# Cosmological bounce from a deformed Heisenberg algebra

Marco Valerio Battisti<EMAIL_ADDRESS>Dipartimento di Fisica (G9) and ICRA
- International Center for Relativistic Astrophysics, Università di Roma
“Sapienza” P.le A. Moro 5, 00185 Rome, Italy

###### Abstract

The implications of a deformed Heisenberg algebra on the Friedmann-Robertson-
Walker cosmological models are investigated. We consider the Snyder non-
commutative space in which the translation group is undeformed and the
rotational invariance preserved. When this framework is implemented to one-
dimensional systems (which is this case) the modifications are uniquely fixed
up to a sign. A cosmological quantum bounce à la loop quantum cosmology is
then obtained. We also get the Randall-Sundrum braneworld scenario and this
way a Snyder-deformed quantum cosmology can be considered as a common
phenomenological description for both theories.

###### pacs:

98.80.Qc; 11.10.Nx; 04.20.Dw

##  I Introduction

The emergence of space-time singularities in Einstein general relativity
undoubtedly implies that such a classical description is valid only on
macroscopic scale HE . One of the most important example is the big-bang
singularity appearing in the standard model of cosmology. However, the
Friedmann dynamics is expected to be modified by quantum effects in the regime
of small scale factor and that such corrections naturally come out from a
quantum theory of gravity. Anyway, it is not yet clear which kind of
modifications may appear and this problem is somehow related to the one of
finding the right (phenomenological) description of the low energy limit of
quantum gravity. In particular, it is argued that this limit should contain
the notion of an other invariant, observer independent, scale (the Planck
scale) RovSmo  and this can be regarded as the main intuition of doubly
special relativity (DSR) AMS .

In this work the (deformed) dynamics of the Friedmann-Robertson-Walker (FRW)
Universe is analyzed in the context of the Snyder non-commutative space Sny .
The Snyder approach is relevant since it can be related to some of DSR models
Kov  and has some motivations from loop quantum gravity LO  as well as two-
time physics tt .

The only deformed commutator in the Snyder framework is the one between the
coordinates, i.e. the translation group is not modified and the rotational
symmetry is preserved. This way, infinitely many generalized uncertainty
relations, underlying deformed Heisenberg algebras, are predicted BatMel08 .
Assuming the FRW phase space as Snyder-deformed (in this case we deal with a
one-dimensional system), the scheme is almost uniquely fixed. A non-singular
bouncing cosmological evolution is then predicted. Notably, the modified
Friedmann equation resembles the one arising in loop quantum cosmology (LQC)
elqc . It is worth noting that, since the Snyder picture is fixed up to a
sign, also the Randall-Sundrum braneworld scenario Roy  is predicted by our
model. In other words, this deformed phase space can be considered, from a
phenomenological point of view, as an effective framework which is able to
describe the results obtained in more general theories. From this perspective,
the different predictions of such approaches can then be understood
considering the opposite sign in the deformation term.

Our model can be regarded as an important step toward the comparison between
deformed and loop quantum cosmology. In fact the quantum behavior of the FRW
Universe, using deformed Heisenberg algebras reproducing the string theory
uncertainty principle String , has been analyzed to describe the fate of the
big-bang singularity BM07 . However, in the previous work the classical
singularity appears to be probabilistically suppressed, but no evidences for a
big-bounce arise. (For a comparison between deformed and polymer-loop quantum
cosmology in the Taub Universe see BM08 .) On the other hand, we now deal with
a more general framework (the previous approach is recovered as a particular
case) and a cosmological bounce à la loop is then allowed.

The paper is organized as follows. In Section II the Snyder space and its
deformed quantum mechanics are analyzed. Section III is devoted to the study
of the modified FRW cosmological dynamics. Finally, in Section IV the wave
packets dynamics is investigated and a comparison with LQC showed. Concluding
remarks follow.

Over the paper we adopt units such that $\hbar=c=1$.

##  II Snyder Space and Deformed Quantum Mechanics

In this Section the modifications induced on the Heisenberg uncertainty
relations by a non-commutative Snyder geometry are described. The Hilbert
space representation and the deformed harmonic oscillator are also
investigated. A relation with the $\kappa$-deformed Poincaré algebra is
reviewed.

###  II.1 Snyder-deformed Heisenberg algebras

Let us start by considering a $n$-dimensional non-commutative (deformed)
Euclidean space such that the commutator between the coordinates has the non-
trivial structure ($\\{i,j,...\\}\in\\{1,...,n\\}$)

$[\tilde{q}_{i},\tilde{q}_{j}]=\alpha M_{ij}\,,$ (1)

where with $\tilde{q}_{i}$ we refer to the non-commutative coordinates and
$\alpha\in\mathbb{R}$ is the deformation parameter with dimension of a squared
length (for more details see BatMel08 ). We then demand that the rotation
generators $M_{ij}=-M_{ji}=i(q_{i}p_{j}-q_{j}p_{i})$ satisfy the ordinary
$SO(n)$ algebra and that the translation group is not deformed, i.e.
$[p_{i},p_{j}]=0$. We also assume that the rotational symmetry is preserved
(the non-commutative coordinates transform as vectors under $SO(n)$
rotations), i.e. the commutators

$\displaystyle[M_{ij},\tilde{q}_{k}]$ $\displaystyle=$
$\displaystyle\tilde{q}_{i}\delta_{jk}-\tilde{q}_{j}\delta_{ik},$ (2)
$\displaystyle[M_{ij},p_{k}]$ $\displaystyle=$ $\displaystyle
p_{i}\delta_{jk}-p_{j}\delta_{ik}$

hold. This way we deal with the (Euclidean) Snyder space Sny . The above
relations however do not uniquely fix the commutators between $\tilde{q}_{i}$
and $p_{j}$. More precisely, there are infinitely many of such commutators
which are all compatible (in the sense that the algebra closes in virtue of
the Jacobi identities) with the above natural requirements.

Such a feature can be understood considering a rescaling of the non-
commutative coordinates $\tilde{q}_{i}$ in terms of ordinary phase space
variables ($q_{i},p_{j}$) Mel . The most general $SO(n)$ covariant realization
for $\tilde{q}_{i}$ is given by

$\tilde{q}_{i}=q_{i}\varphi_{1}(\alpha
p^{2})+\alpha(q_{j}p_{j})p_{i}\varphi_{2}(\alpha p^{2}),$ (3)

where the convention $a_{i}b_{i}=\sum_{i}a_{i}b_{i}$ is adopted and
$\varphi_{1}$ and $\varphi_{2}$ are two finite functions. The boundary
condition we have to impose in order to recover the ordinary Heisenberg
algebra as $\alpha=0$, reads $\varphi_{1}(0)=1$. The rescaling (3) depends on
the adopted algebraic structure, but the two functions $\varphi_{1}$ and
$\varphi_{2}$ are not uniquely fixed. Given any function $\varphi_{1}$
satisfying the boundary condition $\varphi_{1}(0)=1$, the function
$\varphi_{2}$ is thus determined by the relation BatMel08
$\varphi_{2}=(1+2\dot{\varphi}_{1}\varphi_{1})/(\varphi_{1}-2\alpha
p^{2}\dot{\varphi}_{1})$ where $\dot{\varphi}_{1}=d\varphi_{1}/d(\alpha
p^{2})$. The commutator between $\tilde{q}_{i}$ and $p_{j}$ then arises from
the realization (3) and reads

$[\tilde{q}_{i},p_{j}]=i\left(\delta_{ij}\varphi_{1}+\alpha
p_{i}p_{j}\varphi_{2}\right).$ (4)

From this relation we obtain the generalized uncertainty principle underlying
the Snyder non-commutative space as

$\Delta\tilde{q}_{i}\Delta
p_{j}\geq\frac{1}{2}|\delta_{ij}\langle\varphi_{1}\rangle+\alpha\langle
p_{i}p_{j}\varphi_{2}\rangle|$ (5)

and the ordinary framework is recovered in the $\alpha\rightarrow 0$ limit.
Therefore, the deformation of the only commutator between the spatial
coordinates as in (1) leads to infinitely many realizations of the algebra,
and thus generalized uncertainty relations (5), all of them consistent with
the assumptions underlying the model. We also note that, unless
$\varphi_{2}=0$, compatible observables no longer exist.

The most interesting feature to be stressed is that, for one-dimensional
systems, this picture is almost uniquely fixed. In this case the symmetry
group is trivial ($SO(1)=\text{Id}$) and the most general realization is given
by $\tilde{q}=q\varphi(\alpha p^{2})=q\sqrt{1-\alpha p^{2}}$. The commutation
relation (4) reduces to

$[\tilde{q},p]=i\sqrt{1-\alpha p^{2}}$ (6)

and the only freedom relies on the sign of the deformation parameter $\alpha$.
It is worth noting that, when $\alpha>0$ a natural cut-off on the momentum
arises, i.e. $|p|<\sqrt{1/\alpha}$, while as $\alpha<0$ the uncertainty
relation (5) predicts a minimal observable length
$\Delta{\tilde{q}}_{\text{min}}=\sqrt{-\alpha}/2$. In fact, equation (5)
becomes

$\Delta\tilde{q}\Delta p\geq\frac{1}{2}|\langle\sqrt{1-\alpha p^{2}}\rangle|,$
(7)

from which the minimal uncertainty in $\tilde{q}$, if $\alpha<0$, is obtained.
Moreover, at the first order in $\alpha$, the string theory result String
$\Delta q\gtrsim(1/\Delta p+l_{s}^{2}\Delta p)$, in which the string length
$l_{s}$ can be identified with $\sqrt{-\alpha/2}$, is recovered. On the other
hand, if $\alpha>0$ a vanishing uncertainty in the non-commutative coordinate
is allowed and appears as soon as $\Delta p$ reaches the critical value of
$(\Delta p)^{\star}=\sqrt{(1-\alpha\langle p\rangle)/\alpha}$. We can then
conclude that, a maximum momentum or a minimal length are predicted by the
Snyder-deformed relation (6) if $\alpha>0$ or $\alpha<0$, respectively.

