Generalizations of Snyder model to curved spaces

S. Meljanac††† e-mail<EMAIL_ADDRESS>
Rudjer Bošković Institute, Theoretical Physics Division

Bljenička c. 54, 10002 Zagreb, Croatia

and

S. Mignemi††∗ e-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

We consider generalizations of the Snyder algebra to a curved spacetime
background with de Sitter symmetry. As special cases, we obtain the algebras
of the Yang model and of triply special relativity. We discuss the
realizations of these algebras in terms of canonical phase space coordinates,
up to fourth order in the deformation parameters. In the case of triply
special relativity we also find exact realization, exploiting its algebraic
relation with the Snyder model.

1\. Introduction Noncommutative geometries have often been advocated as
plausible candidates for describing physics at the Planck scale [1]. The first
model of noncommutative spacetime was suggested by Snyder [2] in 1947. Soon
after the publication of Snyder’s paper, C.N. Yang proposed a model that
combined noncommutativity with spacetime curvature [3]. Yang’s model was based
on the fifteen-dimensional SO(1,5) algebra. The generators of this algebra
were identified with the coordinates of a phase space with de Sitter symmetry
and with the generators of Lorentz transformation. The remaining generator
rotates positions into momenta, but its physical meaning was not specified.

More recently, Kowalski-Glikman and Smolin [4] proposed a model inspired by
that of Yang, which realizes the same symmetries in a nonlinear way, reducing
to fourteen the number of generators. They called this model triply special
relativity (TSR) because it contains three fundamental constants, identified
with the speed of light, the Planck length and the cosmological constant,
generalizing in this way the idea advanced in doubly special relativity
theories [5] of deforming the Poincaré symmetry by the introduction of a new
fundamental constant. A particularly interesting property of TSR [6] is that
it realizes the Born duality [7] for the exchange of position and momentum
operators. Another interesting consequence of this model is the prediction of
the existence of both a minimal length and a minimal momentum [8].

Later, one of us showed that this model can be realized exactly in terms of
coordinates and momenta only [8], and introduced the alternative denomination
of Snyder-de Sitter (SdS) spacetime. In [8,9] it was also shown that TSR
algebra can be obtained from the Snyder algebra by a nonunitary
transformation.

While we are not aware of other papers dealing with the Yang model, except the
recent proposal of a supersymmetric extension of the algebra [10], a number of
articles have investigated aspects of TSR. Most of them treat its classical
limit, either in a nonrelativistic or relativistic setting [11]. Also the
quantum field theory of a self-interacting scalar field in SdS spacetime has
been investigated in [12].

It is known that a fruitful approach to noncommutative geometry is based on
Hopf algebras [13], that describe the symmetries of the quantum spacetime. A
powerful tool in this formalism are realizations of Hopf algebras in terms of
the Heisenberg algebra, that were introduced in [14-16]. Only recently this
approach has been considered in the context of Yang and TSR models in [17]. In
this paper, it has been proposed that TSR and a slight generalization of the
Yang model can be treated in a unified way in this formalism.

In the present paper, we discuss general perturbative realizations of the
unified model proposed in [17], in terms of the standard Heisenberg algebra.
We also exploit the relation of TSR with the Snyder model to write down some
exact realizations. This results should consent to define a star product and a
twist following the approach of [15-17]. This topic is currently being
investigated.

2\. The model We consider a noncommutative algebra of the form [17]

$[\hat{x}_{\mu},\hat{x}_{\nu}]=i\beta^{2}M_{\mu\nu},\qquad[\hat{p}_{\mu},\hat{p}_{\nu}]=i\alpha^{2}M_{\mu\nu},\qquad[\hat{x}_{\mu},\hat{p}_{\nu}]=ig_{\mu\nu},$
$None$
$[M_{\mu\nu},\hat{x}_{\lambda}]=i(\eta_{\mu\lambda}\hat{x}_{\nu}-\eta_{\nu\lambda}\hat{x}_{\mu}),\qquad[M_{\mu\nu},\hat{p}_{\lambda}]=i(\eta_{\mu\lambda}\hat{p}_{\nu}-\eta_{\nu\lambda}\hat{p}_{\mu}),$
$None$
$[M_{\mu\nu},M_{\rho\sigma}]=i\big{(}\eta_{\mu\rho}M_{\nu\sigma}-\eta_{\mu\sigma}M_{\nu\rho}-\eta_{\nu\rho}M_{\mu\sigma}+\eta_{\nu\sigma}M_{\mu\rho}\big{)}.$
$None$

with real parameters $\alpha$ and $\beta$ and $\eta_{\mu\nu}$ the flat metric.
We interpret the Hermitian operators $\hat{x}_{\mu}=\hat{x}_{\mu}^{\dagger}$
and $\hat{p}_{\mu}=\hat{p}_{\mu}^{\dagger}$ as coordinates of the phase space
and $M_{\mu\nu}=M_{\mu\nu}^{\dagger}$ as generators of Lorentz
transformations. The rank-2 tensor $g_{\mu\nu}$ depends on $\hat{x}_{\mu}$,
$\hat{p}_{\mu}$ and $M_{\mu\nu}$, with $g_{\mu\nu}^{\dagger}=g_{\mu\nu}$. The
algebra (1)-(3) is invariant under Born duality, $\alpha\leftrightarrow\beta$,
$\hat{x}_{\mu}\leftrightarrow\hat{p}_{\mu}$, $M_{\mu\nu}\leftrightarrow
M_{\mu\nu}$, $g_{\mu\nu}\leftrightarrow g_{\nu\mu}$. In the limit $\beta\to 0$
it contains as a subalgebra the de Sitter algebra, in the limit $\alpha\to 0$
the Snyder algebra.

The Jacobi identities imply

$[M_{\mu\nu},g_{\rho\sigma}]=i\big{(}\eta_{\mu\rho}g_{\nu\sigma}-\eta_{\mu\sigma}g_{\nu\rho}-\eta_{\nu\rho}g_{\mu\sigma}+\eta_{\nu\sigma}g_{\mu\rho}\big{)},$
$None$
$[g_{\lambda\mu},\hat{x}_{\nu}]-[g_{\lambda\nu},\hat{x}_{\mu}]=i(\eta_{\mu\lambda}\hat{p}_{\nu}-\eta_{\nu\lambda}\hat{p}_{\mu}),\qquad[g_{\lambda\mu},\hat{p}_{\nu}]-[g_{\lambda\nu},\hat{p}_{\mu}]=i(\eta_{\mu\lambda}\hat{x}_{\nu}-\eta_{\nu\lambda}\hat{x}_{\mu}),$
$None$
$[g_{\mu\nu},g_{\rho\sigma}]=i\Big{(}[\,[g_{\mu\nu},\hat{p}_{\sigma}],\hat{x}_{\rho}]-[\,[g_{\mu\nu},\hat{x}_{\rho}],\hat{p}_{\sigma}]\Big{)}.$
$None$

Depending on the form of $g_{\mu\nu}$, one can recover well known models. For
example, the Yang model is characterized by the choice [3]

$g_{\mu\nu}=h(\hat{x}^{2},\hat{x}{\cdot}\hat{p}+\hat{p}{\cdot}\hat{x},\hat{p}^{2})\,\eta_{\mu\nu},$
$None$

while the TSR model is characterized by [4]

$g_{\mu\nu}=\eta_{\mu\nu}+\alpha^{2}\hat{x}_{\mu}\hat{x}_{\nu}+\beta^{2}\hat{p}_{\mu}\hat{p}_{\nu}+\alpha\beta(\hat{x}_{\mu}\hat{p}_{\nu}+\hat{p}_{\mu}\hat{x}_{\nu}-M_{\mu\nu}).$
$None$

3\. Hermitian realizations We are interested in finding Hermitian realizations
of the above models in phase space, in terms of canonical variables $x_{\mu}$
and $p_{\mu}$, satisfying $[x_{\mu},x_{\nu}]=[p_{\mu},p_{\nu}]=0$,
$[x_{\mu},p_{\nu}]=i\eta_{\mu\nu}$. In particular, we shall look for
representations where the generators $M_{\mu\nu}$ and $g_{\mu\nu}$ can be
written in terms of $x_{\mu}$ and $p_{\mu}$. Therefore, we assume

$\hat{x}_{\mu}={1\over
2}\Big{(}x_{\mu}F+F^{\dagger}x_{\mu}+p_{\mu}G+G^{\dagger}p_{\mu}\Big{)},\qquad\hat{p}_{\mu}={1\over
2}\Big{(}p_{\mu}H+H^{\dagger}p_{\mu}+x_{\mu}K+K^{\dagger}x_{\mu}\Big{)},$
$None$ $M_{\mu\nu}=x_{\mu}p_{\nu}-x_{\nu}p_{\mu},$ $None$
$\eqalign{g_{\mu\nu}=&\
\eta_{\mu\nu}h_{0}+x_{\mu}x_{\nu}h_{1}+h_{1}^{\dagger}x_{\mu}x_{\nu}+p_{\mu}p_{\nu}h_{2}+h_{2}^{\dagger}p_{\mu}p_{\nu}+(x_{\mu}p_{\nu}+p_{\nu}x_{\mu})h_{3}\cr&+h_{3}^{\dagger}(x_{\mu}p_{\nu}+p_{\nu}x_{\mu})+(x_{\nu}p_{\mu}+p_{\mu}x_{\nu})h_{4}+h_{4}^{\dagger}(x_{\nu}p_{\mu}+p_{\mu}x_{\nu}),}$
$None$

where $F$, $G$, $H$, $K$, $h_{i}$ are Lorentz-invariant functions of $x^{2}$,
$\,x{\cdot}p\,+\,p{\cdot}x\,$ and $p^{2}$.

