# The Equivalence Principle as a Noether Symmetry

Salvatore Capozziello1,2,3<EMAIL_ADDRESS>Carmen Ferrara2,3
<EMAIL_ADDRESS>1 Dipartimento di Fisica "E. Pancini", Università
degli Studi di Napoli “Federico II”, Via Cinthia Edificio 6, 80126 Napoli,
Italy
2 Scuola Superiore Meridionale, Largo San Marcellino 10, I-80138 Napoli,
Italy,
3 Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Complesso
Universitario di Monte S. Angelo, Via Cinthia Edificio 6, I-80126 Napoli,
Italy

###### Abstract

The Equivalence Principle is considered in the framework of metric-affine
gravity. We show that it naturally emerges as a Noether symmetry starting from
a general non-metric theory. In particular, we discuss the Einstein
Equivalence Principle and the Strong Equivalence Principle showing their
relations with the non-metricity tensor. Possible violations are also
discussed pointing out the role of non-metricity in this debate.

††preprint: APS/123-QED

##  I Introduction

Gravity is one of the interactions of Nature but its formulation, at
fundamental level, is still matter of debate. In 1915, Einstein published the
theory of General Relativity (GR) as a geometric formulation of gravitational
interaction apt to account for the motion of astrophysical bodies and the
dynamics of the Universe. The foundation of the theory is based upon some
principles Romano and Furnari (2019): the General Covariance, the Equivalence
Principle (EP), and the Principle of Causality. According to them, the space-
time structure is determined by two fundamental fields, a Lorentzian metric
$g$ (fixing the space-time causal structure) and a linear connection $\Gamma$
(defining the free-fall of the bodies). $\Gamma$ and $g$ can be a-priori
independent, but, in GR, $\Gamma$ has to be the Levi-Civita connection
constructed upon the metric tensor $g$ and then defining the locally inertial
observers in agreement with EP).

The gravitational interaction is geometrically encoded in the curvature of a
four-dimensional space-time manifold $(M,g)$. Curvature is locally determined
by the distribution of sources, which are expressed by the four-dimensional
generalization of the ”matter stress-energy tensor”, e.g., a rank-two
(symmetric) tensor $T^{\mu\nu}$. The affine linear connection and parallel
transport enclose their physical interpretation in EP, about which there are
some different versions. Here we mainly focus on two of them: (1) the Einstein
Equivalence Principle (EEP), postulating the equivalence between gravitational
and inertial effects; (2) the Strong Equivalence Principle (SEP), according to
which GR is locally special relativistic, i.e., locally there is no evidence
of gravity. Physically, the EEP means that gravity is the space-time geometry,
so that dynamics is governed by the Local Lorentz invariance (LLI), the Local
Position Invariance (LPI), and the Weak Equivalence Principle (WEP). According
to the so-called Schiff conjecture, the validity of EEP depends on the
simultaneous validity of the three aforementioned conditions Sch (1960). On
the other hand, the SEP states that: _all tests of fundamental physics,
including gravitational physics, are not affected, locally, by the presence of
a gravitational field._

Specifically, the SEP guarantees the local validity of Special Relativity (SR)
in GR, and, since it does not follow from the Einstein field equations, it
provides the geometric interpretation of the metric $g_{\mu\nu}$ beyond the
gravitational dynamics. The EEP allows to unify the Newtonian concepts of
gravity and inertia. In other words, it is the bridge between GR and Newtonian
gravity, as well as the SEP is the bridge between GR and Special Relativity
(SR). Following Refs. Pauli (1921); Brown (2005), the SEP characterizes the
internal structure of GR itself.

However, the SEP results to be valid also for the Teleparallel Equivalent of
GR (TEGR) and the Symmetric Teleparellel Equivalent of GR (STEGR). These
theories belong to the more general class of Metric Affine Theories, in which
the Riemaniann geometry can be extended. The gravitational field is expressed
with respect to other two dynamical variables, the torsion and the non-
metricity tensors, since the connection represents another geometrical field
with its own dynamics, physical implications, and independent of the metric.
In this framework, the Poincaré group can be enlarged to space-time symmetries
giving rise to the so-called Metric Affine Gravity (MAG), where the spin
density, the dilation, and the shear currents are properties of matter
associated to the connection dynamics together with the energy-momentum tensor
Hehl  _et al._ (1976a, b); Obukhov and Tresguerres (1993).

Thus, in TEGR and STEGR, the SEP is no more at the foundation of the theory,
as in GR, but it is a consequence of the Poincaré gauge. These three theories
form the so-called Geometric Trinity of Gravity because, despite of the
differences in the representations, field equations give the same dynamics
Beltrán Jiménez  _et al._ (2019); Capozziello  _et al._ (2022); Beltrán
Jiménez  _et al._ (2018). In general, metric theories of gravity, like GR or
the Jordan-Brans-Dicke one Brans and Dicke (1961); Kajari  _et al._ (2010),
satisfy both EEP and SEP. In these theories, the metric and energy-momentum
tensors can involve also other fields associated with gravity, such as scalar
or vector fields. On the other hand, non-metric theories can be defined with
variable non-gravitational couplings, associated to dynamical fields coupled
to matter. Thus, in non-metric theories, the gravitational coupling to other
non-gravitational fields (i.e., particle fields of the Standard Model) is not
universal, while in metric theories, the coupling to the gravitational field
is universal Will (1993). As a consequence, experimental tests of the EEP are
often performed considering, a priori, the universal coupling of gravity,
where different non-gravitational fields and particles are used to study the
metric aspects of space-time, i.e., the mutual attraction between bodies or
the detection of gravitational waves. Conversely, in the case of SEP, we have
that the observers of an experiment will locally obtain the same results,
independently of the background field. In fact, throughout the universal
coupling to the gravitational field, test particles behave in the same way
both in a gravitational field and in vacuum, independently of their
properties.

In this debate, it is worth noticing the paper Ref.Exirifard and Sheikh-
Jabbari (2008), where a generic metric-affine theory of gravity is considered.
Authors argue that EEP requires metric and metric-affine (Palatini)
formulations to be equivalent. Then, they prove that only for Lovelock
theories, metric and Palatini formulations are equivalent. In this sense, only
Lovelock theories satisfy EEP.

In particular, a universal coupling for the gravitational field has been
assumed in several efforts to unify GR and Quantum Mechanics (QM) in view of
Quantum Gravity (QG). Moreover, the validity of the Superposition Principle,
required in QM as well as in QG, allowed several studies on gravity-matter
entanglement Kay (1998); Oniga and Wang (2016); Paunković and Vojinović
(2018); Bose  _et al._ (2017), indefinite causal orders Oreshkov (2019);
Vilasini (2019), quantum reference frames Giacomini  _et al._ (2019);
Vanrietvelde  _et al._ (2020), and deformations of Lorentz symmetry Amelino-
Camelia (2012); Amelino-Camelia  _et al._ (2020).

Present-day tests of EEP are sensitive only to violations that alter the
spectra of internal energies. Such violations can be expressed in terms of
differences between diagonal elements of operators. In fact, it is assumed
that internal energy operators commute, so that EEP violations regard only
their eigenvalues Zych and Brukner (2018). This led to several formulations of
EEP in the context of QG, and, in particular, of the WEP, which could be
violated for some specific gravitational field superpositions Pipa  _et al._
(2019); Das  _et al._ (2023).

Some space missions have been proposed with the goal to validate EEP, also in
the QM contest: for example, the "MICROSCOPE mission" Touboul  _et al._
(2019), the "Satellite Test of the Equivalence Principle" (STEP) Sumner  _et
al._ (2007), the ”Gamma Ray Astronomy International Laboratory for QUantum
Exploration of Space–Time” (GrailQuest) Burderi and et al. (2021), the ”Space
Atomic Gravity Explorer” (SAGE) Tino  _et al._ (2020), the ”Space-Time
Explorer and QUantum Equivalence Space Test” (STE-QUEST) Altschul and et al.
(2015). In most of them, however, the gravitational coupling is assumed
universal.

In this paper, we want to demonstrate that EEP can be derived from the Noether
theorems and, as consequence, is a Noether symmetry. On the contrary, if the
Noether symmetry does not hold, also the EEP can be violated.

The layout of the paper is the following. In Sec II, we describe the MAG
framework. The role of linear affine connection is highlighted in Sec III. The
two Noether theorems and their geometrical interpretation are reported in Sec.
IV. Finally, in Sec. V, we show that the EP is a Noether symmetry. Conclusions
are drawn in Sec. VI.

Notation: Quantities with tilde are built up in the Levi-Civita connection
(i.e. $\tilde{A}_{\mu}$). We use the Lorentz metric with signature,
$(-,+,+,+)$. It is denoted with $g_{\mu\nu}$ or $g$. Greek indices represent
the space-time, with values $0,1,2,3$. Latin indices indicate general
coordinates (internal and external) not necessarily space-time ones. The flat
metric is indicated by $\eta_{\alpha\beta}=\text{diag}(-1,1,1,1)$.

##  II Metric-Affine Theories of Gravity

GR is based on the Riemaniann geometry, where the metric tensor $g_{\mu\nu}$
plays the role of fundamental field. For this reason, the associated linear
connection is metric-compatible, i.e., length and angle measurements are
integrable and symmetric. The symmetry of the Levi-Civita connection in the
lower indices leads to the closure of infinitesimally small parallelograms
during the parallel transport Eisenhart (1997). Therefore, a natural way to
modify and/or extend the geometry is to relax the Riemannian constraints. In
other words, by starting with a general linear affine connection admitting
both torsion and non-metricity, it is possible to enlarge the geometric
framework, considering non-Riemannian geometries Eisenhart (2012). The
resulting gravity theory is the MAG, that is a class of theories where metric
and affine connection are both fundamental fields Saridakis  _et al._ (2021).
In this framework, the manifold is described by the following invariant
quantities: curvature, torsion, and non-metricity. These quantities are
described by the metric $g$ and affine connection $\Gamma$, having thus the
structure $(M,g,\Gamma)$. The connection coefficients can be uniquely
decomposed, with respect to a given metric, as:

$\Gamma^{\lambda}_{\mu\nu}=\tilde{\Gamma}^{\lambda}_{\mu\nu}+K^{\lambda}_{\mu\nu}+L^{\lambda}_{\mu\nu},$
(1)

where $\tilde{\Gamma}^{\lambda}_{\mu\nu}$ is the Levi-Civita connection,
$K^{\lambda}_{\mu\nu}$ is the contortion tensor, and $L^{\lambda}_{\mu\nu}$ is
the disformation tensor, whose explicit expressions are:

$\displaystyle\tilde{\Gamma}^{\lambda}_{\mu\nu}$
$\displaystyle:=\frac{1}{2}g^{\rho\lambda}(\partial_{\mu}g_{\rho\nu}+\partial_{\nu}g_{\mu\rho}-\partial_{\lambda}g_{\mu\nu}),$
(2a) $\displaystyle K^{\lambda}_{\ \mu\nu}$
$\displaystyle:=\frac{1}{2}(T_{\mu}^{\ \lambda}{}_{\nu}+T_{\nu}^{\
\lambda}{}_{\mu}-T^{\lambda}_{\ \mu\nu}),$ (2b) $\displaystyle L^{\lambda}_{\
\mu\nu}$ $\displaystyle:=\frac{1}{2}(Q^{\lambda}_{\ \mu\nu}-Q_{\mu}^{\
\lambda}{}_{\nu}-Q_{\nu}^{\ \lambda}{}_{\mu}).$ (2c)

Therefore, the three dynamical (geometric) variables, built on $\Gamma$, have
the following explicit expressions in terms of metric and linear connection:

$\displaystyle R^{\alpha}_{\beta\mu\nu}$
$\displaystyle:=\partial_{\mu}\Gamma^{\alpha}_{\beta\nu}-\partial_{\nu}\Gamma^{\alpha}_{\beta\nu}+\Gamma^{\alpha}_{\sigma\mu}\Gamma^{\sigma}_{\beta\nu}-\Gamma^{\alpha}_{\sigma\nu}\Gamma^{\sigma}_{\beta\mu},$
(3a) $\displaystyle T^{\alpha}_{\ \mu\nu}$ $\displaystyle:=\Gamma^{\alpha}_{\
\mu\nu}-\Gamma^{\alpha}_{\ \nu\mu},$ (3b) $\displaystyle Q_{\alpha\mu\nu}$
$\displaystyle:=\partial_{\alpha}g_{\mu\nu}-\Gamma^{\sigma}_{\alpha\mu}g_{\alpha\nu}-\Gamma^{\sigma}_{\alpha\nu}g_{\sigma\mu}.$
(3c)

They affect the parallel transport of a vector defined by the linear affine
connection. During the parallel transport along a closed curve on a non-flat
background, curvature causes a non-null angle when the vector comes back in
its initial position; torsion expresses the antisymmetric behaviour of two
vectors when they and their direction are switched, thus it represents how
much the formed parallelogram is not-closed; non-metricity causes the change
of the norms and the lengths of vectors under the parallel transport Bahamonde
_et al._ (2023); Heisenberg (2023).

