# Towards Continuous Time Finite Horizon LQR Control in SE(3)

Shivesh Kumar1, Andreas Mueller2, Patrick Wensing3 and Frank Kirchner1,4 This
work was supported by RIMA project (FKZ H2020-DT-2018-2020/H2020-DT-2018-1
#824990) funded by European Unions Horizon 2020 research and innovation
programme. The second author acknowledges the support of the LCM-K2 Center
within the framework of the Austrian COMET-K2 program.1Robotics Innovation
Center, German Research Center for Artificial Intelligence (DFKI GmbH),
Bremen, Germany<EMAIL_ADDRESS>of Robotics, Johannes Kepler
University, Linz, Austria<EMAIL_ADDRESS>of Aerospace and
Mechanical Engineering, University of Notre Dame, USA<EMAIL_ADDRESS>of Mathematics and Computer Science, University of Bremen, Germany
<EMAIL_ADDRESS>
###### Abstract

The control of free-floating robots requires dealing with several challenges.
The motion of such robots evolves on a continuous manifold described by the
Special Euclidean Group of dimension 3, known as $SE(3)$. Methods from finite
horizon Linear Quadratic Regulators (LQR) control have gained recent traction
in the robotics community. However, such approaches are inherently solving an
unconstrained optimization problem and hence are unable to respect the
manifold constraints imposed by the group structure of $SE(3)$. This may lead
to small errors, singularity problems and double cover issues depending on the
choice of coordinates to model the floating base motion. In this paper, we
propose the use of canonical exponential coordinates of $SE(3)$ and the
associated Exponential map along with its differentials to embed this
structure in the theory of finite horizon LQR controllers.

##  I Introduction

Methods from Lie Group and Screw theory are becoming increasingly popular in
the robotics community to describe rigid body kinematics and dynamics [1, 2].
They are equally popular in the domain of robot control [3]. Additionally, the
geometric framework has been used in time integration schemes for general
multi-body systems (MBS) [4] including the flexible MBS [5].

MBS motions evolve on a Lie group and their dynamics is naturally described by
differential equations on that Lie group. The most crucial Lie group for
studying rigid body motion is the Special Euclidean Group of dimension 3,
known as $SE(3)$. Its importance is shown by the fact that all possible rigid
body motions are captured by subgroups of $SE(3)$. For the purpose of robot
control or time integration of the robot dynamics, it is quite common to
consider a direct product of the translational group $\mathbb{R}^{3}$ and the
special orthogonal group $SO(3)$ i.e. $SO(3)\times\mathbb{R}^{3}$. This allows
one to use different parameterizations for rotation (e.g., Euler angles,
quaternions, etc.) and translation parts. This configuration space (c-space)
does, however, not account for the intrinsic geometry of rigid body motions in
that rotations and translations are decoupled. However, rigid body motions in
$SE(3)$ are mathematically defined as the semi-direct product between $SO(3)$
and $\mathbb{R}^{3}$ i.e. $SE(3)=SO(3)\ltimes\mathbb{R}^{3}$. In other words,
$SO(3)$ is its quotient or factor group and $\mathbb{R}^{3}$ is its normal
subgroup. In [6], it was shown that the naive use of
$SO(3)\times\mathbb{R}^{3}$ can lead to constraint violations during geometric
integration, which must be corrected by additional constraint stabilization
techniques. By contrast, using $SE(3)=SO(3)\ltimes\mathbb{R}^{3}$ can often
avoid such issues. Note that both $SE(3)$ and $SO(3)\times\mathbb{R}^{3}$
allow for the representation of rigid body configurations. But only $SE(3)$
allows for representing rigid body motions [6].

Similarly, improper treatment of $SE(3)$ leads to various challenges in robot
control. For example, the use of Euler angles to describe the rotation matrix
leads to singularity issues [7, 8]. The use of quaternions addresses the
singularity issue but the solutions may suffer from double cover issue i.e.
multiple quaternion solutions to represent the same rotation matrix [9].
Geometric control [3] attempts to unify the study of mechanics and control
under the setting of differential geometry. In [10], a time varying LQR
controller was developed using a variation-based linearization of the non-
linear systems evolving on various Lie groups such as $SO(3)$ and
$\mathbb{S}^{2}$. In [11], the framework of geometric control was adopted to
plan trajectories in $SO(3)\times\mathbb{R}^{3}$ with large attitude changes
by exploiting the Cayley map of $SO(3)$ which involves the use of its local
(non-canonical) coordinates. An extensive treatment of discrete-time
differential dynamic programming (DDP) on Lie groups has been recently
presented in [12]. Another recent work [13] presents an approach for
trajectory optimization on generic matrix Lie groups using an augmented
Lagrangian-based constrained discrete Differential Dynamic Programming.
Finally, [14] proposes the use of cost function design on the Lie algebra for
control on Lie Groups.

#### Contribution

This paper proposes the use of canonical coordinates (screw coordinates) of
$SE(3)$ to synthesize a finite horizon LQR controller for trajectory
stabilization. By exploiting the exponential map and its differentials, we
derive the linearization of the equation of motion (EOM) of a free-floating
rigid body. The proposed linearization is easy to implement as it avoids the
use of tensors by exploiting the directional derivative of the derivative of
the exponential map (dexp map). Based on this linearization, a time varying
LQR controller is developed. Note that we are currently evaluating the
proposed controller in some numerical experiments and the results will be
presented in the near future.

#### Organization

Section II presents the canonical screw coordinates of $SE(3)$ along with the
exponential map and its differentials. Section III presents the equation of
motion of a free-floating rigid body and its geometric linearization. Section
IV derives the corresponding finite horizon LQR controller. Section V
concludes the paper and highlights our future work.

##  II Canonical Coordinates on $SE(3)$, exp map and its Differentials

This section presents the fundamentals for describing rigid body motion in
$SE(3)$ in terms of canonical screw coordinates via the exponential map exp
and its differential dexp. Further, the directional derivative of the dexp
mapping is presented which is required for linearizing the EOM. For a detailed
treatment, refer to [15].

###  II-A Preliminaries

Let $G$ be a $n$-dimensional Lie group with Lie algebra $\mathfrak{g}$. Lie
algebra elements are denoted $\hat{\mathbf{X}}\in\mathfrak{g}$, and when
represented as vectors, they are denoted $\mathbf{X}\in{\mathbb{R}}^{n}$,
which implies an obvious isomorphism. Let $\exp:\mathfrak{g}\rightarrow G$ be
the exponential map on $G$. Its right-trivialized differential
$\mathrm{dexp}_{\hat{\mathbf{X}}}:\mathfrak{g}\rightarrow\mathfrak{g}$ is
defined as

$\left(\mathrm{D}_{\hat{\mathbf{X}}}\exp\right)(\hat{\mathbf{Y}})=\mathrm{dexp}_{\hat{\mathbf{X}}}(\hat{\mathbf{Y}})\exp(\hat{\mathbf{X}})$
(1)

where
$\left(\mathrm{D}_{\hat{\mathbf{X}}}\exp\right)(\hat{\mathbf{Y}}):=\frac{d}{dt}\exp(\hat{\mathbf{X}}+t\hat{\mathbf{Y}})|_{t=0}$
is the directional derivative
$\mathrm{D}_{\hat{\mathbf{X}}}\exp:\mathfrak{g}\rightarrow
T_{\exp\hat{\mathbf{X}}}G$ of $\exp$ at $\hat{\mathbf{X}}$ in direction of
$\hat{\mathbf{Y}}$. The differential and its inverse admit the series
expansions [16, pp. 26 & 36ff],[17],[18, Theorem 2.14.3.]

$\displaystyle\mathrm{dexp}_{\hat{\mathbf{X}}}(\hat{\mathbf{Y}})$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}\frac{1}{\left(i+1\right)!}\mathrm{ad}_{\hat{\mathbf{X}}}^{i}(\hat{\mathbf{Y}})$
(2) $\displaystyle\mathrm{dexp}_{\hat{\mathbf{X}}}^{-1}(\hat{\mathbf{Y}})$
$\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}\frac{B_{i}}{i!}\mathrm{ad}_{\hat{\mathbf{X}}}^{i}(\hat{\mathbf{Y}})$
(3)

where $B_{i}$ denote the Bernoulli numbers. In vector representation of
$\mathfrak{g}$, the differential mapping attains the form
$\mathbf{dexp}_{\mathbf{X}}\mathbf{Y}$, with matrix
$\mathbf{dexp}_{\mathbf{X}}$. This matrix and its inverse admit the series
expansions

$\displaystyle\mathbf{dexp}_{\hat{\mathbf{X}}}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}\frac{1}{\left(i+1\right)!}\mathbf{ad}_{\hat{\mathbf{X}}}^{i}$
(4) $\displaystyle\mathbf{dexp}_{\hat{\mathbf{X}}}^{-1}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}\frac{B_{i}}{i!}\mathbf{ad}_{\hat{\mathbf{X}}}^{i}.$
(5)

with the little adjoint matrix $\mathbf{ad}_{\mathbf{X}}$ such that
$\mathrm{ad}_{\hat{\mathbf{X}}}(\hat{\mathbf{Y}})=\mathbf{\widehat{\mathbf{ad}_{\mathbf{X}}\mathbf{Y}}}$.

