# Second quantization of time and energy in Relativistic Quantum Mechanics

M. Bauer* and C.A. Aguillón**
*Instituto de Física, **Instituto de Ciencias Nucleares,
Universidad Nacional Autónoma de México
e-mail<EMAIL_ADDRESS>
###### Abstract

Based on Lorentz invariance and Born reciprocity invariance, the canonical
quantization of Special Relativity (SR) has been shown to provide a unified
origin for the existence of Dirac’s Hamiltonian and a self adjoint time
operator that circumvents Pauli’s objection. As such, this approach restores
to Quantum Mechanics (QM) the treatment of space and time on an equivalent
footing as that of momentum and energy. Second quantization of the time
operator field follows step by step that of the Dirac Hamiltonian field. It
introduces the concept of time quanta, in a similar way to the energy quanta
in Quantum Field Theory (QFT). An early connection is found allready in
Feshbach’s unified theory of nuclear reactions. Its possible relevance in
current developments such as Feshbach resonances in the fields of cold atom
systems, of Bose-Einstein condensates and in the problem of time in Quantum
Gravity is noted. .

”One must be prepared to follow up the consequences of theory, and feel that
one just has to accept the consequences no matter where they lead”  P.A.M.
Dirac111Quoted by J. Polchinsky of UCSB in ”23d Solvay Conference - The
Quantum Structure of Space and Time”, World Scientific, 2007

##  1 Introduction

Resolving the Problem of Time is still a central issue in Quantum Mecanics and
Quantm Gravity[1, 2, 3]. Pauli’objection222 ”In the older literature on
quantum mechanics, we often find the operator equation Ht-tH=i$\hslash$. It is
generally not possible, however, to construct a Hermitian operator (e.g. as a
function of P and Q) which satisfies this equation. This is so because, from
the C.R. written above it follows that H possesses continously all eigenvalues
from -$\infty$ to +$\infty$, whereas on the other hand, discrete eigenvalues
of H can be present. We, therefore, conclude that the introduction of an
operator t is basically forbidden and the time t must necessarily be
considered an ordinary number (‘c’ number) in Quantum Mechanics”[4]to the
existence of a time operator canonically conjugate to the Hamiltonian did set
time to remain a parameter (a $c$-number) while space coordinates were
represented by self-adjoint operators ($q$-numbers)[5], foregoing the equal
footing of space and time accorded by Special Relativity (SR), as well as
questioning the existence and interpretation of a time energy uncertainty
relation[6, 7].

However the canonical quantization of SR[8, 9], together with Born’s
reciprocity principle[10, 11], has now been shown to provide a formal basis
for both the Dirac Hamiltonian and the existence of a self-adjoint ”time
operator” in Relativistic Quantum Mechanics (RQM). As the generator of
continuous momentum displacements (there is no gap in the momentum spectrum)
this time operator induces consequently a shift of energy in both branches of
the energy relativistic spectrum, circumventing Pauli’s objection [9, 12, 13].
Its eigenspinors provide an orthonormal basis alternative to the energy
momentum spinor basis provided by the Dirac Hamiltonian, and consequently a
different representation. Then both can be subjected to second quantization to
show that a field can be equivalently expressed as a function of spacetime
coordinates $\psi(x^{\mu})$ or of energy–momentum coordinates
$\phi(p_{{\mu}}),$ the two formulations being related by the ordinary Fourier
transform.

The present paper explores whether this new basic element of RQM provides
additional insights on the dynamics of many particle systems in the ocupation
number representation of Quantum Field Theory (QFT). To quote: ”Physics does
not depend on the choice of basis, but which is the most convenient choice
depends on the physics”333A. Zee, ”Quantum Field Theory in a Nutshell”,
Princeton University Press (2010). Section 2 reviews briefly the derivation of
the time operator. Section 3 presents the second quantization of the time
operator field that follows step by step that of the Dirac Hamiltonian field.
Interpretation and conclusions are included in Section 4.

##  2 First quantization: configuration, momentum, time and energy
representations[8]

Canonical quantization and factorization of the special relativity free
particle invariant $p^{\mu}p_{\mu}=\pi^{2}:=(m_{0}c)^{2}$ yields a constraint
that is satisfied by the linear equation:

$[\gamma^{\nu}\hat{p}_{\nu}-m_{0}c]\left|\Psi\right\rangle=0$ (1)

provided that in the Minkowski metric $\eta^{\mu\nu}=diag(1,-1,-1,-1)$ the
following anticommutation and commutation relations are satisfied:

$\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}=2\eta^{\mu\nu}\mathbf{I}\text{ \ \
\ \ \ }\left[\hat{p}_{\mu},\hat{p}_{\nu}\right]=0$ (2)

where I is the $4\times 4$ identity matrix. Thus the $\gamma$’s satisfy a
Clifford algebra and are represented by matrices. Eq.1 is recognized as the
Lorentz invariant Dirac equation with:

$H_{D}=c\mathbf{\alpha.\hat{p}}+\beta m_{0}c^{2},\text{ \ \
}\alpha^{i}=\gamma^{0}\gamma^{i}\text{ \ \ \ }\beta=\gamma^{0}$

In the same way, canonical quantization and factorization of the special
relativity invariant $x^{\mu}x_{\mu}=s^{2}:=(\tau_{0}c)^{2}$ yields a
constraint that is satisfied by the linear equation:

$[\gamma^{\nu}\hat{x}_{\nu}-\tau_{0}c]\left|\Psi\right\rangle=0$ (3)

provided now that:

$\left\\{\gamma^{\mu},\gamma^{\nu}\right\\}=2\eta^{\mu\nu}\mathbf{I}\text{ \ \
\ \ \ }\left[\hat{x}^{\mu},\hat{x}^{\nu}\right]=0$ (4)

Eq.3 is a Lorentz invariant equation satisfied by the self-adjoint time
operator:

$T=\mathbf{\alpha.\hat{r}/}c+\beta\tau_{0}\text{ \ \ \
}\alpha^{i}=\gamma^{0}\gamma^{i}\text{ \ \ \ }\beta=\gamma^{0}$

introduced earlier in analogy to the Dirac Hamiltonian[9]. It introduces an
intrinsic time characteristic $\tau_{0}$ of the system.

