#  BIASeD: Bringing Irrationality into Automated System Design

Aditya Gulati<EMAIL_ADDRESS>
ELLIS Alicante Miguel Angel Lozano<EMAIL_ADDRESS>
Universidad de Alicante Bruno Lepri<EMAIL_ADDRESS>
Fondazione Bruno Kessler Nuria Oliver<EMAIL_ADDRESS>
ELLIS Alicante

###### Abstract

Human perception, memory and decision-making are impacted by tens of cognitive
biases and heuristics that influence our actions and decisions. Despite the
pervasiveness of such biases, they are generally not leveraged by today’s
Artificial Intelligence (AI) systems that model human behavior and interact
with humans. In this theoretical paper, we claim that the future of human-
machine collaboration will entail the development of AI systems that model,
understand and possibly replicate human cognitive biases. We propose the need
for a research agenda on the interplay between human cognitive biases and
Artificial Intelligence. We categorize existing cognitive biases from the
perspective of AI systems, identify three broad areas of interest and outline
research directions for the design of AI systems that have a better
understanding of our own biases.

##  1 Introduction

A cognitive bias is a systematic pattern of deviation from rationality that
occurs when we process, interpret or recall information from the world, and it
affects the decisions and judgments we make. Cognitive biases may lead to
inaccurate judgments, illogical interpretations and perceptual distortions.
Thus, they are also referred to as _irrational_ behavior [1, 2].

Since the 1970s, scholars in social psychology, cognitive science, and
behavioral economics have carried out studies aimed at uncovering and
understanding these apparently irrational elements in human decision making.
As a result, different theories have been proposed to explain the source of
our cognitive biases.

In 1955, Simon proposed the theory of _bounded rationality_ [3]. It posits
that human decision making is rational, but limited by our computation
abilities which results in sub-optimal decisions because we are unable to
accurately solve the utility function of all the options available at all
times. Alternative theories include the _dual process theory_ and the
_prospect theory_ , both proposed by Kahneman [4, 1].

Even though there is no unified theory of our cognitive biases, it is clear
that we use multiple shortcuts or heuristics111While heuristics typically
refer to a simplifying rule used to make a decision and a cognitive bias
refers to a consistent pattern of deviation in behavior, in this paper both
terms are used interchangeably as both impact human decisions in a similar
way. to make decisions which might lead to sub-optimal outcomes. However and
despite these limitations, cognitive biases and heuristics are a crucial part
of our decision making.

In fact, cognitive biases have traditionally been commercially leveraged in
different sectors to manipulate human behavior. Examples include casinos [5],
addictive apps [6], advertisement and marketing strategies to drive
consumption [7, 2] and social media campaigns to impact the outcome of
elections [8]. However, we advocate in this paper for a constructive and
positive use of cognitive biases in technology, moving from manipulation to
collaboration. We propose that considering our cognitive biases in AI systems
could lead to more efficient human-AI collaboration.

Nonetheless, there has been limited research to date on the interaction
between human biases and AI systems, as recently highlighted by several
authors [9, 10, 11, 12]. In this context, we highlight the work by Akata et
al. [13] who propose a research agenda for the design of AI systems that
collaborate with humans, going beyond a human-in-the-loop setting. They pose a
set of research questions related to how to design AI systems that collaborate
with and adapt to humans in a responsible and explainable way. In their work,
they note the importance of understanding humans and leveraging AI to mitigate
biases in human decisions.

In this paper, we build from previous work by proposing a taxonomy of
cognitive biases that is tailored to the design of AI systems. Furthermore, we
identify a subset of 20 cognitive biases that are suitable to be considered in
the development of AI systems and outline three directions of research to
design cognitive bias-aware AI systems.

##  2 A Taxonomy of Cognitive Biases

Since the early studies in the 1950s, approximately 200 cognitive biases have
been identified and classified [14, 15]. Several taxonomies of cognitive
biases have been proposed in the literature, particularly in specific domains,
such as medical decision making [16, 17], tourism [18] or fire evacuation
[19]. Alternative taxonomies classify biases based on their underlying
phenomenon [20, 21, 22]. However, given that there is no widely accepted
theory of the source of cognitive biases [23], classifying them according to
their hypothesized source might be misleading.

Dimara et al. [24] report similar limitations with existing taxonomies and
propose a new taxonomy of cognitive biases based on the experimental setting
where each bias was studied and with a focus on visualization. While this
taxonomy is of great value for visualization, our focus is the interplay
between AI and cognitive biases. Thus, we propose classifying biases according
to five stages in the human decision making cycle as depicted in Figure 1.

The left part of Figure 1 represents the physical world that we perceive,
interpret and interact with. The right part represents the internal models and
memories that we create based on our experience. As seen in Figure 1, we
propose classifying biases according to five main stages in the human
perception, interpretation and decision making process: _presentation_ biases,
associated with how information or facts are presented to humans;
_interpretation_ biases that arise due to misinterpretations of information;
_value attribution_ biases that emerge when humans assign values to objects or
ideas that are not rational or based on an underlying factual reality;
_recall_ biases associated with how we recall facts from our memory and
_decision_ biases that have been documented in the context of human-decision
making.

Figure 1 also illustrates how AI systems (represented as an orange undirected
graph) may interact with humans in this context. First, AI systems could be
entities in the external world that humans perceive or interact with (e.g.
chatbots, robots, apps…). Second, they may be active participants and assist
humans in their information processing and decision-making processes (e.g.
cognitive assistants, assistive technologies…). Finally, AI systems could be
observers that model our behavior and provide feedback without directly being
involved in the decision making process. Note that these three forms of
interaction with AI systems may occur simultaneously.

Figure 1: Stages of the human perception, interpretation and decision-making
process that are impacted by cognitive biases. AI systems (represented by an
orange undirected graph) could observe our behavior, detect biases and help us
mitigate them.

We also present four representative cognitive biases for each category. These
biases were chosen according to the amount of evidence in the literature about
the existence of the bias and their relevance for the design of AI systems.
Tables 1, 2 and 3 summarize the selected biases, their description, supporting
literature and relevance to AI.

| Bias | Brief Description | Relevance to AI
---|---|---|---
Presentation |  _Decoy effect_ [25, 26, 27, 28] |  Placing deliberately a worse alternative between two choices can reverse the user’s preference |  Could AI systems learn to place decoys effectively while presenting alternatives? Could AI systems learn to identify decoys? [29]
_Framing effect_ [30, 31, 32, 33] |  How a statement is framed can alter its perceived value |  Studies have shown that when humans are placed in human-AI teams, their decisions [34] and trust [35] are impacted by the framing effect. Could AI systems learn to frame explanations to make them more trustworthy?
_Anchoring effect_ [20, 36, 37] |  Human decision making is influenced by certain reference points or anchors |  The use of anchors to alter user preferences has been studied in marketing and recommender systems [38]. Could AI systems automatically identify anchors that humans might be subject to?
_Pseudocertainty effect_ [30, 39, 40] |  Humans incorrectly estimate the certainty of statements in a multi-stage decision making process |  Could AI systems identify situations where humans are likely to be unable to accurately compute the “complete picture”? Could this effect be leveraged by AI algorithms to learn effectively from smaller datasets?
Interpretation |  _Conjunction_ fallacy [41, 42, 43, 44] |  In certain situations, humans see the conjunction of two events as being more likely than any one event individually |  Could AI systems recognize situations where humans are likely to make such errors and provide alternate decisions?
_Base Rate_ fallacy [45, 46] |  Humans have a tendency to ignore the base rate information when making decisions |  Human reasoning does not follow Bayesian reasoning in certain settings. Could AI systems leverage these non-Bayesian computations effectively?
_Gamblers_ fallacy [20, 47, 48, 49] |  Humans tend to overvalue the impact of past events when predicting the outcome of independent future events |  Decision making systems that learn from human decision making –e.g. legal, college admissions or HR decision-making systems– learn from data that reflects the gamblers fallacy. How could this bias be mitigated to design fairer AI-based decision-support systems?
_Hyperbolic discounting_ effect [50, 51, 52] |  Humans tend to choose immediate rewards over rewards that come later in the future |  Studies have shown a link between high social media usage and hyperbolic discounting leading to unhealthy behavior [53, 54]. Could AI systems recognize when we are impacted by this bias and help mitigate it?

Table 1: Selected biases in Presentation and Interpretation categories of the
proposed taxonomy and their relevance to the study of AI systems.

| Bias | Brief Description | Relevance to AI
---|---|---|---
Value Attribution |  _Halo_ effect [55, 56, 57, 58] |  Positive attributes associated with a person in one setting carry over other settings |  Could this effect be utilized to create systems that are easier to trust? Does the halo effect manifest itself when humans interact with chatbots or robots?
_IKEA_ effect [59, 60, 61] |  Humans associate a higher value to their own creations than those of others |  Could this effect be leveraged to provide explanations that users are more likely to accept?
_Risk aversion_ bias [62, 63, 64, 65] |  We tend to avoid risky decisions even if they have a higher net expected utility than less risky options, especially if the uncertainty is high |  Could AI systems support human decision-making by counter-balancing the risk aversion risk?
_Social desirability_ bias [66, 67, 68, 69] |  Humans tend to provide the answers to surveys or questions that they believe are expected from them |  Do people provide socially desirable answers even when they are interacting with or being evaluated by machines? If yes, could the social desirability bias be be leveraged to nudge users to improve their behavior?
Recall |  _False memory_ bias [70, 71, 72] |  Humans incorrectly remember a past event depending on the questions they are asked about the event |  False memories impact how we make decisions. Positive false memories have been shown to have positive consequences [73]. Could this AI systems use this effect to improve user experience?
_Self-reference_ effect [74, 75] |  Events with a direct impact are more likely to be remembered |  Could AI systems leverage this effect to make explanations about their behavior more “memorable”?
_Serial-positioning_ effect [76, 77, 78] |  Items at the start and the end of a list are more memorable than those in the middle |  When providing explanations, could AI systems leverage this bias to have a more effective human interaction?
_Peak-end_ rule [79, 80, 81] |  The value of an event tends to be assessed based on its peak and final values, neglecting other parameters, such as its duration or average value |  Could this bias be leveraged by AI systems to increase the perceived utility of hard tasks? Instead of maximizing the time a user stays on a social media platform, could this bias be used to reduce time spent while also increasing user satisfaction?

