# Balanced Reduced-Order Models
for Iterative Nonlinear Control of Large-Scale Systems

Yizhe Huang and Boris Kramer Y. Huang, was with the Department of Computer
Science and Engineering, University of California San Diego. He is now with
the Department of Information, Risk and Operations Management, University of
Texas at Austin, TX 78712 (e-mail: yizhehuang@utexas.edu)B. Kramer is with the
Department of Mechanical and Aerospace Engineering, University of California
San Diego, (e-mail<EMAIL_ADDRESS>
###### Abstract

We propose a new framework to design controllers for high-dimensional
nonlinear systems. The control is designed through the iterative linear
quadratic regulator (ILQR), an algorithm that computes control by iteratively
applying the linear quadratic regulator on the local linearization of the
system at each time step. Since ILQR is computationally expensive, we propose
to first construct reduced-order models (ROMs) of the high-dimensional
nonlinear system. We derive nonlinear ROMs via projection, where the basis is
computed via balanced truncation (BT) and LQG balanced truncation (LQG-BT).
Numerical experiments are performed on a semi-discretized nonlinear Burgers
equation. We find that the ILQR algorithm produces good control on ROMs
constructed either by BT or LQG-BT, with BT-ROM based controllers
outperforming LQG-BT slightly for very low-dimensional systems.

Model/controller reduction, iterative learning control, large-scale systems,
distributed parameter systems, fluid flow systems.

##  1 INTRODUCTION

High-dimensional nonlinear models are ubiquitous in science and engineering,
arising, e.g., during semi-discretization of partial differential equations
(PDEs), or in large-scale circuits and biological models. Real-time control
and filtering for these systems remains challenging. On the one hand, using
linearization and subsequently linear controllers, such as the linear
quadratic regulator (LQR), leads to suboptimal control for nonlinear systems.
On the other hand, while nonlinear controllers can produce better control,
they quickly become computationally intractable in high dimensions. The use of
nonlinear reduced-order models (ROMs) is a viable alternative in these
applications.

We propose a new framework to design controllers for high-dimensional
nonlinear systems. The approach combines system-theoretic model reduction to
obtain nonlinear ROMs and the iterative linear quadratic regulator (ILQR)
control [1]. ILQR computes local linearizations at every time step, and
combines it with a modified, discrete-time LQR algorithm to find the optimal
controller and the corresponding nominal trajectory. The next iteration then
uses the new trajectory to recompute the local linearizations and repeat the
process, resulting in the iterative nature of ILQR.

The ILQR algorithm has been applied to a wide range of very low-dimensional
problems such as quadrotor UAV with cable-suspended loads [2, 3], lower limb
exoskeleton models [4], and robotic arm models [1]. Moreover, several
extensions to ILQR have been proposed. An iterative LQG method is developed in
[5], and [6] extends ILQR to systems with non-quadratic cost functions and
apply it to differential-drive robots and quadrotor helicopters in
environments with obstacles. The authors in [7] propose a way of learning
local linearization of dynamics from image inputs and tests it on problems
like planar systems (navigation), inverted pendulum, and cart-pole balancing.
In addition, [8] proposes constrained iterative LQR to handle the constraints
in ILQR and applies it to on-road autonomous driving motion planning. However,
while these extensions further expand the range of problems that ILQR can be
applied to, none of these extensions addresses the problem that ILQR is
computationally intractable for high-dimensional systems. This results from
the running time of each iteration in ILQR increases quadratically with the
number of dimensions.

Model reduction provides a mathematical and system-theoretic framework to
reduce the dimensionality of models, see [9, 10, 11] for an overview. Here, we
are interested in designing the controller using the nonlinear ROM, and using
that very control on the original high-dimensional system. One of the most
popular system-theoretic model reduction technique for linear open-loop
systems is the balanced truncation (BT) method [12]. It finds a state-space
transformation in which the controllability and observability Gramians are
equal and diagonal, and then discards the states of the system that are hard
to control and hard to observe. LQG balanced truncation (LQG-BT) [13] is a
modification to BT; it produces ROMs that are suitable for closed-loop systems
[14, 15, 16, 17]. The LQG-BT algorithm does take the matrices of the control
cost function into account in that it solves the LQG Riccati equations instead
of the Lyapunov equations. In this work, we compare the performance of ILQR
controllers based on nonlinear ROMs, which we compute via BT and LQG-BT model
reduction. Our numerical testbed is the semi-discretized high-dimensional
Burgers equation.

This paper is organized as follows. Section 2 briefly introduces the ILQR
algorithm, presents algorithmic complexity results, and the special case of
quadratic-in-state systems. Section 3 presents BT and LQG-BT model reduction.
Section 4 numerically investigates the performance of the BT and LQG-based
controllers. Section 5 offers conclusions and an outlook for future work.

##  2 ITERATIVE LINEAR QUADRATIC REGULATOR

We define the nonlinear control problem in Section 2.1. Then, in Section 2.2
we introduce the ILQR method, first developed in [1], which extends LQR to
nonlinear systems. In Section 2.3 we derive a computational cost analysis of
ILQR, and in Section (2.4) we specifically analyze quadratic-in-state systems
in combination with ILQR.

###  2.1 Nonlinear Control Problem

We consider the nonlinear control problem of minimizing a quadratic cost
subject to a nonlinear dynamical system:

$\displaystyle\min_{u_{0},u_{1},\ldots,u_{N-1}}$
$\displaystyle{\color[rgb]{0,0,0}J(x_{0:N},u_{0:N-1})}$ (1) s.t.
$\displaystyle x_{k+1}=f(x_{k},u_{k}).$

where $x_{k}\in\mathbb{R}^{n}$ is the system state at time step $k$,
$u_{k}\in\mathbb{R}^{m}$ is the control input at time step $k$, $N$ is the
number of time steps,  $x_{0:N}=\\{x_{0},x_{1},\ldots,x_{N}\\}$ and
$u_{0:N-1}=\\{u_{0},u_{1},\ldots,u_{N-1}\\}$ denote the sequence of states and
controls, respectively, and
$f:\mathbb{R}^{n}\times\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$ is a
differentiable function in both arguments that maps the current state and
control to the future state. The cost function is assumed in quadratic form

${\color[rgb]{0,0,0}J(x_{0:N},u_{0:N-1})}=(x_{N}-x^{*})^{\top}Q_{f}(x_{N}-x^{*})+\sum_{k=0}^{N-1}(x_{k}^{\top}Qx_{k}+u_{k}^{\top}Ru_{k}),$
(2)

where $x^{*}\in\mathbb{R}^{n}$ is the target state. The symmetric positive
semi-definite matrix $Q_{f}\in\mathbb{R}^{n\times n}$ defines the final state
cost, and the symmetric positive semi-definite matrix $Q\in\mathbb{R}^{n\times
n}$ defines the intermediate state cost, while the positive definite matrix
$R\in\mathbb{R}^{m\times m}$ defines the control cost.

###  2.2 The ILQR Algorithm

The ILQR algorithm provides a solution for the nonlinear control problem in
(1). The iterative algorithm starts with a nominal control sequence
$u_{0:N-1}$ and nominal trajectory  $x_{0:N}$ that results from that input.
The algorithm then improves  $u_{0:N-1}$ at each iteration so to minimize (2)
while using local linearizations at every time step. We refer to [1] for
details. ILQR iteratively computes a local linearization

$\delta x_{k+1}=A_{k}\delta x_{k}+B_{k}\delta u_{k}$ (3)

given a nominal control sequence  $u_{0:N-1}$ and corresponding trajectory
$x_{0:N}$, where $\delta x_{k}$ and $\delta u_{k}$ denote the deviation from
the nominal trajectory, while $A_{k}=D_{x}f(x_{k},u_{k})$ and
$B_{k}=D_{u}f(x_{k},u_{k})$ are the Jacobians of $f$ with respect to $x$ and
$u$, respectively. The algorithm then uses the local linearization in a
modified LQR algorithm to compute a local optimal control, which is

$\delta u_{k}=-{\color[rgb]{0,0,0}K_{k}}\delta
x_{k}-{\color[rgb]{0,0,0}K_{k}}^{v}v_{k+1}-{\color[rgb]{0,0,0}K_{k}^{u}}u_{k},$
(4)

where
${\color[rgb]{0,0,0}K_{k}^{v}}=(B^{\top}_{k}S_{k+1}B_{k}+R)^{-1}B^{\top}_{k}$
and ${\color[rgb]{0,0,0}K_{k}^{u}}=(B^{\top}_{k}S_{k+1}B_{k}+R)^{-1}R$, and
where

$\displaystyle{\color[rgb]{0,0,0}K_{k}}$
$\displaystyle=(B^{\top}_{k}S_{k+1}B_{k}+R)^{-1}B^{\top}_{k}S_{k+1}A_{k}$
$\displaystyle{\color[rgb]{0,0,0}S_{k}}$
$\displaystyle=A^{\top}_{k}S_{k+1}(A_{k}-B_{k}K_{k})+Q$
$\displaystyle{\color[rgb]{0,0,0}v_{k}}$
$\displaystyle=(A_{k}-B_{k}K_{k})^{\top}v_{k+1}-K_{k}^{\top}Ru_{k}+Qx_{k}.$

Observe that to compute $S_{k}$ and $v_{k}$ we need $S_{k+1}$ and $v_{k+1}$,
yielding a backward-in-time iteration. In addition, we have that
$S_{N}=Q_{f}$, $v_{N}=Q_{f}(x_{N}-x^{*})$. This control is then used to
compute the nominal trajectory in the next iteration, and the process proceeds
until convergence. Algorithm 1 summarizes the ILQR algorithm. We terminate the
algorithm when the relative difference of cost between two consecutive
iterations is small, $|J_{\text{old}}-J|/J_{\text{old}}\leq\text{tol}$.

1 Input: Nonlinear term $f(x_{k},u_{k})$, initial state $x_{0}$, target
$x^{*}$, initial control sequence  $u_{0:N-1}$, number of time steps $N$,
matrices in cost function $Q$, $Q_{f}$, $R$, convergence threshold tol;

2 Output: Control  $u_{0:N-1}$, number of ILQR iterations $l$;

3 $l=0$, $S_{N}=Q_{f}$, $\delta x_{0}=0$;

4 $x_{k+1}=f(x_{k},u_{k})$, $\forall k=0,1,\ldots,N-1$;

5 $J_{\text{old}}=1,\ J=1+2\texttt{tol}$;

6 while  _$|J_{\text{old}}-J|/J_{\text{old}} >\texttt{tol}$_ do

7 $J_{\text{{old}}}=J$;

8 $J=0$;

9 $v_{N}=Q_{f}(x_{N}-x^{*})$;

10 $x_{1:N}^{\text{{old}}}=x_{1:N}$;

11 for  _$k\leftarrow 0$ to $N-1$_ do

12 $A_{k}=D_{x}f(x_{k},u_{k})$, $B_{k}=D_{u}f(x_{k},u_{k})$

13 end for

14 for  _$k\leftarrow N-1$ to 0_ do

15
${\color[rgb]{0,0,0}K_{k}}=(B^{\top}_{k}S_{k+1}B_{k}+R)^{-1}B^{\top}_{k}S_{k+1}A_{k}$;

16
${\color[rgb]{0,0,0}K_{k}^{v}}=(B^{\top}_{k}S_{k+1}B_{k}+R)^{-1}B^{\top}_{k}$;

17 ${\color[rgb]{0,0,0}K_{k}^{u}}=(B^{\top}_{k}S_{k+1}B_{k}+R)^{-1}R$;

18 $S_{k}=A^{\top}_{k}S_{k+1}(A_{k}-B_{k}{\color[rgb]{0,0,0}K_{k}})+Q$;

19
$v_{k}=(A_{k}-B_{k}{\color[rgb]{0,0,0}K_{k}})^{\top}v_{k+1}-{\color[rgb]{0,0,0}K_{k}^{\top}}Ru_{k}+Qx_{k}$;

20

21 end for

22 for  _$k\leftarrow 0$ to $N-1$_ do

23 $\delta u_{k}=-{\color[rgb]{0,0,0}K_{k}}\delta
x_{k}-{\color[rgb]{0,0,0}K_{k}^{v}}v_{k+1}-{\color[rgb]{0,0,0}K_{k}^{u}}u_{k}$;

24 $u_{k+1}=u_{k}+\delta u_{k}$;

25 $x_{k+1}=f(x_{k},u_{k})$;

26 $\delta x_{k+1}=x^{\text{{old}}}_{k+1}-x_{k+1}$;

27 $J=J+x_{k+1}^{\top}Qx_{k+1}+u_{k+1}^{\top}Ru_{k+1}$;

28

29 end for

30 $J=J+x_{N}^{\top}Q_{f}x_{N}$, $l=l+1$;

31

32 end while

Algorithm 1 ILQR Algorithm

###  2.3 Computational Cost of ILQR

In each iteration, Algorithm 1 completes three portions. First, we obtain the
linearizations (step 11-13); second, we compute
${\color[rgb]{0,0,0}K_{k},K^{v}_{k},K^{u}_{k}},S_{k},v_{k}$ (step 14-20);
third, we compute the control increment  $(\delta u)_{0:N-1}$ and the new
trajectory  $x_{0:N}$ (step 21-27). Linearizations are obtained in $N$
Jacobian calls if it is known analytically (e.g., for polynomial models). If
we need to approximate the Jacobian numerically, then we have to march the
system equation once for each dimension of state and control, which requires
evaluating $f(x_{k},u_{k})$ at total of $N(n+m)$ times. The second and third
portion require matrix multiplication and inversion. State-of-the-art
algorithms of those operations have complexity $\mathcal{O}(n^{2.4})$[18],
whereas classical methods scale as $\mathcal{O}(n^{3})$, so the complexity of
the second and third pass is at best $\mathcal{O}(N(n^{2.4}+m^{2.4}))$.

