# Quantum Configuration and Phase Spaces: Finsler and Hamilton Geometries

Saulo Albuquerque<EMAIL_ADDRESS>Physics Department, Federal
University of Paraíba, Caixa Postal 5008, 58059-900, João Pessoa, PB, Brazil.
Valdir B. Bezerra<EMAIL_ADDRESS>Physics Department, Federal
University of Paraíba, Caixa Postal 5008, 58059-900, João Pessoa, PB, Brazil.
Iarley P. Lobo<EMAIL_ADDRESS>Department of Chemistry and Physics,
Federal University of Paraíba, Rodovia BR 079 - Km 12, 58397-000 Areia-PB,
Brazil.  Physics Department, Federal University of Lavras, Caixa Postal 3037,
37200-000 Lavras-MG, Brazil.  Gabriel Macedo<EMAIL_ADDRESS>Physics Department, Federal University of Paraíba, Caixa Postal 5008,
58059-900, João Pessoa, PB, Brazil.  Pedro H. Morais<EMAIL_ADDRESS>Physics Department, Federal University of Paraíba, Caixa Postal 5008,
58059-900, João Pessoa, PB, Brazil.  Ernesto Rodrigues<EMAIL_ADDRESS>Physics Department, Federal University of Paraíba, Caixa Postal 5008,
58059-900, João Pessoa, PB, Brazil.  Luis C. N. Santos<EMAIL_ADDRESS>Physics Department, Federal University of Paraíba, Caixa Postal 5008,
58059-900, João Pessoa, PB, Brazil.  Gislaine Varão<EMAIL_ADDRESS>Physics Department, Federal University of Paraíba, Caixa Postal 5008,
58059-900, João Pessoa, PB, Brazil.

###### Abstract

In this paper, we review two approaches that can describe, in a geometrical
way, the kinematics of particles that are affected by Planck-scale departures,
named Finsler and Hamilton geometries. By relying on maps that connect the
spaces of velocities and momenta, we discuss the properties of configuration
and phase spaces induced by these two distinct geometries. In particular, we
exemplify this approach by considering the so-called $q$-de Sitter-inspired
modified dispersion relation as a laboratory for this study. We finalize with
some points that we consider as positive and negative ones of each approach
for the description of quantum configuration and phases spaces.

##  I Introduction

Since the original works by Bronstein Bronstein:2012zz  that demonstrated
uncertainty in the localization of events when geometrical degrees of freedom
are quantized, it has been argued that attempts to formulate quantum gravity
in a differentiable manifold endowed with smooth geometric quantities would
not be an interesting path to follow if one aims to pursue a fundamental
approach to this problem. Attempts in this direction have accumulated over the
years, having prominent representatives such as loop quantum gravity (LQG)
Rovelli:2008zza  and causal dynamical triangulation Ambjorn:2010rx . These
approaches to quantum gravity predict or describe several effects that should
be manifest at the Planckian regime of length and energy, such as the
discretization of geometry, which requires a language that obviously departs
from the usual Riemannian construction of general relativity. Despite the
elegance of such approaches, with current technology we are far from being
able to concretely address the regime in which such discretization would
become evident. Nevertheless, the notion that spacetime could effectively
behave like a medium formed by ”atoms of space” has led to a rich
phenomenological approach to quantum gravity, which by encoding generic
departures from relativistic equations, can describe common predictions
expected to be present at an intermediate stage between classical and quantum
gravity. Such an approach is encompassed in the area of quantum gravity
phenomenology, which addresses a myriad of effects beyond the one described in
this paragraph, as can be seen in Ref. Amelino-Camelia:2008aez , and in
particular, has found in multimessenger astronomy a fruitful environment to be
explored Addazi:2021xuf .

Usually, the regime, in which this idea is considered, is the regime, in which
the test particle approximation is valid consisting of the approximation, in
which one would have simultaneously faint gravitational and quantum effects,
described by the limits of the gravitational constant, $G\rightarrow 0$, and
the reduced Planck’s constant, $\hbar\rightarrow 0$, however, with the Planck
energy, $E_{P}=\sqrt{c^{5}\hbar/G}$, being finite, with $c$ the speed of
light. This deformed ”Minkowski limit,” which presents departures from
Minskowski spacetime’s structure has been suggested by various quantum gravity
proposals, such as the linearization of the hypersurface deformation algebra
inspired by LQG Amelino-Camelia:2016gfx ; Brahma:2016tsq ; Brahma:2018rrg  and
non-commutative geometry Majid:1994cy ; Lukierski:1991pn ; Lukierski:1992dt ;
Majid:1996kd  (for more details on this Miskowski limit, see Section 3.1.1 of
Ref. Amelino-Camelia:2008aez , and for more references on other theoretical
approaches in which such limit emerges, please refer to Section 2.2 of Ref.
Addazi:2021xuf ). It is expected that the path between the differentiable
Riemannian description of special (and general) relativity and the complete
quantum gravity theory should pass through an intermediate regime, in which
one has departures from the Riemannian character of spacetime but still has
geometric features that could describe a bottom-up phenomenology.

Furthermore, geometry plays an important role in the description of principles
that have guided the developments of relativistic theories; for example, the
principle of covariance is manifest through the use of tensorial equations of
motion, the local relativity principle is a physical manifestation of having
local equations of motion invariant under the Poincaré group (which is the
group of isometries of Minkowski space), the equivalence principle of general
relativity is manifest in the fact that the motion of free particles is
realized through geodesics, and the clock postulate can be expressed by
stating that an observer measures its proper time by the arc-length of its own
trajectory.

An important part of quantum gravity phenomenology is devoted to the question
of whether, in the aforementioned Minkowski limit, the Lorentz invariance, and
consequently, the local relativity principle, is preserved or broken due to
Planck-scale effects Amelino-Camelia:2010lsq . As is known, a length/energy
scale is not invariant under Lorentz transformations, which implies that
either a quantum gravity scale breaks the equivalence of inertial frames in
the aforementioned Minkowski limit, or the Lorentz or Poincaré group only
describes a low energy/large distance approximation of a deeper transformation
between inertial frames. The former possibility is known as a Lorentz
invariance violation (LIV) scenario Mattingly:2005re ; Liberati:2013xla , and
the latter is known as doubly (or deformed) special relativity (DSR) Amelino-
Camelia:2000stu ; Magueijo:2001cr . As the geometrization of special
relativity, due to Minkowski, paved the way to more fundamental descriptions
of nature, we shall follow a similar path, but of geometrizing DSR.

Geometric descriptions of DSR through non-commutative geometry are known
Majid:1994cy ; Lukierski:1991pn ; Lukierski:1992dt ; Majid:1996kd , but we
revise some continuous, differentiable ways of exploring non-Riemannian
degrees of freedom and the possibilities for preserving the aforementioned
principles. This way, we critically analyze two extensions of Riemannian
geometry that are capable of describing aspects of an emergent ”quantum
configuration and phase spaces” that preserve the intuition of those
principles: they are Finsler and Hamilton geometries. Finsler geometry
originally is related to the space of events and velocities (for this reason
we refer to a quantum configuration space), and Hamilton geometry originally
described the space of events and momenta (for this reason, we call it a
quantum phase space). In this paper, we revise the phenomenological
opportunities that emerge from these approaches and the interplay between
them. We also condensate the utility of each of these geometries and their
limitations in the current scenario.

We should also stress that the approaches described in this review, refer to
configuration and phase spaces probed by a single particle. The geometry
probed by a multi-particle system and its interplay with Finsler and Hamilton
languages (or even geometries that go beyond them) should still be further
explored, in which, possibly the intuition gained from the relative locality
framework Amelino-Camelia:2011lvm  would play a prominent role in this
approach.

The paper is organized as follows. Section II revisits the origin of the idea
of describing the effective spacetime probed by a particle that propagates
through a modified dispersion relation (MDR) by the proposal of rainbow
metrics.

Section III revisits how this general idea is realized by the use of Finsler
geometry in the tangent bundle, whose dual version in the cotangent bundle is
discussed in Section IV, which is illustarated by considering the curved non-
trivial case of $q$-de Sitter-inspired Finsler geometry. Section V considers
the situation of deriving the geometry of the cotangent bundle, and, in
Section VI, its dual tangent bundle formalization defined by Hamilton geometry
is considered, which is illustrated by the $q$-de Sitter case. In Section VII,
we comparatively discuss these two approaches and highlight points that we
consider as useful as well as their limitations. Finally, some important
remarks are drawn in Section VIII. Throughout the paper, a system of units
with $c=\hbar=1$ is used, so that the Planck length is the inverse of the
Planck energy: $\sqrt{G}=\ell=E_{\text{P}}^{-1}$.

##  II Preliminaries on Rainbow Geometries

As described above, over the years, the intuition that spacetime would behave
like material media, where instead of atoms of matter, one would have atoms of
spacetime, has been solidified through some approaches of quantum gravity.
Just as occurs in matter, in which one does not need to know the specific
details of the granular structure of a given medium to study the propagation
of particles through it, in spacetime, one can build phenomenology-inspired
ways of modeling how elementary particles interact with discrete gravitational
degrees of freedom while traveling through space, a so-called ”in-vaccuum
dispersion.” One could say that the most popular way of doing this is through
the assumption that particles would obey a modified dispersion relation, whose
corrections are given perturbatively by powers of the quantum gravity scale,
which we could assume as being in the order of Planck units. The dispersion
relation furnishes the group velocity of waves and defines the trajectory that
on-shell particles follow from the Hamilton equations. Actually, when the
interplay between the presence of amplifiers of observables and the
uncertainties of observations allows us to constrain this parameter at a level
close to its Planckian version, we say that we are at Planck-scale sensitivity
Amelino-Camelia:2008aez .

Such behavior also happens in meta-materials proutorov2018finsler , in which
it is possible to describe the motion of particles through it by geodesics in
a given geometry; it also appears in the motion of a charged particle in a
pre-metric formulation of electromagnetism hehl2003foundations , in the
description of seismic waves vcerveny2002fermat , etc.; for a review, see Ref.
Pfeifer:2019wus . Additionally, one could wonder if the motion of particles,
determined by Planck-scale modified dispersion relations, could also be
described by geodesics of a non-Riemannian geometry. Besides, the dispersion
relation itself is usually determined by the norm of the $4$-momentum measured
by a Riemannian metric, which also determines the symmetries observed by
measurements in that spacetime.

This intuition was early realized by the so-called ”rainbow geometries”
Magueijo:2002xx , idealized by João Magueijo and Lee Smolin which aimed to
extend the DSR formulation proposed by them in Ref. Magueijo:2001cr  to curved
spacetimes. In that case, the way found to express local modified dispersion
relations through a norm, consisted in absorbing functions of the particle’s
energy divided by Planck energy, $\epsilon=E/E_{\text{P}}$, such as
$f(\epsilon)$ and $g(\epsilon)$, which would appear in the MDR that follows:

$m^{2}=f^{2}(\epsilon)E^{2}-g^{2}(\epsilon)|\vec{p}|^{2}\,,$ (1)

(with the three-momentum $\vec{p}$) into the definition of new spacetime
tetrads, $\tilde{e}_{(0)}^{\quad\mu}(\epsilon)=f(\epsilon)e_{(0)}^{\quad\mu}$
and $\tilde{e}_{(I)}^{\quad\mu}(\epsilon)=g(\epsilon)e_{(I)}^{\quad\mu}$, such
that the MDR reads

$m^{2}=\eta^{AB}\tilde{e}_{(A)}^{\quad\mu}\tilde{e}_{(B)}^{\quad\nu}p_{\mu}p_{\nu}=\tilde{g}^{\mu\nu}(\epsilon)p_{\mu}p_{\nu}\,,$
(2)

where
$g^{\mu\nu}(\epsilon)=\eta^{AB}\tilde{e}_{(A)}^{\quad\mu}\tilde{e}_{(B)}^{\quad\nu}$
is the rainbow metric, $\eta^{AB}$ is the Minkowski metric $\text{diag}(+---)$
, Greek letters denote four-dimensional indices and take on the values 0
(time) 1, 2, and 3 (space), low-case Latin letters denote the space indices,
and $p_{\mu}$ is the 4-momentum. This description would imply that when an
observer uses the motion of a particle with energy $E$ to probe spacetime,
then the line element assigned to that spacetime is the following:

$ds^{2}=\tilde{g}_{\mu\nu}dx^{\mu}dx^{\nu}=\frac{g_{00}}{f^{2}(\epsilon)}(dx^{0})^{2}+\frac{g_{ij}}{g^{2}(\epsilon)}dx^{i}dx^{j}+2\frac{g_{0i}}{f(\epsilon)g(\epsilon)}dx^{0}dx^{i}\,,$
(3)

where $g_{\mu\nu}$ is the metric found from undeformed tetrads. Thus, in a
nutshell, one identifies the rainbow functions, $f$ and $g$, from a MDR that
is usually inspired by fundamental theories of quantum gravity or by
phenomenological intuition; then, one uses $\tilde{g}_{\mu\nu}$ as an input
into the classical gravitational field equations. Considering modifications of
the stress-energy tensor due to the rainbow functions, one derives what should
be $g_{\mu\nu}$ (since $f$ and $g$ are determined a priori). Usually, this
procedure gives that $g_{\mu\nu}$ is the Riemannian metric found from the
usual gravitational field equations. Therefore, this approach gives basically
the usual metric components of a given theory, just modified by factors of the
rainbow functions as in Equation (3).

