# Searching for Coherent States:
From Origins to Quantum Gravity

Pierre Martin-Dussaud<EMAIL_ADDRESS>Aix Marseille Univ, Université de
Toulon, CNRS, CPT, Marseille, France  Basic Research Community for Physics
e.V.

###### Abstract

We discuss the notion of coherent states from three different perspectives:
the seminal approach of Schrödinger, the experimental take of quantum optics,
and the theoretical developments in quantum gravity. This comparative study
tries to emphasise the connections between the approaches, and to offer a
coherent short story of the field, so to speak. It may be useful for
pedagogical purposes, as well as for specialists of quantum optics and quantum
gravity willing to embed their perspective within a wider landscape.

##  1 Introduction

Coherent states are essential tools in theoretical physics. Since their early
introduction by Schrödinger in 1926, they have served practical purposes in
quantum optics, while several mathematical generalisations of the notion have
been proposed, and some of them applied to quantum gravity. The present paper
was initially motivated by the following observations:

  * •

The existing reviews of coherent states, like [1] or [2], do not deal with
quantum gravity. So, we would like to summarise the various coherent states
introduced in quantum gravity.

  * •

The quantum gravity literature is very technical and does not insist much on
the conceptual motivations behind the definitions. We would like to show that
the semi-classical properties of coherent states are expected rather than
magical.

  * •

The many approaches to coherent states convey the impression of a disparate
field made of arbitrary definitions. On the contrary, we would like to insist
on the unity of the landscape and expose the big picture.

Thus we offer a journey among coherent states, from Schrödinger to quantum
gravity, passing by quantum optics, always triggered both by conceptual
clarity and concision. The resulting paper has this kind of hybrid format,
between the review and the pedagogical introduction, trading exhaustiveness
for clarity. It may be of interest for both communities of quantum gravity and
quantum optics as it explains to the one what has been done by the other.

Along the way, we will notably answer the following puzzles:

  * •

Coherent states are usually introduced for the harmonic oscillator, but can’t
we define them for even simpler systems like the free particle?

  * •

Coherent states are sometimes presented as the states $\ket{\psi}$ such that
$\expectationvalue{\hat{x}(t)}{\psi}$ and
$\expectationvalue{\hat{p}(t)}{\psi}$ satisfy the classical equations of
motion. It is, for instance, the impression conveyed in the seminal paper of
Schrödinger [3], but also in the recent reference book [1]. However such a
property cannot be a characterisation of coherent states whatsoever, since it
is clear, from Ehrenfest theorem, that any time-evolved state $\ket{\psi(t)}$,
coherent or not, satisfies it. Is there a way to make this first intuition of
classicality rigorous?

  * •

Coherent states are also often introduced as eigenstates of the annihilation
operator, but this does not seem to be the best pedagogical way as the
physical motivation of this approach may seem rather obscure at first sight.
Indeed, doing so, the classical properties that can be checked afterwards
appear as magical, rather than expected. What could be a better pedagogical
introduction to the topic?

  * •

Coherent states can also be generated by the action of the Heisenberg group
over the vacuum state. This group is sometimes called the dynamical symmetry
group of the harmonic oscillator (see [4, 5]), although it is very unclear in
which sense the group is ‘dynamical’, a ‘symmetry group’, or even specific to
the harmonic oscillator. Can we make the statement precise?

  * •

Coherent states are wanted to be quasi-classical states, but in quantum
optics, for instance, the coherent states of light are those that maximise the
interference pattern, which is paradoxically regarded as a very quantum
feature, far from being a classical source of light as an incandescent bulb
may be. Is the paradox of designation only superficial?

  * •

The definition of coherent states in quantum gravity is covered by a jungle of
technicalities, far from the experimental point of view of quantum optics. Can
we nevertheless summarise the story to keep the key physical idea and make our
way through the jungle?

To start with, we go back to the initial ideas of Schrödinger in section 2,
and propose a modern follow-up in section 3. Then, in section 4, we enlarge
the discussion with a kinematical characterisation of coherent states in terms
of annihilation operators. We explain the physical meaning of these operators
in quantum optics in section 5, which motivates an algebraic generalisation of
coherent states presented in section 6. In section 7, we present an
independent geometric generalisation, which was later applied in quantum
gravity, as we show in section 8.

##  2 Schrödinger coherent states

Historically, the initial motivation for introducing coherent states is to
demonstrate how classical mechanics can be recovered from quantum mechanics.
It is done in 1926 in a short seminal paper by Erwin Schrödinger [3],
translated in English in [6], entitled The Continuous Transition from Micro-
to Macro-Mechanics. The title is rather explicit about its goal, although one
may discuss whether it has been achieved or not.

Interestingly, Schrödinger does not use the word ‘coherent’ anywhere, but he
aims at constructing mathematically

> a group of proper vibrations [that] may represent a ‘particle’, which is
> executing the ‘motion’, expected from the usual mechanics.

Neither does he use the words ‘quasi-classical’ or ‘semi-classical’, but the
latter would convey his intuition probably better than ‘coherent’. The paper
does not shine by its clarity, but one can understand the overall logic, that
we present below in modernised terms and notations.

Quantum harmonic oscillator.

Let’s consider the quantum harmonic oscillator in one dimension, with mass $m$
and pulsation $\omega$. Its Hilbert space is $L^{2}(\mathbb{R})$ over which
are acting the position operator $\hat{x}\psi(x)=x\psi(x)$ and the momentum
$\hat{p}\psi=-i\hbar\,\partial_{x}\psi$ . The dynamics is provided by the
hamiltonian which reads

$\hat{H}=\frac{\hat{p}^{2}}{2m}+\frac{1}{2}m\omega^{2}\hat{x}^{2}.$ (1)

The eigenstates of $\hat{H}$ form an orthonormal basis $\ket{n}$, indexed by
$n\in\mathbb{N}$, whose coefficients, in the basis of eigenstates $\ket{x}$ of
$\hat{x}$, read

$\innerproduct{x}{n}=\sqrt[4]{\frac{m\omega}{\pi\hbar}}\frac{1}{\sqrt{2^{n}n!}}e^{-\frac{m\omega}{2\hbar}x^{2}}H_{n}\left(\sqrt{\frac{m\omega}{\hbar}}x\right)$
(2)

where $H_{n}(x)$ are Hermite’s polynomials111Wikipedia mentions two
conventions for Hermite’s polynomial. We use the physicist one, i.e.
$H_{n}(x)\overset{\rm def}{=}(-1)^{n}e^{x^{2}}\derivative[n]{x}e^{-x^{2}}.$
(3) and the associate eigenvalues are

$E_{n}=\hbar\omega\left(n+\frac{1}{2}\right).$ (4)

Schrödinger coherent states.

Then, Schrödinger defines222Compared to the strictly original definition of
Schrödinger, we have here chosen to normalise the states, with a factor
$e^{-A^{2}/2}$ in front of the sum, and a phase factor $e^{-i\omega t/2}$ to
match the standard Dirac notation., out of the blue, the following family of
states, indexed by time $t\in\mathbb{R}$ and another parameter
$A\in\mathbb{R}$:

$\ket{Ae^{i\omega t}}\overset{\rm def}{=}e^{-\frac{A^{2}}{2}}e^{-i\omega
t/2}\sum_{n=0}^{\infty}\frac{A^{n}}{\sqrt{n!}}e^{\frac{i}{\hbar}E_{n}t}\ket{n}.$
(5)

It is immediate to see that $\ket{Ae^{i\omega t}}$ is the temporal evolution
of $\ket{A}$ by the unitary operator $e^{\frac{i}{\hbar}\hat{H}t}$, as

$e^{\frac{i}{\hbar}\hat{H}t}\ket{A}=e^{i\omega t/2}\ket{Ae^{i\omega t}}.$ (6)

Then Schrödinger argues that these states approximate the ‘macro-mechanics’,
what we would call in modern language, being semi- or quasi-classical. More
precisely, he highlights three properties:

  1. 1.

First the average position satisfies the law of classical motion:

$\expectationvalue{\hat{x}}=A\cos\omega t.$ (7)

  2. 2.

Secondly, the average energy is almost the classical one:

$\expectationvalue{\hat{H}}\approx A^{2}m\omega^{2}.$ (8)

  3. 3.

Third, he argues (without any explicit computation) that the wave packet does
not ‘spread out’, but ‘remains compact’, like a particle.

The two first properties provide physical meaning to the parameter $\alpha$ as
the amplitude of some corresponding classical wave. Thus the Schrödinger
coherent states are parametrised by an amplitude $\alpha$ and an instant $t$.

Wrong characterisation of quasi-classicality.

The three arguments above appear as a first attempt to formalise the property
of ‘quasi-classicality’, and have been the basis of the later developments of
coherent states. Unfortunately, it has been hardly never noticed that the
first property cannot characterise quasi-classicality in any way. Indeed all
the quantum states of the harmonic oscillator satisfy this property! More
precisely, given any initial state $\ket{\psi_{0}}$, its time evolution will
be so that it satisfies equation (7). It is a consequence of Ehrenfest
theorem, that drives the evolution of the expected value of a time-independent
observable $\hat{A}$ in any state $\ket{\psi(t)}$, according to the equation

$\derivative{\expectationvalue*{\hat{A}}}{t}=\frac{1}{i\hbar}\expectationvalue{[\hat{A},\hat{H}]}.$
(9)

In the case of the harmonic oscillator, the equations for $\hat{x}$ and
$\hat{p}$ are

$\displaystyle\derivative{\expectationvalue*{\hat{x}}}{t}=\frac{1}{m}\expectationvalue{\hat{p}}$
and
$\displaystyle\derivative{\expectationvalue*{\hat{p}}}{t}=-m\omega^{2}\expectationvalue{\hat{x}}.$
(10)

These are actually the classical equations of motion for
$\expectationvalue{\hat{x}}$ and $\expectationvalue{\hat{p}}$, and so all
solutions $\expectationvalue{\hat{x}}$ take the form of equation (7). It is
completely generic and so cannot be used as a characterisation of quasi-
classicality333I am indebted to Federico Zalamea to have made me realised this
fact.. Thus, there is nothing such as a constraining property in Schrödinger’s
first statement, except maybe the implicit demand that the time evolution of a
‘quasi-classical state’ should still be ‘quasi-classical’. It is surprising
that this fact has not been much recognised, and that many recent developments
of coherent states still treat this property as an argument for the
‘peakiness’ of the coherent states. Of course, with a more complicated
hamiltonian, the property is not a trivial statement, but for the harmonic
oscillator, it is.

