# Quantum Cosmology with Dynamical Vacuum in a Minimal-Length Scenario

M. F. Gusson  Universidade Federal do Espírito Santo – Ufes
Vitória, Espírito Santo, 29.075-910, Brasil  A. Oakes O. Gonçalves Actual
adress: Instituto Federal Baiano, Campus Itapetinga, BA. Universidade Federal
do Espírito Santo – Ufes
Vitória, Espírito Santo, 29.075-910, Brasil  R. G. Furtado  Universidade
Federal do Espírito Santo – Ufes
Vitória, Espírito Santo, 29.075-910, Brasil
J. C. Fabris and J. A. Nogueira National Research Nuclear University MEPhI,
Kashirskoe sh. 31, Moscow 115409, Russia.julio.fabris@cosmo-
<EMAIL_ADDRESS>Universidade Federal do Espírito Santo – Ufes
Vitória, Espírito Santo, 29.075-910, Brasil

###### Abstract

> In this work, we consider effects of the dynamical vacuum in quantum
> cosmology in presence of a minimum length introduced by the GUP (generalized
> uncertainty principle) related to the modified commutation relation
> $[\hat{X},\hat{P}]:=\frac{i\hbar}{1-\beta\hat{P}^{2}}$ . We determine the
> wave function of the Universe $\psi_{qp}(\xi,t)$, which is solution of the
> modified Wheeler-DeWitt equation in the representation of the quasi-position
> space, in the limit where the scale factor of the Universe is small.
> Although $\psi_{qp}(\xi,t)$ is a physically acceptable state it is not a
> realizable state of the Universe because $\psi_{qp}(\xi,t)$ has infinite
> norm, as in the ordinary case with no minimal length.
> PACS numbers: 98.80.Qc, 04.60.-m, 04.50.Kd,
> Keywords: Minimal length; generalized uncertainty principle; quantum
> gravity; quantum cosmology; dynamical vacuum.

##  1 Introduction

The Standard Cosmological Model (SCM) is based on an expanding universe which
has a very hot and dense origin. The extrapolation of the SCM for very early
times leads to energy scales deep in the Planck regime and, strictly speaking,
the expanding universe would have a singular initial state. On the other hand,
in order to explain many features of the observed Universe today, a period of
short but rapid, almost exponential, expansion, which is called inflationary
phase, is necessary in the very early Universe. It is expected that in the
primordial Universe quantum effects are relevant and a quantum approach is
necessary to describe the gravitational effects. A quantum approach to gravity
is also necessary in the study of black holes. The features of compact
astrophysical objects like neutron stars, and even white dwarfs, may be also
affected by quantum gravity effects. However, we have no quantum gravity
theory which is fully acceptable, although there are many proposals for it. An
important aspect of almost all proposals for quantization of the gravity is
the prediction of the existence of a minimal length. Thus, an effective
description of the effects of quantum gravity through a phenomenological
approach can be obtained by considering a minimal-length scenario [1, 2, 3,
4]. Some results in the literature indicate that the introduction of a minimal
length could also solve the problems like the initial singularity (big bang
singularity) [5, 6, 7, 8, 9, 10, 11, 12, 13], the black hole complete
evaporation (catastrophic evaporation of a black hole) [3, 14, 15, 16, 17,
18], the cosmological constant [12, 19, 20, 21], the Chandrasekhar limit [4,
18, 20], the instability of the Einstein static universe (emergent universe)
[22], the vanishing of cosmological constant in multiverse theories [19], and
so on.

We can obtain a minimal-length scenario by modifying the Heisenberg
uncertainty principle (HUP) between the position and momentum operators. There
are several different proposals of modification of the HUP which introduce a
minimal-length scenario. Those modified uncertainty principles are called
Generalized Uncertainty Principle (GUP)111Readers interested in finding out
more about minimal-length literature may consult References [17, 23, 24, 25,
26].. GUP’s have been derived from different contexts such as string theory
[27, 28, 29, 30, 31, 32], black hole physics [19, 33, 34, 35] and extra
dimensions [36].

Although the most common modified or deformed commutation relation (MCR)
associated with a GUP (KMM GUP [37]) is quadratic in the momentum operator,
effects of a MCR with a linear term in the momentum operator, which leads to a
minimal length and a maximum momentum [38], have been studied in contexts of
cosmological models [10, 11, 39, 40, 41]. Nevertheless, apparently this GUP
leads to non-unitary theories [40] and it is not compatible with current data
available [42].

In general, the inflationary phase is described by introducing a scalar field.
Hence, in an inflationary quantum cosmology theory it is necessary to perform
the second quantization of that scalar field. Consequently, in a minimal-
length scenario that scalar field and its conjugate momentum have to obey the
modified commutation relations [43, 44, 45, 46]. As it is well known, the
ordinary approach of the third quantization222The 3rd quantization consists of
initially considering the Wheeler-DeWitt equation as a classical equation of a
classical field (the wave function of the Universe in the 2nd quantization)
which then is turned an operator. In this way universes can be created and
annihilated in a multiverses theory. of the Wheeler-DeWitt (WDW) equation
leads to the vanishing of the cosmological constant. However, it might be
possible to obtain a non-zero cosmological constant in a minimal-length
scenario [19].

Many authors have studied effects of a minimal length in cosmology using a
classical approach in which the Poisson brackets are modified according to the
correspondence principle [10, 22, 47, 48, 49, 50, 51]. In the vast majority of
the GUP formulation the parameter related to the minimal length (deformation
parameter) is positive. Although GUP with positive deformation parameter
prevents black holes to evaporate completely, at the same time they remove the
Chandrasekhar limit333The maximum mass of a stable white dwarf star., that is,
a white dwarf star could turn out arbitrarily large [52, 53]. In [18, 20] the
author shows that GUP with a negative deformation parameter can restore the
Chandrasekhar limit and, in spite of allowing a black hole to evaporate
completely, this evaporation takes an infinite amount of time. Even though a
negative deformation parameter is unusual, it is consistent with a description
in which the universe has an underlying crystal lattice-like structure [54].

Mu-In Park proposed that the HUP can be modified to include the cosmology
constant term, which is called extend uncertainty principle (EUP) [35, 55].
Whereas GUP has a quadratic term in the momentum uncertainty, EUP has a
quadratic term in the position uncertainty. GUP that has also a quadratic term
in the position uncertainty is called GEUP. In [4] the authors use GEUP to
show that a non-zero cosmological constant can restore the Chandrasekhar
limit. Effects of GEUP in cosmology has been studied in [56]. Modified
uncertainty principles that induce a maximum length have been employed in
order to describe the cosmological particle horizon [57]. This modified
uncertainty principle has terms proportional to even powers in the position
uncertainty [58]. Cosmological observational data have been used to obtain
constraints for GUP parameters [21, 42, 48, 56, 59].

Even though the de Sitter Cosmology Model describes the rapidly expansion
phase of the Universe, during which vacuum energy dominates, its traditional
treatment considers a system without physical content since there is only a
single degree of freedom and one constraint [60]. Schutz formalism [61, 62],
which describes a relativistic fluid interacting with the gravity field, can
be used to regard the vacuum as a dynamic entity having different degrees of
freedom. This procedure overcomes the difficulties found in de Sitter Model
and it leads to a natural way of introducing a variable playing the role of
time. Then, the Dynamical Vacuum Model can be implemented in a homogeneous and
isotropic universe filled with vacuum fluid whose state equation is $P=-\rho$,
which is classically treated according to the Schutz canonical formalism. This
process implies a linear term in one of the momenta in the Hamiltonian whose
associated degree of freedom will play the role of time [63].

In general, with the intention of avoiding the problems caused by the WDW
equation defined in the superspace444As far as we know, there is no known
general solution for the WDW equation in the superspace. (the space of all
possible three dimensional metrics), the process of quantization is performed
using the mini-superspace approach [64], where an infinite number of degrees
of freedom of the gravity field is frozen555In fact, the mini-superspace
approach is an approximation in which are only considered the largest gravity
field wavelength modes of the order of the size of the Universe. and remaining
degrees of freedom are turned operators.

Of course, if GUP is a fundamental aspect of the nature then not only the
position and the momentum have to obey the GUP but also every other variables
which will be quantized [65]. Since GUP corresponds to a modification of the
commutation relation between the operator and its conjugate operator, the WDW
equation in a minimal-length scenario can be obtained imposing that some or
all of those operators (which have come from the quantization of remaining
degrees of freedom in the mini-superspace approach) and its conjugate momentum
operators satisfy modified commutation relations.

In this work, our primary purpose is to determine the corrections in the wave
function of the Universe due to the use of a specific generalized uncertainty
principle (GUP) in a quantum cosmology model. Hence, we determine the modified
WDW equation up to ${\cal O}(\beta^{2})$ considering a quantum cosmology model
of dynamical vacuum in a minimal-length scenario induced by the commutation
relation proposed by P. Pedram [66, 67],

$[\hat{X},\hat{P}]:=\frac{i\hbar}{1-\beta\hat{P}^{2}},$

which induces a minimal uncertainty in the position, $\Delta
x_{min}=\frac{3\sqrt{3}}{4}\hbar\sqrt{\beta}$, and an upper bound on the
conjugate momentum, $P_{max}=\frac{1}{\sqrt{\beta}}$. In the equation above
$\beta$ is a parameter related to the minimal length. An important aspect of
the GUP associated with this modified commutation relation is that it is not
perturbative (in the minimal length). Moreover, it is consistent with Doubly
Special Relativity (DSR) theories which predicts an upper bound for particle
momentum [59, 68, 69, 70] and it is in agreement with several theoretical
proposals for quantum gravity [66]. We find the wave function of the Universe
for small scale factors in the formal representation of the “position” space,
in fact representation of the scale-factor space666We say representation of
the “position” space keeping in mind that it should be the more correct to
call represetation of the scale-factor space.. However, it is not possible to
obtain from wave function any physical information since the eigenstates of
the “position” operators are not physical states [37]. We overcome this
problem by obtaining the wave function in the representation of the quasi-
position space as a superposition of wave functions in the formal
representation of the “position” space [71]. Even if it may be questioned
whether this procedure is correct, we find the modified WDW equation in the
representation of the quasi-position space and we show that the wave function
in the quasi-position space previously obtained is solution this equation, as
we expected. Although the wave functions in the quasi-position space, which
are eigenfunctions of the cosmological constant, are physically acceptable
states they are not realizable states of the Universe because have infinite
norm. Consequently, it will be necessary to construct wave packets from the
superposition of eigenfunctions with different eigenvalues of the cosmological
constant. Last but not least, it should be said that the quantization process
occurs into the mini-superspace scenario in which the scale factor is the only
degree of freedom. As far as we know, the first applications of a minimal-
length scenario to a mini-superspace dynamics can be found in [5, 7, 8].