Effective energy-dependent spacetimes have emerged in different approaches to
the description of the quantization of gravitational/geometric degrees of
freedom Weinfurtner:2008if ; Assanioussi:2014xmz ; Olmo:2011sw . Along this
line of research, Magueijo-Smolin’s proposal has been applied in a myriad of
contexts, such as in black hole physics Ling:2005bp ; Lobo:2021bag , cosmology
Gorji:2016laj , wormholes Garattini:2015pmo ; Amirabi:2018ncf , cosmic strings
Bezerra:2019vrz , disformal geometries Carvalho:2015omv ; Lobo:2017bfh , and
electrostatic self-interaction of charged particles Santos:2019 . However,
despite its range of applicability and utility in furnishing intuition about
extreme scenarios, this approach presents some conceptual and technical
limitations that seem unavoidable, such as the lack of a rigorous mathematical
framework in which this idea is formulated or the imposition of a preferred
vielbein in which the particle’s energy is measured, which seems in
contradiction with the local DSR intention of this proposal. As shown below,
the solution to these problems is actually coincident, and the search for a
rigorous mathematical formulation for these ideas will be responsible for
giving a framework, in which proper physical questions can be answered and
novel phenomenological opportunities to born. The main issue here is what is
the proper formulation of a geometry that should not only depend on spacetime
points, but also should carry energy dependence of the particle itself that
probes this spacetime. This paper deals with the two main proposals—Finsler
and Hamilton geometries— solving some of the raised problems and also
discusses limitations on their owns.

##  III Geometry of the Tangent Bundle: Finsler Geometry

The 1854 Habilitation Dissertation by Bernhard Riemann presents the germ of
the idea behind what would later be called Finsler geometry. In the second
part of the dissertation, it is said (see Ref. riemann , p. 35):

> ”For Space, when the position of points is expressed by rectilinear co-
> ordinates, $ds=\sqrt{\sum(dx)^{2}}$; Space is therefore, included in this
> simplest case. The next case in simplicity includes those manifoldnesses in
> which the line-element may be expressed as the fourth root of a quartic
> differential expression. The investigation of this more general kind would
> require no really different principles, but would take considerable time and
> throw little new light on the theory of space, especially as the results
> cannot be geometrically expressed; I restrict myself, therefore, to those
> manifoldnesses in which the line-element is expressed as the square root of
> a quadric differential expression.”

The exploration of such more general cases of line elements will be done only
64 years later, in 1918, in the Ph.D. thesis of Paul Finsler book:1130902 ,
where at least from the metric point of view, the distance between points is
(”distance”), please confirm. measured by a 1-homogeneous function
(homogeneous with the degree of 1) Such a metric tensor would be defined in
the tangent bundle of the base manifold, since it would depend not only on the
manifold points, but also on a direction, which is a manifestation of the non-
Pythagorean nature of this space. Later on, the issue of non-linear
connections was further developed and incorporated as a fundamental structure
for the dynamical description of Finsler spaces (for a historical perspective
on Finsler geometry, we refer the reader to the Preface of Ref.
bao2000introduction  and references therein). The case of pseudo-Finsler
geometries, as an arena for describing spacetime, has been recently discussed
Hohmann:2021zbt ; Bernal:2020bul , where, for instance, different definitions
are presented and important theorems regarding its causal structure among
other issues are being derived Minguzzi:2014aua .

In Section II, a glimpse of the non-Riemannian nature of spacetime was
notified emerging as a manifestation of the quantization of gravitational
degrees of freedom. Actually, as one can anticipate, the non-quadratic, i.e.,
non-Pythagorean nature of a dispersion relation is connected to a possible
Finslerian nature of spacetime through an intermediate step that connects the
kinematics of particles in a Hamiltonian to a Lagrangian formulation.
Actually, the MDR corresponds to a Hamiltonian constraint, which physical
particles supposedly obey, the way that the trajectories of free particles,
induced by such a deformed Hamiltonian, capture the propagation of a particle
through a quantized spacetime. For this reason, the Helmholtz action,
associated with such a particle, is naturally given by the functional,

$S[x,p,\lambda]=\int d\mu(\dot{x}^{\alpha}p_{\alpha}-\lambda f(H(x,p),m))\,,$
(4)

where the dot denotes differentiation with respect to the parameter $\mu$,
$p_{\mu}$ is the particle’s momenta, $f$ is a function that is null if the
dispersion relation is satisfied, namely, $H(x,p)=m$, and $\lambda$ is a
Lagrange multiplier. This is a premetric formulation that is actually defined
in the space $T^{*}M\times{\mathbb{R}}$, where $T^{*}M$ is the phase space of
analytical mechanics or cotangent bundle. In order to find an arc-length, and
consequently, a geometric structure, one needs to calculate an equivalent
Lagrangian defined in the configuration space or the tangent bundle $TM$
described by points and velocities (such an observation was firstly presented
in Ref. Girelli:2006fw ). The algorithm for doing so is as follows
Lobo:2020qoa :

  1. 1.

variation with respect to $\lambda$ enforces the dispersion relation;

  2. 2.

variation with respect to $p_{\mu}$ yields an equation
$\dot{x}^{\mu}=\dot{x}^{\mu}(p,\lambda)$, which must be inverted to obtain
$p_{\mu}(x,\dot{x},\lambda)$ to eliminate the momenta $p_{\mu}$ from the
action;

  3. 3.

using $p_{\mu}(x,\dot{x},\lambda)$ in the dispersion relation, one can solve
for $\lambda(x,\dot{x})$; and

  4. 4.

finally, the desired length measure is obtained as
$S[x]=S[x,p(x,\dot{x},\lambda(x,\dot{x})),\lambda(x,\dot{x})]_{H}$.

This is a Legendre transformation, whose conditions of existence and
capability of providing a physical framework are discussed in Refs.
Raetzel:2010je ; Rodrigues:2022mfj . These formal conditions are always
guaranteed when one considers deformations at the perturbative level. This is
crucial because the following algorithm cannot be applied in practice if it is
not possible to invert the velocity function to find the momenta as a function
of the other variables. In general, this cannot be done, especially for
complicated dispersion relations, such as those that depend on sums of
hyperbolic functions Gubitosi:2011hgc . Anyway, since quantum gravity
phenomenology is usually concerned with first order effects, which are those
attainable by experiments nowadays, we shall concentrate on the perturbative
level in order to derive our conclusions.

For example, if this algorithm is applied to a Hamiltonian of the form,

$H(x,p)=g(p,p)+\varepsilon h(x,p)\,,$ (5)

where $g(p,p)=g^{ab}(x)p_{a}p_{b}$ is an undeformed dispersion relation,
$h(x,p)$ is a function of spacetime points and momenta that depends on the
model under consideration, and $\varepsilon$ is the perturbation parameter
that is usually a function of the energy scale of the deformation (such as the
Planck or quantum gravity length scale). As shown in Ref. Lobo:2020qoa , after
the Legendre transformation, the equivalent action takes the form,

$S[x]=m\int
d\mu\sqrt{g(\dot{x},\dot{x})}\left(1-\varepsilon\frac{h(x,\bar{p}(x,\dot{x}))}{2m^{2}}\right)\,,$
(6)

where $\bar{p}_{a}(x,\dot{x})=m{\dot{x}_{a}}/{\sqrt{g(\dot{x},\dot{x})}}$. In
particular, when $h$ is a polynomial function of momenta as (the index is
shifted: $n\rightarrow n+2$, in comparison with Ref. Lobo:2020qoa , such that
now $n$ corresponds to the power of Planck length in the MDR),

$h(x,p)=h^{\mu_{1}\mu_{2}....\mu_{n+2}}(x)p_{\mu_{1}}p_{\mu_{2}}...p_{\mu_{n+2}}\,,$
(7)

and $\varepsilon=\ell^{n}$, one finds an action of the form,

$S[x]=m\int d\mu\sqrt{g(\dot{x},\dot{x})}\left(1-(\ell
m)^{n}\frac{h_{\mu_{1}\mu_{2}....\mu_{n+2}}(x)\dot{x}^{\mu_{1}}\dot{x}^{\mu_{2}}...\dot{x}^{\mu_{n+2}}}{2g(\dot{x},\dot{x})^{\frac{n+2}{2}}}\right)\,,$
(8)

where we lowered the indices of $h$ with the components of $g$. The connection
between the mechanics of free particle and geometry takes place when the above
expression is identified with the arc-length functional, $s[x]$, of a given
geometry, i.e., $s[x]=S[x]/m$. Such an identification makes sense if we want
to state that the trajectories of free particles are extremizing curves or
geodesics in a given geometry, it is related to the preservation of the
equivalence principle even in this Planck-scale deformed scenario.

In this case, the spacetime in which a particle propagates by a MDR is
described by an arc-length functional that generalizes the one of Riemannian
geometry and is given by a function $F(x,\dot{x})$ that is 1-homogeneous in
the velocity $\dot{x}$, such that the arc-length is indeed parametrization
invariant, as it must be:

$s[x]=\int F(x,\dot{x})d\mu\,.$ (9)

Actually, this is the kind of scenario envisaged by Riemann in his
dissertation, and explored by Finsler, that emerges here quite naturally.
There are some definitions of a pseudo-Finsler spacetime in the literature,
but we rely on that given in Ref. Hohmann:2021zbt  (the differences in
comparison to other definitions are discussed in Ref. Hohmann:2021zbt ). First
of all, we are going to work with a smooth manifold, $M$, endowed with a real
valued positive function $L$ that takes values on the tangent bundle $TM$,
described by coordinates $(x,y)$, where $\\{x^{\mu}\\}$ are spacetime
coordinates and $\\{y^{\mu}\\}$ refer to vector or velocity coordinates.
Actually, we shall need the slit tangent bundle $\widetilde{TM}=TM/\\{0\\}$,
in which we remove the zero section, and we also need the projection
$\pi:TM\rightarrow M$. A conic subbundle is a submanifold ${\cal
D}\subset\widetilde{TM}$ such that $\pi({\cal D})=M$ and with the conic
property that states that if $(x,y)\in{\cal D}\Rightarrow(x,\lambda y)\in{\cal
D},\,\forall\lambda>0$.

In a nutshell, a Finsler spacetime is a triple $(M,{\cal D},L)$, where
$L:{\cal D}\rightarrow{\mathbb{R}}$ is a smooth function satisfying the
conditions:

  1. 1.

positive 2-homogeneity: $L(x,\alpha y)=\alpha^{2}L(x,y),\,\forall\alpha>0$;

  2. 2.

at any $(x,y)\in{\cal D}$ and in any chart of $\widetilde{TM}$, the following
Hessian (metric) is non-degenerate:

$g_{\mu\nu}(x,y)=\frac{1}{2}\frac{\partial^{2}}{\partial y^{\mu}\partial
y^{\nu}}L(x,y)\,;$ (10)

  3. 3.

the metric $g_{\mu\nu}$ has a Lorentzian signature.

The function $L$ is actually the square of the Finsler function,
$L(x,y)=F^{2}(x,y)$, and from it the Finsler arc-length is defined as given in
Equation (9). Condition $1$ above guarantees that Equation (9) does not depend
on the parametrization used to describe the curve and that using Euler’s
theorem for homogeneous functions, this expression can be cast as

$s[x]=\int\sqrt{g_{\mu\nu}(x,\dot{x})\dot{x}^{\mu}\dot{x}^{\nu}}d\mu\,.$ (11)

From a coordinate transformation,

$\displaystyle\tilde{x}^{\mu}=\tilde{x}^{\mu}(x)\,,$ (12)
$\displaystyle\tilde{y}^{\mu}=\frac{\partial\tilde{x}^{\mu}}{\partial
x^{\nu}}y^{\nu}\,,$ (13)

the functions $g_{\mu\nu}$ transform according to

$\tilde{g}_{\mu\nu}(\tilde{x},\tilde{y})=\frac{\partial
x^{\alpha}}{\partial\tilde{x}^{\mu}}\frac{\partial
x^{\beta}}{\partial\tilde{x}^{\nu}}g_{\alpha\beta}(x,y)\,.$ (14)

Due the property 14, $g_{\mu\nu}$ is referred here as the components of a
distinguished tensor field (or $d$-tensor field) on the manifold
$\widetilde{TM}$, which follows the notation adopted in Ref. Miron:1994nvt .
The extremization of the arc-lenght functional (9) gives the following
geodesic equation,

$\frac{d^{2}x^{\mu}}{d\mu^{2}}+2G^{\mu}(x,\dot{x})=2\frac{dF}{d\mu}\frac{\partial
F}{\partial\dot{x}^{\mu}}\,,$ (15)

where $G^{\mu}=G^{\mu}(x,\dot{x})$ are the spray coefficients spray-finsler
and are given in terms of the Christoffel symbols, $\gamma^{\alpha}_{\mu\nu}$,
of the metric $g_{\mu\nu}$:

$\displaystyle
G^{\alpha}(x,\dot{x})=\frac{1}{2}\gamma^{\alpha}_{\mu\nu}(x,\dot{x})\dot{x}^{\mu}\dot{x}^{\nu}\,,$
(16)
$\displaystyle\gamma^{\alpha}_{\mu\nu}(x,\dot{x})=\frac{1}{2}g^{\alpha\beta}\left(\frac{\partial
g_{\mu\beta}}{\partial x^{\nu}}+\frac{\partial g_{\nu\beta}}{\partial
x^{\mu}}-\frac{\partial g_{\mu\nu}}{\partial x^{\beta}}\right)\,.$ (17)

If we choose the arc-length parametrization, i.e., the one in which $F=1$, we
have a sourceless geodesic equation. This expression means that the
trajectories generated by a MDR of the form $H(x,\dot{x})=m^{2}$ are,
actually, geodesics of a Finsler metric. The presence of spray coefficients
allows us to construct another quite a useful quantity, the so-called Cartan
non-linear connection, given by (in this paper, we interchange the notation
$\dot{x}\leftrightarrow y$ freely)

$N^{\mu}{}_{\nu}(x,y)=\frac{\partial}{\partial y^{\nu}}G^{\mu}(x,y)\,,$ (18)

that transforms according to

$\tilde{N}^{\mu}{}_{\nu}=\frac{\partial\tilde{x}^{\mu}}{\partial
x^{\alpha}}\frac{\partial
x^{\beta}}{\partial\tilde{x}^{\nu}}N^{\alpha}{}_{\beta}-\frac{\partial^{2}\tilde{x}^{\mu}}{\partial
x^{\alpha}\partial x^{\beta}}\frac{\partial
x^{\beta}}{\partial\tilde{x}^{\nu}}y^{\alpha}\,.$ (19)

The introduction of this quantity allows us to introduce a useful basis of the
tangent space of the tangent bundle at each point. In fact, since according to
the coordinate transformation (12) and (13), the usual coordinate basis
transforms as

$\displaystyle\frac{\partial}{\partial\tilde{x}^{\mu}}=\frac{\partial
x^{\nu}}{\partial\tilde{x}^{\mu}}\frac{\partial}{\partial
x^{\nu}}+\frac{\partial^{2}x^{\nu}}{\partial\tilde{x}^{\mu}\partial\tilde{x}^{\alpha}}\frac{\partial\tilde{x}^{\alpha}}{\partial
x^{\beta}}y^{\beta}\frac{\partial}{\partial y^{\nu}}\,,$ (20)
$\displaystyle\frac{\partial}{\partial\tilde{y}^{\mu}}=\frac{\partial
y^{\nu}}{\partial\tilde{y}^{\mu}}\frac{\partial}{\partial y^{\nu}}\,.$ (21)

In addition, a non-linear connection allows us to define the following frame:

$\displaystyle\frac{\delta}{\delta
x^{\mu}}=\delta_{\mu}=\frac{\partial}{\partial
x^{\mu}}-N^{\nu}{}_{\mu}\frac{\partial}{\partial y^{\nu}}\,,$ (22)
$\displaystyle\dot{\partial}_{\mu}=\frac{\partial}{\partial y^{\mu}}\,.$ (23)

Due to the transformation properties of the non-linear connection, this basis
transforms as

$\displaystyle\tilde{\delta}_{\mu}=\frac{\partial
x^{\nu}}{\partial\tilde{x}^{\mu}}\delta_{\nu}\,,$ (24)
$\displaystyle\tilde{\dot{\partial}}_{\mu}=\frac{\partial
x^{\nu}}{\partial\tilde{x}^{\mu}}\dot{\partial}_{\nu}\,.$ (25)

This means that one is able to split the tangent space of the tangent bundle
into horizontal, $HTM=\text{span}\\{\delta_{\mu}\\}$, and vertical,
$VTM=\text{span}\\{\dot{\partial}_{\mu}\\}$, spaces, such that
$T\widetilde{TM}=HTM\oplus VTM$ in each point $(x,y)$. Similarly, the same
reasoning applies to the cotangent space; i.e., we split
$T^{*}\widetilde{TM}=H^{*}TM\oplus V^{*}TM$ spanned as
$H^{*}TM=\text{span}\\{dx^{\mu}\\}$ and $V^{*}TM=\text{span}\\{\delta
y^{\mu}\\}$, where

$\displaystyle\delta y^{\mu}=dy^{\mu}+N^{\mu}{}_{\nu}dx^{\nu}\,,$ (26)

which transforms as

$\displaystyle d\tilde{x}^{\mu}=\frac{\partial\tilde{x}^{\mu}}{\partial
x^{\nu}}dx^{\nu}\,,$ (27)
$\displaystyle\delta\tilde{y}^{\mu}=\frac{\partial\tilde{x}^{\mu}}{\partial
x^{\nu}}\delta y^{\nu}\,.$ (28)

Such a decomposition of the tangent and cotangent vector spaces implies that a
vector $X$ and a $1$-form $\omega$ with horizontal and vertical terms can read
as

$\displaystyle
X=X^{\mu}\delta_{\mu}+\dot{X}^{\mu}\dot{\partial}_{\mu}=X^{H}+X^{V}\,,$ (29)
$\displaystyle\omega=\omega_{\mu}dx^{\mu}+\dot{\omega}_{\mu}\delta
y^{\mu}=\omega^{H}+\omega^{V}\,.$ (30)

Endowed with this basis, the metric $\mathbb{G}(x,y)$ of the configuration
space is described by the so-called Sasaki-Matsumoto lift of the metric
$g_{\mu\nu}$:

$\mathbb{G}(x,y)=g_{\mu\nu}(x,y)dx^{\mu}\otimes dx^{\nu}+g_{\mu\nu}(x,y)\delta
y^{\mu}\otimes\delta y^{\nu}\,.$ (31)

###### Definition III.1

A tensor field $T$ of type $(m+n,p+q)$ on the manifold $\widetilde{TM}$ is
called a distinguished tensor field (or $d$-tensor field) if it has the
property

$\displaystyle
T\left(\overset{1}{\omega},...,\overset{m}{\omega},\overset{1}{\tau},...,\overset{n}{\tau},\underset{1}{X},...,\underset{p}{X},\underset{1}{Y},...,\underset{q}{Y}\right)=T\left(\overset{1}{\omega}{}^{H},...,\overset{m}{\omega}{}^{H},\overset{1}{\tau}{}^{V},...,\overset{n}{\tau}{}^{V},\underset{1}{X}{}^{H},...,\underset{p}{X}{}^{H},\underset{1}{Y}{}^{V},...,\underset{q}{Y}{}^{V}\right).$
(32)

This definition implies that one can write a $d$-tensor $T$ in the preferred
frame as

$\displaystyle
T=T^{\mu_{1}...\mu_{m}\nu_{1}...\nu_{n}}{}_{\alpha_{1}...\alpha_{p}\beta_{1}...\beta_{q}}\frac{\delta}{\delta
x^{\mu_{1}}}$ $\displaystyle\otimes...\otimes\frac{\delta}{\delta
x^{\mu_{m}}}\otimes\frac{\partial}{\partial
y^{\nu_{1}}}\otimes...\otimes\frac{\partial}{\partial y^{\nu_{n}}}$
$\displaystyle\otimes dx^{\alpha_{1}}\otimes...\otimes
dx^{\alpha_{p}}\otimes\delta y^{\beta_{1}}\otimes...\otimes\delta
y^{\beta_{q}}\,,$ (33)

and that it transforms according to the rule,

$\displaystyle\widetilde{T}^{\mu_{1}...\mu_{m}\nu_{1}...\nu_{n}}{}_{\alpha_{1}...\alpha_{p}\beta_{1}...\beta_{q}}$
(34) $\displaystyle=\frac{\partial\tilde{x}^{\mu_{1}}}{\partial
x^{\epsilon_{1}}}...\frac{\partial\tilde{x}^{\mu_{m}}}{\partial
x^{\epsilon_{m}}}\frac{\partial\tilde{x}^{\nu_{1}}}{\partial
x^{\lambda_{1}}}...\frac{\partial\tilde{x}^{\nu_{n}}}{\partial
x^{\lambda_{n}}}\frac{\partial
x^{\gamma_{1}}}{\partial\tilde{x}^{\alpha_{1}}}...\frac{\partial
x^{\gamma_{p}}}{\partial\tilde{x}^{\alpha_{p}}}\frac{\partial
x^{\rho_{1}}}{\partial\tilde{x}^{\beta_{1}}}...\frac{\partial
x^{\rho_{q}}}{\partial\tilde{x}^{\beta_{q}}}T^{\epsilon_{1}..\epsilon_{n}\lambda_{1}...\lambda_{m}}{}_{\gamma_{1}...\gamma_{p}\rho_{1}...\rho_{q}}\,.$

An example of $d$-tensor field is the metric whose components are given by
Equation (14).

###  III.1 $N$-Linear Connection

Given a linear connection, $D$, on the manifold $\widetilde{TM}$, if it
preserves the parallelism of the horizontal and vertical spaces, i.e., if it
can be written as

$D_{\delta_{\nu}}\delta_{\mu}=L^{\alpha}_{\mu\nu}\delta_{\alpha}\,,\qquad
D_{\delta_{\nu}}\dot{\partial}_{\alpha}=L^{\mu}_{\alpha\nu}\dot{\partial}_{\mu}\,,$
(35)
$D_{\dot{\partial}_{\nu}}\delta_{\mu}=C^{\alpha}_{\mu\nu}\delta_{\alpha}\,,\qquad
D_{\dot{\partial}_{\nu}}\dot{\partial}_{\mu}=C^{\alpha}_{\mu\nu}\dot{\partial}_{\alpha}\,,$
(36)

then is called an $N$-linear connection. Let us consider a coordinate change;
thus, the coefficients (35) and (36) transform as

$\displaystyle\tilde{L}^{\alpha}_{\mu\nu}=\frac{\partial\tilde{x}^{\alpha}}{\partial
x^{\beta}}\frac{\partial x^{\lambda}}{\partial\tilde{x}^{\mu}}\frac{\partial
x^{\epsilon}}{\partial\tilde{x}^{\nu}}L^{\beta}_{\lambda\epsilon}+\frac{\partial^{2}x^{\beta}}{\partial\tilde{x}^{\mu}\partial\tilde{x}^{\nu}}\frac{\partial\tilde{x}^{\alpha}}{\partial
x^{\beta}}\,,$ (37)
$\displaystyle\tilde{C}^{\alpha}_{\mu\nu}=\frac{\partial\tilde{x}^{\alpha}}{\partial
x^{\beta}}\frac{\partial x^{\lambda}}{\partial\tilde{x}^{\mu}}\frac{\partial
x^{\epsilon}}{\partial\tilde{x}^{\nu}}C^{\beta}_{\lambda\epsilon}\,.$ (38)

Endowed with these coefficients, the derivative of a $d$-tensor can be
decomposed into a horizontal and a vertical parts, such that one can apply the
covariant derivative of a tensor $T$ of type $(m+n,p+q)$ in the direction of a
vector $X$ as a direction of a vector $X$ as

$\displaystyle D_{X}T=D_{X^{H}}T+D_{X^{V}}T$
$\displaystyle=\left(T^{\mu_{1}...\mu_{m}\nu_{1}...\nu_{n}}{}_{\alpha_{1}...\alpha_{p}\beta_{1}...\beta_{q}|\epsilon}X^{\epsilon}+T^{\mu_{1}...\mu_{m}\nu_{1}...\nu_{n}}{}_{\alpha_{1}...\alpha_{p}\beta_{1}...\beta_{q}||\epsilon}\dot{X}^{\epsilon}\right)\frac{\delta}{\delta
x^{\mu_{1}}}\otimes...\otimes\frac{\delta}{\delta x^{\mu_{m}}}$
$\displaystyle\otimes\frac{\partial}{\partial
y^{\nu_{1}}}\otimes...\otimes\frac{\partial}{\partial y^{\nu_{n}}}\otimes
dx^{\alpha_{1}}\otimes...\otimes dx^{\alpha_{p}}\otimes\delta
y^{\beta_{1}}\otimes...\otimes\delta y^{\beta_{q}}\,,$ (39)

where

$\displaystyle
T^{\mu_{1}...\mu_{m}\nu_{1}...\nu_{n}}{}_{\alpha_{1}...\alpha_{p}\beta_{1}...\beta_{q}|\epsilon}$
(40) $\displaystyle=\frac{\delta}{\delta
x^{\epsilon}}T^{\mu_{1}...\mu_{m}\nu_{1}...\nu_{n}}{}_{\alpha_{1}...\alpha_{p}\beta_{1}...\beta_{q}}+L^{\mu_{1}}_{\gamma\epsilon}T^{\gamma...\mu_{m}\nu_{1}...\nu_{n}}{}_{\alpha_{1}...\alpha_{p}\beta_{1}...\beta_{q}}+...-L^{\gamma}_{\alpha_{1}\epsilon}T^{\mu_{1}...\mu_{m}\nu_{1}...\nu_{n}}{}_{\gamma...\alpha_{p}\beta_{1}...\beta_{q}}\,,$
$\displaystyle
T^{\mu_{1}...\mu_{m}\nu_{1}...\nu_{n}}{}_{\alpha_{1}...\alpha_{p}\beta_{1}...\beta_{q}||\epsilon}$
(41) $\displaystyle=\frac{\partial}{\partial
y^{\epsilon}}T^{\mu_{1}...\mu_{m}\nu_{1}...\nu_{n}}{}_{\alpha_{1}...\alpha_{p}\beta_{1}...\beta_{q}}+C^{\mu_{1}}_{\gamma\epsilon}T^{\gamma...\mu_{m}\nu_{1}...\nu_{n}}{}_{\alpha_{1}...\alpha_{p}\beta_{1}...\beta_{q}}+...-C^{\gamma}_{\alpha_{1}\epsilon}T^{\mu_{1}...\mu_{m}\nu_{1}...\nu_{n}}{}_{\gamma...\alpha_{p}\beta_{1}...\beta_{q}}\,,$

and the property that the covariant derivative is linear in the direction $X$
is used. The triple $D\Gamma(N,L,C)$ describes the parallel transport and
decomposition of the tangent and cotangent spaces of the tangent bundle into
horizontal and vertical spaces. At this point, we need to comment on some
remarkable $N$-linear connections that are considered in the literature.