###  II.2 Hilbert space representation

The Hilbert space representation of the deformed Heisenberg algebra (6) is
constructed. Such a relation can be represented in the momentum space, where
the $p$ and $\tilde{q}$ operators act on the wave function $\psi(p)=\langle
p|\psi\rangle$ as

$p\,\psi(p)=p\,\psi(p),\qquad\tilde{q}\,\psi(p)=i\sqrt{1-\alpha
p^{2}}\,\partial_{p}\psi(p),$ (8)

on a dense domain $D$ of smooth functions. Further, the self-adjoint
requirement of the position and momentum operators implies a modified measure
in the Hilbert space. In fact, $p$ and $\tilde{q}$ are self-adjoint operators
in the domain $D$ with respect to the scalar product

$\langle\psi|\phi\rangle_{\pm}=\int_{I(\mathbb{R})}\frac{dp}{\sqrt{1-\alpha
p^{2}}}\psi^{\ast}(p)\phi(p),$ (9)

where $I=\left(-1/\sqrt{\alpha}\,,1/\sqrt{\alpha}\right)$ and $\pm$ indicates
the cases $\alpha>0$ and $\alpha<0$ respectively. Here the factor $(1-\alpha
p^{2})^{-1/2}$ in the measure is necessary in order to cancel the
corresponding term of the operator representation of $\tilde{q}$. This way,
the identity operators can be immediately obtained and the scalar product
between momentum eigenstates appears modified as $\langle
p|p^{\prime}\rangle=\sqrt{1-\alpha p^{2}}\delta(p-p^{\prime})$. The deformed
Hilbert spaces $\mathcal{F}_{\pm}$, which are the Cauchy completions with
respect to the inner product (9), can be written as

$\mathcal{F}_{\pm}=L^{2}\left(I(\mathbb{R}),dp/\sqrt{1-\alpha p^{2}}\right).$
(10)

We note that these Hilbert spaces are unitarily inequivalent to each other and
also with respect to the ordinary one $L^{2}(\mathbb{R},dp)$ which appears as
$\alpha\rightarrow 0$. This is not surprising since the deformation of the
canonical commutation relations can be viewed, from the realization (3), as an
algebra homomorphism which is a non-canonical transformation. In particular,
it cannot be implemented at the quantum level as an unitary transformation.
New features are then introduced at both classical and quantum level.

Let us now investigate how the position eigenvectors are modified in this
framework. They satisfy, in the momentum space, the differential equation
$i\sqrt{1-\alpha p^{2}}\,\partial_{p}\psi_{k}(p)=k\psi_{k}(p)$ and explicitly
read

$\psi^{(+)}_{k}(p)=c\exp\left(-i\frac{k}{\sqrt{\alpha}}\sin^{-1}(\sqrt{\alpha}p)\right),\qquad\psi^{(-)}_{k}(p)=c\exp\left(-i\frac{k}{\sqrt{-\alpha}}\sinh^{-1}(\sqrt{-\alpha}p)\right)$
(11)

$c$ being the normalization constant. These states generalize the plane waves
and, as in the ordinary quantum mechanics, they are not normalizable. The
$(+)$ and $(-)$ eigenstates can be obtained each other simply sending
$\alpha\rightarrow-\alpha$, but an important difference between these states
has to be stressed. As we have seen, if $\alpha<0$ a finite minimal
uncertainty in position $\Delta{\tilde{q}}_{\text{min}}>0$ is predicted. This
feature implies that there cannot be any physical state which is a position
eigenstate since an eigenstate of an observable necessarily has vanishing
uncertainty on it Kem . To be more precise, in the ordinary quantum mechanics
a sequence $|\psi_{n}\rangle\in D$ with position uncertainties decreasing to
zero, exists. On the other hand, in presence of a minimal uncertainty, it is
not longer possible to find some $|\psi_{n}\rangle\in D$ such that

$\lim_{n\rightarrow\infty}\left(\Delta\tilde{q}\right)_{|\psi_{n}\rangle}=\lim_{n\rightarrow\infty}\langle\psi|(\tilde{q}-\langle\psi|\tilde{q}|\psi\rangle)^{2}|\psi\rangle=0.$
(12)

Thus, although it is possible to construct position eigenvectors, they are
only formal eigenvectors and not physical states. In this case we have lost
direct information on the position itself (it can be recovered by an analysis
like the one in Kem ). On the other hand, when $\alpha>0$ the position
eigenvectors $\psi^{(+)}_{k}(p)$ in (11) are “proper physical states” in the
sense of the standard quantum theory (a zero uncertainty in position is
allowed), but two remarks are in order. (i) They are generally no longer
orthogonal111The position operator is no longer essentially self-adjoint but
has a one-parameter family of self-adjoint extensions. Considering
$k=\sqrt{\alpha}(2n+\lambda)$ $(n\in\mathbb{Z},\,\lambda\in(-1,1))$, a one-
parameter family of eigenvectors $\psi^{(+)}_{k}$ which explicitly
diagonalizes the position operator is constructed and a lattice spacing
$2\sqrt{\alpha}$ is then introduced.. In fact, the scalar product of these
eigenvectors appear to be
$\langle\psi^{(+)}_{k^{\prime}}|\psi^{(+)}_{k}\rangle_{-}=2c^{2}\sqrt{\alpha}(\pi(k-k^{\prime}))^{-1}\sin\left(\pi(k-k^{\prime})/2\sqrt{\alpha}\right)$,
instead of the Dirac $\delta$-distribution. (ii) They have a finite energy,
namely the mean value of $p^{2}$ between such states is finite, i.e.
$\langle\psi^{(+)}_{k}|p^{2}|\psi^{(+)}_{k}\rangle_{-}=c^{2}\pi/2\alpha^{3/2}$.
Of course, when the limit $\alpha\rightarrow 0$ is taken into account, the
usual results of Heisenberg quantum mechanics are recovered.

###  II.3 The harmonic oscillator

We now investigate a direct physical prediction of the framework discussed
above. The harmonic oscillator is surely one of the most relevant mechanical
systems for testing any quantization scheme and therefore we apply such a
formalism to this model, focusing on the modifications on the energy spectrum
induced by the deformation parameter $\alpha$. Considering the classical
Hamiltonian with a Snyder-deformed quadratic potential

$\mathcal{H}=\frac{p^{2}}{2m}+\frac{1}{2}m\omega^{2}\tilde{q}^{2}$ (13)

and the representation for $p$ and $\tilde{q}$ as reported in (8), we
immediately get the deformed stationary Schrödinger equation for the model

$\psi^{\prime\prime}(p)-\frac{\alpha p}{1-\alpha
p^{2}}\psi^{\prime}(p)+\frac{1}{1-\alpha
p^{2}}\left(\epsilon-d^{4}p^{2}\right)\psi(p)=0,$ (14)

where the prime denotes differentiation with respect to $p$ and we have
defined $\epsilon=2E/m\omega^{2}$ and the characteristic length scale
$d=1/\sqrt{m\omega}$. We note that if $\alpha>0$, i.e. $p\in I$, no
singularities appear in (14). This equation is well-known in mathematics as
the so-called Mathieu equation in the algebraic form AS . Its solution can be
explicitly written in terms of the Mathieu cosine $\mathcal{C}$ and sine
$\mathcal{S}$ as

$\psi(p)=A\mathcal{C}\left(\nu,q,\cos^{-1}(\sqrt{-\alpha}p)\right)+B\mathcal{S}\left(\nu,q,\cos^{-1}(\sqrt{-\alpha}p)\right),$
(15)

where $A,B$ are integration constants and

$\nu=\frac{-d^{4}-2\alpha\epsilon}{2\alpha^{2}},\qquad
q=\frac{d^{4}}{4\alpha^{2}}.$ (16)

Modifications on the ordinary energy eigenvalues $E=E_{n}$, induced by the
deformed algebra, can be easily obtained considering an asymptotic formula for
the $\nu$ coefficients. In fact, we are interested only at first-order
corrections to the spectrum and these appear for $\alpha\rightarrow 0$, or
more precisely when the scale $d^{2}$ of the harmonic oscillator is much
bigger then the deformation scale $\alpha$, i.e. when $\alpha/d^{2}\ll 1$. In
other words, we are interested when $q\gg 1$ and in this case $\nu$ can be
expanded as AS

$\nu=\nu_{n}=-2q+2\sqrt{q}(2n+1)-\frac{2n^{2}+2n+1}{4}+\mathcal{O}\left(\frac{1}{\sqrt{q}}\right).$
(17)

This way, considering the definitions (16), the deformed energy spectrum reads

$E_{n}=\frac{\omega}{2}(2n+1)-\frac{\omega}{8}(2n^{2}+2n+1)\left(\frac{\alpha}{d^{2}}\right)+\mathcal{O}\left(\frac{\alpha^{2}}{d^{4}}\right)$
(18)

and, as expected, for $\alpha/d^{2}\rightarrow 0$ the ordinary eigenvalues are
recovered. Let us discuss such a result. The spectrum in the $\alpha>0$ case
is nothing but the one obtained in the polymer quantum mechanics pol . This
framework relies on a non-standard representation of the canonical commutation
relations (inspired by loop quantum gravity LQG ) which is (unitarily)
inequivalent to the Heisenberg one. A notable fact is that such type of
quantization when applied to the minisuperspace models, leads to LQC LQC .
Differently, the spectrum (18) for $\alpha<0$ appears to be the same as the
one achieved in minimal length quantum mechanics Kem , i.e. considering the
fundamental commutator as $[\tilde{q},p]=i(1+\beta p^{2})$, which can be
considered as the first order approximation in $\alpha=-2\beta$ of (6).

