# Global One–Dimensionality conjecture within Quantum General Relativity

Łukasz Andrzej Glinka111Previously at Bogoliubov Laboratory of Theoretical
Physics of Joint Institute for Nuclear Research in Dubna, Russian Federation
E-mail<EMAIL_ADDRESS>
_International Institute for Applicable_
_Mathematics & Information Sciences,_
_Hyderabad (India) & Udine (Italy),_
_B.M. Birla Science Centre,_
_Adarsh Nagar, 500 063 Hyderabad, India_

###### Abstract

The simple quantum gravity model, based on a new conjecture within the
canonically quantized $3+1$ general relativity, is presented. The conjecture
states that matter fields are functionals of an embedding volume form only,
and reduces the quantum geometrodynamics. By dimensional reduction the
resulting theory is presented in the form of the Dirac equation, and
application of the Fock quantization with the diagonalization procedure yields
construction of the appropriate quantum field theory. The 1D wave function is
derived, the corresponding 3-dimensional manifolds are discussed, and physical
scales are associated with quantum correlations.

Keywords general relativity; 3+1 splitting ; quantum gravity models ; low
dimensional quantum field theories ; quantization methods ; global one-
dimensionality

PACS 04.60.-m ; 05.30.Jp ; 05.70.Ce; 11.10.Kk ; 98.80.Qc

##  1 Introduction

Quantum gravity is one of the fundamental problems of modern theoretical
physics. In spite of the significant efforts and various approaches, we are
still very far of understanding the role of quantized gravitational fields in
physical phenomena at high energies (for different approaches to quantum
gravity see _e.g._ Ref. [1]). In this paper we propose a simple model of
quantum gravity which can be useful for clarifying its some important aspects.

The celebrated field-theoretic formalism yields plausible phenomenology for
numerous experimental data of all areas of physics. In this paper this point
of view is used for construction of a simple quantum gravity model. The $3+1$
splitting of a general relativistic metric tensor and the canonical
quantization of the appropriate action functional are used in the well-
grounded way. In straightforward and strict analogy with the generic
cosmological model [2], the new conjecture within the Wheeler–DeWitt quantum
geometrodynamics is proposed. The model is based on the ansatz composed by the
steps

  1. 1.

global one–dimensionality conjecture, _i.e._ one-dimensional matter fields,

  2. 2.

reduced quantum geometrodynamics, yielding one-dimensional theory,

  3. 3.

dimensional reduction, resulting in the Dirac equation formulation,

and expressing the supposition that the quantum geometrodynamics in itself is
a one-dimensional field theory. The dimensional reduction leads to the
appropriate Dirac equation and the Euclidean Clifford algebra. Its the Fock
quantization with the diagonalization procedure, consisting of the Bogoliubov
transformation and the Heisenberg equations of motion, yields correctly
defined quantum field theory. The resulting model describes quantum gravity in
terms of a quantum field theory formulated in the Fock static operator reper
associated with initial data. The 1D wave function is derived, the
corresponding 3-dimensional manifolds are discussed, and quantum correlations
are associated with physical scales. Mathematically, we employ the one-
dimensional functional integrals, so that the proposing quantum gravity model
is methodologically corresponding to the trend initiated by Hartle and Hawking
in the paper [3].

An organization of the paper is as follows. In the preliminary section 2
historically first quantized 3+1 general relativity is presented. Section 3 is
devoted to the ansatz presentation. Next, the sections 4 and 5 discuss field
quantization and some implications of general formulation, respectively.
Finally, in the section 6 the entire paper’s results are summarized in
condensed way.

##  2 Quantum 3+1 General Relativity

In general relativity (See _e.g._ [4]) a pseudo-Riemannian manifold $(M,g)$
with a metric tensor $g_{\mu\nu}$, the Christoffel symbols
$\Gamma^{\rho}_{\mu\nu}$, the Riemann curvature $R^{\lambda}_{\mu\alpha\nu}$,
the Einstein curvature
$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}{{}^{(4)}}\\!R$, where
$R_{\mu\nu}=R^{\lambda}_{\mu\lambda\nu}$,
${{}^{(4)}}\\!R=R^{\kappa}_{\kappa}$, satisfying the Einstein field
equations222We use the units $c=\hbar=8\pi G/3=1$ in this text.

$G_{\mu\nu}+\Lambda g_{\mu\nu}=3T_{\mu\nu},$ (1)

where $\Lambda$ is cosmological constant and $T_{\mu\nu}$ is stress-energy
tensor, models a spacetime333In (1) the coefficient of $T_{\mu\nu}$ usually
equals $8\pi G/c^{4}$ that is exactly 3 in the our units.. For a compact $M$
with a boundary $(\partial M,h)$ and curvature $K_{ij}$, the usual variational
principle is corrected [5] and (1) arise by the action

$S[g]=\int_{M}d^{4}x\sqrt{-g}\left\\{-\dfrac{1}{6}R+\dfrac{\Lambda}{3}\right\\}+S_{\psi}[g]-\dfrac{1}{3}\int_{\partial
M}d^{3}x\sqrt{h}K,$ (2)

where $K=h^{ij}K_{ij}$, $S_{\psi}[g]$ is Matter fields action,
$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta S_{\psi}[g]}{\delta g^{\mu\nu}}$.
For $\Lambda=0$, a global timelike Killing field on $M$ $\mathcal{K}$ exists,
the foliation $t=const$ is spacelike, $\partial M$ is the Nash embedding [6],
and $3+1$ splitting [7] holds

$\displaystyle g_{\mu\nu}=\left[\begin{array}[]{cc}-N^{2}+N_{i}N^{i}&N_{j}\\\
N_{i}&h_{ij}\end{array}\right]\quad,\quad
h_{ik}h^{kj}=\delta_{i}^{j}\quad,\quad N^{j}=h^{ij}N_{i}$ (5)

For $\Lambda>0$, $\mathcal{K}$ does not exist, spacelike $\partial M$ only
foliates an exterior to the horizons on geodesic lines, (5) is a gauge. In
both cases (2) takes the form

$\displaystyle S[g]=\int dt\int_{\partial
M}d^{3}x\left\\{\pi\dot{N}+\pi^{i}\dot{N_{i}}+\pi_{\psi}\dot{\psi}+\pi^{ij}\dot{h}_{ij}-NH-
N_{i}H^{i}\right\\},$ (6)

where dot means $t$-differentiation, $H$ and $H^{i}$ are defined as

$\displaystyle
H=\sqrt{h}\left\\{K^{2}-K_{ij}K^{ij}+{{}^{(3)}R}-2\Lambda-6\varrho\right\\},\qquad
H^{i}=-2\pi^{ij}_{\leavevmode\nobreak\ ;j}\leavevmode\nobreak\ ,$ (7)

with ${{}^{(3)}R}=h^{ij}R_{ij}$, $\varrho=n^{\mu}n^{\nu}T_{\mu\nu}$,
$n^{\mu}=[1/N,-N^{i}/N]$, and particularly

$\pi^{ij}=-\sqrt{h}\left(K^{ij}-h^{ij}K\right).$ (8)

The curvature $K_{ij}$ satisfies the Gauss–Codazzi equations [8]

$2NK_{ij}=N_{i|j}+N_{j|i}-\dot{h}_{ij}.$ (9)

where stroke means intrinsic covariant differentiation. Time-preservation [9]
of the primary constraints [10] leads to the secondary ones – (scalar)
Hamiltonian constraint yielding dynamics, and (vector) diffeomorphism one
merely reflecting spatial diffeoinvariance

$\pi\approx 0\quad,\quad\pi^{i}\approx 0\quad\longrightarrow\quad H\approx
0\quad,\quad H^{i}\approx 0.$ (10)

DeWitt [10] showed that $H^{i}$ generate the diffeomorphisms
$\widetilde{x}^{i}=x^{i}+\xi^{i}$

$\displaystyle i\left[h_{ij},\int_{\partial M}H_{a}\xi^{a}d^{3}x\right]$
$\displaystyle=$ $\displaystyle-
h_{ij,k}\xi^{k}-h_{kj}\xi^{k}_{\leavevmode\nobreak\
,i}-h_{ik}\xi^{k}_{\leavevmode\nobreak\ ,j}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (11) $\displaystyle i\left[\pi_{ij},\int_{\partial
M}H_{a}\xi^{a}d^{3}x\right]$ $\displaystyle=$
$\displaystyle-\left(\pi_{ij}\xi^{k}\right)_{,k}+\pi_{kj}\xi^{i}_{\leavevmode\nobreak\
,k}+\pi_{ik}\xi^{j}_{\leavevmode\nobreak\ ,k}\leavevmode\nobreak\
\leavevmode\nobreak\ ,$ (12)

and consequently the first-class constraints algebra can be derived

$\displaystyle i\left[H_{i}(x),H_{j}(y)\right]$ $\displaystyle=$
$\displaystyle\int_{\partial M}H_{a}c^{a}_{ij}d^{3}z,$ (13) $\displaystyle
i\left[H(x),H_{i}(y)\right]$ $\displaystyle=$ $\displaystyle
H\delta^{(3)}_{,i}(x,y),$ (14) $\displaystyle i\left[\int_{\partial
M}H\xi_{1}d^{3}x,\int_{\partial M}H\xi_{2}d^{3}x\right]$ $\displaystyle=$
$\displaystyle\int_{\partial
M}H^{a}\left(\xi_{1,a}\xi_{2}-\xi_{1}\xi_{2,a}\right)d^{3}x.$ (15)

Here $H_{i}=h_{ij}H^{j}$, and
$c^{a}_{ij}=\delta^{a}_{i}\delta^{b}_{j}\delta^{(3)}_{,b}(x,z)\delta^{(3)}(y,z)-(x\rightarrow
y)$ are structure constants of diffeomorphism group. The scalar constraint
reduced by (8) with using of the canonical primary quantization [9, 11]

$\displaystyle
i\left[\pi^{ij}(x),h_{kl}(y)\right]=\frac{1}{2}\left(\delta_{k}^{i}\delta_{l}^{j}+\delta_{l}^{i}\delta_{k}^{j}\right)\delta^{(3)}(x,y),$
(16) $\displaystyle
i\left[\pi^{i}(x),N_{j}(y)\right]=\delta^{i}_{j}\delta^{(3)}(x,y)\quad,\quad
i\left[\pi(x),N(y)\right]=\delta^{(3)}(x,y),$ (17)

yields the Wheeler–DeWitt equation [12, 10]

$\left\\{G_{ijkl}\dfrac{\delta^{2}}{\delta h_{ij}\delta
h_{kl}}+h^{1/2}\left({{}^{(3)}R}-2\Lambda-6\varrho\right)\right\\}\Psi[h_{ij},\phi]=0,$
(18)

where $G_{ijkl}$ is the Wheeler metric on superspace $S(\partial M)$ [12, 13]

$G_{ijkl}=\dfrac{1}{2\sqrt{h}}\left(h_{ik}h_{jl}+h_{il}h_{jk}-h_{ij}h_{kl}\right),$
(19)

and first class constraints are conditions on $\Psi[h_{ij},\phi]$

$\pi\Psi[h_{ij},\phi]=0\quad,\quad\pi^{i}\Psi[h_{ij},\phi]=0\quad,\quad
H^{i}\Psi[h_{ij},\phi]=0.$ (20)

In fact quantum general relativity, given by the Wheeler–DeWitt equation,
historically is one of the first attempts of quantum gravity theory
construction. Actually, however, quantum geometrodynamics has became the
motivation for development of the quantum gravity idea and building novel
formulations. We are going to build the our model basing on the Wheeler–DeWitt
theory (18). Strictly speaking, however, the proposing toy model will possess
a reductionist character.