In MAG, some of the three dynamical variables can be trivial and then it is
possible to classify MAGs in the following theories:

  * •

Metric theories defined in terms of Riemannian geometry given by the couple
$(M,g)$. They are the dynamical arena of GR, $f(R)$ gravity, and of other
theories like Brans-Dicke or scalar-tensor theories. Here, dynamics is given
in terms of curvature, constructed only by the metric Capozziello and Faraoni
(2011);

  * •

Metric-affine theories à la Palatini where $g$ and $\Gamma$ are disentangled
and geometry is given by the triplet $(M,g,\Gamma)$. Here, dynamics results
enlarged because both $g$ and $\Gamma$ are fundamental fields. In this case,
while GR is exactly restored, other theories, like $f(R)$ or scalar-tensor
ones result different with respect to the pure metric analogues Olmo (2011);

  * •

Teleparallel geometries, where the parallel transport of vectors is
independent of the path, the fundamental fields are the vierbeins and
affinities are given by the Weitzenböck connection Aldrovandi and Pereira
(2013).

Among teleparallel geometries, it is worth mentioning TEGR and STEGR, since,
together with GR, they constitute the so-called Geometric Trinity of Gravity
Beltrán Jiménez  _et al._ (2019), because it can be proved that they are
dynamically equivalent at Lagrangian, field equations and solutions levels
Capozziello  _et al._ (2022).

On the other hand, symmetric teleparallel theories have been introduced in
terms of differential forms Nester and Yo (1999); Hehl  _et al._ (1995). They
represent a subclass of teleparallel theories, where the constraint related to
the metric postulate is relaxed. In fact, metric and linear affine connection
coefficients transform under a general coordinate map $\xi^{\lambda}\to
x^{\lambda}$ as:

$\displaystyle g_{\mu\nu}(x^{\lambda})$
$\displaystyle=\frac{\partial\xi^{\alpha}}{\partial
x^{\mu}}\frac{\partial\xi^{\beta}}{\partial
x^{\nu}}g_{\alpha\beta}(x^{\lambda}),$ (4)
$\displaystyle\Gamma^{\rho}_{\mu\nu}(x^{\lambda})$
$\displaystyle=\frac{\partial
x^{\rho}}{\partial\xi^{\gamma}}\frac{\partial\xi^{\alpha}}{\partial
x^{\mu}}\frac{\partial\xi^{\beta}}{\partial
x^{\nu}}\Gamma^{\gamma}_{\alpha\beta}(\xi^{\lambda})+\frac{\partial^{2}\xi^{\alpha}}{\partial
x^{\mu\partial x^{\nu}}}\frac{\partial x^{\rho}}{\partial\xi^{\alpha}}.$ (5)

Therefore, it emerges that the curvature scalar associated to the general
curvature tensor (3a) is:

$R=\tilde{R}-Q+B$ (6)

where the non-metricity scalar is defined as:

$Q=\frac{1}{4}\big{(}Q_{\alpha}Q^{\alpha}-Q_{\alpha\beta\gamma}Q^{\alpha\beta\gamma}\big{)}+\frac{1}{2}\big{(}Q_{\alpha\beta\gamma}Q^{\beta\alpha\gamma}-Q_{\alpha}\bar{Q}^{\alpha}\big{)}$
(7)

with $Q_{\alpha}:=Q_{\alpha\lambda}{}^{\lambda}$ and
$\bar{Q}_{\alpha}:=Q^{\lambda}{}_{\lambda\alpha}$. _B_ is a boundary term
referred to the non-metricity scalar Q Jiménez  _et al._ (2018); Bahamonde and
Järv (2022); Capozziello  _et al._ (2022); Beltrán Jiménez  _et al._ (2018).

In the symmetric teleparallel geometry, it is possible to choose a coordinate
system where the connection vanishes globally on the manifold. This particular
system of coordinate is the so-called _coincident gauge_ , since the
connection is trivial. In this case, covariant and partial derivatives
coincide. The physical meaning of the coincident gauge is that the origin of
space-time, whose coordinate are $x^{\mu}$, coincide with the origin of
tangent space, whose coordinate are $\xi^{\alpha}$, namely we apply a Poincaré
(linear affine) transformation. When the coordinate system, corresponding to
the coincident gauge is chosen, then the transformation (5) of the connection
into any other coordinate system $x^{\alpha}$ reduces to:

$\Gamma^{\rho}_{\mu\nu}(x^{\lambda})=\Big{(}\frac{\partial\xi^{\alpha}}{\partial
x^{\rho}}\Big{)}^{-1}\partial_{\mu}\Big{(}\frac{\partial\xi^{\alpha}}{\partial
x^{\nu}}\Big{)},$ (8)

since the following affine (gauge) roto-translational transformation of
coordinates has been performed:

$\xi^{\alpha}:=M^{\alpha}_{\beta}x^{\beta}+\xi^{\alpha}_{0}.$ (9)

In this gauge, the field equations are equivalent to GR because:

$\tilde{R}=Q-\tilde{B}$ (10)

and the boundary term $B$ assumes the form:

$\tilde{B}=\tilde{\nabla}_{\alpha}(Q^{\alpha}-\bar{Q}^{\alpha})\,.$ (11)

See Capozziello  _et al._ (2023) for details.

##  III Connection and Parallel Transport

Gravitation, in the GR picture, is a manifestation of space-time curvature,
and that curvature shows up in the deviation of a geodesic from a nearby one
(i.e., relative acceleration of test particles) Thorne  _et al._ (2000). In
this scenario, the EEP has the role to connect flat and curved space-times. In
fact, according to the Principle of General Covariance Weinberg (1972), an
equation valid in SR holds in presence of a gravitational field if it is
general covariant. Thus, SR laws can be locally recovered when the metric of a
curved space-time is replaced by the Minkowski one, that is
$g_{\mu\nu}\rightarrow\eta_{\mu\nu}$, and the equations do not change when we
perform a general coordinate transformation $x\rightarrow x^{{}^{\prime}}$. In
order to make an equation generally covariant, we have to introduce an extra
structure called connection, which specifies how tensors are transported along
a curve. In general, it represents the inertial properties of the coordinate
system under consideration.

The Principle of General Covariance can be thought as an active version of
EEP: given an equation valid in presence of gravitational interaction, the
corresponding SR equation is locally recovered (at a point or along a
trajectory). At the same time, SR equations must be recovered in a locally
inertial frame (passive version) Sciama (1964).

A general linear affine connection $\nabla$ on a manifold $M$ is a map, which
assigns to every pair smooth vector fields $X,Y$ on $M$ another smooth vector
field $\nabla_{X}Y$. In a local chart $(U,\phi)$ on $M$, with coordinate
$x=\phi(p)$, the two vector fields read as $X=X^{i}e_{i}$ and $Y=Y^{i}e_{i}$,
where $\\{e_{i}\\}=\partial/\partial x^{i}$ is the coordinate basis of the
tangent space at any point of the open set $U$. The coordinate representation
of the map $\nabla$ is:

$\displaystyle\nabla_{X}Y=X^{i}\Big{(}\frac{\partial Y^{k}}{\partial
x^{i}}+Y^{j}\Gamma^{k}_{ij}\Big{)}e_{k}\equiv X^{k}Y^{i}_{;k}e_{i},$ (12)

where $\nabla_{e_{k}}(e_{j})=\Gamma^{i}_{kj}e_{i}$ are the _connection
coefficients_. In particular, they have not to satisfy a priori the symmetry
condition $\Gamma^{k}_{ij}=\Gamma^{k}_{ji}$. The covariant derivative
represents the generalization of directional derivative of functions to
tensors.

Given a curve $\gamma$ on a manifold $M$, let us consider a chart $(U,x^{i})$
on $M$ and the parametric equations of $\gamma$, $\gamma^{i}(\tau)$, with
$\tau$ the affine parameter. We define the tangent vector $\dot{\gamma}$ to
the curve $\gamma$ in the natural basis ${e_{i}}$ as follows:

$\dot{\gamma}\equiv\frac{d\gamma}{d\tau}e_{i}=\frac{d\gamma}{d\tau}\partial_{i}.$
(13)

Let $Y$ be a vector field defined on an open neighborhood of $M$. $Y$ is said
to be parallel transported along $\gamma$ if:

$\nabla_{\dot{\gamma}}Y=\frac{dY}{d\tau}=0,$ (14)

or, in components:

$\nabla_{\dot{\gamma}}Y^{k}=\frac{dY^{k}}{d\tau}+\Gamma^{k}_{ij}\frac{d\gamma^{i}}{d\tau}Y^{j}=0.$
(15)

Notice that $Y(\gamma(\tau))$, obtained by a parallel transport of
$Y(\gamma(\tau_{0}))$ along $\gamma$, depends on $\gamma$, with initial
condition $Y^{i}(\gamma(\tau_{0}))=Y^{i}_{0}$ Romano and Furnari (2019). If
the tangent vector $\dot{\gamma}(\tau)$ is parallel transported along
$\gamma(\tau)$, namely if:

$\nabla_{\dot{\gamma}}\dot{\gamma}=\frac{d^{2}x^{k}}{d\tau^{2}}+\Gamma^{k}_{ij}\frac{dx^{i}}{d\tau}\frac{dx^{j}}{d\tau}=0,$
(16)

with $x^{i}$ coordinates of $\gamma(\tau)$ in the chart $(U,\phi)$, then the
curve $\gamma$ is said to be autoparallel. When the affine connection is the
Levi-Civita one, i.e. in GR, the autoparallel curve is said geodesic. In
teleparallel gravity, they give rise to two different structures, because
autoparallels are related to the affine connection, whereas the geodesics are
related to the metric, since they measure the minimal lengths between two or
more points.

These considerations can be linked together under the standard of the two
Noether theorems. We will show that the same EP is a Noether symmetry under
given conditions.

##  IV The Noether Theorems

In 1918, Noether discovered a relation between (continuous families of)
symmetries of Lagrangian systems and their first integrals Noether (1918).
Conserved quantities allow to reduce and integrate dynamics.

The first Noether theorem states: If a dynamical system is defined on a
manifold $M$ of dimension $m={\rm dim}(M)$, then there exists a $(m-1)-$form
on $M$, called Noether current. This quantity results to be closed along
solutions, implying a continuity equation. Therefore conserved quantities are
defined as the integrals of such currents on a $(m-1)-$volume in $M$.

The continuity equation holding for the Noether current relates the changes of
conserved quantities to the flows at the boundary of the region and some
residual at singularities. In GR, Noether currents are not only closed forms,
but even exact forms along solutions. This introduces a superpotential $S$ for
each Noether current and conserved quantities are obtained by surface
integrals of $S$ Fatibene  _et al._ (2010); Bajardi and Capozziello (2022).

Given a classical $n$-degree-of-freedom dynamical system of particles, an
infinitesimal point transformation maps “points” in configuration space and
time into infinitesimal neighboring “points” Struckmeier and Riedel (2002).
Thus, the first Noether theorem relates the conserved quantities of an
$n$-degree-of-freedom Lagrangian system $L(q,\dot{q},t)$ to infinitesimal
point transformations that leave the Lagrange action $Ldt$ invariant. It
formally states that:

_To every differentiable symmetry, generated by local actions, there
corresponds a conserved current_ , called _Noether current_.

If we consider a general Lagrangian density $\mathcal{L}$, depending on
coordinates $x^{a}$ and fields $\phi^{i}$, we get Bajardi and Capozziello
(2022):

$\displaystyle\mathcal{L}(\phi^{i},\partial_{a}\phi^{i},x^{a})$
$\displaystyle\rightarrow\mathcal{L}(\tilde{\phi}^{i},\partial_{a}\tilde{\phi}^{a},\tilde{x}^{a}),$
(17a) $\displaystyle\tilde{\phi}^{i}$ $\displaystyle=\phi^{i}-\delta\phi^{i},$
(17b) $\displaystyle\tilde{x}^{a}$ $\displaystyle=x^{a}-\delta x^{a}.$ (17c)

In order to find the generator of transformations (17b), (17c), it is possible
to consider the transformed prime derivatives of the field $\phi$, obtaining:

$X^{[1]}=\delta
x^{a}\partial_{a}+\delta\phi^{i}\frac{\partial}{\partial\phi^{i}}+(\partial_{a}\delta\phi^{i}-\partial_{a}\phi^{i}\partial_{b}\delta
x^{b})\frac{\partial}{\partial(\partial_{a}\phi^{i})},$ (18)

where the suffix [1] indicates the first prolongation of the Noether vector,
including only the first derivatives of the fields. If the Euler-Lagrangian
equations are invariant under transformations (17b), (17c), then there exists
a function $h^{a}=h^{a}(x^{a},\phi^{i})$ such that:

$\tilde{\mathcal{L}}=\frac{d\tilde{x}^{a}}{dx^{a}}\mathcal{L}+\partial_{a}h^{a}.$
(19)

The two Lagrangians (i.e., the initial and the transformed ones) differ only
by a four-divergence. Thus, substituting the first prolongation of the Noether
vector (18), we get:

$X^{[1]}\mathcal{L}+\partial_{a}\delta x^{a}\mathcal{L}=\partial_{a}h^{a}.$
(20)

In conclusion, if the Euler-Lagrangian equations are invariant under
transformations (17b), (17c), then the condition (20) is respected.