Table 2: Selected biases in the Value Attribution and Recall categories of the
proposed taxonomy and their relevance to the study of AI systems.

| Bias | Brief Description | Relevance to AI
---|---|---|---
Decision |  _Status quo_ bias [82, 83] |  Humans tend to make decisions that maintain the current state rather than changing it |  Are AI systems that recommend fewer changes more likely to be trusted?
_Shared information_ bias [84, 85, 86] |  In group settings, humans tend to focus the discussion on information everyone already has rather than trying to bring in new information |  Could AI systems leverage this bias to effectively drive group conversations?
_Naive_ allocation [87, 88, 89] |  People tend to allocate resources equally between all options rather than based on the value of the options |  Could AI systems provide alternatives to avoid naive allocation? Alternatively, could AI systems leverage this bias effectively to make decisions in high uncertainty situations?
_Take-the-best_ heuristic [90] |  The decision between two alternatives is made based on the first cue that discriminates them |  Could this heuristic be implemented in AI systems to learn efficiently from small datasets? [91]

Table 3: Selected biases in the Decision category of the proposed taxonomy and
their relevance to the study of AI systems.

##  3 Cognitive Biases and AI: Research Directions

Given the ubiquity of AI-based systems in our daily lives –from recommender
systems to personal assistants and chatbots– and the pervasiveness of our
cognitive biases, there is an opportunity to leverage cognitive biases to
build more efficient AI systems.

In this section, we propose three research directions to further explore the
interplay between cognitive biases and AI: (1) Human-AI interaction, (2)
Cognitive biases in AI algorithms and (3) Computational modeling of cognitive
biases.

###  3.1 Area I. Human-AI Interaction

Cognitive biases have been studied since the 1970’s in experiments where human
participants interacted with other humans, animals or inanimate objects.
However, as Hidalgo et al. [92] note, we do not necessarily perceive, interact
with and evaluate machines in the same way as we do with humans, animals or
objects. Thus, it is unclear today whether these cognitive biases exist when
humans interact with AI systems, and if so with which degree of intensity and
under what circumstances.

This is especially the case with biases related to presentation and decision-
making, as per Figure 1. Previous work has reported that humans are influenced
by observing machine behavior. For example, Hang, Ono, and Yamada [93] showed
that participants who saw a video of robots exhibiting altruistic behavior
were more likely to demonstrate altruistic behavior themselves. Others suggest
that humans regard machines as social entities if they display “sufficient
interactive and social cues” [94]; and a third set of studies propose that
humans view machines as being different from themselves in social interactions
[95]. Given the impact that cognitive biases have in many of our daily tasks
and given the increased presence of AI algorithms to tackle many of these
tasks, it becomes important to understand whether interactions with AI systems
exhibit the same biases as those observed in human-to-human interactions.

For example, according to the _social desirability bias_ , users tend to
respond to surveys with socially expected answers which are not necessarily
their honest responses. Would this bias also emerge when users interact with a
chatbot? Another suitable bias to study in this context is the _halo_ effect.
Does it exist when humans interact with chatbots, robots or avatars? Are
positive traits associated with an AI system in one area carried forward to
other areas as well or are machines viewed as tools with a single purpose and
hence outside of the scope to the halo affect?

In addition to verifying if cognitive biases exist in human-machine
interactions, these biases could be leveraged to design more human-like AI
systems. While it has been reported that humans rely on heuristics when
deciding if they should trust AI decisions [96], would the existence of
cognitive biases in AI systems have an impact on their interpretability and
trustworthiness? Anthropomorphic agents have been shown to increase user
satisfaction and increase acceptance [97]. However, studies on
anthropomorphism tend to focus on physical attributes. We propose exploring
anthropomorphism in the realm of cognitive biases. Exemplary biases that could
be studied include the _framing_ effect, which could inform the fine tuning of
language models in chatbots; the _status quo_ bias, which could increase the
trust in AI systems that suggest small rather than major changes; or the
_halo_ effect, which could lead to humans trusting chatbots, avatars or robots
with certain appearances and attributes more than others independently of
their actual performance.

Another dimension worth exploring is the intersection between cognitive biases
and the explainability of AI systems (XAI). While there is a large body of
previous work in XAI, only a small subset of cognitive biases has been
considered in this research area [98, 99], mainly, with the objective of
mitigating them. Buçinca, Malaya, and Gajos [100] note that many machine
explanations are not useful because users rely on heuristics about when to
trust the machine and when not to, rather than using the provided explanation
to make such a decision. The authors propose cognitive forcing functions to
help users consider machine explanations carefully and show the effectiveness
of these functions in certain scenarios through user studies. We propose
expanding the research agenda to consider the inclusion of cognitive biases in
XAI. Exemplary biases in this context are the _framing effect_ and the _self-
reference effect_ which AI models could potentially leverage to provide more
trustworthy explanations.

Finally, we postulate that it could be valuable to include knowledge about
human cognitive biases when designing AI systems that interact with users.
Examples of biases that could be considered by AI systems include the
_gambler’s fallacy_ , the _anchoring_ effect or the _framing_ effect, by
presenting information to users in a manner that would mitigate the existence
of these biases; the _default_ heuristic, by nudging users to consider all
options or highlighting the existence of alternative possibilities; and the
_shared information_ bias, by performing a topic analysis on human
conversations and providing hints on novel topics to be discussed.

###  3.2 Area II. Cognitive Biases in AI Algorithms

While cognitive biases and heuristics lead to sub-optimal decision making in
certain situations, they are undoubtedly useful decision making aids. These
heuristics are at times as effective as complex decision making rules while at
the same time significantly reduce our cognitive load [101]. Given that humans
benefit greatly from the use of cognitive biases and heuristics, it is worth
exploring how AI could also benefit from them.

Taniguchi, Sato, and Shirakaw [102] work in this direction by building a
modified Naive Bayes classifier that leverages the symmetry [103] and the
mutual exclusion [104] biases. The proposed model is able to perform better
than alternative, state-of-the-art methods on a spam classification task when
the dataset is small and biased. Taniguchi, Sato, and Shirakawa [105] and
Manome et al. [106] extend this idea by incorporating the same biases in
neural networks and learning vector quantization respectively for different
tasks.

Given these successful examples of incorporating two cognitive biases in the
design of AI systems, it is worth considering how other biases could be
leveraged to help design AI systems that would learn faster and from less
data. Additional biases –beyond the symmetry and mutual exclusion biases– that
could be relevant for this purpose include the _take-the-best_ heuristic;
_naive allocation_ ; the _status-quo_ bias which humans use to make effective
decisions in situations with a high degree of uncertainty; and the _fast-and-
frugal_ heuristics [107] which have been shown to be effective decision making
tools in real world scenarios such as medical decision making [108]. They
could potentially be used in the design of AI systems as noted in their recent
work [91].

###  3.3 Area III. Computational Modeling of Cognitive Biases

The third research area addresses the computational modeling of cognitive
biases. Hiatt et al. [9] present a detailed survey of the computational
approaches proposed to date to model human behavior in human-machine
interaction. While they highlight the importance of having models that can
account for “some basic understanding of human reasoning, fallacy and error”,
none of the approaches they present explicitly models cognitive biases, which
we believe is crucial for the design of AI systems going forward.

Interestingly, in the past decade we have witnessed an exponential growth of
research on understanding and mitigating algorithmic biases [109, 110] which
are different from human cognitive biases. However, a failure to recognize the
differences has led to misrepresentations [111, 112].

Previous work has focused on building models of the underlying cognitive
process that explain cognitive biases –such as bounded rationality [113, 114].
Alternatively, scholars have focused on a subset of biases and have proposed a
variety of computational models in a particular task. One of the most
promising modeling frameworks in this context is Bayesian modeling [10, 115,
11, 116, 117, 118]. Beyond Bayesian modeling, Kang and Lerman [119] build on
existing generative models to predict the relevance of an item while
accounting for the position bias [120].

Additional work has been proposed in the medical field: Crowley et al. [121]
define a set of $8$ biases as a sequence of events in computer-based pathology
tasks. McShane, Nirenburg, and Jarrel [122] provide a tool to support doctors
in their diagnoses while mitigating recall biases. Alternative approaches are
based on expert observations of subjects performing certain tasks [123, 124].
However, such methods are difficult to scale.

Proposing a unifying, task-independent AI-based framework to automatically
identify cognitive biases from observed human behavior could have a profound
impact in the design of AI systems. Such a framework could provide a
representation that makes it possible to mitigate cognitive biases
effectively. It could enable the development of personalized systems that
support each individual by providing the most suitable mitigation strategy for
them.

In the persuasive computing literature, it has been observed that different
people respond to different techniques to support them in modifying their
behavior, from simple awareness to social support or competition [125, 126].
Recent work by Kliegr, Bahník, and Fürnkranz [12] has studied 20 cognitive
biases that could potentially impact how humans interpret machine learning
models and proposes several debiasing techniques. While informing users about
their biases is certainly a useful first step, it is generally not enough to
mitigate them [127]. Personalization and persuasive computing methods could
open new doors to more effective cognitive bias mitigation strategies.