###  II.4 Relation with the $\kappa$-Poincaré algebra

The non-commutative Snyder space has been analyzed in the literature from
different points of view Kov ; LO ; tt , but only two particular realizations
of its algebra are known: the Snyder Sny  and the Maggiore Mag  ones. The
original realization of Snyder is recovered as a special case of (3) if
$\varphi_{1}=1$. On the other hand, the Maggiore algebra
$[\tilde{q}_{i},p_{j}]=i\delta_{ij}\sqrt{1-\alpha p^{2}}$, can be regarded as
the particular case of (3) when the condition $\varphi_{2}=0$ is taken into
account. In this case the Snyder framework can be related to the
$\kappa$-Poincaré scheme in the following sense Mag . The $\kappa$-Poincaré
algebra (for reviews see kP ) provides an explicit realization of the Snyder-
deformed algebra, once some identifications are taken into account. If
$\tilde{q}$ is identified with a suitably $\kappa$-deformed Newton-Wigner
position operator and $p$ and $M_{ij}$ as the generators of spatial
translations and rotations of the $\kappa$-Poincaré algebra respectively, the
modified Heisenberg relations are recovered. (For other comparisons between
deformed Heisenberg algebras and $\kappa$-Poincaré see gupk .) Physical
interest in the $\kappa$-Poincaré algebra arises since it is the mathematical
structure of DSR. Moreover, this framework describes the symmetries of fields
living on $\kappa$-Minkowski non-commutative space-times and is widely
expected that the study of (quantum) fields, invariant under such symmetries,
may give physical insights on a flat space limit of quantum gravity Arz .

##  III Deformed Dynamics of the FRW models

We investigate the Snyder-deformed dynamics of the isotropic cosmological
models. The system is studied at classical level searching for the
modifications induced by the deformed Heisenberg algebra. We start reviewing
the ordinary FRW dynamics and then turn to the deformed one.

###  III.1 Ordinary canonical dynamics

The FRW cosmological models are (spatially) isotropic models described by the
line element

$ds^{2}=-N^{2}dt^{2}+a^{2}\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right),$
(19)

where $N=N(t)$ is the lapse function and $a=a(t)$ the scale factor. The lapse
function does not play a dynamical role while the scale factor is the only
degree of freedom of the system describing the expansion of the Universe. The
parameter $k$ can be zero or $\pm 1$ depending on the symmetry group. The
dynamics of such models is summarized in the scalar constraint

$\mathcal{H}=-\frac{2\pi G}{3}\frac{p_{a}^{2}}{a}-\frac{3}{8\pi
G}ak+a^{3}\rho=0,$ (20)

where $G=l_{P}^{2}$ in the gravitational constant and $\rho=\rho(a)$ denotes a
generic energy density we have introduced into the system. Therefore, isotropy
reduces the phase space of general relativity to be 2-dimensional in which the
only non-vanishing Poisson bracket is $\\{a,p_{a}\\}=1$. The Friedmann
equation can be extracted by using the Hamilton equations with respect to the
extended Hamiltonian

$\mathcal{H}_{E}=\frac{2\pi G}{3}N\frac{p_{a}^{2}}{a}+\frac{3}{8\pi G}Nak-
Na^{3}\rho+\lambda\pi,$ (21)

where $\lambda$ is a Lagrange multiplier and the term $\lambda\pi$ is
introduced since $\pi$, the momenta conjugate to $N$, vanishes. We note that
$\dot{N}=\\{N,\mathcal{H}_{E}\\}=\lambda$ and that the scalar constraint (20)
is obtained requiring the constraint $\pi=0$ will be satisfied at all times,
i.e. demanding that the secondary constraint
$\dot{\pi}=\\{\pi,\mathcal{H}_{E}\\}=\mathcal{H}=0$ holds. The remaining
equations of motion with respect to $\mathcal{H}_{E}$ read

$\dot{a}=\\{a,\mathcal{H}_{E}\\}=\frac{4\pi
G}{3}N\frac{p_{a}}{a},\qquad\dot{p}_{a}=\\{p_{a},\mathcal{H}_{E}\\}=N\left(\frac{2\pi
G}{3}\frac{p_{a}^{2}}{a^{2}}-\frac{3}{8\pi
G}k+3a^{2}\rho+a^{3}\frac{d\rho}{da}\right).$ (22)

Making use of the above equations and the scalar constraint (20), we
immediately obtain the equation of motion for the Hubble rate $(\dot{a}/a)$ as

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi G}{3}\rho-\frac{k}{a^{2}},$
(23)

which is the desired Friedmann equation in a synchronous reference frame222In
the synchronous reference frame, defined in the $3+1$ framework by $N=1$ and
$N^{i}=0$, the time coordinate identifies with the proper time at each point
of space.. It is well-known that this equation leads to the big-bang
singularity where the (general-relativistic) description of the Universe is no
longer appropriate and quantum modifications are required.

###  III.2 Deformed canonical dynamics

We now perform the analysis of the deformed dynamics of the FRW models and
therefore we consider the one-dimensional case of the scheme analyzed above.
More precisely, we check the modifications arising from the algebra (6) on the
classical trajectory of the Universe described in the previous Section. A
quantum cosmological bouncing solution is obtained and it resembles the one
achieved in recent issues of LQC (if $\alpha>0$). As $\alpha<0$ the Randall-
Sundrum braneworld scenario is recovered.

The Snyder-deformed classical dynamics is summarized in the modified
symplectic geometry arising from the classical limit of (6), as soon as the
parameter $\alpha$ is regarded as an independent constant with respect to
$\hbar$. It is then possible to replace the quantum-mechanical commutator (6)
via the Poisson bracket

$-i[\tilde{q},p]\Longrightarrow\\{\tilde{q},p\\}=\sqrt{1-\alpha p^{2}}.$ (24)

We stress once again that this relation corresponds exactly to the unique (up
to a sign) possible realization of the Snyder space. In order to obtain the
deformed Poisson bracket, some natural requirements have to be considered. As
a matter of fact, it must possess the same properties as the quantum
mechanical commutator, i.e. it has to be anti-symmetric, bilinear and satisfy
the Leibniz rules as well as the Jacobi identity. This way, the Poisson
bracket (for any two-dimensional phase space function) appears to be

$\\{F,G\\}=\left(\frac{\partial F}{\partial\tilde{q}}\frac{\partial
G}{\partial p}-\frac{\partial F}{\partial p}\frac{\partial
G}{\partial\tilde{q}}\right)\sqrt{1-\alpha p^{2}}.$ (25)

In particular, the time evolution of the coordinate and momentum with respect
to a given deformed Hamiltonian $\mathcal{H}(\tilde{q},p)$, are now specified
by

$\dot{\tilde{q}}=\\{\tilde{q},\mathcal{H}\\}=\frac{\partial\mathcal{H}}{\partial
p}\sqrt{1-\alpha
p^{2}},\qquad\dot{p}=\\{p,\mathcal{H}\\}=-\frac{\partial\mathcal{H}}{\partial\tilde{q}}\sqrt{1-\alpha
p^{2}}.$ (26)

Let us apply this framework to the FRW model in the presence of a generic
matter energy density, namely to the Hamiltonian (21). Therefore we assume the
minisuperspace as Snyder-deformed and then the commutator between the
isotropic scale factor $a$ and its conjugate momentum $p_{a}$ is uniquely
fixed by the relation

$\\{a,p_{a}\\}=\sqrt{1-\alpha p_{a}^{2}}\,,$ (27)

while the equations of motion $\dot{N}=\\{N,\mathcal{H}_{E}\\}=\lambda$ and
$\dot{\pi}=\\{\pi,\mathcal{H}_{E}\\}=\mathcal{H}=0$ remain unchanged. In fact,
the Poisson bracket $\\{N,\pi\\}=1$ is not affected by the deformations
induced by the $\alpha$ parameter on the system. On the other hand, the
equations of motion (22) become modified in such an approach via the relation
(27) and read

$\dot{a}=\\{a,\mathcal{H}_{E}\\}=\frac{4\pi
G}{3}N\frac{p_{a}}{a}\sqrt{1-\alpha
p_{a}^{2}},\qquad\dot{p}_{a}=\\{p_{a},\mathcal{H}_{E}\\}=N\left(\frac{2\pi
G}{3}\frac{p_{a}^{2}}{a^{2}}-\frac{3}{8\pi
G}k+3a^{2}\rho+a^{3}\frac{d\rho}{da}\right)\sqrt{1-\alpha p_{a}^{2}}.$ (28)