###  II-B Rigid Body Motion in Terms of Exponential Map

Pose of the rigid body is expressed in terms of the canonical coordinates
$\mathbf{X}\in{\mathbb{R}}^{6}$ of the first kind with the exponential map
$\mathbf{C}\left(t\right)=\mathbf{C}_{0}\exp\hat{\mathbf{X}}\left(t\right)$,
where $\hat{\mathbf{X}}\in se\left(3\right)$. The closed form can be expressed
for $\mathbf{X}=\left(\mathbf{x},\mathbf{y}\right)$ as

$\displaystyle\exp(\hat{\mathbf{X}})$
$\displaystyle=\left(\begin{array}[]{cc}\mathbf{R}&\ \
\tfrac{1}{\left\|\mathbf{x}\right\|^{2}}(\mathbf{I}-\mathbf{R})\tilde{\mathbf{x}}\mathbf{y}+h\mathbf{y}\\\
\mathbf{0}&1\end{array}\right),\ \mathrm{for\ }\mathbf{x}\neq\mathbf{0}\ $ (8)
$\displaystyle=\left(\begin{array}[]{cc}\mathbf{I}&\ \mathbf{y}\\\
\mathbf{0}&1\end{array}\right),\ \mathrm{for\ }\mathbf{x}=\mathbf{0}$ (11)

where

$\mathbf{R}=\exp\tilde{\mathbf{x}}=\mathbf{I}+\alpha\tilde{\mathbf{x}}+\tfrac{1}{2}\beta\tilde{\mathbf{x}}^{2}$
(12)

with $\alpha:=\mathrm{sinc}\left\|\mathbf{x}\right\|,\ \
\beta:=\mathrm{sinc}^{2}\mathrm{\,}\frac{\left\|\mathbf{x}\right\|}{2}$. The
body twist in body-fixed representation is given in terms of the time
derivative of $\mathbf{X}$ by the _local reconstruction equation_ :

$\mathbf{V}=\mathbf{dexp}_{-\mathbf{X}}\dot{\mathbf{X}}$ (13)

where $\mathbf{dexp}_{\mathbf{X}}:\mathbb{R}^{6}\rightarrow\mathbb{R}^{6}$ is
the matrix form of the right-trivialized differential of the exp map. The
inverse relation is
$\dot{\mathbf{X}}=\mathbf{dexp}_{-\mathbf{X}}^{-1}\mathbf{V}$. For $SE(3)$, it
can be expressed in closed-form as

$\mathbf{dexp}_{\mathbf{X}}^{-1}=\left(\begin{array}[]{cc}\mathbf{dexp}_{\mathbf{x}}^{-1}&\
\ \mathbf{0}\\\
(\mathrm{D}_{\mathbf{x}}\mathbf{dexp}^{-1})\left(\mathbf{y}\right)&\mathbf{dexp}_{\mathbf{x}}^{-1}\end{array}\right)$
(14)

with

$\displaystyle(\mathrm{D}_{\mathbf{x}}\mathbf{dexp}^{-1})\left(\mathbf{y}\right)=-\tfrac{1}{2}\tilde{\mathbf{y}}+\frac{1}{\left\|\mathbf{x}\right\|^{2}}\left(1-\gamma\right)\left(\tilde{\mathbf{x}}\tilde{\mathbf{y}}+\tilde{\mathbf{y}}\tilde{\mathbf{x}}\right)+$
$\displaystyle\tfrac{\mathbf{x}^{T}\mathbf{y}}{\left\|\mathbf{x}\right\|^{4}}\left(\tfrac{1}{\beta}+\gamma-2\right)\tilde{\mathbf{x}}^{2}$

and $\gamma:=\alpha/\beta$ [15]. In vector representation
$\mathbf{X}=(\mathbf{x},\mathbf{y})\in\mathbb{R}^{6}$, the matrix form of the
adjoint operator for $SE(3)$ is

$\mathbf{ad}_{\mathbf{X}}=\left(\begin{array}[]{cc}\tilde{\mathbf{x}}&\mathbf{0}\\\
\tilde{\mathbf{y}}&\tilde{\mathbf{x}}\end{array}\right)\;.$ (15)

Using it, a computationally efficient and numerically stable version of (14)
was presented in [19, 20] and is given by

$\displaystyle\mathbf{dexp}_{\mathbf{X}}^{-1}$
$\displaystyle=\mathbf{I}-\frac{1}{2}\mathbf{ad}_{\mathbf{X}}+\tfrac{1}{\left\|\mathbf{x}\right\|^{2}}\left(2-\tfrac{1+3\alpha}{2\beta}\right)\mathbf{ad}_{\mathbf{X}}^{2}+$
(16)
$\displaystyle\tfrac{1}{\left\|\mathbf{x}\right\|^{4}}\left(1-\tfrac{1+\alpha}{2\beta}\right)\mathbf{ad}_{\mathbf{X}}^{4}\;.$

The singularity in (16) is removable by exploiting the limit
$\lim_{\lVert\mathbf{x}\rVert\to\mathbf{0}}\mathbf{dexp}_{\mathbf{X}}^{-1}=\mathbf{I}-\frac{1}{2}\mathbf{ad}_{\mathbf{X}}$.

###  II-C Differential of the dexp mapping

The directional derivative of the matrix $\mathbf{dexp}^{-1}$ for $SE(3)$ is

$\displaystyle(\mathrm{D}_{\mathbf{X}}\mathbf{dexp}^{-1})\left(\mathbf{U}\right)=$
$\displaystyle\left(\begin{array}[]{cc}(\mathrm{D}_{\mathbf{x}}\mathbf{dexp}^{-1})\left(\mathbf{u}\right)&\mathbf{0}\\\
(\mathrm{D}_{\mathbf{X}}\mathbf{Ddexp}^{-1})\left(\mathbf{U}\right)&(\mathrm{D}_{\mathbf{x}}\mathbf{dexp}^{-1})\left(\mathbf{u}\right)\end{array}\right)$
(19)

where $\mathbf{U}=\left(\mathbf{u},\mathbf{v}\right)$, and the directional
derivative of matrix $\mathbf{Ddexp}^{-1}$ possesses the explicit closed-form

$\displaystyle(\mathrm{D}_{\mathbf{X}}\mathbf{Ddexp}^{-1})\left(\mathbf{U}\right)=-\tfrac{1}{2}\tilde{\mathbf{v}}$
$\displaystyle+\tfrac{1-\gamma}{\left\|\mathbf{x}\right\|^{2}}\left(\tilde{\mathbf{x}}\tilde{\mathbf{v}}+\tilde{\mathbf{v}}\tilde{\mathbf{x}}+\tilde{\mathbf{y}}\tilde{\mathbf{u}}+\tilde{\mathbf{u}}\tilde{\mathbf{y}}+\tfrac{1}{4}(\mathbf{x}^{T}\mathbf{u})(\tilde{\mathbf{x}}\tilde{\mathbf{y}}+\tilde{\mathbf{y}}\tilde{\mathbf{x}})\right)$
$\displaystyle-\tfrac{1}{\left\|\mathbf{x}\right\|^{4}}\Big{(}\left(1-\gamma\right)(\mathbf{x}^{T}\mathbf{u})(2+\gamma)(\tilde{\mathbf{x}}\tilde{\mathbf{y}}+\tilde{\mathbf{y}}\tilde{\mathbf{x}})$
$\displaystyle+(2-\gamma-\tfrac{1}{\beta})(\mathbf{x}^{T}\mathbf{y})(\tilde{\mathbf{x}}\tilde{\mathbf{u}}+\tilde{\mathbf{u}}\tilde{\mathbf{x}})+(\mathbf{x}^{T}\mathbf{v}+\mathbf{y}^{T}\mathbf{u})$
$\displaystyle(2+\tfrac{1}{4}(\mathbf{x}^{T}\mathbf{y})(\mathbf{x}^{T}\mathbf{u})-\gamma-\tfrac{1}{\beta})\tilde{\mathbf{x}}^{2}\Big{)}$
$\displaystyle+\tfrac{1}{\left\|\mathbf{x}\right\|^{6}}(\mathbf{x}^{T}\mathbf{y})(\mathbf{x}^{T}\mathbf{u})\left(8-3\gamma-\gamma^{2}-\tfrac{2}{\beta^{2}}\left(\alpha+\beta\right)\right)\tilde{\mathbf{x}}^{2}\;.$

The above relation can be implemented to cope with
$\lVert\mathbf{x}\rVert=\mathbf{0}$ [15]. An equivalent expression for the
directional derivative of the matrix of the left-trivialized differential,
i.e. with negative argument $\mathbf{X}$, was derived in Appendix A.1 of [5].
Evaluating (19) additionally requires the directional derivative of
$\mathbf{dexp}^{-1}$ for $SO(3)$ which is given by

$\displaystyle(\mathrm{D}_{\mathbf{x}}\mathbf{dexp}^{-1})(\mathbf{y})=$
$\displaystyle-\frac{1}{2}\tilde{\mathbf{y}}+\frac{1}{{\lVert\mathbf{x}\rVert}^{2}}\left(1-\gamma\right)(\tilde{\mathbf{x}}\tilde{\mathbf{y}}+\tilde{\mathbf{y}}\tilde{\mathbf{x}})$
$\displaystyle+\frac{\mathbf{x}^{T}\mathbf{y}}{{\lVert\mathbf{x}\rVert}^{4}}\left(\frac{1}{\beta}+\gamma-2\right)\tilde{\mathbf{x}}^{2}\;.$
(20)

##  III Geometric Linearization of EOM of Floating-Base Systems

This section presents the equation of motion of a free-floating single rigid
body in $SE(3)$, its state space form, and the linearization for inclusion in
finite horizon LQR controllers. Note that we consider a fully actuated free-
floating rigid body. However, without any loss of generality, the derivation
presented below can be extended to underactuated free-floating systems by
including an actuator selection matrix.