##  4 Conclusion

Human perception, memory and decision-making are impacted by cognitive biases
and heuristics that impact our actions and decisions. Despite the
pervasiveness of such biases, they are largely disregarded by today’s AI
systems that model human behavior and interact with humans. However, given the
increased prominence of AI-human collaboration, we believe that it would be
crucial for AI systems to consider this fundamental element of human
cognition.

In this theoretical paper, we have proposed a taxonomy of cognitive biases
from the perspective of AI-human collaboration and have selected four
exemplary biases in each of the five key dimensions of the proposed taxonomy.
We have also proposed three broad research areas in the intersection between
AI and cognitive biases: First, human-AI interaction which focuses on open
questions, such as determining if human-AI interaction exhibits the same
cognitive biases as human-to-human interaction and exploring the potential
value of including cognitive biases in AI systems to make them more
trustworthy and interpretable. Second, cognitive biases in AI systems, focused
on the potential of leveraging the mechanisms behind our cognitive biases and
heuristics to build more robust and efficient machine learning algorithms.
Third, the computational modeling of cognitive biases to achieve a unifying
modeling framework which could be used to design personalized mitigation
strategies to support human decision making.

## Acknowledgements

Aditya Gulati and Nuria Oliver are supported by a nominal grant received at
the ELLIS Unit Alicante Foundation from the Regional Government of Valencia in
Spain (Convenio Singular signed with Generalitat Valenciana, Conselleria
d’Innovació, Universitats, Ciència i Societat Digital, Dirección General para
el Avance de la Sociedad Digital). Aditya Gulati is also supported by a grant
by the Banc Sabadell Foundation.

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# Gauge dependence of spontaneous radiation spectrum in a time-dependent
relativistic non-perturbative Coulomb field

Xue-Nan Chen  Yu-Hang Luo  Xiang-Song Chen  For correspondence:
<EMAIL_ADDRESS>Hubei Key Laboratory of Gravitational and Quantum Physics,
School of Physics, Huazhong University of Science and Technology, Wuhan
430074, China

###### Abstract

The delicacy of gauge choice in calculating atomic transitions was first
raised by Lamb and gave arise to intensive discussion as well as much
controversy. These discussion and controversy focused on choosing a proper
gauge for the electromagnetic wave that interacts with an atom. The issue was
claimed to have been solved, especially by Lamb himself and co-workers, by
favoring a gauge-invariant Hamiltonian for defining the atomic state in the
presence of electromagnetic wave. Here we extend the problem to include a
time-dependent relativistic non-perturbative Coulomb field, which can be
produced by a cluster of relativistic charged particles. If adiabatic
conditions are carefully maintained, such a field must be included along side
the nuclear Coulomb potential when defining the atomic state. We reveal that
when taking the external field approximation, the gauge choice for this time-
dependent relativistic non-perturbative Coulomb field cannot be overcome by
previous method, and leads to considerable gauge-dependence of the transient
spontaneous radiation spectrum. We calculate explicitly with a simple one-
dimensional charged harmonic oscillator that such a gauge-dependence can be of
a measurable magnitude of 10 MHz or larger for the commonly used Coulomb,
Lorentz, and multipolar gauges. Contrary to the popular view, we explain that
this gauge dependence is not really a disaster, but actually an advantage
here: The relativistic bound-state problem is so complicated that a fully
quantum-field method is still lacking, thus the external field approximation
cannot be derived and hence not guaranteed. However, by fitting to the
experimental data, one may always define an effective external field, which
may likely be parameterized with the gauge potential in a particular gauge.
This effective external field would not only be of phenomenological use, but
also shed light on the physical significance of the gauge field.

##  I Introduction

Gauge symmetry is trivial in classical electrodynamics, which is essentially a
theory of the electric field $\vec{E}$ and magnetic field $\vec{B}$. A quantum
formulation of electromagnetic interaction has to utilize the gauge potential
$A^{\mu}$, therefore does not share the absolutely safe gauge-invariance as in
the classical case. It is known that for scattering problems of elementary
particles, gauge symmetry is still well preserved in a quantum theory.
However, the issue seems to be tricky as a bound state is involved. It was
first noticed by Lamb about 70 years ago Lamb52  when studying the renowned
Lamb-shift transition $2S_{1/2}\to 2P_{1/2}$ that a straightforward
calculation can correctly give the observed line-shape in just one particular
gauge, namely the so-called length gauge with the $-q\vec{E}\cdot\vec{r}$
interaction. Here, “straightforward calculation” means that the atomic state
is defined as the eigenstate of the conventional energy operator

${}{\cal E}_{0}=\frac{{\vec{p}}^{2}}{2m}+qV(\vec{x}),~{}~{}{\cal
E}_{0}\left|\psi_{n}^{0}\right>=E_{n}\left|\psi_{n}^{0}\right>$ (1)

where $n$ labels the state, $\vec{p}=-i\vec{\nabla}$ (we set $\hbar=c=1$) is
the canonical momentum operator, $q=-e$ is the electron charge, and
$V(\vec{x})$ is the nuclear Coulomb potential.

Lamb’s observation is somehow surprising, and also puzzling, because the
quantum-mechanical formulation of electromagnetic interaction, for example, in
the Schrödinger equation

${}i\partial_{t}\psi(\vec{x},t)=H(t)\psi=\left\\{\frac{1}{2m}(\vec{p}-q\vec{A}(\vec{x},t))^{2}+q\phi(\vec{x},t)+qV(\vec{x})\right\\}\psi(\vec{x},t),$
(2)

is constructed purposely to be invariant under the joint gauge transformation

${}\psi=U\psi^{\prime}=e^{-iq\Lambda(\vec{x},t)}\psi^{\prime},~{}A_{\mu}=A_{\mu}^{\prime}-\frac{i}{q}U\partial_{\mu}U^{-1}=A_{\mu}^{\prime}+\partial_{\mu}\Lambda(\vec{x},t),$
(3)

where $U=e^{-iq\Lambda(\vec{x},t)}$, with $\Lambda(\vec{x},t)$ an arbitrary
function.

At the root of Lamb’s observation is that although the Schrödinger equation is
gauge-invariant, the energy operator ${\cal E}_{0}$ in Eq. (1) has a gauge-
dependent expectation value. A way out of this difficulty is therefore quite
natural, as many authors argued Yang76 ; Yang81 ; Au84 ; Lamb87 ; Funai19 ,
that in the presence of an electromagnetic field the atomic state should be
defined instead as the eigenstate of the gauge-invariant energy operator

${}{\cal E}_{A}(t)=\frac{{\vec{\pi}}^{2}}{2m}+qV(\vec{x}),~{}~{}{\cal
E}_{A}(t)\left|\psi_{n}^{A}(t)\right>=E_{n}\left|\psi_{n}^{A}(t)\right>,$ (4)

where $\vec{\pi}=\vec{p}-q\vec{A}(\vec{x},t)$ is the mechanical momentum
operator. Accidentally, in the length gauge $\vec{A}=0$, ${\cal E}_{A}$
coincides with ${\cal E}_{0}$, thus the “straightforward calculation” can
work. As Lamb et. al. elaborated in Ref. Lamb87 , a gauge with non-zero
$\vec{A}$ leads to complication, but careful and consistent calculations can
give the same result as in the length gauge. The major cause of complication
is that the operator ${\cal E}_{A}(t)$ is time-dependent, and normally does
not commute at different times, therefore Eq. (4) defines instantaneous
instead of stationary eigenstates $\psi_{n}^{A}(t)$, which are also time-
dependent. (The eigenvalues $E_{n}$, nevertheless, are gauge-invariant and
time-independent. Namely, $E_{n}$ do not depend on the explicit form of
$\vec{A}$. We will come back to this simple but delicate point in the next
Section.)

So far so good. The gauge-choice problem as Lamb raised was thus claimed to
have been solved, especially by Lamb himself and co-workers in a “concluding
paper” in 1987 Lamb87 . However, we would like to remind that Eq. (4) cannot
really be taken for granted, and is rather a conjecture. Especially, what is
the justification for Eq. (4) to include $qV(\vec{x})$ but discard
$q\phi(\vec{x},t)$? Why not just take the total $H(t)$ in Eq. (2) to define
the instantaneous atomic eigenstates in the presence of electromagnetic
interaction? Note  Certainly, the total $H(t)$ is gauge-dependent, and if Eq.
(4) is replaced by

${}H(t)\left|\psi_{n}(t)\right>=E_{n}(t)\left|\psi_{n}(t)\right>,$ (5)

this would make both the eigenvalues and the atomic transition rates gauge-
dependent. But quite interestingly, by Eq. (5) the correct result can still be
obtained in one particular gauge, namely the Coulomb gauge with $\phi=0$,
instead of the length gauge with $\phi=-\vec{E}\cdot\vec{x}$. This time it is
the Coulomb gauge that stands out.