In the following, we shall consider these realizations in a perturbative
expansion in $\alpha$ and $\beta$. At second order, we make the ansatz

$\eqalign{\hat{x}_{\mu}&=x_{\mu}+{a_{1}\over
2}\alpha\beta(x_{\mu}\,x{\cdot}p\,+\,p{\cdot}x\,x_{\mu})+{a_{2}\over
2}\beta^{2}(x_{\mu}p^{2}+p^{2}x_{\mu})+{a_{3}\over
2}\beta^{2}(p_{\mu}\,p{\cdot}x\,+\,x{\cdot}p\,p_{\mu})+{a_{4}\over
2}\alpha\beta(p_{\mu}x^{2}+x^{2}p_{\mu}),\cr\hat{p}_{\mu}&=p_{\mu}+{b_{1}\over
2}\alpha\beta(p_{\mu}\,p{\cdot}x\,+\,x{\cdot}p\,p_{\mu})+{b_{2}\over
2}\alpha^{2}(p_{\mu}x^{2}+x^{2}p_{\mu})+{b_{3}\over
2}\alpha^{2}(x_{\mu}\,x{\cdot}p\,+\,p{\cdot}x\,x_{\mu})+{b_{4}\over
2}\alpha\beta(x_{\mu}p^{2}+p^{2}x_{\mu}),}$ $None$

with constant $a_{i}$, $b_{i}$.

We may also go to the next order, with the ansatz

$\eqalignno{\hat{x}^{(4)}_{\mu}=&\ {c_{1}\over
2}\alpha^{3}\beta(x_{\mu}x^{2}\,x{\cdot}p\,+\,p{\cdot}x\,x^{2}x_{\mu})+{c_{2}\over
2}\alpha^{2}\beta^{2}(x_{\mu}x^{2}p^{2}+p^{2}x^{2}x_{\mu})+{c_{3}\over
2}\alpha^{2}\beta^{2}(x_{\mu}\,x{\cdot}p\,\,p{\cdot}x\,+\,p{\cdot}x\,\,x{\cdot}p\,x_{\mu})&\cr&+{c_{4}\over
2}\alpha\beta^{3}(x_{\mu}\,x{\cdot}p\,p^{2}+p^{2}\,p{\cdot}x\,x_{\mu})+{c_{5}\over
2}\beta^{4}(x_{\mu}p^{4}+p^{4}x_{\mu})+{c_{6}\over
2}\beta^{4}(p_{\mu}\,x{\cdot}p\,p^{2}+p^{2}\,p{\cdot}x\,p_{\mu})&\cr&+{c_{7}\over
2}\alpha\beta^{3}(p_{\mu}p^{2}x^{2}+x^{2}p^{2}p_{\mu})+{c_{8}\over
2}\alpha\beta^{3}(p_{\mu}\,p{\cdot}x\,\,x{\cdot}p\,+\,p{\cdot}x\,\,x{\cdot}p\,p_{\mu})+{c_{9}\over
2}\alpha^{2}\beta^{2}(p_{\mu}x^{2}\,x{\cdot}p\,+\,p{\cdot}x\,x^{2}p_{\mu})&\cr&+{c_{10}\over
2}\alpha^{3}\beta(p_{\mu}x^{4}+x^{4}p_{\mu}),&(13)}$
$\eqalignno{\hat{p}^{(4)}_{\mu}=&\ {d_{1}\over
2}\alpha\beta^{3}(p_{\mu}p^{2}\,p{\cdot}x\,+\,x{\cdot}p\,p^{2}p_{\mu})+{d_{2}\over
2}\alpha^{2}\beta^{2}(p_{\mu}p^{2}x^{2}+x^{2}p^{2}p_{\mu})+{d_{3}\over
2}\alpha^{2}\beta^{2}(p_{\mu}\,p{\cdot}x\,\,x{\cdot}p\,+\,x{\cdot}p\,\,p{\cdot}x\,p_{\mu})&\cr&+{d_{4}\over
2}\alpha^{3}\beta(p_{\mu}\,p{\cdot}x\,x^{2}+x^{2}\,x{\cdot}p\,p_{\mu})+{d_{5}\over
2}\alpha^{4}(p_{\mu}x^{4}+x^{4}p_{\mu})+{d_{6}\over
2}\alpha^{4}(x_{\mu}\,p{\cdot}x\,x^{2}+x^{2}\,x{\cdot}p\,x_{\mu})&\cr&+{d_{7}\over
2}\alpha^{3}\beta(x_{\mu}x^{2}p^{2}+p^{2}x^{2}x_{\mu})+{d_{8}\over
2}\alpha^{3}\beta(x_{\mu}\,x{\cdot}p\,\,p{\cdot}x\,+\,x{\cdot}p\,\,p{\cdot}x\,x_{\mu})+{d_{9}\over
2}\alpha^{2}\beta^{2}(x_{\mu}p^{2}\,p{\cdot}x\,+\,x{\cdot}p\,p^{2}x_{\mu})&\cr&+{d_{10}\over
2}\alpha\beta^{3}(x_{\mu}p^{4}+p^{4}x_{\mu}),&(14)}$

where $c_{i}$, $d_{i}$ are constants. Born dual realizations of (12)-(14) are
obtained by $\hat{x}_{\mu}\leftrightarrow\hat{p}_{\mu}$,
$\alpha\leftrightarrow\beta$ and $x_{\mu}\leftrightarrow p_{\mu}$.

4\. Yang model The original Yang model was characterized by an algebra where
$g_{\mu\nu}$ was considered as an independent generator. Here we adopt instead
the definition (7), where $g_{\mu\nu}$ is a Hermitian operator, written in
terms of a Lorentz-invariant function $h$ of the phase space variables
$\hat{x}_{\mu}$ and $\hat{p}_{\mu}$. Clearly, at zeroth order, $h^{(0)}=1$. At
second order in $\alpha$ and $\beta$ we can set

$h^{(2)}=g_{1}\alpha^{2}x^{2}+g_{2}\alpha\beta(\,x{\cdot}p\,+\,p{\cdot}x\,)+g_{3}\beta^{2}p^{2}.$
$None$

with $g_{i}$ real parameters. One easily sees that the realization (12)
satisfies the Yang algebra if

$a_{2}=b_{2}=-{1\over 2},\quad a_{3}=b_{3}=0,\quad a_{1}+b_{1}=0\quad
a_{4}+b_{4}=0,$ $None$

and

$h^{(2)}=-{1\over 2}\Big{(}\alpha^{2}x^{2}+\beta^{2}p^{2}\Big{)}.$ $None$

At this order, the simplest realization of the Yang algebra is given by the
choice $a_{1}=b_{1}=a_{4}=b_{4}=0$.

To fourth order, we can assume

$\eqalign{h^{(4)}&=e_{1}\alpha^{4}x^{4}+{e_{2}\over
2}\alpha^{3}\beta(x^{2}\,x{\cdot}p\,+\,p{\cdot}x\,x^{2})+{e_{3}\over
2}\alpha^{2}\beta^{2}(x^{2}p^{2}+p^{2}x^{2})\cr&+e_{4}\alpha^{2}\beta^{2}\,x{\cdot}p\,\,p{\cdot}x\,+{e_{5}\over
2}\alpha\beta^{3}(\,x{\cdot}p\,p^{2}+p^{2}\,p{\cdot}x\,)+e_{6}\beta^{4}p^{4}.}$
$None$

Inserting (13)-(14) in the Yang algebra, one gets the independent conditions

$c_{6}=d_{6}=0,\quad c_{6}-4c_{5}={1\over 2},\quad d_{6}-4d_{5}={1\over
2},\quad c_{8}+d_{1}=0,\quad d_{8}+c_{1}=0,$
$c_{1}+d_{4}=a_{1}b_{2}+2a_{4}b_{2},\quad
d_{1}+c_{4}=a_{2}b_{1}+2a_{2}b_{4},\quad 2c_{2}+d_{9}=-a_{1}b_{4},\quad
2d_{2}+c_{9}=-a_{4}b_{1},$ $c_{7}+2d_{10}=-a_{2}b_{4},\quad
d_{7}+2c_{10}=-a_{4}b_{2},\quad c_{3}+d_{3}=-a_{1}b_{1}+2a_{2}b_{2},\quad
c_{9}+d_{9}=-a_{4}b_{1}-a_{1}b_{4}-2a_{4}b_{4}.$

with

$e_{1}=d_{5},\quad e_{2}=c_{1}+d_{4}+a_{1}b_{2},\quad
e_{3}=c_{2}+d_{2}+a_{2}b_{2}-a_{4}b_{4},$ $e_{4}=c_{3}+d_{3}+a_{1}b_{1},\quad
e_{5}=c_{4}+d_{1}+a_{2}b_{1},\quad e_{6}=c_{5}.$