###  III-A EOM of free-floating single rigid body

Let us consider a simple case of free-floating rigid body with its body fixed
reference (BFR) frame located at the center of mass (COM). The EOM of such a
rigid body in $SE(3)$ is given by

$\mathbf{W}=\mathbf{M}\dot{\mathbf{V}}+\mathbf{ad}_{\mathbf{V}}^{T}\mathbf{M}\mathbf{V}$
(21)

where $\mathbf{W}\in se^{*}(3)$ is the net wrench acting on the body,
$\mathbf{V}\in se(3)$ and $\dot{\mathbf{V}}\in\mathbb{R}^{6}$ represent the
twist and acceleration of the moving body respectively - all in body fixed
representation. $\mathbf{M}\in\mathcal{P}(6)$ denotes the $6\times 6$
symmetric and positive-definite mass-inertia matrix of the body with the
following form:

$\mathbf{M}=\left[\begin{matrix}\mathbf{I}_{b}&\mathbf{0}\\\
\mathbf{0}&m\mathbf{I}\\\ \end{matrix}\right]$ (22)

where $\mathbf{I}_{b}\in\mathcal{P}(3)$ is its rotational inertia and
$m\in\mathbb{R}^{+}$ is the mass of the moving body. The expression for
forward dynamics can be obtained by rearranging (21) as

$\dot{\mathbf{V}}=\mathbf{M}^{-1}\left(\mathbf{W}-\mathbf{ad}_{\mathbf{V}}^{T}\mathbf{M}\mathbf{V}\right)$
(23)

which is a $2^{\text{nd}}$ order ordinary differential equation (ODE).

###  III-B Dynamics in state-space representation

Let $\mathbf{S}=(\bm{\eta},\bm{\xi})^{T}\in\mathbb{R}^{6}$ denote the
canonical screw coordinates of $SE(3)$. Together with $\mathbf{V}$, one could
denote the state of the rigid body as
$\mathbf{x}=(\mathbf{S},\mathbf{V})^{T}\in\mathbb{R}^{12}$. Using (23) and
$\mathbf{V}=\mathbf{dexp}_{-\mathbf{S}}\dot{\mathbf{S}}$ (substitute
$\mathbf{X}=\mathbf{S}$ in (13)), its first order time derivative
$\dot{\mathbf{x}}\in\mathbb{R}^{12}$ is given by:

$\dot{\mathbf{x}}=\left[\begin{matrix}\dot{\mathbf{S}}\\\ \dot{\mathbf{V}}\\\
\end{matrix}\right]=\left[\begin{matrix}\mathbf{dexp}_{-\mathbf{S}}^{-1}\mathbf{V}\\\
\mathbf{M}^{-1}\left(\mathbf{W}-\mathbf{ad}_{\mathbf{V}}^{T}\mathbf{M}\mathbf{V}\right)\\\
\end{matrix}\right]=\mathbf{f}(\mathbf{x},\mathbf{W})$ (24)

which captures the dynamics of the free-floating rigid body in the form of a
$1^{\text{st}}$ order ODE. Note that any other choice of coordinates of
$SE(3)$ (Study parameters [21], dual quaternions [22]) would require
resolution of additional algebraic constraints typically leading to a
differential-algebraic equation (DAE).

###  III-C Linearization of the state-space dynamics

Considering the Taylor series expansion of (24), the system dynamics can be
linearized and written in the following state-space form:

$\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{W}$ (25)

where
$\mathbf{A}=\frac{\partial\mathbf{f}}{\partial\mathbf{x}}\in\mathbb{R}^{12\times
12}$ and
$\mathbf{B}=\frac{\partial\mathbf{f}}{\partial\mathbf{W}}\in\mathbb{R}^{12\times
6}$ are the partial derivatives of the dynamics
$\mathbf{f}(\mathbf{x},\mathbf{W})$ with respect to the state vector
$\mathbf{x}$ and wrench acting on the body $\mathbf{W}$ respectively. The
expression for the matrix $\mathbf{A}$ is given by

$\mathbf{A}=\left[\begin{matrix}\frac{\partial\left(\mathbf{dexp}^{-1}_{-\mathbf{S}}\mathbf{V}\right)}{\partial\mathbf{S}}&\mathbf{dexp}^{-1}_{-\mathbf{S}}\\\
\mathbf{0}_{6\times
6}&\mathbf{M}^{-1}\frac{\partial\left(\mathbf{ad}_{\mathbf{V}}^{T}\mathbf{M}\mathbf{V}\right)}{\partial\mathbf{V}}\\\
\end{matrix}\right]\;.$ (26)

Here the top-left block matrix
$\frac{\partial\left(\mathbf{dexp}^{-1}_{-\mathbf{S}}\mathbf{V}\right)}{\partial\mathbf{S}}$
requires the multiplication of first order partial differential of the inverse
of $\mathbf{dexp}$ mapping (which is a tensor) by $\mathbf{V}$. In order to
avoid the computation of this tensor quantity explicitly, one can exploit the
directional derivative expression in (19) with the basis vectors
$\mathbf{e}_{i}$ taken from the $i^{\text{th}}$ column of a $6\times 6$
identity matrix $\mathbf{I}_{6\times 6}$ (for $i\in\\{1,2,\ldots,6\\}$) as

$\displaystyle\frac{\partial\left(\mathbf{dexp}^{-1}_{-\mathbf{S}}\mathbf{V}\right)}{\partial\mathbf{S}}=$
$\displaystyle\left[\begin{matrix}(\mathrm{D}_{\mathbf{-S}}\mathbf{dexp}^{-1})\left(\mathbf{e}_{1}\right)\mathbf{V}&\ldots&(\mathrm{D}_{\mathbf{-S}}\mathbf{dexp}^{-1})\left(\mathbf{e}_{6}\right)\mathbf{V}\end{matrix}\right]_{6\times
6}\;.$

The top-right block matrix can be evaluated with (16). The bottom-right block
matrix
$\mathbf{M}^{-1}\frac{\partial\left(\mathbf{ad}_{\mathbf{V}}^{T}\mathbf{M}\mathbf{V}\right)}{\partial\mathbf{V}}$
requires the use of tensorial quantities but can be computed easily by
multiplying $\mathbf{M}^{-1}$ with the formula for

$\displaystyle\frac{\partial\left(\mathbf{ad}_{\mathbf{V}}^{T}\mathbf{M}\mathbf{V}\right)}{\partial\mathbf{V}}$
$\displaystyle=\mathbf{ad}_{\mathbf{V}}^{T}\mathbf{M}+$
$\displaystyle\left(\mathbf{ad}_{\mathbf{e}_{1}}^{T}\mathbf{M}\mathbf{V}\quad\mathbf{ad}_{\mathbf{e}_{2}}^{T}\mathbf{M}\mathbf{V}\quad\ldots\quad\mathbf{ad}_{\mathbf{e}_{6}}^{T}\mathbf{M}\mathbf{V}\right)\;.$

In the second matrix summand of the above formula, each matrix column
$\mathbf{ad}_{\mathbf{e}_{i}}^{T}\mathbf{M}\mathbf{V}$ is evaluated with the
basis vectors $\mathbf{e}_{i}$ as before. The expression of the matrix
$\mathbf{B}$ is simply given by

$\mathbf{B}=\left[\begin{matrix}\mathbf{0}\\\ \mathbf{M}^{-1}\\\
\end{matrix}\right]\;.$ (27)

##  IV Trajectory stabilization in $SE(3)$

Assume that pre-computed optimal state
$\mathbf{x}_{0}(t)=(\mathbf{S}_{0}(t),\mathbf{V}_{0}(t))^{T}$ and input
$(\mathbf{W}_{0}(t))$ trajectories for the floating base system are described
in terms of canonical coordinates (screw coordinates) on $SE(3)$. The error in
a local coordinate system relative to the nominal trajectory can be defined
as:

$\bar{\mathbf{x}}(t)=\mathbf{x}(t)-\mathbf{x}_{0}(t)\in\mathbb{R}^{12}\;,\quad\bar{\mathbf{W}}(t)=\mathbf{W}(t)-\mathbf{W}_{0}(t)\in\mathbb{R}^{6}$