Therefore, the conserved current, associated to the symmetry transformation,
results to be:

$j^{a}=\frac{\partial\mathcal{L}}{\partial(\partial_{a}\phi^{i})}\delta\phi^{i}-\frac{\partial\mathcal{L}}{\partial(\partial_{a}\phi^{i})}\partial_{b}\phi^{i}\delta
x^{b}+\mathcal{L}\delta x^{a}-h^{a},$ (21)

and the first Noether Theorem can be equivalently recast in terms of the first
prolongation of the Noether vector: _If the condition
$X^{[1]}\mathcal{L}+\partial_{a}\delta x^{a}\mathcal{L}-\partial_{a}h^{a}=0$
holds, then the quantity $j^{a}$ is a first integral of the Equations of
Motion (EoMs)._ Bajardi and Capozziello (2022).

By integrating $\partial_{a}j^{a}$ over the four-volume $\Omega$ and using the
divergence theorem, one obtains the Noether charge:

${\cal
Q}=\int_{\Omega}\partial_{a}j^{a}d\Omega=\int_{\Sigma}j^{a}d\Sigma_{a},$ (22)

which is the flow of $j^{a}$ and a scalar quantity under symmetry
transformations.

The second Noether theorem relates the symmetries of a given action with the
symmetries of the EoMs and it formally states: _Let S be an action having an
infinite-dimensional Lie algebra of infinitesimal symmetries linearly
parameterized by m arbitrary functions, $f^{k}(x^{a}),k=0,1,2,..$ and let
$\Gamma^{m}_{k}\equiv\Gamma^{m}_{k}(x^{a},\phi,\partial_{a}\phi)$ and
$\zeta^{m}_{k}\equiv\zeta^{m}_{k}(x^{a},\phi,\partial_{a}\phi)$ be two generic
functions, depending on the considered transformations. Then, there exists a
set of functions $\Phi_{m}$ such that_:

$\Phi_{m}\Gamma^{m}_{k}=\partial_{a}{\Phi_{m}\zeta^{m}{}_{k}{}^{a}}.$ (23)

If the functions parameterizing the action are independent of the space-time
coordinates, we can connect the first and the second Noether theorems as
follows:

$\partial_{a}(\Phi_{m}\zeta^{m}k{}^{a}\delta f^{k}-h^{a})=0.$ (24)

If $\Phi_{m}$ are a set of $m$ Euler-Lagrange equations, then Eq. (24) is a
conserved quantity; as consequence, the existence of a set of Euler-Lagrange
equations implies the existence of a conserved quantity, without any further
assumption. From this point of view, the first theorem provides the explicit
form of the Noether current, and throughout the second theorem, EoMs are
related to such a current.

###  IV.1 Internal Symmetries

Let us consider now an on-shell internal symmetry transformation, so that
space-time coordinates are not taken into account. Then, transformation laws
(17a), (17b), and (17c) become:

$\displaystyle\tilde{\mathcal{L}}(\tilde{x}^{a},\tilde{\phi}^{i},\partial_{a}\tilde{\phi}^{i})$
$\displaystyle=\mathcal{L}(x^{a},\partial_{a}\phi^{i},\partial_{a}\phi^{i}),$
(25a) $\displaystyle\delta\phi^{i}$ $\displaystyle=\tilde{\phi}^{i}-\phi^{i},$
(25b) $\displaystyle\delta x^{a}$ $\displaystyle=0,$ (25c)

as well as the Lagrangian variation:

$\delta\mathcal{L}=\frac{\partial\mathcal{L}}{\partial\phi^{i}}\delta\phi^{i}+\frac{\partial\mathcal{L}}{\partial(\partial_{a}\phi^{i})}\delta\partial_{a}\phi^{i}=\partial_{a}\Bigg{(}\frac{\partial\mathcal{L}}{\partial(\partial_{a}\phi^{i})}\delta\phi^{i}\Bigg{)}=0.$
(26)

In this case, the Noether current and the Noether vector become:

$\displaystyle j^{a}$
$\displaystyle=\frac{\partial\mathcal{L}}{\partial(\partial_{a}\phi^{i})}\delta\phi^{i},$
(27a) $\displaystyle X$
$\displaystyle=\frac{\partial}{\partial\phi^{i}}\delta\phi^{i}+\frac{\partial}{\partial(\partial_{a}\phi^{i})}\delta\partial_{a}\phi^{i}.$
(27b)

Now, we can derive the cyclic variables of the system $\psi^{1}$, such that
its conjugate momentum is a constant of motion:

$\frac{\partial\mathcal{L}}{\partial(\partial_{a}\psi^{1})}=\pi^{a}_{\psi^{1}}.$
(28)

According to Eq. (27a), $\pi^{a}_{\psi^{1}}$ is related to the Noether
current, if the generator of $\psi^{1}$ is of the order of unity. Therefore,
introducing a general change of variables
$\phi^{i}\rightarrow\psi^{i}(\phi^{j})$, such that $\psi^{1}$ is cyclic, we
can rewrite the Noether vector $X$, obtaining the Noether current, which is
expressed with respect to the fields $\psi^{i}$ as:

$j^{a}=\delta\psi^{i}\frac{\partial\mathcal{L}}{\partial(\partial_{a}\psi^{i})}=i_{X}d\psi^{i}\frac{\partial\mathcal{L}}{\partial(\partial_{a}\psi^{i})}.$
(29)

The equality between the conserved quantity and the conjugate momentum of
$\psi^{1}$, according to Eq (29), implies that $\pi^{a}_{\psi^{1}}=cost$ (see
Ref. Bajardi and Capozziello (2022), for more details).

The condition $X^{\prime}\mathcal{L^{\prime}}=X\mathcal{L}=0$ holds regardless
of the variables considered, so that the Noether symmetry is preserved under
the change of variables. If $T$ is a covariant tensor and $L_{X}T=0$, then we
get the so-called Killing vector fields Manoff (1977); Woolley (1977).

###  IV.2 Invariance conditions

Let $X$ be a vector field on $M$ and $\omega$ a form on $M$. A differential
form is conformal invariant with respect to a vector field $X$ if
$L_{X}\omega=h\omega$, where $h\in\mathcal{C}^{\infty}(M)$. If $h=0$, then
$L_{X}\omega=0$ and $\omega$ is invariant with respect to $X$, namely it does
not change along the trajectories of $X$ Steeb (1978). Therefore, $\omega$ is
the related integral of $X$ Dieudonné (1971); Fels and Olver (1997); Steeb
(1978).

The trajectories of the vector field $Y$ are invariant under a one-parameter
group of transformations generated by a vector field $X$, if $L_{X}Y=[X,Y]=0$,
representing an integrability condition. Finally, by considering functions,
i.e., 0-forms, a function $f$ is an invariant with respect to $X$ (first
integral of the dynamical system $x=X(x)$), if $L_{X}f=0$. Thus, if $X$ is the
generator of a certain symmetry, then the conserved quantities of the Noether
theorem close on the same algebra as the generators:

$\displaystyle\big{[}X^{i},X^{j}\big{]}=if^{ij}_{k}X^{k}$ (30)
$\displaystyle\\{\Sigma^{i},\Sigma^{j}\\}=if^{ij}_{k}\Sigma^{k},$ (31)

where the curly brackets are the Poisson brackets of the conserved quantities
$\Sigma^{i}$.

When the Lagrangian $\mathcal{L}$ is invariant under a transformation
$\delta\mathcal{L}=0$, its internal symmetries are selected by the Noether
vector $X$ and the related Noether current $j^{a}$ is a constant of motion
(cf. Eq. (27a)).

The condition of internal symmetry $X\mathcal{L}=0$ is encoded into the Lie
derivative, since:

$L_{X}\mathcal{L}=\big{[}X,\mathcal{L}\big{]}=X\mathcal{L}=0.$ (32)

Thus, the Lagrangian is invariant with respect to the vector field $X$,
assuring the conserved quantity $\Sigma$, which represents the Noether charge.
Moreover, when external symmetries are considered, we can recast Eq. (20) in
terms of the first prolongation of the Noether vector, generator of the
transformation, and the Lie derivative:

$\displaystyle X^{[1]}\mathcal{L}+\partial_{a}\delta
x^{a}\mathcal{L}=\partial_{a}h^{a}\hookrightarrow$ $\displaystyle
L_{X}\mathcal{L}-\partial_{b}\delta x^{b}\mathcal{T}=\partial_{a}h^{a},$ (33)

where $\mathcal{T}$ is the trace of the energy-momentum tensor. By considering
a time-dependent scalar field, Eq. (33) becomes:

$L_{X}\mathcal{L}+\dot{\xi}\mathcal{H}=\dot{h},$ (34)

with $\mathcal{H}$ the Hamiltonian of the system and $\xi=\delta t$. The
choice of the function $h$ is a gauge choice, therefore if
$\dot{\xi}=\xi_{0}$, then $L_{X}\mathcal{L}=c=costant$. If $c$ is trivial,
i.e. $c=0$, internal symmetries are recovered.

In order to define the symmetries of the metric tensor $g$, we can apply the
above results on it. A vector field $X$ on a (pseudo-)Riemannian manifold
$(M,g)$ is a Killing field if

$L_{X}g=0.$ (35)

The Killing equation holds if and only if $g$ is invariant under the flow of
$X$. More generally in fact:

$L_{X}g_{\mu\nu}=2g_{\mu\nu}\phi(x^{\mu})$ (36)

Therefore, there are three types of Killing vectors, depending on the
properties of $\phi$:

  1. 1.

Proper Killing Vector: $\phi(x^{\mu})\neq 0\rightarrow
L_{X}g_{\mu\nu}=2g_{\mu\nu}\phi(x^{\mu})$;

  2. 2.

Special Killing Vector: $\phi(x^{\mu})=0\rightarrow L_{X}g_{\mu\nu}=0$;

  3. 3.

Homotetic Killing Vector: $\partial_{\mu}\phi(x^{\mu})\rightarrow
L_{X}g_{\mu\nu}\sim g_{\mu\nu}$

The special Killing vector encodes the concept of isometry on a given
manifold.

Let $(M,g)$ be a (pseudo)-Riemannian manifold. A diffeomorphism
$f:M\rightarrow M$ is an isometry if it preserves the metric tensor:

$f^{\ast}g_{f(p)}=g_{p},$ (37)

which implies $g_{f(p)}(f_{\ast}X,f_{\ast}Y)=g_{p}(X,Y)$ for $X,Y\in T_{p}M$.

If $\phi_{t}:M\rightarrow M$ is a one-parameter group of transformations,
which generates the Killing vector field $X$, then, according to the
definition of Killing vector, the local geometry does not change as we move
along $\phi_{t}$. In this sense, the Killing vector fields represent the
direction of the symmetry of a manifold Nakahara (2003).

Finally, the Lie derivative can be used to find and select the symmetries of a
given Lagrangian; in fact by considering Eq. (33), we can classify the Noether
symmetries as Bajardi and Capozziello (2022):

  * •

$L_{X}\mathcal{L}=\psi(x^{a},\phi^{i},\partial_{a}\phi^{i})=\partial_{b}\delta
x^{b}\mathcal{T}+\partial_{a}h^{a}$, General Noether Symmetry;

  * •

$L_{X}\mathcal{L}=cost$, Noether Symmetry for canonical Lagrangians;

  * •

$L_{X}\mathcal{L}=0$, Internal Noether Symmetry.

Thus, the Killing vectors and conserved quantities obey the same rules imposed
by the Lie derivative, applied to the metric tensor, and they can be used as
criteria to determine the symmetry of a given dynamical system Camci and
Yildirim (2015); Grumiller  _et al._ (2014).