Finally the purely imaginary symmetrized invariant
$O^{-}:=(\hat{x}^{\mu}\hat{p}_{\mu}-\hat{p}_{\mu}\hat{x}^{\mu})$ that
satisfies Born’s reciprocity invariance under the transformation
$\hat{x}^{\nu}\rightarrow\hat{p}_{\nu},$
$\hat{p}_{\nu}\rightarrow-\hat{x}^{\nu}$ [10] suggests using Planck’s constant
to accept:

$\hat{x}^{0}\hat{p}_{0}-\hat{p}_{0}\hat{x}^{0}=\hat{x}^{i}\hat{p}_{i}-\hat{p}_{i}\hat{x}^{i}=i\hbar$
(5)

that insures the satisfaction of the constaint:

$[\hat{x}^{\mu}\hat{p}_{\mu}-\hat{p}_{\mu}\hat{x}^{\mu}]\left|\Psi\right\rangle=[(\hat{x}^{0}\hat{p}_{0}-\hat{p}_{0}\hat{x}^{0})-(\hat{x}^{i}\hat{p}_{i}-\hat{p}_{i}\hat{x}^{i})]\left|\Psi\right\rangle=0$

Eqs.5 complement the commutation relations
$\left[\hat{p}_{\mu},\hat{p}_{\nu}\right]=0$ and
$\left[\hat{x}^{\mu},\hat{x}^{\nu}\right]=0$ to yield: a) infinite continous
range of the four-space and four-momentum spectra; b) the known configuration
and momentum representations of the operators $\hat{x}^{\nu}$ and
$\hat{p}_{\nu};$ c) the Fourier transfom relation between the representations
of the system state vector and d) the position-momentum uncertainty
relation[8].

In the configuration representation, $\Psi(\mathbf{r,}x_{0})=\left\langle
x\mid\Psi\right\rangle$, defining $t:=x_{0}/c$, Eq.1 reads:

$i\hbar\frac{\partial\Psi(\mathbf{r},t)}{\partial t}=\\{-i\hbar
c\alpha^{j}\frac{\partial}{\partial x^{j}}+\beta
m_{0}c^{2}\\}\Psi(\mathbf{r},t)$ (6)

recognized as the time dependent Dirac equation for free motion.

In the momentum representation $\Phi(\mathbf{p},p_{0})=\left\langle
p\mid\Psi\right\rangle,$ defining $e:=cp_{0},$ Eq.3 reads:

$i\hbar\frac{\partial\Phi(\mathbf{p},e)}{\partial
e}=\\{i(\hbar/c)\alpha_{j}\frac{\partial}{\partial
p_{j}}+\beta\tau_{0}\\}\Phi(\mathbf{p},e)$ (7)

that clearly relates energy changes to momentum changes.

As a self-adjoint operator, $T$ is the generator of infinitesimal momentum
displacements (Lorentz boosts) $\delta\mathbf{p}=(\delta
e/c)\mathbf{\alpha=}(\delta e/c^{2})c\mathbf{\alpha}$ and thus indirectly
energy displacements, circumventing Pauli’s objection. In the same way,
$H_{D}$ is the generator of infinitesimal space displacements
$\delta\mathbf{r}=c\mathbf{\alpha(\delta}t\mathbf{)}$ where
$c\mathbf{\alpha=}d\mathbf{r}/dt$ is the velocity operator[15]. For a wave
packet $\left\langle c\mathbf{\alpha}\right\rangle=\mathbf{v}_{gp}$, the group
velocity. Thus $\left\langle\delta\mathbf{p}\right\rangle=(\delta
m)\mathbf{v}_{gp}=\gamma m_{0}\mathbf{v}_{gp}$ and
$\left\langle\delta\mathbf{r}\right\rangle=\mathbf{v}_{gp}(\delta t).$

$T$ also generates a phase change $\delta\varphi=\beta(\delta
e)\tau_{0}/\hbar$ while $H_{D}$ generates a phase change
$\delta\chi=\beta(\delta t)m_{0}c^{2}/\hbar.$ These are equal provided $\delta
e=m_{0}c^{2}$ and $\delta t=\tau_{0}$. Furthermore a common finite $2\pi$
phase shift requires $\tau_{0}=h/m_{0}c^{2}$. In conclusion, the dynamical
time operator $T=\mathbf{\alpha.\hat{r}}/c+\beta h/m_{0}c^{2}$ (where the
parameter $\tau_{0}$ is equated to de Broglie period $h/m_{0}c^{2}$),
generates the Lorentz boost that gives rise to the de Broglie wave[18]. This
also supports the de Broglie period as an intrinsic property of matter, in
agreement with experiment[16].

These Dirac energy and time operators satisfy the commutation relation:

$\left[T,H_{D}\right]=i\\{I+2\beta
K\\}\hbar+2\beta\\{\tau_{0}H_{D}-m_{0}c^{2}T\\}$ (8)

where $K=\beta\\{2\mathbf{s.l}/\hbar^{2}+1\\}$[15] is a constant of motion.
The related uncertainties are such that $(\Delta T)(\Delta
H_{D})\simeq(\Delta\mathbf{r})(\Delta\mathbf{p})$[9, 13], sustaining the
interpretation given by Bohr originally: the time uncertainty in the instant
of passage at a certain point is given by the width of the wave packet which
is complementary to the momentum uncertainty and thus to the energy
uncertainty[14].

In the Heisenberg picture:

$\frac{dT(t)}{dt}=\frac{1}{i\hslash}[T,H_{D}]=\\{\mathbf{I}+2\beta
K\\}+\frac{2}{i\hslash}\beta\\{\tau_{0}H_{D}-m_{0}c^{2}T\\}$ (9)

that upon integration yields:

$T(t)=\\{\mathbf{I}+2\beta
K+2\beta(1/i\hbar)\tau_{0}H_{D}\\}t-(1/i\hbar)m_{0}c^{2}\beta\int_{0}^{t}dtT$
(10)

as $\hat{H}_{D}$ is constant. The last term introduces an oscillatory behavior
(Zitterbewegung) about a linear time evolution.

The eigenspinors of the self adjoint energy $H_{D}$ and time $T$ operators
(Appendix A) provide orthogonal basis additional to the continous vector ones
generated by operators $\hat{x}_{\nu}$ and $\hat{p}_{\nu}$444This clarifies
the confusion addressed by Hilgevoord[21] between the coordinates of a point
in space and the position variables of a particle (”clearly fostered by the
notation $x,y,z$ for both concepts”); as well as noting that $t:=p^{0}/c\ $and
$H_{D}$ are not canonical conjugate variables., with the following
characteristics. The energy spectrum goes from $-\infty$ to $+\infty$, with a
$\ 2m_{0}c^{2\text{ \ }}$gap at the origin. The time spectrum goes from
$-\infty$ to $+\infty$, with a $2\tau_{0}$ gap at the origin. As
$\tau_{0}=h/m_{0}c^{2\text{ }}$ (the deBroglie or Compton period[17, 18]) the
gaps are seen to be complementary. To a small energy gap corresponds a large
time gap, and viceversa. This complementarity may provide the first indication
that the electron neutrino mass has to differ from zero as this would send the
time gap to infinity. Also in the same way that the energy spectrum clearly
defines non relativistic $(cp\ll m_{0}c^{2})$ and relativistic energy limits
$(cp\gg m_{0}c^{2})$ , the time spectrum recognizes short $(r/c\ll\tau_{0};$
$\ r\ll c\tau_{0})$ and long $(r/c\gg\tau_{0};$ $\ r\gg c\tau_{0})$ time or
space limits. As in the non relativistic limit Eq.6 yields the two-component
positive energy Schrödinger-Pauli equation[15], Eq. 7 results in a two-
component positive time short range approximation.

