# Literature Review on
Maneuver-Based Scenario Description for
Automated Driving Simulations

Nicole Neis1, Juergen Beyerer2,3 *This publication was written in the context
of the AVEAS research project (www.aveas.org), funded by the German Federal
Ministry for Economic Affairs and Climate Action (BMWK) within the program
“New Vehicle and System Technologies”.1Porsche Engineering Group GmbH, 71287
Weissach, Germany<EMAIL_ADDRESS>IOSB, 76131
Karlsruhe, Germany3Karlsruhe Institute of Technology (KIT), Vision and Fusion
Laboratory (IES), 76131 Karlsruhe, Germany

###### Abstract

The increasing complexity of automated driving functions and their growing
operational design domains imply more demanding requirements on their
validation. Classical methods such as field tests or formal analyses are not
sufficient anymore and need to be complemented by simulations. For
simulations, the standard approach is scenario-based testing, as opposed to
distance-based testing primarily performed in field tests. Currently, the time
evolution of specific scenarios is mainly described using trajectories, which
limit or at least hamper generalizations towards variations. As an
alternative, _maneuver-based approaches_ have been proposed. We shed light on
the state of the art and available foundations for this new method through a
literature review of early and recent works related to maneuver-based scenario
description. It includes related modeling approaches originally developed for
other applications. Current limitations and research gaps are identified.

###### Index Terms:

automated driving, AI, maneuver-based, simulation, scenario-based testing

##  I Introduction

Research in automated driving (AD) functions aims at automated systems taking
over an increasing share of the dynamic driving task with growing operational
design domains and a decreasing need for the driver to take over control over
the vehicle [1, 2]. The fatal Tesla accident of 2016 [3] showcased the
difficulties related to the validation of such AD functions under increasing
complexity: the accident was the result of various factors coinciding in a
singular scenario: the system was not designed to, and did not detect a
crossing semi-trailer truck under the given circumstances. Furthermore,
neither the Tesla driver nor the truck driver paid sufficient attention. In
addition to that, the means by which the Tesla monitored the driver’s
attention and the system’s reaction to it were found inappropriate. This
highlights that testing solely based on the specification of the components is
not sufficient [4, 5]. Instead, the system as a whole needs to be tested in
its realistic, diverse operational environment. Classically, field tests
performed this task. They are mainly based on the principle of distance-based
testing [6, 7]: a certain amount of kilometers is driven under conditions in
which the AD is expected to work in order to draw a statistical conclusion on
a system’s safety. However, to statistically prove that a highway pilot
function operating on German highways is at least as safe as a human driver,
$6.6\times 10^{9}$ km must be driven on German highways [7]. Therefore,
classical validation methods such as field tests [8, 9] need to be
complemented by simulations. A critical factor driving up the amount of
kilometers that theoretically need to be driven is the low probability of
situations that bring the system to its limits [7]. Simulations enable the
step from distance- to scenario-based testing [10, 11, 12, 13, 14, 4]. As
illustrated in Fig. 1, scenario-based testing can explicitly focus on the
critical situations. The long chain of uncritical situations that needs to be
passed in case of distance-based testing can be left out. The key challenge
with scenario-based testing is the selection of a scenario catalogue. The
number and character of the scenarios to test is still an open question and
the content of numerous research papers [15, 16, 17]. However, as outlined in
[18], already the validation based on a limited scenario catalogue can
increase the safety of an AD function, by reducing risk factors and thus the
probability of several such factors coinciding, as in case of the Tesla
accident, thereby decreasing the risk for an accident. To avoid focusing on
unrealistically extreme scenarios while overlooking much more likely and
mundane factors, the scenarios should be representative for reality [12, 7].
To achieve a high coverage of the space of all possible scenarios, and to
avoid overfitting the function to certain selected conditions, a high number
of different scenarios should be tested [12, 7]. A proposed approach, taken by
the AVEAS project, is therefore to start from real data and extract scenarios
from those. These are then parametrized and modified [6, 19, 20, 21, 7] to
achieve higher variety. Current scenario descriptions are mainly trajectory-
based [22, 23, 24, 25, 26]. An advancement of this is the _maneuver-based_
approach [6, 27, 28], which forms the basis for this paper.

Figure 1: Distance- vs. scenario-based testing: scenario-based testing can
concentrate on critical situations while long distances of uncritical
situations have to be passed before encountering the critical ones in case of
distance-based testing

After the general background on scenario-based testing of AD functions in
simulations given in this section, some wording will be discussed in Sec. II.
Then, the results of the literature review will be presented in Sec. III. The
aim is to assess the current state of the art in the field of maneuver-based
scenario description. Available work that can serve as foundation for further
research in this field will be outlined and categorized, and current
limitations of the presented methods, leaving room for further research, will
be identified. Key conclusions are drawn in Sec. IV.

##  II Wording

This work uses the classical definitions of _scene_ and _scenario_ as coined
by Ulbrich et al. [29]. Of special relevance for this paper is the term
_scenery_ , referring to all geo-spatially stationary aspects of a scene such
as the road network, signs, buildings and trees [29].

_Resimulation_ here refers to the simulation of scenarios that are derived
from real-world driving data. Thus, it covers not only a replay of the
recorded scenario but also the simulation of scenarios obtained by variation
of the original scenario.

Within this paper and the ones referenced below, the term _maneuver_ is
considered to be part of the system state, as opposed to part of the
transition between quasi-stationary system states (see [30] for more details).
This is also the understanding dominating in literature. According to Braun et
al. [31], maneuvers can be categorized based on their direction, the presence
of interaction with other traffic participants, their start and end conditions
as well as their hierarchical level (see below). Start conditions can be
either _triggered_ or _implicit_ , end conditions _self-terminating_ or
_indefinite_. Regarding the hierarchical level, a maneuver can be categorized
as either _composed_ or _atomic_. When putting the driver into the center of
consideration, maneuvers can also be classified according to their reason
[32]: they can be _intended_ or _unintended_ (for example caused by
distraction or fatigue). Furthermore, a maneuver can be classified according
to its level of abstraction, also coming from a driver’s perspective: a
maneuver can be _strategic_ (e.g. turning), _tactical_ (e.g. overtaking) or
_operative_ (e.g. braking) [33]. Already this short consideration of the term
maneuver hints at difficulties that arise when trying to set up a maneuver-
based scenario description. Open questions that need to be clarified are:

  * •

Which maneuvers – in the sense of general kind, hierarchical level and level
of abstraction – to choose?

  * •

Which start and end conditions to select?

For the sake of flexibility, an advantageous choice of maneuvers could be a
set of atomic behaviors from which every higher-level maneuver can be derived
as proposed in [30, 31] and [34]. This would partially answer the first
question. However, the decisions on which level of abstraction to define the
maneuvers and which categorizations to apply are yet to be made. Jatzkowski et
al. [30], for example, propose to describe the vehicle’s state at any time
using two overlapping maneuvers, one for the lateral and one for the
longitudinal direction.

##  III Literature Review

This section presents the scenario descriptions currently used in resimulation
and outlines research conducted in the two fields “maneuver detection,
classification and prediction” and “trajectory prediction and modeling” that
are capable of providing methods for the maneuver-based approach.

The motivation behind this selection of research fields is the following: a
key advantage of maneuver-based resimulation (see Sec. III-A2) is the
possibility to easily vary the scenario’s course or its scenery. These
variations require conditional models enabling the derivation of consistent
low-level details (e.g. trajectories) based on high-level maneuvers. Maneuver-
based resimulation is a relatively new tool in the field of virtual validation
for AD. Thus, few models exist that have been directly developed for this
application. However, yet a wide range of models supports similar tasks, in
particular the inverse type of task, such as predicting the upcoming
trajectory or detecting a maneuver based on low-level details. Therefore, this
literature review covers these fields as well.

All papers whose models are presented within this section are categorized
using the scheme defined by Tab. I. The aim is to assess the potential of the
presented methods to be used for maneuver-based scenario descriptions in the
context of AD simulations. The categorizations applied within Tab. I cover
different aspects relevant for this assessment. The row and column a method is
assigned to describe the general model characteristics. The row indicates the
granularity of the fitting data used for parametrization/calibration of the
models and thus indicates down to which level the realistic behavior of the
model is ensured. The columns indicate up to which granularity phenomena are
considered within the model descriptions. Both aspects need to be in-line with
the requirements on the models needed for a maneuver-based scenario
description. On-board details are quantities related to the vehicle’s interior
that are not typically visible from the outside. For the granularity of the
model descriptions they are further separated into vehicle dynamics-related,
human factor-related, or both. Vehicle dynamics-related on-board details
include pedal positions, brake pressure, as well as slip and steering dynamics
of the vehicle; those related to human factors refer to the physical and
psychological state of the driver comprising quantities such as the driver’s
head position, perception thresholds, level of aggressiveness or personality
traits. The color scheme indicates whether or not a method differentiates
between maneuvers, and if so, whether it is able to detect and classify these
maneuvers. The presence of the maneuver concept indicates a familiarity to the
maneuver-based approach and thus bears the potential for reuse. Methods that
are applied in the context of AD simulations are highlighted with bold
letters. Methods of this category are thus already usable within AD
simulations and possibly also when using the maneuver-based approach. All
categorizations are made based on what was actually used in the publication,
as opposed to what could in theory be possible (e.g. fitting the model with
data of lower granularity than the ones used in the paper).