###### Proposition 1

The complexity of the $l$th iteration of the ILQR Algorithm 1 is
$\mathcal{O}(N(n^{2.4}+m^{2.4}))$, where $n$ is the state dimension, $m$ is
the number of controls, and $N$ is the number of time steps.

The previous result analyzes a single iteration of the ILQR algorithm. We note
that the number of iterations of the ILQR algorithm $l$ varies significantly,
depending on the structure of the nonlinear function $f(x_{k},u_{k})$, as we
see next.

###### Proposition 2

When applied to a discrete-time (DT) LTI system, the ILQR algorithm converges
in one iteration. The resulting control is the same as the output of DT-
LQR111Refer to, e.g., [19, Thm. 6.28] for DT-LQR control..

###  2.4 ILQR for Quadratic-in-State Systems

The Euler equations in specific volume form describe a purely quadratic system
and are commonly used in aerospace applications. Other models, such as Navier-
Stokes and Burgers equations are nearly quadratic for small viscous terms.
Thus, we analyze the ILQR algorithm specifically for quadratic-in-state
systems of the form:

$x_{k+1}=G(x_{k}\otimes x_{k})+Bu_{k},$ (5)

where $G\in\mathbb{R}^{n\times n^{2}}$, $B\in\mathbb{R}^{n\times m}$ and
$\otimes$ denotes the Kronecker product, defined as $x\otimes
x=[x_{k}^{(1)}x_{k}^{\top},\ldots,x_{k}^{(n)}x_{k}^{\top}]^{\top}\in\mathbb{R}^{n^{2}}$.

###### Proposition 3

Let $Q=0$ in the cost equation (2) (penalty only on the final state). Suppose
the initial state is $x_{0}=0$, and the initial control sequence is
$u_{0:N-1}$. Then the ILQR algorithm applied to the quadratic system (5) takes
at least $N$ iterations to converge, where $N$ is the number of time steps.

######  Proof 2.1.

Since $x_{0}=0$ and initial control $u_{k}=0,k=1,2,\ldots$, the initial
trajectory of the system (5) is stationary: $x_{k}=0$ for $k=0,1,\ldots,N-1$.
Likewise, in the local linearization $A_{k}=D_{x}(G(x_{k}\otimes
x_{k})+Bu_{k})=G(x_{k}\otimes I_{n}+I_{n}\otimes x_{k})$, therefore, $A_{k}=0$
in all local linearizations and $B_{k}=D_{u}(G(x_{k}\otimes x_{k})+Bu_{k})=B$.
We first analyze the behavior of the ILQR algorithm in the first iteration.
There, except $S_{N}$, all of $S_{k}=0$, since $A_{k}=Q=0$. Therefore, with
the exception of step $N-1$, we have $K=0$, $K_{v}=R^{-1}B^{\top}$ and
$K_{u}=I$. Therefore $(A_{k}-B_{k}K)=0\ \forall k$, which means that all
$v_{k}=0$ except $v_{N}$. Therefore, $\delta u_{k}=0$ except $\delta u_{N-1}$.
In particular, $\delta u_{N-1}=(B^{\top}Q_{f}B+R)^{-1}B^{\top}Q_{f}x^{*}$. As
a result, only $u_{N-1}$ is updated in the first iteration. Similarly, we find
that only $u_{N-2}$ and $u_{N-1}$ are updated in the second iteration, and so
on. It takes $N$ iterations before $u_{0}$ gets updated; thus, it takes at
least $N$ iterations for ILQR algorithm to converge.

The previous theorem points to a computational challenge. If it takes $O(N)$
iterations to converge and since in each iteration the time complexity is
$O(N)$ with respect to the number of time steps (see Proposition 1) the total
time complexity is $O(N^{2})$. For stiff problems and long simulation times
this becomes intractable. To address this problem, it may be helpful to use a
coarse time step on the first few iterations and then use finer timer step
later to speed up convergence. For example, say $N=10$ for the first few
iterations, then all $u_{k}$ can be updated in $10$ iterations, and we can
later switch to $N=1000$. This strategy can be thought of as using the output
from coarse ILQR to initialize fine-grain ILQR. Clearly, the use of ROMs also
speeds up the algorithm since the time needed for each iteration is reduced
significantly.

##  3 SYSTEM-THEORETIC MODEL REDUCTION

We present two system-theoretic model reduction techniques: balanced
truncation [12] in Section 3.1, and LQG-BT [13] in Section 3.2. Both methods
are defined for continuous-time LTI systems $\dot{x}=Ax+Bu,\ y=Cx$ where
$y\in\mathbb{R}^{p}$ is the output and the system matrices are
$A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$ and
$C\in\mathbb{R}^{p\times n}$. We use these methods for discrete-time nonlinear
systems as follows. First, we linearize the nonlinear systems and then convert
the system from discrete-time to continuous-time using a bilinear Tustin
transformation. We then use BT and LQG-BT to compute the state-space
transformations, and use those transformations to reduce the nonlinear
systems. The control inputs keep the system around the
equilibrium/linearization point. Thus, using these methods to obtain the
projection matrices for the nonlinear ROM is more appropriate than in the
open-loop nonlinear setting. More ROM details are given in Section 4.2. Both
concepts are for infinite-time horizon. There are finite-time BT methods [20],
yet we found the infinite-time BT method to outperform the finite-time BT on
our test problems.

###  3.1 Computing the Balanced Truncation Basis

Balanced truncation computes a state-space transformation for stable systems
in such a way that the controllability and observability Gramians of the ROM
are equal and diagonal, e.g., only the easy-to-control and easy-to-observe
states are kept in the ROM. The controllability Gramian $P$, and the
observability Gramian $Q$, satisfy the Lyapunov equations

$AP+PA^{\top}+BB^{\top}=0,\qquad QA+A^{\top}Q+C^{\top}C=0.$

Applying the transformation $x=T\tilde{x}$ to the LTI system yields
$\dot{\tilde{x}}=\tilde{A}\tilde{x}+\tilde{B}u,\ \tilde{y}=\tilde{C}\tilde{x}$
where $\tilde{A}=T^{-1}AT$, $\tilde{B}=T^{-1}B,\tilde{C}=CT$. The Gramians of
the transformed system are $\tilde{P}=T^{-1}PT^{-\top}$ and
$\tilde{Q}=T^{\top}QT$. We want $T$ such that $\tilde{P}=\tilde{Q}=\Sigma$,
hence

$\Sigma^{2}=\tilde{P}\tilde{Q}=T^{-1}PT^{-\top}T^{\top}QT=T^{-1}PQT.$

Therefore, $T$ can be obtained by computing the eigenvectors of $PQ$ and
eigenvalues $\sigma_{i}^{2}$ in
$\Sigma^{2}=\text{diag}(\sigma_{1}^{2},\ldots,\sigma_{n}^{2})$. To ensure
$\tilde{P}=\tilde{Q}$, proper scaling of the eigenvectors is required. The
complete balanced truncation method for stable systems in an efficient
implementation is given in Algorithm 2. This implementation only computes the
leading $r$ columns of $T_{r}$, and $T_{l}$ and is therefore numerically more
stable and efficient.

10: Matrices $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$ and
$C\in\mathbb{R}^{p\times n}$,

ROM dimension $r$

0: Transformation matrices $T_{r},T_{l}$

1: Solve $AP+PA^{\top}+BB^{\top}=0$ for $P=RR^{\top}$

2: Solve $QA+A^{\top}Q+C^{\top}C=0$ for $Q=LL^{\top}$

3: $r=\min(r,\text{rank}(L),\text{rank}(R))$

4: $U,\Sigma,V=\texttt{svd}(L[:,1:r]^{\top}R[:,1:r])$

5: $T_{r}=R[:,1:r]V\Sigma^{-\frac{1}{2}}$

6: $T_{l}=\Sigma^{-\frac{1}{2}}U^{\top}L[:,1:r]^{\top}$

Algorithm 2 Balanced truncation algorithm

###  3.2 Computing the LQG-Balanced Truncation Basis

LQG-BT [13] applies BT to closed-loop (stabilized) systems, and therefore also
applies to unstable systems where, however, $(A,B)$ is stabilizable and
$(A,C)$ is detectable. It balances the unique positive definite solutions
$P,Q$ of the LQG algebraic Riccati equations

$\displaystyle AP+PA^{\top}-PC^{\top}CP+BB^{\top}$ $\displaystyle=0,$
$\displaystyle A^{\top}Q+QA-QBR^{-1}B^{\top}Q+C^{\top}C$ $\displaystyle=0.$

LQG-BT follows the same steps as Algorithm 2, except at Steps 1–2 it uses the
solutions to the LQG AREs. As a byproduct, we directly obtain LQR or LQG
controllers.

##  4 NUMERICAL RESULTS: BURGERS EQUATION

Burgers equation has been widely used in modeling of fluids and traffic flows,
and as a numerical testbed for nonlinear model reduction and control [21, 22,
23, 24].

###  4.1 PDE Model and Semi-Discretized Model

We consider a 1D Burgers equation with the setup as in [25]. The PDE model is

$\dot{z}(\xi,t)=\epsilon
z_{\xi\xi}(\xi,t)-\frac{1}{2}(z^{2}(\xi,t))_{\xi}+\sum_{k=1}^{m}\chi_{[(k-1)/m,k/m]}(\xi)u_{k}(t)$

for $t>0$ and where $\xi$ is the spatial variable, $u_{k}(t)$ is the
distributed control, and $m$ is the dimension of control, $z\in
H_{\text{per}}^{1}(0,1)$, which means that the system has periodic boundary
conditions. Moreover, $z(\xi,0)=z_{0}(\xi)=0.5\sin(2\pi\xi)^{2}$ for
$\xi\in[0,0.5]$ and zero otherwise. The function $\chi_{[a,b]}(x)$ is the
characteristic function over $[a,b]$. Thus, the spatial domain is divided into
$m$ equidistant intervals, and each control is applied on one corresponding
interval. When discretized with linear finite elements, and after an inversion
with the mass matrix, the semi-discretized system is a nonlinear ODE which is
quadratic in the state, and linear in the control:

$\dot{x}=Ax+G(x\otimes x)+Bu,\qquad y=Cx,$ (6)

where $G\in\mathbb{R}^{n\times n^{2}}$, and initial condition $x(0)=x_{0}$. We
choose the following parameters for our numerical experiments: $n=101$ states,
$m=5$ controls, and viscosity $\epsilon=5\times 10^{-3}$ to make the nonlinear
quadratic term dominant. The system is simulated for $t_{f}=5$s, and the time
interval is divided into $N=500$ time steps. The trajectory is computed using
the backward Euler method, which is implemented using the Levenberg-Marquardt
algorithm. The output matrix $C=I_{n}$, the state cost matrices
$Q=Q_{f}=C^{\top}C$, and the input cost matrix is $R=10^{3}\times I_{m}$,
where $I_{n}$ is the $n\times n$ identity matrix.

###  4.2 BT and LQG-BT Reduced-Order Models

We linearize (6) around the zero equilibrium and use the linearized matrices
$A_{\text{lin}},B,C$ in the BT or LQG-BT Algorithm 2 to compute the
transformations $T_{r},T_{l}$. The linearized matrix $A_{\text{lin}}$ is
stable. We then project the full-order model (FOM) in (6) to obtain

$\dot{x}_{r}=A_{r}x_{r}+G_{r}(x_{r}\otimes x_{r})+B_{r}u_{r},\qquad
y_{r}=C_{r}x_{r}$ (7)

where $A_{r}=T_{l}AT_{r}$, $B_{r}=T_{l}B$, $C_{r}=CT_{r}$,
$G_{r}=T_{l}G(T_{r}\otimes T_{r})$ and $x_{r}(0)=T_{l}x_{0}$. Fig. 1 shows the
normalized singular values of the matrix $L^{\top}R$ in the balancing
procedure (see step 4 in Algorithm 2) for both BT and LQG-BT. For the standard
BT algorithm, those are also known as the Hankel singular values. We observe
that the first $20$ Hankel singular values decay fast, and all Hankel singular
values except the first one come in pairs. In contrast, the normalized
singular values in the LQG-BT decay much more slowly, and only level off at
$r>60$. Similar to the Hankel singular values, these singular values except
the first one come in pairs. The fact that the singular values in the BT
algorithm decay much faster indicates that it is likely that BT outperforms
LQG-BT in open-loop simulation accuracy, as when we truncate the transform
matrix we are discarding dimensions that have lower associated singular
values.

Figure 1: Normalized singular values of the matrix $L^{\top}R$. For balanced
truncation, those are the Hankel singular values.