##  3 Second quantization of the energy-momentum and time-space fields

(Note There are many books on quantum field theory. In this section the
presentation of F. Schawbl, ”Advanced Quantum Mechanics”, Chapter 13[19] is
followed, where the representation of the field is given as a superposition of
free solutions in a finite volume $V$. The passage to infinite volume is
achieved with
$\sum_{\mathbf{k}}\left(\frac{m}{Ve_{\mathbf{k}}}\right)^{1/2}\Rightarrow\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\sqrt{m}}{k_{0}}$
where the factor $\sqrt{m}$ is chosen in order to cancel the factor
$1/\sqrt{m}$ in the spinors, so that the limit $m\rightarrow 0$ exists).

a) The time-space representation

The second quantization of the $\Psi(\mathbf{r},t)$ field considers its
expansion in terms of the spinor eigenvector basis
$\\{\left|e^{q}\right\rangle\\}$ (Appendix A) and transforms the expansion
coefficientes into creation and anhilation (particle and antiparticle)
operators:

$\left\langle\mathbf{r}\mid\hat{\Psi}\right\rangle=\hat{\Psi}(\mathbf{r})=\sum_{\mathbf{p},q}\left(\frac{m_{0}c^{2}}{Ve_{p}}\right)^{1/2}(\hat{b}_{q\mathbf{p}}u_{q}^{e_{p}}(p)e^{-i\mathbf{p.r/}\hbar}+\hat{d}_{q\mathbf{p}}^{\dagger}w_{q}^{e_{p}}(p)e^{i\mathbf{p.r}/\hbar})$
(11)

where the ”energy spinors” are:

$\left|e^{q}\right\rangle=u_{q}^{e_{p}}(p)\left|\mathbf{p}\right\rangle\text{
\ \ \ }e_{p}>m_{0}c^{2}\text{ \ and\ \
}\left|e^{q}\right\rangle=w_{q}^{e_{p}}(p)\left|\mathbf{p}\right\rangle\text{
\ \ \ }e<m_{0}c^{2}-\text{\ }$ (12)

$\displaystyle u_{q}^{e_{p}}(p)$ $\displaystyle=$
$\displaystyle\left(\frac{e_{p}+m_{0}c^{2}}{2m_{0}c^{2}}\right)^{1/2}\left(\begin{array}[]{c}\chi_{q}\\\
\frac{c\mathbf{\sigma.p}}{e_{p}+m_{0}c^{2}}\chi_{q}\end{array}\right)\text{\
}$ (15) $\displaystyle w_{q}^{e_{p}}(p)$ $\displaystyle=$
$\displaystyle-\left(\frac{e_{p}+m_{0}c^{2}}{2m_{0}c^{2}}\right)^{1/2}\left(\begin{array}[]{c}\frac{c\mathbf{\sigma.p}}{e_{p}+m_{0}c^{2}}\chi_{q}\\\
\chi_{q}\end{array}\right)$ (18)

with $\ p=\left|\mathbf{p}\right|\ ,\ \
e_{p}=+\sqrt{(c\mathbf{p})^{2}+(m_{o}c^{2})^{2}}$ and for $q=1,2\ \ \
\chi_{{}_{1}}=\left(\begin{array}[]{c}1\\\ 0\end{array}\right),$
$\chi_{2}=\left(\begin{array}[]{c}0\\\ 1\end{array}\right)$. Thus $u_{1,2}(p)$
correspond to positive energy and up and down spin, while $w_{1,2}(p)$
correspond to negative energy and up and down spin, as can be seen clearly in
the rest frame where $\mathbf{p}=0$. The minus sign in $w_{q}^{e_{p}}(p)$
insures that the charge conjugation operation $C$ transforms the spinors
$u_{q}(p)$ into $w_{q}(p)$ and viceversa.

It then follows:

$P^{\mu}=i\int
d^{3}x\\{\bar{\psi}\gamma_{0}\partial^{\mu}\psi\\}=\sum_{\mathbf{p},q}p^{\mu}(\hat{b}_{q\mathbf{p}}^{\dagger}\hat{b}_{q\mathbf{p}}-\hat{d}_{q\mathbf{\hat{p}}}\hat{d}_{q\mathbf{p}}^{\dagger})$
(19)

where:

$P^{0}=\sum_{\mathbf{p},q}cp_{0}(\hat{b}_{q\mathbf{p}}^{\dagger}\hat{b}_{q\mathbf{p}}-\hat{d}_{q\mathbf{\hat{p}}}\hat{d}_{q\mathbf{p}}^{\dagger})=\sum_{\mathbf{p},q}e_{p}(\hat{b}_{q\mathbf{p}}^{\dagger}\hat{b}_{q\mathbf{p}}-\hat{d}_{q\mathbf{\hat{p}}}\hat{d}_{q\mathbf{p}}^{\dagger})$
(20)

Anticommutation of the field operators to satisfy FermiDirac statistics[20]
and normal ordering to avoid zero point terms yields:

$P^{0}:=H=\sum_{\mathbf{p},q}e_{p}(\hat{b}_{q\mathbf{p}}^{\dagger}\hat{b}_{q\mathbf{p}}+\hat{d}_{q\mathbf{p}}^{\dagger}\hat{d}_{q\mathbf{p}})>0$
(21)

i.e., the total energy as the sum of positive energy quanta $e_{p}$ for both
states, now interpreted as representing electrons and positrons respectively.
The total momentum is given as:

$\mathbf{P}=\sum_{\mathbf{p},q}\mathbf{p}(\hat{b}_{q\mathbf{p}}^{\dagger}\hat{b}_{q\mathbf{p}}+\hat{d}_{q\mathbf{p}}^{\dagger}\hat{d}_{q\mathbf{p}})$
(22)

As components of a four vector, $P^{0}$ and $P^{i}$ are part of the energy-
momentum tensor (stress-energy tensor):

$T^{\mu\nu}=i\bar{\psi}\gamma^{\mu}\partial^{\nu}\psi$ (23)

whose other components yield the total (orbital plus spin) angular momentum of
the system.