The time derivative of the state error trajectory is

$\dot{\bar{\mathbf{x}}}(t)=\dot{\mathbf{x}}(t)-\dot{\mathbf{x}}_{0}(t)=\mathbf{f}(\mathbf{x}(t),\mathbf{W}(t))-\mathbf{f}(\mathbf{x}_{0}(t),\mathbf{W}_{0}(t))$

which can be approximated by $1^{\text{st}}$ order Taylor series approximation
as

$\displaystyle\dot{\bar{\mathbf{x}}}(t)$
$\displaystyle\approx\mathbf{f}(\mathbf{x}_{0}(t),\mathbf{W}_{0}(t))+\frac{\partial\mathbf{f}(\mathbf{x}_{0}(t),\mathbf{W}_{0}(t))}{\partial\mathbf{x}}(\mathbf{x}(t)-\mathbf{x}_{0}(t))$
$\displaystyle+\frac{\partial\mathbf{f}(\mathbf{x}_{0}(t),\mathbf{W}_{0}(t))}{\partial\mathbf{W}}(\mathbf{W}(t)-\mathbf{W}_{0}(t))-\mathbf{f}(\mathbf{x}_{0}(t),\mathbf{W}_{0}(t))$
$\displaystyle=\mathbf{A}(t)\bar{\mathbf{x}}(t)+\mathbf{B}(t)\bar{\mathbf{u}}(t)$

where $\mathbf{A}(t)$ and $\mathbf{B}(t)$ can be evaluated with (26) and (27)
respectively. Note that the linearization is time varying and is evaluated
along the nominal trajectory.

Considering a quadratic form on the trajectory following cost

$J=\bar{\mathbf{x}}^{T}(t)\mathbf{Q}_{f}\bar{\mathbf{x}}(t)+\int_{0}^{t_{f}}\left(\bar{\mathbf{x}}^{T}(t)\mathbf{Q}\bar{\mathbf{x}}(t)+\bar{\mathbf{W}}(t)\mathbf{R}\bar{\mathbf{W}}(t)\right)dt$

with $\mathbf{Q}\succeq 0,\mathbf{Q}_{f}\succeq 0$ and $\mathbf{R}\succ 0$,
the optimal tracking controller can be derived by solving differential Riccati
equation

$\displaystyle-\dot{\mathbf{S}}(t)=$
$\displaystyle{\mathbf{S}}(t){\mathbf{A}}(t)+{\mathbf{A}}^{T}(t){\mathbf{S}}(t)$
$\displaystyle-{\mathbf{S}}(t)\mathbf{B}(t){\mathbf{R}}^{-1}\mathbf{B}^{T}(t){\mathbf{S}}(t)+{\mathbf{Q}},$
(28)

with the terminal condition

$\mathbf{S}(t_{f})=\mathbf{Q}_{f}\;.$ (29)

The resulting optimal controller takes the linear form
$\bar{\mathbf{W}}^{*}(t)=-\mathbf{K}(t)\bar{\mathbf{x}}(t)$, or:

$\mathbf{W}^{*}(t)=\mathbf{W}_{0}(t)-\mathbf{K}(t)(\dot{\mathbf{x}}(t)-\dot{\mathbf{x}}_{0}(t))$
(30)

and the corresponding optimal cost-to-go function is given by

$J^{*}(\mathbf{x}(t))=\bar{\mathbf{x}}^{T}(t)\mathbf{S}(t)\bar{\mathbf{x}}(t)\;.$
(31)

Note that it is crucial to respect the symmetric and positive semi-definite
structure of the $\mathbf{S}(t)$ matrix in order to avoid numerical errors in
solving (28). This can be done by using the square-root factorization of the
matrix $\mathbf{S}(t)=\mathbf{P}^{T}(t)\mathbf{P}(t)$ as discussed in [23]. An
alternative approach to ensure this is to use symplectic integrators as shown
in [24].

##  V Conclusion and Outlook

This paper presents a novel geometric linearization of the equations of motion
of a free-floating rigid body in $SE(3)$ in terms of its canonical screw
coordinates. This linearization exploits the differential of the exponential
map and its directional derivative in order to compute the involved partial
derivatives. Subsequently, we use this linearization to develop a finite
horizon LQR controller which can be used to locally stabilize any pre-computed
optimal trajectory. The next step includes numerical validation of the
approach presented in this paper. In the near future, we also plan to extend
our work to other similar control approaches, such as iterative LQR (iLQR) and
differential dynamic programming (DDP).

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# Even the photon propagator must break de Sitter symmetry

Dražen Glavan<EMAIL_ADDRESS>CEICO, Institute of Physics of the Czech Academy
of Sciences, Na Slovance 1999/2, 182 21 Prague 8, Czech Republic  Tomislav
Prokopec<EMAIL_ADDRESS>Institute for Theoretical Physics, Spinoza
Institute & EMME$\Phi$, Utrecht University, Buys Ballot Building,
Princetonplein 5, 3584 CC Utrecht, The Netherlands

###### Abstract

The propagator for the massless vector field in de Sitter space cannot
maintain de Sitter invariance in the general covaraint gauge, except in the
exactly transverse gauge limit. This is due to a previously overlooked Ward-
Takahashi identity that the propagator must satisfy. Here we construct the
propagator that satisfies all the conditions of a consistently quantized
theory. Our solution preserves cosmological symmetries and dilations, but
breaks spatial special conformal transformations. The solution amounts to
adding a homogeneous de Sitter breaking term to previously reported de Sitter
invariant solutions of the propagator equation of motion. Even though the
corrections we report pertain to the gauge sector of the linear theory, they
are relevant and have to be accounted for when interactions are included.

Introduction. Understanding interacting quantum field theory in de Sitter
space is of paramount importance for apprehending the physics of the
primordial inflationary phase of the Universe. Computations of quantum loop
corrections in realistic slow-roll inflation are still prohibitively
difficult, so simplifications are necessary. The de Sitter space is often
taken as an appropriate idealization for two reasons: (i) it is close enough
to slow-roll inflation that is phenomenologically relevant, and (ii) it is a
maximally symmetric space. It is the latter that is very often useful when
describing physical systems – more symmetric they are the simpler the
description. Such is our experience in Minkowski space, where Poincaré
invariance provides an efficient organizational principle for computations. It
is often assumed that de Sitter symmetries provide the same level of
simplifications and organize the computations in an economical manner. Even
though adhering to symmetries is the right approach in many circumstances, it
must not be taken for granted in de Sitter space.

Two-point functions of free fields are essential ingredients for perturbative
computations in quantum field theory. In maximally symmetric spaces it is
natural to assume that they respect symmetries of the background spacetime.
However, it has long been known that issues with this approach arise already
for arguably the simplest system of minimally coupled, massless scalar (MMCS),
whose propagator satisfies the equation of motion,

$\sqrt{-g}\,\kern 1.0pt\vbox{\hrule height=0.4pt\hbox{\vrule width=0.4pt\hskip
3.0pt\vbox{\vskip 6.0pt}\hskip 3.0pt\vrule width=0.9pt}\hrule
height=0.9pt}\kern
1.0pt\,i\Delta(x;x^{\prime})=i\delta^{D}(x\\!-\\!x^{\prime})\,,$ (1)

where $\kern 1.0pt\vbox{\hrule height=0.4pt\hbox{\vrule width=0.4pt\hskip
3.0pt\vbox{\vskip 6.0pt}\hskip 3.0pt\vrule width=0.9pt}\hrule
height=0.9pt}\kern 1.0pt\\!=\\!g^{\mu\nu}\nabla_{\\!\mu}\nabla_{\\!\nu}$
denotes the d’Alembertian (wave) operator. Even though this equation is
invariant under de Sitter symmetries, it does not admit a de Sitter invariant
solution with the appropriate singularity structure [1, 2] in any number $D$
of spacetime dimensions. This letter is devoted to pointing out a similar, but
more subtle, obstruction to maintaining de Sitter symmetry for the massless
vector propagator in the general covariant gauge. Even though the equation of
motion does allow a de Sitter invariant solution, it is the Ward-Takahashi
identity, overlooked thus far for propagators of massless vector fields, that
prevents a de Sitter invariant solution. Here we present a solution for the
photon propagator in $D$-dimensional spacetime, appropriate for dimensionally
regulated quantum loop computations, that accounts for both the equation of
motion and the Ward-Takahashi identity.

Photon in the general covariant gauge. The physical photon is conformally
coupled to gravity in four dimensions, which is no longer true in
$D$-dimensional spacetime,

$S[A_{\mu}]=\int\\!d^{D\\!}x\,\sqrt{-g}\,\biggl{[}-\frac{1}{4}g^{\mu\rho}g^{\nu\sigma}F_{\mu\nu}F_{\rho\sigma}\biggr{]}\,,$
(2)

where $F_{\mu\nu}\\!=\\!\partial_{\mu}A_{\nu}\\!-\\!\partial_{\nu}A_{\mu}$ is
the field strength tensor of the vector potential $A_{\mu}$. We consider the
spacetime to be the expanding Poincaré patch of de Sitter where the metric,
$g_{\mu\nu}\\!=\\!a^{2}(\eta)\eta_{\mu\nu}$, is conformal to the Minkowski
space metric, $\eta_{\mu\nu}\\!=\\!{\rm diag}(-1,1,\dots,1)$, $g\\!=\\!{\rm
det}(g_{\mu\nu})$, and $a(\eta)\\!=\\!1/[1\\!-\\!H(\eta\\!-\\!\eta_{0})]$ is
the scale factor expressed in terms of conformal time $\eta$ and a constant
Hubble parameter $H$. We shall not consider symmetry breaking theories, in
which the vector field can acquire a mass, nor shall we consider spatially
compact global coordinates on de Sitter where the problem of linearization
instability arises [3].