##  3 The Ansatz

For construction of a simple quantum gravity model let us apply the following
three step ansatz.

Global one–dimensionality conjecture. Suppose that Matter fields are one-
variable functionals $\phi=\phi[h]$ where $h$ is a volume form of $\partial M$

$h\equiv\det
h_{ij}=\dfrac{1}{3}\epsilon^{ijk}\epsilon^{lmn}h_{il}h_{jm}h_{kn},$ (21)

and $\epsilon$ is the Levi-Civita symbol. Also we assume, as an element of the
model, that gravitational field is described only variable h. In result a wave
function $\Psi[h_{ij},\phi]$ becomes

$\Psi[h_{ij},\phi]\rightarrow\Psi[h],$ (22)

so that, the proposed quantum gravity model is

$\left\\{-G_{ijkl}\dfrac{\delta^{2}}{\delta h_{ij}\delta
h_{kl}}-h^{1/2}\left({{}^{(3)}R}-2\Lambda-6\varrho[h]\right)\right\\}\Psi[h]=0.$
(23)

In analogy to the generic cosmology [2], (22) describes isotropic
spacetimes444Assumption (22) means that we consider a strata of full
superspace, _i.e._ the DeWitt minisuperspace model where the wave function
depends only one variable $h$.. Reduced quantum geometrodynamics. Using $3+1$
splitting (5) within the Jacobi formula [4]

$\delta g=gg^{\mu\nu}\delta g_{\mu\nu},$ (24)

establishes the Jacobian matrix for for the reduction of variables $h_{ij}$ to
$h$

$\displaystyle N^{2}\delta h=N^{2}hh^{ij}\delta
h_{ij}\longrightarrow\mathcal{J}\left(h_{ij},h\right)=\dfrac{\delta(h)}{\delta(h_{ij})}=\dfrac{\delta
h}{\delta h_{ij}}\equiv hh^{ij}.$ (25)

Because of the approximation (22) the variational derivative $\delta/\delta
h_{ij}$ acts on functional depending only on $h$. It allows us to express the
derivative with respect $h_{ij}$ through the derivative with respect $h$.
Therefore

$\dfrac{\delta\Psi[h]}{\delta h_{ij}}=hh^{ij}\dfrac{\delta\Psi[h]}{\delta h}.$
(26)

Consequently, application of (26) within the differential operator in (23)
gives

$\displaystyle G_{ijkl}\dfrac{\delta^{2}}{\delta h_{ij}\delta
h_{kl}}=G_{ijkl}h^{ij}h^{kl}h^{2}\dfrac{\delta^{2}}{\delta h^{2}}.$ (27)

So that the reduction is given by the double contraction

$\displaystyle
G_{ijkl}h^{ij}h^{kl}=\dfrac{1}{2\sqrt{h}}\left(h_{ik}h_{jl}+h_{il}h_{jk}-h_{ij}h_{kl}\right)h^{ij}h^{kl}=-\dfrac{3}{2}h^{-1/2},$
(28)

where we have used the relations for 3-dimensional embedding
$h^{ab}h_{bc}=h^{a}_{c}$, $h^{a}_{a}=\mathrm{Tr}h_{ab}=3$. Jointing (27) and
(28) one obtains finally the relation555Because the relation (25) arises due
to $3+1$ approximation, so (28) is an approximation within the ansatz.

$\displaystyle G_{ijkl}\dfrac{\delta^{2}}{\delta h_{ij}\delta
h_{kl}}=-\dfrac{3}{2}h^{3/2}\dfrac{\delta^{2}}{\delta h^{2}},$ (29)

which leads to the reduced theory

$\left\\{\dfrac{3}{2}h^{3/2}\dfrac{\delta^{2}}{\delta
h^{2}}+h^{1/2}\left({{}^{(3)}R}-2\Lambda-6\varrho[h]\right)\right\\}\Psi[h]=0.$
(30)

Dimensional reduction. The model (30) can be rewritten as

$\left(\dfrac{\delta^{2}}{\delta{h^{2}}}-m^{2}\right)\Psi=0,$ (31)

where $m^{2}$ is a squared (variable) mass of $\Psi$

$m^{2}=\dfrac{2}{3h}\left({}^{(3)}R-2\Lambda-6\varrho\right)=\dfrac{2}{3h}(K_{ij}K^{ij}-K^{2}),$
(32)

and scalar constraint was used. Eq. (31) arises by stationarity of the
action666Here $S[\Psi]$ is a field-theoretic action functional in $\Psi$ so
that any dependence on $h$ of the mass $m$ does not play a role for equations
of motion $\delta S/\delta\Psi=0$.

$S[\Psi]=\int\delta hL\left(\Psi,\dfrac{\delta\Psi}{\delta
h}\right)\quad,\quad
L=\dfrac{1}{2}\left(\Pi_{\Psi}^{2}+m^{2}\Psi^{2}\right)\quad,\quad\Pi_{\Psi}=\dfrac{\delta\Psi}{\delta
h}.$ (33)

By using $\Pi_{\Psi}$ one rewrites the equation (31) as

$\dfrac{\delta\Pi_{\Psi}}{\delta h}-m^{2}\Psi=0,$ (34)

which together with $\Pi_{\Psi}$ in (33) yields the appropriate Dirac equation

$\left(i\gamma\dfrac{\delta}{\delta
h}-M\right)\Phi=0\quad,\quad\Phi=\left[\begin{array}[]{c}\Pi_{\Psi}\\\
\Psi\end{array}\right]\quad,\quad M=\left[\begin{array}[]{cc}-1&0\\\
0&m^{2}\end{array}\right].$ (35)

The $\gamma$ matrices algebra consists only one element - the Pauli matrix
$\sigma_{y}$

$\gamma=\left[\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right]\quad,\quad\gamma^{2}=I\quad,\quad\left\\{\gamma,\gamma\right\\}=2\delta_{E}\quad,\quad\delta_{E}=\left[\begin{array}[]{cc}1&0\\\
0&1\end{array}\right].$ (36)

The matrix algebra (36) forms the Euclidean Clifford algebra [14]
$\mathcal{C}\ell_{1,1}(\mathbb{R})$ which has a $2D$ complex representation.
Restricting to $Pin_{1,1}(\mathbb{R})$ yield a complex representation of 2D
Pin group (2D spin representation); restricting to $Spin_{1,1}(\mathbb{R})$
splits it onto a sum of two half spin $1D$ representations (1D Weyl
representation). The algebra decomposes into a direct sum of central simple
algebras isomorphic to matrix algebra over $\mathbb{R}$

$\displaystyle\mathcal{C}\ell_{1,1}(\mathbb{R})=\mathcal{C}\ell^{+}_{1,1}(\mathbb{R})\oplus\mathcal{C}\ell^{-}_{1,1}(\mathbb{R})\quad,\quad\mathcal{C}\ell_{1,1}(\mathbb{R})\cong\mathbb{R}(2)\quad,\quad\mathcal{C}\ell^{\pm}_{1,1}(\mathbb{R})\cong\mathbb{R},$
(37)

and moreover has a decomposition into a tensor product

$\displaystyle\mathcal{C}\ell_{1,1}(\mathbb{R})=\mathcal{C}\ell_{2,0}(\mathbb{R})\otimes\mathcal{C}\ell_{0,0}(\mathbb{R})\quad,\quad\mathcal{C}\ell_{0,0}(\mathbb{R})\cong\mathbb{R}.$
(38)

##  4 Quantization

The Dirac equation (35) can be canonically quantized (See _e.g._ [15])

$\displaystyle
i\left[\mathbf{\Pi}_{\Psi}[h^{\prime}],\mathbf{\Psi}[h]\right]=\delta(h^{\prime}-h)\leavevmode\nobreak\
,\leavevmode\nobreak\
i\left[\mathbf{\Pi}_{\Psi}[h^{\prime}],\mathbf{\Pi}_{\Psi}[h]\right]=0\leavevmode\nobreak\
,\leavevmode\nobreak\
i\left[\mathbf{\Psi}[h^{\prime}],\mathbf{\Psi}[h]\right]=0.$ (39)

Using of the Fock space allows to derive the solution in the form

$\mathbf{\Phi}=\mathbb{Q}\mathfrak{B}\quad,\quad\mathbb{Q}=\dfrac{1}{\sqrt{2}}\left[\begin{array}[]{cc}\sqrt{1/|m|}&\sqrt{1/|m|}\\\
-i\sqrt{|m|}&i\sqrt{|m|}\end{array}\right],$ (40)

where $\mathfrak{B}=\mathfrak{B}[h]$ is a dynamical reper

$\mathfrak{B}=\left\\{\left[\begin{array}[]{c}\mathsf{G}[h]\\\
\mathsf{G}^{\dagger}[h]\end{array}\right]:\left[\mathsf{G}[h^{\prime}],\mathsf{G}^{\dagger}[h]\right]=\delta\left(h^{\prime}-h\right),\left[\mathsf{G}[h^{\prime}],\mathsf{G}[h]\right]=0\right\\},$
(41)

and yields non Heisenberg-like dynamics of (41)