Taking into account (16), it follows that at this order there are six further
independent parameters $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$, $c_{7}$ and
$c_{10}$, with

$c_{6}=d_{6}=0,\quad c_{5}=d_{5}=-{1\over 8},\quad c_{8}=-{a_{1}\over
2}-a_{4}+c_{4},\quad c_{9}=a_{1}a_{4}+2a_{4}^{2}+2c_{2},$ $d_{1}={a_{1}\over
2}+a_{4}-c_{4},\quad d_{2}=-a_{4}^{2}-c_{2},\quad d_{3}={1\over
2}+a_{1}^{2}-c_{3},\quad d_{4}=-{a_{1}\over 2}-a_{4}-c_{1},$
$d_{7}={a_{4}\over 2}-2c_{10},\quad d_{8}=-c_{1},\quad
d_{9}=a_{1}a_{4}-2c_{2},\quad d_{10}=-{a_{4}\over 4}-{c_{7}\over 2}.$

and

$e_{1}=e_{6}=-{1\over 8},\quad e_{3}={1\over 4},\quad e_{4}={1\over 2},\quad
e_{5}=-e_{2}=a_{1}+a_{4}.$

The simplest Hermitian realization of the Yang model up to fourth order in
$\alpha$, $\beta$ is therefore

$\eqalignno{\hat{x}_{\mu}&=x_{\mu}-{\beta^{2}\over
4}\Big{(}x_{\mu}p^{2}+p^{2}x_{\mu}\Big{)}-{\beta^{4}\over
16}\Big{(}x_{\mu}p^{4}+p^{4}x_{\mu}\Big{)}+{\alpha^{2}\beta^{2}\over
8}\Big{(}x_{\mu}\,x{\cdot}p\,\,p{\cdot}x\,+\,p{\cdot}x\,\,x{\cdot}p\,x_{\mu}\Big{)},&\cr\hat{p}_{\mu}&=p_{\mu}-{\alpha^{2}\over
4}\Big{(}p_{\mu}x^{2}+x^{2}p_{\mu}\Big{)}-{\alpha^{4}\over
16}\Big{(}p_{\mu}x^{4}+x^{4}p_{\mu}\Big{)}+{\alpha^{2}\beta^{2}\over
8}\Big{(}p_{\mu}\,p{\cdot}x\,\,x{\cdot}p\,+\,x{\cdot}p\,\,p{\cdot}x\,p_{\mu}\Big{)}.&(19)}$

with

$h=1-{1\over 2}\Big{(}\alpha^{2}x^{2}+\beta^{2}p^{2}\Big{)}-{1\over
8}\Big{(}\alpha^{2}x^{2}-\beta^{2}p^{2}\Big{)}^{2}+{\alpha^{2}\beta^{2}\over
2}\,\,x{\cdot}p\,\,p{\cdot}x\,.$ $None$

Note that $h$, $\hat{x}_{\mu}$ and $\hat{p}_{\mu}$ satisfy
$[h,\hat{x}_{\mu}]=i\beta^{2}\hat{p}_{\mu}$ and
$[h,\hat{p}_{\mu}]=-i\alpha^{2}\hat{x}_{\mu}$.

5\. Triply special relativity The TSR algebra is defined by (1)-(3) with
$g_{\mu\nu}$ given by (8). An important relation following from (8) is

$g_{\mu\nu}-g_{\nu\mu}=-2\alpha\beta M_{\mu\nu}.$ $None$

From this, after some manipulations, one can obtain an equivalent form of the
algebra, written explicitly in terms of $\hat{x}_{\mu}$ and $\hat{p}_{\mu}$
only, that was first proposed in [8,9]. In those papers, the Lorentz
generators were defined as

$M_{\mu\nu}={1\over
2}\Big{(}\hat{x}_{\mu}\hat{p}_{\nu}+\hat{p}_{\nu}\hat{x}_{\mu}-\hat{x}_{\nu}\hat{p}_{\mu}-\hat{p}_{\mu}\hat{x}_{\nu}\Big{)},$
$None$

or equivalently, using (21),

$M_{\mu\nu}={1\over
1-i\alpha\beta}(\hat{x}_{\mu}\hat{p}_{\nu}-\hat{x}_{\nu}\hat{p}_{\mu})={1\over
1+i\alpha\beta}(\hat{p}_{\nu}\hat{x}_{\mu}-\hat{p}_{\mu}\hat{x}_{\nu}).$
$None$

It follows that $g_{\mu\nu}$ can be written as

$g_{\mu\nu}=\eta_{\mu\nu}+\alpha^{2}\hat{x}_{\mu}\hat{x}_{\nu}+\beta^{2}\hat{p}_{\nu}\hat{p}_{\mu}+\alpha\beta(\hat{x}_{\nu}\hat{p}_{\mu}+\hat{p}_{\mu}\hat{x}_{\nu}).$
$None$

We call this SdS realization of the TSR algebra. Using (22) and (24) we can
obtain second-order realizations of the SdS algebra in terms of the canonical
Heisenberg algebra by inserting (10) and (12) in the defining relations. It is
easy to see that the SdS algebra is satisfied if

$a_{1}+b_{1}=0,\quad a_{2}=b_{2}=0,\quad a_{3}=b_{3}=1,\quad a_{4}+b_{4}=1.$
$None$

The simplest choice of coefficients satisfying these relations is given by
$a_{1}=b_{1}=0$, $a_{4}=b_{4}={1\over 2}$.

At higher order, using the ansatz (13)-(14), one obtains the following
independent relations among the parameters:

$c_{5}=c_{6}=d_{5}=d_{6}=0,\quad c_{2}+d_{2}=a_{4}b_{4},\quad
c_{3}+d_{3}=a_{3}b_{3}-a_{1}b_{1}$ $c_{1}+d_{4}=a_{4}b_{3},\quad
d_{1}+c_{4}=a_{3}b_{4},\quad c_{1}+d_{8}=a_{1}+b_{3},\quad
d_{1}+c_{8}=b_{1}+a_{3},$ $2c_{7}+4d_{10}=b_{4}-a_{3}b_{4},\quad
2d_{7}+4c_{10}=a_{4}-a_{4}b_{3},\quad 2c_{2}+d_{9}=2b_{4}-a_{1}b_{4},\quad
2d_{2}+c_{9}=2a_{4}-a_{4}b_{1}.$

It follows that, taking into account (25), one has, in analogy with the Yang
model, six new independent parameters, say $c_{1}$, $c_{2}$, $c_{3}$, $c_{4}$,
$c_{7}$, $c_{10}$ and the relations

$c_{5}=c_{6}=0,\quad c_{8}=-a_{1}+a_{4}+c_{4},\quad
c_{9}=a_{1}a_{4}+2a_{4}^{2}+2c_{2},\quad d_{1}=1-a_{4}-c_{4},$
$d_{2}=a_{4}-a_{4}^{2}-c_{2},\quad d_{3}=1-a_{1}b_{1}-c_{3},\quad
d_{4}=a_{4}-c_{1},\quad d_{5}=d_{6}=0,$ $d_{7}=-2c_{10},\quad
d_{8}=1+a_{1}-c_{1},\quad d_{9}=2-a_{1}-2a_{4}+a_{1}a_{4}-2c_{2},\quad
d_{10}=-{c_{7}\over 2}.$

A simple realization of the SdS algebra up to fourth order with symmetric
$\hat{x}$ and $\hat{p}$ is therefore given by

