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TO APPEAR IN PHYSICAL REVIEW D
Coherent States of Quantum Space-Time,
de Sitter and Black Holes
Diego J. Cirilo-Lombardo (a)and Norma G. Sanchez (b)
(a)M. V. Keldysh Institute of the Russian Academy of Sciences,
Federal Research Center-Institute of Applied Mathematics, Miusskaya sq. 4, 125047 Moscow,
Russian Federation and CONICET-Universidad de Buenos Aires, Departamento de Fisica,
Instituto de Fisica Interdisciplinaria y Aplicada (INFINA), Buenos Aires, Argentina.
(b)International School of Astrophysics Daniel Chalonge - Hector de Vega,
CNRS, INSU-Institut National des Sciences de l’Univers,
Sorbonne University, 75014 Paris, France.
(Dated: October 23, 2023)
Abstract: We provide a group theory approach to coherent states describing
quantum space-time and its properties. This provides a relativistic framework for
the metric of a Riemmanian space with bosonic and fermionic coordinates, its
continuum and discrete states, and a kind of ”quantum optics” for the space-time.
New results of this paper are: (i) The space-time is described as a physical coherent
state of the complete covering of the SL(2C) group, eg the Metaplectic group
Mp(n). (ii) (The discrete structure arises from its two irreducible: even (2n) and
odd (2n+ 1) representations, (n= 1,2,3... ), spanning the complete Hilbert
space H=Hodd ⊕ Heven. Such a global or complete covering guarantees the CPT
symmetry and unitarity. Large nyields the classical and continuum manifold,
as it must be. (iii) The coherent and squeezed states and Wigner functions
of quantum-space-time for black holes and de Sitter, and (iv) for the quantum
space-imaginary time (instantons), black holes in particular. They encompass
the semiclassical space-time behaviour plus high quantum phase oscillations, and
notably account for the classical- quantum gravity duality and trans-Planckian
domain. The Planck scale consistently corresponds to the coherent state eigenvalue
α= 0 (and to the n= 0 level in the discrete representation). It is remarkable the
3
CONTENTS
I. Introduction and Results 3
II. Quantum Coherent States 7
A. Momentum representation 9
B. Wigner function quasiprobability 10
III. Metaplectic Group M P (n), Algebraic interpretation of the Metric and the Square
Root Hamiltonian 12
A. Mp(2), SU(1,1) and Sp(2) 14
B. Geometrical Spinorial SL(2C) description of the Zitterbewegung 16
IV. Relativistic Wave Equation and the Complete Hilbert Space 17
A. Mp (2) - Coherent basic and bilinear states 19
B. The Mp(2) Squeezed Vacuum and Physical States 21
C. Discrete representation 24
V. Quantum Space-Time : de Sitter and Black Hole Coherent States 28
A. Quantum Space-Time 28
B. QST deS and BH. Minimal uncertainty and Mp(2) vacuum 30
C. Continuum and Discrete deS and BH Coherent states 32
VI. Imaginary Time. Coherent States of Quantum Gravitational Instantons 36
VII. Discussion 39
VIII. Remarks and Conclusions 41
References 44
I. INTRODUCTION AND RESULTS
Quantum space-time is a key concept both for quantum theory in its own and for a
quantum gravity theory. Coherent states are a fundamental part in quantum physics with
multiple theoretical and practical realizations from mathematical physics to quantum optics
4
and wave packet experiments, see for example Refs [1] to [7] and refs therein. In this
paper, within a group theory approach, we construct generalized coherent states to describe
quantum space-time.
We describe quantum space-time as arising from a mapping P(G, M) between the quan-
tum phase space manifold of a group Gand the real space-time manifold M. The metric
gab on the phase space group manifold determines the space-time metric of Mafter iden-
tification of one component of the momentum Poperator with the time T. The signature
of the metric depends on the compact or non compact nature of the group, but in the most
cases of physical interest, the real space-time signature and its hyperbolic structure require
non compact groups.
A group theory approach, a quantum algebra, reveals a key part in the quantum space-
time description in order to obtain the line element associated to a discrete quantum struc-
ture of the space-time. Such an emergent metric is obtained here from a Riemmanian phase
space and described as a physical coherent state of the underlying covering of the group
SL(2C): Interestingly, it appears necessary to consider the complete covering of the symplec-
tic group, that is the Metaplectic group Mp(n), its spectrum for all nleading in particular
for very large nthe continuum space-time. This approach allows us to construct here co-
herent states of the coset type for the quantum space-time and describe with them coherent
de Sitter and black hole states.
This quantum description is based on the phase space of a relativistic particle in the su-
perspace with bosonic and fermionic coordinates, allowing to conserve at the quantum level
the square root forms of the geometrical operators (eg the Hamiltonian or Lagrangian). The
discrete spacetime structure arises from the basic states of the Metaplectic representation
with one interesting feature to remark here: The decomposition of the SO(2,1) group into
two irreducible representations span even |2n⟩and odd |2n+ 1 ⟩states, (n= 1,2,3... ),
respectively, whose totality is covered by the Metapletic group. In the Metaplectic represen-
tation the general or complete states must be the sum of the two kind of states : even and
odd nstates spanning respectively the two Hilbert sectors H1/4and H3/4, whose complete
covering is H1/4⊕H3/4. This yields the relativistic quantum space-time metric with discrete
structure . For increasing number of levels n, the metric solution goes to the continuum and
to a classical manifold as it must be. Such a global or complete covering with the sum of
5
the two sectors, even and odd states to have the complete Hilbert space reflects the CPT
symmetry and unitarity of the description.
As we know, the Metaplectic group Mp(2) acts irreducibly on each of the subspaces H1/4,
H3/4(even and odd) by which the total Hilbert space (namely H) is divided according to
the Casimir operator:
K2=K2
3−K2
1−K2
2=k(k−1 ) = −3
16 I
giving precisely the values k= 1/4,3/4. Then,
H1/4= Span { | neven ⟩:n= 0,2,4,6, ... }(1.1)
H3/4= Span { | nodd ⟩:n= 1,3,5,7, .... }(1.2)
Based on the highest eigenvalue of the number operator T3|n⟩=−1
2n+1
2|n⟩
occurring in H ≡ H1/4⊕ H3/4, the two unitary irreducible representations (UIR) Dof
Mp(2) are denoted as:
(UIR) restricted to H1/4→ D1/4∈Mp(2) (1.3)
(UIR) restricted to H3/4→ D3/4∈Mp(2) (1.4)
One of the clear examples of the group theory approach presented here is the quantum
space-time derived from the phase space of the harmonic oscillator (Refs [8], [9], [10]), and
the mapping (X, P )→(X , T ), in the case of the inverted (imaginary frequency) oscillator,
or alternatively →(X, iT ), in the normal (real frequency) oscillator. The inverted oscillator
in its different representations does appear in a variety of interesting physical situations from
particle physics to black holes and modern cosmology as inflation and today dark energy, eg
Refs [14] to [26].
The group theory framework presented here to describe quantum space-time and its
coherent states allows to correlate and extend the approachs of Refs [8], [9], [10] and [27],
[29], [30] to obtain new results. Novel results of this paper are:
(i) the generalization of the quantum light-cone to include fermionic coordinates,
(ii) the construction of coherent and squeezed states of quantum space-time, their prop-
erties and interpretation, their continuum and discrete representations, and for both, de
Sitter and black hole space-times.
6
(iii) The coherent states for the quantum space-imaginary time instantons, for black
holes in particular.
(iv) We find that coherent states encompass the space-time behaviour in the semiclassical
and classical de Sitter and black hole regions, exhibit high quantum phase oscillations of the
space-time, and account for the classical-quantum gravity duality and the trans-Planckian
scales.
•It is remarkable the power of coherent states in describing both continuum and discrete
space-time, even in the Planckian and trans-Planckian domains:
•The Planck scale consistently corresponds here to the continuum coherent state eigen-
value α= 0, (and to the fundamental state n= 0 in the discrete representation).
Higher values of αin the quantum gravity (trans-Planckian) domain account for the
smaller and sub-Planckian sizes and higher excitations.