The aim of this paper to extend the previous studies, and discuss a more
serious case that in Eq. (2) the time-dependent scalar potential
$\phi(\vec{x},t)$ can be adiabatic and comparable to $V(\vec{x})$ in effect,
therefore must be treated at the same footing as $V(\vec{x})$. Then, Eq. (4)
could not possibly apply, and Eq. (5) is a more reasonable option. This would
give rise to significant gauge-dependence that could not be overcome by
existing methods. The paper is organized as follows: In Section II, we
introduce our physical system: a cluster of relativistic charged particles
passing by a one-dimensional oscillator. Parameters can be adjusted to make
$\phi(\vec{x},t)$ of the moving charge cluster to be adiabatic and non-
perturbative for the oscillator, whose instantaneous eigenstates have to be
calculated with Eq. (5) instead of Eq. (4). The solutions differ significantly
for the commonly used Coulomb, Lorentz, and multipolar gauges. Then in Section
III we compute explicitly the transient spontaneous radiation spectrum of such
a system, and find again significant gauge-dependence. In the last Section we
summarize our results and discuss their physical implications.

##  II A time-dependent relativistic non-perturbative Coulomb field acting
adiabatically on a quantum oscillator

FIG. 1 shows a schematic design of our physical system. The electromagnetic
field is produced by a cluster of relativistic protons of a huge number $N$,
like a bunch from an accelerator. (Certainly electrons and heavy ions may also
be utilized.) This charge cluster passes by a one-dimensional oscillator,
formed of an electron moving in a nanowire or carbon nanotube along the $x$
axis, and constrained by two other nearby negative charges, placed at the
coordinates $(x,y)=(l,0)$ and $(-l,0)$, respectively. The charge cluster moves
in the same direction, with an impact distance of $Y$. If excited, the
oscillator can emit a photon by spontaneous radiation. It can be expected that
by enlarging the parameters $N$, $l$, and $Y$, the scalar potential
$\phi(\vec{x},t)$ of the charge cluster can be non-perturbative and adiabatic
for the oscillator for a duration which is sufficiently long for the excited
oscillator to emit a transient photon. This emission must then be computed by
including both $\phi(\vec{x},t)$ and $V(\vec{x})$ when defining the
instantaneous eigenstates of the oscillator. Namely, we have to employ Eq. (5)
instead of Eq. (4), as we remarked above in Section I.

Figure 1: The schematic design of the physical system. The big solid circle
represents the proton cluster. Two small solid circles represent stationay
charges. The small hollow circle represents the electron. See the text for
description of parameters.

For simplicity, we treat the proton cluster as a point charge, and leave the
task of considering the actual spatial distribution and possible dispersing of
the cluster to a future study. This would influence some quantitative detail
but not the main concern of this paper about gauge-dependence. The cluster
velocity $\beta=v/c$ should be chosen large enough to produce a considerable
difference for $\phi(\vec{x},t)$ among various gauges, and at the same time
small enough to allow for a rough external-field approximation. (This point
will be commented on further in the last Section.)

Upon solving Eq. (5) for our one-dimensional oscillator, we first note that it
shares the same feature as Eq. (4) that the eigenvalues do not depend
explicitly on $\vec{A}$. This is a simple fact, but may often cause confusion,
so we elaborate a little bit here.

We may always apply to Eq. (5) the following unitary transformation:

${}\left|\psi_{n}(t)\right>=U\left|\tilde{\psi}_{n}(t)\right>,~{}H(t)=U^{-1}\tilde{H}(t)U,$
(6)

then Eq. (5) becomes

${}\tilde{H}(t)\left|\tilde{\psi}_{n}(t)\right>=E_{n}(t)\left|\tilde{\psi}_{n}(t)\right>.$
(7)

Note that although the factor $U$ in Eq. (6) can be the same as in Eq. (3),
Eq. (6) is not a gauge transformation. It is just a mathematical technique,
after fixing a gauge, for the convenience of solving the eigen-equation.
Especially, the new operator

$\displaystyle\tilde{H}(t)=UH(t)U^{-1}$
$\displaystyle=\frac{1}{2m}(\vec{p}-q\vec{A}(\vec{x},t)+q\vec{\nabla}\Lambda(\vec{x},t))^{2}+q\phi(\vec{x},t)+qV(\vec{x})$
(8)
$\displaystyle=\frac{1}{2m}(\vec{p}-q\vec{A}^{\prime}(\vec{x},t))^{2}+q\phi(\vec{x},t)+qV(\vec{x}),$

is just a mathematical tool without much physical meaning. It is not the
gauge-transformed Hamiltonian $H^{\prime}(t)$. The latter should be obtained
by applying $\psi=U\psi^{\prime}$ to the time-evolution equation (2) instead
of the eigen-equation (5). The result is familiar:

$H^{\prime}(t)=UH(t)U^{-1}-iU^{-1}\partial_{t}U=\frac{1}{2m}(\vec{p}-q\vec{A}^{\prime}(\vec{x},t))^{2}+q\phi^{\prime}(\vec{x},t)+qV(\vec{x}),$
(9)

where both the vector and scalar potentials are the gauge-transformed ones. In
contrast, $\tilde{H}(t)$ looks like a hybrid-gauge expression: the vector
potential is transformed, while the scalar potential remains the same.

It is exactly the hybrid feature of $\tilde{H}$ in Eq. (8) that makes the
unitary transformation (6) advantageous when solving Eq. (5): For a one-
dimensional problem, the vector potential $\vec{A}^{\prime}$ may always be set
to zero, and $\tilde{H}$ got simplified. After obtaining the simpler solution
$\tilde{\psi}_{n}$, the original $\psi_{n}$ is easily got by multiplying the
factor $U$, and the eigenvalue $E_{n}(t)$ is unchanged. Hence, the vector
potential $\vec{A}$ is trivial when solving Eq. (5), as we just commented
above, and we need only to consider the scalar potential $\phi(\vec{x},t)$.

Since in our system the moving cluster and the fixed charges have comparable
effects on the electron, we include all their contributions into
$\phi(\vec{x},t)$. The expression is easy to calculate. For the Lorentz gauge,

$\phi_{L}=\frac{1}{4\pi}\left(\frac{-e}{l+x}+\frac{-e}{l-x}+\frac{Ne}{\sqrt{(x-L(t))^{2}+\left(1-\beta^{2}\right)Y^{2}}}\right),$
(10)

and for Coulomb gauge,

$\phi_{C}=\frac{1}{4\pi}\left(\frac{-e}{l+x}+\frac{-e}{l-x}+\frac{Ne}{\sqrt{(x-L(t))^{2}+Y^{2}}}\right).$
(11)

Here, $x$ is the coordinate of the electron, $L(t)$ is the coordinate of the
charge cluster at time $t$. The subscripts $L,C$ refer to expressions in the
Lorentz and Coulomb gauges, respectively, and a subscript $G$ will denote a
general gauge.

By applying the PZW transformation Power59 ; Wolley71 :

$U_{G}=\exp\left[ie\int_{0}^{1}d\lambda\vec{A}_{G}\left(\lambda\vec{x}+(1-\lambda)\vec{x}_{0},t\right)\cdot\left(\vec{x}-\vec{x}_{0}\right)\right],$
(12)

one obtains $\phi_{M}$ in the multipolar gauge,

$\phi_{M}=\left\\{\phi_{G}(\vec{x}_{0},t)-\int_{0}^{1}d\lambda\vec{E}\left(\lambda\vec{x}+(1-\lambda)\vec{x}_{0},t\right)\cdot\left(\vec{x}-\vec{x}_{0}\right)\right\\}.$
(13)

To further simplify our calculation, we approximate the electron motion as a
harmonic oscillator, with the equilibrium point influenced by the charge
cluster. To this end, we Taylor-expand $\phi_{G}$ at $x_{0}$ to the second
order:

$\phi_{G}(x,t)\approx\phi_{G}(x_{0},t)+\partial_{x}\phi_{G}(x,t)|_{x=x_{0}}\cdot(x-x_{0})+\frac{1}{2}\partial_{x}^{2}\phi_{G}(x,t)|_{x=x_{0}}\cdot(x-x_{0})^{2},$
(14)

then the value of $x_{0}$ is found by solving
$\partial_{x}\phi_{G}(x,t)|_{x=x_{0}}=0$. For the Lorentz gauge,

$\displaystyle\phi_{L}$
$\displaystyle\approx\frac{1}{4\pi}\left\\{\frac{-e}{x_{0}+l}-\frac{-e}{x_{0}-l}+\frac{Ne}{\sqrt{\left(x_{0}-L(t)\right)^{2}+\left(1-\beta^{2}\right)Y^{2}}}\right.$
(15)
$\displaystyle-\left(\frac{-e}{\left(x_{0}+l\right)^{2}}-\frac{-e}{\left(x_{0}-l\right)^{2}}+\frac{Ne\left(x_{0}-L(t)\right)}{\sqrt{\left(x_{0}-L(t)\right)^{2}+\left(1-\beta^{2}\right)Y^{2}}^{3}}\right)\left(x-x_{0}\right)$
$\displaystyle+\left(\frac{-e}{\left(x_{0}+l\right)^{3}}-\frac{-e}{\left(x_{0}-l\right)^{3}}\right.$
$\displaystyle\left.\left.-\frac{1}{2}\left(\frac{Ne}{\sqrt{\left(x_{0}-L(t)\right)^{2}+\left(1-\beta^{2}\right)Y^{2}}^{3}}-\frac{3Ne\left(x_{0}-L(t)\right)^{2}}{\sqrt{\left(x_{0}-L(t)\right)^{2}+\left(1-\beta^{2}\right)Y^{2}}^{5}}\right)\right)\left(x-x_{0}\right)^{2}\right\\}.$