$\dfrac{\delta\mathfrak{B}}{\delta
h}=\mathbb{X}\mathfrak{B}\quad,\quad\mathbb{X}=\left[\begin{array}[]{cc}-im&\dfrac{1}{2m}\dfrac{\delta
m}{\delta h}\\\ \dfrac{1}{2m}\dfrac{\delta m}{\delta h}&im\end{array}\right].$
(42)

Supposing that there is an other reper $\mathfrak{F}$ determined by the
Bogoliubov transformation and the Heisenberg equations of motion

$\displaystyle\mathfrak{F}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}u&v\\\
v^{\ast}&u^{\ast}\end{array}\right]\mathfrak{B},\qquad|u|^{2}-|v|^{2}=1,$ (45)
$\displaystyle\dfrac{\delta\mathfrak{F}}{\delta h}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}-i\Omega&0\\\
0&i\Omega\end{array}\right]\mathfrak{F},$ (48)

where $u$, $v$, $\Omega$ are functionals of $h$, one obtains

$\dfrac{\delta\mathbf{b}}{\delta
h}=\mathbb{X}\mathbf{b}\quad,\quad\mathbf{b}=\left[\begin{array}[]{c}u\\\
v\end{array}\right],$ (49)

and $\Omega\equiv 0$, so that $\mathfrak{F}$ is the Fock static reper with
respect to initial data ($I$)

$\mathfrak{F}=\left\\{\left[\begin{array}[]{c}\mathsf{G}_{I}\\\
\mathsf{G}^{\dagger}_{I}\end{array}\right]:\left[\mathsf{G}_{I},\mathsf{G}^{\dagger}_{I}\right]=1,\left[\mathsf{G}_{I},\mathsf{G}_{I}\right]=0\right\\},$
(50)

and vacuum state $\left|\mathrm{VAC}\right\rangle$ is correctly defined

$\mathsf{G}_{I}\left|\mathrm{VAC}\right\rangle=0\quad,\quad
0=\left\langle\mathrm{VAC}\right|\mathsf{G}_{I}^{\dagger}.$ (51)

Integrability of Eqs. (49) is crucial. The transformation (45) suggests
employing the superfluid parametrization which yield777In (52) the functional
measure $\delta h$ for the case of a fixed space configuration transits into
the Riemann–Lebesgue measure $dh$. However, $h$ in general is a smooth
function of space parameters, $\delta h$ is a total variation and has a sense
of the Stieltjes measure.

$u=\dfrac{1+\lambda}{2\sqrt{\lambda}}\exp\left\\{im_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\lambda^{\prime}}\right\\}\quad,\quad
v=\dfrac{1-\lambda}{2\sqrt{\lambda}}\exp\left\\{-im_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\lambda^{\prime}}\right\\},$ (52)

where $\lambda\equiv\lambda[h]$, $\lambda^{\prime}=\lambda[h^{\prime}]$ is a
length scale _i.e._ inverted mass scale $\mu=m/m_{I}=1/\lambda$. Consequently
the integrability problem is solved by

$\mathbf{\Phi}=\mathbb{Q}\mathbb{G}\mathfrak{F},$ (53)

where $\mathbb{G}$ is the monodromy matrix

$\mathbb{G}=\left[\begin{array}[]{cc}\dfrac{\lambda+1}{2\sqrt{\lambda}}\exp\left\\{-im_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\lambda^{\prime}}\right\\}\vspace*{10pt}&\dfrac{\lambda-1}{2\sqrt{\lambda}}\exp\left\\{im_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\lambda^{\prime}}\right\\}\\\
\dfrac{\lambda-1}{2\sqrt{\lambda}}\exp\left\\{-im_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\lambda^{\prime}}\right\\}&\dfrac{\lambda+1}{2\sqrt{\lambda}}\exp\left\\{im_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\lambda^{\prime}}\right\\}\end{array}\right].$ (54)

One sees now that the presented model expresses quantum gravity as a quantum
field theory, where the quantum field associated with a space configuration is
given by the relation (53). In this manner one can write out some
straightforward conclusions following form the simple model.

##  5 Some implications of general formulation

The proposed field-theoretic model was solved. However, still we do not know
what it the role of the 1D wave function given by the equation (23). The same
problem is to define any geometric quantities related to this model. The
quantum field theory (53) has also unclear significance. Let us present now
some conclusions arising from the previous section’s model, which will clarify
these questions in some detail.
Global 1D wave function. The Dirac equation (35) can be rewritten in the form
of Schrödinger-like evolution equation

$\dfrac{\delta\Phi}{\delta h}=H\Phi\quad,\quad
H=-\left[\begin{array}[]{cc}0&\dfrac{m_{I}^{2}}{\lambda^{2}}\\\
1&0\end{array}\right],$ (55)

yielding unitary evolution operator $U=U(h,h_{I})=\exp\int_{h_{I}}^{h}H\delta
h$ given by

$U=\left[\begin{array}[]{cc}\cosh
f[h,h_{I}]&\left(-m_{I}^{2}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\lambda^{\prime 2}}\right)\dfrac{\sinh f[h,h_{I}]}{f[h,h_{I}]}\\\
(h_{I}-h)\dfrac{\sinh f[h,h_{I}]}{f[h,h_{I}]}&\cosh
f[h,h_{I}]\end{array}\right],$ (56)

where $f[h,h_{I}]=|m_{I}|\sqrt{{(h-h_{I})\int_{h_{I}}^{h}\frac{\delta
h^{\prime}}{\lambda^{\prime 2}}}}$, so that Eq. (55) is solved by

$\Phi[h]=U(h,h_{I})\Phi[h_{I}].$ (57)

Straightforward elementary algebraic manipulations allow to determine the
global one-dimensional wave function as

$\Psi=\Psi^{I}\cosh
f[h,h_{I}]-\Pi_{\Psi}^{I}(h-h_{I})\mathrm{sgn}(h-h_{I})\dfrac{\sinh
f[h,h_{I}]}{f[h,h_{I}]},$ (58)

and similarly the canonical conjugate momentum is

$\Pi_{\Psi}=\Pi_{\Psi}^{I}\cosh
f[h,h_{I}]-\Psi^{I}m_{I}^{2}\mathrm{sgn}(h-h_{I})\left(\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\lambda^{\prime 2}}\right)\dfrac{\sinh f[h,h_{I}]}{f[h,h_{I}]},$
(59)

where $\Psi^{I}=\Psi[h_{I}]$ and
$\Pi_{\Psi}^{I}=\Pi_{\Psi}[h_{I}]=\left.\dfrac{\delta\Psi}{\delta
h}\right|_{h=hI}$ are initial data. In this manner the probability density in
the classical reduced model is

$\displaystyle\Omega[h]$ $\displaystyle=$
$\displaystyle(\Psi^{I})^{2}\cosh^{2}f[h,h_{I}]+(\Pi_{\Psi}^{I})^{2}(h-h_{I})^{2}\left[\dfrac{\sinh{f[h,h_{I}]}}{f[h,h_{I}]}\right]^{2}-$
(60) $\displaystyle-$ $\displaystyle
2\Psi^{I}\Pi_{\Psi}^{I}(h-h_{I})\mathrm{sgn}(h-h_{I})\dfrac{\sinh{2f[h,h_{I}]}}{2f[h,h_{I}]},$

and $\Psi^{I}$ and $\Pi_{\Psi}^{I}$ are determined by the normalization
condition

$\int_{h_{I}}^{\infty}\Omega[h^{\prime}]\delta
h^{\prime}=1\quad\longrightarrow\quad
C(\Pi_{\Psi}^{I})^{2}-2B\Psi^{I}\Pi_{\Psi}^{I}+A(\Psi^{I})^{2}-1=0,$ (61)

where the constants $A$, $B$, $C$ are given by the integrals

$\displaystyle A$ $\displaystyle=$
$\displaystyle\int_{h_{I}}^{\infty}\cosh^{2}f[h^{\prime},h_{I}]\delta
h^{\prime},$ (62) $\displaystyle B$ $\displaystyle=$
$\displaystyle\int_{h_{I}}^{\infty}(h^{\prime}-h_{I})\mathrm{sgn}(h^{\prime}-h_{I})\dfrac{\sinh{2f[h^{\prime},h_{I}]}}{2f[h^{\prime},h_{I}]}\delta
h^{\prime},$ (63) $\displaystyle C$ $\displaystyle=$
$\displaystyle\int_{h_{I}}^{\infty}(h^{\prime}-h_{I})^{2}\left[\dfrac{\sinh{f[h^{\prime},h_{I}]}}{f[h^{\prime},h_{I}]}\right]^{2}\delta
h^{\prime},$ (64)

The equation (61) can be solved straightforwardly. In result one obtains

$\displaystyle\Pi_{\Psi}^{I}=\dfrac{B}{C}\Psi^{I}\pm\sqrt{{\left[\left(\dfrac{B}{C}\right)^{2}-\dfrac{A}{C}\right](\Psi^{I})^{2}+\dfrac{1}{C}}}.$
(65)

Using $\Pi_{\Psi}^{I}=\dfrac{\delta\Psi^{I}}{\delta h_{I}}$ in (65) yields
differential equation for $\Psi^{I}$, with the solution

$\Psi^{I}=f_{\pm}^{(-1)}\left(\pm\dfrac{h_{I}}{C}+C_{1}\right),$ (66)

where $C_{1}$ is integration constant, and $f_{\pm}(h_{I})$ are the functions

$\displaystyle
f_{\pm}(h_{I})=\dfrac{B}{AC}\Bigg{\\{}\mathrm{artanh}\dfrac{Bh_{I}}{\sqrt{{\left(B^{2}-AC\right)h_{I}^{2}+C}}}\pm\ln\sqrt{{Ah_{I}^{2}-1}}-$
$\displaystyle-\dfrac{\sqrt{{B^{2}-AC}}}{B}\ln\left[\left(B^{2}-AC\right)h_{I}+\sqrt{{B^{2}-AC}}\sqrt{{\left(B^{2}-AC\right)h_{I}^{2}+C}}\right]\Bigg{\\}}.$
(67)

