# Deep Spiking Neural Networks
with Resonate-and-Fire Neurons

Badr AlKhamissi
Independent
badr [at] khamissi.com
&Muhammad ElNokrashy
Microsoft EGDC
muhammad.nael [at] gmail.com
&D. Bernal-Casas
U Barcelona, Spain

###### Abstract

In this work, we explore a new _Spiking Neural Network_ (SNN) formulation with
_Resonate-and-Fire_ (RAF) neurons (Izhikevich, 2001) trained with gradient
descent via back-propagation. The RAF-SNN, while more biologically plausible,
achieves performance comparable to or higher than conventional models in the
Machine Learning literature across different network configurations, using
similar or fewer parameters. Strikingly, the RAF-SNN proves robust against
noise induced at testing/training time, under both _static_ and _dynamic_
conditions. Against CNN on MNIST, we show $25\%$ higher absolute accuracy with
$\mathcal{N}(0,0.2)$ induced noise at testing time. Against LSTM on N-MNIST,
we show $70\%$ higher absolute accuracy with $20\%$ induced noise at training
time.

##  1 Introduction

Artificial intelligence traces many of its roots back to research in
neuroscience and psychology—research seeking to understand the human brain,
from the neuronal to the behavioral level. For example, Convolutional Neural
Networks (CNNs) were inspired by the hierarchical feed-forward structure of
the visual cortex (Hubel & Wiesel, 1968; Fukushima, 1980). Reinforcement
Learning (RL) branched out from psychology research on animal conditioning
(Rescorla & Wagner, 1972). Interestingly, there are many instances where AI
progress has inspired new brain theories that were later empirically verified.
For example, RL research inspired a reward-based learning theory of
dopaminergic function (Schultz et al., 1997). While deep CNNs trained on
natural images were shown to reproduce neurophysiological patterns observed in
animals (Lindsay, 2020). Both fields have mutually benefited one another in
what has been recently called “a virtuous circle” (Hassabis et al., 2017).

In this light, the scientific community has been exploring Spiking Neural
Networks (SNNs). SNNs employ simplified models that approximate neuronal
mechanisms we believe the brain uses to process discrete spatio-temporal
events (the spikes). One prominent such model is the _Leaky-Integrate-and-
Fire_ (LIF) neuron. In LIF, the neuron integrates the inputs over time, firing
when the _potential_ passes a set threshold. Hardware architectures have been
developed to exploit this event-based behaviour (Merolla et al., 2014; Ankit
et al., 2017; Davies et al., 2018). Their results are auspicious for achieving
ultra-low power processing of event-based data streams. For example, in deep
learning architectures with spiking neurons, it was observed that the number
of spikes drops significantly at deeper layers, reducing the computation
requirements for neuromorphic hardware (Rueckauer et al., 2017; Sengupta et
al., 2019). A caveat of these approaches is that the neuron model utilized
(the LIF neuron) can not reproduce relevant features of cortical neurons
(Izhikevich, 2001).

Towards overcoming this limitation, we consider a novel SNN with a more
biologically plausible neuron model: the _Resonate-and-Fire_ (RAF) model
(Izhikevich, 2001). The RAF neuron can model more neurodynamics given its
characteristic as a resonator model, and research has shown it to replicate
biological neural data, despite its simplicity compared to other resonator
neuronal models (Torikai & Hishiki, 2009; Pauley et al., 2018).

##  2 Methods and Experiments

(a)

(b)

Figure 1: (a) The forward pass of the (Upward)222We consider RAF-SNNs with
only 1 kind of inter-neuron connections: From neurons in layer $\ell$ to
neurons in layer $\ell+1$. Biologically, this is not a known constraint. RAF-
SNN, unrolled in time. (b) The RAF2 neuron in the style of conventional
recurrent cells such as the LSTM.

Figure 1(a) visualizes an Upward2 RAF-SNN unrolled in time. A time unit $dt$
is defined as a single step forward for each cell in the network. Thus, notice
that input at $t=0$ propagates through the network such that the corresponding
output would be at $t=L-1$, for depth $L$, not at $t=0$. This formulation is
useful for dealing with the dynamical system of the RAF neuron, but is
otherwise functionally equivalent to conventional RNN cells, like the LSTM,
with the caveat that observed outputs are shifted in time by $+L-1$. The self-
transforms ${\bm{v}}^{\ell}_{t-1}\mapsto{\bm{v}}^{\ell}_{t}$ (green)
correspond to ${\bm{h}}^{\ell}_{\tau-1}\mapsto{\bm{h}}^{\ell}_{\tau}$. The
depth-transforms ${\bm{v}}^{\ell-1}_{t-1}\mapsto{\bm{v}}^{\ell}_{t}$ (blue)
correspond to ${\bm{h}}^{\ell-1}_{\tau}\mapsto{\bm{h}}^{\ell}_{\tau}$, and so
on.

###  2.1 The RAF Neuron Model

Equation 1 shows the RAF neuron model. It describes the dynamics of the
membrane potential of a neuron with the equation of motion of a forced, damped
harmonic oscillator:

$\displaystyle\frac{d^{2}v}{dt^{2}}+2\xi\frac{dv}{dt}+\omega^{2}v$
$\displaystyle=I_{\text{ext}}[t]$ (1) $\displaystyle V_{t}$
$\displaystyle\approx V_{t-1}+dt\odot u_{t-1},$ (2) $\displaystyle u_{t}$
$\displaystyle\approx u_{t-1}+dt\odot\left(I_{\text{ext}}-2\xi
u_{t-1}-\omega^{2}V_{t-1}\right).\dot{V}$ $\displaystyle=u_{t-1},$ (3)
$\displaystyle\dot{u}$ $\displaystyle=I_{\text{ext}}+-\omega^{2}V_{t-1}-2\xi
u_{t-1},$ (4) $\displaystyle x_{t}$
$\displaystyle=x_{t-1}+\dot{x}dt\quad\forall x\in\\{u,V\\}.$ (5)

where $v$ is the membrane potential, $\xi$ is the damping factor, $\omega$ is
the natural frequency, and $I_{\text{ext}}$ is any external current to the
neuron (i.e., the weighted summation of pre-synaptic spikes).

###  2.2 Datasets

We evaluate our formulation on both the static and the dynamic (or
neuromorphic) versions of the MNIST dataset (LeCun et al., 1998; Orchard et
al., 2015). The static version comprises $28\times 28$ gray-scale images of
hand written digits, while the neuromorphic version (“N-MNIST”) was derived
from the static dataset by panning and tilting a Dynamic Vision Sensor (DVS)
(Lichtsteiner et al., 2008) in front of a screen displaying the digits. Each
sample consists of a $300\text{\,}\mathrm{ms}$ period of ON and OFF events
that represent increases or decreases in pixel intensity. The data-stream is
pre-processed to be a sequence of $34\times 34\times 2$ tensors, similar to
Lee et al. (2016), with a sampling time of $10\text{\,}\mathrm{ms}$.

###  2.3 Experimental Setup

#### Models

We consider $4$ models: RAF vs. LSTM on N-MNIST, and RAF-CNN vs. CNN on MNIST.
RAF models follow the recurrence in Equations 8, 9 & 10. Beyond that, the RAF-
CNN substitutes a convolutional layer for the linear at $f^{(\ell)}(s)$.
Architecture parameters (e.g. depth and width) are matched for LSTM & RAF
networks, and CNN & RAF-CNN. Further details in Appendix A.1.

#### Poisson Encoding

For the RAF-CNN model, the static inputs are first encoded as Poisson-
distributed spike trains where the intensity of each pixel defines the event
rate—a standard practice for SNNs (O’Connor et al., 2013; Diehl et al., 2015).

#### Perturbing Images

In the _static_ case, we add Gaussian noise of a certain std-dev to the pixel
values before Poisson Encoding. In the _dynamic_ case, we flip individual
binary pixels with some probability $p$ at every time step.

#### Optimization

Since SNNs deal with discontinuous spikes (binary activations), we employ the
method proposed by Yanguas-Gil (2020) to enable training with back-
propagation. The forward pass follows Equation 6, while the gradients are
based on the smooth approximation in Equation 7:

$\displaystyle s^{{\ell}j}_{t}$
$\displaystyle=H(v^{{\ell}j}_{t}-\theta^{{\ell}j}),$ (6)
$\displaystyle\partial~{}s^{{\ell}j}_{t}$
$\displaystyle=\partial~{}\sigma(\beta^{{\ell}j}(v^{{\ell}j}_{t}-\theta^{{\ell}j})).$
(7)

where $s^{{\ell}j}_{t}$ is the output spike of neuron $j$ in layer $\ell$ at
time $t$, $v^{{\ell}j}_{t}$ is its membrane potential, $\theta^{{\ell}j}$ is
the firing threshold, $\beta^{{\ell}j}$ is a regularization parameter
(controls the steepness of the sigmoid approximation), $H(\cdot)$ is the
Heaviside function, and $\sigma(\cdot)$ is the sigmoid function. Per neuron,
$\\{\omega,\xi,\theta,\beta\\}$ are learnable. Equations 8 & 9 are first-order
approximations of Equation 1 in vector form. $f^{(\ell)}(\cdot)$ is a learned
forward transform, such as a linear or convolutional layer.