##  V The Equivalence Principle as a Noether Symmetry

Previous considerations, in particular exploiting the second Noether theorem
formulated with respect to the Lie derivative as in Eq. (33), can be specified
in view to obtain the EP as a Noether symmetry. Let us consider the Lagrangian
$\mathcal{L}$ of a free particle

$\mathcal{L}=g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=g_{\mu\nu}u^{\mu}u^{\nu},$
(38)

where $u^{\mu}$ is a four-velocity. The Lie derivative of a function $f$ is
defined as:

$L_{X}f=X\cdot\nabla f$ (39)

and represents the directional derivative of $f$ along $X$, namely the rate of
change of $f$ measured by a comoving observer. By applying the Lie derivative
to the Lagrangian (38), we have

$L_{X}\mathcal{L}=X\cdot\nabla\mathcal{L}=X^{\alpha}\nabla_{\alpha}(g_{\mu\nu}u^{\mu}u^{\nu})=X^{\alpha}\nabla_{\alpha}(u^{\mu}u_{\mu}),$
(40)

which leads to

$L_{X}\mathcal{L}=X^{\alpha}\nabla_{\alpha}g_{\mu\nu}u^{\mu}u^{\nu}+2X^{\alpha}u_{\mu}\nabla_{\alpha}u^{\mu}.$
(41)

When we work in a metric affine theory, we consider the most general linear
affine connection, defined in Eq. (1), with its related dynamical variable of
non-metricity, given by Eq. (3c). Therefore, when the non-metricity tensor is
considered, we can rewrite Eq. (41) as:

$L_{X}\mathcal{L}=X^{\alpha}Q_{\alpha\mu\nu}u^{\mu}u^{\nu}+2X^{\alpha}u_{\mu}\nabla_{\alpha}u^{\mu}$
(42)

The covariant derivative of the metric tensor can be expressed as follows:

$\displaystyle Q_{\alpha\mu\nu}=$
$\displaystyle\partial_{\alpha}g_{\mu\nu}-\Gamma^{\lambda}_{\alpha\mu}g_{\lambda\nu}-\Gamma^{\lambda}_{\alpha\nu}g_{\lambda\mu}=$
$\displaystyle\partial_{\alpha}g_{\mu\nu}-\tilde{\Gamma}^{\lambda}_{\alpha\mu}g_{\lambda\nu}-K^{\lambda}_{\alpha\mu}g_{\lambda\nu}-L^{\lambda}_{\alpha\mu}g_{\lambda\nu}-$
$\displaystyle\tilde{\Gamma}^{\lambda}_{\alpha\nu}g_{\lambda\mu}-K^{\lambda}_{\alpha\nu}g_{\lambda\mu}-L^{\lambda}_{\alpha\nu}g_{\lambda\mu}=$
$\displaystyle-$ $\displaystyle
K^{\lambda}_{\alpha\mu}g_{\lambda\nu}-L^{\lambda}_{\alpha\mu}g_{\lambda\nu}-K^{\lambda}_{\alpha\nu}g_{\lambda\mu}-L^{\lambda}_{\alpha\nu}g_{\lambda\mu}$
(43)

Eq. (V) can be inserted into Eq. (42) obtaining:

$L_{X}\mathcal{L}=2X^{\alpha}u_{\mu}\nabla_{\alpha}u^{\mu}-2X^{\alpha}(K^{\lambda}_{\alpha\mu}+L^{\lambda}_{\alpha\mu})u^{\mu}u_{\lambda}.$
(44)

According to Eq. (44), it is possible to reconstruct the geodesic, that is the
autoparallel equations. In particular, since the four-velocity is the tangent
vector to the worldline $x^{\mu}$, and $u_{\mu}=dx_{\mu}/d\tau$, with $\tau$
the proper time, the Lie vector $X^{\mu}$ exactly corresponds to the four-
velocity $u^{\mu}$. Thus, we have:

$\displaystyle L_{X}\mathcal{L}$
$\displaystyle=2u^{\alpha}u_{\mu}\nabla_{\alpha}u^{\mu}-2u^{\alpha}(K^{\lambda}_{\alpha\mu}+L^{\lambda}_{\alpha\mu})u^{\mu}u_{\lambda}\rightarrow$
(45) $\displaystyle L_{X}\mathcal{L}$
$\displaystyle=2u^{\alpha}u_{\mu}\nabla_{\alpha}u^{\mu}-2u^{\alpha}(\Gamma^{\lambda}_{\mu\nu}-\tilde{\Gamma}^{\lambda}_{\mu\nu})u^{\mu}u_{\lambda},$
(46)

the Lie derivative of a free particle Lagrangian for a general affine
connection represents how much the theory is different from GR. Moreover, we
can define:

$\displaystyle a^{\mu}$ $\displaystyle:=u^{\lambda}\nabla_{\lambda}u^{\mu},$
(47a) $\displaystyle\tilde{a}_{\mu}$
$\displaystyle:=u^{\lambda}\nabla_{\lambda}u_{\mu}=a_{\mu}+Q_{\lambda\nu\mu}u^{\lambda}u^{\nu},$
(47b)

where $a^{\mu}$ is an _acceleration_ , whereas $\tilde{a}_{\mu}$ is an
_anomalous acceleration_. We recover the STEGR results Capozziello  _et al._
(2022) since:

$\displaystyle L_{X}\mathcal{L}=$ $\displaystyle
Q_{\alpha\mu\nu}u^{\alpha}u^{\mu}u^{\nu}+2u^{\alpha}u_{\mu}\nabla_{\alpha}u^{\mu}=$
$\displaystyle
Q_{\alpha\mu\nu}u^{\alpha}u^{\mu}u^{\nu}+2u_{\mu}a^{\mu}=a^{\mu}u_{\mu}+\tilde{a}_{\mu}u^{\mu}$
(48)

The non-metricity tensor expresses how much the anomalous acceleration
deviates from the standard acceleration, and it is also responsible of how
much the acceleration departs from the spatial hypersurface orthogonal to the
four-velocity. Moreover, the Lie derivative of the free-particle Lagrangian
encodes the feature that the non-metricity does not preserve the norm and the
length of a vector and, as consequence, its behaviour during the parallel
transport. Furthermore, developing the covariant derivative of $a^{\nu}$,
according to Eq. (47a), we also have:

$u^{\alpha}\nabla_{\alpha}u^{\nu}=u^{\alpha}\Big{[}\frac{\partial
u^{\nu}}{\partial x^{\alpha}}+\Gamma^{\nu}_{\alpha\lambda}u^{\lambda}\Big{]}$
(49)

and inserting Eq. (49) into Eq. (V), we obtain:

$Q_{\alpha\mu\nu}u^{\alpha}u^{\mu}u^{\nu}+2\Big{[}\frac{d^{2}x^{\nu}}{d\tau^{2}}+\Gamma^{\nu}_{\alpha\lambda}u^{\alpha}u^{\lambda}\Big{]}u_{\nu}=a^{\nu}u_{\nu}+\tilde{a}_{\nu}u^{\nu}$
(50)

If we develop non-metricity tensor according to Eq. (V), we obtain:

$\displaystyle
Q_{\alpha\mu\nu}u^{\alpha}u^{\mu}u^{\nu}+2\Big{[}\frac{d^{2}x^{\nu}}{d\tau^{2}}+\Gamma^{\nu}_{\alpha\lambda}u^{\alpha}u^{\lambda}\Big{]}u_{\nu}=$
$\displaystyle(\partial_{\alpha}g_{\mu\nu}-\Gamma^{\lambda}_{\alpha\mu}g_{\lambda\nu}\Gamma^{\lambda}_{\alpha\nu}g_{\mu\lambda}+2\Big{[}\frac{d^{2}x^{\nu}}{d\tau^{2}}+\Gamma^{\nu}_{\alpha\lambda}u^{\alpha}u^{\lambda}\Big{]}u_{\nu}=$
$\displaystyle\frac{d^{2}x^{\nu}}{d\tau^{2}}u_{\nu}+\tilde{\Gamma}^{\nu}_{\alpha\mu}u^{\alpha}u^{\mu}u_{\nu}=\frac{1}{2}\Big{(}a^{\nu}u_{\nu}+\tilde{a}_{\nu}u^{\nu}\Big{)}$
(51)

Thus, we have the autoparallel equation of STEGR for a trivial acceleration
$a^{\mu}=0$, that is Eq. (47a):

$L_{X}\mathcal{L}=Q_{\alpha\mu\nu}u^{\alpha}u^{\mu}u^{\nu}=\tilde{a}_{\nu}u^{\nu}.$
(52)

As consistency check, we rewrite Eq. (V) as

$\displaystyle\frac{d^{2}x^{\nu}}{d\tau^{2}}u_{\nu}+\tilde{\Gamma}^{\nu}_{\alpha\mu}u^{\alpha}u^{\mu}u_{\nu}=\frac{1}{2}\tilde{a}_{\nu}u^{\nu}$
(53)
$\displaystyle\frac{d^{2}x^{\nu}}{d\tau^{2}}u_{\nu}+\tilde{\Gamma}^{\nu}_{\alpha\mu}u^{\alpha}u^{\mu}u_{\nu}=\frac{1}{2}Q_{\alpha\mu\nu}u^{\alpha}u^{\mu}u^{\nu}$
(54)
$\displaystyle\frac{d^{2}x^{\nu}}{d\tau^{2}}u_{\nu}+\tilde{\Gamma}^{\nu}_{\alpha\mu}u^{\alpha}u^{\mu}u_{\nu}+[K^{\nu}_{\alpha\mu}+L^{\nu}_{\alpha\mu}]u^{\alpha}u^{\mu}u_{\nu}=0$
(55)

Therefore, in the general case, the Lie derivative of the Lagrangian
$\mathcal{L}$ is equal to the autoparallel equation of a metric affine theory:

$L_{X}\mathcal{L}=\frac{d^{2}x^{\nu}}{d\tau^{2}}+\Gamma^{\nu}_{\alpha\mu}u^{\alpha}u^{\mu}=0$
(56)

GR is recovered if we have that Eq. (42) implies the condition:

$\displaystyle L_{X}\mathcal{L}$ $\displaystyle=0$ (57a) $\displaystyle
Q_{\alpha\mu\nu}=0$ $\displaystyle\iff X^{\alpha}\nabla_{\alpha}u^{\mu}=0,$
(57b)

since the covariant derivative along the direction of the vector $X$ is zero
and we again recover the geodesic transport equation:

$\displaystyle X^{\alpha}\nabla_{\alpha}u^{\mu}\equiv$
$\displaystyle\frac{dx^{\alpha}}{d\lambda}\Big{[}\frac{\partial
u^{\mu}}{\partial
X^{\alpha}}+\tilde{\Gamma}^{\mu}_{\alpha\rho}u^{\rho}\Big{]}=\frac{du^{\mu}}{d\lambda}+\tilde{\Gamma}^{\mu}_{\alpha\rho}u^{\rho}X^{\alpha}.$
(58)

In other words, we restore the norm conservation of a vector and the geodesic
of GR:

$\frac{d^{2}x^{\mu}}{d\tau^{2}}+\tilde{\Gamma}^{\mu}_{\rho\nu}x^{\rho}x^{\nu}=0$
(59)

Thus, the EEP (and SEP) is derived by the Noether symmetry of Lagrangian (38).
On the contrary, the SEP (and EEP) violation is related to the non-metricity
tensor thanks to Eqs. (45) and (46). For a non-metric theory, the only way to
recover EEP is to choose the coincident gauge, nullifying the general affine
connection. In this case, Eq. (56) reduces to:

$L_{X}\mathcal{L}=\frac{d^{2}x^{\nu}}{d\tau^{2}}=0\,.$ (60)

We can conclude that the SEP is a Noether symmetry since we locally recovered
the SR laws. This is a necessary and sufficient condition. As a consequence
also EEP is a Noether symmetry and, also in this case, it is a necessary and
sufficient condition.

It is worth noticing that the above two accelerations can be recovered also
considering the Raychaudhuri equation in spacetimes with torsion and non-
metricity as reported in Ref.Iosifidis  _et al._ (2018). In that case, they
are called path (because related to autoparallels) and hyper-acceleration
respectively. The latter corresponds to our anomalous acceleration.

##  VI Discussion and Conclusions

In this work, we showed that the EEP is a Noether Symmetry for metric-affine
theories of gravity. In particular, this statement holds for GR.

We introduced MAG as possible extensions of GR. In this general picture,
dynamics can be related to torsion $T^{\alpha}_{\mu\nu}$ and non-metricity
$Q_{\alpha\mu\nu}$, besides curvature $R^{\alpha}_{\beta\mu\nu}$. In
particular, we focused on the dynamical equivalence of GR, TEGR, and STEGR and
the role acquired by EEP in the geometric trinity of gravity. It is recovered
both in TEGR and STEGR through a gauge choice of the affine connection.
Moreover, symmetric-teleparallel theories of gravity, more general than STEGR,
can be characterized by a non-trivial non-metricity tensor.

Considering connection and parallel transport, EEP is strictly related to the
Principle of General Covariance and the presence of autoparallel curves. These
geometric concepts can be reinterpreted thanks to the two Noether theorems
which allow to find conserved quantities related to internal and space-time
symmetries.

In this framework, it is possible to demonstrate that the EP (in particular
the EEP and SEP) is a Noether symmetry in agreement with the formulation with
respect to the Fermi coordinates (Fermi-Walker transport). Specifically,
considering local coordinates, we can take into account the Lagrangian
formalism and follow the trajectory of particles along world lines. We obtain
that, in a general metric-affine context, the parallel transport does not
preserve orthonormal vectors so that one cannot ask for the conservation of
metric along timelike geodesics. However, it is possible to recover EEP when
the coincident gauge is chosen.

The validity of SEP implies the validity of EEP. In fact SEP is an extensions
of EEP including gravitational phenomena. In absence of gravity, the physics
reduces to SR, giving a local Minkowskian space-time structure. According to
EEP, a general affine connection $\Gamma$ is locally indistinguishable from
the flat Levi–Civita connection $\tilde{\Gamma}$ of the Minkowski metric
$\eta_{\alpha\beta}$. In GR case, this means that the $\Gamma$ itself is the
Levi-Civita connection. Here, we demonstrated that this approach can be
developed also for the symmetric teleparallel theories through the coincident
gauge Hohmann (2021). It is worth noticing that such a gauge allows also to
recover gravitational waves strating from general non-metric theories of
gravity Capozziello  _et al._ (2024).

In this perspective, several fundamental issues can be investigated. For
example, since SEP can be recovered in TEGR and STEGR, any possible violation
(e.g. at quantum level) Altschul and et al. (2015); Tino  _et al._ (2020),
would make TEGR and STEGR more fundamental than GR, because they do not
require EP as a basic principle. Moreover, it is essential to establish the
true number of degrees of freedom of gravitational field. This issue is still
matter of debate and should be fully investigated by precision experiments Di
Virgilio  _et al._ (2021), gravitational wave astronomy Abedi and Capozziello
(2018), and precision cosmology observations Cai  _et al._ (2016); Abdalla
_et al._ (2022); Capozziello and Shokri (2022).

