# Formation of Quiescent Prominence Magnetic Field by Supergranulations

Qingjun Liu  School of Physics and Astronomy, Yunnan University
Kunming 650500, China  Chun Xia  School of Physics and Astronomy, Yunnan
University
Kunming 650500, China  National Astronomical Observatories, Chinese Academy of
Sciences
Beijing 100101, China

###### Abstract

To understand the formation of quiescent solar prominences, the origin of
their magnetic field structures, i.e., magnetic flux ropes (MFRs), must be
revealed. We use three-dimensional magnetofriction simulations in a spherical
subdomain to investigate the role of typical supergranular motions in the
long-term formation of a prominence magnetic field. Time-dependant horizontal
supergranular motions with and without the effect of Coriolis force are
simulated on the solar surface via Voronoi tessellation. The vortical motions
by the Coriolis effect at boundaries of supergranules inject magnetic helicity
into the corona. The helicity is transferred and accumulated along the
polarity inversion line (PIL) as strongly sheared magnetic field via helicity
condensation. The diverging motions of supergranules converge opposite
magnetic polarities at the PIL and drive the magnetic reconnection between
footpoints of the sheared magnetic arcades to form an MFR. The magnetic
network, negative-helicity MFR in the northern hemisphere, and fragmented-to-
continuous formation process of magnetic dip regions are in agreement with
observations. Although diverging supergranulations, differential rotation, and
meridional flows are included, the simulation without the Coriolis effect can
not produce an MFR or sheared arcades to host a prominence. Therefore Coriolis
force is a key factor for helicity injection and the formation of magnetic
structures of quiescent solar prominences.

Solar prominences — Supergranulation — Solar magnetic fields

††journal: ApJL††software: MPI-AMRVAC, PDFI_SS

##  1 Introduction

Quiescent prominences and filaments are long thin structures of dense and cool
plasma that remain stable in the solar corona for days or weeks along the
polarity inversion lines (PILs) of quiescent regions (Parenti, 2014). Their
magnetic fields have special structures, so-called filament channels, which
support the dense mass against gravity (Mackay et al., 2010). The aligned
chromospheric fibrils at two sides of a filament (Foukal, 1971) and dominant
magnetic field components along the PILs of prominences (Leroy et al., 1983)
indicate that filament channels have strongly sheared magnetic fields. The
overall magnetic topology of quiescent filament channels is most likely a
helical magnetic flux rope (MFR) (Xia et al., 2014b, a; Xia & Keppens, 2016)
with plenty of observational evidence, such as the elliptical shape of coronal
cavities around quiescent prominences (Gibson et al., 2010), concentric rings
of Doppler velocity in coronal cavities (Bak-SteAlicka et al., 2013),
persistent swhirling motions of coronal plasma around the center of coronal
cavities above prominences (Wang & Stenborg, 2010), and the inverse polarity
of prominence magnetic field whose component perpendicular to the PIL is
opposite to the one of a potential field (Bommier et al., 1994). Dipped
sheared arcades with concave-up magnetic fields (DeVore & Antiochos, 2000) may
be the magnetic structure of short filament channels around active regions,
but not long ones in quiescent regions (Patsourakos et al., 2020).

Two mechanisms were proposed to explain the formation of MFRs in the corona.
The first mechanism suggests that the emergence of the upper part of an MFR
from the conversion zone into the corona leads to sheared magnetic arcades
(SMAs), arcade-like magnetic loops without concave-up magnetic fields, which
are then transformed into an MFR through magnetic reconnection (Fan, 2001).
However, this flux emergence mechanism does not apply to the filaments in
quiescent regions where no large-scale magnetic flux emergence was found
(Mackay et al., 2008). Statistical study of filament observations reveals that
over 90% of filaments lie above PILs external to conjugate bipolar fields by
flux emergence (Mackay et al., 2008). The second mechanism relies on the
shearing, converging, and cancellation of opposite-polarity magnetic flux on
the photosphere, which transforms SMAs into a helical MFR by magnetic
reconnection in the lower solar atmosphere (Van Ballegooijen & Martens, 1989).
This flux cancellation mechanism has been supported by many numerical
simulations from magnetofriction models (Mackay & Van Ballegooijen, 2006) and
zero-beta magnetohydrodynamic (MHD) models (Amari et al., 1999) to isothermal
MHD models (Xia et al., 2014b). However, these models were confronted with
three difficulties. First, these models use smooth magnetic flux
distributions, which contradicts the observational fact that the photospheric
magnetic flux under quiescent filaments is discrete and concentrated as small
elements at supergranular boundaries (Zhou et al., 2021). Second, the models
relied on large-scale systematic converging flows or equivalent magnetic flux
diffusion towards the PIL on the photosphere to drive the flux cancellation.
But observations have not found such large-scale flows, instead,
supergranular-scale converging flows between diverging supergranules were
found both inside and outside filament channels (Rondi et al., 2007; Schmieder
et al., 2014). Third, the models required large-scale shearing motions, such
as differential rotation, to get shear arcades. However, the differential
rotation injects helicity of the opposite sign at west-east oriented PILs in
contrast to the observed hemispheric preference (van Ballegooijen et al.,
1998), which manifests that filaments with negative (positive) helicity
dominate in the northern (southern) hemisphere (Ouyang et al., 2017).

Antiochos (2013) proposed the helicity condensation theory to explain the
formation of filament channels. In the theory, photospheric vortical motions
between the convective cells inject magnetic helicity into magnetic flux tubes
and coronal magnetic reconnections between neighboring flux tubes transfer the
injected twist towards the outer periphery of the flux tubes, leading to an
inverse cascade of magnetic helicity from small scales to the largest scale at
the periphery of the whole magnetic flux system along the PIL, where the
helicity condenses and SMAs appear. The theory was demonstrated by ideal MHD
simulations based on topological coronal models between two parallel plates
(Zhao et al., 2015; Knizhnik et al., 2015) and a Cartesian coronal model with
a circular PIL (Knizhnik et al., 2017). These numerical models ignored the
primary diverging flows of supergranular cells and simplified the observed
cyclonic vortices between the anticyclonic flows of supergranules (Duvall &
Gizon, 2000; Langfellner et al., 2015) caused by the Coriolis force (Hathaway,
1982; Egorov et al., 2004), as many annular vortical flows to inject magnetic
helicity into corona and produced the SMAs along PILs. Using a large-scale
averaged representation of the small-scale vortical motions in magnetofriction
simulations, Mackay et al. (2014) found that the helicity condensation can
overcome the incorrect sign of helicity injection from differential rotation
on a west-east oriented PIL and help to form MFRs in filament channels. In
this Letter, we investigate the role of supergranular flows in the formation
of the magnetic field of quiescent prominences via magnetofriction
simulations.

##  2 Numerical Methods

The simulation domain is a spherical subdomain extending over radii
$r\in[1R_{\odot},1.5R_{\odot}]$, colatitudes
$\theta\in[39.6^{\circ},90^{\circ}]$, and longitudes
$\phi\in[0^{\circ},60^{\circ}]$ with a 4-level adaptive mesh refinement (AMR)
to get $256^{3}$ effective resolution. From a smooth bipolar photospheric
magnetogram, we use the spherical potential field extrapolation module in the
PDFI_SS software (Fisher et al., 2020) to extrapolate the initial potential
magnetic field. We solve the magnetofriction equations (Guo et al., 2016)
without explicit resistivity using the code MPI-AMRVAC (Porth et al., 2014;
Xia et al., 2018). Magnetic reconnection here is caused by numerical
resistivity. The magnetofrictional velocity
$v=\mathbf{J}\times\mathbf{B}/(\nu_{0}B^{2})$, where $\nu_{0}=10^{-15}$ s cm-2
is the viscous coefficient, has a smooth decay to zero towards the photosphere
(Cheung & DeRosa, 2012) and an upper limit of $30$ km s-1 (Pomoell et al.,
2019). We use the constrained transport (CT) scheme (Gardiner & Stone, 2005),
with the HLL flux for the electric field, on the staggered AMR mesh (Olivares
et al., 2019) to keep magnetic field divergence-free. Boundary conditions are
periodic on the longitudinal boundaries, open on the outer radial boundary,
and closed on the latitudinal boundaries. On the photospheric boundary, we use
zero normal velocity with the supergranular horizontal velocity field
described as follows.