•One of the new features, for the space-imaginary time instantons is the emergence of
amaximum eigenvalue αcharacterizing the coherent states due to the minimal non-
zero quantum radius because of the minimal quantum uncertainty ∆ X∆T=/2,
in particular in the central and regular black hole quantum region. The coherent state
instanton remarkably accounts for this quantum gravity feature and determines the
radius being
R0(lP, tP)2=1
√π1
lP
+lP
,
lPbeing the Planck length. The origin is flurried or smoothed within this constant
and bounded curvature region.
•In the quantum space-time description, there is no any space -time singularity as it
must be. The consistent description by coherent states of such quantum scales does
appear here as a result of the classical-quantum gravity duality across the Planck
scale, and reflected here in the double covering of the SL(2C) group or Metaplectic
symmetry.
This paper is organized as follows: In Section (II) we construct the generalized or
coset group coherent states and squeezed states and the corresponding Wigner quasiprobality
functions. In Section (III), we describe the Mp(n) general group approach including bosonic
7
and fermionic coordinates, in particular Mp(2) and the geometrical interpretation of high
quantum oscillatory effects in this context. Section (IV) describes the Mp (n) associated
relativistic wave equation, the complete Hilbert space and the discrete representations. The
physical states, the Mp(2) squeezed vacuum and the direct sum of the both odd and even
states, necessary to uncover the complete space-time are discussed in this section. In Section
(V) we find the coherent states for the de Sitter and black hole space-times, their properties
and interpretation. Section (VI) deals with the coherent states of quantum (imaginary
time) instantons, for black holes in particular and its new effects. In Section (VII) provides
a discussion in the context of our results and other Refs, and Section VIII summarizes our
remarks and conclusions.
II. QUANTUM COHERENT STATES
We construct first coherent states within a group theory approach of the Klauder-
Perelomov type (Refs [2], [4]) or coset group coherent states and then in Sections and we
describe them in terms of the Metaplectic Mp(n) group and the associated relativistic wave
equation. For this purpose we define the coset generators as the generalized displacement
operators by means of the creation and annihilation operators aand a+. These operators
are analogous to those corresponding to quadratic Hamiltonians but the changement of sign
for the generalized coordinate introduces the imaginary frequency into the definition, (by
analogy to the generalized inverted oscillator), namely
a=iz
2|z|1/2√mω q +p
√mω ,(2.1)
a+=iz∗
2|z|1/2√mω q −p
√mω
Precisely, the change in the character of the frequency introduces the global phase factor
eiπ/4.
Consequently, the general displacement operator D(α) for any general complex parameter
αand ztakes the following form
D(α)S(z) = exp α a+−α∗a×exp 1
2za+2 −z∗a2= (2.2)
= exp "i
2|z|1/2√m ω d (α, z)q−p
√m ω d(α, z)#exp −i|z|
2(q p +p q ),
8
d(α, z)≡α√z∗−α∗√z(2.3)
The displacement operator D(α) is a unitary operator. In the coordinate representation
p=−i∂q,the displacement operator takes the form
D(α)S(z) = (2.4)
= exp "i
2|z|1/2√m ω d (α, z)q+i
√m ω d(α, z)d
dq #exp − |z|qd
dq +1
2,
(2.5)
The operator D(α) Eq. (2.4) acts on the vacuum of the inverted oscillator, namely
⟨q|0⟩inv−osc =imω
π1/4
exp −imω
2q2(2.6)
Therefore, we obtain the generalized coherent states with the following form:
ψα,z (q) = imω
π1/4
exp −1
2|z|+|α|2e−2|z|+e−|z|
|z|α2z∗cosh |z| − α∗2zsinh |z|·
(2.7)
exp "−imω
2q2+s2imω
|z|α√z∗cosh |z| − α∗√zsinh |z|e−|z|q#(2.8)
These states are squeezed because the quantum uncertainty in space and momentum coor-
dinates is not equally distributed in the both directions. In particular, we can test it putting
z= 0, and we obtain the coherent state for the inverted harmonic oscillator:
ψα(q)inv−osc =i m ω
π1/4
e−1
2|α|2exp −i m ω
2q2+rm ω
2( 1 + i)α q(2.9)
It is convenient to consider this type of coherent states as being based in a Lie group Gwith
a unitary, irreducible representation Tacting on some Hilbert space H. If we take a fixed
vector ψ0of H, we define the coherent state system {T, ψ0}to be the set of vectors ψ∈ H
such that ψ=T(g)ψ0for some g∈G. Then, generalized coherent states are defined as the
states |ψ⟩corresponding to these vectors in H.
We can see that in the definition of the coherent state of the Klauder-Perelomov type,
the general displacement operator contains, in the exponential representation of the coset,
9
a linear part in the annihilation and creation operators and another quadratic part corre-
sponding to the ”squeezed” sector, eg see Eq. (2.2): The latter belongs in this representation
to Mp (2). Therefore, at least for this purely squeezed part in the a2, a+2 representation the
complete vacuum state is
j+ka+⟨q|0⟩inv−osc
where (j, k) are constants determined by the normalization of states and the boundary
conditions. This is a consequence of the action of the Metaplectic group which increases the
spectrum of physical states: ∆n=±2, because the complete states are spanded by both:
the H1/4states (eg. even (2n) states ), and the H3/4states (eg odd (2n+ 1) states). Thus,
the lowest level (n= 0) is in H1/4, while in H3/4it is n= 1.
Consequently, being the complete vacuum under the action of an element of M p (2), Eq.
(2.6) would take as a wave function the precise form:
ψvacuum |Mp(2) →i m ω
π1/4
e−imω
2q21 + ei π/4rm ω
4q(2.10)
A. Momentum representation
Analogously to the case of the representation of coordinates, the generalized coherent
states are calculated in the same way but taking into account that in the moment represen-
tation premains the same but q=i∂pin all the operators, from those of annihilation and
creation, namely
ap=i z
2|z|1/2√mω i ∂p+p
√mω ,(2.11)
a+
p=i z∗
2|z|1/2−√mω i ∂p+p
√mω
as well as in the operators of displacement D(α)S(z) Eq.(2.4) acts on the vacuum of the
inverted oscillator in the momentum representation, namely
⟨p|0⟩inv−osc =i
πmω 1/4
exp −i p2
2mω (2.12)
10
Therefore, we obtain the generalized coherent states with the following form:
ψα,z (p) = i
πmω 1/4
exp −1
2|z|+|α|2e−2|z|+e−|z|
|z|α2z∗cosh |z| − α∗2zsinh |z|·
exp "−ip2
2mω +s2i
|z|mω α√z∗cosh |z| − α∗√zsinh |z|e−|z|p#
These states are squeezed because the quantum uncertainty in the space and momentum
coordinates is not equally distributed. In particular, we can test it putting z= 0, and we
obtain the coherent state for the inverted harmonic oscillator in the prepresentation:
ψα(p)inv−osc =i
πmω 1/4
e−1
2|α|2exp −i p2
2m ω +r1
2mω ( 1 + i)α p!
B. Wigner function quasiprobability
As we have made mention, the inverted Hamiltonian is formally obtainable from the stan-
dard harmonic oscillator by the change ω→ ± i ω and it corresponds to the Hamiltonian
of the harmonic oscillator with purely imaginary frequency. Therefore, this replacement
transforms the eigenfunctions of the harmonic oscillator into generalized eigenvectors of the
inverted harmonic oscillator, which, from the spectral point of view, leads us to a discrete
purely imaginary spectrum: Einv−osc =±i E h−osc. =iω(n+ 1/2 ). Notice that the
replacement ω→ ± i ω generates in the fiducial or fundamental states of the inverted
oscillator the following forms:
⟨q|0⟩inv−osc =i mω
π1/4
exp −i mω
2q2and
⟨q|0⟩inv−osc =−i mω
π1/4
exp i mω
2q2.