As in the canonical case, the equation of motion for the Hubble rate can be
obtained solving the constraint (20) with respect to $p_{a}$ and then
considering the first equation of (28). Explicitly it becomes (taking $N=1$)

$\left(\frac{\dot{a}}{a}\right)^{2}=\left(\frac{8\pi
G}{3}\rho-\frac{k}{a^{2}}\right)\left[1-\frac{3\alpha}{2\pi
G}a^{2}\left(a^{2}\rho-\frac{3}{8\pi G}k\right)\right].$ (29)

We refer to this equation as the deformed Friedmann equation as it entails the
modification arising from the Snyder-deformed Heisenberg algebra (6). It is
interesting to consider the flat FRW Universe, i.e. the $k=0$ model. In this
case the deformed equation (29) appears to be

$\left(\frac{\dot{a}}{a}\right)^{2}_{k=0}=\frac{8\pi
G}{3}\rho\left(1-\text{sgn}\,\alpha\frac{\rho}{\rho_{c}}\right),$ (30)

where $\rho_{c}=(2\pi G/3|\alpha|)\rho_{P}$ is the critical energy density,
$\rho_{P}$ being the Planck one. In the last step we have assumed the
existence of a fundamental minimal length. In fact, as widely accepted, one of
the most peculiar consequences of all promising quantum gravity theories is
the existence of a fundamental cut-off length, which should be related to the
Planck one (for a review see gar ). Therefore, although this minimal length
appears differently in distinct contexts, it is reasonable that the scale
factor (the energy density) has a minimum (maximum) at the Planck scale.

The modifications arising from the deformed Heisenberg algebra on the
Friedmann equation (30) are manifested in the form of a $\rho^{2}$-term. Such
factor is relevant in high energy regime and, if $\alpha>0$ and $\rho$ reaches
the critical value $\rho_{c}$, the Hubble rate vanishes and the Universe
experiences a bounce (or more generally a turn-around) in the scale factor.
For energy density much smaller then the critical one the standard Friedmann
dynamics, equation (23) for $k=0$, is recovered. In the same way, when the
deformation parameter $\alpha$ vanishes, the correction term disappears and
the ordinary behavior of the Hubble parameter is obtained.

The interesting feature to be stressed is the equivalence, at phenomenological
level, between the deformed Friedmann equation (30) in the $\alpha>0$ case and
the one obtained considering the effective dynamics of LQC elqc . On the other
hand, the string inspired Randall-Sundrum braneworld scenario leads to a
modified Friedmann equation as in (30) with $\alpha<0$ Roy . The opposite sign
of the $\rho^{2}$-term in such an equation, is the well-known key difference
between the loop cosmology and the Randall-Sundrum framework. In fact, the
former approach leads to a non-singular bouncing cosmology while in the
latter, because of the positive sign, $\dot{a}$ cannot vanish and a
cosmological bounce cannot take place. Of course, to obtain a bounce the
correction should be negative, i.e. make a repulsive contribution. This can
occur also in the Randall-Sundrum scheme as soon as the extra dimension of the
bulk space-time is considered to be time-like sha1 . However, the minus sign
in this approach remains an open question and no definitive answers are given
sha2 .

##  IV Physical Considerations on the Model

A peculiar model is investigated in the framework of the Snyder-deformed
minisuperspace with particular attention to the evolution of the relative
fluctuations of the scale factor. Comparison with other similar approaches is
then showed.

###  IV.1 Flat isotropic model filled with a scalar field

An isotropic flat Universe filled with a massless scalar field $\phi$ is
analyzed in the context of the previous discussion. Such a model deserves
interest since it is the one most studied in the framework of LQC. The energy
density of this scalar field takes the form $\rho=p_{\phi}^{2}/2a^{6}$, where
$p_{\phi}$ denotes the momentum canonically conjugate to $\phi$, and then the
scalar constrain (20) becomes

$\mathcal{H}=-\frac{2\pi
G}{3}\frac{p_{a}^{2}}{a}+\frac{p_{\phi}^{2}}{2a^{3}}=0.$ (31)

The phase space is $4$-dimensional, with coordinates $(a,p_{a},\phi,p_{\phi})$
and, since $p_{\phi}$ is a constant of motion, each classical trajectory is
specified in the $(a,\phi)$-plane. The scalar field $\phi$ can be then
regarded as an internal clock for the dynamics and this condition can be
imposed requiring the time gauge $\dot{\phi}=1$, i.e. $N=a^{3}/p_{\phi}$. In
this case the deformed Friedmann equation (29) rewrites as

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{4\pi
G}{3}\left(1-\frac{3\alpha}{4\pi G}\frac{p_{\phi}^{2}}{a^{2}}\right),$ (32)

whose solution is given by $a(\phi)\sim e^{-\sqrt{4\pi
G/3}\phi}(\alpha\tilde{p}_{\phi}^{2}+e^{2\sqrt{4\pi G/3}\phi})$, where
$\tilde{p}_{\phi}^{2}=9p_{\phi}^{2}/16\pi^{4}G^{2}$. This equation clearly
predicts a big-bounce if $\alpha>0$ (from now on we consider only this case).
The ordinary solutions $a(\phi)\sim e^{\pm\sqrt{4\pi G/3}\phi}$ are recovered
at late times, i.e. at $|\phi|\rightarrow\infty$. Fixing the lapse function as
before, the (effective) Hamiltonian in the internal time $\phi$ description is
given by $H=\sqrt{4\pi G/3}\,p_{a}a$. The time evolution of any observable
$\mathcal{O}$ can then be realized with respect to such a Hamiltonian, i.e.
the equation of motion for the expectation value
$d\langle\mathcal{O}\rangle/d\phi=-i\langle[\mathcal{O},H]\rangle$ holds. The
equations of motion

$\frac{d}{d\phi}\langle a\rangle=\sqrt{\frac{4\pi G}{3}}\left\langle
a\sqrt{1-\frac{3\alpha}{4\pi
G}\frac{p_{\phi}^{2}}{a^{2}}}\right\rangle,\qquad\frac{d}{d\phi}\langle
p_{a}\rangle=-\sqrt{\frac{4\pi G}{3}}\left\langle
p_{a}\sqrt{1-\frac{3\alpha}{4\pi G}\frac{p_{\phi}^{2}}{a^{2}}}\right\rangle$
(33)

are thus immediately obtained and these trajectories are in exact agreement
with the classical ones. We are now interested to investigate the semi-
classical proprieties of such a Snyder-deformed quantum Universe. To be
precise, with semi-classical requirement for an observable $\mathcal{O}$ we
refer to the requirement that its expectation value be close to the classical
one and that the relative fluctuations
$(\Delta\mathcal{O})^{2}/\langle\mathcal{O}\rangle^{2}\ll 1$. We now analyze
the evolution of the scale factor relative fluctuations which are governed by
the equation

$\frac{d}{d\phi}\left(\frac{(\Delta a)^{2}}{\langle
a\rangle^{2}}\right)=\sqrt{\frac{16\pi G}{3}}\frac{1}{\langle
a\rangle^{2}}\left(\left\langle a^{2}\sqrt{1-\frac{3\alpha}{4\pi
G}\frac{p_{\phi}^{2}}{a^{2}}}\right\rangle-\frac{\langle a^{2}\rangle}{\langle
a\rangle}\left\langle a\sqrt{1-\frac{3\alpha}{4\pi
G}\frac{p_{\phi}^{2}}{a^{2}}}\right\rangle\right).$ (34)

As we can see, in the ordinary framework ($\alpha=0$), such a quantity is
conserved during the whole evolution (the only fluctuations $(\Delta a)^{2}$
are not constants and neither bounded) and thus the semi-classicity of an
initial state is preserved. Such a propriety is also valid in the deformed
scheme at late times $|\phi|\rightarrow\infty$, i.e. for large scale factor
$a\gg\sqrt{\alpha}p_{\phi}/l_{P}$. We note that at the bouncing time, i.e.
when the scale factor reaches its minimum value
$a_{\text{min}}=\sqrt{3\alpha/4\pi G}\,p_{\phi}$, the derivative of this
uncertainty vanishes.