A Voronoi tessellation of solar surface resembles (super)granular segmentation
in both topology and statistics (Schrijver et al., 1997). We use a weight
function $\omega=3|\sin(\pi t/\tau+\xi)|+0.7$ to generate Voronoi
tessellation, in which $\tau$ is the lifetime of a supergranular cell with a
Gaussian random distribution centered on 1.6 days (Hirzberger et al., 2008)
and $\xi$ is the random initial phase. We use 1 as the lower limit of the
weights for living cells, smaller weights are set to zero to remove the
corresponding cells from the tessellation map. So the simulated supergranular
cells are changing with time with their area proportional to their weights. We
adopt the dimensionless diverging velocity within supergranular cells as
$v_{r}(r)=2r^{2}/r_{0}\exp(-4r^{2}/r_{0}^{2})$ (Gudiksen & Nordlund, 2005),
where $r$ is the spherical distance to the cell center and $r_{0}$ is the
radius of a circle with area equal to the supergranular cell. The vortical
velocity due to Coriolis force acting on the diverging horizontal flows is
simplified as $v_{t}(r)=2r/r_{0}v_{r}(r)$ neglecting the latitude-dependence
of Coriolis force. We use a normalization factor to make sure the maximal
supergranular speed is initially 500 m s-1. The velocities of differential
rotation and meridional flow (Mackay et al., 2014) are also included. We
multiply the total driving speed by 5 to speed up the long-term evolution.
Dimensionless values in our results have the time unit of 8.3 hours and the
magnetic field unit of 2 Gauss. We have run a model SR with supergranules
rotating under Coriolis force and a model SS without the Coriolis effect.

##  3 Results

Initially (Figure 1 (a)), the magnetic field is a bipolar arcade rooted in the
smooth positive (red) and negative (blue) polarity regions separated by a
straight north-south directed PIL. Then, the supergranular diverging flows
transport and concentrate magnetic flux to the boundaries of supergranules
forming a magnetic network, and the differential rotation gradually shifts the
northern part of magnetic flux to the east and the southern part to the west
making the PIL tilted anticlockwise. At time 30 (Figure 1 (b)), short magnetic
loops near the PIL have skewed away from the initial potential field state.
The skew angle, defined as the angle between the horizontal component of the
magnetic field above the PIL and the horizontal direction perpendicular to the
PIL (Mackay & van Ballegooijen, 2001), is about 45 degrees. Long loops, which
root in the central region of each polarity far from the PIL, remain in an
approximately potential state with neglilible skew angles during the whole
evolution. At time 60 (Figure 1 (c)), low-lying magnetic loops near the PIL
are almost aligned with the PIL with large skew angles and some of them are
changed into helical field lines (red line) along the PIL. At time 90 (Figure
1 (d)), more helical field lines form and enwind the earlier-born helical
lines assembling a low-lying thin MFR along the PIL. Later at time 120 (Figure
1 (e)), which is after 41.5 days, the MFR matures with a larger height and
cross-section size suitable for hosting a quiescent prominence. Different from
earlier MFR models in which helical field lines root in two compact regions,
the footpoints of this MFR distribute extensively along the PIL. In contrast
to model SR, model SS does not produce any MFR or SMAs until time 120 (Figure
1 (f)), and all magnetic loops stay nearly potential for another 120 time.

Figure 1: Magnetic field lines of model SR with Coriolis effect at time 0 (a),
30 (b), 60 (c), 90 (d), and 120 (e) in contrast to model SS without Coriolis
effect at time 120 (f). The photosphere is colored by the radial magnetic
field in red-to-blue colors saturated at $\pm$18 G.

To understand the origin of the sheared magnetic loops, we illustrate the
build-up process of the sheared loops of model SR in the left column of Figure
2. For comparison, a similar representation of model SS at the same time is
plotted in the right column. In panels (a)-(f), the magnetic loops crossing
over the cyan PIL at different heights are plotted in rainbow colors. At time
15, the lowest blue loop in model SR has a short length and a small skew angle
to the PIL, while the loops in model SS have nearly zero skew angles. For
model SR at later times 45 and 60, several loops close to the PIL are sheared
and the lowest loop has the fastest growth rate in the skew angle and length.
Further away from the PIL, loops with higher altitudes have smaller skew
angles. In contrast, loops in model SS at all heights and distances from the
PIL, remain roughly perpendicular to the PIL. The sheared loops with large
skew angles in model SR are not created by persistent photospheric shearing
flows, because such shearing flows along the PIL do not exist. Instead, only
vortical diverging and converging flows by supergranules exist as shown by
grey arrows in panels (g) and (h). A logical explanation of the strongly
sheared loops around the PIL in model SR is referable to the helicity
condensation theory (Antiochos, 2013). Vortical converging flows towards the
junctions of several supergranules inject negative magnetic helicity into the
magnetic flux tubes with footpoints at strong magnetic flux on the
photosphere. Component magnetic reconnection happens on the interface of
adjacent twisted flux tubes, which transports the twisted flux to the common
periphery of the flux tubes and leaves behind the untwisted flux in the
central regions. This process inversely cascades from small scales to large
scales until reaching the biggest flux tube representing the whole bipolar
region with part of the periphery folded around the PIL, where twisted fluxes
with negative helicity condense as SMAs.

Figure 2: Comparative views of model SR and SS at time 15, 45, and 60. In
panel (a)-(f), magnetic field lines, in rainbow colors, are integrated from
uniformly sampled seed points along a radial white line starting from the cyan
PIL at a fixed latitude. The photosphere is colored by radial magnetic field
saturated at $\pm$20 G. Zoom-in views at time 15 are shown in panel (g) and
(h), where grey arrows present the photospheric horizontal velocity field.

Then we present how much the magnetic helicity is injected into the corona and
how the magnetic free energy is stored in our models. The flux of (relative)
magnetic helicity through the photosphere is defined by the equation with zero
resistivity (Berger & Field, 1984):

$\frac{dH_{R}}{dt}=-2\int\mathbf{A}^{P}\times\mathbf{E}\cdot
d\mathbf{S}=-2\int(A^{P}_{\theta}E_{\phi}-A^{P}_{\phi}E_{\theta})dS$ (1)

where $\mathbf{A}^{P}$ is the vector potential of the potential magnetic field
which matches the instantaneous photospheric $B_{r}$, $\mathbf{E}$ is the
electric field and quantified as the electric field from the CT scheme, and
$d\mathbf{S}$ is the surface element with normal direction $\mathbf{\hat{r}}$.
Figure 3 (a) shows the time evolution of the magnetic helicity fluxes through
the photosphere in model SR as the total (solid line), the negative (dashed
line), and the positive flux (dotted line). The positive helicity flux is
about 20 to 30 times larger than the negative helicity flux. Magnetic helicity
fluxes through other boundaries are negligible. The photospheric distribution
of the integrand in equation (1) is shown in Figure 3 (d) with horizontal
velocity vectors and contours of $B_{r}$ overplotted. The positive values of
the integrand are distributed in the outer regions of supergranules in donut
shapes, induced by the clockwise rotating diverging flows of supergranules.
The negative values are distributed sporadically along the borders of several
supergranules and caused by irregular counterclockwise converging flows
towards strong magnetic flux. For model SS as shown in panel (b), the positive
and negative helicity flux almost cancel out with a small positive total flux.
Panel (e) for model SS shows that positive values alternates with negative
values in the outer region of supergranules. Figure 3 (c) presents the time
evolution of the magnetic free energy, which is obtained by subtracting the
potential field energy from the total magnetic energy. Both models have an
increasing magnetic free energy, while the magnetic free energy in model SR
increases about three times faster than it in model SS.

Figure 3: (a) Time evolution of the magnetic helicity fluxes as the total
(solid line), the negative (dashed line), and the positive (dotted line) ones
through the photosphere of model SR. (b) Similar curves as (a) for model SS.
(c) Time evolution of the magnetic free energy of model SR (solid line) and SS
(dashed line). (d) Magnetic helicity flux density on the photosphere overlaid
with horizontal velocity field as yellow arrows and contours of $B_{r}$ at
$\pm 16$ G for model SR at time 36. (e) Similar plot as (d) for model SS.

