# Superquantization rule for multistability in driven-dissipative quantum
systems

Nikolett Német<EMAIL_ADDRESS>HUN-REN Wigner RCP, H-1525
Budapest, P.O. Box 49., Hungary  Árpád Kurkó<EMAIL_ADDRESS>HUN-REN Wigner RCP, H-1525 Budapest, P.O. Box 49., Hungary  András Vukics
<EMAIL_ADDRESS>HUN-REN Wigner RCP, H-1525 Budapest, P.O. Box
49., Hungary  Péter Domokos<EMAIL_ADDRESS>HUN-REN Wigner
RCP, H-1525 Budapest, P.O. Box 49., Hungary

###### Abstract

We present a superquantization rule which indicates the possible robust
stationary states of a generic driven-dissipative quantum system.
Multistability in a driven cavity mode interacting with a qudit is revealed in
this way within a simple intuitive picture. The accuracy of the
superquantization approach is confirmed by numerical simulations of the
underlying quantum model. In the case when the qudit is composed of several
two-level emitters coupled homogeneously to the cavity, we demonstrate the
robustness of the superquantized steady states to single-emitter decay.

Cavity QED, multistability, photon blockade breakdown, driven-dissipative
phase transition, superquantization

###### pacs:

<pacs codes>

Out-of-equilibrium quantum phases can exist in a generic class of microscopic
systems formed by a single driven-dissipative harmonic oscillator coupled to a
small, finite dimensional quantum system 1; 2; 3; 4; 5; 6; 7; 8; 9. The
external coherent drive balanced by the loss to the environment leads to a
steady state with a stationary current of quanta through the system that
provides for a measurable signal 10; 11; 12; 13. The components of the steady
state are unambiguously identified through the current outcoupled from the
oscillator mode by a classical measurement apparatus. The defining feature of
these quantum phases is that they show resilience to the back-action of
measurement 14. All this is possible because one of the two components is a
harmonic oscillator which, when dressed by the coupling to the small system,
forms highly excited stationary quasi-coherent states 15; 16.

A typical implementation can be an electromagnetic radiation mode in a
resonator, being the oscillator, while a set of (artificial) atoms coupled to
this mode constitutes a non-linear quantum medium 17; 18; 19; 20. The
measurement signal is provided by the field outcoupled from the resonator
which is recorded by a photo-detector. While the harmonic oscillator ensures
the robustness of the stationary states, the number and characteristics of the
possible out-of-equilibrium phases are determined by the properties of the
small finite-dimensional system strongly coupled to the oscillator 15; 21, as
is shown in this paper through a simple example.

The simplest possible scenario is when a two-level system is coupled to a
coherently driven harmonic oscillator. The quantum mechanical description of
such a setup is given by the driven Jaynes-Cummings model 22. This system is
readily realized in cavity-QED settings, i.e., as a qubit coupled to a single
mode of a resonator. Previously, two components of the steady-state have been
identified in such a system, that can exists in a mixture in a “bistability”
domain of the parameter space 23; 5; 24; 6; 25; 26. One of them is referred to
as the “dim” state, where the photon transmission through the resonator is
suppressed due to the lack of resonance with the uneven energy-level spacings
of the low-energy part of the spectrum. The other one is referred to as the
“bright” state where the cavity-QED system is highly transmissive, emitting a
strong photo-current from the driven resonator.

The two-component steady state manifests on quantum trajectories as stochastic
alternation between the two metastable states, resulting in a telegraph signal
in the resonator intensity 27. The mechanism of the transition from the dim to
the bright state has been termed the breakdown of photon blockade 5; 24; 27;
26; 28; 29. It has also been shown that the switching behavior is a finite-
size precursor of a first-order dissipative phase transition 5. Increasing the
strength of the coupling between the qubit and the mode leads toward a
thermodynamic limit where the dwell times of the metastable states and their
separation in mode intensity become infinite 5; 27; 26; 13. The macroscopic
switching of output intensities has been experimentally demonstrated in
circuit-QED with transmon artificial atoms 10; 13. Transmons are intrinsically
multilevel structures; therefore, realistic characterization potentially
requires multiple energy levels, and circuit QED systems are non-linear par
excellence 30; 31; 32; 33; 34; 35; 36; 37; 38.

Figure 1: Left: a spin-3/2 is composed of three qubits coupled with the same
strength $g$ to a driven resonator mode with linewidth $\kappa$. a) In the
spectrum of the Hamiltonian without the drive ($\eta=0$ in eq. 1), the first
two excited states split into a doublet and a triplet, respectively. b) Higher
up on the ladder they split into quadruplets. c) When driving is considered,
the detuning of the drive from the mode and spin transition frequencies
determines which eigenstates are involved in shaping the resonator output
intensity.

In this paper, we consider a spin-3/2 coupled to a single-mode resonator field
(cf. fig. 1) in order to reveal the connection between the structure of the
non-linear atomic system and the possible multistable solutions 39. We will
show that these solutions can be determined from a simple algebraic self-
consistency equation, i.e., a “super-quantization” rule. This equation is
derived on the basis of an intuitive picture of the emergent quasi-coherent
states and proves to be very accurate, as it is also demonstrated using
quantum-trajectory simulations of the full quantum dynamics of the driven-
dissipative system using the C++QED framework 40; *cppqedgithub. In the case
when the spin-3/2 is composed of three qubits coupled homogeneously to the
mode, we can add single-qubit decay to the quantum trajectories, and
demonstrate the robustness of the quasi-coherent states to this decoherence
channel. This is a remarkable result, since the quasi-coherent states, being
derived solely from the Hamiltonian, reside in the completely symmetric
subspace, which is not closed against single-emitter decay.

The system Hamiltonian is the driven Tavis-Cummings model 42; 43; 44 that in a
frame rotating with the drive frequency reads

$\hat{H}=-\Delta\left(\hat{a}^{\dagger}\hat{a}+\frac{\hat{S}_{z}}{2}\right)+ig\left(\hat{a}^{\dagger}\hat{S}^{-}-\hat{S}^{+}\hat{a}\right)+i\eta\left(\hat{a}^{\dagger}-\hat{a}\right),$
(1)

where $\hat{a}$ and $\hat{a}^{\dagger}$ are the mode annihilation and creation
operators, the $\hat{S}$s are the spin operators, and $g$ is the spin-
resonator coupling strength. The resonator mode is tuned to the spin
transition frequency. Finally, $\eta$ is the amplitude of the coherent
resonator driving, with detuning $\Delta$ from both the frequencies of the
spin and the mode. We consider photon loss from the resonator described by the
Liouvillian operator

$\mathcal{L}\left[\hat{\rho}\right]=\kappa\left(2\hat{a}\hat{\rho}\hat{a}^{\dagger}-\hat{a}^{\dagger}\hat{a}\hat{\rho}-\hat{\rho}\hat{a}^{\dagger}\hat{a}\right)$
(2)

where $2\kappa$ is the rate of resonator decay. The loss of excitations are
compensated by the coherent field that drives the system into a steady state.

In order to obtain an intuitive picture of the dynamics, we first consider
only the undriven Hamiltonian, that is, we take $\eta=0$ in eq. 1. The
eigenstates of this system have been studied in detail 45; 46. Since the
Hamiltonian conserves the total excitation number $n$, it is enough to perform
the diagonalization on a subspace spanned by
$\ket{n-\left(m+\tfrac{3}{2}\right)}\ket{m}$, where
$m\in\left\\{-\tfrac{3}{2},-\tfrac{1}{2},\tfrac{1}{2},\tfrac{3}{2}\right\\}$
is the spin-projection quantum number. The eigenvalues read (cf. fig. 1[a-b])

$\displaystyle\lambda_{-3}(n)=$
$\displaystyle-\Delta\,n-g\sqrt{(5+4\sqrt{1+\epsilon^{2}})(n-1)},$ (3a)
$\displaystyle\lambda_{-1}(n)=$
$\displaystyle-\Delta\,n-g\sqrt{(5-4\sqrt{1+\epsilon^{2}})(n-1)},$ (3b)
$\displaystyle\lambda_{1}(n)=$
$\displaystyle-\Delta\,n+g\sqrt{(5-4\sqrt{1+\epsilon^{2}})(n-1)},$ (3c)
$\displaystyle\lambda_{3}(n)=$
$\displaystyle-\Delta\,n+g\sqrt{(5+4\sqrt{1+\epsilon^{2}})(n-1)},$ (3d)

where $\epsilon=\frac{3}{4(n-1)}$ vanishes for large enough photon numbers, so
that even for $n\sim 3$, $\epsilon^{2}\ll 1$, and the eigenvalues can be well
approximated by

$\lambda_{u}(n)\approx-\Delta\,n+u\,g\sqrt{n},$ (4)

with dressed-state ladder indices $u\in\left\\{-3,-1,1,3\right\\}$. The lower
part of the spectrum is “qubit-dominated”, meaning that it is highly non-
linear. For high photon numbers $n$, the frequency difference between the
dressed states corresponding to given $u$ in adjacent manifolds $n$ and $n+1$
is

$\displaystyle\lambda_{u}(n+1)-\lambda_{u}(n)$
$\displaystyle\approx-\Delta+\frac{u\,g}{2\sqrt{n}}.$ (5)

The function $1/\sqrt{n}$ is slowly varying such that there is a range in
which the difference is almost constant, i.e., the dressed states with
identical $u$ in the high $n$ manifolds span a closely equidistant ladder.
This part of the spectrum hosts quasi-coherent states that can be steadily
maintained by the external drive $\eta$.

This is the ground for forming an intuitive picture. The equidistant ladders
at high numbers $n$ can be considered as harmonic oscillators. The well-known
solution of an externally driven, lossy harmonic oscillator leads then to the
self-consistent equation for the mean photon number,

$n=\frac{\eta^{2}}{\kappa^{2}+\left[\lambda_{u}(n+1)-\lambda_{u}(n)\right]^{2}}=\frac{\eta^{2}/\kappa^{2}}{1+\left(\delta-\frac{u\,g}{2\kappa\sqrt{n}}\right)^{2}},$
(6)

with $\delta=\Delta/\kappa$. Since eq. 6 leads to a quadratic polynomial
equation in $\sqrt{n}$, at a given detuning, there can exist two distinct
solutions $n$ for each ladder $u$. However, it can be shown that the secondary
root is non-physical, since it decreases with increasing drive amplitude 111A
monotonically decreasing intra-cavity intensity vs. drive amplitude function
means that if for any finite drive amplitude the intensity is positive, then
the intensity remains non-zero for zero drive, which is manifestly non-
physical. On the other hand, negative intensity (for any drive strength) is
also clearly non physical..

