Searching for Coherent States:
From Origins to Quantum Gravity
Pierre Martin-Dussaud
1,2
1
Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France
2
Basic Research Community for Physics e.V.
We discuss the notion of coherent states from
three different perspectives: the seminal ap-
proach of Schrödinger, the experimental take
of quantum optics, and the theoretical devel-
opments in quantum gravity. This compara-
tive study tries to emphasise the connections
between the approaches, and to offer a coher-
ent short story of the field, so to speak. It
may be useful for pedagogical purposes, as well
as for specialists of quantum optics and quan-
tum 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 quan-
tum optics, while several mathematical generalisa-
tions 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 mo-
tivations behind the definitions. We would like
to show that the semi-classical properties of co-
herent states are expected rather than magical.
• The many approaches to coherent states convey
the impression of a disparate field made of arbi-
trary 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 pedagogi-
cal introduction, trading exhaustiveness for clarity. It
Pierre Martin-Dussaud: pmd@cpt.univ-mrs.fr
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 |ψi such that hψ|ˆx(t)|ψi and hψ|ˆp(t)|ψi
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 can-
not be a characterisation of coherent states what-
soever, since it is clear, from Ehrenfest theorem,
that any time-evolved state |ψ(t)i, 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 ex-
pected. What could be a better pedagogical in-
troduction to the topic?
• Coherent states can also be generated by the ac-
tion of the Heisenberg group over the vacuum
state. This group is sometimes called the dy-
namical symmetry group of the harmonic oscil-
lator (see [4, 5]), although it is very unclear in
which sense the group is ‘dynamical’, a ‘symme-
try group’, or even specific to the harmonic oscil-
lator. 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 be-
ing a classical source of light as an incandescent
bulb may be. Is the paradox of designation only
superficial?
Accepted in Quantum 2021-01-23, click title to verify. Published under CC-BY 4.0. 1
arXiv:2003.11810v4 [quant-ph] 25 Jan 2021
• The definition of coherent states in quantum
gravity is covered by a jungle of technicalities,
far from the experimental point of view of quan-
tum 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 al-
gebraic 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 co-
herent states is to demonstrate how classical mechan-
ics 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 repre-
sent a ‘particle’, which is executing the ‘mo-
tion’, expected from the usual mechanics.
Neither does he use the words ‘quasi-classical’ or
‘semi-classical’, but the latter would convey his in-
tuition 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 ω. Its
Hilbert space is L
2
(R) over which are acting the po-
sition operator ˆxψ(x) = xψ(x) and the momentum
ˆpψ = −i~ ∂
x
ψ . The dynamics is provided by the
hamiltonian which reads
ˆ
H =
ˆp
2
2m
+
1
2
mω
2
ˆx
2
. (1)
The eigenstates of
ˆ
H form an orthonormal basis |ni,
indexed by n ∈ N, whose coefficients, in the basis of
eigenstates |xi of ˆx, read
hx|ni =
4
r
mω
π~
1
√
2
n
n!
e
−
mω
2~
x
2
H
n
r
mω
~
x
(2)
where H
n
(x) are Hermite’s polynomials
1
and the as-
sociate eigenvalues are
E
n
= ~ω
n +
1
2
. (4)
Schrödinger coherent states.
Then, Schrödinger defines
2
, out of the blue, the
following family of states, indexed by time t ∈ R and
another parameter A ∈ R:
Ae
iωt
def
= e
−
A
2
2
e
−iωt/2
∞
X
n=0
A
n
√
n!
e
i
~
E
n
t
|ni. (5)
It is immediate to see that
Ae
iωt
is the temporal
evolution of |Ai by the unitary operator e
i
~
ˆ
Ht
, as
e
i
~
ˆ
Ht
|Ai = e
iωt/2
Ae
iωt
. (6)
Then Schrödinger argues that these states approxi-
mate the ‘macro-mechanics’, what we would call in
modern language, being semi- or quasi-classical. More
precisely, he highlights three properties:
1. First the average position satisfies the law of clas-
sical motion:
hˆxi = A cos ωt. (7)
2. Secondly, the average energy is almost the clas-
sical one:
D
ˆ
H
E
≈ A
2
mω
2
. (8)
3. Third, he argues (without any explicit computa-
tion) that the wave packet does not ‘spread out’,
but ‘remains compact’, like a particle.
The two first properties provide physical meaning
to the parameter α as the amplitude of some corre-
sponding classical wave. Thus the Schrödinger coher-
ent states are parametrised by an amplitude α and an
instant t.
Wrong characterisation of quasi-classicality.
The three arguments above appear as a first at-
tempt to formalise the property of ‘quasi-classicality’,
and have been the basis of the later developments of
coherent states. Unfortunately, it has been hardly
1
Wikipedia mentions two conventions for Hermite’s polyno-
mial. We use the physicist one, i.e.
H
n
(x)
def
= (−1)
n
e
x
2
d
n
dx
n
e
−x
2
. (3)
2
Compared 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
−iωt/2
to match
the standard Dirac notation.
Accepted in Quantum 2021-01-23, click title to verify. Published under CC-BY 4.0. 2
never noticed that the first property cannot charac-
terise quasi-classicality in any way. Indeed all the
quantum states of the harmonic oscillator satisfy this
property! More precisely, given any initial state |ψ
0
i,
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
ˆ
A in any state |ψ(t)i, accord-
ing to the equation
d h
ˆ
Ai
dt
=
1
i~
D
[
ˆ
A,
ˆ
H]
E
. (9)
In the case of the harmonic oscillator, the equations
for ˆx and ˆp are
d hˆxi
dt
=
1
m
hˆpi and
d hˆpi
dt
= −mω
2
hˆxi. (10)
These are actually the classical equations of motion
for hˆxi and hˆpi, and so all solutions hˆxi take the form
of equation (7). It is completely generic and so can-
not be used as a characterisation of quasi-classicality
3
.
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 recog-
nised, 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 den-
sity 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
(R). The Dirac delta function δ(x)
describes the state of a particle perfectly well localised
at x = 0. The uncertainty about its position is zero:
3
I am indebted to Federico Zalamea to have made me re-
alised this fact.
∆ˆx = 0. For that reason, it may seem a good candi-
date for being a quasi-classical state.
However, this first attempt fails because the parti-
cle does not remain localised as time goes by. Indeed,
the hamiltonian of the free particle
ˆ
H =
ˆp
2
2m
, (11)
drives the time-evolution of δ(x) to
ψ(x, t) =
r
m
2π~|t|
e
−i sgn(t)
π
4
e
i
mx
2
2~t
. (12)
The probability distribution |ψ(x, t)|
2
is now com-
pletely spread, ∆ˆx = ∞, and not even normalised!
Thus, a wave function which is infinitely well localised
at initial time, turns instantaneously into an infinitely
spread state
4
.