Let us consider the deformed Wheeler-DeWitt (WDW) equation for this model.
Regarding the massless scalar field $\phi$ as a relational time for the
evolution and using the representation (8), the constraint (31) takes the form

$\partial_{\phi}^{2}\Psi(p_{a},\phi)=-\Theta\Psi(p_{a},\phi),\qquad\Theta=-\frac{4\pi
G}{3}p_{a}^{2}\left((1-\alpha p_{a}^{2})\partial_{p_{a}}^{2}-\alpha
p_{a}\partial_{p_{a}}\right).$ (35)

As usual the WDW equation has the same form as the Klein-Gordon equation where
$\Theta$ plays the role of the ordinary Laplacian. In order to have an
explicit Hilbert space, we perform the natural frequencies decomposition of
the solution of (35) and focus on the positive frequency sector. The wave
function $\Psi_{\omega}(p_{a},\phi)=e^{i\omega\phi}\psi_{\omega}(p_{a})$ is
thus of positive frequency with respect to $\phi$ and satisfies the positive
frequency (square root) of the quantum constraint (35), i.e. we deal with a
Schödinger-like equation $-i\partial_{\phi}\Psi=\sqrt{\Theta}\Psi$ with a non-
local Hamiltonian $\sqrt{\Theta}$ (here $\omega^{2}$ denotes the spectrum of
$\Theta$ and covers the interval $(0,\pi G/3)$). The wave function
$\psi_{\omega}$ is explicitly expressed in terms of the Gauss hypergeometric
functions $F$ and reads

$\psi_{\omega}(p_{a})=Ap_{a}^{(1-\gamma)/2}F\left(\frac{1}{4}(1-\gamma),\frac{1}{4}(1-\gamma),1-\frac{\gamma}{2};\alpha
p_{a}^{2}\right)+Bp_{a}^{(1+\gamma)/2}F\left(\frac{1}{4}(1+\gamma),\frac{1}{4}(1+\gamma),1+\frac{\gamma}{2};\alpha
p_{a}^{2}\right),$ (36)

where $\gamma=\sqrt{1-3\omega^{2}/\pi G}$. It is worth stressing that this
function is well defined (is not divergent) since we remember the existence of
a cut-off on the momentum, i.e. $p_{a}\in I$. This way, taking a weighting
function $A(\omega)$, we can construct a wave packet which has the general
form

$\Psi(p_{a},\phi)=\int d\omega A(\omega)\psi_{\omega}(p_{a})e^{i\omega\phi}.$
(37)

As last step of our analysis we note that, although the relative fluctuations
$(\Delta a)^{2}/\langle a\rangle^{2}$ are in general not constant during the
evolution (equation (34)), the difference in the asymptotic values

$D=\lim_{\phi\rightarrow\infty}\left|\left(\frac{(\Delta a)^{2}}{\langle
a\rangle^{2}}\right)_{-\phi}-\left(\frac{(\Delta a)^{2}}{\langle
a\rangle^{2}}\right)_{\phi}\right|$ (38)

vanishes. This consideration can be realized since the fluctuations $(\Delta
a)^{2}(\phi)$ and the mean value $\langle a\rangle(\phi)$ are symmetric in
time. In fact, given any real $A(\omega)$ (for example a Gaussian weighting
function $e^{-(\omega-\omega_{0})^{2}/2\sigma^{2}}$), the mean value of any
self-adjoint operator $\mathcal{O}$ with respect to the states (37) is
invariant under time inversion $\phi\rightarrow-\phi$. Therefore, starting
with a Gaussian semi-classical state such that $(\Delta a)^{2}/\langle
a\rangle^{2}\ll 1$ at late times, this propriety is satisfied on the other
side of the bounce when the Universe approaches large scale
($a\gg\sqrt{\alpha}p_{\phi}/l_{P}$).

###  IV.2 Comparison with other approaches

Our model can be regarded as an attempt to mimic the original LQC system by a
simpler one and in this respect it has to be compared with analogous existing
approaches. There are essentially two (related) ways to capture the essential
features of the original quantum system by an exactly solvable one and both
these frameworks regard the cosmological model described by (31). The first
approach Boj1  is based on replacing the connection by its sine (the
connection itself cannot be directly implemented as an operator in the loop
Hilbert space) and then rewriting the Hamiltonian in term of non-canonical
variables. This model serves as a perturbative basis for realistic bounce
scenarios and allows a precise analysis of the evolution of dynamical coherent
states. The second approach Cor  relies on a reduction of the so-called
improved dynamics and allows to clearly define in which sense the WDW theory
approximates LQC. Both models agree, although this is matter of current debate
Boj2 , with the claim that semi-classicality is preserved across the
cosmological quantum bounce BC . This feature is in agreement with our
approach. However two remarks are in order. (i) The (simplified) LQC theory is
based on a Weyl representation of the canonical commutation relations which
turns out to be inequivalent to the Schödinger representation. On the other
hand, as explained before, the Snyder-deformed algebra cannot be obtained by a
canonical transformation of the ordinary Poisson brackets of the system.
(ii)333I thank Martin Bojowald for stressing me this point. The
$\rho^{2}$-term in the modified Friedmann equation (30) is not the only
correction from LQC unless the only matter source is a massless scalar field.
If it has mass or is self-interacting, there are infinitely many other
correction terms which also involve pressure Boj3 . In our model the form of
(30) is independent of the precise matter content.

##  V Concluding remarks

In this paper we have shown that a bouncing cosmology is predicted by a
Snyder-deformed Friedmann dynamics. In particular, we have implemented a
Snyder non-commutative geometry, which can be related to DSR as well as to the
$\kappa$-Poincaré algebra, in the FRW minisuperspace arena. Since we deal with
a one-dimensional system, the deformation is almost uniquely fixed and a
cosmological bounce is then obtained. Our deformed Friedmann equation has the
same form of the LQC one. Notably, also the effective cosmological dynamics of
the Randall-Sundrum braneworld scenario is allowed because of the freedom in
the sign of the deformation term. Such a result is also corroborated by the
analysis of the Snyder-deformed harmonic oscillator. The LQC-like framework is
the one in which a cut-off on the momentum is predicted. On the other hand, in
the braneworld-like one, a minimal observable length arises and the string
theory uncertainty relation is recovered. Summarizing, a non-commutative
(deformed) picture which leads, at phenomenological level, to the predictions
of more general theories can be formulated. The validity of such an approach
in more general, and physically interesting, frameworks will be subject of
future investigations.

Acknowledgments. I thank Giovanni Montani for having encouraged this work.
Francesco Cianfrani and Orchidea Maria Lecian are thanked for several
discussions and for a critical reading of the manuscript.

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# Open-quantum-systems approach to in-medium heavy quarkonium dynamics

Alexander Rothkopf  Faculty of Science and Technology, University of
Stavanger, 4021 Stavanger, Norway<EMAIL_ADDRESS>
###### Abstract

Heavy quarkonium continues to play a central role in the study of nuclear
matter under extremes of temperature and density in relativistic heavy-ion
collisions. In this talk I report on recent developments in the theoretical
description of quarkonium in-medium dynamics using the open-quantum systems
approach. Not only does it provide a clear interpretation of the imaginary
part of the complex heavy-quark potential but also reveals that a subtle
interplay between screening and wavefunction decoherence is responsible for
the melting of heavy quarkonium states.

##  1 Introduction

According to the standard model of relativistic heavy-ion collisions (HIC)
(for a review see e.g. [1]) the time evolution of matter in the collision
center can be roughly divided into three stages. The earliest stages just
after the projectile nuclei hit, are characterized by strong coherent color
electric and color magnetic fields, the so called glasma. The energy stored in
these field is quickly converted into quark and gluon particles which
efficiently exchange energy and momentum to form a locally thermal quark-gluon
plasma (QGP) after around $t\sim 1$fm. This QGP expands and cools over a a
range of around $\Delta t\sim 5-10$fm, well described by relativistic
hydrodynamics. Reaching the transition temperature of strongly interacting
matter, predicted by quantum chromodynamics (QCD) at around $155$MeV, colored
partons have to come together to form color neutral hadrons at hadronization
after which the produced particles freeze out chemically and kinetically
before flying out into the surrounding detectors.

Heavy quarkonium, the bound states of a heavy quark and antiquark (charm or
bottom) are ideal probes of the quark-gluon plasma created in HICs (for a
review see e.g. [2]). Their life cycle spans that of the HIC, with production
of $Q\bar{Q}$ pairs in the earliest instances that may or may not form bound
states before finding themselves immersed in the hot QGP. As the QGP cools,
quarkonium states are affected by the environment and depending on how deeply
they are bound, may survive until hadronization or will dissociate. At
hadronization either surviving bound states emerge or formerly dissociated
pairs of heavy quarks may find a chance to recombine before flying off into
the detectors. Our goal here is to understand from first principles the
evolution of a fully formed quarkonium state immersed in the QGP.

Quarkonium dilepton decay channels provide clean signals in experiment [3] and
an inherent separation of scales simplifies their theory treatment [4]. The
fact that the heavy quark mass $m_{Q}$ is larger than both the environment
temperature $T$ and the intrinsic scale of quantum fluctuations in QCD
$\Lambda_{\rm QCD}$ allows one to describe the quarks and anti-quarks
separately in the effective field theory of non-relativistic QCD (NRQCD). If
the quarkonium state is of small extent relative to environment scales it can
be described by color singlet and octet wavefunction via potential NRQCD
(pNRQCD). While the singlet channel can sustain bound states, the octet
channel is associated with a repulsive interaction. Note that the separation
in terms of energy scales also translates into a separation of time-scales.

Figure 1: Nuclear modification factor $R_{AA}$ for the vector channel (left)
charmonium ground state $J/\Psi$ [5] and (right) for the two lowest lying
states of bottomonium $\Upsilon(1S)$ and $\Upsilon(2S)$ [6]. Note that
multiple phenomenological models, based on significantly different physics
processes, are able to capture the data reasonable well, indicating the need
for genuine 1st principle insight.

One may ask why a genuine ab-initio understanding of quarkonium dynamics is
called for? Phenomenological modelling based on the rate equation (mainly for
charmonium) and either the Boltzmann equation or the deterministic Schrödinger
equation (mainly for bottomonium) have successfully postdicted the nuclear
modification factor of several quarkonium species as shown in fig. 1. In the
absence of ab-initio insight on the parameters of the models, one however
mixes the determination of heavy-quarkonium properties and those of the
medium. I.e. inference of QGP properties alone requires first-principles
insight into $Q\bar{Q}$ dynamics.