Let us now analyse the two other properties, $3$ and $2$, which at first may
disappoint us with their vague formulation. When they are made precise, we
show that each of them, alone, is a sufficient condition that fully
characterises the family of coherent states.

##  3 Dynamical characterisation

A characteristic feature of quantum mechanics is the fact that the position of
a particle is not given by a classical trajectory, but rather by a probability
density that evolves with time. Thus, a ‘quasi-classical’ state could be one
for which a quantum particle is well localised in space, and remains localised
as time goes by. Let us try to formalise it, and see how this programme fails
in the case of the free particle and succeeds for the harmonic oscillator.

Free particle.

Consider the free particle in one dimension. Its Hilbert space is
$L^{2}(\mathbb{R})$. The Dirac delta function $\delta(x)$ describes the state
of a particle perfectly well localised at $x=0$. The uncertainty about its
position is zero: $\Delta\hat{x}=0$. For that reason, it may seem a good
candidate for being a quasi-classical state.

However, this first attempt fails because the particle does not remain
localised as time goes by. Indeed, the hamiltonian of the free particle

$\hat{H}=\frac{\hat{p}^{2}}{2m},$ (11)

drives the time-evolution of $\delta(x)$ to

$\psi(x,t)=\sqrt{\frac{m}{2\pi\hbar|t|}}e^{-i\,{\rm
sgn}(t)\,\frac{\pi}{4}}e^{i\frac{mx^{2}}{2\hbar t}}.$ (12)

The probability distribution $|\psi(x,t)|^{2}$ is now completely spread,
$\Delta\hat{x}=\infty$, and not even normalised! Thus, a wave function which
is infinitely well localised at initial time, turns instantaneously into an
infinitely spread state444This matter of fact seems even to contradict the
postulate according to which two successive measurements should give the same
result. But these pathologies can be imputed to the already suspicious Dirac
delta function..

So consider instead a more reasonable initial state, like a gaussian curve

$\psi_{0}(x)=\frac{1}{\sqrt{\sigma\sqrt{2\pi}}}e^{-\frac{x^{2}}{4\sigma^{2}}},$
(13)

which is spread as $\Delta_{0}\hat{x}=\sigma$. It evolves as a free particle
to

$\psi(x,t)=\frac{1}{\left(2\pi(\sigma^{2}+i\hbar
t/m)\right)^{1/4}}e^{-\frac{x^{2}}{4(\sigma^{2}+i\hbar t/m)}}.$ (14)

It is also a gaussian which is spread like

$\Delta\hat{x}=\sqrt{\sigma^{2}+\frac{t^{2}\hbar^{2}}{4m^{2}\sigma^{2}}},$
(15)

so that it irrevocably spreads with time and looses its initially compact
shape.

In fact, whatever the initial state $\psi_{0}$ at time $t=0$, it evolves as a
free particle to a state $\psi(x,t)$ which satisfies555A proof can be found in
[7] p. 104.:

$(\Delta\hat{x})^{2}=(\Delta_{0}\hat{x})^{2}+\frac{(\Delta_{0}\hat{p})^{2}}{m^{2}}t^{2}\\\
+\frac{t}{m}\left(\expectationvalue{\hat{x}\hat{p}+\hat{p}\hat{x}}_{0}-2\expectationvalue{\hat{x}}_{0}\expectationvalue{\hat{p}}_{0}\right).$
(16)

It is a second order polynomial in $t$. A necessary condition to prevent the
time spreading would be to have $(\Delta_{0}\hat{p})^{2}=0$, but this implies,
through Heisenberg inequality, that $\Delta_{0}\hat{x}=\infty$, i.e. a
maximally spread states in space… So, for the free particle, the spreading is
unavoidable. From this perspective, there is no ‘quasi-classical state’ for
the free particle.

Harmonic oscillator.

Let us now consider the more sophisticated hamiltonian of the harmonic
oscillator:

$\hat{H}=\frac{\hat{p}^{2}}{2m}+\frac{1}{2}m\omega^{2}\hat{x}^{2}.$ (17)

A priori, there is more chance to find coherent states now, because we have
added a potential well in the hamiltonian that can help to confine the wave-
function and prevent it from spreading. In this case, the spreading of a
general solution $\psi(x,t)$ is given by666Proof. Using Ehrenfest theorem we
have
$\begin{array}[]{l}\frac{d}{dt}\expectationvalue{\hat{x}}=\expectationvalue{\hat{p}}/m\\\
\frac{d}{dt}\expectationvalue{\hat{p}}=-m\omega^{2}\expectationvalue{\hat{x}}\\\
\frac{d}{dt}\expectationvalue{\hat{x}^{2}}=\expectationvalue{\hat{x}\hat{p}+\hat{p}\hat{x}}/m\\\
\frac{d}{dt}\expectationvalue{\hat{p}^{2}}=-m\omega^{2}\expectationvalue{\hat{x}\hat{p}+\hat{p}\hat{x}}\\\
\frac{d}{dt}\expectationvalue{\hat{x}\hat{p}+\hat{p}\hat{x}}=2\expectationvalue{\hat{p}^{2}}/m-2m\omega^{2}\expectationvalue{\hat{x}^{2}}\end{array}$
(18) from which we show the differential equation
$\frac{d^{4}}{dt^{4}}\expectationvalue{\hat{x}^{2}}=-4\omega^{2}\frac{d^{2}}{dt^{2}}\expectationvalue{\hat{x}^{2}}$
(19) which is finally solved easily, and leads to our expression for
$(\Delta\hat{x})^{2}\overset{\rm
def}{=}\expectationvalue{\hat{x}^{2}}-\expectationvalue{\hat{x}}^{2}$. $\Box$

$(\Delta\hat{x})^{2}=\frac{1}{2}\left((\Delta_{0}\hat{x})^{2}+\frac{(\Delta_{0}\hat{p})^{2}}{m^{2}\omega^{2}}\right)\\\
-\frac{1}{2}\left(\frac{(\Delta_{0}\hat{p})^{2}}{m^{2}\omega^{2}}-(\Delta_{0}\hat{x})^{2}\right)\cos(2\omega
t)\\\
+\frac{1}{2m\omega}\left(\expectationvalue{\hat{x}\hat{p}+\hat{p}\hat{x}}_{0}-2\expectationvalue{\hat{x}}_{0}\expectationvalue{\hat{p}}_{0}\right)\sin(2\omega
t)$ (20)

It is noticeable that the spreading oscillates. There is no irresistible
increasing of the spreading. Instead, whatever the state we start with, the
wave packet will stay confined within a finite range, and even come back
periodically to its initial spreading $\Delta_{0}\hat{x}$.

So a first lesson to draw from this computation is that one should not be
surprised by the fact that Schrödinger coherent states do not spread out, as a
free particle would do, because no state of the harmonic oscillator does it!

Constant and minimal.

Then, one can try to express the third property of Schrödinger in more precise
terms. For instance, one can look for states for which $\Delta\hat{x}$ is
constant in time. This requires the two conditions

$\left\\{\begin{array}[]{l}(\Delta_{0}\hat{p})^{2}=m^{2}\omega^{2}(\Delta_{0}\hat{x})^{2}\\\
\expectationvalue{\hat{x}\hat{p}+\hat{p}\hat{x}}_{0}=2\expectationvalue{\hat{x}}_{0}\expectationvalue{\hat{p}}_{0}\end{array}\right.$
(21)

One can check that the Schrödinger coherent states do satisfy these
conditions. However, these two conditions are not sufficient to characterise
them. For instance all the eigenstates of the hamiltonian, $\ket{n}$, also
satisfy these conditions. Another condition is still required: the
minimisation of $\Delta_{0}\hat{x}$.

The family of states for which $\Delta\hat{x}$ is constant in time is foliated
by the value of $\Delta_{0}\hat{x}$, with a minimal value being strictly
positive. Indeed, from the Heisenberg inequality

$\Delta_{0}\hat{x}\Delta_{0}\hat{p}\geq\frac{\hbar}{2}.$ (22)

and from the first condition in (21), we have

$\Delta_{0}\hat{x}\geq\sqrt{\frac{\hbar}{2m\omega}}.$ (23)

Now one can show that the only states minimising this inequality are the
coherent states of Schrödinger! We have thus found a characterisation of them:
they are these states whose spreading in position $\Delta\hat{x}$ is constant
and minimal. Both conditions are important. Otherwise, there are states whose
spreading is momentarily smaller but will grow later to a larger value. There
are also states whose spreading is constant, but not minimal (like the
$\ket{n}$). Geometrically, the two conditions select a $2$-dimensional
submanifold out of the infinite-dimensional space $L^{2}(\mathbb{R})$.

Minimal time average.

There is another way to make the third property of Schrödinger more precise.
Consider the time average of $\Delta\hat{x}$:

$T\left[\Delta\hat{x}\right]=\sqrt{\frac{(\Delta_{0}\hat{x})^{2}}{2}+\frac{(\Delta_{0}\hat{p})^{2}}{2m^{2}\omega^{2}}}.$
(24)

Now, from Heisenberg inequality, this time average is bounded by

$T\left[\Delta\hat{x}\right]\geq\sqrt{\frac{\hbar}{2m\omega}}.$ (25)

This inequality is saturated for the coherent states and only for them. So we
have a second characterisation of coherent states as the states which minimise
$\Delta\hat{x}$ on average (as time goes by).

‘Almost classical energy’.