The momentum operator $P^{\mu}$ is furthermore shown to be the generator of
space-time displacements, i.e.:

$e^{ia_{\mu}P^{\mu}}\Psi(x)e^{-ia_{\mu}P^{\mu}}=\Psi(x+a)$ (24)

b) The energy-momentum representation

The above procedure can also be applied to the energy-momentum representation
$\Phi(p)=\left\langle p\mid\Psi\right\rangle=\Phi(\mathbf{p},p_{0})$ and its
adjoint. Expanding the field $\Phi(\mathbf{p},E)$ in the spinor
$\\{\left|t_{r}\right\rangle\\}$ basis (Appendix A) yields555Here a finite
momentum volume is considered, yielding an unfamliar discretization of space.
However in the infinite volume limit $\ \sum_{\mathbf{r}}$ goes to the
familiar $\int d\mathbf{r}$.:

$\left\langle\mathbf{p}\mid\hat{\Psi}\right\rangle=\hat{\Phi}(\mathbf{p})=\sum_{\mathbf{r},q}\left(\frac{\tau_{0}}{Vt_{r}}\right)^{1/2}(\hat{a}_{q\mathbf{r}}u_{q}^{t_{r}}(r)e^{-i\mathbf{p.r}/\hbar}+\hat{c}_{q\mathbf{r}}^{\dagger}w_{q}^{t_{r}}(r)e^{i\mathbf{p.r}/\hbar})$
(25)

with eigenvectors, now ”time spinors”:

$\displaystyle u_{q}^{t_{r}}(r)$ $\displaystyle=$
$\displaystyle\left(\frac{t_{r}+\tau_{0}}{2\tau_{0}}\right)^{1/2}\left(\begin{array}[]{c}\chi_{q}\\\
\frac{\mathbf{\sigma.r/}c}{t_{r}+\tau_{0}}\chi_{q}\end{array}\right)\text{...}t>0$
(28) $\displaystyle w_{q}^{t_{r}}(r)$ $\displaystyle=$
$\displaystyle\left(\frac{t_{r}+\tau_{0}}{2\tau_{0}}\right)^{1/2}\left(\begin{array}[]{c}\frac{\mathbf{\sigma.r/}c}{t_{r}+\tau_{0}}\chi_{q}\\\
\chi_{q}\end{array}\right)\text{...}t<0$ (31)

with $t_{r}=+\sqrt{(\mathbf{r}/c)^{2}+\tau_{0}^{2}}$ and again $q=1,2$ ,$\
\chi_{{}_{1}}=\left(\begin{array}[]{c}1\\\ 0\end{array}\right)$,
$\chi_{{}_{2}}=\left(\begin{array}[]{c}0\\\ 1\end{array}\right)$. Thus
$u_{1,2}^{t_{r}}(\mathbf{r})$ correspond to positive time and up and down
spin, while $w_{1,2}^{t_{r}}(\mathbf{r})$ correspond to negative time and up
and down spin.

One now obtains:

$T^{\mu}=\sum_{\mathbf{r},q}(x^{\mu}/c)(\hat{a}_{q\mathbf{r}}^{\dagger}\hat{a}_{q\mathbf{r}}-\hat{c}_{q\mathbf{r}}\hat{c}_{q\mathbf{r}}^{\dagger})$
(32)

As before, requiring anticommutation and normal ordering yields:

$T^{0}=\sum_{\mathbf{r},q}t_{r}(\hat{a}_{q\mathbf{r}}^{\dagger}\hat{a}_{q\mathbf{r}}+\hat{c}_{q\mathbf{r}}^{\dagger}\hat{c}_{q\mathbf{r}})>0\text{
\ \ \ }t_{r}=+\sqrt{(\mathbf{r}/c)^{2}+\tau_{0}^{2}}>0$ (33)

i.e., the time operator field contains only positive times. This development
introduces time quanta $t_{r}=+\sqrt{(\mathbf{r}/c)^{2}+\tau_{0}^{2}}>0$ that
are created and destroyed. by the operators $a$ ,$a^{\dagger}$ and $c$
,$c^{\dagger}$. In analogy to Eq.17, Eq.25 represents a total intrinsic time
associated with the system.

One also obtains from Eq.23 the vector relation:

$\mathbf{T}=\sum_{\mathbf{r},q}(\mathbf{r}/c)(\hat{a}_{q\mathbf{r}}^{\dagger}\hat{a}_{q\mathbf{r}}+\hat{c}_{q\mathbf{r}}^{\dagger}\hat{c}_{q\mathbf{r}})$
(34)

or$:$

$c\mathbf{T}=\sum_{\mathbf{r},q}\mathbf{r}(\hat{a}_{q\mathbf{r}}^{\dagger}\hat{a}_{q\mathbf{r}}+\hat{c}_{q\mathbf{r}}^{\dagger}\hat{c}_{q\mathbf{r}})$
(35)

In a similar way $T^{0}$ and $T^{i}$ as components of a four vector are part
of a time-space tensor,

$\tilde{T}^{\mu\nu}=i\bar{\psi}\gamma^{\mu}\partial^{p_{\nu}}\psi$ (36)

whose other components yield the space boosts of the system as in analogy with
Eq.20 the time operator $T^{\mu}$ can be shown to be the generator of energy-
momentum displacements , i.e.:

$e^{iq_{\mu}T^{\mu}}\Phi(p)e^{-iq_{\mu}T^{\mu}}=\Phi(p+q)$ (37)

##  4 Interpretation and Conclusion

The inclusion of the self adjoint time operator and associated representation
restores in QM the equal footing of time and energy accorded by SR,
circumventing Pauli’s objection and providing an extended insight on the
problem of time. The time space representation exhibits a total energy which
is the sum of positive energy quanta for both particles and antiparticles
present (Eq.17). Particles with negative energies are interpreted as
antiparticles with positive energies. On the other hand, the energy-momentum
representation exhibits a total intrinsic time as the sum of positive time
quanta for both particles and antiparticles (Eq.25), giving a formal support
to Feynman’s extraordinary identification of the negative energy solutions
evolving backward as positrons evolving forward in time.

