mu}=(-\alpha,\alpha\vec{n})$, with $\alpha$ varying between 0 and 1. This
draws a curve $y^{\mu}$ in space-time, parameterized by $\alpha$, with

$y^{0}(\alpha)=\frac{\alpha}{1-2\alpha},\qquad\qquad\vec{y}(\alpha)=\frac{1-\alpha}{1-2\alpha}\vec{n}.$
(2.54)

This curves begins at the space-like point $y=(0,\vec{n})$, and ends in the
past light cone at $y=(-1,\vec{0})$. Note that $y$ never crosses a light cone:
the image of a point $x$ is never null under a special conformal
transformation unless $x$ is itself null, since $x^{\prime 2}=x^{2}/(1-2x\cdot
b+x^{2}b^{2})$. Instead, we have

$y^{2}(\alpha)=\frac{1}{1-2\alpha}\neq 0.$ (2.55)

What happens is that the point travels all the way to space-like infinity at
$\alpha=\frac{1}{2}$, and comes back from past infinity. Clearly, special
conformal transformation break causality!

The resolution of this puzzle is that conformal transformations do not act
directly on Minkowski space, but rather on its universal cover that is
isomorphic to the Lorentzian cylinder described in figure 3. Evolution on that
cylinder is given by the Hamiltonian $H=\frac{1}{2}(P^{0}-K^{0})$, but this
differs from the Minkowski time evolution generated by $P^{0}$. On any given
slice of the Lorentzian cylinder, space is compactified in such a way that the
notion of infinite distance is unequivocal: space-like infinity corresponds to
a point on the sphere, antipodal to the origin. If one takes any other point
of that sphere and applies finite translations using the generator $P^{\mu}$,
then this defines a compact _Poincaré patch_. The full Lorentzian cylinder is
a patchwork of Poincaré patches, but every local observer only has access to
one.777As Lüscher and Mack put it: _In picturesque language, [the superworld]
consists of Minkowski space, infinitely many “spheres of heaven” stacked above
it and infinitely many “circles of hell” below it_ [6].

The lesson that we must learn is that _only the infinitesimal form of special
conformal transformations can be used in Minkowski space-time_ : any finite
special conformal transformation brings part of space-time into another patch
on the cylinder. This is sometimes called _weak conformal invariance_.

###  2.6 Conformal symmetry in classical field theory

So far we have only been discussing conformal transformations of the
coordinates. The next step is to consider a field theory (for the moment a
classical one) that has conformal symmetry built in. The simplest example is
the free, massless scalar field, defined by the action

$S=\int d^{d}x\left[-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi\right].$
(2.56)

We shall see in the next section that the action principle can in fact be
dropped in CFT, but for now it is a convenient starting point.

In this context, a conformal transformation is a transformation _of the
fields_. There are two distinct and complementary perspectives one can adopt.
It is often convenient to think of the metric tensor as a field in its own
right, and to define a conformal transformation as a (position-dependent)
scale transformation of the field

$\phi(x)\to e^{\Delta\sigma(x)}\phi(x),$ (2.57)

combined with a Weyl transformation of the metric

$g_{\mu\nu}(x)\to e^{2\sigma(x)}g_{\mu\nu}(x).$ (2.58)

$\Delta$ is the _scaling dimension_ of the field $\phi$. In a free theory it
coincides with the dimension of the field in units of energy (inverse units of
length), namely

$\Delta=\frac{d-2}{2}.$ (2.59)

$\sigma(x)$ is an infinitesimal scale factor that satisfies
$\partial_{\mu}\partial_{\nu}\sigma=0$. The advantage of this perspective is
that the conformal transformations are simple, multiplicative transformations
of the fields. The disadvantage is that it requires thinking of the theory in
curved space-time. This means that the metric that is implicit in the action
(2.56) must be made explicit, but also that the action can be supplemented
with a term depending on the scalar curvature tensor $R$ as

$S=\int
d^{d}x\,\sqrt{\left|g\right|}\left[-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+\alpha
R\phi^{2}\right].$ (2.60)

Since $R$ vanishes in flat space, it looks like this additional term could
appear with an arbitrary coefficient $\alpha$ without modifying the original
flat-space action, but this is not the case.

Exercise 2.4 Verify that there is a unique value of $\alpha$ for which this
action is invariant under the infinitesimal conformal transformations (2.57)
and (2.58). What is it?

For this reason, it is also convenient to consider the opposite perspective in
which conformal transformations are transformations of the dynamical fields
and of the coordinates, but not of the metric. In this case the conformal
transformation can be defined as

$\phi(x)\to e^{\Delta\sigma(x)}\phi(x+\varepsilon),$ (2.61)

where the parameters $\sigma$ and $\varepsilon$ are related by the conformal
Killing equation (2.14), i.e.
$\sigma=\frac{1}{d}\partial_{\mu}\varepsilon^{\mu}$. In infinitesimal form,
this transformation becomes

$\phi(x)\to\left[1+\frac{d-2}{2d}(\partial_{\nu}\varepsilon^{\nu})+\varepsilon^{\nu}\partial_{\nu}\right]\phi(x).$
(2.62)

Exercise 2.5 Show that under the transformation (2.62), the Lagrangian of the
free scalar field

$\mathscr{L}=-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi,$

is shifted by a total derivative term

$\delta\mathscr{L}=\partial_{\mu}\left(-\frac{1}{2}\varepsilon^{\mu}\partial_{\nu}\phi\partial^{\nu}\phi-\frac{d-2}{2d}\partial_{\nu}\varepsilon^{\nu}\phi\partial^{\mu}\phi\right),$

hence proving that this is a symmetry of the action. You will need to use the
fact that $\varepsilon$ is at most quadratic in $x$.

Note that Poincaré symmetry is a special case of this transformation,
corresponding to constant $\varepsilon$ (and thus $\sigma=0$).

By Noether’s theorem, whenever an action is invariant under some
transformation of the field

$\phi\to\phi+\delta_{\varepsilon}\phi,$ (2.63)

i.e. whenever the Lagrangian varies by a total derivative term,

$\mathscr{L}\to\mathscr{L}+\partial_{\mu}\Lambda_{\varepsilon}^{\mu},$ (2.64)

then there exists a conserved current

$J_{\varepsilon}^{\mu}=\Lambda^{\mu}_{\varepsilon}-\frac{\partial\mathscr{L}}{\partial(\partial_{\mu}\phi)}\delta_{\varepsilon}\phi.$
(2.65)

In our example, this conserved current is therefore

$J_{\varepsilon}^{\mu}=\varepsilon_{\nu}\left(\partial^{\mu}\phi\partial^{\nu}\phi-\frac{1}{2}g^{\mu\nu}\partial_{\rho}\phi\partial^{\rho}\phi\right)\equiv\varepsilon_{\nu}T_{c}^{\mu\nu}.$
(2.66)

The 2-index tensor on the right-hand side is called the _canonical energy-
momentum tensor_. Its divergence satisfies

$\partial_{\nu}T_{c}^{\mu\nu}=\partial^{\mu}\phi\partial^{2}\phi,$ (2.67)

therefore vanishing by the equation of motion for the free field,
$\partial^{2}\phi=0$. This implies in turn that the Noether current (2.66) is
conserved _for constant $\varepsilon$_. If on the contrary $\varepsilon$
depends on space(-time), then we have

$\partial_{\mu}J_{\varepsilon}^{\mu}=(\partial_{\nu}\varepsilon_{\mu})T_{c}^{\mu\nu}.$
(2.68)

In our example the canonical energy-momentum tensor is symmetric in its
indices $\mu$ and $\nu$, and therefore we can write

$\partial_{\mu}J_{\varepsilon}^{\mu}=\frac{1}{2}(\partial_{\mu}\varepsilon_{\nu}+\partial_{\nu}\varepsilon_{\mu})T_{c}^{\mu\nu}=\sigma
g_{\mu\nu}T_{c}^{\mu\nu}=-\frac{d-2}{2}\,\sigma\,\partial_{\mu}\phi\partial^{\mu}\phi.$
(2.69)

In dimensions $d>2$, this current is only conserved when $\sigma=0$. This is
surprising, because we just showed that scale and special conformal
transformations are also symmetries of the action, so why is the Noether
current not conserved?

The reason is that the version of Noether’s theorem given above does not
straightforwardly apply to the case of a space-time dependent parameter
$\varepsilon$. In fact, the energy-momentum tensor that we computed in this
example is not unique: one can always add to it a piece proportional to

$\left(\partial^{\mu}\partial^{\nu}-g^{\mu\nu}\partial^{2}\right)\phi^{2},$
(2.70)

without affecting the conservation equation (2.67), but changing the value of
its trace. The combination

$T^{\mu\nu}=T^{\mu\nu}_{c}+\frac{d-2}{2(d-1)}\left(\partial^{\mu}\partial^{\nu}-g^{\mu\nu}\partial^{2}\right)\phi^{2}$
(2.71)

is for instance traceless in any $d$. It turns out that it is always possible
in a field theory with conformal symmetry to construct an energy-momentum
tensor that is:

  * •

symmetric ($T^{\mu\nu}=T^{\nu\mu}$),

  * •

traceless ($g_{\mu\nu}T^{\mu\nu}=0$), and

  * •

conserved once the equation of motions are imposed
($\partial_{\nu}T^{\mu\nu}\stackrel{{\scriptstyle e.o.m.}}{{=}}0$).

This is a non-trivial fact, but we will skip its proof (as already mentioned,
we are interested in theories that are not necessarily defined through an
action).

Strictly speaking, Noether’s theorem only applies to theories that have a
Lagrangian description, but we will assume the existence of a traceless
energy-momentum tensor in all cases (in some sense this is going to be one of
the “axioms” of conformal field theory). From this assumption, we can deduce
that the theory is invariant under conformal transformations: a conserved
current can always be built from the energy-momentum tensor and a conformal
Killing vector $\varepsilon$ as

$J^{\mu}=\varepsilon_{\nu}T^{\mu\nu}\qquad\Rightarrow\qquad\partial_{\mu}J^{\mu}=\varepsilon_{\nu}\partial_{\mu}T^{\mu\nu}\stackrel{{\scriptstyle
e.o.m.}}{{=}}0.$ (2.72)

As always, conserved charges can be constructed as the integral of the time
component of a conserved current over space. The simplest example is

$P^{\mu}=-i\int d^{d-1}\vec{x}\,T^{0\mu}(x).$ (2.73)

This is in principle a function of $x^{0}$, but it is in fact constant over
time, since

$\partial_{0}P^{\mu}=-i\int d^{d-1}\vec{x}\,\partial_{0}T^{0\mu}(x)=i\int
d^{d-1}\vec{x}\,\partial_{i}T^{i\mu}(x)=0.$ (2.74)

This conserved charge is the _momentum_ , associated with translation
symmetry. Similarly, there are conserved charges associated with Lorentz
transformations,

$M^{\mu\nu}=-i\int
d^{d-1}\vec{x}\,\left[x^{\mu}T^{0\nu}(x)-x^{\nu}T^{0\nu}(x)\right]$ (2.75)

with scale transformations,

$D=-i\int d^{d-1}\vec{x}\,x_{\mu}T^{0\mu}(x),$ (2.76)

and with special conformal transformations

$K^{\mu}=-i\int
d^{d-1}\vec{x}\,\left[2x^{\mu}x_{\nu}T^{0\nu}(x)-x^{2}T^{0\mu}(x)\right].$
(2.77)

Exercise 2.6 Show that the charges (2.75), (2.76) and (2.77) are conserved in
time.