Let us now turn to the second property underlined by Schrödinger: ‘the average
energy is almost classical’. As we said before, the classical behaviour of
$\expectationvalue{\hat{x}}$ cannot reasonably be taken as evidence for the
classicality of coherent states. So one may wonder whether the same holds for
$\expectationvalue*{\hat{H}}$. It is easy to see that

$\expectationvalue{\hat{H}(\hat{x},\hat{p})}=H(\expectationvalue{\hat{x}},\expectationvalue{\hat{p}})\\\
+\frac{1}{2m}(\Delta\hat{p})^{2}+\frac{1}{2}m\omega^{2}(\Delta\hat{x})^{2},$
(26)

where

$H(x,p)\overset{\rm def}{=}\frac{1}{2m}p^{2}+\frac{m\omega^{2}}{2}x^{2}$ (27)

is the classical hamiltonian, function over the phase space $\mathbb{R}^{2}$
with coordinates $(x,p)$. From Heisenberg inequality one deduces then that

$\expectationvalue{\hat{H}(\hat{x},\hat{p})}-H(\expectationvalue{\hat{x}},\expectationvalue{\hat{p}})\geq\frac{\hbar\omega}{2}.$
(28)

One can say in precise terms that a state is quasi-classical with respect to
the energy if it saturates this inequality. And happily, this condition alone
is sufficient to define the coherent states!

Conclusion.

We have cleaned up the properties of coherent states underlined by Schrödinger
in his seminal paper. First, we have realised that the first property was very
generic and not specific to coherent states. Secondly, a careful analysis of
the two other properties has led us to formulate three equivalent definitions
of coherent states of the harmonic oscillator:

  1. 1.

Constant and minimal $\Delta\hat{x}$

  2. 2.

Minimal temporal average of $\Delta\hat{x}$

  3. 3.

Minimal
$\left(\expectationvalue*{\hat{H}(\hat{x},\hat{p})}-H(\expectationvalue{\hat{x}},\expectationvalue{\hat{p}})\right)$.

We regard these definitions as better suited for a pedagogical introduction to
coherent states, compared to the abstract definition as eigenstates of the
annihilation operator that one finds in most textbooks.

These three definitions are dynamical in the sense that they make use of the
temporal evolution of states or the hamiltonian. A priori, if another
hamiltonian is used, like for the free particle, another family of states will
be found. In this sense, one can talk indeed of the coherent states of the
harmonic oscillator, and not of the free particle. The stability of the family
of coherent states under temporal evolution is made obvious with the two first
definitions, but not so much with the third one.

In an attempt of generalisation of the notion of coherent states, we are now
going to relax this dynamical aspect of the definitions and propose a purely
kinematical characterisation.

##  4 Kinematical characterisation

Let’s start all over again, from a general quantum system. Its states form a
Hilbert space $\mathcal{H}$, and we consider the problem of finding the states
which are ‘quasi-classical’ in a sense to be determined.

Geometrical formulation.

As regards its kinematical features, the departure of quantum mechanics from
classical mechanics can be understood geometrically, through the so-called
geometrical formulation of quantum mechanics [8, 9].

In quantum mechanics, we are used to systems whose states are taken to be
vectors of a Hilbert space $\mathcal{H}$ endowed with a scalar product
$\innerproduct{.}{.}$, and the algebra of observables
$\mathcal{B}_{\mathbb{R}}(\mathcal{H})$ consists of the (bounded) self-adjoint
linear operators over $\mathcal{H}$. In fact, we only consider the normalised
vectors of $\mathcal{H}$, up to a global phase, so that the space of physical
states really is the projective Hilbert space $P\mathcal{H}$. The geometric
features of this space enable a fair comparison to the classical phase space.
$P\mathcal{H}$ is a Khäler manifold which means that it is naturally endowed
with two geometric structures: a symplectic $2$-form $\omega$ (coming from the
imaginary part of $\innerproduct{.}{.}$) and a Riemannian metric $g$ (from the
real part of $\innerproduct{.}{.}$). Then, the algebra of observables
$\mathcal{B}_{\mathbb{R}}(\mathcal{H})$ can be recast as the space of
functions of $\mathcal{C}^{\infty}(P\mathcal{H},\mathbb{R})$ which preserve
both geometric structures, i.e. whose hamiltonian vector fields are also
Killing vector fields.

Although it may look a bit abstract, this formulation frames quantum mechanics
in very similar terms to classical mechanics, where the space of states is a
symplectic manifold $(\mathcal{P},\omega)$, and the observables are functions
of $\mathcal{C}^{\infty}(\mathcal{P},\mathbb{R})$. In this framework, both
classical and quantum space of states are symplectic manifolds, but the
quantum case bears the additional structure of a Riemannian manifold. One does
not need to know the details of the geometrical formulation to understand the
point we want to make, that is, classical mechanics can be seen as the
particular case of quantum mechanics when the Riemannian structure is trivial,
i.e. $g=0$!

This fact is of importance because the Riemannian metric gives precisely a
measure of the uncertainty of observables. An observable
$\hat{A}\in\mathcal{B}_{\mathbb{R}}(\mathcal{H})$, defines a function $A$ over
$P\mathcal{H}$ through $A:\ket{\psi}\mapsto\expectationvalue{A}$, and thus a
hamiltonian vector field $X_{A}$. One can prove that

$\Delta\hat{A}=g(X_{A},X_{A}).$ (29)

In the classical case ($g=0$), we have $\Delta\hat{A}=0$ for any observable
$\hat{A}$ and state $\ket{\psi}$. Classical mechanics is quantum mechanics
without uncertainty.

One may wonder whether it was necessary to appeal to the abstract geometrical
formulation to reach this conclusion. Indeed the result goes along very well
with the intuitive idea that the quantum is fuzzy, while the classical is
peaked. However, the geometrical formulation brings clarity and precision to
the debate, and points towards a definite mathematical direction where to look
for classicality inside the quantum realm.

The quest for ‘quasi-classical’ states can now be reformulated in the
following terms. Are there states $\ket{\psi}$ for which $\Delta\hat{A}=0$ for
any observable $\hat{A}$?

Eigenstates.

Start considering a single observable $\hat{A}$. What are the states that
satisfy $\Delta\hat{A}=0$? It is easily shown that they are all, and only, the
eigenstates $\ket{a}$ of $\hat{A}$. Thus, an eigenstate shows some classical
features, which is not a surprise after all: the eigenstate $\ket{a}$ of
$\hat{A}$ is very peaked with respect to $\hat{A}$. Similarly, $\ket{x}$ is
classical in the sense that $\Delta\hat{x}=0$, i.e. it is very peaked with
respect to $\hat{x}$, which was indeed our first attempt to define ‘quasi-
classical state’ in section 3. So the last question of the previous paragraph,
admits a direct answer: no. Because if $\Delta\hat{A}=0$ for all $\hat{A}$,
then $\ket{\psi}$ is an eigenstate of all $\hat{A}$ which is not possible.

Our expectations have to be qualified, and one can look instead for states
which satisfy $\Delta\hat{A}=0$ for some observables $\hat{A}$, i.e. a common
eigenstate of a subset
$\mathcal{A}\subset\mathcal{B}_{\mathbb{R}}(\mathcal{H})$. Such an eigenstate
can be said ‘quasi-classical’ with respect to $\mathcal{A}$. If $\mathcal{A}$
is commutative, then its common eigenstates form a basis of $\mathcal{H}$.
Such are the eigenstates of a CSCO (complete set of commuting observables)
which may be regarded in this respect as the most classical states of a given
quantum system. However, if $\mathcal{A}$ is non-commutative, there will be
generically no common eigenstates. The question is now shifted to the
definition of ‘quasi-classical’ states with respect to a non-commutative set
of observables.

Squeezed coherent states.

Let’s consider two non-commutative observables $\hat{A}$ and $\hat{B}$.
Generically, they do not share any common eigenstates, so that we cannot have
both $\Delta\hat{A}=0$ and $\Delta\hat{B}=0$. One has to find instead a fair
trade-off between $\Delta\hat{A}$ and $\Delta\hat{B}$, so that they are both
small, although non zero. The trade-off is ruled by Heisenberg inequality
which reads

$\Delta\hat{A}\Delta\hat{B}\geq\frac{1}{2}\left|\expectationvalue{[\hat{A},\hat{B}]}\right|.$
(30)

One can show777See for instance [7] p. 244. that this inequality is saturated
precisely when $\ket{\psi}$ is an eigenstate either of $\hat{A}$, or of
$\hat{B}$, or of

$\hat{A}+i\gamma\hat{B}$ (31)

with $\gamma\in\mathbb{R}$. The meaning of this $\gamma$ is understood with
the following corollary:

$\gamma=\frac{\Delta\hat{A}}{\Delta\hat{B}}$ (32)

It ponders the respective weight of $\hat{A}$ and $\hat{B}$. The eigenstates
of $\hat{A}+i\gamma\hat{B}$ are called the $\gamma$-squeezed coherent states
with respect to $\hat{A}$ and $\hat{B}$.

Application to $\hat{x}$ and $\hat{p}$.

Let’s apply the result to $\hat{A}=\hat{x}$ and $\hat{B}=\hat{p}$. Heisenberg
inequality reads

$\Delta\hat{x}\Delta\hat{p}\geq\frac{\hbar}{2}.$ (33)

The normalised eigenstate of $\hat{x}+i\gamma\hat{p}$, with eigenvalue
$z\in\mathbb{C}$, is

$\psi_{z,\gamma}(x)=\frac{e^{-\frac{(\imaginary
z)^{2}}{2\gamma\hbar}}}{\sqrt[4]{\pi\gamma\hbar}}e^{-\frac{(x-z)^{2}}{2\gamma\hbar}}$
(34)

The normalisation is only possible for $\gamma>0$. Thus, the squeezed coherent
states (with respect to $\hat{x}$ and $\hat{p}$) form a $3$-dimensional
submanifold of $PL^{2}(\mathbb{R})$, parametrised by $\gamma$ and $z$.

The Schrödinger coherent states of equation (5) are recovered by fixing
$\gamma=\frac{1}{m\omega}$. More precisely, we have

$\innerproduct{x}{Ae^{i\omega t}}=e^{i\frac{A^{2}}{2}\sin(2\omega
t)}\psi_{z,\gamma}(x)$ (35)

with

$\displaystyle\gamma=\frac{1}{m\omega}$ and $\displaystyle
z=\sqrt{\frac{2\hbar}{m\omega}}Ae^{i\omega t}.$ (36)

We have found an equivalent definition of Schrödinger coherent states: they
are the states which minimise $\Delta\hat{x}\Delta\hat{p}$, with equal
weight888I.e. $\Delta\hat{p}=m\omega\Delta\hat{x}$. The constant $m\omega$
guarantees the homogeneity of the physical dimension. for $\hat{x}$ and
$\hat{p}$.