The first connection is the metrical Cartan connection,
$C\Gamma(N^{\mu}{}_{\nu},L^{\alpha}_{\mu\nu},C^{\alpha}_{\mu\nu})$. In this
case, $N^{\mu}{}_{\nu}$ is given by the canonical Cartan non-linear
connection, defined by the spray coefficients (18). The coefficients
$L^{\alpha}_{\mu\nu}$ and $C^{\alpha}_{\mu\nu}$ are given, respectively, by

$\displaystyle
L^{\alpha}_{\mu\nu}=\frac{1}{2}g^{\alpha\beta}\left(\frac{\delta
g_{\mu\beta}}{\delta x^{\nu}}+\frac{\delta g_{\nu\beta}}{\delta
x^{\mu}}-\frac{\delta g_{\mu\nu}}{\delta x^{\beta}}\right)\,,$ (42)
$\displaystyle
C^{\alpha}_{\mu\nu}=\frac{1}{2}g^{\alpha\beta}\left(\frac{\delta
g_{\mu\beta}}{\delta y^{\nu}}+\frac{\delta g_{\nu\beta}}{\delta
y^{\mu}}-\frac{\delta g_{\mu\nu}}{\delta y^{\beta}}\right)\,.$ (43)

This connection is metrical (i.e., without non-metricity tensors) considering
both horizontal and vertical covariant derivatives of the Finsler metric.

Besides, the Berwald connection is given by the triple
$B\Gamma(N^{\mu}{}_{\nu},\partial N^{\alpha}{}_{\mu}/\partial y^{\nu},0)$ and
presents horizontal and vertical non-metricities. The Chern–Rund connection,
$R\Gamma(N^{\mu}{}_{\nu},L^{\alpha}_{\mu\nu},0)$, is horizontally metrical,
but represents vertical non-metricity. Additionally, the Hashiguchi
connection, $H\Gamma(N^{\mu}{}_{\nu},\partial N^{\alpha}{}_{\mu}/\partial
y^{\nu},C^{\alpha}_{\mu\nu})$, represents horizontal non-metricity, but it is
vertically metrical. In these expressions, $N$ is the canonical Cartan non-
linear connection (18), $L$ is given by Equation (42), and $C$ is given by
Equation $\eqref{c-cartan}$.

###  III.2 Symmetries

Geometrical language naturally realizes the concept of symmetry of physical
equations. General relativity given in terms of Riemannian geometry
encompasses the invariance under general coordinate transformations, and the
isometries of the Minkowski space describe the Poincaré transformations
(actually, one can further apply this technique for maximally symmetric
spaces, including de Sitter and anti-de Sitter ones). Finsler geometry, as we
have been using, allows us to go beyond this scope and to define deformed
Lorentz/Poincaré transformations that present Planck scale corrections even in
the presence of a local modified dispersion relation. One can see how this
will naturally emerge, since the invariance of the arc-length (9) is
compatible with the invariance of the action in the Hamiltonian formulation
(4), from which such an arc-length was derived. This idea was firstly noticed
in Ref. Girelli:2006fw  and later explicitly explored in Refs. Amelino-
Camelia:2014rga ; Lobo:2016xzq . The master equation for this purpose is the
one that follows from the invariance of the Finslerian interval $ds^{2}$, as
done in Appendix A of Ref. Amelino-Camelia:2014rga . From this invariance, the
Finslerian killing equation for the killing vector was found, with components
$\xi^{\alpha}$, which should be solved in order to derive the deformed
symmetries in the DSR context,

$\xi^{\alpha}\partial_{\alpha}g_{\mu\nu}+g_{\alpha\nu}\partial_{\mu}\xi^{\alpha}+g_{\mu\alpha}\partial_{\nu}\xi^{\alpha}+y^{\alpha}\partial_{\alpha}\xi^{\beta}\dot{\partial}_{\beta}g_{\mu\nu}=0\,.$
(44)

###  III.3 Finsler–q-de Sitter (Tangent Bundle Case)

As an example that presents a non-trivial non-linear connection, we shall
consider the case of a Finsler geometry inspired by the so-called $q$-de
Sitter deformed relativity. This case has been previously studied in the
literature, e.g., in Refs. Barcaroli:2015eqe ; Letizia:2016lew ; Lobo:2016xzq
; Lobo:2016lxm , and can be described by an algebra that deforms the one of
Poincaré in a way that gives the de Sitter symmetry when a quantum gravity
parameter goes to zero, and on the other hand, gives the so-called
$\kappa$-Poincaré algebra (that deforms the Poincaré one by an energy scale
parameter, supposedly the Planck energy) when the de Sitter curvature
parameter goes to zero. Therefore, it corresponds to an authentic realization
of a deformed relativity scenario, even in the presence of what can be
interpreted as spacetime curvature. In this subsection, we initially consider
results that were originally presented in Ref. Letizia:2016lew  in $1+1$
dimensions.

The MDR related to this algebra (in a given basis) can be perturbed to first
order in the Planck length and de Sitter curvature parameters $\ell$ and $H$,
respectively, as

${\cal H}(x,p)=p_{0}^{2}-p_{1}^{2}(1+\ell p_{0})(1-2Hx^{0})\,.$ (45)

By using the action given by Equation (4) and the algorithm that follows it,
the following Finsler function can be obtained:

$F(x,\dot{x})=\sqrt{(\dot{x}^{0})^{2}-(1-2Hx^{0})(\dot{x}^{1})^{2}}+\ell\frac{m}{2}\frac{(1-2Hx^{0})\dot{x}^{0}(\dot{x}^{1})^{2}}{(\dot{x}^{0})^{2}-(1-2Hx^{0})(\dot{x}^{1})^{2}}\,,$
(46)

from which the Finsler metric can be found from Equation (10):

$\displaystyle
g_{\mu\nu}^{F}(x,\dot{x})=\begin{pmatrix}1+\frac{3a^{4}m\ell\dot{x}^{0}(\dot{x}^{1})^{4}}{2[(\dot{x}^{0})^{2}-a^{2}(\dot{x}^{1})^{2}]^{5/2}}&\frac{m\ell
a^{4}(\dot{x}^{1})^{3}[a^{2}(\dot{x}^{1})^{2}-4(\dot{x}^{0})^{2}]}{2[(\dot{x}^{0})^{2}-a^{2}(\dot{x}^{1})^{2}]^{5/2}}\\\
\frac{m\ell
a^{4}(\dot{x}^{1})^{3}[a^{2}(\dot{x}^{1})^{2}-4(\dot{x}^{0})^{2}]}{2[(\dot{x}^{0})^{2}-a^{2}(\dot{x}^{1})^{2}]^{5/2}}&-a^{2}+\frac{m\ell
a^{2}(\dot{x}^{0})^{3}[2(\dot{x}^{0})^{2}+a^{2}(\dot{x}^{1})^{2}]}{2[(\dot{x}^{0})^{2}-a^{2}(\dot{x}^{1})^{2}]^{5/2}}\end{pmatrix}\,,$
(47)

where $a=a(t)=e^{Ht}=1+Ht+{\cal O}(H^{2})$ (in this paper, the terms that grow
with higher orders of $H$ and $\ell$ are discarded). The geodesic equation is
found from the extremization of the Finsler arc-length defined by $F$, from
which Christoffel symbols and spray coefficients can be calculated. Actually,
the $\gamma^{\alpha}_{\mu\nu}(x,\dot{x})$ are given, for an arbitrary
parametrization, by the set of Equations (44) of Ref. Letizia:2016lew , from
which the spray coefficients are given by

$\displaystyle G^{0}(x,\dot{x})=\frac{1}{8}a^{2}H(\dot{x}^{1})^{2}$
$\displaystyle\left[4-\frac{\ell
m\dot{x}^{0}}{\left[(\dot{x}^{0})^{2}-a^{2}(\dot{x}^{1})^{2}\right]^{7/2}}\left(-28a^{6}(\dot{x}^{1})^{6}+12a^{2}(\dot{x}^{0})^{4}(\dot{x}^{1})^{2}\right.\right.$
$\displaystyle\left.\left.+a^{2}\left(17a^{2}+28\right)(\dot{x}^{0})^{2}(\dot{x}^{1})^{4}+16(\dot{x}^{0})^{6}\right)\right]\,,$
(48) $\displaystyle G^{1}(x,\dot{x})=H\dot{x}^{0}\dot{x}^{1}+\ell$
$\displaystyle\left[\frac{a^{2}Hm(\dot{x}^{1})^{3}\left(a^{6}(\dot{x}^{1})^{6}-6a^{4}(\dot{x}^{0})^{2}(\dot{x}^{1})^{4}+3a^{2}(\dot{x}^{0})^{4}(\dot{x}^{1})^{2}-28(\dot{x}^{0})^{6}\right)}{4\left((\dot{x}^{0})^{2}-a^{2}(\dot{x}^{1})^{2}\right)^{7/2}}\right]\,.$
(49)

As can be seen, these coefficients are $2$-homogeneous in the velocities, as
expected. The Cartan non-linear connection coefficients read:

$\displaystyle N^{0}{}_{0}(x,\dot{x})=$ $\displaystyle\frac{H\ell
m(\dot{x}^{1})^{4}\left(-28(\dot{x}^{1})^{6}-33(\dot{x}^{1})^{4}(\dot{x}^{0})^{2}+240(\dot{x}^{1})^{2}(\dot{x}^{0})^{4}+136(\dot{x}^{0})^{6}\right)}{8\left((\dot{x}^{0})^{2}-(\dot{x}^{1})^{2}\right)^{9/2}}\,,$
(50) $\displaystyle N^{0}{}_{1}(x,\dot{x})=$ $\displaystyle
H\dot{x}^{1}-\frac{H\ell
m\dot{x}^{1}\dot{x}^{0}}{8\left((\dot{x}^{0})^{2}-(\dot{x}^{1})^{2}\right)^{9/2}}\left(28(\dot{x}^{1})^{8}-179(\dot{x}^{1})^{6}(\dot{x}^{0})^{2}+306(\dot{x}^{1})^{4}(\dot{x}^{0})^{4}\right.$
$\displaystyle\left.+128(\dot{x}^{1})^{2}(\dot{x}^{0})^{6}+32(\dot{x}^{0})^{8}\right),$
$\displaystyle N^{1}{}_{0}(x,\dot{x})=$ $\displaystyle
H\dot{x}^{1}+\frac{H\ell
m(\dot{x}^{1})^{3}\dot{x}^{0}\left(5(\dot{x}^{1})^{6}+18(\dot{x}^{1})^{4}(\dot{x}^{0})^{2}+159(\dot{x}^{1})^{2}(\dot{x}^{0})^{4}+28(\dot{x}^{0})^{6}\right)}{4\left((\dot{x}^{0})^{2}-(\dot{x}^{1})^{2}\right)^{9/2}}\,,$
(51) $\displaystyle N^{1}{}_{1}(x,\dot{x})=$ $\displaystyle
H\dot{x}^{0}-\frac{H\ell
m(\dot{x}^{1})^{2}\left(2(\dot{x}^{1})^{8}-9(\dot{x}^{1})^{6}(\dot{x}^{0})^{2}+36(\dot{x}^{1})^{4}(\dot{x}^{0})^{4}+97(\dot{x}^{1})^{2}(\dot{x}^{0})^{6}+84(\dot{x}^{0})^{8}\right)}{4\left((\dot{x}^{0})^{2}-(\dot{x}^{1})^{2}\right)^{9/2}}\,,$
(52)

where the worldlines are autoparallel curves of this non-linear connection.
Let us note that some terms of the connection are only present due to the
coupling between the spacetime curvature parameter, $H$, and the one that
gives a non-trivial velocity space, $\ell$. Some curvature-triggered effects
in quantum gravity have been recently analyzed Amelino-Camelia:2020bvx .

Endowed with these coefficients, the preferred frames that induce the
horizontal and vertical decomposition can be immediately found, in addition
the $N$-linear connection coefficients $L^{\alpha}_{\mu\nu}$ and
$C^{\alpha}_{\mu\nu}$, as discussed in Section II. Till now, only kinematical
properties were discussed, but the choice of the given connection should be
given either by physical conditions imposed on the dynamics of the spacetime
or by possible effective gravitational field equations for a quantum
configuration space.

