###  8.2 Formal quasi-differential operators

A quantisation of the cotangent bundle $T^{*}M$ determines a star product on
$\mathcal{S}(M)[\hbar]$. There is a unique deformation quantisation of the
cotangent bundle $T^{*}M$ whose star product on
$C^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$

  * (1)

is differential, and

  * (2)

reduces to the star product on $\mathcal{S}(M)[\hbar]$.

In fact, since the star product $\star$ is assumed to be differential, it is
fixed entirely by its action on symbols, i.e. on the subspace
$\mathcal{S}(M)\subset C^{\infty}(T^{*}M)$. Note that the corresponding star
product on the cotangent budle is natural due to our assumption on
differential quantisations. This deformation quantisation of the cotangent
bundle (in the sense of Section 7) will be called the formal extension of the
quantisation of the cotangent bundle (in the sense of Section 4). For
instance, the bilinear map (80) provides a formal extension of the normal
quantisation (59).282828See [19, 26] for more details on the geometric
construction of the homogeneous Fedosov star products on the cotangent bundle
generalising the normal star product (see also [27]). Two different
quantisations of the cotangent bundle would nevertheless lead to equivalent
star products on $\mathcal{S}(M)[\hbar]$ since the composition product in
$\mathcal{D}(M)$ remains the same.

Consider a deformation $C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$
of the Poisson algebra $C^{\infty}(T^{*}M)$ of functions on the cotangent
bundle $T^{*}M$ via a star product $\star$ arising from a formal extension of
a quantisation of the cotangent bundle $T^{*}M$. Its elements will be called
formal quasi-differential operators on the manifold $M$.292929As shown in [26,
29], the analogue of the Gelfand–Naimark–Segal construction applies for such
formal deformations $C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$.
This establishes on firm ground the quantum mechanical interpretation of these
elements as operators acting on a Hilbert space. This is not explored here but
could be important in the future for looking for a suitable real form of the
algebra of formal quasi-differential operators considered here. An equivalence
class of such deformations
$C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$ with respect to the
equivalence of star products, will be called a formal algebra of quasi-
differential operators on $M$, denoted $\mathcal{Q}\mathcal{D}(M)$. Similarly,
two such formal algebras of quasi-differential operators over the same
manifold $M$ will be considered isomorphic iff their start products are
equivalent. By construction, the algebra $\mathcal{D}(M)$ of differential
operators is a subalgebra of the formal algebra $\mathcal{Q}\mathcal{D}(M)$ of
quasi-differential operators. A corollary of well-known results in symplectic
geometry and deformation quantisation is the uniqueness property of the formal
algebra of quasi-differential operators. Strictly speaking, within our
technical abilities we were only able to prove it under a mild assumption on
the topology of the manifold $M$ but we conjecture that it must be valid for
any manifold $M$ and provide some arguments below.

Proposition (Uniqueness) : When the second Betti number of the manifold $M$
vanishes ( $b_{2}(M)=0$ ), the formal algebra $\mathcal{Q}\mathcal{D}(M)$ of
quasi-differential operators on a manifold $M$ is unique, up to automorphisms.

In such case, the algebra $\mathcal{Q}\mathcal{D}(M)$ is entirely determined
by $M$, as the terminology and notation suggests ($\hbar$ is absent).

Proof: An old theorem of Lichnerowicz [28] asserts that the differential star
product is unique (up to equivalences) on any symplectic manifold
$\mathcal{M}$ whose second Betti number vanishes $b_{2}(\mathcal{M})=0$. Note
that any vector bundle is homotopically equivalent to its base manifold $M$.
Therefore, the Betti numbers of the cotangent bundle $T^{*}M$ are equal to the
ones of its base manifold $M$ : $b_{i}(T^{*}M)=b_{i}(M)$. $\square$

Remark: In the general case (when the second Betti number of the manifold $M$
does not vanish), the vector spaces of equivalence classes of symplectic two-
forms and of star products are characterised by the de Rham cohomology in
degree two. Nevertheless, one should remember that the star product on
$C^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$ has been required to reduce to
the associative product on $\mathcal{S}(M)[\hbar]$ obtained from a
quantisation map
$Q:\mathcal{S}(M)\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{D}(M)$ of the
cotangent bundle. This requirement implies that the symplectic two-form on
$T^{*}M$ is the canonical one. Therefore, one may expect that the admissible
star products for the construction of $\mathcal{Q}\mathcal{D}(M)$ are in the
same equivalence class as the homogeneous Fedosov star product considered in
[19].303030For the sake of completeness, note that the corresponding star
products for a non-canonical symplectic two-form on the cotangent bundle have
been studied in [29]. Another strategy for proving the uniqueness of the
algebra $\mathcal{Q}\mathcal{D}(M)$ would be to show that the star products on
$C^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$ arising from the formal
extension of distinct quantisations of the cotangent bundle $T^{*}M$ are
isomorphic. This is to be expected since two different quantisations of the
cotangent bundle lead to equivalent star products on $\mathcal{S}(M)[\hbar]$.

The star product of a function $X(x,p)\in C^{\infty}(T^{*}M)$ on the cotangent
bundle with a function $f(x)\in C^{\infty}(M)$ on the base manifold, evaluated
at vanishing momenta, produces a formal function on the base manifold. In this
sense, formal quasi-differential operators can indeed be interpreted as
“operators”, i.e. as linear maps on $C^{\infty}(M)\llbracket\hbar\rrbracket$.
They are “formal” in the sense that they are formal power series in $\hbar$
but, more importantly, they are “quasi-differential” in the sense that they
can be interpreted as formal power series in $\hbar$ with coefficients which
are differential operators.

Proposition (Formal quantisation map) : Let
$Q:\mathcal{S}(M)\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{D}(M)$ be a
compatible quantisation of the cotangent bundle $T^{*}M$. There is an
$\hbar$-linear surjective anchor
$\hat{\bullet}_{\hbar}:C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket\twoheadrightarrow\mathcal{D}(M)\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
of the deformation of the algebra of functions on the cotangent bundle, that
extends the compatible quantisation and whose image is the algebra of almost-
differential operators.