Kinematics vs dynamics.

This new characterisation of coherent states differs from the previous one of
Schrödinger in a central aspect: it only refers to the kinematics, and not to
the dynamics. Indeed, the previous definitions were involving the specific
form of the hamiltonian $\hat{H}$ of the harmonic oscillator, while now the
definition only uses two observables $\hat{x}$ and $\hat{p}$ acting on the
Hilbert space $L^{2}(\mathbb{R})$.

The move is noticeable because, for instance, the free particle and the
harmonic oscillator have the same kinematics, and only differ by their
dynamics. From this new kinematical characterisation, the previous coherent
states of the harmonic oscillator could be equally called coherent states of
the free particle, while the original dynamical characterisation doesn’t allow
such a possibility.

To be clear, the family of coherent states, as a whole, can be fully
characterised by kinematical considerations, but it will only exhibit nice
dynamical properties in a particular case. Indeed, the family is stable under
the harmonic oscillator evolution, whereas it is not with the free particle
one.

The transitional role from the dynamical to the kinematical perspective is
played by the operator

$\hat{a}\overset{\rm
def}{=}\sqrt{\frac{m\omega}{2\hbar}}\left(\hat{x}+\frac{i}{m\omega}\hat{p}\right).$
(37)

The task of minimising Heisenberg inequalities has been shown to reduce to
that of finding the eigenstates of this operator, which are precisely the
Schrödinger coherent states. This way, the operator $\hat{a}$ has arisen
through purely kinematical considerations. However, the same operator plays an
important role in the dynamics of the harmonic oscillator, where it is known
as the annihilation operator, for it acts destructively over the eigenstates
of $\hat{H}$:

$\hat{a}\,\ket{n}=\sqrt{n}\,\ket{n-1}.$ (38)

It is important to keep in mind this double role of $\hat{a}$ to understand
better later generalisations of coherent states.

##  5 Optical coherence

In the previous section, we have been looking for quasi-classical states and
ended up with an abstract definition of coherent states as eigenstates of the
so-called annihilation operator $\hat{a}$. This definition is the one used in
many textbooks to define coherent states at first, but it is disappointing to
see that it is often not motivated by physical considerations. We have shown
how it could be motivated in a rather abstract way, from the geometrical
formulation and the minimisation of Heisenberg inequalities. We are now going
to show a more experimentally grounded way to introduce it, which is also the
path that was followed historically. It is the way of Glauber when he revived
coherent states, thirty years after Schrödinger, in the concrete context of
quantum optics. Besides, we will understand why the adjective ‘coherent’ can
be preferred to ‘quasi-classical’. The formulas of this section are taken from
the book of recollections [10].

Quantum optics.

Quantum optics describes light using the theory of quantum electrodynamics
(QED). The electromagnetic field is described by a state in a Hilbert space,
and the observable quantities are described by the electric and magnetic
hermitian operators, $\hat{\vec{E}}(\vec{r},t)$ and $\hat{\vec{B}}(\vec{r},t)$
(we use Heaviside-Lorentz units). In absence of any sources, the time
evolution of states is driven by the hamiltonian

$\hat{H}=\frac{1}{2}\int\differential\vec{r}\,(\hat{\vec{E}}^{2}+\hat{\vec{B}}^{2}).$
(39)

In the time gauge, these observables can in fact be derived from a vector
potential $\hat{\vec{A}}$ such that

$\displaystyle\hat{\vec{E}}=-\frac{1}{c}\partialderivative{\hat{\vec{A}}}{t}$
and $\displaystyle\hat{\vec{B}}=\nabla\times\hat{\vec{A}}.$ (40)

Assuming the field is confined within a cubic box of side $L$, the vector
potential $\hat{\vec{A}}$ can be decomposed into a superposition of modes $k$
such that

$\hat{\vec{A}}(\vec{r},t)=c\sum_{k}\left(\frac{\hbar}{2L^{3}\omega_{k}}\right)^{1/2}(\hat{a}_{k}\,\vec{e}_{\lambda}e^{i(\vec{k}\cdot\vec{r}-\omega_{k}t)}\\\
+\hat{a}_{k}^{\dagger}\,\vec{e}_{\lambda}e^{-i(\vec{k}\cdot\vec{r}-\omega_{k}t)}),$
(41)

where the sum is made over an index $k$, used as a shorthand for
$(\lambda,\vec{k})$, where $\vec{e}_{\lambda}$ ($\lambda\in\\{1,2\\}$) is the
polarisation vector, perpendicular to $\vec{k}$, and $\vec{k}$ ranges over a
discrete set of values permitted by the boundary conditions. Then
$\hat{a}_{k}$ and $\hat{a}^{\dagger}_{k}$ are operators associated
respectively to the positive and negative frequency part of $\hat{\vec{A}}$.
They satisfy

$\displaystyle[\hat{a}_{k},\hat{a}_{k^{\prime}}]=[\hat{a}^{\dagger}_{k},\hat{a}^{\dagger}_{k^{\prime}}]=0$
and
$\displaystyle[\hat{a}_{k},\hat{a}^{\dagger}_{k^{\prime}}]=\delta_{kk^{\prime}}.$
(42)

The hamiltonian can be rewritten as

$\hat{H}=\sum_{k}\hbar\omega_{k}(\hat{a}_{k}^{\dagger}\hat{a}_{k}+\frac{1}{2}).$
(43)

Thus, in absence of sources, the electromagnetic field is mathematically
equivalent to an assembly of one-dimensional harmonic oscillators (one per
mode $k$), so that $\hat{a}_{k}$ and $\hat{a}^{\dagger}_{k}$ are properly
annihilation and creation operators. The basis of eigenstates of $\hat{H}$ is
immediately deduced:

$\bigotimes_{k}\ket{n_{k}}$ (44)

where $n_{k}$ is the number of photons in the mode $k$.

Interacting theory.

The basis of stationary states of the free theory, equation (44), is not the
best suited for the description of states of light coming out of photon beams.
Instead, the family of coherent states is much more convenient, and it appears
naturally once one considers interactions.

So far, we have described the free theory of the electromagnetic field. But
light is created by sources, like lamps or antennae, which consist of moving
charges that excite the electromagnetic field. It is thus crucial to describe
the interaction of light with charged matter, and notably to model the photon
field radiated by a classical electric current. A classical current
$\vec{j}(\vec{r},t)$ is assumed to interact with the vector potential through
the following hamiltonian of interaction:

$\hat{H}_{I}=\frac{1}{c}\int\vec{j}\cdot\hat{\vec{A}}\,\differential\vec{r}.$
(45)

Starting at initial time in the vaccum state $\ket{0}$, the field gets
excited, and ends up at time $t$ in a state

$\ket{t}=e^{i\phi(t)}e^{\frac{i}{\hbar}\int_{0}^{t}\hat{H}_{I}(t^{\prime})\differential
t^{\prime}}\ket{0}$ (46)

where the phase $\phi(t)$ admits a definite expression, but irrelevant for our
purposes. It can be rewritten

$\ket{t}=e^{i\phi(t)}\bigotimes_{k}\ket{\alpha_{k}(t)},$ (47)

where $\ket{\alpha_{k}(t)}$ is the coherent state with

$\alpha_{k}(t)=\frac{i}{\sqrt{2L^{3}\hbar\omega_{k}}}\int_{0}^{t}\vec{j}\cdot\vec{e}_{\lambda}e^{-i(\vec{k}\cdot\vec{r}+\omega_{k}t^{\prime})}\differential\vec{r}\differential
t^{\prime}.$ (48)

This hamiltonian of interaction is a good model for most of the macroscopic
sources where radiation is generated by a charged current $\vec{j}(\vec{r},t)$
whose expression is known. In practice, lasers indeed produce coherent states
of light, but incandescent bulbs do not, for they consist of many independent
and chaotic sources which break the overall coherence.

Thus coherent states have appeared as the most natural states of the
electromagnetic field when it is minimally coupled to a classical source. In
addition to this special role in the production of light, coherent states
exhibit major features from the point of view of its detection, which provides
a clearer operational meaning to coherence, as we now explain.

Maximising interference.

To detect light, one uses a photon counter. Typically, a photon counter is
sensible to the intensity of the electric field $E$, that we now assume to be
only a scalar field, for simplicity. If you assume that the electromagnetic
field is in a state $\ket{\psi}$, then the intensity in $x=(r,t)$ (the
spacetime point where the detector is) is on average

$I(x)=\expectationvalue{\hat{E}^{(-)}(x)\hat{E}^{(+)}(x)}{\psi}$ (49)

where $\hat{E}^{(-)}$ is the negative frequency part of the electric field,
conjugate to the positive frequency part $\hat{E}^{(+)}$, which is (from
equations (40) and (41))

$\hat{E}^{+}(\vec{r},t)=i\sum_{k}\left(\frac{\hbar\omega_{k}}{2L^{3}}\right)^{1/2}\hat{a}_{k}\,e^{i(\vec{k}\cdot\vec{r}-\omega_{k}t)}.$
(50)

Equation (49) is a fancy way of writing that the energy of the electric field
is $|E|^{2}$.