In short, in this paper we intend to study a dynamical vacuum, using a
description given by the Schutz formalism, in a quantum cosmological scenario
where the minimal-length proposal is explicitly considered. Our main aim is to
verify how GUP affects the solutions for the WDW equation with a dynamical
vacuum state, in the mini-superspace, in comparison with the usual approach.

The paper is organized in the following way. In Section 2, we describe the
minimal-length scenario used, presenting its main results. In Section 3, we
describe the ordinary cosmology model, that is, in a scenario without minimal
length. We obtain the gravity action and the fluid action according to the
Schutz formalism. In Section 4, we determine the modified WDW equation in the
formal representation of the “position” space and we find its solution for
small values of the scale factor. We also discuss about the validity range for
which the approach employed here is consistent. Then, we find the physically
acceptable solutions, that is, the wave function of the Universe in the
representation of the quasi-position space. Last we show that the norm of that
wave function is infinite, as in the ordinary case with no minimal length. In
Section 5, we present our comments and conclusions.

##  2 Minimal-Length Scenario

In a quantum approach, a minimal length can be introduced by modifying the HUP
in order to implement a non-zero minimal uncertainty in the position777It is
not trivial to show that a non-zero minimal uncertainty in position can be
interpreted as a minimal length [23, 37, 72].. There are many proposals for
modification of HUP [14, 37, 38, 50, 57, 66, 67, 73].

In this work we concern with the GUP

$\Delta X\Delta P\geq\frac{\frac{\hbar}{2}}{1-\beta\left[(\Delta
P)^{2}+\langle\hat{P}\rangle^{2}\right]},$ (1)

proposed by P. Pedram [66, 67], which induces a non-zero minimal uncertainty
in the position given by

$\Delta x_{min}=\frac{3\sqrt{3}}{4}\hbar\sqrt{\beta},$ (2)

where $\beta$ is a parameter related to the minimal length. We choose the GUP
(1) because it is consistent with several proposals for quantum gravity, such
as string theory, loop quantum gravity, and it also introduces a maximal
measured momentum888It is worth noting that this GUP does not induce a maximum
uncertainty in the conjugate momentum.,

$P_{max}=\frac{1}{\sqrt{\beta}},$ (3)

which is in agreement with the DSR. Another important aspect this GUP is that
it is not perturbative. Consequently, if $\beta$ is small then (1) can be
expanded until any order in $\beta$ we wish.

Since,

$\Delta X\Delta
P\geq\frac{1}{2}\left|\left\langle[\hat{X},\hat{P}]\right\rangle\right|,$ (4)

then related to the GUP (1) we have the modified commutation relation,

$[\hat{X},\hat{P}]:=\frac{i\hbar}{1-\beta\hat{P}^{2}}.$ (5)

Although, the representation of the operators:

$\hat{X}=\hat{x},$ (6)
$\hat{P}=\hat{p}+\frac{\beta}{3}\hat{p}^{3}+\frac{\beta^{2}}{3}\hat{p}^{5}+\frac{4\beta^{3}}{9}\hat{p}^{7}+\dots,$
(7)

where $\hat{x}$ and $\hat{p}$ are the ordinary operators of position and
momentum satisfying the canonical commutation relation
$[\hat{x},\hat{p}]:=i\hbar$, is not an exact representation of the algebra
(5), it preserves the ordinary form of the position operator. Thus, in this
representation of “position” space we have,

$\langle x|\hat{X}|\psi(t)\rangle=x\psi(x,t),$ (8) $\langle
x|\hat{P}|\psi(t)\rangle=$ $\left(-i\hbar\frac{\partial}{\partial
x}+i\hbar^{3}\frac{\beta}{3}\frac{\partial^{3}}{\partial
x^{3}}-i\hbar^{5}\frac{\beta^{2}}{3}\frac{\partial^{5}}{\partial
x^{5}}+\dots\right)\psi(x,t),$ (9)

where $\lvert x\rangle$ are the state eigenvectors of the position operator.

This representation of “position” space is only formal since the eigenvalues
of the position operator (6) are not physical states and consequently they do
not belong to the Hilbert space. This is because the position operator
uncertainty vanishes when it is calculated in any of its eigenstates. But that
is physically impossible since $\Delta
x\geq\frac{3\sqrt{3}}{4}\hbar\sqrt{\beta}$ for all physically allowable state
in a minimal-length scenario. However, all information on position can be
accessible through the maximal localization states, defined as,

$\langle\psi_{\xi}^{ml}|\hat{X}|\psi_{\xi}^{ml}\rangle=\xi$ (10)

and

$\left(\Delta x\right)_{\lvert\psi_{\xi}^{ml}\rangle}=\left(\Delta
x\right)_{min}.$ (11)

In the DGS (Detournay, Gabriel and Spindel) approach [74] the maximal
localization states are found to be,999Note that it is in the representation
of the ordinary momentum space.

$\psi_{\xi}^{ml}(p)=\sqrt{\frac{2\sqrt{3}}{2}}\exp\left(-\frac{i}{\hbar}\xi
p\right)\cos\left(\frac{3\pi}{4}\sqrt{\beta}p\right).$ (12)

The representation of quasi-position space is obtained by projecting the state
vectors onto the maximal localization states,

$\langle\psi_{\xi}^{ml}|\psi(t)\rangle=\psi_{qp}(\xi,t).$ (13)

The action of the position and the momentum operators on the quasi-position
space are given by,

$\langle\psi_{\xi}^{ml}|\hat{X}|\psi(t)\rangle=\left[\xi+ib\sqrt{\beta}\tan\left(-ib\sqrt{\beta}\frac{\partial}{\partial\xi}\right)\right]\psi_{qp}(\xi,t)$
(14)

and

$\langle\psi_{\xi}^{ml}|\hat{P}|\psi(t)\rangle=\langle\psi_{\xi}^{ml}|\left(\hat{p}+\frac{\beta}{3}\hat{p}^{3}+\frac{\beta^{2}}{3}\hat{p}^{5}+\dots\right)|\psi(t)\rangle,$
(15)

with

$\langle\psi_{\xi}^{ml}|\hat{p}|\psi(t)\rangle=-i\hbar\frac{\partial\psi_{qp}(\xi,t)}{\partial\xi},$
(16)

and

$b:=\frac{3\pi\hbar}{4}.$ (17)

Hence, up to ${\cal O}(\beta^{2})$ we have

$\langle\psi_{\xi}^{ml}|\hat{X}|\psi(t)\rangle=\left(\xi+\beta
b^{2}\frac{\partial}{\partial\xi}-\beta^{2}\frac{b^{4}}{3}\frac{\partial^{3}}{\partial\xi^{3}}\right)\psi_{qp}(\xi,t),$
(18)

and

$\langle\psi_{\xi}^{ml}|\hat{P}|\psi(t)\rangle=$
$\left(-i\hbar\frac{\partial}{\partial\xi}+i\hbar^{3}\frac{\beta}{3}\frac{\partial^{3}}{\partial\xi^{3}}-i\hbar^{5}\frac{\beta^{2}}{3}\frac{\partial^{5}}{\partial\xi^{5}}\right)\psi_{qp}(\xi,t).$
(19)

Finally, one can show that the wave function in the quasi-position space is a
superposition of the wave functions in the formal “position” space, given by

$\psi_{qp}(\xi)=\frac{1}{\sqrt{2}}\left[\psi(\xi+b\sqrt{\beta})+\psi(\xi-b\sqrt{\beta})\right].$
(20)

##  3 The Model

We will now consider the de Sitter Cosmology Model, which describes the phase
of the Universe with an exponential expansion, during which the vacuum energy
dominates the energy density and gives rise the term corresponding to the
cosmological constant $\Lambda$.

###  3.1 Gravity Action

The more general gravity action leading up to second order differential
equations is given by,101010In units so that $c=1$ and $\frac{16\pi
G}{c^{4}}=1$.

$S_{g}=\int_{M}d^{4}x\sqrt{-g}(R-2\Lambda)+2\int_{\partial M}d^{3}x\sqrt{h}K,$
(21)

where $R$ is the Ricci curvature scalar, $K$ is the trace of the extrinsic
curvature $K_{ij}$, $g$ is the determinant of the metric $g_{\mu\nu}$, $h$ is
the determinant of the induced metric over the three-dimensional spatial
hypersurface and $\partial M$ is the bounded of the four-dimensional manifold
$M$. The cosmological constant $\Lambda$ appears as a geometric term and its
value is arbitrary. In fact, the first integral in the l.h.s. of (21)
constitutes the most general geometrical Lagrangian in four dimensions leading
to second order differential equations according to the Lovelock theorem [75].

According to the Cosmological Principle, that is, for a homogeneous and
isotropic universe, the metric is that of FLRW (Friedmann-Lemaître-Robertson-
Walker),

$ds^{2}=-N^{2}(t)dt^{2}+$ (22)
$a^{2}(t)\left[\frac{dr^{2}}{\left(1-kr^{2}\right)}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}\right],$

where $N(t)$ is the lapse function, $k=-1,0,+1$ for hyperbolic open, flat open
and close universe, respectively, and $a(t)$ is the scale factor of universe.