So consider instead a more reasonable initial state,
like a gaussian curve
ψ
0
(x) =
1
p
σ
√
2π
e
−
x
2
4σ
2
, (13)
which is spread as ∆
0
ˆx = σ. It evolves as a free
particle to
ψ(x, t) =
1
(2π(σ
2
+ i~t/m))
1/4
e
−
x
2
4(σ
2
+i~t/m)
. (14)
It is also a gaussian which is spread like
∆ˆx =
r
σ
2
+
t
2
~
2
4m
2
σ
2
, (15)
so that it irrevocably spreads with time and looses its
initially compact shape.
In fact, whatever the initial state ψ
0
at time t = 0,
it evolves as a free particle to a state ψ(x, t) which
satisfies
5
:
(∆ˆx)
2
= (∆
0
ˆx)
2
+
(∆
0
ˆp)
2
m
2
t
2
+
t
m
(hˆxˆp + ˆpˆxi
0
− 2 hˆxi
0
hˆpi
0
) . (16)
It is a second order polynomial in t. A necessary con-
dition to prevent the time spreading would be to have
(∆
0
ˆp)
2
= 0, but this implies, through Heisenberg in-
equality, that ∆
0
ˆx = ∞, i.e. a maximally spread
states in space... So, for the free particle, the spread-
ing is unavoidable. From this perspective, there is no
‘quasi-classical state’ for the free particle.
4
This 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.
5
A proof can be found in [7] p. 104.
Accepted in Quantum 2021-01-23, click title to verify. Published under CC-BY 4.0. 3
Harmonic oscillator.
Let us now consider the more sophisticated hamil-
tonian of the harmonic oscillator:
ˆ
H =
ˆp
2
2m
+
1
2
mω
2
ˆx
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 ψ(x, t) is given by
6
(∆ˆx)
2
=
1
2
(∆
0
ˆx)
2
+
(∆
0
ˆp)
2
m
2
ω
2
−
1
2
(∆
0
ˆp)
2
m
2
ω
2
− (∆
0
ˆx)
2
cos(2ωt)
+
1
2mω
(hˆxˆp + ˆpˆxi
0
− 2 hˆxi
0
hˆpi
0
) sin(2ωt) (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 ∆
0
ˆx.
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 har-
monic 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 ∆ˆx is constant in time.
This requires the two conditions
(∆
0
ˆp)
2
= m
2
ω
2
(∆
0
ˆx)
2
hˆxˆp + ˆpˆxi
0
= 2 hˆxi
0
hˆpi
0
(21)
One can check that the Schrödinger coherent states
do satisfy these conditions. However, these two con-
ditions are not sufficient to characterise them. For
instance all the eigenstates of the hamiltonian, |ni,
also satisfy these conditions. Another condition is still
required: the minimisation of ∆
0
ˆx.
The family of states for which ∆ˆx is constant in
time is foliated by the value of ∆
0
ˆx, with a minimal
6
Proof. Using Ehrenfest theorem we have
d
dt
hˆxi = hˆpi/m
d
dt
hˆpi = −mω
2
hˆxi
d
dt
ˆx
2
= hˆxˆp + ˆpˆxi /m
d
dt
ˆp
2
= −mω
2
hˆxˆp + ˆpˆxi
d
dt
hˆxˆp + ˆpˆxi = 2
ˆp
2
/m − 2mω
2
ˆx
2
(18)
from which we show the differential equation
d
4
dt
4
ˆx
2
= −4ω
2
d
2
dt
2
ˆx
2
(19)
which is finally solved easily, and leads to our expression for
(∆ˆx)
2
def
=
ˆx
2
− hˆxi
2
.
value being strictly positive. Indeed, from the Heisen-
berg inequality
∆
0
ˆx∆
0
ˆp ≥
~
2
. (22)
and from the first condition in (21), we have
∆
0
ˆx ≥
r
~
2mω
. (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 ∆ˆx is
constant and minimal. Both conditions are impor-
tant. Otherwise, there are states whose spreading is
momentarily smaller but will grow later to a larger
value. There are also states whose spreading is con-
stant, but not minimal (like the |ni). Geometrically,
the two conditions select a 2-dimensional submanifold
out of the infinite-dimensional space L
2
(R).
Minimal time average.
There is another way to make the third property of
Schrödinger more precise. Consider the time average
of ∆ˆx:
T [∆ˆx] =
r
(∆
0
ˆx)
2
2
+
(∆
0
ˆp)
2
2m
2
ω
2
. (24)
Now, from Heisenberg inequality, this time average is
bounded by
T [∆ˆx] ≥
r
~
2mω
. (25)
This inequality is saturated for the coherent states
and only for them. So we have a second characterisa-
tion of coherent states as the states which minimise
∆ˆx 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 classi-
cal’. As we said before, the classical behaviour of hˆxi
cannot reasonably be taken as evidence for the classi-
cality of coherent states. So one may wonder whether
the same holds for h
ˆ
Hi. It is easy to see that
D
ˆ
H(ˆx, ˆp)
E
= H(hˆxi, hˆpi)
+
1
2m
(∆ˆp)
2
+
1
2
mω
2
(∆ˆx)
2
, (26)
where
H(x, p)
def
=
1
2m
p
2
+
mω
2
2
x
2
(27)
is the classical hamiltonian, function over the phase
space R
2
with coordinates (x, p). From Heisenberg
inequality one deduces then that
D
ˆ
H(ˆx, ˆp)
E
− H(hˆxi, hˆpi) ≥
~ω
2
. (28)
One can say in precise terms that a state is quasi-
classical with respect to the energy if it saturates this
Accepted in Quantum 2021-01-23, click title to verify. Published under CC-BY 4.0. 4
inequality. And happily, this condition alone is suffi-
cient to define the coherent states!
Conclusion.
We have cleaned up the properties of coherent
states underlined by Schrödinger in his seminal pa-
per. 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. Constant and minimal ∆ˆx
2. Minimal temporal average of ∆ˆx
3. Minimal
h
ˆ
H(ˆx, ˆp)i − H(hˆxi, hˆpi)
.
We regard these definitions as better suited for a ped-
agogical introduction to coherent states, compared to
the abstract definition as eigenstates of the annihila-
tion 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 co-
herent states, we are now going to relax this dynami-
cal aspect of the definitions and propose a purely kine-
matical characterisation.
4 Kinematical characterisation
Let’s start all over again, from a general quantum
system. Its states form a Hilbert space 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 geo-
metrical 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
H endowed with a scalar product h.|.i, and the alge-
bra of observables B
R
(H) consists of the (bounded)
self-adjoint linear operators over H. In fact, we only
consider the normalised vectors of H, up to a global
phase, so that the space of physical states really is
the projective Hilbert space P H. The geometric fea-
tures of this space enable a fair comparison to the
classical phase space. P H is a Khäler manifold which
means that it is naturally endowed with two geomet-
ric structures: a symplectic 2-form ω (coming from
the imaginary part of h.|.i) and a Riemannian met-
ric g (from the real part of h.|.i). Then, the algebra
of observables B
R
(H) can be recast as the space of
functions of C
∞
(P H, R) which preserve both geomet-
ric structures, i.e. whose hamiltonian vector fields are
also Killing vector fields.