To finalize this Section, let us discuss the symmetries of the spacetime. A
deep analysis of the killing vectors of the $H\rightarrow 0$ limit of this
Finsler framework was carried out in Ref. Barcaroli:2015eqe . Even in that
simplified scenario, the equations are quite lengthy which we omit here.
However, some properties should be mentioned. Firstly, the transformations
generated by the killing vectors seem to not exactly preserve the line
element, but contribute with a term that is given by a total derivative in the
action parameter; therefore, the kinematical results of these two line
elements coincide. Secondly, the results found are compatible with the
$\kappa$-Poincaré scenario that inspired this approach. From the Finsler
perspective, it is possible to derive more general results, but they reduces
to those of the bicrossproduct basis of $\kappa$-Poincaré by an appropriate
choice of free functions and parameters. The third point is that a finite
version of transformations that preserve the $\kappa$-Poincaré dispersion
relation was recently made in Ref. Lobo:2021yem  through an alternative
approach, which does not rely on the killing vectors but is determined by the
Finsler function and the definition of momentum (explored in Section IV
below); however, a complete integration of the finite isometry and a
comparison between these approaches is still missing in the literature. To
finalize, the case of $H\neq 0$ was investigated in Ref. Lobo:2016xzq , but in
conformal coordinates (which are not the ones that are considered in this
application), and was not done in so much detail as the flat case, but a
generator of the corresponding curved boost transformation was made explicit
in Equation (25) of Ref. Lobo:2016xzq .

##  IV The Cotangent Bundle Version of Finsler Geometry

As was discussed in Ref. Girelli:2006fw , by mapping the velocity of the
particle to its momentum, it is possible to find the version of the Finsler
metric defined in the cotangent bundle or phase space. Already from the
definition of the 4-momentum,

$p_{\mu}=m\frac{\partial F}{\partial y^{\mu}}\,,$ (53)

when it is possible to invert this expression to find $y=y(p)$, one can
substitute this result in the Finsler metric as
$h^{F}_{\mu\nu}(x,p)=g_{\mu\nu}^{F}(x,y(p))$. This metric is defined on the
slit cotangent bundle, $\widetilde{T^{*}M}=T^{*}M/\\{0\\}$, where we also
remove the zero section in each spacetime point for the same technical reasons
as discussed in Section III above. Since the quantities are now defined in the
cotangent bundle, we need to also address some issues that were raised in
Section III concerning the tangent bundle. This Section’s notation is applied
according to Ref. Miron:1994nvt . For instance, under a change of coordinates,
the spacetime and momentum variables transformed according to

$\displaystyle\tilde{x}^{\mu}$ $\displaystyle=\tilde{x}^{\mu}(x)\,,$ (54)
$\displaystyle\tilde{p}_{\mu}$ $\displaystyle=\frac{\partial
x^{\nu}}{\partial\tilde{x}^{\mu}}p_{\nu}\,,$ (55)

which means that the frame $(\partial/\partial x^{\mu},\partial/\partial
p_{\nu})$ transforms as

$\displaystyle\frac{\partial}{\partial\tilde{x}^{\mu}}$
$\displaystyle=\frac{\partial
x^{\nu}}{\partial\tilde{x}^{\mu}}\frac{\partial}{\partial
x_{\nu}}+\frac{\partial
p_{\nu}}{\partial\tilde{x}^{\mu}}\frac{\partial}{\partial p_{\nu}}\,,$ (56)
$\displaystyle\frac{\partial}{\partial\tilde{p}_{\mu}}$
$\displaystyle=\frac{\partial\tilde{x}^{\mu}}{\partial
x^{\nu}}\frac{\partial}{\partial p_{\nu}}\,.$ (57)

On the other hand, the natural coframe $(dx^{\mu},dp_{\nu})$ changes as

$\displaystyle d\tilde{x}^{\mu}$
$\displaystyle=\frac{\partial\tilde{x}^{\mu}}{\partial x^{\nu}}dx^{\nu}\,,$
(58) $\displaystyle d\tilde{p}_{\mu}$ $\displaystyle=\frac{\partial
x^{\nu}}{\partial\tilde{x}^{\mu}}dp_{\nu}+\frac{\partial^{2}x^{\nu}}{\partial\tilde{x}^{\mu}\partial\tilde{x}^{\lambda}}p_{\nu}d\tilde{x}^{\lambda}\,.$
(59)

Simlarly to that in Section III, the presence of a nonlinear connection,
$O_{\mu\nu}$, allows one to split the cotangent bundle into a horizontal and a
vertical subbundle. Inspired by the consideration of the Hamilton case
considered in Ref. Barcaroli:2015xda  (discussed below), we propose the
following dual non-linear connection (constructed in Appendix A):

$O_{\mu\nu}(x,p)=-m\left[N^{\alpha}{}_{\mu}\frac{(g_{\alpha\nu}-p_{\alpha}p_{\nu}/m^{2})}{F}-\partial_{\mu}\dot{\partial}_{\nu}F\right]\Bigg{|}_{(x,y(p))}\,,$
(60)

where $p=p(y)$ is the kinematical map defined by Equation (53). By
construction, these symbols have the transformation properties of a nonlinear
connection,

$\tilde{O}_{\mu\nu}=\frac{\partial
x^{\lambda}}{\partial\tilde{x}^{\mu}}\frac{\partial
x^{\epsilon}}{\partial\tilde{x}^{\nu}}O_{\lambda\epsilon}+\frac{\partial^{2}x^{\beta}}{\partial\tilde{x}^{\mu}\partial\tilde{x}^{\nu}}p_{\beta}\,.$
(61)

Endowed with a nonlinear connection $O_{\mu\nu}$, one can decompose the
tangent bundle of the cotangent bundle by the Whitney sum in each point
$T_{u}\widetilde{T^{*}M}=O_{u}\oplus V_{u},\,\forall u\in\widetilde{T^{*}M}$.
The subbundle $O_{u}$ is called horizontal space and is spanned by the frame,

$\frac{\delta}{\delta x^{\mu}}=\delta_{\mu}=\frac{\partial}{\partial
x^{\mu}}+O_{\mu\nu}\frac{\partial}{\partial p_{\nu}}\,,$ (62)

and the subbundle $V_{u}$ is called vertical space and is spanned by the frame
in each point of $\widetilde{T^{*}M}$:

$\bar{\partial}^{\mu}=\frac{\partial}{\partial p_{\mu}}\,,$ (63)

such that
$T_{u}\widetilde{T^{*}M}=\text{span}\\{\delta_{\mu},\bar{\partial}^{\nu}\\}$.
The transformation properties of the nonlinear connection are implied in the
following rule for transforming this basis:

$\displaystyle\frac{\delta}{\delta\tilde{x}^{\mu}}=\tilde{\delta}_{\mu}=\frac{\partial
x^{\nu}}{\partial\tilde{x}^{\mu}}\frac{\delta}{\delta x^{\nu}}=\frac{\partial
x^{\nu}}{\partial\tilde{x}^{\mu}}\delta_{\nu}\,,$ (64)
$\displaystyle\frac{\partial}{\partial\tilde{p}_{\mu}}=\tilde{\bar{\partial}}^{\mu}=\frac{\partial\tilde{x}^{\mu}}{\partial
x^{\nu}}\frac{\partial}{\partial
p_{\nu}}=\frac{\partial\tilde{x}^{\mu}}{\partial
x^{\nu}}\bar{\partial}^{\nu}\,.$ (65)

Equivalently, with the nonlinear connection, we can decompose the cotangent
space $T^{*}_{u}\widetilde{T^{*}M}=\text{span}\\{dx^{\mu},\delta p_{\nu}\\}$,
where

$\delta p_{\mu}=dp_{\mu}-O_{\nu\mu}dx^{\nu}\,.$ (66)

Therefore, the dual basis transforms as

$\displaystyle d\tilde{x}^{\mu}$
$\displaystyle=\frac{\partial\tilde{x}^{\mu}}{\partial x^{\nu}}dx^{\nu}\,,$
(67) $\displaystyle\delta\tilde{p}_{\mu}$ $\displaystyle=\frac{\partial
x^{\nu}}{\partial\tilde{x}^{\mu}}\delta p_{\nu}\,.$ (68)

Similarly to what has been done for the tangent bundle case, such a
decomposition allows us to express a vector and a $1$-form via horizontal and
vertical components, where now, the vertical component is considered along
momenta instead of velocities,

$\displaystyle
X=X^{\mu}\delta_{\mu}+\bar{X}_{\mu}\bar{\partial}^{\mu}=X^{H}+X^{V}\,,$ (69)
$\displaystyle\omega=\omega_{\mu}dx^{\mu}+\bar{\omega}^{\mu}\delta
p_{\mu}=\omega^{H}+\omega^{V}\,.$ (70)

Besides, the metric $\mathbb{H}(x,p)$ of the configuration space is defined as
follows. Given a metric $h^{\mu\nu}(x,p)$, and the nonlinear connection
$O_{\mu\nu}(x,p)$, the quantum phase space presents metrical properties given
by the tensor,

$\mathbb{H}(x,p)=h_{\mu\nu}(x,p)dx^{\mu}\otimes dx^{\nu}+h^{\mu\nu}(x,p)\delta
p_{\mu}\otimes\delta p_{\nu}\,.$ (71)

We refer to the tensor $\mathbb{H}$ as the $N$-lift to $\widetilde{T^{*}M}$ of
the metric $h_{\mu\nu}$. The map between $y$ and $p$ cannot be done, in
general, involving quantities that are parametrization-dependent because $p$
itself is parametrization-invariant, whereas $y$ is not. That is why one can
only assume $y(p)$ for the definition of the metric $h^{F}_{\mu\nu}$.

Endowed with these quantities, one can just extend the definition of
$d$-tensors III.1 to the cotangent case, in which one only needs to consider
the use of the nonlinear connection $O_{\mu\nu}$ and the adapted basis defined
in this Section.

The above implies that a $d$-tensor $T$ of type $(m+q,n+p)$ can be rewritten
in the preferred basis as

$\displaystyle
T=T^{\mu_{1}...\mu_{m}}{}_{\nu_{1}...\nu_{n}\alpha_{1}...\alpha_{p}}{}^{\beta_{1}...\beta_{q}}$
$\displaystyle\frac{\delta}{\delta
x^{\mu_{1}}}\otimes...\otimes\frac{\delta}{\delta
x^{\mu_{m}}}\otimes\frac{\partial}{\partial
p_{\nu_{1}}}\otimes...\otimes\frac{\partial}{\partial p_{\nu_{n}}}$
$\displaystyle\otimes dx^{\alpha_{1}}\otimes...\otimes
dx^{\alpha_{p}}\otimes\delta p_{\beta_{1}}\otimes...\otimes\delta
p_{\beta_{q}}\,,$ (72)

whose components transform according to usual linear transformation rules, as
the one of Equation (34).

###  IV.1 N-Linear Connection

Equivalently, the notion of differentiation can be defined in the cotangent
bundle through the $N$-linear connection $D$, which has the following
coefficients in the frame $(\delta_{\mu},\bar{\partial}^{\nu})$ (see Theorem
4.9.1 in Ref. Miron:1994nvt ):

$\displaystyle
D_{\delta_{\nu}}\delta_{\mu}=H^{\alpha}_{\mu\nu}\delta_{\alpha}\,,\qquad
D_{\delta_{\nu}}\bar{\partial}^{\mu}=-H^{\mu}_{\alpha\nu}\bar{\partial}^{\alpha}\,,$
(73) $\displaystyle
D_{\bar{\partial}^{\nu}}\delta_{\mu}=C^{\alpha\nu}_{\mu}\delta_{\alpha}\,,\qquad
D_{\bar{\partial}^{\nu}}\bar{\partial}^{\mu}=-C_{\alpha}^{\mu\nu}\bar{\partial}^{\alpha}\,.$
(74)

Otherwise, in the frame $(dx^{\mu},\delta p_{\nu})$ one has (see Proposition
4.9.1 in Ref. Miron:1994nvt )

$\displaystyle
D_{\delta_{\nu}}dx^{\mu}=-H^{\mu}_{\alpha\nu}dx^{\alpha}\,,\qquad
D_{\delta_{\nu}}\delta p_{\mu}=H^{\alpha}_{\mu\nu}\delta p_{\alpha}\,,$ (75)
$\displaystyle
D_{\bar{\partial}^{\nu}}dx^{\mu}=-C^{\mu\nu}_{\alpha}dx^{\alpha}\,,\qquad
D_{\bar{\partial}^{\nu}}\delta p_{\mu}=C_{\mu}^{\alpha\nu}\delta
p_{\alpha}\,.$ (76)

Considering a $N$-linear connection $D$ with set of coefficients,
$D\Gamma(N)=(H^{\alpha}_{\mu\nu},C^{\alpha}_{\mu\nu})$, one can add to it a
nonlinear connection, $N_{\mu\nu}$, that is in general independent of the
coefficients of $D$, such that the new set is
$D\Gamma=(N_{\mu\nu},H^{\alpha}_{\mu\nu},C^{\alpha}_{\mu\nu})$. For this
reason, the derivative of a $d$-tensor in the cotangent bundle presents
similar usual rules for dealing with up and down indices:

$\displaystyle
T^{\mu_{1}...\mu_{m}}{}_{\nu_{1}...\nu_{n}\alpha_{1}...\alpha_{p}}{}^{\beta_{1}...\beta_{q}}{}_{|\epsilon}$
(77) $\displaystyle=\frac{\delta}{\delta
x^{\epsilon}}T^{\mu_{1}...\mu_{m}}{}_{\nu_{1}...\nu_{n}\alpha_{1}...\alpha_{p}}{}^{\beta_{1}...\beta_{q}}+H^{\mu_{1}}_{\gamma\epsilon}T^{\gamma...\mu_{m}}{}_{\nu_{1}...\nu_{n}\alpha_{1}...\alpha_{p}}{}^{\beta_{1}...\beta_{q}}+...-H^{\gamma}_{\nu_{1}\epsilon}T^{\mu_{1}...\mu_{m}}{}_{\gamma...\nu_{n}\alpha_{1}...\alpha_{p}}{}^{\beta_{1}...\beta_{q}}\,,$
$\displaystyle
T^{\mu_{1}...\mu_{m}}{}_{\nu_{1}...\nu_{n}\alpha_{1}...\alpha_{p}}{}^{\beta_{1}...\beta_{q}}{}^{||\epsilon}$
(78) $\displaystyle=\frac{\partial}{\partial
p_{\epsilon}}T^{\mu_{1}...\mu_{m}}{}_{\nu_{1}...\nu_{n}\alpha_{1}...\alpha_{p}}{}^{\beta_{1}...\beta_{q}}{}+C^{\mu_{1}\epsilon}_{\gamma}T^{\gamma...\mu_{m}}{}_{\nu_{1}...\nu_{n}\alpha_{1}...\alpha_{p}}{}^{\beta_{1}...\beta_{q}}{}+...-C^{\gamma\epsilon}_{\nu_{1}}T^{\mu_{1}...\mu_{m}}{}_{\gamma...\nu_{n}\alpha_{1}...\alpha_{p}}{}^{\beta_{1}...\beta_{q}}{}\,.$

Let us note that from the kinematical map relating velocities and momenta, the
coefficients $H^{\alpha}_{\mu\nu}(x,y(p))$ and $C^{\alpha}_{\mu\nu}(x,y(p))$
can be found as been parametrization-invariant.

###  IV.2 Finsler–q-de Sitter (Cotangent Bundle Case)

Here, we again consider the $q$-de Sitter-inspired case. Then, using the
Finsler function (46), the momentum is given by Equation: (53)

$\displaystyle
p_{0}=\frac{m\dot{x}^{0}}{\sqrt{(\dot{x}^{0})^{2}-a^{2}(\dot{x}^{1})^{2}}}-\ell\frac{m^{2}a^{2}(\dot{x}^{1})^{2}(a^{2}(\dot{x}^{1})^{2}+(\dot{x}^{0})^{2})}{2[(\dot{x}^{0})^{2}-a^{2}(\dot{x}^{1})^{2})]^{2}}\,,$
(79) $\displaystyle
p_{1}=-\frac{ma^{2}\dot{x}^{1}}{\sqrt{(\dot{x}^{0})^{2}-a^{2}(\dot{x}^{1})^{2}}}+\ell\frac{m^{2}a^{2}(\dot{x}^{0})^{3}\dot{x}^{1}}{((\dot{x}^{0})^{2}-a^{2}(\dot{x}^{1})^{2}))^{2}}\,,$
(80)

which furnishes a helpful expression that is throughout this Section and is a
common trick when trying to find momentum-dependent quantities from the
Finsler approach:

$\displaystyle\frac{m\dot{x}^{0}}{\sqrt{(\dot{x}^{0})^{2}-a^{2}(\dot{x}^{1})^{2}}}=p_{0}+\ell\frac{a^{-2}(p_{1})^{2}(a^{-2}(p_{1})^{2}+(p_{0})^{2})}{2m^{2}}\,,$
(81)
$\displaystyle\frac{ma\dot{x}^{1}}{\sqrt{(\dot{x}^{0})^{2}-a^{2}(\dot{x}^{1})^{2}}}=-a^{-1}p_{1}\left(1+\ell\frac{(p_{0})^{3}}{m^{2}}\right)\,.$
(82)

The above expressions allow us to express the Finsler metric through its
momentum dependence:

$\displaystyle
g_{\mu\nu}^{F}(x,\dot{x}(p))=h_{\mu\nu}^{F}(x,p)=\begin{pmatrix}1+\frac{3\ell
p_{0}(p_{1})^{4}}{m^{4}}&-\frac{\ell
a(p_{1})^{3}[(p_{1})^{2}-4(p_{0})^{2}]}{2m^{4}}\\\ -\frac{\ell
a(p_{1})^{3}[(p_{1})^{2}-4(p_{0})^{2}]}{2m^{4}}&-a^{2}+\frac{\ell
a^{2}(p_{0})^{3}[2(p_{0})^{2}+(p_{1})^{2}]}{m^{4}}\end{pmatrix}\,,$ (83)

which can be called a ”Finsler-rainbow metric.”

One can also find the induced non-linear connection in the cotangent bundle
through the definition (60) to read as

$\displaystyle O_{00}(x,p)=$
$\displaystyle-\frac{H\ell(p_{1})^{2}}{8m^{10}}\left[4(p_{0})^{10}+44(p_{0})^{8}(p_{1})^{2}+190(p_{0})^{6}(p_{1})^{4}-196(p_{0})^{4}(p_{1})^{6}\right.$
$\displaystyle\left.+31(p_{0})^{2}(p_{1})^{8}+32(p_{1})^{10}\right]\,,$ (84)
$\displaystyle O_{01}(x,p)=$ $\displaystyle Hp_{1}-\frac{\ell
Hp_{0}p_{1}}{8m^{10}}\left[-4m^{8}(p_{0})^{2}+8(p_{0})^{10}+32(p_{0})^{8}(p_{1})^{2}+206(p_{0})^{6}(p_{1})^{4}\right.$
$\displaystyle\left.-212(p_{0})^{4}(p_{1})^{6}+43(p_{0})^{2}(p_{1})^{8}+28(p_{1})^{10}\right]\,,$
(85) $\displaystyle O_{10}(x,p)=$ $\displaystyle Hp_{1}-\frac{H\ell
p_{0}p_{1}}{8m^{10}}\left(-4m^{8}(p_{0})^{2}+4(p_{0})^{10}+140(p_{0})^{8}(p_{1})^{2}+2(p_{0})^{6}(p_{1})^{4}\right.$
$\displaystyle\left.-106(p_{0})^{4}(p_{1})^{6}+61(p_{0})^{2}(p_{1})^{8}+4(p_{1})^{10}\right)\,,$
(86) $\displaystyle O_{11}(x,p)=$ $\displaystyle
Hp_{0}+\frac{H\ell}{8m^{10}}\left(4(p_{0})^{2}(p_{1})^{2}\left(m^{8}+3(p_{1})^{8}\right)+8m^{8}(p_{1})^{4}-8(p_{0})^{12}-124(p_{0})^{10}(p_{1})^{2}\right.$
$\displaystyle\left.-30(p_{0})^{8}(p_{1})^{4}+138(p_{0})^{6}(p_{1})^{6}-89(p_{0})^{4}(p_{1})^{8}-4(p_{1})^{12}\right)\,.$
(87)

From these expressions, one can construct the decomposition of the tangent and
cotangent spaces of the cotangent bundle into horizontal and vertical parts,
accordingly.

##  V Geometry of the Cotangent Bundle: Hamilton Geometry

Besides the Finsler geometry, another interesting proposal for building a
natural geometry for propagation of particles that probe a modified dispersion
relation consists of the so-called Hamilton geometry. In this case, different
from the Finsler geometry, we start with a geometric structure defined in the
cotangent bundle (the definitions used in this metric follow that in the book
Miron:1994nvt  and in papers Barcaroli:2015xda ; Barcaroli:2016yrl ;
Barcaroli:2017gvg ; Pfeifer:2018pty ).

A Hamilton space is a pair, $(M,H(x,p))$, where $M$ is a smooth manifold and
$H:T^{*}M\rightarrow{\mathbb{R}}$ is a continuous function on the cotangent
bundle that satisfies the following properties:

  1. 1.

$H$ is smooth on the manifold $\widetilde{T^{*}M}$;

  2. 2.

the Hamilton metric, $h_{H}$, with components,

$h_{H}^{\mu\nu}(x,p)=\frac{1}{2}\frac{\partial}{\partial
p_{\mu}}\frac{\partial}{\partial p_{\nu}}H(x,p)\,,$ (88)

is nondegenerate.

Since one does not have an arc-length functional, worldlines as extremizing
curves are an absent concept in this approach. Instead, the equations of
motion of a particle that obeys a given Hamiltonian are given by the Hamilton
equations of motion:

$\displaystyle\dot{x}^{\mu}$ $\displaystyle=\frac{\partial H}{\partial
p_{\mu}}\,,$ (89) $\displaystyle\dot{p}_{\mu}$ $\displaystyle=-\frac{\partial
H}{\partial\dot{x}^{\mu}}\,.$ (90)

Since this is just another metric structure defined in the cotangent bundle,
the same results regarding the tools for coordinate transformations given by
Equation (54) are applicable here. As the case of Hamiltonian mechanics, the
definition of Poisson brackets is useful enough for our purposes. For two real
valued functions $F(x,p)$ and $G(x,p)$, their Poisson brackets are given in
Barcaroli:2015xda  (the geometry of the cotangent bundle with deformed
Hamiltonian can also be described with the language of symplectic geometry,
which is reviewed in Ref. Nozari:2014qja ):

$\\{F(x,p),G(x,p)\\}=\partial_{\mu}F\bar{\partial}^{\mu}G-\partial_{\mu}G\bar{\partial}^{\mu}F\,.$
(91)

As above, in order to divide the tangent and cotangent spaces of the cotangent
bundle into horizontal and vertical spaces, a non-linear connection is
necessary, and the canonical choice is given in Theorem 5.5.1 of Ref.
Miron:1994nvt  and Definition 2 of Ref. Barcaroli:2015xda  as

$O_{\mu\nu}(x,p)=\frac{1}{4}(\\{h_{\mu\nu}^{H},H\\}+h^{H}_{\mu\alpha}\partial_{\nu}\bar{\partial}^{\alpha}H+h^{H}_{\nu\alpha}\partial_{\mu}\bar{\partial}^{\alpha}H)\,,$
(92)

where $h_{\mu\nu}^{H}$ is the inverse of the metric $h^{\mu\nu}_{H}$. This
non-linear connection allows us to use the basis
$\delta_{\mu}=\partial_{\mu}+O_{\mu\nu}\bar{\partial}^{\nu}$ and
$\bar{\partial}^{\mu}$ as a special basis of $T_{(x,p)}\widetilde{T^{*}M}$,
and to use the basis $dx^{\mu}$ and $\delta
p_{\mu}=dp_{\mu}-O_{\nu\mu}dx^{\nu}$ as a special basis of
$T^{*}_{(x,p)}\widetilde{T^{*}M}$, which transforms according to Equations
(64), (65) and (67), (68).

Endowed with these coefficients, following Theorem 5.6.1 of Ref. Miron:1994nvt
, there exists a unique $N$-linear connection
$D\Gamma(O)=(H^{\alpha}_{\mu\nu},C_{\alpha}^{\mu\nu})$ such that:

  1. 1.

$O_{\mu\nu}$ is the canonical non-linear connection;

  2. 2.

the metric $h^{\mu\nu}_{H}$ is $h$-covariant constant (no horizontal non-
metricity):

$D_{\delta_{\alpha}}h_{H}^{\mu\nu}=0\,;$ (93)

  3. 3.

the metric $h^{\mu\nu}_{H}$ is $v$-covariant constant (no vertical non-
metricity):

$D{\bar{\partial}^{\alpha}}h_{H}^{\mu\nu}=0\,;$ (94)

  4. 4.

$D\Gamma(N)$ is horizontally torsion free:

$T^{\alpha}_{\ \mu\nu}=H^{\alpha}_{\mu\nu}-H^{\alpha}_{\nu\mu}=0\,;$ (95)

  5. 5.

$D\Gamma(N)$ is vertically torsion free:

$S_{\alpha}^{\ \mu\nu}=C_{\alpha}^{\mu\nu}-C_{\alpha}^{\nu\mu}=0\,;$ (96)

  6. 6.

the triple $(O_{\mu\nu},H^{\alpha}_{\mu\nu},C_{\alpha}^{\mu\nu})$ has
coefficients given by

$\displaystyle
O_{\mu\nu}(x,p)=\frac{1}{4}(\\{h_{\mu\nu}^{H},H\\}+h^{H}_{\mu\alpha}\partial_{\nu}\bar{\partial}^{\alpha}H+h^{H}_{\nu\alpha}\partial_{\mu}\bar{\partial}^{\alpha}H)\,,$
(97) $\displaystyle
H_{\alpha}^{\mu\nu}=\frac{1}{2}h_{H}^{\alpha\beta}(\delta_{\mu}h^{H}_{\beta\nu}+\delta_{\nu}h^{H}_{\beta\mu}-\delta_{\beta}h^{H}_{\mu\nu})\,,$
(98) $\displaystyle
C_{\alpha}^{\mu\nu}=-\frac{1}{2}h_{\alpha\beta}^{H}\bar{\partial}^{\mu}h_{H}^{\beta\nu}\,.$
(99)

This is called a Cartan $N$-linear covariant derivative. Equivalently, the
notion of $d$-tensors and their derivatives discussed in Section IV.1 are
applicable.