$\eqalignno{\hat{x}_{\mu}=&\ x_{\mu}+\Big{(}{\beta^{2}\over
2}p_{\mu}\,p{\cdot}x\,+{\alpha\beta\over
4}p_{\mu}x^{2}+{\alpha^{2}\beta^{2}\over
16}x_{\mu}x^{2}p^{2}+{\alpha^{2}\beta^{2}\over
4}x_{\mu}\,x{\cdot}p\,\,p{\cdot}x\,+{\alpha\beta^{3}\over
4}x_{\mu}\,x{\cdot}p\,p^{2}&\cr&+{\alpha\beta^{3}\over
2}p_{\mu}\,p{\cdot}x\,\,x{\cdot}p\,+{3\alpha^{2}\beta^{2}\over
8}p_{\mu}x^{2}\,x{\cdot}p\,+{\rm h.c.}\Big{)}&(26)\cr\hat{p}_{\mu}=&\
p_{\mu}+\Big{(}{\alpha^{2}\over 2}x_{\mu}\,x{\cdot}p\,+{\alpha\beta\over
4}x_{\mu}p^{2}+{\alpha^{2}\beta^{2}\over
16}p_{\mu}p^{2}x^{2}+{\alpha^{2}\beta^{2}\over
4}p_{\mu}\,p{\cdot}x\,\,x{\cdot}p\,+{\alpha^{3}\beta\over
4}p_{\mu}\,p{\cdot}x\,x^{2}&\cr&+{\alpha^{3}\beta\over
2}x_{\mu}\,x{\cdot}p\,\,p{\cdot}x\,+{3\alpha^{2}\beta^{2}\over
8}x_{\mu}p^{2}\,p{\cdot}x\,+{\rm h.c.}\Big{)}&(27)}$

6\. Further developments The relation of the TSR algebra with the Snyder
algebra was first noticed in [8,9]. Here we exploit it using a different
derivation. We proceed as follows: the expression (8) of $g_{\mu\nu}$ can be
written as

$g_{\mu\nu}=\eta_{\mu\nu}+(\alpha\hat{x}_{\mu}+\beta\hat{p}_{\mu})(\alpha\hat{x}_{\nu}+\beta\hat{p}_{\nu})-\alpha\beta
M_{\mu\nu}=\eta_{\mu\nu}+\beta^{2}P_{\mu}P_{\nu}-\alpha\beta M_{\mu\nu},$
$None$

where we have defined $P_{\mu}=\hat{p}_{\mu}+{\alpha\over\beta}\hat{x}_{\mu}$.
Since $\hat{x}^{\dagger}_{\mu}=\hat{x}_{\mu}$,
$\hat{p}^{\dagger}_{\mu}=\hat{p}_{\mu}$, $M^{\dagger}_{\mu\nu}=M_{\mu\nu}$, it
follows that $P^{\dagger}_{\mu}=P_{\mu}$, $g^{\dagger}_{\mu\nu}=g_{\mu\nu}$,
and consequently from (28) $[P_{\mu},P_{\nu}]=0$. Hence,

$[\hat{x}_{\mu},P_{\nu}]=i(\eta_{\mu\nu}+\beta^{2}P_{\mu}P_{\nu}).$ $None$

Together with $[\hat{x}_{\mu},\hat{x}_{\nu}]=i\beta^{2}M_{\mu\nu}$ this
recalls the commutation relations of the Snyder model [2]. One can hence
derive a realization of the SdS algebra from the realizations of the Snyder
model discussed in [15,18], in terms of canonical variables $X_{\mu}$,
$P_{\nu}$ satisfying $[X_{\mu},X_{\nu}]=[P_{\mu},P_{\nu}]=0$,
$[X_{\mu},P_{\nu}]=i\eta_{\mu\nu}$, namely,

$\eqalignno{\hat{x}_{\mu}&=X_{\mu}+{\beta^{2}\over
2}\Big{(}X{\cdot}P\,P_{\mu}+P_{\mu}\,P{\cdot}X\Big{)},&\cr\hat{p}_{\mu}&=P_{\mu}-{\alpha\over\beta}\hat{x}_{\mu}=P_{\mu}-{\alpha\beta\over
2}\left(X{\cdot}PP_{\mu}+P_{\mu}P{\cdot}X\right)-{\alpha\over\beta}X_{\mu}.&(30)}$

This realization is not included in the class investigated in the previous
section, because $\hat{p}_{\mu}$ contains terms proportional to $X_{\mu}$ that
were neglected in the ansatz (12), but is exact, although it is not symmetric
in $X$ and $P$ and is not well defined in the limit $\beta\to 0$.

A realization that is regular for vanishing $\beta$, but singular for
$\alpha\to 0$ can be obtained by duality, starting from a representation of de
Sitter algebra in Beltrami coordinates [19]. One has

$\eqalignno{\hat{p}_{\mu}&=P_{\mu}+{\alpha^{2}\over
2}\Big{(}P{\cdot}X\,X_{\mu}+X_{\mu}\,X{\cdot}P\Big{)},&\cr\hat{x}_{\mu}&=X_{\mu}-{\beta\over\alpha}\hat{p}_{\mu}=X_{\mu}-{\beta\alpha\over
2}\left(P{\cdot}XX_{\mu}+X_{\mu}X{\cdot}P\right)-{\beta\over\alpha}P_{\mu},&(31)}$

where $X_{\mu}$ and $P_{\mu}$ still satisfy canonical commutation relations,
but are not the same as in (30).

Further realizations can be obtained by similarity transformations starting
from the ones found above. In fact, let us consider the commutation relations
(1)-(3) and act on them with a unitary operator $S$ from the left and $S^{-1}$
from the right, defining

$\hat{x}^{\prime}_{\mu}=S\hat{x}_{\mu}S^{-1},\qquad\hat{p}^{\prime}_{\mu}=S\hat{p}_{\mu}S^{-1},\qquad
M^{\prime}_{\mu\nu}=SM_{\mu\nu}S^{-1}=M_{\mu\nu},\qquad
g^{\prime}_{\mu\nu}=Sg_{\mu\nu}S^{-1}.$ $None$

Then $\hat{x}^{\prime}_{\mu}$ and $\hat{p}^{\prime}_{\mu}$ satisfy the same
commutation relations as $\hat{x}_{\mu}$ and $\hat{p}_{\mu}$.

We may write $S=e^{iG}$, with
$G=G(x^{2},\,x{\cdot}p\,\\!+\\!\,p{\cdot}x\,,p^{2})$, $G^{\dagger}=G$ and
$[M_{\mu\nu},G]=0$. In this way we generate infinitely many realizations of
$\hat{x}$ and $\hat{p}$ in terms of $x$ and $p$, satisfying the same algebra.

7\. Conclusions We have discussed a general quantum algebra that depends on
two parameters $\alpha$ and $\beta$, usually identified with a minimum length
and the cosmological constant, and includes as special cases Yang [3] and TSR
[4] algebras. This algebra is relevant for quantum gravity research, because
it combines the effects of non- commutativity with those of the curvature of
spacetime, a subject that has attracted a large interest recently [21].

We have found realizations of these quantum algebras on canonical phase space.
The form of the algebra (1)-(3) is much more general than the special cases we
have considered, and we are now constructing more general models of this
class. In particular, even considering algebras not more than quadratic in the
generators, several possibilities are available.

A more difficult problem is to construct a quasi-Hopf algebra associated to
these models. It seems that star product and twist have not been constructed
so far for Yang and TSR models. Star products should be nonassociative as for
the Snyder model [15,17,18]. Star products, related to noncommutative
coordinates $\hat{x}_{\mu}$ whose realizations depend on the parameters
$\alpha$ and $\beta$ and on $x_{\mu}$, $p_{\mu}$ are under construction,
exploiting the method proposed in [15,18,20]. This construction implies a
generalization of the Hopf algebroid approach [22].

An interesting field of application of our results is QFT. A field theory
based on the SdS algebra has been discussed in [12], where it was also
remarked its similitude with the Grosse-Wulkenhaar model [23]. This model is
of primary relevance because it gives rise to a renormalizable and exactly
solvable theory, which, in analogy with SdS field theory, can be thought as a
field theory in noncommutative curved space [24].

Ackowledgements We wish to thank J. Lukierski and M. Woronowicz for
interesting discussions. S. Mignemi acknowledges support from GNFM and COST
action CA18108.

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# Transient Optoplasmonic Detection of Single Proteins with Sub-Microsecond
Resolution

Martin D. Baaske  N. Asgari  D. Punj  Michel Orrit  Huygens-Kamerlingh Onnes
Laboratory, Leiden University, Postbus 9504, 2300 RA Leiden, The Netherlands

Optoplasmonic methods capable of single protein detection so far rely on
analyte immobilization in order to facilitate detection1, 2, 3, 4, 5, 6. These
detection schemes, even if they facilitate transient single-molecule
detection7, 8 via consequent formation and cleavage of chemical bonds,
typically exhibit time resolutions on the order of milliseconds. The need for
analyte immobilisation is a direct consequence of the minuscule dimensions of
plasmonic near fields typically providing sub-attolitre-sized detection
volumes which in turn demand sub-microsecond temporal resolution for the
direct detection of proteins in motion. Here we show that such temporal
resolution can indeed be achieved. We demonstrate the observation of single
proteins as small as Hemoglobin (molecular weight: $\mathbf{64}$ kDa) as they
traverse plasmonic near fields of gold nanorods and interact with their
surface, all while maintaining signal-to-noise ratios larger than $5$ and an
unprecedented temporal resolution well below microseconds. This method enables
the label-free observation of single-molecule dynamics on previously
unaccessible timescales.