Furthermore, the equivalence among different theories could be restored by
considering appropriate boundary terms, which, again, are fixed by gauge
choices. In particular, $f(R)$, $f(T,B)$, and $f(Q,B)$ can be compared as
fourth order theories when appropriate boundary terms $B$ are defined in each
gravity framework Bahamonde  _et al._ (2015); Bahamonde and Capozziello
(2017); Bahamonde and Järv (2022); Capozziello  _et al._ (2018, 2023). Thus,
SEP and the boundary terms $B$ could be essential to establish the correct
relation among gravity theories.

Finally, since possible violations of the EEP (and SEP) are related to non-
metricity, the experimental investigation of non-metricity tensor, and its
related quantities, could be the key ingredient to establish the validity of
EP both at classical and quantum level.

## Acknowledgements

The Authors thank Vittorio De Falco for the useful comments and discussions on
the topics and acknowledge the Istituto Italiano di Fisica Nucleare (INFN)
iniziative specifiche QGSKY and MOONLIGHT2. This paper is based upon work from
COST Action CA21136 Addressing observational tensions in cosmology with
systematics and fundamental physics (CosmoVerse) supported by COST (European
Cooperation in Science and Technology).

## References

  * Romano and Furnari (2019) A. Romano and M. M. Furnari,  _The Physical and Mathematical Foundations of the Theory of Relativity: A Critical Analysis_ (Springer Nature, 2019).
  * Sch (1960) American Journal of Physics 28, 340 (1960).
  * Pauli (1921) W. Pauli,  _Relativitätstheorie_ (Vieweg+Teubner Verlag, 1921).
  * Brown (2005) H. Brown, Physical relativity / Harvey R. Brown. Oxford: Clarendon Press, ISBN 0-19-927583-1, 978-0-19-927583-0, 2005, XIII+225 pp. (2005), 10.1093/0199275831.001.0001.
  * Hehl  _et al._ (1976a) F. W. Hehl, G. D. Kerlick, and P. von der Heyde, Zeitschrift für Naturforschung A 31, 111 (1976a).
  * Hehl  _et al._ (1976b) F. W. Hehl, G. D. Kerlick, and P. von der Heyde, Zeitschrift für Naturforschung A 31, 823 (1976b).
  * Obukhov and Tresguerres (1993) Y. N. Obukhov and R. Tresguerres, Phys. Lett. A 184, 17 (1993), arXiv:gr-qc/0008013 .
  * Beltrán Jiménez  _et al._ (2019) J. Beltrán Jiménez, L. Heisenberg, and T. S. Koivisto, Universe 5, 173 (2019), arXiv:1903.06830 [hep-th] .
  * Capozziello  _et al._ (2022) S. Capozziello, V. De Falco, and C. Ferrara, European Physical Journal C 82, 865 (2022), arXiv:2208.03011 [gr-qc] .
  * Beltrán Jiménez  _et al._ (2018) J. Beltrán Jiménez, L. Heisenberg, and T. S. Koivisto, JCAP 08, 039 (2018), arXiv:1803.10185 [gr-qc] .
  * Brans and Dicke (1961) C. Brans and R. H. Dicke, Phys. Rev. 124, 925 (1961).
  * Kajari  _et al._ (2010) E. Kajari, N. Harshman, E. Rasel, S. Stenholm, G. Süssmann, and W. P. Schleich, Applied Physics B 100, 43 (2010).
  * Will (1993) C. M. Will,  _Theory and Experiment in Gravitational Physics_ (1993).
  * Exirifard and Sheikh-Jabbari (2008) Q. Exirifard and M. M. Sheikh-Jabbari, Phys. Lett. B 661, 158 (2008), arXiv:0705.1879 [hep-th] .
  * Kay (1998) B. S. Kay, Classical and Quantum Gravity 15, L89 (1998).
  * Oniga and Wang (2016) T. Oniga and C. H.-T. Wang, Phys. Rev. D 93, 044027 (2016).
  * Paunković and Vojinović (2018) N. Paunković and M. Vojinović, Classical and Quantum Gravity 35, 185015 (2018).
  * Bose  _et al._ (2017) S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht, M. Toroš, M. Paternostro, A. A. Geraci, P. F. Barker, M. S. Kim, and G. Milburn, Phys. Rev. Lett. 119, 240401 (2017).
  * Oreshkov (2019) O. Oreshkov, Quantum 3, 206 (2019).
  * Vilasini (2019) V. Vilasini (2019).
  * Giacomini  _et al._ (2019) F. Giacomini, E. Castro-Ruiz, and v. Brukner, Nature Commun. 10, 494 (2019), arXiv:1712.07207 [quant-ph] .
  * Vanrietvelde  _et al._ (2020) A. Vanrietvelde, P. A. Hoehn, F. Giacomini, and E. Castro-Ruiz, Quantum 4, 225 (2020).
  * Amelino-Camelia (2012) G. Amelino-Camelia, Symmetry 4, 344 (2012).
  * Amelino-Camelia  _et al._ (2020) G. Amelino-Camelia, M. Palmisano, M. Ronco, and G. D’Amico, International Journal of Modern Physics D 29, 2050017 (2020), https://doi.org/10.1142/S0218271820500170 .
  * Zych and Brukner (2018) M. Zych and Č. Brukner, Nature Physics 14, 1027 (2018).
  * Pipa  _et al._ (2019) F. Pipa, N. Paunković, and M. Vojinović, Journal of Cosmology and Astroparticle Physics 2019, 057 (2019).
  * Das  _et al._ (2023) S. Das, M. Fridman, and G. Lambiase, Commun. Phys. 6, 198 (2023), arXiv:2307.09632 [gr-qc] .
  * Touboul  _et al._ (2019) P. Touboul _et al._ (MICROSCOPE), Class. Quant. Grav. 36, 225006 (2019), arXiv:1909.10598 [gr-qc] .
  * Sumner  _et al._ (2007) T. Sumner, J. Anderson, J.-P. Blaser, A. Cruise, T. Damour, H. Dittus, C. Everitt, B. Foulon, Y. Jafry, B. Kent, N. Lockerbie, F. Loeffler, G. Mann, J. Mester, C. Pegrum, R. Reinhardt, M. Sandford, A. Scheicher, C. Speake, and P. Worden, Advances in Space Research 39, 254 (2007).
  * Burderi and et al. (2021) L. Burderi and et al., Experimental Astronomy 51, 1255 (2021).
  * Tino  _et al._ (2020) G. M. Tino, L. Cacciapuoti, S. Capozziello, G. Lambiase, and F. Sorrentino, Progress in Particle and Nuclear Physics 112, 103772 (2020), arXiv:2002.02907 [gr-qc] .
  * Altschul and et al. (2015) B. Altschul and et al., Advances in Space Research 55, 501 (2015).
  * Eisenhart (1997) L. P. Eisenhart,  _Riemannian geometry_ , Vol. 51 (Princeton university press, 1997).
  * Eisenhart (2012) L. P. Eisenhart,  _Non-riemannian geometry_ (Courier Corporation, 2012).
  * Saridakis  _et al._ (2021) E. N. Saridakis, R. Lazkoz, V. Salzano, and et al., arXiv e-prints , arXiv:2105.12582 (2021), arXiv:2105.12582 [gr-qc] .
  * Bahamonde  _et al._ (2023) S. Bahamonde, K. F. Dialektopoulos, C. Escamilla-Rivera, and et al., Reports on Progress in Physics 86, 026901 (2023), arXiv:2106.13793 [gr-qc] .
  * Heisenberg (2023) L. Heisenberg, arXiv e-prints , arXiv:2309.15958 (2023), arXiv:2309.15958 [gr-qc] .
  * Capozziello and Faraoni (2011) S. Capozziello and V. Faraoni,  _Beyond Einstein Gravity: A Survey of Gravitational Theories for Cosmology and Astrophysics_ (Springer, Dordrecht, 2011).
  * Olmo (2011) G. J. Olmo, Int. J. Mod. Phys. D 20, 413 (2011), arXiv:1101.3864 [gr-qc] .
  * Aldrovandi and Pereira (2013) R. Aldrovandi and J. G. Pereira,  _Teleparallel Gravity: An Introduction_ (Springer, 2013).
  * Nester and Yo (1999) J. M. Nester and H.-J. Yo, Chin. J. Phys. 37, 113 (1999), arXiv:gr-qc/9809049 .
  * Hehl  _et al._ (1995) F. W. Hehl, J. D. McCrea, E. W. Mielke, and Y. Ne’eman, Phys. Rept. 258, 1 (1995), arXiv:gr-qc/9402012 .
  * Jiménez  _et al._ (2018) J. B. Jiménez, L. Heisenberg, and T. Koivisto, Phys. Rev. D 98, 044048 (2018).
  * Bahamonde and Järv (2022) S. Bahamonde and L. Järv, Eur. Phys. J. C 82, 963 (2022), arXiv:2208.01872 [gr-qc] .
  * Capozziello  _et al._ (2023) S. Capozziello, V. De Falco, and C. Ferrara, Eur. Phys. J. C 83, 915 (2023).
  * Thorne  _et al._ (2000) K. S. Thorne, J. A. Wheeler, and C. W. Misner,  _Gravitation_ (Freeman San Francisco, CA, 2000).
  * Weinberg (1972) S. Weinberg,  _Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity_ (John Wiley and Sons, New York, 1972).
  * Sciama (1964) D. W. Sciama, Rev. Mod. Phys. 36, 463 (1964).
  * Noether (1918) E. Noether, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1918, 235 (1918).
  * Fatibene  _et al._ (2010) L. Fatibene, M. Francaviglia, and S. Mercadante, Symmetry 2, 970 (2010).
  * Bajardi and Capozziello (2022) F. Bajardi and S. Capozziello,  _Noether Symmetries in Theories of Gravity: With Applications to Astrophysics and Cosmology_, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2022).
  * Struckmeier and Riedel (2002) J. Struckmeier and C. Riedel, Phys. Rev. E 66, 066605 (2002).
  * Manoff (1977) S. Manoff, Acta Physica Austriaca 48, 41 (1977).
  * Woolley (1977) M. L. Woolley, Journal of Physics A: Mathematical and General 10, 2107 (1977).
  * Steeb (1978) W. H. Steeb, Zeitschrift Naturforschung Teil A 33, 742 (1978).
  * Dieudonné (1971) J. Dieudonné, “Éléments d’analyse. Tome IV,” Cahiers scientifique. Fasc. 34. Paris: Gauthier-Villars, XVIII. (1971).
  * Fels and Olver (1997) M. E. Fels and P. J. Olver, Mathematische Annalen 308, 701 (1997).
  * Nakahara (2003) M. Nakahara,  _Geometry, topology and physics_ (CRC Press, 2003).
  * Camci and Yildirim (2015) U. Camci and A. Yildirim, International Journal of Geometric Methods in Modern Physics 12, 1550120 (2015).
  * Grumiller  _et al._ (2014) D. Grumiller, M. Irakleidou, I. Lovrekovic, and R. McNees, Phys. Rev. Lett. 112, 111102 (2014).
  * Iosifidis  _et al._ (2018) D. Iosifidis, C. G. Tsagas, and A. C. Petkou, Phys. Rev. D 98, 104037 (2018), arXiv:1809.04992 [gr-qc] .
  * Hohmann (2021) M. Hohmann, Phys. Rev. D 104, 124077 (2021).
  * Capozziello  _et al._ (2024) S. Capozziello, M. Capriolo, and S. Nojiri, arXiv e-prints , arXiv:2401.06424 (2024), arXiv:2401.06424 [gr-qc] .
  * Di Virgilio  _et al._ (2021) A. D. Di Virgilio _et al._ , Eur. Phys. J. C 81, 400 (2021), [Erratum: Eur.Phys.J.C 81, 457 (2021)], arXiv:2101.08179 [gr-qc] .
  * Abedi and Capozziello (2018) H. Abedi and S. Capozziello, Eur. Phys. J. C 78, 474 (2018), arXiv:1712.05933 [gr-qc] .
  * Cai  _et al._ (2016) Y.-F. Cai, S. Capozziello, M. De Laurentis, and E. N. Saridakis, Rept. Prog. Phys. 79, 106901 (2016), arXiv:1511.07586 [gr-qc] .
  * Abdalla  _et al._ (2022) E. Abdalla _et al._ , JHEAp 34, 49 (2022), arXiv:2203.06142 [astro-ph.CO] .
  * Capozziello and Shokri (2022) S. Capozziello and M. Shokri, Phys. Dark Univ. 37, 101113 (2022), arXiv:2209.06670 [gr-qc] .
  * Bahamonde  _et al._ (2015) S. Bahamonde, C. G. Böhmer, and M. Wright, Phys. Rev. D 92, 104042 (2015), arXiv:1508.05120 [gr-qc] .
  * Bahamonde and Capozziello (2017) S. Bahamonde and S. Capozziello, European Physical Journal C 77, 107 (2017), arXiv:1612.01299 [gr-qc] .
  * Capozziello  _et al._ (2018) S. Capozziello, M. Capriolo, and M. Transirico, International Journal of Geometric Methods in Modern Physics 15, 1850164 (2018), arXiv:1804.08530 [gr-qc] .