Using the FLRW metric, Eq. (22), the gravity action becomes

$S_{g}=\int dt\left(-\frac{6\dot{a}^{2}a}{N}+kNa-2Na^{3}\Lambda\right).$ (23)

Since $S_{g}=\int dtL_{g}$ we have that the gravity Lagrangian is given by

$L_{g}=-\frac{6\dot{a}^{2}a}{N}+kNa-2Na^{3}\Lambda.$ (24)

Hence the gravity Hamiltonian, $NH_{g}=\dot{a}p_{a}-L_{g}$, reads

$H_{g}=-\frac{p_{a}^{2}}{24a}-ka+2\Lambda a^{3},$ (25)

where

$p_{a}=\frac{\partial L_{g}}{\partial\dot{a}}=-\frac{12\dot{a}a}{N}.$ (26)

###  3.2 Fluid Action

In our model the Universe is fully composed of perfect fluid. Now, we will
employ the Schutz’s formalism [61, 62] for the dynamics description the fluid
interacting with the gravitational field [63]. For the fluid which the state
equation is,

$p=\alpha\rho,$ (27)

where $\rho$ is the density and $\alpha$ is the equation of state parameter
depending on the type of fluid, the pressure is given by,

$p=\alpha\left(\frac{\mu}{\alpha+1}\right)^{1+\frac{1}{\alpha}}e^{-\frac{S}{\alpha}},$
(28)

where $\mu$ is the specific enthalpy and $S$ is the entropy.

In the Schutz’s formalism the four-velocity is defined by means of six
potentials:

$U_{\nu}:=\frac{1}{\mu}\left(\phi,_{\nu}+\epsilon\eta,_{\nu}+\theta
S,_{\nu}\right).$ (29)

Since $\epsilon$ and $\eta$ are associated with rotation movement, they
vanish. From the normalization condition,

$U^{\nu}U_{\nu}=-1,$ (30)

we obtain,

$\mu=\frac{1}{N}\left(\dot{\phi}+\theta\dot{S}\right).$ (31)

The fluid action is given by

$S_{F}=\int_{M}d^{4}x\sqrt{-g}p.$ (32)

Using Eq. (31) into Eq. (28) and since the spatial section is homogeneous and
isotropic, we have,

$S_{F}=\int
dtNa^{3}\alpha\left[\frac{\dot{\phi}+\theta\dot{S}}{N(\alpha+1)}\right]^{1+\frac{1}{\alpha}}e^{-\frac{S}{\alpha}}.$
(33)

Thus, the fluid Lagrangian is,

$L_{F}=\frac{\alpha
a^{3}}{N^{\frac{1}{\alpha}}(1+\alpha)^{1+\frac{1}{\alpha}}}\left(\dot{\phi}+\theta\dot{S}\right)^{1+\frac{1}{\alpha}}e^{-\frac{S}{\alpha}}.$
(34)

The conjugate canonical momenta to $\phi$ and $S$ are,

$p_{\phi}=\frac{\partial
L_{F}}{\partial\dot{\phi}}=\frac{a^{3}\left(\dot{\phi}+\theta\dot{S}\right)^{\frac{1}{\alpha}}e^{-\frac{S}{\alpha}}}{N^{\frac{1}{\alpha}}(1+\alpha)^{\frac{1}{\alpha}}}$
(35)

and

$p_{S}=\frac{\partial L_{F}}{\partial\dot{S}}=\theta p_{\phi}.$ (36)

Therefore, the fluid Hamiltonian,
$NH_{F}=\dot{\phi}p_{\phi}+\dot{S}p_{S}-L_{F}$, is given by,

$H_{F}=\frac{p_{\phi}^{1+\alpha}e^{S}}{a^{3\alpha}}.$ (37)

From Eqs. (36) and (25), it follows that the total Hamiltonian,
$H=H_{g}+H_{F}$, is

$H=-\frac{p_{a}^{2}}{24a}-ka+2\Lambda
a^{3}+\frac{p_{\phi}^{1+\alpha}e^{S}}{a^{3\alpha}}.$ (38)

Now, performing the canonical transformation (re-parametrization)

$T:=-p_{S}e^{-S}p_{\phi}^{-(1+\alpha)},$ (39)

and

$p_{T}:=p_{\phi}^{1+\alpha}e^{S},$ (40)

the total Hamiltonian becomes,

$H=-\frac{p_{a}^{2}}{24a}-ka+2a^{3}\Lambda+\frac{p_{T}}{a^{3\alpha}}.$ (41)

Note that $p_{T}$ is linear in Eq. (41). When the quantization is implemented
this fact allows us to consider $t=-T$ as playing role of time and to obtain a
Schroedinger-like equation.

##  4 The Quantum Cosmology Model in a Minimal-Length Scenario

###  4.1 The modified Wheeler-DeWitt equation

The quantization process takes place in the framework of mini-superspace and
according with the Wheeler-DeWitt quantization scheme,

$p_{a}\longrightarrow\hat{p}_{a}$ (42)

and

$p_{T}\longrightarrow\hat{p}_{T}\equiv-i\frac{\partial}{\partial T},$ (43)

such that $H\longrightarrow\hat{H}$, implying the WDW equation,

$\hat{H}\lvert\psi\rangle=0.$ (44)

In order to obtain a minimal-length scenario we demand that

$[\hat{a},\hat{p}_{a}]:=\frac{i\hbar}{1-\beta\hat{p}_{a}^{2}}.$ (45)

with the representation of the operators as follow:

$\hat{a}=\hat{x}$ (46)

and

$\hat{p}_{a}\equiv\hat{p}+\frac{\beta}{3}\hat{p}^{3}+\frac{\beta^{2}}{3}\hat{p}^{5}+{\cal
O}\left(\beta^{3}\right),$ (47)

where the $\hat{x}$ and $\hat{p}$ operators satisfy the canonical commutation
relation,

$[\hat{x},\hat{p}]:=i\hbar.$ (48)

Using the representation (46) and (47), the modified WDW equation turns out

$-\frac{\hat{x}^{(3\alpha-1)}}{24}\left[\hat{p}^{2}+\frac{2}{3}\beta\hat{p}^{4}+\frac{7}{9}\beta^{2}\hat{p}^{6}+{\cal
O}\left(\beta^{3}\right)\right]\lvert\psi(t)\rangle$
$+\left[-k\hat{x}^{(1+3\alpha)}+2\Lambda\hat{x}^{3(1+\alpha)}\right]\lvert\psi(t)\rangle=-i\frac{\partial}{\partial
t}\lvert\psi(t)\rangle,$ (49)

where we have made $T=-t$.

Now, projecting Eq. (50) onto the formal representation of the “position”
space we have,

$\langle x|\hat{H}|\psi(t)\rangle=0,$ (50)
$-\frac{x^{(3\alpha-1)}}{24}\left[-\frac{\partial^{2}}{\partial
x^{2}}+\frac{2}{3}\beta\frac{\partial^{4}}{\partial
x^{4}}-\frac{7}{9}\beta^{2}\frac{\partial^{6}}{\partial x^{6}}+{\cal
O}\left(\beta^{3}\right)\right]\psi(x,t)$ $+\left[-kx^{(1+3\alpha)}+2\Lambda
x^{3(1+\alpha)}\right]\psi(x,t)=-i\frac{\partial}{\partial t}\psi(x,t),$ (51)

where we have used Eqs. (8) and (9).

Assuming that the solution of the above equation can been written as
$\psi(x,t)=\varphi(x)\tau(t)$, we find the stationary solutions,

$\psi_{\omega}(x,t)=e^{-i\omega t}\varphi_{\omega}(x),$ (52)

and the time independent equation,

$\left[\frac{d^{2}}{dx^{2}}-\frac{2}{3}\beta\frac{d^{4}}{dx^{4}}+\frac{7}{9}\beta^{2}\frac{d^{6}}{dx^{6}}+{\cal
O}\left(\beta^{3}\right)\right]\varphi_{\omega}(x)$
$+\biggl{[}24x^{(1-3\alpha)}\omega-24kx^{2}+48\Lambda
x^{4}\biggr{]}\varphi_{\omega}(x)=0,$ (53)

where $\omega$ is a constant.

The equation of state parameter $\alpha$ defines the matter content in a
single fluid model. The case $\alpha=-1$ corresponds to the vacuum energy
equation of state. It comes from the quantum field contributions for the
energy associated with the vacuum state. Its precise value is the object of
discussions today, and it can be related to the dark energy component in the
Universe. Here, we consider it as a dynamic fluid component with internal
degrees of freedom. Remark that the contribution of this vacuum component for
the matter content, given by $\alpha=-1$, is similar to the geometric
cosmological constant $\Lambda$, but the physical meaning and origin are very
different.

Lastly, considering a flat universe111111We choose a flat universe because it
is in agreement with current cosmological observations [76, 77]. ($k=0$),
without any other matter content, fully completed by a dynamical vacuum with
an equation of state,

$P=-\rho,$ (54)

and $\Lambda=0$,121212In the same way as in [63] we hope that the cosmological
constant arises from the degrees of freedom of the vacuum and not being
postulated from the start as a geometric term. Note also that if from the
start $\Lambda>0$ we can always redefine $\omega$ as
$\omega\prime:=\omega+2\Lambda$. Eq. (53) up to ${\cal
O}\left(\beta^{2}\right)$ becomes,131313From now on, for the sake of
simplicity we are going to omit the subscript $\omega$.

$\frac{d^{2}\varphi}{dx^{2}}-\frac{2}{3}\beta\frac{d^{4}\varphi}{dx^{4}}+\frac{7}{9}\beta^{2}\frac{d^{6}\varphi}{dx^{6}}+24\omega
x^{4}\varphi=0.$ (55)

###  4.2 Solution of the modified Wheeler-DeWitt equation

It is not possible to solve Eq. (55) analytically since we do not know all
initial or boundary conditions141414In Ref. [2] the author has solved a
modified WDW equation of forth order $\left(\frac{d^{4}}{dx^{4}}\right)$ using
the Sommerfeld polynomial method. However, the solution holds dependent on
four parameters that must be determined by the initial conditions, which are
not known., that is, $\varphi^{(N)}(x_{0})$, for $N=0,1,...,5$. A way to get
around this problem is considering $x$ small. This is well reasonable since we
are interested in solutions describing the Universe in its quantum regime,
that is, in the initial phases of the Universe, when the scale factor was
small151515Later on, we will discuss more carefully the issue if $x$ can
rigorously describe the scale factor. and as it is known quantum effects are
significant only for small values of the scale factor.