The 3-dimensional manifolds. The model (31) can be rewritten as

$\left(\dfrac{\delta^{2}}{\delta{h^{2}}}-\dfrac{2}{3h}{{}^{(3)}R}\right)\Psi[h]=-\dfrac{4}{h}\left(\varrho[h]+\dfrac{\Lambda}{3}\right)\Psi[h].$
(68)

and considered as the equation for the 3-dimensional scalar curvature
${{}^{(3)}\\!R}$

$^{(3)}\\!R=-6\left(\varrho[h]+\dfrac{\Lambda}{3}\right)+\varphi(\Psi)h\quad,\quad\varphi(\Psi)=\dfrac{3}{2}\dfrac{1}{\Psi}\dfrac{\delta^{2}\Psi}{\delta{h^{2}}}.$
(69)

In the vacuum case, _i.e._ for the conditions $\left(\varrho[h]\equiv
0\cap\Lambda\equiv 0\right)$ or $\varrho[h]=-\dfrac{\Lambda}{3}$, one obtains
from (69) that

$^{(3)}\\!R=\varphi_{n}h,$ (70)

where $\varphi_{n}$ in an eigenvalue determined by the equation

$\dfrac{\delta^{2}\Psi}{\delta{h^{2}}}-\dfrac{2}{3}\varphi_{n}\Psi=0.$ (71)

Supposing analytical form of $\Psi$ one establishes $\varphi_{n}$

$\Psi=\sum_{n=0}^{\infty}a_{n}(h-h_{I})^{n}\longrightarrow\varphi_{n}=\dfrac{3}{2}\left.\left(\dfrac{\delta^{n}}{\delta
h^{n}}\left(\dfrac{m_{I}^{2}}{\lambda^{2}[h]}\Psi[h]\right)\left/\dfrac{\delta^{n}\Psi[h]}{\delta
h^{n}}\right.\right)\right|_{h=h_{I}}.$ (72)

Let us assume that there are generalized functional Fourier transforms

$\widetilde{\Psi}[s]=\int\delta he^{-2\pi
ish}\Psi[h]\quad,\quad\widetilde{\dfrac{1}{\lambda^{2}}}[s]=\int\delta
he^{-2\pi ish}\dfrac{1}{\lambda^{2}[h]},$ (73)

as well as the generalized Leibniz product formula for functional derivatives

$\dfrac{\delta^{n}}{\delta
h^{n}}\left(\dfrac{1}{\lambda^{2}[h]}\Psi[h]\right)=\sum_{r=0}^{n}\binom{n}{r}\left(\dfrac{\delta^{r}}{\delta
h^{r}}\dfrac{1}{\lambda^{2}[h]}\right)\left(\dfrac{\delta^{n-r}}{\delta
h^{n-r}}\Psi[h]\right).$ (74)

Using (73), (74) and $\sum_{r=0}^{n}\binom{n}{r}x^{r}=(1+x)^{n}$, within (72)
yields

$\varphi_{n}=\dfrac{3}{2}m_{I}^{2}\iint\delta s^{\prime}\delta
se^{2i\pi(s^{\prime}+s)h_{I}}\left(1+\dfrac{s^{\prime}}{s}\right)^{n}\widetilde{\dfrac{1}{\lambda^{2}}}[s^{\prime}]\widetilde{\Psi}[s],$
(75)

so that applying the inverted Fourier transforms

$\Psi[h]=\int\delta se^{2\pi
ish}\widetilde{\Psi}[s]\quad,\quad\dfrac{1}{\lambda^{2}[h]}=\int\delta
se^{2\pi ish}\widetilde{\dfrac{1}{\lambda^{2}}}[s],$ (76)

within the relation (75) one receives finally

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\varphi_{n}$ $\displaystyle=$
$\displaystyle\dfrac{3}{2}\iint\delta h\delta
h\mathcal{G}(h-h_{I})\dfrac{m_{I}^{2}}{\lambda^{2}[h]}\Psi[h]=\dfrac{3}{2}\iint\delta
h\delta h\mathcal{G}(h-h_{I})\dfrac{\delta^{2}\Psi[h]}{\delta h^{2}}=$ (77)
$\displaystyle=$ $\displaystyle\dfrac{3}{2}\iint\delta h\delta
h\mathcal{G}(h-h_{I})\dfrac{\delta\Pi_{\Psi}[h]}{\delta
h}=-\dfrac{3}{2}\iint\delta h\delta h\dfrac{\delta}{\delta
h}\mathcal{G}(h-h_{I})\Pi_{\Psi}[h],$

where we have used equations of motion, partial integration method. In (77)
the kernel $\mathcal{G}(h-h_{I})$ and its derivative can be derived
straightforwardly as

$\displaystyle\mathcal{G}(h-h_{I})$ $\displaystyle=$ $\displaystyle\iint\delta
s^{\prime}\delta
se^{-2i\pi(s^{\prime}+s)(h-h_{I})}\left(1+\dfrac{s^{\prime}}{s}\right)^{n},$
(78) $\displaystyle\dfrac{\delta}{\delta h}\mathcal{G}(h-h_{I})$
$\displaystyle=$ $\displaystyle-\iint\delta s^{\prime}\delta
s\dfrac{e^{-2i\pi(s^{\prime}+s)(h-h_{I})}}{2i\pi(s^{\prime}+s)}\left(1+\dfrac{s^{\prime}}{s}\right)^{n}.$
(79)

Estimation of the functional integrals (78) or (79), and using of (58) or
(59), leads to (77), which is a crucial for the relation (70).

Quantum correlations. With using of the matrices (54) and (40), and the
relation (53) one derives the quantum field

$\displaystyle\mathbf{\Psi}[h]=\frac{\lambda[h]}{2\sqrt{2m_{I}}}\left(\exp\left\\{-im_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\lambda[h^{\prime}]}\right\\}\mathsf{G}_{I}+\exp\left\\{im_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\lambda[h^{\prime}]}\right\\}\mathsf{G}_{I}^{\dagger}\right).$
(80)

Taking into account the $n$-field one-point quantum states determined as

$\displaystyle|h,n\rangle\equiv\mathbf{\Psi}^{n}\left|\mathrm{VAC}\right\rangle=\left(\frac{\lambda}{2\sqrt{2m_{I}}}\exp\left\\{im_{I}\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\lambda^{\prime}}\right\\}\right)^{n}\mathsf{G}^{\dagger
n}_{I}\left|\mathrm{VAC}\right\rangle,$ (81)

yields two-point correlators
$\mathrm{Cor}_{n^{\prime}n}(h^{\prime},h)\equiv\langle
n^{\prime},h^{\prime}|h,n\rangle$ or explicitly

$\displaystyle\mathrm{Cor}_{n^{\prime}n}(h^{\prime},h)$ $\displaystyle=$
$\displaystyle\dfrac{\lambda^{\prime
n^{\prime}}\lambda^{n}}{\left(\sqrt{8m_{I}}\right)^{n^{\prime}+n}}\exp\left\\{im_{I}\left(n^{\prime}\int_{h^{\prime}}^{h_{I}}+n\int_{h_{I}}^{h}\right)\dfrac{\delta
h^{\prime\prime}}{\lambda^{\prime\prime}}\right\\}\times$ (82)
$\displaystyle\times$
$\displaystyle\left\langle\mathrm{VAC}\right|\mathsf{G}_{I}^{n^{\prime}}\mathsf{G}^{\dagger
n}_{I}\left|\mathrm{VAC}\right\rangle,$

where $\lambda^{\prime}\equiv\lambda[h^{\prime}]$ and so on. Basically one
obtains

$\displaystyle\mathrm{Cor}_{00}(h,h)=\mathrm{Cor}_{00}(h^{\prime},h)=\mathrm{Cor}_{00}(h_{I},h_{I})=\left\langle\mathrm{VAC}|\mathrm{VAC}\right\rangle,$
(83)
$\displaystyle\mathrm{Cor}_{11}(h_{I},h_{I})=\dfrac{1}{8m_{I}}\quad,\quad\dfrac{\mathrm{Cor}_{n^{\prime}n}(h_{I},h_{I})}{\left[\mathrm{Cor}_{11}(h_{I},h_{I})\right]^{(n^{\prime}+n)/2}}=\left\langle\mathrm{VAC}\right|\mathsf{G}_{I}^{n^{\prime}}\mathsf{G}^{\dagger
n}_{I}\left|\mathrm{VAC}\right\rangle,$ (84)

so by elementary algebraic manipulations one receives

$\displaystyle\mathrm{Cor}_{11}(h^{\prime},h)=\dfrac{\sqrt{{\mathrm{Cor}_{11}(h^{\prime},h^{\prime})\mathrm{Cor}_{11}(h,h)}}}{\mathrm{Cor}_{11}(h_{I},h_{I})}\exp\left\\{im_{I}\int_{h^{\prime}}^{h}\dfrac{\delta
h^{\prime\prime}}{\lambda^{\prime\prime}}\right\\},$ (85)
$\displaystyle\dfrac{\mathrm{Cor}_{nn}(h^{\prime},h)}{\mathrm{Cor}_{00}(h_{I},h_{I})}=\left[\dfrac{\mathrm{Cor}_{11}(h^{\prime},h)}{\mathrm{Cor}_{00}(h_{I},h_{I})}\right]^{n}\quad,\quad\dfrac{\mathrm{Cor}_{11}(h,h)}{\mathrm{Cor}_{00}(h_{I},h_{I})}=\lambda^{2}\mathrm{Cor}_{11}(h_{I},h_{I}).$
(86)

Straightforwardly from (86) one relate a size scale with quantum correlations

$\displaystyle\lambda=\sqrt{{\dfrac{\mathrm{Cor}_{11}(h,h)}{\mathrm{Cor}_{11}(h_{I},h_{I})\mathrm{Cor}_{00}(h_{I},h_{I})}}},$
(87)

that consequently leads to the formulas

$\dfrac{\mathrm{Cor}_{n^{\prime}n}(h,h)}{\mathrm{Cor}_{n^{\prime}n}(h_{I},h_{I})}=\lambda^{n^{\prime}+n}\exp\left\\{-im_{I}(n^{\prime}-n)\int_{h_{I}}^{h}\dfrac{\delta
h^{\prime}}{\lambda^{\prime}}\right\\},$ (88)
$\dfrac{\mathrm{Cor}_{11}(h^{\prime},h)}{\mathrm{Cor}_{00}(h_{I},h_{I})\mathrm{Cor}_{11}(h_{I},h_{I})}=\lambda^{\prime}\lambda\exp\left\\{im_{I}\int_{h^{\prime}}^{h}\dfrac{\delta
h^{\prime\prime}}{\lambda^{\prime\prime}}\right\\},$ (89)
$\dfrac{\mathrm{Cor}_{nn}(h^{\prime},h)}{\mathrm{Cor}_{00}(h_{I},h_{I})}=\lambda^{\prime
n}\lambda^{n}[\mathrm{Cor}_{11}(h_{I},h_{I})]^{n}\exp\left\\{im_{I}n\int_{h^{\prime}}^{h}\dfrac{\delta
h^{\prime\prime}}{\lambda^{\prime\prime}}\right\\}.$ (90)

A whole information on the system is contained in $\lambda$, $\mu$, and
$m_{I}$. Quantum correlations are determined by these quantities only. By
measurement of quantum correlations one deduces $\lambda$, $\mu$, $m_{I}$.