For the Coulomb gauge

$\displaystyle\phi_{C}$
$\displaystyle\approx\frac{1}{4\pi}\left\\{\frac{-e}{x_{0}+l}-\frac{-e}{x_{0}-l}+\frac{Ne}{\sqrt{\left(x_{0}-L(t)\right)^{2}+Y^{2}}}\right.$
(16)
$\displaystyle-\left(\frac{-e}{\left(x_{0}+l\right)^{2}}-\frac{-e}{\left(x_{0}-l\right)^{2}}+\frac{Ne\left(x_{0}-L(t)\right)}{\sqrt{\left(x_{0}-L(t)\right)^{2}+Y^{2}}^{3}}\right)\left(x-x_{0}\right)$
$\displaystyle\left.+\left(\frac{-e}{\left(x_{0}+l\right)^{3}}-\frac{-e}{\left(x_{0}-l\right)^{3}}-\frac{1}{2}\left(\frac{Ne}{\sqrt{\left(x_{0}-L(t)\right)^{2}+Y^{2}}^{3}}-\frac{3Ne\left(x_{0}-L(t)\right)^{2}}{\sqrt{\left(x_{0}-L(t)\right)^{2}+Y^{2}}^{5}}\right)\right)\left(x-x_{0}\right)^{2}\right\\}.$

And for the multipolar gauge,

$\displaystyle\phi_{M}$
$\displaystyle\approx\phi_{G}(\vec{x}_{0},t)+\frac{1}{4\pi}\left\\{-\left(\frac{-e}{\left(x_{0}+l\right)^{2}}-\frac{-e}{\left(x_{0}-l\right)^{2}}+\frac{Ne\left(1-\beta^{2}\right)\left(x_{0}-L(t)\right)}{\sqrt{\left(x_{0}-L(t)\right)^{2}+\left(1-\beta^{2}\right)Y^{2}}^{3}}\right)\left(x-x_{0}\right)\right.$
(17)
$\displaystyle+\left(\frac{-e}{\left(x_{0}+l\right)^{3}}-\frac{-e}{\left(x_{0}-l\right)^{3}}\right.$
$\displaystyle\left.\left.-\frac{1}{2}\left(\frac{Ne\left(1-\beta^{2}\right)}{\sqrt{\left(x_{0}-L(t)\right)^{2}+\left(1-\beta^{2}\right)Y^{2}}^{3}}-\frac{3Ne\left(1-\beta^{2}\right)\left(x_{0}-L(t)\right)^{2}}{\sqrt{\left(x_{0}-L(t)\right)^{2}+\left(1-\beta^{2}\right)Y^{2}}^{5}}\right)\right)\left(x-x_{0}\right)^{2}\right\\}.$

From the above expressions, it is clear that for a time-dependent
electromagnetic field the so-obtained equilibrium point $x_{0}$ is gauge-
dependent, and generally does not coincide with the point where $\vec{E}=0$,
except in the multipolar gauge with $\vec{A}=0$. We therefore add the gauge
label, and write the Hamiltonian of the oscillator approximately as

$H_{0G}(t)\approx\frac{1}{2m}(p-i\partial_{x}U^{\dagger}_{G})^{2}+\frac{1}{2}k_{G}(t)(x-x_{0G}(t))^{2}.$
(18)

Here, $k_{G}(t)$ is the strength of the harmonic potential, and varies with
the equilibrium point. $U_{G}$ is the PZW-transformation factor given by Eq.
(12). For the multipolar gauge $U_{G}$ is simply unity. With this Hamiltonian
(18), we can solve the instantaneous eigenvalues and the instantaneous quantum
states as

$\displaystyle E_{n,G}(t)$ $\displaystyle=(n+\frac{1}{2})\omega_{G}(t),$ (19)
$\displaystyle|n_{G}(t)\rangle$
$\displaystyle=U^{\dagger}_{G}N_{nG}(t)\exp[-\frac{1}{2}\gamma_{G}(t)^{2}(x-x_{0G}(t))^{2}]\mathrm{H}_{n}[\gamma_{G}(t)(x-x_{0G}(t))].$

Here, $\omega_{G}(t)=\sqrt{k_{G}(t)/m}$,
$\gamma_{G}(t)=\sqrt{m\omega_{G}(t)}$,
$N_{nG}(t)=\left[\gamma_{G}(t)/\sqrt{\pi}2^{n}n!\right]^{1/2}$, and
$\mathrm{H}_{n}$ is Hermite polynomial.

In what follows, we set up the parameters and compute the numerical results.
We take the physical mass of the electron, and keep in mind that its effective
mass might be different in an actual system. After some rough estimation, we
find that the following values suffice our study: $N=10^{12}$, $\beta=0.1$,
$l=6.33\mathrm{nm}$, and $Y/l=10^{6}$. The cluster moves from $x=-100Y$ to
$x=100Y$.

FIG. 2 gives the multipolar-gauge results of the instantaneous equilibrium
position and the frequency of the oscillator. As we commented above, this is
the gauge that essentially sets the equilibrium position at $\vec{E}=0$. FIG.
3 is an enlarged view of FIG. 2 for the period when the cluster acts
significantly. FIG. 4 gives the corresponding results in the Lorentz and
Coulomb gauges, expressed as the deviation from the values in the multipolar
gauge.

(a) $x_{0M}$ (b) $\omega_{M}$

Figure 2: The result of the instantaneous equilibrium position (a) and the
frequency (b) of the oscillator in multipolar gauge. To visualize the
adiabatic chacacter of the system, we divide the equilibrium coordinate by
$c$, and plot it in unit of time.

(a) $x_{0M}$ (b) $\omega_{M}$

Figure 3: The enlarged view of FIG. 2 for the period when the cluster acts
significantly.

(a) $x_{0G}-x_{0M}$ (b) $\omega_{G}-\omega_{M}$

Figure 4: The reults of the instantaneous equilibrium position (a) and the
frequency (b) of the oscillator in the Coulomb and Lorentz gauges, displayed
as deviation from the corresponding values in the multipolar gauge.

Before solving the eigen-equations, we must check that with our chosen
parameters, $\phi(\vec{x},t)$ is indeed an adiabatic potential for the
oscillator, in all three gauges we use. The criterion for adiabatic
approximation is the parameter

$r_{nm}=\left|\frac{\langle
n_{G}(t)|\dot{H}_{0G}(t)|m_{G}(t)\rangle}{(E_{mG}(t)-E_{nG}(t))^{2}}\right|,$
(20)

where $n,m$ label two different states. FIG. 5 gives the results of $r_{nm}$
for the lowest two states, which shows that for the Lorentz, Coulomb, and
multipolar gauges, we all have $r_{01}=r_{10}\ll 1$. Thus, if prepared in the
excited state $\left|1\right>$ before the cluster moves in, the oscillator
will largely stay in the state $\left|1_{G}(t)\right>$ during the whole
process as the cluster moves through.

Figure 5:  The results of the criterion parameter $r_{01}=r_{10}$ for the
adiabatic approximation in all three gauges we use.

With the adiabatic conditions justified, the gauge-dependence of the
instantaneous energy levels, as we just displayed above for our designed
quantum oscillator, may in principle be tested experimentally. A possible
observable, which is cleanest theoretically, is the spectrum of spontaneous
radiation, as we will compute in the next Section.

##  III The transient spontaneous radiation spectrum and its gauge-dependence

We now add into our system the coupling to the background vacuum
electromagnetic field $A^{\mu}_{B}=(\vec{A}_{B},\phi_{B})$. For the
convenience of imbedding our one-dimensional oscillator into a three-
dimensional formulation, we introduce an artificial harmonic potential
$V(y,z)=\frac{m}{2}(\omega_{y}^{2}y^{2}+\omega_{z}^{2}z^{2})$, with
$\omega_{y},\omega_{z}\rightarrow\infty$. The entire Hamiltonian is now
expressed as

${}H(t)=\frac{1}{2m}[\vec{p}-q(\vec{A}(\vec{x},t)+\vec{A}_{B}(\vec{x},t))]^{2}+q(\phi(\vec{x},t)+\phi_{B}(\vec{x},t))+V(y,z)+H_{B}.$
(21)

Here, $H_{B}$ is the Hamiltonian of the background photon:

${}H_{B}=\int\frac{d^{3}k}{(2\pi)^{3}}\sum_{\lambda}\omega_{\vec{k}}a_{\vec{k},\lambda}^{\dagger}a_{\vec{k},\lambda}.$
(22)

The expression of $A^{\mu}_{B}$ also depends on the gauge. For the Coulomb
gauge,

${}\vec{A}_{B}=\int\frac{d^{3}k}{(2\pi)^{3}}\sum_{\lambda}\sqrt{\frac{1}{2\omega_{\vec{k}}}}(\vec{e}_{\lambda}a_{{\vec{k}},\lambda}e^{i\vec{k}\cdot\vec{x}-i\omega_{\vec{k}}t}+\vec{e}^{*}_{\lambda}a^{\dagger}_{{\vec{k}},\lambda}e^{-i\vec{k}\cdot\vec{x}+i\omega_{\vec{k}}t}),\quad\phi_{B}=0\quad(\vec{e}_{\lambda}\cdot\vec{k}=0).$
(23)

For the Lorentz gauge,

${}A_{B}^{\mu}=\int\frac{d^{3}k}{(2\pi)^{3}}\sum_{\lambda}\sqrt{\frac{1}{2\omega_{\vec{k}}}}\left(e_{\lambda}^{\mu}a_{\vec{k},\lambda}e^{i\vec{k}\cdot\vec{x}-i\omega_{\vec{k}}t}+e_{\lambda}^{\mu*}a_{\vec{k},\lambda}^{\dagger}e^{-i\vec{k}\cdot\vec{x}+i\omega_{\vec{k}}t}\right).$
(24)

Since we shall only consider the emission of a physical photon, and only
compute this effect to leading order, in the Lorentz gauge the part of
$A_{B}^{\mu}$ that really works actually agrees with Eq. (23). We therefore
omit the gauge label in the background potentials.