Initially, we take into account Eq. (55) in the limit $\beta=0$, which we call
ordinary WDW equation,

$\frac{d^{2}\varphi_{0}}{dx^{2}}+24\omega x^{4}\varphi_{0}=0.$ (56)

Making $\varphi_{0}=\sqrt{x}y(x)$ and the change of variable
$\rho:=\sqrt{\frac{8\omega}{3}}x^{3}$, Eq. (55) turns the Bessel equation,

$\frac{d^{2}y}{d\rho^{2}}+\frac{1}{\rho}\frac{dy}{d\rho}+\left[1-\frac{\left(\frac{1}{6}\right)^{2}}{\rho^{2}}\right]y=0.$
(57)

Then, the solution of Eq. (57) is

$\varphi_{0}(x)=A\sqrt{x}J_{1/6}\left(K_{0}x^{3}\right)+B\sqrt{x}N_{1/6}\left(K_{0}x^{3}\right),$
(58)

where $K_{0}:=\sqrt{\frac{8\omega}{3}}$, $J_{1/6}(X)$ and $N_{1/6}(X)$ are the
Bessel and the Neumann functions of order $1/6$, respectively, and $A$ and $B$
are constants.

It is easy to see that in the limit for small $x$,

$J_{1/6}(X)\approx\sqrt{x}$ (59)

and

$N_{1/6}(X)\approx\frac{1}{\sqrt{x}}.$ (60)

Now, imposing the condition161616That condition is obtained by demanding that
the Hamiltonian $H$ must be self-adjoint [63]. That condition should not
change, even though it is probably that boundary conditions satisfied by
derivatives change. Note that condition does not avoid the initial singularity
[8, 9]. $\varphi_{0}(0)=0$, we get $B=0$, and the solution is given by,

$\varphi_{0}(x)=A\sqrt{x}J_{1/6}\left(K_{0}x^{3}\right).$ (61)

Remembering that

$J_{\nu}(X)=\sum_{N=0}^{\infty}\frac{(-1)^{N}}{N!\Gamma(N+\nu+1)}\left(\frac{X}{2}\right)^{2N+\nu},$
(62)

and retaining the first three significant terms of $\varphi_{0}(x)$ we get,

$\varphi_{0}(x)=C_{1}x+C_{2}x^{7}+C_{3}x^{13}.$ (63)

It is appropriate to rewrite the solution (63) by absorbing $C_{1}$ in a
normalization constant, which we are going to omit without affecting the
results:

$\varphi_{0}(x)=x-\frac{4}{7}\omega x^{7}+\frac{8}{91}\omega^{2}x^{13}.$ (64)

We can obtain an approximate solution for Eq. (55), in the range of $x$ small,
using Eq. (64) into ${\cal O}\left(\beta\right)$ and ${\cal
O}\left(\beta^{2}\right)$ terms. Thus,

$\frac{d^{4}\varphi_{0}}{dx^{4}}=-480\omega
x^{3}+\frac{10560}{7}\omega^{2}x^{9}$ (65)

and

$\frac{d^{6}\varphi_{0}}{dx^{6}}=-2880\omega
x+\frac{760320}{7}\omega^{2}x^{7}.$ (66)

With the approximation $\varphi\approx x$ into Eqs. (65) and (66), Eq. (55)
becomes,

$\frac{d^{2}\varphi}{dx^{2}}+24\omega x^{4}\varphi+\beta\left(320\omega
x^{2}\varphi-\frac{7040}{7}\omega^{2}x^{8}\varphi\right)+$
$\beta^{2}\left(-2240\omega\varphi+591360\omega^{2}x^{6}\varphi\right)=0.$
(67)

Subsequently, we are going to find the approximate solution of the above
equation in two different ways.

####  4.2.1 Solution: first method

Since $x=0$ is an ordinary point of Eq. (67) we can find a solution in power
series of $x$. Then, replacing $\varphi(x)=\sum_{n=0}^{\infty}a_{n}x^{n}$ into
Eq. (67) and retaining only significant terms, we have,

$\varphi(x)=x-\frac{4}{7}\omega x^{7}+\frac{8}{91}\omega^{2}x^{13}+$
$\beta\left(-16A\omega
x^{5}+\frac{5504}{385}B\omega^{2}x^{11}\right)+\beta^{2}\frac{1120}{3}C\omega
x^{3}.$ (68)

The parameters $A$, $B$ and $C$ are determined by requiring that (68)
satisfies Eq. (67) up to order chosen in $x$. Then, substituting (68) into
(67) we have $A=1$, $B=\frac{38}{43}$ and $C=\frac{3}{7}$. Therefore, the
solution reads as,

$\varphi(x)=x-\frac{4}{7}\omega x^{7}+\frac{8}{91}\omega^{2}x^{13}+$
$\beta\left(-16\omega
x^{5}+\frac{4864}{385}\omega^{2}x^{11}\right)+\beta^{2}160\omega x^{3}.$ (69)

####  4.2.2 Solution: second method

The same previous solution can be obtained by taking the approximation
following,

$\frac{d^{2}\varphi}{dx^{2}}=-24\omega
x^{4}\varphi_{0}+\frac{2}{3}\beta\frac{d^{4}\varphi_{0}}{dx^{4}}-\frac{7}{9}\beta^{2}\frac{d^{6}\varphi_{0}}{dx^{6}}.$
(70)

Integrating twice the above equation we obtain,

$\frac{d^{2}\varphi}{dx^{2}}=-24\omega\int dx\int dxx^{4}\varphi_{0}+$
$\frac{2}{3}\beta\frac{d^{2}\varphi_{0}}{dx^{2}}-\frac{7}{9}\beta^{2}\frac{d^{4}\varphi_{0}}{dx^{4}}+\alpha_{1}x+\alpha_{2},$
(71)

where $\alpha_{1}$ and $\alpha_{2}$ are integration constants. After that,
using (64) into (71) we obtain,

$\varphi(x)=\alpha_{2}+\alpha_{1}x-\frac{4}{7}\omega
x^{7}+\frac{8}{91}\omega^{2}x^{13}+$ $\beta\left(-16A^{\prime}\omega
x^{5}+\frac{64}{7}B^{\prime}\omega^{2}x^{11}\right)+\beta^{2}\frac{120}{3}C^{\prime}\omega
x^{3}.$ (72)

The constants $\alpha_{1}$ and $\alpha_{2}$ can be determined by demanding
that $\varphi(x)\longrightarrow\varphi_{0}(x)$ when $\beta\longrightarrow 0$.
In a similar way, the parameters $A^{\prime}$, $B^{\prime}$ and $C^{\prime}$
are found requiring that the solution (72) satisfies Eq. (67). Consequently,
we again have the solution (69), as we expected.

###  4.3 Validity range

In order that the solution (69) works consistently the first disregarded term
in $\varphi_{0}(x)$ must be much smaller than the last kept terms in ${\cal
O}(\beta)$ and ${\cal O}\left(\beta^{2}\right)$, that is,
$x\ll\left(\frac{\beta}{\omega}\right)^{\frac{1}{8}}$. Since $\beta$ is much
small, this drastically reduces the validity range for our solution. We can
improve this if terms of higher powers in $x$ are taken account in order
${\cal O}\left(\beta^{0}\right)$ part of the solution. Therefore we can
consider $\sqrt{x}J_{1/6}\left(K_{0}x^{3}\right)$ as the part in ${\cal
O}\left(\beta^{0}\right)$ of $\varphi(x)$, that is, $\varphi_{0}(x)$. It is
clear that terms of order such that $x^{6n-2}<\beta^{2}\omega^{1-n}$ lie far
outside the validity range and they must throw away.

The same reasoning applies in the case of terms in ${\cal
O}\left(\beta^{2}\right)$ when compared to terms in ${\cal O}(\beta)$. This
way, with the purpose of increasing the validity range of our solution we take
account the two first significant terms in ${\cal O}(\beta)$. In conclusion,
in light of the forgoing the solution turns out to be,171717Note that solution
is non-normalized.

$\varphi(x)=\sqrt{x}J_{1/6}\left(K_{0}x^{3}\right)+$ $\beta\left(-16\omega
x^{5}+\frac{4864}{385}\omega^{2}x^{11}\right)+\beta^{2}160\omega x^{3}.$ (73)

In dealing with expansions of small parameters (in this case $\beta$) it is
necessary to take care to consistently work when $x$ goes to an extremely
small value. It is easy to see that terms in $\beta^{0}$ are like
$\omega^{N}x^{1+6N}$, terms in $\beta$ are like $\beta\omega^{N}x^{5+6N}$ and
terms in $\beta^{2}$ are like $\beta^{2}\omega^{N}x^{3+6N}$. Therefore, the
terms in $\beta\omega x^{5}$ and $\beta\omega^{2}x^{11}$ are negligible when
$x<\sqrt{\beta}$ and the term $\beta\omega^{2}x^{11}$ is negligible when
$\sqrt{\beta}<x<\beta^{1/8}$. Note that the term in $\beta^{0}x$ always will
be greater than the term in $\beta^{2}\omega x^{3}$ for $x$ small.

Keep in mind that in fact solution (73) is $\varphi_{\omega}(x)$, that is,
eigenfunctions whose associated eigenvalues are values of the cosmology
constant.

###  4.4 Physically acceptable solutions

As we have already said, we can not obtain directly from $\varphi(x)$ physical
results because the $\hat{x}$ operator eigenstates do not belong to the Hibert
space. Nevertheless, the projections of the state vectors
$\lvert\varphi\rangle$ on the maximal localization states, that is, the wave
functions in the representation of the quasi-position space do. The wave
functions in the quasi-position space can easily be obtained from (73) by
using Eq. (20). Therefore, replacing (73) into (20) we have181818Remember that
$b:=\frac{3\pi}{4}$, Eq. (17).

