# Quantum Configuration and Phase Spaces: Finsler and Hamilton Geometries

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## Abstract

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## 1. Introduction

## 2. Preliminaries on Rainbow Geometries

## 3. Geometry of the Tangent Bundle: Finsler Geometry

”For Space, when the position of points is expressed by rectilinear co-ordinates, $ds=\sqrt{\sum {\left(dx\right)}^{2}}$; 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.”

- variation with respect to $\lambda $ enforces the dispersion relation;
- 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;
- using ${p}_{\mu}(x,\dot{x},\lambda )$ in the dispersion relation, one can solve for $\lambda (x,\dot{x})$; and
- finally, the desired length measure is obtained as $S\left[x\right]=S{[x,p(x,\dot{x},\lambda (x,\dot{x})),\lambda (x,\dot{x})]}_{H}$.

- positive 2-homogeneity: $L(x,\alpha y)={\alpha}^{2}L(x,y),\phantom{\rule{0.166667em}{0ex}}\forall \alpha >0$;
- at any $(x,y)\in \mathcal{D}$ and in any chart of $\tilde{TM}$, the following Hessian (metric) is non-degenerate:$${g}_{\mu \nu}(x,y)=\frac{1}{2}\frac{{\partial}^{2}}{\partial {y}^{\mu}\partial {y}^{\nu}}L(x,y)\phantom{\rule{0.166667em}{0ex}};$$
- the metric ${g}_{\mu \nu}$ has a Lorentzian signature.

**Definition 1.**

#### 3.1. N-Linear Connection

#### 3.2. Symmetries

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

## 4. The Cotangent Bundle Version of Finsler Geometry

#### 4.1. N-Linear Connection

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

## 5. Geometry of the Cotangent Bundle: Hamilton Geometry

- H is smooth on the manifold $\tilde{{T}^{*}M}$;
- the Hamilton metric, ${h}_{H}$, with components,$${h}_{H}^{\mu \nu}(x,p)=\frac{1}{2}\frac{\partial}{\partial {p}_{\mu}}\frac{\partial}{\partial {p}_{\nu}}H(x,p)\phantom{\rule{0.166667em}{0ex}},$$

- ${O}_{\mu \nu}$ is the canonical non-linear connection;
- the metric ${h}_{H}^{\mu \nu}$ is h-covariant constant (no horizontal non-metricity):$${D}_{{\delta}_{\alpha}}{h}_{H}^{\mu \nu}=0\phantom{\rule{0.166667em}{0ex}};$$
- the metric ${h}_{H}^{\mu \nu}$ is v-covariant constant (no vertical non-metricity):$$D{\overline{\partial}}^{\alpha}{h}_{H}^{\mu \nu}=0\phantom{\rule{0.166667em}{0ex}};$$
- $D\Gamma \left(N\right)$ is horizontally torsion free:$${T}_{\phantom{\rule{4pt}{0ex}}\mu \nu}^{\alpha}={H}_{\mu \nu}^{\alpha}-{H}_{\nu \mu}^{\alpha}=0\phantom{\rule{0.166667em}{0ex}};$$
- $D\Gamma \left(N\right)$ is vertically torsion free:$${S}_{\alpha}^{\phantom{\rule{4pt}{0ex}}\mu \nu}={C}_{\alpha}^{\mu \nu}-{C}_{\alpha}^{\nu \mu}=0\phantom{\rule{0.166667em}{0ex}};$$
- the triple $({O}_{\mu \nu},{H}_{\mu \nu}^{\alpha},{C}_{\alpha}^{\mu \nu})$ has coefficients given by$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {O}_{\mu \nu}(x,p)=\frac{1}{4}(\{{h}_{\mu \nu}^{H},H\}+{h}_{\mu \alpha}^{H}{\partial}_{\nu}{\overline{\partial}}^{\alpha}H+{h}_{\nu \alpha}^{H}{\partial}_{\mu}{\overline{\partial}}^{\alpha}H)\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {H}_{\alpha}^{\mu \nu}=\frac{1}{2}{h}_{H}^{\alpha \beta}({\delta}_{\mu}{h}_{\beta \nu}^{H}+{\delta}_{\nu}{h}_{\beta \mu}^{H}-{\delta}_{\beta}{h}_{\mu \nu}^{H})\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& {C}_{\alpha}^{\mu \nu}=-\frac{1}{2}{h}_{\alpha \beta}^{H}{\overline{\partial}}^{\mu}{h}_{H}^{\beta \nu}\phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}$$

#### 5.1. Symmetries

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

## 6. The Tangent-Bundle Version of Hamilton Geometry

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

## 7. Advantages and Difficulties of Each Formalism

#### 7.1. Finsler Geometry

#### 7.1.1. 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 $m\to 0$ in the geodesic Equation [49,50]. 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 [61,62] (for reviews, see [4,5]). 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 [63,64].

**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. [49] 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:

**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. [67]), 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 [43,55] through the search for signatures in particle accelerators and cosmic rays.

#### 7.1.2. 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 [24,25]. 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 $m\to 0$ 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. [43,49,50,53]. 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 [50,53]. 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 was 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. [55] from the momentum space perspective, but further issues are being currently faced by some authors of the present paper.

#### 7.2. Hamilton Geometry

#### 7.2.1. 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. [57,58].

**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 [59] and references therein).

#### 7.2.2. 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. [56], 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.

## 8. Final Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DSR | Doubly/Deformed Special Relativity, |

LQG | Loop Quantum Gravity, |

LIV | Lorentz Invariance Violation, |

MDR | Modified Dispersion Relation |

## Appendix A. Dual Finsler Nonlinear Connection

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Albuquerque, S.; Bezerra, V.B.; Lobo, I.P.; Macedo, G.; Morais, P.H.; Rodrigues, E.; Santos, L.C.N.; Varão, G.
Quantum Configuration and Phase Spaces: Finsler and Hamilton Geometries. *Physics* **2023**, *5*, 90-115.
https://doi.org/10.3390/physics5010008

**AMA Style**

Albuquerque S, Bezerra VB, Lobo IP, Macedo G, Morais PH, Rodrigues E, Santos LCN, Varão G.
Quantum Configuration and Phase Spaces: Finsler and Hamilton Geometries. *Physics*. 2023; 5(1):90-115.
https://doi.org/10.3390/physics5010008

**Chicago/Turabian Style**

Albuquerque, Saulo, Valdir B. Bezerra, Iarley P. Lobo, Gabriel Macedo, Pedro H. Morais, Ernesto Rodrigues, Luis C. N. Santos, and Gislaine Varão.
2023. "Quantum Configuration and Phase Spaces: Finsler and Hamilton Geometries" *Physics* 5, no. 1: 90-115.
https://doi.org/10.3390/physics5010008