Second quantization, also referred to as occupation number representation, is
a formulation that allows to describe states with varying numbers of particles
either free or bound by an external or a self consistent Hartree-Fock
potential arising from their interactions. In the each case the energy quanta
are the single particle energies, which comprise the discrete and continuum
energy spectra in the single particle potential $\\{e_{i,}e(\mathbf{p})\\}.$
Then Eq.17 represent the total energy of the independent particle
approximation, where the energy quanta are the single particle energies. In a
description where the ground state is the full Fermi sea, this energy accounts
for the number of paticle-hole (electrons and positrons or protons and
neutrons) states that define an excited state of the system. In the same way,
Eq.25 represents a total intrinsic time summing up the corresponding
individual time quanta expectation values, which are $\pm\tau_{0\text{ \
}}$for bound single particle states666In the RQM description of atom and
nuclear bound states appear as a consequence of attractive potentials. If
these depend solely on position, e.g., the Coulomb potential in atoms, the
shell model self consistent potential in nuclei, the time operator
$T=\mathbf{\alpha.r}/c+\beta\tau_{0\text{ }}$satisfies the same commutation
relation as with the free particle Hamiltonian[9], The expectation value of
$T$ in a bound state $n,.$is:  $\left\langle T\right\rangle_{n}=\left\langle
n\mid T\mid n\right\rangle=\left\langle
n\mid\mathbf{\alpha.r}/c+\beta\tau_{0\text{ }}\mid
n\right\rangle=\pm\tau_{0\text{ }}$ as the first term is zero,
$\mathbf{\alpha}$ being a non diagonal matrix.
. Thus the intrinsic time increases with the number of particle-hole pairs
involved in an excited state of the system.

An early connection is found in Feshbach’s unified theory of nuclear
reactions[25, 26] where sharp energy resonances are shown to be due to
compound nucleus states. These are tipically many particle-hole configurations
of mainly single particle bound states in the common potential. Unbound single
particle states connecting to the open reaction channels will appear with very
small amplitudes, leading to long delay times and long lifetimes[26, 28, 27],
wheras in the present formulation they are shown to carry large intrinsic
times.

To conclude, the purpose of this paper stresses the extension of basic RQM
with the existance a self-ajoint time operator and the additional basis it
provides. Some applications have been allready identified. Observable effects
in experiments that simulate the Dirac equation[22] and in tunneling in
attosecond optical ionization[23] have allready been associated with the
existence of the time operator in the first quantization of SR. The time
operator also provides support for the conditional interpretation of time in
QG[24]. However its full impact in the extensive development and applications
of QFT remains to be explored, both theoretically and experimentally. To be
noted is that Feshbach resonances are currently subject of extensive reseach
in the fields of cold atom systems[29] (where they provide the essential tool
to control the interaction between atoms), and of Bose-Einstein
condensates[30, 31]. The many facets of the problem of time in QG[2, 3] are
also to be considered in the present context.

##  5 Appendix A Time operator eigenvalues and eigenvectors

A plane wave solution of the form
$\Psi(\mathbf{r},x_{0})=e^{-ipx/\hbar}\psi(\mathbf{r})=e^{-i\\{p_{0}x_{0}-\mathbf{p.r}//\hbar\\}}\psi(\mathbf{r})$
in Eq.8 yields the eigenvalue equation:

$\\{-i\hbar c\alpha^{i}\frac{\partial}{\partial x^{i}}+\beta
m_{0}c^{2}\\}\psi(\mathbf{r})=e\psi(\mathbf{r})$ (A.1)

with positive and negative eigenvalues:

$e=\pm\sqrt{(c\mathbf{p})^{2}+(m_{0}c^{2})^{2}}$ (A.2)

and eigenvectors (”energy” spinors):

$\left|e^{q}\right\rangle=u_{q}(p)\left|\mathbf{p}\right\rangle\text{ \ \ \
}e>m_{0}c^{2}\text{ \ and\ \
}\left|e^{q}\right\rangle=w_{q}(p)\left|\mathbf{p}\right\rangle\text{ \ \ \
}e<-m_{0}c^{2}-\text{\ }$ (A.3)

where

(40)

where $\ e_{p}=+\sqrt{(c\mathbf{p})^{2}+(m_{0}c^{2})^{2}}\ \ \ $and$-$for
$q=1,2$ , $\chi_{{}_{1}}=\left(\begin{array}[]{c}1\\\ 0\end{array}\right)$ and
$\chi_{{}_{2}}=\left(\begin{array}[]{c}0\\\ 1\end{array}\right)$. Note that
the negative energy spinors have opposite spin projection to the corresponding
positive energy spinors. (”Given this definition, the charge conjugation
operation $C$ transforms the spinors $u_{q}(p)$ into $w_{q}(p)$ and
viceversa”).

For a particle in an attractive $r-$dependent potential with bound states,
these spinors satisfy the equation

$\\{-i\hbar c\alpha^{i}\frac{\partial}{\partial x_{i}}+V(r)+\beta
m_{0}c^{2}\\}\psi(\mathbf{r})=e\psi(\mathbf{r})$

where $\\{e\\}=\\{E_{i},E\\}$ , the $E_{i}^{q}$ being the bound states
$\varphi_{i}^{q}(\mathbf{r})$ eigenvalues and $E^{q}$ the energy continuum of
distorted eigenfunctions.

The expectation value of the time operator in a bound state is then

$\left\langle T\right\rangle_{i}=\int
dr\varphi_{i}^{q\ast}(\mathbf{r})\\{\mathbf{\alpha.r/c}+\beta\tau_{0}\\}\varphi_{i}^{q}(\mathbf{r})=\int
dr\varphi_{i}^{q\ast}(\mathbf{r})\beta\tau_{0}\\}\varphi_{i}^{q}(\mathbf{r})=\pm\tau_{0}$

as $\mathbf{\alpha.}$is not a diagonal operator in spin space.

In the same way, a plane wave solution of the form
$\Phi(\mathbf{p},p_{0})=e^{-ipx/}\phi(\mathbf{p})=e^{-i\\{\hat{p}_{0}x_{0}-p.x//\hbar\\}}\phi(\mathbf{p})$
in Eq.15 yields the eigenvalue equation:

$\\{i(\hbar/c)\alpha^{i}\frac{\partial}{\partial
p_{i}}+\beta\tau_{0}\\}\phi(\mathbf{p})=t\phi(\mathbf{p})$ (A.5)

with positive and negative eigenvalues:

$t=\pm\sqrt{(\mathbf{r}/c)^{2}+(\tau_{0})^{2}}$ (A.6)

and eigenvectors (”time” spinors):

$\left|t^{q}\right\rangle=u(r)_{q}\left|\mathbf{r}\right\rangle\text{ \ \ \
}t>\tau_{0}\text{ \ and\ \
}\left|t^{q}\right\rangle=w(r)\left|\mathbf{r}\right\rangle\text{ \ \ \
}t<-\tau_{0}\text{\ }$ (A.7)

with

$\displaystyle u_{q}(r)$ $\displaystyle=$
$\displaystyle\left(\frac{t_{r}+\tau_{0}}{2\tau_{0}}\right)^{1/2}\left(\begin{array}[]{c}\chi_{q}\\\
\frac{\mathbf{\sigma.r/}c}{t_{r}+\tau_{0}}\chi_{q}\end{array}\right)$ (50)
$\displaystyle w_{q}(r)$ $\displaystyle=$
$\displaystyle-\left(\frac{t_{r}+\tau_{0}}{2\tau_{0}}\right)^{1/2}\left(\begin{array}[]{c}\frac{\mathbf{\sigma.r/}c}{t_{r}+\tau_{0}}\chi_{q}\\\
\chi_{q}\end{array}\right)\TCItag{A.8}$ (53)

where $t_{r}=+\sqrt{(\mathbf{r}/c)^{2}+(\tau_{0})^{2}}$ and for $q=1,2$ ,
$\chi_{{}_{1}}=\left(\begin{array}[]{c}1\\\ 0\end{array}\right)$ ,
$\chi_{{}_{2}}=\left(\begin{array}[]{c}0\\\ 1\end{array}\right)$.