###  V.1 Symmetries

Hamilton geometry also allows one to encompass a DSR language, as was the case
for Finsler geometry discussed in Section III.2. However, its realization does
not come from the invariance of an interval $ds^{2}$, since one does not have
it, but from the invariance of the Hamiltonian function $H(x,p)$. The
approach, which we highlight here, was done starting from Definition 4 of
Section II-D of Ref. Barcaroli:2015xda . In a Hamilton space $(M,H)$ with
manifold $M$, and Hamiltonian $H$, let $X=\xi^{\mu}\partial_{\mu}$ be a vector
field in the basis manifold $M$ and
$X^{C}=\xi^{\mu}\partial_{\mu}-p_{\nu}\partial_{\mu}\xi^{\nu}\bar{\partial}^{\mu}$
be the so-called complete lift of $X$ to $\widetilde{T^{*}M}$. A symmetry of
the Hamiltonian is a transformation generated by $X^{C}$, whose components
satisfy

$X^{C}(H)\xi^{\mu}\partial_{\mu}H-p_{\nu}\partial_{\mu}\xi^{\nu}\bar{\partial}^{\mu}H=0\,.$
(100)

If one derivates this expression twice with respect to momenta, one gets the
following result:

$0=\frac{1}{2}\bar{\partial}^{\mu}\bar{\partial}^{\nu}X^{C}(H)=\xi^{\alpha}\partial_{\alpha}h_{H}^{\mu\nu}-h_{H}^{\mu\alpha}\partial_{\alpha}\xi^{\nu}-h_{H}^{\nu\alpha}\partial_{\alpha}\xi^{\mu}-p_{\beta}\partial_{\alpha}\xi^{\beta}\bar{\partial}^{\alpha}h_{H}^{\mu\nu}\,.$
(101)

This is just the generalization of the killing equation to a general Hamilton
space. In general, if $h_{H}$ does not depend on momenta, then it reduces to
the standard Riemannian case. Besides, from the expression of the Poisson
brackets (91), it can verified that such symmetries give rise to conserved
charges $\xi^{\mu}p_{\mu}$; i.e., that Poisson commutes with the Hamiltonian:

$\\{\xi^{\mu}p_{\mu},H\\}=0.$ (102)

These are the charges that, at an algebraic level, can generate translations,
boosts, and rotations, for instance.

###  V.2 Hamilton–q-de Sitter (Cotangent Bundle Case)

As an example, we rely on the results presented in Ref. Barcaroli:2015xda ,
which are as well inspired by the $q$-de Sitter Hamiltonian (45). In this
case, the Hamilton metric, defined by Equation (88), reads:

$\displaystyle h^{\mu\nu}_{H}(x,p)=\begin{pmatrix}1&-\ell p_{1}(1+2Hx^{0})\\\
-\ell p_{1}(1+2Hx^{0})&-(1+2Hx^{0})(1+\ell p_{0})\end{pmatrix}\,,$ (103)

which, as can be seen, acquires a shape much simpler than the rainbow-Finsler
one (83) due to the much direct way in which it is calculated.

The non-linear connection can be read from Equation (92) and can be cast in a
matrix form due to its simplicity:

$\displaystyle O_{\mu\nu}(x,p)=\begin{pmatrix}H\ell p_{1}^{2}&Hp_{1}\\\
Hp_{1}&Hp_{0}(1-\ell p_{0})\end{pmatrix}\,.$ (104)

As expected, it coincides with the case (IV.2) in the Riemannian case, i.e.,
when $\ell=0$.

The Hamilton equations of motion can be found from Equation (89) and read:

$\displaystyle\dot{x}^{0}-2p_{0}+\ell p_{1}^{2}(1+2Hx^{0})=0\,,$ (105)
$\displaystyle\dot{x}^{1}+2p_{1}(1+Hx^{0})+2\ell p_{0}p_{1}(1+2Hx^{0})=0\,,$
(106) $\displaystyle\dot{p}_{0}-2Hp_{1}^{2}-2H\ell p_{0}p_{1}^{2}=0\,,$ (107)
$\displaystyle\dot{p}_{1}=0\,.$ (108)

The autoparallel (horizontal) curves of the non-linear connection satisfy (see
Equation (8.2) in Ref. Miron:1994nvt )

$\dot{p}_{\mu}-O_{\nu\mu}\dot{x}^{\nu}=0\,,$ (109)

and, as can be seen from Equation (104) for $O_{\mu\nu}$, the worldlines,
defined from the Hamilton equations of motion, are not autoparallels of the
non-linear connection.

The symmetries have also been analyzed in Ref. Barcaroli:2015xda , where it
has been noticed that the conserved charges that generate translations and the
boost coincide with the results from Ref. Barcaroli:2015eqe  that do not rely
on the geometrical approach used in this paper.

##  VI The Tangent-Bundle Version of Hamilton Geometry

Endowed with Hamilton equations of motion (89), one has a map between the
momenta and velocities from $\dot{x}^{\mu}=y^{\mu}=\partial H/\partial
p_{\mu}$. When it is possible to invert this map to find $p_{\mu}=p_{\mu}(y)$
(as done in Appendix B of Ref. Barcaroli:2017gvg ), one derives an interesting
map between the cotangent and tangent space version of Hamilton geometry.
Indeed, using this map, a Hamilton metric defined in the tangent bundle reads:

$\displaystyle g_{H}^{\mu\nu}(x,y)\doteq h_{H}^{\mu\nu}(x,p(y))\,.$ (110)

The dual non-linear connection in this case has been discussed in Appendix C
of Ref. Barcaroli:2015xda , and is given by

$N(x,y)^{\mu}{}_{\nu}=2O(x,p(y))_{\nu\alpha}h_{H}^{\alpha\mu}(x,p(y))-(\partial_{\nu}\bar{\partial}^{\mu}H)|_{p=p(y)}\,.$
(111)

Its main property is the preservation of the horizontal tangent spaces of the
cotangent and tangent bundle connected through the kinematical map
$y^{\mu}=\partial H/\partial p_{\mu}$.

With this map, it is possible to define the dual non-linear and $N$-linear
connections, now defined in the tangent bundle. It should be stressed that
although this gives geometrical quantities defined in the tangent bundle, this
does not represent a Finsler geometry, since there is no arc-length functional
and the Hamilton metric is not, in general, 0-homogeneous to start with.

### Hamilton-$\kappa$-Poincaré (Tangent Bundle Case)

The kinematical map that allows us to describe $y=y(p)$ is found by inverting
the relation $y^{\mu}=\partial H/\partial p_{\mu}$ for the $q$-de Sitter
Hamiltonian, given by

$\displaystyle p_{0}=\frac{y^{0}}{2}+\ell\frac{(y^{1})^{2}}{8}\,,$ (112)
$\displaystyle
p_{1}=-\frac{y^{1}}{2}+H\frac{x^{0}y^{1}}{2}+\ell\frac{y^{0}y^{1}}{4}\,.$
(113)

The metric in the tangent bundle reads:

$\displaystyle
g^{\mu\nu}_{H}(x,y)=\begin{pmatrix}1&\ell(Hx^{0}y^{1}+y^{1})/2\\\
\ell(Hx^{0}y^{1}+y^{1})/2&-(1+2Hx^{0})(1+\ell y^{0}/2)\end{pmatrix}\,.$ (114)

The dual non-linear connection reads

$\displaystyle N^{\mu}{}_{\nu}(x,y)=\begin{pmatrix}-H\ell(y^{1})^{2}/2&\ell
hy^{0}y^{1}+hy^{1}\\\ Hy^{1}&-hy^{0}-3\ell h(y^{1})^{2}/4\end{pmatrix}\,.$
(115)

In Section VII below, some key points of each approach are discussed while
comparing the descriptions of configuration and phase spaces.

##  VII Advantages and Difficulties of Each Formalism

The approaches considered—Finsler and Hamilton spaces—present the points that
can be considered positive or negative. In this Section, we highlight some of
those points which look to be most important from theoretical and
phenomenological points of view.

###  VII.1 Finsler Geometry

Let us emphasize that here not a complete list of positive or negative points
are given, and, certainly, the points listed represent just our view on the
subject under scrutiny and some points we are classifying in one way or
another can be seen by others completely differently.

####  VII.1.1 Advantages

Preservation of the equivalence principle. Due to the presence of an arc-
length functional, the extremizing geodesics of the Finsler function are the
same worldlines of the Hamiltonian, from which the arc-length was derived.
This means that, in the Finslerian language, the equivalence principle is
satisfied, as soon as the worldlines are trajectories of free particles in
this spacetime. There is a fundamental difference in comparison to the special
or general relativity formulation, since these trajectories are now mass-
dependent, since the Finsler function and the metric carry the mass of the
particle due to Planck-scale effects. Intriguingly, although the metric does
not present a massless limit (which is discussed below), it is possible to
find trajectories of massless particles, which are compatible with the
Hamiltonian formulation, by taking the limit $m\rightarrow 0$ in the geodesic
Equation Amelino-Camelia:2014rga ; Lobo:2016xzq . This finding leads to some
effects due to modifications of the trajectories of particles. For instance,
one of the most explored avenues of quantum gravity phenomenology (maybe
competing with threshold effects) is the time delay until particles with
different energies might arrive at a detector after a (almost) simultaneous
emission Jacob:2008bw ; Zhu:2022blp  (for reviews, see Amelino-Camelia:2008aez
; Addazi:2021xuf ). This kind of experimental investigation is not exhausted,
and novelties have arrived in the analysis of sets of gamma-ray bursts and
candidate neutrinos emitted from them in the multimessenger astronomy approach
Amelino-Camelia:2016ohi ; Amelino-Camelia:2022pja .

Preservation of the relativity principle. This formalism allows one to derive
and solve the killing equation, which furnishes infinitesimal symmetry
transformations of the metric. It has been shown in Ref. Amelino-
Camelia:2014rga  that generators of these transformations can be constructed
and identified with the transformations that are generally depicted in the
doubly special relativity. The latter implies, in a preservation of the
relativity principle that inertial frames should assign the same MDR to a
given particle which, in its turn, implies that the deformation scale of
quantum gravity is observer-independent, i.e., two observers would not assign
different values, in the same system of units, to the quantum gravity scale.
This preservation has important phenomenological consequences, such as the
point that the threshold constraints on the quantum gravity parameter do not
apply in the DSR scenario. The reason is that, accompanied by the deformation
of the Lorentz (Poincaré) symmetries, comes a deformation of the composition
law of momenta of particles (for instance $p$ and $q$), such that the nature
of interaction vertices to not get modified when transforming from one frame
to another:

$\Lambda(p\oplus q)=\Lambda(p)\oplus\Lambda(q)\,,$ (116)

where $\Lambda$ is a deformed Lorentz transformation induced by the killing
vectors and $\oplus$ represents a modified composition of components of the
involved momenta (this covariance condition usually needs a back-reaction on
the boost parameter, but we do not dwell on that here; for more details, see
Majid:2006xn ; Lobo:2021yem  and references therein). Threshold constraints,
such as the one placed in Ref. HAWC:2019gui , assumes that the composition of
momenta is undeformed, although the dispersion relation is modified in a
Lorentz invariance violation (LIV) scenario. When this is the case, processes
that are forbidden in special relativity, such as the decay of the photon into
an electron–positron pair, becomes kinematicaly allowed for a given threshold
energy. The no observation of such decays allows one to place constraints on
the quantum gravity parameter. When the dispersion relation is modified as
well, what happens is that generally these kinds of processes remain forbidden
or modifications in the threshold energies are so minute that they are
unobservable for a quantum gravity parameter in the order of the Planck energy
Lobo:2021yem . This is an important feature of "deforming" instead of
"violating" the Lorentz symmetry.

Preservation of the clock postulate. The availability of an arc-length
functional leads to a possibility to analyze the consequences of having the
proper time of a given particle given by it. If this is the case, then the
worldlines or geodesics are just paths that extremize the proper time an
observer measures in spacetime, similar to that in special relativity. One of
the consequences of this feature consists of the possibility of connecting the
time elapsed in the comoving frame of a particle during its lifetime (which is
its lifetime at rest) and the coordinate time, which is the one that is
assigned to this phenomenon in the laboratory coordinates. Using this
expression, one can investigate the relativistic time dilation (responsible
for the "twin paradox") or the so-called first clock effect (for further
details on the first and also on the second clock effect, which can appear in
theories with a non-metricity tensor; see Ref. Lobo:2018zrz ), in which, for
instance, the lifetime of a particle is dilated in comparison to the one
assigned in the laboratory. Due to Finslerian corrections, the lifetime of a
particle in the laboratory would receive Planckian corrections, which,
actually, is a novel avenue of phenomenological investigation that is being
currently carried out Lobo:2020qoa ; Lobo:2021yem  through the search for
signatures in particle accelerators and cosmic rays.