A map with the above properties will be called a formal quantisation map,

$\hat{\bullet}_{\hbar}\,:\,C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket\twoheadrightarrow\mathcal{D}(M)\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}\,:\,X\mapsto\hat{X}_{\hbar}\,,$
(112)

Concretely, it is defined as

$\hat{X}_{\hbar}[f]\,:=\,\zeta^{*}\Big{(}\,X\,\star\,\,\tau^{*}(f)\,\Big{)}\in
C^{\infty}(M)\llbracket\hbar\rrbracket\,.$ (113)

Proof: Firstly, the map (112) takes values in
$\mathcal{D}\llbracket\hbar\rrbracket$. Indeed, remember that star products
have been assumed to be differential, hence the definition (113) implies that
$\hat{X}_{\hbar}\in\mathcal{D}\llbracket\hbar\rrbracket$. Secondly, the map
(112) is $\hbar$-linear. This is by construction because $\zeta^{*}$ should be
understood in (113) as the $\hbar$-linear extension of (23). Thirdly, the map
(112) is surjective. In fact, the situation is similar to Subsection 8.1. The
image of a principal symbol $X_{r}(x,p)\in\mathcal{S}^{r}(M)$ of degree $r$ in
the momenta is a $r$th-order differential operator
$\hat{X}_{\hbar}\in\mathcal{D}^{r}(M)\,\hbar^{r}$ of degree $r$ in $\hbar$
whose principal symbol is $X(x,p)\,\hbar^{r}\,$. Therefore, the images of
formal quasi-differential operators of the form
$X_{r}(x,p)\in\mathcal{S}^{r}(M)\hbar^{s}$ (for all $r$ and $s$) will span the
whole
$\mathcal{D}(M)\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$.
Finally, the map (112) is a morphism of algebras. Indeed, the definition (113)
mimicks (63). The quantisation of the cotangent bundle is assumed compatible,
which means that the quantisation map (54) is compatible with the definition
(63). Therefore, the $\hbar$-linearity ensures that this remains true for the
extension (112) of (54).

Example (Normal quantisation) : Since

$\hat{X}_{\hbar}[f]\,:=\,\big{(}\,X(x,p)\,\stackrel{{\scriptstyle
N}}{{\star}}f(x)\,\big{)}\,\big{|}_{p=0}=\sum\limits_{n=0}^{\infty}\frac{\hbar^{n}}{n!}\,\frac{\partial^{n}X(x,0)}{\partial
p_{\mu_{1}}\cdots\partial p_{\mu_{n}}}\,\frac{\partial^{n}f(x)}{\partial
x^{\mu_{1}}\cdots\partial x^{\mu_{n}}}\,,$ (114)

the normal star product (80) leads to the formal quantisation map

$\hat{\bullet}^{N}_{\hbar}\,:\,X(x,p)\mapsto\hat{X}_{\hbar}=\sum\limits_{n=0}^{\infty}\frac{\hbar^{n}}{n!}\,\frac{\partial^{n}X(x,0)}{\partial
p_{\mu_{1}}\cdots\partial
p_{\mu_{n}}}\,\partial_{\mu_{1}}\cdots\partial_{\mu_{n}}\,\,,$ (115)

in agreement with the quantisation map (111).

The idea behind the introduction of formal (quasi-)differential operators is
that, although the subspace $\mathcal{S}(M)[\hbar]$ (or
$C^{\infty}(T^{*}M)[\hbar]$ ) of elements polynomial in $\hbar$ reduces to the
space of symbols (or of functions on the cotangent bundle) if one would set
the formal variable to $\hbar=1$, this is not true for the space
$C^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$ of formal quasi-differential
operators because the evaluation at $\hbar=1$ can be divergent. Nevertheless,
one can interpret the formal algebra $\mathcal{Q}\mathcal{D}(M)$ of quasi-
differential operators as a completion of the almost-commutative algebra
$\mathcal{D}(M)$ of differential operators. Metaphorically it would produce
the desired algebra of strict quasi-differential operators, if one were
allowed to set $\hbar=1$ in the formulae. This will be good enough for our
purpose.

##  9 Higher-spin diffeomorphisms

Let us stress that if $\hbar$ is treated as a real parameter (rather than a
formal variable), then it becomes natural to consider only isomorphisms and
automorphisms that are $\hbar$ -linear.

###  9.1 Looking for higher-spin diffeomorphisms

The known general results on star product (and their equivalence) ensure that
deformation quantisation provides a possible cure of the formal exponentiation
of higher-spin Lie derivatives. The Appendix B of [26] provides a very concise
review of the mathematically rigorous results on Heisenberg-picture time
evolution of operators in deformation quantisation.

One can define the higher-spin Lie derivative of a differential operator
$\hat{Y}(x,p)\in\mathcal{D}(M)$ along a differential operator
$\hat{X}\in\mathcal{D}(M)$ as a trivial inner derivation of the deformation
$C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$ associated to the
corresponding symbols $X,Y\in\mathcal{S}(M)$. This defines the one-parameter
group of inner self-equivalences:

$\displaystyle Y(x,p)$ $\displaystyle\mapsto$ $\displaystyle
Y_{t}(x,p\,;\hbar)=\exp\Big{(}\,t\,\,{}^{\star}ad_{X}\,\Big{)}\,Y(x,p)$
$\displaystyle=\exp_{\star}\big{(}\,+t\,X(x,p)\,\big{)}\,\star\,Y(x,p)\,\,\star\,\,\exp_{\star}\big{(}\,-t\,X(x,p)\,\big{)}\,.$

where

$\exp_{\star}A\,:=\,\sum\limits_{n=0}^{\infty}\frac{1}{n!}\underbrace{A\star\cdots\star
A}_{n\,\,\text{factors}}\,.$ (117)

It remains true that if $X(x,p)\in\mathcal{S}^{r}(M)$ is a symbol of degree
$r>1$ then its star adjoint action on the algebra of symbols

${}^{\star}ad_{X}\,:\,\mathcal{S}^{q}(M)\to\mathcal{S}^{q+r-1}(M)[\hbar]$
(118)

increases the degree in momenta by $r-1>0$. However the difference now is that
each such higher-spin Lie derivative brings at least one power of $\hbar$.
Therefore,

$Y_{t}(x,p\,;\hbar)\in\mathcal{S}(M)\llbracket\hbar\rrbracket\qquad\text{if}\qquad
Y(x,p)\in\mathcal{S}(M)\,.$ (119)

Consequently, inner self-equivalences generated by higher-order differential
operators $\hat{X}\in\mathcal{D}(M)$ are inner automorphisms of the subalgebra
$\mathcal{S}_{\star}(M)\llbracket\hbar\rrbracket\subset
C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$ denoting the space of
symbols $\mathcal{S}(M)$ endowed with the star product. Unfortunately, these
inner automorphisms are trivial automorphisms of
$\mathcal{S}_{\star}(M)\llbracket\hbar\rrbracket$. In fact, their classical
limit is the identity:

$Y_{t}(x,p\,;\hbar\to 0)=Y(x,p)$ (120)

for all $t\in\mathbb{R}$. This is consistent with the Grabowski-Poncin no-go
theorem.