Although it may look a bit abstract, this formula-
tion frames quantum mechanics in very similar terms
to classical mechanics, where the space of states is
a symplectic manifold (P, ω), and the observables
are functions of C
∞
(P, 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 formula-
tion 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 struc-
ture 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
ˆ
A ∈ B
R
(H), defines a
function A over P H through A : |ψi 7→ hAi, and thus
a hamiltonian vector field X
A
. One can prove that
∆
ˆ
A = g(X
A
, X
A
). (29)
In the classical case (g = 0), we have ∆
ˆ
A = 0 for
any observable
ˆ
A and state |ψi. Classical mechanics
is quantum mechanics without uncertainty.
One may wonder whether it was necessary to ap-
peal 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 geomet-
rical formulation brings clarity and precision to the
debate, and points towards a definite mathematical
direction where to look for classicality inside the quan-
tum realm.
The quest for ‘quasi-classical’ states can now be re-
formulated in the following terms. Are there states
|ψi for which ∆
ˆ
A = 0 for any observable
ˆ
A?
Eigenstates.
Start considering a single observable
ˆ
A. What are
the states that satisfy ∆
ˆ
A = 0? It is easily shown that
they are all, and only, the eigenstates |ai of
ˆ
A. Thus,
an eigenstate shows some classical features, which is
not a surprise after all: the eigenstate |ai of
ˆ
A is very
peaked with respect to
ˆ
A. Similarly, |xi is classical in
the sense that ∆ˆx = 0, i.e. it is very peaked with
respect to ˆx, 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 ∆
ˆ
A = 0 for all
ˆ
A, then |ψi is
an eigenstate of all
ˆ
A which is not possible.
Our expectations have to be qualified, and one can
look instead for states which satisfy ∆
ˆ
A = 0 for some
Accepted in Quantum 2021-01-23, click title to verify. Published under CC-BY 4.0. 5
observables
ˆ
A, i.e. a common eigenstate of a subset
A ⊂ B
R
(H). Such an eigenstate can be said ‘quasi-
classical’ with respect to A. If A is commutative, then
its common eigenstates form a basis of H. Such are
the eigenstates of a CSCO (complete set of commut-
ing observables) which may be regarded in this respect
as the most classical states of a given quantum sys-
tem. However, if 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
ˆ
A
and
ˆ
B. Generically, they do not share any common
eigenstates, so that we cannot have both ∆
ˆ
A = 0
and ∆
ˆ
B = 0. One has to find instead a fair trade-off
between ∆
ˆ
A and ∆
ˆ
B, so that they are both small, al-
though non zero. The trade-off is ruled by Heisenberg
inequality which reads
∆
ˆ
A∆
ˆ
B ≥
1
2
D
[
ˆ
A,
ˆ
B]
E
. (30)
One can show
7
that this inequality is saturated pre-
cisely when |ψi is an eigenstate either of
ˆ
A, or of
ˆ
B,
or of
ˆ
A + iγ
ˆ
B (31)
with γ ∈ R. The meaning of this γ is understood with
the following corollary:
γ =
∆
ˆ
A
∆
ˆ
B
(32)
It ponders the respective weight of
ˆ
A and
ˆ
B. The
eigenstates of
ˆ
A + iγ
ˆ
B are called the γ-squeezed co-
herent states with respect to
ˆ
A and
ˆ
B.
Application to ˆx and ˆp.
Let’s apply the result to
ˆ
A = ˆx and
ˆ
B = ˆp. Heisen-
berg inequality reads
∆ˆx∆ˆp ≥
~
2
. (33)
The normalised eigenstate of ˆx + iγ ˆp, with eigenvalue
z ∈ C, is
ψ
z,γ
(x) =
e
−
(Im z)
2
2γ~
4
√
πγ~
e
−
(x−z)
2
2γ~
(34)
The normalisation is only possible for γ > 0. Thus,
the squeezed coherent states (with respect to ˆx and
ˆp) form a 3-dimensional submanifold of P L
2
(R),
parametrised by γ and z.
The Schrödinger coherent states of equation (5) are
recovered by fixing γ =
1
mω
. More precisely, we have
x
Ae
iωt
= e
i
A
2
2
sin(2ωt)
ψ
z,γ
(x) (35)
7
See for instance [7] p. 244.
with
γ =
1
mω
and z =
r
2~
mω
Ae
iωt
. (36)
We have found an equivalent definition of Schrödinger
coherent states: they are the states which minimise
∆ˆx∆ˆp, with equal weight
8
for ˆx and ˆp.
Kinematics vs dynamics.
This new characterisation of coherent states differs
from the previous one of Schrödinger in a central as-
pect: it only refers to the kinematics, and not to the
dynamics. Indeed, the previous definitions were in-
volving the specific form of the hamiltonian
ˆ
H of the
harmonic oscillator, while now the definition only uses
two observables ˆx and ˆp acting on the Hilbert space
L
2
(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 pre-
vious 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 con-
siderations, but it will only exhibit nice dynamical
properties in a particular case. Indeed, the fam-
ily 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
ˆa
def
=
r
mω
2~
ˆx +
i
mω
ˆp
. (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 ˆa has arisen
through purely kinematical considerations. However,
the same operator plays an important role in the dy-
namics of the harmonic oscillator, where it is known
as the annihilation operator, for it acts destructively
over the eigenstates of
ˆ
H:
ˆa |ni =
√
n |n − 1i. (38)
It is important to keep in mind this double role of ˆa
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
8
I.e. ∆ˆp = mω∆ˆx. The constant mω guarantees the homo-
geneity of the physical dimension.
Accepted in Quantum 2021-01-23, click title to verify. Published under CC-BY 4.0. 6
definition of coherent states as eigenstates of the so-
called annihilation operator ˆa. 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 min-
imisation of Heisenberg inequalities. We are now go-
ing 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 pre-
ferred 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 electromag-
netic field is described by a state in a Hilbert space,
and the observable quantities are described by the
electric and magnetic hermitian operators,
ˆ
~
E(~r, t) and
ˆ
~
B(~r, t) (we use Heaviside-Lorentz units). In absence
of any sources, the time evolution of states is driven
by the hamiltonian
ˆ
H =
1
2
Z
d~r (
ˆ
~
E
2
+
ˆ
~
B
2
). (39)
In the time gauge, these observables can in fact be
derived from a vector potential
ˆ
~
A such that
ˆ
~
E = −
1
c
∂
ˆ
~
A
∂t
and
ˆ
~
B = ∇ ×
ˆ
~
A. (40)
Assuming the field is confined within a cubic box of
side L, the vector potential
ˆ
~
A can be decomposed into
a superposition of modes k such that
ˆ
~
A(~r, t) = c
X
k
~
2L
3
ω
k
1/2
(ˆa
k
~e
λ
e
i(
~
k·~r−ω
k
t)
+ ˆa
†
k
~e
λ
e
−i(
~
k·~r−ω
k
t)
), (41)
where the sum is made over an index k, used as a
shorthand for (λ,
~
k), where ~e
λ
(λ ∈ {1, 2}) is the po-
larisation vector, perpendicular to
~
k, and
~
k ranges
over a discrete set of values permitted by the bound-
ary conditions. Then ˆa
k
and ˆa
†
k
are operators asso-
ciated respectively to the positive and negative fre-
quency part of
ˆ
~
A. They satisfy
[ˆa
k
, ˆa
k
0
] = [ˆa
†
k
, ˆa
†
k
0
] = 0 and [ˆa
k
, ˆa
†
k
0
] = δ
kk
0
. (42)
The hamiltonian can be rewritten as
ˆ
H =
X
k
~ω
k
(ˆa
†
k
ˆa
k
+
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 ˆa
k
and ˆa
†
k
are properly annihilation and creation
operators. The basis of eigenstates of
ˆ
H is immedi-
ately deduced:
O
k
|n
k
i (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 descrip-
tion 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 consid-
ers interactions.