$\varphi_{qp}(\xi)=\sqrt{\xi}J_{1/6}\left(K_{0}\xi^{3}\right)+$
$\beta\left[-\left(16+12b^{2}\right)\omega\xi^{5}+\left(\frac{4864}{385}+\frac{48}{7}b^{2}\right)\omega^{2}\xi^{11}\right]+$
$\beta^{2}\left(160-160b^{2}-20b^{4}\right)\omega\xi^{3}.$ (74)

Hence, above quasi-position wave function represents the probability amplitude
for the Universe being maximally localized around the position $\xi$.

The reader may be questioning if the correct procedure would be to determine
the solution of the modified WDW equation in the representation of the quasi-
position space,

$\langle\psi_{\xi}^{ml}|\left(-\hat{p}_{a}^{2}+24\omega\hat{a}^{4}\right)|\psi(t)\rangle=0.$
(75)

Using Eqs. (18) and (19) we obtain

$\frac{d^{2}\varphi_{qp}}{d\xi^{2}}+24\omega\xi^{4}\varphi_{qp}+$ $\beta
b^{2}\left[-\frac{2}{3b^{2}}\frac{d^{4}\varphi_{qp}}{d\xi^{4}}+144\omega\xi^{2}\varphi_{qp}+96\omega\xi^{3}\frac{d\varphi_{qp}}{d\xi}\right]+$
$\beta^{2}b^{4}\left[-\frac{7}{9b^{4}}\frac{d^{6}\varphi_{qp}}{d\xi^{6}}+24\omega\varphi_{qp}+96\omega\xi\frac{d\varphi_{qp}}{d\xi}\right]+$
$\beta^{2}b^{4}\left[-24\omega\xi^{2}\frac{d^{2}\varphi_{qp}}{d\xi^{2}}-40\omega\xi^{3}\frac{d^{3}\varphi_{qp}}{d\xi^{3}}\right]=0.$
(76)

As it can easily be checked (after some algebra) the solution (74) satisfies
Eq. (76) up to the considered validity order.

###  4.5 Norm of the Wave Function

In the ordinary case the norm of the wave function solution of the WDW
equation is infinite [63]. This means that stationary states of the Universe
with well-defined values of the cosmological constant are not realizable and
it is necessary the construction of wave packets.

Now, we are going to calculate the norm of the wave function (74). Since the
determination of the norm of the wave function in the quasi-position space can
be laborious, it is convenient to perform first the calculation formally in
the “position” space and after to obtain the normalized wave function (if it
is normalized) in the quasi-position space using Eq. (20).

In the “position” space the inner product does not change, so

$\left\langle\psi_{1}\mid\psi_{2}\right\rangle=\int_{0}^{\infty}\psi_{2}^{*}(x)\psi_{1}(x)dx.$
(77)

On the basis of solution (69) we assume that $\psi_{1}^{(n)}(0)=0$ for
$n=0,2,4,\dots$. Thus, it is easy to show that the operator,

$\hat{D}:=\frac{d^{2}}{dx^{2}}-\frac{2}{3}\beta\frac{d^{4}}{dx^{4}}+\frac{7}{9}\beta^{2}\frac{d^{6}}{dx^{6}}$
(78)

is self-adjoint191919Note that $\hat{D}$ is not defined on the whole Hilbert
space ${\cal H}$ because $\hat{D}$ is an unbounded linear operator. Thus, the
domain of $\hat{D}$ is a dense subspace into ${\cal H}$..

However, the operator $\hat{H}:=\hat{D}+24\omega x^{4}$ is not self-adjoint in
the inner product (77), but rather in the inner product

$\left\langle\psi_{1}\mid\psi_{2}\right\rangle=\int_{0}^{\infty}x^{4}\psi_{2}^{*}(x)\psi_{1}(x)dx,$
(79)

where $x^{4}=w(x)$ is a weight function.

Some caution is necessary with the calculation of the norm of the solution
(73) because the solution (73) is only valid for small values of $x$.
Therefore it can not be used throughout the whole positive $x$-axis. Thus, we
consider that the effects of the existence of a minimal length are negligible
for values of $x$ greater than those of the validity range of the solution
(73), say $x_{M}$. Consequently, the solution turns to the ordinary
$\varphi_{0}(x)$ for $x>x_{M}$. In this way,

$\left\langle\varphi\mid\varphi\right\rangle=\int_{0}^{x_{M}}x^{4}\mid\varphi(x)\mid^{2}dx+\int_{x_{M}}^{\infty}x^{4}\mid\varphi_{0}(x)\mid^{2}dx.$
(80)

Note that if $x_{M}\approx\left(\frac{\beta}{\omega}\right)^{\frac{1}{8}}$
then $\varphi(x)$ of the first integral in the l.h.s. of (80) is given by Eq.
(69), according to subsection 4.3. Because of the second integral in the
l.h.s. of (80) the norm is infinite. This result was already expected since
the divergence takes place for $x\rightarrow\infty$ where the effects of a
minimal length are negligible. In the ordinary case, the problem with the
infinite norm is solved by constructing a wave packet. The construction of a
wave packet when the minimal length is presented is much less simple due to
the modifications given by the existence of a minimal length near the origin.

Although the wave function (74) can not be normalized, it is interesting to
note that the leading correction term in ${\cal O}(\beta)$ of
$|\varphi_{qp}(\xi)|^{2}$ is

$-2\beta\left(16+12b^{2}\right)\omega\xi^{6}.$

Consequently, $|\varphi_{qp}|^{2}$ grows more slowly than $|\varphi_{0}|^{2}$.
In [47] the author202020The author has considered that the variable
$u:=a^{\frac{2}{3}}$ and its conjugate momentum
$p_{u}=\frac{2}{3}a^{-\frac{1}{2}}p_{a}$ satisfy the commutation relation
proposed by Kempf, Mangano and Mann [37] when turned operators. has found that
the leading correction term is positive implying a more rapid growth of
$|\varphi|^{2}$.

##  5 Conclusion

In this work, we performed the study of the primordial Universe by
implementing a quantum approach in the gravitation effects. It follows the
need to implement a minimal-length scenario, which is carried out by using a
GUP. Since we chose the GUP (1) we can expand the representation of the
momentum until ${\cal O}(\beta^{2})$ and thus to obtain a modified WDW
equation up to ${\cal O}(\beta^{2})$, too.

We found the modified WDW equation in the formal representation of “position”
space, Eq. (55), and its solution $\varphi(x)$, Eq. (73), because it is
simpler than in the representation of quasi-position space. However, we can
not obtain directly from $\varphi(x)$ physical results. Consequently we
obtained the wave function of the Universe in the representation of quasi-
position space, $\varphi_{qp}(\xi)$, as a superposition of two wave functions
of the Universe in the formal representation of “position” space,
$\varphi_{qp}(\xi)=\frac{1}{\sqrt{2}}\left[\varphi(\xi+x_{min})+\varphi(\xi-
x_{min})\right]$. With the aim of insuring our result, we found the modified
WDW equation in the representation of quasi-position space, Eq. (76), and we
checked that $\varphi_{qp}(\xi)$ is actually its solution.

The ignorance of the initial or boundary conditions, that is, of the
derivatives of the wave function does not allow us to find an exact solution
for modified WDW equation, what forced us to seek a solution for small values
of the scale factor of the Universe.

Since in the ordinary case the divergence in the norm of the wave function
takes place for $x\rightarrow\infty$, the introduction of a minimal length in
the theory can not change this, because its effects are only significant for
small $x$. Therefore realizable states of the Universe are wave packets
constructed by superposition of the stationary states of different values of
$\omega$, that is, of the cosmological constant.

We should note that different changes in the WDW equation due to different
GUP’s have been found in the literature. For example, a linear GUP [38] has
led to the emergence of third and fourth order derivatives in the modified WDW
equation [2, 11, 12, 39], whereas the KMM GUP [37] has only induced to fourth
order derivatives [5, 19, 47, 78]. Those results are in agreement with our
modified WDW equation which also displays a term of fourth order derivative in
${\cal O}(\beta)$.

Fluid radiation is another model of interest to study since the radiation-
dominated era has been presented in the early Universe when the quantum
effects were relevant. For a radiation fluid $\alpha=\frac{1}{3}$ and the
modified WDW equation is given by

$\frac{d^{2}\varphi}{dx^{2}}-\frac{2}{3}\beta\frac{d^{4}\varphi}{dx^{4}}+\frac{7}{9}\beta^{2}\frac{d^{6}\varphi}{dx^{6}}+24\omega\varphi=0.$
(81)

At first sight we could easily obtain an analytical solution of the above
equation. For this we suppose $\varphi(x)=e^{kx}$ in order to get an algebraic
equation of 6th order which can be transformed in one of 3rd order. However,
we do not know the initial or boundary conditions in order to determine the 6
constants of the general solution. Again, we need to appeal to the
approximation for small $x$.

In next works, we are going to study the effects on the scale factor
evolution, applications of the obtained results and comparisons with others
models and GUP’s.

## Acknowledgements

We would like to thank FAPES, CAPES and CNPq (Brazil) for financial support.