A proper refinement of the previous attempt is to consider instead the
mapping:

$Y(x,p)\mapsto
Y^{\hbar}_{t}(x,p)\,:=\exp\Big{(}\,t\,\mathcal{L}^{\hbar}_{X}\,\Big{)}\,Y(x,p)\,.$
(121)

where $\mathcal{L}^{\hbar}_{X}$ denotes a non-trivial inner derivation(95) of
the deformation $C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$. More
generally, a derivation of the form (95) with $X\in
C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$ will be called a formal
higher-spin Lie derivative along a quasi-differential operator. Locally, any
$\hbar$ -linear derivation of the deformation
$C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$ is a formal higher-spin
Lie derivative. The classical limit of (121) is the symplectomorphism

$Y(x,p)\mapsto Y^{\hbar\to
0}_{t}(x,p)=\exp\Big{(}\,t\,\\{\,X(x,p)\,,\,\,\,\\}\,\Big{)}\,Y(x,p)$ (122)

generated by the Hamiltonian vector field with $X(x,p)$ as Hamiltonian. Let us
stress again that the subspace $\mathcal{S}(M)\subset C^{\infty}(T^{*}M)$ of
symbols is not preserved by such symplectomorphisms, consistently with the
Grabowski-Poncin no-go theorem. However, the transformations (122) are well-
defined on the whole Poisson algebra $C^{\infty}(T^{*}M)$. In other words,
generically $Y^{\hbar\to 0}_{t}\in C^{\infty}(T^{*}M)$ even when
$X,Y\in\mathcal{S}(M)$.

###  9.2 Formal higher-spin diffeomorphisms

An $\hbar$ -linear isomorphism of associative algebras between two
deformations $C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$ and
$C_{\star^{\prime}}^{\infty}(T^{*}M^{\prime})\llbracket\hbar\rrbracket$ will
be called a formal higher-spin diffeomorphism between the manifold $M$ and the
manifold $M^{\prime}$.

A corollary of the known results for star products on symplectic manifolds
(see Proposition 9.4 of [25]) is that any formal higher-spin diffeomorphism
$\Phi:C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket\stackrel{{\scriptstyle\sim}}{{\to}}C_{\star^{\prime}}^{\infty}(T^{*}M^{\prime})\llbracket\hbar\rrbracket$
between $M$ and $M^{\prime}$ is the composition $\Phi=F^{*}\circ T$ of its
classical limit
$F^{*}:C_{\star}^{\infty}(T^{*}M)\stackrel{{\scriptstyle\sim}}{{\to}}C_{\star^{\prime}}^{\infty}(T^{*}M^{\prime})$
(i.e. an isomorphism of Poisson algebras, associated to a symplectomorphism
$F:T^{*}M^{\prime}\stackrel{{\scriptstyle\sim}}{{\to}}T^{*}M$ between the
corresponding cotangent bundles) and a quantum correction
$T:C_{\star}^{\infty}(T^{*}M)\stackrel{{\scriptstyle\sim}}{{\to}}C_{\star}^{\infty}(T^{*}M)$
(i.e. a self-equivalence of star product).

Therefore, there is a one-to-one correspondence between:

  1. 1.

symplectomorphisms from the cotangent bundle $T^{*}M^{\prime}$ to the
cotangent bundle $T^{*}M$,

  2. 2.

classical limit of formal higher-spin diffeomorphisms between the manifolds
$M$ and $M^{\prime}$,

  3. 3.

equivalence classes of $\hbar$ -linear isomorphisms between the associative
algebras $C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$ and
$C_{\star^{\prime}}^{\infty}(T^{*}M^{\prime})\llbracket\hbar\rrbracket$ of
formal quasi-differential operators with respect to the equivalence of star
products,

  4. 4.

$\hbar$ -linear isomorphisms between the formal algebras
$\mathcal{Q}\mathcal{D}(M)$ and $\mathcal{Q}\mathcal{D}(M^{\prime})$ of quasi-
differential operators.

###  9.3 Formal higher-spin flows

An $\hbar$ -linear algebra automorphism of the deformation
$C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$ will be called a formal
higher-spin diffeomorphism of the manifold $M$. They form a group which will
be denoted $HSDiff(M)$. The formal higher-spin diffeomorphisms exhaust all
automorphisms of the associative algebra
$C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$ of formal quasi-
differential operators, up to spurious perturbative redefinitions of $\hbar$.
Locally, any non-trivial formal higher-spin diffeomorphism of $M$ is a non-
trivial inner automorphism of
$C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$.

Any symplectomorphism of the cotangent bundle $T^{*}M$, connected to the
identity by a path of symplectomorphisms, admits an extension to a formal
higher-spin diffeomorphisms of $M$. Moreover, this extension is (locally)
unique up to self-equivalences. In fact, formal higher-spin diffeomorphisms on
a manifold $M$ can be thought of as “quantum-corrected” symplectomorphisms on
the cotangent bundle $T^{*}M$.

An action of the additive group $\mathbb{R}$ on the deformation
$C^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$ will be called a (globally-
defined) formal higher-spin flow on the manifold $M$. One will also admit
locally-defined formal higher-spin flows, i.e. actions of a Lie subgroup
$I\subset\mathbb{R}$ of the additive group $\mathbb{R}$ on the associative
algebra $C^{\infty}(T^{*}N)\llbracket\hbar\rrbracket$ of formal differential
operators on a submanifold $N\subset M$. This allows to formulate a solution
to the main problem addressed in this paper.

Yes-go proposition 2 (Formal quasi-differential operator) : Any formal quasi-
differential operator $X\in
C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$ is integrable to a formal
higher-spin flow on $M$, i.e. a group morphism

$\exp(\bullet\mathcal{L}^{\hbar}_{X})\,:\,I\to
HSDiff(N)\,:\,t\mapsto\exp(\,t\,\mathcal{L}^{\hbar}_{X})\,,$ (123)

defined for an open subset $I\subseteq\mathbb{R}$ (e.g. an open interval
$I=]a,b[$ ) and submanifold $N\subseteq M$.

More precisely, the formal higher-spin flow is the combination

$\exp(\,t\,\mathcal{L}^{\hbar}_{X})=T_{t}\circ\exp(\,t\,\mathcal{L}^{\hbar\to
0}_{X})$ (124)

of the corresponding Hamiltonian flow on $T^{*}M$, i.e. the group morphism

$\exp(\bullet\mathcal{L}^{\hbar\to 0}_{X})\,:\,I\to
Diff(T^{*}N)\,:\,t\mapsto\exp(\,t\,\mathcal{L}^{\hbar\to 0}_{X})\,,$ (125)

with the quantum correction map

$T_{\bullet}\,:\,{\mathbb{R}}\to\exp\big{(}\,\hbar\,\mathcal{D}(M)\llbracket\hbar\rrbracket\,\big{)}\,:\,t\mapsto
T_{t}$ (126)

where the perturbative redefinitions

$T_{t}=id+\sum\limits_{n=1}^{\infty}\stackrel{{\scriptstyle(n)}}{{T}}_{t}\hbar^{n}\quad\text{with}\quad\stackrel{{\scriptstyle(n)}}{{T}}_{t}\in\mathcal{D}^{2n}(T^{*}M)\,,$
(127)

have coefficients which are differential operators on the cotangent bundle
$T^{*}M$ of order equal to twice the corresponding degree in
$\hbar$.313131This last property holds because the star product is natural.
Detailed statements about the quantum correction to the Heisenberg-picture
time evolution can be found in Appendix B of [26].