So far, we have described the free theory of the elec-
tromagnetic field. But light is created by sources, like
lamps or antennae, which consist of moving charges
that excite the electromagnetic field. It is thus cru-
cial to describe the interaction of light with charged
matter, and notably to model the photon field radi-
ated by a classical electric current. A classical current
~
j(~r, t) is assumed to interact with the vector potential
through the following hamiltonian of interaction:
ˆ
H
I
=
1
c
Z
~
j ·
ˆ
~
A d~r. (45)
Starting at initial time in the vaccum state |0i, the
field gets excited, and ends up at time t in a state
|ti = e
iφ(t)
e
i
~
R
t
0
ˆ
H
I
(t
0
)dt
0
|0i (46)
where the phase φ(t) admits a definite expression, but
irrelevant for our purposes. It can be rewritten
|ti = e
iφ(t)
O
k
|α
k
(t)i, (47)
where |α
k
(t)i is the coherent state with
α
k
(t) =
i
√
2L
3
~ω
k
Z
t
0
~
j ·~e
λ
e
−i(
~
k·~r+ω
k
t
0
)
d~rdt
0
. (48)
This hamiltonian of interaction is a good model for
most of the macroscopic sources where radiation is
generated by a charged current
~
j(~r, t) whose expres-
sion 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.
Accepted in Quantum 2021-01-23, click title to verify. Published under CC-BY 4.0. 7
Maximising interference.
To detect light, one uses a photon counter. Typi-
cally, 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 elec-
tromagnetic field is in a state |ψi, then the intensity
in x = (r, t) (the spacetime point where the detector
is) is on average
I(x) = hψ|
ˆ
E
(−)
(x)
ˆ
E
(+)
(x)|ψi (49)
where
ˆ
E
(−)
is the negative frequency part of the elec-
tric field, conjugate to the positive frequency part
ˆ
E
(+)
, which is (from equations (40) and (41))
ˆ
E
+
(~r, t) = i
X
k
~ω
k
2L
3
1/2
ˆa
k
e
i(
~
k·~r−ω
k
t)
. (50)
Equation (49) is a fancy way of writing that the en-
ergy of the electric field is |E|
2
.
Let’s define the first-order (or two-point) correla-
tion function as
G(x
1
, x
2
)
def
= hψ|
ˆ
E
(−)
(x
1
)
ˆ
E
(+)
(x
2
)|ψi. (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
:
ˆ
E(x) ∝
ˆ
E(x
1
) +
ˆ
E(x
2
). (52)
In this equation, x, x
1
and x
2
are related by the condi-
tion 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 Re 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 ob-
served. In fact, the visibility of the fringes is given
by
v
def
=
I
max
− I
min
I
max
+ I
min
=
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
≤ 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
)| =
p
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 E(x)
so that
G(x
1
, x
2
) = E
∗
(x
1
)E(x
2
). (57)
G(x
1
, x
2
) factorises, and the state |ψi 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 suf-
ficient condition for the factorisability of G(x
1
, x
2
) is
that |ψi is an eigenstate of
ˆ
E
(+)
(x) for all x. From
equation (50), we see that it is equivalent to say that
|ψi 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 as-
sembly of independent harmonic oscillators. Here the
definition is motivated on strong physical ground: the
maximisation of the inference pattern or say differ-
ently the factorisation of the 2-point correlation func-
tion!
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 ex-
perimentally measures the coincidence rate between
many detectors
9
.
Coherence, Classicality, and Purity.
At this stage, the origin of the word ‘coherent’ has
been brought to light: the state |ψi is such that the
values of the field at different points of space-time
‘conspire’ together to maximise the interference pat-
tern. Meanwhile, we have lost sight of the sense in
which they can be seen as ‘quasi-classical’. Even
worse, coherence and classicality may seem contra-
dictory. 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 lay-
ers 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 poten-
tial
ˆ
A and the electric field
ˆ
E play now respectively
the role of the position ˆx and the momentum ˆp. Co-
herent states of light are quasi-classical in the sense
that they minimise ∆
ˆ
A and ∆
ˆ
E together.
The second layer of classicality is the difference be-
tween pure and mixed states. Classical physics is usu-
ally very noisy, that is very mixed, due to the diffi-
culty 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
9
However, 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.
Accepted in Quantum 2021-01-23, click title to verify. Published under CC-BY 4.0. 8
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 ac-
tually 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
ρ =
1
2π
Z
2π
0
|α|e
iθ
|α|e
iθ
dθ, (58)
although it is optically coherent (it factorises the 2-
point correlation function). This distinction is some-
times 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 solv-
ing 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:
|α = 0i = |n = 0i. (59)
For this reason, there is no ambiguity and one can
write |0i
10
. In the previous section, we have seen how
the other coherent states are generated from the vac-
uum |0i by the unitary evolution of a simple hamilto-
nian of interaction
ˆ
H
I
. Equation (46) can be rewrit-
ten
|ti = e
iφ(t)
Y
k
ˆ
D
k
(α
k
(t)) |0i (60)
with α
k
(t) given by equation (48), and
ˆ
D a unitary
operator-valued function over C, called the displace-
ment operator, and defined by
ˆ
D(α)
def
= e
αˆa
†
−α
∗
ˆa
. (61)
It is easy to check indeed that
|αi =
ˆ
D(α) |0i. (62)
Heisenberg group.
The set of displacement operators almost form a
group:
ˆ
D(α + β) = e
−i Im(αβ
∗
)
ˆ
D(α)
ˆ
D(β) (63)
10
Whereas it would be ambiguous to write for instance |1i,
since |α = 1i = e
−1/2
P
∞
n=0
1
√
n!
|ni 6= |n = 1i.
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 ˆx and ˆp
generate a Lie algebra called the Heisenberg algebra
(or also the Weyl algebra). It is the smallest alge-
bra generated by ˆx and ˆp, with linear combination
and Lie bracketing i[., .]. It is a 3-dimensional non-
commutative real algebra, denoted h and consisting
of elements of the form
aˆx + bˆp + c
ˆ
I a, b, c ∈ R, (64)
where
ˆ
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(aˆx+bˆp+c
ˆ
I)
a, b, c ∈ R. (65)
The displacement operators are just a subset of this
group, such that
ˆ
D(α) = e
i
√
2 Im αˆx−i
√
2 Re αˆp
. (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 |0i 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 |αi, and not only |0i. 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 H, which was first proposed in paral-
lel by Perelomov [2, 12] and Gilmore [4].