The presented conclusions have a formal character, however, they show a
general feature of the proposed simple model of quantum gravity. We have
solved the model by the 1D wave function (58). We have discussed the
3-dimensional manifolds (70) corresponding to this model, and we have found
the relation between quantum correlations and physical scales (87).

##  6 Summary

This paper discussed some consequences of the global one–dimensionality
conjecture within the Wheeler–DeWitt theory. We have applied a field theory
for formulation of the simple model of quantum gravity. The model was
constructed by the following steps

  1. 1.

We have started from general relativity of compact manifold with boundaries;
its action was written in $3+1$ splitting and the scalar constraint was
canonically quantized. Resulting theory was the Wheeler–DeWitt model of
quantum gravity;

  2. 2.

Next stage we have proposed the ansatz based on the global one-dimensionality
conjecture;

  3. 3.

With using of the ansatz the quantum geometrodynamics was reduced to one-
dimensional global evolution, with the dimension being an embedding volume
form;

  4. 4.

Employing canonical formalism, we have used a field-theoretic action
corresponding to the model, and by dimensional reduction the appropriate Dirac
equation was received;

  5. 5.

Finally the Fock quantization was applied. Static reper of creators and
annihilators was found by using of the diagonalization procedure employing the
Bogoliubov transformation and the Heisenberg equations of motion.
Consequently, the proposed model is defining quantum gravity as a quantum
field theory. The quantum field was derived in a straightforward way (53).

In result, we have obtained correctly defined integrability problem, which
allowed to study its formal consequences. Particularly, we have discussed

  1. 1.

The integrability problem and global one-dimensional wave function. It was
shown that by integration of the model in the Schrödinger equation form there
is a possibility to obtain an exact solutions of the model.

  2. 2.

3-dimensional manifolds corresponding to the model. It was shown that the
model is defining a 3-dimensional manifolds ${}^{(3)}\\!R=\phi_{n}h$, where
for given wave function the parameter $\phi_{n}$ can be derived by the
assumption of the appropriate Fourier transforms and its inverted transforms.

  3. 3.

Relation between quantum correlations and physical scales. We have connected
one-point quantum correlations with size and mass scales.

The presented conclusions are partial, but they show possible physical and
geometric implications following from the simple quantum gravity model. From a
mathematical point of view we have applied a strategy of one-dimensional
integration, with the Riemann–Lebesgue measure for fixed space configuration
or the Stieltjes measure in general case. Both physical and mathematical sides
of the model were emphasized in this paper.

## Acknowledgements

Special thanks are directed to Prof. I. L. Buchbinder for very constructive
discussion and his helpful comments to primary notes of the author. The author
benefitted also many valuable discussions from Profs. A. B. Arbuzov, I. Ya.
Aref’eva, B. M. Barbashov, K. A. Bronnikov, and V. N. Pervushin.

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# Psychometric Analysis and Coupling of Emotions Between State Bulletins and
Twitter in India during COVID-19 Infodemic

Baani Leen Kaur Jolly<EMAIL_ADDRESS>Indraprastha Institute
Information Technology, Delhi ,  Palash Aggrawal<EMAIL_ADDRESS>Indraprastha Institute Information Technology, Delhi ,  Amogh Gulati
<EMAIL_ADDRESS>Indraprastha Institute Information Technology, Delhi ,
Amarjit Singh Sethi<EMAIL_ADDRESS>Guru Nanak Dev University, Amritsar ,
Ponnurangam Kumaraguru<EMAIL_ADDRESS>Indraprastha Institute Information
Technology, Delhi and  Tavpritesh Sethi<EMAIL_ADDRESS>Indraprastha Institute Information Technology, Delhi

###### Abstract.

COVID-19 infodemic has been spreading faster than the pandemic itself. The
misinformation riding upon the infodemic wave poses a major threat to people’s
health and governance systems. Since social Media is the largest source of
information, managing the infodemic not only requires mitigating of
misinformation but also an early understanding of psychological patterns
resulting from it. During the COVID-19 crisis, Twitter alone has seen a sharp
45% increase in the usage of its curated events page, and a 30% increase in
its Direct messaging usage, since March 6th
2020.111https://blog.twitter.com/en_us/topics/company/2020/An-update-on-our-
continuity-strategy-during-COVID-19.html
In this study, we analyze the psychometric impact and coupling of the COVID-19
infodemic with the official bulletins related to COVID-19 at the national and
state level in India. We look at these two sources with a psycho-linguistic
lens of emotions and quantified the extent and coupling between the two. We
modified Empath, a deep skip-gram based open sourced lexicon builder for
effective capture of health related emotions. We were then able to capture the
time-evolution of health-related emotions in social media and official
bulletins. An analysis of lead-lag relationships between the time series of
extracted emotions from official bulletins and social media using Granger’s
causality showed that state bulletins were leading the social media for some
emotions such as Medical Emergency. Further insights that are potentially
relevant for the policymaker and the communicators actively engaged in
mitigating misinformation are also discussed. Our paper also introduces
CoronaIndiaDataset222http://precog.iiitd.edu.in/resources.html, the first
social media based COVID-19 dataset at national and state levels from India
with over 5.6 million national and 2.6 million state level tweets. Finally, we
present our findings as COVibes333http://covibes.tavlab.iiitd.edu.in/, an
interactive web application capturing psychometric insights captured upon the
CoronaIndiaDataset, both at a national and state level.

##  1\. Introduction

”We’re not just fighting an epidemic; We’re fighting an infodemic”. These
words spoken by the WHO Director-General at the Munich Security Conference on
15 February 2020444https://www.who.int/dg/speeches/detail/munich-security-
conference, sums up the challenges faced by our society due to COVID-19.
Infodemic comes from the root words information and epidemic. It refers to an
excessive amount of information which is made publicly available, consisting
of both accurate and not so accurate information, which makes it harder to
find reliable, trustworthy and accurate guidance when
needed.555https://www.who.int/docs/default-source/coronaviruse/situation-
reports/20200415-sitrep-86-covid-19.pdf.
Social media is one of the most popular media for the diffusion of information
during an infodemic. Twitter, a micro-blogging site, is one of the most widely
used social media platforms and it has seen a sharp 45% increase in the usage
of its curated events page, and a 30% increase in its Direct messaging usage,
since March 6th 2020, during the COVID-19
emergency.666https://blog.twitter.com/en_us/topics/company/2020/An-update-on-
our-continuity-strategy-during-COVID-19.html. Social media platforms are not
only easily accessible and have a global reach, they provide a virtual space
to the users distant from their real world. Due to this, they provide a medium
to help people discuss many taboo topics which they might not in their real
social circles, like mental healthcare, domestic violence, sexual assault etc.
This provides an opportunity to use data mining & natural language processing
techniques to analyse the web data from the standpoint of psychology.

The recent outbreak of COVID-19 (COrona VIrus Disease 2019) has the world in
its grips. With exponentially rising cases, WHO declared it as a global
pandemic on March 11, 2020. Following health advisories and the virus rapidly
spreading, most countries have declared national emergencies, closed borders
and restricted public movement. Most countries in the world have gone into
national lockdown to contain the pandemic.

Naturally, such a strong event’s impact has affected the daily lives of people
all over the world, something which has also impacted people’s social media
usage. In recent times of social distancing, social media has become a popular
platform for people to express their thoughts & opinions. Social Media is not
only being used by people to express how their lives have been affected and
appeal to everyone to understand the importance of the situation, but trending
hashtags like #CoronaOutbreak #COVID19, are used by authorities disseminate
important information and health advisories.

A lot of research has previously been done on social media analysis related to
pandemics. Ritterman et al. (Ritterman2009, ) showed how prediction market
models and social media analysis can be used to model public sentiment on the
spread of a pandemic. Signorini et al. (signorini2011use, ) examined Twitter
based-information to track the swiftly-evolving public sentiment regarding
Swine Flu, in 2011 as well as correlate the H1N1 related activity to
accurately track reported disease levels in the US. Jain et
al.(jain2015effective, ) used Twitter as a surveillance system to track the
spread of 2015 H1N1 pandemic in India as well as the general public awareness
towards it. In this paper, we examine the use of social media during this
ongoing time of n-CoV2019 pandemic. As the impact of this pandemic is growing
at an exponential rate, we try to model this rapidly evolving public sentiment
and dig deeper to understand how it changes daily, using time-series based
analysis. To this end, we have curated dataset of more than 5.6 million tweets
and retweets, specific to India. This dataset was collected using two
approaches - Content-Based and Location-Based queries, as explained in Section
3. We model the public sentiment using sentiment analysis and the Empath
psycho-linguistic features.