###  4.3 Performance of ROM-based ILQR Controllers

We use the ROMs to compute controllers $u_{r}^{\text{BT}}$ and
$u_{r}^{\text{LQGBT}}$ via the ILQR Algorithm 1 with tol=$3\times 10^{-5}$. We
note that the Jacobians of (7), which are used to obtain the local
linearization of the system in the ILQR algorithm, are
$D_{x_{r}}\dot{x}_{r}=A_{r}+G_{r}(x_{r}\otimes I_{n_{r}}+I_{n_{r}}\otimes
x_{r})$ and $D_{u_{r}}\dot{x}_{r}=B_{r}$, where $n_{r}$ is the number of
dimensions of the state $x_{r}$ in the ROM. The availability of the Jacobians
in explicit form significantly speeds up the computational routine, and should
be used whenever available, c.f. Proposition 1.

Fig. 2 shows the ROM cost $J(x_{r},u_{r})$ computed as in (2) yet with state
$x_{r}$, controls $u_{r}$ and cost matrix $Q_{r}=Q_{f,r}=C_{r}^{\top}C_{r}$,
versus the iterations of the ILQR algorithm. Moreover, the figure shows the
FOM cost $J(x,u_{r})$ from (2) when the ROM controller is used for the FOM
(thin horizontal lines). Both ROMs had dimension $r=5$. Note that the ILQR
algorithm converges monotonically for both models terminating requiring 170
iterations with the LQG-BT model and 223 iterations with the BT model. With
the final controllers the cost function evaluated for the FOM, $J(x,u_{r})$,
is 68.9 for BT and 63.6 for LQG-BT showing a slightly better performance of
LQG-BT. The norm of the controls obtained from ILQR is shown in Fig. 3, and
Fig. 4 shows the corresponding output. Due to the periodic boundary conditions
and the large penalty on the control, the state shows a low-magnitude periodic
behavior.

Fig. 5 shows the open-loop behavior of the system when excited with the
initial condition $x_{0}$ and no inputs. The initial condition gets convected
in a nonlinear fashion to the right. In contrast, the controlled systems with
the ROM-based ILQR controllers are shown in Fig. 6(a) for BT and Fig. 6(b) for
LQG-BT. In both cases, the control signal (Fig. 3) reduces the initial
condition significantly, showing the efficacy of the control.

Figure 2: ROM cost vs number of iterations in the ILQR algorithm for $r=5$.
The FOM cost when using the converged ILQR controller shown in thin lines,
which is 68.9 for BT and 63.6 for LQG-BT. Figure 3: Norm of ILQR control
$\|u\|_{2}$ computed using the two nonlinear ROMs with $r=5$. Figure 4: Norm
of outputs $\|y\|_{2}$ in the closed-loop FOM, with the ILQR controller
computed using the two nonlinear ROMs with $r=5$. Figure 5: FOM simulation:
initial condition $x_{0}$ and zero input.

(a) Standard BT

(b) LQG-BT

Figure 6: FOM simulation: initial condition $x_{0}$, controlled by
$u_{r}^{\text{BT}}$ and $u_{r}^{\text{LQGBT}}$ with $r=5$.

Finally, while the above results were obtained for $r=5$, Table 1 gives an
overview of results for different dimensions $r$ of the nonlinear ROMs. We
note that while there is an anomaly at $r=9$ for the BT-ROM-based controller,
in general, BT-ROM controllers seem to outperform the LQG-BT ROM controllers,
evidenced by a lower number of iterations, as well as FOM cost, which appears
to be a result of the slow singular value decay of the LQG-BT singular values
in Fig. 1.

Table 1: Comparison of cost of ROM controllers on FOM, $J(x,u_{r})$, on ROM, $J(x_{r},u_{r})$, and number of iterations of ILQR for increasing dimension of the ROM. | LQG-BT | LQG-BT | BT | BT | LQG-BT | BT
---|---|---|---|---|---|---
$r$ | $J(x_{r},u_{r})$ | $J(x,u_{r})$ | $J(x_{r},u_{r})$ | $J(x,u_{r})$ | iter. | iter.
2 | 56.8 | 197.3 | 85.9 | 245.0 | 89 | 58
3 | 71.2 | 136.6 | 44.5 | 101.8 | 58 | 278
4 | 45.8 | 89.8 | 46.6 | 83.0 | 170 | 277
5 | 47.4 | 63.6 | 53.8 | 68.9 | 170 | 222
6 | 50.5 | 69.0 | 44.3 | 47.7 | 1328 | 966
7 | 50.8 | 61.3 | 43.9 | 46.5 | 1208 | 861
8 | 54.2 | 63.3 | 43.5 | 45.7 | 3647 | 930
9 | 53.3 | 59.1 | 272.7 | 272.1 | 9070 | 101
10 | 53.8 | 58.8 | 51.5 | 51.2 | 2395 | 2687
11 | 60.4 | 60.2 | 48.8 | 48.7 | 7927 | 4165

##  5 CONCLUSIONS AND FUTURE DIRECTIONS

We presented a new control framework for high-dimensional nonlinear systems by
combining the iterative linear quadratic regulator (ILQR) framework and
system-theoretic model reduction. As shown in the numerical experiments, both
standard balanced truncation model reduction and LQG balanced truncation
allowed us to compute well-performing controllers from ROMs with about $95\%$
reduction of model order compared to the full-order model. Further work may be
done by using nonlinear system-theoretic model reduction, such as [26, 27,
28], which despite being open-loop techniques, can provide more accurate ROMs
for nonlinear systems. One would also need to investigate the observed
correlation between larger state dimensions and increased iteration numbers
for ILQR to understand scalability of ILQR. Lastly, convergence of ILQR for
increasing $N$ remains an open problem.

## ACKNOWLEDGMENT

B.K. thanks Prof. Hermann Mena for discussions on ILQR.

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# Optimisation of future long baseline neutrino experiments

Olga Mena
IEEC/CSIC, Bellaterra, Spain and
IFIC/CSIC, Valencia, Spain
E-mail It is a pleasure to thank all my long-baseline collaborators and to
Enrique Fernández Martínez who has produced the sensitivity plots shown in
this talk. O. M is supported by a _Ramón y Cajal_ contract from MEC, Spain.
<EMAIL_ADDRESS>
###### Abstract:

The aim of this _talk_ is to review near and far future long baseline neutrino
experiments as superbeams, $\beta$-Beams and neutrino factories, comparing
their sensitivities to the unknown parameters in the neutrino oscillation
sector. We focus on the extraction of the neutrino mass hierarchy, exploring
alternatives to the commonly used neutrino-antineutrino comparison. Special
attention to a new concept of neutrino factory design, the _low energy
neutrino factory_ , is given.

##  1 Introduction

During the last several years the physics of neutrinos has achieved remarkable
progress. The present data require two large ($\theta_{12}$ and $\theta_{23}$)
and one small ($\theta_{13}$) angles in the neutrino mixing matrix, and at
least two mass squared differences, $\Delta m_{ij}^{2}\equiv
m_{j}^{2}-m_{i}^{2}$ (where $m_{j}$’s are the neutrino masses), one driving
the atmospheric ($\Delta m_{23}^{2}$) and the other one the solar ($\Delta
m_{12}^{2}$) neutrino oscillations. The mixing angles $\theta_{12}$ and
$\theta_{23}$ control the solar and the atmospheric neutrino oscillations,
while $\theta_{13}$ is the angle which connects the atmospheric and solar
neutrino realms.

A recent global fit [1] provides the following $3\sigma$ allowed ranges for
the atmospheric mixing parameters $|\mbox{$\Delta m_{23}^{2}$}|=(2-3.2)\times
10^{-3}$ eV2 and $0.32<\sin^{2}\theta_{23}<0.64$. The sign of $\Delta
m_{23}^{2}$, sign$(\mbox{$\Delta m_{23}^{2}$})$, cannot be determined with the
existing data. The two possibilities, $\mbox{$\Delta m_{23}^{2}$}>0$ or
$\mbox{$\Delta m_{23}^{2}$}<0$, correspond to two different types of neutrino
mass ordering: normal hierarchy and inverted hierarchy. In addition,
information on the octant in which $\theta_{23}$ lies, if
$\sin^{2}2\theta_{23}\neq 1$, is beyond the reach of present experiments. The
solar neutrino oscillation parameters lie in the low-LMA (Large Mixing Angle)
region, with best fit values [1] $\mbox{$\Delta m_{12}^{2}$}=7.9\times
10^{-5}~{}{\rm eV^{2}}$ and $\sin^{2}\theta_{12}=0.30$. A combined 3-neutrino
oscillation analysis of the solar, atmospheric, reactor and long-baseline
neutrino data [1] constrains the third mixing angle to be
$\sin^{2}\theta_{13}<0.04$ at the $3\sigma$ C.L.

The future goals for the study of neutrino properties is to precisely
determine the already measured oscillation parameters and to obtain
information on the unknown ones: namely $\theta_{13}$, the CP–violating phase
$\delta$ and the type of neutrino mass hierarchy (or equivalently
sign$(\mbox{$\Delta m_{23}^{2}$})$).

##  2 The golden channels

The most promising way to determine the unknown parameters $\delta$ and
$\theta_{13}$ is through the detection of the subleading transitions
$\mbox{$\nu_{e}$}(\mbox{$\overline{\nu}_{e}$})\leftrightarrow\mbox{$\nu_{\mu}$}(\mbox{$\overline{\nu}_{\mu}$})$.
These channels have been named as _golden channels_ [2]. Defining
$\Delta_{ij}\equiv\frac{\Delta m^{2}_{ij}}{2E}$, a convenient and precise
approximation is obtained by expanding to second order in the following small
parameters: $\theta_{13}$, $\Delta_{12}/\Delta_{23}$, $\Delta_{12}/A$ and
$\Delta_{12}\,L$ [2, 3]

$\displaystyle P_{\nu_{e}\nu_{\mu}(\bar{\nu}_{e}\bar{\nu}_{\mu})}$
$\displaystyle=$ $\displaystyle
s_{23}^{2}\sin^{2}2\mbox{$\theta_{13}$}\,\left(\frac{\mbox{$\Delta_{23}$}}{\tilde{B}_{\mp}}\right)^{2}\,\sin^{2}\left(\frac{\tilde{B}_{\mp}\,L}{2}\right)\,+\,c_{23}^{2}\sin^{2}2\theta_{12}\,\left(\frac{\Delta_{12}}{A}\right)^{2}\,\sin^{2}\left(\frac{A\
L}{2}\right)$ (1) $\displaystyle+$
$\displaystyle\tilde{J}\;\frac{\Delta_{12}}{A}\,\frac{\mbox{$\Delta_{23}$}}{\tilde{B}_{\mp}}\,\sin\left(\frac{AL}{2}\right)\,\sin\left(\frac{\tilde{B}_{\mp}L}{2}\right)\,\cos\left(\pm\delta-\frac{\mbox{$\Delta_{23}$}\,L}{2}\right)\,,$

where $L$ is the baseline, $\tilde{B}_{\mp}\equiv|A\mp\mbox{$\Delta_{23}$}|$
and the matter parameter $A$ is defined in terms of the average electron
number density, $n_{e}(L)$, as $A\equiv\sqrt{2}\,G_{F}\,n_{e}(L)$, and
$\tilde{J}$ is defined as

$\displaystyle\tilde{J}\equiv\cos\theta_{13}\;\sin 2\theta_{13}\;\sin
2\theta_{23}\;\sin 2\theta_{12}~{}.$ (2)

The golden transitions Eqs. (1) are sensitive to all the unknown parameters
quoted in the introductory section. They are clearly sensitive to the mixing
angle $\theta_{13}$. These channels are also sensitive to the CP–violating
phase (via the third term or the _interference_ term, the only term which
differs for neutrinos and antineutrinos). The golden transitions allow us also
to extract the sign of the atmospheric mass difference exploiting matter
effects. In the presence of matter effects, the neutrino (antineutrino)
oscillation probability gets enhanced [4] for the normal (inverted) hierarchy.
Making use of the different matter effects for neutrinos and antineutrinos
seems, in principle, the most promising way to distinguish among the two
possibilities: normal versus inverted hierarchy.

##  3 Degenerate solutions

We can ask ourselves whether it is possible to unambiguously determine
$\delta$ and $\theta_{13}$ by measuring the golden transition probabilities,
$\mbox{$\nu_{e}$}\to\mbox{$\nu_{\mu}$}$ and
$\mbox{$\overline{\nu}_{e}$}\to\mbox{$\overline{\nu}_{\mu}$}$, Eqs. (1) (or
equivalently, $\mbox{$\nu_{\mu}$}\to\mbox{$\nu_{e}$}$ and
$\mbox{$\overline{\nu}_{\mu}$}\to\mbox{$\overline{\nu}_{e}$}$) at fixed
neutrino energy $E$ and at just one baseline $L$. The answer is no. By
exploring the full (allowed) range of the $\delta$ and $\theta_{13}$
parameters, that is, $-180^{\circ}<\delta<180^{\circ}$ and
$0^{\circ}<\mbox{$\theta_{13}$}<10^{\circ}$, one finds, at fixed neutrino
energy and at fixed baseline, the existence of degenerate solutions
($\theta^{{}^{\prime}}_{13}$, $\delta^{{}^{\prime}}$), that we label
_intrinsic degeneracies_ , which give the same oscillation probabilities than
the set ($\theta_{13}$, $\delta$) chosen by nature [5]. It has also been
pointed out that other fake solutions might appear from unresolved
degeneracies in two other oscillation parameters:

  1. 1.