Consequently, in this approach and in order to have functions to be truly L2, one must
consider eφ∗
inv−osc.φinv−osc⟩to take the square norms, (e.g. a biorthonormalization condi-
tion). Note that according to these symmetries, both for the oscillator states and for the
coherent states obtained here, it is fulfilled:
eφ∗
inv−osc (q) = φ(q) for the inverted oscillator
11
e
ψ∗
α(q) = ψα(q) for coherent states
and this is a consequence of the underlying symmetry of M p(2) since the generator T1=
1
4(qp +pq) = i
4(a+2 −a2) is the one that produces the rotation or mapping on the states
of the harmonic oscillator, e.g.
φinv−osc (q) = e−π
2T1φHO (q)
eφinv−osc (q) = eπ
2T1φHO (q)
Taking this fact and symmetries into account, the Wigner function quasiprobability will
be defined as
W(q, p) = Zdv e−ip v
e
ψ∗
αq−v
2ψαq+v
2
Obtaining explicitly:
W(q, p) = exp −mω
q2+p2
mω −2rmω
α q +|α|2
Similarly, in the momentum representation, the Wigner function will be defined as
W(q, p) = Zdu e i q w
e
ψ∗
α,0p−u
2ψα,0p+u
2
Explicitly:
W(q, p) = exp −mω
q2−p2
mω −2rmω
α p +|α|2
We can also consider the Wigner function for the pure squeezed case, namely
ψz(q) = imω
π1/4
e−1
2|z|exp −imω
2q2(2.13)
Again, the Wigner function in this case will be defined as
Wsq (q, p) = Zdv e−ip v
e
ψ∗
zq−v
2ψzq+v
2
with the result:
Wsq (q, p) = e− |z|exp −mω
q2−p2
mω (2.14)
12
which shows a purely Gaussian behaviour without the α−shift tail.
As is it known the spectrum of the inverted oscillator gives rise to complex generalized
eigenvalues because from the point of view of the potential this is unbounded from below.
Here we saw that the fact of considering the mapping given by the T1generator of the group
Mp(2) allows to treat in equal footing the respective solutions of the inverted oscillator and
the standard harmonic one. It is worth mentioning that using quasi-hermiticity techniques
[31] the problem can be analogously solved by considering a scaling operator proportional
to T1.
On the other hand, solutions for the singular case (x-coordinates) lead to solutions of
the parabolic cylinder type. These solutions only reflect the symmetries of the metaplectic
vacuum, due that they are factorized, by means of hypergeometric type functions, in an
even part and an odd part corresponding precisely to the two irreducible sectors of Mp(2).
It is useful to remember that the generators of M p (2) are the following ones
T1=1
4(q p +p q ) = i
4a+2 −a2,(2.15)
T2=1
4p2−q2=−1
4a+2 +a2,
T3=−1
4p2+q2=−1
4a+a+a a+
With the following commutation relations,
[T3, T1] = i T2,[T3, T2] = −i T1,[T1, T2] = −i T3
being (q, p), alternatively (a, a+), the variables of the standard harmonic oscillator, as usual.
In the following Section we describe the generalized coherent states in terms of the Meta-
plectic group MP(n), its metric constructed in phase space and its associated relativistic
particle field equation.
III. METAPLECTIC GROUP MP (n), ALGEBRAIC INTERPRETATION OF
THE METRIC AND THE SQUARE ROOT HAMILTONIAN
One of the basis of the dynamical description is the Hamiltonian or Lagrangian of the
square root type, that is, a non-local and non-linear operator in principle. This is because
the invariance under reparametrizations as a Lagrangian and as an associated Hamiltonian,
13
generates the correct physical spectrum. The essential guidelines of our approach here are
based on the items specifically described in the sequel:
•(i) The elementary distance function (positive square root of the line element) is taken
as the fundamental geometric object of the space-time-matter structure, the geometric
Lagrangian (functional action) of the theory.
•(ii) From (i) the geometric Hamiltonian is obtained in the usual way : this will be the
fundamental classical-quantum operator.
•(iii) This universal Hamiltonian (square root Hamiltonian) contains a zero moment P0
characteristic of the complete phase space at the maximum level, from the point of
view of the physical states. The inclusion of a P0prevents the arbitrary nullification of
the Hamiltonian, a fact that occurs in the proper time system in which the evolution
coincides with the time coordinate: in this case time ”disappears” from the dynamic
equations.
The method that we use to preliminarily expand the phase space to determine later
here (via Hamilton equations) the physical role of P0is the Lanczos method [34], which
is the most geometrically consistent and mathematically simplest.
•(iv) The Hamiltonian, when rewritten in differential form, defines a new relativistic
wave equation of second order and degree 1/2. This can be reinterpreted as a Dirac-
Sudarshan type equation of positive energies and internal variables (e.g. oscillator
type variables), having a para-Bose or para-Fermi interpretation of the solution-states
of the system.
•(v) The spectrum will be formed by states that are bilinear in fundamental functions,
which in the case of Mp (2) are f1/4and f3/4having a spin weight s= 1/4 and 3/4
supported and connected by a vector representation of the generators of Mp (n), or
those covered by this, e.g. SU (p, n −p), SL(nR ), etc. A characteristic physical
state of Mp (2) is of the form Φµ=⟨s|Lµ|s′⟩with (s, s′= 1/4,3/4 ), and Lµ
being the vector representation of one of the generators of Mp (2).
14
A. Mp(2), SU(1,1) and Sp(2)
Following on the items (i) to (v) above, we use as a base the line element in a N=
1 superspace with differential forms. Consequently, we extend our manifold to include
fermionic coordinates. Geometrically, we take as starting point the functional action that
describes the world-line (measure on a superspace) of the superparticle as follows :
S=x, θ, θ =−mZτ2
τ1
dτ r◦
ωµ◦ωµ+γ.
θα.
θα−γ∗
.
θ
.
α.
θ.
α(3.1)
where ◦
ωµ=.
xµ−i(.
θ σµθ−θ σµ
.
θ), and the dot indicates derivative with respect to the
parameter τas usual; the complex constant γallows generality to characterize the states
and describing limiting cases
The above Lagrangian is constructed considering the line element (e.g.: the measure,
positive square root of the interval) of the non-degenerated supermetric introduced in [27]
ds2=ωµωµ+γ ωαωα−γ∗ω˙αω˙α,
where the bosonic term and the Majorana bispinor compose a superspace (1,3|1), with
coordinates (t, xi, θα,¯
θ˙α), and where the Cartan forms of the supersymmetry group are
described by: ωµ=dxµ−i(dθσµ¯
θ−θσµd¯
θ), ωα=dθα, ω ˙α=dθ ˙α(obeying
evident supertranslational invariance).
The generalized momenta from the geometric Lagrangian are computed in the usual way:
Pµ=∂L/∂xµ=m2/L ◦
ωµ(3.2)
Pα=∂L/ .
∂θα=iPµ(σµ)α.
βθ
.
β+m2γ/ L .
θα(3.3)
P.
α=∂L/
.
∂θ
.
α=iPµθα(σµ)α.
α−m2γ/L .
θ.
α(3.4)
We write them in a canonical form
Πα=Pα+iPµ(σµ)α.
βθ
.
β(3.5)
Π.
α=P.
α−iPµθα(σµ)α.
α(3.6)
(Pαand Pµbeing defined from the Lagrangian, as usual). Then, we start with the equation
(which will become the wave equation) :
S[Ψ] = HsΨ = sm2− P0P0−PiPi+1
γΠαΠα−1
γ∗Π.
αΠ.
αΨ (3.7)
15
As we have extended our manifold to include fermionic coordinates, it is natural to extend
also the concept of a point particle trajectory to the superspace. To do this, we take the
coordinates x(τ), θα(τ) and θ
.
α(τ) depending on the evolution parameter τ.
Consequently, there exist an algebraic interpretation of the pseudo-differential operator
(square root) in the case of an underlying Metaplectic group structure Mp (n) :
√F | Ψ⟩ ≡ sm2− P0P0−PiPi+1
γΠαΠα−1
γ∗Π.
αΠ.
α|Ψ⟩= 0 (3.8)
n[F]α
β(Ψ Lα)oΨβ≡(m2− P0P0−PiPi+1
γΠαΠα−1
γ∗Π.
αΠ.