This section continues with a review of all models listed in Tab. I and
concludes with a synopsis in Sec. III-D that presents the insights that can be
gained from Tab. I against the background of the conducted literature review.

TABLE I: Paper categorization Granularity of fitting data down to… |  Granularity of model descriptions down to…
---|---
Map-referenced trajectories |  On-board details
vehicle dynamics |  vehicle dynamics & human factors |  human factors
Traffic densities |  [35] |  |  [36] |  [37] [38] [39] [40] [41] [42]
Map-referenced trajectories |  [6] [22] [23] [24] [25] [26] [27] [28] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [34] [65] [66] [67] |  [68] |  [69] |  [70] [71] [72]
On-board details |  [73] |  [74] |  [75][76] |  [77] [78] [79]
|  Method does not differentiate between maneuvers $|$ Method differentiates
between maneuvers but is not able to detect and classify them
---
Method differentiates between maneuvers and is able to detect and classify
them $|$ Bold letters indicate methods applied in AD simulations

###  III-A Resimulation

This section explains the scenario description approaches currently used in
resimulations and presents their advantages as well as limitations.

####  III-A1 Trajectory-based approach

The majority of literature published in the field of scenario-based simulation
uses a trajectory-based scenario description. Hereby, a georeferenced
trajectory is generated and precisely reconstructed in a simulation
environment based on the measured coordinates and relative positions of the
ego vehicle and the surrounding traffic participants [22, 23, 24, 25, 26]. On
the one hand, this enables an easy generation of trajectory-based scenarios,
only requiring the ability to track the GNSS coordinates (or comparable) of a
traffic participant. The accuracy of the resulting trajectory is only limited
by the measurement accuracy and can include detailed lateral offset as well as
velocity profiles [6]. Therefore, the trajectory-based approach is suitable in
case the aim is to exactly resimulate the course of a recorded scenario. On
the other hand, the utilization of coordinates entails the limitation that, by
this, the scenario is bound to the road geometry of the scenery in which it
was recorded [28]. Only the remaining aspects of the scenery such as
buildings, colors or the weather can be changed. Moreover, a scenario
described via a trajectory is difficult to vary as the trajectory’s internal
consistency and the consistency with the scenery must be maintained [6]. This
makes it difficult to change a vehicle’s driving behavior based on the
surrounding traffic or scenery elements. Some of the approaches also introduce
the term “maneuver”, e.g. to define variations of the trajectory [26] or limit
the scope of the extracted trajectory [22].

####  III-A2 Maneuver-based approach

In case of the maneuver-based approach, a recorded trajectory is generalized
into maneuvers. This allows the transfer of a scenario to another start
position within a given scenery, and from one scenery to another. The road
geometry can be varied. Exemplary transfers of interest are listed below:

  * •

shift of a scenario from a rural to an urban environment, potentially
characterized by other lighting effects such as dazzling lights and different
classes of traffic participants, for example pedestrians

  * •

replay of a cut-in scenario within a curve instead of a straight section of a
highway, as occurring global velocity distributions and occlusion effects
might differ

  * •

use of a left-hand instead of right-hand traffic scenery to test if the system
is still able to operate safely in countries with left-hand traffic

Besides the transferability, the maneuver-based approach allows for easier
parametrization and modification [6]. In this work, parametrization refers to
a variation of a quantity describing a maneuver such as the longitudinal
distance to the vehicle in front before initiating a lane change. A
modification, instead, changes the general course of a scenario by adding or
deleting maneuvers or changing their order. Thus, by using a maneuver-based
approach changes are not limited to small local variations as in case of the
trajectory-based approach. Instead, it allows for variations of the scenario’s
general course – in accordance with the possibly changed scenery or
surrounding traffic. Due to the previously outlined need for creating a
variety of scenarios for scenario-based testing, the maneuver-based approach
bears the potential of providing a better and more profound foundation for the
validation and verification of AD functions. However, the generalization
introduced by the maneuver-based approach also has limitations. Besides the
two open questions identified earlier, the description as such is non-trivial.
It requires semantics capable of describing any scenario observable in
reality. First approaches in this direction are presented in [80] and given by
the ASAM OpenSCENARIO111https://www.asam.net/standards/detail/openscenario/
standard. Moreover, there is the trade-off between generalization on the one
hand and the ability to accurately describe occurring _submicroscopic_
phenomena such as velocity distributions and the vehicle’s lateral offset
within its lane on the other hand. A more detailed explanation of the level
identifier “submicroscopic” will be given in Sec. III-A3.

Possibilities to apply the maneuver-based approach are presented in [6] and
[28]. The approach of [6] is taken up in [27] and transferred from highway to
intersection scenarios. In all cases, the recorded trajectory is abstracted
into a sequence of maneuvers without gaps. In [6] and [27] the approach
proposed in [30] is applied, and a vehicle’s state is at any time defined by
two overlapping maneuvers: one for the lateral and one for the longitudinal
direction. In case of [28], instead, exactly one maneuver is assigned to a
vehicle at any time. In [28] and [6] distances and absolute positions are used
as maneuver trigger conditions. In addition to that, timestamps are used in
[6]. While positions and distances could reflect the level at which also a
driver takes decisions and starts maneuvers, this does not apply for pure
timestamps. When using absolute positions as trigger conditions, the
conditions need to be modified as well when transferring the scenario from one
scenery to another or within a scenery. The methods presented in [6] and [28]
both simplify the submicroscopic behavior (see Sec. III-A3). Nevertheless,
even the transfer of a simplified lateral offset profile of a scenario from
one scenery to another yielded non-realistic behavior, as the lateral offset
behavior depends on the curvature of the road and is therefore not
transferable one-to-one [28]. This addresses the general open question of how
to perform a realistic parametrization or modification of a scenario. As long
as a pure resimulation is performed, as done in [6], no issue arises. However,
if the scenario is transferred to a scenery where for example the distance to
be travelled during a maneuver is longer, the maneuver has to reflect this.
Possibilities to achieve this are a stretching or a repetition of the maneuver
or the increase of the velocity. However, the representativity of these
approaches with respect to real-world driving behavior is thus far not
substantiated by a systematic evaluation on real-world data.

####  III-A3 Microscopic and submicroscopic traffic simulation

An alternative approach to describe the behavior of vehicles within
simulations is the use of models as used in microscopic traffic simulations
such as SUMO [37], Vissim [38], Aimsun [40], Paramics [39] and DRACULA [41].
Even though not commonly considered a maneuver-based approach, they can
provide applicable models. The route a vehicle takes through the road network
is determined on a high level through its origin and destination [37, 38, 39].
The description of the vehicle’s microscopic behavior is commonly split in a
lane-changing model for the lateral movement between lanes, and a car-
following model for the longitudinal direction [40, 37, 38, 41] such as the
Wiedemann model in Vissim [38] or the Krauß model in SUMO [37]. In
longitudinal direction, a vehicle chooses its velocity depending on the
velocity of the leading vehicle, the distance to it and further ego vehicle-
specific quantities such as its maximum deceleration ability [37, 38] as well
as infrastructure-related criteria [37, 40] and the surrounding traffic [40].
To model the imperfection of a human driver, different strategies are applied.
In SUMO [37] a stochastic deceleration is introduced, in Vissim [38] the
following vehicle does not accelerate and decelerate directly but after
predefined perception thresholds are reached. However, in all cases collisions
between vehicles are excluded by design. The lane change behavior of a vehicle
is modeled based on the tactical and strategic decisions of a driver: a
vehicle changes the lane if this is required to reach its destination on the
prescribed route or if it has an advantage by changing the lane (e.g.
proceeding faster) [37, 38, 39, 40, 41]. For the modeling of the lane change
itself, a simplified trajectory is used [38, 39]. Alternatives to the
approaches presented so far apply machine learning methods to learn the models
describing the longitudinal and lateral movement of a vehicle [42, 34]. Apart
from the lane changes, the majority of the presented methods neglects lateral
movements with respect to the road’s center line [40, 37, 41, 39]. An
exception is Vissim [38] where the lateral position of a vehicle within its
lane is based on the maximum longitudinal time-to-collision to the vehicle in
front and a lateral safety distance to vehicles on the neighboring lanes.
Under free driving conditions, it can be preset to rather left, right, center
or random. Moreover, there are papers that explicitly deal with modelling the
lateral displacement of a vehicle within its lane [67]. To improve the realism
on this level, so called _submicroscopic_ or _nanoscopic_ simulators are
developed [81, 82, 68, 36, 69]. The work in this field can be split into two
parts: the first deals with physics-based models for increased realism in
vehicle dynamics [83, 82, 68, 36, 69], while the second aims for setting up
realistic behavior models for the driver by taking into account psychological
driver characteristics such as the level of aggressiveness [82, 36, 69]. In
all cases the submicroscopic models are coupled with microscopic ones. While
the latter ones determine the long-term behavior of the vehicles, the former
ones define the details of execution.