Proof: Any formal quasi-differential operator $X\in
C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$ defines a formal higher-
spin Lie derivative (95) along $X$ whose classical limit
$\mathcal{L}^{\hbar\to 0}_{X}$ is a Hamiltonian vector field on $T^{*}M$ for
the Hamiltonian $X|_{\hbar=0}\in C^{\infty}(T^{*}M)$. As any vector field on a
manifold, this vector field is integrable to a local flow. In particular, the
Hamiltonian vector field $\mathcal{L}^{\hbar\to 0}_{X}$ on $T^{*}M$ is
integrable to a local Hamiltonian flow $\exp(\bullet\mathcal{L}^{\hbar\to
0}_{X})$ on $T^{*}M$, defined for an open subset $I\subseteq\mathbb{R}$ and
submanifold $N\subseteq M$. A theorem of Fedosov (Proposition 5.5.6 of [24])
ensures that each symplectomorphisms $\exp(\,t\,\mathcal{L}^{\hbar\to 0}_{X})$
for $t\in I$ can be extended to a formal higher-spin diffeomorphism of $M$,
which can be denoted $\exp(\,t\,\mathcal{L}^{\hbar}_{X})$.  $\square$

##  10 Quotient algebra of almost-differential operators

As was shown in Section 9, the algebra of (formal) quasi-differential
operators actually meets our goal in that it allows to define (formal) higher-
spin diffeomorphisms. Nevertheless, this completion of the algebra of
differential operators is quite huge. Two smaller algebras are actually
available: one is a subalgebra and one is a quotient algebra of the algebra of
formal quasi-differential operators.

###  10.1 Subalgebra

There is an algebra sitting in between the two algebras
$\mathcal{S}_{\star}(M)\llbracket\hbar\rrbracket$ and
$C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$: it is spanned by formal
power series in $\hbar$ whose coefficients are smooth functions of the base
$M$ of the cotangent bundle $T^{*}M$ but analytic functions of the momenta
(see [19] for a proof). This small completion is of interest. However, it is
another one that we will investigate here.

###  10.2 Quotient algebra

#### Infinite-order contact ideal of the zero section.

Consider the Poisson subalgebra
$\mathcal{I}^{\infty}\big{(}\,\zeta(M)\,\big{)}\subset C^{\infty}(T^{*}M)$
spanned by all functions on the cotangent bundle vanishing on the zero section
together with all their derivatives along momenta. More concretely, all
derivatives along momenta vanish when evaluated at zero momenta:323232For
instance, if the base manifold is the Euclidean space $M={\mathbb{R}}^{n}$,
then the function $f(x)\exp(-1/\vec{p}\,{}^{2})$ is such that its Taylor
series along the direction of momenta vanishes on the zero section
$\vec{p}=\vec{0}$.

$X(x,p)\in\mathcal{I}^{\infty}\big{(}\,\zeta(M)\,\big{)}\quad\Longleftrightarrow\quad
X(x,p)\in C^{\infty}(T^{*}M)\quad\text{and}\quad\frac{\partial
X(x,0)}{\partial p_{\mu_{1}}\cdots\partial p_{\mu_{k}}}=0\,,\quad\forall
k\in{\mathbb{N}}\,.$ (128)

The subalgebra $\mathcal{I}^{\infty}\big{(}\,\zeta(M)\,\big{)}$ is an ideal of
$C^{\infty}(T^{*}M)$ for both the pointwise product and the Poisson bracket.
It will be called the infinite-order contact ideal of the zero section of the
cotangent bundle. Note that this ideal does not contain any non-trivial
symbol, i.e.

$\mathcal{I}^{\infty}\big{(}\,\zeta(M)\,\big{)}\,\cap\,\mathcal{S}(M)\,=\,\\{0\\}$
(129)

#### Infinitesimal neighborhood of the zero-section.

The quotient

$\mathcal{J}^{\infty}\big{(}\,\zeta(M)\,\big{)}\,:=\,C^{\infty}\big{(}\,T^{*}M\,\big{)}\,/\,\mathcal{I}^{\infty}\big{(}\,\zeta(M)\,\big{)}$
(130)

of the Poisson algebra of functions the cotangent bundle $T^{*}M$ by the
infinite-order contact ideal of the zero section $\zeta(M)\subset T^{*}M$, is
a Poisson algebra, whose elements will be called jet fields on the
infinitesimal neighborhood of the cotangent bundle zero-section. They can be
thought of as Taylor series at the origin of cotangent spaces (i.e. at zero
momenta) of smooth functions on the cotangent bundle:

$X(x\,;p)=\sum\limits_{r=0}^{\infty}X^{\mu_{1}\cdots\mu_{r}}(x)\,p_{\mu_{1}}\cdots
p_{\mu_{1}}\,.$ (131)

#### Deformation quantisation.

For any differential star product $\star\,$, the subspace
$\mathcal{I}^{\infty}\big{(}\,\zeta(M)\,\big{)}\llbracket\hbar\rrbracket\subset
C^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$ is an ideal of the deformed
algebra, i.e. with respect to the star product. As such, it will be denoted
$\mathcal{I}_{\star}^{\infty}\big{(}\,\zeta(M)\,\big{)}\llbracket\hbar\rrbracket\subset
C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$. The quotient

$\mathcal{J}^{\infty}_{\star}\big{(}\,\zeta(M)\,\big{)}\llbracket\hbar\rrbracket\,:=\,C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket\,/\,\mathcal{I}_{\star}^{\infty}\big{(}\,\zeta(M)\,\big{)}\llbracket\hbar\rrbracket$
(132)

of the deformation $C_{\star}^{\infty}(T^{*}M)\llbracket\hbar\rrbracket$ by
the ideal
$\mathcal{I}_{\star}^{\infty}\big{(}\,\zeta(M)\,\big{)}\llbracket\hbar\rrbracket$
is an associative algebra, which is a deformation of the Poisson algebra
$\mathcal{J}^{\infty}\big{(}\,\zeta(M)\,\big{)}$ of jet fields on the
infinitesimal neighborhood of the cotangent bundle zero-section. One may
interpret this construction as a deformation quantisation of the infinitesimal
neighborhood of the cotangent bundle zero-section.