αα
β
(ΨLα))Ψβ= 0
(3.9)
Then, both structures can be identified, e.g:
√F | Ψ⟩ ↔ n[F]α
β(Ψ Lα)oΨβ,(3.10)
being the state Ψ the square root of a spinor Φ (where the ”square root” Hamiltonian acts)
such that it can be bilinearly defined as Φ = Ψ LαΨ.
The operability of the pseudo-differential ”square root” Hamiltonian can be clearly in-
terpreted if it acts on the square root of the physical states. In the case of the Metaplectic
group, the square root of a spinor certainly exist [35], [36], [37], [38] making the identification
Eqs (3.8)- (3.9) fully consistent both: from the relativistic and group theoretical viewpoints.
Is also possible to describe a complete multiplet spanning spins from ( 0,1/2,1,3/2,2 ).
This is so because with the fundamental states and the allowed vectorial generators, the
tower of states is finite and the states involved are all physical, as it must be from the
physical viewpoint.
The choice of Eq.(3.1) as a functional action in superspace is justified because from
the point of view of symmetries, it contains the largest symmetry algebra of the harmonic
oscillator with 3 quadratic generators in aand a+(B0: even sector) and the two generators
in the B1: odd sector, describing the superalgebra Osp (1/2, R) with its 5 generators.
It is notable that in the general case, Sp(2m) can be embedded somehow in a larger alge-
bra as Sp(2m) + R2madmitting an Hermitian structure with respect to which it becomes
the orthosymplectic superalgebra Osp(2m, 1). Consequently the metaplectic representation
16
of Sp(2m) extends to an irreducible representation (IR) of Osp(2m, 1) which can be realized
in terms of the space Hof all holomorphic functions h:Cm→C/ R|h(z)|2e−|z|2dλ (z)<∞
with λ(z) the Lebesgue measure on Cm. The restriction of the M p (n) representation to
Sp(2m), implies that the two irreducible sectors are supported by the subspaces H±of H,
where H+and H−are the spans (closed) of the set of functions zn≡(zn1
1, ...., znm
m) with
nθ∈Z,|n|=Pnθ,even and odd respectively.
B. Geometrical Spinorial SL(2C) description of the Zitterbewegung
Let us briefly analyze in an algebraic description, the origin of the quantum relativistic
effects as the prolongated highly oscillations effect or so called ”Zitterbewegung”. There are
two types of states: the basic (non-observable) states and the observable physical states. The
basic states are coherent states corresponding to the double covering of the SL(2C), eg the
Metaplectic group [33], [38] responsible for projecting the symmetries of the 6 dimensional
Mp(4) group space to the 4 dimensional space-time by means of a bilinear combination of
the Mp(4) generators. The supermultiplet solution for the geometric lagrangian is given by
gab(0, λ) = ⟨ψλ(t)|Lab |ψλ(t)⟩
gab (0, λ) = exp [ A] exp [ ξϱ (t) ] χf⟨ψλ( 0 ) |( [c]c)ab |ψλ( 0 ) ⟩,
A(t) = −m
|γ|2
t2+c1t+c2,(c1, c2)∈C(3.11)
where we have written the corresponding indices for the simplest supermetric state so-
lution, being Lab the corresponding generators ∈M p (n), and χfcoming from the odd
generators of the big covering group of the symmetries of the specific model. Consider-
ing for simplicity the ‘square’ solution for the three compactified dimensions (spin λfixed,
ξ≡ −ξ
.
α−ξα), the exponential even fermionic part is given by:
ϱ(t)≡◦
ϕαα eiωt/2+β e−iωt/2−σ0α
.
αα eiωt/2−β e−iωt/2 (3.12)
+2i
ωσ0.
β
αZ.
β+σ0α
.
αZα(3.13)
◦
ϕα, Zα, Z .
βbeing constant spinors, and αand βC-numbers (the constant c1∈Cdue
to the obvious physical reasons and the chirality restoration of the superfield solution. By
consistency, (and as in the string case), two geometric-physical options are related to the
17
orientability of the superspace trajectory: α=±β. We take without loss of generality
α= +βthen, exactly, there are two possibilities:
(i) The compact case which is associated to the small mass limit (or |γ|>> 1) :
ϱ(t) =
◦
ϕαcos ( ω t/2 ) + 2
ωZα
−
◦
ϕ·
αsin ( ωt/2 ) −2
ωZ.
α
(3.14)
(ii) And the non-compact case, which can be associated to the imaginary frequency ( ω→
i ω generalized inverted oscillators) case:
ϱ(t) =
◦
ϕcosh ( ω t /2 ) + 2
ωZα
−
◦
ϕ·
αsinh ( ω t/2 ) −2
ωZ.
α
(3.15)
Obviously (in both cases), this solution represents a Majorana fermion where the C(or
hypercomplex) symmetry wherever the case) is inside the constant spinors.
The spinorial even part of the superfield solution in the exponent becomes:
ξ ϱ (t) = θα◦
ϕαcos (ωt/2) + 2
ωZα−θ
·
α−
◦
ϕ·
αsin (ωt/2) −2
ωZ.
α(3.16)
We easily see that in the above expression there appear a type of continuous oscillation
between the chiral and antichiral part of the bispinor ϱ(t), or Zitterbewegung as shown
qualitatively in Fig.(1) for suitable values of the group parameters. This oscillation reflects
in our context the underlying chiral and antichiral quantum structure of the spacetime.
Thus, the physical meaning of such a relativistic oscillation (Zitterbewegung) does appear
here as an underlying geometrical supersymmetric effect, namely a kind of duality between
supersymmetric and relativistic effects.
In the next Section we provide more details about how the quantum dynamics and space-
time structure emerge from this principle of symmetry.
IV. RELATIVISTIC WAVE EQUATION AND THE COMPLETE HILBERT
SPACE
The importance of illustrating with this model based on the simplest N = 1 supergroup is
that it has a formal equivalence with known cases containing the Poincare group, generally
18
FIG. 1. Oscillation between the chiral and antichiral part of the bispinor ϱ(t), or Zitterbewegung,
for suitable values of the group parameters. This oscillation reflects the underlying chiral and
antichiral quantum structure of the spacetime.
coming from symmetry breaking models with minimum group manifolds SO(1,4 ), SO(2,3)
as characteristic examples , SU SYN=1 ∼SO(1,4).Recalling the geometric Lagrangian
constructed from the line element from the Maurer Cartan forms induced via pullback (e.g.
nonlinear realization for example) of a fundamental symmetry group:
S=Zτ2
τ1
dτ L (x) = −mZτ2
τ1
dτ pωAB ωAB (4.1)
A, B = 0, ...., 5. The line element is based on the Cartan forms of the symmetry group, for
which it is induced and reflected in the geometric Lagrangian. Consequently, for SO(1,4),
for example, we have 10 that agree with the number of generators of the group, as it
must be, the indices of the forms run from 0 to 4. If by some process, the symmetry is
preferably dynamically broken, the Cartan forms from the point of view of the algebra,
are divided into the 6 generators of SO(1,3) plus 4 generators of the Cartan forms, namely
→ωAB →ωµν, ωµ4∼√λ θµ( Poincare - tetrad fields ), µ , ν = 0, ..., 3
S=Zτ2
τ1
dτ L (x) = −mZτ2
τ1
dτpωµν ωµν +ωµ4ωµ4
with ωµ4ωµ4=λ θµθµ. Following on the arguments given in the precedent paragraphs,
19
we are going to see how wave equations for physical states emerge from the very spacetime
structure.
A. Mp (2) - Coherent basic and bilinear states
Now we will demonstrate how the sector of the metaplectic group becomes determinant in
the problem of determining the geometric structure and symmetries of the interplay between
physical states and spacetime. To this end, we know from the so-called positive energy
equations, that these types of equations should emerge. We introduce the transformation
(evolution-type ansatz)
Φγ(t) = e[B(t) + pixi+ξ ϱ (t) ] Φγ( 0 )
Note that, in contrast to the case where only σ0=I2comes into play, here we include the
parameters piin order to generate the complete and less trivial matrix structure. Conse-
quently,
|γ|2∂2
0−∂2
i+∆+−∆−
4+m21/2
|Ψ⟩= 0 (4.2)
(|γ|2∂2
0−∂2
i+∆+−∆−
4+m2α
β|Φα⟩)1/2
= 0 (4.3)
where ∆±≡[∂η∓∂ξ±i σµ∂µ(η±ξ) ]2and we consider the equivalence at the level
of operators between the square root on the basic state of the metaplectic |Ψ⟩defined as
an independent coherent state in each even or odd irreducible sector, and the radicand on
the bilinear Φα=⟨Ψ|Lα|Ψ⟩written in the ket usual form : |Φα⟩=
a
a+
α
|Ψ⟩.