Let G be a Lie group, and T a unitary irreducible
representation (irrep) of G over a Hilbert space H.
Choose |ψ
0
i ∈ H, and denote H the subgroup which
stabilises |ψ
0
i up to a phase, i.e.
H
def
=
g ∈ G | ∃φ ∈ R, T (g) |ψ
0
i = e
iφ
|ψ
0
i
.
(67)
The family of generalised coherent states is defined as
the orbit of |ψ
0
i under the action of the (left) quotient
space G/H. More precisely, for each class x ∈ G/H,
choose a representative g(x) ∈ G, and define the gen-
eralised coherent states as
|xi = T (g(x)) |ψ
0
i. (68)
Thus, the generalisation of coherent states depends a
priori on many choices: a group G, a unitary irrep
T , a vector |ψ
0
i and a set of representatives g(x). Of
Accepted in Quantum 2021-01-23, click title to verify. Published under CC-BY 4.0. 9
course, when it is projected down to the projective
Hilbert space P H, the set of coherent states does not
depend on the choice of the representatives g(x). The
choice of initial state |ψ
0
i 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 |αi, the
group is H
3
, with stabiliser U(1), the initial state
is the ground state |0i of the harmonic oscillator,
and the representatives are the displacement opera-
tors D(α). We see again the interplay between the
kinematical and dynamical sides of coherent states:
the dynamics lies in the choice of the ground state
|0i, and the full set of |αi 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 su(2), spanned by the (imaginary)
Pauli matrices (iσ
1
, iσ
2
, iσ
3
). In other words, any
u ∈ SU(2) can be written as
u = e
i~α·~σ
, α ∈ R
3
. (69)
The unitary irreps of SU(2) are Hilbert spaces H
j
,
labelled by a spin j ∈ N/2 and spanned by the mag-
netic basis |j, mi, m ∈ {−j, ..., j}, which diagonalises
J
3
and
~
J
2
(in physicists notations, J
i
def
=
σ
i
2
), such as
J
3
|j, mi = m |j, mi,
~
J
2
|j, mi = j(j + 1) |j, mi.
(70)
As an initial ‘vacuum’ state we choose
11
|j, −ji.
Then, we can show that the stabiliser is U (1), and we
have the following diffeomorphism SU(2)/U(1)
∼
=
S
2
.
The unit sphere S
2
can be parametrised by a complex
number ζ ∈ C (except for one point), by (the inverse
of) the stereographical projection
~n(ζ) =
1
1 + |ζ|
2
−ζ − ζ
∗
i(ζ
∗
− ζ)
1 − |ζ|
2
. (71)
The representative u ∈ SU(2) for each class ~n(ζ) ∈ S
2
is given by
u(ζ)
def
=
1
p
1 + |ζ|
2
1 ζ
−ζ
∗
1
. (72)
Finally we define the SU(2) coherent states as
|j, ~n(ζ)i
def
= u(ζ) |j, −ji. (73)
In terms of the magnetic basis, one can show that
|j, ~n(ζ)i =
1
(1 + |ζ|
2
)
j
j
X
m=−j
2j
j + m
1
2
ζ
j+m
|j, mi.
(74)
11
The choice |j, ji is often made too.
Among the important properties that these states
satisfy, we should note that the SU (2) coherent states
are eigenstates of ~n ·
~
J
~n ·
~
J |j, ~ni = −j |j, ~ni. (75)
Also they saturate the following Heisenberg inequality
∆J
1
∆J
2
≥
1
2
|hJ
3
i|. (76)
Finally, they satisfy the following resolution of the
identity
2j + 1
4π
Z
S
2
d~n |j, ~nihj, ~n| = 1, (77)
with d~n being the usual measure on the unit sphere
S
2
. The Bloch states have latter been used in quan-
tum 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 evolu-
tion of states. In quantum optics, one deals typically
with some effective model of perturbed hamiltonian:
ˆ
H =
ˆ
H
0
+
ˆ
H
pert
. (78)
The initial state is chosen to be the ground state of
ˆ
H
0
, and the coherent states are generated through
the time evolution induced by the perturbation
ˆ
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 (P, ω) and a hamiltonian H. Souriau
defines a dynamical group as any Lie group G acting
as a symplectomorphism (canonical transformation)
over P. Then one can consider the symmetry of the
hamiltonian, i.e. the functions C
i
∈ C
∞
(P, 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 symplec-
tomorphism and for that reason, it is sometimes em-
phasised 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
Accepted in Quantum 2021-01-23, click title to verify. Published under CC-BY 4.0. 10
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 |0i, 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 coher-
ent states, by Perelomov [12] and Gilmore [4], using
Lie groups. In this section, we explore a second and
independent path of generalisation in more geometri-
cal 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
∗
R, endowed with the usual symplectic structure
given by the determinant. The quantum analogue is
the Hilbert space L
2
(R), with the usual scalar prod-
uct. The family of coherent states constitute a 2-
dimensional submanifold of L
2
(R), parametrised by
amplitude and time (A, t), or the complex number
z = Ae
iωt
. It could be as well-parametrised by po-
sition x = A cos ωt and momentum p = −Aω sin ωt,
which exhibit an explicit diffeomorphism between the
classical phase space and the family of coherent states.
Any point in phase space determines uniquely a co-
herent state, and conversely. This fact is not a coin-
cidence, 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
P into the Hilbert space 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 H from a phase
space P. Well, this is the whole point of quantiza-
tion, and so one should know that the subject is not
easy. Nevertheless geometric quantization
12
is a pro-
totypical such method, rather technical, but we can
skip the details and keep the general idea.
Geometric quantisation.
Start with a configuration space M and build the
phase space P = T
∗
M. There are actually many ways
in which M can be seen as a subspace of T
∗
M. Each
way consists in choosing what is called a polarisation.
12
Not to be confused with the previously discussed geomet-
rical formulation of quantum mechanics [8].
Then one can build a complex line bundle over T
∗
M,
denoted L. Locally, we have L ≈ T
∗
M × C. The
prequantum Hilbert space, PH, is the space of equiva-
lence classes of square-integrable sections of L, where
two sections are said equivalent when they are equal
almost everywhere. Roughly, PH ≈ L
2
(T
∗
M). It is
much too big to be a good quantum Hilbert space. So
a choice of polarisation enables to select a subspace
of PH and to build the good quantum Hilbert space
H
∼
=
L
2
(M). This construction really enable to see
the Hilbert space H as a subspace of L
2
(T
∗
M). Thus
a state |ψi ∈ H can be seen as a complex function ψ
over T
∗
M.
Then, for each phase space point z ∈ T
∗
M, we
define the coherent state |zi as the unique state in H
such that
∀|ψi ∈ H, hz|ψi = ψ(z). (80)
This definition is both elegant and confusing. Ele-
gant, because the equation is very simple, confusing,
because it is too simple. On the LHS, we have a scalar
product between two states in H, while on the RHS
we have the evaluation of a function ψ in one point z
of the phase space T
∗
M. A practical example should
clarify the matter.