#### Injective anchor.

The formal quantisation map $\hat{\bullet}_{\hbar}$ in (112) is surjective but
not injective. In fact, it is not injective because it has a non-trivial
kernel. The kernel of $\hat{\bullet}_{\hbar}$ is precisely the
$\infty$-contact ideal
$\mathcal{I}^{\infty}\big{(}\zeta(M)\big{)}\llbracket\hbar\rrbracket$ of the
zero section of the cotangent bundle.333333This can be checked explicitly for
the normal star product in Darboux coordinates, via the formula (115).
Nevertheless, the conclusion is coordinate-free and remains valid for any
equivalent differential star product. Therefore, the quotient map

$\hat{\bullet}_{\hbar}\,:\,\mathcal{J}^{\infty}_{\star}\big{(}\,\zeta(M)\,\big{)}\llbracket\hbar\rrbracket\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{D}(M)\,\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}\,:\,X(x\,;p,\hbar)\mapsto\hat{X}_{\hbar}(x\,;\partial)\,\,.$
(133)

is an isomorphism of associative algebras between
$\mathcal{J}^{\infty}_{\star}\big{(}\,\zeta(M)\,\big{)}\llbracket\hbar\rrbracket$
and the algebra
$\mathcal{D}(M)\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$.
Therefore, the elements of
$\mathcal{J}^{\infty}_{\star}\big{(}\,\zeta(M)\,\big{)}\llbracket\hbar\rrbracket$
can be thought as almost-differential operators (although the realisation is
rather different).

#### Example (Normal quantisation).

The formal quantisation map (115) takes exactly the same form if one replaces
the functions $X(x,p)$ on the cotangent bundle by Taylor series (131) at the
zero section. The analogue of the map (111) would be the following embedding

$\displaystyle Q_{N}\circ i$ $\displaystyle:$
$\displaystyle\mathcal{J}^{\infty}_{\star}\big{(}\,\zeta(M)\,\big{)}\hookrightarrow\mathcal{D}(M)\,\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
(134) $\displaystyle:$
$\displaystyle\sum\limits_{r=0}^{\infty}\frac{1}{r!}\,X^{\mu_{1}\cdots\mu_{r}}(x)\,p_{\mu_{1}}\cdots
p_{\mu_{r}}\,\mapsto\,\sum\limits_{r=0}^{\infty}\frac{\hbar^{r}}{r!}\,X^{\mu_{1}\cdots\mu_{r}}(x)\,\partial_{\mu_{1}}\cdots\partial_{\mu_{r}}\,,$
(135)

whose $\hbar\,$-linear extension would reproduce the isomorphism
$\hat{\bullet}^{N}_{\hbar}:\mathcal{J}^{\infty}_{\star}\big{(}\,\zeta(M)\,\big{)}\llbracket\hbar\rrbracket\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{D}(M)\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
. Its inverse is the map

$\displaystyle(\hat{\bullet}^{N}_{\hbar})^{-1}$ $\displaystyle:$
$\displaystyle\mathcal{D}(M)\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{J}^{\infty}_{\star}\big{(}\,\zeta(M)\,\big{)}\llbracket\hbar\rrbracket$
(136) $\displaystyle:$ $\displaystyle\hat{X}_{\hbar}\mapsto
X(x\,;p,\hbar)\,=\,\exp(-\tfrac{1}{\hbar}\,p_{\mu}x^{\mu})\,\hat{X}[\,\exp(\tfrac{1}{\hbar}\,p_{\mu}x^{\mu})\,]$
(137)

#### Quotient algebra of almost differential operators.

The algebra

$\mathcal{D}(M)\,\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}\cong\mathcal{J}^{\infty}_{\star}\big{(}\,\zeta(M)\,\big{)}\llbracket\hbar\rrbracket$
(138)

of almost-differential operators is a completion with a reasonable size (since
the quotient throws away all elements which are invisible in the operatorial
interpretation) of the algebra $\mathcal{D}(M)$ of differential operators.
Unfortunately, the deformation
$\mathcal{J}^{\infty}_{\star}\big{(}\,\zeta(M)\,\big{)}\llbracket\hbar\rrbracket$
of the Poisson algebra $\mathcal{J}^{\infty}\big{(}\,\zeta(M)\,\big{)}$ of jet
fields on the infinitesimal neighborhood of the cotangent bundle zero-section
is constructed as a quotient (cf. (132) ) and, as such, requires more care
than the algebra
$\mathcal{D}(M)\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
of quasi-differential operators. For instance, the zero section $\zeta(M)$ is
not preserved by generic symplectomorphisms of the cotangent bundle, so the
quotient algebra
$\mathcal{J}^{\infty}_{\star}\big{(}\,\zeta(M)\,\big{)}\llbracket\hbar\rrbracket$
would meet the same conceptual problem as the Schouten algebra
$\mathcal{S}(M)$ of symbols, as far as automorphisms are concerned. This
implies that only those formal higher-spin diffeomorphisms on $M$ whose
classical limit are symplectomorphism of $T^{*}M$ sending the zero section
$\zeta(M)\subset T^{*}M$ into itself will descend from automorphism of the
algebra $C_{\star}^{\infty}\big{(}\,T^{*}M\,\big{)}\llbracket\hbar\rrbracket$
of quasi-differential operators to automorphisms of the algebra
$\mathcal{J}^{\infty}_{\star}\big{(}\,\zeta(M)\,\big{)}\llbracket\hbar\rrbracket$
of almost-differential operators.343434From the point of view of higher-spin
gravity, the status of such a strong condition on higher-spin diffeomorphisms
is unclear since it would remove the Maxwell gauge symmetries of the spin-one
sector (since they correspond to vertical automorphisms of the cotangent
bundle that do not preserve the zero section). Nevertheless, they appear as a
reasonable candidate subclass of higher-spin symmetries which could be
compatible with some weak notion of locality.

#### Zeroth-order contact ideal of the zero section.

The zeroth-order contact ideal
$\mathcal{I}^{0}\big{(}\,\zeta(M)\,\big{)}=\text{Ker}\,\zeta^{*}\subset
C^{\infty}(T^{*}M)$ of the cotangent bundle zero-section is spanned by all
functions $X$ on the cotangent bundle vanishing on the zero section, i.e. such
that $X_{0}=\zeta^{*}(X)=0$. It is an ideal for both the pointwise product and
the Poisson bracket. The infinite-order contact ideal
$\mathcal{I}^{\infty}\big{(}\,\zeta(M)\,\big{)}\subset\mathcal{I}^{0}\big{(}\,\zeta(M)\,\big{)}$
remains an ideal inside the zeroth-order one. The quotient algebra

$\mathcal{I}\big{(}\,\zeta(M)\,\big{)}\,:=\,\mathcal{I}^{0}\big{(}\,\zeta(M)\,\big{)}\,/\,\mathcal{I}^{\infty}\big{(}\,\zeta(M)\,\big{)}$
(139)

of the zeroth-order ideal of the zero section $\zeta(M)\subset T^{*}M$ by the
infinite-order one, is a Poisson algebra. Its elements can be thought of as
Taylor series (131) vanishing at zero momenta (i.e. the sum starts at $r=1$).
Due to the isomorphism (138), a corollary of the yes-go proposition in
Subsection 6.2 is that any element
$X\in\mathcal{I}\big{(}\,\zeta(M)\,\big{)}\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
is locally integrable to a one-parameter group of non-trivial automorphisms
$\exp(-t\,\mathcal{L}^{\hbar}_{X}\,)$ of the algebra
$\mathcal{J}^{\infty}_{\star}\big{(}\,\zeta(M)\,\big{)}[[\hbar]]$ of almost-
differential operators.