Progress in this direction has been made in the past years through the open
quantum systems paradigm [7], originally developed in condensed matter theory.
It represents a general formalism to treat a small probe system immersed in an
environment not necessarily in equilibrium. The overall system, consisting of
probe and environment is closed and its density matrix $\rho_{\rm tot}$
evolves unitarily $\partial_{t}\rho_{\rm tot}=-i[H_{\rm tot},\rho]$, according
to a hermitian Hamiltonian $H_{\rm tot}^{\dagger}=H_{\rm tot}$. In order to
avoid having to treat the surrounding medium explicitly, we focus on the
evolution of the probe d.o.f. and may formally trace out the environment. In
turn we obtain the reduced probe density matrix ${\rm Tr}_{\rm
medium}[\rho_{\rm tot}]=\rho_{Q\bar{Q}}$, whose dynamics are in general
dissipative, as it remains in contact with the surrounding bath.

The so called master equation, i.e. the equation of motion that governs
$\rho_{Q\bar{Q}}$ takes on different forms, depending on the separation of
three relevant timescales. The environmental relaxation scale $\tau_{E}$ tells
us how quickly perturbations among the environment d.o.f. die down. The
intrinsic probe scale $\tau_{S}\sim 1/|\omega-\omega^{\prime}|$ is related to
the inverse level spacing of the probe and encodes how quickly information is
updated in the small system. The third scale is the $Q\bar{Q}$ relaxation
scale $\tau_{\rm rel}$ which encodes how quickly information travels between
the medium and the probe system.

For a system in which memory effects can be neglected, i.e. for
$\tau_{E}\ll\tau_{\rm rel}$, so-called Markovian time evolution, it has been
proven that the most general master equation can be expressed in the Lindblad
form

$\displaystyle\frac{d}{dt}\rho_{Q\bar{Q}}=-i[\bar{H}_{Q\bar{Q}},\rho_{Q\bar{Q}}]+\sum_{k}\gamma_{k}\Big{(}L_{k}\rho_{Q\bar{Q}}L_{k}^{\dagger}-\frac{1}{2}\\{L^{\dagger}_{k}L_{k},\rho_{Q\bar{Q}}\\}\Big{)}.$
(1)

Here $\bar{H}_{Q\bar{Q}}$ denotes the in-medium probe Hamiltonian with a real-
valued potential inducing coherent dynamics, while the Lindblad operators
$L_{k}$ together with the relaxation rate $\gamma_{k}$ encode the residual
interactions between probe and environment. They can be associated with
fluctuations (energy flows into the probe system) and dissipation (energy
flows out of the probe system).

##  2 Open Quantum Systems approaches for heavy quarkonium

Figure 2: The systematic chain of approximations that the open quantum systems
approach has built between ab-intio theory QCD, the effective field theories
NRQCD and pNRQCD and the phenomenological approaches used in the past. At
intermediate steps different genuine quantum descriptions of quarkonium,
applicable in regimes with appropriate time-scale separation, have been
established.

Over the past seven years the open quantum systems approach has been used to
derive from first principles QCD several Lindblad equations for heavy
quarkonium (for a review see e.g. [8]), applicable in different regimes of
scale separation (see fig. 2). One part of the community uses the effective
field theory pNRQCD as starting point, which, when evaluated in the weak-
coupling limit, gives rise to a Lindblad equation [9]. In the quantum optical
limit and under the assumption of fast classicalization this equation be
reduced to the Boltzmann equation [10]. Assuming that the system is close to
equilibrium, it may further be reduced to the rate equation used in
phenomenological modelling. This establishes a systematic chain of
approximations with well defined ranges of validity from QCD to phenomenology.
To overcome the weak-coupling approximation, another research branch focuses
on very tightly bound quarkonium states (Coulombically bound) [11], in which
the dynamics of the system can be conveniently captured in two non-
perturbative transport coefficients $\kappa$ and $\gamma$ [12]. The former is
nothing but the heavy-quark diffusion constant and the latter amounts to a
correction term for the coherent dynamics. Both quantities are being actively
investigated using lattice QCD simulations (see e.g. [13]).

Another active research direction within the community considers the open-
quantum systems approach for quarkonium based on the effective field theory
NRQCD, in which the quark and antiquark remain as individual entities. If one
considers a weakly-coupled hot environment, one can argue that the information
within the quarkonium probe refreshes much more slowly than among the medium
d.o.f., leading ref. [14] to establish a quantum Brownian motion Lindblad
equation. It is known in the open quantum systems community that the dynamics
of a Lindblad equation can always be unravelled, i.e. reexpressed in terms of
an ensemble of wavefunctions, evolving under a non-linear stochastic
Schrödinger equation [15]. This for the first time allowed to connect models
with such a non-linear equation of motion (see e.g. [16]) to first principles
QCD. Interestingly one can now carry out further controlled approximations to
arrive at other phenomenological models previously deployed in the literature.
In the dissipationless limit, the non-linear stochastic Schrödinger equation
reduces to a linear one [17, 18]. And if in turn one uses the adiabatic
approximation, a deterministic Schrödinger equation with an complex valued
potential ensues (see e.g. [19]). It is interesting to note that in this NRQCD
based approach it is the real- and imaginary part of the complex interquark
potential, originally discovered in ref. [20] and currently investigated on
the lattice (see e.g. [21, 22, 23, 24]), that characterizes the evolution.

If one focuses on systems where the quarkonium decoheres very quickly [25],
one may also go over to a semi-classical description [26] in terms of a
Langevin equation for the quarks and antiquarks, which offers insight into
e.g. the mechanism of recombination.

##  3 The Quantum Brownian Motion Lindblad equation

Let me focus on the quantum Brownian motion Lindblad equation for heavy
quarkonium, as it has revealed that two distinct processes compete in the
evolution of the bound state in a hot medium (see discussion in [18]). In the
standard form of eq. 1 we get that

$\displaystyle\bar{H}_{Q\bar{Q}}=\frac{{\bf p}^{2}}{m_{Q}}+V_{Q\bar{Q}}({\bf
r}),\quad V_{Q\bar{Q}}({\bf r})=-\alpha_{S}\frac{e^{-m_{D}|{\bf r}|}}{|{\bf
r}|}+c={\rm Re}[V_{\rm EFT}({\bf r})],$ (2)

which indicates that Debye-screening of the interactions with Debye mass
$m_{D}$ will affect the binding of the quark antiquark pair. This mechanism of
destabilization was known since the seminal works of ref. [27]. Let us take a
closer look at the Lindblad operators themselves. Since behind the scenes,
each interaction of the quarkonium system with a medium gluon leads to a
momentum exchange and a color rotation we expect these phenomena to be
represented in the $L$’s, which indeed carry a momentum and color label

$\displaystyle L_{{\bf k},a}=$ $\displaystyle\sqrt{\frac{D({\bf
k})}{2}}\Big{[}1-\frac{{\bf k}}{4m_{Q}T}\cdot\Big{(}\frac{1}{2}{\bf P}_{\rm
CM}+{\bf p}_{\rm rel}\Big{)}\Big{]}e^{i{\bf k}\cdot{\bf
r}/2}\big{(}T^{a}\otimes 1\big{)}$ (3) $\displaystyle-\sqrt{\frac{D({\bf
k})}{2}}\Big{[}1-\frac{{\bf k}}{4m_{Q}T}\cdot\Big{(}\frac{1}{2}{\bf P}_{\rm
CM}+{\bf p}_{\rm rel}\Big{)}\Big{]}e^{i{\bf k}\cdot{\bf r}/2}\big{(}1\otimes
T^{a}\big{)}.$ (4)

In NRQCD quark and antiquark are separate entities and hence we find two
terms, with opposite signs. The color rotations on either one are implemented
via the Gell-Mann matrices $T^{a}$. The momentum independent term $1$ in the
square brackets encodes the physics of fluctuations, i.e. it allows energy
from the medium to enter the quarkonium system and heat up. The momentum
dependent terms on the other hand are associated with dissipation, which
allows quarkonium to shed energy back to the medium. Obviously only the
correct interplay between the two will allow the system to thermalize. The
overall strength of the interaction with the medium is governed by a function
$D({\bf k})$, which, as it turns out, is intimately connected in coordinate
space with the imaginary part of the complex interquark potential $D({\bf
r})\sim{\rm Im}[V_{\rm EFT}({\bf r})]$.

The function $D({\bf r})$ exhibits a peaked behavior in coordinate space,
which allows us to identify a new scale in the system, the medium correlation
length $\ell_{\rm corr}$. With this new scale comes a new mechanism for the
destabilization of heavy quarkonium: wavefunction decoherence. It is thus the
interplay of screening and decoherence that governs the survival of heavy
quarkonium.

As temperature increases, the screening length $r_{s}=1/m_{D}$ reduces. As
long as it is larger than the interquark distance, the gluons exchanged
between the constituent quark antiquark pair are not impacted. As the
screening length shrinks and reaches the characteristic extent of the bound
state, gluon exchange is impeded and eventually binding cannot be sustained.