We can roughly estimate that for our system the wavelength of the possibly
detected photon is much larger than the geometric size of the oscillator, so
we approximate $A_{B}^{\mu}$ as the value at the equilibrium point $x_{0}$ of
the oscillator:

$\vec{A}_{B}\approx\int\frac{d^{3}k}{(2\pi)^{3}}\sum_{\lambda}\sqrt{\frac{1}{2\omega_{\vec{k}}}}(\vec{e}_{\lambda}a_{\vec{k},\lambda}e^{i\vec{k}\cdot\vec{x}_{0G}(t)-i\omega_{\vec{k}}t}+\vec{e}^{*}_{\lambda}a^{\dagger}_{\vec{k},\lambda}e^{-i\vec{k}\cdot\vec{x}_{0G}(t)+i\omega_{\vec{k}}t}),\quad\phi_{B}=0\quad(\vec{e}_{\lambda}\cdot\vec{k}=0).$
(25)

Note that $x_{0}$ depends on the gauge for the external field, so we still
need a gauge label here, $G=C,L$. Using the PZW transformation, we can get the
multipolar-gauge expression in the dipole approximation:

$\vec{A}_{B}\approx
0\quad\phi_{B}\approx(\vec{x}-\vec{x}_{0M}(t))\cdot\int\frac{d^{3}k}{(2\pi)^{3}}\sum_{\lambda}\sqrt{\frac{\omega_{\vec{k}}}{2}}i(\vec{e}_{\lambda}a_{\vec{k},\lambda}e^{i\vec{k}\cdot\vec{x}_{0M}(t)-i\omega_{\vec{k}}t}-\vec{e}^{*}_{\lambda}a^{\dagger}_{\vec{k},\lambda}e^{-i\vec{k}\cdot\vec{x}_{0M}(t)+i\omega_{\vec{k}}t})\quad(\vec{e}_{\lambda}\cdot\vec{k}=0).$
(26)

To compute the radiation spectrum, we now have two possible sources of gauge-
dependence, one from the external field and the other from the background
field. For the non-perturbative, adiabatic, and relativistic external field,
we just explained in the previous Section that the gauge-choice seriously
affects the energy levels of our quantum system. As we will shortly show, this
indeed leads to significant gauge dependence of the spectrum. For the
background field, since it does not contribute a non-perturbative scalar
potential, its gauge-dependence can be handled by Eq. (4), which agrees well
with the measured atomic spectroscopy, as Lamb et. al. elaborated in Ref.
Lamb87 . We can expect that this method still applies when coupling the
background field to our non-perturbative time-dependent system. (Although our
comments in the paragraph around Eq. (5) still hold here.) So, analogous to
Eq. (19), the instantaneous eigenstate under the influence of the background
electromagnetic field is defined by applying another unitary transformation:

${}\left|n_{BG}(t)\right>=U_{BG}\left|n_{G}(t)\right>,$ (27)

where

$U_{BM}=1,\quad
U_{BG}=\exp\\{-ie\vec{A}_{B}(\vec{x}_{0G}(t),t)\cdot(\vec{x}-\vec{x}_{0G}(t))\\}~{}(G=L,C).$
(28)

Using the above state, gauge-dependence from the background electromagnetic
field is removed. Certainly, the calculation is the simplest in the multipolar
gauge with $\vec{A}_{B}=0$ and $U_{BM}=1$, then
$\left|n_{BG}(t)\right\rangle=\left|n_{G}(t)\right\rangle$, and the total
Hamiltonian (21) becomes

$H_{G}(t)\approx
H_{0G}(t)+e\vec{E}_{B}(\vec{x}_{0G}(t),t)\cdot(\vec{x}-\vec{x}_{0G}(t))+H_{B}.$
(29)

We would like to remark that for our system the gauge-dependence from the
background field is actually not serious anyhow, as compared with that from
the external field. Even if we do not apply the transformation in Eq. (27),
and go ahead in any gauge with $\left|n_{G}(t)\right\rangle$ and the minimal-
coupling Hamiltonian

$H(t)\approx
H_{0G}(t)+\frac{1}{m}\vec{A}_{B}(\vec{x}_{0G}(t),t)\cdot(\vec{p}+e\vec{A}(\vec{x},t))+H_{B},$
(30)

the result would differ from that using Eq. (29) by an undetectable amount, as
we will check at the end of this Section.

Note that our quantum oscillator is essentially one-dimensional, so we only
need to consider the physical photon with $e_{x}$ polarization, and the
background photon Hamiltonian (22) reduces to

$H_{B}=\int\frac{d^{3}k}{(2\pi)^{3}}\omega_{\vec{k}}a_{{\vec{k}},x}^{\dagger}a_{\vec{k},x}.$
(31)

Moreover, due to the constraint $\vec{e}_{\lambda}\cdot\vec{k}=0$, the photon
momentum-space integration is reduced as well:

${}\int
d^{3}k\Rightarrow\int_{0}^{\infty}\omega_{\vec{k}}^{2}d\omega_{\vec{k}}\int_{0}^{\pi}\sin\theta_{\vec{k}}d\theta_{\vec{k}}\delta(\theta_{\vec{k}}-\frac{\pi}{2})\int_{0}^{2\pi}d\phi_{\vec{k}}=2\pi\int_{0}^{\infty}d\omega_{\vec{k}}\omega_{\vec{k}}^{2},$
(32)

where $\theta_{\vec{k}}$ is the angle between the photon momentum $\vec{k}$
and the $x$ axis, and $\varphi_{\vec{k}}$ is the azimuthal angle of $\vec{k}$
in the $y-z$ plane.

To facilitate our discussion, we write collectively in all gauges the
interaction term between the background field and the time-dependent quantum
system as
$\int\frac{d^{3}k}{(2\pi)^{3}}H_{I}(a_{\vec{k},x},a^{\dagger}_{\vec{k},x},t)$.
We use energy-level raising and lowering operators $\sigma_{+}$ and
$\sigma_{-}$ to represent the position operator $x-x_{0G}(t)$ and the
mechanical-momentum operator $p_{x}+eA_{x}$:

$x-x_{0G}(t)=\frac{1}{\sqrt{2}\gamma_{G}(t)}(\sigma_{+}+\sigma_{-}),\quad
p_{x}+eA_{x}=i\frac{\gamma_{G}(t)}{\sqrt{2}}(\sigma_{+}-\sigma_{-}),$ (33)

where
$\sigma_{+}|n_{G}(t)\rangle=\sqrt{n+1}|(n+1)_{G}(t)\rangle,\sigma_{-}|n_{G}(t)\rangle=\sqrt{n}|(n-1)_{G}(t)\rangle$.
By the standard practice of rotating-wave approximation, the interaction
Hamiltonian can be expressed as

$H_{I}(a_{\vec{k},x},a^{\dagger}_{\vec{k},x},t)\approx
H_{I}(\sigma_{+},a_{\vec{k},x},t)+H_{I}(\sigma_{-},a^{\dagger}_{\vec{k},x},t).$
(34)

The explicit expressions, if the multipolar gauge is employed for the
background field, are

$\begin{aligned}
H_{I}(\sigma_{+},a_{\vec{k},x},t)&=\frac{i}{2}\int\frac{d^{3}k}{(2\pi)^{2}}\frac{\sqrt{\omega_{\vec{k}}}}{\gamma_{G}(t)}\sigma_{+}a_{\vec{k},x}e^{i\vec{k}\cdot\vec{x}_{0G}(t)-i\omega_{\vec{k}}t}\\\
H_{I}(\sigma_{-},a^{\dagger}_{\vec{k},x},t)&=-\frac{i}{2}\int\frac{d^{3}k}{(2\pi)^{2}}\frac{\sqrt{\omega_{\vec{k}}}}{\gamma_{G}(t)}\sigma_{-}a^{\dagger}_{\vec{k},x}e^{-i\vec{k}\cdot\vec{x}_{0G}(t)+i\omega_{\vec{k}}t}\end{aligned}\quad(G=M,L,C).$
(35)

Note that the gauge imprint from the external field still persists here. If
the Lorentz or Coulomb gauge is employed for the background field, then

$\begin{aligned}
H_{I}(\sigma_{+},a_{\vec{k},x},t)&=\frac{i}{2}\int\frac{d^{3}k}{(2\pi)^{2}}\frac{\gamma_{G}(t)}{\sqrt{\omega_{\vec{k}}}}\sigma_{+}a_{\vec{k},x}e^{i\vec{k}\cdot\vec{x}_{0G}(t)-i\omega_{\vec{k}}t}\\\
H_{I}(\sigma_{-},a^{\dagger}_{\vec{k},x},t)&=-\frac{i}{2}\int\frac{d^{3}k}{(2\pi)^{2}}\frac{\gamma_{G}(t)}{\sqrt{\omega_{\vec{k}}}}\sigma_{-}a^{\dagger}_{\vec{k},x}e^{-i\vec{k}\cdot\vec{x}_{0G}(t)+i\omega_{\vec{k}}t}\end{aligned}\quad(G=L,C).$
(36)

The whole state function of the time-dependent oscillator plus the possibly
emitted photon can be written as

$|\psi(t)\rangle=\sum_{n}a_{n}(t)|n_{G}(t),0\rangle+\sum_{n}\int\frac{d^{3}k}{(2\pi)^{3}}a_{n,\vec{k}}(t)|n_{G}(t),\gamma_{\vec{k},x}\rangle+\cdots.$
(37)