The conservation of all these charges relies on the vanishing divergence of
the energy-momentum tensor, $\partial_{\nu}T^{\mu\nu}$, which itself relies on
the equation of motion being satisfied. This is certainly true in the absence
of sources. But when source terms are added to the action, the equation of
motion is modified. In our example of the free scalar field theory, adding to
the action (2.56) a source term of the form

$S_{\text{source}}=\int d^{d}x\,J(x)\phi(x),$ (2.78)

modifies the equation of motion to

$\partial^{2}\phi+J=0.$ (2.79)

Instead of the conservation equation (2.67) for the canonical energy-momentum
tensor, we must replace it with

$\partial_{\nu}T^{\mu\nu}(x)=-J\partial^{\mu}\phi.$ (2.80)

In the presence of such a source, the charges (2.73)–(2.77) are not conserved
anymore. However, if the source is local, say $J(x)=\delta^{d}(x-x_{\odot})$,
so that

$\partial_{\nu}T^{\mu\nu}(x)=-\delta^{d}(x-x_{\odot})\partial^{\mu}\phi(x_{\odot}),$
(2.81)

then we can determine the change in one of the charge — say $P^{\mu}$ —
between a time $x^{0}<x_{\odot}^{0}$ anterior to the local source, and a time
$x^{0}>x_{\odot}^{0}$ posterior to it, and call this difference the momentum
of the source, $P^{\mu}_{\odot}$. By our definition, this is equal to

$P^{\mu}_{\odot}=-i\int
d^{d-1}\vec{x}\,T^{0\mu}\Big{|}_{x^{0}>x^{0}_{\odot}}+i\int
d^{d-1}\vec{x}\,T^{0\mu}\Big{|}_{x^{0}<x^{0}_{\odot}}.$ (2.82)

Since the two surfaces of integration meet at spatial infinity, they can be
viewed as the two sides of a closed surface $\partial\Sigma$ surrounding the
point $x_{\odot}$, hence

$P^{\mu}_{\odot}=-i\int\limits_{\partial\Sigma}d^{d-1}n_{\nu}T^{\mu\nu}.$
(2.83)

By the divergence theorem, this is equal to

$P^{\mu}_{\odot}=-i\int\limits_{\Sigma}d^{d}x\,\partial_{\nu}T^{\mu\nu}(x)=i\int\limits_{\Sigma}d^{d}x\,\delta^{d}(x-x_{\odot})\partial^{\mu}\phi(x)=i\partial^{\mu}\phi(x_{\odot}).$
(2.84)

Note that this result does not depend on the choice of surface
$\partial\Sigma$, as long as it encloses the point $x_{\odot}$. In technical
terms, $P^{\mu}$ is a _topological charge_. We chose in eq. (2.73) to use a
standard definition of $P^{\mu}$ in which the energy-momentum tensor is
integrated along a surface of constant time. But if we work in Euclidean
space, this is conformally equivalent to integrating over the surface of a
sphere, or in fact over any other closed surface.

Eq. (2.84) is very important: it says that the charge $P^{\mu}$ associated
with a local source for the field $\phi$ is equal to $i\partial^{\mu}\phi$.
This is strikingly similar to the action of the generator (2.30) on functions
of the coordinates. In fact, it is easy to verify that the other charges
(2.75), (2.76), and (2.77) also act on the classical field $\phi(x)$ exactly
like the generators (2.31), (2.32), and (2.33) respectively. We worked here
with the free scalar field theory as an example, but our discussion can be
generalized to arbitrary classical field theories. The important lesson is
that a traceless energy-momentum tensor can be used to give a field-
theoretical realization of the conformal generators discussed before.

##  3 Conformal quantum field theory

Let us now turn to quantum field theory and study the implications of
conformal symmetry in that context. The standard approach to quantum field
theory is to think of a classical field theory, in which we now have a basic
understanding of what conformal symmetry does, and then quantize it by
promoting the fields to operators acting on some Hilbert space. But this is
not the approach that we will take here. There will still be states and
operators, but the latter will not necessarily be associated with fields
appearing in a Lagrangian.

###  3.1 Non-perturbative quantum field theory

To define a quantum field theory non-perturbatively, we need the following
ingredients:

  1. 1.

Hilbert space: The Minkowski space-time is foliated into surfaces of equal-
time, and to each time slice we associate a Hilbert space of quantum states.

  2. 2.

Local operators: There are a number of (in fact infinitely many) _local_
operators that act on this Hilbert space. For instance, let us take $\phi(x)$
to be an operator acting on the Hilbert space at time $t=x^{0}$. We call this
operator _local_ because we require that it commutes with any other local
operator inserted at a distinct point $\vec{x}^{\prime}\neq\vec{x}$ on the
same time slice:888For operators of half-integer spin, the commutator must be
replaced by an anti-commutator.

$\left[\phi(x),\phi(x^{\prime})\right]=0,\qquad\qquad x^{0}=x^{\prime
0},\qquad\vec{x}\neq\vec{x}^{\prime}.$ (3.1)

  3. 3.

Symmetries: One of the local operators of the theory is the energy-momentum
tensor, and from it we can define conserved charges, including $P^{\mu}$ and
$M^{\mu\nu}$ as in eqs. (2.73) and (2.75). In a generic QFT the energy-
momentum tensor needs not be traceless, so the charges $D$ and $K^{\mu}$
cannot be considered. $P^{\mu}$ and $M^{\mu\nu}$ are conserved in time, so
they are valid operators on all Hilbert spaces at every time $t$. Their value
changes however every time an operator is inserted at some point $x$. In
analogy with eq. (2.84), we require that this change is encoded in the
commutator

$\left[P^{\mu},\phi(x)\right]=i\partial^{\mu}\phi(x).$ (3.2)

Since this equation is solved by

$\phi(x)=e^{-ix\cdot P}\phi(0)e^{ix\cdot P},$ (3.3)

we say that $P^{\mu}$ is the generator of translations, which act as _unitary_
transformations on the operators (note that $P^{\mu}$ is Hermitian).

Lorentz transformations are similarly realized as unitary transformations,
generated by the charge $M^{\mu\nu}$. We can choose to decompose local
operators inserted at the origin of space-time into irreducible
representations of the Lorentz group and denote these with $\phi^{a}(0)$, with
$a$ standing for a collection of Lorentz indices, so that

$\left[M^{\mu\nu},\phi^{a}(0)\right]=i\left(\mathcal{S}^{\mu\nu}\right)^{a}_{\leavevmode\nobreak\
b}\phi^{b}(0),$ (3.4)

where $\left(\mathcal{S}^{\mu\nu}\right)^{a}_{\leavevmode\nobreak\ b}$ is a
matrix that satisfies the Lorentz algebra. For a scalar operator,
$\mathcal{S}^{\mu\nu}$ vanishes; for a vector operator with one Lorentz index,
it is given by

$\left(\mathcal{S}^{\mu\nu}\right)_{ab}=\delta^{\mu}_{a}\delta^{\nu}_{b}-\delta^{\mu}_{b}\delta^{\nu}_{a},$
(3.5)

and so on. When combined with eq. (3.3), and requiring that $P^{\mu}$ and
$M^{\mu\nu}$ satisfy the Poincaré algebra (2.34), this implies

$\left[M^{\mu\nu},\phi^{a}(x)\right]=-i\left(x^{\mu}\partial^{\nu}-x^{\nu}\partial^{\mu}\right)\phi^{a}(x)+i\left(\mathcal{S}^{\mu\nu}\right)^{a}_{\leavevmode\nobreak\
b}\phi^{b}(0).$ (3.6)

Note that $P^{\mu}$ and $M^{\mu\nu}$ are not local operators, but their
commutator with any local operator is again local (the same will later be true
of the generators $D$ and $K^{\mu}$).

  4. 4.

Vacuum state: The Hilbert space includes a vacuum state
$\left|0\right\rangle$, which we assume to be invariant under Poincaré
transformations (and later conformal transformations),

$P^{\mu}\left|0\right\rangle=M^{\mu\nu}\left|0\right\rangle=0.$ (3.7)

Other states of the theory are obtained acting with products of local
operators on the vacuum (see below for a more precise statement). There is in
principle one Hilbert space for each time slice, but since time translation is
a symmetry generated by $P^{0}$, the evolution operator $U(t)=e^{itP^{0}}$ is
unitary and all Hilbert spaces are equivalent. We also require that the vacuum
is the lowest-energy state in the Hilbert space. This means that if we can
construct an eigenstate of energy,

$P^{0}\left|\Psi\right\rangle=E\left|\Psi\right\rangle,$ (3.8)

then its eigenvalue must satisfy $E\geq 0$.

These four points essentially give the most general non-perturbative
definition of quantum field theory. They are nearly equivalent to the so-
called _Wightman axioms_ (but presented here without much mathematical rigor).
One notable difference is that the Wightman axioms do not rely on the
existence of an energy-momentum tensor, but assume directly that the Poincaré
transformations are realized as unitary transformations on the Hilbert space.

The locality condition (3.1) is often formulated in the Lorentz-invariant way

$\left[\phi(x),\phi(y)\right]=0\qquad\text{if}\leavevmode\nobreak\
(x-y)^{2}>0,$ (3.9)

stating that local operators commute as long as they are space-like separated,
which is known as the _micro-causality_ axiom. Similarly, when it is possible
to work with eigenstates of energy and momentum,

$P^{\mu}\left|\Psi\right\rangle=p^{\mu}\left|\Psi\right\rangle,$ (3.10)

then the Lorentz-invariant condition on the positivity of energy becomes

$p^{0}\geq\left|\vec{p}\right|\qquad\Leftrightarrow\qquad p^{2}\leq
0\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ p^{0}\geq 0,$ (3.11)

i.e. the momentum $p^{\mu}$ in contained in the forward light cone. Such
eigenstates of $P^{\mu}$ can be constructed from the Fourier transform of
local operators,

$\widetilde{\phi}(p)=\int d^{d}x\,e^{ip\cdot x}\phi(x)$ (3.12)

and acting on the vacuum. This implies that

$\widetilde{\phi}(p)\left|0\right\rangle=0\qquad\qquad\text{if}\leavevmode\nobreak\
p^{0}<\left|\vec{p}\right|,$ (3.13)

as well as its generalization to the product of multiple local operators,

$\widetilde{\phi}_{1}(p_{1})\cdots\widetilde{\phi}_{n}(p_{n})\left|0\right\rangle=0\qquad\qquad\text{if}\leavevmode\nobreak\
p_{1}^{0}+\ldots+p_{n}^{0}<\left|\vec{p}_{1}+\ldots+\vec{p}_{n}\right|.$
(3.14)

This is sometimes called the _spectral condition_.

Exercise 3.1 Using the definition (3.3) and integration by parts, show that

$P^{\mu}\widetilde{\phi}(p)\left|0\right\rangle=p^{\mu}\widetilde{\phi}(p)\left|0\right\rangle.$

###  3.2 Wightman functions

Starting from these axioms, the next thing we can do is compute the vacuum
expectation value of products of local operators,

$\left\langle
0\right|\phi_{1}(x_{1})\cdots\phi_{n}(x_{n})\left|0\right\rangle.$ (3.15)

This can be viewed as an overlap of the vacuum state, $\left\langle 0\right|$,
with a state created acting on the vacuum with a sequence of local operators.
Note that these operators need not be ordered in time: the time-evolution
operator is unitary, so it can go both ways. This object is therefore
different from time-ordered correlation functions obtained from the path
integral.

Correlators of this type are called _Wightman functions_. They are the
fundamental observables in non-perturbative quantum field theory. In fact, it
is even possible to completely define a quantum field theory just by its
Wightman functions: the Wightman reconstruction theorem states that the
Hilbert space of a quantum field theory can be constructed from all its
Wightman functions. A convenient perspective is therefore to forget about the
Hilbert space and focus on correlation functions.

The symmetry properties of these correlation functions are encoded in “Ward
identities”: given a conserved charge $G$ that annihilates the vacuum,
$G\left|0\right\rangle=0$ (this could be $P^{\mu}$ or $M^{\mu\nu}$), the
following equation must be satisfied

$\displaystyle\left\langle
0\right|\left[G,\phi_{1}(x_{1})\right]\phi_{2}(x_{2})\cdots\phi_{n}(x_{n})\left|0\right\rangle$
$\displaystyle+\left\langle
0\right|\phi_{1}(x_{1})\left[G,\phi_{2}(x_{2})\right]\cdots\phi_{n}(x_{n})\left|0\right\rangle$
$\displaystyle+\ldots$ $\displaystyle+\left\langle
0\right|\phi_{1}(x_{1})\,\phi_{2}(x_{2})\cdots\left[G,\phi_{n}(x_{n})\right]\left|0\right\rangle=0.$
(3.16)

This is an equation that is obvious in the Hilbert space picture, but it is
also valid as a differential equation for the Wightman function, since each
commutator is again related to the local operator. Let us see some examples.

The simplest Wightman function involves a single scalar operator,

$\left\langle 0\right|\phi(x)\left|0\right\rangle.$ (3.17)

In this case the Ward identity associated with translations implies

$\left\langle
0\right|\left[P^{\mu},\phi(x)\right]\left|0\right\rangle=0\qquad\Rightarrow\qquad\frac{\partial}{\partial
x^{\mu}}\left\langle 0\right|\phi(x)\left|0\right\rangle=0,$ (3.18)

or in other words that the vacuum expectation value of the operator is a
constant over all of space-time. Lorentz symmetry does not give more
information about that constant, but it forbids vacuum expectation values for
all operators transforming non-trivially under the Lorentz group.