$\displaystyle{\bm{v}}_{t+1}$
$\displaystyle={\bm{v}}_{t}+dt\odot{\bm{u}}_{t},$ (8)
$\displaystyle{\bm{u}}_{t+1}$
$\displaystyle={\bm{u}}_{t}+dt\odot\left({\bm{I}}_{\text{ext}}[t]-2{\bm{\xi}}\odot{\bm{u}}_{t}-{\bm{\omega}}^{2}\odot{\bm{v}}_{t}\right),$
(9) $\displaystyle{\bm{I}}_{\text{ext}}^{\ell}[t]$
$\displaystyle=f^{(\ell)}({\bm{s}}_{t}^{\ell-1}).$ (10)

We use the AdamW optimizer (Loshchilov & Hutter, 2019) with an initial
learning rate of $0.01$. The model is trained until the validation accuracy
fails to improve for 6 consecutive epochs, where each epoch enumerates the
training set in random order. The learning rate is halved when the validation
loss does not improve for $2$ epochs. The mini-batch size is $32$. Neuron
parameters are initialized by uniform sampling:
$\omega\sim\mathcal{U}(0,1.1\times 2\pi)$, $\xi\sim\mathcal{U}(0,2.5)$,
$\theta\sim\mathcal{U}(0,2.5)$. While $\beta_{\text{init}}=5$.

#### Objective Function

In the static case, optimizing the cross-entropy loss on the last observed
membrane potential ${v}_{t=\text{last}}$ yields the best results. In the
dynamic case, we compute the cross-entropy loss on the number of spikes in the
last $15$ time steps as values to the $\mathrm{softmax}$.

##  3 Results

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2: Test accuracy on the MNIST dataset (static). Comparison of RAF-CNNs
vs. CNNs as a function of: (a) Noise std-dev at test, (b) Noise std-dev at
train, (c) Training set size, (d) Batch size4, (e) Depth4, and (f) Width4. The
red line is a visual anchor at $98\%$ accuracy.

* Each data point is the average of $8$ runs, to account for the probabilistic noise and the Poisson encoder.

In the static (or synthesized) case (Figure 2), we observe model behavior when
adding Gaussian noise to the base input images across a range of std-dev
values. With up to $\sigma\\!=\\!0.2$ noise added at testing time for a
network trained with clean input: The RAF-CNN maintains performance, while the
CNN degrades by up to $25\%$ accuracy. Both networks, however, maintain
performance when fed increasingly noisy inputs at training time, as measured
on the clean test set. Both networks also present similar performance
trajectories when varying the training data size, although the RAF-CNN manages
to lead very slightly. In addition, we tested across a range of values for
batch size, network width, and depth—both models behaved similarly.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: Test accuracy on the N-MNIST dataset (dynamic). Comparison of RAF
vs. LSTM-networks as a function of: (a) Noise $p$ at test, (b) Noise $p$ at
train, (c) Network depth444The vertical green lines indicate the baseline
model used to generate each plot with one varying parameter., (d) Training set
size, (e) Batch size4, and (f) Network width4. The red line is a visual anchor
at $95\%$ accuracy.

In the dynamic (or neuromorphic) case (Figure 3), both the RAF and LSTM
degrade as we add more noise at _testing_ time to networks trained with clean
input. At $p\\!=\\!0.2$, The RAF network gets $32.73\%$ accuracy, twice that
of the LSTM. When adding noise to the _training_ data and testing on the clean
test set, the RAF maintains performance, while the LSTM degrades to nearly
$1/4$th of its baseline. As in the static case, performance remains largely
comparable between the two models across a range of batch sizes, training set
sizes, number of layers, and number of hidden units per layer. Noteworthy are
the sudden dips for RAF at $4$ hidden layers, and for LSTM at $1024$ hidden
units. See Appendix A.1 for further details on all four models.

##  4 Conclusions

In this paper, we presented a novel SNN implementation with certain advantages
over conventional ML models: (a) It utilizes a more biologically plausible
neuron model: the RAF neuron—able to model an extensive repertoire of
experimental observations (Izhikevich, 2001), (b) It achieves performance
similar to homologous deep networks across a range of batch sizes, training
sizes, and network widths and depths, with only 23.8% of the competing model
size in the dynamic case, and a comparable size in the static case (see
Appendix A.1), and (c) It copes more robustly with high noise levels in
train/test time compared to homologous approaches in both static and dynamic
scenarios (see Figures 2(a), 3(a) & 3(c)). In conclusion, with this approach,
we propose a novel SNN that can outperform standard methods, especially in
noisy environments. Importantly, it is an interpretable SNN in terms of
neuronal parameters, with the long-term goal of using it in neuroscience
research. We are also interested in investigating the robustness of RAF
networks to adversarial attacks, which is a major shortcoming of conventional
ML models.

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##  Appendix A Appendix

###  A.1 Architecture

Figure 4: Above (red): Encoding and model pipeline for the RAF-CNN, trained on
MNIST data. Below (green): Model pipeline for the N-MNIST-based experiments
with RAF and LSTM networks.

We consider $4$ different models: RAF vs. LSTM, and RAF-CNN vs. CNN. A plain
RAF network is analogous to a fully-connected (FC) network with one hidden
layer but uses RAF neurons instead and is therefore recurrent. This is
compared to an LSTM (Hochreiter & Schmidhuber, 1997) with a similar
configuration. Table 1 shows the baseline architecture for each along with the
number of trainable parameters. Note that the RAF model contains significantly
fewer parameters than its counterpart yet outperforms it as discussed in
Section 3.

Table 1: Comparing the RAF vs. LSTM models RAF | LSTM
---|---
Layer Type | Dimension | Layer Type | Dimension
Input | 34x34x2 | Input | 34x34x2
Flatten | 2312 | Flatten | 2312
RAF | 512 | LSTM | 512
Linear | 10 | Linear | 10
# of Params | 297,768 | # of Params | 1,251,594

Table 2 compares the RAF-CNN model with its non-spiking counterpart. They both
follow LeNet architecture as in Lee et al. (2020). The RAF-CNN employs
convolutional layers in place of FC layers in the RAF. The “Spiking Average
Pooling” layer follows the same spatial-pooling method as in Lee et al. (2020)
but uses the HardSoft activation function (Yanguas-Gil, 2020) instead as
described in Section 2.3 with the $\beta$ and thresholds made learnable. The
“Convolution” layer in the CNN uses a ReLU activation function. The RAF-CNN
contains more trainable parameters because each neuron has a set of
parameters: the damping factor ($\xi$), the natural frequency ($\omega$), the
firing threshold ($\theta$), and the steepness of the sigmoid ($\beta$)—used
to approximate the gradient in the backward pass.

Table 2: Comparing the RAF-CNN vs. CNN models

RAF-CNN | CNN
---|---
Layer Type | Kernel Size | Dimension | Stride | Layer Type | Kernel Size | Dimension | Stride
Input |  | 28x28x1 |  | Input |  | 28x28x1 |
RAF Convolution | 5x5 | 128 | 1 | Convolution | 5x5 | 128 | 1
Spiking Avg Pooling | 2x2 |  | 2 | Avg Pooling | 2x2 |  | 2
RAF Convolution | 5x5 | 128 | 1 | Convolution | 5x5 | 128 | 1
Spiking Avg Pooling | 2x2 |  | 2 | Avg Pooling | 2x2 |  | 2
Flatten |  | 6272 |  | Flatten |  | 6272 |
Spiking Linear |  | 200 |  | Linear + ReLU |  | 200 |
Spiking Linear |  | 10 |  | Linear |  | 10 |
# of Parameters | 1,671,576 | # of Parameters | 1,669,200


# Spinorial Snyder and Yang Models From Superalgebras And Noncommutative
Quantum Superspaces

Jerzy Lukierski, Mariusz Woronowicz
Institute for Theoretical Physics, University of Wroclaw,
pl. Maxa Borna 9, 50-205 Wroclaw, Poland

###### Abstract

The relativistic Lorentz-covariant quantum space-times obtained by Snyder can
be described by the coset generators of (anti) de-Sitter algebras. Similarly,
the Lorentz-covariant quantum phase spaces introduced by Yang, which contain
additionally quantum curved fourmomenta and quantum-deformed relativistic
Heisenberg algebra, can be defined by suitably chosen coset generators of
conformal algebras. We extend such algebraic construction to the respective
superalgebras, which provide quantum Lorentz-covariant superspaces (SUSY
Snyder model) and indicate also how to obtain the quantum relativistic phase
superspaces (SUSY Yang model). In last Section we recall briefly other ways of
deriving quantum phase (super)spaces and we compare the spinorial Snyder type
models defining bosonic or fermionic quantum-deformed spinors.