####  VII.1.2 Difficulties

Absence of massless rainbow Finsler metric. The Finsler approach had emerged
as an opportunity to describe in a consistent way the intuition that the
quantum spacetime probed by a high-energy particle would present some energy-
momentum (of the particle itself) corrections, which is justified by different
approaches to quantum gravity Assanioussi:2014xmz ; Weinfurtner:2008if . Since
then, proposals of rainbow metrics have considered a smooth transition from
massive to massless cases, not only from the point of view of the
trajectories, but from the metric itself. This is not the case for the Finsler
approach presented here. Although the trajectories and symmetries are defined
for both massive and massless cases by considering the $m\rightarrow 0$ limit,
the rainbow metric of Finsler geometry, given by Equation (83), is certainly
not defined for massless particles. The reason for this is the point that when
passing from the Hamiltonian to the Lagrangian formalism, we defined an arc-
length functional, which is not a legitimate action functional for massless
particles. In other words, a crucial step for deriving the Finsler function is
the handling of the Lagrange multiplier $\lambda$ of action (4), which can
only be solved if the particle is massive, as can be found in Refs. Amelino-
Camelia:2014rga ; Lobo:2016xzq ; Lobo:2016lxm ; Lobo:2020qoa . A possibility
that has been explored consisted of not solving the equation for $\lambda$ and
defining a metric that depends on $\lambda$ and on velocities from a Polyakov-
like action for free particles (instead of the Nambu–Goto one given by the
arc-length), which turned out to be out of the Finsler geometry scope
Lobo:2016xzq ; Lobo:2016lxm . However, this possibility has not been further
explored beyond preliminary investigations. The issue of the absence of a
massless rainbow-Finsler metric could be circumvented by proposing a different
kind of geometry, which from the very beginning started from the momenta
formulation, like the other possibility described in this paper, namely the
Hamilton geometry.

Definition only through perturbations. the Finsler geometry has been
considered in this paper in this context at most perturbatively around the
quantum-gravity-length scale (or inverse of energy scale), which may be
considered as a negative point if one aims to make it at a more fundamental or
theoretical level. Nevertheless, from the pragmatic perspective of
phenomenology, since such effects, if they exist, are minute, then the
perturbative approach is enough for proposing new effects that could serve as
avenues of experimental investigation.

The handling of finite symmetries. Another issue that can be problematic is
the handling of finite symmetries in the Finsler context. Up to today, the
connection between Finsler geometry and quantum gravity phenomenology has not
faced the issue of integrating the symmetries and finding finite versions of
deformed Lorentz transformations. Some initial investigations were carried out
in Ref. Lobo:2021yem  from the momentum space perspective, but further issues
are being currently faced by some authors of the present paper.

###  VII.2 Hamilton Geometry

The descriptions of the points in this section will be a bit shorter than the
previous ones, because some universal points we already described above;
therefore, we instruct the reader to check on them when that is the case.

####  VII.2.1 Advantages

Presence of a massless rainbow Hamilton metric. Differently from the Finsler
case, the Hamilton geometry does not need an arc-length functional; instead,
it only needs a given Hamiltonian, from which the metric, non-linear
connection, and symmetries are derived. This means that from the very
beginning, the massless limit of geometrical quantities exists.

Does not require perturbative methods. Another positive point about the
Hamilton geometry is the finding that one can handle with the exact form of
the proposed Hamiltonian, and it does not need to consider perturbations
around a certain scale. Instead, independently of the form of the (smooth)
dispersion relation that arises from de facto approaches to quantum gravity,
the geometry can be handled, as has been considered, e.g., in Refs.
Barcaroli:2016yrl ; Barcaroli:2017gvg .

Preservation of the relativity principle and the handling of symmetries. Due
to the proximity of this approach to the way that the DSR formalism generally
handles with Planck scale corrections, i.e., from the point of view of
momentum space and Hamilton equations, the handling of symmetries is
facilitated in this approach. For instance, it is straightforward to find the
conserved charges from the killing vectors, which generate finite
transformations that are momenta-dependent without tedious terms in the
denominator of the equations when one is working in velocity space, as Finsler
geometry is initially formulated (or without mass terms in the denominator in
the Finsler version of the phase space).

Generalization to curved spacetimes. This approach is considered in more
curved space cases, beyond the $q$-de Sitter exemplified in this paper; for
instance, its spherically symmetric and cosmological versions were explored
giving rise to interesting phenomenological opportunities, from the point of
view of time delays and gravitational redshift, among others (for some
applications of Hamilton geometry in this context, see Pfeifer:2018pty  and
references therein).

####  VII.2.2 Difficulties

Non-geodesic trajectory. An issue that may be considered problematic is the
point that the worldlines of particles, given by the Hamilton equations, are
not geodesics of the non-linear connection that means that there exists a
force term in the geodesic equation, which is in contrast with the Finsler
case. This is a property of the Hamilton geometry, as has been shown in Ref.
Barcaroli:2015xda , and is not specific to the $q$-de Sitter case analyzed
here.

Absence of the arc-length. The Hamilton geometry does not dwell with an arc-
length functional that means that the only geodesics present are those of the
non-linear or of the $N$-linear connections and there are no extremizing ones.
The absence of a function that allows one to measure distances in spacetime
can be seen as a difficulty of this geometry; if the distances cannot be
calculated, one could wonder what such a metric means. Even if the norm of a
tangent vector can be integrated, this integral would not be, in general,
parametrization-independent, which is also a drawback of this tentative.
Besides, the absence of an arc-length limits the phenomenology of the
preservation of the clock postulate that was discussed in the Finsler case.

##  VIII Final Remarks

We revised two proposals that have been considered as candidates for
describing the quantum configuration and phase spaces probed by particles
whose kinematics are modified by a length scale identified as the quantum
gravity scale.

Finsler geometry starts from a configuration space framework that presents
applications on its own in biology, thermodynamics, and modified gravity; and
it finds a natural environment in quantum gravity phenomenology due to its
power to describe a scenario in which important principles that guided physics
in the XXth century, such as the relativity principle, are preserved even at a
Planckian regime. Besides its traditional description in terms of the couple
spacetime and velocity space (configuration space), we also explored its
development in terms of the induced couple spacetime and momentum space (phase
space), which is actually more appropriate for a pure quantum description than
the configuration space. Some points that we consider positive and negative
and which are consequences of the requirements for using the Finsler language,
the derivation of an arc-length functional defined in the slit tangent bundle,
are discussed in Section VII.

The second case of the present study is the Hamilton geometry, whose
properties are derived directly from the Hamiltonian itself, without the need
to go through the definition of an arc-length. Actually, in general, the
Hamilton metric does not even define a curve-parametrization-invariant length
measure which brings some limitations to phenomenological investigations of
this subject in quantum gravity. On the other hand, this issue circumvents
some intrinsic difficulties of Finsler geometry, which were also discussed in
Section VII.

The goal of this paper was to review some topics of these two important
geometries by using kinematical descriptions of particles whose behavior might
present departures from special relativity results due to the effective
quantum gravity influence. We also aimed to bring some points that we consider
as under-explored perspectives on the subject by explicitly presenting some
geometric quantities that are dual to those, in which those quantities were
originally presented, such as the dual metrics and non-linear connections
(whose Finslerian one was proposed in this paper, by inspiration of
definitions in the Hamilton geometry literature) of Finsler and Hamilton
geometries in the cotangent and tangent bundles, respectively.

At least two global points could be considered insufficiently explored or
unexplored in this subject. One is the geometry probed by an (non-)interacting
multi-particle system. Some challenges of this problem can be found, for
instance, in Ref. Hossenfelder:2014ifa , but the relations between the
approaches there described and Finsler/Hamilton geometries remains unclear.
Another point that remains unexplored consists of the dynamics of the
configuration/phase space in a way that is compatible with quantum gravity
phenomenology-inspired approaches. For instance, one could wonder if there
exists a gravitational field theory defined in Finsler or Hamilton spaces that
has $q$-de Sitter or other proposals as solutions, and how matter would
interact in this scenario. The exploration of this topic might shed light on
the one regarding a multi-particle system. These are more challenges that
might be subjects of the future research in this area and which may help to
build a bridge between quantum and modified gravities.

## Ackowledgements

I.P.L. was partially supported by the National Council for Scientific and
Technological Development—CNPq grant 306414/2020-1, and by the grant
3197/2021, Paraíba State Research Foundation (FAPESQ). I.P.L. would like to
acknowledge the contribution of the COST Action CA18108. L.C.N.S. would like
to thank Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)
for partial financial support through the research project no. 164762/2020-5.
V.B.B. is partially supported by CNPq through the research project no.
307211/2020-7. P.H.M. and S.A. thank Coordenação de Aperfeiçoamento de Pessoal
de Nível Superior—Brazil (CAPES)—Finance Code 001, for financial support.
G.V.S thank Conselho Nacional de Desenvolvimento Científico e Tecnológico
(CNPq) for financial support. E.R. and G.M were supported by the PIBIC program
of the Federal University of Paraíba.

##  Appendix A Dual Finsler Nonlinear Connection

The momenta of a particle in Finsler geometry, given by the following
expression,

$p_{\mu}=m\frac{\partial F}{\partial y^{\mu}}\equiv m\dot{\partial}_{\mu}F\,,$
(117)

defines a kinematical map between velocity and momenta variables at each given
point in the base manifold $M$. We refer to such a map as

$\displaystyle\flat:\quad\widetilde{TM}\rightarrow\widetilde{T^{\ast}}M$ (118)
$\displaystyle(x,y)\mapsto\flat(x,y)=(x,m\dot{\partial}F(x,y))=(x,p(x,y))\,.$
(119)

Inspired by the construction of Ref. Barcaroli:2015xda , the condition that a
nonlinear connection in the tangent bundle is dual to one in the cotangent
bundle by a kinematical map, $\flat$, is that such an application maps the
tangent space of the tangent bundle onto the tangent space of the cotangent
bundle. This means that the differential of such a map maps the preferred
basis of one tangent space,
$\delta_{\mu}=\partial_{\mu}-N^{\nu}{}_{\mu}\dot{\partial}_{\nu}$, onto the
other,
$d\,\flat(\delta_{\mu})=\delta^{\prime}_{\mu}=\partial_{\mu}-O_{\mu\nu}\dot{\partial}^{\nu}$.
This means that the action of this differential on a vector
$X=X^{\mu}\partial_{\mu}+\dot{X}^{\mu}\dot{\partial}_{\mu}$ is given by

$\displaystyle d\,\flat_{(x,y)}:\quad T_{(x,y)}\widetilde{TM}$
$\displaystyle\rightarrow T_{\flat(x,y)}\widetilde{T^{*}M}\,,$ (120)
$\displaystyle X=X^{\mu}\partial_{\mu}+\dot{X}^{\mu}\dot{\partial}_{\mu}$
$\displaystyle\mapsto
d\,\flat_{(x,y)}(X)=X^{\mu}d\,\flat_{(x,y)}(\partial_{\mu})+\dot{X}^{\mu}d\,\flat_{(x,y)}(\dot{\partial}_{\mu})$
(121)
$\displaystyle=X^{\mu}(\partial_{\mu}+m\partial_{\mu}\dot{\partial}_{\nu}F\bar{\partial}^{\nu})+m\dot{X}^{\mu}\dot{\partial}_{\mu}\dot{\partial}_{\nu}F\bar{\partial}^{\nu}\,.$
(122)

By acting on the basis vectors
$\delta_{\mu}=\partial_{\mu}-N^{\nu}{}_{\mu}\dot{\partial}_{\nu}$, one finds:

$\displaystyle
d\,\flat_{(x,y)}(\delta_{\mu})=d\,\flat_{(x,y)}(\delta_{\mu})-N^{\nu}{}_{\mu}d\,\flat_{(x,y)}(\dot{\partial}_{\nu})=\partial_{\mu}+m\partial_{\mu}\dot{\partial}_{\nu}F\bar{\partial}^{\nu}-mN^{\nu}{}_{\mu}\dot{\partial}_{\nu}\dot{\partial}_{\alpha}F\bar{\partial}^{\alpha}\,.$
(123)

In order to simplify this expression, the relation
$2g_{\nu\alpha}=\dot{\partial}_{\nu}\dot{\partial}_{\alpha}F^{2}=\dot{\partial}_{\nu}(2F\dot{\partial}_{\alpha}F)$
is used that leads to

$\dot{\partial}_{\nu}\dot{\partial}_{\alpha}F=\frac{g_{\nu\alpha}-p_{\nu}p_{\alpha}/m^{2}}{F}\,.$
(124)

From this expression, one finds that

$d\,\flat_{(x,y)}(\delta_{\mu})=\partial_{\mu}-m\left[N^{\alpha}{}_{\mu}\frac{(g_{\alpha\nu}-p_{\alpha}p_{\nu}/m^{2})}{F}-\partial_{\mu}\dot{\partial}_{\nu}F\right]\bar{\partial}^{\nu}=\partial_{\mu}+O_{\mu\nu}\bar{\partial}^{\nu}\,,$
(125)

which leads to the dual nonlinear connection,

$O_{\mu\nu}(x,p)=-m\left[N^{\alpha}{}_{\mu}\frac{(g_{\alpha\nu}-p_{\alpha}p_{\nu}/m^{2})}{F}-\partial_{\mu}\dot{\partial}_{\nu}F\right]\Bigg{|}_{(x,y(p))}\,.$
(126)

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