Let us consider next a Wightman 2-point function of identical scalar
operators,

$\left\langle 0\right|\phi(x)\phi(y)\left|0\right\rangle.$ (3.19)

In this case, translation symmetry tells us that

$\left(\frac{\partial}{\partial x^{\mu}}+\frac{\partial}{\partial
y^{\mu}}\right)\left\langle 0\right|\phi(x)\phi(y)\left|0\right\rangle=0.$
(3.20)

If we think of the correlator as being a function of $x+y$ and $x-y$, then
this Ward identities establishes that there is no dependence on the former,
i.e.

$\left\langle 0\right|\phi(x)\phi(y)\left|0\right\rangle=W(y-x),$ (3.21)

where $W$ denotes a function that is so far arbitrary.

In general, the consequences of translation symmetry are easier to see in
momentum space, using the Fourier transform of the local operators. By eq.
(3.12), we can establish that

$\big{[}P^{\mu},\widetilde{\phi}(p)\big{]}=p^{\mu}\,\widetilde{\phi}(p),$
(3.22)

and therefore the Fourier transform of the Wightman 2-point function obeys

$(p^{\mu}+q^{\mu})\left\langle
0\right|\widetilde{\phi}(p)\widetilde{\phi}(q)\left|0\right\rangle=0.$ (3.23)

From this, we conclude that the 2-point function is proportional to a Dirac
delta function:

$\left\langle
0\right|\widetilde{\phi}(p)\widetilde{\phi}(q)\left|0\right\rangle=(2\pi)^{d}\delta^{d}(p+q)\widetilde{W}(q).$
(3.24)

The numerical factor $(2\pi)^{d}$ is merely a convention. As the notation
suggests, $\widetilde{W}$ is actually the Fourier transform of $W$,

$\widetilde{W}(q)=\int d^{d}x\,e^{ip\cdot x}W(x).$ (3.25)

Taking into account Lorentz symmetry, one can also establish that the Wightman
function $W(x)$ can only depend on the Lorentz-invariant distance $x^{2}$,
although there is a subtlety: this is a different function depending whether
$x$ is space-like or time-like (future- or past-directed), as Lorentz
transformations act separately on each of these regions. In momentum space,
the same arguments says that $\widetilde{W}(q)$ must be a function of $q^{2}$.
In this case, the condition that only states of positive energy exist requires
that $\widetilde{W}(q)$ vanishes unless $q^{0}\geq\left|\vec{q}\right|$, and
therefore we can unambiguously write

$\widetilde{W}(q)=2\pi\,\theta\left(q^{0}-\left|\vec{q}\right|\right)\rho(-q^{2}),$
(3.26)

where $\rho$ is a function of the positive quantity $-q^{2}$, and $\theta$ is
the Heaviside step function.999It satisfies $\theta(a)=0$ for $a<0$, and
$\theta(a)=1$ for $a>1$.

The fact that the Wightman 2-point function (3.24) is proportional to a delta
function raises an important concern: in spite of their names, Wightman
functions are _not_ functions but rather distributions (this is also part of
the Wightman axioms: they are in fact _tempered distributions_). Note also
that the same function computes the overlap between the two states

$\widetilde{\phi}(q)\left|0\right\rangle\qquad\text{and}\qquad\widetilde{\phi}(-p)\left|0\right\rangle$
(3.27)

(Hermitian conjugation flips the sign of momenta). Therefore, the limit
$p\to-q$ corresponds to the norm of either of these states. But this limit is
clearly discontinuous, or the norm of the state infinite. The resolution of
this issue is that the objects $\phi(x)$ and its Fourier transform
$\widetilde{\phi}(p)$ are not operators, but rather _operator-valued
distributions_. In other words, $\phi(x)\left|0\right\rangle$ and
$\widetilde{\phi}(p)\left|0\right\rangle$ are _not_ states of the theory, as
they have in fact infinite norm. Formally, these operator-valued distributions
only make sense when they are integrated against test functions, defining

$\phi[f]=\int d^{d}x\,f(x)\phi(x),$ (3.28)

or

$\widetilde{\phi}[\tilde{f}]=\int d^{d}p\,\tilde{f}(p)\widetilde{\phi}(p),$
(3.29)

where $f$ and $\widetilde{f}$ are Schwartz-class test functions (smooth
functions decaying faster than any power at infinity). When acting on the
vacuum, these smeared operators give well-defined states, with finite norms.
For instance, we have

$\displaystyle\left\|\widetilde{\phi}[\tilde{f}]\left|0\right\rangle\right\|^{2}$
$\displaystyle=\int d^{d}pd^{d}q\tilde{f}^{*}(p)\tilde{f}(q)\left\langle
0\right|\widetilde{\phi}(-p)\widetilde{\phi}(q)\left|0\right\rangle$
$\displaystyle=(2\pi)^{d+1}\int\limits_{q^{0}>\left|\vec{q}\right|}\\!\\!d^{d}q\left|\tilde{f}(q)\right|^{2}\rho(-q^{2}).$
(3.30)

Test functions will not appear further in these lectures. For physicists, they
are mostly an annoyance that we prefer to avoid. However, it is important to
know that there exists a mathematically rigorous way of dealing with Wightman
functions. For one thing, this gives a proper justification of why it is
always fine to take the Fourier transform between the position- and momentum-
space representation, as tempered distributions always admit a Fourier
transform. But bear in mind that this is only true of Wightman functions, not
of time-ordered correlators.

###  3.3 Spectral representation

The norm (3.30) is also giving away important information: it can only be
positive for any test function $f$ if the function $\rho$ is positive,

$\rho(\mu^{2})\geq 0,\qquad\quad\forall\,\mu^{2}>0.$ (3.31)

$\rho(\mu^{2})$ is in fact the _spectral density_ encountered in standard
quantum field theory textbooks, where we can often find it in the form

$\left\langle
0\right|\phi(x)\phi(y)\left|0\right\rangle=2\pi\int\frac{d^{d}k}{(2\pi)^{d}}\int\limits_{0}^{\infty}d\mu^{2}\,e^{ik\cdot(x-y)}\theta(k^{0})\delta(k^{2}+\mu^{2})\rho(\mu^{2}).$
(3.32)

The spectral density is an essential tool in non-perturbative quantum field
theory. It can for instance be used in the construction of the time-ordered
correlation function

$\langle\phi(x)\phi(y)\rangle_{T}\equiv\theta(x^{0}-y^{0})\left\langle
0\right|\phi(x)\phi(y)\left|0\right\rangle+\theta(y^{0}-x^{0})\left\langle
0\right|\phi(y)\phi(x)\left|0\right\rangle.$ (3.33)

Unlike the Wightman function, this is not a tempered distribution, because the
$\theta$ function is not differentiable at the origin. Nevertheless, the time-
ordered product admits the simple representation

$\langle\phi(x)\phi(y)\rangle_{T}=\int\frac{d^{d}k}{(2\pi)^{d}}\int\limits_{0}^{\infty}d\mu^{2}\,e^{ik\cdot(x-y)}\frac{i}{-k^{2}-\mu^{2}+i\varepsilon}\,\rho(\mu^{2})$
(3.34)

where the limit $\varepsilon\to 0_{+}$ is understood. This is the _Källen-
Lehmann representation_ for the time-ordered 2-point function.

Exercise 3.2 Derive the Källen-Lehmann representation. An elegant derivation
is to first show that the time-ordered 2-point function can be written as the
difference between the Wightman function and the vacuum expectation value of a
retarded commutator,

$\langle\phi(x)\phi(y)\rangle_{T}=W(y-x)-\theta(y^{0}-x^{0})\left\langle
0\right|\left[\phi(x),\phi(y)\right]\left|0\right\rangle.$

The next step is to Fourier transform both terms in $y$ after setting $x=0$.
We know that the Wightman function (3.24) has a nice Fourier transform

$\widetilde{W}(q)=2\pi\int\limits_{0}^{\infty}d\mu^{2}\,\theta(q^{0})\delta(q^{2}+\mu^{2})\rho(\mu^{2}),$

which can be equivalently written

$\widetilde{W}(q)=\int\limits_{0}^{\infty}d\mu^{2}\,\theta(q^{0})\left[\frac{i}{q^{2}+\mu^{2}+i\varepsilon}-\frac{i}{q^{2}+\mu^{2}-i\varepsilon}\right]\rho(\mu^{2}).$

The retarded commutator is only non-zero in the forward light cone in $y$, and
therefore it also admits a Fourier transform that converges provided that we
give an imaginary part to $q$. Compute this Fourier transform, and show that
at real $q$ it is equal to

$\int\limits_{0}^{\infty}d\mu^{2}\,\left[\theta(q^{0})\frac{i}{q^{2}+\mu^{2}+i\varepsilon}+\theta(-q^{0})\frac{i}{q^{2}+\mu^{2}-i\varepsilon}\right]\rho(\mu^{2}).$

The difference between these last two integrals can then easily be turned into
the Källen-Lehmann representation (3.34).

The spectral representation for the 2-point function gives familiar results in
non-interacting theories. A massive scalar field has for instance the spectral
density

$\rho(\mu^{2})=\delta(\mu^{2}-m^{2}),$ (3.35)

and from this we recover the known massive propagator

$\langle\phi(x)\phi(y)\rangle_{T}=\int\frac{d^{d}k}{(2\pi)^{d}}e^{ik\cdot(x-y)}\frac{i}{-k^{2}-m^{2}+i\varepsilon}.$
(3.36)

In an interacting theory, the spectral density will get contributions
corresponding to particle production above a certain threshold (see figure 4).

Figure 4: Left: spectral density in the theory of a free massive field, and
typical contributions above the threshold for particle production (dashed
line). Right: possible spectral densities in a scale-invariant theory.

The discussion applied so far to a generic quantum field theory without
conformal symmetry. Let us now examine the role of scale and special conformal
invariance, starting with the former.

###  3.4 Scale symmetry

The assumption of scale symmetry coincides with the existence of a third
conserved charge besides $P^{\mu}$ and $M^{\mu\nu}$, namely the operator $D$.
We obtained before the commutator of a generic local operator with
$M^{\mu\nu}$ assuming that it transforms in some irreducible representation of
the Lorentz group at the origin $x=0$ (an operator inserted at some other
point is not in an irreducible representation because $P^{\mu}$ does not
commute with $M^{\mu\nu}$). Since $M^{\mu\nu}$ commutes with $D$, the same
assumption can be made about scale: any local operator can be decomposed
further into irreducible representations of the group of scale
transformations, meaning that we can write

$\left[D,\phi(0)\right]=-i\Delta\phi(0).$ (3.37)

$\Delta$ is called the _scaling dimension_ of the operator $\phi$ (each local
operator of the theory has its own scaling dimension). The factor of $i$
ensures that $\Delta$ is a real number when $\phi$ is a real operator. As
before, we can use eq. (3.3) to obtain the commutator at any other point $x$:

$\left[D,\phi(x)\right]=-i\left(x^{\mu}\partial_{\mu}+\Delta\right)\phi(x).$
(3.38)

Note that this is consistent with the transformation rule (2.62) for a
classical field: the scaling dimension $\Delta$ coincides with the mass
dimension of the operator $\phi$ in a free theory.

Using this new commutator and assuming that the vacuum state is invariant
under scale transformations, a new Ward identity can be obtained for the
Wightman 2-point function,

$\left(x^{\mu}\frac{\partial}{\partial
x^{\mu}}+y^{\mu}\frac{\partial}{\partial y^{\mu}}+2\Delta\right)\left\langle
0\right|\phi(x)\phi(y)\left|0\right\rangle=0,$ (3.39)

or equivalently, using eq. (3.21),

$\left(x^{\mu}\frac{\partial}{\partial x^{\mu}}+2\Delta\right)W(x)=0.$ (3.40)

The corresponding condition on the momentum-space 2-point function is obtained
from the Fourier transform, using integration by parts:

$\left(-q^{\mu}\frac{\partial}{\partial
q^{\mu}}+2\Delta-d\right)\widetilde{W}(q)=0.$ (3.41)

The solution to this equation consistent with the form (3.26) is unique, up to
a multiplicative constant $C$,

$\widetilde{W}(q)=2\pi
C\,\theta\left(q^{0}-\left|\vec{q}\right|\right)(-q^{2})^{\Delta-d/2},$ (3.42)

implying that the spectral density is a power of the energy,

$\rho(\mu^{2})=C\,(\mu^{2})^{\Delta-d/2}.$ (3.43)

This kind of spectral density, shown in figure 4, is very different from that
of a massive interacting theory: the operator $\phi$ creates states of all
energies. At the same time, its simplicity is striking: it is characterized by
a single parameter $\Delta$, and a normalization constant that does not carry
physical information (the operators can be re-defined to absorb this
constant).