The possible maximum number of stationary bright states is thus 4 in the case
when the qudit interacting with the mode is a spin-3/2. From the way this
result has been derived above, it is clear that this number equals the number
of dimensions of the qudit. For a given drive amplitude, however, not all of
these solutions are realized over the full quantum dynamics of the driven-
dissipative system, since the solution has to fulfill other conditions that we
detail below. The maximum intra-cavity photon number is $\eta^{2}/\kappa^{2}$,
reached at exact resonance, $\delta={u\,g}/{2\eta}$. In this case, the qudit
tunes the cavity back into resonance with the driving field.

From the same intuitive picture of the driven-dissipative steady state of what
is approximately a harmonic oscillator, we can obtain a self-consistent
equation for the complex intra-cavity field amplitude, that is more general
than eq. 6, as it gives information also on the phase 15; 48:

$\alpha_{u}=\frac{\eta/\kappa}{1-i\left(\delta-\frac{u\,g}{2\kappa|\alpha_{u}|}\right)}.$
(7)

This is a kind of quantization rule, i.e., a “super-quantization” for the
quasi-coherent field amplitudes. Straightforward calculation for the imaginary
and real parts,

$\alpha_{u}\equiv|\alpha_{u}|\left(\cos{\varphi_{u}}+i\sin{\varphi_{u}}\right),$

leads to the following expressions

$\displaystyle|\alpha_{u}|$
$\displaystyle=\frac{\eta}{\kappa}\cos{\varphi_{u}}$ (8a)
$\displaystyle\cos{\varphi_{u}}$
$\displaystyle=\frac{1}{\sqrt{1+\delta^{2}}}\left(\zeta_{u}\delta+\sqrt{1-\zeta_{u}^{2}}\right)$
(8b) $\displaystyle\sin{\varphi_{u}}$
$\displaystyle=\frac{1}{\sqrt{1+\delta^{2}}}\left(-\zeta_{u}+\delta\sqrt{1-\zeta_{u}^{2}}\right)$
(8c)

where we introduced the scaled drive strength parameter

$\displaystyle\zeta_{u}=\frac{u\,g/2\eta}{\sqrt{1+\delta^{2}}}\,.$ (9)

Figure 2: Map of the available solutions according to the intuitive approach,
illustrated in the plane of the external control parameters of the drive
$\eta$ and $\delta$. The lines represent the boundaries above which the
solutions with the applied colours become physical.

In contrast to the implicit expression for the mean photon number in eq. 6,
here we got an explicit result for the cavity field amplitude determined by
the external control parameters, i.e., by the drive amplitude $\eta$ and the
detuning $\delta$. Such a result may correspond to a valid physical steady
state provided the condition $0\leq\cos{\varphi_{u}}\leq 1$ is satisfied. This
is summarized in the solution diagram of fig. 2. Clearly, the drive strength
parameter $\zeta_{u}$ must stay below a threshold to ensure that the solution
8a is both real and positive.

  * •

In the case of $\zeta_{u}\delta\geq 0$ this is $\absolutevalue{\zeta_{u}}\leq
1$

  * •

and for $\zeta_{u}\delta<0$, $\absolutevalue{\zeta_{u}}\sqrt{1+\delta^{2}}\leq
1$.

In cases where these conditions are not satisfied, the solution for the intra-
cavity photon number corresponding to $u$ does not exist. Furthermore, even if
the solution exists according to eq. 8, it is not necessarily consistent with
the assumption of large enough photon numbers and, accordingly, a quasi-
equidistant ladder $u$.

Let us now turn to the question how relevant the intuitive solutions are for
the fully quantum driven-dissipative dynamics. We are generating quantum
trajectories based on the quantum-jump Monte Carlo method implemented in the
C++QED framework.

Figure 3: Quantum trajectories for the cavity field amplitude. Numerically
calculated time evolution of the cavity photon number normalized to the empty
cavity photon number (blue, upper half of the panels), and phase (orange,
lower panels). The fixed values $\zeta_{u}/u=0.6$ and $\delta=-2$ determine
the intuitive solutions in eq. 8, indicated by the horizontal dotted lines
with color codes. The subplots (a)-(d) correspond to different coupling
strength $g$ and drive amplitudes $\eta$ such that $g/\eta$ is constant. The
four panels display examples for the photon transmission in the weak (a), in
the intermediate (b), and in the strong coupling regime with bistability (c)
and multistability (d). The inset (e) shows the chosen work point on the map
of possible quasi-coherent solutions, cf. fig. 2.

Sample quantum trajectories are presented in Figure 3 for fixed values of
parameters $\zeta_{u}/u=0.6$ and $\delta=-2$ which determine the intuitive
solutions, indicated by dotted horizontal lines in the panels. All the
solutions except for $u=3$ are possible in this case, however, the solution
for $u=1$ is at very low photon numbers, contradicting the premise of the
intuitive approach. The different panels are devoted to illustrate the
emergence of the predicted quasi-coherent solutions as the coupling strength
increases, together with the drive strength $\eta$ such that $g/\eta$ is
constant. With this scaling the intuitive solutions are fixed relative to the
empty-cavity intensity, but the higher drive strength can inject higher photon
numbers in the cavity. Accordingly, the quantum trajectories exhibit
substantially different behaviour regarding the cavity transmission for the
four selected values of the coupling constant $g$ in the panels (a)-(d).

For the case of weak coupling, shown in panel (a), the low-lying excitation
spectrum does not lead to transmission blockade 49; 50 at all due to weak
anharmonicity. The qudit, in its low excitation regime being a linear
polarizability, amounts to a simple dispersive shift of the mode resonance
that results in an approximately steady intra-cavity photon number
$\langle\hat{a}^{\dagger}\hat{a}\rangle/I_{0}(\Delta)=2.8$. This average over
the numerical trajectory is fit well by the value calculated from the
dispersive shift, represented as the blue solid horizontal line in panel (a),
though this shift is estimated by a leading order perturbative calculation.
Throughout, we use for normalization the empty-cavity intensity
$I_{0}(\Delta)=\frac{\eta^{2}}{\kappa^{2}+\Delta^{2}}$. None of the potential
quasi-coherent solutions for the fixed value of $\zeta_{u}/u=0.6$ play any
role in this parameter regime. The “super-quantization rule” 7 is applicable
only at high photon numbers which is not met in panel (a). The range of photon
numbers actually encountered in the time evolution is indicated by the maximum
photon number $n_{\text{max}}$ in the row of parameters above the panels,
i.e., $n_{\text{max}}=0.25$ for (a). The photon number is not high enough in
panel (b), either, which represents an intermediate range of the coupling,
$g=2.73\kappa$. There is still no well-resolved photon blockade, and the
intuitive solutions do not appear, either. The system does not find a robust
attractor state and thus strong fluctuations of the photon number fill the
full range of intensity between vacuum level and the $n_{\text{max}}=5.23$.

Going toward the strong coupling regime 51, in panel (c), the intuitive
solution with $u=-3$ emerges clearly as a metastable solution for macroscopic
durations of time, alternating with photon blockade periods in a random
switching signal. This is a similar telegraph signal of bistability as the one
studied with a single qubit 27. The quasi-coherent field state according to
the super-quantization rule is formed here at photon numbers around $\sim 14$.
With still stronger coupling, in panel (d), multiple solutions can stabilize
and can be observed in the intra-cavity intensity. The appearance of these
metastable solutions with the increasing parameter $g$ confirms that the
strong-coupling “thermodynamic limit” 5 holds for the breakdown of photon
blockade in a qudit-mode system as well. The numerical study furthermore
verifies that the phase of cavity field, when the system state stabilizes in
one of the available quasi-coherent states, follows accurately the analytical
value in eq. 8.

The demonstration of the adequacy of the intuitive picture in the strong-
coupling regime is completed in fig. 4. Here, the coupling constant $g$ is
fixed at a given detuning $\delta=1$, and the drive strength $\eta$ is
increased in order to address increasingly higher lying dressed states. Long
quantum trajectories have been generated, and the extracted photon-number
histograms are displayed in color code.

Figure 4: Histograms showing the probabilities of normalized resonator
intensity (with respect to the detuned empty-cavity intensity) (a-b) and phase
values (c-d). The parameters $g=7.2\kappa$ and $\Delta=\kappa$ are fixed,
while the drive strength $\eta$ is varied, corresponding to points on the
right side of (e). Qubit spontaneous emission is considered in (b) and (d)
with rate $\gamma=\kappa/100$. The field observables gather around the
solutions of eq. 8 which gradually appear as solid lines, the critical drive
strengths associated with the emergence of different $u$ solutions are
depicted as vertical dashed lines. The white dotted lines connect the time
averages of the corresponding observable at the given drive intensities.

For weak drive the cavity is in a dim state due to the drive being out of
resonance with the dressed excitation levels in the strong coupling regime.
This is the photon blockade regime which is broken down at a certain drive
strength. A new peak at around
$\langle\hat{a}^{\dagger}\hat{a}\rangle/I_{0}(\Delta)=2$ appears in the
histogram, cf. fig. 4(a). Vertical dashed delimit the onset of a new intuitive
solution associated with a ladder $u$, where this critical value of $\eta$ is
determined by the condition $\zeta_{u}\leq 1$
($\absolutevalue{\zeta_{u}}\sqrt{1+\delta^{2}}\leq 1$) for $u>0$ ($u<0$). On
increasing the drive intensity, the numerical simulation confirms the
emergence of further new peaks in the photon number distribution. For very
strong drive, i.e. the right side of the panels, all the four possible
solutions predicted by the intuitive picture for $\delta>0$, cf. fig. 2, are
occupied in a genuine multistability. A sample quantum trajectory with cavity
intensity switching among the four possible values is shown in fig. 5. The
position of the peaks in the photon number cf. fig. 4(a) as well as in the
phase distribution cf. fig. 4(c) fit very accurately on the intuitive solution
represented by solid lines with various colours corresponding to the different
ladders $u$.

The stability of the quasi-coherent states is due to their robustness against
the cavity photon loss, just as a true coherent state of an oscillator is
robust against a photon-loss quantum jump. In our case, a quantum jump
corresponding to the loss of a photon mixes negligibly the ladders
corresponding to different $u$s. This is why the drive (coherent displacement)
and the dissipation establish a stable Poissonian distribution of excitations
in one of the ladders.