Here,
$|n_{G}(t),0\rangle=|n_{G}(t)\rangle\otimes|0\rangle,|n_{G}(t),\gamma_{\vec{k},x}\rangle=|n_{G}(t)\rangle\otimes|\gamma_{\vec{k},x}\rangle$.
$|n_{G}(t)\rangle$ is the instantaneous eigenstate of the oscillator as we
constructed in the previous Section, and
$|0\rangle,|\gamma_{\vec{k},x}\rangle$ are the Fock states of the photon. We
only consider the single-photon process and the lowest two levels of the
oscillator. Then, the whole state is approximated as

$|\psi(t)\rangle\approx\sum_{n=0,1}a_{n}(t)|n_{G}(t),0\rangle+\sum_{n=0,1}\int\frac{d^{3}k}{(2\pi)^{3}}a_{n,\vec{k}}|n_{G}(t),\gamma_{\vec{k},x}\rangle.$
(38)

To handle the coefficients $a_{n}$ and $a_{n,\vec{k}}$, we introduce the
dynamic phase $\theta_{n}=-\int^{t}_{0}E_{n}(s)ds$ and adiabatic phase
$\gamma_{n}=i\int^{t}_{0}\langle n(s)|\dot{n}(s)\rangle$:

$\displaystyle a_{n}(t)$
$\displaystyle=\exp[i(\gamma_{n}(t)+\theta_{n}(t))]c_{n}(t),$ (39)
$\displaystyle a_{n,\vec{k}}(t)$
$\displaystyle=\exp[i(\gamma_{n}(t)+\theta_{n}(t))]c_{n,\vec{k}}(t).$

Using the state-evolution equation
$i\partial_{t}|\psi(t)\rangle=H(t)|\psi(t)\rangle$, we obtain differential
equations for the new coefficients $c_{n}$ and $c_{n,\vec{k}}$:

$\displaystyle\dot{c}_{0}(t)$ $\displaystyle=-\langle
0_{G}(t),0|\dot{1}_{G}(t),0\rangle
c_{1}(t)\exp[i(\gamma_{1}(t)+\theta_{1}(t))-i(\gamma_{0}(t)+\theta_{0}(t))],$
(40) $\displaystyle\dot{c}_{0,\vec{k}}(t)$
$\displaystyle=-\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\langle
0_{G}(t),\gamma_{\vec{k},x}|\dot{1}_{G}(t),\gamma_{\vec{k^{\prime}},x}\rangle
c_{1,\vec{k}^{\prime}}(t)\exp[i(\gamma_{1}(t)+\theta_{1}(t))-i(\gamma_{0}(t)+\theta_{0}(t))]$
$\displaystyle-i\langle
0_{G}(t),\gamma_{\vec{k},x}|H_{I}(\sigma_{-},a_{\vec{k},x}^{\dagger},t)|1_{G}(t),0\rangle
c_{1}(t)\exp[i(\gamma_{1}(t)+\theta_{1}(t))-i(\gamma_{0}(t)+\theta_{0}(t))],$
$\displaystyle\dot{c}_{1}(t)$ $\displaystyle=-\langle
1_{G}(t),0|\dot{0}_{G}(t),0\rangle
c_{0}(t)\exp[-i(\gamma_{1}(t)+\theta_{1}(t))+i(\gamma_{0}(t)+\theta_{0}(t))]$
$\displaystyle-i\int\frac{d^{3}k}{(2\pi)^{3}}\langle
1_{G}(t),0|H_{I}(\sigma_{+},a_{\vec{k},x},t)|0_{G}(t),\gamma_{\vec{k},x}\rangle
c_{0,\vec{k}}(t)\exp[-i(\gamma_{1}(t)+\theta_{1}(t))+i(\gamma_{0}(t)+\theta_{0}(t))],$
$\displaystyle\dot{c}_{1,\vec{k}}(t)$
$\displaystyle=-\int\frac{d^{3}k^{\prime}}{(2\pi)^{3}}\langle
1_{G}(t),\gamma_{\vec{k},x}|\dot{0}_{G}(t),\gamma_{\vec{k^{\prime}},x}\rangle
c_{0,\vec{k}^{\prime}}(t)\exp[-i(\gamma_{1}(t)+\theta_{1}(t))+i(\gamma_{0}(t)+\theta_{0}(t))].$

The notations are $\langle n_{G}(t),0|\dot{m}_{G}(t),0\rangle=\langle
n_{G}(t)|\otimes\langle 0|\partial_{t}|m_{G}(t)\rangle\otimes|0\rangle=\langle
n_{G}(t)|\partial_{t}|m_{G}(t)\rangle\otimes\langle 0|0\rangle=\langle
n_{G}(t)|\partial_{t}|m_{G}(t)\rangle\otimes\mathbf{I}$ and $\langle
n_{G}(t),\gamma_{\vec{k},x}|\dot{m}_{G}(t),\gamma_{\vec{k}^{\prime},x}\rangle=\langle
n_{G}(t)|\otimes\langle\gamma_{\vec{k},x}|\partial_{t}|m_{G}(t)\rangle\otimes|\gamma_{\vec{k}^{\prime},x}\rangle=\langle
n_{G}(t)|\partial_{t}|m_{G}(t)\rangle\otimes\langle\gamma_{\vec{k},x}|\gamma_{\vec{k}^{\prime},x}\rangle=(2\pi^{3})\delta^{(3)}(\vec{k}-\vec{k}^{\prime})\langle
n_{G}(t)|\partial_{t}|m_{G}(t)\rangle\otimes\mathbf{I}$ . It would be very
complicated to calculate the complete evolution process. Fortunately, we
demonstrated in Section II that the moving charge cluster acts adiabatically,
therefore the transition terms which conserve the photon number can be
neglected, and we only need to consider the emission and absorption of photon
caused by coupling with the background electromagnetic field. Then Eqs. (40)
simplify greatly to

$\displaystyle\dot{c}_{0,\vec{k}}(t)$ $\displaystyle=-i\langle
0_{G}(t),\gamma_{\vec{k},x}|H_{I}(\sigma_{-},a_{\vec{k},x}^{\dagger},t)|1_{G}(t),0\rangle
c_{1}(t)\exp[i(\gamma_{1}(t)+\theta_{1}(t))-i(\gamma_{0}(t)+\theta_{0}(t))],$
(41) $\displaystyle\dot{c}_{1}(t)$
$\displaystyle=-i\int\frac{d^{3}k}{(2\pi)^{3}}\langle
1_{G}(t),0|H_{I}(\sigma_{+},a_{\vec{k},x},t)|0_{G}(t),\gamma_{\vec{k},x}\rangle
c_{0,\vec{k}}(t)\exp[-i(\gamma_{1}(t)+\theta_{1}(t))+i(\gamma_{0}(t)+\theta_{0}(t))].$

To handle the decay of the excited state, we follow a method similar to the
Weisskopf-Wigner approximation Weisskopf30 :

$\displaystyle\dot{c}_{1}(t)$
$\displaystyle=-\int\frac{d^{3}k}{(2\pi)^{3}}\langle
1_{G}(t),0|H_{I}(\sigma_{+},a_{\vec{k},x},t)|0_{G}(t),\gamma_{\vec{k},x}\rangle\exp[-i(\gamma_{1}(t)+\theta_{1}(t))+i(\gamma_{0}(t)+\theta_{0}(t))]$
(42) $\displaystyle\times\int^{t}_{0}dt^{\prime}\langle
0_{G}(t^{\prime}),\gamma_{\vec{k},x}|H_{I}(\sigma_{-},a_{\vec{k},x}^{\dagger},t^{\prime})|1_{G}(t^{\prime}),0\rangle
c_{1}(t^{\prime})\exp[i(\gamma_{1}(t^{\prime})+\theta_{1}(t^{\prime}))-i(\gamma_{0}(t^{\prime})+\theta_{0}(t^{\prime}))].$

To proceed with the computation in a clearer form, we denote
$\Delta_{G}(t)=E_{1G}(t)-E_{0G}(t)-i\langle 1_{G}(t)|\dot{1}_{G}(t)+i\langle
0_{G}(t)|\dot{0}_{G}(t)\rangle$. Since the emission spectrum typically has a
peak frequency, we can effectively perform the frequency integration in the
range $(-\infty,+\infty)$, and get

${}\int_{-\infty}^{\infty}d\omega_{k}e^{i\left(\omega-\omega_{\vec{k}}\right)\left(t-t^{\prime}\right)}=2\pi\delta\left(t-t^{\prime}\right).$
(43)

Using Eqs. (32), (42) and (43), we obtain

$\dot{c}_{1}(t)=-\Gamma_{G}(t)c_{1}(t),$ (44)

where the time-dependent decaying rate is,

$\displaystyle\Gamma_{G}(t)$
$\displaystyle=\frac{\Delta^{2}_{G}(t)}{2\pi}\langle
1_{G}(t),0|H_{I}(\sigma_{+},a_{\Delta_{G}(t),x},t)e^{i\Delta_{G}(t)t}|0_{G}(t),\gamma_{\Delta_{G}(t),x}\rangle$
(45) $\displaystyle\times\langle
0_{G}(t),\gamma_{\Delta_{G}(t),x}|H_{I}(\sigma_{-},a^{\dagger}_{\Delta_{G}(t),x},t)e^{-i\Delta_{G}(t)t}|1_{G}(t),0\rangle.$

Then, to obtain the solutions for $c_{1}(t)$ and $c_{0,\vec{k}}(t)$, we just
combine Eq. (44) with the first line of Eqs. (41), compute the relevant matrix
element, and perform the numerical integration step by step. Finally, the
observed emission spectrum, accumulated till a time $t_{f}$, is computed via:

${}\int
d\omega_{\vec{k}}S(\omega_{\vec{k}},t_{f})=\int\frac{d^{3}k}{(2\pi)^{3}}|c_{0,\vec{k}}(t_{f})|^{2}\delta(\theta_{\vec{k}}-\frac{\pi}{2})=\int
d\omega_{\vec{k}}\frac{\omega_{\vec{k}}^{2}}{(2\pi)^{3}}\int
d\theta_{\vec{k}}d\varphi_{\vec{k}}\sin\theta_{\vec{k}}|c_{0,\vec{k}}(t_{f})|^{2}\delta(\theta_{\vec{k}}-\frac{\pi}{2}),$
(46)

Explicitly,

${}S(\omega_{\vec{k}},t_{f})=\frac{\omega_{\vec{k}}^{2}}{(2\pi)^{2}}|c_{0,\vec{k}}(t_{f})|^{2},~{}~{}~{}({\theta_{\vec{k}}=\frac{\pi}{2}}).$
(47)

FIG. 6 gives our calculated results of the spontaneous-emission spectrum
$S(\omega_{\vec{k}},t_{f})$, and FIG. 7 shows an enlarged view around the peak
frequency.