It is instructive to compute the time-ordered function using the Källen-
Lehmann representation: performing the integral over $\mu^{2}$, we find

$\langle\phi(x)\phi(y)\rangle_{T}=\frac{i\pi
C}{\sin\left[\pi\left(\Delta-\frac{d}{2}\right)\right]}\int\frac{d^{d}k}{(2\pi)^{d}}e^{ik\cdot(x-y)}\left(k^{2}-i\varepsilon\right)^{\Delta-d/2}.$
(3.44)

The term $\left(k^{2}-i\varepsilon\right)^{\Delta-d/2}$ in the integral looks
like the propagator for a massless scalar field raised to a non-integer power.
This is in fact what we expect in perturbation theory when the
$\beta$-function has a non-trivial fixed point: the renormalized 2-point
function has logarithms that can be re-summed into a power controlled by the
anomalous dimension $\gamma$ of the field $\phi$,

$\frac{i}{-q^{2}}\left[1+\gamma\log(-q^{2})+\ldots\right]\approx
i(-q^{2})^{-1+\gamma}.$ (3.45)

In this case, the scaling dimension of the scalar operator corresponding to
the renormalized field is

$\Delta=\frac{d-2}{2}+\gamma.$ (3.46)

In the limit $\gamma\to 0$, we recover the free propagator mentioned above.
Note however that the factor in front of the integral diverges in this limit,
unless the coefficient $C$ satisfies

$C\propto\gamma=\Delta-\frac{d-2}{2}.$ (3.47)

Assuming that it is the case (see below), the spectral density obeys101010This
limit can be verified by integrating both sides in $\mu^{2}$ and taking the
limit $\gamma\to 0$ afterward.

$\rho(\mu^{2})\propto\gamma\,(\mu^{2})^{-1+\gamma}\xrightarrow{\gamma\to
0}\delta(\mu^{2}).$ (3.48)

This is precisely the spectral density that is expected in the free scalar
field theory. In this case (and this case only!), the operator $\phi$
describes a massless scalar particle.

It turns out that $\Delta=\frac{d-2}{2}$ is the lowest possible value for
$\Delta$: for any $\Delta$ below that value, the spectral density is not
integrable in the limit $\mu^{2}\to 0$. One might also worry about the
opposite limit $\mu^{2}\to\infty$ in the integral: for any
$\Delta>\frac{d}{2}$, the spectral density grows with $\mu^{2}$. However,
remember that this spectral density is in fact a Wightman function, i.e. a
tempered distribution that should be understood as integrated against test
functions that decay faster than any power at large $q^{2}$. Therefore,
arbitrarily large values of $\Delta$ are possible, but there is a lower bound
on $\Delta$ below which the states smeared with test functions have infinite
norm. The inequality

$\Delta\geq\frac{d-2}{2}.$ (3.49)

is known in the literature as the _unitarity bound_ for scalar
operators.111111It is usually derived using the action of special conformal
transformations, but we have just seen here that it applies to scale-invariant
theories as well. Note that any scalar operator that saturates this unitarity
bound has

$\left\langle
0\right|\widetilde{\phi}(p)\widetilde{\phi}(q)\left|0\right\rangle\propto\delta(q^{2}),$
(3.50)

which implies

$q^{2}\left\langle
0\right|\widetilde{\phi}(p)\widetilde{\phi}(q)\left|0\right\rangle=0,$ (3.51)

or in position space

$\left\langle 0\right|\phi(x)\partial^{2}\phi(y)\left|0\right\rangle=0.$
(3.52)

Since this is true for any $x$ and $y$, it implies that

$\partial^{2}\phi(x)=0$ (3.53)

is true as an operator equation. Since this is the equation of motion for a
free field, a theory in which $\Delta=\frac{d-2}{2}$ is a free field theory.

The simplicity of the momentum-space 2-point function in a scale-invariant
theory also means that it can easily be Fourier transformed back to position
space using

$W(x)=\int\frac{d^{d}q}{(2\pi)^{d}}e^{-iq\cdot x}\widetilde{W}(q).$ (3.54)

Exercise 3.3 Perform the Fourier transform explicitly. You can use the fact
that the integral is Lorentz invariant to determine that $W(x)$ is in fact a
function of $x^{2}$. Moreover, since the integrand only has support for $q$ in
the forward light-cone, this defines a function of $x$ that is analytic in $x$
as long as $\operatorname{\text{im}}x$ is contained in the future light cone:
the integrand is damped by the exponential
$e^{q\cdot\operatorname{\text{im}}x}$ with $q\cdot\operatorname{\text{im}}x<0$
in that case (this domain of analyticity is known as the “future tube”). This
means that we are free to evaluate the integral at a point $x=(i\tau,0)$, and
then use $\tau^{2}=x^{2}$ to recover the general solution. The integral at
that point is convergent for all $\Delta$ satisfying the unitarity bound, and
you should find

$W(\tau)=C\frac{2^{2\Delta}\Gamma(\Delta)\Gamma\left(\Delta-\frac{d-2}{2}\right)}{(4\pi)^{d/2}}\,\tau^{-2\Delta}$

The result of this integral can be written as

$W(x)=\frac{C^{\prime}}{\left[-(x^{0}+i\varepsilon)^{2}+\vec{x}^{2}\right]^{\Delta}}$
(3.55)

where the limit $\varepsilon\to 0_{+}$ is understood to make sense of the case
in which $x^{2}\leq 0$, namely

$W(x)=\frac{C^{\prime}}{\left|x^{2}\right|^{\Delta}}\times\left\\{\begin{array}[]{ll}e^{-i\pi\Delta}&\text{if}\leavevmode\nobreak\
x^{0}<-\left|\vec{x}\right|,\\\ 1&\text{if}\leavevmode\nobreak\
-\left|\vec{x}\right|<x^{0}<\left|\vec{x}\right|,\\\
e^{i\pi\Delta}&\text{if}\leavevmode\nobreak\
x^{0}>\left|\vec{x}\right|.\end{array}\right.$ (3.56)

The coefficient $C^{\prime}$ and $C$ are related by

$C=\frac{(4\pi)^{d/2}}{2^{2\Delta}\Gamma(\Delta)\Gamma\left(\Delta-\frac{d-2}{2}\right)}C^{\prime}.$
(3.57)

Note that the proportionality factor is positive for all $\Delta$ satisfying
the unitary bound (3.49). This implies that the 2-point correlation function
is always decreasing with the distance and not the other way around. It is in
fact customary in conformal field theory to normalize the scalar operator
$\phi$ so that $C^{\prime}=1$, in which case $C$ vanishes as in eq. (3.47) in
the limit $\Delta\to\frac{d-2}{2}$.

Finally, let us conclude the analysis of scale symmetry with a comment about
one-point functions. We saw in the previous section that a constant vacuum
expectation value for a scalar operator was compatible with Poincaré symmetry.
However, the commutator (3.38) requires then that $\Delta=0$, which violates
the unitarity bound. We conclude that all one-point functions must vanish in a
scale-invariant theory.

###  3.5 Special conformal symmetry

As with scale symmetry, the presence of special conformal symmetry is
associated with the existence of the conserved charges $K^{\mu}$, organized in
a $d$-dimensional vector. Unlike $D$, however, $K^{\mu}$ does not commute with
$M^{\mu\nu}$, so it cannot be diagonalized at the point $x=0$. Nevertheless,
we can use the conformal algebra to establish that, if $\phi$ is a local
operator with scaling dimension $\Delta$, then $\left[K^{\mu},\phi\right]$ has
scaling dimension $\Delta-1$:

$\displaystyle\big{[}D,[K^{\mu},\phi(0)]\big{]}$
$\displaystyle=\big{[}K^{\mu},[D,\phi(0)]\big{]}+\big{[}[D,K^{\mu}],\phi(0)\big{]}$
$\displaystyle=\big{[}K^{\mu},-i\Delta\phi(0)\big{]}+\big{[}iK^{\mu},\phi(0)\big{]}=-i(\Delta-1)[K^{\mu},\phi(0)].$
(3.58)

This is similar to the observation that the commutator
$\left[P^{\mu},\phi(x)\right]$ has scaling dimension $\Delta+1$,

$\big{[}D,[P^{\mu},\phi(0)]\big{]}=-i(\Delta+1)[P^{\mu},\phi(0)],$ (3.59)

consistent with the fact that the derivative $\partial^{\mu}$ has mass
dimension $+1$. The fact that $K^{\mu}$ lowers the scaling dimensions appears
to be in contradiction with our findings of the last section stating that
$\Delta$ is bounded below: given any local operator, one can always construct
other local operators with arbitrarily smaller scaling dimension.

The only way out of this apparent paradox is to assume that at some point the
action of $K^{\mu}$ annihilates the operator. In other words, there must exist
some local operator such that

$\left[K^{\mu},\phi(0)\right]=0.$ (3.60)

We call this local operator a _primary_. Any other local operator can be
obtained acting on a primary with $P^{\mu}$, and we call it a _descendant_.
Since the action of $P^{\mu}$ coincides with taking derivatives, a primary
operator is simply an operator that cannot be written as the derivative of
some other operator. Unless specified otherwise, we shall from now on only
consider Wightman correlation functions of primary operators. Descendants will
be explicitly denoted with a derivative.

The transformation of a primary operator away from the origin can once again
be obtained from eq. (3.3). Note that since the commutator of $P^{\mu}$ and
$K^{\mu}$ involves both $D$ and $M^{\mu\nu}$, this transformation depends on
the scaling dimension and on the Lorentz representation of the operators, i.e.
on the eigenvalues $\Delta$ and $\mathcal{S}^{\mu\nu}$. We find

$\left[K^{\mu},\phi(x)\right]=-i\left(2x^{\mu}x^{\nu}\partial_{\nu}-x^{2}\partial^{\mu}+2\Delta
x^{\mu}-2\mathcal{S}^{\mu\nu}x_{\nu}\right)\phi(x).$ (3.61)

This equation also defines the commutator of a momentum-space operator: using
integration by parts in the definition (3.12), one can show that this amounts
to replacing $\partial_{\mu}\to-iq_{\mu}$ and $x^{\mu}\to-i\partial/\partial
q_{\mu}$, so that

$\big{[}K^{\mu},\widetilde{\phi}(q)\big{]}=\left[2\frac{\partial^{2}}{\partial
q_{\mu}\partial q_{\nu}}q_{\nu}-\frac{\partial^{2}}{\partial q_{\nu}\partial
q^{\nu}}q^{\mu}-2\Delta\frac{\partial}{\partial
q_{\mu}}+2\mathcal{S}^{\mu\nu}\frac{\partial}{\partial
q^{\nu}}\right]\widetilde{\phi}(q),$ (3.62)

or after permuting the derivatives with $q$,

$\big{[}K^{\mu},\widetilde{\phi}(q)\big{]}=\left[2q^{\nu}\frac{\partial^{2}}{\partial
q_{\mu}\partial q^{\nu}}-q^{\mu}\frac{\partial^{2}}{\partial q_{\nu}\partial
q^{\nu}}+2(d-\Delta)\frac{\partial}{\partial
q_{\mu}}+2\mathcal{S}^{\mu\nu}\frac{\partial}{\partial
q^{\nu}}\right]\widetilde{\phi}(q).$ (3.63)

This is now a second-order differential acting on the operator expressed in
momentum space.

The first thing we can do with this commutator is to examine the related Ward
identity for the Wightman 2-point function. Remember that this function can be
written as

$W(x)=\left\langle 0\right|\phi(0)\phi(x)\left|0\right\rangle.$ (3.64)

The commutator acts trivially on the operator inserted at the origin, so that
we must have (note that $\mathcal{S}^{\mu\nu}=0$ for scalar operators)

$\left(2x^{\mu}x^{\nu}\partial_{\nu}-x^{2}\partial^{\mu}+2\Delta
x^{\mu}\right)W(x)=0.$ (3.65)

Let us check at space-like $x$: using $W(x)=1/(x^{2})^{\Delta}$, we have
$\partial_{\mu}W(x)=-2\Delta W(x)x^{\mu}/x^{2}$, and therefore the
differential equation is readily satisfied. The same can be verified in
momentum space: by definition, we have

$\widetilde{W}(q)=\left\langle
0\right|\phi(0)\widetilde{\phi}(q)\left|0\right\rangle,$ (3.66)

where only the operator on the right is Fourier transformed while the one on
the left is kept fixed at the origin in position space, and so the commutator
above implies that

$\left[2q^{\nu}\frac{\partial^{2}}{\partial q_{\mu}\partial
q^{\nu}}-q^{\mu}\frac{\partial^{2}}{\partial q_{\nu}\partial
q^{\nu}}+2(d-\Delta)\frac{\partial}{\partial
q_{\mu}}\right]\widetilde{W}(q)=0.$ (3.67)

With $\widetilde{W}(q)=(-q^{2})^{\Delta-d/2}$, this equation is again
satisfied.