Consequently, the sector B0(Bose) generates the system
|γ|2··
B+·
B
2
−p2
i+m2= 0
where the function B is determined by
B= ln [ c2cos b(t) ] , b (t) = sm2
|γ|2−p2
i(t−t0)
It is important to notice that in the general case B= ln [ c2cos b(t) + c′
2sin b(t) ] we
take without losing generality c′
2= 0 because we concentrate on the Mp(2) part. It is easy
20
to see that if c′
2=ic2, the solution for Bis proportional to qm2
|γ|2−p2
i(t−t0) and also to
the Gaussian resolvent packet with the factor ( m2
|γ|2+p2
i) instead of just m2
|γ|2.
The sector B1(Fermi N = 1) gives us the equation
|γ|2ξ··
ρ+ 2 ·ρ·
B= 0
with a general solution of the form
ϱ(t) = 1
c2
2qm2
|γ|2−p2
i
◦
ϕααtan b(t)−βσ0α
.
αsec b(t)(4.4)
The two parts are not independent (in chiral and antichiral zones). Therefore, the equation
reduces finally to:
−ipz−·
B−ipx−py
−ipx+pyipz−·
B
α
β
|Φα⟩= 0 (4.5)
Knowing that: (i) |Φα⟩=Lα|Ψ⟩is the generator in vector representation based on
annihilation and creation operators, and that: (ii) It transforms as a spinor under the group
SO(1,2), SU(1,1) and Mp(2) (with the respective mappings between them), it is shown that
|Ψ⟩is the coherent state formed by two separate even and odd coherent states of the
considered metaplectic group. We explicitly have
−ipz−·
B−ipx−py
−ipx+pyipz−·
B
α
β
a
a+
α
|Ψ⟩= 0 (4.6)
which have exactly the same appearance as the equations of the type of internal variables
and positive energies of Majorana and Dirac for example. This is easily seen by introducing
the choice of parameters: pz=−i ϵ, px= 0, py=p:
ϵ+·
B p
−p−ϵ+·
B
α
β
a
a+
α
|Ψ⟩= 0 (4.7)
Notice that ·
Bwould take the formal role of ”mass” and the transformations are just of
the squeezed type.
21
B. The Mp(2) Squeezed Vacuum and Physical States
The displacement operator in the case of the vacuum squeezed is an element of Mp(2)
written in the respective variables of the canonical annihilation and creation operators.
S(ξ) = exp 1
2ξ∗a2−ξ a+2 ∈Mp (2 ) (4.8)
Seeing Eqs. (4.6) and (4.7) the relationship is shown directly:
a
a+
→S(ξ)
a
a+
S−1(ξ) =
λ µ
µ∗λ∗
a
a+
(4.9)
From Eqs (4.8) (4.9), we see that the dynamics of these ”square root” fields of Φγ, in the
particular representation that we are interested in, is determined by considering these fields
as coherent states in the sense that they are eigenstates of a2via the action of the Mp(2)
group that is of the type:
Ψ1/4(0, ξ, q)=
+∞
X
k=0
f2k(0, ξ)|2k⟩=
+∞
X
k=0
f2k( 0, ξ )a†2k
p( 2k) ! |0⟩(4.10)
Ψ3/4( 0, ξ, q)=
+∞
X
k=0
f2k+1 ( 0, ξ )|2k+ 1 ⟩=
+∞
X
k=0
f2k+1 (0, ξ)a†2k+1
p( 2k+ 1 ) ! |0⟩
For simplicity, we will take all normalization and fermionic dependence or possible
fermionic realization, into the functions f(ξ). Explicitly, at t= 0, the states are:
Ψ1/4( 0, ξ, q )=f(ξ)|α+⟩
Ψ3/4( 0, ξ, q )=f(ξ)|α−⟩(4.11)
where |α±⟩are the CS basic states in the subspaces λ=1
4and λ=3
4of the full Hilbert
space. In other words, the action of an element of Mp (2) keeps them invariant (coherent),
ensuring the irreducibility of such subspace, e.g :
H ∼
H1/4
H3/4
Consequently, the two symmetric and antisymmetric combinations (±) of the two sets of
states (1/4,3/4 ) will span all the Hilbert space: H:
|Ψ±⟩=Ψ1/4±Ψ3/4,|±⟩ =|+⟩ ± |−⟩ (4.12)
22
And the general bilinear states are of the type:
⟨±|Lα|∓⟩ and ⟨±|Lα|±⟩
where:
Lα=
α
α∗
,Lα=
a2
(a+)2
α
;Ψ1/4=|+⟩,Ψ3/4=|−⟩
For example, we have for the states with the explicit form:
Φα(t, λ) = ⟨Ψλ(t)|Lα|Ψλ(t)⟩=eA(t)eξ ϱ (t)⟨Ψλ(0) |
a2
(a+)2
α
|Ψλ(0) ⟩(4.13)
Φα(t, λ) = eA(t)eξ ϱ (t)|f(ξ)|2
α2
λ
α∗2
λ
α
(4.14)
λbeing the helicity label or the spanned subspace, e.g. (±), and A(t) is given by
A=−m
|γ|2
t2+c1t+c2; ( c1, c2)∈C(4.15)
The ”square root” solution takes the following form
Ψλ(t) = e1
2A(t)eξ ϱ(t)
2|f(ξ)|
α
α∗
λ
(4.16)
where λ= (1/4,3/4). Notice the difference with the case of the Heisenberg-Weyl realization
for the states Ψ :
|Ψ⟩=f(ξ)
2(|α+⟩+|α−⟩) = f(ξ)|α⟩(4.17)
where, the linear combination of the states |α+⟩and |α−⟩span now the full Hilbert space,
being for this CS basis λ=1
2. The ”square” states at t= 0 are
Φα(0) = ⟨Ψ (0) |Lα|Ψ (0) ⟩=f∗(ξ)f(ξ)
α
α∗
α
(4.18)
The square state and the obtained square root state at time tare:
Φγ(t) = eAeξϱ(t)|f(ξ)|2
α
α∗
α
,Ψ (t) = e1
2Aeξ ϱ (t)
2|f(ξ)|
α1/2
α∗1/2
(4.19)
Let us discuss the obtained results:
23
(i) We can see that the algebra, carrying the topological information of the group
manifold, is ”mapped” over the spinors solutions through the eigenvalues αand α∗from the
dynamical viewpoint. The constants in the exponential functions of the Gaussian type in
the solutions come from the action of a unitary operator over the respective coherent basic
states in each Irreducible representation.
(ii) The Osp (1/2,R) supergroup allows a metaplectic representation containing the
complete superalgebra in functions of a single complex variable zexactly coinciding with
the example treated here: it contains SU (1,1) as subgroup which can lead or explain the
fermionic factors of the type [ exp ( ξ ϱ (t)
2) ] × | f(ξ)|in the solutions.
(iii) The K±and K0generators operate over the Bose states (B0sector). The B1sector
of the algebra given by aand a+operates over the fermionic part. In this case, the coherent
and squeezed states that can be constructed are eigenstates of the displacement and squeezed
operators respectively (as in the standard case) but they cannot minimize simoultaneously
the dispersion of the quadratic Casimir operator, such that they are not minimum uncer-
tainty states. This is so because the only states which minimize the Schrodinger uncertainty
relation are those obtained by applying the displacement or squeezed operator on the lowest
normalized state.