In the field of microscopic traffic simulation, the driver’s behavior and the
decisions behind it are described on a high level. However, one problem that
the approaches referenced before have in common is that they are calibrated to
get a representative macroscopic traffic flow [37, 40, 41], while the
individual vehicle behavior might not be realistically modeled. To solve this
issue, some recent approaches use trajectory data for calibration [70, 71].
Another aspect revealing a lack of realism is the complete absence of
collisions enforced through the model design. Consequently, these approaches
are not sufficient when aiming for the immediate safety evaluation of an AD
function such as a cut-in detection. When using a simulator with vehicles
always driving in the center of the lane for the training of a cut-in
detection, the algorithm would learn that whenever a vehicle is slightly
leaving the center of the lane, it will definitely change the lane.
Consequently, using such a model for validation would fail to evaluate the
actual capabilities of a given detector, leaving substantial uncertainty to
what degree the function is safe under realistic traffic conditions.
Similarly, it is not possible to draw a conclusion on whether a system could
have avoided an accident if these were in beforehand excluded by the model.

###  III-B Maneuver detection, prediction and classification

The aim of research performed in this field is the detection of maneuvers in
data such as real-world vehicle trajectories and their classification. Both
tasks can be either performed online while the maneuver takes place, or
offline and retrospectively on the full driving data. If the goal is to detect
a maneuver before it starts or immediately after it has started, it is also
referred to as maneuver prediction [43, 77, 78, 64]. Insights gained and
methods developed in the field of maneuver detection and classification could
provide a fruitful ground for the advancement of the maneuver-based approach.
Therefore, the state of the art is assessed within this section. Methods for
maneuver detection and classification are developed since driver assistance
systems need to have high situational awareness of the ego vehicle’s
environment as well as the driver’s state [43]. For this, it is important for
the system to predict maneuvers of the ego vehicle as well as surrounding
vehicles. By this, an AD function is enabled to align its reactions with the
intended behavior of the driver [44, 79], to remind the driver of actions that
need to be taken [77], to warn him or her in case the intended maneuver cannot
be executed safely [43] or just keep the driver informed about the environment
[84]. Examples for maneuvers whose detection and classification is desired
apart from the road following maneuver are turning maneuvers [77, 45, 46, 43,
79], lane change maneuvers [64, 79, 77, 44, 85, 47, 48, 46, 49, 50, 51, 52,
43, 75, 55, 53], braking [50], and drifts into another lane [54]. For the
detection and classification of these maneuvers, a wide variety of methods
such as clustering algorithms [46, 53, 73], support vector machines [44, 47]
hidden Markov models [49, 54], neural networks [52, 43, 79, 47, 48, 79, 64,
56], dynamic Bayesian networks [57, 55], decision trees [78] and random forest
[47] are applied. Just as the used algorithms, also the data fed to those in
order to detect and classify maneuvers are diverse. The predominantly used
features are physical quantities, especially the longitudinal and lateral
coordinates of the vehicle for which the maneuver is expected [45, 86, 50, 51,
44, 53, 85, 55], referred to as target vehicle. Beside the position
information, also the longitudinal and lateral velocity and acceleration of
the target vehicle are of importance [45, 49, 46, 51, 85, 48, 75, 55]. Further
physical and vehicle-related quantities used for the detection and
classification of maneuvers are the yaw angle [45, 48, 55], the yaw rate [48,
57, 75, 55], turn and brake indicators [57, 75], pedal positions [75] and
dimensions [55] of the target vehicle. Features that can be fed into the model
to also include the road geometry and topology are the curvature of the road
and the existence of a left or right lane next to the target vehicle [57, 79,
55]. To take into account the surrounding traffic, relative positions,
velocities and accelerations between vehicles [53, 46, 57, 75] are considered.
A step further is made in [50], [56] and [51] where also the overall traffic
situation is considered for the maneuver detection and classification. In [64]
the surrounding traffic as well as road conditions are included implicitly by
using video data. When only focusing on the detection and classification of
ego vehicle maneuvers, further features of the driver such as physiological
signals [77], the head movement [79, 75] and personality traits [78] can be
used.

As outlined, a great variety of maneuver sets, methods for their detection and
classification and features fed to these methods exists in literature. The
used maneuvers are characterized by specific patterns that can be found in the
features. This is exploited by the applied methods. To achieve a realistic
modeling of the vehicle’s submicroscopic behavior in a simulation, these
characteristic patterns need to be reproduced under the given circumstances
such as the ego vehicle’s state, the surrounding traffic and the road
structure. The maneuvers used in the cited papers can give orientation for the
selection of a maneuver set for the scenario description and the distinction
between maneuvers. Thus, they can help to answer the open questions identified
in Sec. II. For instance, it needs to be clarified when a lane change starts
and ends. While there is a common understanding that the “middle” of the lane
change is the crossing of the lane marking [64, 48], there are different
understandings concerning its start and end. In some cases, a lane change is
defined by a time window around the crossing of the lane marking [48], in
others by a spatial area [34] or the start and end of lateral movement [63].

###  III-C Trajectory prediction and modeling

Trajectory prediction is closely related to maneuver detection and
classification. In contrast to maneuver prediction, trajectory prediction does
not aim for predicting the next maneuver of a vehicle out of a discrete
maneuver set, but its detailed upcoming course [87]. This, in turn, usually
depends on the maneuver: while a lane keeping maneuver yields a trajectory
staying on the vehicle’s current lane, an upcoming lane change needs to be
followed by a trajectory continuing on a neighboring lane. Therefore,
trajectory prediction is often preceded by a maneuver prediction which
determines the selection of the maneuver-dependant trajectory prediction
algorithm [53]. Also these techniques find application in AD functions. They
are for example applied to target vehicles to plan the ego vehicle’s
trajectory accordingly [53, 58, 48, 86, 54, 59, 35] or to the ego vehicle,
e.g. to adapt the driving style of an AD function to the one of the driver to
increase the driver’s comfort [60, 63]. Besides the field of trajectory
_prediction_ for real-world situations, there is also the field of trajectory
_modeling_ , aiming for generating a realistic vehicle trajectory within a
simulation [76, 72, 74]. A frequently applied method for trajectory prediction
and modeling are long short-term memory networks [53, 72, 86, 61, 74],
however, also other variants of neural networks [48, 60, 58, 59, 74] as well
as Gaussian Mixture Models [54, 63], polynomial fitting [62] and stochastic
methods [76, 35, 65, 66] are used. Depending on the application, in some cases
the goal is to construct an exact trajectory [86, 61, 62, 76, 76, 72, 74], for
example when aiming for a personalized driving function or a trajectory within
a simulation, in other cases the aim is to return a set of possible
trajectories, combined with a probability for each of them, or a fused
trajectory [53, 54, 35, 65, 66]. This becomes important when the aim is to
predict the trajectory of another vehicle to plan the ego vehicle’s behavior.
The prediction of the upcoming trajectory is based on the previous trajectory
[48, 53, 86, 59, 61, 65] as well as velocity and acceleration data [60, 58,
59, 61, 62, 63, 65]. As in the case of maneuver prediction, also in this case
surrounding traffic [58, 61, 63, 66] and the road structure [59, 62, 65] are
taken into account. There are also approaches in which the modeling goes down
to driver [76] and vehicle level [76, 74] meaning that for example the
driver’s perception and pedal position are taken into account. The outcome of
the trajectory prediction is manifold. Apart from a direct prediction of the
next position(s) [48, 61, 62], there exist methods predicting the lateral and
longitudinal trajectory separately [58] or delivering a chain of velocities or
accelerations from which the positions can be derived [54, 61, 66]. Due to the
usual dependence of the trajectory on the upcoming maneuver, there are also
methods dealing with trajectory prediction for a certain maneuver such as a
lane change [60, 59, 63].