#### Distinct operators:

The fact that $W(x)$ and $\widetilde{W}(q)$ readily satisfy the constraint
imposed by special conformal symmetry is very specific to identical scalar
operators. In every other case, special conformal symmetry adds more
constraints than Poincaré and scale symmetry alone. The simplest example is
that of a 2-point function of distinct scalar operators,

$\left\langle 0\right|\phi_{1}(x)\phi_{2}(y)\left|0\right\rangle.$ (3.68)

This is still a function of $(x-y)^{2}$ by Poincaré symmetry. But there are
now two distinct scaling dimensions $\Delta_{1}$ and $\Delta_{2}$
corresponding to the operators $\phi_{1}$ and $\phi_{2}$, and the Ward
identity for scale symmetry becomes (for simplicity setting $\phi_{1}$ at the
origin)

$\left(x^{\mu}\frac{\partial}{\partial
x^{\mu}}+\Delta_{1}+\Delta_{2}\right)\left\langle
0\right|\phi_{1}(0)\phi_{2}(x)\left|0\right\rangle=0.$ (3.69)

The solution is fixed up to a multiplicative constant to be (assuming $x$
space-like for simplicity)

$\left\langle
0\right|\phi_{1}(0)\phi_{2}(x)\left|0\right\rangle=\frac{C_{12}}{(x^{2})^{(\Delta_{1}+\Delta_{2})/2}}.$
(3.70)

The Ward identity for special conformal transformation is obtained from the
commutator (3.61), giving

$\left(2x^{\mu}x^{\nu}\partial_{\nu}-x^{2}\partial^{\mu}+2\Delta_{2}x^{\mu}\right)\left\langle
0\right|\phi_{1}(0)\phi_{2}(x)\left|0\right\rangle=0.$ (3.71)

Using
$\partial_{\mu}\left[(x^{2})^{-(\Delta_{1}+\Delta_{2})/2}\right]=-(\Delta_{1}+\Delta_{2})x^{\mu}/x^{2}$,
this implies

$\left(\Delta_{2}-\Delta_{1}\right)C_{12}\frac{x^{\mu}}{(x^{2})^{(\Delta_{1}+\Delta_{2})/2}}=0.$
(3.72)

If the scaling dimensions are different ($\Delta_{1}\neq\Delta_{2}$), then
$C_{12}$ must vanish. This is an important lesson: in conformal field theory,
only primary operators of identical scaling dimensions can have non-zero
2-point functions.

In fact, if there are several scalar operators with the same scaling dimension
$\phi_{i}$ with $i=1,\ldots,N$, then

$\left\langle
0\right|\phi_{i}(0)\phi_{j}(x)\left|0\right\rangle=\frac{C_{ij}}{(x^{2})^{\Delta}},$
(3.73)

where $C_{ij}$ is a symmetric $N\times N$ matrix. By unitarity, this matrix
must be positive-definite: if this were not the case, then one could define a
negative-norm state by taking an appropriate linear combination of the
$\phi_{i}$ and smearing. Therefore, it is always possible to choose a basis of
operators in which $C_{ij}$ is diagonal. Moreover, the operators can be
normalized so that $C_{ij}=\delta_{ij}$. From now on, we will therefore always
assume that the only non-zero 2-point functions are those involving identical
operators.

#### Operators with spin:

The other situation in which special conformal symmetry plays an essential
role is when the operators carry spin. Let us take the simplest example of a
vector operator $A^{\mu}(x)$, and denote its 2-point function by

$W^{\mu\nu}(x)=\left\langle 0\right|A^{\mu}(0)A^{\nu}(x)\left|0\right\rangle.$
(3.74)

As with scalars, one can also take the Fourier transform of this tempered
distribution, defining

$\widetilde{W}^{\mu\nu}(p)=\int d^{d}x\,e^{ip\cdot x}\left\langle
0\right|A^{\mu}(0)A^{\nu}(x)\left|0\right\rangle=\left\langle
0\right|A^{\mu}(0)\widetilde{A}^{\nu}(p)\left|0\right\rangle.$ (3.75)

Again, this function corresponds to the momentum-space correlation function
without the delta-function imposing momentum conservation, namely

$\left\langle
0\right|\widetilde{A}^{\mu}(p)\widetilde{A}^{\nu}(q)\left|0\right\rangle=(2\pi)^{d}\delta^{d}(p+q)\widetilde{W}^{\mu\nu}(q).$
(3.76)

By Lorentz symmetry, this function $\widetilde{W}$ of a single momentum can be
decomposed into two different tensor structures multiplying scalar functions,

$\widetilde{W}^{\mu\nu}(p)=(p^{\mu}p^{\nu}-p^{2}\eta^{\mu\nu})\widetilde{W}_{1}(p)+p^{\mu}p^{\nu}\widetilde{W}_{0}(p).$
(3.77)

Moreover, using scale symmetry and energy positivity, we can infer that the
functions $\widetilde{W}_{1,0}$ are just powers of $p^{2}$ over the forward
light cone,

$\widetilde{W}_{1,0}(p)=\theta\left(p^{0}-\left|\vec{p}\right|\right)(-p^{2})^{\Delta-d/2-1}C_{1,0},$
(3.78)

where $\Delta$ is the scaling dimension of the operator $A^{\mu}$ and $C_{1}$,
$C_{0}$ two constants that cannot be related by scale and Poincaré symmetry
only.

There is a good reason for using precisely the two tensor structures in eq.
(3.77) and not, say, $\eta^{\mu\nu}$ and $p^{\mu}p^{\nu}$. Thanks to energy
positivity, it is always possible to choose a Lorentz frame in which
$\vec{p}=0$.121212This is similar to going to a massive particle’s rest frame.
In this frame, the momentum is invariant under the group $\text{SO}(d-1)$ of
spatial rotations, and therefore the 2-point function can be decomposed into
irreducible representations of that group. The part proportional to $C_{0}$
only appears in the component $\widetilde{W}^{00}$, and it transforms like a
scalar under rotations. Conversely, the part proportional to $C_{1}$ only has
non-zero entries for spatial Lorentz indices $\widetilde{W}^{ij}$; it is in
fact proportional to the identity in the $d-1$ subspace, i.e. it is the
invariant tensor for the vector representation of $\text{SO}(d-1)$. In
particle physics language, we would call these two parts respectively
_longitudinal_ and _transverse_.

Being able to use irreducible representations of $\text{SO}(d-1)$ is an
advantage of working in momentum space: there is no obvious Lorentz frame in
which such a decomposition can be made in position space since the 2-point
function has support over all of Minkowski space-time. The disadvantage of
working in momentum space is that the Ward identity for special conformal
transformation is a second-order differential equation in $p$, while it is a
first-order differential in position space. This Ward identity can
nevertheless be straightforwardly applied to eq. (3.77), and it yields a
relation between the longitudinal and transverse parts, i.e. between the
coefficients $C_{0}$ and $C_{1}$, given by (see exercise)

$C_{0}=\frac{\Delta-d+1}{\Delta-1}\,C_{1}.$ (3.79)

This is a very important consequence of special conformal symmetry: while in a
scale-invariant theory the longitudinal and transverse polarizations are
independent, in a conformal theory they are related.

Exercise 3.4 Using the definition (3.75) for the function
$\widetilde{W}^{\mu\nu}(p)$, show that it satisfies the special conformal Ward
identity

$\left(2p^{\beta}\frac{\partial^{2}}{\partial p_{\alpha}\partial
p^{\beta}}-p^{\alpha}\frac{\partial^{2}}{\partial p_{\beta}\partial
p^{\beta}}+2(d-\Delta)\frac{\partial}{\partial
p_{\alpha}}+2\mathcal{S}^{\alpha\beta}\frac{\partial}{\partial
p_{\beta}}\right)\widetilde{W}^{\mu\nu}(p)=0,$

where $\mathcal{S}^{\alpha\beta}$ is given by eq. (3.5), and use this to prove
the relation (3.79).

This has consequences on the possible values that $\Delta$ can take. As
before, this 2-point function computes the norm of a state, and its positivity
requires:

  * •

$\Delta>\frac{d}{2}$ so that the 2-point function is integrable at $p^{2}\to
0$;

  * •

$C_{0}$ and $C_{1}$ both positive, so that the norm is positive for any choice
of external polarization vector (i.e. the tensor $\widetilde{W}^{\mu\nu}$ must
be positive-definite). This requires $\Delta-1$ and $\Delta-d+1$ to have the
same sign.

The combination of these two conditions in $d>2$ dimensions (spin is treated
differently in $d=2$) implies

$\Delta\geq d-1.$ (3.80)

This is known as the unitarity bound for a vector operator.

As in the scalar case, something special happens when the unitarity bound is
saturated ($\Delta=d-1$). In this case the 2-point function has no
longitudinal component, $C_{0}=0$, and $\widetilde{W}^{\mu\nu}$ vanishes when
contracted with $p_{\mu}$ or $p_{\nu}$. This implies that the longitudinal
part of the state is null,

$p_{\mu}\widetilde{A}^{\mu}(p)\left|0\right\rangle=0,$ (3.81)

or equivalently that $\widetilde{A}^{\mu}(p)$ is an operator that only creates
transverse-polarization states. The equivalent statement in position-space is

$\partial_{\mu}A^{\mu}(x)\left|0\right\rangle=0.$ (3.82)

In other words, $A^{\mu}$ is a conserved current. The equivalence goes both
way: any vector operator with $\Delta=d-1$ is a conserved current, and any
conserved current must have scaling dimension $\Delta=d-1$. This also shows
that conserved currents are primary operators: they cannot be written as
$\partial^{2}$ acting on another vector operator (that operator would have
$\Delta=d-3$, below the unitarity bound), nor as $\partial^{\mu}$ acting on a
scalar operator $\phi$, because the conservation requirement would then imply
$\partial^{2}\phi=0$, which is only possible if $\phi$ has scaling dimensions
$(d-2)/2,$ and thus the current $\Delta=d/2$ (again below the unitarity
bound).

The fact that 2-point functions of primary operators are completely fixed by
conformal symmetry up to a choice of normalization is not specific to scalar
and vector operators. In fact, any local operator specified by a
representation under the Lorentz group and a scaling dimension defines an
irreducible representation of the conformal group $\text{SO}(d,2)$, and as
such its 2-point function is fixed by group theory. This also explains on more
general grounds why 2-point functions of distinct operators vanish. The
construction of all unitary representations of the conformal group in $d=4$
dimensions was performed by Mack in 1975 [7], and similar constructions can be
done in other dimensions. Some Lorentz representations are specific to a given
dimension $d$, and others exist in any $d$, like the scalar and the symmetric,
traceless representations with $\ell$ Lorentz indices (the vector discussed
above is a special case corresponding to $\ell=1$). All such symmetric tensors
satisfy the unitarity bound

$\Delta\geq d-2+\ell,$ (3.83)

and they are in general described by $\ell+1$ distinct polarizations
(irreducible representations of the rotation group), except when the bound is
saturated, in which case there is just a single, transverse polarization and
the operator is a higher-spin conserved current. In terms of representations
of the conformal group, generic operators are said to belong to _long
multiplets_ , whereas special cases such as scalars with $\Delta=(d-2)/2$ or
symmetric tensors with $\Delta=d-2+\ell$ are said to be in _short multiplets_
(they contain fewer descendants).

Exercise 3.5 Construct explicitly the 2-point function of a 2-index symmetric
traceless operator $B^{\mu\nu}(x)$. As a starting point, let us decompose the
momentum-space correlation functions into tensors that transform covariantly
under rotations in the rest frame. Using the transverse projector

$\eta_{\perp}^{\mu\nu}=\eta^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{p^{2}}$

satisfying $p_{\mu}\eta_{\perp}^{\mu\nu}=0$, this can be done as

$\displaystyle\left\langle
0\right|B^{\mu\nu}(0)\widetilde{B}^{\rho\sigma}(p)\left|0\right\rangle=\bigg{[}$
$\displaystyle\frac{1}{2}\left(\eta_{\perp}^{\mu\rho}\eta_{\perp}^{\nu\sigma}+\eta_{\perp}^{\mu\sigma}\eta_{\perp}^{\nu\rho}-\text{traces}\right)C_{2}$
$\displaystyle+\frac{1}{4}\left(\eta_{\perp}^{\mu\rho}\frac{p^{\nu}p^{\sigma}}{p^{2}}+\text{permutations}\right)C_{1}$
$\displaystyle+\frac{p^{\mu}p^{\nu}p^{\rho}p^{\sigma}}{(p^{2})^{2}}C_{0}\bigg{]}\theta\left(p^{0}-\left|\vec{p}\right|\right)(-p^{2})^{\Delta-d/2+1}.$

Then write down the Ward identity for special conformations (including the
spin operator for a 2-index tensor, which you have to determine), and show
that it leads to the conditions

$\displaystyle C_{1}$ $\displaystyle=2\frac{\Delta-d}{\Delta}\,C_{0},$
$\displaystyle C_{2}$
$\displaystyle=\frac{d}{d-1}\frac{(\Delta-d)(\Delta-d+1)}{\Delta(\Delta-1)}\,C_{0}.$

Argue that this gives rise to the unitarity bound $\Delta\geq d$, in agreement
with eq. (3.83). Conclude that the energy-momentum tensor $T^{\mu\nu}$ is a
primary operator with $\Delta=d$.