The sign of the atmospheric mass difference $\Delta m_{23}^{2}$ may remain
unknown. In this particular case,
$P(\theta^{{}^{\prime}}_{13},\delta^{{}^{\prime}},-\Delta
m_{23}^{2})=P(\theta_{13},\delta,\Delta m_{23}^{2})$ [6, 7].

  2. 2.

Disappearance experiments only give us information on $\sin^{2}2\theta_{23}$:
is $\theta_{23}$ in the first octant, or is it in the second one,
$(\pi/2-\theta_{23}$)? . In terms of the probabilities,
$P(\theta^{{}^{\prime}}_{13},\delta^{{}^{\prime}},\frac{\pi}{2}-\mbox{$\theta_{23}$})=P(\theta_{13},\delta,\mbox{$\theta_{23}$})$
[7, 8].

All these ambiguities complicate the experimental determination of $\delta$
and $\theta_{13}$. The situation has been dubbed the _eight-fold degeneracy_.
A lot of work has been devoted to resolve the degeneracies by exploiting the
different neutrino energy and baseline dependence of two (or more) LBL
experiments. A complete list of references is beyond the scope of this talk. I
suggest to see Ref. [9] and references therein.

##  4 The tree level approach: superbeams

The next generation of long baseline $\nu_{e}$ neutrino appearance experiments
will be the so-called superbeam experiments. The major goal of superbeam
experiments is to set a non-zero value of the small mixing angle $\theta_{13}$
(or, in the absence of a positive signal, to improve the current upper bound
on this mixing angle). A superbeam experiment consists, basically, of a higher
intensity version of a conventional neutrino (antineutrino) beam. Superbeams
represent the logical next step in accelerator-based neutrino physics. There
are two possible strategies regarding the neutrino beam. The _off-axis_
technique produces a neutrino spectrum very narrow in energy (nearly
monochromatic, $\Delta E/E\sim 15-25\%$), which peaks at lower energies with
respect to the on-axis one. The off-axis technique allows a discrimination
between the peaked $\nu_{e}$ oscillation signal and the intrinsic $\nu_{e}$
background which has a broad energy spectrum. In addition, the off-axis
technique reduces significantly the background resulting from neutral current
interactions of higher energy neutrinos with a $\pi^{0}$ in the final state.
Unfortunately, off axis experiments are counting experiments in which one has
only two measurements (the number of neutrinos and the number of antineutrino
events) and resolving degeneracies becomes an impossible task. This is the
technique exploited by the $\nu_{e}$ appearance experiments T2K [10] and
NO$\nu$A [11].

The _wide band beam (WBB)_ technique exploits the spectral information of the
signal, being sensitive to many $E/L$’s at the same time. The neutrino beam is
on-axis and therefore the fluxes and the beam energies are higher than the
ones exploited in the off-axis case. Higher beam energies imply longer
distances, and therefore larger detectors. The WBB technique requires Mton
class detectors with extremely good energy resolution and optimal neutral-
current background rejection. See Refs. [12] for the physics opportunities
with a WBB at a Deep Underground Science and Engineering Laboratory (DUSEL).

The authors of [13] have studied carefully the two possible techniques,
finding (for the same exposure) the WBB option better for the mass hierarchy
extraction, and the NO$\nu$A off-axis experiment better for CP–violation
searches.

##  5 The race for the hierarchy

Typically, the proposed near term LBL neutrino oscillation experiments
(superbeams) have a single far detector and plan to run with the beam in two
different modes, muon neutrinos and muon antineutrinos. Suppose we compute the
oscillation probabilities $P(\nu_{\mu}\to\nu_{e})$ and
$P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})$ for a given set of oscillation parameters
and the CP–phase $\delta$ is varied between $0$ and $2\pi$: we obtain a closed
CP trajectory (an ellipse) in the bi–probability space of neutrino and
antineutrino conversion [14]. Matter effects are responsible for the departure
of the center of the ellipses from the diagonal line in the bi–probability
plane for normal and inverted hierarchy. In Fig. 1, we have illustrated the
case for $E=2.0$ GeV and $L=810$ km, which roughly correspond to those of the
NO$\nu$A experiment. The distance between the center of the ellipse for the
normal hierarchy (lower blue) and that for the inverted hierarchy (upper red)
is governed by the size of the matter effects. Notice that the ellipses
overlap for a significant fraction of values of the CP–phase $\delta$ for
every allowed value of $\sin^{2}2\theta_{13}$. This makes the determination of
sign$(\mbox{$\Delta m_{23}^{2}$})$ extremely difficult, i. e., the
sign$(\mbox{$\Delta m_{23}^{2}$})$-extraction is not free of degeneracies and
it is highly dependent on the value of $\delta$.

|  |
---|---|---
Figure 1: Left panel: bi–probability plot for $P(\nu_{\mu}\to\nu_{e})$ versus
$P(\bar{\nu}_{\mu}\to\bar{\nu}_{e})$ at a baseline of 810 km and an energy of
2.0 GeV for the normal (blue) and the inverted (red) hierarchies. The smaller,
lower (larger, upper) ellipses are for $\sin^{2}2\theta_{13}=0.02$
($~{}0.10$). Medium panel: bi–probability plot for $P(\nu_{\mu}\to\nu_{e})$
versus $P(\nu_{\mu}\to\nu_{e})$ with baselines 295 km and 810 km. The mean
neutrino energies are chosen such that the $\langle E\rangle/L$ for the two
experiments are approximately identical. The right panel is the bi–probability
plot for $P(\nu_{\mu}\to\nu_{e})$ versus $P(\bar{\nu}_{e}\to\bar{\nu}_{\mu})$
for the normal (blue) and the inverted (red) hierarchies. The baseline and
mean neutrino energy for both experiments are 810 km and $\sim$ 2 GeV.

Following the line of thought developed by Minakata, Nunokawa and Parke [15],
we exploited [16] the neutrino data only from two experiments at different
distances and at different off-axis locations, such that the $\langle
E\rangle/L$ is the same for the two experiments (see also Refs. [17, 18, 19,
20]). In the case of bi–probability plots for neutrino–neutrino modes at
different distances (which will be referred as near (N) and far (F)), the
CP–trajectory is also elliptical. In Fig. 1 (medium panel) we present the
bi–probability plot for the mean energies and baselines of the $\nu_{e}$
appearance experiments T2K and NO$\nu$A. The overlap of the two ellipses,
which implies the presence of a degeneracy of the type of hierarchy with other
parameters, is determined by their width and the difference in the slopes. The
ratio of the slopes, at first order in the matter parameter, and assuming that
the $\langle E\rangle/L$ of the near and far experiments is the same, reads

$\displaystyle\frac{\alpha_{+}}{\alpha_{-}}\simeq 1+2\left(A_{\rm N}L_{\rm
N}-A_{\rm F}L_{\rm
F}\right)\left(\frac{1}{\Delta_{13}L/2}-\frac{1}{\tan(\Delta_{13}L/2)}\right)~{},$
(3)

where $\alpha_{+}$ and $\alpha_{-}$ are the slopes of the center of the
ellipses as one varies $\theta_{13}$ for normal and inverted hierarchies,
$A_{\rm F}$ and $A_{\rm N}$ are the matter parameters, and $L_{\rm F}$ and
$L_{\rm N}$ are the baselines for the two experiments. The separation between
the center of the ellipses for the two hierarchies increases as the difference
in the matter parameter times the path length, ($AL$), for the two experiments
increases. However the width of the ellipses is crucial: even when the
separation between the central axes of the two regions is substantial, if the
ellipses for the normal and inverted hierarchy overlap, the hierarchy cannot
be resolved for values of the CP–phase, $\delta$, for which there is overlap.
The width of the ellipses is determined by the difference in the $\langle
E\rangle/L$ of the two experiments.

In the case of bi–probability plots for the $\nu_{\mu}\to\nu_{e}$ and its CPT
conjugated channel $\bar{\nu}_{e}\to\bar{\nu}_{\mu}$ at the same energy
divided by baseline,$\langle E\rangle/L$, the CPT–trajectory collapses to a
line (see Fig. 1, right panel). As for the neutrino-neutrino case, assuming
that the $\langle E\rangle/L$ of the CPT conjugated channels is the same (to
minimize the ellipses width), at first order, the ratio of the slopes reads
[15]

$\displaystyle\frac{\alpha_{+}}{\alpha_{-}}\simeq 1+2\left(AL+A_{\rm
CPT}L_{\rm
CPT}\right)\left(\frac{1}{\Delta_{13}L/2}-\frac{1}{\tan(\Delta_{13}L/2)}\right)~{},$
(4)

where $\alpha_{+}$ and $\alpha_{-}$ are the slopes of the center of the
ellipses as one varies $\theta_{13}$ for normal and inverted hierarchies, $A$
and $A_{\rm CPT}$ are the matter parameters and $L$ and $L_{\rm CPT}$ are the
baselines for the two experiments which exploit the $\nu_{\mu}\to\nu_{e}$ and
its CPT conjugated channel ($\bar{\nu}_{e}\to\bar{\nu}_{\mu}$). Notice that,
compared to the neutrino–neutrino case given by Eq. (3), the separation
between the center of the ellipses for the two hierarchies increases as the
sum of the matter parameter times the baseline, $AL$, for both experiments
does. Here the ratio of the slopes is enhanced by matter effects for both
$\nu_{\mu}\to\nu_{e}$ and its CPT conjugated channel
$\bar{\nu}_{e}\to\bar{\nu}_{\mu}$. Figure 1 (right panel) shows the
bi–probability curves for the combination of these two flavor transitions,
assuming that the two experiments are performed at the same mean energy and
baseline. If the $\langle E\rangle/L$ of both experiments is the same, the
ellipses will become lines with a negligible width. The separation of the
lines for the normal and inverted hierarchy grows as the matter effects for
both experiments increase. Consequently, the comparison of CPT conjugated
channels is more sensitive to the neutrino mass hierarchy than the
neutrino–neutrino one, see Ref. [21].

##  6 Higher order corrections: $\beta$-beams and neutrino factories

Precision lepton flavor physics requires powerful machines and extremely pure
neutrino beams. Future LBL experiments which exploits pure $\nu_{e}$
($\bar{\nu}_{e}$) neutrino beams are $\beta$-beams and neutrino factories. A
$\beta$-beam experiment [22] exploit ions which are accelerated to high
Lorentz factors, stored and then $\beta$-decay, producing a collimated
electron neutrino beam. The typical neutrino energies are in the $200$ MeV-GeV
range, requiring detectors with hundred-of-MeV thresholds and good energy
resolution. The only requirement is good muon identification in order to
detect the appearance of muon neutrinos (or muon antineutrinos) from the
initial electron neutrino (or antineutrino) beam. No magnetisation is required
and therefore several detectors technologies (water Cherenkov, totally active
scintillator (TASD), liquid argon and non-magnetised iron calorimeter) could
be used, depending on the peak energy.

The initial $\beta$-beam setup [22] considers a _low_ -$\gamma$ machine which
accelerates ${}^{6}He$ ($\bar{\nu}_{e}$ emitter) and ${}^{18}Ne$ ($\nu_{e}$
emitter) up to $\gamma\sim 100$. In order to tune the $E/L$ at the vacuum
oscillation maximum, a large water Cherenkov detector is located at a distance
$\mathcal{O}(100)$ km. The first exciting option to improve this initial
$\beta$-beam scenario was presented in Ref. [23], where the possibility of
using higher $\gamma$ factors was first suggested. The second exciting option,
see Ref. [24], proposes to accelerate alternative ions, as ${}^{8}Li$
($\bar{\nu}_{e}$ emitter) and ${}^{8}B$ ($\nu_{e}$ emitter), with higher
Q-values. A plethora of setups have been proposed in the literature (see Ref.
[9] for a complete list of references).

A neutrino factory (NF) [25, 26] consists, essentially, of a muon storage ring
with long straight sections along which the muons decay. These muons provide
high intensity and extremely pure neutrino beams. Hence, the NF provides
$\nu_{e}$ and $\overline{\nu}_{e}$ beams in addition to $\nu_{\mu}$ and
$\overline{\nu}_{\mu}$ beams, with minimal systematic uncertainties on the
neutrino flux and spectrum. One of the most important advantages of the NF,
compared to the $\beta$-beam, is its ability to measure with high precision
the atmospheric mixing parameters via the disappearance channels
($\mbox{$\nu_{\mu}$}(\mbox{$\overline{\nu}_{\mu}$})\to\mbox{$\nu_{\mu}$}(\mbox{$\overline{\nu}_{\mu}$})$)

The NF exploits the golden signature of the _wrong-sign muon_ events [2]. What
is a “wrong sign muon” event? Suppose, for example, that positive charged
muons have been stored in the ring. These muons will decay as $\mu^{+}\to
e^{+}+\mbox{$\nu_{e}$}+\mbox{$\overline{\nu}_{\mu}$}$. The muon antineutrinos
will interact in the detector to produce positive muons. Then, any _wrong-sign
muons_ (negatively-charged muons) detected are an unambiguous proof of
electron neutrino oscillations in the $\mbox{$\nu_{e}$}\to\mbox{$\nu_{\mu}$}$
channel. A magnetized detector with good muon charge identification is
mandatory.