Quantization of the theory requires fixing a gauge. Choosing the general
covariant gauge,

$S_{\rm
gf}[A_{\mu}]=\int\\!d^{D\\!}x\,\sqrt{-g}\,\biggl{[}-\frac{1}{2\xi}\bigl{(}\nabla^{\mu}\\!A_{\mu}\bigr{)}^{\\!2}\biggr{]}\,,$
(3)

allows to maintain de Sitter symmetries of the dynamics. However, it breaks
conformal coupling of the photon even in four spacetime dimensions, and for
any choice of the gauge-fixing parameter $\xi$.

Central objects in perturbative nonequilibrium quantum field theory are the
two-point functions determined from the free theory. The relevant ones are the
Feynman propagator and the positive frequency Wightman function,

$\displaystyle
i\bigl{[}\tensor{*}[_{\mu}]{\Delta}{{}_{\nu}}\bigr{]}(x;x^{\prime})={}$
$\displaystyle\bigl{\langle}\Omega\bigr{|}\mathcal{T}\big{[}\hat{A}_{\mu}(x)\hat{A}_{\nu}(x^{\prime})\big{]}\bigl{|}\Omega\bigr{\rangle}\,,$
(4) $\displaystyle
i\bigl{[}\tensor{*}[_{\mu}^{\scriptscriptstyle-\\!}]{\Delta}{{}_{\nu}^{\scriptscriptstyle\\!+}}\bigr{]}(x;x^{\prime})={}$
$\displaystyle\bigl{\langle}\Omega\bigr{|}\hat{A}_{\mu}(x)\hat{A}_{\nu}(x^{\prime})\bigl{|}\Omega\bigr{\rangle}\,,$
(5)

where $\bigl{|}\Omega\bigr{\rangle}$ is the state, $\mathcal{T}$ stands for
time ordering, and $\hat{A}_{\mu}(x)$ is the free photon field operator. The
propagator equation of motion in this gauge is,

$\sqrt{-g}\,\mathcal{D}^{\mu\nu}\,i\bigl{[}\tensor{*}[_{\nu}]{\Delta}{{}_{\alpha}}\bigr{]}(x;x^{\prime})=\delta^{\mu}_{\alpha}\,i\delta^{D}(x\\!-\\!x^{\prime})\,,$
(6)

where we set $\hbar=1$, and the kinetic operator is,

$\mathcal{D}^{\mu\nu}=g^{\mu\nu}\kern 1.0pt\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt\hskip 3.0pt\vbox{\vskip 6.0pt}\hskip
3.0pt\vrule width=0.9pt}\hrule height=0.9pt}\kern
1.0pt-\Bigl{(}1\\!-\\!\frac{1}{\xi}\Bigr{)}\nabla^{\mu}\nabla^{\nu}-R^{\mu\nu}\,,$
(7)

where $R^{\mu\nu}\\!=\\!(D\\!-\\!1)H^{2}g^{\mu\nu}$ is the Ricci tensor in de
Sitter. Due to the exchange symmetry $(\mu,x)\leftrightarrow(\nu,x^{\prime})$,
the photon propagator (4) obeys an equation analogous to (6) on the other leg
$(\nu,x^{\prime})$. The Wightman function obeys the same equation of motion
(6), but without the local term on the right-hand-side.

The photon two-point functions in covariant gauges (3) have been considered in
several works over the last decades, starting from the seminal work of Allen
and Jacobson [4], who reported many results, among which the de Sitter space
covariant gauge propagator for $\xi\\!=\\!1$ in $D$ spacetime dimensions.
Subsequent works have extended and generalized this result. Tsamis and Woodard
[5] reported the transverse massive vector propagator, whose massless limit
reduces to the photon propagator in the Landau gauge ($\xi\\!\to\\!0$) in $D$
dimensions; Youssef [6] computed the propagator for arbitrary $\xi$ in
$D\\!=\\!4$ spacetime dimensions; and Fröb and Higuchi [7] reported the result
for a massive vector propagator, with the massless limit producing the photon
propagator for arbitrary $\xi$ and arbitrary $D$. The last of these
encompasses all the previously reported results as special cases. Only
propagators from [4] and [5] were used for loop computations in de Sitter. The
former was rederived and used to study scalar electrodynamics [10], while the
latter was utilized for both scalar electrodynamics [8, 9, 10, 11], and for
quantum gravity interacting with electromagnetism [12, 13].

In this letter we are interested only in the massless vector (photon)
propagators. All of the photon propagators reported in previous works satisfy
the equation of motion (6). However, it was pointed out recently in [15] that
photon propagators should satisfy additional subsidiary conditions dictated by
the consistent canonical quantization in average/multiplier gauges. Among them
is the condition that the double divergence of both the Feynman propagator and
the Wightman function should vanish off-coincidence. But the reported results
violate this condition, 111 The Wightman function exhibits the same problem as
in (8), except that the local term on the right-hand-side is absent.

$\nabla^{\mu}\nabla^{\prime\nu}i\bigl{[}\tensor{*}[_{\mu}]{\Delta}{{}_{\nu}}\bigr{]}(x;x^{\prime})\overset{!}{=}-\xi\frac{i\delta^{D}(x\\!-\\!x^{\prime})}{\sqrt{-g}}-\xi\frac{H^{D}\,\Gamma(D)}{(4\pi)^{\frac{D}{2}}\,\Gamma\bigl{(}\frac{D}{2}\bigr{)}}\,,$
(8)

except in the limit $\xi\\!\to\\!0$ when the offending term vanishes,
suggesting than only the Tsamis-Woodard result is consistent. This is a
problem that needs to be addressed. In the companion paper [16] to this letter
we consider the problem from the first principles of canonical quantization
and construct the photon propagator as a sum over modes. This letter is
devoted to resolving the problem in (8) in an elegant manner by considering
the Becchi-Rouet-Stora-Tyutin (BRST) quantization [17]. Our approach is
similar to the one employed in [18] for studying retarded and advanced Green’s
functions for massive vector fields. However, the problem in (8) is not
encountered for these Green’s functions, as for them the problematic term is
absent.

BRST quantization. In addition to introducig the gauge-fixing term (3), the
BRST formalism requires the inclusion of the Faddeev-Popov ghost action for
Grassmann fields $c$ and $\overline{c}$,

$S_{\rm
gh}[c,\overline{c}]=\int\\!d^{D\\!}x\sqrt{-g}\,g^{\mu\nu}\bigl{(}\nabla_{\mu}\overline{c}\bigr{)}\bigl{(}\nabla_{\nu}c\bigr{)}\,,$
(9)

to the complete gauge-fixed action, $S_{\star}\\!=\\!S\\!+\\!S_{\rm
gf}\\!+\\!S_{\rm gh}$, which is invariant under infinitesimal BRST
transformations,

$A_{\mu}\to A_{\mu}+\theta\xi\partial_{\mu}c\,,\quad\
\overline{c}\to\overline{c}-\theta\nabla^{\mu}A_{\mu}\,,\quad\ c\to c\,,$ (10)

parametrized by $\theta$, that are generated by the associated conserved BRST
charge,

${\rm
Q}=\int\\!d^{D-1}x\,a^{D-4}\Bigl{[}a^{2}\bigl{(}\nabla^{\mu}\\!A_{\mu}\bigr{)}\partial_{0}c+\xi
F_{0i}\partial_{i}c\,\Bigr{]}\,.$ (11)

The ghost propagator,
$i\Delta_{c}(x;x^{\prime})\\!=\\!\bigl{\langle}\Omega\bigl{|}\mathcal{T}\,\bigl{[}\hat{\overline{c}}(x)\hat{c}(x^{\prime})\bigr{]}\bigr{|}\Omega\bigr{\rangle}$,
satisfies,

$\sqrt{-g}\,\kern 1.0pt\vbox{\hrule height=0.4pt\hbox{\vrule width=0.4pt\hskip
3.0pt\vbox{\vskip 6.0pt}\hskip 3.0pt\vrule width=0.9pt}\hrule
height=0.9pt}\kern
1.0pt\,i\Delta_{c}(x;x^{\prime})=i\delta^{D}(x\\!-\\!x^{\prime})\,.$ (12)

This equation implies that the ghost propagator equals the MMCS propagator,
$i\Delta_{c}(x;x^{\prime})\\!=\\!i\Delta(x;x^{\prime})$. Thus, the ghost
propagator must break de Sitter symmetry. The natural choice for the MMCS
propagator in the Poincaré patch of de Sitter is the one preserving
cosmological symmetries, but breaking dilations and special spatial conformal
transformations [19]. For our purposes it is best to write it as a limit [20],