Segal-Bargmann transform.
The classical phase space of the harmonic oscil-
lator is T
∗
R
∼
=
C, so that the prequantum Hilbert
space is roughly L
2
(C). By using the Kähler polarisa-
tion, the quantum Hilbert space finally obtained is the
Segal-Bargmann space
13
[14], denoted SB. It is made
of functions over C which are both holomorphic and
square-integrable, with the (gaussian) scalar product
(f
1
, f
2
) =
Z
C
¯
f
1
(z)f
2
(z)
i
2π~
e
−|z|
2
/~
dz ∧ d¯z (81)
SB is isomorphic to L
2
(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
˜
φ(z) =
Z
R
K(z, x)φ(x)dx (82)
with the kernel
K(z, x) =
4
r
mω
π~
e
1
~
z
2
2
−(
√
mω
2
x−z)
2
(83)
This isomorphism is called the Segal-Bargmann trans-
form. Its inverse is given by
φ(x) =
Z
C
K(z, x)
˜
φ(z)
i
2π~
e
−|z|
2
/~
dz ∧ d¯z (84)
13
Also called the Fock-Bargmann space in [1].
Accepted in Quantum 2021-01-23, click title to verify. Published under CC-BY 4.0. 11
Equation (82) is in fact a concrete instantiation of
equation (80), so that the kernel K is actually a co-
herent state! More precisely, it matches the expres-
sion of equation (34), provided a rescaling of z, and
up to a phase and a normalisation factor:
K (z, x) = e
i
~
Re z Im z
e
1
~
|z|
2
ψ
√
2
mω
z,
1
mω
(x). (85)
For this reason the Segal-Bargmann transform is also
called the coherent-state transform.
In the case of Schrödinger coherent states, the alge-
braic approach (with the Heisenberg group H
3
) gen-
erates the same coherent states as the geometric ap-
proach (with the phase space T
∗
R). Both approaches
are secretly linked by the fact that the Heisenberg
group H
3
is naturally obtained by exponentiating the
quantised coordinates ˆx and ˆp of the phase space T
∗
R.
However, the two methods do not always match. In
the case of the phase space S
2
, the exponentiation of
the quantised coordinates σ
1
, σ
2
, σ
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
1 =
1
π
Z
C
|αihα| d Re α d Im α. (86)
This equation should be understood in the sense of
weak convergence, that is, for any two given states
|φ
1
i and |φ
2
i,
hφ
1
|φ
2
i =
1
π
Z
C
hφ
1
|αihα|φ
2
i d Re α d Im α. (87)
This resolution of the identity is similar to the more
familiar one of any orthonormal basis of H, like
1 =
X
n
|nihn|, (88)
up to the crucial difference that the coherent states
are parametrised by a continuous parameter α 7→ |αi.
The (strong) continuity of the map α 7→ |αi to-
gether with the (weak) resolution of identity are so im-
portant that they are often regarded as the two prop-
erties that coherent states should have to deserve such
a designation. It is a bit surprising because it seems
too generic
14
, 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 us-
ing the Khäler polarisation, the coherent states finally
14
In the same manner as the condition of being an orthonor-
mal basis is far from being a sufficient property to define the
|ni basis.
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 ρ
t
(x) is the solution of the heat
equation
dρ
dt
= ∂
2
x
ρ (89)
that satisfies ρ
0
(x) = δ(x). Its explicit expression is
15
ρ
t
(x) =
1
√
4πt
e
−
x
2
4t
(90)
The kernel K can be rewritten in term of the heat
kernel ρ
t
(x) such that
K
z
√
~, x
r
~
2mω
!
=
4
r
2mω
~
ρ
1
2
(x − z)
q
ρ
1
2
(x)
. (91)
This observation has suggested a construction of
coherent states when the configuration space is a con-
nected compact Lie group G, instead of R [14]. The
heat equation over G reads
dρ
dt
= ∆ρ, (92)
where ∆ is the Casimir operator, and ρ a function of
t ∈ R and g ∈ 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
ρ
t
(g) =
X
π
dim π e
−λ
π
t
χ
π
(g), (93)
where the sum is taken over all irreps π (up to uni-
tary equivalence), and λ
π
is the characteristic Casimir
(non-negative) number of the representation, and χ
π
is the character. For instance, in the case of SU (2)
one gets
ρ
o
t
(g) =
X
j∈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
C
of the Lie
group G. For instance, for SU(2) it is SL
2
(C). From
this he shows that there is a unique analytic continua-
tion of the heat kernel ρ
t
from G to G
C
. In analogy
16
with equation (91), the coherent states are defined as
the functions over G, indexed by x ∈ G
C
, as
K
x,t
(g)
def
= ρ
t
(x
−1
g), with g ∈ G. (95)
15
The heat equation is just Schrödinger equation with com-
plex time, so that the solution can be easily recovered from
(12). However, the solutions in (12) were discarded as coher-
ent states. It seems to be a pure coincidence that the same
equation with complex time now gives proper coherent states.
16
The 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.
Accepted in Quantum 2021-01-23, click title to verify. Published under CC-BY 4.0. 12
In [15], it is shown how the heat kernel construc-
tion with a group G matches the geometric quanti-
sation approach over the phase space T
∗
G. The lat-
ter approach shows how the coherent states provide a
natural embedding of the phase space within the cor-
responding Hilbert space, and in this sense, it points
towards their quasi-classical properties. The heat ker-
nel 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 exper-
imental control of the latter has motivated many the-
oretical developments, like the coherent states, which
can now be reinvested into the former. At least, hav-
ing 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 inter-
ference patterns and addressed different kind of ques-
tions, 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 gravi-
tational field, whereas quantum optics deals with the
electromagnetic field. The question here is to under-
stand how general relativity can be recovered, in some
limit, from quantum gravity, i.e. how a classical ge-
ometry 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 charac-
terisation 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 rele-
vant observables were the vector potential
ˆ
A and the
electric field
ˆ
E. General relativity can be formulated
in close analogy to electromagnetism and this sug-
gests to take as observables the so-called holonomy
and flux, which carry the geometrical meaning of cur-
vature 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 ker-
nel) approaches to coherent states have been applied
in LQG. Historically, first has come the heat kernel
method, starting with an extension of Hall’s construc-
Figure 1: Example of a finite directed graphs Γ, with 8 links
and 4 nodes.
tion 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
H
LQG
=
M
Γ
L
2
(SU (2)
L
)/G
Γ
, (96)
where the direct sum is made over all the different
abstract finite directed graphs Γ, with different num-
ber 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 func-
tions Ψ(g
l
1
, ..., g
l
L
) over SU(2)
L
. It is quotiented by
the so-called Gauss constraint G
Γ
, which amounts to
imposing the following invariance property, given any
element u
n
∈ SU (2) per node n of Γ,
Ψ(g
l
1
, ..., g
l
L
) = Ψ(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.