In Ref. [9] a complete study of possible near and far future LBL facilities
has been performed, including superbeams, $\beta$-beams and NF. The optimal
setup is found to be a $20$ GeV NF delivering $5\cdot 10^{20}$ muon decays per
year, baseline and polarity. The running time assumed is 5 years per polarity.
Two iron calorimeter detectors of $50$ kton are placed at two different
baselines, at $\mathcal{O}(4000)$ km and at $\mathcal{O}(7000)$ km (the so-
called _magic baseline_ [27]). The oscillated data from detector at the
largest baseline helps enormously in resolving the mass hierarchy degeneracy.
Once that the sign($\Delta m_{23}^{2}$) degeneracy is resolved, leptonic
CP–violation can be measured unambiguously using the data from the detector
located at $\mathcal{O}(4000)$ km.

###  6.1 The low energy neutrino factory

The optimal $20$ GeV plus two detectors NF setup described in the previous
section outperforms any other planned scenario so far, as we will shortly
show. However, such an aggressive setup could be extremely challenging to
construct (a $\mathcal{O}(7000)$ km baseline would require the construction of
a decay tunnel with an inclination of $\sim 30^{\circ}$ 111I would like to
thank C. Quigg for making this observation). More important, such an
aggressive scenario might not be needed if
$\sin^{2}2\theta_{13}>10^{-4}-10^{-3}$. The reason for that is simple: if the
mixing angle $\theta_{13}$ is not so small, there is no need to go to very
long baselines to amplify it. Shorter baselines require lower energies. Lower
energy Neutrino Factories (LENF), which store muons with energies $<10$ GeV,
require a detector technology that can detect lower energy muons. In previous
studies [28, 29], we have considered a LENF with an energy of about $4$ GeV
providing $5\cdot 10^{20}$ muon decays per year. The detector exploited was a
magnetized TASD [29] of $20$ kton, with a muon energy detection threshold of
$500$ MeV, located at a distance of $1480$ km (Fermilab to Henderson mine).
The results are similar for a baseline of $1280$ km (Fermilab to Homestake).
The intrinsic background fraction is $10^{-3}$. Here, we improve the LENF
setup in two ways. First, the detector energy resolution would be $dE/E\sim
10\%$ 222Based on NO$\nu$A results, we expect the TASD dE/E to be better than
$6\%$ at 2 GeV.. Second, and more interesting, since it seems possible to
measure in a magnetized TASD the electron charge [30], apart of exploiting the
$\mbox{$\nu_{e}$}(\mbox{$\overline{\nu}_{e}$})\to\mbox{$\nu_{\mu}$}(\mbox{$\overline{\nu}_{\mu}$})$
channels, their _T_ -conjugate channels
$\mbox{$\nu_{\mu}$}(\mbox{$\overline{\nu}_{\mu}$})\to\mbox{$\nu_{e}$}(\mbox{$\overline{\nu}_{e}$})$
channels are also added to the analysis. These extra _T_ -conjugate channels
will help enormously in resolving degeneracies. We assume here that the
electron charge identification is constant in energy and equal to $50\%$ (a
detailed analysis will be presented elsewhere [30]).

Figure 2 shows the 3-$\sigma$ $\theta_{13}$ discovery potential and the
sensitivities to the mass hierarchy to CP–violation expected from data at a
future LENF with the characteristics quoted above (the exposure is $10^{23}$
kton-decays). As a comparison, we show as well the expected sensitivities
exploiting data from the future LBL facilities presented in Ref. [9]. Notice
that the high-$\gamma$ $\beta$-beam [23], labelled as BB$350$, provides a
slightly better sensitivity than the LENF to both CP violation and to
$\theta_{13}$ due to its lower energy and its huge statistics. The $20$ GeV NF
with two baselines ($4000$ km+$7000$ km) is unbeatable, but we might only need
such an aggressive scenario if $\sin^{2}2\theta_{13}<10^{-4}-10^{-3}$. To
conclude, the low energy neutrino factory (LENF) [28, 29] provides a
compromise between super precision machines and feasible setups, and it could
provide an ideal and realistic scenario for precision lepton physics.

|  |
---|---|---
Figure 2: The left, medium and right panels show the 3-$\sigma$ $\theta_{13}$
discovery potential, the mass hierarchy sensitivity and the CP-violation
sensitivity, respectively, expected from future data at a LENF. We present as
well the expected sensitivities from future data at the different LBL
experiments presented in Ref. [9]. Figure produced using the GLOBES software
[31].

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Noncommutative Yang model and its generalizations

S. Meljanac111e-mail<EMAIL_ADDRESS>
Rudjer Bošković Institute, Theoretical Physics Division

Bljenička c. 54, 10002 Zagreb, Croatia

and

S. Mignemi222e-mail<EMAIL_ADDRESS>
Dipartimento di Matematica, Università di Cagliari

via Ospedale 72, 09124 Cagliari, Italy

and INFN, Sezione di Cagliari

Cittadella Universitaria, 09042 Monserrato, Italy

Abstract

Long time ago, C.N. Yang proposed a model of noncommutative spacetime that
generalized the Snyder model to a curved background. In this paper we review
his proposal and the generalizations that have been suggested during the
years. In particular, we discuss the most general algebras that contain as
subalgebras both de Sitter and Snyder algebras, preserving Lorentz invariance,
and are generated by a two-parameter deformation of the canonical Heisenberg
algebra. We also define their realizations on quantum phase space, giving
explicit examples, both exact and in terms of a perturbative expansion in the
deformation parameters.

##  1 Introduction

At the Planck scale, gravitational and quantum effects have comparable
strength and can affect the structure of spacetime. In this regime,
noncommutative geometries may play a relevant role [1]. The first example of
noncommutative spacetime was proposed by Snyder[2] in 1947, but at the time
did not receive much attention. In spite of the assumed granular structure of
spacetime, this model is characterized by the preservation of the Lorentz
invariance.

A first generalization of Snyder’s idea was advanced by C.N. Yang, who
combined noncommutativity with spacetime curvature [3], in terms of a fifteen-
dimensional SO(1,5) algebra of symmetries of phase space. The generators of
this algebra were identified with the generators of the de Sitter algebra and
with the coordinates of the de Sitter spacetime. The remaining generator $h$
rotates positions into momenta, but its physical interpretation was not
specified. More recently, this model was slightly generalized by Khruschev and
Leznov (KL) [4].

Later, in ref. [5] was proposed a model inspired by that of Yang, which
realizes the same symmetries in a nonlinear way, reducing to fourteen the
number of independent generators of the algebra, thus eliminating the
unphysical generator $h$. This model was dubbed triply special relativity
(TSR) because it contains three fundamental constants, identified with the
speed of light, the Planck mass and the cosmological constant, generalizing in
this way the idea advanced in doubly special relativity theories [6] of
deforming the Poincaré symmetry by the introduction of a new fundamental
constant proportional to the Planck mass. In this model, however, the Lorentz
symmetry is preserved and only translation symmetries are deformed.

It was then shown in [7] that the TSR model can be realized exactly in terms
of coordinates and momenta only. This particular realization in phase space
was called Snyder-de Sitter (SdS) spacetime. In [7, 8] it was also shown that
the SdS algebra can be obtained from the Snyder algebra by a nonunitary
transformation.

The previous models of quantum phase space have the common property of
realizing explicitly the duality introduced by Born [9] for the exchange of
position and momentum operators. Recently, the possibility of studying general
quantum phase spaces displaying the structure described above has been
advanced in [10] and studied in more detail in [11]. These papers employed a
widespread approach to noncommutative geometry based on the formalism Hopf
algebras [12], that aims to describe the symmetries of the quantum spacetime.
A powerful tool in this formalism are the realizations of Hopf algebras in
terms of the Heisenberg algebra, that were introduced in [13, 14, 15].

The Snyder model has been extensively discussed in the literature, see e.g.
[13, 14, 16, 17], in several theoretical and phenomenological aspects, like
algebraic representations, relativistic field equations, quantum field theory,
relations with Hopf algebras, but also its nonelativistic limit has been
widely investigated, including generalized uncertainty principles, simple
models like the harmonic oscillator, etc. To our knowledge, instead, besides
ref. [4], the Yang model has been considered only in [18], where its
supersymmetric extensions were analyzed. More authors discuss aspects of TSR,
see e.g. [7, 8, 11, 19]. Most of them treat its classical limit, either in a
nonrelativistic or relativistic setting, but also the quantum field theory of
a self-interacting scalar field in SdS spacetime has been investigated in
[20].

In the present paper, we discuss general perturbative realizations of the
unified model proposed in [10], in terms of the canonical Heisenberg algebra,
extending the results of ref. [11]. In sect. 2 we review the generalized
Snyder spaces and their realizations. In sect. 3 we discuss the de Sitter
algebras obtained by duality from the Snyder ones. In sect. 4 a formalism
unifying the two algebras is introduced, that describes a curved
noncommutative spacetime and includes Yang, KL and TSR models as special
cases. Sect. 5 analyzes the perturbative realizations of these models, while
in sect. 6 we exploit the relation of TSR with the Snyder model to write down
some exact realizations.

##  2 Generalized Snyder space

In this section we review some generalizations of Snyder’s original proposal
for a deformation of the momentum space, that leads to noncommutativity of
spacetime. Most of the results of this section were obtained in [13, 14, 17].

We define a generalized Snyder space introducing a Lorentz-invariant
deformation parameter $\beta\sim L_{P}$, and postulate the commutation
relations

$[\hat{x}_{\mu},\hat{x}_{\nu}]=i\epsilon\beta^{2}M_{\mu\nu}\;\psi^{p}(\epsilon\beta^{2}p^{2}),\qquad[p_{\mu},p_{\nu}]=0,\qquad[\hat{x}_{\mu},p_{\nu}]=i\phi_{\mu\nu}^{p}(\epsilon\beta^{2}p^{2}),$
(1)

where $\epsilon=\pm 1$ and $M_{\mu\nu}=M_{\mu\nu}^{\,\dagger}$ are the
generators of the Lorentz algebra, that we assume to satisfy the standard
relations,

$\displaystyle[M_{\mu\nu},M_{\rho\sigma}]=i\big{(}\eta_{\mu\rho}M_{\nu\sigma}-\eta_{\mu\sigma}M_{\nu\rho}-\eta_{\nu\rho}M_{\mu\sigma}+\eta_{\nu\sigma}M_{\mu\rho}\big{)},$
(2)
$\displaystyle[M_{\mu\nu},\hat{x}_{\lambda}]=i(\eta_{\mu\lambda}\hat{x}_{\nu}-\eta_{\nu\lambda}\hat{x}_{\mu}),\qquad[M_{\mu\nu},p_{\lambda}]=i(\eta_{\mu\lambda}p_{\nu}-\eta_{\nu\lambda}p_{\mu})$
(3)

with $\eta_{\mu\nu}$ the flat metric, $\eta_{\mu\nu}={\rm diag}(-1,1,1,1)$,
and the functions $\psi^{p}(\epsilon\beta^{2}p^{2})$ and
$\phi_{\mu\nu}^{p}(\epsilon\beta^{2}p^{2})$ are constrained so that the Jacobi
identities hold. The commutation relations (1)-(2) generalize those originally
investigated in [2], which are recovered for $\psi^{p}={\rm const}$ and
$\phi^{p}_{\mu\nu}=\eta_{\mu\nu}+\beta^{2}p_{\mu}p_{\nu}$.

In the following, we shall look for realizations of the algebra (1)-(2) in
terms of the Heisenberg algebra ${\cal H}$: in its undeformed version the
Heisenberg algebra is generated by commutative coordinates $x_{\mu}$ and
momenta $p_{\mu}$ that satisfy

$[x_{\mu},x_{\nu}]=[p_{\mu},p_{\nu}]=0,\qquad[x_{\mu},p_{\nu}]=i\eta_{\mu\nu},$
(4)

with $x_{\mu}^{\dagger}=x_{\mu}$ and $p_{\mu}^{\dagger}=p_{\mu}$.

The action of $x_{\mu}$ and $p_{\mu}$ on functions $f(x)$ belonging to the
enveloping algebra ${\cal A}$ generated by the $x_{\mu}$ is defined as

$x_{\mu}\triangleright f(x)=x_{\mu}f(x),\qquad p_{\mu}\triangleright
f(x)=-i{\partial f(x)\over\partial x^{\mu}}.$ (5)

Noncommutative coordinates $\hat{x}_{\mu}$ and Lorentz generators $M_{\mu\nu}$
can be expressed in terms of commutative coordinates $x_{\mu}$ and momenta
$p_{\mu}$ as

$\displaystyle\hat{x}_{\mu}=x_{\mu}\varphi^{p}_{1}(\epsilon\beta^{2}p^{2})+\epsilon\beta^{2}x{\cdot}p\,p_{\mu}\varphi^{p}_{2}(\epsilon\beta^{2}p^{2})+\epsilon\beta^{2}p_{\mu}\chi^{p}(\epsilon\beta^{2}p^{2}),$
(6) $\displaystyle M_{\mu\nu}=x_{\mu}p_{\nu}-x_{\nu}p_{\mu},\qquad
M_{\mu\nu}=M^{\dagger}_{\mu\nu}.$ (7)

In terms of the realizations (6), the functions $\phi_{\mu\nu}^{p}$ in (1)
read

$\phi^{p}_{\mu\nu}=\eta_{\mu\nu}\varphi^{p}_{1}+\epsilon\beta^{2}p_{\mu}p_{\nu}\varphi^{p}_{2}.$
(8)

The Jacobi identities are satisfied if

$\psi^{p}=-2\varphi^{p}_{1}(\varphi^{p}_{1})^{\prime}+\varphi^{p}_{1}\varphi^{p}_{2}-2\epsilon\beta^{2}p^{2}(\varphi^{p}_{1})^{\prime}\varphi^{p}_{2},$
(9)

where a prime denotes a derivative with respect to $\epsilon\beta^{2}p^{2}$.
The function $\psi^{p}$ is independent of $\chi^{p}$.