Now at ]2550 Engineering, Göteborg, Sweden

# Orthonormal and periodic levels for Quantum Cascade Laser simulation

Zakaria Mohamed  D. Ekin Önder  [  Andreas Wacker
<EMAIL_ADDRESS>https://www.matfys.lu.se/staff/faculty/andreas-
wacker/ Mathematical Physics and NanoLund, Lund University, Box 118, 22100
Lund, Sweden

###### Abstract

A Python package to evaluate Wannier, Wannier-Stark, and EZ (both Energy and
location Z resolved) levels for Quantum Cascade Lasers is presented. We
provide the underlying theory in detail with a focus on the orthonormality and
periodicity of the generated states.

††preprint: Levels for Quantum Cascade Lasers

##  I Introduction

The operation of Quantum Cascade Lasers (QCLs)[1, 2, 3] is based on optical
transitions between quantized levels in the conduction band of a semiconductor
heterostructure. Further such levels are employed for an effective population
of the upper laser level and the emptying of the lower laser level, which is
facilitated by a sequence of tunneling and scattering processes at the
designated bias. In order to provide light amplification over a larger spatial
range, a carefully designed sequence of layers, referred to as module in the
following, is repeated periodically about 30-300 times. Each of these modules
contains a laser transition, so that the electron flow resembles a cascade.

The design of QCLs requires reliable simulation schemes, see Refs. 4, 5 for an
overview. In a first step this needs to establish the energy levels for a
given sequence of semiconductor layers. Here the periodic repetition of the
module is typically reflected by the requirement that the levels in a given
module are periodically repeated in the neighboring modules (periodicity
criterion). Alternatively, few modules embedded between contacts can be
simulated[6]. Furthermore, the orthonormality of the levels is highly
relevant, as this is a common requirement for basis states and often tacitly
assumed in concepts such as Fermi’s golden rule.

The periodicity criterion and orthonormality are actually difficult to achieve
simultaneously. The reason is that common approaches require boundary
conditions to avoid escape to infinity. In this case, levels located in
different modules do not satisfy the periodicity criterion, as their distance
to the boundary differs. Alternatively, a central module is identified and
levels in neighboring modules are defined by shifting the levels from the
central module. In this case, levels in neighboring modules are solutions of a
setup with shifted boundaries and thus intermodule orthogonality is not
guaranteed. Albeit these orthogonality violations may be small[7] they
constitute a matter of concern.

A second issue affecting the orthogonality of levels is the common use of
energy-dependent effective masses, see, e.g., Refs. 6, 8, 9. These are
particular important for infrared QCLs with level energies of several 100 meV
above the conduction band edge where the conduction band becomes non-
parabolic, see e.g. Ref. 10 and references cited therein. If the Hamiltonian
contains energy-dependent components, the solutions obtained at different
energies are not orthogonal.

Over the last decades our group developed simulation techniques for QCLs,
where we strived to overcome the issues mentioned above. From the very start
we used a basis of Wannier states[11] to combine the periodicity condition
with orthogonality. Later, we implemented non-parabolicity [12] by applying a
two-band model [13, 14, 15] explicitly. In this article we give a
comprehensive discussion of the underlying concepts. We also carefully
overhauled our codes and provide a new Python-based simulation package
resource_QCL to generate sets of orthonormal and periodic levels for a variety
of QCL simulations. The package is provided in the supplemental material and
can be used under a CC-BY license.

##  II Generating Wannier states

To avoid boundary conditions, which are incompatible with the periodicity
condition, we start by calculating Bloch functions for an infinite periodic
repetition of the modules without bias. These are mapped on Wannier functions
$w^{\nu n}$ by a unitary transformation. Here, $n$ specifies the module to
which they are localized and $\nu$ distinguishes different levels in a module.
Most importantly, they satisfy the periodicity criterion and constitute a
basis, which is complete within the energy range covered by the Bloch bands
considered. In particular, we obtain an explicit matrix representation (18) of
the Hamiltonian for the heterostructure. Fig. 1 shows an example for these
Wannier levels. Here, we consider the sample of Ref. 16, which achieved room-
temperature CW operation by double-phonon extraction using two extraction
levels with a spacing of the optical phonon energy.

Nonparabolicity is treated by a two-band model and keeping both the conduction
and valence band components $(w^{\nu n}_{c}(z),w^{\nu
n}_{v}(z))^{\textrm{tr}}$. Thus orthonormality is ensured, see Eq. (17) below.
This task is performed by the routine resource_QCL.calcWannier1 which
establishes the relevant data based on the following physical input
parameters:

thlist

list of layer thicknesses $w_{i}$ in one module

Eclist

list of conduction band offsets $V_{i}$ for the layers

mclist

list of effective masses $m_{c}^{*}$ at the $\Gamma$ point

E_Kane

the common Kane energy $E_{\textrm{Kane}}$

In this section the underlying theory and numerical issues are detailed.

Figure 1:  Wannier states for the QCL of Ref. 16. The band gap is depicted by
the grey/blue area. The lines show the probability density
$w^{\nu,n}_{c}(z)^{2}+w^{\nu,n}_{v}(z)^{2}$ for the 14 lowest Wannier states
(label $\nu$), where the baseline is shifted by the respective level energy.
Thick colored lines indicate levels related to (from top to bottom) the upper
laser, the injector, the lower laser, and the two extraction levels. Full
lines refer to levels in the central module ($n=0$) ranging from 0 to 59.8 nm.
Dashed and dot-dashed lines are periodic repetitions to the right and left
module ($n=\pm 1$), respectively.

###  II.1 Two-band formalism

The two-band model considers states $(\Psi_{c}(z),\Psi_{v}(z))^{\textrm{tr}}$,
which contain a conduction and valence band contribution in order to include
non-parabolicity in the conduction band for planar systems. This model is
applicable only along a single spatial direction, typically the growth
direction denoted as $z$. In the absence of an external potential, the two-
band Hamiltonian is given by the matrix [13]

$\hat{H}^{\textrm{het}}=\begin{pmatrix}E_{c}(z)&\frac{p_{cv}}{m_{e}}\frac{\hbar}{{\rm
i}}\frac{\partial}{\partial z}\\\ \frac{p_{vc}}{m_{e}}\frac{\hbar}{{\rm
i}}\frac{\partial}{\partial z}&E_{v}(z)\end{pmatrix}\,.$ (1)

$E_{c}(z)$ is the conduction-band offset, which depends on the specific
material at position $z$. Correspondingly, $E_{v}(z)$ mimics the valence band
offset. The stationary Schrödinger equation reads

$\hat{H}^{\textrm{het}}\begin{pmatrix}\Psi_{c}(z)\\\
\Psi_{v}(z)\end{pmatrix}=E\begin{pmatrix}\Psi_{c}(z)\\\
\Psi_{v}(z)\end{pmatrix}$ (2)

where the lower component provides

$\Psi_{v}(z)=\frac{1}{E-E_{v}(z)}\frac{p_{vc}}{m_{e}}\frac{\hbar}{{\rm
i}}\frac{\partial\Psi_{c}(z)}{\partial z}\,.$ (3)

Inserting into the upper component gives the Schrödinger-like equation

$\left[E_{c}(z)-\frac{\partial}{\partial
z}\frac{\hbar^{2}}{2m_{c}(E,z)}\frac{\partial}{\partial
z}\right]\Psi_{c}(z)=E\Psi_{c}(z)$ (4)

with the energy-dependent mass

$m_{c}(E,z)=\frac{m_{e}^{2}[E-E_{v}(z)]}{2|p_{cv}|^{2}}\,,$ (5)

which also depends on the material at position $z$. Identifying the known
effective mass $m^{*}_{c}$ at the $\Gamma$ point as $m_{c}(E_{c})$, we find
the relation

$m^{*}_{c}(z)=\frac{m_{e}^{2}(E_{c}(z)-E_{v}(z))}{2|p_{cv}|^{2}}$

so that the values of $E_{c}(z),E_{v}(z),p_{cv}$ and $m^{*}_{c}(z)$ are
intrinsically related to each other within the two-band model. The value of
$p_{cv}$ is required to be the same for all materials involved in a given
heterostructure within the two band model. Its value is provided by the Kane
energy[15] $E_{\textrm{Kane}}=2|p_{vc}|^{2}/m_{e}$, which is used as an input
parameter. Focusing on the conduction band, we use literature values for
$m^{*}_{c}(z)$ and $E_{c}(z)$ for the respective material at $z$ as further
input parameters. Hence we apply

$E_{v}(z)=E_{c}(z)-2|p_{vc}|^{2}\frac{m^{*}_{c}(z)}{m_{e}^{2}}=E_{c}(z)-E_{\textrm{Kane}}\frac{m^{*}_{c}(z)}{m_{e}}\,,$
(6)

for the effective valence band offset. This approximation deviates from the
actual material value, as expected within the framework of the two-band model,
which simplifies the intricate multi-band structure of the valence band. Here,
its sole purpose is to quantify the energy-dependent effective mass
$m_{c}(E,z)$ from Eq. (5). Finally, the phase of the momentum matrix elements
is chosen as

$p_{vc}={\rm i}|p_{vc}|={\rm i}\sqrt{\frac{m_{e}E_{\textrm{Kane}}}{2}}\textrm{
and consequently }p_{cv}=-{\rm i}|p_{vc}|\,,$

which provides a real Hamiltonian (1) and consequently the option to establish
real Wannier functions below.

Note, that the solutions $\Psi^{n}_{c}(z)$ of the effective Schrödinger
equation (4) at different eigenenergies $E_{n}$ are not necessarily orthogonal
due to the energy-dependent mass. However, adding the valence band component
from Eq. (3) provides after normalization

$\int{\rm
d}z\,\left[\Psi^{m*}_{c}(z)\Psi^{n}_{c}(z)+\Psi^{m*}_{v}(z)\Psi^{n}_{v}(z)\right]=\delta_{mn}$

as they are actually solutions of the energy-independent Hamiltonian (1). The
main concept in the following is to start with othonormal Wannier functions,
see Eq. (17), and inherit this property to other sets by unitary
transformations.

###  II.2 Bloch States

Now we consider a QCL, where each module consists of a sequence of $N$ layers
with thicknesses $w_{0},w_{1},\ldots w_{N-1}$ extending from $z_{i}$ to
$z_{i+1}=z_{i}+w_{i}$. Here we set $z_{0}=0$ and denote $z_{N}=d$. As the
modules are periodically repeated, we have a periodic structure with period
$d=\sum_{i=0}^{N-1}w_{i}$. Consequently, for vanishing external potential, the
eigenstates are Bloch states
$(\Psi^{q\nu}_{c}(z),\Psi^{q\nu}_{v}(z))^{\textrm{tr}}$ as characterized by
the Bloch vector $q$ (with $-\pi/d<q\leq\pi/d$ covering the one-dimensional
Brillouin zone) and the band index $\nu$. The Bloch condition reads

$\begin{pmatrix}\Psi^{q\nu}_{c}(z+d)\\\ \Psi^{q\nu}_{v}(z+d)\end{pmatrix}={\rm
e}^{{\rm i}qd}\begin{pmatrix}\Psi^{q\nu}_{c}(z)\\\
\Psi^{q\nu}_{v}(z)\end{pmatrix}$ (7)

for all $z$. In order to calculate the Bloch states we note that in a region
$z_{i}<z<z_{i+1}$ of constant $E_{c}(z)=V_{i}$ and constant
$m_{c}(E,z)=m_{i}(E)$, Eq. (4) is solved by

$\begin{split}&\Psi^{i}_{c}(z)=C_{i}\cos\left[k_{i}\left(z-z_{i}-\frac{w_{i}}{2}\right)\right]+\frac{D_{i}}{k_{i}}\sin\left[k_{i}\left(z-z_{i}-\frac{w_{i}}{2}\right)\right]\\\
&\textrm{with }k_{i}=\frac{\sqrt{2m_{i}(E)(E-V_{i})}}{\hbar}\quad\textrm{for
}z_{i}<z<z_{i+1}\end{split}$ (8)

This provides a finite nonvanishing result for the $D_{i}$-term at $k_{i}=0$,
where we implement $\frac{1}{k_{i}}\sin(k_{i}z)=z\;\textrm{sinc}(k_{i}z/\pi)$
with the normalized sinc function $\textrm{sinc}(x)=\sin(\pi x)/(\pi x)$ as
defined in the NumPy library of Python.

For $E<V_{i}$, $k_{i}={\rm i}\lambda_{i}$ is purely imaginary and we get the
equivalent representation

$\begin{split}&\Psi^{i}_{c}(z)=C_{i}\cosh\left[\lambda_{i}\left(z-z_{i}-\frac{w_{i}}{2}\right)\right]+\frac{D_{i}}{\lambda_{i}}\sinh\left[\lambda_{i}\left(z-z_{i}-\frac{w_{i}}{2}\right)\right]\\\
&\textrm{with }\lambda_{i}=\frac{\sqrt{2m_{i}(E)(V_{i}-E)}}{\hbar}\end{split}$

Our choice for the middle point of the layers $z_{i}+\frac{w_{i}}{2}$ as
reference position limits the exponential growth of the hyperbolic functions,
which reduces numerical instabilities.