Long-lived prominence plasma should be supported by magnetic tension force
against downward gravity in locally concave-up magnetic fields, i.e., magnetic
dips. We locate the magnetic dip regions (MDRs) where the radial component of
the curvature of magnetic field lines is positive and the radial component of
the magnetic field has less than 10% proportion. Figure 4 presents snapshots
of the formation of magnetic dips from the top views in upper panels (a)-(b)
and from the side views in lower pannels (a1)-(b1). At time 61.2, small
isolated MDRs distribute along the PIL with supergranular scale intervals.
Later at time 82.8, the small MDRs grow horizontally along the PIL and connect
to form three long MDRs. Helical field lines tangential to the photosphere
outline the periphery of an MFR above the PIL and the MDRs naturally appear in
the lower half of the MFR. At time 120, further growth and connection of MDRs
result in a long slab-like MDR with a growing height reaching 28,000 km and a
length of about 480,000 km. The MFR surrounding the MDR has helical field
lines winding 1 to 2 turns around a common axis. If the MDRs are filled with
prominence plasma, the formation process of MDRs is consistent with filament
observations showing that quiescent filaments form from aligned fragments to a
continuous body (Pevtsov & Neidig, 2005).

Figure 4: Formation of magnetic dip regions shown as isosurfaces colored by
solar radius $r$ from top views (a)-(c) and from side views (a1)-(c1) at time
61.2, 82.8, and 120. Surrounding magnetic field lines in different colors and
the main PIL in cyan are plotted in (a1)-(c1) additionally.

To understand the transition from SMAs to MFRs, we find the sites of magnetic
reconnection and flux cancellation on the photosphere. In Figure 5 (a), the
footpoints of the pink and the sky blue field line, in the middle of the
panel, converge to the white PIL by adjacent supergranular flows indicated by
the purple arrows, and the two field lines are about to be reconnected at the
PIL to form a helical field line like the yellow one. Similarly, the blue and
the red field line will be reconnected to form a helical field line like the
green one in the lower right region. Figure 5 (b) presents a zoom-out side
view of these field lines above the photospheric supergranular cells. The
sites of magnetic reconnection are at the borders of supergranular cells on
two sides of the PIL. Some parts of the PIL crossing through supergranular
cells are bald patches where the MFR touches the photosphere. Evidence of
magnetic flux cancellation is shown in Figure 5 (c) in which the time
evolution of the total unsigned magnetic flux on the photosphere is plotted
for both models. After a short period of flux dispersion without many
cancellations, the unsigned magnetic flux of model SR decreases linearly
losing about 20 % until time 120, which indicates continuous flux
cancellation. The unsigned magnetic flux of model SS decreases less than 3 %
until time 70 and then slightly increases with small oscillation and slow flux
cancellation. The slight increase is caused by numerical error of the magnetic
boundary condition.

Figure 5: Linking sheared loops to helical field lines via photospheric
magnetic reconnection along the white PIL. (a) Top view of 6 selected magnetic
field lines above the photospheric magnetogram with arrows presenting
horizontal velocity. (b) Side view of the same magnetic field lines in (a)
above a photospheric color map presenting supergranular cells. (c) Evolution
of total unsigned magnetic flux on photosphere for SR (solid line) and SS
(dashed line) model.

##  4 Discussion and Conclusions

The only difference in model setup between model SR and SS is the vortical
motions by Coriolis force, which leads to strikingly different results,
namely, an MFR in model SR and potential arcades in model SS. The
counterclockwise vortical flows in strong magnetic flux regions around
supergranular boundaries inject negative magnetic helicity, which is
accumulated along the PIL via helicity condensation to form a dextral filament
channel in the northern hemisphere. Therefore Coriolis force is a key factor
for helicity injection and formation of quiescent prominences, while effective
magnetic flux diffusion by supergranular flows (Leighton, 1964) and
differential rotation are not the main reasons for the formation of filament
channels. We will include the latitude dependence of the Coriolis effect
(Duvall & Gizon, 2000) in the future expecting faster MFR formation at higher
latitudes.

The helicity condensation theory descibes how the injected magnetic helicity
at small scale is transferred and condensed to the PIL, which explain the
origin of strong axial magnetic flux along quiescent filament channels. But
this theory can only lead to SMAs, which contradict the observational evidence
(Bak-SteAlicka et al., 2013; Wang & Stenborg, 2010) of quiescent prominences.
Previous numerical simulations on helicity condensation (Zhao et al., 2015;
Knizhnik et al., 2015, 2017) did not produce MFRs because of the absent of the
primary diverging motions of supergranules, which not only generate the
magnetic network but also converge opposite magnetic polarities at the PIL and
drive the magnetic reconnection between footpoints of SMAs, the magnetic flux
cancellation in forming filament channels (Wang & Muglach, 2007), and the
formation of MFRs.

During the formation of the coherent MFR in model SR, a chain of small MFRs
first forms along the PIL with fragmented small MDRs to host prominence
plasma. Later on, these small MFRs and MDRs grow and merge with neighbors to
form the mature MFR, which is consistent with the “head-to-tail” conceptual
model of prominence formation proposed by Martens & Zwaan (2001). The isolated
MDRs consist of piled-up magnetic dips from the photosphere may correspond to
the observed pillars or legs of quiescent prominences (Li & Zhang, 2013; Zhou
et al., 2021). We run models starting from west-to-east PILs to simulate high-
latitude quiescent prominences and formed MFRs to be reported in a follow-up
paper.

This research was supported by the Basic Research Program of Yunnan Province
(2019FB140, 202001AW070011), the National Natural Science Foundation of China
(11803031, 12073022). We acknowledge discussions with Mark C. M. Cheung,
George H. Fisher, and Shangbin Yang. The numerical simulations were conducted
on the Yunnan University Astronomy Supercomputer.

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[1] K. Sravan Kumar

These authors contributed equally to this work.

These authors contributed equally to this work.

[1]Institute of Cosmology and Gravitation, U. Portsmouth, Dennis Sciama
Building, Burnaby Road, United Kingdom, Portsmouth, PO1 3FX, United Kingdom

2]Departamento de Física, Centro de Matemática e Aplicações (CMA-UBI),
Universidade da Beira Interior, Rua Marquês D Ávila e Bolama, Covilhã,
6200-001, Portugal

# Hawking radiation with pure states

<EMAIL_ADDRESS>João Marto<EMAIL_ADDRESS>*  [

###### Abstract

Hawking’s seminal work on black hole radiation highlights a critical issue in
our understanding of quantum field theory in curved spacetime (QFTCS),
specifically the problem of unitarity loss (where pure states evolve into
mixed states). In this paper, we examine a recent proposal for a direct-sum
QFTCS, which maintains unitarity through a novel quantization method that
employs superselection rules based on discrete spacetime transformations. This
approach describes a quantum state as components that evolve within
superselection sectors of the complete Hilbert space, adhering to the discrete
symmetries of a Schwarzschild black hole. Consequently, it represents a
maximally entangled pure state as a direct-sum of two components in the
interior and exterior regions of the black hole, thereby preserving the
unitarity of Hawking radiation by keeping it in the form of pure states.