##  1 Introduction

In order to obtain quantum gravity models which reconcile two basic theories
in physics, namely general relativity (GR) and quantum mechanics (QM), it
appears desirable to introduce noncommutative (NC) quantum space-times (see
e.g. [1]-[5]) and look for also the quantum deformations of quantum-mechanical
relativistic phase space algebra with quantum NC fourmomenta (see e.g.
[6]-[9]). Further, because the spinorial variables are even more fundamental
that the vectorial ones (see e.g. quark model of hadrons or Penrose twistor
theory) it is interesting to look for the algebraic systems providing quantum-
deformed spinorial variables.

In this short paper we consider firstly various NC Lorentz-covariant models of
quantum space-times and quantum-deformed relativistic phase spaces, which
originate from the ones proposed firstly in 1947 by Snyder [1] and Yang [6].
These historically first NC structures of quantum space-times and quantum
relativistic phase spaces are directly associated with the Lie algebras
describing D=4 space-time symmetries which contain $D=4$ Lorentz algebra as
its subalgebra. Further, these models will be naturally extended to
supersymmetric ones, with supercharges promoted to fundamental quantum-
deformed fermionic spinorial variables.

In standard $D=4$ Snyder models one introduces NC space-time as described by
the coset generators $O(4,1)/O(3,1)$ (dS Snyder model) or $O(3,2)/O(3,1)$ (AdS
Snyder model)111We assume that the NC space-time coordinates and Lorentz
$\hat{O}(3,1)$ generators are algebraically independent. Recently such models
were studied under the name of extended Snyder models [10],[11]. We add that
dS Snyder model is also called ”Snyder model” and AdS Snyder model as ”anti-
Snyder model (see e.g.[12]).. Each of these two models can be extended in two-
fold way into Yang type models describing quantum-deformed relativistic phase
spaces

  1. i)

dS Snyder model can be embedded into $O(5,1)$ or $O(4,2)$ Yang models, with
quantum relativistic phase space variables $(\hat{x}_{\mu},\hat{p}_{\mu})$ as
described by the generators of cosets $O(5,1)/O(3,1)\otimes O(2)$ or
$O(4,2)/O(3,1)\otimes O(1,1)$;

  2. ii)

AdS Snyder model can be embedded into $O(4,2)$ or $O(3,3)$ Yang models with
quantum relativistic phase space sector described by the coset generators of
$O(4,2)/O(3,1)\otimes O(2)$ or $O(3,3)/O(3,1)\otimes O(1,1)$222$\hat{o}(4,2)$
provides $D=4$ conformal algebra and $\hat{o}(5,1)$ can be considered as $D=4$
Euclidean conformal algebra. $\hat{o}(3,3)$ algebra is physically more exotic,
represents the conformal extension of $\hat{o}(2,2)$ algebra
$(\hat{o}(2,2)=\hat{o}(2,1)\oplus\hat{o}(2,1))$, which appears as a
counterpart of Lorentz algebra in models with two times (see e.g. [13],[14]).
Further we consider only $\hat{o}(5,1)$ and $\hat{o}(4,2)$ Yang models..

The aim of this paper is to consider the supersymmetric extensions of four Lie
algebras $\hat{o}(4,1)$, $\hat{o}(3,2)$, $\hat{o}(5,1)$, $\hat{o}(4,2)$ and
construct respective supersymmetric extensions of Snyder quantum space-times
and Yang quantum relativistic phase spaces. For that purpose we will study the
superalgebras containing as bosonic subalgebras the spinorial coverings of Lie
groups describing $D=4$ space-time symmetries which we mentioned above333The
N-component quaternionic real spinors can be described as the pair of
N-component complex spinors satisfying symplectic SU(2)-Mayorana subsidiary
conditions [15]-[17].

$\displaystyle\overline{O(4,1)}=U(1,1|H)\simeq
USp(2,2),\quad\overline{O(3,2)}=Sp(4|R)$ (1.1)
$\displaystyle\overline{O(5,1)}=SL(2|H)\simeq
SU^{\star}(4),\quad\overline{O(4,2)}=SU(2,2)$ (1.2)

The way of constructing NC spaces and NC superspaces from Lie-algebraic and
Lie-superalgebraic generators firstly proposed with the use of linear formulae
by Snyder [1] and Yang [6], was further generalized to nonlinear relations by
Madore, Gazeau and Buric (see e.g. [18]-[20]). Promoting the symmetry
generators to quantum space-times or quantum phase space variables we will
call ”Snyderization procedure”. For that purpose let us split the (super)Lie
algebra $\hat{g}$ into the (super)Riemanian symmetric pair

$\hat{g}=\hat{k}\oplus\hat{h}$ (1.3)

where $\hat{g}=\hat{k}\oplus\hat{h}$ and444Because superalgebra $\hat{g}$ is
split into bosonic and fermionic sector
$\hat{g}=\hat{g}_{(0)}\oplus\hat{g}_{(1)}$, by $[\cdot,\cdot\\}$ we denote
graded commutator $(i,j=0,1$ mod $2)$
$[g_{(i)},g_{(j)}\\}=g_{(i)}g_{(j)}-(-1)^{i\cdot j}g_{(j)}g_{(i)}.$ (1.4)

$[\hat{k},\hat{k}\\}\subset\hat{h}\qquad[\hat{h},\hat{k}\\}\subset\hat{k}\qquad[\hat{k},\hat{h}\\}\subset\hat{h}.$
(1.5)

The sub(super)algebra $\hat{h}$ enters as the covariance (super)algebra, and
$\hat{o}(3,1)\subset\hat{h}$555We will consider here only $D=4$ relativistic-
covariant Lorentz Snyder models; if $\hat{o}(4)\subset\hat{h}$ one gets the
class of $D=4$ Euclidean Snyder models (see e.g. [21]).; the generators
$\hat{k}$ via Snyderization procedure are defining quantum (super) space
coordinates, and to quantum phase (super)space if we are able to embedd in
$\hat{k}$ some quantum-deformation of relativistic Heisenberg algebra.

We observe that in our Snyderization procedure of superalgebra $\hat{g}$ all
supercharges belonging to $\hat{k}$ will be promoted to quantum-deformed
spinors, which will transform under the spinorial covering groups (see e.g.
(1.1)-(1.2)). In the case of semisimple superalgebras $\hat{g}$ the fermionic
odd spinorial generators form an algebraic basis of $\hat{g}$ because for such
Lie superalgebras all the bosonic generators can be described as bilinear
products of supercharges. Subsequently, in our quantum-deformed covariant
superspaces the spinorial quantum coordinates are primary, i.e. they describe
the algebraic basis of $\mathcal{U}(\hat{g})$. That property inclined us to
use the names ”spinorial Snyder” or ”spinorial Yang” models for those obtained
by the Snyderization of supercharges in semisimple Lie superalgebras.

We add that the quantum (super) spaces obtained via described above
Snyderization procedure have two important properties:

i) The algebraic construction of quantum (super)spaces inherits from Lie
(super)algebras the validity of Jacobi identities i.e. the respective quantum
(super)spaces are described by the associative (super)algebras.

ii) Because underlying classical Lie (super)algebras are endowed with Hopf-
algebraic structure, the quantum (super)spaces inherit as well the coalgebraic
structure, described for classical Lie (super)algebra generators by primitive
coproducts.

The plan of this paper is the following. Firstly in Sect. 2 we recall briefly
the algebraic construction of quantum D=4 Snyder space-time and quantum D=4
Yang relativistic phase spaces. In Sect. 3 we consider two $D=4$
supersymmetric Snyder models of quantum dS and quantum AdS superspaces.
Further, in Sect. 4 we describe three types of quantum Yang phase superspaces
(one in $D=3$, two in $D=4$), which can be linked with quantum-deformed
supersymmetric Heisenberg algebras. In last Section we provide outlook and
final remarks.

It should be mentioned that the idea of using Lie (super)algebra relations for
description of NC quantum superspaces was already considered earlier in the
literature (see e.g. [23]-[24]), but these examples studied in the literature
were not providing physically the most interesting physically cases of $D=4$
Lorentz-covariant quantum superspaces with AdS and dS quantum fermionic
spinors describing their odd sectors. One can add however that interest in
Snyder and Yang type models is increasing; in particular recently Zoupanos et
al [9],[25] applied the ideas based on Yang type models in order to describe
the dynamics of $D=4$ fundamental interactions, with included quantum gravity
sector.