In order to achieve such resolution we have improved the confocal system we
had previously used to detect single metal nanoparticles in Brownian motion9.
Specifically we achieve gains in signal-to-noise ratio via optimization of
incident and analyzed polarization, probe wavelength and Gouy phase for
individual plasmonic gold nanorods. The setup is shown in Fig. 1a and was
specifically designed to maintain linear polarization states. For each NR we
determine its orientation via rotation of linear incident and analyzer
polarisation and obtain its white-light scattering spectra, which allow us to
determine whether the overlap of the NRs spectrum with the tuning range of our
probe laser $(785\pm 20)$ nm is suited. We then align the incident and
analyzer polarisations parallel with the NR’s axis and select the wavelength
which exhibits the highest slope in the scattering spectra. The detected
intensity
$I_{det}\propto|E_{r}|^{2}+|E_{s}|^{2}+2|E_{r}||E_{s}|\cos(\Delta\phi)$ is a
result of the interference between the scattered $E_{s}$ and reflected $E_{r}$
fields projected onto the analyzer’s axis and the phase difference between
both fields is $\Delta\phi$. NRs with different scattering cross sections
require different experimental parameters in order to maximize signal-to-
noise. To address this requirement we utilize photothermal spectroscopy10, 11,
12, 13 as a convenient probe of an individual NR’s sensitivity. This is done
by heating the nanorod with the intensity-modulated $532$ nm laser (modulation
frequency $\nu_{H}=1.3$ MHz) while probing it with the intensity-modulated
wavelength-tunable probe laser (modulation frequency $\nu_{P}=1$ MHz). Heating
the NR gives rise to minute changes of the surrounding medium’s refractive
index which in turn causes the NR’s localized surface plasmon resonance (LPSR)
to shift. This shift is then detected as a change of $I_{Det}$ at the
modulation frequency $\nu_{H}$. The modulation of the probe beam further
enables us to directly determine the relative photothermal amplitude
$A_{PT}=\frac{A(\nu_{H})}{A(\nu_{P})}$, where $A(\nu)$ denotes the root-mean-
squared (rms) amplitude at the respective frequency $\nu$, with minimal
contributions from $1/f$-noise sources. $A_{PT}$ is proportional to the
relative change of intensity as a response to the photothermal refractive
index change over the total detected intensity and therefore a direct measure
of the NR’s sensitivity, to be optimized by the adjustment of such
experiment’s parameters as probe wavelength, probe power and the NR’s position
in the focal volume. For our measurements we use nanorods with dimensions of
$25\times 80$ and $40\times 110$ nm and typical LSPR’s of $1.6$ eV ($775$ nm).

Figure 1: Experimental setup with photothermal signal optimization: Panel a
shows the optical setup. Panel b shows sections of white-light scattering
spectra normalized to the reflection on the glass slide ($(s/r)^{2}$, green
line) and the corresponding relative photothermal amplitude $A_{PT}$ values
(blue dots) obtained while varying the tunable laser’s wavelength for NRs with
various diameters.

For NRs with $\approx 40$ nm diameter the on-resonance scattering coefficient
$s=E_{s}/E_{0}$, where $E_{0}$ is the incident field, significantly exceeds
the reflection coefficient $r=E_{r}/E_{0}$ of the glass-water interface and
the interference term is therefore negligible. In this case high $A_{PT}$
values are found at a wavelength coinciding with the flanks of the NR’s LSPR
spectrum i.e. where the slopes of the Lorentzian are highest (see Fig.1 b,
left). For the $20-25$ nm diameter NRs $s\approx r$ and high $A_{PT}$ values
are found closer to the LSPR center (see Fig.1 b, center and right). We find
that the interplay between contributions of scattering cross section and phase
changes upon heating is more complex and also dependent on the Gouy phase i.e.
the NR’s distance to the focal plane. In order to streamline the alignment
process we follow a simple recipe for all NRs: We first center the NR in the
focus (xyz) by maximizing the scattered intensity with crossed polarizers.
Then we optimize $A_{PT}$ with parallel polarizers aligned along the NR’s long
axis by first tuning the wavelength and then adjusting the NR position along
the focal axis (z).

Single-molecule measurements are typically performed by recording intensity
time traces of $10$ ms length with a sampling rate of $100$ MHz. These traces
are then de-noised by applying a running $10$-point median filter.
Specifically we detect Glucose Oxidase (GOx, molecular weight $\approx 160$
kDa) from Aspergillus niger (Fig. 2a) and Hemoglobin (Hem, $\approx 64$ kDa)
from bovine blood (Fig. 2b) molecules as they move through the NR’s near
field. These molecules produce transient shifts of the LSPR, which are
recognized as changes of the detected intensity. These changes appear on the
intensity traces as patterns of two different types: 1) Short spike-like
perturbations (comp. Fig. 2a I-1, II-1 left, III-2 and 2b II-1, II-2, III-2)
which we interpret as protein molecules moving directly in and out of the the
near-field. 2) Level-transition patterns (comp. Fig. 2a I-1 and III-1 center
and Fig. 2b I, III-1 and IV-1) which we think are caused by protein molecules
moving through the aqueous medium into the near field and then dwelling at the
NR’s surface due to attractive forces until the attractive potential is
eventually overcome and they again move out of the near field. We also
observed a few binding and unbinding events without their respective
counterparts in the same trace. This suggests that sticking lasting longer
than $10$ ms occurs.

Figure 2: Intensity perturbations due to GOx (blue) and Hem (red) molecules
moving in a NR’s near field. a) Trace (top) and shorter timescale subtraces (I
- IV) showing perturbations caused by Glucose Oxidase molecules observed with
a $25$ nm diameter NR and a DC-coupled detector. b) Trace (top) and shorter
timescale subtraces (I - IV) showing perturbations caused by Hemoglobin
molecules observed with a $40$ nm diameter NR and an AC-coupled detector. Most
perturbations exhibit sub-µs rise- and fall-times. Longer perturbations as a
I-1 and III-1 center, b I, III-1 and IV-1) are likely caused by attractive
interactions between analyte and NR. All traces were recorded with $\Delta
t=10$ ns and de-noised with a $10$-point median filter. Protein concentrations
were $500$ nM (a, GOx) and $30$ nM (b, Hem)

In order to obtain statistics we count fluctuations of intensity traces as
events if their amplitudes exceed $5\sigma$, where $\sigma$ is the standard
deviation of the whole trace. This does not include rare single step events
without counterpart in the same trace (GOx). In the case of (Hem) level
transitions with durations longer than $0.1$ ms were excluded from analysis in
order to avoid artifacts due to the low-frequency cut-off ($1.5$ kHz) of the
AC-coupled detector. For each event we determine the maximum amplitudes
$\Delta I_{max}$ and the following temporal properties: $\tau_{rise}$
($\tau_{fall}$) the rise (fall) time i.e. the time it takes from the beginning
(end) of the event to rise to half the maximum and the duration between these
points i.e. the full duration at half maximum (FDHM).

Figure 3: Statistics of rise and fall times, characterizing the molecular
diffusion in the abscence of sticking as well as immediately before and after
sticking, for GOx (a) and Hem (b). Each panel (a,b) shows the $\tau_{rise}$
(bottom) and $\tau_{fall}$ (right) distributions alongside their respective
2D-histograms (center). Solid lines (a, light blue and b, dark red) in the
distributions show fits to mono-exponential time distributions plotted on
logarithmic scale of times.

For both Hem and GOx we find rise- and fall-times in the range from $10$ to
$1000$ ns (see Fig. 3a, b). All $\tau_{rise}$ and $\tau_{fall}$ distributions
fit well to mono-exponential decays (i.e. to $f(t)=N\frac{t}{\tau}e^{-t/\tau}$
for the logarithmically spaced distributions) and we find rise (fall) decay
times of $101\pm 8$ ns ($108\pm 6$ ns) for GOx and $153\pm 11$ ns ($155\pm 11$
ns) for Hem. Using these values as diffusion times for spheres with
hydrodynamic radii of $4.45$ nm (GOx)14, $3.11$ nm (Hem)15 we find rms
displacements of $\approx 6$ nm (GOx) and $8$ nm (Hem) which match well with
the half decay length of optical near fields. The similarity between rise- and
fall-time distributions suggests that entry and the exit processes of both
proteins into and out of the NR’s sensitive volume are subject to equivalent
interactions. The 2D-histograms (Fig. 3a and 3b, center) further show no
apparent correlation between $\tau_{rise}$ and $\tau_{fall}$ for individual
events further suggesting that entry and exit processes into and out of near
field are mutually independent as we would expect for Brownian motion. The
rise and fall times for GOx are shorter than the ones we find for Hem although
GOx exhibits the higher molecular weight. We speculate that this difference is
due to the (in comparison to the globular Hem) more anisotropic shape of GOx16
and may well reflect an additional contribution from rotational diffusion. For
GOx we estimate a rotational diffusion time in the order of $50$ ns which
falls well into our temporal resolution. Differences in the autocorrelation
measurements for both molecules further support this hypothesis (see suppl.
info. section S1). Nonetheless further work beyond the scope of this
manuscript is needed to confirm this.