(iv) Geometrically, in the description of any physical system through SU(1,1) coherent
states (CS) or squeezed states (SS), the orbits will appear as the intersections of constant-
energy surfaces with one sheet of a two sheeted hyperboloid - the curved phase space of
SU (1,1) or Lobachevsky plane - in the space of averaged algebra generators. The group
containing SU (1,1) as subgroup linear and bilinear functions of the algebra generators, can
factorize operators as the Hamiltonian or the Casimir operator (when averaged with respect
to the group CS or SS): this defines corresponding curves in the averaged algebra space. If
the exact dynamics is confined to the SU (1,1) hyperboloid, the validity of the Ehrenfest’s
theorem for the coherent or squeezed states implies that it necessarily coincides with the
variational motion that derives from the Euler- Lagrange equations for the Lagrangian
L=⟨z|ib
∂
∂t −b
H|z⟩,
that will be different if |z⟩=|α⟩or |z⟩=|α±⟩, as it is evident.
24
C. Discrete representation
Eq.(4.16) describes a standard coherent state (eigenstate of the operator (a) as a linear
combination of two states belonging to H1/4and H3/4respectively, (which are two inde-
pendent coherent states as eigenstates of (a2). The corresponding Metaplectic vacuum as
fiducial vector of the physical system is:
|z0⟩Mp(2) =M(1 + M2a+)|0⟩(4.20)
M ≡ [m2−ϵ2+p2sign ( ϵ2−m2) ]1/4(4.21)
Notice that this vacuum is not singular at m→ϵbut is analytically continued into the
complex plane where it is defined. Then, the solution for Eq.(4.7) is the following:
|Ψ⟩Mp(2) ≡S(t, A, p, ϵ )|z0⟩Mp(2) (4.22)
|Ψ⟩Mp(2) =1 + p2sign E
|E | 1/4
ep/2
(m+ϵ)(a+)2"1 + 1 + p2sign E
|E | 1/2
a+#|0⟩
E ≡ ϵ2−(·
A)2(4.23)
being S(t, A, p, ϵ )∈M p (2) the operator Eq.(4.8) for the set of parameters and functions
in Eq. (4.7). The total solution of the system Eqs.(4.2),(4.3) for these parameters being
G |Ψ⟩Mp(2) with G ≡ e(A+ξ ρ )( t, m, p, ϵ )e(p y −iϵz).
The Bargmann representation of Hassociates an entire analytic function f(z) of a com-
plex variable z, with each vector |φ⟩ ∈ H in the following manner:
|φ⟩ ∈ H → f(z) =
∞
X
n= 0 ⟨n|φ⟩zn
√n!(4.24)
⟨φ|φ⟩ ≡ ||φ||2=
∞
X
n= 0 |⟨n|φ⟩|2=Zd2z
πe−|z|2|f(z)|2(4.25)
where the integration is over the entire complex plane. The above association can be com-
pactly written in terms of the normalized coherent states. Consequently:
(i) The H1/4states occupy the sector even of the full Hilbert space Hand we describe
them as:
f(+) (z, ω) = 1− |ω|21/4eωz2/2=1− |ω|21/4X
n= 0,1,2,..
(ω/2)2n
(2n)! z2n(4.26)
25
Then, in the vector representation we have:
Ψ(+) (ω)=1− |ω|21/4X
n= 0,1,2,..
(ω/2)2n
p(2n) ! |2n⟩(4.27)
Consequently, the H1/4(or even) number representation is obtained as:
⟨2nΨ(+) (ω)=1− |ω|21/4(ω/2)2n
p(2n) ! ,⟨2n+ 1 Ψ(+) (ω)≡0 (4.28)
(ii) The H3/4states occupy the odd sector of the full Hilbert space Hand we similarly
describe them as for H1/4:
f(−)(z, ω) = 1− |ω|23/4z e ω z2/2=1− |ω|23/4X
n= 0,1,2,..
(ω/2)2n+1
(2n+ 1)! z2n+1 (4.29)
and the vector representation is:
Ψ(−)(ω)=1− |ω|23/4X
n= 0,1,2,..
(ω/2)2n+1
p(2n+ 1) ! |2n+ 1 ⟩(4.30)
The H3/4(or odd ) number representation is consequently:
⟨2n+ 1 Ψ(−)(ω)=1− |ω|23/4(ω/2)2n+1
p(2n+ 1) ! ,⟨2nΨ(−)(ω)≡0 (4.31)
(iii) The full Hilbert space, defined by the direct sum H=H1/4⊕H3/4,is the following:
f(z, ω) = f(+) (z, ω) + f(−)(z, ω) (4.32)
f(z, ω) = 1− |ω|21/4X
n= 0,1,2,..
(ω/2)2n
(2n)! z2n"1 + 1− |ω|21/2
(2n+ 1) z#(4.33)
Then, in complete analogy with their even and odd subspaces, the corresponding states are:
Ψ (ω) = Ψ(+) (ω)+Ψ(−)(ω) (4.34)
Ψ (ω) = 1− |ω|21/4X
n= 0,1,2,..
(ω/2)2n
p(2n)! "1 + 1− |ω|21/2
(2n+ 1) a+#|2n⟩(4.35)
⟨n|Ψ (ω)⟩=
1− |ω|21/4(ω/2)2n
(2n) ! √2n! even states
1− |ω|23/4(ω/2)2n+1
(2n+1) ! p(2n+ 1) ! odd states
(4.36)
26
FIG. 2. The H1/4discrete (number representation) states occupy the even sector of the full Hilbert
space H. This Irreducible representation of the Mp (2) group is not dense (in a topological sense)
but it contains the ground state |0⟩.
where the link between the physical observables and the group parameters is given by the
following expression (measure):
1 + p2sign ( ϵ2−m2)
|m2−ϵ2|1/4
→1− |ω|21/4(4.37)
Fig.(2) and Fig.(3) display the discrete spectra in the number representation of the co-
herent states in H1/4(even n) and H3/4(odd n).
27
FIG. 3. The H3/4discrete (number representation) states occupy the sector odd of the full Hilbert
space H. This Irreducible representation of Mp (2) is not dense (in a topological sense) but its
lower or fundamental state is the first excited state |1⟩.
The Limit ϵ→m:
This is precisely the limit |ω|2→1, which from the point of view of the Metaplectic
analysis corresponds to the edge of the complex disc. As we could easily see, the state
solutions span the full spectrum corresponding to H. What happens is that in the limit
ϵ→mthe density of states corresponding to H1/4is greater than that of the odd states
belonging to H3/4. It is for this reason that the states belonging to H1/4, will survive in this
limit.
28
V. QUANTUM SPACE-TIME : DE SITTER AND BLACK HOLE COHERENT
STATES
A. Quantum Space-Time
We restrict in the sequel to the purely bosonic space-time and consider the (X, T ) quan-
tum space and time dimensions which are relevant to the quantum space-time structure.
The remaining spatial transverse dimensions X⊥are not considered here as fully quantum
non-commuting coordinates. Notice that although the transverse spatial dimensions ⊥have
zero commutators they can fluctuate. This corresponds to quantize the two-dimensional
space-time surface which is relevant to determine the light-cone structure. This is enough
for considering the novel features arising in the global quantum space-time and the quantum
light cone.
The relevant quantum space-time (X, T ) structure is described essentially by a quantum
inverted oscillator type algebra with discrete hyperbolic levels (X2−T2)n= (2n+ 1), n =
0,1,2, .... The zero point energy (n= 0) being the Planck energy level. The truly quantum
gravity (trans-Planckian) vacuum in the quantum space time is delimitated by the four
quantum hyperbolae X2−T2=±1 (in Planck units) of the Planck scale (n= 0) level. This
is precisely a constant curvature de Sitter vacuum.
The de Sitter space-time can be described as a (inverted, ie with imaginary frequency)
harmonic oscillator, the oscillator constant and length being [9],[12]:
κosc =H2, H =r( 8π G Λ)
3=c / losc (5.1)
The oscillator length losc is classically the Hubble radius, the Hubble constant H=κ
being the surface gravity, as the black hole surface gravity is the inverse of (twice) the black
hole radius.