As eigenvectors of self-adjoint energy and time operators, energy and time
spinors constitute complete orthogonal sets
$\\{\left|e^{q}\right\rangle\\},\\{\left|t^{q}\right\rangle\\}$ as: (eq.
6.3.15 )

$\displaystyle\bar{u}_{q}^{e}(p)u_{s}^{e}(p)$ $\displaystyle=$
$\displaystyle\delta_{qs}\text{ \ \ \ }\bar{u}_{q}^{e}(p)w_{s}^{e}(p)=0$
$\displaystyle\bar{w}_{q}^{e}(p)\bar{w}e_{s}^{e}(p)$ $\displaystyle=$
$\displaystyle-\delta_{qs}\text{ \ \ \
}\bar{w}_{q}^{e}(p)u_{s}^{e}(p)=0\TCItag{A.9}$ (58)

and similar for the time spinors. They both provide representations for a
system state vector.

## References

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# Float, borosilicate and tellurites as cover glasses in Si photovoltaics:
optical properties and performances under sunlight

M. P. Belançon<EMAIL_ADDRESS>M. Sandrini  H. S. Muniz
Departamento de Física, Universidade Tecnológica Federal do Paraná, Pato
Branco, PR 85503-390, Brazil  L. S. Herculano  G. V. B. Lukasievicz
Departamento de Física, Universidade Tecnológica Federal do Paraná,
Medianeira, PR 85884-000, Brazil  E. L. Savi  O. A. Capeloto  L. C. Malacarne
N. G. C. Astrath  M. L. Baesso  Departamento de Física, Universidade Estadual
de Maringá, Maringá, PR 87020-900, Brazil  G. J. Schiavon  A. A. Silva Junior
Mestrado Profissional em Inovações Tecnológicas, Universidade Tecnológica
Federal do Paraná, Campo Mourão, PR 87301-899, Brazil  J. D. Marconi  Centro
de Engenharia, Modelagem e Ciências Sociais Aplicadas, Universidade Federal do
ABC, Santo André, SP 09210-580, Brazil

###### Abstract

One of the most significant materials in a solar panel is the glass, which
provides transparency, UV protection as well as mechanical and chemical
resistance. In this work, we describe the production of prototypes of four
solar modules made using borosilicate, zinc-tellurite, Pr3+ doped zinc-
tellurite, and float glass as cover materials. The performance of these
prototypes was evaluated under a solar simulator, and a device was developed
to monitor all prototypes under real conditions. A comparison between indoor
and outdoor measurements shows that outdoor results are fundamental to
evaluate the performance of modified solar modules as the ones considered in
this study. In addition, we demonstrate the fundamental role played by the
refractive index of cover glasses in the performance of the prototypes, and
discuss how this feature could be explored to achieve enhanced devices, as
well as other benefits that may arise from this field of research.

††preprint: AIP/123-QED

In the 1950s two breakthroughs made today’s mainstream solar energy technology
possible: Silicon solar cells [1] and the float glass method to produce flat
glasses [2]. Before that time, high-quality flat glass was produced by the
metal plate manufacturing process. Due to the need of grinding and polishing,
this method wasted about 20% of the glass mass. It demanded high capital and
operating costs, which motivated the development of the float glass process,
where a ribbon of glass floats on the surface of a molten tin bath while
continuously moving and cooling. This process produces highly homogeneous flat
glasses, making grinding and polishing unnecessary. The technology was ready
to dominate the market at the end of that decade, after about 7 years under
development at the Pilkington industries [2]. A modern floating line produces
flat glass at speeds of $\sim$4000 m2/h and coatings can be cheaply deposited
inline by chemical vapor deposition (CVD) at fast rates [3] ($\sim$100 nm/s),
providing for example anti-reflective or self-cleaning capabilities [4].
Silicon photovoltaics are profiting from float glass transparency, mechanical
quality, chemical resistance, and UV protection, which are fundamental to
produce panels that achieve a lifespan of several decades.

Silicon photovoltaics (PV) constituted 95% of the photovoltaic market in 2020,
which corresponds to about 200 GWp in annual production capacity [5]. The
power per area in such devices is $\sim 200$ W/m2, and roughly 1 billion m2 of
solar panels are produced every year. Even though it seems impressive, our
needs are orders of magnitude beyond this number, which imposes many
challenges towards the tera-watt deployment of PV [6]. One of them is the
expansion of production of float glass [7]. The thickness of the cover glass
in most of the PV on the market today is above 3 mm, and bi-facial PV [8]
growth is expected to increase [5] and speed up the flat glass demand.
Pilkington’s method was decisive to enable this massive deployment of solar
energy, and glasses may contribute even further to tackle the climate crisis.

About 80 million tonnes of solar panels are going to reach their end-of-life
(EOL) in the next 30 years [9]. Glasses account for more than 50% of this
total weight [10], and glass science may contribute to reducing this amount of
EOL products. There is a demand to reduce the thickness [11] (and weight) of
cover glasses in PV’s, as well as to attenuate their degradation [12], which
consequently would expand PV’s lifespan even further. This would also reduce
the demand for metals, which has been pointed out as a constraint to PV
expansion and sustainability [13, 14, 15, 16, 17]. Although the cover glass
may seem one of the simplest components of a PV panel, several complex
features are demanded of a material to be viable in practical applications.
Besides high transparency, chemical, mechanical, and UV resistance [11], other
properties include coatability [18], which is fundamental to reduce reflection
losses in the glass/air interface and may even be explored to provide self-
cleaning capabilities to the surface [19].