Figure 6: The transient spontaneous-emission spectrum of our physical system.
The subscript “$0$” represents the gauge selection of the electromagnetic
field produced by a cluster of relativistic charged particles, and the
subscript “$B$” represents the gauge selection of the background vacuum
electromagnetic field. The last two lines in the legend represent the
unperturbed spectrum in the absence of the charge cluster calculated in two
different gauge selections of the background vacuum electromagnetic field.
Figure 7: The enlarged view of FIG. 6 around the peak frequency.

These results show clearly that the computed spontaneous-emission spectrum
depends considerably on the gauge, in both the peak frequency and the emission
intensity. Even with a relatively mild velocity of $\beta=0.1$ for the charge
cluster, the computed peak frequency in the Lorentz gauge is higher than that
in the multipolar gauge by a detectable amount $\sim 10\mathrm{MHz}$, and
higher than that in the Coulomb gauge by $\sim 60\mathrm{MHz}$.

For another comparison, we also plot in FIG. 6 and FIG. 7 the unperturbed
spectrum in the absence of the charge cluster, whose peak frequency is around
$6.3369\times 10^{13}\mathrm{Hz}$, and is higher than that of the perturbed
spectrum in the Lorentz gauge by about $120\mathrm{MHz}$. This is a cross-
check that the charge cluster indeed acts non-perturbatively on the quantum
oscillator.

To close this Section, we would like to comment that the significant gauge-
dependence we revealed above comes mainly from the non-perturbative, adiabatic
scalar potential produced by the relativistic charge cluster. In comparison,
the gauge-dependence similar to that as Lamb originally noticed Lamb52 , is
quantitatively negligible in our case, as we show in FIG. 8.

Figure 8: The “Lamb-type” gauge-dependence, displayed as the spectrum
deviation from the multipolar-gauge result.

##  IV Summary and discussion

In this paper, we proposed a physical system that a cluster of relativistic
charges produces a non-perturbative Coulomb field which acts adiabatically on
a quantum system. This is a case that we have to face sharply the long-
standing concerns about what is actually the physical significance of the
scalar potential when it is time-dependent, and how the electromagnetic field
actually interacts with a quantum system, especially, how to quantize a bound
state in the presence of a time-dependent, non-perturbative, and adiabatic
scalar potential. We calculated the spontaneous-radiation spectrum of such a
quantum system, and find significant gauge-dependence which cannot be cured by
existing methods.

Whenever some gauge-dependence stands in the way, there is often an intent and
attempt to discard $(\vec{A},\phi)$ and stick to $(\vec{E},\vec{B})$,
especially in the early practices and debates. However, a consensus was
gradually reached that this is not possible, at least in a local formulation.
For example, the multipolar-gauge expressions involve only
$(\vec{E},\vec{B})$, but in terms of path-integrals. It should especially be
reminded that such integrals are in general path-dependent. Namely, the
multipolar gauge itself is not unique.

Another intent of long history is to distinguish between energy and
hamiltonian for a non-conservative system, and refuse assigning the time-
dependent $\phi(\vec{x},t)$ any serious physical significance. We do not claim
against such an intent, and just bring a case that the existing methods have
to utilize $\phi(\vec{x},t)$, with a gauge-dependent result.

There also exists an intent in the opposite way, especially after the work of
Aharanov and Bohm AB , to think of $(\vec{A},\phi)$ as physical reality.
Another support for this idea is that the Coulomb field always accompanies a
charge, and can never be stripped away. We do not claim either that our
findings add to that intent. In fact, we must be clear-minded that our
starting point, namely the semi-classical Hamiltonian (5) or (21) under the
external-field approximation, is known to be unsafe if the external source is
relativistic Wein95 , while a safe and fully quantum-field method is still
lacking. In such a circumstance, the gauge-dependence we just found is not a
disaster, but an advantage: Since the Hamiltonian (5) or (21) is not justified
anyway, there is no sense in sticking to the actual $(\vec{A},\phi)$ generated
by the external source. Instead, one may always try to define an effective
external field which, when plugged into the Hamiltonian (5) or (21), may fit
well the experimental data. This is indeed quite probable, as we saw that by
just tuning the gauge, we may fit the peak frequency of the radiation
spectrum; and another overall factor may be multiplied to fit the radiation
intensity. Namely, the effective external field may likely be parameterized
with the gauge potential in a particular gauge. This gauge is not necessarily
among the most common ones, and might possibly be individualized condition for
the field of each charge, instead of the usual conditions for the overall
field.

Such an effective external field is not merely of important phenomenological
use, but might also shed light on the physical significance of the gauge
potential. Should it turn out to be closer to the multipolar-gauge expression,
it might suggest that $\vec{E}$ is indeed more essential than $\phi$. Should
it turn out to be closer to the expression of the Lorentz or Coulomb gauge,
and deny that of the multipolar gauge, however, we would find an extremely
interesting phenomenon which may be termed “kinematic scalar Aharanov-Bohm
effect”: Due to Lorentz contraction, a charged particle close to the speed of
light produces negligible electric field along its trajectory. However, its
scalar potential could possibly be felt by a quantum system. We may even put
this in a science-fiction style: The classical bodies can never feel a
projectile flying directly to us at nearly the speed of light, no matter how
crazily energetic, until the (too-late) moment it crashes in; while a quantum
system may act as an alerting whistle. Pitifully, we human beings may belong
to the “classical bodies”. Such a projectile might be a particle, or even a
planet. Should we expect that gravity display a similar effect, then even a
massive black hole flying directly to us at nearly the speed of light can only
be detected beforehand by a quantum system, but not by classical bodies.

We should compare our physical system with related phenomena. The key elements
of our design include: a relativistic charge cluster of a huge number, a
quantum oscillator of relatively large spatial extension and low frequency,
and adiabatic conditions for the oscillator to complete transient spontaneous
radiation during the interacting period. These elements have been encountered
individually in some studies. For example, Coulomb excitation Bayman05 ;
Dasso06  may also utilize a relativistic projectile, but in a perturbative
and/or non-adiabatic way. The AC-starkAutler55  effect may be adiabatic and
non-perturbative, but the electric field is from an electromagnetic wave, not
relativistic particles. The term “strong-field” typically refers to an
intensive laser field, not a relativistic Coulomb field. It is our study that
combines all these key elements, and brings sharply a “Coulomb-type” gauge
dependence. We invent this terminology to refer to the gauge-choice problem
for the Coulomb field of charged particles, especially relativistic ones.
Naturally, the gauge-choice problem for an electromagnetic wave or the vacuum
field may be termed “Lamb-type”.

Since our system has very low energy, the decaying rate is rather small. This
also reveals clearly that the gauge-dependence we encounter comes mainly from
defining the quantum basis, and is not the Lamb-type. To further address this
new type of gauge dependence, other quantum systems of large spatial extension
may also be employed, especially the fancy Rydberg atoms, for which the
Coulomb field of a relativistic charged cluster may become non-perturbative
relatively easily.

In this paper, we displayed gauge-dependence for the accumulated radiation
spectrum without time-resolution. Registering accurately the arrival time of
the emitted photons would reveal more detailed information. If time-resolution
is applied, one may also consider induced transitions, either absorption or
emission.

The design we proposed also makes a tunable system to study the relativistic
bound-state problems. The usual relativistic bound states often go to two
extremes: either all particles are relativistic and the problem got over
complicated, or, the acted particle is relativistic while the acting particles
non-relativistic, which can be easily handled by the Dirac equation. In our
design, the acting cluster is relativistic, while the acted particle is non-
relativistic. (One may feel like to turn the situation around by going to the
rest frame of the acting cluster. But this idea would not work, since we may
easily arrange two or more acting clusters with different velocities.)
Moreover, the design makes both the relativity parameter $\beta$ and the
interaction strength continuously adjustable.

The ultimate goal of this study is to thoroughly examine how the gauge field
actually interacts with a quantum system, and explore more closely the
physical significance of the gauge potential. A similar system as we proposed
may also be designed to address gravitational gauge dependence, by studying
quantum transitions in the presence of a time-dependent relativistic non-
perturbative Newtonian field.

##  V Acknowledgements

This is a long-time work that almost all our group members have joined the
discussion, including our graduated fellows. We also benefited from fruitful
discussions with many HUST colleagues, especially Jian-Wei Cui, Wei-Tian Deng,
Lin Li, and Yi-Qiu Ma, et al. The work had been, and was partly supported by
the China NSF via Grants No. 11275077 and No. 11535005.

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