Figure 4: Statistics of dwell times and amplitudes as obtained for GOx (a) and
Hem (b). Each panel (a,b) shows the FDHM (bottom) and maximum amplitude
(right) distributions alongside their respective 2D-histograms (center). Solid
lines in the FDHM distributions show fits to mono- (a, light blue) and bi-
exponential time distributions, plotted on logarithmic scale of times (b, dark
red).

We also find similar FDHMs distributions for both proteins species. These
times represent the dwell time of individual proteins in the NR’s sensitive
volume and are centered around $\approx 1$ µs and distributed over ranges from
$100$ ns up to few tens of microseconds i.e. significantly longer than the
rise- and fall-times alone (see Fig. 3b). Here the stretch towards long
timescales is likely due to attractive interactions between the proteins and
the NR’s surface, presumably unspecific sticking. The FDHM distribution of Hem
(Fig.4 b, bottom) is more stretched towards long times than the one of GOx
(Fig.4 a, bottom). Specifically we find that the FDHM distribution for GOx
fits well to a mono-exponential decay as expected for a Langmuir adsorption
process governed by a single time constant for which we find $\tau=0.9$ µs
from the respective fit. For Hem this is not the case as the single rate fit
clearly deviates from the data (not shown). This suggests contributions from
multiple processes with different rates. Already for two rates
($\tau_{1}=0.83$ µs and $\tau_{2}=4.8$ µs) we find a much better match. This
is rather unsurprising due to the larger set of fitting parameters. We think
it is more likely that a broader spectrum of rates exists rather than just
two. The maximum amplitude vs. FDHM distribution for Hem (Fig.4 b, center)
also reveals a stronger correlation between larger amplitudes and longer times
as compared with GOx (Fig. 4a, center). We hypothesize the differences between
Hem and GOx, specifically the existence of multiple rate constants could arise
due to the following reasons: 1) Hemoglobin exhibits surface areas which
possess different affinity to gold and therefore shows different sticking
dynamics dependent on the protein’s contact area with the surface. The
correlation of high amplitudes with long FDHMs would then arise due to
increased overlap between protein sections with high polarizability (i.e.
Hem’s iron-complex) and the NR’s near field. 2) Hemoglobin has different
affinities to different types of gold crystal facets. In this case the
correlation of higher maximum amplitudes with longer FDHMs would imply that
facets which offer higher affinities coincide with zones of higher near-field
intensity. Which hypothesis is right or whether there is a process we have not
covered will require further study beyond the scope of this work. We want to
point out that GOx and Hem measurements shown here (Fig.2, 3 and 4) were
performed with different detectors, i.e., the amplitude values given are not
directly comparable. To obtain a direct comparison of perturbation amplitudes
we determine the relative intensity changes $\Delta I_{rel}=\left|\frac{\Delta
I_{max}}{\overline{I_{det}}}\right|$ caused by GOx and Hem from measurements
performed with the same DC-coupled detector. We find average values of
$\Delta\overline{I_{rel}}=(3.4\pm 1.5)\%$ for GOx and $(1.55\pm 0.45)\%$ for
Hem. The ratio
$\Delta\overline{I_{rel}}(GOx)/\Delta\overline{I_{rel}}(Hem)=2.2\pm 1$ matches
the corresponding molecular weight ratio of $2.5$ within the errors. We would
also like to put our time resolution into context: None of the events
discussed above - even the longest (FDHM$\approx 100$ µs), would be resolved
with other state-of-the-art nanoplasmonic assays 1, 2, 3, 4, 7, 5, 8, 17, 18,
19, 6, 20, which typically exhibit time resolutions in the order of
milliseconds, i.e., see the whole extent of the traces displayed in Fig.
2a(top) and 2b(top) as either a single or a set of few points. We think being
able to resolve and analyse such short unspecific interactions opens up a
whole new pathway for fast molecular fingerprinting. We envision that
previously not resolvable differences in the interaction dynamics between
specific proteins sub-domains and small weakly bonding receptor molecules on
the NR’s surface may be used as means to distinguish between proteins of
similar mass. In the simplest case the average number of formed bonds
determines the dwell time. This would further allow to gain structural
information on analytes, especially unknown ones therefore mitigating the need
for target-specific receptors. In addition physical analyte properties such as
Stokes radii and molecular weight may be determined via $\tau_{rise}$,
$\tau_{fall}$ and relative amplitudes following NR calibration with a suited
standard. In conclusion, we have demonstrated the transient detection of
single proteins with masses as low as $64$ kDa traversing the subattoliter
volumes spanned by plasmonic near fields during times as short as $100$ ns and
with a signal-to-noise ratio exceeding $5$. We found good agreement between
relative signal amplitudes and molecular weights. We have used our
unprecedented temporal resolution to observe protein motion alongside the
dynamics of unspecific protein surface interactions. We found distinctive
differences in sticking behaviors of Glucose Oxidase and Hemoglobin and found
initial evidence that suggests that our method may resolve rotational
diffusion. We think this result offers but a glimpse of the additional
information that may be gained on physical and biochemical processes on the
timescales now made accessible by fast optoplasmonic detection.

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##  I Acknowledgements

This work was supported by the Netherlands Organisation for Scientific
Research (NWO) and has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie Grant
Agreement no. 792595 (MDB).

##  II Author Contributions:

MDB and MO conceived the idea. MDB planned the experiments. MDB and NA built
the optical setup. MDB performed experiments and data analysis. DP obtained
and analysed SEM micrographs. MDB and MO wrote the manuscript. All authors
commented on the manuscript.

##  III Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

##  IV Methods

###  IV.1 Optical Setup(s):

Here we list the components used in our measurements.

  * Objective: Olympus UPLFLN100XOP

  * Tube lens: Olympus Super Wide Tube Lens Unit

  * Lasers: Toptica DL pro 785nm & Cobolt Samba 532nm

  * APD: Thorlabs APD430A/M (Thorlabs) (DC-Coupled)

  * APD: Helix-902-200 (Excelitas Technologies) (AC-Coupled)

  * 10:90 Beamsplitter BSN11 (Thorlabs)

  * Glan-Thompson Polarizer GTH10M-B (Thorlabs)

  * Piezo Translator P-561.3CD (Phyisk Instrumente GmbH & Co KG)

  * White-light source: EQ-99XFC (Energetiq)

  * Spectrometer: QE-65000 (Ocean Optics)

  * Reference Photodiodes 1&2: PDA36A2 (Thorlabs)& HCA-S-200M (Femto)

  * EOM: Amplitude Modulator AM532 (Jenoptik)

  * AOM: MT110-A1-IR (AA Opto-Electronic)

  * Achromatic $\lambda/2$-plate: RAC 4.2.10 (B. Halle)

  * Notch Filter: ZET532NF (Chroma)

Traces were digitized with an oscilloscope (WaveSurfer 24MXs-B, Teledyne
Lecroy) and streamed to a PC.

###  IV.2 Slide preparation:

CTAB-capped gold nanorods were purchased from Nanopartz. GNR stock solutions
containing $10$ mM CTAB were sonicated (20 min./ Branson 2510) and then
deposited onto glass slides (Borosilicate glass diameter $25$ mm thickness
No.1, VWR ), which were previously sonicated in ethanol and Milli-Q ($30$ min.
each/ Branson 2510), via spin-coating (Specialty coating Systems Spin Coater
6700). The CTAB-layer was consequently removed via UV-cleaning ($10-60$ min.,
Ossila) and the slide was rinsed with Milli-Q water.

###  IV.3 Protein measurements:

Solutions containing various concentrations of Hem and GOx were prepared
freshly before the start of each measurement. For GOx we used aqueous
solutions (Milli-Q water) containing $20$ mM of Sodium Chloride and for Hem we
used $1\times$PBS (Phosphate Buffered Saline) buffer solutions. All chemicals
and proteins were purchased from Sigma-Aldrich.


#  ELEPHANT-HUMAN CONFLICT MITIGATION: AN AUTONOMOUS UAV APPROACH

Weiyun Jiang, Yukai Yang11footnotemark: 1
Electrical and Computer Engineering
University of California, Santa Barbara, CA
{weiyunjiang<EMAIL_ADDRESS>
Faculty Advisor:
Yogananda Isukapalli These two authors contributed equally.

##  1 ABSTRACT

Elephant-human conflict (EHC) is one of the major problems in most African and
Asian countries. As humans overutilize natural resources for their
development, elephants’ living area continues to decrease; this leads
elephants to invade the human living area and raid crops more frequently,
costing millions of dollars annually. To mitigate EHC, in this paper, we
propose an original solution that comprises of three parts: a compact custom
low-power GPS tag that is installed on the elephants, a receiver stationed in
the human living area that detects the elephants’ presence near a farm, and an
autonomous unmanned aerial vehicle (UAV) system that tracks and herds the
elephants away from the farms. By utilizing proportional–integral–derivative
controller and machine learning algorithms, we obtain accurate tracking
trajectories at a real-time processing speed of 32 FPS. Our proposed
autonomous system can save over 68% cost compared with human-controlled UAVs
in mitigating EHC.