###### keywords:

Quantum field theory in curved spacetime, Black holes, Hawking radiation,
Quantum gravity

##  1 Introduction

The seminal papers of Hawking’s on black hole radiation [1, 2] have exposed
explicitly the incompatibility in our understanding of gravity and quantum
mechanics. To be precise, the issue of understanding the behavior of fields in
black hole spacetime was first discussed by Einstein and Rosen in 1935 [3],
even before the formal development of quantum field theory. The important
conundrum is the loss of unitarity, which involves the evolution of pure
states into mixed states. In other words, Hawking radiation is formed by mixed
states, and the (maximally) entangled partner is always left behind the SBH
horizon [4, 5]. This would mean an observer outside the SBH would access
states whose fate depends on what happens to the partner states behind the
horizon. This opens a serious conceptual conundrum: the observer’s quantum
physics depends on what is happening in the region of spacetime that the
observer has no causal access to. Since an asymptotic observer would never
witness any particle (quantum state) crossing the horizon, the violation of
unitarity results in an incomplete understanding of quantum physics for the
asymptotic observer. Often, it is interpreted as an acceptable outcome to
treat black holes as special objects in the Universe that would violate
unitarity [6]. But this interpretation is not the solution [7], and even if it
is, one is left with the prediction, by D. Page, that the entanglement entropy
of Hawking quanta would at some point exceed the thermodynamic entropy of the
black hole [8, 9, 10]. Often (Planck scale) quantum gravity proponents take
solace in an unknown ultraviolet (UV) complete theory to resolve the problem
of unitarity in black hole physics [11, 12]. This raises the logical question
of why Planck-scale quantum gravity is necessary to address a fundamental
problem that arises from understanding quantum fields in curved spacetime.
Along with unitarity loss, there is also a related issue of information loss,
which carries an additional question on the universal nature of Hawking
radiation that leaves no knowledge of what has formed the black hole.
Understanding the formation of a black hole (quantum mechanically) is another
hard question that requires a robust formulation of QFT in a dynamically
collapsing geometry, which we do not have yet. Hawking’s actual calculation
involves quantization in Schwarzschild spacetime, even though he framed the
discussion in the context of a black hole formed by collapse. Despite numerous
attempts [13, 14] to address the information paradox issue, Einstein-Rosen’s
initial question of particle problem in general relativity remains and needs
to be taken very seriously. Recent ongoing efforts by Gerard ’t Hooft[15, 16,
17, 18, 19, 20, 21, 22], building on the work of Sanchez and Whiting and the
gravitational backreaction calculations by Dray and ’t Hooft [23], underscore
the importance of resolving the information paradox through fundamental
physics before turning to an unknown theory of quantum gravity.

The recently proposed direct-sum quantum field theory in curved spacetime
presents an elegant new approach to addressing the issues of unitarity and
information loss in black hole physics [24, 25, 26]. The direct-sum QFTCS has
also led to the promising explanation for the long-standing CMB anomalies in
the context of inflationary cosmology [27, 28]. The framework of direct-sum
QFTCS is based on the intricate understanding of discrete spacetime symmetries
and separating the notions of classical and quantum mechanical time. In this
review, we explore how this framework provides a unitary description of
Hawking radiation and offers significant insights into extracting information
about the black hole interior through a fundamental approach. Black holes and
de Sitter spacetime share certain analogies related to event horizons, as
explored by Gibbons and Hawking.[29]. Thus, we briefly address a similar
unitarity issue in de Sitter space and the potential resolution offered by
direct-sum QFTCS.

This review is organized as follows: in Sec. 2, we discuss the parity
($\mathcal{P}$) and time reversal ($\mathcal{T}$) symmetries in quantum
mechanics and Minkowski QFT. We then present the foundational framework of
direct-sum QFT. In Sec. 3, we discuss the Schwarzchild spacetime and the basic
assumptions in Hawking’s original calculation. We then present the details of
’t Hooft’s quantum algebra that emerges from the effects of gravitational
backreaction. In Sec. 4, we summarize the results of direct-sum QFT in BH
spacetime and its approach to obtaining Hawking radiation in the form of pure
states. We also show the QFT extension of ’t Hooft quantum algebra and the
resulting guidelines to extract information from Hawking radiation. In Sec. 5,
we conclude with a qualitative discussion and present outlook for future
investigations.

Throughout the manuscript, we use the metric signature mostly positive
$\left(-,\,+,\,+,\,+\right)$ and $\hbar=c=1$.

##  2 $\mathcal{P}\mathcal{T}$ symmetry and direct-sum quantum theory

Quantum mechanics is recognized for its time symmetry [30, 31], a property
linked to the anti-unitary nature of the time reflection operation . In
quantum theory, time is treated as a parameter rather than an operator, unlike
spatial position. This distinct treatment of time sets it apart in quantum
theory compared to classical theory. Quantum Field Theory (QFT) in Minkowski
spacetime represents a unification of special relativity and quantum
mechanics, achieved by enforcing the commutation of field operators at space-
like separations. In the context of Klein-Gordon field this is expressed as

$\left[\hat{\phi}(x),\,\hat{\phi}(y)\right]=0,\quad\left(x-y\right)^{2}>0\,.$
(1)

The above condition together with Lorentz invariance dictates the expansion of
Klien-Gordon (KG) (real scalar) field operator as

$\hat{\phi}(x)=\int\frac{d^{3}k}{\left(2\pi\right)^{3/2}}\frac{1}{\sqrt{2|k_{0}|}}\left[a_{\textbf{k}}e^{ik\cdot
x}+a_{\textbf{k}}^{\dagger}e^{-ik\cdot x}\right]$ (2)

where $k\cdot x=-k_{0}t+\textbf{k}\cdot\textbf{x}$. The connection with
quantum mechanics stems from the positive energy state definition, defined by

$|\Psi\rangle_{t}=e^{-iEt}|\Psi\rangle_{0}$ (3)

assuming the arrow of time to be $t:-\infty\to\infty$. We can see that the
expression of KG field operator (2) is a combination of positive and negative
energy solutions. These are associated with creation and annihilation
operators $\left(a_{\textbf{k}},\,a_{\textbf{k}}^{\dagger}\right)$ which
satisfy the canonical commutation relations and the Minkowski vacuum is
defined by

$\left[a_{\textbf{k}},\,a_{\textbf{k}}^{\dagger}\right]=1,\quad
a_{\textbf{k}}|0_{M}\rangle=0.$ (4)

Another equivalent way of defining a positive energy state (3) is

$|\Psi\rangle_{t}=e^{iEt}|\Psi\rangle_{0}$ (5)

with the presumption on the arrow of time $t:\infty\to-\infty$, the entire
quantum theory is constructed based on this convention. As noted in [32], one
can redefine the entire framework of quantum theory by reversing the
definition of positive energy and the arrow of time, effectively replacing $i$
with $-i$ throughout the theory.

Direct-sum quantum theory starts with a construction that combine both arrows
of time together with parity operation to express a single quantum state.
According to this, a single quantum state is expressed by joining two
components through a direct-sum operation

$|\Psi\rangle=\frac{1}{\sqrt{2}}\left(\
|\Psi_{+}\rangle\oplus|\Psi_{-}\rangle\right)=\frac{1}{\sqrt{2}}\begin{pmatrix}|\Psi_{+}\rangle\\\
|\Psi_{-}\rangle\end{pmatrix}$ (6)

$|\Psi_{\pm}\rangle$ are orthogonal states [33] of the direct-sum Hilbert
space

$\mathcal{H}=\mathcal{H}_{+}\oplus\mathcal{H}_{-}$ (7)

Here $\mathcal{H}_{\pm}$ are called the superselection sectors, states in
these sectors cannot be superposed coherently with each other. Here $\pm$ mean
the parity conjugate regions of physical space. The states
$|\Psi_{\pm}\rangle$ evolve according the following direct-sum Schrödinger
equation

$i\frac{\partial|\Psi\rangle}{\partial
t_{p}}=\begin{pmatrix}\hat{{H}}_{+}&0\\\
0&-\hat{{H}}_{-}\end{pmatrix}|\Psi\rangle$ (8)

where $\hat{H}_{\pm}\left(\hat{x}_{\pm},\,\hat{p}_{\pm}\right)$ give the
Hamiltonians of the system $\hat{H}=\hat{H}_{+}\oplus\hat{H}_{-}$, which are
functions of position and momenta operators of the target space defined by

$\displaystyle\hat{p}_{+}$ $\displaystyle=-i\frac{d}{dx_{+}},\quad
x_{+}=x\gtrsim 0$ (9) $\displaystyle\hat{p}_{-}$
$\displaystyle=i\frac{d}{dx_{-}},\quad x_{-}=x\lesssim 0\,.$

The canonical non-zero commutation relations are111The remaining relations are
$\left[\hat{x}_{+},\,\hat{x}_{-}\right]=\left[\hat{p}_{+},\,\hat{p}_{-}\right]=\left[\hat{x}_{+},\,\hat{p}_{-}\right]=\left[\hat{p}_{+},\,\hat{x}_{-}\right]=0$.