###  3.6 UV/IR divergences and anomalies

The discussion has been focused so far on Wightman functions. Besides having a
Hilbert-space interpretation and satisfying conformal Ward identities that
give strong constraints on their possible form, Wightman functions are also
free of divergences. In momentum space, regularity at small momenta (IR) is
enforced by the unitarity bound, whereas the power-law growth at large momenta
(UV) is compatible with the damping provided by test functions. In position
space, the apparent singularity at short distance (UV) is resolved by the
$i\varepsilon$ that provides an unequivocal prescription for deforming any
contour of integration.

These properties are not found in time-ordered products. Let us consider once
again the scalar 2-point function. Using the standard CFT normalization (3.57)
and the Källen-Lehmann representation, we find

$\langle\phi(x)\phi(y)\rangle_{T}=-i\frac{(4\pi)^{d/2}\Gamma\left(\frac{d}{2}-\Delta\right)}{2^{2\Delta}\Gamma(\Delta)}\int\frac{d^{d}k}{(2\pi)^{d}}e^{ik\cdot(x-y)}\left(k^{2}-i\varepsilon\right)^{\Delta-d/2}.$
(3.84)

This expression diverges whenever $\Delta=\frac{d}{2}+n$ with integer $n$ due
to the $\Gamma$-function multiplying the integral, indicating that the Fourier
transform does not exist. In this case, the 2-point function has scaling
dimension $d+2n$, and is therefore compatible with contact terms of the form

$(\partial^{2})^{n}\delta^{d}(x-y).$ (3.85)

In path integral language, this is the situation in which the source field $J$
for the operator $\phi$ has scaling dimension $d-\Delta=\frac{d}{2}-n$, and
therefore contact terms of the form $J(\partial^{2})^{n}J$ can (and must) be
added to the action. These contact terms are obviously covariant under
Poincaré and scale transformations, so they can be added to the correlation
function without affecting the Ward identities at separated points. (3.85) is
the only possible contact term appearing in a scalar 2-point function, but
more terms can appear in other correlation functions.

When Fourier transformed to momentum space, all such contact terms become
polynomials in the momenta. In the scalar 2-point function case, these are
$(p^{2})^{n}$. Polynomial terms are incompatible with the positive-energy
condition of Wightman functions, as they have support over all causal regions.
But there are allowed (and in fact required) in time-ordered products. Time-
ordered products are defined in position space as the product of Wightman
functions, which are tempered distributions, with step functions, which are
not: therefore they do not necessarily have a Fourier transform. Contact terms
can be seen as a way of “fixing” the time-ordered product so that they can be
Fourier transformed.

This can be understood by analytic continuation in scaling dimension $\Delta$.
Let us assume that our scalar operator has scaling dimension
$\Delta=\frac{d}{2}+n-\epsilon$, and take the limit $\epsilon\to 0$ (note that
this $\epsilon$ is different from the one in the $i\varepsilon$ limit). Then
the time-ordered 2-point function in momentum space takes the form

$\Gamma(-n+\epsilon)(p^{2})^{n-\epsilon}+Z(p^{2})^{n},$ (3.86)

where the first term with a pole in $\epsilon$ comes from the expression
(3.84) valid at $x\neq y$, and the second term is the counterterm added to the
action. Choosing $Z\propto\epsilon^{-1}$ allows to cancel the divergence as
$\epsilon\to 0$, but it also gives rise to a logarithm,

$(p^{2})^{n}\log\left(\frac{p^{2}}{\mu^{2}}\right).$ (3.87)

The fact that we need to introduce a dimensionful quantity is the sign of a
_conformal anomaly_ : a 2-point function of this form does not satisfy the
Ward identity for scale transformations (3.41),

$\left[-p^{\mu}\frac{\partial}{p^{\mu}}+2n\right](p^{2})^{n}\log\left(\frac{p^{2}}{\mu^{2}}\right)=-2(p^{2})^{n}\neq
0.$ (3.88)

However, the anomalous term on the right-hand side is a polynomial in the
momenta, corresponding to a contact term, indicating that the anomaly is
local. In QFT language, this is typical of a (renormalized) UV divergence.

Note that the presence of contact terms is associated with a very special type
of operator of (half-)integer scaling dimensions. Such operators are
generically absent in an interacting conformal field theory, where the scaling
dimensions take irrational values. However, an exception to this rule concerns
conserved currents in even space-time dimension $d$. For instance, a conserved
current $J^{\mu}$ in $d=4$ has scaling dimensions $\Delta=3$; it is therefore
associated in the path integral language with a source $a_{\mu}$ that carries
dimension 1. This source is also subject to a gauge symmetry, $a_{\mu}\sim
a_{\mu}+\partial_{\mu}\alpha$, and therefore a possible contact term is
$f_{\mu\nu}f^{\mu\nu}$, where
$f_{\mu\nu}=\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}$. As in the scalar
case, this term must be used as a counterterm to cancel a divergence arising
in the time-ordered correlation function, leading generically to logarithms in
correlation functions involving the transverse polarization of the current,

$\langle
J^{\mu}(p)J^{\nu}(q)\rangle_{T}\propto(2\pi)^{4}\delta^{4}(p+q)(q^{\mu}q^{\nu}-q^{2}\eta^{\mu\nu})\log\left(\frac{q^{2}}{\mu^{2}}\right).$
(3.89)

As in the scalar case, this is the manifestation of a UV divergence. However,
the logarithm also implies that the limit $q^{2}\to 0$ diverges: this is what
we would call an IR divergence in quantum field theory. In the context of CFT,
there is no clear distinction between UV and IR divergences, as the two are
closely related. They both arise from an ambiguity in taking the Fourier
transform. UV divergences can be cured with local counterterms, at the price
of introducing a reference scale, but IR divergences are physical.

Note finally that time-ordered correlation functions involving a conserved
current do not only have transverse polarizations: while the conservation
condition implies the vanishing of the state

$\partial_{\mu}J^{\mu}(x)\left|0\right\rangle=0,$ (3.90)

and thus of Wightman correlation functions constructed from that state, this
is not true of the divergence of $J^{\mu}$ appearing in a time-ordered
product: the conservation equation $\partial_{\mu}J^{\mu}(x)=0$ is only true
as an _operator equation_ , namely away from coincident points. On general
grounds, one expects

$\displaystyle\langle\phi_{1}(x_{1})\cdots\phi_{n}(x_{n})\partial_{\mu}J^{\mu}(y)\rangle_{T}$
$\displaystyle=-\delta^{d}(x_{1}-y)\langle\delta\phi_{1}(x_{1})\cdots\phi_{n}(x_{n})\rangle_{T}$
$\displaystyle\quad-\ldots$
$\displaystyle\quad-\delta^{d}(x_{n}-y)\langle\phi_{1}(x_{1})\cdots\delta\phi_{n}(x_{n})\rangle_{T},$
(3.91)

where $\delta\phi_{i}$ indicates the charge of the field $\phi_{i}$ under
$J^{\mu}$.131313It is conventional to normalize the conserved current $J$ so
that eq. (3.91) is true, which means that the 2-point function of $J$ cannot
be arbitrarily normalized like a scalar operator in eq. (3.57). The same is
true of the energy-momentum tensor, and of higher-spin conserved currents. For
all these operators the normalization of the 2-point function involves a
coefficient of physical relevance.

##  4 Conformal correlation functions

The method presented in the last section using conformal Ward identities and
the Wightman axioms could in principle be used to determine 3- and higher-
point correlation functions.141414The conformal Ward identities give
differential equations and the Wightman axioms boundary conditions. Together,
these are sufficient to constrain the conformal correlation functions. Note
however that energy positivity alone is not sufficient: the micro-causality
axiom is required as well (see ref. [8] for a discussion at the level of
3-point functions). Alternatively, for time-ordered or Euclidean correlators,
the boundary condition can be given by an OPE limit [9] (see below). However,
it is quite inconvenient, and it hides the simplicity of the result. To put
things in perspective, note that the Wightman 3-point function of scalar
operators in momentum space was only constructed in 2019 [10,
11],151515Results for the Euclidean momentum-space 3-point function appeared
in 2013 already [12, 13]. whereas the position-space correlator has been known
since the work of Polyakov in 1970 [14].

###  4.1 From Minkowksi space-time to Euclidean space

For a start, let us go back to the Wightman 2-point function of scalar
operators, given in position space by

$W(x)=\frac{1}{\left[-(x^{0}+i\varepsilon)^{2}+\vec{x}^{2}\right]^{\Delta}}.$
(4.1)

The very definition of this 2-point function with its $i\varepsilon$
prescription suggests that $x^{0}$ should be thought of as a complex variable:
as a function of a complex $x^{0}$, $W$ is analytic in the upper-half complex
plane. Taking $x^{0}$ to be purely imaginary, i.e.

$x^{0}=i\tau,\qquad\qquad\tau>0,$ (4.2)

we obtain the _Schwinger function_

$\langle\phi(0)\phi(x_{E})\rangle=\frac{1}{(\tau^{2}+\vec{x}^{2})^{\Delta}}\equiv\frac{1}{(x_{E}^{2})^{\Delta}}.$
(4.3)

We denote with $x_{E}=(\tau,\vec{x})$ the Euclidean vector that is contracted
with the $d$-dimensional Euclidean metric. Schwinger functions will always be
written using the “average” notation $\langle\cdots\rangle$, as opposed to the
Wightman functions $\left\langle 0\right|\cdots\left|0\right\rangle$ that can
be interpreted as vacuum expectation values; since the two types of functions
cannot be confused, we shall denote the Euclidean coordinates by $x$ instead
of $x_{E}$, even though the latter is implied.

Note that the Schwinger function transforms covariantly under the Euclidean
conformal group $\text{SO}(d+1,1)$ that is obtained by replacing the Minkowski
metric by the Euclidean one. This includes both translations and
$\text{SO}(d)$ rotations, and as a consequence we have

$\langle\phi(0)\phi(x)\rangle\stackrel{{\scriptstyle\text{rotation}}}{{=}}\langle\phi(0)\phi(-x)\rangle\stackrel{{\scriptstyle\text{translation}}}{{=}}\langle\phi(x)\phi(0)\rangle,$
(4.4)

exhibiting a new symmetry under the exchange of the order of the two
operators. Also note that the Schwinger 2-point function is positive, and that
it is not defined at the point $x_{E}=0$, unlike the Wightman function whose
$i\varepsilon$ prescription indicates how to approach any null point.

Exercise 4.1 There is another way to get to the same result, starting from the
momentum-space representation of the 2-point function. The Wightman function
is not well-suited to do so (at least not without exploring its analyticity
properties following from the micro-causality axiom), but the time-ordered
function is: starting from the Källen-Lehmann representation (3.84), which in
momentum space becomes (using a notation in which the momentum-conservation
delta function is implicit)

$\langle\phi(-p)\phi(p)\rangle_{T}=-i\frac{(4\pi)^{d/2}\Gamma\left(\frac{d}{2}-\Delta\right)}{2^{2\Delta}\Gamma(\Delta)}\left(p^{2}-i\varepsilon\right)^{\Delta-d/2},$

one can perform a Wick rotation in which $x^{0}$ and $p^{0}$ are
simultaneously rotated in opposite directions in the complex plane, to arrive
at the Euclidean result

$\langle\phi(-p_{E})\phi(p_{E})\rangle=\frac{(4\pi)^{d/2}\Gamma\left(\frac{d}{2}-\Delta\right)}{2^{2\Delta}\Gamma(\Delta)}\left(p_{E}^{2}\right)^{\Delta-d/2}.$

Perform the Fourier transform in $p_{E}$ to recover the Schwinger function
(4.3). Note that you will need to make assumptions about $\Delta$ for the
Fourier integral to converge. The result is however analytic in $\Delta$, so
you can argue a posteriori that it must be valid for all scaling dimensions.