##  2 Snyder quantum space-times and Yang quantum phase spaces

###  2.1 Snyder quantum space-times

D=4 dS and AdS algebras are described by the following five-dimensional
orthogonal algebras (A=0,1,2,3,4)

$[M_{AB},M_{CD}]=i(\eta_{AD}M_{BC}+\eta_{BC}M_{AD}-\eta_{AC}M_{BD}-\eta_{BD}M_{AC})$
(2.1)

with signature $\eta_{AB}=diag(-1,1,1,1,\epsilon)$ and $\epsilon=\eta_{44}=\pm
1$, ($\epsilon=1$ for dS algebra and $\epsilon=-1$ for AdS algebra). If
following the Snyderization procedure we postulate taht $M_{\mu
4}=\frac{1}{\lambda}\hat{x}_{\mu}$, where $\lambda$ is an elementary length in
quantum physics given usually by the Planck length
$\lambda_{p}=\sqrt{\frac{\hbar G}{c^{3}}}\simeq 1.6\cdot 10^{-33}$cm, we
obtain besides the Lorentz algebra generators $M_{\mu\nu}$ ($\mu=0,1,2,3)$ the
relations describing NC Snyder space-times666In the paper we choose
$\hslash=c=1$ with only lenght or mass dimensionalities taken into
consideration.

$\displaystyle[M_{\mu\nu},\hat{x}_{\rho}]=i(\eta_{\nu\rho}\hat{x}_{\mu}-\eta_{\mu\rho}\hat{x}_{\eta})$
(2.2) $\displaystyle[\hat{x}_{\mu},\hat{x}_{\nu}]=i\epsilon\beta
M_{\mu\nu}\qquad\beta=\lambda^{2}>0$ (2.3)

The difference between dS and AdS Snyder space-times consists only in
difference of sign on rhs of relation (2.3). The algebra with the basis
($M_{\mu\nu},\hat{x}_{\rho}$) introduces algebraically an elementary
relativistic quantum system, with D=4 quantum (A)dS space-times
$\hat{x}_{\mu}$ as Lorentz algebra module and Lorentz transformations
providing the D=4 relativistic covariance of Snyder equations (2.2-2.3).
Snyder models (see (2.1-2.3) are Born-dual
($\hat{x}_{\mu}\leftrightarrow\hat{p}_{\mu}$, $M_{\mu\nu}$ unchanged,
$\lambda\to\frac{1}{R}$) to the momentum space realizations
$(M_{\mu\nu},\hat{p}_{\mu})$ of $\hat{o}(4,1)$ or $\hat{o}(3,2)$ algebras with
generators describing the automorphisms of five-dimensional pseudospheres

$x^{A}\eta_{AB}x^{B}=-x^{2}_{0}+x^{2}_{1}+x^{2}_{2}+x^{2}_{3}+\epsilon
x^{2}_{4}=R^{2}$ (2.4)

where $M_{\mu 4}=R\hat{p}_{\mu}$ describe the generators of curved
translations on the pseudospheres $\frac{O(4,1)}{O(3,1)}$ or
$\frac{O(3,2)}{O(3,1)}$ described by (2.4)777The model realized in
fourmomentum space which is Born-dual to Snyder original construction was
already in sixties applied to the description of dynamics in de-Sitter
Universe ($\epsilon=1$), with R describing the cosmological de-Sitter radius
(see e.g. [26]-[27])..

In fact Snyder constructed his model as aimed at the description of NC
geometry at ultra short (Planckian) distances, in order to regularize
geometrically the ultraviolet divergencies in renormalization procedure of
quantized fields. Born duality formalizes a physical as well as some
philosophical concept that one can relate the micro and macro world phenomena
- the first ones of quantum nature described by NC geometry, and the second
linked with classical de-Sitter dynamics of general relativity at very large
cosmological distances.

###  2.2 Yang $D=4$ quantum phase spaces

Already in 1947 C.N. Yang observed that by considering D=6 rotations algebras
(i.e. putting in (2.1) $A=0,1,2,3,4,5$) one can interprete the presence of
rotation generators $M_{5\mu}$ as adding to Snyder model the NC fourmomenta
$\hat{p}_{\mu}$. The sixth dimension can be added to dS or AdS Snyder model in
two-fold way, by postulating that $\eta_{55}=\epsilon^{\prime}=\pm 1$.
Assuming that $M_{\mu 5}=R\hat{p}_{\mu}$ one gets the following extension of
Snyder equations (2.2)-(2.3):

$[M_{\mu\nu},\hat{p}_{\rho}]=i(\eta_{\nu\rho}\hat{p}_{\mu}-\eta_{\mu\rho}\hat{p}_{\mu})$
(2.5) $[\hat{p}_{\mu},\hat{p}_{\nu}]=i\epsilon^{\prime}\gamma
M_{\mu\nu}\qquad\gamma=\frac{1}{R^{2}}$ (2.6)

Additionally besides (2.3) and (2.6) one gets the quantum-deformed canonical
phase space commutator

$[\hat{x}_{\mu},\hat{p}_{\nu}]=i\frac{\lambda}{R}\eta_{\mu\nu}M_{45}=i\eta_{\mu\nu}\hat{d}$
(2.7)

with operator-valued substitution of Planck constant $\hbar$ by rescaled
generator $\hat{d}=\frac{\lambda}{R}M_{45}$, which is a $D=4$ Lorentz scalar
$([M_{\mu\nu},\hat{d}]=0)$. The generator $M_{45}$ commutes with
$\hat{x}_{\mu}$ and $\hat{p}_{\mu}$ as follows

$[\hat{d},\hat{x}_{\mu}]=i\epsilon\lambda R\hat{p}_{\mu}$ (2.8)

$[\hat{d},\hat{p}_{\mu}]=i\frac{\epsilon^{\prime}}{\lambda R}\hat{x}_{\mu}$
(2.9)

and describes in $D=4$ $o(2)$ (if $\epsilon=-\epsilon^{\prime}$) or $o(1,1)$
(if $\epsilon=\epsilon^{\prime}$) internal symmetries.

As we mentioned in Introduction we can obtain in such a way four types of Yang
models, with two dychotomic parameters $\epsilon=\pm 1$ (see(2.3)) and
$\epsilon^{\prime}=\pm 1$ (see (2.6)), which could be also called dSdS, dSAdS,
AdSdS, AdSAdS Yang models.

##  3 From D=4 SUSY Snyder model to quantum superspaces

###  3.1 Quantum $D=4$ AdS superspace from $D=4$ SUSY AdS Snyder model

In $\hat{osp}(1|4)$ superalgebra the generators $M_{\mu\nu}$ and
$\hat{x}_{\mu}=\lambda M_{\mu 4}$ form the $D=4$ AdS $\hat{o}(2,3)\simeq
Sp(4)$ subalgebra, with the generators $M_{\mu\nu}$ describing its $D=4$
Lorentz subalgebra. The $\hat{osp}(1|4)$ superalgebra is obtained by adding to
$D=4$ AdS algebra the four additional real Majorana spinor supercharges
$Q_{\alpha}$ which due to ”Snyderization procedure” will be interpreted as
quantum NC fermionic real components of D=4 AdS spinors $\hat{\xi}_{\alpha}$.
The $\hat{osp}(1|4)$ superalgebra is described by the following supersymmetric
extension of relations (2.1)-(2.3) with $\beta\rightarrow-\beta$
$(\beta=L^{2})$

$\displaystyle[M_{\mu\nu},M_{\rho\sigma}]$ $\displaystyle=$ $\displaystyle
i(\eta_{\mu\sigma}M_{\nu\rho}+\eta_{\nu\rho}M_{\mu\sigma}-\eta_{\mu\rho}M_{\nu\sigma}-\eta_{\nu\sigma}M_{\mu\rho})$
(3.1) $\displaystyle[M_{\mu\nu},\hat{x}_{\rho}]$ $\displaystyle=$
$\displaystyle i(\eta_{\nu\rho}\hat{x}_{\mu}-\eta_{\mu\rho}\hat{x}_{\nu})$
(3.3) $\displaystyle[\hat{x}_{\mu},\hat{x}_{\nu}]$ $\displaystyle=$
$\displaystyle-i\beta M_{\mu\nu},$ (3.5)
$\displaystyle\left\\{{\hat{\xi}}_{\alpha},\hat{\xi}_{\beta}\right\\}$
$\displaystyle=$
$\displaystyle-2(C\gamma^{\mu})_{\alpha\beta}\hat{x}_{\mu}+\beta^{\frac{1}{2}}\left(C\gamma^{\mu\nu}\right)_{\alpha\beta}M_{\mu\nu}$
(3.7) $\displaystyle[M_{\mu\nu},\hat{\xi}_{\alpha}]$ $\displaystyle=$
$\displaystyle-{i\over
2}\hat{\xi}_{\beta}\left(\gamma_{\mu\nu}\right)^{\beta}_{\ \alpha}$ (3.9)
$\displaystyle[\hat{x}_{\mu},\hat{\xi}_{\alpha}]$ $\displaystyle=$
$\displaystyle-{i\over
2}\beta^{\frac{1}{2}}\hat{\xi}_{\beta}\left(\gamma_{\mu}\right)^{\beta}_{\
\alpha}.$ (3.11)

where the quantum spinors $\hat{\xi}_{\alpha}$ appearing in the place of
supercharges have (see(3.7)) the length dimensionality
$[\hat{\xi}_{\alpha}]=L^{\frac{1}{2}}$. The parameter $\beta$ is the $AdS_{4}$
Planckian lenght square and $\gamma_{\mu}$ are $D=4$ Dirac $O(3,1)$ matrices
in real Majorana representation; $\gamma_{\mu\nu}={1\over
2}(\gamma_{\mu}\gamma_{\nu}-\gamma_{\nu}\gamma_{\mu})$. Further, by
$C_{\alpha\beta}=(\gamma_{0})_{\alpha\beta}$ we denote the charge conjugation
matrix with the properties
$C^{T}=-C,~{}(\gamma^{\mu}C^{-1})^{T}=-\gamma^{\mu}C^{-1},C^{2}=-1$