$[\hat{x}_{+},\,\hat{p}_{+}]=i,\quad[\hat{x}_{-},\,\hat{p}_{-}]=-i\,,$ (10)

The complete description of a quantum state, in the entire physical space, is
given by $|\Psi\rangle$

$\Psi(x)=\frac{1}{\sqrt{2}}\begin{pmatrix}\langle x_{+}|\quad\langle
x_{-}|\end{pmatrix}\begin{pmatrix}|\Psi_{+}\rangle_{0}e^{-iEt}\\\
|\Psi_{-}\rangle_{0}e^{iEt}\end{pmatrix}\implies\begin{cases}\frac{1}{\sqrt{2}}\Psi_{+}\left(x_{+}\right)e^{-iEt},\quad
x_{+}=x\gtrsim 0\\\ \frac{1}{\sqrt{2}}\Psi_{-}\left(x_{-}\right)e^{iEt},\quad
x_{-}=x\lesssim 0\,.\end{cases}$ (11)

The square integrability and the probabilities of states is given by

$\int_{-\infty}^{\infty}dx\langle\Psi|\Psi\rangle=\frac{1}{2}\int^{0}_{-\infty}dx_{-}\langle\Psi_{-}|\Psi_{-}\rangle+\frac{1}{2}\int^{\infty}_{0}dx_{+}\langle\Psi_{+}|\Psi_{+}\rangle=1\,.$
(12)

Figure 1: This is a pictorial description of how a quantum harmonic oscillator
is described by direct-sum quantum theory by means of two quantum states,
evolving with two opposite arrows of time at parity conjugate points in
physical space.

The description of a harmonic oscillator, in the current approach, can be
hinted from Fig. 1. The wave function of the parity conjugate region is
described by

$\Psi_{n}(x)=\langle
x|\Psi_{n}\rangle\equiv\begin{cases}\frac{1}{\sqrt{2^{n+1}n!}}\left(\frac{1}{\pi}\right)^{1/4}e^{-\frac{1}{2}x_{+}^{2}}H_{n}\left(x_{+}\right)e^{-iE_{n}t_{p}},\quad
x_{+}\gtrsim 0\\\
\frac{1}{\sqrt{2^{n+1}n!}}\left(\frac{1}{\pi}\right)^{1/4}e^{-\frac{1}{2}x_{-}^{2}}H_{n}\left(x_{-}\right)e^{iE_{n}t_{p}},\quad
x_{-}\lesssim 0\end{cases}$ (13)

where $E_{n}$ corresponds to energy state for each $n$ of the Hermite
polynomials $H_{n}$. We can see that in the limit $x_{\pm}\to 0_{\pm}$, the
two-component states $|\Psi_{\pm}\rangle$ match automatically. The point $x=0$
is not a special point since under the translation $x\to x+L$, the
construction remains the same because it is based on $\mathcal{P}\mathcal{T}$,
a discrete transformation. The orthogonality of states corresponding to
different energy levels is given by

$\int_{-\infty}^{\infty}\langle\Psi_{n}|\Psi_{m}\rangle
dx=\int_{0}^{\infty}dx_{+}\langle\Psi_{n+}|\Psi_{m+}\rangle+\int_{-\infty}^{0}dx_{-}\langle\Psi_{n-}|\Psi_{m-}\rangle
dx_{-}=\delta_{n,m}$ (14)

Furthermore, an observer can only measure the quantum state in either of the
parity conjugate points, and the arrow of time in quantum theory is not
observable for any classical observer.

Minkowski spacetime is $\mathcal{P}\mathcal{T}$ symmetric, thus one can extend
the direct-sum Schrödinger equation to relativistic case by direct-sum QFT
where a KG field operator is expressed as direct-sum of two components as
function of $\mathcal{P}\mathcal{T}$ conjugate points

$\hat{\phi}(x)=\frac{1}{\sqrt{2}}\begin{pmatrix}\hat{\phi}_{+}&0\\\
0&\hat{\phi}_{-}\end{pmatrix}$ (15)

where

$\displaystyle\hat{\phi}_{+}(x)$
$\displaystyle=\int\frac{d^{3}k}{\left(2\pi\right)^{3/2}}\frac{1}{\sqrt{2|k_{0}|}}\left[a_{(+)\textbf{k}}e^{ik\cdot
x}+a_{(+)\textbf{k}}^{\dagger}e^{-ik\cdot x}\right]$ (16)
$\displaystyle\hat{\phi}_{-}(-x)$
$\displaystyle=\int\frac{d^{3}k}{\left(2\pi\right)^{3/2}}\frac{1}{\sqrt{2|k_{0}|}}\left[a_{(-)\textbf{k}}e^{-ik\cdot
x}+a_{(-)\textbf{k}}^{\dagger}e^{ik\cdot x}\right]$

The creation and annihilation operators obey

$\left[a_{(\pm)\textbf{k}},\,a^{\dagger}_{(\pm)\textbf{k}}\right]=1,\quad\left[a_{(\pm)\textbf{k}},\,a^{\dagger}_{(\mp)\textbf{k}}\right]=\left[a_{(\pm)\textbf{k}},\,a_{(\mp)\textbf{k}}\right]=0\,.$
(17)

which imply a new causality condition

$\left[\hat{\phi}_{+}(x),\,\hat{\phi}_{-}(-y)\right]=0\,.$ (18)

Here we notice that, in our notation, $\hat{\phi}_{\pm}$ are field operators
defined for parity conjugate points in physical space, which mean the first
line in (16) is defined only for $\textbf{x}\gtrsim 0$ where as the second
line is defined only for $\textbf{x}\lesssim 0$, the direct-sum of these two
operators define the quantum field (15) everywhere in Minkowski spacetime.
Since the construction is based on $\mathcal{P}\mathcal{T}$, any Lorentz
transformation on (15) preserve the structure of $\mathcal{P}\mathcal{T}$
symmetric Minkowski vacuum given by

$|0_{M}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}|0_{M+}\rangle\\\
|0_{M-}\rangle\end{pmatrix},\quad a_{(+)\textbf{k}}|0_{M+}\rangle=0,\quad
a_{(-)\textbf{k}}|0_{M-}\rangle=0\,.$ (19)

The two-point function and propagator in direct-sum QFT given by

$\displaystyle\langle
0|\hat{\phi}\left(x\right)\hat{\phi}\left(x^{\prime}\right)|0\rangle$
$\displaystyle=\frac{1}{2}\langle
0_{+}|\hat{\phi}_{+}\left(x\right)\hat{\phi}_{+}\left(x^{\prime}\right)|0_{+}\rangle+\frac{1}{2}\langle
0_{-}|\hat{\phi}_{-}\left(x\right)\hat{\phi}_{-}\left(x^{\prime}\right)|0_{-}\rangle$
(20) $\displaystyle\langle
0|T\hat{\phi}\left(x\right)\hat{\phi}\left(x^{\prime}\right)0|\rangle$
$\displaystyle=\frac{1}{2}\langle
0_{+}|T\hat{\phi}_{+}\left(x\right)\hat{\phi}_{+}\left(x^{\prime}\right)|0_{+}\rangle+\frac{1}{2}\langle
0_{-}|T\hat{\phi}_{-}\left(x\right)\hat{\phi}_{-}\left(x^{\prime}\right)|0_{-}\rangle\,,$

where $T$ represents time ordering. All the interactions are divided into
direct-sum components. For example, cubic interaction like following

$\frac{\lambda}{3}\hat{\phi}^{3}=\frac{\lambda}{3}\begin{pmatrix}\hat{\phi}_{+}^{3}&0\\\
0&\hat{\phi}_{-}^{3}\end{pmatrix}$ (21)

shows that there will never be any mixing between $\hat{\phi}_{+}$ and
$\hat{\phi}_{-}$. Therefore, all the standard QFT calculations can be
straightforwardly extended to DQFT but we do not see any change in results
because Minkowski spacetime is $\mathcal{P}\mathcal{T}$ symmetric (See [24,
26] for more details. According to DQFT standard model vacuum and the degrees
of freedom of particles ($|SM\rangle$) and antiparticles
($|\overline{SM}\rangle$ are represented by