Let’s define the first-order (or two-point) correlation function as

$G(x_{1},x_{2})\overset{\rm
def}{=}\expectationvalue{\hat{E}^{(-)}(x_{1})\hat{E}^{(+)}(x_{2})}{\psi}.$
(51)

In a double-slit experiment, the interference pattern observed on the screen
is a measure of the intensity $I(x)=G(x,x)$ along the screen. In fact the
electric field $E(x)$ on the screen is the linear superposition of the
electric field $E(x_{1})$ and $E(x_{2})$ that was emitted by each of the two
slits at spacetime points $x_{1}$ and $x_{2}$:

$\hat{E}(x)\propto\hat{E}(x_{1})+\hat{E}(x_{2}).$ (52)

In this equation, $x$, $x_{1}$ and $x_{2}$ are related by the condition that
spherical waves emitted at spacetime points $x_{1}$ and $x_{2}$ intersect in
$x$. As a consequence,

$I(x)=G(x_{1},x_{1})+G(x_{2},x_{2})+2\real G(x_{1},x_{2}).$ (53)

The two first terms are the independent contributions from each slit. The last
term is responsible for the interference. When $G(x_{1},x_{2})=0$, no fringes
are observed. In fact, the visibility of the fringes is given by

$v\overset{\text{def}}{=}\frac{I_{max}-I_{min}}{I_{max}+I_{min}}=\frac{2|G(x_{1},x_{2})|}{G(x_{1},x_{1})+G(x_{2},x_{2})}$
(54)

Now one can show the inequality:

$|G(x_{1},x_{2})|^{2}\leq G(x_{1},x_{1})G(x_{2},x_{2})$ (55)

so that, keeping $G(x_{1},x_{1})$ and $G(x_{2},x_{2})$ fixed, the maximum of
interference is obtained for

$|G(x_{1},x_{2})|=\sqrt{G(x_{1},x_{1})G(x_{2},x_{2})}.$ (56)

When this condition is assumed to be valid for all $x_{1}$ and $x_{2}$, one
can show that there exists a function $\mathcal{E}(x)$ so that

$G(x_{1},x_{2})=\mathcal{E}^{*}(x_{1})\mathcal{E}(x_{2}).$ (57)

$G(x_{1},x_{2})$ factorises, and the state $\ket{\psi}$ is said to be
optically coherent. This notion of coherence recovers one that already existed
in classical electromagnetism, prior to quantum optics.

It is easy to see from the definition (51) that a sufficient condition for the
factorisability of $G(x_{1},x_{2})$ is that $\ket{\psi}$ is an eigenstate of
$\hat{E}^{(+)}(x)$ for all $x$. From equation (50), we see that it is
equivalent to say that $\ket{\psi}$ is an eigenstate of $a_{k}$ for all $k$.
And here we land on our feet! This is indeed the definition of coherent states
that was given previously but applied to an assembly of independent harmonic
oscillators. Here the definition is motivated on strong physical ground: the
maximisation of the inference pattern or say differently the factorisation of
the 2-point correlation function!

However, being a coherent state is only a sufficient, and not a necessary
condition to be optically coherent, i.e. to factorise the 2-point correlation
function. The coherent states can do much more: they factorise all of the
higher-order correlation functions, which experimentally measures the
coincidence rate between many detectors999However, it is unknown to the author
whether a pure state that factorises all of the higher-order correlation
functions is necessarily a coherent state of harmonic oscillator..

Coherence, Classicality, and Purity.

At this stage, the origin of the word ‘coherent’ has been brought to light:
the state $\ket{\psi}$ is such that the values of the field at different
points of space-time ‘conspire’ together to maximise the interference pattern.
Meanwhile, we have lost sight of the sense in which they can be seen as
‘quasi-classical’. Even worse, coherence and classicality may seem
contradictory. Indeed, an example of coherent light is that produced by a
laser, which is usually presented as a very quantum device, far from anything
classical. On the contrary, ordinary light produced by incandescent bulbs,
close to black-body radiation, is optically very incoherent, while it seems to
be much more ‘classical’ than lasers. Where is the catch?

The paradox arises from the confusion of two layers of ‘classicality’. The
first layer is classicality as the minimisation of the uncertainties of some
non-commuting observables. Compared to the previous example of the harmonic
oscillator, the vector potential $\hat{A}$ and the electric field $\hat{E}$
play now respectively the role of the position $\hat{x}$ and the momentum
$\hat{p}$. Coherent states of light are quasi-classical in the sense that they
minimise $\Delta\hat{A}$ and $\Delta\hat{E}$ together.

The second layer of classicality is the difference between pure and mixed
states. Classical physics is usually very noisy, that is very mixed, due to
the difficulty to control interactions with the environment. For instance, the
light of an incandescent bulb is a very mixed state (black-body radiation is
maximally mixed), so that it is tempting to say that it is more classical with
respect to a pure state, which is much more difficult to create in a lab.

The two layers, coherent/incoherent and pure/mixed, shall not be confused and
are actually independent. Many pure states are incoherent, while some mixed
states can be coherent. For instance, the state of an ideal laser is actually
a mixed state, which reads

$\rho=\frac{1}{2\pi}\int_{0}^{2\pi}\outerproduct{|\alpha|e^{i\theta}}{|\alpha|e^{i\theta}}d\theta,$
(58)

although it is optically coherent (it factorises the $2$-point correlation
function). This distinction is sometimes overlooked, especially in the context
of quantum gravity, where one usually exclusively considers pure coherent
states. This state of research may surprise as it is reasonable to believe
that quantum states of space are horrendously hard to isolate. Considering
mixed states instead may have some relevance in solving some of the hard
problems in the field [11].

##  6 Algebraic approach

Displacement operator.

The vacuum state is the only coherent state which is also an eigenstate of of
the harmonic oscillator:

$\ket{\alpha=0}=\ket{n=0}.$ (59)

For this reason, there is no ambiguity and one can write
$\ket{0}$101010Whereas it would be ambiguous to write for instance $\ket{1}$,
since
$\ket{\alpha=1}=e^{-1/2}\sum_{n=0}^{\infty}\frac{1}{\sqrt{n!}}\ket{n}\neq\ket{n=1}$..
In the previous section, we have seen how the other coherent states are
generated from the vacuum $\ket{0}$ by the unitary evolution of a simple
hamiltonian of interaction $\hat{H}_{I}$. Equation (46) can be rewritten

$\ket{t}=e^{i\phi(t)}\prod_{k}\hat{D}_{k}(\alpha_{k}(t))\ket{0}$ (60)

with $\alpha_{k}(t)$ given by equation (48), and $\hat{D}$ a unitary operator-
valued function over $\mathbb{C}$, called the displacement operator, and
defined by

$\hat{D}(\alpha)\overset{\rm
def}{=}e^{\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a}}.$ (61)

It is easy to check indeed that

$\ket{\alpha}=\hat{D}(\alpha)\ket{0}.$ (62)

Heisenberg group.

The set of displacement operators almost form a group:

$\hat{D}(\alpha+\beta)=e^{-i\imaginary(\alpha\beta^{*})}\hat{D}(\alpha)\hat{D}(\beta)$
(63)

In other words they form a group up to a phase. More precisely, they form a
subset of a group, called the Heisenberg group. Let’s see what it is.

The position and momentum operators $\hat{x}$ and $\hat{p}$ generate a Lie
algebra called the Heisenberg algebra (or also the Weyl algebra). It is the
smallest algebra generated by $\hat{x}$ and $\hat{p}$, with linear combination
and Lie bracketing $i[.,.]$. It is a $3$-dimensional non-commutative real
algebra, denoted $\mathfrak{h}$ and consisting of elements of the form

$a\hat{x}+b\hat{p}+c\hat{I}\quad a,b,c\in\mathbb{R},$ (64)

where $\hat{I}$ is the identity operator.

Exponentiating the Heisenberg algebra gives a $3$-dimensional real Lie group,
which is called, wisely, the Heisenberg group (or also the Weyl group),
denoted $H_{3}$. It consists of elements of the form

$e^{i(a\hat{x}+b\hat{p}+c\hat{I})}\quad a,b,c\in\mathbb{R}.$ (65)

The displacement operators are just a subset of this group, such that

$\hat{D}(\alpha)=e^{i\sqrt{2}\imaginary\alpha\hat{x}-i\sqrt{2}\real\alpha\hat{p}}.$
(66)

Any element of $H_{3}$ can be written as a displacement operator times a
phase. But since the global phase of states is irrelevant, the action of
$H_{3}$ on $\ket{0}$ generates exactly the family of coherent states!

In fact, from equation (63), it is easy to see that one can generate the whole
family of coherent states starting from any $\ket{\alpha}$, and not only
$\ket{0}$. One says that the action of the Heisenberg group is transitive: any
two coherent states are related by a transformation of the Heisenberg group.

Generalisation.

The previous analysis motivates a generalisation of coherent states for any
Lie group $G$ acting over a Hilbert space $\mathcal{H}$, which was first
proposed in parallel by Perelomov [12, 2] and Gilmore [4].

Let $G$ be a Lie group, and $T$ a unitary irreducible representation (irrep)
of $G$ over a Hilbert space $\mathcal{H}$. Choose
$\ket{\psi_{0}}\in\mathcal{H}$, and denote $H$ the subgroup which stabilises
$\ket{\psi_{0}}$ up to a phase, i.e.

$H\overset{\text{def}}{=}\left\\{g\in G\mid\exists\phi\in\mathbb{R},\quad
T(g)\ket{\psi_{0}}=e^{i\phi}\ket{\psi_{0}}\right\\}.$ (67)

The family of generalised coherent states is defined as the orbit of
$\ket{\psi_{0}}$ under the action of the (left) quotient space $G/H$. More
precisely, for each class $x\in G/H$, choose a representative $g(x)\in G$, and
define the generalised coherent states as

$\ket{x}=T(g(x))\ket{\psi_{0}}.$ (68)

Thus, the generalisation of coherent states depends a priori on many choices:
a group $G$, a unitary irrep $T$, a vector $\ket{\psi_{0}}$ and a set of
representatives $g(x)$. Of course, when it is projected down to the projective
Hilbert space $P\mathcal{H}$, the set of coherent states does not depend on
the choice of the representatives $g(x)$. The choice of initial state
$\ket{\psi_{0}}$ is a priori arbitrary, but can be motivated by another
criterion, like being a ground state or minimising some uncertainty relations.

In the case of Schrödinger coherent states $\ket{\alpha}$, the group is
$H_{3}$, with stabiliser $U(1)$, the initial state is the ground state
$\ket{0}$ of the harmonic oscillator, and the representatives are the
displacement operators $D(\alpha)$. We see again the interplay between the
kinematical and dynamical sides of coherent states: the dynamics lies in the
choice of the ground state $\ket{0}$, and the full set of $\ket{\alpha}$ is
generated by the group $H_{3}$, which contains the kinematical aspects.

Bloch states.