Interestingly, the description of de Sitter space-time as an (inverted, classical and quan-
tum) harmonic oscillator derives from three results:
(i) From the Einstein Equations on the one hand , [9], [10], [13], [14],
(ii) From the de Sitter geometrical description on the other hand: an hyperboloid em-
bedded in flat Minkowski space-time with one more spatial dimension :
−T2+X2+X2
i+Z2=L2
QG (5.2)
29
LQG = ( LQ+LG) = lPH
hP
+hP
H,(5.3)
LQG is the complete length allowing to describe both the classical, semiclassical and quantum
(trans-Planckian) gravity domains, lPthe constant Planck length:
LQ=l2
P/ LG, lP= ( 2 G/ c3)1/2, hP=c / lP(5.4)
(iii) From the hyperbolic quantum space-time structure which delimitates a purely quantum
trans-Planckian central region of constant curvature , [8], [9] [10]
In the Anti-de Sitter space-time, the description is the same but with −T2+X2+X2
i+
Z2=−L2
QG, and therefore Anti- de Sitter background is associated to a real frequency
(non inverted) harmonic oscillator. Also, the propagation of fields and linearized pertur-
bations in the de Sitter vacuum all satisfy equations which are like the inverted oscillator
equations, [15], [16], [40], or the normal oscillator equations in Anti de Sitter space-time.
In the (Schwarzschild) black hole space-time: (quantum interior constant curvature vac-
uum; semiclassical and classical exterior regions), the physical magnitudes as the oscillator
constant H2and the typical oscillator length losc are related to the black hole mass M:
H=c / losc =hPmP
M,Λ = λPmP
M2, λP= 3 h2
P/ c4(5.5)
Classical space-time regions or regimes are described by the low values of Λ and of the
gravitational density ρG, and the large classical gravitational sizes LG>> lP:
LG=lPrλP
Λ=lPM
mP(5.6)
Truly Quantum gravitational regimes, eg in the trans-Planckian domaine of very small sub-
Planckian sizes, very high quantum density ρQand very high vacuum values ΛQ:
LQ=lPrΛ
λP
=lPmP
M,ΛQ=λ2
P
Λ(5.7)
Consistently, the high value of the classical/semiclassical gravitational entropy SGis equal
(in Planck units) to such high ΛQvalue. This is clearly explicitated by the following classical-
quantum gravity duality relations in this context:
ρG
ρP
=lP
LG2
=mP
M2=SQ
sP(5.8)
30
ρQ
ρP
=lP
Λ=M
mP2
=SG
sP(5.9)
ρP= 3 h2
P/8πG, sP=πκB
The last r.h.s. of Eqs.(5.8)-(5.9) show the link to the gravitational entropy: quantum
gravitational entropy SQand classical/semiclassical SGentropy. (This last is the Bekenstein-
Hawking-Gibbons entropy [41] - [43]). Lower case magnitudes with subscript Pdenote the
corresponding Planck scale fundamental constant magnitudes.
The external BH region is precisely a classical gravity dilute vacuum, which (Λ, ρG)BH
values in the present universe cannot be larger than the observed very low values of (Λ, ρG)
Refs [17] to [25]. Their quantum duals provide an upper bound to the high values (ΛQ, ρQ)
in the quantum central BH vacuum region as determined by Eqs. (5.8)-(5.9)
B. QST deS and BH. Minimal uncertainty and Mp(2) vacuum
For the quantum space-time (QST) de Sitter states, the oscillator parameters entering
in the coherent states and their representations Section II are the following:
As discussed above, de Sitter space-time is described by an inverted oscillator with oscil-
lator length losc =c/H, and the generic quantum coherent states built in Section II, have
in particular the inverted oscillator length losc =p/mω.
The de Sitter quantum space-time coherent states are described by the states Eqs (2.2)
- (2.4), Eq.2.7) with the corresponding oscillator constant given by:
l−2
osc dS =mω
dS =H2=Λ
3(5.10)
In the (Schwarzschild) black hole space-time, the physical magnitudes as the oscillator
constant and the oscillator length are related to the black hole mass M:
l−2
osc BH =mω
BH =l−2
PmP
M2(5.11)
lP= ( 2 G/ c3)1/2, hP=c / lP
The complete length LQG in Eq.(5.2) covers both the classical, semiclassical and quantum
(trans- Planckian) gravity domains. Quantum space-time derives from the quantum non
commutative space and momentum (phase space) operators with the mapping of momentum
31
into time, Refs ( [8], [9], [10], [11]). As a consequence, quantum space-time described by
coherent states have minimal and equally distributed uncertainty: ∆X∆T=/2
( ∆X)2=
2m ω ,( ∆T)2=m ω
2(5.12)
Therefore, coherent states of quantum de Sitter space-time have the spatial and temporal
uncertainty:
( ∆X)2deS =
2m ω deS
=
2H2(5.13)
( ∆T)2
deS =H2
2(5.14)
And for the Black Hole coherent states, the quantum uncertainty in space and time is:
( ∆X)2
BH =
2m ω BH
=l2
P
2M
mP2
(5.15)
( ∆T)2
BH =t2
P
2mP
M2(5.16)
(lPand tPbeing the Planck length and time). The de Sitter and black hole coherent
states derive from the explicit expressions Eqs (2.2) - (2.4), Eq.(2.7) with the respective
(deS) and (BH) physical magnitudes given by Eqs (5.10) and Eqs (5.11). In particular, the
quantum metaplectic Mp ( 2 ) vacuum is given by:
ψvacuum |MP (2) deS =i
π1/4√H e−i
2(H X)2H X
2√2( 1 + i) + i H X
2√2(5.17)
ψvacuum |MP (2) BH =i
π1/4√2Ke−i
2( 2 KX)2KX
√2( 1 + i) + iKX
√2
where:
K= 1 /( 2 RBH ) = 1 /( 4 G M ) (5.18)
Both vacuum states are expressed in terms of the surface gravity (Hor K) respectively, or
similarly in terms of the de Sitter or BH radius. Both states are totally regular, as it must be
for quantum space-time. For X >> RBH ,(RBH being the BH radius), and asymptotically
for very large X, the quantum coherent state consistently accompasses the quantum space
classicalization, as such exterior BH regions are semiclassical and classical. We discuss below
the excited (α) states.
32
C. Continuum and Discrete deS and BH Coherent states
Quantum space-time de Sitter and black hole coherent states follow from Eqs (2.2) - (2.4)
and Eq (2.7) with the physical magnitudes and uncertainty relations Eqs (5.13 - (5.16). The
quantum space-time deS coherent states have the following expressions :
ψα(X)deS =i
π1/4√H e−1
2|α|2exp α H X
√2(1 + i)−i H2X2
2(5.19)
ψα(T)deS =i
π1/41
√He−1
2|α|2exp α T
√2H( 1 + i)−i T 2
22H2(5.20)
A similar coherent state expression holds for the BH space-time with the corresponding BH
factor 2 Kinstead of H, being Kthe surface gravity Eq. (5.18).
αis the complex constant number, eigenvalue of the displacement operator D(α), which
characterizes the coherent state excitations (displacement from the vacuum), and their con-
tinuum spectrum.
(i) The quantum space-time coherent states Eqs (5.19) - (5.20) clearly display an ex-
ponential de Sitter expansion term α T/(√2H) plus a phase (linear and quadratic) in
[( α T / (√2H)], which is simply T /( 2 √2π TH,THbeing the Hawking de Sitter Temper-
ature. The quantum space-time exhibits an accelerated expansion plus quantum oscillations
of the same sign (linear term), and of different sign ( quadratic term). The presence of these
oscillations is a new feature of quantum space-time.
(ii) The continuum (α)-coherent states Eqs.(5.19) - (5.20) describe semi-classical, (or
semi-quantum), space- time regimes and in agreement with the space-time described by
quantum oscillators. Quantum discrete space-time becames more and more continuous for
large nin agreement with its description by continuum coherent states. Consistently, the
continuum coherent states are characterized by the Hawking temperature which is a semi-
classical (or semiquantum) temperature.