As in case of trajectory prediction, also when deriving models for the
submicroscopic vehicle behavior, the aim is to describe the vehicle’s dynamic
quantities with a high temporal and spatial resolution. However, the focus of
both methods differs. In case of trajectory prediction, the aim is to get a
high accuracy for the immediately upcoming vehicle behavior while the goal of
a submicroscopic behavior model is to get a realistic behavior of the vehicle
for the full duration of a maneuver. Furthermore, prediction tasks typically
aim to optimize extrinsic quantities, such as providing a trajectory of
minimum expected error w.r.t. all possible outcomes, or a conservative
estimate maximizing safety for the AD function, whereas, when used for
simulation, submicroscopic behavior models aim at sampling realistic
candidates from the modeled distribution, including candidates with low
likelihood. Measuring the degree of realism of modeled behavior, and
determining the required degree of realism for a given simulation task, is an
open question in its own. As the methods developed within the field of
trajectory prediction are limited to short-term prediction, they are not
suitable for predicting the course of a vehicle for the full duration of a
road following maneuver. However, they can be used to vary the course a
vehicle takes during a maneuver that – in case of a simulation – is known in
advance, to get a realistic imperfect submicroscopic driving behavior. Due to
the comparatively low duration of a lane change maneuver, these maneuvers
could be an exception and trajectory prediction models might be applicable.
The methods for trajectory modeling directly address the task of providing a
representative trajectory and therefore are promising candidates to build on
further work in the field of submicroscopic behavior models.

###  III-D Synopsis

All methods presented within this section have been placed in Tab. III-D.
Almost every combination of model granularity and fitting data granularity is
covered. The predominant share of papers differentiates between maneuvers.
Within those, the number of methods being able perform the detection and
classification themselves and those not being able to do so is almost equal.
As soon as the model granularity goes down to on-board details, all papers
apply a maneuver differentiation. This could indicate that the relevance of
maneuvers increases if on-board details are considered within the model. More
than half of the reviewed papers’ models reach down to a granularity of map-
referenced trajectories and were fitted with data of that same granularity.
This is exactly the minimum model and fitting data granularity needed for
advancing the maneuver-based approach, as map-referenced trajectories
represent the level visible to an AD function. To ensure realistic behavior of
the model at this granularity, the fitting data should have at least the same
granularity. Considering also human factors and vehicle dynamics-related
phenomena in models for the maneuver-based approach and enriching the fitting
data with on-board details could be useful to relate the characteristics
observed in resulting trajectories to their in-vehicle origin and increase
their representativity for real-world driving behavior.

Only three of the papers listed in Tab. I use the maneuver-based approach for
resimulation – with the limitations identified earlier. This hints at a
research gap in the field of maneuver-based resimulation. However, as outlined
earlier and demonstrated by the classification in Tab. I, alternative
resimulation approaches and methods for maneuver detection and classification
and trajectory prediction and modeling address similar issues and thus provide
a valuable foundation for an advancement of the maneuver-based approach.

##  IV Conclusion

This paper presents the approach of using a maneuver-based scenario
description within AD simulations and highlights its advantages as well as
current limitations. The maneuver-based approach offers more degrees of
freedom when varying a recorded scenario but its description is non-trivial.
Prerequisites for a maneuver-based scenario description are the selection of a
maneuver set and respective trigger conditions. To exploit the full potential
of the maneuver-based approach the performed literature review indicates the
need to use high-level maneuvers at the level on which a driver plans
maneuvers, combined with submicroscopic behavior models and trigger conditions
derived from the static environment or surrounding traffic participants. Based
on this research gap, related research fields that can serve as starting point
for further research were identified and categorized.

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# Minimal Lengths in 3D via the Generalized Uncertainty Principle

Michael Bishop<EMAIL_ADDRESS>Mathematics Department,
California State University Fresno, Fresno, CA 93740  Joey Contreras
<EMAIL_ADDRESS>Physics Department, California State University
Fresno, Fresno, CA 93740  Peter Martin<EMAIL_ADDRESS>Physics
Department, California State University Fresno, Fresno, CA 93740  Piero
Nicolini<EMAIL_ADDRESS>Dipartimento di Fisica, Università degli
Studi di Trieste, Trieste, Italy
Frankfurt Institute for Advanced Studies (FIAS), Frankfurt am Main, Germany
Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität-Frankfurt
am Main, Frankfurt am Main, Germany  Douglas Singleton
<EMAIL_ADDRESS>Physics Department, California State University
Fresno, Fresno, CA 93740  Kavli Institute for Theoretical Physics, University
of California Santa Barbara, Santa Barbara, CA 93106, USA

###### Abstract

We investigate an extension of the Generalized Uncertainty Principle (GUP) in
three dimensions by modifying the three dimensional position and momentum
operators in a manner that remains coordinate-independent and retains as much
of the standard position-momentum commutators as possible. Moreover, we bound
the physical momentum which leads to an effective minimal length in every
coordinate direction. The physical consequences of these modified operators
are explored in two scenarios: (i) when a spherically-symmetric wave function
is ‘compressed’ into the smallest possible volume; (ii) when the momentum is
directed in a single direction. In case (ii), we find that the three
dimensional GUP exhibits interesting phenomena that do not occur in one
dimension: the minimal distance in the direction parallel to a particle’s
momentum is different from the minimal distance in the orthogonal directions.

##  I Introduction

The Generalized Uncertainty Principle (GUP) and its associated minimal length
scale, or maximum energy-momentum scale, are considered fruitful
phenomenological approaches to quantum gravity vene ; gross ; amati2 ; amati ;
maggiore ; garay ; KMM ; scardigli ; adler-1999 . The GUP approach to quantum
gravity has led to testable predictions and observations, e.g. explanations
for photon dispersion in vacuum AC-nature  and in various table-top laboratory
experiments vagenas ; vagenas2 . While many works mention the three
dimensional generalization of the position-momentum commutators and the
associated three dimensional uncertainty relationships, they generally
restrict their GUPs to one spatial dimension. There are a few exceptions
including KMM  and more recently the work in todrinov , which looked at a
fully relativistic GUP in four space-time dimensions.

In this work, we want to explore the GUP in three spatial dimensions and note
some new features which are present in three spatial dimensions, but are
absent in a single spatial dimension. This is similar to the theory of special
relativity where boosts were introduced in only one spatial dimension and
later expanded to three dimensions. Looking at special relativity in three
spatial dimensions leads not only to a more complex form for the boosts, but
also gives rise to the mixing of boost transformations, rotations, and related
effects such as Thomson precession. In a similar way to Lorentz boosts, we
find that certain variations of three dimensional GUPs lead to an interplay
between the different coordinate directions, resulting in different minimal
distances in the directions orthogonal to the direction of momentum.

It is interesting to notice that a similar behavior has already been observed
in the context of string inspired noncommutative geometry carroll ; smailagic
; spallucci . Noncommutative relations determine a foliation of the spacetime
in noncommutative planes. If, on the one hand, the foliation is an invariant
character, on the other hand, on each of such planes the minimal length
transforms like a antisymmetric second rank tensor, i.e., the Kalb-Ramond
field – see e.g. seiberg .

##  II Minimal Lengths in one dimension

In this section we will review the general construction of a one dimensional
GUP, and introduce a specific form of the GUP that we will focus on in this
work. The specific GUP studied here will modify the position and momentum
operators, while trying to retain the standard commutator between conventional
position and momentum.

Most works on GUP start with a modified commutator of the form

$[{\hat{X}},{\hat{P}}]=i\hbar
F\left(\frac{{\hat{x}}}{x_{m}},\frac{{\hat{p}}}{p_{M}}\right),$ (1)

where $F\left(\frac{{\hat{x}}}{x_{m}},\frac{{\hat{p}}}{p_{M}}\right)$ is a
function of the canonical position and momentum operators which satisfy
$[{\hat{x}},{\hat{p}}]=i\hbar$; and the capitalization on $\hat{X}$ and
$\hat{P}$ is used to denote the modified operators. The constants $x_{m}$
($p_{M}$) set the small distance (large momentum) cut-off scale. When $\Delta
x\gg x_{m}$ and/or $\Delta p\ll p_{M}$, the ansatz function has the limit
$\langle
F\left(\frac{{\hat{x}}}{x_{m}},\frac{{\hat{p}}}{p_{M}}\right)\rangle\to 1$,
and the modified commutator becomes the standard one for large distances
and/or small momenta. A common example of such a modified commutator is one
from KMM  where
$F\left(\frac{{\hat{x}}}{x_{m}},\frac{{\hat{p}}}{p_{M}}\right)=1+\beta{\hat{p}}^{2}=1+\frac{{\hat{p}}^{2}}{p_{M}^{2}}$,
with $\beta\equiv p_{M}^{-2}$, to connect our notation with that of KMM .

With a modified commutator defined, it is now necessary to find corresponding
modified quantum operators that lead to that modified commutator. An
acceptable modification would be to modify the operators in momentum space to
have the form

${\hat{X}}=i\hbar
G\left(\frac{{\hat{x}}}{x_{m}},\frac{{\hat{p}}}{p_{M}}\right)\partial_{p}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}{\hat{P}}=H\left(\frac{{\hat{x}}}{x_{m}},\frac{{\hat{p}}}{p_{M}}\right)~{}{\hat{p}}{}~{},$
(2)

and choose $G$ and $H$ so as to yield (1).