##  11 Conclusion

General results in deformation quantisation provide cures of the obstruction
to the integration of differential operators to one-parameter groups of inner
automorphisms. One does so by considering some larger algebras (which can be
thought of as a sort of completion of the algebra of differential operators)
on which “formal” higher-spin diffeomorphisms are well-defined. The latter are
nothing but a fancy name for quantum symplectomorphisms addressed in the realm
of deformation quantisation.

The obstruction to the integration of higher-degree Hamiltonian vector fields
to one-parameter groups of inner automorphisms of the Schouten algebra of
symbols was bypassed in Section 3 by enlarging the latter to the Poisson
algebra of functions on the cotangent bundle. Similarly, the obstruction to
the integration of higher-spin Lie derivatives to one-parameter groups of
inner automorphisms of the almost-commutative algebra of differential
operators was bypassed in Section 9 by enlarging the latter to the associative
algebra of formal quasi-differential operators. Moreover, by properly taking
into account the equivalence relations underlying this extension, there is a
one-to-one correspondence between the symmetries of their classical limits,
i.e. between symplectomorphisms of the cotangent bundle and the classical
limit of formal higher-spin diffeomorphisms.

Overcoming the obstacle was done at the price of considering formal
deformations. It would be nice to see if similar results hold for some
associative algebra of strict quasi-differential operators. Nevertheless, the
results obtained from deformation quantisation will be taken as an indication
that strict higher-spin diffeomorphisms can be defined rigorously. The table 4
summarises and compares the symmetries of the classical vs quantum tangent
bundle. As a sign of optimism, no explicit distinction was made between
“strict” and “formal” in the table.

| Classical | Quantum
---|---|---
Algebra | Poisson algebra (symplectic) | Associative algebra (central)
| $C^{\infty}(T^{*}M)$ | $\mathcal{Q}\mathcal{D}(M)$
Elements | Functions on the cotangent bundle | Quasi-differential operators
| $X(x,p)$ | $\hat{X}(x,\partial)$
Finite | Symplectomorphisms | Higher-spin
automorphisms |  of $T^{*}M$ |  diffeomorphisms of $M$
Flow of inner | Hamiltonian flow | Higher-spin flow
automorphisms |  on $T^{*}M$ |  on $M$
Infinitesimal | Symplectic vector field | Infinitesimal higher-spin
automorphism |  on $T^{*}M$ |  diffeomorphism of $M$
Inner | Hamiltonian vector | Higher-spin
derivation |  field on $T^{*}M$ |  Lie derivative on $M$
Table 4: Automorphisms of classical versus quantum algebras of functions on
the cotangent bundle

## Acknowledgments

Thomas Basile, Kevin Morand and Paolo Saracco are thanked for useful
discussions. The author also thank the Erwin Schrödinger International
Institute for Mathematics and Physics (Vienna) for hospitality during the
thematic programme “Geometry for Higher Spin Gravity: Conformal Structures,
PDEs, and Q-manifolds” (August 23rd — September 17th, 2021) during which the
final version of this work was completed.

## Proof of technical lemma

One possible strategy for the proof of the “technical lemma” in Subsection 6.1
is to start by showing that the lemma holds in the particular case of
homogeneous elements $a(\hbar)=a_{n}\,\hbar^{n}$ for all $n\geqslant 1$, and
then to conclude the proof by showing that the corresponding inner
automorphisms generate the generic case via composition. The proof is
organised slightly differently but follows the same logic.

Firstly, it is clear that the lemma holds in the particular case
$a(\hbar)=a_{1}\,\hbar$ with $a_{1}\in\mathcal{A}_{1}$. Indeed,
$\exp(t\,ad^{\hbar}_{a})=\exp(t\,ad_{a_{1}})$ is an automorphism of the
$\hbar$-filtered associative algebra
$\mathcal{A}\,\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
since $\exp(t\,ad_{a_{1}})$ is, by assumption, an automorphism of the almost-
commutative algebra $\mathcal{A}$ (i.e. it preserves the filtration).

Secondly, one can check by explicit computation that the lemma also holds for
any element
$b(\hbar)\in\mathcal{A}\,\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}\cap\hbar^{2}\,\mathcal{A}\llbracket\hbar\rrbracket$,
that is to say an element of the form
$b(\hbar)=\sum^{\infty}_{n=2}b_{n}\,\hbar^{n}$ with $b_{n}\in\mathcal{A}_{n}$:

$\displaystyle\exp(t\,ad^{\hbar}_{b})$ $\displaystyle=$
$\displaystyle\exp\Big{(}\,t\sum^{\infty}_{n=2}\hbar^{n-1}\,ad_{b_{n}}\Big{)}$
(140) $\displaystyle=$
$\displaystyle\sum\limits_{k=0}^{\infty}\frac{t^{k}}{k!}\sum\limits_{n_{1},\cdots,\,n_{k}\geqslant
2}\hbar^{n_{1}+\cdots+n_{k}-k}\,ad_{{}_{b_{{}_{n_{1}}}}}\cdots
ad_{{}_{b_{{}_{n_{k}}}}}$ (141)

The main point to observe is that, at each order in $\hbar$, there will only
be only a finite sum of products $ad_{{}_{b_{{}_{n_{1}}}}}\cdots
ad_{{}_{b_{{}_{n_{m}}}}}$ (since each $n_{i}$ contributes to a strictly
positive number in the exponent). This ensures that the exponential (140) is a
well-defined $\hbar\,$-linear map on
$\mathcal{A}\,\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
. Moreover, since it is an exponential of a derivation of
$\mathcal{A}\,\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$,
it is automatically an algebra automorphism.