On panels (b) and (d) of fig. 4, a new piece of physics is added. For these
plots, we considered the spin-3/2 as being composed of 3 qubits with identical
transition frequencies, and homogeneously coupled to the mode, that is
$\hat{S}^{-}=\sum_{i=1}^{3}\hat{\sigma}^{-}_{i}$ in eq. 1. In this case, it
becomes a relevant question what the effect of single-qubit decay is on the
dynamics, that we can simulate by supplementing the dissipator 2 with

$\mathcal{L}_{\text{single
emitter}}\left[\hat{\rho}\right]=\gamma\sum_{i=1}^{3}\left(2\hat{\sigma}^{-}_{i}\hat{\rho}\hat{\sigma}^{+}_{i}-\hat{\sigma}^{+}_{i}\hat{\sigma}^{-}_{i}\hat{\rho}-\hat{\rho}\hat{\sigma}^{+}_{i}\hat{\sigma}^{-}_{i}\right).$
(10)

What is remarkable is that this dissipator brings out of the subspace
determined by the spin quantum number $3/2$, that is, where the state of the
qubits is permutationally symmetric. Single-emitter decay has the potential of
jeopardizing the stability of the quasi-coherent states by strongly mixing the
different $u$ ladders. As displayed in fig. 4(b,d), however, for a limited
value of $\gamma$, the intuitive solutions still strongly govern the dynamics.
The enhanced mixing of these quasi-coherent states is nevertheless evident in
the diminished dwell times on the trajectories, and in that the average intra-
cavity photon number (depicted by the white dotted line in fig. 4(a) and (b),
converges to the detuned empty resonator population in the high-driving limit.

Figure 5: Example trajectory where all the solutions are present. Parameters:
$\gamma=0,g=7.2\kappa,\eta=18\kappa$ and $\Delta=\kappa$

In conclusion, we gained clear insight into the multistability exhibited by
the strongly coupled quantum system composed of a driven resonator and a
qudit. Beyond the photon blockade regime, where the greatly anharmonic
spectrum prevents the system from absorbing excitations, the coupled system
can be excited to bright stationary states which can be described as quasi-
coherent states with well-defined amplitude and phase that have been
determined analytically. The number of the possible stationary states equals
the number of the dimensions of the qudit. Identification of them can be a
useful resource in sensing applications: the considered small quantum system,
when impacted by a weak external signal, can be triggered to undergo a sharp
transition between the stationary states with a monitoring output 52; 53; 54.

While preparing this manuscript, we became aware of the related work by
Karmstrand et al. (arXiv:2403.02417 [2024]), exploring multistability in a
similar system in the case of resonant driving, when, in contrast to the
offresonant case studied here, the quasi-energies are known exactly.

###### Acknowledgements.

This research was supported by the Ministry of Culture and Innovation and the
National Research, Development and Innovation Office within the Quantum
Information National Laboratory of Hungary (Grant No.
2022-2.1.1-NL-2022-00004). We are grateful to the HUN-REN Cloud for providing
us with suitable computational infrastructure for the simulations.

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Fundamental role of Dirac supersingletons in particle theory

Felix M Lev

Email<EMAIL_ADDRESS>
###### Abstract

As shown in the famous Dyson’s paper ”Missed Opportunities”, even from purely
mathematical considerations (without any physics) it follows that Poincare
quantum symmetry is a special degenerate case of de Sitter quantum symmetries.
Then the question arises why in particle physics Poincare symmetry works with
a very high accuracy. The usual answer to this question is that a theory in de
Sitter space becomes a theory in Minkowski space in the formal limit when the
radius of de Sitter space tends to infinity. However, de Sitter and Minkowski
spaces are purely classical concepts, and in quantum theory the answer to this
question must be given only in terms of quantum concepts. At the quantum
level, Poincare symmetry is a good approximate symmetry if the eigenvalues of
the representation operators $M_{4\mu}$ of the anti-de Sitter algebra are much
greater than the eigenvalues of the operators $M_{\mu\nu}$
($\mu,\nu=0,1,2,3$). We show that explicit solutions with such properties
exist within the framework of the approach proposed by Flato and Fronsdal
where particles that are considered elementary in the standard theory are
bound states of two Dirac singletons.

Keywords: irreducible representations; de Sitter supersymmetry; Dirac
supersingletons; accuracy of Poincare symmetry

##  1 Problem statement

In the literature, relativistic (Poincare) symmetry in quantum field theory
(QFT) is usually explained as follows. Since Poincare group is the group of
motions of Minkowski space, the system under consideration should be described
by unitary representations of this group. This implies that the representation
generators commute according to the commutation relations of the Poincare
group Lie algebra:

$\displaystyle[P^{\mu},P^{\nu}]=0,\quad[P^{\mu},M^{\nu\rho}]=-i(\eta^{\mu\rho}P^{\nu}-\eta^{\mu\nu}P^{\rho}),$
$\displaystyle[M^{\mu\nu},M^{\rho\sigma}]=-i(\eta^{\mu\rho}M^{\nu\sigma}+\eta^{\nu\sigma}M^{\mu\rho}-\eta^{\mu\sigma}M^{\nu\rho}-\eta^{\nu\rho}M^{\mu\sigma})$
(1)

where $\mu,\nu=0,1,2,3$, $\eta^{\mu\nu}=0$ if $\mu\neq\nu$,
$\eta^{00}=-\eta^{11}=-\eta^{22}=-\eta^{33}=1$, $P^{\mu}$ are the four-
momentum operators and $M^{\mu\nu}$ are the Lorentz angular momentum
operators. This approach is in the spirit of the Erlangen Program proposed by
Felix Klein in 1872 when quantum theory did not yet exist. However, although
the Poincare group is the group of motions of Minkowski space, the description
(1) does not involve this group and this space.

As noted in [1, 2], in quantum theory, background space is only a mathematical
concept because here each physical quantity should be described by an operator
but there are no operators for the coordinates of background space. There is
no law that every physical theory must involve a background space. For
example, it is not used in nonrelativistic quantum mechanics and in
relativistic quantum theory for describing irreducible representations (IRs)
for elementary particles. In particle theory, transformations from the
Poincare group are not used because, according to the Heisenberg $S$-matrix
program, it is possible to describe only transitions of states from the
infinite past when $t\to-\infty$ to the distant future when $t\to+\infty$. In
this theory, systems are described by observable physical quantities — momenta
and angular momenta. So, symmetry at the quantum level is defined not by a
background space and its group of motions but by commutation relations of the
symmetry algebra (see [1, 2] for more details). In particular, Eqs. (1) can be
treated as the definition of Poincare invariance at the quantum level.

In his famous paper ”Missed Opportunities” [3] Dyson notes that:

  * •

a) Relativistic quantum theories are more general than nonrelativistic quantum
theories even from purely mathematical considerations because Poincare group
is more symmetric than Galilei one: the latter can be obtained from the former
by contraction $c\to\infty$.

  * •

b) de Sitter (dS) and anti-de Sitter (AdS) quantum theories are more general
than relativistic quantum theories even from purely mathematical
considerations because dS and AdS groups are more symmetric than Poincare one:
the latter can be obtained from the former by contraction $R\to\infty$ where
$R$ is a parameter with the dimension $length$, and the meaning of this
parameter will be explained below.

  * •

c) At the same time, since dS and AdS groups are semisimple, they have a
maximum possible symmetry and cannot be obtained from more symmetric groups by
contraction.

As noted above, symmetry at the quantum level should be defined in terms of
Lie algebras, and in [2], the statements a)-c) have been reformulated in such
terms. It has also been shown that the fact that quantum theory is more
general than classical theory follows even from purely mathematical
considerations because formally the classical symmetry algebra can be obtained
from the symmetry algebra in quantum theory by contraction $\hbar\to 0$. For
these reasons, the most general description in terms of ten-dimensional Lie
algebras should be carried out in terms of quantum dS or AdS symmetry.

The definition of those symmetries is as follows. If $M^{ab}$
($a,b=0,1,2,3,4$, $M^{ab}=-M^{ba}$) are the angular momentum operators for the
system under consideration, they should satisfy the commutation relations:

$[M^{ab},M^{cd}]=-i(\eta^{ac}M^{bd}+\eta^{bd}M^{ac}-\eta^{ad}M^{bc}-\eta^{bc}M^{ad})$
(2)

Here the tensor $\eta^{ab}$ is such that $\eta^{ab}=\eta_{ab}$, $\eta^{ab}=0$
if $a\neq b$, $\eta^{00}=-\eta^{11}=-\eta^{22}=-\eta^{33}=1$, $\eta^{44}=\mp
1$ for the dS and AdS symmetries, respectively, and this tensor is used to
raise and lower the indices of operators $M^{ab}$.

Although the dS and AdS groups are the groups of motions of dS and AdS spaces,
respectively, the description in terms of (2) does not involve those groups
and spaces, and it is a definition of dS and AdS symmetries at the quantum
level (see the discussion in [1, 2]).

The procedure of contraction from dS or AdS symmetry to Poincare one is
defined as follows. If we define the momentum operators $P^{\mu}$ as
$P^{\mu}=M^{4\mu}/R$ ($\mu=0,1,2,3$) then in the formal limit when
$R\to\infty$, $M^{4\mu}\to\infty$ but the quantities $P^{\mu}$ are finite,
Eqs. (2) become Eqs. (1). Here R is a parameter which has nothing to do with
the dS and AdS spaces. As seen from Eqs. (2), quantum dS and AdS theories do
not involve the dimensional parameters $(c,\hbar,R)$ because $(kg,m,s)$ are
meaningful only at the macroscopic level.

At the classical (non-quantum) level, the transition from dS or AdS symmetry
to Poincare one is explained as follows. When the radius $R$ of dS or AdS
space becomes infinitely large, the angular momentum $M$ of a particle moving
in this space also becomes infinitely large. In the formal limit $R\to\infty$
when dS or AdS space transforms into flat Minkowski space, the motion of a
particle in such space must be described by momentum $p=M/R$ which is finite
in this limit.

One can raise the question why Poincare symmetry works with high accuracy in
particle physics. At the classical level, the explanation of this fact is that
we live in dS or AdS space whose radius $R$ is very large. However, as noted
above, the concept of background space is purely classical. Therefore, the
question arises whether the answer to the above question can be given within
the framework of purely quantum theory, without involving classical concepts.
As follows from the above definition of contraction from dS or AdS algebra to
Poincare one, Poincare invariance works with a high accuracy, provided that in
nature such states play the major role in which the eigenvalues of the
operators $M^{4\mu}$ are much greater than the eigenvalues of the operators
$M^{\mu\nu}$ ($\mu,\nu$=0,1,2,3). In this paper, we propose a scenario that
describes such a situation.