## References

  * [1] S. Hossenfelder _“A note on theories with a minimal length”_ , Class. Quantum Grav. 23, 1815 (2006). doi.org/10.1088/0264-9381/23/5/N01
  * [2] M. Kober, _“Generalized Uncertainty Principle in Canonical Quantum Gravity and Application to Quantum Cosmology”_ , Int. J. Mod. Phys. A 27(20), 1250106 (2012). doi.org/10.1142/S0217751X12501060
  * [3] Y. Sabri and Kh. Nouicer, _“Phase transitions of a GUP-corrected Schwarzschild black hole within isothermal cavities”_ , Class. Quantum Grav. 29(21), 215015 (2012). doi:10.1088/0264-9381/29/21/215015.
  * [4] Y. C. Ong and Y. Yao _“Generalized uncertainty principle and white dwarfs redux: How the cosmological constant protects the Chandrasekhar limit”_ , Phys. Rev. D 98(12), 126018 (2018). doi.org/10.1103/PhysRevD.98.126018
  * [5] B. Vakili and H. R. Sepangi, _“Generalized uncertainty principle in Bianchi type I quantum cosmology”_ , Phys. Lett. B 651(2-3), 79 (2007). doi.org/10.1016/j.physletb.2007.06.015
  * [6] M. V. Battisti and G. Montani, _“The Big-Bang singularity in the framework of a Generalized Uncertainty Principle”_ , Phys. Lett. B 656(1-3), 96 (2007). doi.org/10.1016/j.physletb.2007.09.012.
  * [7] M. V. Battisti and G. Montani, _“Minisuperspace dynamics in a generalized uncertainty principle framework”_ , AIP Conf. Proc. 966, 219Phys (2007). doi.org/10.1063/1.2836998
  * [8] M. V. Battisti and G. Montani, _“Quantum dynamics of the Taub universe in a generalized uncertainty principle framework”_ , Phys. Rev. D 77(2), 023518 (2008). doi.org/10.1103/PhysRevD.77.023518
  * [9] M. V. Battisti and G. Montani, _“Quantum Cosmology with a Minimal Length”_ , Int. J. Mod. Phys. A 23(8), 1257 (2008). doi.org/10.1142/S0217751X08040184
  * [10] A. F. Ali and B. Majumder, _“Towards a cosmology with minimal length and maximal energy”_ , Class. Quantum Grav. 31(21), 215007 (2014). doi.org/10.1088/0264-9381/31/21/215007
  * [11] M. Faizal, _“Deformation of the Wheeler DeWitt equation”_ , Int. J. Mod. Phys. A 29(20), 1450106 (2014). doi.org/10.1142/S0217751X14501061
  * [12] R. Garattini and M. Faizal, _“Cosmological constant from a deformation of the Wheeler-DeWitt equation”_ , Nucl. Phys. B 905, 313 (2016). dx.doi.org/10.1016/j.nuclphysb.2016.02.023
  * [13] O. Ökcü, C. Corda and E. Aydiner, _“Modified Friedmann equations from DSR-GUP”_ , Europhys. Lett. 129(5), 50002 (2020). doi.org/10.1209/0295-5075/129/50002
  * [14] Kh. Nouicer, _“Quantum-corrected black hole thermodynamics to all orders in the Planck length”_ , Phys. Lett. B 626(2-3), 63 (2007). doi.org/10.1016/j.physletb.2006.12.072
  * [15] Kh. Nouicer, _“Black hole thermodynamics to all orders in the Planck length in extra dimensions”_ , Class. Quantum Grav. 24(23), 5917 (2007). doi.org/10.1088/0264-9381/24/23/014
  * [16] W. Kim, E. J. Son and M. Yoon, _“Thermodynamics of a black hole based on a generalized uncertainty principle”_ , J. High Energy Phys. 2008, 035 (2008). doi.org/10.1088/1126-6708/2008/01/035
  * [17] Kh. Nouicer, _“Quantum gravity effects from generalized uncertainty principle”_ , Conference paper (2009). https://www.researchgate.net/publication/280384349
  * [18] Y. C. Ong, _“An effective black hole remnant via infinite evaporation time due to generalized uncertainty principle”_ , J. High Energ. Phys. 2018, 95 (2018). doi.org/10.1007/JHEP10(2018)195
  * [19] M. Faizal, _“Deformation of second and third quantization”_ , Int. J. Mod. Phys. A 30(9), 1550036 (2015). doi.org/10.1142/S0217751X15500360
  * [20] Y. C. Ong, _“Generalized uncertainty principle, black holes, and white dwarfs: a tale of two infinities”_ , J. Cosmo. Astro. Phys, 09, 015 (2015). ddoi.org/10.1088/1475-7516/2018/09/015
  * [21] A. M. Diab and A. N. Tawfik, _“A Possible Solution of the Cosmological Constant Problem based on Minimal Length Uncertainty and GW170817 and PLANCK Observations”_ , [arXiv:gr-qc/2005.03999].
  * [22] M. Khodadi, K. Nozari and E. N. Saridakis _“Emergent universe in theories with natural UV cutoffs”_ , Class. Quant. Grav. 35(1), 015010 (2018). doi.org/10.1088/1361-6382/aa95aa
  * [23] L. N. Chang, Z. Lewis, D. Minic and T. Takeuchi, _“On the Minumum Length Uncertainty Relation and the Foundations of String Theory”_ , Adv. High Energy Phys. 2011, 493514 (2011). doi.org/10.1155/2011/493514
  * [24] S. Hossenfelder, _“Minimal Length Scale Scenarios for Quantum Gravity”_ , Living. Rev. Rel. 16, 2 (2013). doi.org/10.12942/lrr-2013-2
  * [25] A. Tawfik and A. Diab, _“Generalized uncertainty principle: Approaches and applications”_ , Int. J. Mod. Phys. D 23(12), 1430025 (2014). doi.org/10.1142/S0218271814300250
  * [26] A. Tawfik and A. Diab, _“Review on Generalized Uncertainty Principle”_ , Rept. Prog. Phys. 78(12), 126001 (2015). doi.org/10.1088/0034-4885/78/12/126001
  * [27] G. Veneziano, _“A Stringy Nature Needs Just Two Constants ”_ , Europhys. Lett. 2(3), 199 (1986). doi.10.1209/0295-5075/2/3/006
  * [28] D. Amati, M. Ciafaloni and G. Veneziano, _“Superstring collisions at planckian energies”_ , Phys. Lett. B 197(1-2), 81 (1987). doi.org/10.1016/0370-2693(87)90346-7
  * [29] D. Amati, M. Ciafaloni and G. Veneziano, _“Can Space-Time Be Probed Below The String Size?”_ , Phys. Lett. B 216(1-2), 41 (1989). doi.org/10.1016/0370-2693(89)91366-X
  * [30] D. J. Gross and P. F. Mende, _“The high-energy behavior of string scattering amplitudes”_ , Phys. Lett. B 197(1-2), 129 (1987). doi.org/10.1016/0370-2693(87)90355-8
  * [31] D. J. Gross and P. F. Mende, _“String theory beyond the Planck scale”_ , Nucl. Phys. B 303(3), 407 (1988). doi.org/10.1016/0550-3213(88)90390-2
  * [32] K. Konishi, G. Paffuti and P. Provero, _“Minimum physical length and the generalized uncertainty principle in string theory”_ , Phys. Lett. B 234(3), 276 (1990). doi.org/10.1016/0370-2693(90)91927-4
  * [33] M. Maggiore, _“A Generalized uncertainty principle in quantum gravity”_ , Phys. Lett. B 304(1-2), 65 (1993). doi.org/10.1016/0370-2693(93)91401-8
  * [34] F. Scardigli, _“Generalized uncertainty principle in quantum gravity from micro-black hole gedanken experiment”_ , Phys. Lett. B 452(1-2), 39 (1999). doi.org/10.1016/S0370-2693(99)00167-7
  * [35] Mu-In Park, _“The generalized uncertainty principle in (A)dS space and the modification of Hawking temperature from the minimal length”_ , Phys. Lett. B 659(3), 698 (2008). doi.org/10.1016/j.physletb.2007.11.090
  * [36] F. Scardigli, _“Generalized uncertainty principle, extra dimensions and holography”_ , Class. Quant. Grav. 20(18), 3915 (2003). doi.org/10.1088/0264-9381/20/18/305
  * [37] A. Kempf, G. Mangano and R. B. Mann, _“Hilbert Space Representation Of The Minimal Length Uncertainty Relation”_ , Phys. Rev. D 52(2), 1108 (1995). doi.org/10.1103/PhysRevD.52.1108
  * [38] A. F. Ali, S. Das and E. C. Vagenas, _“Discreteness of space from the generalized uncertainty principle”_ , Phys. Lett. B 678(5), 497 (2009). doi.org/10.1016/j.physletb.2009.06.061
  * [39] B. Majumder, _“Effects of GUP in quantum cosmological perfect fluid models”_ , Phys. Lett. B 699(5), 315 (2011). doi.org/10.1016/j.physletb.2011.04.030.
  * [40] B. Majumder, _“Quantum black hole and the modified uncertainty principle”_ , Phys. Lett. B 701(4), 384 (2011). doi.org/10.1016/j.physletb.2011.05.076
  * [41] B. Majumder, _“The Generalized Uncertainty Principle and the Friedmann equations”_ , Astrophys. Space Sci. Lett. 336, 331 (2011). doi.org/10.1007/s10509-011-0815-6
  * [42] S. Giardino and V. Salzano, _“Cosmological constraints on GUP from modified Friedmann equations”_ , Eur. Phys. J. C 81, 110 (2021). doi.org/10.1140/epjc/s10052-021-08914-2
  * [43] A. Kempf, _“Mode generating mechanism in inflation with a cutoff”_ , Phys. Rev. D 63(8), 083514 (2001). doi.org/10.1103/PhysRevD.63.083514
  * [44] S. F. Hassan and M. S. Sloth _“Trans-Planckian effects in inflationary cosmology and the modified uncertainty principle”_ , Nucl. Phys. B 674(1-2), 434 (2003). oi.org/10.1016/j.nuclphysb.2003.09.041
  * [45] A. Ashoorioon, A. Kempf and R. B. Mann, _“Minimum length cutoff in inflation and uniqueness of the action”_ , Phys. Rev. D 71(2), 023503 (2005). doi.org/110.1103/PhysRevD.71.023503
  * [46] A. Kempf and L. Lorenz, _“Exact solution of inflationary model with minimum length”_ , Phys. Rev. D 74(10), 023503 (2006). doi.org/10.1103/PhysRevD.74.103517.
  * [47] B. Vakili and H. R. Sepangi, _“Cosmology with Minimal Length Uncertainty Relations”_ , Int. J. Mod. Phys. D 18(7), 1059 (2009). doi.org/10.1142/S0218271809014935
  * [48] A. N. Tawfik and A. M. Diab _“Generalized Uncertainty Principle and Recent Cosmic Inflation Observations”_ , Electron. J. Theor. Phys. 12(32), 9 (2015)
  * [49] A. Mohammadi, A. Farag Ali, T. Golanbari, A. Aghamohammadi, Kh. Saaidi and M. Faizal, _“Inflationary universe in the presence of a minimal measurable length”_ , Annals Phys. 385, 214 (2017). doi.org/10.1016/j.aop.2017.08.008
  * [50] K. Atazadeh and F. Darabi, _“Einstein static universe from GUP”_ , Phys. Dark Universe 16, 87 (2017). doi.org/10.1016/j.dark.2017.04.008
  * [51] M. Moumni and A. Fouhal _“Minimal length effects on Friedmann equations”_ , Int. J. Mod. Phys. A 35(2-3), 2040043 (2015). doi.org/10.1142/S0217751X20400436
  * [52] M. Moussa, _“Effect of Generalized Uncertainty Principle on Main-Sequence Stars and White Dwarfs”_ , Adv. High Energ. Phys. 2015, 343284 (2015). doi.org/10.1155/2015/343284
  * [53] R. Rashidi, _“Generalized uncertainty principle and the maximum mass of ideal white dwarfs”_ , Annals Phys. 374, 434 (2016). doi.org/10.1016/j.aop.2016.09.005
  * [54] P. Jizba, H. Kleinert and F. Scardigli, _“Uncertainty relation on a world crystal and its applications to micro black holes”_ , Phys. Rev. D 81(8), 084030 (2010). doi.org/10.1103/PhysRevD.81.084030
  * [55] R. N Costa Filho, J. P. M. Braga, J. H. S. Lira and J. S. Andrade Jr. _“Extended uncertainty from first principles”_ , Phys. Lett. B 755, 367 (2016). doi.org/10.1016/j.physletb.2016.02.035
  * [56] S. Kouwn, _“Implications of minimum and maximum length scales in cosmology”_ , Phys. Dark Universe 21, 76 (2018). doi.org/10.1016/j.dark.2018.07.001
  * [57] L. Perivolaropoulos _“Cosmological horizons, uncertainty principle, and maximum length quantum mechanics”_ , Phys. Rev. D 95(10), 103523 (2017). doi.org/10.1103/PhysRevD.95.103523
  * [58] L. Perivolaropoulos _“Primordial power spectra of cosmological fluctuations with generalized uncertainty principle and maximum length quantum mechanics”_ , Phys. Rev. D 100(12), 123527 (2019). doi.org/10.1103/PhysRevD.100.123527
  * [59] S. Jalalzadeh, M. Ali Gorji and K. Nozari, _“Deviation from the standard uncertainty principle and the dark energy problem”_ , Gen. Relat. Gravit. 46(1), 1632 (2014). doi.org/10.1007/s10714-013-1632-8
  * [60] A. Vilenkin, _“Approaches to quantum cosmology”_ , Phys. Rev. D 50(4), 2581 (1994). doi.org/10.1103/PhysRevD.50.2581
  * [61] B. F. Shutz, _“Perfect Fluids in General Relativity: Velocity Potentials and a Variational Principle”_ , Phys. Rev. D 2(12), 2762 (1970). doi.org/10.1103/PhysRevD.2.2762
  * [62] B. F. Shutz, _“Hamiltonian Theory of a Relativistic Perfect Fluid”_ , Phys. Rev. D 4(12), 27262 (1970). doi.org/10.1103/PhysRevD.4.3559
  * [63] F. G. Alvarenga and N. A. Lemos, _“Dynamical Vacuum in Quantum Cosmology”_ , Gen. Relat. Gravit. 30(5), 681 (1998). doi.org/10.1023/A:1018896900336
  * [64] B. S. DeWitt, _“Quantum Theory of Gravity. I. The Canonical Theory”_ , Phys. Rev. 160, 1113 (1967). doi.org/10.1103/PhysRev.160.1113
  * [65] P. Pedram, _“Generalized uncertainty principle and the conformally coupled scalar field quantum cosmology”_ , Phys. Rev. D 91(6), 063517 (2015). doi.org/10.1103/PhysRevD.91.063517
  * [66] P. Pedram, _“A higher order GUP with minimal length uncertainty and maximal momentum”_ , Phys. Lett. B 714(2-5), 317 (2012). doi.org/10.1016/j.physletb.2012.07.005
  * [67] P. Pedram, _“A higher order GUP with minimal length uncertainty and maximal momentum II: Applications”_ , Phys. Lett. B 718(2), 638 (2012). doi.org/10.1016/j.physletb.2012.10.059
  * [68] G. Amelino-Camelia, _“Relativity in Spacetimes with Short-Distance Structure Governed by an Observer-Independent (Planckian) Length Scale”_ , Int. J. Mod. Phys. D 11(1), 35 (2002). doi.org/10.1142/S0218271802001330
  * [69] G. Amelino-Camelia, _“Special treatment”_ , Nature 418(1), 34 (2002). doi.org/10.1038/418034a
  * [70] J. Magueijo and L. Smolin _“Lorentz Invariance with an Invariant Energy Scale”_ , Phys. Rev. Lett. 88(19), 190403 (2002). doi.org/10.1103/PhysRevLett.88.190403
  * [71] P. Pedram, _“On the Boundary Conditions in Deformed Quantum Mechanics with Minimal Length Uncertainty”_ , Adv. High Energ. Phys. 2013, 853696 (2013). doi.org/10.1155/2013/853696
  * [72] C. A. Mead, _“Possible connection between gravitation and fundamental length”_ , Phys. Rev. 135, B849 (1964). doi.org/10.1103/PhysRev.135.B849.
  * [73] H. Shababi and W. S. Chung, _“A new type of GUP with commuting coordinates”_ , Mod. Phys. Lett. A 35(6), 2050018 (2020).doi.org/10.1142/S0217732320500182
  * [74] S. Detournay, C. Gabriel and P. Spindel, _“About maximally localized states in quantum mechanics”_ , Phys. Rev. D66(12), 125004 (2002). doi.org/10.1103/PhysRevD.66.125004
  * [75] D. Lovelock, _“The Einstein Tensor and Its Generalizations”_ , J. Math. Phys. 12(3), 498 (1971). https://doi.org/10.1063/1.1665613
  * [76] F. Bianchini et al, _“Constraints on Cosmological Parameters from the 500 deg 2 SPTPOL Lensing Power Spectrum”_, Astrophys. J. 888(2), 119 (2020). https://doi.org/10.3847/1538-4357/ab6082
  * [77] G. Efstathiou and S. Gratton _“The Einstein Tensor and Its Generalizations”_ , Mon. Not. Roy. Astron. Soc. 496(1), L91 (2020). https://doi.org/10.1093/mnrasl/slaa093
  * [78] K. Zeynali, F. Darabi and H. Motavalli _“Multi-dimensional cosmology and GUP”_ , J. Cosmology Astroparticle Phys. 12, 033 (2012). https://doi.org/10.1088/1475-7516/2012/12/033