This description of quantum states of space might
seem very abstract and one needs to draw a connec-
tion to some intuitive understanding in terms of ge-
ometric quantities. To make it short, the quantum
space over a single link, L
2
(SU (2)), is the quanti-
sation of the phase space T
∗
SU (2). Actually, since
SU (2) is a Lie group, its cotangent bundle can be triv-
ialised as T
∗
SU (2)
∼
=
SU (2) × su(2), and the coordi-
nates (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
Accepted in Quantum 2021-01-23, click title to verify. Published under CC-BY 4.0. 13
the coherent states of LQG to be indexed by elements
of the complexification of SU(2), that is SL
2
(C).
Coherent spin-networks.
The rigorous construction of coherent states for
LQG was done in [22–25], where they are called ‘com-
plexifier 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 Γ. They
are denoted Ψ
(Γ,H
l
,τ
l
)
, parametrised by one element
H
l
∈ SL
2
(C) and one positive number τ
l
∈ R
+
per
link l of Γ. To each link l is associated the coher-
ent state ρ
o
τ
l
(g
l
H
−1
l
), with ρ
o
τ
(g) the analytic contin-
uation of (94) to SL
2
(C). All these terms are then
multiplied together, and finally projected it down to
H
LQG
by an SU(2) integration over each node to im-
pose the Gauss constraint. All this fits in the following
formula:
Ψ
(Γ,H
l
,τ
l
)
(g
l
)
def
=
Z
SU(2)
N
Y
l
ρ
o
τ
l
(u
t
l
g
l
u
−1
s
l
H
−1
l
)
!
[du
n
]. (98)
The formal jungle of techniques that helps to de-
fine these coherent states in [22–25] 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 quan-
tum Hilbert space. The embedding is parametrised by
H
l
∈ SL
2
(C)
∼
=
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
E
8πG~γ
τ
, (99)
with h ∈ SU(2) and E ∈ su(2) being respectively
the holonomy and the flux over a link, while γ ∈ R
is a constant, called the Immirzi parameter, and G
and ~ 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(ζ
s
) e
−izσ
3
u(ζ
t
) (100)
with ζ
s
, ζ
t
∈ C, and u(ζ) ∈ SU (2) given by equation
(72), and
z = ξ + i a τ (101)
with ξ, a ∈ R.
With this parametrisation, the state is peaked on
a discrete kind of geometry, called twisted geome-
try [29]. The phase space T
∗
SU (2) on each link is
parametrised by (~n(ζ
s
), ~n(ζ
t
), ξ, a), with the map ~n(ζ)
given by equation (71). Thus each node is surrounded
by a set of unit vectors ±~n
i
, one per attached link,
with the sign σ = ± chosen depending on whether
the link is ingoing or outgoing. At the semi-classical
level, the Gauss constraint imposes the closure of the
vectors ~n
i
surrounding a node, such as
X
l∈n
σ
l
a
l
~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 σ
l
~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 ξ can be used
to encode the extrinsic curvature on the common face,
when the polyhedra are embedded into 4-dimensional
spacetime
17
.
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 |j, ~ni (see equation
(73)), which are very familiar to quantum optics for
very different reasons.
The Hilbert space H
LQG
can be seen as a subspace
of some direct sum of the Hilbert spaces 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 Γ, the idea is to select one Bloch state
|j
l
, ~n
l
i per link l attached to n, to tensor them all and
to project them down by imposing the Gauss con-
straint. One obtains the so-called coherent intertwin-
ers |{j, ~n}i
n
[21]:
|{j, ~n}i
n
def
=
Z
SU(2)
O
l∈n
D
j
l
(g) |j
l
, ~n
l
i
!
dg. (103)
Over a graph Γ, the tensor product of coherent inter-
twiners (one per node n of the graph),
O
n∈Γ
|{j, ~n}i
n
, (104)
can be seen as a state
Ψ
Γ,j
l
,~n
l
∈ L
2
(SU (2)
L
)/G
Γ
. (105)
17
The issue is delicate, see [32].
Accepted in Quantum 2021-01-23, click title to verify. Published under CC-BY 4.0. 14
The semi-classical interpretation of Ψ
Γ,j
l
,~n
l
is the jux-
taposition 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 Ψ
(Γ,H
l
,τ
l
)
.
For large values of a, the two are related by
Ψ
(Γ,H
l
,τ
l
)
(g
l
)
∼
X
j
l
Y
l
(2j
l
+ 1)e
−τ
l
(j
l
−
a
l
2
+
1
2
)
2
e
−iγθ
l
j
l
!
× Ψ
Γ,j
l
,~n
l
(g
l
), (106)
(see [26] for details).
We have given a quick overview of the two main def-
initions 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 un-
der investigation like the U(N) coherent states [34],
the SO
∗
(2N) coherent states [35], or the coherent in-
tertwiners 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 co-
herent state of the free particle.
• We proposed the kinematical/dynamical di-
chotomy 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 liter-
ature in the use of the term of ‘dynamical sym-
metry group’.
• We summarised the genealogy of coherent states
from Schrödinger to quantum gravity, enlight-
ing 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 di-
rections (see [1] or [2] for more exhaustive treatment,
although quantum gravity is absent in them). Never-
theless, 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 de-
velopments are, to some extent, independent, but we
have shown how they mingle together to form a beau-
tiful 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, sug-
gestions, encouragements and proofreadings.
References
[1] Jean-Pierre Gazeau. Coherent States in Quan-
tum Physics. Wiley-VCH, 2009. ISBN 978-3-
527-40725-5. DOI: 10.1002/9783527628285.
[2] Askold Perelomov. Generalized Coherent States
and Their Applications. Springer-Verlag, Berlin
Heidelberg, 1986. ISBN 978-3-540-15912-4. DOI:
10.1007/978-3-642-61629-7.
[3] Erwin Schrödinger. Der stetige Übergang von
der Mikro- zur Makromechanik. Die Natur-
wissenschaften, 14(28):664–666, 1926. DOI:
10.1007/BF01507634.
[4] Robert Gilmore. On the properties of Coherent
States. Revista Mexicana de Física, 23:143–187,
1974.
[5] Da Hsuan Feng, Robert Gilmore, and Wei-Min
Zhang. Coherent states: Theory and some appli-
cations. Reviews of Modern Physics, 62(4):867,
1990. DOI: 10.1103/RevModPhys.62.867.
[6] Erwin Schrödinger. Collected Papers On Wave
Mechanics. Blackie & Son Limited, 1928.
[7] Brian C. Hall. Quantum Theory for Mathemati-
cians, volume 267 of Graduate Texts in Mathe-
matics. Springer New York, New York, NY, 2013.
DOI: 10.1007/978-1-4614-7116-5.
[8] Troy A. Schilling. Geometry of Quantum Me-
chanics. PhD thesis, The Pennsylvania State
University, 1996.
[9] Federico Zalamea. The Twofold Role of Ob-
servables in Classical and Quantum Kinemat-
ics. Foundations of Physics, 48(9):1061–1091,
November 2018. DOI: 10.1007/s10701-018-0194-
8. arXiv: 1711.06914.