During the last decade, a significant effort has been made to develop spectral
converters (SC), which could be built-in the cover glass or deposited as a
thin film over or below it, to improve the spectral mismatch between sunlight
radiation and Si solar cells spectral sensitivity. SC’s provide by this way a
mechanism to expand the efficiency limit of PVs beyond the Shockley-Queisser
limit [20]. This can be achieved by processes such as downconversion (DC) [21,
22, 23] or upconversion [24, 25]. The downshifting (DF) of photons, a
phenomenon that can be classified as a subcategory of DC [23], consists in the
conversion of high energy photons into low energy photons, with a quantum
efficiency lower than 1. Pr3+ doped materials, for instance, offer one of the
simplest ways to achieve DF, due to the well-known emission lines of this ion
around 600 nm, 650 nm and 1050 nm under blue light excitation [26]. Yet
interesting scientific findings have been reported on several SC materials,
most of them are unfeasible at the needed scale [27, 28, 23, 29, 30]. In this
way, some authors have investigated small variations in the composition of the
float glass [11], which should not interfere in the production process and
would be readily available at the industry level.

Several works have demonstrated high quantum efficiencies of tellurite
glasses, a glass family that we have been investigating in the last few years
[26, 31, 32], and could potentially be used as SCs [33, 34]. A few prototypes
made with glasses of this family were already investigated [35] and, to the
best of our knowledge, they were not evaluated under sunlight or compared to
other prototypes made with commercially available glasses, which is the goal
of this work. Here we report on the evaluation of tellurite glasses as cover
materials of PV prototypes, where they are directly compared to similar
prototypes made with two commercially available glasses. To accomplish this
comparative study, we also developed an affordable tool to monitor the power
output of several prototypes simultaneously, which was used to evaluate their
performances under real sunlight. The following glasses were selected for this
investigation: low iron float glass; borofloat 33; the TZNL
(73.3%Te-19.6%Zn-4.9%Na-2.2%La [32]) and Praseodymium doped (0.1%Pr6O11) TZNL
glasses. Some key properties of these samples are shown in Table 1.

Table 1:  Properties of the glasses investigated in this work. TZNL data that includes an error estimate were obtained by the thermal mirror method as described in Ref. 36. Glass |  $\rho$ (g/cm3) |  $n$111Linear refractive index at 633 nm | Thickness (mm) | k (Wm-1K-1) |  $c$ (J kg-1K-1) | D ($10^{-7}$m2s-1) |  $\alpha(10^{-6}$K${}^{-1})$
---|---|---|---|---|---|---|---
Float glass222Data from Saint-Gobain | 2.5 | 1.52 | 4.0 | 1.06 | 800 | 5.3 | 9
Borosilicate 3.3333Data from Schott | 2.23 | 1.47 | 1.54 | 1.2 | 830 | 6.8 | 3.3
TZNL | 5.34$\pm 0.01$ | 1.98 (Ref. 37) | 4.2 | 1.1 (Ref. 38) | 801$\pm 85$ | 2.6$\pm 0.2$ | 11.8$\pm 0.8$

A calibrated fiber-coupled spectrometer (LaserLine LSP-2) was used to measure
the incident and transmitted spectral flux through the glass sample, under
sunlight illumination. The results are shown in Fig. 1, where the high quality
of the thinner borosilicate sample is evidenced. In the Pr3+ doped TZNL sample
we could detect four ground state absorption bands from this dopant and an
enhanced overall transmission of light. This may happen due to a small
decrease in the refractive index and possibly due to reactions induced by the
dopant [32]. A small band at 645 nm was also detected, and attributed to the
3P0-3F2 decay of Pr3+ ions. This result confirms that we have archived DF,
still the ion is much more effective in absorbing than emitting photons.

Figure 1: Spectral flux under sunlight illumination for the samples
investigated in this study.

The glasses were used to build prototypes. Four identical AR coated p-type Si
solar cells (2.0x3.9 cm) were encapsulated at 85°C using ethylene-vinyl
acetate (EVA) as the encapsulant and Tedlar-Pet-Tedlar (TPT) as the backsheet.
The obtained prototypes were named Float, Boro, TZ, and TZPr, referring to the
glass used in each one. A solar simulator (ScienceTech SS2.5kW) was used to
measure IxV curves under standard conditions (AM1.5G spectrum, 1000 W/m2 and
cell temperature at 25°C), which were used to obtain the power as shown in
Fig. 2.

Figure 2: Power of four solar cells encapsulated with different glasses

The power achieved by TZ and TZPr was 95 mW, while Boro and Float have reached
peaks of 106 mW and 116 mW, respectively. It is important to consider that the
ASTM standard for solar simulators [39] is designed to compare silicon cells
or modules under indoor testing. An AM1.5G spectrum in a class AAA simulator
should reproduce the solar irradiance within a margin of $\pm$25% in six bands
of the spectrum (five 100 nm wide and the last one between 900-1100 nm). For
optical spectroscopy, a solar simulator does not necessarily reproduce the
solar spectrum. Also, other glass properties may affect the performance of the
prototypes, and to properly evaluate them it is critical to measure their
performances under real conditions, simultaneously. We developed a system to
perform such measurements, which is described next.

From the data in Fig. 2, the circuit load to achieve peak power of $\sim$ 2
Ohms is obtained. Precision resistors (1 Ohm each) were used to obtain this
load and Hall sensors (WCS2702) were used to measure the currents. A
multimeter HP 34401a was used to obtain the sensitivity of the current sensors
($0.90\pm 0.02$ mV/mA) and to measure the total load in each prototype
($2.08\pm 0.05$ Ohm). Voltages were measured directly by the built-in analog-
to-digital converter of a microcontroller (STM32103C8T6, resolution 0.8056
mV/level), which averaged current and voltage values over 192 measurements
that took about 5 milliseconds each. The averaged values were sent to a
computer via serial communication.

The cells have to be cleaned and soldered to tinned copper wires before the
encapsulation, where the components are heated at 85°C and slightly pressured.
In the construction of the prototypes, all the steps may potentially introduce
problems. In order to evaluate our ability to reproduce all these steps, as
well as the reliability of our measurement system, several prototypes were
produced using float glass as cover. The power of four of these prototypes was
simultaneously measured under sunlight illumination, and the results were
consistent inside a margin error lower than 3%. Damaged prototypes were
intentionally evaluated and the power drop observed was very high if the
silicon cell was cracked ($\sim$50-90%), and measurable ($\sim$10%) if air
bubbles remained in the EVA below the cover glass. Figure 3 (a) shows data
obtained in a measurement performed on February 5 (2021, location
-26.228°,-52.670°). The plane of the prototypes was set to be perpendicular to
the sun around noon. The sunrise/sunset time was 06:09/19:21, which resulted
in an angular change of about 0.2272°/min during 792 minutes on that day. The
inset in this figure highlights the power loss due to clouds, which lowered
momentarily the prototype temperatures. As a result, all devices reached their
peak power on that day just after 14:25. To interpret the data, the Fresnel
transmittance of several surfaces in the prototypes was analyzed. Fig. 3 (b)
shows the S-polarization transmittance for all prototypes.