For example, with
$F\left(\frac{{\hat{x}}}{x_{m}},\frac{{\hat{p}}}{p_{M}}\right)=1+\beta{\hat{p}}^{2}$
the operators can be defined as ${\hat{X}}=i\hbar(1+\beta p^{2})\partial_{p}$
and ${\hat{P}}={\hat{p}}=p$. Here, the function $F$ depends only on the
momentum and not the position, and the momentum operator is simply the
canonical momentum operator i.e. multiplication by $p$. From here, we will
take this specialized case where the ansatz functions modify the position and
momentum operators which leads to a commutator that depends only on the
canonical momentum, $p$.111Also from this point onward we will not be overly
careful about distinguishing ${\hat{p}}$ from the momentum input variable $p$
since we consider these operators as acting on wave functions defined in
momentum space, i.e. $\psi(p)$. Another acceptable set of modified operators
which gives the same modified commutator from (1) is
${\hat{X}}={\hat{x}}=i\hbar\partial_{p}$ and
${\hat{P}}=p\left(1+\frac{1}{3}\beta p^{2}\right)$,

In aiken-2019 ; BLS  it was shown that there are a host of different ways to
modify the position and momentum operators that lead to the same modified
commutator. However, in some cases these different modifications of the
operators, while giving the same modified commutator, did not lead to a
minimum distance. For this reason, the works BLS ; BJLS  assert that it is
primarily the form of modification of the operators, rather than the form of
the modified commutator, that determines a minimal length scale in GUP models.
This contrasts with the more common approach of first choosing the modified
commutator and then choosing the (non-unique) modified operators which give
this modified commutator.

Reference BJLS  takes the ansatz functions $G$ and $H$ in (2) such that
${\hat{X}}$ and ${\hat{P}}$ satisfied the standard commutator relationship

$[{\hat{X}},{\hat{P}}]=i\hbar.$ (3)

$H$ was chosen such that when $p\ll p_{M}$,
$\hat{P}=p~{}H\left(\frac{p}{p_{M}}\right)\approx p$; and such that when $p\gg
p_{M}$, $\hat{P}=p~{}H\left(\frac{p}{p_{M}}\right)\approx p_{M}$, t cap
momentum at $p_{M}$.

For the ansatz function $G\left(\frac{p}{p_{M}}\right)$, we require
$G\left(\frac{p}{p_{M}}\right)\to 1$ for $\Delta p\ll p_{M}$. The limit of
$G\left(\frac{p}{p_{M}}\right)$ for $p\gg p_{M}$ is fixed below by requiring
that the modified position and momentum operators still satisfy the standard
commutator (3). These choices for the ansatz functions still give a minimum in
position because of the following:

  1. (i)

The general form of the ansatz function $H$ gives ${\hat{P}}\leq p_{M}$.

  2. (ii)

Since $G\left(\frac{p}{p_{M}}\right)$ and $H\left(\frac{p}{p_{M}}\right)$ are
fixed to maintain $[{\hat{X}},{\hat{P}}]=i\hbar$, this yields the standard
uncertainty principle $\Delta X\Delta P\geq\frac{\hbar}{2}$.

  3. (iii)

Finally, this leads to a minimum uncertainty in position, $\Delta
X\geq\frac{\hbar}{2p_{M}}$.

Now, one obtains a condition on $G$ and $H$,

$G\left(\frac{p}{p_{M}}\right)\partial_{p}\left[H\left(\frac{p}{p_{M}}\right)~{}p\right]=1.$
(4)

In BJLS , two specific sets of ansatz functions where given which satisfied
the conditions above,

$G\left(\frac{p}{p_{M}}\right)=\cosh^{2}\left(\frac{p}{p_{M}}\right)~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}H\left(\frac{p}{p_{M}}\right)=\frac{p_{M}}{p}\tanh\left(\frac{p}{p_{M}}\right)~{},$
(5)

and

$G\left(\frac{p}{p_{M}}\right)=\left[1+\left(\frac{\pi
p}{2p_{M}}\right)^{2}\right]~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}H\left(\frac{p}{p_{M}}\right)=\frac{2p_{M}}{\pi
p}\arctan\left(\frac{\pi p}{2p_{M}}\right)~{}.$ (6)

The ansatz functions in (5) and (6) satisfy condition (4).

In both cases, the modified operators, ${\hat{X}}$ and ${\hat{P}}$, formulated
by using the $G$’s and $H$’s in (2) satisfy the standard commutator
$[{\hat{X}},{\hat{P}}]=i\hbar$, and the related standard uncertainty $\Delta
X\Delta P\geq\frac{\hbar}{2}$. One might think recovering the standard
commutator (3) would not lead to a minimum length, but due to the modified
behavior of ${\hat{P}}$, and therefore $\Delta P$, one still finds a non-zero
minimum distance. In conclusion, the crux of the question is if operators are
capped or not. Physical operators must display ultraviolet
convergence/finiteness. Uncapped operators have only a mathematical meaning,
although can be used as auxiliary variables in calculations. The algebra of
operators is per se irrelevant to discriminate the short distance nature of
observables.

##  III Generalizing to Three Dimensions

In this section, we generalize the above construction from one spatial
dimension to three spatial dimensions. The modified operators must now become
three-vector operators – ${\hat{X}}\to{\hat{X}}_{i}$ and
${\hat{P}}\to{\hat{P}}_{i}$, where $i=1,2,3$. This can be done by applying
different conditions to the ansatz functions, $G$ and $H$ as we now discuss in
detail.

For the hyperbolic ansatz functions of (5), one could try letting
$H\left(\frac{p}{p_{M}}\right)=\frac{p_{M}}{p_{i}}\tanh\left(\frac{p_{i}}{p_{M}}\right)$,
which gives a modified momentum of the form
${\hat{P}}_{i}=p_{i}H\left(\frac{p}{p_{M}}\right)=p_{M}\tanh\left(\frac{p_{i}}{p_{M}}\right)$.
This would have a low momentum limit of ${\hat{P}}_{i}\approx p_{i}$, and a
high momentum limit of $p_{M}$ in the direction of $p_{i}$. The vector
character of the modified position operator arises through
$\partial_{p}\to\partial_{p_{i}}$, and the ansatz function becomes
$G\left(\frac{p_{i}}{p_{M}}\right)=\cosh^{2}\left(\frac{p_{i}}{p_{M}}\right)$.
In order to satisfy (4), the $\cosh^{2}$ term for the modified position
operator has to have the same functional dependence as the $\tanh$ term in the
modified momentum operator. Note that condition (4) becomes
$G\left(\frac{p_{i}}{p_{M}}\right)\partial_{p_{i}}\left[p_{j}H\left(\frac{p_{j}}{p_{M}}\right)\right]=\delta_{ij}$
in this three dimensional case.

This approach has a fundamental problem, though. $G$ and $H$ are functions of
$p_{i}$, which spoils the rotational symmetry of the modified position and
momentum operators, as discussed in section 6.2 of KMM .

In order to preserve rotational symmetry and still retain the feature that the
momentum operators have upper bounds of the form given in section II, we need
ansatz functions of the form

$G\left(\frac{|{\vec{p}}|}{p_{M}}\right)=\cosh^{2}\left(\frac{|{\vec{p}}|}{p_{M}}\right)~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}H\left(\frac{|{\vec{p}}|}{p_{M}}\right)=\frac{p_{M}}{|{\vec{p}}|}\tanh\left(\frac{|{\vec{p}}|}{p_{M}}\right)~{},$
(7)

or for the arctan modification

$G\left(\frac{|{\vec{p}}|}{p_{M}}\right)=\left[1+\left(\frac{\pi|{\vec{p}}|}{2p_{M}}\right)^{2}\right]~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}H\left(\frac{|{\vec{p}}|}{p_{M}}\right)=\frac{2p_{M}}{\pi|{\vec{p}}|}\arctan\left(\frac{\pi|{\vec{p}}|}{2p_{M}}\right)~{}.$
(8)

These ansatz functions in (7) and (8) lead to modified three dimensional
position and momentum operators of the form

${\hat{X}}_{i}=i\hbar\cosh^{2}\left(\frac{|{\vec{p}}|}{p_{M}}\right)\partial_{p_{i}}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}{\hat{P}}_{i}=p_{i}~{}H\left(\frac{|{\vec{p}}|}{p_{M}}\right)=p_{i}\frac{p_{M}}{|{\vec{p}}|}\tanh\left(\frac{|{\vec{p}}|}{p_{M}}\right)~{},$
(9)

or for the arctan modification

${\hat{X}}_{i}=i\hbar\left[1+\left(\frac{\pi|{\vec{p}}|}{2p_{M}}\right)^{2}\right]\partial_{p_{i}}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}{\hat{P}}_{i}=p_{i}~{}H\left(\frac{|{\vec{p}}|}{p_{M}}\right)=p_{i}\frac{2p_{M}}{\pi|{\vec{p}}|}\arctan\left(\frac{\pi|{\vec{p}}|}{2p_{M}}\right)~{}.$
(10)

The modified position and momentum operators in (9) and (10) are the three
dimensional generalizations of (5) and (6), which satisfy rotational symmetry
and still have capped momentum.