$|0_{SM}\rangle=\begin{pmatrix}|0_{SM+}\rangle\\\
|0_{SM-}\rangle\end{pmatrix}\quad|SM\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}|SM_{+}\rangle\\\
|SM_{-}\rangle\end{pmatrix}\quad|\overline{SM}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}|\overline{SM}_{+}\rangle\\\
|\overline{SM}_{-}\rangle\end{pmatrix}$ (22)

Note that we apply the same super-selection-rule based on
$\mathcal{P}\mathcal{T}$ for all Fock spaces of the standard model degrees of
freedom, i.e., the parity conjugate regions are uniquely defined for the all
the particle and antiparticle states of the standard model. All the Standard
model calculations remain the same because all the interactions terms become

$\mathcal{L}_{c}\sim\mathcal{O}_{SM}^{3}=\begin{pmatrix}\mathcal{O}_{SM_{+}}^{3}&0\\\
0&\mathcal{O}_{SM_{-}}^{3}\end{pmatrix}\quad\mathcal{L}_{q}\sim\mathcal{O}_{SM}^{4}=\begin{pmatrix}\mathcal{O}_{SM_{+}}^{4}&0\\\
0&\mathcal{O}_{SM_{-}}^{4}\end{pmatrix}$ (23)

where $\mathcal{O}_{SM}$ is any operator in the Standard Model involving
quantum fields and their derivatives. Therefore, direct-sum QFT is a framework
that does not alter the QFT calculations in Minkowski due to the spacetime
being $\mathcal{P}\mathcal{T}$ symmetric.

However, this new understanding provides a fresh perspective on QFT in curved
spacetime, ultimately leading to unitarity and observer complementarity in
black hole spacetime, as proposed by Susskind [34]. What was previously only a
hypothesis is now supported by our framework, offering a fundamental approach
to achieving these principles.

##  3 On the assumptions behind Hawking’s calculation and ’t Hooft
Gravitational backreaction

In this section, we discuss the crucial observations of Hawking calculation
[1, 2] that has lead to the unitarity loss and information paradox [25].
Hawking’s calculation is concerned with the quantization of scalar field in
Schwarzschild BH spacetime given by

$ds^{2}=-\left(1-\frac{2GM}{r}\right)dt^{2}+\frac{1}{\left(1-\frac{2GM}{r}\right)}dr^{2}+r^{2}d\Omega^{2}\,,$
(24)

SBH spacetime is static and spherically symmetric

$\mathcal{T}:t\to-t,\quad\mathcal{P}:\left(\theta,\,\phi\right)\to\left(\pi-\theta,\,\pi+\varphi\right)$
(25)

and asymptotically Minkowski i.e., $r\to\infty$. The coordinate singularity at
$r=2GM$ makes the Schwarzschild coordinates unfit for understanding quantum
fields (which was first found by Einstein-Rosen in 1935 [3]). This is why
Kruskal-Szekers (KS) coordinates are often used in the literature instead. In
those coordinates, the SBH metric is given by

$ds^{2}=-\frac{2GM}{r}e^{1-\frac{r}{2GM}}dUdV+r^{2}d\Omega^{2}\,,$ (26)

where

$UV=16G^{2}M^{2}e^{\frac{r}{2GM}}\left(1-\frac{r}{2GM}\right)$ (27)

The relation between KS coordinates (U, V) and the Schwarzschild coordinates
is given by

$r>2GM\implies\begin{cases}U=-4GMe^{-\frac{u}{4GM}}<0&V=4GMe^{\frac{v}{4GM}}>0\\\
U=4GMe^{-\frac{u}{4GM}}>0&V=-4GMe^{\frac{v}{4GM}}<0\,.\end{cases}$ (28)

and

$r<2GM\implies\begin{cases}U=4GMe^{-\frac{u}{4GM}}>0&V=4GMe^{\frac{v}{4GM}}>0\\\
U=-4GMe^{-\frac{u}{4GM}}<0&V=-4GMe^{\frac{v}{4GM}}<0\,.\end{cases}$ (29)

where $u=t-r_{\ast}$ and $v=t+r_{\ast}$ with
$r_{\ast}=r+2GM\ln\Bigg{|}\frac{r}{2GM}-1\Bigg{|}$ being the so-called
tortoise coordinate which asymptote to $r_{\ast}\to-\infty$ as $r\to 2GM$.
Thus

$r\to 2GM\implies\begin{cases}U\to 0^{\mp}&V\to 0^{\pm}\,\text{for}\quad
r>2GM\\\ U\to 0^{\pm}&V\to 0^{\pm}\,\text{for}\quad r<2GM\end{cases}$ (30)

As (24) coincides with Minkowski spacetime in the limit $r\to\infty$, and if
one assumes an arrow of time $t:-\infty\to\infty$ for the quantum fields in
this asymptotic Minkowski spacetime, we find that Hawking considers
$U<0,\,V>0$ to be the only physical region in the exterior $r>2GM$. Thus, the
other possibility $U>0,\,V<0$ is considered to be a parallel or an unphysical
Universe. In a similar way, $U<0,\,V<0$ (often called white hole) of $r<2GM$
is declared to be unphysical too. This is because the spacetime, long before
the black hole has formed, is assumed to be Minkowski with an arrow of time
$t:-\infty\to\infty$. Thus, the formation of spacetime with $U<0,\,V<0$ was
thought to be an impossible scenario from a collapsing geometry. There are two
caveats in these assumptions

  * •

We must be very careful with taking classical intuitions into formulating the
description of quantum fields in curved spacetime. Historically quantum
physics is known to be a counter intuitive formulation.

  * •

Understanding the formation of BH quantum mechanically is a profound open
question; thus, declaring $U<0,\,V<0$ as irrelevant for quantum fields in SBH
spacetime has to be taken with great care.

  * •

Symmetries are a guiding concept in physics, and one must be careful in
discarding them. Given that the following (discrete) spacetime transformations
on SBH metric (27)

$\left(U,\,V\right)\to\left(-U,\,-V\right),\quad\left(\theta,\,\varphi\right)\to\left(\pi-\theta,\,\pi+\varphi\right)$
(31)

leave the metric (24) invariant, there is an ambiguity on the choice of the
sign of the coordinates, and it was pointed out by Einstein and Rosen as an
important ordeal for combining gravity and quantum mechanics.

Unitarity loss is the hypothetical result of restricting quantum fields to
only certain regions of the KS coordinates $\left(U<0,\,V>0\right)$ and
$\left(U>0,\,V<0\right)$ whereas the other regions $\left(U>0,\,V<0\right)$
and $\left(U<0,\,V<0\right)$ are equally allowed by symmetry. Indeed, the
Hartle-Hawking vacuum, obtained from the gravitational path integral (in
Euclidean time), gives the relation between the quantum states connecting the
previously omitted regions $\left(U>0,\,V<0\right)$ and
$\left(U<0,\,V<0\right)$ [35]. It is not a surprising revelation because
allowing a complex time coordinate $t\to it$ would render all arrows of time
possible.

Another assumption in the Hawking calculation is the commutation between the
ingoing and outgoing quantum states. Following Eq. (2.4) of [2] the radial
component of KG field operator222Defined by expansion in terms of spherical
harmonics
$\phi\left(U,\,V\right)=\sum_{\ell,\,m}\frac{\Phi(U,\,V)}{r}Y_{\ell}^{m}\left(\theta,\,\varphi\right)$
(32) is expanded as the sum of interior and exterior parts as

$\hat{\Phi}=\hat{\Phi}_{in}+\hat{\Phi}_{out}=\sum_{i}\left(p_{i}\textbf{b}_{i}+p_{i}^{\ast}\textbf{b}_{i}^{\dagger}\right)+\left(q_{i}\textbf{c}_{i}+q_{i}^{\ast}\textbf{c}_{i}^{\dagger}\right)$
(33)

where $p_{i}$ describe purely outgoing states ($r>2GM$) and $q_{i}$ describe
purely ingoing states, that end up inside $r<2GM$. In [2] the following
commutation relations are assumed

$\left[\hat{\Phi}_{in},\,\hat{\Phi}_{out}\right]=0\implies\left[\textbf{b}_{i},\,\textbf{c}_{i}\right]=\left[\textbf{b}_{i},\,\textbf{c}^{\dagger}_{i}\right]=0\,,$
(34)

The points worth to be paid attention here are:

  * •

The commutation relation (34) is driven by the intution that an interior state
cannot affect the exterior state.