The paper is organized as follows. In Sec. 2 we explain why supersymmetric AdS
symmetry is more general (fundamental) than standard AdS symmetry. In Sec. 3
we describe how the CPT transformation works at the quantum level. In Sec. 4
we explicitly describe the construction of the IR for Dirac supersingletons
and explicit relations between representation operators in this IR and
representation operators of the AdS algebra. Then in Sec. 5 it is explicitly
shown that there exist scenarios when Poincare symmetry works with a high
accuracy.

##  2 Supersymmetry

A problem discussed in a wide literature is that supersymmetric generalization
exists in the AdS case but does not exist in the dS one. As shown in [4], in
standard quantum theory (over the field of complex numbers), dS symmetry is
more general than AdS one, and it may be a reason why supersymmetry has not
been discovered yet. However, as shown in [2], standard quantum theory is a
special degenerate case of a quantum theory over a finite ring of field (FQT)
of characteristic $p$ in a formal limit $p\to\infty$, and in FQT, dS and AdS
symmetries are equivalent.

From the formal point of view, representations of the AdS so(2,3) superalgebra
osp(1,4) are similar to representations of the Poincare superalgebra because
they are also described by 14 operators — ten representation operators of the
so(2,3) algebra and four fermionic operators. There are three types of
relations: the operators of the so(2,3) algebra commute with each other as
usual (see Eqs. (2)), anticommutators of the fermionic operators are linear
combinations of the so(2,3) operators and commutators of the latter with the
fermionic operators are their linear combinations. However, in fact,
representations of the osp(1,4) superalgebra can be described exclusively in
terms of the fermionic operators. The matter is as that in the general case,
the anticommutators of four operators form ten independent linear
combinations. Therefore, ten bosonic operators can be expressed in terms of
fermionic ones. This is not the case for the Poincare superalgebra since the
Poincare algebra operators are obtained from the so(2,3) ones by contraction.
One can say that the representations of the osp(1,4) superalgebra is an
implementation of the idea that supersymmetry is the extraction of the square
root from the usual symmetry (by analogy with the treatment of the Dirac
equation as a square root from the Klein-Gordon equation).

We use
$(d_{1}^{\prime},d_{2}^{\prime},d_{1}^{\prime\prime},d_{2}^{\prime\prime})$ to
denote the fermionic operators of the osp(1,4) superalgebra. They should
satisfy the following relations. If $(A,B,C)$ are any fermionic operators,
[…,…] is used to denote a commutator and $\\{...,...\\}$ to denote an
anticommutator then

$[A,\\{B,C\\}]=F(A,B)C+F(A,C)B$ (3)

where the form $F(A,B)$ is skew symmetric, $F(d_{j}^{\prime},d_{j}")=1$
$(j=1,2)$ and the other independent values of $F(A,B)$ are equal to zero.

As shown by various authors (see e.g., [2, 5]), the operators $M^{ab}$ in Eqs.
(2) can be expressed through bilinear combinations of the fermionic operators
as follows:

$\displaystyle
h_{1}=\\{d_{1}^{\prime},d_{1}^{\prime\prime}\\},\,\,h_{2}=\\{d_{2}^{\prime},d_{2}^{\prime\prime}\\},\,\,M_{04}=h_{1}+h_{2},\,\,M_{12}=L_{z}=h_{1}-h_{2}$
$\displaystyle
L_{+}=\\{d_{2}^{\prime},d_{1}^{\prime\prime}\\},\,\,L_{-}=\\{d_{1}^{\prime},d_{2}^{\prime\prime}\\},\,\,M_{23}=L_{x}=L_{+}+L_{-}$
$\displaystyle
M_{31}=L_{y}=-i(L_{+}-L_{-}),\,\,M_{14}=(d_{2}^{\prime\prime})^{2}+(d_{2}^{\prime})^{2}-(d_{1}^{\prime\prime})^{2}-(d_{1}^{\prime})^{2}$
$\displaystyle
M_{24}=i[(d_{1}^{\prime\prime})^{2}+(d_{2}^{\prime\prime})^{2}-(d_{1}^{\prime})^{2}-(d_{2}^{\prime})^{2}]$
$\displaystyle
M_{34}=\\{d_{1}^{\prime},d_{2}^{\prime}\\}+\\{d_{1}^{\prime\prime},d_{2}^{\prime\prime}\\},\,\,M_{30}=-i[\\{d_{1}^{\prime\prime},d_{2}^{\prime\prime}\\}-\\{d_{1}^{\prime},d_{2}^{\prime}\\}]$
$\displaystyle
M_{10}=i[(d_{1}^{\prime\prime})^{2}-(d_{1}^{\prime})^{2}-(d_{2}^{\prime\prime})^{2}+(d_{2}^{\prime})^{2}]$
$\displaystyle
M_{20}=(d_{1}^{\prime\prime})^{2}+(d_{2}^{\prime\prime})^{2}+(d_{1}^{\prime})^{2}+(d_{2}^{\prime})^{2}$
(4)

where ${\bf L}=(L_{x},L_{y},L_{z})$ is the standard operator of three-
dimensional rotations.

The fact that the representation of the osp(1,4) superalgebra is fully defined
by Eqs. (3,4) and the properties of the form $F(.,.)$, shows that osp(1,4) is
a special case of the superalgebra.

We require the existence of the generating vector $e_{0}$ satisfying the
conditions :

$\displaystyle
d_{j}^{\prime}e_{0}=d_{2}^{\prime}d_{1}^{\prime\prime}e_{0}=0,\quad
d_{j}^{\prime}d_{j}^{\prime\prime}e_{0}=q_{j}e_{0}\quad(j=1,2)$ (5)

These conditions are written exclusively in terms of the $d$ operators. The
full representation space can be obtained by successively acting by the
fermionic operators on $e_{0}$ and taking all possible linear combinations of
such vectors. The theory of self-adjoint IRs of the osp(1,4) algebra has been
developed by several authors (see e.g., [5]), and in [2] this theory has been
generalized to the case of FQT.

##  3 CPT transformation in osp(1,4) invariant theory

In Poincare invariant particle theory, the CPT transformation is considered
the most general discrete spacetime transformation. Based on what was said in
Sec. 1, at the quantum level, this transformation should be considered not
from the point of view of Minkowski space, but at the operator level. We use
$\theta$ to denote the operator corresponding to the classical CPT
transformation. As Wigner noted [6], since the sign of the energy must remain
positive under the $\theta$ transformation, the operator $\theta$ must be not
unitary, but antiunitary, that is, it can be represented as $\theta=\beta K$
where $\beta$ is a unitary operator, and $K$ is the complex conjugation
operator. As shown by Schwinger [7], the problem of the sign of energy can
also be solved if instead of the antiunitary transformation the transpose
operation is used. In this work we use Wigner’s approach.

As shown by Wigner [6] (see also [8]), transformation $\theta$ transforms the
representation operators of the Poincare algebra in Eq. (1) as

$\theta P_{\mu}\theta^{-1}=P_{\mu},\quad\theta
M_{\mu\nu}\theta^{-1}=-M_{\mu\nu}$ (6)

The question arises of how to generalize these relationships to the case of dS
and AdS theories and, as noted in Sec. 1, this generalization should not
involve dS and AdS spaces. The issue of CPT transformation in such theories
has been considered by many authors. However, to the best of our knowledge,
these authors considered the CPT transformation only from the point of view of
transformations of fields on the dS and AdS spaces and did not consider a
direct generalization of Eq. (6). Moreover, as noted in Sec. 2, the
superalgebra osp(1,4) is a generalization of the algebra so(2,3) to the case
of supersymmetry, and, to the best of our knowledge, Eqs. (6) have not been
generalized to representations of this superalgebra.

For this purpose, it is necessary to define how the operator $\theta$
transforms the operators
$(d_{1}^{\prime},d_{2}^{\prime},d_{1}^{\prime\prime},d_{2}^{\prime\prime})$.
We define such a transformation as follows:

$\theta d_{1}^{\prime}\theta^{-1}=-id_{2}^{\prime},\,\,\theta
d_{2}^{\prime}\theta^{-1}=id_{2}^{\prime},\,\,\theta
d_{1}^{\prime\prime}\theta^{-1}=id_{2}^{\prime\prime},\,\,\theta
d_{2}^{\prime\prime}\theta^{-1}=-id_{1}^{\prime\prime}$ (7)

It is easy to see that the second of these relations follows from the first,
and the fourth follows from the third, taking into account the fact that the
operator $\theta$ is antiunitary.

Now, based on Eqs. (4) and (7) we conclude that

$\theta M_{4\mu}\theta^{-1}=M_{4\mu},\quad\theta
M_{\mu\nu}\theta^{-1}=-M_{\mu\nu},\quad\mu,\nu=0,1,2,3$ (8)

and this is a generalization of Eq. (6) to the case of representations of the
algebra so(2,3) because, as noted in Sec. 1, when contracting representations
of the algebra AdS into representations of the Poincare algebra, the operators
$M_{\mu\nu}$ are not affected, and the operators $M_{4\mu}$ go into $P_{\mu}$.
This result is also natural from the observation that, as it is easy to see,
Eqs. (2) are invariant under substitutions

$M_{4\mu}\to M_{4\mu},\quad M_{\mu\nu}\to-M_{\mu\nu},\quad i\to-i$

That these substitutions involve $i\to-i$ follows from the fact that the
operator $\theta$ is antiunitary.

##  4 Dirac supersingleton

When describing elementary particles within the framework of AdS symmetry, the
following problems arise.

If $m$ is the mass of a particle in Poincare invariant theory then its mass
$\mu$ in AdS theory is dimensionless and the relation between $\mu$ and $m$ is
$\mu=mR$ where $R$ is the contraction parameter for the transition from AdS to
Poincare symmetry. As explained in [4], $R$ has nothing to do with the radius
of classical AdS space, $R$ is fundamental to the same extent as $c$ and
$\hbar$, and the problem why the value of $R$ is as is does not arise. The
data on cosmological acceleration show that, at the present stage of the
universe, $R$ is of the order of $10^{26}m$ [4]. Therefore, even for
elementary particles, the AdS masses are very large. For example, the AdS
masses of the electron, the Earth and the Sun are of the order of $10^{39}$,
$10^{93}$ and $10^{99}$, respectively. The fact that even the AdS mass of the
electron is so large might be an indication that the electron is not a true
elementary particle. In addition, the present upper level for the photon mass
is $10^{-17}ev$ or less. This value seems to be an extremely tiny quantity.
However, the corresponding AdS mass is of the order of $10^{16}$ and so, even
the mass which is treated as extremely small in Poincare invariant theory
might be very large in AdS invariant theory.