The superalgebra (3.1)-(3.11) describes graded associative quantum superspace
$(\hat{x}_{\mu}^{AdS};\hat{\xi}_{\alpha})$

$\mathbb{X}^{(4,4)}_{AdS}=(\hat{x}_{\mu}^{AdS};\hat{\xi}_{\alpha}|M_{\mu\nu})$
(3.12)

where the generators $M_{\mu\nu}$ describe the Lorentz covariance algebra (see
(3.3) and (3.9)). Alternatively, one can Snyderize in $\hat{osp}(1;4)$ only
the supercharges, and obtain in such a way purely spinorial model of Snyder
type $(\hat{\xi}_{\alpha}|M_{AB})$, with anticommuting $D=4$ AdS quantized
spinors, covariant under the $D=4$ AdS transformations.

###  3.2 Quantum $D=4$ dS superspace from $D=4$ SUSY dS Snyder model

For $D=4$ dS algebra $\hat{o}(4,1)$ the supercharges transform as fundamental
spinor realizations of the quaternionic spinorial covering
$\overline{O(4,1)}=U(1,1;H)\equiv\hat{osp}(1,2;H)=\hat{usp}(2,2)$ [10]-[13].
The supersymmetrization of $D=4$ dS algebra requires a pair of quaternionic
supercharges, which can be equivalently represented by the pair of four-
component complex spinors $\hat{Q}_{A}^{i}(i=1,2;A=1\dots 4)$ with their
quaternionic structure represented by symplectic $SU(2)$ Majorana condition
[15]-[17],[35].

The simple $(N=1)$ $D=4$ dS quaternionic superalgebra is described as the
intersection of two complex superalgebras888We use the notation, where N
denotes the number of 2-component quaternionic D=4 dS supercharges (see e.g.
[15], [17]).

$\hat{uu}_{\alpha}(1,1;1|H)=\hat{su}(2,2;2)\cap\hat{osp}(4;2|C)$ (3.13)

with bosonic sector
$\hat{u}(1,1;H)\oplus\hat{u}_{\alpha}(1;H)\equiv\hat{usp}(2,2)\oplus\hat{o}(2)$.
Using complex spinors notation the superalgebra (3.13) is described by the
following set of (anti)commutators
$(A,B=0,1,2,3,4;~{}\alpha,\beta=1,2,3,4;~{}i,j=1,2)$

$\displaystyle[M_{AB},M_{CD}]$ $\displaystyle=$ $\displaystyle
i(\eta_{AD}M_{BC}+\eta_{BC}M_{AD}-\eta_{AC}M_{BD}-\eta_{BD}M_{AC})$ (3.14)
$\displaystyle\left\\{\hat{Q}^{i}_{\alpha},\hat{Q}^{j}_{\beta}\right\\}$
$\displaystyle=$
$\displaystyle\delta^{ij}(\Sigma_{AB}C)_{\alpha\beta}M^{AB}+\epsilon^{ij}C_{\alpha\beta}T$
(3.16) $\displaystyle[M_{AB},\hat{Q}^{i}_{\alpha}]$ $\displaystyle=$
$\displaystyle-(\Sigma_{AB})_{\alpha\beta}\hat{Q}_{\beta}^{i}$ (3.18)
$\displaystyle[M_{AB},T]$ $\displaystyle=$ $\displaystyle 0$ (3.20)
$\displaystyle[T,\hat{Q}^{i}_{\alpha}]$ $\displaystyle=$
$\displaystyle-\epsilon^{ij}\hat{Q}^{j}_{\alpha}$ (3.22)

where $\eta_{AB}=diag(1,1,1,1,-1)$, $M_{AB}$ are $\hat{o}(4;1)$ generators,
$T$ is a scalar internal $o(2)$ symmetry generator and
$\Sigma_{AB}=\frac{1}{2}[\gamma_{A},\gamma_{B}]$ represents the 4$\times$4
complex matrix realization of $\hat{o}(4,1)$ algebra (3.14). The complex
$\hat{o}(4,1)$ Dirac matrices can be chosen for A=0,1,2,3 as real
$(\gamma^{dS}_{\mu}=\gamma_{\mu})$ and the choice for $A=4$
$\gamma^{dS}_{4}=i\gamma_{5}$ is purely imaginary. The fermionic supercharger
$\hat{Q}^{i}_{\alpha}$ satisfy the following quaternionic $SU(2)$-symplectic
Majorana condition

$\hat{Q}^{i}_{\alpha}=\epsilon^{ij}(\gamma_{5}\hat{\bar{Q}}^{j})_{\alpha},\qquad\hat{\bar{Q}}=\hat{Q}^{\dagger}C.$
(3.23)

In the Snyderization procedure we replace the generators in the coset
$\frac{UU_{\alpha}(1,1;1|H)}{sl(2;\mathbb{C})}$ by quantum D=4 dS superspace
coordinates as follows

$\hat{\psi}_{\alpha}=\sqrt{i}\beta^{-\frac{1}{4}}\hat{Q}^{1}_{\alpha},\qquad\hat{\psi}^{\dagger}_{\alpha}=-\sqrt{i}\beta^{-\frac{1}{4}}(\gamma_{5}\hat{Q}^{2})_{\alpha},\qquad\hat{x}_{\mu}=\beta^{\frac{1}{2}}M_{\mu
4},$ (3.24)

and we obtain the following superalgebra defining quantum D=4 dS superspace

$\displaystyle[M_{\mu\nu},M_{\rho\sigma}]$ $\displaystyle=$ $\displaystyle
i(\eta_{\mu\sigma}M_{\nu\rho}+\eta_{\nu\rho}M_{\mu\sigma}-\eta_{\mu\rho}M_{\nu\sigma}-\eta_{\nu\sigma}M_{\mu\rho})$
(3.25) $\displaystyle[M_{\mu\nu},\hat{x}_{\rho}]$ $\displaystyle=$
$\displaystyle i(\eta_{\nu\rho}\hat{x}_{\mu}-\eta_{\mu\rho}\hat{x}_{\nu})$
(3.27) $\displaystyle[\hat{x}_{\mu},\hat{x}_{\nu}]$ $\displaystyle=$
$\displaystyle i\beta M_{\mu\nu}$ (3.29)
$\displaystyle\left\\{\hat{\psi}_{\alpha},\hat{\psi}_{\beta}\right\\}$
$\displaystyle=$
$\displaystyle-i(\gamma^{\mu}C)_{\alpha\beta}\hat{x}_{\mu}+i\beta\left(\gamma^{\mu\nu}C\right)_{\psi\beta}M_{\mu\nu}$
(3.31)
$\displaystyle\left\\{\hat{\psi}^{\star}_{\alpha},\hat{\psi}^{\star}_{\beta}\right\\}$
$\displaystyle=$
$\displaystyle-i(\gamma^{\mu}C)_{\alpha\beta}\hat{x}_{\mu}-i\beta\left(\gamma^{\mu\nu}C\right)_{\alpha\beta}M_{\mu\nu}$
(3.33)
$\displaystyle\left\\{\hat{\psi}_{\alpha},\hat{\psi}^{\star}_{\beta}\right\\}$
$\displaystyle=$ $\displaystyle-(\gamma_{5})_{\alpha\beta}T$ (3.35)
$\displaystyle[M_{\mu\nu},\hat{\psi}_{\alpha}]$ $\displaystyle=$
$\displaystyle-\left(\gamma_{\mu\nu}\right)^{\beta}_{\
\alpha}\hat{\psi}_{\beta}$ (3.37)
$\displaystyle[\hat{x}_{\mu},\hat{\psi}_{\alpha}]$ $\displaystyle=$
$\displaystyle i\beta^{\frac{1}{2}}\left(\gamma_{\mu}\right)^{\beta}_{\
\alpha}\hat{\psi}_{\beta}$ (3.39) $\displaystyle[T,\hat{\psi}_{\alpha}]$
$\displaystyle=$ $\displaystyle\gamma_{5}\hat{\psi}^{\star}_{\alpha}$ (3.41)

with the lenght dimensionalities

$[M_{\mu\nu}]=0,\qquad[\hat{x}_{\mu}]=1,\qquad[\hat{\psi}_{\alpha}]=[\hat{\psi}_{\alpha}]=\frac{1}{2},\quad[T]=1.$
(3.42)

The superalgebraic relations define the quantum $D=4$ dS superspace

$\mathbb{X}_{dS}^{(5;4+\bar{4})}=(\hat{x}_{\mu}^{dS};\hat{\psi}_{\alpha},\hat{\psi}_{\alpha}^{\star}|M_{\mu\nu},T)$
(3.43)

where $M_{\mu\nu}$ are the Lorentz generators and $T$ describes the internal
$O(2)$ symmetries. generator $T$.