# A Broad Study on the Transferability of Visual Representations with
Contrastive Learning

Ashraful Islam1, Chun-Fu Chen2,3, Rameswar Panda2,3, Leonid Karlinsky3,
Richard Radke1, Rogerio Feris2,3
1Rensselaer Polytechnic Institute 2MIT-IBM Watson AI Lab 3IBM Research
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>This work was
done while the author was an intern at IBM.

###### Abstract

Tremendous progress has been made in visual representation learning, notably
with the recent success of self-supervised contrastive learning methods.
Supervised contrastive learning has also been shown to outperform its cross-
entropy counterparts by leveraging labels for choosing where to contrast.
However, there has been little work to explore the transfer capability of
contrastive learning to a different domain. In this paper, we conduct a
comprehensive study on the transferability of learned representations of
different contrastive approaches for linear evaluation, full-network transfer,
and few-shot recognition on 12 downstream datasets from different domains, and
object detection tasks on MSCOCO and VOC0712. The results show that the
contrastive approaches learn representations that are easily transferable to a
different downstream task. We further observe that the joint objective of
self-supervised contrastive loss with cross-entropy/supervised-contrastive
loss leads to better transferability of these models over their supervised
counterparts. Our analysis reveals that the representations learned from the
contrastive approaches contain more low/mid-level semantics than cross-entropy
models, which enables them to quickly adapt to a new task. Our codes and
models will be publicly available to facilitate future research on
transferability of visual representations.
111https://github.com/asrafulashiq/transfer_broad

##  1 Acknowledgement

This material is based upon work supported by the Defense Advanced Research
Projects Agency (DARPA) under Contract No. FA8750-19-C-1001. Any opinions,
findings and conclusions or recommendations expressed in this material are
those of the author(s) and do not necessarily reflect the views of the Defense
Advanced Research Projects Agency (DARPA).