##  2\. Literature Survey

Prior work done on understanding the public sentiment during the Ebola medical
crisis by Lazard et al. (lazard2015detecting, ) detected the ongoing themes in
the ongoing discourse on the live Twitter chat by Centers for Disease Control
and Prevention. Wong et al. (wong2017local, ) further studied the tweets by
the local health departments related to Ebola. The work done by Kim et al.
(kim2016topic, ) presents a topic-based sentiment analysis of the Ebola virus
on twitter and in the news.

With the world’s efforts focused on battling COVID-19, research efforts from
many fields including online social media & textual analysis has vigorously
started in this direction. Chen et al. (Chen2020, ) provide the first Twitter
dataset by collecting tweets related to #COVID19. This is an ongoing
collection, started from January 22, 2019. Haouari et al.
(haouari2020ArCov-19, ) also present ArCov-19, a large Arabic Twitter dataset
collected from January 27,2019 to March 31, 2019.

During this infodemic, Zhao et al.(Zhao2020, ) analyse the attention of
Chinese public to COVID-19 by analysing search trends on Sina Microblog and
evaluating public opinion through word frequency and sentiment analysis.
Cinelli et al. (cinelli2020covid, ) provide insights into the evolution of the
global COVID-19 discourse on Twitter, Instagram, Reddit, YouTube and Gab.
Alhajji et al. (Alhajji2019, ) analysed public sentiment of Saudi Arabia by
collecting upto 20,000 tweets within 48 hours of key events in Saudi Arabia’s
timeline, using transfer learning to do sentiment analysis. Kayes et al.
(Kayes2020, ) attempt to measure community acceptance of social distancing in
Australia, reporting that majority of tweets were in favour of social
distancing. Li et al. (Li2020, ) try to extract psychological profiles of
active Weibo users during the time of COVID-19 spread in China to analyse
linguistic, emotional and cognitive indicators.

Analysing user behavior and response can provide a critical understanding of
what policies and decision worked. Hou et al. (Hou2020, ) assess the public’s
response to the situation and the government guidelines in terms of attention,
risk perception, emotional and behavioral response by analysing search trends,
shopping trends and blog posts on popular Chinese services. Li et al.
(Li2020b, ) analyse how information regarding COVID-19 was disseminating,
suggesting useful insights into the need for information. Work by Schild et
al. (Schild2020, ) shows how the current pandemic situation has caused an
unfortunate rise in Sinophobic behavior on the web.

##  3\. Dataset

We curated the CoronaIndiaDataset(CID), collected using Twitter’s official
Tweepy API, from 1st March to 27th April 2020. CoronaIndiaDataset contains
56,24,066 tweets talking about COVID-19 in Indian settings as well as
26,55,503 state-specific tweets. Table 1 shows the state-wise frequency of
tweets available in our dataset.

###  3.1. Twitter Data Collection

The data was collected using 2 separate approaches in parallel – Content Based
Query and Location Based Query.

####  3.1.1. Content-based Query:

To collect the relevant Twitter data, we explored the trending and most
popular hashtags for each of the Indian states and manually curated the list
of hashtags related to COVID-19. We also went through several n-CoV2019
related tweets manually to find and subsequently mined the most popular
hashtags(list given in appendix) related to the same, which may not be
trending. Further, in order to automate the collection of relevant tweets, we
tried to formulate generic queries like ‘corona ¡state¿’, etc. and collected
the state-wise n-CoV2019 related twitter data using the same.
This approach focuses on getting all tweets which are talking about COVID-19
in the context of India or the Indian states. Multiple queries were built by
joining terms related to COVID-19 with the name or common aliases of the
region and data was collected from 1st March to 23rd April 2020. Some of the
terms used were - ‘corona’, ‘covid19’, ‘coronavirus’, ‘lockdown’, etc. In
addition, popular hashtags like #coronavirusin¡region¿, #coronain¡region¿,
#corona¡region¿, #¡region¿fightscorona were used, where ‘¡region¿‘ is replaced
by the name or popular alias of India or various states. Examples of a popular
alias are ‘Orissa’ for Odisha, ‘TN’ for Tamil Nadu, ‘UP’ for Uttar Pradesh, or
spelling mistakes like ‘chatisgarh’ for Chhattisgarh.

Table 1. CoronaIndiaDataset Statistics. Region | Tweets Available | Region | Tweets Available | Region | Tweets Available
---|---|---|---|---|---
India | 56,24,066 | Chhattisgarh | 32,058 | JnK | 15,923
Delhi | 5,57,458 | Madhya Pradesh | 82,290 | Jharkhand | 24,276
Karnataka | 2,09,582 | Maharashtra | 3,89,982 | Himachal Pradesh | 13,017
Tamil Nadu | 2,08,608 | Uttar Pradesh | 2,33,507 | Assam | 16,707
Telangana | 1,75,444 | Goa | 11,943 | Meghalaya | 2,641
Haryana | 95,197 | Rajasthan | 1,18,571 | Mizoram | 1,547
West Bengal | 93,861 | Gujarat | 97,478 | Arunanchal Pradesh | 1,388
Odisha | 50,704 | Bihar | 93,623 | Manipur | 2,796
Kerala | 47,644 | Uttarakhand | 17,992 | Tripura | 4,078
Punjab | 43,426 | Sikkim | 1,126 | Nagaland | 3,013
Andhra Pradesh | 18,759 |  |  |  |

####  3.1.2. Location-based Query:

Tweets are collected for globally trending COVID-19 related hashtags, and then
filtering tweets based on the User Location. Tweets for the following hashtags
and keywords are collected - #COVID19, ‘CoronaVirusUpdates’, ‘coronavirus’,
‘corona virus outbreak’, ‘corona wuhan’, #Coronavirus, #NCOV19,
#CoronavirusOutbreak, #coronaviruschina, #coronavirus, ‘COVID19’ from 14th
March to 27th April 2020. This resulted in a collection of a total of 12
million tweets from all over the world. We also create a list of location
filters for various states. These are the state names, aliases (as explained
above) and name of popular cities in those states. Using these, we first
filter out all tweets having ‘india’ in their user location, and then sort
them based on keyword matches of tokens in the user location with the above
list. The User Location was lowercased before matching.

We analysed using a chatterplot, the most frequent occurrences in our dataset,
as shown in Figure 1. The plot shows the top 200 words, arranged by their
frequency and Bing Sentiments (liu2012sentiment, ). +1 is positive, -1 is
negative and the group of words in the middle have no sentiment value
associated. In this plot, we have removed the outlier ’India’ for a better
representation of other terms.

Figure 1. Chatterplot for COVID-19 India-Related Tweets. The X axis is the
Bing Sentiment and Y axis is the frequency.
Figure 2. Frequency of tweets available in our Dataset.

###  3.2. Indian State Government n-CoV2019 related bulletins

DataMeet Community has curated a database of COVID-19 related government
bulletins from Indian States 777http://projects.datameet.org/covid19/
888https://github.com/datameet/covid-19-indian-state-reports. These bulletins
have statistics about COVID-19 cases in the state, government’s response to
them, advisories and other useful information. We have analysed all the
reports that are in English language and belong to the states of Delhi, West
Bengal, Punjab, Tamil Nadu, Odisha and Kerala.

##  4\. Methodology

###  4.1. Preprocessing Data

We followed the below steps to pre-process the data and reduce the noise:

  1. (1)

Lowercasing

  2. (2)

Tokenisation

  3. (3)

Links Removal

  4. (4)

Removal of all alphanumeric characters, allowing only standard English
alphabets

  5. (5)

Stop Words Removal:
Performed Stop Words removal, using NLTK, for English and 12 other languages
like French, Spanish, German, which share the same script with English.

  6. (6)

Lemmatisation:
We used Wordnet Lemmatiser instead of Porter Stemmer, since the latter leads
to the reduction of words to a form wherein they are no longer real words,
however, the former ensures that each word is reduced to a real word in
English dictionary. For example, ‘studies’ and ‘studying’ get converted to
‘studi’ and ‘study’ by Porter Stemmer, while a Lemmatiser matches both of them
to a common lemma ‘study’.

###  4.2. Quantitative Empath Analysis

Empath (Empath-Fast2016, ) is an open vocabulary based tool to generate and
validate lexical categories. It is based on deep skip-gram model to draw
correlation between many words and phrases starting from a small set of seed
words. It has some inbuilt categories, including emotions, which can be used
to identify the emotion associated with a text.

Empath provides 3 types of datasets to build the lexicon from - ‘reddit’
(social media), ‘nytimes’ (news articles) and ‘fiction’, and models a category
by finding the words closest to the ”seed words” of that category. But, the
text data used for either of them is outdated and does not have enough
information about the language used in the current scenario. Preliminary
analysis using the Empath library showed that the current lexicon was
inadequate to properly analyse the current situation. A case in point would be
that the word ‘positive’ had a connotation with positive emotion in the Empath
categories, however, in the COVID-19 scenario, it was often used in the
context of ‘tested positive’, which by itself was neither a positive nor a
negative emotion, and rather hinted at the activity of testing positive for
COVID-19

To rectify this, we manually examined the most frequent unigrams and bigrams
in the collected data as well as some common bigrams in the given context
which may be classified incorrectly, and manually annotated them into the most
relevant categories or created new categories to help better analyse the
emotional content of the tweets. Some important modifications are shown in
Table 2.

Table 2. Important modifications to Empath categories. Empath Category |  Tokens added (Lemmatized)
---|---
medical emergency |  ‘case’, ‘positive’, ‘positive case’, ‘test_positive’, ‘pandemic’, ‘lockdown’, ‘spread’
health |  ‘test_positive’, ‘test_negative’
healing |  ‘vaccine’
government |  ‘cm’, ‘pm’, ‘prime_minister’, ‘minister’, ‘govt’
movement |  ‘socialdistancing’, ‘social_distancing’, ‘awareness’
fight, war |  ‘eradicate’, ‘contain’, ‘overcome’, ‘prevent’
business |  ‘startup’

We analyse Empath scores of emotions related to Positive Sentiment, Negative
Sentiment, Country and Government, the pandemic caused by COVID-19 and the
fight against COVID-19. Details of the specific categories used can be found
in Table 3.