As shown in [4], in standard quantum theory, dS symmetry is more general than
AdS one but in the framework of dS symmetry it is not possible to describe
neutral elementary particles, i.e., particles which are equivalent to the
their antiparticles. In FQT, dS and AdS symmetries are equivalent and, as
shown in [2], in this theory also there are no neutral elementary particles.
In particular, even the photon is not elementary.

This problem has been discussed by several authors. In Standard Model (based
on Poincare invariance) only massless particles are treated as elementary.
However, as shown in the seminal paper by Flato and Fronsdal [9] (see also
[10]), in standard AdS theory, each massless IR can be constructed from the
tensor product of two singleton IRs discovered by Dirac in his paper [11]
titled ”A Remarkable Representation of the 3 + 2 de Sitter group”, and the
authors of [9] believe that this is indeed a truly remarkable property.

The IR describing the supersingleton is constructed as follows. In Eq. (5), we
choose $q_{1}$ and $q_{2}$ the same and equal $q_{0}$ where $q_{0}=1/2$ in
standard theory over complex numbers and $q_{0}=(p+1)/2$ in FQT, where $p$ is
the characteristic of the field or ring and, in the latter case, $p$ is odd.

The authors of [9] and other publications treat singletons as true elementary
particles because their weight diagrams has only a single trajectory (that’s
why the corresponding IRs are called singletons). However, one should answer
the following questions:

  * •

a) Why singletons have not been observed yet.

  * •

b) Why such massless particles as photons and others are stable and their
decays into singletons have not been observed.

There exists a wide literature (see e.g., [12, 13] and references therein)
where this problem is investigated from the point of view of standard AdS QFT.
For example, in AdS QFT, singleton fields live on the boundary at infinity of
the AdS bulk (boundary which has one dimension less than the bulk). However,
as noted in Sec. 1, the explanation in the framework of quantum theory should
not involve classical spaces.

In standard theory, an IR characterized by $(q_{1},q_{2})$ can be constructed
from tensor products of two IRs characterized by $(q_{1}^{(1)},q_{2}^{(1)})$
and $(q_{1}^{(2)},q_{2}^{(2)})$ only if $q_{1}\geq(q_{1}^{(1)}+q_{1}^{(2)})$
and $q_{2}\geq(q_{2}^{(1)}+q_{2}^{(2)})$. Since no interaction is assumed, a
problem arises whether a particle constructed from a tensor product of other
two particles will be stable. In standard theory, a particle with the mass $m$
can be a stable composite state of two particles with the masses $m_{1}$ and
$m_{2}$ only if $m<(m_{1}+m_{2})$ and the quantity $(m_{1}+m_{2}-m)c^{2}$ is
called the binding energy. The greater the binding energy is the more stable
is the composite state with respect to external interactions.

As argued in [2], in FQT, the properties of Dirac singletons are even more
remarkable than in standard theory. Here the eigenvalues of the operators
$h_{1}$ and $h_{2}$ for singletons in FQT are $(p+1)/2,(p+3)/2,(p+5)/2...$,
i.e., huge numbers if $p$ is huge. Hence Poincare limit for Dirac singletons
in FQT has no physical meaning and they cannot be observable.

For answering question b) we note the following. In standard theory, the
binding energy is a measure of stability: the greater the binding energy is,
the greater is the probability that the bound state will not decay into its
components under the influence of external forces.

If a massless particle is a composite state of two Dirac singletons, and the
eigenvalues of the operators $h_{1}$ and $h_{2}$ for the Dirac singletons in
FQT are $(p+1)/2,(p+3)/2,(p+5)/2...$ then, since in FQT the eigenvalues of
these operators should be taken modulo $p$, the corresponding eigenvalues for
the massless particle are $1,2,3...$. Hence an analog of the binding energy
for the operators $h_{1}$ and $h_{2}$ is $p$, i.e., a huge number. This
phenomenon can take place only in FQT: although, from the formal point of
view, the Dirac singletons comprising the massless state do not interact with
each other, the analog of the binding energy for the operators $h_{1}$ and
$h_{2}$ is huge. In other words, the fact that all the quantities in FQT are
taken modulo $p$ implies a very strong effective interactions between the
singletons. It explains why the massless state does not decay into Dirac
singletons and why free Dirac singletons effectively interact pairwise for
creating their bound state.

As noted in the literature on singletons (see e.g., the review [12] and
references therein), the possibility that only singletons are true elementary
particles but they are not observable has some analogy with quarks. However,
the analogy is not full. According to Quantum Chromodynamics, forces between
quarks at large distances prevent quarks from being observable in free states.
In FQT, Dirac singletons cannot be in free states even if there is no
interaction between them; the effective interaction between Dirac singletons
arises as a consequence of the fact that FQT is based on arithmetic modulo
$p$. In addition, quarks and gluons are used for describing only strongly
interacting particles while in standard AdS theory and in FQT, quarks, gluons,
leptons, photons, W and Z bosons can be constructed from Dirac singletons.

In standard AdS theory, there exist four Dirac singletons which in the
literature are called Di singleton, Rac singleton and their antiparticles. In
the case of supersymmetry, Di and Rac singletons are combined into one
superparticle - the Dirac supersingleton, so that there are two
supersingletons - the Dirac supersingleton and its antiparticle. However, in
FQT those supersingletons are combined into one object and so there is only
one supersingleton. Here, one of the remarkable properties of supersingletons
is the following. The physical meaning of division comes from classical
physics, which assumes that every object can be divided into any arbitrarily
large number of arbitrarily small parts. However, standard division loses its
standard physical meaning when we reach the level of elementary particles
since, for example, the electron cannot be divided into two, three, and so on
parts. As shown in [2], in FQT, the theory of singletons can be built over a
ring in which there is no division, but only addition, subtraction and
multiplication.

As shown in [2], the operators $d_{1}^{\prime\prime}$ and
$d_{2}^{\prime\prime}$ commute in the space of the supersingleton IR. The
basis of this IR can be chosen as [2]
$e(j,k)=(d_{1}^{\prime\prime})^{j}(d_{2}^{\prime\prime})^{k}e_{0}$ where
$j,k=0,1,...\infty$ in standard theory and $j,k=0,1,...p-1$ in FQT. Then it
can be shown [2] that

$d_{1}^{\prime}e(j,k)=\frac{1}{2}je(j-1,k),\,\,d_{2}^{\prime}e(j,k)=\frac{1}{2}ke(j,k-1)$
(9)

in standard theory, and $\frac{1}{2}$ should be replaced by $(p+1)/2$ in FQT.

The important property of supersingletons is that the above results can be
immediately generalized to the case of higher dimensions. Consider a
superalgebra defined by the set of operators
$(d_{j}^{\prime},d_{j}^{\prime\prime})$ where $j=1,2,...J$ and formally any
triplet of the operators $(A,B,C)$ satisfies the commutation-anticommutation
relation Eq. (3) where the form $F(A,B)$ is skew symmetric,
$F(d_{j}^{\prime},d_{j}^{\prime\prime})=1$ $(j=1,2,...J)$ and the other
independent values of $F(A,B)$ are equal to zero. The higher-dimensional
analog of the supersingleton IR can now be defined such that the
representation space contains a vector $e_{0}$ satisfying the conditions

$d_{j}^{\prime}e_{0}=0,\quad
d_{j}^{\prime}d_{j}^{\prime\prime}e_{0}=\frac{1}{2}e_{0}\quad(j=1,2,...J)$

in standard theory and $\frac{1}{2}$ should be replaced by $(p+1)/2$ in FQT.
The basis of the representation space can be chosen in the form

$e(n_{1},n_{2},...n_{J})=(d_{1}^{\prime\prime})^{n_{1}}(d_{2}^{\prime\prime})^{n_{2}}\cdots(d_{J}^{\prime\prime})^{n_{J}}e_{0}$

where the operators $(d_{1}^{\prime\prime},...d_{J}^{\prime\prime})$ mutually
commute on the representation space. The fact that singleton physics can be
directly generalized to the case of higher dimensions has been indicated by
several authors (see e.g., [12] and references therein). It is interesting to
explore the possibility that spatial and internal quantum numbers are combined
within the framework of the theory of supersingletons in which $J>2$.

##  5 Why Poincare symmetry in particle theory works with high accuracy

Now we can consider the problem posed in Sec. 1: why in particle theory, the
eigenvalues of the operators $M_{4\mu}$ are much greater than the eigenvalues
of the operators $M_{\mu\nu}$ ($\mu,\nu$=0,1,2,3). As noted in Sec. 1, this
problem must be solved exclusively within the framework of quantum theory,
without involving such classical concepts as dS or AdS space.

Let us first consider the case of neutral particles, which are considered
elementary in the standard theory. As noted in Sec. 4, the standard theory has
problems describing such particles, and in FQT neutral particles cannot be
elementary. Therefore, based on the comments in Sec. 4, it is natural to
consider the case when such particles are bound states of two sigletons. In
this case, it is natural to treat the sigletons in the bound state are
antiparticles for each other.

Therefore, if the first singleton is considered a particle for which
representations of the AdS algebra are described by the operators (2), then
the second singleton is naturally considered an antiparticle for which
representations of the AdS algebra are described by the operators (2)
transformed using the operator $\theta$, i.e., by the operators (8).

The osp(1,4) superalgebra representation for a system of two supersingletons
is the tensor product of the representations for supersingletons 1 and 2, and
the representation operators are the sums of the corresponding operators:
$M_{ab}=M_{ab}^{(1)}+M_{ab}^{(2)}$. This means that if $\Psi_{1}$ is the state
of the first supersingleton, and $\Psi_{2}$ is the state of the second
supersingleton, which is the antiparticle for the first, then the operator
$M_{ab}$ acts on the tensor product of these supersingletons as

$M_{ab}(\Psi_{1}\times\Psi_{2})=(M_{ab}^{(1)}\Psi_{1}\times\Psi_{2})+(\Psi_{1}\times
M_{ab}^{(2)}\Psi_{2})$ (10)

The question arises under what conditions a system consisting of such
supersingletons can be described in the framework of Poincare symmetry with a
high accuracy. As noted above, a necessary condition for this is that for the
system as a whole, the eigenvalues of the operators $M_{4\mu}$ must be much
greater than the eigenvalues of the operators $M_{\mu\nu}$. Since the
operators $M_{ab}^{(1)}$ and $M_{ab}^{(2)}$ are selfadjoined, all their
eigenvalues are real. As follows from Eq. (8), if $M_{ab}^{(2)}=\theta
M_{ab}^{(1)}\theta^{-1}$ then the operators $M_{4\mu}^{(1)}$ and
$M_{4\mu}^{(2)}$ have the same set of eigenvalues and the sets of eigenvalues
for the operators $M_{\mu\nu}^{(1)}$ and $M_{\mu\nu}^{(2)}$ differ by sign.