We will now show that the position and momentum operators’ dependence on
$|{\vec{p}}|$ causes the standard commutator (i.e.,
$[{\hat{X}}_{i},{\hat{P}}_{j}]=i\hbar\delta_{ij}$) to no longer necessarily be
maintained in all directions. The standard commutator can be maintained in one
direction but at the cost of altering the commutator in the two orthogonal
directions, resulting in there being different minimal distances in different
directions. In one dimension, the condition for the commutator to retain its
standard form was given by (4). To generalize (4) to three dimensions one
would like to require

$G\left(\frac{|{\vec{p}}|}{p_{M}}\right)\partial_{p_{i}}\left[p_{j}~{}H\left(\frac{|{\vec{p}}|}{p_{M}}\right)\right]\stackrel{{\scriptstyle?}}{{=}}\delta_{ij}~{},$
(11)

where the question mark indicates we are asking if it is possible for the left
hand side to equal the right hand side.

The complication that arises with (11), that is not found in (4), is that the
momentum derivative operates on all the components of momentum “hidden” in the
function in square brackets due to the $|{\vec{p}}|$ dependence of $H$. This
leads to $p_{i}$ and $p_{j}$ terms that are not found in the one dimensional
case.

For instance, for the $\tanh$ modification of the momentum, condition (11)
becomes

$\displaystyle
G\left(\frac{|{\vec{p}}|}{p_{M}}\right)\partial_{p_{i}}\left[p_{j}~{}H\left(\frac{|{\vec{p}}|}{p_{M}}\right)\right]$
$\displaystyle=$
$\displaystyle\left[\frac{p_{M}}{|{\vec{p}}|}\sinh\left(\frac{|{\vec{p}}|}{p_{M}}\right)\cosh\left(\frac{|{\vec{p}}|}{p_{M}}\right)\left(\delta_{ij}-\frac{p_{j}p_{i}}{|{\vec{p}}|^{2}}\right)+\frac{p_{j}p_{i}}{|{\vec{p}}|^{2}}\right]$
(12) $\displaystyle=$
$\displaystyle\delta_{ij}+\mathcal{O}\left(\frac{|\vec{p}|^{2}}{p_{M}^{2}}\right)~{},$

and the $\arctan$ modification of the momentum, condition (11) becomes

$\displaystyle
G\left(\frac{|{\vec{p}}|}{p_{M}}\right)\partial_{p_{i}}\left[p_{j}~{}H\left(\frac{|{\vec{p}}|}{p_{M}}\right)\right]$
$\displaystyle=$
$\displaystyle\left[\left(1+\left(\frac{\pi|{\vec{p}}|}{2p_{M}}\right)^{2}\right)\frac{2p_{M}}{\pi|{\vec{p}}|}\arctan\left(\frac{\pi|{\vec{p}}|}{2p_{M}}\right)\left(\delta_{ij}-\frac{p_{j}p_{i}}{|{\vec{p}}|^{2}}\right)+\frac{p_{j}p_{i}}{|{\vec{p}}|^{2}}\right]$
(13) $\displaystyle=$
$\displaystyle\delta_{ij}+\mathcal{O}\left(\frac{|\vec{p}|^{2}}{p_{M}^{2}}\right)~{}.$

In the second lines of (12) and (13), we expanded the functions in square
brackets on the right hand side to first order in $|{\vec{p}}|/p_{M}$. To
first order, one can recover the standard commutators in three dimensions.
However, to second order, there is a difference with the one dimensional case,
and the expressions in (12) and (13) are no longer equal to solely
$\delta_{ij}$.

Using the modified position and momentum from (9) and (10), and the results of
(12) and (13), we find that the modified commutator for the $\tanh$
modification, to second order, is

$[{\hat{X}}_{i},{\hat{P}}_{j}]\approx
i\hbar\left[\delta_{ij}+\frac{|{\vec{p}}|^{2}}{2p_{M}^{2}}\left(\delta_{ij}-\frac{p_{j}p_{i}}{|{\vec{p}}|^{2}}\right)\right]~{}.$
(14)

For the $\arctan$ modification there is a bit if a subtlety. From (13), the
term we want to expand is of the form $(1+x^{2})\frac{\arctan(x)}{x}$, with
$x\equiv\frac{\pi|{\vec{p}}|}{2p_{M}}$. Due to the $x$ in the denominator of
this expression, one needs to expand $\arctan$ to ${\cal O}(x^{3})$ namely
$\arctan(x)\approx x-\frac{x^{3}}{3}$. In this way (13) to ${\cal O}(x^{2})$
is

$[{\hat{X}}_{i},{\hat{P}}_{j}]\approx
i\hbar\left[\delta_{ij}+\frac{2}{3}\left(\frac{\pi|{\vec{p}}|}{2p_{M}}\right)^{2}\left(\delta_{ij}-\frac{p_{j}p_{i}}{|{\vec{p}}|^{2}}\right)\right]~{}.$
(15)

The second order approximations in (14) and (15), are similar to the modified
commutators in KMM  in that the modifications are quadratic in $|{\vec{p}}|$.
However, unlike the one dimensional case, one can no longer recover the
standard commutator (3) with the modified operators from (9) and (10).

##  IV Physical consequences of three dimensional GUP

We now want to investigate the physical consequences of the modified
commutators of the previous section on the resulting uncertainty principle.
The left hand side of the uncertainty principle will be the product of the
variation of the modified position and momentum operators, $\Delta X_{i}\Delta
P_{j}$, where for a general operator ${\hat{A}}$ the variation is $\Delta
A=\sqrt{\langle{\hat{A}}^{2}\rangle-\langle{\hat{A}}\rangle^{2}}$. The right
hand side of the uncertainty relationship involves taking the expectation of
the right hand side of (12) or (13). However, the presence of the hyperbolic
functions in (12) or the inverse tangent function in (13) makes it hard to
take the expectation values and get an analytical form. In order to more
easily calculate the expectation values, we will instead use the second order
commutators given in (14) and (15). Even after this, we will need to make
another approximation to obtain the final form of the uncertainty principle.

We will look at two different cases for evaluating the physical consequences
of the 3D GUP: (a) We assume a spherically symmetric wave function so that the
average of the modified momentum and position operators is zero so that
$\Delta X_{i}=\sqrt{\langle{\hat{X}_{i}}^{2}\rangle}$, and $\Delta
P_{j}=\sqrt{\langle{\hat{P}_{j}}^{2}\rangle}$. (b) We assume that the system
has a non-zero, average momentum in one direction (e.g. the $i=1$ direction)
so that $\langle{\hat{P}}_{i}\rangle\neq 0$).

For both cases, the minimum distance will be proportional to the inverse of
the cut-off scale $\Delta X^{min}\propto\frac{1}{p_{M}}$, up to numerical
factors of ${\cal O}(1)$. For the second case, with an average momentum in one
direction, we find that the minimum position uncertainty in the direction of
the momentum is smaller than in the two perpendicular directions.

###  IV.1 Spherically Symmetric Case

In this subsection we study the case where the wave function is spherically
symmetric in momentum space. We use this to explore how focused/concentrated a
stationary particle can get in position space in these theories. We we utilize
the spherical symmetry to make the substitution $\langle
p_{j}p_{i}\rangle=\delta_{ij}\langle p_{i}^{2}\rangle$.

Starting with the hyperbolic case (14), the modified uncertainty relationship
is

$\Delta X_{i}\Delta
P_{j}\geq\frac{\hbar}{2}\left(\delta_{ij}+\frac{\delta_{ij}}{2p_{M}^{2}}\langle|{\vec{p}}|^{2}-p_{i}^{2}\rangle\right)=\frac{\hbar}{2}\delta_{ij}\left(1+\frac{\langle
p_{\perp}^{2}\rangle}{2p_{M}^{2}}\right)~{},$ (16)

where $\langle p_{\perp}^{2}\rangle=\langle|{\vec{p}}|^{2}-p_{i}^{2}\rangle$
is the expectation of the square of the perpendicular part of the momentum. We
can use the symmetry to write $\langle p_{\perp}\rangle^{2}=(d-1)\langle
p_{i}^{2}\rangle$, where $d$ is the number of spatial dimensions. In this
case, $d=3$ and $|p_{\perp}|^{2}=2|p_{i}|^{2}$ so (16) becomes

$\Delta X_{i}\Delta
P_{j}\geq\frac{\hbar}{2}\delta_{ij}\left(1+\frac{\langle|p_{i}|^{2}\rangle}{p_{M}^{2}}\right)~{}.$
(17)

Next, we need to estimate $\Delta P_{j}$,

$\Delta
P_{j}=\sqrt{\langle({\hat{P}}_{j})^{2}\rangle-\langle{\hat{P}}_{j}\rangle^{2}}\leq\sqrt{\langle({\hat{P}}_{j})^{2}\rangle}=\sqrt{\left\langle\frac{{\hat{p}}_{j}^{2}p_{M}^{2}}{|\vec{p}|^{2}}\tanh^{2}\left(\frac{|{\vec{p}}|}{p_{M}}\right)\right\rangle}$
(18)

where we take $\langle{\hat{P}}_{j}\rangle=0$; and bounding $\tanh^{2}(x)\leq
x^{2}$, equation (18) yields

$\Delta P_{j}\leq\sqrt{\left\langle\hat{p}_{j}^{2}\right\rangle}=\Delta
p_{j}~{},$ (19)

where the uncertainty of the modified momentum is bound by the uncertainty of
the canonical momentum. Combining (17) and (19) gives a $\Delta X_{i}$ of

$\Delta X_{i}\geq\frac{\hbar}{2}\left(\frac{1}{\Delta p_{i}}+\frac{\Delta
p_{i}}{p_{M}^{2}}\right)~{}.$ (20)

The minimum of (20) occurs at $\Delta p_{i}=p_{M}$ which then gives

$\Delta X^{min}_{i}=\frac{\hbar}{p_{M}}.$ (21)

The structure for this 3D GUP given by (17) (20) and (21) is very similar to,
but not exactly identical to, the original KMM model KMM .