At the boundary $z_{n+1}$ between two layers, the functions $\Psi_{c}(z)$ and
$\Psi_{v}(z)$ should be continuous. From Eq. (3) we find, that this is
equivalent to the two equations

$\begin{split}\Psi^{n}_{c}(z_{n+1}-0^{+})&=\Psi^{n+1}_{c}(z_{n+1}+0^{+})\\\
\frac{1}{m_{n}}\frac{\partial\Psi^{n}_{c}(z)}{\partial
z}_{|z=z_{n+1}-0^{+}}&=\frac{1}{m_{n+1}}\frac{\partial\Psi^{n+1}_{c}(z)}{\partial
z}_{|z=z_{n+1}+0^{+}}\,,\end{split}$

where $0^{+}$ stands for an infinitesimal positive quantity (more formally to
be replaced by an $\epsilon>0$ with $\lim_{\epsilon\to 0}$). Inserting Eq. (8)
provides the relation

$\begin{split}&\begin{pmatrix}C_{n+1}\\\ D_{n+1}\end{pmatrix}=\tilde{\cal
M}_{n}\begin{pmatrix}C_{n}\\\ D_{n}\end{pmatrix}\quad\textrm{with }\tilde{\cal
M}_{n}=\begin{pmatrix}c_{n}c_{n+1}-\beta_{n}k_{n}^{2}s_{n}s_{n+1}&s_{n}c_{n+1}+\beta_{n}c_{n}s_{n+1}\\\
-c_{n}s_{n+1}k_{n+1}^{2}-\beta_{n}k_{n}^{2}s_{n}c_{n+1}&-k_{n+1}^{2}s_{n}s_{n+1}+\beta_{n}c_{n}c_{n+1}\end{pmatrix}\\\
&\textrm{where}\quad\beta_{n}=\frac{m_{n+1}}{m_{n}}\qquad
c_{n}=\cos\left(\frac{k_{n}w_{n}}{2}\right)\qquad
s_{n}=\frac{1}{k_{n}}\sin\left(\frac{k_{n}w_{n}}{2}\right)=\frac{w_{n}}{2}\textrm{sinc}\left(\frac{k_{n}w_{n}}{2\pi}\right)\end{split}$

Here the values $c_{n}$ and $s_{n}$ are real both for real and (purely)
imaginary $k_{n}$, so that the entire matrix is real. Note that the matrices
$\tilde{\cal M}_{n}$ depend on $E$ via $k_{n}$ and $m_{n}$. Applying the Bloch
condition (7) for the conduction band component111The valence band component
$\Psi^{q\nu}_{v}(z)$ satisfies the same condition due to Eq. (3).
$\Psi^{q\nu}_{c}(z+d)={\rm e}^{{\rm i}qd}\Psi^{q\nu}_{c}(z)$ for $z_{0}\leq
z<z_{1}$, where $z_{N}\leq z+d<z_{N+1}$ is in region $N$, which has the same
properties as region 0 (i.e. $m_{N}=m_{0}$ and $V_{N}=V_{0}$) requires

${\rm e}^{{\rm i}qd}\begin{pmatrix}C_{0}\\\
D_{0}\end{pmatrix}=\begin{pmatrix}C_{N}\\\ D_{N}\end{pmatrix}=\tilde{\cal
M}\begin{pmatrix}C_{0}\\\ D_{0}\end{pmatrix}$ (9)

with $\tilde{\cal M}=\tilde{\cal M}_{N-1}\tilde{\cal M}_{N-2}\ldots\tilde{\cal
M}_{1}\tilde{\cal M}_{0}$. This only allows for non-vanishing $C_{0},D_{0}$ if
$\textrm{det}\\{\tilde{\cal M}-{\rm e}^{{\rm i}qd}{\cal I}\\}=0$. As
$\tilde{\cal M}$ depends on energy this provides a set of energies
$E_{\nu}(q)$ with $\nu=1,2,\ldots$ for each value of $q$. (Restricting to
conduction band states, we consider only solutions, where $E$ is larger than
the minimal value of the set $\\{V_{n}\\}$.)

Straightforward evaluation provides det$\\{\tilde{\cal M}_{n}\\}=\beta_{n}$.
Thus det$\\{\tilde{\cal M}\\}=m_{N}/m_{0}=1$ due to the periodicity. This
provides

$\begin{split}0=&\textrm{det}\\{\tilde{\cal M}-{\rm e}^{{\rm i}qd}{\cal
I}\\}=1-(M_{00}+M_{11}){\rm e}^{{\rm i}qd}+{\rm e}^{2{\rm i}qd}\\\ =&{\rm
e}^{{\rm i}qd}\left[2\cos(qd)-(M_{00}+M_{11})\right]\\\
\Leftrightarrow&M_{00}+M_{11}=2\cos(qd)\end{split}$

For the specific energies $E_{\nu}(q)$, a solutions of Eq. (9) is given by

$\begin{pmatrix}C_{0}\\\ D_{0}\end{pmatrix}=\begin{pmatrix}\tilde{M}_{11}-{\rm
e}^{{\rm i}qd}\\\
-\tilde{M}_{10}\end{pmatrix}+0.5\begin{pmatrix}-\tilde{M}_{01}\\\
\tilde{M}_{00}-{\rm e}^{{\rm i}qd}\end{pmatrix}$ (10)

where we added solutions for the first and second row in order to avoid
problems with accidental small values. Furthermore, we directly get all
coefficients

$\begin{pmatrix}C_{n}\\\ D_{n}\end{pmatrix}=\tilde{\cal M}_{n-1}\tilde{\cal
M}_{n-2}\ldots\tilde{\cal M}_{0}\begin{pmatrix}C_{0}\\\ D_{0}\end{pmatrix}$

which allows us to construct the wave function $\Psi^{n}_{c}(z)$ for arbitrary
$z$ in the interval $0<z\leq d$ by evaluation of Eq. (8) in the appropriate
region. [Outside this central module, the periodicity condition (7) is applied
to evaluate $\Psi^{n}_{c}(z)$ based on the result in the central module.]
Furthermore, $\Psi^{n}_{v}(z)$ is obtained from Eq. (3). By rescaling
$C_{0}^{\nu}(q)$ and $D_{0}^{\nu}(q)$ the normalization

$\int_{0}^{d}{\rm d}z\left[|\Psi^{\nu q}_{c}(z)|^{2}+|\Psi^{\nu
q}_{v}(z)|^{2}\right]=1$ (11)

is achieved. This procedure defines the initial Bloch functions. As the
Hamiltonian is real we have $E_{\nu}(-q)=E_{\nu}(q)$ and our choice (10)
guarantees $C_{n}(-q)=C_{n}(q)^{*}$ and = $D_{n}(-q)=D_{n}(q)^{*}$ so that

$\begin{pmatrix}\Psi^{q\nu}_{c}(z)\\\
\Psi^{q\nu}_{v}(z)\end{pmatrix}^{*}=\begin{pmatrix}\Psi^{-q\nu}_{c}(z)\\\
\Psi^{-q\nu}_{v}(z)\end{pmatrix}$ (12)

Furthermore, the chosen Bloch funtions are periodic in $q$ and continuous at
$q=\pm\pi/d$.

In practice, we calculate the Bloch functions for a finite number $(N_{q})$ of
$q$-values

$q_{j}=-\frac{\pi}{d}-\frac{\pi}{N_{q}d}+\frac{2\pi}{N_{q}d}j\quad\textrm{for
}j=1,2,\ldots N_{q}\,,$ (13)

which are chosen such that they cover the entire Brillouin zone with equal
spacing. Furthermore for each $j$, we have a value $k=N_{q}+1-j$, so that both
$q_{j}$ and $q_{k}=-q_{j}$ are in the set of chosen $q$-values. The choice of
$q_{j}$ values restricts to the eigenstates of a system of length $L=N_{q}d$
with boundary conditions $\Psi(x+L)=(-1)^{(N_{q}+1)}\Psi(x)$. (For odd $N_{q}$
this is the standard Born von-Karman boundary condition). Thus, the
eigenstates are orthogonal. Based on normalization for a single period before,
we thus find

$\int_{L/2}^{L/2}{\rm
d}z\,\begin{pmatrix}\Psi^{q_{j}\nu}_{c}(z)&\Psi^{q_{j}\nu}_{v}(z)\end{pmatrix}^{*}\begin{pmatrix}\Psi^{q_{k}\mu}_{c}(z)\\\
\Psi^{q_{k}\mu}_{v}(z)\end{pmatrix}=N_{q}\delta_{\nu\mu}\delta_{jk}$ (14)

As the Bloch states are eigenstates of the Hamiltonian
$\hat{H}^{\textrm{het}}$ for the ideal layer sequence (1), we can write its
matrix elements of

$\int_{L/2}^{L/2}{\rm
d}z\,\begin{pmatrix}\Psi^{q_{j}\nu}_{c}(z)&\Psi^{q_{j}\nu}_{v}(z)\end{pmatrix}^{*}\hat{H}^{\textrm{het}}\begin{pmatrix}\Psi^{q_{k}\mu}_{c}(z)\\\
\Psi^{q_{k}\mu}_{v}(z)\end{pmatrix}\\\
=E_{\nu}(q)N_{q}\delta_{\nu\mu}\delta_{jk}$ (15)

In the continuum limit $L=N_{q}d\to\infty$, one needs to replace
$N_{q}\delta_{jk}\to 2\pi\delta(q_{j}-q_{k})$ at the right-hand of Eqs.
(14,15).

###  II.3 Wannier States

The Wannier functions are given by

$\begin{split}\begin{pmatrix}w_{c}^{\nu,n}(z)\\\
w_{v}^{\nu,n}(z)\end{pmatrix}=&\frac{d}{2\pi}\int_{-\pi/d}^{\pi/d}{\rm
d}q\,{\rm e}^{-{\rm i}qnd}\begin{pmatrix}\Psi^{q\nu}_{c}(z)\\\
\Psi^{q\nu}_{v}(z)\end{pmatrix}\\\ \approx&\frac{1}{N_{q}}\sum_{q}{\rm
e}^{-{\rm i}qnd}\begin{pmatrix}\Psi^{q\nu}_{c}(z)\\\
\Psi^{q\nu}_{v}(z)\end{pmatrix}\end{split}$ (16)

and satisfy $w^{\nu,n}(z)=w^{\nu,0}(z-nd)$, so that it is sufficient to
calculate the $n=0$ terms. Here the $q$ sum runs over the values specified in
Eq. (13). For the normalization (11) they satisfy

$\int{\rm
d}z\begin{pmatrix}w_{c}^{\nu,n}(z)&w_{v}^{\nu,n}(z)\end{pmatrix}\begin{pmatrix}w_{c}^{\mu,m}(z)\\\
w_{v}^{\mu,m}(z)\end{pmatrix}=\delta_{nm}\delta_{\nu\mu}$ (17)

as given in Ref. 18. [This also holds exactly for the approximation with a
finite set of $q_{j}$ due to Eq. (14) as long as the Wannier functions become
vanishingly small for $|z|>L/2$.] Note that the Wannier functions are real due
to Eq. (12), which actually requires the combined presence of $q_{j}$ and
$-q_{j}$ in the sum for a finite number of values.

Based on Eq. (15) we can express the Hamiltonian in the basis of the Wannier
states as

$H^{\textrm{het}}_{\nu n,\mu m}=E_{\nu|m-n|}\delta_{\nu\mu}\textrm{ with
}E_{\nu h}=\frac{1}{N_{q}}\sum_{q}E_{\nu}(q)\cos(hqd)$ (18)

This provides the energies $E_{\nu 0}$ of the Wannier levels, as shown in Fig.
1, which are just the average energy of the respective Bloch band. The
interperiod couplings $E_{\nu 1}$, $E_{\nu 2},\ldots$ reflect the width and
further details in the dispersion of the Bloch band. In most cases, one can
restrict to $E_{\nu 1}$ and possibly $E_{\nu 2}$. (Exceptions exist for
particular cases where the gaps between minibands vanish for energies above
the barriers [19].)

The Wannier functions evaluated by Eq. (16) depend on the phases chosen for
the Bloch functions, which can be modified by an arbitrary q-dependent phase
factor ${\rm e}^{{\rm i}\varphi_{q}}$ with the restriction
$\varphi_{-q}=-\varphi_{q}$. W. Kohn demonstrated already in 1959 [20] that
choosing the Wannier functions real at symmetry points provides often the best
localization of the Wannier functions. While superlattices have such
characteristic points with inversion symmetry, things are more complicated for
QCLs, where such symmetry are lacking. However, choosing all Bloch states real
and positive at certain points is often still a good choice, resulting in
Wannier functions localized close to the chosen point.