Also, the question arises what the states of two supersingletons must be to
combine them into one particle, which is considered elementary in the standard
theory. Since a system of two supersingletons is described by the tensor
product of single supersingleton representations, then, from the point of view
of standard theory, supersingletons do not interact with each other, but, as
noted in Sec. 4, in FQT they actually interact. Nevertheless, it is natural to
think that supersigletons in a bound state have close four momenta and are
close to each other, that is, they have approximately the same coordinates.

As noted in Sec. 1, if dS or AdS symmetry works with good accuracy, then the
operators $P_{\mu}$ are obtained from $M_{4\mu}$ by contraction. Therefore, it
is natural to expect that for supersingletons in the bound state, the
operators $M_{4\mu}$ have close eigenvalues. In particular, if
$\Psi_{2}=\theta\Psi_{1}$ and for some value of $\mu$,
$M_{4\mu}^{(1)}\Psi_{1}=\lambda_{1}\Psi_{1}$ then, as follows from Eqs. (8),
$M_{4\mu}^{(2)}\Psi_{2}=\lambda_{1}\Psi_{2}$ and therefore, as follows from
Eq. (10)

$M_{4\mu}(\Psi_{1}\times\Psi_{2})=2\lambda_{1}(\Psi_{1}\times\Psi_{2})$ (11)

Also, $\Psi_{1}$ and $\Psi_{2}=\theta\Psi_{1}$ can be treated as the states
having the same coordinates. As discussed e.g., in [14], the position
operators can be constructed from the operators $M_{\mu\nu}$. Therefore if for
some values of $\mu$ and $\nu$, $M_{\mu\nu}^{(1)}\Psi_{1}=\lambda_{2}\Psi_{1}$
then, as follows from Eqs. (8),
$M_{\mu\nu}^{(2)}\Psi_{2}=-\lambda_{2}\Psi_{2}$ and therefore, as follows from
Eq. (10)

$M_{\mu\nu}(\Psi_{1}\times\Psi_{2})=0$ (12)

Now we have a natural explanation of the fact that, for a system consisting
from supersingleton and antisupersingleton in a bound state, the eigenvalues
of the operators $M_{4\mu}$ are much greater than the eigenvalues of the
operators $M_{\mu\nu}$: as follows from Eqs. (8,11,12), the eigenvalues of
operators $M_{4\mu}$ for individual supersingletons are included in the two-
particle operators $M_{4\mu}$ with the same signs, while the eigenvalues of
the operators $M_{\mu\nu}$ \- with different ones. Therefore, we have a
natural explanation of the fact that for a particle consisting from
supersingleton and antisupersingleton, Poincare symmetry works with high
accuracy.

Let us now consider the case when supersingletons entering a bound state are
not antiparticles for each other. Then the IR of the AdS algebra for each
supersingleton in a bound state is described by operators satisfying the
relations (2) and there is no need to use the operator $\theta$. Even when we
work in FQT and consider states of supersingletons in which the quantum
numbers $j,k$ are much less than $p$, then, with high accuracy, we can apply
standard mathematics. We assume that, although the numbers $j,k$ can be very
large, they are still much less than $p$. Therefore, in what follows we
consider Dirac supersingletons only within the framework of standard
mathematics.

We now treat
$(d_{1}^{\prime},d_{2}^{\prime},d_{1}^{\prime\prime},d_{2}^{\prime\prime})$ as
the operators in the Hilbert space related by Hermitian conjugation as
$(d_{1}^{\prime})^{*}=d_{1}^{\prime\prime}$ and
$(d_{2}^{\prime})^{*}=d_{2}^{\prime\prime}$. Then, as follows from Eq. (9),
the norm squared of $e(j,k)$ equals

$||e(j,k)||^{2}=\frac{j!k!}{2^{j+k}}$ (13)

and the normalized basis vectors can be defined asn

${\tilde{e}}(j,k)=(\frac{2^{j+k}}{j!k!})^{1/2}e(j,k)$ (14)

In particle scattering experiments, the four-momenta and coordinates of
initial and final particles are known with great accuracy. Therefore, these
particles are described with good accuracy in the semiclassical approximation.
In this case, the supersingleton wave functions
$\sum_{jk}c(j,k){\tilde{e}}(j,k)$ are such that the coefficients $c(j,k)$ are
not equal to zero only at $j\in(j_{1},j_{2}),\,\,k\in(k_{1},k_{2})$ where
$j_{2}-j_{1}\ll j_{1},\,\,k_{2}-k_{1}\ll k_{1}$ and the values of $|c(j,k)|$
at such $j,k$ are approximately the same. We define the angular dependence of
the coefficients as $c(j,k)=|c(j,k)|exp[i(j+k)\chi+i(j-k)\varphi]$. Then
taking into account Eqs. (4,9,14) and the definition of the basis elements and
the coefficients $c(j,k)$, direct calculation shows that, in semiclassical
approximation, the operators $M_{ab}$ can be replaced by their numerical
values:

$\displaystyle
L_{x}=2(jk)^{1/2}cos(2\varphi),\,\,L_{y}=-2(jk)^{1/2}sin(2\varphi),\,\,L_{z}=j-k$
$\displaystyle
M_{10}=jsin(2\varphi+2\chi)+ksin(2\varphi-2\chi),\,\,M_{04}=j+k+1$
$\displaystyle M_{20}=jcos(2\varphi+2\chi)+kcos(2\varphi-2\chi)$
$\displaystyle M_{30}=2(jk)^{1/2}sin(2\chi),\,\,M_{34}=2(jk)^{1/2}cos(2\chi)$
$\displaystyle M_{14}=kcos(2\varphi-2\chi)-jcos(2\varphi+2\chi)$
$\displaystyle M_{24}=jsin(2\varphi+2\chi)-ksin(2\varphi-2\chi)$ (15)

As can be seen from these expressions, for a single supersingleton there is no
scenario when the eigenvalues of the operators $M_{4\mu}$ are much greater
than the eigenvalues of the operators $M_{\mu\nu}$ ($\mu,\nu$=0,1,2,3). This
is an additional argument why singletons cannot exist in free states.

The eigenvalues of the operators in Eq. (15) satisfy the property that when
one applies the transformations

$j\leftrightarrow
k,\,\,\chi\rightarrow-\chi,\,\,\varphi\rightarrow\varphi+\pi/2$ (16)

then all the eigenvalues of the operators $M_{4\mu}$ do not change while all
the eigenvalues of the operators $M_{\mu\nu}$ change their sign.

As noted above, for a system of two free supersingletons 1 and 2, the AdS
superalgebra representation for such a system is the tensor product of the
representations for supersingletons 1 and 2, and the representation operators
are the sums of the corresponding operators:
$M_{ab}=M_{ab}^{(1)}+M_{ab}^{(2)}$.

If the eigenvalues of $M_{ab}^{(1)}$ are described by Eqs. (15) with the
parameters $(j,k,\chi,\varphi)=(j_{1},k_{1},\chi_{1},\varphi_{1})$ and the
eigenvalues of the operators $M_{ab}^{(2)}$ are described by Eqs. (15) with
the parameters $(j,k,\chi,\varphi)=(j_{2},k_{2},\chi_{2},\varphi_{2})$ then,
as follows from the remarks after Eq. (16), if

$j_{2}\approx k_{1},\,\,k_{2}\approx
j_{1},\,\,\chi_{2}\approx-\chi_{1},\,\,\varphi_{2}\approx\varphi_{1}+\pi/2$
(17)

then the eigenvalues of the operators $M_{4\mu}^{(1)}$ and $M_{4\mu}^{(2)}$
will be approximately equal for each $\mu$ while for each $\mu,\nu$ the
eigenvalues of the operators $M_{\mu\nu}^{(1)}$ and $M_{\mu\nu}^{(2)}$ will
approximately differ by sign. Therefore, for the operators describing the
tensor product, the eigenvalues of the operators $M_{4\mu}$ will be much
greater than the eigenvalues of the operators $M_{\mu\nu}$, and this
guarantees that Poincare symmetry will be a good approximate symmetry.

##  6 Conclusion

As shown in the famous Dyson’s paper [3], even from purely mathematical
considerations (without any physics) it follows that Poincare quantum symmetry
is a special degenerate case of de Sitter quantum symmetries. As shown in [2,
4], in standard quantum theory (over the field of complex numbers), dS
symmetry is more general than AdS one but in the framework of dS symmetry it
is not possible to describe neutral elementary particles, i.e., particles
which are equivalent to the their antiparticles. In particular, even the
photon cannot be a neutral elementary particle. This problem has been
discussed by several authors.

In Standard Model (based on Poincare invariance) only massless particles are
treated as elementary. However, as shown by Flato and Fronsdal [9] (see also
[10]), in standard AdS theory, each massless IR can be constructed from the
tensor product of two singleton IRs discovered by Dirac in his seminal paper
[11]. As explained in Sec. 2, AdS theory based on supersymmetry is more
general (fundamental) than standard AdS theory.

Therefore, the question arises why in particle physics, Poincare symmetry
works with a very high accuracy. The usual answer to this question is that a
theory in de Sitter space becomes a theory in Minkowski space in the formal
limit when the radius of de Sitter space tends to infinity. However, de Sitter
and Minkowski spaces are purely classical concepts, and in quantum theory the
answer to this question must be given only in terms of quantum concepts.

As noted in Sec. 1, at the quantum level, Poincare symmetry is a good
approximate symmetry if the eigenvalues of the operators $M_{4\mu}$ are much
greater than the eigenvalues of the operators $M_{\mu\nu}$
($\mu,\nu=0,1,2,3$). As shown in Sec. 5, for a single supersingleton there is
no scenario when these conditions are met but explicit solutions with such
properties exist when:

  * •

a particle which in the standard theory is considered a neutral elementary
particle consists of a supersingleton and its antiparticle, and the result
follows from Eqs. (8,11,12).

  * •

a particle which in the standard theory is considered elementary, consists of
two supersingletons satisfying the semiclassical approximation, and their
states satisfy the conditions (17).

Acknowledgments. The author is grateful to Vladimir Karmanov for useful
discussions.