One could also do a rougher, but simpler, approximation and get close to the
same result as (21): instead of (17), simply take $\Delta X_{i}\Delta
P_{j}\geq\frac{\hbar}{2}\delta_{ij}$; and instead of (19) one could take
$\Delta P_{j}\leq p_{M}$. This yields $\Delta
X_{i}^{min}=\frac{\hbar}{2p_{M}}$ which is of the same order as (21). There is
not a big difference between this rough estimate for $\Delta X_{i}^{min}$ and
the more refined one in (21).

We now repeat this analysis for the $\arctan$ modification given by (13). At
first glance, it seems this is straight forward, but there is a subtle issue
as compared to the previous $\tanh$ case. For the $\tanh$ modification, in
going from (12) to the approximation in (14), we note that the right hand side
of (14) is a lower bound to (12), thus in (16) we are justified in using
$\geq$. However since the expansion for $\arctan$ has alternating signs, the
approximation in (15) is not always a lower bound on (13). This makes getting
the uncertainty relationship coming from (13) more complicated.

One could make a similar approximation to the rough approximations mentioned
after (21). Note that the $(1+x^{2})\frac{\arctan(x)}{x}$ term from (13) (here
$x=\frac{\pi|{\vec{p}}|}{2p_{M}}$) has a lower bound of $1$, i.e.
$(1+x^{2})\frac{\arctan(x)}{x}\geq 1$. Replacing this term in (13) by $1$
gives $[X_{i},P_{j}]=i\hbar\delta_{ij}$, which then gives $\Delta X_{i}\Delta
P_{j}\geq\delta_{ij}\frac{\hbar}{2}$. As was for the $\tanh$ modification, the
modified momentum uncertainty for the $\arctan$ modification has $\Delta
P_{j}\leq p_{M}$, which then gives the same minimum as for the $\tanh$
modification. Namely, $\Delta X^{min}_{i}=\frac{\hbar}{2p_{M}}$. The fact that
this rough estimate for $\Delta X^{min}_{i}$ is the same for both $\tanh$ and
$\arctan$ modification is not surprising. To this level of approximation, the
right hand side of the commutator is the standard one, $i\hbar\delta_{ij}$.
$\Delta X^{min}_{i}$ ultimately comes from the cut-off in the momentum
uncertainty, which is the same in both cases, namely $\Delta P_{j}\leq p_{M}$.

To get a better estimate for $\Delta X^{min}_{i}$, we need a finer lower bound
on $(1+x^{2})\frac{\arctan(x)}{x}$. This is given by the function
$(1+x^{2})^{1/2}$, i.e. $(1+x^{2})\frac{\arctan(x)}{x}\geq(1+x^{2})^{1/2}$.
The fact that $(1+x^{2})^{1/2}$ provides a lower bound to
$(1+x^{2})\frac{\arctan(x)}{x}$ is not obvious (there are other possible
functions that may provide a tighter lower bound), but can be verified by
expanding both functions, or by graphing them.

In any case, replacing $(1+x^{2})\frac{\arctan(x)}{x}$ by $(1+x^{2})^{1/2}$ in
(13) and expanding to second order in $x=\frac{\pi|{\vec{p}}|}{2p_{M}}$ gives

$[{\hat{X}}_{i},{\hat{P}}_{j}]\approx
i\hbar\left[\delta_{ij}+\frac{1}{2}\left(\frac{\pi|{\vec{p}}|}{2p_{M}}\right)^{2}\left(\delta_{ij}-\frac{p_{j}p_{i}}{|{\vec{p}}|^{2}}\right)\right]~{}.$
(22)

The commutator in (22) yields the following uncertainty relationship

$\Delta X_{i}\Delta
P_{j}\geq\frac{\hbar}{2}\left(\delta_{ij}+\frac{\delta_{ij}\pi^{2}}{8p_{M}^{2}}\langle|{\vec{p}}|^{2}-p_{i}^{2}\rangle\right)=\frac{\hbar}{2}\delta_{ij}\left(1+\frac{\pi^{2}\langle
p_{\perp}^{2}\rangle}{8p_{M}^{2}}\right)~{}.$ (23)

In going from the (22) to (23) we have again used spherical symmetry to write
the expectation of the last term in (22) as $\frac{\pi^{2}\langle
p_{j}p_{i}\rangle}{8p_{M}^{2}}=\delta_{ij}\frac{\pi^{2}\langle
p_{i}^{2}\rangle}{8p_{M}^{2}}$, such that the expectation is zero unless
$i=j$; and as before, we define $\langle
p_{\perp}^{2}\rangle=\langle|{\vec{p}}|^{2}-p_{i}^{2}\rangle$. From (23) we
now repeat all the steps going from (17) to (19) to obtain the following
expressions for the $\arctan$ modification

$\Delta X_{i}\geq\frac{\hbar}{2}\left(\frac{1}{\Delta
p_{i}}+\frac{\pi^{2}\Delta p_{i}}{8p_{M}^{2}}\right)~{},$ (24)

The minimum of (20) occurs at $\Delta p_{i}=p_{M}$ which then gives

$\Delta X^{min}_{i}=\frac{\pi\hbar}{2\sqrt{2}p_{M}}.$ (25)

Equations (24) and (25) are the $\arctan$ modification analogs of (20) and
(21), only they differ by factors of order $1$. Notice, there is not that big
a difference between the minima from (21) and (25), and the rougher estimate
obtained when the minimum in position uncertainty which came entirely from
$\Delta P_{i}$ being bounded by $p_{M}$, where $\Delta
X^{min}_{i}=\frac{\hbar}{2p_{M}}$.

###  IV.2 Case with momentum in one direction

We now look at the physical consequences on the modified uncertainty
relationship when the system has some overall average momentum in one
direction, e.g. ${\vec{p}}=(p_{1},0,0)$. For this choice in momentum, the
right hand side of (12) and (13) simply becomes $1$ when $i=j=1$, which gives
the standard commutator $[X_{1},P_{1}]=i\hbar$ for both cases. In the $1$
direction, the uncertainty relationship becomes $\Delta X_{1}\Delta
P_{1}\geq\frac{\hbar}{2}$, which is just the standard uncertainty but
constructed with modified operators.

For both $\tanh$ and $\arctan$ modifications, one has $\Delta P_{1}\leq
p_{M}$; the same position uncertainty minimum in the $1$ direction of $\Delta
X^{min}_{1}\geq\frac{\hbar}{2p_{M}}$. Note that this minimum in the
$1$-direction is the same for both types of modified operators when the right
hand side is bounded by the standard commutator.

We now turn to the uncertainty in the two orthogonal directions $2$ and $3$.
For the modified operators $\hat{X}_{j}$ and $\hat{P}_{j}$ with $j=2$ or $3$
for the $\tanh$ modification, the commutator becomes

$[{\hat{X}}_{j},{\hat{P}}_{j}]=i\hbar\left[\frac{p_{M}}{|{\vec{p}}|}\sinh\left(\frac{|{\vec{p}}|}{p_{M}}\right)\cosh\left(\frac{|{\vec{p}}|}{p_{M}}\right)\right]=i\hbar\left[\cosh^{2}\left(\frac{|\vec{p}|}{p_{M}}\right)\frac{p_{M}}{|{\vec{p}}|}\tanh\left(\frac{|\vec{p}|}{p_{M}}\right)\right]~{}.$
(26)

The associated uncertainty is then

$\Delta X_{j}\Delta
P_{j}\geq\frac{\hbar}{2}\left\langle\frac{p_{M}}{|{\vec{p}}|}\sinh\left(\frac{|{\vec{p}}|}{p_{M}}\right)\cosh\left(\frac{|{\vec{p}}|}{p_{M}}\right)\right\rangle\geq\frac{\hbar}{2}\left(1+\frac{\langle|{\vec{p}}|^{2}\rangle}{2p_{M}^{2}}\right)~{},$
(27)

where in the last step we expanded the hyperbolic functions to second order.
For the $1$-direction modified momentum uncertainty, we take the rough
estimate that the momentum uncertainties in the $2,3$-directions are bounded
by $p_{M}$, i.e. $\Delta P_{j}\leq p_{M}$. Again, we assume $\langle
p_{j}\rangle=0$ and $\langle P_{j}\rangle=0$. With this, the modified position
uncertainty minimum is

$\Delta
X^{min}_{j}=\frac{\hbar}{2p_{M}}\left(1+\frac{\langle|{\vec{p}}|^{2}\rangle}{2p_{M}^{2}}\right)\geq\frac{\hbar}{2p_{M}}\left(1+\frac{p_{1}^{2}}{2p_{M}^{2}}\right)~{}.$
(28)

In the last expression in (28), we used $\langle|{\vec{p}}|^{2}\rangle\geq
p_{1}^{2}$ since we assume that the momentum is all in the $1$-direction. This
leads to the conclusion that the minimum uncertainty for the modified position
in the $1$-direction is smaller than in the orthogonal, $2,3$-directions; such
that $\Delta X^{min}_{2,3}>\Delta X^{min}_{1}$.