##  2 INTRODUCTION

Elephant-human conflict (EHC) has been one of the most significant problem in
most African and Asian countries. EHC is extremely prevalent because nearly
1.2 billion people in the world live in African and Asian elephant range
countries [1]. Crop raiding is one of the most common types of EHC [2]. C.
Mackenzie, et al. found that elephants around Kibale National Park, Uganda,
damage various crops, such as maizes, beans, sweet potatoes, and so on, and
lead to over US$3500 of total economic loss in a village of 145 households
during 6 months [3]. These crops are the main source of income for these
villagers in the village, where the median household capital asset wealth was
only US$5033. Crop raiding causes huge economical loss to not only small
vegetation, but also large plantation. Riau, the largest palm oil producing
province in Indonesia, loses millions of dollars during to crop-raiding [4].
Besides the tremendous economic loss, EHC also leads to casualties of both
human and elephants. EHC in India leads to approximately 400 human deaths and
100 elephant deaths annually [5].

Traditional methods of mitigating EHC are limited to human guarding, fire,
beating drums, scare shooting, dogs, and so on [6]. These methods are not
cost-effective and efficient because almost all these traditional methods
require expensive human labor. In addition, elephants begin to get used to
these methods after people have used them many times. Recently, researchers in
[7] founds that UAVs can mimic honeybees’ humming sound, which is known to
annoy elephants. These small bees love to sting elephants’ sensitive areas
such as eyes, ears, and noses [7]. N. Hahn, et al. have also proven the
effectiveness of drones by hiring rangers to manually control UAVs to herd
elephants away from the farms [8], in this paper we build on this idea and
propose an autonomous UAV system.

(a) Elephant at the border

(b) Elephant invades the farm

(c) UAV navigates to the elephant

(d) Camera turns on

(e) Elephant herded away

Figure 1: Our Autonomous UAV Illustration

###  A. Paper Contributions

Our contribution in this paper can be summarized as follows:

  * •

Our solution comprises of three parts: a compact custom low-power GPS tag
installed on the elephants, a receiver stationed in the human living area that
detects the elephants’ presence near a farm or village, and an autonomous UAV
system that tracks and herds the elephants away from the farms.

  * •

By utilizing PID controller and machine learning techniques, we obtain
accurate tracking trajectories at a real-time processing speed of 32 FPS.

  * •

Our proposed autonomous system can save over 68% cost compared with human-
controlled UAVs in mitigating EHC.

##  3 Overall Approach

Our autonomous UAV solution can be summarized as follows:

  * •

To begin with, we place a GPS tag on the elephant (ankle or neck) by
tranquilizing it. And then, a ground base is set up on the farm, where
protection is needed from elephants’ invasions. The ground base will emit
signals that the GPS tag will pick up when the signal is strong enough. So the
range of the signal from the ground base defines the perimeter of the
protected area we want to prevent elephants from entering.

  * •

When the elephant with the GPS tag goes into the range of the signal of the
ground base, the GPS tag will be woken up from the sleep mode, and it will
start to send out real-time coordinates of the elephant.

  * •

Once the UAV (situated in the farm) receives the coordinates of the elephants,
it navigates to the location of the elephant.

  * •

When the UAV is closer to the coordinates of the elephant, the onboard camera
of the UAV is enabled, and the live video feed is then processed by vision
algorithms on NVIDIA Jetson Nano to track the elephant effectively.

  * •

Finally, the drone will herd the elephant away from the ground base and
prevent elephants from damaging crops since the UAV emits sounds similar to
honeybees that elephants are afraid of. Elephants tend to stay away from bees
because these small bees love to sting elephants’ sensitive areas, such as
eyes and ears [9].

Figure 1 illustrates the process of our proposed autonomous UAV system in more
details. The yellow shaded circle represents the region of the signal covered
by the ground station. And the three little houses in the middle of the circle
represent the ground station. In addition, the yellow shaded circle is the
region where the signal from the base station will trigger and wake up the GPS
tag on the elephant. Initially, when the elephant is outside the yellow
region, which is the protected region of the farm, the drone is on the ground
and ready to take off.

When the elephant comes into the protected region around the ground base, the
Xbee module receives the signal from the ground base, which triggers and wakes
up the GPS tag on the elephant. Then the GPS coordinate of the elephant is
sent out by the Xbee module on the GPS tag and is then received by the Xbee
module on the drone. The drone immediately takes off and uses waypoint
navigation to fly to the coordinate of the elephant.

When the drone is close enough to the position of the elephant, the onboard
camera will turn on. GPS signals only provide a coarse location of the
elephant; the vision algorithm enables the drone to track the elephant’s
movement in a more fine-grained fashion. Finally, the drone maneuvers and
herds the elephants until they are outside the protected area.

##  4 Hardware Design

Figure 2: System Block Diagram

In this section, we discuss our hardware building blocks shown in Figure 2,
which include the custom UAV and the GPS tag installed on elephants.

Our UAV is a standard hexacopter with six motors and electronic speed
controllers (ESC) complemented by an onboard computer, a camera, and a
communication module.

We select Jetson Nano as the onboard processor due to its low power
consumption and compactness. Further, its integrated NVIDIA GPU allows us to
perform neural network inference with hardware acceleration. Figure 3(a) shows
our completed UAV prototype, with the DJI N3 flight controller, Xbee RF
module, and the camera mounted. DJI N3 has multiple built-in sensors such as
GPS and IMU that support stable waypoint navigation and control. The Xbee RF
module is responsible for receiving the location of the elephants sent by the
GPS tag. To ensure our UAV can complete the round-trip herding task, we
utilize a high capacity 12000 mAh LiPo battery to power the entire UAV system.
The LiPo battery allows flight time as long as 20 minutes and enables the UAV
to travel up to 4 miles without recharging.

As shown in Figure 3(b), our GPS tags are made of four major components: a
Xbee RF module, a GPS module, a SD card reader and a ATmega328P micro-
controller. Further, we also include a button cell for the GPS module. This
button cell prevents the GPS module from shutting down completely. It takes
more than 10 minutes for the GPS module to find a fixed location after
shutting down completely. Thus, the included button cell helps the GPS module
to find a fixed and accurate location coordinate quickly.

The GPS tag is a 4-layer board of 83 mm $\times$ 83 mm and weighs only 26.3 g.
Thus, elephants will barely notice the extra weight if we install our GPS tags
on them. The electric power consumption of our GPS tag is 10.5 mW. If we power
our GPS tag with a 10000 mAh battery bank, our GPS tags can last for about 200
days on the elephants.

(a) UAV Prototype

(b) GPS Tag PCB

Figure 3: Hardware Design

##  5 Algorithm Design

In this section, we discuss our algorithms that run onboard the UAV and how we
meet the real-time processing constraint. Our onboard processing has two major
subcomponents, visual tracking and control. When we launch our control
program, it will spawn the tracker as a separate process and communicate with
it through a socket. The tracker will send the bounding box coordinates of the
elephant to the control program at each frame. Upon receiving the bounding box
coordinates, the control program will generate control signals for the UAV.

###  A. Detection

The rapid advancement of Convolutional Neural Network (CNN) has brought object
detection to a new level. SORT [10], a similar framework to ours, adopts
Faster-RCNN, an accurate yet expensive two-stage model. To deal with the
compute constraint, we utilize SSD [11], a single-stage and highly efficient
detector. We select $300\times 300$ pixels as the input size and Inception V2
as the backbone. The advantage of SSD is that when quantized using TensorRT,
we obtained near real-time inference speed (around 25 FPS) in our environment.
Since we are interested in persons, vehicles, and wildlife, we trained the
model on COCO [12], a widely used detection dataset.

Our experiments found that even though SSD has comparable accuracy with two-
stage detectors on large and medium-sized objects, it struggles to detect
objects with smaller pixel-size consistently. The reason is that SSD only
regresses bounding boxes on downsampled feature maps in the last few layers of
the CNN backbone. However, it is impractical for us to increase the input size
or tweak the SSD architecture due to the hardware constraint.

Therefore, we utilize tiling as a workaround. We divide the $1280\times 720$
video frame into 6 $300\times 300$ tiles. The tiles overlap with each other by
$25\%$ to cover objects near the edges. Ideally, we should be able to process
all 6 tiles in parallel through batching. However, we are once again limited
by the compute constraint of our environment. As a result, we have to process
the tiles sequentially, and we introduce a simple attention mechanism by
weighting the tiles containing a large number of objects more than empty
tiles. To refrain from starvation, we also adopt an aging mechanism that keeps
track of the number of frames since the tile was last processed. Lastly, we
set the confidence threshold of SSD to $0.5$ to filter out possible false
positives.