Thirdly, the composition product of two automorphisms $\exp(ad^{\hbar}_{a})$
and $\exp(ad^{\hbar}_{b})$, with $a(\hbar)=a_{1}\hbar$ and
$b(\hbar)=\sum^{\infty}_{n=2}b_{n}\,\hbar^{n}$ as above, is well-defined
(since each of these two automorphisms is well-defined). Therefore, one should
only check that this product takes the form $\exp(ad^{\hbar}_{c})$ for some
element $c$ in
$\mathcal{A}\,\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
. The Baker–Campbell–Hausdorff formula guarantees that the element $c$ belongs
to $\mathcal{A}\,\llbracket\hbar\rrbracket$ and is well-defined (since one
knows that the above composition product is well-defined). A closer look at
each term in its expansion as nested commutators ensures that
$c\in\mathcal{A}\,\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
. In fact, it is well-known that $\exp(ad_{T})=Ad_{{}_{\exp(T)}}$ for any
endomorphism $T$, hence the operation
$\exp(ad^{\hbar}_{a})\circ\exp(ad^{\hbar}_{b})=\exp(ad^{\hbar}_{c})$ is
equivalent to a conjugation $Ad_{g}$ by the element
$g=\exp(\tfrac{1}{\hbar}\,a)\exp(\tfrac{1}{\hbar}\,b)=\exp(\tfrac{1}{\hbar}\,c)$,
where the Dynkin version of the Baker–Campbell–Hausdorff formula gives

$\displaystyle c$
$\displaystyle=\,\,\hbar\,\log\Big{(}\,\exp(\tfrac{1}{\hbar}\,a)\exp(\tfrac{1}{\hbar}\,b)\,\Big{)}$
(143)
$\displaystyle=a+b-\sum\limits_{n=2}^{\infty}\frac{(-1)^{n}}{n}\sum\limits_{r_{i}+s_{i}>0}\underbrace{k(r_{1},s_{1},\ldots,r_{n},s_{n})}_{\in\mathbb{R}}\,\underbrace{(ad^{\hbar}_{a})^{r_{1}}(ad^{\hbar}_{b})^{s_{1}}\cdots(ad^{\hbar}_{a})^{r_{n}}(ad^{\hbar}_{b})^{s_{n}-1}b}_{=:\,\,c(r_{1},s_{1},\ldots,r_{n},s_{n})}\,.$

The explicit expression of the coefficients
$k(r_{1},s_{1},\cdots,r_{n},s_{n})$ in (143) is well-known (see e.g. the book
[30], p.117) but is not necessary for the proof. It is enough to check that
each term $c(r_{1},s_{1},\cdots,r_{n},s_{n})$ belongs to
$\mathcal{A}\,\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
. This is true because, for any almost-commutative algebra $\mathcal{A}$, one
has that
$ad^{\hbar}_{\alpha}\beta\in\mathcal{A}\,\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
for any
$\alpha,\beta\in\mathcal{A}\,\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
(see Subsection 6.1).

Fourthly, the Zassenhaus formula gives

$\exp\Big{(}\,\tfrac{1}{\hbar}\,(a+b)\,\Big{)}\,=\,\underbrace{\exp(\tfrac{1}{\hbar}\,a)\exp(\tfrac{1}{\hbar}\,b)}_{=\exp\big{(}\tfrac{1}{\hbar}\,c\big{)}}\exp(\tfrac{1}{\hbar}\,d_{1})\exp(\tfrac{1}{\hbar}\,d_{2})\cdots\,,$
(144)

where the expressions of $d_{i}$ in terms of nested commutators of $a$ and $b$
is known recursively (see e.g. footnote 14 in [30], p.365). Their qualitative
form is enough to check that
$d_{i}\in\mathcal{A}\,\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}\cap\hbar^{2}\,\mathcal{A}\llbracket\hbar\rrbracket$
(for $a(\hbar)$ and $b(\hbar)$ as above). Therefore, the second and third step
of the proof guarantee that each factor in the product

$\exp(ad^{\hbar}_{a+b})\,=\,\exp(ad^{\hbar}_{c})\circ\exp(ad^{\hbar}_{d_{1}})\circ\exp(ad^{\hbar}_{d_{2}})\circ\cdots$
(145)

is an automorphism of
$\mathcal{A}\,\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\hbar\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
. This ends the proof.

## References

  * [1] M. A. Vasiliev, “Higher spin gauge theories in four-dimensions, three-dimensions, and two-dimensions,” Int. J. Mod. Phys. D 5 (1996) 763 [hep-th/9611024]; “Higher spin gauge theories in various dimensions,” Fortsch. Phys. 52 (2004) 702 [hep-th/0401177]; “Higher spin gauge theories in any dimension,” Comptes Rendus Physique 5 (2004) 1101 [hep-th/0409260];
X. Bekaert, S. Cnockaert, C. Iazeolla and M. A. Vasiliev, “Nonlinear higher
spin theories in various dimensions,” hep-th/0503128;
V. E. Didenko and E. D. Skvortsov, “Elements of Vasiliev theory,”
arXiv:1401.2975 [hep-th];
M. A. Vasiliev, “Higher-spin theory and space-time metamorphoses,” Lect. Notes
Phys. 892 (2015) 227 [arXiv:1404.1948 [hep-th]].

  * [2] D. Sorokin, “Introduction to the classical theory of higher spins,” AIP Conf. Proc. 767 (2005) 172 [hep-th/0405069];
X. Bekaert, N. Boulanger and P. Sundell, “How higher-spin gravity surpasses
the spin two barrier: no-go theorems versus yes-go examples,” Rev. Mod. Phys.
84 (2012) 987 [arXiv:1007.0435 [hep-th]];
R. Rahman, “Higher Spin Theory - Part I,” PoS ModaveVIII (2012) 004
[arXiv:1307.3199 [hep-th]]. R. Rahman and M. Taronna, “From Higher Spins to
Strings: A Primer,” arXiv:1512.07932 [hep-th];
A. Bengtsson, Higher Spin Field Theory (Concepts, Methods and History) Volume
1: Free Theory (De Gruyter, 2020).

  * [3] R. Argurio, G. Barnich, G. Bonelli and M. Grigoriev (eds), Higher Spin Gauge Theories (International Solvay Institutes, 2004);
L. Brink, M. Henneaux and M. A. Vasiliev (eds), Higher Spin Gauge Theories
(World Scientific, 2017).

  * [4] A. Y. Segal, “Conformal higher spin theory,” Nucl. Phys. B 664 (2003) 59 [arXiv:hep-th/0207212 [hep-th]].
  * [5] X. Bekaert, E. Joung and J. Mourad, “On higher spin interactions with matter,” JHEP 05 (2009) 126 [arXiv:0903.3338 [hep-th]];
X. Bekaert and E. Meunier, “Higher spin interactions with scalar matter on
constant curvature spacetimes: conserved current and cubic coupling generating
functions,” JHEP 11 (2010) 116 [arXiv:1007.4384 [hep-th]].

  * [6] L. Bonora, M. Cvitan, P. Dominis Prester, S. Giaccari, M. Paulišić and T. Štemberga, “Worldline quantization of field theory, effective actions and $L_{\infty}$ structure,” JHEP 04 (2018) 095 [arXiv:1802.02968 [hep-th]];
M. Cvitan, P. Dominis Prester, S. Giaccari, M. Paulišić and I. Vuković,
“Gauging the higher-spin-like symmetries by the Moyal product,” JHEP 06 (2021)
144 [arXiv:2102.09254 [hep-th]].