$i\Delta(x;x^{\prime})=\lim_{\lambda\to\nu+1}i\Delta_{\lambda}(x;x^{\prime})\,,\qquad\nu=\frac{D\\!-\\!3}{2}\,,$
(13)

of a propagator with an effectively slightly tachyonic mass
$M^{2}\\!=\\![(D\\!-\\!1)^{2}/4\\!-\\!\lambda^{2}]H^{2}\\!<\\!0$,

$i\Delta_{\lambda}(x;x^{\prime})=\mathcal{F}_{\lambda}(y)+\mathcal{W}_{\lambda}(u)\,,$
(14)

which consists of a de Sitter invariant part,

$\displaystyle\mathcal{F}_{\lambda}(y)={}$
$\displaystyle\frac{H^{D-2}}{(4\pi)^{\frac{D}{2}}}\frac{\Gamma\bigl{(}\frac{D-1}{2}\\!+\\!\lambda\bigr{)}\,\Gamma\bigl{(}\frac{D-1}{2}\\!-\\!\lambda\bigr{)}}{\Gamma\bigl{(}\frac{D}{2}\bigr{)}}$
$\displaystyle\hskip
14.22636pt\times{}_{2}F_{1}\Bigl{(}\tfrac{D-1}{2}\\!+\\!\lambda,\tfrac{D-1}{2}\\!-\\!\lambda,\tfrac{D}{2},1\\!-\\!\tfrac{y}{4}\Bigr{)}\,,$
(15)

dependent on a de Sitter invariant distance,

$y\\!=\\!aa^{\prime}H^{2}\Bigl{[}\|\vec{x}\\!-\\!\vec{x}^{\,\prime}\|^{2}\\!-\\!\bigl{(}|\eta\\!-\\!\eta^{\prime}|\\!-\\!i\varepsilon\bigr{)}^{\\!2}\Bigr{]}\,,$
(16)

and the de Sitter breaking part [20],

$\mathcal{W}_{\lambda}=\frac{H^{D-2}}{(4\pi)^{\frac{D}{2}}}\frac{\Gamma(2\lambda)\,\Gamma(\lambda)}{\Gamma\bigl{(}\frac{D-1}{2}\bigr{)}\,\Gamma\bigl{(}\frac{1}{2}\\!+\\!\lambda\bigr{)}}\frac{e^{(\lambda-\frac{D-1}{2})u}}{\lambda\\!-\\!\frac{D-1}{2}}\Bigl{(}\frac{k_{0}}{H}\Bigr{)}^{\\!D-1-2\lambda},\,$
(17)

dependent on $u\\!=\\!\ln(aa^{\prime})$, where $k_{0}<H$ is some infrared
scale.

Subsidiary condition. We derive the subsidiary condition for the photon
propagator [21] by considering the expectation value of an anticommutator of
the BRST charge operator $\hat{\rm Q}$ in Eq. (11), with a judiciously chosen
product of the vector potential and anti-ghost operators,

$\bigl{\\{}\hat{\rm
Q},\hat{\overline{c}}(x)\hat{A}_{\nu}(x^{\prime})\bigr{\\}}=i\Bigl{[}\nabla^{\mu}\\!\hat{A}_{\mu}(x)\hat{A}_{\nu}(x^{\prime})+\xi\,\hat{\overline{c}}(x)\partial^{\prime}_{\nu}\hat{c}(x^{\prime})\Bigr{]}\,.$
(18)

Since the BRST charge operator annihilates physical states, $\hat{\rm
Q}\bigl{|}\Omega\bigr{\rangle}\\!=\\!0$, the expectation value of this
anticommutator must vanish. This produces the desired subsidiary condition,

$\nabla^{\mu}\,i\bigl{[}\tensor{*}[_{\mu}]{\Delta}{{}_{\nu}}\bigr{]}(x;x^{\prime})=-\xi\partial^{\prime}_{\nu}\,i\Delta(x;x^{\prime})\,,$
(19)

relating the photon propagator to the Faddeev-Popov ghost propagator. The
photon propagator must therefore satisfy both the equation of motion (6) and
the subsidiary condition (19). The latter has seemingly gone unnoticed thus
far, apart from the exact transverse limit $\xi\\!\to\\!0$ [5]. This
subsidiary condition is the Ward-Takahashi identity of the free theory. It is
the massless limit of the previously derived identity for massive vector
fields [18] adapted to the Feynman propagator and Wightman functions. The
crucial observation is that this condition does not admit a de Sitter
invariant solution for the photon propagator, and the reason behind it is the
MMCS propagator appearing on the left hand side. After the derivative is acted
on it,

$\displaystyle\nabla^{\mu}i\bigl{[}\tensor{*}[_{\mu}]{\Delta}{{}_{\nu}}\bigr{]}(x;x^{\prime})={}$
$\displaystyle-\xi\bigl{(}\partial^{\prime}_{\nu}y\bigr{)}\frac{\partial\mathcal{F}_{\nu+1}}{\partial
y}$
$\displaystyle-\xi\bigl{(}\partial^{\prime}_{\nu}u\bigr{)}\frac{H^{D-1}\,\Gamma(D\\!-\\!1)}{(4\pi)^{\frac{D}{2}}\,\Gamma\bigl{(}\frac{D}{2}\bigr{)}},$
(20)

the right hand side of the subsidiary condition still breaks de Sitter
symmetry. It preserves homogeneity, isotropy, and dilations, but it breaks
spatial special conformal transformations.

Solving for the propagator. Consider first the equation of motion (6), which
simplifies upon plugging in the Ward-Takahashi identity (19) for the middle
term,

$\displaystyle\Bigl{[}\kern 1.0pt\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt\hskip 3.0pt\vbox{\vskip 6.0pt}\hskip 3.0pt\vrule
width=0.9pt}\hrule height=0.9pt}\kern
1.0pt-(D\\!-\\!1)H^{2}\Bigr{]}i\bigl{[}\tensor{*}[_{\mu}]{\Delta}{{}_{\nu}}\bigr{]}(x;x^{\prime})$
$\displaystyle\hskip
28.45274pt=g_{\mu\nu}\frac{i\delta^{D}(x\\!-\\!x^{\prime})}{\sqrt{-g}}+(1\\!-\\!\xi)\partial_{\mu}\partial^{\prime}_{\nu}i\Delta(x;x^{\prime})\,.$
(21)

Upon decomposing the propagator, 222 The Ward-Takahashi identity (19) forbids
contributions in (22) that are transverse on one leg and longitudinal on the
other.

$i\bigl{[}\tensor{*}[_{\mu}]{\Delta}{{}_{\nu}}\bigr{]}(x;x^{\prime})=i\bigl{[}\tensor{*}[_{\mu}]{\Delta}{{}^{\scriptscriptstyle\\!T}_{\nu}}\bigr{]}(x;x^{\prime})+i\bigl{[}\tensor{*}[_{\mu}]{\Delta}{{}^{\scriptscriptstyle\\!L}_{\nu}}\bigr{]}(x;x^{\prime})\,,$
(22)

into a transverse part,

$\nabla^{\mu}\,i\bigl{[}\tensor{*}[_{\mu}]{\Delta}{{}^{\scriptscriptstyle\\!T}_{\nu}}\bigr{]}(x;x^{\prime})=\nabla^{\prime\nu}\,i\bigl{[}\tensor{*}[_{\mu}]{\Delta}{{}^{\scriptscriptstyle\\!T}_{\nu}}\bigr{]}(x;x^{\prime})=0\,,$
(23)

and a longitudinal part,

$i\bigl{[}\tensor{*}[_{\mu}]{\Delta}{{}^{\scriptscriptstyle\\!L}_{\nu}}\bigr{]}(x;x^{\prime})=\partial_{\mu}\partial^{\prime}_{\nu}L(x;x^{\prime})\,,$
(24)

the equation of motion (6) breaks up into the transverse,

$\displaystyle\Bigl{[}\kern 1.0pt\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt\hskip 3.0pt\vbox{\vskip 6.0pt}\hskip 3.0pt\vrule
width=0.9pt}\hrule height=0.9pt}\kern
1.0pt-(D\\!-\\!1)H^{2}\Bigr{]}i\bigl{[}\tensor{*}[_{\mu}]{\Delta}{{}^{\scriptscriptstyle\\!T}_{\nu}}\bigr{]}(x;x^{\prime})$
$\displaystyle\hskip
42.67912pt=g_{\mu\nu}\frac{i\delta^{D}(x\\!-\\!x^{\prime})}{\sqrt{-g}}+\partial_{\mu}\partial^{\prime}_{\nu}i\Delta(x;x^{\prime})\,,$
(25)

and the longitudinal equation,

$\partial_{\mu}\partial^{\prime}_{\nu}\kern 1.0pt\vbox{\hrule
height=0.4pt\hbox{\vrule width=0.4pt\hskip 3.0pt\vbox{\vskip 6.0pt}\hskip
3.0pt\vrule width=0.9pt}\hrule height=0.9pt}\kern
1.0ptL(x;x^{\prime})=-\xi\partial_{\mu}\partial^{\prime}_{\nu}i\Delta(x;x^{\prime})\,.$
(26)

The Ward-Takahashi identity (19) constrains only the longitudinal part,

$\partial^{\prime}_{\nu}\kern 1.0pt\vbox{\hrule height=0.4pt\hbox{\vrule
width=0.4pt\hskip 3.0pt\vbox{\vskip 6.0pt}\hskip 3.0pt\vrule
width=0.9pt}\hrule height=0.9pt}\kern
1.0ptL(x;x^{\prime})=-\xi\,\partial^{\prime}_{\nu}i\Delta(x;x^{\prime})\,.$
(27)

The transverse equation (25) is (the massless limit of) the equation solved by
Tsamis and Woodard [5], and corresponds to the propagator in the
$\xi\\!\to\\!0$ limit. Even though the MMCS propagator in the source on the
right-hand side of (25) breaks de Sitter symmetry, the two derivatives acting
on it annihilate the de Sitter breaking part.