It follows from relation (4.31) and traceless $\gamma_{5}$ matrix that by
putting $\alpha=\beta$ in (4.31) one gets
$\sum_{\alpha=1}^{4}|\hat{\psi}_{\alpha}|^{2}=0$. The nonvanishing quantum
spinors can be therefore only realized in Hilbert-Krein space of states with
indefinite metric (see e.g. [29],[30]) and local gauging of quaternionic
superalgebra (3.13) leads to $D=4$ dS supergravity with appearing necessarily
gauge ghost fields [36].

##  4 From $D=4$ $N=2$ SUSY Yang models to quantum phase superspaces

###  4.1 Quantum $D=3$ SUSY AdS phase space from $D=3$ AdS Yang model

In order to obtain $D=3$ SUSY AdS Yang model to the $\hat{o}(3,2)$ generators
$M_{AB}$ (see (2.1), $\epsilon=-1$) describing $D=3$ AdS Snyder model we add
the pair of real $O(3,2)$ spinorial supercharges $Q^{i}_{\alpha}$
$(i=1,2;\alpha=1\dots 4)$ and $\hat{o}(2)$ internal symmetry generator $T$.
The underlying $\hat{osp}(2;4)$ superalgebra looks as follows (see e.g. [38])

$\displaystyle\\{Q_{\alpha}^{i},Q_{\beta}^{j}\\}=\delta^{ij}(\gamma^{AB}C)_{\alpha\beta}M_{AB}-\epsilon^{ij}C_{\alpha\beta}T$
(4.1)
$\displaystyle[M_{AB},Q_{\alpha}^{i}]=-(\gamma_{AB})_{\alpha}^{~{}\beta}Q_{\beta}^{i}$
(4.2) $\displaystyle[T,Q_{\alpha}^{i}]=-\epsilon^{ij}Q_{\alpha}^{j}$ (4.3)

where $\gamma_{AB}=\frac{1}{2}[\gamma_{A},\gamma_{B}]$ and $\gamma_{A}$
denotes the real $O(3,2)$ Dirac-Majorana matrices. Part of the generators in
(4.1)-(4.3) describe the $D=3$ covariance generators

$M_{rs}\oplus M_{34}\oplus
T\simeq\hat{o}(2,1)\oplus\hat{o}(1,1)\oplus\hat{o}(2)$ (4.4)

and the remaining ones are Snyderized as follows

$M_{3r}\oplus
M_{4r}\oplus\tilde{Q}^{1}_{\alpha}\oplus\tilde{Q}^{2}_{\alpha}\simeq\frac{1}{\lambda}\hat{x}_{r}\oplus
R\hat{p}_{r}\oplus\hat{\xi}_{\alpha}\oplus\hat{\pi}_{\alpha}$ (4.5)

where $(\hat{x}_{r},\hat{p}_{r})$ and
$(\hat{\xi}_{\alpha},\hat{\pi}_{\alpha})$ describe even and odd canonical
pairs of vectorial and spinorial positions and momenta. Denoting $O(1,1)$
generator $M_{45}=\frac{R}{\lambda}\hat{d}$ we obtain the following SUSY-
extended quantum-deformed Heisenberg algebra

  * i)

odd-odd relations

$\displaystyle\\{\hat{\xi}_{\alpha},\hat{\xi}_{\beta}\\}=\\{\hat{\pi}_{\alpha},\hat{\pi}_{\beta}\\}=(\gamma^{rs}C)_{\alpha\beta}M_{rs}+\frac{1}{\lambda}(\gamma^{3r}C)_{\alpha\beta}\hat{x}_{r}$
(4.6)
$\displaystyle\qquad\qquad\qquad\qquad\qquad+R(\gamma^{4r}C)_{\alpha\beta}\hat{p}_{r}+(\gamma^{34}C)_{\alpha\beta}\hat{d}$
$\displaystyle\\{\hat{\pi}_{\alpha},\hat{\xi}_{\beta}\\}=C_{\alpha\beta}T$
(4.7)

  * ii)

even-even relations

$\displaystyle[\hat{x}_{r},\hat{x}_{s}]=\lambda^{2}M_{rs},\qquad[\hat{p}_{r},\hat{p}_{s}]=\frac{1}{R^{2}}M_{rs}$
$\displaystyle[\hat{p}_{r},\hat{x}_{s}]=i\eta_{rs}\hat{d}$ (4.8)

  * iii)

crossed even-odd relations

$\displaystyle[\hat{x}_{r},\hat{\xi}_{\alpha}]=(\gamma_{3r})_{\alpha}^{~{}~{}\beta}\hat{\xi}_{\beta}\qquad[\hat{p}_{r},\hat{\xi}_{\alpha}]=(\gamma_{4r})_{\alpha}^{~{}~{}\beta}\hat{\xi}_{\beta}$
$\displaystyle[\hat{x}_{r},\hat{\pi}_{\alpha}]=-(\gamma_{3r})_{\alpha}^{~{}~{}\beta}\hat{\pi}_{\beta}\qquad[\hat{p}_{r},\hat{\pi}_{\alpha}]=-(\gamma_{4r})_{\alpha}^{~{}~{}\beta}\hat{\pi}_{\beta}.$
(4.9)

We see that the relations (4.7)-(4.9) depend on the quantum superspace
coordinates (4.5) as well as the covariance symmetry generators (4.4). In
order to solve eq. (4.7)-(4.9) one can introduce in place of generators (4.4)
their irreducible algebraic or fundamental matrix realizations.

###  4.2 $D=4$ SUSY AdS phase space from $N=2$ $D=4$ SUSY AdS Yang model

Using $N=2$ superextension of $D=4$ Minkowskian conformal algebra

$\hat{o}(4,2)\simeq\hat{su}(2,2)\xrightarrow{\text{SUSY}}\hat{su}(2,2;2)$
(4.10)

by Snyderization of the coset $\frac{SU(2,2;2)}{SU(2,2)\times U(2)}$
generators one can define the spinorial coordinates and momenta which span the
odd sector of quantum-deformed supersymmetric Heisenberg algebra. The
covariance algebra is described by the following extension od $D=4$ Lorentz
algebra $(\mu,\nu=0,1,2,3)$

$M_{\mu\nu}\rightarrow(M_{\mu\nu},M_{45},I,I_{r})\simeq\hat{o}(3,1)\oplus\hat{o}(1,1)\oplus\hat{o}(2)\oplus\hat{o}(3)$
(4.11)

where $(I,I_{r})$ $(r=1,2)$ describe the internal $u(2)$ symmetries commuting
with conformal $o(4,2)$ algebra and the Snyderization procedure should be
applied to the remaining generators in order to introduce the quantum-deformed
superspace coordinates:

$(M_{3\mu},M_{4\mu},\tilde{Q}_{\alpha}^{a},\tilde{S}_{\alpha}^{a})\xrightarrow{S}(\frac{1}{\lambda}\hat{x}_{\mu},R\hat{p}_{\mu},\hat{\psi}^{a}_{\alpha},\hat{\pi}^{a}_{\alpha}).$
(4.12)

The fermionic sector of $\hat{su}(2,2;2)$ can be conveniently described by two
pairs of four component real Majorana supercharges
$Q^{a}_{\alpha},S_{\beta}^{a}$ $(a=1,2;\alpha,\beta=1\dots 4)$, satisfying the
following basic superalgebraic relations (we use standard basis
$(M_{\mu\nu},P_{\mu},K_{\mu},D)$ for the $D=4$ conformal algebra generators;
see [32]-[33])