This construction of a Euclidean function by analytic continuation from the
Wightman function can in fact be generalized to any number of operators.
Consider the $n$-point Wightman function parameterized as

$\left\langle
0\right|\phi_{n}(x_{n}-x_{n-1})\phi_{n-1}(x_{n-1}-x_{n-2})\cdots\phi_{2}(x_{2}-x_{1})\phi_{1}(x_{1})\left|0\right\rangle,$
(4.5)

and complexify all time components $x_{i}^{0}$. This function of several
complex variables (and many more real ones) is in fact analytic in every
upper-half complex plane in $x_{1}^{0}$ to $x_{n}^{0}$, because it can be
written as the Fourier transform of a function that only has support when the
dual variable $p_{1}^{0}$ to $p_{n}^{0}$ are all positive.161616If we let all
components of $x_{i}^{\mu}$ be complex, then the primary domain of analyticity
is the so-called _future tube_ , defined by
$\operatorname{\text{im}}x_{i}^{0}>\left|\operatorname{\text{im}}\vec{x}_{i}\right|$.
Therefore, going to purely imaginary times $x_{i}^{0}=i\tau_{i}$, one obtains
the Schwinger $n$-point function

$\langle\phi_{n}(x_{n}-x_{n-1})\cdots\phi_{2}(x_{2}-x_{1})\phi_{1}(x_{1})\rangle,$
(4.6)

in which the Euclidean times all satisfy $\tau_{i}>0$, i.e. the operators are
ordered along the Euclidean time direction. This latter observation is however
irrelevant because the ordering of operators does not matter in a Schwinger
function: the analytic continuation can be performed starting with a
configuration in which all real Minkowski times are equal, in which case the
operators commute by micro-causality. The observation made on the 2-point
function is therefore valid more generally:

  * •

Schwinger functions are symmetric under the exchange of operators.

The other observations made before are also true in general:

  * •

Schwinger functions transform covariantly under the Euclidean conformal group
$\text{SO}(d+1,1)$. This property will be very useful because it means that we
can use finite conformal transformations that act nicely in Euclidean space
(including $\infty$ as a point) to simplify our computations.

  * •

Schwinger functions are not defined at coincident points. This is not a bug
but a feature: using functions that are only defined at separated points means
that we do not need to worry about contact terms and UV divergences.171717It
is in fact possible to describe contact terms (local and semi-local ones)
using a generalization of the embedding formalism described in the next
section [15]. But these contact terms are ambiguous: they carry more
information than the Schwinger or Wightman functions themselves, and as such
are unphysical. It means however that we cannot simply take the Fourier
transform of these functions to obtain momentum-space Schwinger functions.

  * •

Schwinger functions enjoy a property called _reflection positivity_ : if the
operators are organized in a configuration that is invariant under reflection
across a plane (e.g. four points at the corners of a square; or trivially any
two points), then the correlator is positive.181818This property would not be
needed if we had been studying Euclidean conformal field theory from the
start. Indeed, there are interesting critical fixed points in condensed matter
or statistical physics that are described by CFTs that are not reflection-
positive (often called non-unitary). However, the conformal bootstrap
described in section 6 relies on this property in an essential way.

We have seen that the Wightman functions let us define Schwinger functions by
analytic continuation. But it turns out that the opposite is also true: the
Osterwalder-Schrader theorem states that the properties of Schwinger functions
listed above are sufficient to reconstruct Wightman functions.191919There is
in fact another property that is needed in the proof, called _linear growth
condition_. This property is very difficult to establish in quantum field
theory. But within conformal field theory the linear growth condition is not
necessary if one works instead with a set of “Euclidean CFT axioms”, which are
otherwise equivalent to the Osterwalder-Schrader axioms, and from which the
Wightman axioms can be recovered, at least for correlation functions of up to
4 operators [16, 17]. This means that we can in fact focus on the Euclidean
Schwinger functions for all our purposes, as any other physical observables
can be reconstructed from them (we do not claim that this reconstruction is
easy, though).

The only piece of information that we shall take for granted from the analysis
of section 3 is the unitarity bound on the scaling dimension of operators.
Working out these bounds purely from the Schwinger functions is also possible
[18], but we will not discuss this procedure here.

###  4.2 From Euclidean space to embedding space

Once we are dealing with correlation functions that transform under the
Euclidean conformal group $\text{SO}(d+1,1)$, it makes a lot of sense to make
use of an analogy with the Lorentz group in $d+2$ dimensions to gain
mileage.202020This idea dates back to Dirac in 1936 [19]. We already
introduced in section 2 a set of coordinates in this $(d+2)$-dimensional,
_embedding_ space, and a metric obeying

$\eta_{MN}dX^{M}dX^{N}=(dX^{\mu})^{2}+(dX^{d+1})^{2}-(dX^{d+2})^{2}.$ (4.7)

Note that the directions labeled with indices $\mu$ are all space-like: the
time-like direction is $X^{d+2}$.

The question is how do we go from $X^{M}\in\mathbb{R}^{d+1,1}$ to
$x^{\mu}\in\mathbb{R}^{d}$ without explicitly breaking the $(d+2)$-dimensional
Lorentz symmetry. This can be done in two steps:

  1. 1.

Restrict our attention to the future light-cone $X^{2}=0$ with $X^{d+2}>0$,
which is an invariant subspace.

  2. 2.

Identify any two points related by a scale transformation on this light cone,
i.e. $X^{M}\sim\lambda X^{M}$ with $\lambda>0$.

This means that we are essentially considering a map between a point $x^{\mu}$
in $d$-dimensional Euclidean space and a light ray in a $(d+2)$-dimensional
Minkowski space-time. To make the map explicit, we choose a section of the
cone, which we will take to be $X^{d+1}+X^{d+2}=1$, and identify

$X^{\mu}=x^{\mu},\qquad X^{d+1}=\frac{1-x^{2}}{2},\qquad
X^{d+2}=\frac{1+x^{2}}{2}.$ (4.8)

Conformal transformations now act linearly in embedding space,

$X^{M}\to X^{\prime M}=\Lambda^{M}_{\leavevmode\nobreak\ N}X^{N}$ (4.9)

To obtain the action on $x^{\mu}$, we first map it to our preferred section of
the cone using eq. (4.8), apply the Lorentz transformation on $X^{M}$, then
perform a (space-time dependent) rescaling $X^{\prime
M}\to\lambda(X^{\prime})X^{\prime M}$ to get back on the preferred section,
and read off $x^{\prime\mu}=\lambda(X^{\prime})X^{\prime\mu}$.

Local operators in Euclidean space must also be lifted to the embedding space,
or at least to the null cone. On our preferred section, we declare

$\phi(X)\equiv\phi(x),\qquad\left(X^{2}=0,\leavevmode\nobreak\
X^{d+1}+X^{d+2}=1\right).$ (4.10)

Then primary operators are defined on the rest of the cone by the scaling rule

$\phi(\lambda X)=\lambda^{-\Delta}\phi(X).$ (4.11)

Note that this rule can only apply to _primary_ operators: descendants
obtained acting with derivatives do not satisfy the same scaling property.
This choice gives the right transformation rules for primary operators under
all infinitesimal conformal transformations. This can be verified explicitly
(see the exercise below), or argued as follows: a conformal transformation is
a composition of a $(d+2)$-dimensional Lorentz transformation, which locally
acts on the operator like a rotation or a boost, followed by a position-
dependent scale transformation to get back to the preferred section, which by
the rule (4.11) amounts to a local scale transformation with weight given by
the scaling dimension $\Delta$ of the operator. The combination of these two
local transformations is precisely what we expect from the conformal
transformation of an operator.

Exercise 4.2 Verify that a Lorentz boost in the direction of $X^{d+1}$
corresponds to a scale transformation in Euclidean space, in agreement with
eq. (2.39), and that operators transform accordingly.

A bit more care is required to define operators that carry spin on the
projective null cone, as the additional Lorentz indices in the directions of
$X^{d+1}$ and $X^{d+2}$ imply that there are additional degrees of freedom
that need to be constrained [20]. This can be achieved by imposing
transversality in embedding space, together with a gauge symmetry condition.
But we do want to dive into this level of technical detail here. We will
therefore restrict our attention to correlation functions of scalar primary
operators only.

Once the transformation properties of local operators are clear, the
construction of correlation functions of $n$ points $X_{1}$ to $X_{n}$ follows
two simple rules:

  * •

The correlators must depend on Lorentz-invariant quantities in embedding
space, i.e. scalar products of the form $X_{i}\cdot X_{j}$. Since
$X_{i}^{2}=0$ on the null cone, only scalar products with $i\neq j$ can
appear.

  * •

Applying a local scale transformation $X^{M}\to\lambda(X)X^{M}$ under which
all primary operators $\phi_{i}$ with scaling dimension $\Delta_{i}$ satisfy

$\phi_{i}(X_{i})\to\lambda(X_{i})^{-\Delta_{i}}\phi_{i}(X_{i}),$ (4.12)

the correlator must transform homogeneously as

$\langle\phi_{1}(X_{1})\cdots\phi_{n}(X_{n})\rangle\to\lambda(X_{1})^{-\Delta_{1}}\cdots\lambda(X_{n})^{-\Delta_{n}}\langle\phi_{1}(X_{1})\cdots\phi_{n}(X_{n})\rangle.$
(4.13)

These rules imply immediately that there cannot be one-point functions, as
there simply is no corresponding Lorentz-invariant quantity on the projective
null cone. The simplest non-trivial case is that of a 2-point function, which
must obey

$\langle\phi(X_{1})\phi(X_{2})\rangle\propto(X_{1}\cdot X_{2})^{-\Delta}.$
(4.14)

Note that this is only consistent with the homogeneous scaling rule if both
operators have the same scaling dimension, another property that was derived
in the hard way in section 3. To recover the dependence on the $d$-dimensional
Euclidean coordinates, we simply use the identification (4.8), yielding

$X_{1}\cdot X_{2}=-\frac{1}{2}(x_{1}-x_{2})^{2}.$ (4.15)

Upon fixing the proportionality factor in the above equation, one recovers the
expected result

$\langle\phi(x_{1})\phi(x_{2})\rangle=\frac{1}{\left[(x_{1}-x_{2})^{2}\right]^{\Delta}}.$
(4.16)

###  4.3 3-point functions

The embedding space formalism becomes very interesting when examining the
3-point function

$\langle\phi_{1}(X_{1})\phi_{2}(X_{2})\phi_{3}(X_{3})\rangle,$ (4.17)

where the 3 scalar operators have now possibly distinct scaling dimensions
$\Delta_{1}$, $\Delta_{2}$ and $\Delta_{3}$. Solving the conformal Ward
identities for this 3-point function would be an annoying task. Instead, by
the above rules, we immediately know that this is a function of the 3
invariant quantities

$X_{1}\cdot X_{2},\qquad X_{1}\cdot X_{3},\qquad X_{2}\cdot X_{3}.$ (4.18)

Moreover, by the homogeneous scaling rule, the only possible form of the
3-point function is

$\langle\phi_{1}(X_{1})\phi_{2}(X_{2})\phi_{3}(X_{3})\rangle\propto(X_{1}\cdot
X_{2})^{\alpha_{12}}(X_{1}\cdot X_{3})^{\alpha_{13}}(X_{2}\cdot
X_{3})^{\alpha_{23}}$ (4.19)

with the exponents satisfying

$\displaystyle\alpha_{12}+\alpha_{13}$ $\displaystyle=-\Delta_{1},$
$\displaystyle\alpha_{12}+\alpha_{23}$ $\displaystyle=-\Delta_{2},$ (4.20)
$\displaystyle\alpha_{13}+\alpha_{23}$ $\displaystyle=-\Delta_{3}.$

This system of equations admits as unique solution

$\displaystyle\alpha_{12}$
$\displaystyle=-\frac{\Delta_{1}+\Delta_{2}-\Delta_{3}}{2},$
$\displaystyle\alpha_{13}$
$\displaystyle=-\frac{\Delta_{1}+\Delta_{3}-\Delta_{2}}{2},$ (4.21)
$\displaystyle\alpha_{23}$
$\displaystyle=-\frac{\Delta_{2}+\Delta_{3}-\Delta_{1}}{2}.$ (4.22)

In terms of Euclidean coordinates, this can be written

$\langle\phi_{1}(x_{1})\phi_{2}(x_{2})\phi_{3}(x_{3})\rangle=\frac{\lambda_{123}}{(x_{12}^{2})^{\Delta_{12,3}}(x_{13}^{2})^{\Delta_{13,2}}(x_{23}^{2})^{\Delta_{23,1}}},$
(4.23)

where we have introduced the compact notation

$x_{ij}^{2}=(x_{i}-x_{j})^{2},$ (4.24)

and

$\Delta_{ij,k}=\frac{\Delta_{i}+\Delta_{j}-\Delta_{k}}{2}.$ (4.25)

Eq. (4.23) is truly special. It should be compared with the most general
3-point function invariant under Poincaré and scale symmetry only: in this
case, any term of the form

$(x_{12}^{2})^{\alpha}(x_{13}^{2})^{\beta}(x_{23}^{2})^{\gamma}$ (4.26)

with

$\alpha+\beta+\gamma=\frac{\Delta_{1}+\Delta_{2}+\Delta_{3}}{2}$ (4.27)

satisfies the symmetry requirement, so that the 3-point function could take
the form

$\langle\phi_{1}(x_{1})\phi_{2}(x_{2})\phi_{3}(x_{3})\rangle=\sum_{i}\frac{c_{i}}{(x_{12}^{2})^{\alpha_{i}}(x_{13}^{2})^{\beta_{i}}(x_{23}^{2})^{\gamma_{i}}},$
(4.28)

with infinitely many free coefficients $c_{i}$. Instead, the conformal 3-point
function (4.23) is fixed up to a _unique_ multiplicative coefficient
$\lambda_{123}$.