We follow the procedure of Ref. 18, which requires the initial phases to be
chosen such that the Bloch functions are continuous in $q$ including the
periodicity at $q=\pm\pi/d$. Then we calculate Eq. (12, 13) of Ref. 18

$\begin{split}X_{\nu}(q)=&{\rm i}\int_{0}^{d}{\rm d}z\,{\rm e}^{{\rm
i}qz}\begin{pmatrix}\Psi^{q\nu}_{c}(z)&\Psi^{q\nu}_{v}(z)\end{pmatrix}^{*}\frac{\partial}{\partial
q}\left[{\rm e}^{-{\rm i}qz}\begin{pmatrix}\Psi^{q\nu}_{c}(z)\\\
\Psi^{q\nu}_{v}(z)\end{pmatrix}\right]\\\ \textrm{and
}&x_{\nu}=\frac{d}{2\pi}\int_{-\pi/d}^{\pi/d}{\rm d}q\,X_{\nu}(q)\end{split}$

and Eq. (20) of Ref. 18 determines the phases

$\phi_{\nu}^{\textrm{ML}}(q)=\int_{0}^{q}{\rm
d}q^{\prime}[X_{\nu}(q^{\prime})-x_{\nu}]$

Finally, we multiply the Bloch states with index $\nu,q$ by the factor ${\rm
e}^{{\rm i}\phi_{\nu}^{\textrm{ML}}(q)}$ and use these in Eq. (16) to obtain
maximally localized Wannier functions.

This procedure turns out to be robust for Wannier states derived from Bloch
bands with energies below the conduction band edge of the barrier. For states
with higher energy, large numbers of $N_{q}$ are required and orthonormality
becomes less precise.

##  III States at finite bias

Under operation, an applied bias provides an additional potential with the
two-band Hamiltonian
$\hat{H}^{\varphi}=-e\varphi(z){\tiny\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}}$.
This tilts the energy landscape, as can be seen in Fig. 2. The matrix elements
in the Wannier basis read:

$H^{\varphi}_{\nu n,\mu m}=-e\int{\rm d}z\,\varphi(z)(w^{\nu n}_{c}(z)w^{\mu
m}_{c}(z)+w^{\nu n}_{v}(z)w^{\mu m}_{v}(z))$ (19)

The diagonal elements shift the energies of the Wannier levels as illustrated
in Fig. 2. This manifests their denotation (such as upper and lower laser
levels) as provided in the caption of Fig. 1.

Figure 2:  Wannier levels as in Fig. 1 under a constant electric field
corresponding to a bias drop of 260 mV per module.

Here we assume a constant electric field $F$ providing $\phi(z)=Fz$ ($F$ is
counted positive, if electrons with charge $-e$ are accelerated towards the
right). The mean-field from the electron distribution and the ionized dopands
provides a further contribution to $\phi(z)$, which is periodically repeated
for the modules and can be directly included in the package. However, its
implementation requires to solve a kinetic equation for electron populations
separately. Most importantly, we assume a constant bias drop $Fd$ per module,
which is the main parameter in the following. (This excludes the formation of
field domains[21, 22, 23].)

###  III.1 Wannier-Stark States

We have an explicit expression for the matrix elements of the total
Hamiltonian $\hat{H}^{F}=\hat{H}^{\textrm{het}}+\hat{H}^{\varphi}$ in the
basis of Wannier levels, see Eqs. (18,19). This Hamiltonian can be
diagonalized and the corresponding energy eigenstates are related to the
Wanneir levels by a unitary transformation. These levels are orthonormal by
construction.

The matrix elements of $\hat{H}^{F}$ in Wannier basis have the symmetry

$H^{F}_{\nu n+h,\mu m+h}=H^{F}_{\nu n,\mu m}-eFd\delta_{n,m}\delta_{\nu,\mu}$

This implies the labeling of the eigenlevels as $\psi^{\alpha
n}(z)=\psi^{\alpha 0}(z-nd)$ and $E^{\textrm{WS}}_{\alpha
n+h}=E^{\textrm{WS}}_{\alpha 0}-neFd$ for the infinite structure. These are
known as Wannier-Stark (WS) levels [24]. In our code, we repeat the central
module Nper times in each direction and choose the levels $\psi^{\alpha 0}(z)$
where the expectation value of $z$ is located within the central module. For
other values of $n$, we apply the periodicity condition above for the
functions and energies. We check explicitly the orthonormality between
neighboring modules (default accuracy $10^{-4}$ in resource_QCL.calcZmat).
Otherwise, Nper should be increased. Similar to the Wannier and Bloch levels,
these functions $\psi(z)$ have a valence and conduction band component.

Fig. 3 shows the result. Comparing with the Wannier levels at the same bias
drop per period in Fig. 2, the WS levels get typically repelled from each
other resembling avoided crossings. The upper and lower extraction level are
now approximately one and two optical phonon energies below the lower laser
level, respectively, as targeted by the design.

Figure 3:  Wannier-Stark (WS) levels obtained by diagonalising the matrix of
the Hamiltonian in the basis of Wannier levels from Fig. 2. See Fig. 1 for
labeling.

Now we want to point out, how this cumbersome approach to establish energy
eigenstates via Bloch and Wannier functions is able to solve the problem of
combined orthonormality and periodicity addressed in the introduction. The key
problem in conventional calculations under applied bias is the need for a
spatial boundary. Otherwise, the wave functions leak out to infinity. Here we
work in the basis of Wannier functions, where we applied a cut-off in energy
by restricting to a finite number of Bloch bands. This cut-off is relative to
the conduction band edge and thus tilted under applied bias. Therefore, the
states cannot leak out to infinity as this would require the coupling to
states at high energy above the conduction band (which would be incorrect
within the two-band model anyway).

###  III.2 EZ states

If WS levels are close in energy, they sometimes appear to be linear
combinations of more localised states. Examples can be seen around 210 meV and
-5 meV in Fig. 3. Such pairs of levels have to be treated with care, as they
require a concise treatment of nondiagonal elements in the density matrix
[25]. Thus it is helpful to disentangle theses states into a pair of localized
states. This, however, provides a nondiagonal Hamiltonian containing tunneling
matrix element between these levels.

Based on Ref. 26 this can be achieved by the following procedure: WS-levels
that are close in energy are sorted into a multiplet. The maximal energy
distance is set as gamma in resource_QCL.calcEZ with a default value of 5 meV.
Within each multiplet the matrix of the Z-operator $\hat{z}$ is calculated and
diagonalized. This provides a unitary transformation for the states within the
multiplet. As the levels in the multiplet have approximately the same energy,
the resulting diagonal elements of the Hamiltonian are rather similar, so that
the energy $E$ of the levels are reasonably well defined, suggesting the
denomination as EZ levels. The resulting levels of this procedure are shown in
Fig. 4 for the device of Ref. 16. This demonstrates the spatial splitting of
the extended states close to degeneracies.

Figure 4:  EZ levels obtained from the WS levels in Fig. 3 by unitary
transformations within multiplets of levels, which are close in energy (here
with gamma$=10$ meV). See Fig. 1 for labeling.

##  IV Conclusion

We presented a concise way to establish orthonormal sets of states for QCLs
which satisfy the periodicity criterion. The presented Python package
resource_QCL provides subroutines to establish Wannier, WS and, EZ states via
unitary transformations from the initial Bloch states of the heterostructure
without electrical potential. The respective two-component wave functions, the
matrices for the Hamiltonian in the respective basis, and the
$\hat{z}$-operator, as well as further quantities of interest are stored
within a class for further usage. An example file
example_to_use_resource_QCL.py is provided to demonstrate its usage.

The software has been tested for a large variety of QCL structures operating
either in the infrared or THz range. The included plotting routine can be used
to visualize the levels for a given structure. Equally important, the strictly
orthonormal basis states establish a firm ground to calculate scattering
matrix elements, optical transition strengths, etc, as required for quantum
kinetic calculations.

## References

  * Faist  _et al._ [1994] J. Faist, F. Capasso, D. L. Sivco, C. Sirtori, A. L. Hutchinson, and A. Y. Cho, “Quantum cascade laser,” Science 264, 553–556 (1994).
  * Faist [2013] J. Faist,  _Quantum Cascade Lasers_ (Oxford University Press, Oxford, 2013).
  * Botez and Belkin [2023] D. Botez and M. A. Belkin, eds.,  _Mid-Infrared and Terahertz Quantum Cascade Lasers_ (Cambridge University Press, 2023).
  * Jirauschek and Kubis [2014] C. Jirauschek and T. Kubis, “Modeling techniques for quantum cascade lasers,” Appl. Phys. Rev. 1, 011307 (2014).
  * Wacker [2023] A. Wacker, “Simulating quantum cascade lasers: The challenge to quantum theory,” in  _Mid-Infrared and Terahertz Quantum Cascade Lasers_, edited by D. Botez and M. A. Belkin (Cambridge University Press, 2023) p. 153â€“172.
  * Kubis  _et al._ [2009] T. Kubis, C. Yeh, P. Vogl, A. Benz, G. Fasching, and C. Deutsch, “Theory of nonequilibrium quantum transport and energy dissipation in terahertz quantum cascade lasers,” Phys. Rev. B 79, 195323 (2009).
  * Burnett  _et al._ [2018] B. A. Burnett, A. Pan, C. O. Chui, and B. S. Williams, “Robust density matrix simulation of terahertz quantum cascade lasers,” IEEE Trans. THz Sci. Techn. 8, 492–501 (2018).
  * Baranov and Teissier [2015] A. N. Baranov and R. Teissier, “Quantum cascade lasers in the InAs/AlSb material system,” IEEE Journal of Selected Topics in Quantum Electronics 21, 85–96 (2015).
  * Kato and Souma [2019] T. Kato and S. Souma, “Study of an application of non-parabolic complex band structures to the design for mid-infrared quantum cascade lasers,” Journal of Applied Physics 125, 073101 (2019).
  * Kotera [2013] N. Kotera, “Energy dependence of electron effective mass and effect of wave function confinement in a nanoscale In0.53Ga0.47As/In0.52Al0.48As quantum well,” Journal of Applied Physics 113, 234314 (2013).
  * Lee and Wacker [2002] S.-C. Lee and A. Wacker, “Nonequilibrium Green’s function theory for transport and gain properties of quantum cascade structures,” Phys. Rev. B 66, 245314 (2002).
  * Lindskog, Winge, and Wacker [2013] M. Lindskog, D. O. Winge, and A. Wacker, “Injection schemes in THz quantum cascade lasers under operation,” Proc. SPIE 8846, 884603 (2013).
  * White and Sham [1981] S. R. White and L. J. Sham, “Electronic properties of flat-band semiconductor heterostructures,” Phys. Rev. Lett. 47, 879 (1981).
  * Bastard [1981] G. Bastard, “Superlattice band structure in the envelope-function approximation,” Phys. Rev. B 24, 5693–5697 (1981).
  * Sirtori  _et al._ [1994a] C. Sirtori, F. Capasso, J. Faist, and S. Scandolo, “Nonparabolicity and a sum rule associated with bound-to-bound and bound-to-continuum intersubband transitions in quantum wells,” Phys. Rev. B 50, 8663–8674 (1994a).
  * Beck  _et al._ [2002] M. Beck, D. Hofstetter, T. Aellen, J. Faist, U. Oesterle, M. Ilegems, E. Gini, and H. Melchior, “Continuous wave operation of a mid-infrared semiconductor laser at room temperature,” Science 295, 301 (2002).
  * Note [1] The valence band component $\Psi^{q\nu}_{v}(z)$ satisfies the same condition due to Eq. (3).
  * Bruno-Alfonso and Nacbar [2007] A. Bruno-Alfonso and D. R. Nacbar, “Wannier functions of isolated bands in one-dimensional crystals,” Phys. Rev. B 75, 115428 (2007).
  * Sirtori  _et al._ [1994b] C. Sirtori, F. Capasso, D. L. Sivco, and A. Y. Cho, “Formation of new energy bands and minigap suppression by hybridization of barrier and well resonances in semiconductor superlattices,” Applied Physics Letters 64, 2982–2984 (1994b).
  * Kohn [1959] W. Kohn, “Analytic properties of Bloch waves and Wannier functions,” Phys. Rev. 115, 809–821 (1959).
  * Lu  _et al._ [2006] S. L. Lu, L. Schrottke, S. W. Teitsworth, R. Hey, and H. T. Grahn, “Formation of electric-field domains in GaAs - AlxGa1-xAs quantum cascade laser structures,” Phys. Rev. B 73, 033311 (2006).
  * Dhar  _et al._ [2014] R. S. Dhar, S. G. Razavipour, E. Dupont, C. Xu, S. Laframboise, Z. Wasilewski, Q. Hu, and D. Ban, “Direct nanoscale imaging of evolving electric field domains in quantum structures,” Sci. Rep. 4, 7183 (2014).
  * Almqvist  _et al._ [2019] T. Almqvist, D. O. Winge, E. Dupont, and A. Wacker, “Domain formation and self-sustained oscillations in quantum cascade lasers,” Eur. Phys. J. B 92, 72 (2019).
  * Wannier [1960] G. H. Wannier, “Wave functions and effective Hamiltonian for Bloch electrons in an electric field,” Phys. Rev. 117, 432–439 (1960).
  * Callebaut and Hu [2005] H. Callebaut and Q. Hu, “Importance of coherence for electron transport in terahertz quantum cascade lasers,” J. Appl. Phys. 98, 104505 (2005).
  * Rindert, Önder, and Wacker [2022] V. Rindert, E. Önder, and A. Wacker, “Analysis of high-performing terahertz quantum cascade lasers,” Phys. Rev. Applied 18, L041001 (2022).