  * •

The expansion (33) treats the interior and exterior states as part of the same
Hilbert space, with the same notion of time (in the QFT language), therefore
ingoing and outgoing quanta belong to the same Fock space where time is
uniquely defined.

There are caveats in the above assumptions

  * •

Following Dray and ’t Hooft’s gravitational backreaction [23], an ingoing
particle in Schwarzschild spacetime moving along the direction $V$ with
momentum $P_{i}$ creates a shock wave that causes a shift in the position of
the outgoing particle that is approaching in the $U$ direction. The position
of the outgoing particle is affected by ingoing momenta, and the position of
the ingoing particle is affected by outgoing momenta in the following way333We
note that the ’tHooft considerations ignore interior region of black hole
spacetime [17, 22]. In our approach, we consider both interior and exterior
regions of SBH in the near horizon approximation, and we take the
gravitational backreaction effects between interior and exterior quanta [25]
to define our direct-sum QFT that leads to unitarity. Thus our construction is
technically and conceptually very different from that of ’tHooft [36]. [37,
15, 36] (See [25] for more details)

$V_{i}=-\frac{8\pi G}{\left(\ell^{2}+\ell+1\right)}P_{e},\quad
U_{e}=\frac{8\pi G}{\left(\ell^{2}+\ell+1\right)}P_{i}\,.$ (35)

where the subscripts $P_{i},\,P_{e}$ and $V_{e},\,U_{i}$ represent interior
and exterior (or ingoing and outgoing) momentum and positions that are defined
by spherical harmonics (See Eq. (5.1) of [36])

$\displaystyle V_{e}$ $\displaystyle\to\sum_{\ell,\,m}V_{e}Y_{\ell
m}\left(\theta,\,\phi\right),\quad U_{i}\to\sum_{\ell,\,m}U_{i}Y_{\ell
m}\left(\theta,\,\phi\right)$ (36) $\displaystyle P_{e}$
$\displaystyle\to\sum_{\ell,\,m}P_{e}Y_{\ell,\,m}\left(\theta,\,\phi\right),\quad
P_{i}\to\sum_{\ell m}P_{i}Y_{\ell,\,m}\left(\theta,\,\phi\right)$

It is important to note that $V_{i}$ is the position of ingoing state (or the
interior state at $r\lesssim 2GM$) defined for the KS coordinates $U<0,\,V>0$
where are $U_{e}$ is the position of outgoing state (or the exterior state at
$r\gtrsim 2GM$) for the KS coordinates $U>0,\,V>0$. Similarly, $V_{i},\,U_{e}$
can be equally defined for the choice of KS coordinates $U>0,\,V<0$ and
$U<0,\,V>0$.

One can now apply position and momentum uncertainty relations to the operators
of interior and exterior states as [36]

$\left[\hat{V}_{i},\,\hat{P}_{i}\right]=\left[\hat{U}_{e},\,\hat{P}_{e}\right]=i$
(37)

to (35) yields us

$\left[\hat{V}_{i},\,\hat{U}_{e}\right]=i\frac{8\pi G}{\ell^{2}+\ell+1}$ (38)

Uplifting the above relations to the context of interior and exterior field
operators we get [25]

$\left[\hat{\Phi}_{in},\,\hat{\Phi}_{out}\right]=i\frac{8\pi
G}{R^{2}\left(\ell^{2}+\ell+1\right)}\,,$ (39)

where $R=2GM$. Thus (34) is valid only if we ignore the effects of
gravitational backreaction. In the next section, we discuss the construction
of direct-sum QFTCS, which takes into account the discrete symmetries of SBH
metric (31) and the result of gravitational backreaction algebra (39).

##  4 Direct-sum QFT in curved spacetime and Hawking radiation with pure
states

Direct-sum quantum theory, presented in Sec. 2, forms a new basis for
understanding quantum fields in SBH spacetime. With the direct-sum QFT in
Minkowski spacetime, we can address the issue of two arrows of time associated
with $U<0,\,V>0$ and $U>0,\,V<0$ of SBH metric (26). A simple thumb rule of
direct-sum QFT is to form superselection sectors for the Hilbert space (or
Fock space) subjected to all the discrete spacetime transformations. In the
context of SBH, the metric (31) form guidelines for the description of quantum
fields in the sectorial Fock spaces represented by the regions in the
conformal diagram in Fig. 2.

In our framework, we expand the field operator as direct-sum of interior
(”int”) and exterior (”ext”) components as

$\displaystyle\hat{\Phi}$
$\displaystyle=\hat{\Phi}_{int}\oplus\hat{\Phi}_{ext}$ (40)
$\displaystyle=\frac{1}{\sqrt{2}}\left(\hat{\Phi}^{I}_{int}\oplus\hat{\Phi}^{II}_{int}\right)\oplus\frac{1}{\sqrt{2}}\left(\hat{\Phi}^{I}_{ext}\oplus\hat{\Phi}^{II}_{ext}\right)\,$

which are defined in the total Fock space, which is a direct-sum corresponding
to the different spatial regions and arrows of time ($r\lesssim$ and $r\gtrsim
2GM$) (See Fig. 2

$\mathcal{F}_{BH}=\left(\mathcal{F}_{Iint}\oplus\mathcal{F}_{IIint}\right)\oplus\left(\mathcal{F}_{Iext}\oplus\mathcal{F}_{IIext}\right)$
(41)

For an asymptotic observer at $r\to\infty$ ($r_{\ast}\to\infty$) of SBH space-
time (24), the quantum field is in Minkowski space-time and it becomes

$\hat{\Phi}_{ext}=\frac{1}{\sqrt{2}}\left(\hat{\Phi}_{I\infty}\oplus\hat{\Phi}_{II\infty}\right)\,,$
(42)

corresponding to an asymptotic vacuum

$|0\rangle_{\infty}=|0\rangle_{I\infty}\oplus|0\rangle_{II\infty},$ (43)

defined by the annihilation operators

$\hat{a}_{I\tilde{\omega}}|0\rangle_{I\infty}=0,\quad\hat{a}_{II\,\tilde{\omega}}|0\rangle_{II\infty}=0\,.$
(44)

The vacuum for quantum field $\hat{\Phi}_{ext}$ in the near horizon
approximation $r\to 2GM$ is

$|0\rangle_{ext}=|0\rangle_{Iext}\oplus||0\rangle_{IIext}$ (45)

defined by the annihilation operators

$\hat{b}_{I\tilde{\omega}}|0\rangle_{Iext}=0,\quad\hat{a}_{II\,\tilde{\omega}}|0\rangle_{II\infty}=0\,.$
(46)

Figure 2: This is conformal diagram of direct-sum QFTCS in SBH spacetime (26)
in the Kruskal coordinates $\left(U,\,V\right)$. The yellow and green shared
regions indicate parity conjugate points $\left(\theta,\,\varphi\right)$ and
$\left(\pi-\theta,\,\pi+\varphi\right)$ respectively. The regions labeled
$I_{ext},\,II_{ext}$ represent $r>2GM$ where quantum field components lead to
states evolving forward and backward in time $T=\frac{U+V}{2}$ whereas
$I_{int},\,II_{int}$ represent interior $r<2GM$ where quantum field components
lead to the states with opposite time evolutions. The red lines indicate the
Schwarzchild singularity at $r=0$, which are identified in the regions
$I_{int}$ and $II_{int}$. This conformal diagram does not contain any white
hole or parallel Universe. All the regions are physical; they quantum
mechanically indicate the evolution of quantum fields in SBH spacetime.