The medium correlation length $\ell_{\rm corr}$ encodes over which distances
color rotations act coherently. As long as $\ell_{\rm corr}$ is larger than
the size of quarkonium, color rotation act in sync on the quark and antiquark
constituent, turning a color singlet state into another color singlet state.
As the correlation length diminishes, color rotations may act independently on
the quark or antiquark, leading to an increasing probability to realize a
color octet configuration that is not attractively bound. This can happen at a
temperature where screening has not yet impacted the exchange of gluons
between the constituents of the quarkonium particle.

Figure 3: First results of the quantum Brownian motion Lindblad equation in
one dimension. (left) The evolution of a quarkonium vacuum bound state analog
in 1d evolving in a thermal medium [15]. Red data denote ground state survival
probabilities, blue data the first excited state. Independent of initial
conditions the two simulations shown approach a universal steady state.
(right) The distribution of states in the late-time steady state, plotted
against energy at different temperatures [28].

The quantum Brownian motion Lindblad equation was first implemented in a one-
dimensional setting in ref. [15] using quantum state diffusion and in ref.
[28] via a direct simulation of the master equation. In both cases the
projection onto the singlet sector was taken apriori. In the left panel of
fig. 3 we show a central result of these studies, where the one-dimensional
analog of a quarkonium vacuum bound state at initial time was evolved in a
medium with finite screening and correlation length. The red data represent
the survival probability of the ground state, while the blue points describe
the first excited state. Two simulations are shown, one where we start with
$100\%$ ground and one with $100\%$ excited state at initial time. At late
time both simulations approach the same steady state, which is independent of
the initial conditions. To ascertain whether the steady state is related to a
genuine themalized system we show in the right panel of fig. 3 three spectra
from simulations at late times, where we plot the survival probability against
the energy of each state. The data points clearly align on a straight line
when plotted on logarithmic scale, which agrees with the exponential falloff
expected from the thermal Boltzmann weight. A fit to the inverse slope reveals
values for the inverse slope, which are very close to the temperature of the
medium in which we made the quarkonium particle evolve.

These results are very encouraging, as they confirm that the quantum Brownian
motion Lindblad equation is able to thermalize a quarkonium state in a genuine
quantum fashion. So far this is the only quantum approach on the market that
has been demonstrated to correctly thermalize quarkonium. In order for
thermalization to occur fluctuations and dissipation need to be in balance.
Our simulations show that dissipative effects, contrary to intuition, can have
a significant effect on the ground state survival probability at early times.
This insight was obtained by a direct comparison of the full Lindblad dynamics
and the reduced dynamics in the dissipationless limit. Simulations including
the color singlet and octet d.o.f. explicitly have been carried out by the
Osaka group recently [29] and the extension of the simulations to three
dimensions is work in progress.

##  4 Conclusion

The open quantum systems approach has provided a new perspective on quarkonium
dynamics in relativistic heavy-ion collisions. It has made possible to derive
systematic chains of approximations that for the first time link
phenomenological models with first principles QCD using the effective field
theories NRQCD or pNRQCD at intermediate steps. The quantum Brownian motion
Lindblad equation derived from NRQCD revealed the interplay of screening and
wavefunction decoherence as two relevant mechanism for the dynamic
destabilization of quarkonium in a hot environment and was shown to correctly
thermalize the quark antiquark pair with its surroundings in a genuine quantum
fashion.

##  5 Acknowledgments

A.R. is supported by the Research Council of Norway under the FRIPRO Young
Research Talent grant 286883.

## References

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#  : Harvesting API Information from Various Online Sources

Ferdian Thung  Singapore Management University
Singapore
<EMAIL_ADDRESS>Kisub Kim  Singapore Management University
Singapore
<EMAIL_ADDRESS>Ting Zhang  Ivana Clairine Irsan  Singapore Management
University
Singapore
<EMAIL_ADDRESS>Singapore Management University
Singapore
<EMAIL_ADDRESS>Ratnadira Widyasari  Singapore Management University
Singapore
<EMAIL_ADDRESS>Zhou Yang  David Lo  Singapore Management
University
Singapore
<EMAIL_ADDRESS>Singapore Management University
Singapore
<EMAIL_ADDRESS>
###### Abstract

Using APIs to develop software applications is the norm. APIs help developers
to build applications faster as they do not need to reinvent the wheel. It is
therefore important for developers to understand the APIs that they plan to
use. Developers should also make themselves aware of relevant information
updates about APIs. In order to do so, developers need to find and keep track
of relevant information about the APIs that they are concerned with. Yet, the
API information is scattered across various online sources, which makes it
difficult to track by hand. Moreover, identifying content that is related to
an API is not trivial. Motivated by these challenges, in this work, we
introduce a tool named that aims to ease the process of finding API
information from various online sources. is built on works that link APIs or
libraries to various online sources. It supports finding API information on
GitHub repositories, Stack Overflow’s posts, tweets, YouTube videos, and
common vulnerability and exposure (CVE) entries; and is extensible to support
other sources.

###### Index Terms:

API mining, API information, multi-source

##  I Introduction

APIs are indispensable components of modern software development [1]. APIs
allow developers to focus on developing the main functionality of their
software as they can rely on APIs for common functionalities, which
consequently lowers software production time and improves developer
productivity. Due to its importance, developers should understand the APIs
that they plan to use well. They can read API-related information in online
sources, such as related code in GitHub, related posts in Stack Overflow, or
related tweets in Twitter.

While developers can find the API information themselves, manually finding
them could be tedious and time-consuming. First, the API information is
scattered across various online sources. Thus, developers would need to open
many different websites and perform a search on each one of them. Second, the
search functionality in the website is typically not targeted for API search.
Thus, developers may need to go through a considerable number of search
results before finding the content that is related to the API they search for.
Each content may consist of code or text that contains possible mentions of
the API they search for. These mentions may be ambiguous, especially if the
API name is generic (e.g., read), which may refer to APIs from different
libraries or simply an English word read.

Existing works have dealt with the challenges of linking content that is
related to either an API or a library while handling the ambiguity that may
exist in the possible mentions of API [2, 3, 4, 5]. Luong et al. [2] proposed
an approach that leverages semantic and syntactical analysis to determine
whether a Stack Overflow thread is related to an API. Asyrofi et al. [3]
developed an approach that enables a more accurate code search on GitHub by
performing type resolution on the retrieved code and returns only the code
related to a particular API. Zhang et al. [4] investigated the effectiveness
of several pre-trained models in the task of determining whether a tweet is
related to a library. They found that RoBERTa [6] performs the best. Haryono
et al. [5] investigated several eXtreme Multi-label Learning (XML) techniques
to identify libraries from CVE entries, with LightXML [7] achieving the best
performance.

Although the aforementioned work can be used to identify API content, they can
only be used individually. As such, the overall experience is not much of a
departure from opening multiple websites at once. Moreover, due to the nature
of some contents (e.g., tweets), a direct mention of an API is either
extremely rare or non-existent altogether. For such content, it is likely that
only a library linking approach is available. In such a case, an extra step of
identifying a unique library name that an API belongs to is needed before the
available linking approach can be utilized to find API information.

In this work, we built a tool named that can integrate the existing
API/library linking approaches in one interface. It aims to streamline the
process of finding APIs from many different contents. For content that only
has a library linking approach, can automatically map an API to its library in
order to leverage the library linking approach. It also supports content that
has no linking approach available and defaults to a generic text search. With
these features, is extensible to support various types of content.

The rest of the paper is structured as follows: Section II discusses the
preliminaries that contain some API/library linking approaches that we build
our tool upon. Section III provides the details of our proposed tool. Section
IV describe the evaluation of . Finally, related work and conclusion are
presented in Section V and Section VI, respectively.

##  II Preliminaries

We describe API/library linking approaches that we leverage in . These
approaches can be used to identify whether a content is related to an API or a
library.

###  II-A ARSeek

ARSeek [2] is a method for determining if a post on Stack Overflow is
discussing a specific API by analyzing the natural language paragraphs and the
code snippets in the post. It has two components: DATYS+ and API Relevance
Classifier. The first component, DATYS+, takes in potential posts, which are
posts that contain a word matching the simple name of the given API method,
and API candidates, which are API methods with the same simple name as the
given API method. It then calculates a confidence score for the given API
method being referred to in the thread. The second component, API Relevance
Classifier, converts natural language paragraphs and code snippets in a Stack
Overflow thread; and the comment and implementation code of the given API
method into an API Relevance Embeddings. These embeddings are then used to
determine the likelihood that the thread is discussing the given API method. A
score is then calculated from these embeddings. If the score is above a
certain threshold, the thread is considered relevant for the given API method.

###  II-B AUSearch

Asyrofi et al. [3] created AUSearch, an approach that uses type system and API
method signatures, which consists of a class name, simple name, and
parameters, to improve the precision of GitHub Code
Search.111https://github.com/search This approach solves the problem of
ambiguous API invocations in GitHub code examples, which was a limitation of
the GitHub Code Search. By implementing type resolution, AUSearch eliminates
irrelevant code examples and significantly increases GitHub Code Search’s
accuracy for finding code containing the API.

###  II-C Pretrained Models for Library-related Tweets

Zhang et al. [4] explored several pre-trained models to identify whether a
tweet is related to a library. They experimented with both general-purpose
(i.e., BERT [8], RoBERTa [6], and XLNet [9]) and domain-specific pre-trained
models (i.e., BERTweet [10] and BERTOverflow [11]). They fine-tuned these
models to the task of classifying library-related tweets. They found that
RoBERTa [6], which is a robustly optimized BERT [8], performs the best and
beats existing state-of-the-art.