When we apply the method to the basic Lie group $SU(2)$, we obtain what
quantum opticians call the Bloch states. $SU(2)$ is the exponentiation of the
real Lie algebra $\mathfrak{su}(2)$, spanned by the (imaginary) Pauli matrices
$(i\sigma_{1},i\sigma_{2},i\sigma_{3})$. In other words, any $u\in SU(2)$ can
be written as

$u=e^{i\vec{\alpha}\cdot\vec{\sigma}},\quad\alpha\in\mathbb{R}^{3}.$ (69)

The unitary irreps of $SU(2)$ are Hilbert spaces $\mathcal{H}_{j}$, labelled
by a spin $j\in\mathbb{N}/2$ and spanned by the magnetic basis $\ket{j,m}$,
$m\in\\{-j,...,j\\}$, which diagonalises $J_{3}$ and $\vec{J}^{2}$ (in
physicists notations, $J_{i}\overset{\rm def}{=}\frac{\sigma_{i}}{2}$), such
as

$\begin{array}[]{l}J_{3}\ket{j,m}=m\ket{j,m},\\\
\vec{J}^{2}\ket{j,m}=j(j+1)\ket{j,m}.\end{array}$ (70)

As an initial ‘vacuum’ state we choose111111The choice $\ket{j,j}$ is often
made too. $\ket{j,-j}$. Then, we can show that the stabiliser is $U(1)$, and
we have the following diffeomorphism $SU(2)/U(1)\cong S^{2}$. The unit sphere
$S^{2}$ can be parametrised by a complex number $\zeta\in\mathbb{C}$ (except
for one point), by (the inverse of) the stereographical projection

$\vec{n}(\zeta)=\frac{1}{1+|\zeta|^{2}}\begin{pmatrix}-\zeta-\zeta^{*}\\\
i(\zeta^{*}-\zeta)\\\ 1-|\zeta|^{2}\end{pmatrix}.$ (71)

The representative $u\in SU(2)$ for each class $\vec{n}(\zeta)\in S^{2}$ is
given by

$u(\zeta)\overset{\rm
def}{=}\frac{1}{\sqrt{1+|\zeta|^{2}}}\begin{pmatrix}1&\zeta\\\
-\zeta^{*}&1\end{pmatrix}.$ (72)

Finally we define the $SU(2)$ coherent states as

$\ket{j,\vec{n}(\zeta)}\overset{\rm def}{=}u(\zeta)\ket{j,-j}.$ (73)

In terms of the magnetic basis, one can show that

$\ket{j,\vec{n}(\zeta)}=\frac{1}{(1+|\zeta|^{2})^{j}}\sum_{m=-j}^{j}\binom{2j}{j+m}^{\frac{1}{2}}\zeta^{j+m}\ket{j,m}.$
(74)

Among the important properties that these states satisfy, we should note that
the $SU(2)$ coherent states are eigenstates of $\vec{n}\cdot\vec{J}$

$\vec{n}\cdot\vec{J}\ket{j,\vec{n}}=-j\ket{j,\vec{n}}.$ (75)

Also they saturate the following Heisenberg inequality

$\Delta J_{1}\Delta J_{2}\geq\frac{1}{2}|\expectationvalue{J_{3}}|.$ (76)

Finally, they satisfy the following resolution of the identity

$\frac{2j+1}{4\pi}\int_{S^{2}}\differential\vec{n}\ket{j,\vec{n}}\bra{j,\vec{n}}=\mathbb{1},$
(77)

with $\differential\vec{n}$ being the usual measure on the unit sphere
$S^{2}$. The Bloch states have latter been used in quantum gravity as we shall
see in section (8).

Dynamical group.

From the perspective of experimentalists, the Lie group $G$ is not something
abstract but something very concrete, for its action drives the unitary time
evolution of states. In quantum optics, one deals typically with some
effective model of perturbed hamiltonian:

$\hat{H}=\hat{H}_{0}+\hat{H}_{pert}.$ (78)

The initial state is chosen to be the ground state of $\hat{H}_{0}$, and the
coherent states are generated through the time evolution induced by the
perturbation $\hat{H}_{pert}$, which can be due to the coupling to some
classical current as in equation (45).

In this context, the group $G$ is sometimes called a dynamical symmetry group,
so that for instance $H_{3}$ is said to be the dynamical (symmetry) group of
the harmonic oscillator [4, 5]. This naming is confusing because it conflicts
both with the notion of ‘dynamical group’, as defined by Souriau in [13], and
with the usual notion of ‘symmetry group’ of a hamiltonian.

Usually, a classical physical system is given by a phase space
$(\mathcal{P},\omega)$ and a hamiltonian $H$. Souriau defines a dynamical
group as any Lie group $G$ acting as a symplectomorphism (canonical
transformation) over $\mathcal{P}$. Then one can consider the symmetry of the
hamiltonian, i.e. the functions
$C_{i}\in\mathcal{C}^{\infty}(\mathcal{P},\mathbb{R})$ such that

$\\{C_{i},H\\}=0.$ (79)

They generate a Lie algebra of conserved quantities, which can be
exponentiated into a Lie group, which is called the symmetry group of the
hamiltonian $H$. This group is acting over the phase space as
symplectomorphism and for that reason, it is sometimes emphasised as the
dynamical symmetry group of $H$. In this sense, $H_{3}$ is not the dynamical
symmetry group of the harmonic oscillator!

This should not be too much of a surprise because, as we said earlier, the
Schrödinger coherent states have little to do with the dynamics of the
harmonic oscillator. What might be regarded as dynamical in them is the
initial choice of the ground state $\ket{0}$, but the Heisenberg group that
further generates them is built by from a choice of coordinates $(x,p)$ over
the phase space, that is, independently of the specific form of the
hamiltonian of the harmonic oscillator.

##  7 Geometric approach

In the two previous sections, we have seen how some initial quantum optical
work by Glauber in the 1960s [10], has lead in the 1970s, to a generalisation
of coherent states, by Perelomov [12] and Gilmore [4], using Lie groups. In
this section, we explore a second and independent path of generalisation in
more geometrical terms, which was proposed in the 1990s by Hall [14, 15],
based on some earlier works by Segal [16] and Bargmann [17] in the 1960s. Both
approaches have their relevance for quantum gravity, as we shall see in
section 8.

Phase space vs Hilbert space.

The classical phase space of the harmonic oscillator is $T^{*}\mathbb{R}$,
endowed with the usual symplectic structure given by the determinant. The
quantum analogue is the Hilbert space $L^{2}(\mathbb{R})$, with the usual
scalar product. The family of coherent states constitute a $2$-dimensional
submanifold of $L^{2}(\mathbb{R})$, parametrised by amplitude and time
$(A,t)$, or the complex number $z=Ae^{i\omega t}$. It could be as well-
parametrised by position $x=A\cos\omega t$ and momentum $p=-A\omega\sin\omega
t$, which exhibit an explicit diffeomorphism between the classical phase space
and the family of coherent states. Any point in phase space determines
uniquely a coherent state, and conversely. This fact is not a coincidence, but
rather a crucial aspect of coherent states that shed further light on their
classicality.

It is a general feature of coherent states that they define a natural
injection of the classical phase space $\mathcal{P}$ into the Hilbert space
$\mathcal{H}$. A priori, there are many possible such injections, but coherent
states provide a natural one. To work this out, we shall first see how one can
build a Hilbert space $\mathcal{H}$ from a phase space $\mathcal{P}$. Well,
this is the whole point of quantization, and so one should know that the
subject is not easy. Nevertheless geometric quantization121212Not to be
confused with the previously discussed geometrical formulation of quantum
mechanics [8]. is a prototypical such method, rather technical, but we can
skip the details and keep the general idea.

Geometric quantisation.

Start with a configuration space $\mathcal{M}$ and build the phase space
$\mathcal{P}=T^{*}\mathcal{M}$. There are actually many ways in which
$\mathcal{M}$ can be seen as a subspace of $T^{*}\mathcal{M}$. Each way
consists in choosing what is called a polarisation. Then one can build a
complex line bundle over $T^{*}\mathcal{M}$, denoted $L$. Locally, we have
$L\approx T^{*}\mathcal{M}\times\mathbb{C}$. The prequantum Hilbert space,
$\mathcal{PH}$, is the space of equivalence classes of square-integrable
sections of $L$, where two sections are said equivalent when they are equal
almost everywhere. Roughly, $\mathcal{PH}\approx L^{2}(T^{*}\mathcal{M})$. It
is much too big to be a good quantum Hilbert space. So a choice of
polarisation enables to select a subspace of $\mathcal{PH}$ and to build the
good quantum Hilbert space $\mathcal{H}\cong L^{2}(\mathcal{M})$. This
construction really enable to see the Hilbert space $\mathcal{H}$ as a
subspace of $L^{2}(T^{*}\mathcal{M})$. Thus a state $\ket{\psi}\in\mathcal{H}$
can be seen as a complex function $\psi$ over $T^{*}\mathcal{M}$.

Then, for each phase space point $z\in T^{*}\mathcal{M}$, we define the
coherent state $\ket{z}$ as the unique state in $\mathcal{H}$ such that

$\forall\ket{\psi}\in\mathcal{H},\quad\innerproduct{z}{\psi}=\psi(z).$ (80)

This definition is both elegant and confusing. Elegant, because the equation
is very simple, confusing, because it is too simple. On the LHS, we have a
scalar product between two states in $\mathcal{H}$, while on the RHS we have
the evaluation of a function $\psi$ in one point $z$ of the phase space
$T^{*}\mathcal{M}$. A practical example should clarify the matter.

Segal-Bargmann transform.