(iii) We see that:
|Ψα(X)deS |2− |Ψα(T)deS |2=
33
=1
√πexp− | α|2Hexp [ c(α)H X ]−1
Hexp [ c(α)T
H](5.21)
c(α) = √2 ( Re α−Im α)
which reflects the quantum hyperbolic space-time structure. Let us define:
Rα(X, T )2≡ | Ψα(X)deS |2− | Ψα(T)deS |2
(i) For α= 0 :
R0(X, T )2=1
√πH−1
H(5.22)
which can be also expressed in terms of the quantum uncertainties:
R0(X, T )2=r2
π∆T−∆X
(5.23)
(ii) For:
Rα(X, T )2= 0 → |Ψα(X)deS |=± |Ψα(T)deS |(5.24)
Clearly, α= 0, corresponds to H= 1/√, that is the Planck scale. We see the power of
coherent states in describing space-time and even accounting for the Planck scale, at which:
∆X=/√2,∆T= 1 /√2 ( Planck scale) (5.25)
Obviously, for coherent states it satisfies ∆ X∆T=/2.
For the squeezed states, particularly interesting is the Wigner quasi probability func-
tion, which have here the following expression:
Wsq (X T )deS = exp −H2X2−T2
2H2 (5.26)
This clearly shows the hyperbolic structure of quantum space-time. The characteristic light-
cone structure is manifest here because there is no any α-deformation in this case.
Fig. 4 displays the space-time squeezed state Wigner function and its light- cone hyper-
bolic structure.
(iii) The discrete quantum space-time (Planckian and trans-Planckian) regimes are
described by discrete states, eg. the discrete coherent states of subsection (IV C). The
discrete spectrum of these states describes the different quantum space-time excitation levels,
the less excited (fundamental, n = 0) level corresponding to the Planck scale, (the crossing
34
FIG. 4. The squeezed quasi-probability Wigner function W(X, T ) of quantum space-time. W(X , T )
clearly shows the hyperbolic light-cone space-time structure and with symmetric form. For the
coherent states, Wα(X, T ) endowes the hyperbolic structure but with a linear αtail deformation
in Xor T.
or transition scale). Interestingly, as seen in Section (IV) and (IV C) the Metaplectic group
states with its both sectors and discrete representations, |2n⟩and |2n+ 1 ⟩,even and odd
states, fully cover the complete Hilbert space H
H=H(+) ⊕ H(−)(5.27)
The ( ±) symmetric and antisymmetric sum of the two kind (even and odd) states provides
the complete covering of the Hilbert space and of the space-time mapped from it:
Ψ (n) = Ψ(+) (2n) + Ψ(−)(2n+ 1) (5.28)
where Ψ(+) and Ψ(−)are obtained from Eqs (4.34) - (4.36). For de Sitter space, both sets
35
of states are given by:
Ψ(+) (2n)deS =1−H41/4(H2/2)2n
p(2n) ! (5.29)
Ψ(−)(2n+ 1)deS =1−H43/4(H2/2 )2n+1
p(2n+ 1) ! (5.30)
where we take into account that in the fully quantum trans-Planckian de Sitter phase [9]:
the quantum His H > 1 and thus the analytic covering in this phase. In addition, the
quantum discrete levels of Hare [9] : HQ n =√2n(even levels), and HQ n =√2n+ 1,
(odd levels), which leads :
Ψ(+) (2n)deS =1−4n21/4( 2n)2n
22np(2n) ! (5.31)
Ψ(−)(2n+ 1)deS = [ 4n(n+ 1) ]3/4( 2n+ 1 )2n+1
22n+1 p(2n+ 1) ! (5.32)
It is worth mentioning that independently of this Mp(n) coherent state framework, we
obtained in Refs [8], [9], [11], similar discrete levels in terms of the global cart X, or the local
ones xconstructed from the global (complete) classical - quantum duality including gravity
[11]. In such levels, the two kind of sectors and their global (±) covering do appear, which
reflects somekind of relation between the Mp(n) symmetry and classical-quantum duality:
Xn=√2n+ 1 ,or xn±= [ Xn±pX2
n−1 ], n = 0,1,2, .... (5.33)
The condition X2
n≥1 simply corresponds to the whole spectrum n≥0 :
xn±= [ √2n+ 1 ±√2n] (5.34)
xn=0 (+) = xn=0 (−) = 1 : Planck scale,
which complete all the levels. The (±) branches consistently reflect :
•The classical- quantum duality properties of the global space-time.
•The two p(2n+ 1) and √2n,even and odd (local) sectors. Each symmetric or an-
tisymmetric sum is necessary to cover the whole manifold. The corresponding ( ±)
global states are complete, CPT and unitary, the levels n= 0,1,2, ...., cover the whole
Hilbert space H=H(+) ⊕ H(−)and all space-time regimes.
•The total nstates range over all scales from the lowest excited levels to the highest
excited ones covering the two dual branches (+) and (−) or Hilbert space sectors.
36
VI. IMAGINARY TIME. COHERENT STATES OF QUANTUM
GRAVITATIONAL INSTANTONS
Taking imaginary time T=iT,t=iτ, yields to the elliptic (or circular) structure of
space-time and of the phase space, eg this corresponds in particular to the normal (non
inverted) oscillator description. That is to say, quantum space-imaginary time instantons
correspond to the real frequency quantum oscillators of phase space. They describe in
particular, quantum tunneling effects between different states or different vaccua, or different
phase (space) regions. Besides being saddle points in an euclidean quantum gravity path
integral, they can describe thermal features if the imaginary time endowes periodicity.
In the classical (non-quantum) BH space-time, the identification T=iT,t=i τ, trans-
forms the hyperbolic space-time structure into a circular structure: The classical horizon
X=±,Tcollapses to the origin X=±T = 0. In the classical (non-quantum) BH instan-
ton, the interior is cutted, no horizon, and no central curvature singularity, does appear: The
classical BH instanton is regular but not complete : The interior BH region is not covered
by the classical instanton.
In the complete quantum BH space-time, the quantum hyperbole ( X2−T2=l2
P)
replace the characteristic lines due to the non-zero [X, T ] conmutators, and in the cor-
responding quantum BH instanton the horizon does not collapse to the origin but to the
Planck scale circle ( X2+T2=l2
P). The complete quantum BH instanton includes the
usual classical/semiclassical BH instanton for radius larger than the Planck length, plus a
new central highly dense quantum core of Planck length radius and high constant and finite
curvature corresponding to the black-hole interior, Ref [10] which is absent in the classical
BH instanton.
Particularly interesting here is the Wigner quasiprobability function for the squeezed
states, which for the BH have the following expression:
Wsq (XT)BH = 2 exp −4K2X2+T2
42K2 (6.1)
where the BH oscillatory space-time parameters are expressed in terms of the BH surface
gravity KEq. (5.18). Wsq (XT) clearly shows the circular structure of the quantum space-
37
imaginary time instantons. The circular structure is manifest here without deformation
because the αtail present for the coherent states is absent in this case.
The coherent states for the quantum gravitational instanton, here we explicitate for the
BH, follow similar expressions as Eqs. (5.19), (5.20) but with the BH factor 2 K:
Ψα(X)BH =i
π1/4√2Ke−1
2|α|2exp h√2αKX( 1 + i)−2iK2X2i(6.2)
Ψα(T)BH =i
π1/4r1
2Ke−1
2|α|2exp −αT
2√2K( 1 −i) + iT2
82K2(6.3)
Therefore:
|Ψα(X)BH |2+|Ψα(T)BH |2=
=e− | α|2
√π2Kexp [ 2 c(α)KX] + 1
2Kexp [ c(α)T
2K](6.4)
c(α) = √2 ( Re α−Im α)
which reflects the quantum elliptic (circular) structure of the space-imaginary time instanton.
We define:
Rα(X, T)2≡ |Ψα(X)BH |2+|Ψα(T)B H |2(6.5)
We see that: For α= 0:
R0(X, T)2=1
√π2K+1
2K(6.6)
For (X, T)→0 :
Rα( 0 )2=e− | α|2R2
0(6.7)
The origin is flurred or erased within a quantum circular core of radius Rα( 0 ). This
confirms with a coherent state approach, the regular (non singular) quantum internal BH
region obtained in Ref [10] by using quantum Schwarschild-Kruskal coordinates.
At the Planck scale: ( X, T)→(lP, tP),