The longitudinal equation is, interestingly, a derivative of the subsidiary
condition (19). Thus, any solution satisfying the subsidiary condition will
automatically satisfy the longitudinal equation of motion (27). But,
importantly, not all solutions of the equation of motion will satisfy the
subsidiary condition! It is the second derivative in the longitudinal equation
that enables a de Sitter-invariant solution for the longitudinal part.
However, as can be seen from (20), the Ward-Takahashi identity (19)
necessitates breaking of de Sitter symmetry.

One solves Eq. (27) by requiring,

$\kern 1.0pt\vbox{\hrule height=0.4pt\hbox{\vrule width=0.4pt\hskip
3.0pt\vbox{\vskip 6.0pt}\hskip 3.0pt\vrule width=0.9pt}\hrule
height=0.9pt}\kern 1.0ptL(x;x^{\prime})=-\xi\,i\Delta(x;x^{\prime})\,,$ (28)

which is the equation for the so-called integrated propagator, that is solved
by [22],

$L(x;x^{\prime})=\frac{\xi}{2\lambda}\frac{\partial}{\partial\lambda}i\Delta_{\lambda}(x;x^{\prime})\bigg{|}_{\lambda\to\nu+1}\,.$
(29)

Combining the transverse part worked out in [5] with the longitudinal part
worked out here, the propagator in the general covariant gauge can be written
in a convenient covariant basis that emphasizes our main point,

$\displaystyle
i\bigl{[}\tensor{*}[_{\mu}]{\Delta}{{}_{\nu}}\bigr{]}(x;x^{\prime})={}$
$\displaystyle\bigl{(}\partial_{\mu}\partial^{\prime}_{\nu}y\bigr{)}\,\mathcal{C}_{1}+\bigl{(}\partial_{\mu}y\bigr{)}\bigl{(}\partial^{\prime}_{\nu}y\bigr{)}\,\mathcal{C}_{2}$
$\displaystyle+\bigl{(}\partial_{\mu}u\bigr{)}\bigl{(}\partial^{\prime}_{\nu}u\bigr{)}\,\mathcal{C}_{4}\,.$
(30)

The first two terms are composed out of de Sitter invariant tensor structures
multiplied by de Sitter invariant scalar structure functions, which depend on
$y$ only,

$\displaystyle\mathcal{C}_{1}={}$ $\displaystyle\frac{-1}{2\nu
H^{2}}\biggl{[}\Bigl{(}\nu\\!+\\!\frac{1}{2}\Bigr{)}\mathcal{F}_{\nu}+\Bigl{(}1\\!-\\!\frac{\xi}{\xi_{s}}\Bigr{)}\frac{\partial}{\partial
y}\frac{\partial}{\partial\nu}\mathcal{F}_{\nu+1}\biggr{]}\,,$ (31)
$\displaystyle\mathcal{C}_{2}={}$ $\displaystyle\frac{-1}{2\nu
H^{2}}\biggl{[}\frac{1}{2}\frac{\partial}{\partial
y}\mathcal{F}_{\nu}+\Bigl{(}1\\!-\\!\frac{\xi}{\xi_{s}}\Bigr{)}\frac{\partial^{2}}{\partial
y^{2}}\frac{\partial}{\partial\nu}\mathcal{F}_{\nu+1}\biggr{]}\,,$ (32)

where $\nu$ is defined in (13), and $\xi_{s}\\!=\\!(D\\!-\\!1)/(D\\!-\\!3)$ is
what we refer to as the simple covariant gauge. The last term in (30) consists
of a de Sitter breaking tensor structure multiplied by a constant,

$\mathcal{C}_{4}=\xi\times\frac{H^{D-4}}{(4\pi)^{\frac{D}{2}}}\frac{\Gamma(D\\!-\\!1)}{(D\\!-\\!1)\,\Gamma\bigl{(}\frac{D}{2}\bigr{)}}\,,$
(33)

implying that our propagator preserves dilations, but breaks special spatial
conformal transformations.

All the de Sitter invariant photon propagator results reported in the
literature [4, 5, 6, 7] are captured by the $\mathcal{C}_{1}$ and
$\mathcal{C}_{2}$ parts of the solution. The nonvanishing constant
$\mathcal{C}_{4}$ in (33) however, has been overlooked thus far. But it is
this part that guarantees that the Ward-Takahashi identity and the equation of
motion are simultaneously satisfied, now producing the correct expression,

$\nabla^{\mu}\nabla^{\prime\nu}\,i\bigl{[}\tensor{*}[_{\mu}]{\Delta}{{}_{\nu}}\bigr{]}(x;x^{\prime})=-\xi\frac{i\delta^{D}(x\\!-\\!x^{\prime})}{\sqrt{-g}}\,,$
(34)

instead of (8). Thus, expressions (30–33) constitute a complete solution for
the Feynman propagator for the massless vector field, that satisfies both the
equation of motion (6) and the subsidiary condition (19).

The positive frequency Wightman function is now easily inferred from the
solution for the Feynman propagator by simply changing the $i\varepsilon$
prescription for $y$ in (30)–(32) to the appropriate one,
$y_{\scriptscriptstyle-+}\\!=\\!aa^{\prime}\bigl{[}\|\vec{x}\\!-\\!\vec{x}^{\,\prime}\|^{2}\\!-\\!(\eta\\!-\\!\eta^{\prime}\\!-\\!i\varepsilon)^{2}\bigr{]}$.
This ensures that the Wightman function satisfies a homogeneous equation of
motion,

$\sqrt{-g}\,\mathcal{D}^{\mu\nu}\,i\bigl{[}\tensor{*}[_{\nu}^{\scriptscriptstyle-\\!}]{\Delta}{{}_{\alpha}^{\scriptscriptstyle\\!+}}\bigr{]}(x;x^{\prime})=0\,,$
(35)

and an appropriate subsidiary condition,

$\nabla^{\mu}\,i\bigl{[}\tensor{*}[^{\scriptscriptstyle-\\!}_{\mu}]{\Delta}{{}_{\nu}^{\scriptscriptstyle\\!+}}\bigr{]}(x;x^{\prime})=-\xi\partial^{\prime}_{\nu}\,i\bigl{[}\tensor{*}[^{\scriptscriptstyle\\!-\\!}]{\Delta}{{}^{\scriptscriptstyle\\!+\\!}}\bigr{]}(x;x^{\prime})\,,$
(36)

containing the positive-frequency Wightman function for the ghost, obtained
from (13) and (14) by the same substitution
$y\\!\to\\!y_{\scriptscriptstyle-+}$. Thus, the double divergence vanishes,

$\nabla^{\mu}\nabla^{\prime\nu}\,i\bigl{[}\tensor{*}[_{\mu}^{\scriptscriptstyle-\\!}]{\Delta}{{}_{\nu}^{\scriptscriptstyle\\!+}}\bigr{]}(x;x^{\prime})=0\,,$
(37)

resolving the problems reported in [15].

This result agrees with the independent mode sum analysis in our companion
paper [16]. Apart from correctly accounting for the subsidiary condition, the
missing de Sitter breaking term resolves some issues that we outline in the
remainder of the letter.

Infrared behavior. The missing term in the photon two-point function we report
here does not only solve the issue of non-vanishing double divergence, which
has gone unnoticed for a long time, but also addresses the concern regarding
the infrared behavior of the photon two-point function. It was reported by
Youssef [6] (in $D\\!=\\!4$) and by Rendell [23] (in $D$ dimensions) that the
two-point function does not vanish in the deep infrared,

$i\bigl{[}\tensor{*}[_{\mu}]{\Delta}{{}_{\nu}}\bigr{]}(x;x^{\prime})\,\underset{!}{\overset{|y|\to\infty}{\scalebox{1.8}[1.0]{{\hbox{\sim}}}}}-\xi\,\frac{H^{D-2}aa^{\prime}\delta_{\mu}^{0}\delta_{\nu}^{0}\,\Gamma(D\\!-\\!1)}{(4\pi)^{\frac{D}{2}}(D\\!-\\!1)\,\Gamma\bigl{(}\frac{D}{2}\bigr{)}}\,.$
(38)

Even though this is a gauge dependent statement [6], since $\xi$ ranges on the
entire real line, it nonetheless is not innocuous, as it is a consequence of
failing to account for the Ward-Takahashi identity (19). This behavior is
precisely removed by the missing term (33) that we report, so that in the deep
infrared the photon two-point function vanishes,

$i\bigl{[}\tensor{*}[_{\mu}]{\Delta}{{}_{\nu}}\bigr{]}(x;x^{\prime})\xrightarrow{|y|\to\infty}0\,.$
(39)

While this is immaterial at the linear level, it does influence the loops.