## References

  * [1] Machine learning in python.  https://scikit-learn.org/stable/.
  * [2] Hossein Azizpour, Ali Sharif Razavian, Josephine Sullivan, Atsuto Maki, and Stefan Carlsson.  Factors of transferability for a generic convnet representation.  IEEE transactions on pattern analysis and machine intelligence, 38(9):1790–1802, 2015.
  * [3] John S Bridle.  Probabilistic interpretation of feedforward classification network outputs, with relationships to statistical pattern recognition.  In Neurocomputing, pages 227–236. Springer, 1990.
  * [4] Mathilde Caron, Ishan Misra, Julien Mairal, Priya Goyal, Piotr Bojanowski, and Armand Joulin.  Unsupervised learning of visual features by contrasting cluster assignments.  arXiv preprint arXiv:2006.09882, 2020.
  * [5] Ken Chatfield, Karen Simonyan, Andrea Vedaldi, and Andrew Zisserman.  Return of the devil in the details: Delving deep into convolutional nets.  In Proceedings of the British Machine Vision Conference. BMVA Press, 2014.
  * [6] Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton.  A simple framework for contrastive learning of visual representations.  arXiv preprint arXiv:2002.05709, 2020.
  * [7] Xinlei Chen, Haoqi Fan, Ross Girshick, and Kaiming He.  Improved baselines with momentum contrastive learning.  arXiv preprint arXiv:2003.04297, 2020.
  * [8] Gong Cheng, Junwei Han, and Xiaoqiang Lu.  Remote sensing image scene classification: Benchmark and state of the art.  Proceedings of the IEEE, 105(10):1865–1883, 2017.
  * [9] Mircea Cimpoi, Subhransu Maji, Iasonas Kokkinos, Sammy Mohamed, and Andrea Vedaldi.  Describing textures in the wild.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3606–3613, 2014.
  * [10] Noel Codella, Veronica Rotemberg, Philipp Tschandl, M Emre Celebi, Stephen Dusza, David Gutman, Brian Helba, Aadi Kalloo, Konstantinos Liopyris, Michael Marchetti, et al.  Skin lesion analysis toward melanoma detection 2018: A challenge hosted by the international skin imaging collaboration (isic).  arXiv preprint arXiv:1902.03368, 2019.
  * [11] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei.  Imagenet: A large-scale hierarchical image database.  In 2009 IEEE Conference on Computer Vision and Pattern Recognition, pages 248–255. IEEE, 2009.
  * [12] Carl Doersch, Abhinav Gupta, and Alexei A Efros.  Unsupervised Visual Representation Learning by Context Prediction.  In ICCV, 2015.
  * [13] Jeff Donahue, Yangqing Jia, Oriol Vinyals, Judy Hoffman, Ning Zhang, Eric Tzeng, and Trevor Darrell.  Decaf: A deep convolutional activation feature for generic visual recognition.  In International Conference on Machine Learning, pages 647–655, 2014.
  * [14] Mark Everingham, Luc Van Gool, Christopher KI Williams, John Winn, and Andrew Zisserman.  The pascal visual object classes (voc) challenge.  International journal of computer vision, 88(2):303–338, 2010.
  * [15] Robert Geirhos, Patricia Rubisch, Claudio Michaelis, Matthias Bethge, Felix A. Wichmann, and Wieland Brendel.  Imagenet-trained CNNs are biased towards texture; increasing shape bias improves accuracy and robustness.  In International Conference on Learning Representations, 2019.
  * [16] Spyros Gidaris, Praveer Singh, and Nikos Komodakis.  Unsupervised Representation Learning by Predicting Image Rotations.  In ICLR. arXiv, 3 2018.
  * [17] Chuan Guo, Geoff Pleiss, Yu Sun, and Kilian Q Weinberger.  On calibration of modern neural networks.  arXiv preprint arXiv:1706.04599, 2017.
  * [18] Yunhui Guo, Noel C Codella, Leonid Karlinsky, James V Codella, John R Smith, Kate Saenko, Tajana Rosing, and Rogerio Feris.  A broader study of cross-domain few-shot learning.  In European Conference on Computer Vision, pages 124–141. Springer, 2020.
  * [19] Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick.  Momentum contrast for unsupervised visual representation learning.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 9729–9738, 2020.
  * [20] Patrick Helber, Benjamin Bischke, Andreas Dengel, and Damian Borth.  Eurosat: A novel dataset and deep learning benchmark for land use and land cover classification.  IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 12(7):2217–2226, 2019.
  * [21] Olivier J Hénaff, Aravind Srinivas, Jeffrey De Fauw, Ali Razavi, Carl Doersch, SM Eslami, and Aaron van den Oord.  Data-efficient image recognition with contrastive predictive coding.  arXiv preprint arXiv:1905.09272, 2019.
  * [22] Dan Hendrycks, Steven Basart, Norman Mu, Saurav Kadavath, Frank Wang, Evan Dorundo, Rahul Desai, Tyler Zhu, Samyak Parajuli, Mike Guo, et al.  The many faces of robustness: A critical analysis of out-of-distribution generalization.  arXiv preprint arXiv:2006.16241, 2020.
  * [23] Dan Hendrycks and Thomas Dietterich.  Benchmarking neural network robustness to common corruptions and perturbations.  arXiv preprint arXiv:1903.12261, 2019.
  * [24] Dan Hendrycks, Kevin Zhao, Steven Basart, Jacob Steinhardt, and Dawn Song.  Natural adversarial examples.  arXiv preprint arXiv:1907.07174, 2019.
  * [25] Minyoung Huh, Pulkit Agrawal, and Alexei A Efros.  What makes imagenet good for transfer learning?  arXiv preprint arXiv:1608.08614, 2016.
  * [26] Ziyu Jiang, Tianlong Chen, Ting Chen, and Zhangyang Wang.  Robust pre-training by adversarial contrastive learning.  In H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 16199–16210. Curran Associates, Inc., 2020.
  * [27] Prannay Khosla, Piotr Teterwak, Chen Wang, Aaron Sarna, Yonglong Tian, Phillip Isola, Aaron Maschinot, Ce Liu, and Dilip Krishnan.  Supervised contrastive learning.  arXiv preprint arXiv:2004.11362, 2020.
  * [28] Youngsung Kim, Jinwoo Shin, Eunho Yang, and Sung Ju Hwang.  Few-shot visual reasoning with meta-analogical contrastive learning.  In H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 16846–16856. Curran Associates, Inc., 2020.
  * [29] Simon Kornblith, Honglak Lee, Ting Chen, and Mohammad Norouzi.  What’s in a loss function for image classification?  arXiv preprint arXiv:2010.16402, 2020.
  * [30] Simon Kornblith, Mohammad Norouzi, Honglak Lee, and Geoffrey Hinton.  Similarity of neural network representations revisited.  arXiv preprint arXiv:1905.00414, 2019.
  * [31] Simon Kornblith, Jonathon Shlens, and Quoc V Le.  Do better imagenet models transfer better?  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 2661–2671, 2019.
  * [32] Brenden M Lake, Ruslan Salakhutdinov, and Joshua B Tenenbaum.  Human-level concept learning through probabilistic program induction.  Science, 350(6266):1332–1338, 2015.
  * [33] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick.  Microsoft coco: Common objects in context.  In European Conference on Computer Vision, pages 740–755. Springer, 2014.
  * [34] Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu.  Towards deep learning models resistant to adversarial attacks.  In International Conference on Learning Representations, 2018.
  * [35] Sharada P Mohanty, David P Hughes, and Marcel Salathé.  Using deep learning for image-based plant disease detection.  Frontiers in plant science, 7:1419, 2016.
  * [36] Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew Ng.  Reading digits in natural images with unsupervised feature learning.  NIPS Workshop on Deep Learning and Unsupervised Feature Learning 2011, pages 722–729, 2011.
  * [37] Maria-Elena Nilsback and Andrew Zisserman.  Automated flower classification over a large number of classes.  In 2008 Sixth Indian Conference on Computer Vision, Graphics & Image Processing, pages 722–729. IEEE, 2008.
  * [38] Alex Olsen, Dmitry A. Konovalov, Bronson Philippa, Peter Ridd, Jake C. Wood, Jamie Johns, Wesley Banks, Benjamin Girgenti, Owen Kenny, James Whinney, Brendan Calvert, Mostafa Rahimi Azghadi, and Ronald D. White.  DeepWeeds: A Multiclass Weed Species Image Dataset for Deep Learning.  Scientific Reports, 9(2058), 2 2019.
  * [39] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala.  Pytorch: An imperative style, high-performance deep learning library.  In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems 32, pages 8024–8035. Curran Associates, Inc., 2019.
  * [40] Deepak Pathak, Philipp Krahenbuhl, Jeff Donahue, Trevor Darrell, and Alexei A. Efros.  Context Encoders: Feature Learning by Inpainting.  In CVPR, volume 2016-December, pages 2536–2544. IEEE Computer Society, 4 2016.
  * [41] Ali Sharif Razavian, Hossein Azizpour, Josephine Sullivan, and Stefan Carlsson.  Cnn features off-the-shelf: an astounding baseline for recognition.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition workshops, pages 806–813, 2014.
  * [42] Yonglong Tian, Chen Sun, Ben Poole, Dilip Krishnan, Cordelia Schmid, and Phillip Isola.  What makes for good views for contrastive learning.  arXiv preprint arXiv:2005.10243, 2020.
  * [43] Yingtao Tian, Chikahiko Suzuki, Tarin Clanuwat, Mikel Bober-Irizar, Alex Lamb, and Asanobu Kitamoto.  Kaokore: A pre-modern japanese art facial expression dataset.  arXiv preprint arXiv:2002.08595, 2020.
  * [44] Yonglong Tian, Yue Wang, Dilip Krishnan, Joshua B Tenenbaum, and Phillip Isola.  Rethinking few-shot image classification: a good embedding is all you need?  arXiv preprint arXiv:2003.11539, 2020.
  * [45] Oriol Vinyals, Charles Blundell, Timothy Lillicrap, Daan Wierstra, et al.  Matching networks for one shot learning.  In Advances in neural information processing systems, pages 3630–3638, 2016.
  * [46] Haohan Wang, Songwei Ge, Eric P. Xing, and Zachary C. Lipton.  Learning robust global representations by penalizing local predictive power.  arXiv preprint arXiv:1905.13549, 2019.
  * [47] Xiaosong Wang, Yifan Peng, Le Lu, Zhiyong Lu, Mohammadhadi Bagheri, and Ronald M Summers.  Chestx-ray8: Hospital-scale chest x-ray database and benchmarks on weakly-supervised classification and localization of common thorax diseases.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 2097–2106, 2017.
  * [48] Yuxin Wu, Alexander Kirillov, Francisco Massa, Wan-Yen Lo, and Ross Girshick.  Detectron2.  https://github.com/facebookresearch/detectron2, 2019.
  * [49] Xiaohua Zhai, Avital Oliver, Alexander Kolesnikov, and Lucas Beyer.  S4l: Self-supervised semi-supervised learning.  In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 1476–1485, 2019.
  * [50] Richard Zhang, Phillip Isola, and Alexei A. Efros.  Colorful Image Colorization.  In ECCV, volume 9907 LNCS, pages 649–666. Springer Verlag, 3 2016\.
  * [51] Nanxuan Zhao, Zhirong Wu, R. Lau, and Stephen Lin.  What makes instance discrimination good for transfer learning?  arXiv preprint arXiv:2006.06606, 2020.