We now repeat the above calculations for the $\arctan$ modification. In the
$1$-direction one has as before $\Delta X^{min}_{1}=\frac{\hbar}{2p_{M}}$, by
taking the rough bound $\Delta P_{1}\leq p_{M}$. Now for the two orthogonal
directions, $j=2,3$ the commutator becomes

$[{\hat{X}}_{j},{\hat{P}}_{j}]=i\hbar\left[\left(1+\left(\frac{\pi|{\vec{p}}|}{2p_{M}}\right)^{2}\right)\frac{2p_{M}}{\pi|{\vec{p}}|}\arctan\left(\frac{\pi|{\vec{p}}|}{2p_{M}}\right)\right]~{}.$
(29)

This commutator in (29) now gives the modified uncertainty relationship for
$j=2,3$

$\displaystyle\Delta X_{j}\Delta P_{j}$ $\displaystyle\geq$
$\displaystyle\frac{\hbar}{2}\left\langle\left(1+\left(\frac{\pi|{\vec{p}}|}{2p_{M}}\right)^{2}\right)\left(\frac{2p_{M}}{\pi|\vec{p}|}\right)\arctan\left(\frac{\pi|{\vec{p}}|}{2p_{M}}\right)\right\rangle$
(30) $\displaystyle\geq$
$\displaystyle\frac{\hbar}{2}\left\langle\sqrt{\left(1+\left(\frac{\pi|{\vec{p}}|}{2p_{M}}\right)^{2}\right)}\right\rangle\geq\frac{\hbar}{2}\left(1+\frac{\pi^{2}\langle|{\vec{p}}|^{2}\rangle}{8p_{M}}\right)~{}.$

In going from the first line to the second line in (30), we have employed the
previously used non-obvious lower bound
$(1+x^{2})\frac{\arctan(x)}{x}\geq\sqrt{1+x^{2}}$, and taken a Taylor
expansion $\sqrt{1+x^{2}}\approx 1+\frac{1}{2}x^{2}$. If we again take the
rough bound of $\Delta P_{j}\leq p_{M}$ for the modified momentum in the two
orthogonal directions we obtain a minimum in the modified position uncertainty
of

$\Delta
X^{min}_{j}=\frac{\hbar}{2p_{M}}\left(1+\frac{\pi^{2}\langle|{\vec{p}}|^{2}\rangle}{8p_{M}}\right)\geq\frac{\hbar}{2p_{M}}\left(1+\frac{\pi^{2}p_{1}^{2}}{8p_{M}}\right)~{}.$
(31)

Again, we have used $\langle|{\vec{p}}|^{2}\rangle\geq p_{1}^{2}$ in arriving
at the last expression in (31).

As for the $\tanh$ modification, the term in parentheses in (31) is greater
than $1$. So, as before, the uncertainty in the orthogonal $2,3$-directions is
larger than in the $1$-direction - the direction of the momentum of the
system. This can be compared to when the electric field of a boosted, charged
particle is stronger in the directions orthogonal to the momentum of the
particle.

##  V Summary and conclusions

In this paper we formulated 3D GUP models based on earlier 1D GUP models
studied in aiken-2019 ; BLS ; BJLS . Our aim in 1D was to find a GUP model
which gave a minimal length which only modified the position and momentum
operators, while leaving the commutator of these modified operators
unmodified. In section II, we summarized the 1D GUP models. In section III, we
extended these 1D models to 3D GUP models so that they were rotationally
invariant (coordinate-independent) by requiring the modifications to be
functions of the total momentum $|\vec{p}|$. Consequently, the 3D commutator
must be modified in at least one direction. This can be seen explicitly in
(14) and (15) via the second order corrections to the the position-momentum
commutators which had a form
$\propto\delta_{ij}-\frac{p_{j}p_{i}}{|{\vec{p}}|^{2}}$. In 1D GUP models, the
term $\delta_{ij}-\frac{p_{j}p_{i}}{|{\vec{p}}|^{2}}$ is always zero. We then
investigated the uncertainty relationships produced by the modified 3D
operators and modified commutators. In general, we found that the minimum
uncertainty in position was approximately $\Delta
X_{i}^{min}\sim\frac{\hbar}{p_{M}}$ \- essentially the same result as in 1D.

We also considered the minimum distance in two cases. In the first case, we
used spherically-symmetric wave functions to analyze how small of a volume the
wave function could be compressed; we found that it was the minimal distance
cubed, up to a factor of order one. In the second case, we considered a system
with a large, non-zero momentum in some particular direction. Here we found
that the minimum distance in the direction of the average momentum was
different from the minimum distance in the orthogonal directions. For either
the $\tanh$ and $\arctan$ GUPs, the positional uncertainty in the orthogonal
directions was larger than in the direction of the non-zero momentum. A
similar result occurs in the GUP model in reference KMM  i.e $\Delta
X_{2,3}^{min}>\Delta X_{1}^{min}$.

The difference between the minimum position uncertainties in the second case
above hints that there may be a violation of Lorentz symmetry. Moreover, the
existence of minimal distances would naively imply violation of Lorentz
symmetry. However in BJLS , a GUP model was formulated which gave a minimal
length and preserved the special relativistic energy and momentum
relationship, $E^{2}-|{\vec{p}}|^{2}=m^{2}$, and thus preserved some aspects
of the Lorentz symmetry. A complementary issue is the connection of our work
to a breaking of the isotropy of space, as observed for other GUP models husin
. In future work, we plan to study the issue of the fate of the Lorentz
symmetry of these modified position and momentum operators.

The standard Lorentz algebra is generated by the angular momentum operators
and boost operators in terms of the time-position operators and energy-
momentum operators. One major question to address is how the modified position
and momentum operators affect the observables associated with modified angular
momentum and boost operators. To obtain the modified boost operators we will
need to also define modified time and energy operators that fit in with the
$\tanh$ or $\arctan$ operators of equations (9) and (10).

Another related issue is the modification of commutation relations of position
and momentum operators among themselves. The standard result is
$[{\hat{x}}_{i},{\hat{x}}_{j}]=0$ and $[{\hat{p}}_{i},{\hat{p}}_{j}]=0$. While
this is still true for the modified momentum operators (9) and (10), it is no
longer true for the modified position operators i.e.,
$[{\hat{P}}_{i},{\hat{P}}_{j}]=0$ but $[{\hat{X}}_{i},{\hat{X}}_{j}]\neq 0$.
This non-commutativity of the modified position operators provides a potential
link between GUP models and non-commutative geometry models piero ; euro
which is another approach to minimal lengths.

Also, the modified position operators from (9) and (10) satisfy
$[{\hat{X}}_{i},{\hat{X}}_{j}]\propto\epsilon_{ijk}{\hat{l}}_{k}$, so that the
commutators are proportional to the angular momentum operators. For the
related uncertainty relationships, this implies that $\Delta X_{i}\Delta
X_{j}\propto\epsilon_{ijk}\langle{\hat{l}}_{k}\rangle$; the uncertainty in
area is related to the expectation of the angular momentum operator. This
provides a potential connection between the 3D GUP models in this work and
spin foam models baez ; alejandro  which have a similar connection between
area and angular momentum.

We leave both of these questions – the modification of the Lorentz algebra and
the connection of 3D GUP models to non-commutative geometry and spin foam
models – for a subsequent, companion paper.

Acknowledgment: DS is a 2023-2024 KITP Fellow at the Kavli Institute for
Theoretical Physics and his work was partially supported in part by the
National Science Foundation under Grant No. NSF PHY-1748958. The work of PN is
partially been supported by GNFM, Italy’s National Group for Mathematical
Physics. The work of MB and DS were supported through a Fresno State 2023-2024
RSCA grant.

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