By applying the technique of Bogoliubov transformation, we obtain the number
density of particles created by operators
$\hat{a}^{\dagger}_{I\,k},\,\hat{a}^{\dagger}_{II\,k}$ in the vacuum
$|0\rangle_{BH}$ which is what exactly asymptotic observer see as Hawking
particles. Following [38] we can compute

$\displaystyle a_{I\tilde{\omega}}$
$\displaystyle=\int_{0}^{\infty}d\omega\left(\alpha_{\omega\tilde{\omega}}b_{I\,k}+\beta_{\omega\tilde{\omega}}b^{\dagger}_{I\,k}\right)$
(47) $\displaystyle a_{II\tilde{\omega}}$
$\displaystyle=\int_{0}^{\infty}d\omega\left(\tilde{\alpha}_{\omega\tilde{\omega}}b_{II\,k}+\tilde{\beta}_{\omega\tilde{\omega}}b^{\dagger}_{II\,k}\right)$

where

$\displaystyle\alpha_{\omega\tilde{\omega}}$
$\displaystyle=\sqrt{\frac{\tilde{\omega}}{\omega}}\int_{0}^{\infty}\frac{du}{2\pi}e^{i\tilde{\omega}u-i\tilde{\omega}U}\quad\beta_{\omega\tilde{\omega}}$
$\displaystyle=\sqrt{\frac{\tilde{\omega}}{\omega}}\int_{0}^{\infty}\frac{du}{2\pi}e^{i\tilde{\omega}u+i\tilde{\omega}U}$
(48) $\displaystyle\tilde{\alpha}_{\omega\tilde{\omega}}$
$\displaystyle=\sqrt{\frac{\tilde{\omega}}{\omega}}\int_{0}^{\infty}\frac{du}{2\pi}e^{-i\tilde{\omega}u+i\tilde{\omega}U}\quad\tilde{\beta}_{\omega\tilde{\omega}}$
$\displaystyle=\sqrt{\frac{\tilde{\omega}}{\omega}}\int_{0}^{\infty}\frac{du}{2\pi}e^{-i\tilde{\omega}u-i\tilde{\omega}U}$

Thus, the asymptotic observer witness Hawking radiation by the follow number
of quanta

$\displaystyle N_{\tilde{\omega}}$ $\displaystyle=\frac{1}{2}{}_{BH}\langle
0|\left(\hat{a}_{I\,\tilde{\omega}}^{\dagger}\oplus\hat{a}_{II\,\tilde{\omega}}^{\dagger}\right)\left(\hat{a}_{I\,\tilde{\omega}}\oplus\hat{a}_{II\,\tilde{\omega}}\right)|0\rangle_{BH}$
(49)
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}d\omega\left(|\beta_{\omega\tilde{\omega}}|^{2}+|\tilde{\beta}_{\omega\tilde{\omega}}|^{2}\right)\,$
$\displaystyle=\frac{1}{e^{8\pi GM\omega/\hbar}-1}\,,$

which is thermal distribution with a temperature $T=\frac{\hbar}{8\pi GM}$.
But here in our context Hawking radiation is in the form of pure states.
Because of the direct-sum structure of exterior and interior field operators
(40) which is different from the one of Hawking’s consideration (33) any
maximally entangled non-seperable pure state becomes a direct-sum of two pure
state components

$\displaystyle|\tilde{\psi}_{12}\rangle=\sum_{m,n}\tilde{c}_{mn}|\tilde{\phi}_{m1}\rangle\otimes|\tilde{\phi}_{n2}\rangle$
$\displaystyle=\frac{1}{\sqrt{2}}\sum_{m,n}\tilde{c}_{mn}\left(|\tilde{\phi}^{\rm
ext}_{m1}\rangle\otimes\tilde{\phi}^{\rm
ext}_{n2}\rangle\right)\oplus\left(|\tilde{\phi}^{\rm
int}_{m1}\rangle\otimes\tilde{\phi}^{\rm int}_{n2}\rangle\right)$ (50)
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}|\tilde{\psi}^{ext}_{12}\rangle\\\
|\tilde{\psi}^{int}_{12}\rangle\end{pmatrix}$

where $\tilde{c}_{mn}\neq\tilde{c}_{n}\tilde{c}_{m}$,
$|\phi_{1}\rangle=\sum_{m}\tilde{c}_{m}|\phi_{m1}\rangle$ and
$|\phi_{2}\rangle=\sum_{n}\tilde{c}_{n}|\phi_{n2}\rangle$. This would render
any density matrix of pure states to become two direct-sum counterparts which
are pure states on their own.

$\rho=\frac{\rho_{ext}}{2}\oplus\frac{\rho_{int}}{2}=\frac{\rho_{ext}^{2}}{2}\oplus\frac{\rho_{int}^{2}}{2}=\rho^{2}\,.$
(51)

This means any asymptotic observer, irrespective of what is beyond the
horizon, would witness the pure states evolving into pure states in his/her
super-selection-sector Hilbert space. Furthermore, states behind the horizon
are related by discrete spacetime transformation $\mp U\to\pm U,\,\pm V\to\pm
V$ (See region Fig. 2), one can reconstruct the interior state of SBH.
Furthermore, because of non-commutative relation (39), the exterior hawking
quanta are not independent of the interior counterparts leading to information
reconstruction. This forms the first step towards resolving the information
paradox.

##  5 Conclusions and outlook

Even after decades of developments in quantum gravity research, the deepest
problem in theoretical physics is at a scale much below the Planck scale,
which is the unitarity problem at the gravitational horizons [39]. Quantum
field theory in curved spacetime requires much stronger foundations, which
would help to improve our efforts toward building quantum gravity at the
Planck scales.

Drawing on foundational work by Hawking and ’t Hooft, we propose a QFT
framework that incorporates discrete spacetime transformations, acknowledging
the status of time as a parameter (i.e., not an operator in quantum theory).
This approach, tested in various spacetimes such as Rindler [26], de Sitter,
Minkowski, and inflationary universe [24, 28, 40]. The framework has been
successful in explaining the long-standing CMB anomalies, which are largely
associated with parity asymmetry [28]. In this paper, we presented a
discussion of how direct-sum QFTCS promises new insights into BH physics [25].
The proposed quantization approach, incorporating ’t Hooft’s gravitational
backreaction effects, has unveiled a novel framework for understanding quantum
fields in black hole spacetime, specifically within the context of
Schwarzschild black holes. In this review, we contemplated the idea that the
black hole under the rules of direct-sum QFTCS allows us to redefine
entanglement between interior and exterior Hawking quanta in the form of pure
states. Concurrently, we observed that exterior radiation is intrinsically
linked to the interior due to new non-commutative relations resulting from ’t
Hooft’s gravitational backreaction algebra. This phenomenon, absent from
Hawking’s original 1975 paper, marks a significant departure from traditional
understandings.

We have explored how direct-sum QFT provides a robust framework for
maintaining unitarity and observer complementarity (by means of pure states
evolving into pure states) in the presence of spacetime horizons. By
decomposing the Hilbert space into a direct sum of sectorial Hilbert spaces,
each associated with different superselection sectors (linked to
$\mathcal{P}\mathcal{T}$ symmetry), DQFT allows for a more nuanced
understanding of information conservation. This framework could allow
information retrieval beyond the horizons by splitting Hilbert or Fock space
into superselection sectors.

In conclusion, the study of Hawking radiation with pure states remains a
profound area of research within theoretical physics, particularly concerning
the interplay between quantum mechanics and gravity. Hawking’s initial
proposal highlighted the intrinsic challenges, such as the information
paradox, unitarity violation, and predictability breakdown, which arise from
quantizing a scalar field in Schwarzschild black hole spacetime. These issues,
traceable to early observations made by Einstein and Rosen on the fundamental
conflict between gravity and quantum mechanics [3], continue to spur
significant advancements in the field of quantum gravity. Direct-sum quantum
theory in curved spacetime by reinstating the unitarity would initiate a new
foundational approach towards quantum gravity.

## Acknowledgements

KSK thanks the support of the Royal Society for the Newton International
Fellowship. This research was funded by Fundação para a Ciência e a Tecnologia
grant number UIDB/MAT/00212/2020. KSK would like to thank the organizers of
the EREP 2023 for the kind invitation and opportunity to give a talk on the
subject related to this paper. Authors would like to thank E. Gaztanaga and
Mathew Hull for the useful discussions.

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