From (9) it follows that the $\hat{x}_{\mu}$ are commutative for

$\varphi^{p}_{2}={2\varphi^{p}_{1}(\varphi_{1}^{p})^{\prime}\over\varphi^{p}_{1}-2\epsilon\beta^{2}p^{2}(\varphi^{p}_{1})^{\prime}},$

and correspond to Snyder space if

$\varphi^{p}_{2}={1+2\varphi^{p}_{1}(\varphi_{1}^{p})^{\prime}\over\varphi_{1}^{p}-2\epsilon\beta^{2}p^{2}(\varphi_{1}^{p})^{\prime}}.$

Some interesting special cases arise for particular values of the functions
$\varphi_{i}^{p}$. The Snyder realization [2, 14] is recovered for
$\varphi^{p}_{1}=\varphi^{p}_{2}=1$ and the Maggiore realization [21, 14] for
$\varphi^{p}_{1}=\sqrt{1-\epsilon\beta^{2}p^{2}}$ and $\varphi^{p}_{2}=0$.
Another interesting case of realization of Snyder space [17] is for
$\varphi^{p}_{1}=s=$ const, $\chi^{p}=0$,

$\hat{x}_{\mu}=x_{\mu}+{\epsilon\beta^{2}s\over 4}K_{\mu},$ (10)

where $K_{\mu}=x_{\mu}p^{2}-2x{\cdot}p\,p_{\mu}$ are the generators of
conformal transformations in momentum space, with $[K_{\mu},K_{\nu}]=0$.

The algebra (1)-(2) unifies commutative space, $\psi^{p}=0$, and Snyder space,
$\psi^{p}=1$. Since the Lorentz transformations are not deformed, its Casimir
operator is ${\cal C}^{p}=p^{2}$. The algebra (1)-(2) is invariant under
Hemitian conjugation if $p_{\mu}^{\dagger}=p_{\mu}$,
$\hat{x}_{\mu}^{\dagger}=\hat{x}_{\mu}$, $M_{\mu\nu}^{\dagger}=M_{\mu\nu}$.

Hermitian realizations of $\hat{x}_{\mu}(\beta)$ are obtained from (6) as

$\hat{x}_{\mu}(\beta)={1\over
2}\Big{[}x_{\mu}\varphi^{p}_{1}+\varphi^{p}_{1}x_{\mu}+\epsilon\,\beta^{2}(x{\cdot}p\,p_{\mu}\varphi^{p}_{2}+\varphi^{p}_{2}p_{\mu}p{\cdot}x)\Big{]}+\epsilon\,\beta^{2}p_{\mu}\chi^{p},$
(11)

where $\varphi_{1}^{p}$, $\varphi_{2}^{p}$, and $\chi^{p}$ are real functions.

##  3 Noncommutative momenta of generalized Snyder type

By duality, one can obtain different Snyder-like algebras with noncommutative
momenta, that coincide with generalized de Sitter algebras and can be
associated to the symmetries of spacetimes of constant curvature, namely de
Sitter or anti-de Sitter. The relevance of de Sitter spacetime in general
relativity and quantum gravity is well-known [22], especially in cosmological
context. The algebra of symmetries of de Sitter spacetime is also well-known,
see e.g. [23]. Also deformations of special relativity based on the de Sitter
algebra have been investigated [24]. The models presented here correspond to
unusual parametrizations of the de Sitter manifold, which are isotropic in the
4-dimensional spacetime [25].

To obtain these algebras, we introduce a new Lorentz-invariant deformation
parameter $\alpha$ of dimension inverse length, so that $\alpha^{2}$ may be
identified with the cosmological constant, and assume

$[\hat{p}_{\mu},\hat{p}_{\nu}]=i\epsilon^{\prime}\alpha^{2}M_{\mu\nu}\psi^{x}(\epsilon^{\prime}\alpha^{2}x^{2}),\qquad[x_{\mu},x_{\nu}]=0,\qquad[x_{\mu},\hat{p}_{\nu}]=i\phi_{\mu\nu}^{x}(\epsilon^{\prime}\alpha^{2}x^{2}),$
(12)

with $\epsilon^{\prime}=\pm 1$. Moreover, the Lorentz generators satisfy

$[M_{\mu\nu},\hat{p}_{\lambda}]=i(\eta_{\mu\lambda}\hat{p}_{\nu}-\eta_{\nu\lambda}\hat{p}_{\mu}),$
(13)

and the other standard relations in (2). The functions
$\psi^{x}(\epsilon^{\prime}\alpha^{2}x^{2})$ and
$\phi_{\mu\nu}^{x}(\epsilon^{\prime}\alpha^{2}x^{2})$ are chosen so that the
Jacobi identities are satisfied.

The noncommutative momenta $\hat{p}_{\mu}$ and the Lorentz generators
$M_{\mu\nu}$ can then be expressed in terms of commutative coordinates and
momenta as

$\displaystyle\hat{p}_{\mu}=p_{\mu}\varphi^{x}_{1}(\epsilon^{\prime}\alpha^{2}x^{2})+\epsilon^{\prime}\alpha^{2}p{\cdot}x\,x_{\mu}\varphi^{x}_{2}(\epsilon^{\prime}\alpha^{2}x^{2})+\epsilon^{\prime}\alpha^{2}x_{\mu}\chi^{x}(\epsilon^{\prime}\alpha^{2}x^{2}),$
(14) $\displaystyle M_{\mu\nu}=x_{\mu}p_{\nu}-x_{\nu}p_{\mu},\qquad
M_{\mu\nu}=M^{\dagger}_{\mu\nu}.$ (15)

In terms of this realization, the functions $\phi^{x}_{\mu\nu}$ read

$\phi^{x}_{\mu\nu}=\eta_{\mu\nu}\varphi^{x}_{1}+\epsilon^{\prime}\alpha^{2}x_{\mu}x_{\nu}\varphi^{x}_{2},$
(16)

and the Jacobi identities are satisfied if

$\psi^{x}=-2\varphi^{x}_{1}(\varphi^{x}_{1})^{\prime}+\varphi^{x}_{1}\varphi^{x}_{2}-2\epsilon^{\prime}\alpha^{2}x^{2}(\varphi^{x}_{1})^{\prime}\varphi^{x}_{2},$
(17)

where now a prime denotes derivative with respect to
$\epsilon^{\prime}\alpha^{2}x^{2}$. The function $\psi^{x}$ does not depend on
$\chi^{x}$.

Since the Lorentz transformations are not deformed, the Casimir operator of
this algebra is ${\cal C}=x^{2}$. The algebra (12)-(13) is invariant under
Hemitian conjugation if $\hat{p}_{\mu}^{\dagger}=\hat{p}_{\mu}$,
$\hat{x}_{\mu}^{\dagger}=\hat{x}_{\mu}$, $M_{\mu\nu}^{\dagger}=M_{\mu\nu}$.

Hermitian realizations of $\hat{p}_{\mu}(\alpha)$ are obtained from (14) as

$\hat{p}_{\mu}(\alpha)={1\over
2}\big{[}p_{\mu}\varphi_{1}^{x}+\varphi_{1}^{x}p_{\mu}+\epsilon^{\prime}\alpha^{2}(p{\cdot}x\,x_{\mu}\varphi^{x}_{2}+\varphi^{x}_{2}x_{\mu}x{\cdot}p)\big{]}+\epsilon^{\prime}\alpha^{2}x_{\mu}\chi^{x}.$
(18)

##  4 Quantum deformed phase spaces depending on two parameters

The deformed phase spaces of the previous sections can be unified by
introducing both fundamental Lorentz-invariant parameters $\alpha$ and $\beta$
so that [10, 11]

$[\hat{x}_{\mu},\hat{x}_{\nu}]=i\epsilon\,\beta^{2}M_{\mu\nu}\;\psi^{p}(\epsilon\,\beta^{2}p^{2}),\qquad[\hat{p}_{\mu},\hat{p}_{\nu}]=i\epsilon^{\prime}\alpha^{2}M_{\mu\nu}\;\psi^{x}(\epsilon^{\prime}\alpha^{2}x^{2}),\qquad[\hat{x}_{\mu},\hat{p}_{\nu}]=ig_{\mu\nu},$
(19)

where

$g_{\mu\nu}=h_{0}\eta_{\mu\nu}+\alpha^{2}X_{\mu\nu}+\beta^{2}P_{\mu\nu}+\alpha\beta
H_{\mu\nu},$ (20)

with
$X_{\mu\nu}=h_{1}\hat{x}_{\mu}\hat{x}_{\nu}+\hat{x}_{\nu}\hat{x}_{\mu}h_{1}^{\dagger}$,
$P_{\mu\nu}=h_{2}\hat{p}_{\mu}\hat{p}_{\nu}+\hat{p}_{\nu}\hat{p}_{\mu}h_{2}^{\dagger}$,
$H_{\mu\nu}=h_{3}(\hat{x}_{\mu}\hat{p}_{\nu}+\hat{p}_{\nu}\hat{x}_{\mu})+h_{4}(\hat{x}_{\nu}\hat{p}_{\mu}+\hat{p}_{\mu}\hat{x}_{\nu})+{\rm
h.c.}$, and the $h_{i}$ are Lorentz-invariant functions of $x_{\mu}$ and
$p_{\mu}$, depending on $\hat{x}^{2}$, $\hat{p}^{2}$ and $D={1\over
2}(\hat{x}{\cdot}\hat{p}+\hat{p}{\cdot}\hat{x})$.

If $\hat{x}_{\mu}$, $\hat{p}_{\mu}$ and $M_{\mu\nu}$ are Hermitian operators,
then $g^{\dagger}_{\mu\nu}=g_{\mu\nu}$, $h^{\dagger}=h$,
$X^{\dagger}_{\mu\nu}=X_{\mu\nu}$, $P^{\dagger}_{\mu\nu}=P_{\mu\nu}$ and
$H^{\dagger}_{\mu\nu}=H_{\mu\nu}$. The operators $g_{\mu\nu}$, $X_{\mu\nu}$,
$P_{\mu\nu}$ and $H_{\mu\nu}$ transform as second rank tensors under Lorentz
transformations, while $h_{0}$ is invariant, $[M_{\mu\nu},h_{0}]=0$. We also
require that all Jacobi identities hold. For simplicity, from now on we set
$\epsilon=\epsilon^{\prime}=1$.

Under suitable conditions on the $h_{i}$, the algebra (19)-(20) satisfies the
Born duality, defined as invariance for $\beta\leftrightarrow\alpha$,
$\hat{x}_{\mu}\rightarrow-\hat{p}_{\mu}$,
$\hat{p}_{\mu}\rightarrow\hat{x}_{\mu}$, $M_{\mu\nu}\leftrightarrow
M_{\mu\nu}$ and $g_{\mu\nu}\leftrightarrow g_{\nu\mu}$ [11]. Moreover, it is
easy to see that

$g_{\mu\nu}-g_{\nu\mu}=\alpha\beta(H_{\mu\nu}-H_{\nu\mu})=\alpha\beta
FM_{\mu\nu},$ (21)

with $[M_{\mu\nu},F]=0$.

The above quantum deformed phase spaces include as special cases Yang [3] and
TSR [5, 7] models and their generalizations [4, 11]. The algebras with
$X_{\mu\nu}=P_{\mu\nu}=0$ and $H_{\mu\nu}$ proportional to $M_{\mu\nu}$ reduce
to Lie algebras, generated by $\hat{x}_{\mu}$, $\hat{p}_{\mu}$ $M_{\mu\nu}$
and $h_{0}$. These algebras were introduced in [4] and are defined by

$\displaystyle[\hat{x}_{\mu},\hat{x}_{\nu}]=i\beta^{2}M_{\mu\nu},\qquad[\hat{p}_{\mu},\hat{p}_{\nu}]=i\alpha^{2}M_{\mu\nu},\qquad[\hat{x}_{\mu},\hat{p}_{\nu}]=i(h_{0}\eta_{\mu\nu}-2\alpha\beta\rho
M_{\mu\nu}),$ (23)
$\displaystyle[h_{0},\hat{x}_{\mu}]=i(\beta^{2}\hat{p}_{\mu}+2\alpha\beta\rho\,\hat{x}_{\mu}),\qquad[h_{0},\hat{p}_{\mu}]=-i(\alpha^{2}\hat{x}_{\mu}+2\alpha\beta\rho\,\hat{p}_{\mu}),$

where $\alpha$, $\beta$ and $\rho$ are real parameters. For $\rho=0$ one gets
the Yang model, and for $\alpha=0$ or $\beta=0$ the Snyder or de Sitter
algebra, respectively.

If instead $X_{\mu\nu}$ or $P_{\mu\nu}$ do not vanish, as in the case of TSR
or SdS, the algebras (19)-(20) are not Lie algebras. In the KL model
$X_{\mu\nu}=P_{\mu\nu}=0$, but $H_{\mu\nu}$ is proportional to $M_{\mu\nu}$,
and this gives rise to a Lie algebra.