Figure 3: (a) Power in each solar module prototype and (b) Fresnel
transmittance of the s component (down) for several surfaces, as indicated.
The time in (a) is in scale with the corresponding angle in (b).

The high refractive index of the TZNL glass introduces a significant loss to
the TZ and TZPr prototypes by the total internal reflection between TZNL and
the EVA ($\theta_{c}\cong 50^{o}$). Consequently, for direct sunlight angle
higher than 50o, TZ and TZPr prototypes allow only diffuse light to reach the
silicon cell. Thus, under this condition, TZ and TZPr achieved only half of
the power produced by the Float glass prototype. The Fresnel transmittance is
shown in Fig. 3 (b) highlighting the inflection point around the critical
angle. In previous work, we have already discussed the challenges to use a
high refractive index material as cover in a solar panel [31], and the results
shown here confirm that. When the incident angle is low the performance of the
prototypes remains dictated by the reflection losses, which are quite high in
the TZNL matrix due to its high refractive index.

To achieve the total electrical energy produced by each device during the
experiment, the power presented in Fig. 3 (a) was integrated for each
prototype. A summary of the electrical output data obtained is shown in Table
2.

Table 2: Peak power P(W) per area A(m2) and energy produced E (J) during February 5 experiment, compared to the peak power in the solar simulator measurements (Ps) per area (A). | P/A (W/m2) | E (J) | Ps/A (W/m2)
---|---|---|---
Float | 143 (100%) | 3058 (100%) | 148 (100%)
Boro | 137 (95%) | 2826 (92%) | 135 (86%)
TZ | 110 (76%) | 2005 (65%) | 121 (82%)
TZPr | 117 (82%) | 2288 (74%) | 121 (82%)

Float, boro, TZ and TZPr prototypes produced 3058, 2826, 2005, and 2288
Joules, respectively. As one can see, the TZ delivered only 65% of the total
power delivered by the Float, which is significantly lower than the 82%
observed under the solar simulator. This reduction in total energy is due to
reflection losses in the TZNL/EVA interface, as we can see in Fig. 3 (b).
While the s-polarization transmittance is near 1.0 for the Float/EVA interface
during the entire day, transmittance at the TZNL/EVA interface is dramatically
lower in the morning (before $\sim$10:00) and the afternoon (after
$\sim$15:30). As a result, the power ratio between TZPr/Float increases from
53% at the beginning of the measurement to 80% around 10:00, which is just 2%
lower than what was achieved in the indoor testing. Such difference could be a
result of different equilibrium temperatures in each module, once parameters
such as thermal diffusivity and thickness should affect the heat dissipation,
and consequently the silicon temperature and the performance of the devices.
Such thermal investigation will be considered in future work.

The results shown here demonstrate the importance to investigate these cover
glasses and the prototypes made with them under sunlight. The power
measurement under the solar simulator is an important indicator of the
prototype power in a real situation, though, as we have seen, the total energy
produced may be significantly different. In our case, we could explain such
differences mostly due to the s-polarization transmittance, as seen in Fig. 3,
and even though the TZPr prototype performed better than the TZ, our results
do not indicate that Pr3+ and the DF related to it are playing an important
role. On the other hand, the introduction of this dopant made the glass
slightly more transparent, mainly in the infrared region where Si solar cells
are more sensitive.

Glasses certainly may play a role to boost silicon solar cells. Nonetheless,
if we aim to develop a practical solution to this challenge we cannot focus on
spectral modifications only. For example, even if we could obtain a high
quantum efficiency spectral converter based on a high refractive index
material, such as the TZNL glass, it would be necessary to reduce reflection
losses. This should require the use of a high refractive index polymer [40],
for instance, which is likely to be unfeasible at the photovoltaic industry
level. Besides that, several other requirements were not evaluated for the
TZNL glass, such as its chemical or thermal strengthening, and the resulting
mechanical resistance. TZNL is also an expensive glass, based in Tellurium
oxide, which is a quite rare commodity [41] and the mass production of this
glass may not be feasible. However, in some specific applications, even
expensive SC may be economically viable, as in concentrated photovoltaics
[42], where sunlight is concentrated several hundred times in a small solar
cell, and cover glass can be used.

Some authors have even discussed the rare-earth cost to produce a SC [43], but
such values are meaningless if we are not accounting for the host and the
process in which such SC would be produced and added to the silicon module.
Allsopp et al. [11] have analyzed the performance of PV’s prototypes made with
several glass compositions quite similar to the float glass (doped versions).
However, only solar simulator measurements were performed. As we have
discussed, even a class AAA solar simulator does not exactly reproduce the
sunlight spectrum, which could be important in order to evaluate a SC. As
showed here, evaluating performances under real sunlight was possible using
our homemade measurement system, and it could provide important and
complementary information for PV characterization.

Pilkington describes in detail the challenges to develop the float process. In
his words [2]: “The principles of the process perhaps sound very simple,…How
complex the development work was is illustrated by the fact that it took us
seven years and four million pounds to make any saleable glass and a total of
seven million pounds before the process could achieve the set objective of
replacing plate”. Such an effort and industrial success of the float process
were not achieved by chance, and after 50 years nothing more suitable to
produce massive amounts of high-quality flat glass was discovered.

In this context, the Pilkington method will continue to be fundamental for the
tera-watt deployment of solar energy, still we could think of a modified glass
to be produced by the same process. The method has already been used to
fabricate other glasses, such as borosilicates [44] and aluminosilicates [45,
46], which are required in specific applications. Silicon solar cells are
reaching a “practical Schockley-Queisser limit” [28, 47], and the projections
made by industry [5] do not indicate any prospect that another technology will
replace Si PVs. However, to achieve and sustain a multi tera-watt scale PV
plant worldwide, the materials and processes involved should become more
sustainable and cheaper.

To the best of our knowledge, researchers have not yet explored the
possibility of a low refractive index and low melting temperature glass. To
produce 1 ton of float glass, about 5.5 GJ of energy is needed [48], and a
reduced melting temperature would result in energy savings as well as lower
emissions of greenhouse gases. If such material could achieve the chemical and
mechanical resistance required for a cover glass in PVs, such devices would
benefit from lower reflection losses in the air/glass interface that could be
helpful at low incidence angles of light. SCs are an interesting technology,
but at the industry level, we are restricted to use the float glass or the CVD
process to obtain SCs. In future work, we will pursue different pathways to
expand these possibilities.

###### Acknowledgements.

The authors acknowledge the support from the Brazilian agencies CAPES, CNPq,
Fundação Araucária and FINEP. Companhia Paranaense de Energia (COPEL) and
Agência Nacional de Energia Elétrica (Aneel) are also acknowledged for the
financial support through the contract P&D 2866-0466/2017.

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

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