The classical phase space of the harmonic oscillator is
$T^{*}\mathbb{R}\cong\mathbb{C}$, so that the prequantum Hilbert space is
roughly $L^{2}(\mathbb{C})$. By using the Kähler polarisation, the quantum
Hilbert space finally obtained is the Segal-Bargmann space131313Also called
the Fock-Bargmann space in [1]. [14], denoted $\mathcal{SB}$. It is made of
functions over $\mathbb{C}$ which are both holomorphic and square-integrable,
with the (gaussian) scalar product

$(f_{1},f_{2})=\int_{\mathbb{C}}\bar{f_{1}}(z)f_{2}(z)\frac{i}{2\pi\hbar}e^{-|z|^{2}/\hbar}dz\wedge
d\bar{z}$ (81)

$\mathcal{SB}$ is isomorphic to $L^{2}(\mathbb{R})$, which is not a big
surprise after all, since all Hilbert spaces of the same dimension are
isomorphic. What is more interesting is that there is actually an isometry
between them two, given by

$\tilde{\phi}(z)=\int_{\mathbb{R}}K(z,x)\phi(x)dx$ (82)

with the kernel

$K(z,x)=\sqrt[4]{\frac{m\omega}{\pi\hbar}}e^{\frac{1}{\hbar}\left(\frac{z^{2}}{2}-(\sqrt{\frac{m\omega}{2}}x-z)^{2}\right)}$
(83)

This isomorphism is called the Segal-Bargmann transform. Its inverse is given
by

$\phi(x)=\int_{\mathbb{C}}\overline{K(z,x)}\tilde{\phi}(z)\frac{i}{2\pi\hbar}e^{-|z|^{2}/\hbar}dz\wedge
d\bar{z}$ (84)

Equation (82) is in fact a concrete instantiation of equation (80), so that
the kernel $K$ is actually a coherent state! More precisely, it matches the
expression of equation (34), provided a rescaling of $z$, and up to a phase
and a normalisation factor:

$K\left(z,x\right)=e^{\frac{i}{\hbar}\real z\imaginary
z}e^{\frac{1}{\hbar}|z|^{2}}\psi_{\sqrt{\frac{2}{m\omega}}z,\frac{1}{m\omega}}(x).$
(85)

For this reason the Segal-Bargmann transform is also called the coherent-state
transform.

In the case of Schrödinger coherent states, the algebraic approach (with the
Heisenberg group $H_{3}$) generates the same coherent states as the geometric
approach (with the phase space $T^{*}\mathbb{R}$). Both approaches are
secretly linked by the fact that the Heisenberg group $H_{3}$ is naturally
obtained by exponentiating the quantised coordinates $\hat{x}$ and $\hat{p}$
of the phase space $T^{*}\mathbb{R}$. However, the two methods do not always
match. In the case of the phase space $S^{2}$, the exponentiation of the
quantised coordinates $\sigma_{1},\sigma_{2},\sigma_{3}$, generates $SU(2)$.
However the coherent states obtained from geometric quantisation of $S^{2}$
[18] are different from the $SU(2)$ coherent states of equation (73).

Resolution of the identity.

The fact that the coherent-state transform is an isometry is equivalent to the
following resolution of identity for the coherent states

$\mathbb{1}=\frac{1}{\pi}\int_{\mathbb{C}}\outerproduct{\alpha}{\alpha}\,\differential\real\alpha\,\differential\imaginary\alpha.$
(86)

This equation should be understood in the sense of weak convergence, that is,
for any two given states $\ket{\phi_{1}}$ and $\ket{\phi_{2}}$,

$\innerproduct{\phi_{1}}{\phi_{2}}=\frac{1}{\pi}\int_{\mathbb{C}}\innerproduct{\phi_{1}}{\alpha}\innerproduct{\alpha}{\phi_{2}}\,\differential\real\alpha\,\differential\imaginary\alpha.$
(87)

This resolution of the identity is similar to the more familiar one of any
orthonormal basis of $\mathcal{H}$, like

$\mathbb{1}=\sum_{n}\outerproduct{n}{n},$ (88)

up to the crucial difference that the coherent states are parametrised by a
continuous parameter $\alpha\mapsto\ket{\alpha}$.

The (strong) continuity of the map $\alpha\mapsto\ket{\alpha}$ together with
the (weak) resolution of identity are so important that they are often
regarded as the two properties that coherent states should have to deserve
such a designation. It is a bit surprising because it seems too
generic141414In the same manner as the condition of being an orthonormal basis
is far from being a sufficient property to define the $\ket{n}$ basis., but
this is the point of view defended for instance by Klauder in his collection
of papers [19].

Heat kernel.

When the geometric quantisation is performed using the Khäler polarisation,
the coherent states finally obtained by equation (80) can be expressed in
terms of the more baroque notion of the heat kernel [15]. We explain it below
for it has played a role in quantum gravity as we will see in the next
section.

The heat kernel $\rho_{t}(x)$ is the solution of the heat equation

$\frac{d\rho}{dt}=\partial_{x}^{2}\rho$ (89)

that satisfies $\rho_{0}(x)=\delta(x)$. Its explicit expression is151515The
heat equation is just Schrödinger equation with complex time, so that the
solution can be easily recovered from (12). However, the solutions in (12)
were discarded as coherent states. It seems to be a pure coincidence that the
same equation with complex time now gives proper coherent states.

$\rho_{t}(x)=\frac{1}{\sqrt{4\pi t}}e^{-\frac{x^{2}}{4t}}$ (90)

The kernel $K$ can be rewritten in term of the heat kernel $\rho_{t}(x)$ such
that

$K\left(z\sqrt{\hbar},x\sqrt{\frac{\hbar}{2m\omega}}\right)=\sqrt[4]{\frac{2m\omega}{\hbar}}\frac{\rho_{\frac{1}{2}}(x-z)}{\sqrt{\rho_{\frac{1}{2}}(x)}}.$
(91)

This observation has suggested a construction of coherent states when the
configuration space is a connected compact Lie group $G$, instead of
$\mathbb{R}$ [14]. The heat equation over $G$ reads

$\frac{d\rho}{dt}=\Delta\rho,$ (92)

where $\Delta$ is the Casimir operator, and $\rho$ a function of
$t\in\mathbb{R}$ and $g\in G$. It can be shown the existence of a smooth and
strictly positive solution, called the heat kernel, which is a delta over the
identity at $t=0$. It admits the following expansion

$\rho_{t}(g)=\sum_{\pi}\dim\pi\,e^{-\lambda_{\pi}t}\,\chi^{\pi}(g),$ (93)

where the sum is taken over all irreps $\pi$ (up to unitary equivalence), and
$\lambda_{\pi}$ is the characteristic Casimir (non-negative) number of the
representation, and $\chi^{\pi}$ is the character. For instance, in the case
of $SU(2)$ one gets

$\rho^{o}_{t}(g)=\sum_{j\in\mathbb{N}/2}(2j+1)\,e^{-j(j+1)t}\,\Tr D^{j}(g),$
(94)

with $D^{j}(g)$ the Wigner matrix of $g$ in the spin-$j$ irrep. Then Hall
defines the complexification $G_{\mathbb{C}}$ of the Lie group $G$. For
instance, for $SU(2)$ it is $SL_{2}(\mathbb{C})$. From this he shows that
there is a unique analytic continuation of the heat kernel $\rho_{t}$ from $G$
to $G_{\mathbb{C}}$. In analogy161616The square-root in equation (91) is only
a normalisation factor, of which one can get rid of, provided a good choice of
measure in the definition of the scalar product. with equation (91), the
coherent states are defined as the functions over $G$, indexed by $x\in
G_{\mathbb{C}}$, as

$K_{x,t}(g)\overset{\rm def}{=}\rho_{t}(x^{-1}g),\quad\text{with}\,g\in G.$
(95)

In [15], it is shown how the heat kernel construction with a group $G$ matches
the geometric quantisation approach over the phase space $T^{*}G$. The latter
approach shows how the coherent states provide a natural embedding of the
phase space within the corresponding Hilbert space, and in this sense, it
points towards their quasi-classical properties. The heat kernel approach
presents the advantage of offering more analytical formulas like (95),
compared to (80), but it is then harder to see how the quasi-classical
properties can arise.

##  8 Quantum Gravity

Quantum gravity is still at a very speculative stage compared to quantum
optics, but the two fields of research can speak to one another. Indeed, the
experimental control of the latter has motivated many theoretical
developments, like the coherent states, which can now be reinvested into the
former. At least, having in mind the concrete set-up of quantum optics may
help theoreticians of quantum gravity to keep their feet on the ground, so to
speak.

Schrödinger had introduced the coherent states as an attempt to recover the
macroscopic physics from quantum mechanics. This motivation was somehow lost
in quantum optics, which rather focused on interference patterns and addressed
different kind of questions, closely related to technological achievements,
such as lasers. In quantum gravity, the emphasis is put back on the classical
properties of coherent states. This time, the coherent states are states of
the gravitational field, whereas quantum optics deals with the electromagnetic
field. The question here is to understand how general relativity can be
recovered, in some limit, from quantum gravity, i.e. how a classical geometry
of space-time can arise from quantum states of the gravitational field. Thus,
coherent states are used as a tool to check the consistency of the theory in
an experimentally well-tested macroscopic regime.

Taking the perspective of the kinematical characterisation described in
section 4, one wonders which observables are to be chosen for the coherent
states to be peaked upon a classical configuration of space-time. In quantum
optics, we have seen that the relevant observables were the vector potential
$\hat{A}$ and the electric field $\hat{E}$. General relativity can be
formulated in close analogy to electromagnetism and this suggests to take as
observables the so-called holonomy and flux, which carry the geometrical
meaning of curvature and area.

Although any theory of quantum gravity is likely to make use of coherent
states, we focus in this section on the Loop approach to Quantum Gravity
(LQG). Both the algebraic (Lie group) and the geometric (heat kernel)
approaches to coherent states have been applied in LQG. Historically, first
has come the heat kernel method, starting with an extension of Hall’s
construction to diffeomorphism invariant gauge theories [20]. Bloch states
were later used in [21] and helped in providing a semi-classical picture of
the geometry of space. But let’s start recalling briefly the essential
mathematical structure of LQG.

LQG Hilbert space.

In covariant LQG, space is described by quantum states that form the following
Hilbert space

$\mathcal{H}_{LQG}=\bigoplus_{\Gamma}L^{2}(SU(2)^{L})/\mathcal{G}_{\Gamma},$
(96)

where the direct sum is made over all the different abstract finite directed
graphs $\Gamma$, with different number of links $L$ and nodes $N$. An example
of such a graph is given in figure 1. $L^{2}(SU(2)^{L})$ is the Hilbert space
onsisting of the square-integrable complex functions
$\Psi(g_{l_{1}},...,g_{l_{L}})$ over $SU(2)^{L}$. It is quotiented by the so-
called Gauss constraint $\mathcal{G}_{\Gamma}$, which amounts to imposing the
following invariance property, given any element $u_{n}\in SU(2)$ per node $n$
of $\Gamma$,

$\Psi(g_{l_{1}},...,g_{l_{L}})=\Psi(u_{t_{1}}g_{l_{1}}u^{-1}_{s_{1}},...,u_{t_{L}}g_{l_{L}}u^{-1}_{s_{L}}),$
(97)

with $t_{i}$ and $s_{i}$ being respectivelly the target and the source of the
link $i$.