Energy-momentum tensor. Not accounting for the de Sitter breaking part (33) of
the photon propagator can lead to inconsistencies in how photons source
gravity. There are two definitions possible for the photon energy-momentum
tensor, that have to coincide on-shell. We can either define it as a variation
of the gauge-invariant action,

$T_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\delta S}{\delta
g^{\mu\nu}}=\bigl{(}\delta_{\mu}^{\rho}\delta_{\nu}^{\sigma}\\!-\\!\tfrac{1}{4}g_{\mu\nu}g^{\rho\sigma}\bigr{)}g^{\alpha\beta}F_{\rho\alpha}F_{\sigma\beta}\,,$
(40)

or as a variation of the gauge-fixed action,

$\displaystyle T_{\mu\nu}^{\star}={}$
$\displaystyle\frac{-2}{\sqrt{-g}}\frac{\delta S_{\star}}{\delta
g^{\mu\nu}}=T_{\mu\nu}-\frac{2}{\xi}A_{(\mu}\nabla_{\nu)}\nabla^{\rho}\\!A_{\rho}$
$\displaystyle+\frac{g_{\mu\nu}}{\xi}\Bigl{[}A_{\rho}\nabla^{\rho}\nabla^{\sigma}\\!A_{\sigma}+\tfrac{1}{2}\bigl{(}\nabla^{\rho}\\!A_{\rho}\bigr{)}^{\\!2\,}\Bigr{]}$
$\displaystyle-2\bigl{(}\partial_{(\mu}\overline{c}\bigr{)}\bigl{(}\partial_{\nu)}c\bigr{)}+g_{\mu\nu}g^{\rho\sigma}\bigl{(}\partial_{\rho}\overline{c}\bigr{)}\bigl{(}\partial_{\sigma}c\bigr{)}\,.$
(41)

Classically the two give the same answer as they differ by BRST-exact terms
only, which all vanish on-shell 333 BRST-exact objects are the ones that can
be written as (anti)commutators of other objects with the BRST charge. They
vanish on-shell by the definition of the formalism. . The quantized theory has
to maintain this property at the level of expectation values. Operators
associated to the two definitions of the energy-momentum tensor are defined by
Weyl ordered (symmetrized) products of field operators, and the difference
between their expectation values is best expressed in terms of the Wightman
function and derivatives acting on it,

$\displaystyle\bigl{\langle}\hat{T}_{\mu\nu}^{\star}(x)\bigr{\rangle}-\bigl{\langle}\hat{T}_{\mu\nu}(x)\bigr{\rangle}=\biggl{\\{}\frac{g_{\mu\nu}}{2\xi}\nabla^{\rho}\nabla^{\prime\sigma}i\bigl{[}\tensor{*}[_{\rho}^{\scriptscriptstyle-\\!}]{\Delta}{{}_{\sigma}^{\scriptscriptstyle\\!+}}\bigr{]}(x;x^{\prime})$
$\displaystyle-\frac{1}{\xi}\Bigl{[}\delta^{\rho}_{(\mu}\nabla^{\prime}_{\nu)}\nabla^{\prime\sigma}+\delta^{\sigma}_{(\mu}\nabla_{\nu)}\nabla^{\rho}\Bigr{]}i\bigl{[}\tensor{*}[_{\rho}^{\scriptscriptstyle-\\!}]{\Delta}{{}_{\sigma}^{\scriptscriptstyle\\!+}}\bigr{]}(x;x^{\prime})$
$\displaystyle+\frac{g_{\mu\nu}}{2\xi}\Bigl{[}\nabla^{\prime\rho}\nabla^{\prime\sigma}+\nabla^{\sigma}\nabla^{\rho}\Bigr{]}i\bigl{[}\tensor{*}[_{\rho}^{\scriptscriptstyle-\\!}]{\Delta}{{}_{\sigma}^{\scriptscriptstyle\\!+}}\bigr{]}(x;x^{\prime})$
$\displaystyle-\Bigl{[}2\nabla_{\mu}\nabla^{\prime}_{\nu}-g_{\mu\nu}\nabla^{\rho}\nabla^{\prime}_{\rho}\Bigr{]}i\bigl{[}\tensor{*}[^{\scriptscriptstyle\\!-\\!}]{\Delta}{{}^{\scriptscriptstyle\\!+\\!}}\bigr{]}(x;x^{\prime})\biggr{\\}}{\bigg{|}}_{x^{\prime}\to
x}\,.$ (42)

The terms in the last three lines above cancel on the account of a derivative
of the Ward-Takahashi identity,

$\nabla_{\rho}\nabla^{\mu}i\bigl{[}\tensor{*}[_{\mu}^{\scriptscriptstyle-\\!}]{\Delta}{{}_{\nu}^{\scriptscriptstyle\\!+}}\bigr{]}(x;x^{\prime})=-\xi\partial_{\rho}\partial^{\prime}_{\nu}\,i\bigl{[}\tensor{*}[^{\scriptscriptstyle\\!-\\!}]{\Delta}{{}^{\scriptscriptstyle\\!+\\!}}\bigr{]}(x;x^{\prime})\,,$
(43)

which is insensitive to the de Sitter breaking part that drops out because of
the second derivative. The remaining term from the first line,

$\bigl{\langle}\hat{T}_{\mu\nu}^{\star}(x)\bigr{\rangle}-\bigl{\langle}\hat{T}_{\mu\nu}(x)\bigr{\rangle}\\\
=\frac{g_{\mu\nu}}{2\xi}\nabla^{\rho}\nabla^{\prime\sigma}i\bigl{[}\tensor{*}[_{\rho}^{\scriptscriptstyle-\\!}]{\Delta}{{}_{\sigma}^{\scriptscriptstyle\\!+}}\bigr{]}(x;x^{\prime})\Big{|}_{x^{\prime}\to
x}\,,$ (44)

has to vanish on its own. This is precisely in the form of the problematic
expression (8) we started with. If we were to disregard the de Sitter breaking
part (33) of the photon propagator, this difference would not vanish,

$\bigl{\langle}\hat{T}_{\mu\nu}^{\star}(x)\bigr{\rangle}-\bigl{\langle}\hat{T}_{\mu\nu}(x)\bigr{\rangle}\\\
\overset{!}{=}-\frac{g_{\mu\nu}}{2}\frac{H^{D}\,\Gamma(D)}{(4\pi)^{\frac{D}{2}}\,\Gamma\bigl{(}\frac{D}{2}\bigr{)}}\,.$
(45)

but would have a cosmological constant form. Since this contribution is
independent of the gauge-fixing parameter $\xi$, it would be difficult to
recognize it as an unphysical answer, even though it clearly originates from
not accounting for the gauge sector constraints. In fact, when one uses the de
Sitter breaking two-point function, inserting (37) into (44) shows that the
two definitions give the same answer,

$\bigl{\langle}\hat{T}_{\mu\nu}^{\star}(x)\bigr{\rangle}-\bigl{\langle}\hat{T}_{\mu\nu}(x)\bigr{\rangle}=0\,.$
(46)

When computed in dimensional regularization the expectation value of the
energy-momentum tensor in $D\\!=\\!4$ vanishes,
$\langle\hat{T}_{\mu\nu}(x)\rangle\\!=\\!0$, as shown in [16].

Discussion. The photon propagator in de Sitter in the general covariant gauge
takes the form (30) with the three structure functions given in (31–33). The
de Sitter invariant parts in (31–32) have been derived in previous works [4,
5, 6, 7]. It is the nonvanishing constant $\mathcal{C}_{4}$ in (33),
overlooked thus far, that is our main contribution. This contribution breaks
de Sitter symmetry. In particular, it breaks special spatial conformal
transformations, while preserving dilations, spatial homogeneity and isotropy.
Even though this term is a homogeneous solution of the propagator equation of
motion (6), it cannot be discarded. It is the Ward-Takahashi identity (19),
necessary for the construction of a consistent propagator, that requires it.
The de Sitter breaking can be traced back to the Faddeev-Popov ghost
propagator, which satisfies the equation for the massless, minimally coupled
scalar (12), which does not admit a de Sitter invariant solution. This ghost
propagator appears in the Ward-Takahashi identity (19), whose solution allows
a dilation-preserving photon propagator. At the linear level, this necessary
modification of the photon propagator is of little physical significance, as
it is confined to the pure gauge sector. However, it can be of paramount
importance when interactions are considered, as failure to implement it, in
general leads to incorrect results.

Acknowledgements. D.G. was supported by the Czech Science Foundation (GAČR)
grant No. 20-28525S. This work is part of the Delta ITP consortium, a program
of the Netherlands Organisation for Scientific Research (NWO) that is funded
by the Dutch Ministry of Education, Culture and Science (OCW) – NWO project
number 24.001.027.

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