## References

  * [1] F.M. Lev, de Sitter Symmetry and Quantum Theory. Phys. Rev. D85, 065003 (2012).
  * [2] F.M. Lev, Finite Mathematics as the Foundation of Classical Mathematics and Quantum Theory. With Application to Gravity and Particle theory. ISBN 978-3-030-61101-9, Springer, Cham (2020).
  * [3] F.G. Dyson, Missed Opportunities. Bull. Amer. Math. Soc. 78, 635-652 (1972).
  * [4] F.M. Lev, Solving Particle–Antiparticle and Cosmological Constant Problems. Axioms 13, 138 (2024).
  * [5] W. Heidenreich, All Linear Unitary Irreducible Representations of de Sitter Supersymmetry with Positive Energy. Phys. Lett. B110, 461-464 (1982).
  * [6] E. Wigner, Normal Form of Antiunitary Operators, J. Math. Phys. 1, 409–412 (1960).
  * [7] J. Schwinger, The Theory of Quantized Fields. II Phys. Rev. 91 713 (1953).
  * [8] Yu. V. Novozhilov, Introduction to Elementary Particle Theory. International Series of Monographs in Natural Philosophy 78, B01K2IQ5L2 Pergamon (1975).
  * [9] M. Flato and C. Fronsdal, One Massles Particle Equals two Dirac Singletons. Lett. Math. Phys. 2, 421-426 (1978).
  * [10] W. Heidenreich, Tensor Product of Positive Energy Representations of ${\tilde{S}}(3,2)$ and ${\tilde{S}}(4,2)$. J. Math. Phys. 22, 1566-1574 (1981).
  * [11] P.A.M. Dirac, A Remarkable Representation of the 3 + 2 de Sitter group. J. Math. Phys. 4, 901-909 (1963).
  * [12] M. Flato, C. Fronsdal and D. Sternheimer, Singleton Physics. hep-th/9901043 (1999).
  * [13] X. Bekaert, Singletons and Their Maximal Symmetry Algebras. arXiv:1111.4554 (2011).
  * [14] F.M. Lev, Fundamental Quantal Paradox and its Resolution. Physics of Particles and Nuclei Letters (published by Springer) 14, 444-452 (2017).


# Scaling properties of the action in the Riemann-Liouville fractional
standard map

J. A. Méndez-Bermúdez  Instituto de Física, Benemérita Universidad Autónoma de
Puebla, Puebla 72570, Mexico  R. Aguilar-Sánchez  Facultad de Ciencias
Químicas, Benemérita Universidad Autónoma de Puebla, Puebla 72570, Mexico
José M. Sigarreta  Facultad de Matemáticas, Universidad Autónoma de Guerrero,
Carlos E. Adame No.54 Col. Garita, Acalpulco Gro. 39650, Mexico  Edson D.
Leonel  Universidade Estadual Paulista (UNESP) - Departamento de Física, Av.
24A, 1515 – Bela Vista – CEP: 13506-900 – Rio Claro – SP – Brazil

###### Abstract

The Riemann-Liouville fractional standard map (RL-fSM) is a two-dimensional
nonlinear map with memory given in action-angle variables $(I,\theta)$. The
RL-fSM is parameterized by $K$ and $\alpha\in(1,2]$ which control the strength
of nonlinearity and the fractional order of the Riemann-Liouville derivative,
respectively. In this work, we present a scaling study of the average squared
action $\left<I^{2}\right>$ of the RL-fSM along strongly chaotic orbits, i.e.
for $K\gg 1$. We observe two scenarios depending on the initial action
$I_{0}$, $I_{0}\ll K$ or $I_{0}\gg K$. However, we can show that
$\left<I^{2}\right>/I_{0}^{2}$ is a universal function of the scaled discrete
time $nK^{2}/I_{0}^{2}$ ($n$ being the $n$th iteration of the RL-fSM). In
addition, we note that $\left<I^{2}\right>$ is independent of $\alpha$ for
$K\gg 1$. Analytical estimations support our numerical results.

##  I Preliminaries

The kicked rotor represents a free rotating stick in an inhomogeneous field
that is periodically switched on in instantaneous pulses, see e.g. O08 . It is
described by the second order differential equation

$\ddot{\theta}+K\sin(\theta)\sum_{j=0}^{\infty}\delta\left(\frac{t}{T}-j\right)=0.$
(1)

Here, $\theta\in[0,2\pi]$ is the angular position of the stick, $K$ is the
kicking strength, $T$ is the kicking period (that we set to one from now on),
and $\delta$ is Dirac’s delta function. By replacing the second-order
derivative in the equation of motion of the kicked rotor with a Riemann-
Liouville (RL) derivative of fractional order $\alpha$ SKM93 ; KST06 , the RL
fractional kicked rotor (fKR) is obtained TZ08 ; ET09 :

${}_{0}D_{t}^{\alpha}\theta+K\sin(\theta)\sum_{j=0}^{\infty}\delta\left(t-j\right)=0,\quad
1<\alpha\leq 2.$ (2)

Above SKM93 ; KST06 ,

${}_{0}D_{t}^{\alpha}\theta(t)=D_{t}^{m}{{}_{0}{\cal
I}}_{t}^{m-\alpha}\theta(t)$
$\displaystyle=\frac{1}{\Gamma(m-\alpha)}\frac{d^{m}}{dt^{m}}\int_{0}^{t}\frac{\theta^{\tau}d\tau}{(t-\tau)^{\alpha-m+1}},\quad
m-1<\alpha\leq m,$

with $D_{t}^{m}=d^{m}/dt^{m}$, ${}_{0}{\cal I}_{t}^{m}f(t)$ is a fractional
integral given by

${}_{0}{\cal
I}_{t}^{m}f(t)=\frac{1}{\Gamma(m)}\int_{0}^{t}(t-\tau)^{\alpha-1}f(\tau)d\tau,$

and $\Gamma$ is the Gamma function.

The RL-fKR has a stroboscopic version, a two-dimensional nonlinear map with
memory, which is well known as the RL fractional standard map (RL-fSM) ET09 :

$\begin{array}[]{ll}I_{n+1}=I_{n}-K\sin(\theta_{n}),\\\
\theta_{n+1}=\displaystyle{\frac{1}{\Gamma(\alpha)}\sum_{i=0}^{n}I_{i+1}V^{1}_{\alpha}(n-i+1)},\quad\mbox{mod}~{}(2\pi),\end{array}$
(3)

where $I(t)\equiv{{}_{0}D}_{t}^{\alpha-1}\theta(t)$, $n$ is the discrete time,
and $V^{k}_{\alpha}(m)=m^{\alpha-k}-(m-1)^{\alpha-k}$. Then, the RL-fSM, given
in action-angle variables $(I,\theta)$, is parameterized by $K$ and
$\alpha\in(1,2]$ which control the strength of nonlinearity and the fractional
order of the RL derivative, respectively. In fact, for $\alpha=2$, the RL-fSM
reproduces the celebrated Chirikov’s standard map (CSM) C69 .

Compared with CSM, which presents the generic transition to chaos (in the
context of Kolmogorov–Arnold–Moser theorem), depending on the parameter pair
($K,\alpha$), the RL-fSM shows richer dynamics: It generates attractors (fixed
points, asymptotically stable periodic trajectories, slow converging and slow
diverging trajectories, ballistic trajectories, and fractal-like structures)
and/or chaotic trajectories ET09 ; ET13 ; E19 ; E11 ; MPSL23 . Moreover,
trajectories may intersect and attractors may overlap E11 .

Among the available studies on the RL-fKR, see e.g. ET09 ; ET13 ; E19 ; E11 ;
MPSL23 , the analysis of strongly chaotic orbits has been left unexplored.
Therefore, here we undertake this task and characterize the dynamics of the
RL-fSM by computing the squared average action $\left<I^{2}_{n}\right>$ when
$K\gg 1$.

Figure 1: Average squared action $\left<I^{2}_{n}\right>_{\mbox{\tiny int}}$
as a function of the discrete time $n$ for (a) $\alpha=1.1$, (b) $\alpha=1.5$,
and (c) $\alpha=1.9$. Open symbols (full symbols) correspond to $I_{0}\ll K$
($I_{0}\gg K$). The blue dashed lines, plotted to guide the eye, are
proportional to $n$. The average is over 100 orbits with initial random phases
in the interval $0<\theta_{0}<2\pi$. Red full lines are Eq. (9). Figure 2:
(a,b) Average squared action $\left<I^{2}_{n}\right>_{\mbox{\tiny int}}$ as a
function of $n$ for (a) $K=10^{3}$ and several values of $I_{0}$ ($2\times
10^{3}$, $4\times 10^{3}$, $10^{4}$, $2\times 10^{4}$, $4\times 10^{4}$,
$10^{5}$, and $2\times 10^{5}$, from bottom to top) and (b) $I_{0}=10^{5}$ and
several values of $K$ ($4\times 10^{2}$, $10^{3}$, $2\times 10^{3}$, $4\times
10^{3}$, $10^{4}$, $2\times 10^{4}$, and $4\times 10^{4}$, from bottom to
top). As in Fig. 1, the average is taken over 100 orbits with initial random
phases in the interval $0<\theta_{0}<2\pi$. All data in (a,b) correspond to
$\alpha=1.1$. The horizontal red dashed lines in (a,b) indicate
$\left<I^{2}_{n}\right>_{\mbox{\tiny int}}=I_{0}^{2}$ with (a) $I_{0}=4\times
10^{4}$ and (b) $I_{0}=10^{5}$, respectively. The transversal red dashed lines
in (a,b) are fittings of $\left<I^{2}_{n}\right>_{\mbox{\tiny int}}={\cal C}n$
to the data represented by asterisks (for $n\geq 10^{4}$) with fitting
constants (a) ${\cal C}=294658$ and (b) ${\cal C}=3984348$. (c,d) Crossover
time $n^{*}$ (a) as a function of $I_{0}$ for constant $K$ ($K=10^{3}$) and
(b) as a function of $K$ for constant $I_{0}$ ($I_{0}=10^{5}$). In (c,d),
three values of $\alpha$ are reported: $\alpha=1.1$, 1.5, and 1.9. Red dashed
lines in (c,d) are power-law fittings to the data of the form (a)
$n^{*}\propto I_{0}^{\gamma_{1}}$ with $\gamma_{1}\approx 2$ and (b)
$n^{*}\propto K^{\gamma_{2}}$ with $\gamma_{2}\approx-2$. Blue dot-dashed
lines in (c,d) are Eq. (11). Figure 3: Normalized average squared action
$\left<I^{2}_{n}\right>_{\mbox{\tiny int}}/I_{0}^{2}$ as a function of the
normalized time $nK^{2}/I_{0}^{2}$. Same data sets as in Fig. 1. The red line
is Eq. (10).