  * [7] A. Fotopoulos, N. Irges, A. C. Petkou and M. Tsulaia, “Higher-Spin Gauge Fields Interacting with Scalars: The Lagrangian Cubic Vertex,” JHEP 10 (2007) 021 [arXiv:0708.1399 [hep-th]].
  * [8] M. A. Vasiliev, “Actions, charges and off-shell fields in the unfolded dynamics approach,” Int. J. Geom. Meth. Mod. Phys. 3 (2006) 37 [arXiv:hep-th/0504090 [hep-th]];
M. Grigoriev, “Off-shell gauge fields from BRST quantization,” [arXiv:hep-
th/0605089 [hep-th]];
X. Bekaert, “Comments on higher-spin symmetries,” Int. J. Geom. Meth. Mod.
Phys. 6 (2009) 285 [arXiv:0807.4223 [hep-th]];
E. Sezgin and P. Sundell, “Geometry and Observables in Vasiliev’s Higher Spin
Gravity,” JHEP 1207 (2012) 121 [arXiv:1103.2360 [hep-th]];
M. Grigoriev, “Parent formulations, frame-like Lagrangians, and generalized
auxiliary fields,” JHEP 12 (2012) 048 [arXiv:1204.1793 [hep-th]];
C. Iazeolla, E. Sezgin and P. Sundell, “On Exact Solutions and Perturbative
Schemes in Higher Spin Theory,” Universe 4 (2018) 5 [arXiv:1711.03550 [hep-
th]].

  * [9] I. Bars, “Survey of two time physics,” Class. Quant. Grav. 18 (2001) 3113 [arXiv:hep-th/0008164 [hep-th]];
R. Bonezzi, E. Latini and A. Waldron, “Gravity, Two Times, Tractors, Weyl
Invariance and Six Dimensional Quantum Mechanics,” Phys. Rev. D 82 (2010)
064037 [arXiv:1007.1724 [hep-th]];
X. Bekaert, M. Grigoriev and E. D. Skvortsov, “Higher Spin Extension of
Fefferman-Graham Construction,” Universe 4 (2018) 17 [arXiv:1710.11463 [hep-
th]].

  * [10] J. Grabowski, N. Poncin, “Automorphisms of quantum and classical Poisson algebras”, Compositio Math. 140 (2004) 511 [arXiv:math/0211175 [math.RA]].
  * [11] J. Grabowski, N. Poncin, “Derivations of the Lie algebras of differential operators”, Indag. Mathem. N.S. 16 (2005) 181 [ arXiv:math/0312162 [math.DG]].
  * [12] C. Fronsdal, “Massless Fields with Integer Spin,” Phys. Rev. D 18 (1978) 3624; “Singletons and Massless, Integral Spin Fields on de Sitter Space (Elementary Particles in a Curved Space. 7),” Phys. Rev. D 20 (1979) 848.
  * [13] C. Fronsdal, “Some open problems with higher spins,” in P. van Nieuwenhuizen and D.Z. Freedman, Supergravity (North-Holland, 1979) 245.
  * [14] D. Francia and A. Sagnotti, “On the geometry of higher spin gauge fields,” Class. Quant. Grav. 20 (2003) S473 [arXiv:hep-th/0212185 [hep-th]]; “Higher-spin geometry and string theory,” J. Phys. Conf. Ser. 33 (2006) 57 [arXiv:hep-th/0601199 [hep-th]];
N. Bouatta, G. Compere and A. Sagnotti, “An introduction to free higher-spin
fields,” [arXiv:hep-th/0409068 [hep-th]];
D. Francia, “Low-spin models for higher-spin Lagrangians,” Prog. Theor. Phys.
Suppl. 188 (2011) 94 [arXiv:1103.0683 [hep-th]]; “Aspects of metric-like
higher-spin geometry,” AIP Conf. Proc. 1483 (2012) 118.

  * [15] M. Dubois-Violette and P. W. Michor, “A Common generalization of the Frohlicher-Nijenhuis bracket and the Schouten bracket for symmetric multivector fields,” Indag. Mathem. N.S. 6 (1995) 51 [arXiv:alg-geom/9401006 [math.AG]].
  * [16] K. Krasnov, E. Skvortsov and T. Tran, “Actions for Self-dual Higher Spin Gravities,” [arXiv:2105.12782 [hep-th]].
  * [17] P. Xu, “Quantum groupoids,” Commun. Math. Phys. 216 (2001) 539 [arXiv:math/9905192 [math.QA]].
  * [18] F. A. Berezin, “Some remarks about the associated envelope of a Lie algebra,” Funct. Anal. Appl. 1 (1967) 91.
  * [19] M. Bordemann, N. Neumaier and S. Waldmann, “Homogeneous Fedosov star products on cotangent bundles: I. Weyl and standard ordering with differential operator representation,” Commun. Math. Phys. 198 (1998) 363 [arXiv:q-alg/9707030 [math.QA]].
  * [20] H. Weyl, “Quantum mechanics and group theory,” Z. Phys. 46 (1927) 1;
E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys.
Rev. 40 (1932) 749.

  * [21] M. A. Rieffel, Deformation Quantization for Actions of $\mathbb{R}^{d}$, Memoirs of the American Mathematical Society 106 (American Mathematical Society, 1993).
  * [22] X. Saint Raymond, Elementary Introduction to the Theory of Pseudodifferential Operators (CRC Press, 1991);
L. Hörmander, The Analysis of Linear Partial Differential Operators III:
Pseudo-differential operators (Springer, 1994);
M. M. Wong, An Introduction to Pseudo-differential Operators (WorldScientic,
1999).

  * [23] S. Gutt, “Deformation Quantization: an introduction,” lectures given for 3rd cycle students at Monastir (Tunisie, 2005) cel-00391793.
  * [24] B. Fedosov, Deformation quantization and index theory, Mathematical Topics 9 (Akademie Verlag, 1996).
  * [25] S. Gutt and J. Rawnsley, “Equivalence of star products on a symplectic manifold; an introduction to Deligne’s Cech cohomology classes,” Journ. Geom. Phys. 29 (1999) 347.
  * [26] M. Bordemann, N. Neumaier and S. Waldmann, “Homogeneous Fedosov star products on cotangent bundles: II. GNS representations, the WKB expansion, traces, and applications,” J. Geom. Phys. 29 (1999) 199 [arXiv:q-alg/9711016 [math.QA]].
  * [27] M. Pflaum, “The normal symbol on Riemannian manifolds,” New York J. Math. 4 (1998) 95 [arXiv:dg-ga/9612011].
  * [28] A. Lichnerowicz, “Existence and equivalence of twisted products on a symplectic manifold,” Lett. Math. Phys. 3 (1979) 495.
  * [29] M. Bordemann, N. Neumaier, M. J. Pflaum, S. Waldmann, “On representations of star product algebras over cotangent spaces on Hermitian line bundles,” J. Funct. Anal. 199 (2003) 1 [arXiv:math/9811055 [math.QA]].
  * [30] A. Bonfiglioli and R. Fulci, Topics in Noncommutative Algebra: The Theorem of Campbell, Baker, Hausdorff and Dynkin, Lecture Notes in Mathematics 2034 (Springer, 2012).