###  4.1 General Hermitian realizations

The most general Hermitian realizations of the deformed phase spaces (19)-(20)
are given by

$\hat{x}_{\mu}(\alpha,\beta)=e^{iG_{1}}\hat{x}_{\mu}(\beta)\,e^{-iG_{1}},\qquad\hat{p}_{\mu}(\alpha,\beta)=e^{iG_{2}}\hat{p}_{\mu}(\alpha)\,e^{-iG_{2}},$
(24)

where $\hat{x}_{\mu}(\beta)$ and $\hat{p}_{\mu}(\alpha)$ are given in (11) and
(18) respectively, and

$G_{1}=G_{1}^{\,\dagger}=G_{1}(\alpha^{2}x^{2},\alpha\beta
D,\beta^{2}p^{2}),\qquad
G_{2}=G_{2}^{\,\dagger}=G_{2}(\alpha^{2}x^{2},\alpha\beta D,\beta^{2}p^{2}),$
(25)

with $D={1\over 2}(x{\cdot}p+p{\cdot}x)$.

In order for the Jacobi identities to hold, the functions $\varphi^{p}_{1}$,
$\varphi^{p}_{2}$ and $\varphi^{x}_{1}$, $\varphi^{x}_{2}$ have to satisfy the
relation (9) and (17) respectively.

In these realizations, the functions $X_{\mu\nu}$, $P_{\mu\nu}$, $H_{\mu\nu}$
and $h$ depend on $\varphi^{p}_{1}$, $\varphi^{p}_{2}$, $\varphi^{x}_{1}$,
$\varphi^{x}_{2}$ and $G_{1}$, $G_{2}$. The Born duality applied to the above
realization generates new ones.

###  4.2 Hermitian realization of the Yang model

An Hermitian realization for the Yang model is given by

$\displaystyle\hat{x}_{\mu}(\beta)$ $\displaystyle=$ $\displaystyle{1\over
2}\left(x_{\mu}\sqrt{1-\beta^{2}p^{2}}+\sqrt{1-\beta^{2}p^{2}}\,x_{\mu}\right)+\beta^{2}p_{\mu}\chi^{p}(\beta^{2}p^{2}),$
(26) $\displaystyle\hat{p}_{\mu}(\alpha)$ $\displaystyle=$
$\displaystyle{1\over
2}\left(p_{\mu}\sqrt{1-\alpha^{2}x^{2}}+\sqrt{1-\alpha^{2}x^{2}}\,p_{\mu}\right)+\alpha^{2}x_{\mu}\chi^{x}(\alpha^{2}x^{2}).$
(27)

The Yang model is obtained for $G_{1}$, $G_{2}$ such that

$[\hat{x}_{\mu},\hat{p}_{\nu}]=i\eta_{\mu\nu}h_{0},\quad[h_{0},\hat{x}_{\mu}]=i\beta^{2}\hat{p}_{\mu},\quad[h_{0},\hat{p}_{\mu}]=-i\alpha^{2}\hat{x}_{\mu}$
(28)

hold. The explicit form at fourth order for $\chi^{p}=\chi^{x}=0$ is reported
in [11]. In the limit $\alpha=\beta=0$, $\hat{x}_{\mu}(\alpha,\beta)=x_{\mu}$
and $\hat{p}_{\mu}(\alpha,\beta)=p_{\mu}$.

Similar constructions can be applied to TSR and SdS models, where

$\displaystyle\hat{x}_{\mu}(\beta)=x_{\mu}+{\beta^{2}\over
2}(\,x{\cdot}p\,p_{\mu}+p_{\mu}\,p{\cdot}x\,)+\beta^{2}p_{\mu}\chi^{p}(\beta^{2}p^{2}),$
(29) $\displaystyle\hat{p}_{\mu}(\alpha)=p_{\mu}+{\alpha^{2}\over
2}(\,p{\cdot}x\,x_{\mu}+x_{\mu}\,x{\cdot}p\,)+\alpha^{2}x_{\mu}\chi^{x}(\alpha^{2}x^{2}).$
(30)

##  5 Perturbative expansion of Hermitian realizations

We shall now consider Hermitian realizations in a perturbative expansion in
$\alpha$ and $\beta$ of the two-parameter model introduced in the previous
section, extending the results of [11].

###  5.1 Second-order expansion

At second order in $\alpha$ and $\beta$ we use the ansatz333In $\hat{x}_{\mu}$
we could add also $\alpha\beta x_{\mu}$, $\alpha^{2}x_{\mu}x^{2}$,
$\beta^{2}p_{\mu}$ and in $\hat{p}_{\mu}$ also $\alpha\beta p_{\mu}$,
$\beta^{2}p_{\mu}p^{2}$, $\alpha^{2}x_{\mu}$.

$\hat{x}_{\mu}=x_{\mu}+{a_{1}\over
2}\alpha\beta(x_{\mu}\,x{\cdot}p\,+\,p{\cdot}x\,x_{\mu})+{a_{2}\over
2}\beta^{2}(x_{\mu}p^{2}+p^{2}x_{\mu})+{a_{3}\over
2}\beta^{2}(p_{\mu}\,p{\cdot}x\,+\,x{\cdot}p\,p_{\mu})+{a_{4}\over
2}\alpha\beta(p_{\mu}x^{2}+x^{2}p_{\mu}),$ (31)
$\hat{p}_{\mu}=p_{\mu}+{b_{1}\over
2}\alpha\beta(p_{\mu}\,p{\cdot}x\,+\,x{\cdot}p\,p_{\mu})+{b_{2}\over
2}\alpha^{2}(p_{\mu}x^{2}+x^{2}p_{\mu})+{b_{3}\over
2}\alpha^{2}(x_{\mu}\,x{\cdot}p\,+\,p{\cdot}x\,x_{\mu})+{b_{4}\over
2}\alpha\beta(x_{\mu}p^{2}+p^{2}x_{\mu}),$ (32)

where $a_{i}$, $b_{i}$ are real constants.

Substituting (31)-(32) in the relations
$[\hat{x}_{\mu},\hat{x}_{\nu}]=i\beta^{2}M_{\mu\nu}$ and
$[\hat{p}_{\mu},\hat{p}_{\nu}]=i\alpha^{2}M_{\mu\nu}$, we find
$a_{3}-2a_{2}=1$, $b_{3}-2b_{2}=1$, while calculating
$[\hat{x}_{\mu},\hat{p}_{\nu}]$ we get

$\displaystyle g_{\mu\nu}$ $\displaystyle=$ $\displaystyle\
\eta_{\mu\nu}\left(1+2\tau\alpha\beta
D+a_{2}\beta^{2}p^{2}+b_{2}\alpha^{2}x^{2}\right)+b_{3}\alpha^{2}x_{\mu}x_{\nu}+a_{3}\beta^{2}p_{\mu}p_{\nu}$
(34)
$\displaystyle+\alpha\beta\Big{[}\tau(x_{\mu}p_{\nu}+p_{\nu}x_{\mu})+\rho(x_{\nu}p_{\mu}+p_{\mu}x_{\nu})\Big{]},$

where $\rho=a_{4}+b_{4}$, $\tau={1\over 2}(a_{1}+b_{1})$, and $D={1\over
2}(x{\cdot}p+p{\cdot}x)$.

Hence,

$\displaystyle h_{0}=1+2\tau\alpha\beta
D+a_{2}\beta^{2}p^{2}+b_{2}\alpha^{2}x^{2},\qquad
X_{\mu\nu}=b_{3}x_{\mu}x_{\nu},\qquad P_{\mu\nu}=a_{3}p_{\mu}p_{\nu},$ (35)
$\displaystyle
H_{\mu\nu}=\tau(x_{\mu}p_{\nu}+p_{\nu}x_{\mu})+\rho(x_{\nu}p_{\mu}+p_{\mu}x_{\nu}).$
(36)

In the following, for simplicity, we shall consider the symmetric solutions
$a_{i}=b_{i}$.

For KL models, $a_{2}=b_{2}=-{1\over 2}$, $a_{3}=b_{3}=0$,
$a_{1}=b_{1}=-\rho$, $a_{4}=b_{4}={\rho\over 2}$, $\tau=-\rho$ and

$g_{\mu\nu}=\eta_{\mu\nu}\left(1-2\rho\alpha\beta D-{1\over
2}(\alpha^{2}x^{2}+\beta^{2}p^{2})\right)-2\rho\alpha\beta
M_{\mu\nu}=\eta_{\mu\nu}h_{0}-2\rho\alpha\beta M_{\mu\nu},$ (37)

with $h_{0}=1-2\rho\alpha\beta D-{1\over 2}(\alpha^{2}x^{2}+\beta^{2}p^{2})$,
and

$[h_{0},\hat{x}_{\mu}]=i(2\rho\alpha\beta
x_{\mu}+\beta^{2}p_{\mu}),\qquad[h_{0},\hat{p}_{\mu}]=-i(2\rho\alpha\beta
p_{\mu}+\alpha^{2}x_{\mu}).$ (38)

For Yang models $\rho=0$ and $h_{0}=1-{1\over
2}(\alpha^{2}x^{2}+\beta^{2}p^{2})$. For TSR models $a_{2}=b_{2}=0$,
$a_{3}=b_{3}=1$, $a_{1}=b_{1}=0$, $a_{4}=b_{4}={1\over 2}$, $\tau=0$ and
$\rho=1$.

###  5.2 Fourth-order expansion

At fourth order we use the ansatz444Also in this case we have omitted some
terms that do not contribute significantly, like
$\alpha^{3}\beta\,p_{\mu}x^{4}$, $\alpha^{4}x_{\mu}x^{4}$ and
$\beta^{4}p_{\mu}p^{2}$ in $\hat{x}_{\mu}$ and analogous terms in
$\hat{p}_{\mu}$.

$\displaystyle\hat{x}^{(4)}_{\mu}$ $\displaystyle=$ $\displaystyle\
{c_{1}\over
2}\alpha^{3}\beta(x_{\mu}x^{2}\,x{\cdot}p\,+\,p{\cdot}x\,x^{2}x_{\mu})+{c_{2}\over
2}\alpha^{2}\beta^{2}(x_{\mu}x^{2}p^{2}+p^{2}x^{2}x_{\mu})+{c_{3}\over
2}\alpha^{2}\beta^{2}(x_{\mu}\,x{\cdot}p\,\,p{\cdot}x\,+\,p{\cdot}x\,\,x{\cdot}p\,x_{\mu})$
(42) $\displaystyle+{c_{4}\over
2}\alpha\beta^{3}(x_{\mu}\,x{\cdot}p\,p^{2}+p^{2}\,p{\cdot}x\,x_{\mu})+{c_{5}\over
2}\beta^{4}(x_{\mu}p^{4}+p^{4}x_{\mu})+{c_{6}\over
2}\beta^{4}(p_{\mu}p^{2}\,p{\cdot}x\,+\,x{\cdot}p\,p^{2}p_{\mu})$
$\displaystyle+{c_{7}\over
2}\alpha\beta^{3}(p_{\mu}p^{2}x^{2}+x^{2}p^{2}p_{\mu})+{c_{8}\over
2}\alpha\beta^{3}(p_{\mu}\,p{\cdot}x\,\,x{\cdot}p\,+\,p{\cdot}x\,\,x{\cdot}p\,p_{\mu})$
$\displaystyle+{c_{9}\over
2}\alpha^{2}\beta^{2}(p_{\mu}\,p{\cdot}x\,x^{2}+x^{2}\,x{\cdot}p\,p_{\mu}),$
$\displaystyle\hat{p}^{(4)}_{\mu}$ $\displaystyle=$ $\displaystyle\
{d_{1}\over
2}\alpha\beta^{3}(p_{\mu}p^{2}\,p{\cdot}x\,+\,x{\cdot}p\,p^{2}p_{\mu})+{d_{2}\over
2}\alpha^{2}\beta^{2}(p_{\mu}p^{2}x^{2}+x^{2}p^{2}p_{\mu})+{d_{3}\over
2}\alpha^{2}\beta^{2}(p_{\mu}\,p{\cdot}x\,\,x{\cdot}p\,+\,x{\cdot}p\,\,p{\cdot}x\,p_{\mu})$
(46) $\displaystyle+{d_{4}\over
2}\alpha^{3}\beta(p_{\mu}\,p{\cdot}x\,x^{2}+x^{2}\,x{\cdot}p\,p_{\mu})+{d_{5}\over
2}\alpha^{4}(p_{\mu}x^{4}+x^{4}p_{\mu})+{d_{6}\over
2}\alpha^{4}(x_{\mu}x^{2}\,x{\cdot}p\,+\,p{\cdot}x\,x^{2}x_{\mu})$
$\displaystyle+{d_{7}\over
2}\alpha^{3}\beta(x_{\mu}x^{2}p^{2}+p^{2}x^{2}x_{\mu})+{d_{8}\over
2}\alpha^{3}\beta(x_{\mu}\,x{\cdot}p\,\,p{\cdot}x\,+\,x{\cdot}p\,\,p{\cdot}x\,x_{\mu})$
$\displaystyle+{d_{9}\over
2}\alpha^{2}\beta^{2}(x_{\mu}\,x{\cdot}p\,p^{2}+p^{2}\,p{\cdot}x\,x_{\mu}),$

where $c_{i}$, $d_{i}$ are real constants.