###  B. Tracking

(a) Our tracking module

(b) Our detection module

Figure 4: Tracker Design

SORT [10] assumes that the detector runs at every frame and only relies on the
Kalman filter to process the detections, which is not realistic on an edge
device. Furthermore, we can process a tile $1/6$ of the time on average, or
worse, when the detector cannot achieve real-time inference because of
sequential tile processing. Therefore, we need to track objects from frame to
frame when SSD does not process a specific tile for a large number of frames.

Most correlation and CNN-based trackers are thus unfit for this purpose since
their speed scales poorly with the number of objects. Instead, we utilize
optical flow, a lightweight classical algorithm that tracks feature points on
the objects. For each object, we use ShiTomashi corners and sample a fixed
density of them inside the bounding box. We model bounding box transformation
as affine and compute a transformation matrix from optical flow feature
matches to estimate a new bounding box for the next frame. We also estimate a
homography transformation from background feature matches, which is helpful to
compensate for camera motion during Kalman filtering [13].

###  C. Motion Model

We adopt the standard Kalman filter and a constant velocity model to alleviate
optical flow’s undesirable performance when the targets are partially
occluded. Since Kalman filter models the motion of objects, the next bounding
box locations can be roughly predicted based on velocity when track
measurements are incorrect (occluded). Our Kalman filter state for each object
is defined on the eight dimensional state space:

$\boldsymbol{x}=\begin{bmatrix}x_{tl}&y_{tl}&x_{br}&y_{br}&\dot{x}_{tl}&\dot{y}_{tl}&\dot{x}_{br}&\dot{y}_{br}\end{bmatrix}^{T}$
(1)

where $x_{tl}$, $y_{tl}$ represent horizontal and vertical pixel position of
the top left corner of the bounding box, $x_{br}$, $y_{br}$ represent
horizontal and vertical pixel position of the bottom right corner of the
bounding box.
$\left(\dot{x}_{tl},\dot{y}_{tl},\dot{x}_{br},\dot{y}_{br}\right)$ are the
corresponding velocities. The track bounding boxes are used as measurements to
update the state of Kalman filter. Note that our measurement space only
consists of pixel positions, $\left(x_{tl},y_{tl},x_{br},y_{br}\right)$.

Using only the pixel positions of bounding box corners instead of its width
and height like [10] and [14] enables us to easily transform Kalman filter
state to compensate for camera motion. However, the two corners of the
bounding box are modeled independently, which can cause the bounding box to
drift away from its groundtruth size over time. To tackle this issue, we
update the positions of the two corners jointly when applying the constant-
velocity prediction:

$x_{tl}=x_{tl}+(\lambda\dot{x}_{tl}+(1-\lambda)\dot{x}_{br})\Delta t$ (2)
$y_{tl}=y_{tl}+(\lambda\dot{y}_{tl}+(1-\lambda)\dot{y}_{br})\Delta t$ (3)

where $\lambda$ is the coupling factor that measures the correlation between
the two corners. We update $x_{br}$ and $y_{br}$ in a similar fashion. Through
empirical experiment, we set $\lambda=0.6$.

Figure 5: Unrolled Tracker Pipeline ($N=3$) Figure 6: PID Control Errors

###  D. Data Association

To associate detections to tracks, we make use of both the state
$\boldsymbol{x}$ and the estimated covariance of the state $\boldsymbol{P}$.
We partially follow the approach in Deep SORT [14] by computing the squared
Mahalanobis distance between the Kalman filter state and the detection:

$d(i,j)=\left(\boldsymbol{d}_{j}-\boldsymbol{\hat{x}}_{i}\right)^{\mathrm{T}}\boldsymbol{\hat{P}}_{i}^{-1}\left(\boldsymbol{d}_{j}-\boldsymbol{\hat{x}}_{i}\right)$
(4)

where $(\boldsymbol{\hat{x}}_{i},\boldsymbol{\hat{P}}_{i})$ is the i-th track
state $(\boldsymbol{x}_{i},\boldsymbol{P}_{i})$ projected onto the measurement
space and $\boldsymbol{d}_{j}$ is the j-th bounding box detection. The
Mahalanobis distance factors in state uncertainty by measuring how many
standard deviations the detection is away from the mean track position.

We filter out unlikely associations with large $d(i,j)$ and threshold
Intersection Over Union (IOU) at 0.3. IOU measures the percentage of overlap
between the detection and track bounding boxes. The Hungarian algorithm is
used obtain the optimal $d(i,j)$ assignment. Finally, the remaining
unassociated detections are registered as new tracks.

Table 1: Cost Summary Item |  | Unit Cost (US$)
---
| Annual Cost (US$)
---
Custom UAV Kits |  |
Camera | 23.5 ($\times 3$) | 70.5
Flight Controller | 419 ($\times 3$) | 1257
Jetson Nano | 100 ($\times 3$) | 300
Frame | 170 ($\times 3$) | 510
Motors | 180 ($\times 3$) | 540
ESCs | 126 ($\times 3$) | 378
Propellers | 60 ($\times 3$) | 180
LiPo Battery | 83 ($\times 3$) | 249
Subtotal Kits | 1161.5 | 3484.5
Custom GPS Tags |  |
GPS Module | 13 ($\times 10$) | 130
Xbee Module | 25 ($\times 10$) | 250
SD Card | 14 ($\times 10$) | 140
Atmega328p | 2.5 ($\times 10$) | 25
Battery Bank | 15 ($\times 10$) | 150
Subtotal GPS Tags | 69.5 | 695
Elephant Anesthesia |  | 500
Total |  | 4679.5

###  E. Technical Approach

In this section, we provide a detailed overview of our visual tracking
algorithm. Figure 4(a) shows our tracking module, which is run at almost every
frame. For each target, optical flows takes as input the previous frame, the
current frame, and Kalman filter’s last state
$\left(\boldsymbol{x}_{i}^{n-1},\boldsymbol{P}_{i}^{n-1}\right)$ to estimate a
bounding box for the target in the current frame and outputs it as a new
track. Each track is associated with a track ID to identify the track. After
Kalman filter predicts a new state based on the constant-velocity model, the
new track will be used to update (correct) Kalman filter’s new state to
produce a final state output,
$\left(\boldsymbol{x}_{i}^{n},\boldsymbol{P}_{i}^{n}\right)$.

Figure 4(b) shows our detection module, which is run every $N$ frames due to
its large latency. The SSD detector takes as input the current frame and
outputs a set of detections. Each detection will be associated to a track
using the approach in D. Similarly, each track output by association will be
used to update Kalman filter’s predicted state to produce a final state
output, $\left(\boldsymbol{x}_{i}^{n},\boldsymbol{P}_{i}^{n}\right)$.

Figure 5 shows our unrolled tracker pipeline that combines both modules, where
the detection module is inserted every $3$ frames starting from $frame_{1}$.

###  F. Control

We incorporate proportional–integral–derivative (PID) controllers for yaw,
pitch, and roll velocities based on 2D bounding boxes output by our tracker.
In Figure 6, we define three control errors, $\Delta x$, $\Delta y$, and
$\Delta A$. $\Delta x$ and $\Delta y$ are the horizontal and vertical
distances, respectively, between the centroid of the bounding box and that of
the entire video frame. We also define a reference bounding box (blue) with a
fixed area. $\Delta A$ is the area difference between the current bounding box
and the reference bounding box, which roughly indicates a change in spatial
distance. We use $\Delta x$ to control yaw and roll, while $\Delta y$ and
$\Delta A$ are used to perform PID on the pitch of the UAV.

##  6 RESULTS

The proposed tracker achieves high tracking precision at 32 FPS on Jetson
Nano, which is fast enough to control the UAV in real-time. To evaluate the
control algorithm, in Figure 7, we simulate the flight trajectories in the DJI
Flight Simulator. To retrieve the ground truth 3D trajectory, we project the
2D image coordinates of the elephant of interest (green bounding box) in the
video to the ground plane. As a result, the trajectory of the UAV has 80%
match with the ground truth trajectory given an error margin of a 1-meter
radius.

Figure 7: Simulation Result

N. Hahn et al. investigated the cost of human-controlled UVAs on the borders
of Tanzanian Parks, which is about US$15,000 annually [8]. To cover the area
of interest with our solution, we estimate 10 GPS tags and 3 UAVs are required
on average per year. According to Table 1, our proposed autonomous system only
costs US$4679.5 annually, which saves over 68% cost compared with human-
controlled UAVs in mitigating EHC.

##  7 CONCLUSIONS

In this paper, we proposed an autonomous system that uses telemetry to track
and herd elephants away from the villages. We design and implement all three
parts of our solution: a compact custom low-power GPS tag, a receiver
stationed in the human living area, and an autonomous UAV. Our real-time
tracking algorithm achieves 32 FPS on the mobile edge device, NVIDIA Jetson
Nano. And our proposed autonomous system is highly cost-effective compared
with human-controlled solutions, saving more than 68% per year.

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