Figure 1: Example of a finite directed graphs $\Gamma$, with $8$ links and $4$
nodes.

This description of quantum states of space might seem very abstract and one
needs to draw a connection to some intuitive understanding in terms of
geometric quantities. To make it short, the quantum space over a single link,
$L^{2}(SU(2))$, is the quantisation of the phase space $T^{*}SU(2)$. Actually,
since $SU(2)$ is a Lie group, its cotangent bundle can be trivialised as
$T^{*}SU(2)\cong SU(2)\times\mathfrak{su}(2)$, and the coordinates $(h,E)$ are
respectively called holonomy and flux. These quantities appear in general
relativity, where they can be roughly understood as the curvature and the area
of pieces of space. Subtlety left aside, this description of phase space
suggests that it is possible to apply directly the heat kernel method
described in section 7, with the Lie group $SU(2)$! Thus we expect the
coherent states of LQG to be indexed by elements of the complexification of
$SU(2)$, that is $SL_{2}(\mathbb{C})$.

Coherent spin-networks.

The rigorous construction of coherent states for LQG was done in [22, 23, 24,
25], where they are called ‘complexifier coherent states’, but we prefer to
call them coherent spin-network states as in [26, 27]. They are not properly
coherent states for the full theory, but only for its truncation on a fixed
graph $\Gamma$. They are denoted $\Psi_{(\Gamma,H_{l},\tau_{l})}$,
parametrised by one element $H_{l}\in SL_{2}(\mathbb{C})$ and one positive
number $\tau_{l}\in\mathbb{R}^{+}$ per link $l$ of $\Gamma$. To each link $l$
is associated the coherent state $\rho^{o}_{\tau_{l}}(g_{l}\,H_{l}^{-1})$,
with $\rho^{o}_{\tau}(g)$ the analytic continuation of (94) to
$SL_{2}(\mathbb{C})$. All these terms are then multiplied together, and
finally projected it down to $\mathcal{H}_{LQG}$ by an $SU(2)$ integration
over each node to impose the Gauss constraint. All this fits in the following
formula:

$\Psi_{(\Gamma,H_{l},\tau_{l})}(g_{l})\\\ \overset{\rm
def}{=}\int_{SU(2)^{N}}\left(\prod_{l}\rho^{o}_{\tau_{l}}(u_{t_{l}}\,g_{l}\,u^{-1}_{s_{l}}\,H_{l}^{-1})\right)[\differential
u_{n}].$ (98)

The formal jungle of techniques that helps to define these coherent states in
[22, 23, 24, 25] should not make us forget about the underlying physical
intuition that originally motivates them: they constitute a natural embedding
of the classical phase space into the quantum Hilbert space. The embedding is
parametrised by $H_{l}\in SL_{2}(\mathbb{C})\cong T^{*}SU(2)$ which admits two
possible semi-classical interpretations.

Holonomy and flux.

The first and original interpretation is based on the polar decomposition of
$H$ as

$H=h\,e^{i\frac{E}{8\pi G\hbar\gamma}\tau},$ (99)

with $h\in SU(2)$ and $E\in\mathfrak{su}(2)$ being respectively the holonomy
and the flux over a link, while $\gamma\in\mathbb{R}$ is a constant, called
the Immirzi parameter, and $G$ and $\hbar$ respectively Newton’s and (reduced)
Planck’s constant. They are peaked on the classical holonomy-flux
configuration $(h_{l},E_{l})$ on each link $l$ [23, 28].

Twisted geometry.

The second interpretation is based instead on the decomposition

$H=u(\zeta_{s})\,e^{-iz\sigma_{3}}\,u(\zeta_{t})$ (100)

with $\zeta_{s},\zeta_{t}\in\mathbb{C}$, and $u(\zeta)\in SU(2)$ given by
equation (72), and

$z=\xi+i\,a\,\tau$ (101)

with $\xi,a\in\mathbb{R}$.

With this parametrisation, the state is peaked on a discrete kind of geometry,
called twisted geometry [29]. The phase space $T^{*}SU(2)$ on each link is
parametrised by $(\vec{n}(\zeta_{s}),\vec{n}(\zeta_{t}),\xi,a)$, with the map
$\vec{n}(\zeta)$ given by equation (71). Thus each node is surrounded by a set
of unit vectors $\pm\vec{n}_{i}$, one per attached link, with the sign
$\sigma=\pm$ chosen depending on whether the link is ingoing or outgoing. At
the semi-classical level, the Gauss constraint imposes the closure of the
vectors $\vec{n}_{i}$ surrounding a node, such as

$\sum_{l\in n}\sigma_{l}\,a_{l}\,\vec{n}_{l}=0$ (102)

where the sum is done on all the links $l$ surrounding a node $n$. It is a
theorem by Minkowski that there is a unique convex polyhedron such that its
faces have unit exterior-pointing normals $\sigma_{l}\,\vec{n}_{l}$ and areas
$a_{l}$ satisfying the closure condition (102) [30, 31]. So each node of the
coherent state can really be seen, in the semi-classical picture of twisted
geometry, as a polyhedron.

Two neighbouring polyhedra are glued along a link, that is a face of same
area, although the shape of the face may not match (reason for which the
geometry is said to be twisted). Finally, the number $\xi$ can be used to
encode the extrinsic curvature on the common face, when the polyhedra are
embedded into $4$-dimensional spacetime171717The issue is delicate, see [32]..

Coherent intertwiners.

The previous parametrisation is also interesting for it connects with the
algebraic approach developed in section 6. In quantum gravity, the Lie group
that generates the coherent states is $SU(2)$. It is not too much of a
surprise since $SU(2)$ plays a central role in the description of the geometry
of space. Thus, one recovers the so-called Bloch states $\ket{j,\vec{n}}$ (see
equation (73)), which are very familiar to quantum optics for very different
reasons.

The Hilbert space $\mathcal{H}_{LQG}$ can be seen as a subspace of some direct
sum of the Hilbert spaces $\mathcal{H}_{j}$ in which the Bloch states live.
This suggests a second way to build coherent states for quantum gravity. Given
a node $n$ of $\Gamma$, the idea is to select one Bloch state
$\ket{j_{l},\vec{n}_{l}}$ per link $l$ attached to $n$, to tensor them all and
to project them down by imposing the Gauss constraint. One obtains the so-
called coherent intertwiners $\ket{\\{j,\vec{n}\\}}_{n}$ [21]:

$\ket{\\{j,\vec{n}\\}}_{n}\overset{\rm
def}{=}\int_{SU(2)}\left(\bigotimes_{l\in
n}D^{j_{l}}(g)\ket{j_{l},\vec{n}_{l}}\right)\differential g.$ (103)

Over a graph $\Gamma$, the tensor product of coherent intertwiners (one per
node $n$ of the graph),

$\bigotimes_{n\in\Gamma}\ket{\\{j,\vec{n}\\}}_{n},$ (104)

can be seen as a state

$\Psi_{\Gamma,j_{l},\vec{n}_{l}}\in L^{2}(SU(2)^{L})/\mathcal{G}_{\Gamma}.$
(105)

The semi-classical interpretation of $\Psi_{\Gamma,j_{l},\vec{n}_{l}}$ is the
juxtaposition of all polyhedra, i.e. without the extrinsic angle that glues
them together. For this reason they are called intrinsic coherent states by
Rovelli [33], as opposed to the extrinsic coherent states
$\Psi_{(\Gamma,H_{l},\tau_{l})}$. For large values of $a$, the two are related
by

$\Psi_{(\Gamma,H_{l},\tau_{l})}(g_{l})\\\
\sim\sum_{j_{l}}\left(\prod_{l}(2j_{l}+1)e^{-\tau_{l}(j_{l}-\frac{a_{l}}{2}+\frac{1}{2})^{2}}e^{-i\gamma\theta_{l}j_{l}}\right)\\\
\times\Psi_{\Gamma,j_{l},\vec{n}_{l}}(g_{l}),$ (106)

(see [26] for details).

We have given a quick overview of the two main definitions of coherent states
in the context of quantum gravity. The main point was to show that both lines
of investigation exposed previously, the algebraic and the geometric
approaches, have spawned the field of quantum gravity. There are also other
proposals under investigation like the $U(N)$ coherent states [34], the
$SO^{*}(2N)$ coherent states [35], or the coherent intertwiners of [36], but
we postpone their review to future work.

##  9 Conclusion

The main results of this paper are the followings:

  * •

We formalised in precise terms the hand-waving definition of the seminal paper
of Schrödinger.

  * •

We investigated what it would mean to be a coherent state of the free
particle.

  * •

We proposed the kinematical/dynamical dichotomy as a relevant conceptual shift
towards the generalisation of the initial Schrödinger’s idea.

  * •

We noticed and resolved the paradox that the coherent states of light may
appear as very non-classical states.

  * •

We noticed and clarified a confusion in the literature in the use of the term
of ‘dynamical symmetry group’.

  * •

We summarised the genealogy of coherent states from Schrödinger to quantum
gravity, enlighting the existence of two main parallel lines of thoughts, that
were dubbed the algebraic and the geometric approaches.

Despite the broad range of our exploration, we have only been scratching the
surface of the general subject of coherent states, which is now branching in
many directions (see [1] or [2] for more exhaustive treatment, although
quantum gravity is absent in them). Nevertheless, we have gathered in few
pages, the core ideas of coherent states, as they have emerged in the head of
Schrödinger, later fuelled by quantum optics, and applied to quantum gravity.
These three steps of developments are, to some extent, independent, but we
have shown how they mingle together to form a beautiful and consistent
landscape.

## Acknowledgements

I would like to thank Alexander Thomas, Andrea Di Biagio, Andrea Calcinari,
Carlo Rovelli, Emanuele Polino, Farshid Soltani, Federico Zalamea, John
Schliemann, Klaus Mølmer, Lautaro Amadei, Pietro Dona and Simone Speziale, for
their comments, suggestions, encouragements and proofreadings.

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