###  II-D XML Techniques for Identifying Libraries in CVE

Haryono et al. [5] explored several eXtreme Multi-label Learning (XML)
techniques to identify libraries that are mentioned in CVE entries. They
evaluated five traditional models, which are FastXML [12], DiSMEC [13],
Parabel [14], Bonsai [15], and ExtremeText [16]. They also evaluated two deep-
learning based models, which are XML-CNN [17] and LightXML [7]. These models
are trained on a CVE dataset where the labels are the libraries mentioned
across all CVE entries in the dataset. Bonsai and LightXML were shown to be
the most effective, with LightXML leading the pack.

##  III Tool Architecture

###  III-A Overview

Figure 1: The architecture of

Figure 1 shows the overall architecture of . consists of three main
components: (1) API Database Builder, (2) API Map Builder, and (3) API
Information Retrieval Engine. API Database Builder and API Map Builder belong
to the Data Preparation phase, while API Information Retrieval Engine belongs
to the Deployment phase.

In the Data Preparation phase, populates the API Database and API Map Database
with API knowledge from Input Content and Libraries, respectively. API
Database holds contents from various sources that have been linked to APIs,
while API Map Database holds mappings from APIs to their libraries. API
Database Builder uses existing API/library linking approaches (if available)
to label the Input Content. It only stores the Input Content to the API
Database if the Input Content is labelled as relevant by existing API/library
approaches. API Map Builder adds the mapping from APIs to libraries if only
the library linking approach is available to extract information from Input
Content. In the Deployment phase, API Information Retrieval Engine accepts an
API query and makes use of the API Database and the API Map Database to return
the relevant API content. We further describe components in detail in the
following subsections.

###  III-B API Database Builder

Given an Input Content, the API Database Builder first identifies the content
source. If there exists an API linking approach that supports the content
source, API Database Builder runs the corresponding API linking approach on
the Input Content. If the Input Content is labelled as relevant to API(s), the
Input Content is stored in the API Database with links to all relevant APIs.
Similarly, if there exists only a library linking approach that supports the
content source, API Database Builder runs the corresponding library linking
approach and stores the Input Content to the API Database, with links to all
relevant libraries. On the other hand, if there is no API/library linking
approach that supports the content source, the Input Content is stored without
any links. In all of the above cases, API Database Builder runs a script that
adapts the approach’s output format to the database format.

###  III-C API Map Builder

If the Input Content is from a source that is only supported by a library
linking approach, API Database Builder sends Libraries that are outputted by
the library linking approach to API Map Builder. API Map Builder scans all of
these libraries and extracts all public APIs from each library (e.g., by using
tools such as
javap222https://docs.oracle.com/javase/7/docs/technotes/tools/windows/javap.html).
It then stores the library name and the public APIs’ names in the API Map
Database.

###  III-D API Information Retrieval Engine

Given an API query, API Information Retrieval Engine first passes the query to
API Mapper. The API Mapper queries the API Map Database for the name of the
library that belongs to the API in the query. It then passes both the API name
and the library name to the API Content Retriever. API Content Retriever goes
through all the available API information sources (e.g., Stack Overflow,
GitHub, etc.) recorded in the API Database. For each source, it queries the
API Database using either the API or the library name (depending on the
linking approaches used for the source). If the API linking approach was used,
it queries the database using the API name. Similarly, if library linking
approach was used, it queries using the library name. On the other hand, if
neither linking approaches were used, API Content Retriever leverages a
classical information retrieval strategy to retrieve relevant contents. It
computes and ranks BM25 score [18] between all contents from the source and
the API name. After API Content Retriever obtains relevant contents from each
source, it aggregates all contents and returns them as the API Content.

Figure 2: The user interface of

Figure 2 shows the user interface when searching contents related to an API
method com.google.common.base.Object.hashCode(). The interface allows users to
type the API name into the search bar. Once the user ceases typing, the API
name is sent to API Information Retrieval Engine, which retrieves the related
contents and show them to the user. The contents from different sources are
organized in the form of tabs: StackOverflow, GitHub, Tweet, CVE, and YouTube.
Each tab contains the relevant API content from each source (if any). The
figure shows examples of relevant content in StackOverflow for the given API
name.

##  IV Evaluation

###  IV-A API Information Sources

For evaluation and prototype, we integrate the following information sources
into : Stack Overflow posts, GitHub code snippets, tweets, CVE entries, and
YouTube videos. We run ARSeek [2] and AUSearch [3] to link APIs to Stack
Overflow posts and GitHub code snippets, respectively. Similarly, we run Zhang
et al. approach [4] and Haryono et al. approach [5] to link software libraries
to the tweets and CVE entries, respectively. We use the dataset from their
studies and input the test set as the Input Content. We also use the
implementation provided in their replication package. For each source, we
wrote a script that adapts the data format in the linked content to the
database format. Note that we also need to write such a script if we want to
extend to support new API/library linking approaches.

For YouTube videos, we randomly selected 100 classes from Java Standard
Library.333https://docs.oracle.com/javase/7/docs/api/overview-summary.html We
then query YouTube Search API and take the top-100 video metadata returned by
the API. We then wrote a script to input these metadata into the API Database.
As there is no available API/library linking approaches here, would perform an
information retrieval based on the BM25 score.

###  IV-B Usability

We evaluate in the terms of its usability. We do not evaluate the linking
effectiveness as it is inherently inherited from the respective API/library
linking approaches. We invited 3 Ph.D. students in Computer Science and 2
Research Engineers in Computer Science to evaluate the usability of . All of
them have a programming experience of more than 5 years. We provide them
access to and asked them to score three aspects of whether: is easy-to-use, is
useful, and they would like to use in the future. The score ranges from 1 to
5, which indicates strongly disagree, disagree, neutral, agree, and strongly
agree. The evaluators consider to be easy-to-use (4.8 out of 5) and useful
(4.6 out of 5) and are willing to use in the future (4 out of 5).

##  V Related Work

###  V-A API Linking Approaches

API linking approaches aim to link an API to content that is related to it. A
group of approaches aims to link an API to a code snippet [19, 20, 21, 3]. To
identify API mentions in code snippets, Baker [19] iteratively performs a
deductive linking analysis. Both COSTER [20] and STATTYPE [21] work by
capturing and learning the tokens surrounding a code element and associating
these tokens with similar tokens that they have encountered before. On the
other hand, AUSearch [3] leverages type resolution to resolve the API that is
being called from within a code.

Another group of approaches aims to link an API to software engineering texts
such as Stack Overflow posts [22, 23, 24, 25, 2]. In this line of work,
several studies [22, 23, 24, 25] make use of classical information retrieval
techniques and/or some heuristics. Miler, which was developed by Bacchelli et
al. [24], used string matchings and IR techniques to link emails to source
code entities in software systems such as classes in object-oriented systems
and functions in procedural language systems. Dagenais and Robillard [25] used
filtering heuristics to link APIs mentioned in software support channels
(e.g., mailing lists and forums). ARSeek, which is developed by Luong et al.
[2], performs a semantic and a syntactical analysis on a Stack Overflow post
to determine if the post is related to an API.

###  V-B Library Linking Approaches

Library linking approaches aim to link a library to content that is related to
it. A group of approaches can be used to link a library to microblogs such as
tweets [26, 27, 4]. Prasetyo et al. [26] proposed an approach to identify
whether a microblog is relevant to software. They extracted features from
textual content and URLs in tweets. They then trained a classifier using
Support Vector Machine (SVM), which is used to classify whether software-
related tweets. Sulistya et al. [27] proposed an approach to exploit knowledge
from rich platforms such as Stack Exchange and Stack Overflow to be used in
other platforms such as Twitter. They extracted word embeddings from Stack
Exchange and Stack Overflow; and used these embeddings as features in a
classifier that identifies software-related tweets. These two approaches can
be used to identify library-related tweets by filtering software-related
tweets that contain library names. More recently, Zhang et al. [4] explored
the capability of pre-trained models to identify library-related tweets and
found them to be more effective than baseline approaches.

Another group of approaches aims to link a library to CVE entries [28, 5].
Chen et al. [28] is the first to propose to automatically link a library that
is related to vulnerabilities mentioned in CVE entries. They cast the problem
as eXtreme Multi-label Learning (XML), where libraries are the labels, and
used FastXML [12] to identify libraries used in CVE entries. Haryono et al.
[5] continued this direction by evaluating many XML algorithms and found that
LightXML [7] is the best XML algorithm for this task.

##  VI Conclusion and Future Work

API information is scattered in various online sources. Linking them to APIs
is not a trivial task as some mentions of APIs may be ambiguous due to the
fact that some APIs have the same method name. Although approaches have been
developed to perform the linking, they are typically designed to target only a
specific source. For some sources, there is also no available API linking
approach. Due to these issues, we present , a tool to find API information
from various online sources and present the API information in a single
interface. makes use of both API and library linking approaches to find API
information. It also accommodates finding API information for all kinds of
sources, regardless of the availability of the API/library linking approach.
Our preliminary evaluation supports the usability of . In the future, we plan
to add more online sources by leveraging more API/library linking approaches.
We release the source code of at https://github.com/soarsmu/APIHarvest.

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