Table 3. Empath Categories. Domain |  Empath Categories
---|---
Positive Sentiment |  ‘help’, ‘healing’, ‘positive emotion’, ‘optimism’, ‘sympathy’
Negative Sentiment |  ‘fear’, ‘negative emotion’, ‘sadness’, ‘nervousness’, ‘confusion’
Country |  ‘government’, ‘economics’, ‘business’, ‘occupation’
Pandemic Situation |  ‘medical emergency’, ‘health’, ‘hygiene’
Fighting n-CoV2019 |  ‘war’, ‘fight’, ‘movement’
---
Figure 3. Empath Analysis for Indian Twitter: medical_emergency & government are the most frequently talked about Empath categories. |  |
---|---|---
(a) Delhi | (b) West Bengal | (c) Punjab
|  |
(d) Tamil Nadu | (e) Odisha | (f) Kerala
Figure 4. Empath analysis of tweets particular to Indian states: The most
prominent categories across all states were medical_emergency & government.

##  5\. Analysis

###  5.1. Twitter Content

####  5.1.1. National Level

Using Empath, we analysed the tweets collected in March 2020, along various
psycho-linguistic attributes, as shown in Fig 3. The most common categories
being discussed in the Tweets were government, health and medical emergency,
which reflects that while discussing about the pandemic, the public is
bringing the government in the discourse, be it referring to some government
policy or some information released by the government. Another observation is
that the negative emotion is fairly high amongst these tweets. A positive
indicator is that the confusion level indicated by the Empath analysis is
significantly low, while the frequency of linguistic features related to
positive emotions, healing and optimism are much higher.

|  |
---|---|---
(a) Government-related | (b) Government-related | (c)Health emergency-related
Figure 5. Sample tweets belonging to the popular Empath categories.

####  5.1.2. State Level

We analysed the psycho-linguistic features of COVID-19 related discourses on
Twitter at a state-level, as shown in Figure 4. It is interesting to note that
although the magnitudes of different psycho-linguistic features vary across
different states, their structure remains very similar.
We observed that while in some states like Delhi, Punjab, Odisha tweets
talking about government related words were the most on Twitter, few states
like Tamil Nadu and West Bengal talked more about medical emergency. Kerala,
on the other hand had an equal frequency of words related to medical_emergency
and government. West Bengal and Kerala also have a higher frequency of words
related to negative emotion compared to other states.

West Bengal showed higher levels of healing and positive emotions in the
tweets. Interestingly, it also showed an even higher frequency of war or fight
related words.

###  5.2. Twitter Time Series

|  |
---|---|---
(a) Hygiene | (b) Nervousness | (c) Business
|  |
(d) Optimism | (e) Health | (f) Government
Figure 6. Time Series analysis of emotions in India related Tweets. Clear
patterns of consistent decline/increase or peak-dip can be seen in the time
series plots of various emotions, which correlate to on ground events related
to COVID-19.

We analysed the collected Twitter data in the Indian context over a period of
2 months (March and April 2020), by looking from the lens of various psycho-
linguistic attributes as shown in Figure 6. We observe that while the
frequency of ‘hygiene’ & ‘nervousness’ related words has decreased over time,
since the start of COVID-19 crisis in India, words related to ‘business’ &
‘optimism’ have become more frequent. The categories of ‘health’ and
‘government’ have been one of the most popular categories for the Indian
twitter data, and while the presence of ‘health’ category in tweets observes a
sharp dip on March 28, ‘government’ related words have a sharp rise on the
same date.

####  5.2.1. Relation of public sentiment to the rapidly changing on-ground
situation

We observed that the presence of optimism related keywords in the tweets has
increased over time, with the highest frequency of optimistic words from
18th-21st April. It is interesting to note that on 18th April, due to the
imposition of a nationwide lockdown, the time taken for doubling COVID-19
cases came down from every 3 days to every 8 days.
We also observe that the discussion regarding certain aspects of COVID-19
discourses on twitter have reduced over time, especially those related to
‘Hygiene’ & ‘movement’, which became very popular near the time when the
lockdown first got imposed, however, over time the frequency has declined
possibly hinting at normalisation of certain aspects of the COVID-19 narrative
on Indian twitter.
The frequency of nervousness related words has sharply declined over time,
with the peak around the time when COVID-19 started becoming popular. We also
observe that frequency of business-related words increase over time, with the
peak being observed around 20th April, the day when the government allowed
certain relaxation for shops, etc. to re-open up for the first time post the
COVID-19 lockdown.
The frequency of ‘health’ related tweets take a sharp dip near 28th March,
wherein the frequency of tweets related to ‘government’ observes its peak. It
is interesting to note than on 28th March, India crossed a total number of
1,000 confirmed COVID-19 cases. We thus, observe that the rapidly evolving
public sentiment are reflective of public’s response to the on-ground
n-CoV2019 situation and the government response.

####  5.2.2. The changing discourse around n-CoV2019

As can be observed from Figure 6 the discussion regarding certain aspects of
COVID-19 discourses on twitter reduced over time, especially those related to
Hygiene, movement and nervousness, while the discussions regarding business
and optimism increased. We observe that hygiene-related COVID-19 discourses on
Twitter became very popular near the time when the lockdown first got imposed.
However, over time discussions regarding hygiene have reduced on Twitter,
which might be due to it being normalised over time.

|  |
---|---|---
(a) Delhi | (b) West Bengal | (c) Punjab
|  |
(d) Tamil Nadu | (e) Odisha | (f) Kerala
Figure 7. Empath analysis of Government Bulletins related to COVID-19 in
India.

###  5.3. Government Bulletins Content

We observe that the government bulletins(as described in Section 3.2) shared
by the governments of all the six states, as shown in Figure 7 frequently use
words related to medical emergency and health. The government bulletins
released on n-CoV2019, by the Delhi and West Bengal
government999https://github.com/datameet/covid-19-indian-state-reports, have a
higher frequency of linguistic features related to the topic ‘healing’ than
other states. An interesting observation is that the state government
bulletins in Odisha have a significantly higher inclination towards using
words related to government, while for most other states, the primary focus is
towards medical emergency. Also, all government bulletins show no ‘fear’ or
‘confusion’ related psycho-linguistic markers.

###  5.4. Granger’s Causality Analysis

As a pre-requisite for studying causal mechanism between the time series on
Delhi Bulletin and Delhi Tweets, both the sets of data were subjected to
Augmented Dickey-Fuller (ADF) test of unit root (so as to see whether the
series are stationary or not). The formulation adopted for the ADF test was:

(1) $\triangle{Y_{t}}=\tau
Y_{t-1}+\alpha_{i}\sideset{}{}{\sum}_{i=1}^{m}\triangle{Y_{t-i}}+u_{t}$

where t stands for the time variable; $\triangle{}$ for the difference
operator; and ut for the disturbance terms. The null and alternative
hypotheses for the test are:
$\triangle{H_{0}}:\tau=0$(meaning that the series possesses a unit root and
is, therefore, non-stationary);
$\triangle{H_{1}}:\tau<1$ (meaning that the series does not possess a unit
root and is, therefore, stationary).
For both the time series, the test was performed at levels as well as at first
difference; Table 4. As per the table, value of the test statistic $\tau$ for
the time series (at level) on Help in respect of Delhi Bulletin was computed
to be -1.596, which failed to reach the critical values ( -1.951 at 5% and
-2.623 at 1% level of significance). Accordingly, we could not reject the null
hypothesis of the presence of a unit root in the series. In other words, the
series on ‘Help’ was non-stationary. However, at the first difference, the
series was detected to be free from a unit root ($\tau$ = -6.081) and was,
therefore, stationary in nature. A large majority of the time series on rest
of the variables (except for Fear, Sadness, Nervousness, Confusion, Fun,
Positive Emotion and Economics, wherein stationarity was present at levels
itself) of Delhi Bulletin showed this very type of behaviour. Notably, in
respect of Delhi Tweet, the series on the entire set of variables were
observed to be non-stationary at levels but stationary at the first
difference. Consequently, for examining causality behaviour, we have uniformly
considered the first-differenced series on all the variables in respect of
both data sets. Here, we may mention that ADF test could not be performed on
the variable Sympathy in respect of the first set of data (because of the
absence of variability) and was, therefore, left-out for the subsequent
analysis on Granger’s causality.
For examining causality, each of the corresponding pairs of variables from the
first-differenced data sets was subjected to the estimation of Equations 2 and
3:

(2)
$Y_{t}=\alpha_{01}+\sideset{}{}{\sum}_{i=1}^{p}\alpha_{1i}Y_{t-i}+\sideset{}{}{\sum}_{i=1}^{p}\beta_{1i}X_{t-i}+u_{1t}$
(3) $Y_{t}=\alpha_{02}+\sideset{}{}{\sum}_{i=1}^{p}\alpha_{2i}Y_{t-i}+u_{2t}$

which were then compared for their predictive power through Wald’s test. If
Equation 2 turns out to be statistically superior to the Equation 1 (thus
implying that current value of Y can be better predicted through its own past
values as well as the past values of X than through the past values of Y
alone), then we say X Granger causes Y. The series Y and X were then
interchanged and the process repeated so as to examine if Y Granger causes X.
The optimum number p of lagged terms to be included was decided through min
AIC criterion which, in the present analysis, turned out to be 1, in general.

As per the results through the analysis (Table 5), it was observed that in
respect of the variable ‘Help’, the data set on Delhi Tweet (TWT) failed to
induce any causality on the data set on Delhi Bulletin (BLT), because the
F-value (= 1.194 at 1 & 30 d.f.) associated with Wald’s test turned out to be
statistically non-significant (p = 0.2832). On interchanging the two data
sets, the finding remained virtually similar, thus implying that no causal
linkage could be detected between the two data sets with respect to ‘Help’.

In respect of ‘Medical Emergency’, the data set on Tweet induced significant
causality (at 5% probability level) on such a data such on Bulletin. On
interchanging the two data sets, strength of causality (from Bulletin to
Tweet) became all the more robust (at 0.1% probability level). We may thus say
that although there was an indication of bi-directional (or, equivalently,
feedback) causality between the two sets of data in respect of ‘Medical
Emergency’, yet the strength of causality was more pronounced from Bulletin to
Tweet. Bi- directional causality (at 5% level) between the two sets of data
was observed in respect of ‘Health’. Very strong causality (at ¡ 0.1%
probability level) from Tweet to Bulletin was indicated in respect of the
variables ‘Hygiene’, ‘Leisure’, ‘Fun’ and ‘Government’. Significant (at 5%
probability level) unidirectional causality (from Bulletin to Tweet) was also
detected in respect of each of ‘Death’ and ‘War’. But for rest of the
variables, causal linkages between the two sets of data could failed to be
established. Thus, on the whole, direction and strength of causality between
the two sets of data were peculiar to the variable under consideration.