Inserting (42) and (46) into the relations
$[\hat{x}_{\mu},\hat{x}_{\nu}]=i\beta^{2}M_{\mu\nu}$ and
$[\hat{p}_{\mu},\hat{p}_{\nu}]=i\alpha^{2}M_{\mu\nu}$, we find
$c_{6}=4a_{2}^{2}+4c_{5}+a_{2}$, $d_{6}=4b_{2}^{2}+4d_{5}+b_{2}$.

For KL models, assuming $d_{i}=c_{i}$, one has $c_{1}=c_{4}=c_{6}=c_{8}=0$,
$c_{2}={\rho^{2}\over 8}$, $c_{3}={1\over 4}-{\rho^{2}\over 2}$,
$c_{5}=-{1\over 8}$, $c_{7}={\rho\over 4}$, $c_{9}={\rho^{2}\over 4}$, and
calculating $[\hat{x}_{\mu},\hat{p}_{\nu}]$ one obtains

$\displaystyle h_{0}$ $\displaystyle=$ $\displaystyle\ 1-2\rho\alpha\beta
D-{1\over 2}(\alpha^{2}x^{2}+\beta^{2}p^{2})-{1\over
8}(\alpha^{4}x^{4}+\beta^{4}p^{4})+{\rho\over
4}\alpha^{3}\beta(x^{2}x{\cdot}p+p{\cdot}x\,x^{2})$ (48)
$\displaystyle+{\rho\over
4}\alpha\beta^{3}(p^{2}p{\cdot}x+x{\cdot}p\,p^{2})+{1\over
8}\alpha^{2}\beta^{2}(x^{2}p^{2}+p^{2}x^{2})+{1\over
2}\alpha^{2}\beta^{2}D^{2},$

and

$\displaystyle[h_{0},\hat{x}_{\mu}]$ $\displaystyle=$ $\displaystyle\
i\Big{[}2\rho\alpha\beta x_{\mu}+\beta^{2}p_{\mu}+\left({\rho^{2}\over
2}-{1\over 4}\right)\alpha^{2}\beta^{2}(x^{2}p_{\mu}+p_{\mu}x^{2})-{\rho\over
2}\alpha\beta^{3}(\,x{\cdot}p\,p_{\mu}+p_{\mu}\,p{\cdot}x\,)$ (50)
$\displaystyle-\rho^{2}\alpha^{2}\beta^{2}(\,p{\cdot}x\,x_{\mu}+x_{\mu}\,x{\cdot}p\,)\Big{]}=i(\beta^{2}\hat{p}_{\mu}+2\alpha\beta\rho\,\hat{x}_{\mu}),$
$\displaystyle[h_{0},\hat{p}_{\mu}]=$ $\displaystyle-i\Big{[}2\rho\alpha\beta
p_{\mu}+\alpha^{2}x_{\mu}+\left({\rho^{2}\over 2}-{1\over
4}\right)\alpha^{2}\beta^{2}(p^{2}x_{\mu}+x_{\mu}p^{2})-{\rho\over
2}\alpha^{3}\beta(\,p{\cdot}x\,x_{\mu}+x_{\mu}\,x{\cdot}p\,)$ (52)
$\displaystyle-\rho^{2}\alpha^{2}\beta^{2}(\,p{\cdot}x\,p_{\mu}+p_{\mu}\,p{\cdot}x\,)\Big{]}=-i(\alpha^{2}\hat{x}_{\mu}+2\alpha\beta\rho\,\hat{p}_{\mu})$

in accordance with (23). The Yang model is obtained for $\rho=0$.

For SdS, the results are reported in [11]. In particular, for $c_{i}=d_{i}$,
the coefficients depend on the free parameter $c_{1}$, with
$c_{5}=c_{6}=c_{7}=0$, $c_{2}={1\over 8}$, $c_{3}={1\over 2}$, $c_{4}={1\over
2}-c_{1}$, $c_{8}=1-c_{1}$, $c_{9}={3\over 4}$.

##  6 Exact results on generalized TSR

In this section, we present some exact realizations of generalized TSR
obtained by exploiting a method proposed in [7, 8].

Let us start with the Snyder algebra

$[\hat{x}_{\mu},\hat{x}_{\nu}]=i\beta^{2}M_{\mu\nu},\qquad\beta\neq 0.$ (53)

A class of realizations of $\hat{x}_{\mu}$ is given by

$\hat{x}_{\mu}=X_{\mu}\varphi_{1}(\beta^{2}P^{2})+\beta^{2}X{\cdot}P\,P_{\mu}\varphi_{2}(\beta^{2}P^{2}),\qquad
M_{\mu\nu}=X_{\mu}P_{\nu}-X_{\nu}P_{\mu}=M_{\mu\nu}^{\dagger},$ (54)

with $\hat{x}^{\dagger}\neq\hat{x}$, where $X_{\mu}$ and $P_{\mu}$ satisfy

$[X_{\mu},X_{\nu}]=[P_{\mu},P_{\nu}]=0,\qquad[X_{\mu},P_{\nu}]=i\eta_{\mu\nu},$
(55)

and

$\varphi_{2}={1+2\varphi_{1}\varphi^{\prime}_{1}\over\varphi_{1}-2\beta^{2}P^{2}\varphi_{1}^{\prime}},\qquad{\rm
with}\ \varphi^{\prime}_{1}={d\varphi_{1}\over
d(\beta^{2}P^{2})},\quad\varphi(0)=1.$ (56)

Let us define

$\hat{p}_{\mu}=P_{\mu}-\epsilon{\alpha\over\beta}\hat{x}_{\mu},\qquad\epsilon^{2}=1,\
\alpha\neq 0,$ (57)

and using $[\hat{x}_{\mu},P_{\nu}]=[\hat{x}_{\nu},P_{\mu}]$ we obtain

$[\hat{p}_{\mu},\hat{p}_{\nu}]=i\alpha^{2}M_{\mu\nu},\qquad[\hat{x}_{\mu},\hat{p}_{\nu}]=ig_{\mu\nu}=i\eta_{\mu\nu}\varphi_{1}(\beta^{2}P^{2})+\beta^{2}P_{\mu}P_{\nu}\varphi_{2}(\beta^{2}P^{2})-\epsilon\,\alpha\beta
M_{\mu\nu}.$ (58)

Hence, $g_{\mu\nu}-g_{\nu\mu}=-2\epsilon\,\alpha\beta M_{\mu\nu}$.

Note that

$P_{\mu}=\hat{p}_{\mu}+\epsilon{\alpha\over\beta}\hat{x}_{\mu},$ (59)

and

$M_{\mu\nu}=X_{\mu}P_{\nu}-X_{\nu}P_{\mu}=(\hat{x}_{\mu}P_{\nu}-\hat{x}_{\nu}P_{\mu}){1\over\varphi_{1}(\beta^{2}P^{2})}=\left(\hat{x}_{\mu}\hat{p}_{\nu}-\hat{x}_{\nu}\hat{p}_{\mu}-2i\epsilon\,\alpha\beta
M_{\mu\nu}\right){1\over\varphi_{1}(\beta^{2}P^{2})}.$ (60)

Hence,

$M_{\mu\nu}={1\over
2}(\hat{x}_{\mu}\hat{p}_{\nu}-\hat{x}_{\nu}\hat{p}_{\mu}+\hat{p}_{\nu}\hat{x}_{\mu}-\hat{p}_{\mu}\hat{x}_{\nu}){1\over\varphi_{1}(\beta^{2}P^{2})}.$
(61)

The algebra generated by $\hat{x}_{\mu}$, $\hat{p}_{\mu}$ and $M_{\mu\nu}$ and
all its realizations are invariant under the Born duality
$\alpha\leftrightarrow\beta$, $\hat{x}_{\mu}\rightarrow-\hat{p}_{\mu}$,
$\hat{p}_{\mu}\rightarrow\hat{x}_{\mu}$, $M_{\mu\nu}\leftrightarrow
M_{\mu\nu}$ and $g_{\mu\nu}\leftrightarrow g_{\nu\mu}$. The relation between
$\hat{x}_{\mu}$ and $\hat{p}_{\mu}$ can be written as

$\hat{p}_{\mu}=-\epsilon{\alpha\over\beta}\,S\hat{x}_{\mu}S^{-1},$ (62)

where

$S=\exp\left(-{iZ\over 2\epsilon\,\alpha\beta}\right),\qquad{dZ\over
d(\beta^{2}P^{2})}={1\over\varphi_{1}(\beta^{2}P^{2})+\beta^{2}P^{2}\varphi_{2}(\beta^{2}P^{2})}.$
(63)

Note that $Z^{\dagger}=Z$, $S^{-1}=S^{\dagger}$.

Clearly, Hermitian realizations of $\hat{x}_{\mu}$ and $\hat{p}_{\mu}$ are
given by

$\hat{x}_{\mu}^{H}={1\over
2}(\hat{x}_{\mu}+\hat{x}_{\mu}^{\dagger}),\qquad\hat{p}_{\mu}^{H}={1\over
2}(\hat{p}_{\mu}+\hat{p}_{\mu}^{\dagger}).$ (64)

###  6.1 Special cases

For $\varphi_{1}=\varphi_{2}=1$, we obtain SdS [7, 11]

$\displaystyle[\hat{x}_{\mu},\hat{p}_{\nu}]=i(\eta_{\mu\nu}+\beta^{2}P_{\mu}P_{\nu}-\epsilon\alpha\beta
M_{\mu\nu})=i\big{(}\eta_{\mu\nu}+(\alpha\hat{x}_{\mu}+\beta\hat{p}_{\mu})(\alpha\hat{x}_{\nu}+\beta\hat{p}_{\nu})-\epsilon\alpha\beta
M_{\mu\nu}\big{)},$ (65) $\displaystyle M_{\mu\nu}={1\over
2}\left(\hat{x}_{\mu}\hat{p}_{\nu}-\hat{x}_{\nu}\hat{p}_{\mu}-\hat{p}_{\mu}\hat{x}_{\nu}+\hat{p}_{\nu}\hat{x}_{\mu}\right).$
(66)

In this case,

$Z=\ln(1+\beta^{2}P^{2}),\qquad S=\exp\left(-{i\over
2\epsilon\alpha\beta}\ln(1+\beta^{2}P^{2})\right).$ (67)

For $\varphi_{1}=\sqrt{1-\beta^{2}P^{2}}$, $\varphi_{2}=0$, we obtain

$\displaystyle[\hat{x}_{\mu},\hat{p}_{\nu}]=i\left(\eta_{\mu\nu}\sqrt{1-\beta^{2}P^{2}}-\epsilon\alpha\beta
M_{\mu\nu}\right),$ (68) $\displaystyle M_{\mu\nu}={1\over
2}\left(\hat{x}_{\mu}\hat{p}_{\nu}-\hat{x}_{\nu}\hat{p}_{\mu}-\hat{p}_{\mu}\hat{x}_{\nu}+\hat{p}_{\nu}\hat{x}_{\mu}\right){1\over\sqrt{1-\beta^{2}P^{2}}}.$
(69)

In this case,

$Z=-2\sqrt{1-\beta^{2}P^{2}},\qquad
S=\exp\left({i\over\epsilon\alpha\beta}\sqrt{1-\beta^{2}P^{2}}\right).$ (70)

This example corresponds to a special case of KL model [4] with
$h_{0}=\sqrt{1-\beta^{2}P^{2}}$, $\epsilon=2\rho=\pm 1$, and

$[h_{0},\hat{x}_{\mu}]=i(\beta^{2}\hat{p}_{\mu}+\epsilon\,\alpha\beta\hat{x}_{\mu}),\qquad[h_{0},\hat{p}_{\mu}]=-i(\alpha^{2}\hat{x}_{\mu}+\epsilon\,\alpha\beta\hat{p}_{\mu}).$
(71)

##  7 Conclusions

Models of noncommutative geometry in curved spacetime have recently attracted
much interest because of possible applications to astrophysical observations
[26] and to the measurement of time delay in the propagation of photons by
cosmic sources.

In this paper we have examined a class of these models, characterized by a
high degree of symmetry, which generalize an early proposal by Yang [3], and
include TSR [5, 7] among others. The main feature of these models is that the
defining algebra contains both the Snyder [2] and the de Sitter [23] algebra,
and in particular, the Lorentz invariance is preserved. They may therefore be
relevant in a low-energy limit of quantum gravity, for which theoretical
arguments suggest that both a noncommutative parameter and a cosmological
constant should be relevant [5].

We have presented quantum realizations of these algebras in canonical phase
space, starting from the simpler cases of Snyder or its de Sitter dual.
However, their structure, that involves the full phase space, renders
problematic the definition of a star product or of a Hopf algebroid structure,
like those introduced in [27]. More general mathematical constructions should
be introduced if one wishes to include analogous notions in this formalism.

A possible area of application of our results is quantum field theory. A field
theory based on the SdS algebra was discussed in [20], based on rough
approximations. In that paper it was noted its similitude with the Grosse-
Wulkenhaar model [28], a renormalizable and exactly solvable theory, which, in
analogy with SdS field theory, can be interpreted as a field theory in curved
noncommutative space [29]. An investigation of the field theory on spacetimes
of Yang type that exploit the general results obtained in this paper would
therefore be a promising development of the present research.

##  8 Ackowledgements

S. Mignemi acknowledges support from GNFM and COST action CA18108.

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