##  II Numerical results

For the numerical study we wrote a program in FORTRAN90 to compute the orbits
of map (3) by straightforward iteration. Moreover, since the memory property
in map (3) forbids the use of a large number of orbit realizations, instead of
investigating $\left<I^{2}_{n}\right>$ directly, to smooth the curves
$\left<I^{2}_{n}\right>$ vs. $n$, we compute its cumulative-normalized value

$\left<I^{2}_{n}\right>_{\mbox{\tiny
int}}=\frac{1}{n}\int^{n}_{n_{0}=0}\left<I^{2}_{n^{\prime}}\right>dn^{\prime}.$
(4)

Specifically, we compute $\left<I^{2}_{n}\right>_{\mbox{\tiny int}}$ for map
(3) following two steps: First we calculate the average squared action over
the orbit associated with the initial condition $j$ as
$\left<I^{2}_{n,j}\right>=(n+1)^{-1}\sum^{n}_{i=0}I^{2}_{i,j}$, where $i$
refers to the $i$-th iteration of the map. Then,
$\left<I^{2}_{n}\right>_{\mbox{\tiny int}}$ is defined as the average over
$M=100$ independent realizations of the map (by randomly choosing values of
$\theta_{0}$ in the interval $0<\theta_{0}<2\pi$):
$\left<I^{2}_{n}\right>_{\mbox{\tiny
int}}(I_{0},K,\alpha)=M^{-1}\sum^{M}_{j=1}\left<I^{2}_{n,j}\right>$.

In Fig. 1 we plot $\left<I^{2}_{n}\right>_{\mbox{\tiny int}}$ as a function of
the discrete time $n$ for representative values of $\alpha$ in the interval
$(1,2]$: (a) $\alpha=1.1$, (b) $\alpha=1.5$, and (c) $\alpha=1.9$. Several
combinations of parameter pairs $(I_{0},K)$ are considered, as indicated in
the r.h.s. of the figure. From this figure, we observe two scenarios depending
on the initial action $I_{0}$ as compared with $K$: $I_{0}\ll K$ (open
symbols) or $I_{0}\gg K$ (full symbols). Specifically, when $I_{0}\ll K$,
$\left<I^{2}_{n}\right>_{\mbox{\tiny int}}\propto n$ for all $n$ (see the blue
dashed lines). While for $I_{0}\gg K$, first,
$\left<I^{2}_{n}\right>_{\mbox{\tiny int}}$ remains approximately constant and
proportional to $I^{2}_{0}$ up to a crossover time $n^{*}$, after which
$\left<I^{2}_{n}\right>_{\mbox{\tiny int}}$ grows proportional to $n$.

From Fig. 1, it can also be seen that the crossover time $n^{*}$ depends on
both $I_{0}$ and $K$, while the dependence of $n^{*}$ with $\alpha$ is not
evident. Then, to look for the dependence of $n^{*}$ on the map parameters, in
Fig. 2(a) [Fig. 2(b)] we plot $\left<I^{2}_{n}\right>_{\mbox{\tiny int}}$ vs.
$n$ for several values of $I_{0}$ [$K$] and fixed $K$ [$I_{0}$]. In both
figures, we use $\alpha=1.1$. We numerically extract $n^{*}$ as the crossing
point between the functions $\left<I^{2}_{n}\right>_{\mbox{\tiny
int}}=I_{0}^{2}$ and $\left<I^{2}_{n}\right>_{\mbox{\tiny int}}={\cal C}n$
(which is the fitting to the data in the growing regime); as examples, see the
horizontal and transversal dashed lines in Figs. 2(a,b), respectively. Thus,
in Figs. 2(c,d) we plot the obtained values of $n^{*}$ for $\alpha=1.1$ but
also for $\alpha=1.5$ and 1.9. Figures 2(c,d) reveal the power-law dependence

$n^{*}\propto I_{0}^{\gamma_{1}}K^{\gamma_{2}}$ (5)

and the independence of $n^{*}$ on $\alpha$. Power-law fittings of the data in
Figs. 2(c,d) provide $\gamma_{1}\approx 2$ and $\gamma_{2}\approx-2$, see red
dashed lines.

Equation (5) together with the observation that
$\left<I^{2}_{n}\right>_{\mbox{\tiny int}}\approx I^{2}_{0}$ for $n<n^{*}$
allow us to scale the curves $\left<I^{2}_{n}\right>_{\mbox{\tiny int}}$ vs.
$n$. Indeed, in Fig. 3 we plot $\left<I^{2}_{n}\right>_{\mbox{\tiny
int}}/I_{0}^{2}$ as a function of the normalized time $nK^{2}/I_{0}^{2}$ (i.e.
$n/n^{*}$) and observe the colapse of all curves on top of a universal
function.

##  III Analytical estimation

Now, to support and better understand the scaling performed above, we derive
an analytical estimation for $\left<I^{2}_{n}\right>_{\mbox{\tiny int}}$, see
e.g. MOL16 .

From the first line of map (3) we have that
$I^{2}_{n+1}=I_{n}^{2}-2KI_{n}\sin(\theta_{n})+K^{2}\sin^{2}(\theta_{n})$, so
we can write

$\left<I^{2}_{n+1}\right>=\left<I_{n}^{2}\right>-\
2K\left<I_{n}\right>\left<\sin(\theta_{n})\right>+K^{2}\left<\sin^{2}(\theta_{n})\right>.$

Since for chaotic orbits we can assume that $\left<\sin(\theta_{n})\right>=0$,
the term $2K\left<I_{n}\right>\left<\sin(\theta_{n})\right>$ can be
eliminated. Therefore,

$\left<I^{2}_{n+1}\right>=\left<I_{n}^{2}\right>+\frac{K^{2}}{2},$ (6)

where we have used $\left<\sin^{2}(\theta_{n})\right>=1/2$. Then, by noticing
that

$\left<I^{2}_{n+1}\right>-\left<I_{n}^{2}\right>=\frac{\left<I^{2}_{n+1}\right>-\left<I_{n}^{2}\right>}{(n+1)-n}\approx\frac{dJ}{dn}\
,$

we rewrite Eq. (6) as the first order differential equation:

$\frac{dJ}{dn}=\frac{K^{2}}{2},$ (7)

where $J\equiv\left<I_{n}^{2}\right>$. Therefore, by solving (7), we can write

$\left<I^{2}_{n}\right>=I_{0}^{2}+\frac{K^{2}}{2}n,$ (8)

where we have used $J_{0}=\left<I_{0}^{2}\right>=I_{0}^{2}$ and $n_{0}=0$.
Finally, by substituting (8) into Eq. (4) we can also write down an explicit
expression for $\left<I^{2}_{n}\right>_{\mbox{\tiny int}}$:

$\left<I^{2}_{n}\right>_{\mbox{\tiny int}}=I_{0}^{2}+\frac{K^{2}}{4}n.$ (9)

Indeed, Eq. (9) reproduces well our numerical data as can be seen in Fig. 1
where we have included Eq. (9) as red lines.

##  IV Discussion and conclusions

Given the good correspondence of Eq. (9) and the numerical data, it is clear
that it reproduces the scaling laws reported in Sec. II, which can be
summarized as

$\left<I^{2}_{n}\right>_{\mbox{\tiny int}}=\left\\{\begin{array}[]{ll}\propto
K^{2}n\ ,&\quad\mbox{when}\quad I_{0}\ll K,\\\
\left.\begin{array}[]{ll}\approx I_{0}^{2}\ ,&n<n^{*}\\\ \propto K^{2}n\
,&n>n^{*}\end{array}\right\\}&\quad\mbox{when}\quad I_{0}\gg
K.\end{array}\right.$

Moreover, Eq. (9) can also be used to demonstrate that the ratio
$\left<I^{2}_{n}\right>_{\mbox{\tiny int}}/I_{0}^{2}$ is a simple universal
function of the variable $\overline{n}=n/n^{*}$:

$\frac{\left<I^{2}_{n}\right>_{\mbox{\tiny int}}}{I_{0}^{2}}=1+\overline{n},$
(10)

where the crossover time $n^{*}$ is now naturally defined as

$n^{*}\equiv 4I_{0}^{2}K^{-2},$ (11)

in agreement with Eq. (5). Finally, in Fig. 2(c,d) and Fig. 3 we plot Eq. (11)
and Eq. (10) (see dot-dashed blue lines and red full lines), respectively, and
observe an excellent agreement with the numerical data.

It is relevant to notice that for strongly chaotic orbits, $K\gg 1$, the
average squared action $\left<I^{2}_{n}\right>$ for the RL-fSM does not depend
on the order $\alpha$ of the fractional derivative. Indeed, the panorama
reported here for $\left<I^{2}_{n}\right>$ vs. $n$ is equivalent to that of
CSM MOL16 ; LS07  as well as that of the discontinuous standard map MOL16 ;
MA12 , both with $K\gg 1$. This could be understood from the analytical
estimation of Sec. III by noticing that to obtain the expression for
$\left<I^{2}_{n}\right>$ we mainly used the first equation of map (3) which
does not contain the parameter $\alpha$; i.e. the property of memory,
parametrized by $\alpha$, is only present in the equation for $\theta$ which
is a ciclic variable. So, when $K\gg 1$, $\left<I^{2}_{n}\right>$ must be
independent of $\alpha$. That is, to observe effects of $\alpha$ on the
dynamics of the RL-fSM, $K\sim 1$ should be set, see e.g. ET09 ; ET13 ; E19 ;
E11 ; MPSL23 .

We stress that similar studies can be carried out for other types of
nonlinearity (not just the continuous sine-shaped nonlinearity in the first
equation of the RL-fSM) and for other types of fractional derivatives. This,
in fact, will be the subject of future investigations.

Finally, we want to add that our work falls within the scope of the General
Fractional Dynamics (GFDynamics), a line of research recently introduced in
Ref. T21 .

## Acknowledgements

J.A.M.-B. thanks support from CONAHCyT-Fronteras (Grant No. 425854) and VIEP-
BUAP (Grant No. 100405811-VIEP2024), Mexico. The research of J.M.S. is
supported by a grant from Agencia Estatal de Investigación
(PID2019-106433GB-I00/AEI/10.13039/501100011033), Spain. E.D.L. acknowledges
support from CNPq (No. 301318/2019-0) and FAPESP (No. 2019/14038-6), Brazilian
agencies.

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