[10] Roy J. Glauber. Quantum Theory of Optical Co-
herence. Wiley-VCH, Weinheim, 2007. ISBN
978-3-527-40687-6.
[11] Lautaro Amadei and Alejandro Perez. Hawk-
ing’s information puzzle: A solution realized in
loop quantum cosmology. November 2019. arXiv:
1911.00306.
Accepted in Quantum 2021-01-23, click title to verify. Published under CC-BY 4.0. 15
[12] A M Perelomov. Coherent States for Arbitrary
Lie Group. Commun. math. Phys., 26:222–236,
1972. DOI: 10.1007/BF01645091.
[13] Jean-Marie Souriau. Structure of Dynamical Sys-
tems. Birkhäuser Boston, Boston, MA, 1997.
ISBN 978-1-4612-6692-1. DOI: 10.1007/978-1-
4612-0281-3.
[14] Brian C. Hall. The Segal-Bargmann ‘coherent
state’ transform for compact lie groups. Jour-
nal of Functional Analysis, 122(1):103–151, 1994.
DOI: 10.1006/jfan.1994.1064.
[15] Brian C Hall. Geometric quantization and
the generalized Segal-Bargmann transform for
Lie groups of compact type. Communications
in Mathematical Physics, 226:233–268, 2002.
DOI: 10.1007/s002200200607. arXiv: quant-
ph/0012105.
[16] Irving E. Segal. Mathematical Problems of Rela-
tivistic Physics. In Mark Kac, editor, Lectures
in Applied Mathematics, volume II. American
Mathematical Society, 1960.
[17] V. Bargmann. On a Hilbert Space of An-
alytic Functions and an Associated Integral
Transform. Communications on Pure and Ap-
plied Mathematics, XIV:187–214, 1961. DOI:
10.1002/cpa.3160140303.
[18] Brian C. Hall and Jeffrey J. Mitchell. Co-
herent states on spheres. Journal of Mathe-
matical Physics, 43(3):1211–1236, 2002. DOI:
10.1063/1.1446664. arXiv: quant-ph/0109086.
[19] John R. Klauder and Bo-Sture Skagerstam. Co-
herent States - Applications in Physics and
Mathematical Physics. World Scientific, 1985.
DOI: 10.1142/0096.
[20] Abhay Ashtekar, Jerzy Lewandowski, Donald
Marolf, Jose Mourao, and Thomas Thiemann.
Coherent state transforms for spaces of connec-
tions. J. Funct. Anal., 135:519–551, 1996. DOI:
10.1006/jfan.1996.0018. arXiv: gr-qc/9412014.
[21] Etera R. Livine and Simone Speziale. New spin-
foam vertex for quantum gravity. Physical Re-
view D - Particles, Fields, Gravitation and Cos-
mology, D76(8):84028, 2007. DOI: 10.1103/Phys-
RevD.76.084028. arXiv: 0705.0674.
[22] Thomas Thiemann. Gauge Field Theory Coher-
ent States (GCS): I. General properties. Classical
and Quantum Gravity, 18(11):2025–2064, 2001.
DOI: 10.1088/0264-9381/18/11/304. arXiv: hep-
th/0005233.
[23] T. Thiemann and O. Winkler. Gauge Field
Theory Coherent States (GCS): III. Ehrenfest
Theorems. Class. Quantum Grav., 18(21):4629–
4681, November 2001. DOI: 10.1088/0264-
9381/18/21/315. arXiv: hep-th/0005234.
[24] T. Thiemann and O. Winkler. Gauge Field The-
ory Coherent States (GCS): IV. Infinite Tensor
Product and Thermodynamical Limit. Class.
Quantum Grav., 18(23):4997–5053, December
2001. DOI: 10.1088/0264-9381/18/23/302.
arXiv: hep-th/0005235.
[25] T. Thiemann and O. Winkler. Gauge Field The-
ory Coherent States (GCS): II. Peakedness Prop-
erties. Class. Quantum Grav., 18(14):2561–2636,
July 2001. DOI: 10.1088/0264-9381/18/14/301.
arXiv: hep-th/0005237.
[26] Eugenio Bianchi, Elena Magliaro, and Clau-
dio Perini. Coherent spin-networks. Phys.
Rev., D82:24012, 2010. DOI: 10.1103/Phys-
RevD.82.024012. arXiv: 0912.4054.
[27] Eugenio Bianchi, Elena Magliaro, and Clau-
dio Perini. Spinfoams in the holomorphic rep-
resentation. Phys. Rev., D82:124031, 2010.
DOI: 10.1103/PhysRevD.82.124031. arXiv:
1004.4550.
[28] Hanno Sahlmann, Thomas Thiemann, and
Oliver Winkler. Coherent states for canonical
quantum general relativity and the infinite ten-
sor product extension. Nucl. Phys., B606:401–
440, 2001. DOI: 10.1016/S0550-3213(01)00226-7.
arXiv: gr-qc/0102038.
[29] Laurent Freidel and Simone Speziale. Twisted
geometries: A geometric parametrization of
SU(2) phase space. Physical Review D - Par-
ticles, Fields, Gravitation and Cosmology, 82(8):
1–16, 2010. DOI: 10.1103/PhysRevD.82.084040.
arXiv: 1001.2748.
[30] H. Minkowski. Allgemeine Lehrsätze über die
konvexe Polyeder. Nachr. Ges. Wiss. Goettingen,
pages 198–219, 1897.
[31] Eugenio Bianchi, Pietro Donà, and Si-
mone Speziale. Polyhedra in loop quan-
tum gravity. Phys. Rev., D83:44035, 2011.
DOI: 10.1103/PhysRevD.83.044035. arXiv:
1009.3402.
[32] Fabio Anzà and Simone Speziale. A note
on the secondary simplicity constraints in loop
quantum gravity. 32(19):195015, October 2015.
DOI: 10.1088/0264-9381/32/19/195015. arXiv:
1409.0836.
[33] Carlo Rovelli and Francesca Vidotto. Covariant
Loop Quantum Gravity. Cambridge University
Press, 2014. DOI: 10.1017/CBO9781107706910.
[34] Laurent Freidel and Etera R. Livine. U(N) co-
herent states for loop quantum gravity. Jour-
nal of Mathematical Physics, 52(5), 2011. DOI:
10.1063/1.3587121. arXiv: 1005.2090.
[35] Florian Girelli and Giuseppe Sellaroli. SO*(2N)
coherent states for loop quantum gravity. Jour-
nal of Mathematical Physics, 58(7), 2017. DOI:
10.1063/1.4993223. arXiv: 1701.07519.
[36] Laurent Freidel and Jeff Hnybida. A Discrete
and Coherent Basis of Intertwiners. Class.
Quantum Grav., 31(1):015019, January 2014.
DOI: 10.1088/0264-9381/31/1/015019. arXiv:
1305.3326.
Accepted in Quantum 2021-01-23, click title to verify. Published under CC-BY 4.0. 16