To give another point of comparison, let us examine a 3-point function that
involves a descendant operator. Since the focus is on scalar operators, let us
act on the first operator in (4.23) with $\partial_{\mu}\partial^{\mu}$:

$\displaystyle\langle\partial^{2}\phi_{1}(x_{1})\phi_{2}(x_{2})\phi_{3}(x_{3})\rangle=4\lambda_{123}\bigg{[}$
$\displaystyle\frac{\left(\Delta_{1}-\frac{d-2}{2}\right)\Delta_{12,3}}{(x_{12}^{2})^{\Delta_{12,3}+1}(x_{13}^{2})^{\Delta_{13,2}}(x_{23}^{2})^{\Delta_{23,1}}}$
$\displaystyle+\frac{\left(\Delta_{1}-\frac{d-2}{2}\right)\Delta_{13,2}}{(x_{12}^{2})^{\Delta_{12,3}}(x_{13}^{2})^{\Delta_{13,2}+1}(x_{23}^{2})^{\Delta_{23,1}}}$
$\displaystyle+\frac{\Delta_{12,3}\Delta_{13,2}}{(x_{12}^{2})^{\Delta_{12,3}+1}(x_{13}^{2})^{\Delta_{13,2}+1}(x_{23}^{2})^{\Delta_{23,1}-1}}\bigg{]}.$
(4.29)

Unlike the correlation function of primary operators, this is now the sum of
three terms with distinct powers of the distances $x_{ij}^{2}$, all of which
are individually consistent with Poincaré and scale symmetry.

The coefficients multiplying all three terms are peculiar: there are actually
special cases in which this 3-point function takes the general form of eq.
(4.23), i.e. that of correlator involving primary operators. These special
cases are not accidental, but correspond to physically interesting situations:

  * •

If $\Delta_{1}=\left|\Delta_{2}-\Delta_{3}\right|$, then either
$\Delta_{12,3}=0$ or $\Delta_{13,2}=0$, and in both cases two terms on the
right-hand side of eq. (4.29) vanish. This is a very special situation in
which the primary 3-point function factorizes into a product of 2-point
function, e.g. when $\Delta_{3}=\Delta_{1}+\Delta_{2}$,

$\langle\phi_{1}(x_{1})\phi_{2}(x_{2})\phi_{3}(x_{3})\rangle=\frac{\lambda_{123}}{(x_{13}^{2})^{\Delta_{1}}(x_{23}^{2})^{\Delta_{2}}}\propto\langle\phi_{1}(x_{1})\phi_{1}(x_{3})\rangle\langle\phi_{2}(x_{2})\phi_{2}(x_{3})\rangle.$
(4.30)

This situation is realized in generalized free field theory (definition in
section 6 below), where $\phi_{3}$ is a composite operator
$\phi_{3}\approx\phi_{1}\phi_{2}$.

  * •

If $\Delta_{1}=\frac{d-2}{2}$, then the first two terms in eq. (4.29) vanish.
As we saw in section 3, only a free scalar field satisfying the equation of
motion $\partial^{2}\phi_{1}(x)=0$ can have such a scaling dimension. Since
then obviously $\left[K^{\mu},\partial^{2}\phi_{1}(x)\right]=0$, it is natural
that the correlator involving the equation of motion takes the form of a
primary 3-point function. However, we also know that it must vanish
identically, which implies that either the 3-point coefficient $\lambda_{123}$
vanishes, or that the additional condition
$\Delta_{1}=\left|\Delta_{2}-\Delta_{3}\right|$ is satisfied: this is for
instance the case of a 3-point function involving the primary composite
operator $\phi^{2}$,

$\langle\phi(x_{1})\phi(x_{2})\phi^{2}(x_{3})\rangle=\frac{\lambda}{(x_{13}^{2})^{(d-2)/2}(x_{23}^{2})^{(d-2)/2}},$
(4.31)

which is non-zero but vanishes under the action of $\partial^{2}/(\partial
x_{1})^{2}$.

Finally, let us go back to Minkowski space-time through an analytic
continuation in the opposite direction of what we did before. It is not hard
to see that the Wightman function of scalar primary operators satisfies

$\left\langle
0\right|\phi_{1}(x_{1})\phi_{2}(x_{2})\phi_{3}(x_{3})\left|0\right\rangle=\frac{\lambda_{123}}{(x_{12}^{2})^{\Delta_{12,3}}(x_{13}^{2})^{\Delta_{13,2}}(x_{23}^{2})^{\Delta_{23,1}}},$
(4.32)

where now the Minkowski distances between two points are defined by

$x_{ij}^{2}=-(x_{i}^{0}-x_{j}^{0}-i\varepsilon)+(\vec{x}_{i}-\vec{x}_{j})^{2}.$
(4.33)

Note that $x_{ij}^{2}\neq x_{ji}^{2}$, and as a consequence, the Wightman
3-point function is not symmetric under the exchange of operators.
Nevertheless, for real operators satisfying
$\phi_{i}(x)^{\dagger}=\phi_{i}(x)$, we must have

$\left\langle
0\right|\phi_{1}(x_{1})\phi_{2}(x_{2})\phi_{3}(x_{3})\left|0\right\rangle=\left\langle
0\right|\phi_{3}(x_{3})\phi_{2}(x_{2})\phi_{1}(x_{1})\left|0\right\rangle^{*},$
(4.34)

which is only compatible with eq. (4.32) if the coefficient $\lambda_{123}$ is
real. This property will in fact be essential in the conformal bootstrap
discussed in section 6.

###  4.4 4-point functions

The embedding space technique can be used for higher-point functions as well,
but things are getting more complicated. To avoid dealing with four distinct
operators and as many scaling dimensions, let us focus our attention on the
case of 4 identical scalar operators,

$\langle\phi(X_{1})\phi(X_{2})\phi(X_{3})\phi(X_{4})\rangle.$ (4.35)

In this case there are 6 Lorentz invariants $X_{i}\cdot X_{j}$ with $i\neq j$.
It is easy to see that a term of the form

$\frac{1}{(X_{1}\cdot X_{2})^{\Delta}(X_{3}\cdot X_{4})^{\Delta}}$ (4.36)

satisfies all the constraints of conformal symmetry. It is however not unique:
so does

$\frac{1}{(X_{1}\cdot X_{3})^{\Delta}(X_{2}\cdot X_{4})^{\Delta}}.$ (4.37)

This shows right away that there is no hope of constraining the 4-point
function as much as we did with the 3-point function. In fact, one can
construct two _invariant_ quantities out of the $X_{i}$:

$u=\frac{(X_{1}\cdot X_{2})(X_{3}\cdot X_{4})}{(X_{1}\cdot X_{3})(X_{2}\cdot
X_{4})},\qquad\qquad v=\frac{(X_{1}\cdot X_{4})(X_{2}\cdot X_{3})}{(X_{1}\cdot
X_{3})(X_{2}\cdot X_{4})}.$ (4.38)

These are called _conformal cross-ratios_. Any function of $u$ and $v$ is
conformally invariant, and the most general 4-point function can be of the
form

$\langle\phi(X_{1})\phi(X_{2})\phi(X_{3})\phi(X_{4})\rangle\propto\frac{g(u,v)}{(X_{1}\cdot
X_{2})^{\Delta}(X_{3}\cdot X_{4})^{\Delta}}.$ (4.39)

In terms of Euclidean coordinates, this can be written

$\langle\phi(x_{1})\phi(x_{2})\phi(x_{3})\phi(x_{4})\rangle=\frac{g(u,v)}{\left(x_{12}^{2}x_{34}^{2}\right)^{\Delta}}$
(4.40)

with

$u=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}},\qquad\qquad
v=\frac{x_{14}^{2}x_{23}^{2}}{x_{14}^{2}x_{23}^{2}}.$ (4.41)

To see that this is the most general result, let us consider the following
argument based on a sequence of finite conformal transformations:

  1. 1.

Using translations, it is always possible to choose a reference frame in which
$x_{1}=0$.

  2. 2.

Using a special conformal transformation, one can then send $x_{4}\to\infty$
without moving $x_{1}$ away from the origin.

  3. 3.

A rotation can then be used to place $x_{3}$ along some chosen direction,
followed by a scale transformation to get $x_{3}=(0,\ldots,0,1)$, without
touching the origin nor the point at infinity.

  4. 4.

Finally, there is still a subset of rotations that do not affect $x_{3}$, that
can be used to move the point $x_{2}$ to the position
$x_{2}=(b,0,\ldots,0,a)$, shown later in figure 5.

Note that the first 3 steps can be used to fix entirely the kinematics of a
3-point function, which explains why its only freedom is in the form of a
multiplicative coefficient. Instead, with a 4-point function we are left with
two quantities, $a$ and $b$, which are in one-to-one correspondence with the
two conformal cross ratios. It is in fact convenient to replace these two real
numbers with a complex $z=a+ib$ and its conjugate $\bar{z}=a-ib$. The cross-
ratios are then related to $z$ and $\bar{z}$ by

$u=z\bar{z},\qquad\qquad v=(1-z)(1-\bar{z}).$ (4.42)

Schwinger functions are analytic at all non-coincident configurations of
points, therefore the function $g(z)$ is a single-valued function over the
complex plane minus the points $\\{0,1,\infty\\}$. Note that $g$ is also
subject to the _crossing symmetry_ of the 4-point function, which
requires212121It is standard in the CFT literature to call “crossing symmetry”
the property of Schwinger functions to be symmetric under the exchange of
operators, in analogy with scattering amplitudes. The two types of crossing
are related, but they are not quite the same.

$g(u,v)=g\left(\frac{u}{v},\frac{1}{v}\right)$ (4.43)

upon exchanging the operators $\phi(x_{1})$ and $\phi(x_{2})$, as well as

$g(u,v)=\left(\frac{u}{v}\right)^{\Delta}g(v,u),$ (4.44)

upon $\phi(x_{1})\leftrightarrow\phi(x_{3})$.

The situation is more complicated in Minkowski space-time. One can still
define cross ratios by eq. (4.41), with Minkowski distances defined through
$i\varepsilon$ prescriptions as in (4.33). But the Wightman functions are not
analytic (they lie at the boundary of the domain of analyticity in
complexified coordinates), and therefore $g$ is a multi-valued function. Its
values can be reached by analytic continuation from a configuration in which
all 4 operators live on a constant-time slice, in which case it coincides with
the Schwinger function. For instance, keeping $x_{1}$, $x_{3}$ and $x_{4}$ on
that time slice but letting the time component $b$ of $x_{2}$ become
imaginary, one ends up in a configuration in which $z$ and $\bar{z}$ are both
real but distinct. This procedure is in general tedious, and it is fair to say
that the current understanding of conformal Wightman 4-point functions is
still incomplete.222222Working in momentum space is not helping, even though
energy positivity reduces the number of non-trivial configurations: since the
Ward identity for special conformal transformation is a second-order
differential equation, the solution cannot be formulated in terms of
invariants. The fact that there exist conformal invariants in position space
is intimately connected with Ward identities being first-order differential
equations.

##  5 State-operator correspondence and OPE

The construction of correlation functions could be continued beyond 4 points,
but there is a good reason to stop here (at least in this course). We shall
now see that any $n$-point function can be reduced to a $(n-1)$-point function
using the operator product expansion (OPE). The procedure can be iterated
until everything is expressed in terms of 2- and 3-point functions. The
4-point function will be examined as a typical situation in which this OPE can
be applied, and higher-point functions will not be considered.

###  5.1 The OPE in quantum field theory