Table 4. Results in respect of Augmented Dickey-Fuller test of Unit Root.

| Bulletin | Tweet
---|---|---
Variable | At Levels | First Differenced | At Levels | First Differenced
| $\tau$ | Significance | $\tau$ | Significance | $\tau$ | Significance | $\tau$ | Significance
Help | -1.5957 | NS | -6.0811 | ** | -0.0196 | NS | -6.7529 | **
Medical Emergency | -0.7589 | NS | -4.2722 | ** | -0.3510 | NS | -4.8404 | **
Health | -0.708 | NS | -4.4264 | ** | -0.1305 | NS | -5.3220 | **
Hygiene | -1.0035 | NS | -3.9099 | ** | -0.8163 | NS | -5.5162 | **
Fear | -1.9712 | * | -4.0759 | ** | -1.6783 | NS | -5.2862 | **
Death | -1.0259 | NS | -5.5868 | ** | -1.0432 | NS | -3.9519 | **
Negative Emotion | -0.9951 | NS | -5.3831 | ** | -1.1707 | NS | -4.9076 | **
Sadness | -3.6667 | ** | -6.8719 | ** | -1.4853 | NS | -3.9579 | **
Nervousness | -2.4119 | * | -4.1097 | ** | -2.8012 | NS | -4.4155 | **
Confusion | -2.2471 | * | -6.1171 | ** | -0.3505 | NS | -5.1352 | **
War | -0.7392 | NS | -4.1929 | ** | -0.6573 | NS | -3.6262 | **
Fight | -1.1623 | NS | -5.831 | ** | -0.6296 | NS | -3.4056 | **
Healing | -0.7563 | NS | -5.2625 | ** | 0.0691 | NS | -4.0565 | **
Movement | -0.5542 | NS | -3.4916 | ** | -0.7894 | NS | -4.4361 | **
Leisure | -1.4084 | NS | -4.4319 | ** | -0.7163 | NS | -6.4206 | **
Fun | -1.9783 | * | -5.1959 | ** | -0.7581 | NS | -2.9320 | **
Positive Emotion | -2.705 | ** | -5.7106 | ** | -0.9994 | NS | -3.8429 | **
Optimism | -0.2408 | NS | -3.4171 | ** | -0.7702 | NS | -3.8225 | **
Sympathy | NC | NC | NC | NC | -1.6691 | NS | -5.8757 | **
Family | -1.579 | NS | -4.7658 | ** | -0.1853 | NS | -4.3916 | **
Government | -0.6655 | NS | -4.9432 | ** | 0.2586 | NS | -3.5532 | **
Economics | -2.2898 | * | -5.8496 | ** | 0.0531 | NS | -4.1381 | **
Business | -0.7952 | NS | -4.7221 | ** | 0.2435 | NS | -2.9480 | **
Occupation | -0.7609 | NS | -6.5216 | ** | 0.0550 | NS | -4.3939 | **

  * *

Critical Values of $\tau$: -1.951 at 5% and -2.623 at 1% level of
significance;
**: Significant at 0.1% probability level; **: Significant at 1% probability
level; *: Significant at 5% probability level; NS: Non-significant; NC: Non-
computable.

Table 5. Results in respect of Granger’s Causality Test.

Variable | Caused Set (Y) | Causal Set(X) | F-Value from Wald’s Test# | p-Value | Remark$
---|---|---|---|---|---
Help | Bulletin | Tweet | 1.194 | 0.2832 | NS
Tweet | Bulletin | 1.455 | 0.2372 | NS
Medical Emergency | Bulletin | Tweet | 6.932 | 0.0133 | *
Tweet | Bulletin | 16.673 | 0.0003 | ***
Health | Bulletin | Tweet | 4.988 | 0.0331 | *
Tweet | Bulletin | 5.99 | 0.0205 | *
Hygiene | Bulletin | Tweet | 10.004 | 0.0036 | **
Tweet | Bulletin | 2.71 | 0.1102 | NS
Fear | Bulletin | Tweet | 0.014 | 0.9078 | NS
Tweet | Bulletin | 0.115 | 0.7364 | NS
Death | Bulletin | Tweet | 0.157 | 0.6947 | NS
Tweet | Bulletin | 5.787 | 0.0225 | *
Negative Emotion | Bulletin | Tweet | 0.036 | 0.8505 | NS
Tweet | Bulletin | 2.558 | 0.1202 | NS
Sadness | Bulletin | Tweet | 1.872 | 0.1814 | NS
Tweet | Bulletin | 0.093 | 0.7622 | NS
Nervousness | Bulletin | Tweet | 0.828 | 0.37 | NS
Tweet | Bulletin | 0.034 | 0.8542 | NS
Confusion | Bulletin | Tweet | 0.011 | 0.9164 | NS
Tweet | Bulletin | 3.355 | 0.077 | NS
War | Bulletin | Tweet | 0.031 | 0.8616 | NS
Tweet | Bulletin | 6.425 | 0.0167 | *
Fight | Bulletin | Tweet | 3.937 | 0.0564 | NS
Tweet | Bulletin | 0.32 | 0.5756 | NS
Healing | Bulletin | Tweet | 0.146 | 0.7049 | NS
Tweet | Bulletin | 0.693 | 0.4117 | NS
Movement | Bulletin | Tweet | 0.603 | 0.4436 | NS
Tweet | Bulletin | 2.228 | 0.146 | NS
Leisure | Bulletin | Tweet | 16.293 | 0.0004 | ***
Tweet | Bulletin | 7.044 | 0.0126 | *
Fun | Bulletin | Tweet | 36.026 | 0.0001 | ***
Tweet | Bulletin | 5.561 | 0.0251 | *
Positive Emotion | Bulletin | Tweet | 0.231 | 0.6344 | NS
Tweet | Bulletin | 0.238 | 0.6292 | NS
Optimism | Bulletin | Tweet | 1.734 | 0.1978 | NS
Tweet | Bulletin | 1.263 | 0.2701 | NS
Family | Bulletin | Tweet | 0.419 | 0.5224 | NS
Tweet | Bulletin | 1.492 | 0.2314 | NS
Government | Bulletin | Tweet | 9.097 | 0.0052 | **
Tweet | Bulletin | 2.387 | 0.1328 | NS
Economics | Bulletin | Tweet | 0.071 | 0.7916 | NS
Tweet | Bulletin | 0.195 | 0.6622 | NS
Business | Bulletin | Tweet | 0.479 | 0.4941 | NS
Tweet | Bulletin | 1.484 | 0.2326 | NS
Occupation | Bulletin | Tweet | 2.573 | 0.1192 | NS
Tweet | Bulletin | 3.497 | 0.0713 | NS

  * *

#:At 1 & 30 d.f. $: ***: Significant at 0.1% probability level; **:
Significant at 1% probability level; *: Significant at 5% probability level;
NS: Non-significant.

##  6\. Conclusion

We present novel dataset consisting of more than 5.6 million n-CoV2019 related
Indian tweets, with special emphasis on the tweets related to each of the
Indian states. We further analyse the tweets to find the important topics and
psycho-linguistic features discussed and compare it both between various
states, as well as using a time-series based approach. We further try to link
the rapidly changing psycho-linguistic attributes of the public sentiment to
the real-life on-ground situations arising due to COVID-19. We further
designed an interactive web portal
COVibes101010http://covibes.tavlab.iiitd.edu.in/, which displays psychometric
insights gained both at a national and state level, from the
CoronaIndiaDataset. This dataset and analysis technique can be used for
further research into understanding the public perceptions and taking more
effective policy decisions. We restricted our work to the analysis of tweets
as well as government bulletins in English. Future work in this direction can
be done to increase the scope of the analysis to various Indian languages.

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##  Appendix A Appendix A: Hashtags used to collect Tweets

###  A.1. Content-based query:

We manually curated the following list of hashtags, based on trending and
popular hashtags used at India level as well as each of the states:
#CoronainMaharastra, #CoronainKolkata, #CoronavirusReachesDelhi,
#CoronainDelhi, #CoronainBengal, #CoronaDelhi, #COVID19-Delhi, #COVIDDelhi,
#coronavirusindelhi, #CoronainPunjab, #CoronaPunjab, #COVID19Punjab,
#COVIDPunjab, #CoronaBengaluru, #CoronaKarnataka, #CoronainKarnataka,
#COVID19Bangalore, #CoronaBangalore, #CoronavirusBangalore,
#CoronavirusinBangalore, #CoronaInKerala, #CoronaVirusAgra, #CoronaVirusUp,
#Coronaintamilnadu, #coronatamilnadu, #COVIDLadakh, #coronagurgaon,
#coronatelangana, #COVID19telangana, #COVIDtelangana, #coronahyderabad,
#COVIDhyderabad, #COVID19hyderabad, #coronavirusinhyderabad,
#coronavirushyderabad, #CoronainUP, #CoronainNoida, #COVID19Noida,
#CoronaRajasthan, #CoronainRajasthan, #CoronaBihar, #CoronainBihar,
#coronauttarkhand, #coronaOdisha, #coronaOdisha, #COVID19Odisha, #COVIDOdisha,
#COVIDAndhra-Pradesh, #covidinindia, #coronavirusindia, #coronainindia,
#coronaindia, #covidindia, #covid19india.

Based on the above list, we identified a certain set of patters with region
names or aliases. We used the following prefixes to add before each region
name or it’s aliases (‘term ¡region¿’) to run the queries - ‘corona ’,
‘lockdown ’, ‘coronain ’, ‘covidin ’, ‘stayathome ’, ‘covid ’, ‘coronavirus ’.
In addition, we use these prefixes before region names to create hashtags
(#prefix¡region¿) - #covid, #corona, #coronavirus, #covidin, #coronain,
#coronavirusin, #covid19, #covid2019, #¡region¿fightscorona.