$\displaystyle\\{Q_{\alpha}^{a},Q_{\beta}^{b}\\}=2\delta^{ab}(\gamma^{\mu}C)_{\alpha\beta}P_{\mu}$
$\displaystyle\\{S_{\alpha}^{a},S_{\beta}^{b}\\}=-2\delta^{ab}(\gamma^{\mu}C)_{\alpha\beta}K_{\mu}$
(4.13)
$\displaystyle\\{Q_{\alpha}^{a},S_{\beta}^{b}\\}=\delta^{ab}[(\gamma^{\mu\nu}C)_{\alpha\beta}M_{\mu\nu}+2iC_{\alpha\beta}D]$
$\displaystyle\qquad\qquad\qquad\qquad+\epsilon^{ab}C_{\alpha\beta}I_{2}+i\tau^{(ab)}_{k}(\gamma_{5}C)_{\alpha\beta}I_{k}$

where $k=0,1,2,3$ and $2\times 2$ symmetric matrices
$\tau^{(ab)}_{k}=(1_{2},\sigma_{1},\sigma_{3})$ and generators
$(I=I_{0},I_{1},I_{2},I_{3})$ describe internal symmetry algebra $u(2)\simeq
o(2)\oplus o(3)$. In order to reexpress $P_{\mu},K_{\mu}$ by generators
$M_{3\mu},M_{4\mu}$ we use before employment of the Snyderization procedure
the following formulae

$M_{4\mu}=\frac{1}{\sqrt{2}}(RP_{\mu}+\frac{1}{\lambda}K_{\mu})\xrightarrow{S}\frac{1}{\sqrt{2}\lambda}\hat{x}_{\mu}$
(4.14)
$M_{5\mu}=\frac{1}{\sqrt{2}}(RP_{\mu}-\frac{1}{\lambda}K_{\mu})\xrightarrow{S}\frac{R}{\sqrt{2}}\hat{p}_{\mu}.$
(4.15)

The supercharges $\tilde{Q}^{a}_{\alpha},\tilde{S}^{a}_{\alpha}$ employed in
the Snyderization procedure (see (4.12)) are defined in terms of supercharges
$Q^{a}_{\alpha},S^{a}_{\alpha}$ (see (4.13)) as follows
$([Q^{a}_{\alpha}]=[\tilde{Q}^{a}_{\alpha}])=-\frac{1}{2},[S^{a}_{\alpha}]=[\tilde{S}^{a}_{\alpha}])=-\frac{1}{2}$

$\tilde{Q}^{a}_{\alpha}=\frac{1}{\sqrt{2}}(Q^{a}_{\alpha}+(\lambda
R)^{-\frac{1}{2}}S^{a}_{\alpha})\qquad\tilde{S}^{a}_{\alpha}=\frac{1}{\sqrt{2}}(S^{a}_{\alpha}-(\lambda
R)^{\frac{1}{2}}Q^{a}_{\alpha}).$ (4.16)

The algebra (4.13) after using relations (4.14)-(4.16) can be rewritten in
term of the supercharger (4.16) in the following way

$\displaystyle\\{\tilde{Q}_{\alpha}^{a},\tilde{Q}_{\beta}^{b}\\}=\delta^{ab}[(\gamma^{\mu}C)_{\alpha\beta}\frac{1}{R}M_{5\mu}+(\gamma^{\mu\nu}C)_{\alpha\beta}M_{\mu\nu}]+\epsilon^{ab}C_{\alpha\beta}I_{2}$
(4.17)
$\displaystyle\\{\tilde{S}_{\alpha}^{a},\tilde{S}_{\beta}^{b}\\}=-\delta^{ab}[(\gamma^{\mu}C)_{\alpha\beta}\lambda
M_{4\mu}+(\gamma^{\mu\nu}C)_{\alpha\beta}M_{\mu\nu}]-\epsilon^{ab}C_{\alpha\beta}I_{2}$
(4.18)
$\displaystyle\\{\tilde{Q}_{\alpha}^{a},\tilde{S}_{\beta}^{b}\\}=-\delta^{ab}C_{\alpha\beta}D+i\tau^{(ab)}_{k}(\gamma_{5}C)_{\alpha\beta}I_{k}.$
(4.19)

After Snyderization (see (4.12)) we obtain the following fermionic odd-odd
sector of algebra (4.17)-(4.19)

$\displaystyle\\{\psi_{\alpha}^{a},\psi_{\beta}^{b}\\}=\delta^{ab}[(\gamma^{\mu}C)_{\alpha\beta}\hat{p}_{\mu}+(\gamma^{\mu\nu}C)_{\alpha\beta}M_{\mu\nu}]+\epsilon^{ab}C_{\alpha\beta}I_{2}$
(4.20)
$\displaystyle\\{\pi_{\alpha}^{a},\pi_{\beta}^{b}\\}=-\delta^{ab}[(\gamma^{\mu}C)_{\alpha\beta}\hat{x}_{\mu}+(\gamma^{\mu\nu}C)_{\alpha\beta}M_{\mu\nu}]-\epsilon^{ab}C_{\alpha\beta}I_{2}$
(4.21)
$\displaystyle\\{\pi_{\alpha}^{a},\psi_{\beta}^{b}\\}=\delta^{ab}C_{\alpha\beta}D+i\tau^{(ab)}_{k}(\gamma_{5}C)_{\alpha\beta}I_{k}.$
(4.22)

We see that in the above relations besides the the ”bosonic” and ”fermionic”
phase space coordinates
$(\hat{x}_{\mu},\hat{p}_{\mu})$,$(\psi_{\alpha}^{a},\pi_{\alpha}^{a})$ enter
all the generators of covariance algebra (4.11) (we recall that $D=M_{45}$).

###  4.3 $D=4$ SUSY dS phase space from $D=4$ SUSY dS Yang model

Following the supersymmetrization of $\hat{o}(5,1)\simeq sl(2;H)$ algebra, in
dS Yang model one should consider the following $N=1$ quaternionic $(N=2$
complex) superalgebras

$sl(2|H)\simeq su^{\star}(4)\xrightarrow{\text{SUSY}}sl(2;1|H)\simeq
su^{\star}(4,2)$ (4.23)

with bosonic sector

$sl(2|H)\otimes gl(1|H)\simeq su^{\star}(4)\otimes u^{\star}(2)$ (4.24)

where $su^{\star}(4)\simeq\hat{o}(5,1)$ and
$u^{\star}(2)=\hat{o}(2)\oplus\hat{o}(2,1)$. We see that due to quaternionic
structure in complex notation we will employ for our $D=4$ Yang models with
complex superalgebra $su^{\star}(4,2)$.

The superalgebras $su^{\star}(4;2N)$ can be obtained by so called Weyl trick
(see e.g. [31]) from the $SU(4;2N)$ superalgebra which supersymetrizes
$SU(4)\simeq\hat{o}(6)$. For $SU(4;2N)$ algebra one can introduce the
following $Z_{4}$-grading:

$\begin{array}[]{cccc}L_{0}&L_{1}&L_{2}&L_{3}\\\ USp(4)\oplus
USp(2N)&Q^{+}&\frac{SU(4)}{USp(4)}\oplus\frac{U(2N)}{USp(2N)}&Q^{-}\end{array}$
(4.25)

where (see footnote 4)

$[L_{r},L_{s}\\}=L_{r+s}\quad\quad mod~{}4.$ (4.26)

One passes from $SU(4;2N)$ to $SU^{\star}(4;2N)$ by multiplication of the
generators from sectors (4.25) according to the following compact formula

$SU(4;2N)\rightsquigarrow SU^{\star}(4;2N)\leftrightarrow
L_{r}\rightsquigarrow\exp(\frac{ir\pi}{2})L_{r}.$ (4.27)

We choose for $D=4$ SUSY dS phase space the following covariance algebra
(compare with (4.11))

$(M_{rs}\oplus M_{45}\oplus
I\oplus\tilde{I}_{r})\simeq\hat{o}(3,1)\oplus\hat{o}(2)\oplus\hat{o}(2)\oplus\hat{o}(2,1).$
(4.28)

The remaining generators, which include all supercharges, define via
Snyderization procedure the $D=4$ SUSY dS phase superspace

$(M_{3\mu},M_{4\mu},z^{a}_{\alpha},u^{b}_{\beta})\xrightarrow{S}(\frac{1}{\lambda}\hat{x}_{\mu},R\hat{p}_{\mu},\chi^{a}_{\alpha},\rho^{a}_{\alpha})$
(4.29)

where due to quaternionic $SU(2)$-symplectic Majorana condition (see e.g.
(3.23)) it follows that independent degrees of freedom are described by
$z_{\alpha}\equiv z^{1}_{\alpha},\bar{z}_{\alpha}\sim z^{2}_{\alpha}$ and
$u_{\alpha}\equiv u^{1}_{\alpha},\bar{u}_{\alpha}\sim u^{2}_{\alpha}$ and
after Snyderization by
$\chi_{\alpha}\equiv\chi_{\alpha}^{1},\bar{\chi}_{\alpha}\sim\chi^{2}_{\alpha}$
and
$\rho_{\alpha}\equiv\rho_{\alpha}^{1},\bar{\rho}_{\alpha}\sim\rho^{2}_{\alpha}$.

The $D=5$ dS superalgebra is the same as $D=4$ Euclidean conformal
superalgebra, which has been studied in explicit form (see e.g.
[15],[26],[33]). The fermionic odd-odd sector
$(A=B=1,2,\dots,5;\bar{\Sigma}_{AB}=\frac{1}{2}[\gamma_{A},\gamma_{B}]C$,
where $C$ is a charge conjugation matrix)

$\displaystyle\\{z_{\alpha},\bar{u}_{\beta}\\}=2[(\bar{\Sigma}_{AB}\gamma_{5})_{\alpha\beta}M^{AB}+C_{\alpha\beta}(\tilde{I}_{1}+i\tilde{I}_{2})]$
(4.30)
$\displaystyle\\{z_{\alpha},\bar{z}_{\beta}\\}=\\{u_{\alpha},\bar{u}_{\beta}\\}=0$
(4.31)
$\displaystyle\\{z_{\alpha},u_{\beta}\\}=i(\gamma_{5}C)_{\alpha\beta}(I+\tilde{I}_{3})$
(4.32)

where $I\oplus I_{r}~{}(r=1,2,3)$ describe the internal summetry $O(2)\oplus
O(2,1)$.

After Snyderization given by formulae (4.29) we get the relations for the
complex spinors $\chi_{\alpha},\rho_{\alpha}$ and
$\bar{\chi}_{\alpha},\bar{\rho}_{\alpha}$ depending on the covariance algebra
generators (4.28) and bosonic phase space coordinates
$\hat{x}_{\mu},\hat{p}_{\mu}$.

It should be mentioned that putting $\alpha=\beta$ in relations (4.31) one
gets that (see [37])
$\sum_{\alpha}|z_{\alpha}|^{2}=\sum_{\alpha}|u_{\alpha}|^{2}=0$ or
$\sum_{\alpha}|\chi_{\alpha}|^{2}=\sum_{\alpha}|\rho_{\alpha}|^{2}=0$ and one
can conclude that after quantization, similarly as in Sect. 3.2, the local
gauging of superalgebra $su^{\star}(4;2)$ leads to $D=5$ dS supergravity with
ghost gauge fields.

##  5 Outlook

The Snyder dS and AdS models of NC Lorentz-covariant quantum space-time
coordinates $\hat{x}_{\mu}$ are described by the algebras

$O_{x}(4,1)\xrightarrow{S}\frac{1}{\lambda}{\hat{x}}^{dS}_{\mu}\oplus
M_{\mu\nu}\qquad
O_{x}(3,2)\xrightarrow{S}\frac{1}{\lambda}\hat{x}^{AdS}_{\mu}\oplus
M_{\mu\nu}$ (5.1)

Performing semi-dual Born mapp
$\hat{x}_{\mu}\leftrightarrow\hat{p}_{\mu},~{}\lambda\leftrightarrow\frac{1}{R}$
one gets analogous algebraic structure with Lorentz-covariant quantum NC four-
momenta ${\hat{p}}_{\mu}$

$O_{p}(4,1)\xrightarrow{S}R\hat{p}_{\mu}^{dS}\oplus M_{\mu\nu}\qquad
O_{p}(3,2)\xrightarrow{S}R\hat{p}_{\mu}^{dS}\oplus M_{\mu\nu}$ (5.2)

Subsequently in dS and AdS Yang models the pairs of algebras (5.1), (5.2) are
embedded in D=6 pseudo-orthogonal quantum deformed Yang algebras
$\hat{o}_{H}(5,1)$ and $\hat{o}_{H}(4,2)$, where subindex $H$ denotes that
they contain basic generators $\hat{x}_{\mu},\hat{p}_{\mu}$ defining quantum-
deformed Heisenberg algebras. We have the following two diagrams denoting the
chains of subalgebras

$\begin{array}[]{cc}\begin{array}[]{c}\hat{o}_{H}(5,1)\\\\[-10.0pt]
{\begin{array}[]{c}{\rotatebox{-25.0}{$\Huge\bigcup$}}\par\qquad\qquad\begin{array}[]{c}\phantom{x}\\\\[-4.0pt]
{\rotatebox{35.0}{$\Huge\bigcup$}}\end{array}\end{array}}\\\
\hat{o}_{x}(4,1)\supset\hat{o}(3,1)\subset\hat{o}_{p}(4,1)\end{array}\quad\begin{array}[]{c}\hat{o}_{H}(4,2)\\\\[-10.0pt]
\begin{array}[]{c}{\rotatebox{-25.0}{$\Huge\bigcup$}}\qquad\qquad\begin{array}[]{c}\phantom{x}\\\\[-4.0pt]
{\rotatebox{30.0}{$\Huge\bigcup$}}\end{array}\end{array}\\\
\hat{o}_{x}(3,2)\supset\hat{o}(3,1)\subset\hat{o}_{p}(3,2)\end{array}\end{array}$
(5.3)

In this paper we introduced in Sect.4 the $N=2$ supersymmetric extensions of
$\hat{o}_{H}(5,1)$ and $\hat{o}_{H}(4,2)$ algebras, which contain pairs of
$N=1$ SUSY Snyder models, with the same (self-dual) Lorentz algebra sectors
and dual choices of Lorentz algebra modules.

In the outlook we would like to comment on some possible directions of future
studies, namely:

  * i)

Our method of passing from Snyder models describing quantum space-times to
Yang models providing quantum-deformed Heisenberg algebra relations is quite
general, what also permits to extend supersymmetrically Yang models in order
to obtain quantum-deformed supersymmetric extension of Heisenberg algebra. It
should be recalled however the old Snyder idea of adding “by hand” to quantum
space-time coordinates $\hat{x}_{\mu}$ the commuting fourmomenta $p_{\mu}$. In
such an axiomatic method we postulate further the general covariant formula
for the quantum-deformed basic commutator (for $\beta$ see (2.3)) [22]

$[\hat{x}_{\mu},p_{\nu}]=i\eta_{\nu\nu}F(\beta p^{2})+\beta
p_{\mu}p_{\nu}G(\beta p^{2})$ (5.4)

and impose all required Jacobi identities. Such a problem posed for $D=4$ AdS
Snyder models has a general solution with one of the functions $F$,$G$
remaining arbitrary. Such axiomatic approach can be proposed also for
supersymmetric extension of Snyder model in order to specify the
anticommutator relations for fermionic counterparts
$(\hat{\xi}_{\alpha},\hat{\pi}_{\alpha})$ of the quantum-deformed Heisenberg
algebra basis $(\hat{x}_{\mu},\hat{p}_{\mu})$999One of the authors (JL) would
like to acknowledge that in 2019 using ”axiomatic approach” there were also
undertaken with E.A.Ivanov the attempts to construct the spinorial version of
Snyder model, however without the use of supersymmetry..

  * ii)

If one uses superalgebras for obtaining via Snyderization the spinorial
degrees of freedom, the spinors will appear necessarily as Grasmannian, what
is desirable in the framework of QFT. One can however employ the ”bosonic”
coset of matrix groups with employment of spinorial covering groups - in such
a case one gets curved bosonic spinors. A good example is provided by the case
of conformal Penrose twistors $(t_{A}\in\mathbb{C};A=1\cdots 4)$, which are
the fundamental representations of $D=4$ conformal ”bosonic” group. Describing
twistors by bosonic cosets or as $N=1$ superconformal odd cosets one gets the
following two different choices (see e.g. [34])

$\begin{array}[]{ccc}\text{Penrose twistors}~{}~{}t_{A}^{(B)}&&\text{Fermionic
twistors}~{}~{}t_{A}^{(F)}\\\ t_{A}^{(B)}=\frac{SU(2,3)}{SU(2,2)\otimes
U(1)}&\leftrightarrow&t_{A}^{(F)}=\frac{SU(2,2;1)}{SU(2,2)\otimes
U(1)}.\end{array}$ (5.5)

  * iii)

It is known since eighties [36],[37] that local gauging of $D=4$ and $D=5$
quaternionic dS superalgebras leads to the appearance of gauge ghost fields.
Recently however it appeared proposals (see e.g. [39]) that $D=4$ dS
supergravity without ghost fields can be obtained by suitably spontaneously
broken $D=4$ superconformal gravity. Further algebraic understanding of this
idea is still desired.

## Acknowledgements

J.L. and M.W. have been supported by Polish National Science Center, project
2017/27/B/ST2/01902. The main results have been presented at 2021 Corfu Summer
Institute, Workshop Sept. 21-27.

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