# Quantization as a Categorical Equivalence

Benjamin H. Feintzeig
<EMAIL_ADDRESS>Department of Philosophy
University of Washington

###### Abstract

We demonstrate that, in certain cases, quantization and the classical limit
provide functors that are “almost inverse” to each other. These functors map
between categories of algebraic structures for classical and quantum physics,
establishing a categorical equivalence.

##  1 Introduction

The purpose of this paper is to develop tools for assessing the extent to
which mathematical models of classical and quantum physics share common
structure. To do this, we analyze functors between categories of classical and
quantum models. It is well known [2] that different properties of functors
provide information about relations between models in different categories.
Previously, Landsman [23] has proposed understanding quantization as a
functor;111Others [28, 14, 4] have also provided results concerning senses in
which quantization is a functor. however, he does not analyze the properties
of this functor. The current paper analyzes the properties of quantization
functors that bear on the shared structure of classical and quantum physics.

One standard for when a functor demonstrates structural equivalence between
mathematical models is when the functor exhibits a _categorical equivalence_
between the corresponding categories of models. This is the standard we appeal
to in this paper. We show that a quantization functor can be supplemented by a
classical limit functor, which serves as an “almost inverse,” and which
together yield a categorical equivalence between certain categories of models
of classical and quantum physics.

Unlike Landsman [21, 22, 23], who considers categories whose arrows are
Hilbert bimodules, instead we define categories whose arrows are certain types
of *-homomorphisms that directly preserve the algebraic structure of a model.
We take this as a first step for defining the actions of quantization and
classical limit functors on arrows, although it forces us to significantly
restrict the arrows in the categories we consider. It would be much more
desirable to extend the functoriality of quantization, and the categorical
equivalence, to more interesting morphisms than the ones considered here. The
current paper thus should be understood as a proof of concept, which we hope
to apply in future work to the categories Landsman considers by defining the
classical limit of a Hilbert bimodule. The main result of this paper is thus
the framework of using the classical limit functor to establish that
quantization is a categorical equivalence.

In what follows, we investigate the quantization of two types of models of
classical physics. In §2, we introduce background on strict deformation
quantization of C*-algebras and the classical limit. In §3 we analyze the
quantization of the C*-Weyl algebra, which applies to physical systems with
linear phase spaces. In §4 we analyze Rieffel’s quantization, which applies to
physical systems, whose phase space carries an action of $\mathbb{R}^{d}$. In
each case, we establish that corresponding quantization and classical limit
functors exhibit a categorical equivalence. In §5, we conclude with some
discussion of the results and future directions.

##  2 Quantization and the Classical Limit

We will investigate two functors $Q$ and $L$ corresponding to the processes of
quantization and the classical limit. We will present general definitions for
each process and then establish that for certain classes of physical
systems—corresponding to particular categories of models of classical and
quantum physics—these functors form a categorical equivalence.

To quantize a model of classical physics, we begin with a commutative
C*-algebra of functions on a phase space and continuously deform the product
operation to arrive at a noncommutative C*-algebra of bounded operators on a
Hilbert space. The resulting family of C*-algebras indexed by the parameter
$\hbar$ forms a structure called a continuous bundle.222For more details on
continuous bundles of C*-algebras, see Dixmier [7, Ch. 10], Kirchberg and
Wasserman [15], or Landsman [19, §II.1.2].

######  Definition 1 (continuous bundle of C*-algebras).

A _(uniformly) 333See Steeger and Feintzeig [31, Appendix B] for more details
on different continuity conditions for bundles of C*-algebras. All bundles
considered in this paper are uniformly continuous. continuous bundle of
C*-algebras_ over a locally compact _base space_ 444It is possible to include
more general locally compact metric spaces as base spaces. See Steeger and
Feintzeig [31]. $I\subseteq\mathbb{R}$ (where $I$ contains $0$ as an
accumulation point) is:

  * •

a family of C*-algebras $\\{\mathfrak{A}_{\hbar}\\}_{\hbar\in I}$ called
_fibers_ ;

  * •

a C*-algebra $\mathfrak{A}$ of _continuous sections_ ; and

  * •

a family of surjective *-homomorphisms
$\\{\phi_{\hbar}:\mathfrak{A}\to\mathfrak{A}_{\hbar}\\}_{\hbar\in I}$ called
_evaluation maps_.

Together, these structures must satisfy for each $a\in\mathfrak{A}$,

  1. (i)

$\lVert a\rVert=\sup_{\hbar\in I}\lVert\phi_{\hbar}(a)\rVert_{\hbar}$, where
$\lVert\cdot\rVert_{\hbar}$ denotes the norm on the fiber algebra
$\mathfrak{A}_{\hbar}$;

  2. (ii)

for each uniformly continuous and bounded function $f:I\to\mathbb{C}$, there
is a section $fa\in\mathfrak{A}$ such that
$\phi_{\hbar}(fa)=f(\hbar)\phi_{\hbar}(a)$;

  3. (iii)

the map $\hbar\mapsto\lVert\phi_{\hbar}(a)\rVert_{\hbar}$ is uniformly
continuous and bounded.

Such continuous bundles can be constructed by deforming the product in the
direction of the Poisson bracket of a classical phase space. One can do so
with the following notion of a quantization map.

######  Definition 2 (strict deformation quantization).

A _strict deformation quantization_ (over $I\subseteq\mathbb{R}$ with $0\in
I$) of a manifold $M$ with a *-algebra $\mathcal{P}\subseteq C_{b}(M)$ of
continuous, bounded functions carrying a Poisson bracket is:

  * •

a family of C*-algebras $\\{\mathfrak{A}_{\hbar}\\}_{\hbar\in I}$; and

  * •

a family of linear _quantization maps_
$\\{\mathcal{Q}_{\hbar}:\mathcal{P}\to\mathfrak{A}_{\hbar}\\}_{\hbar\in I}$,
where $\mathcal{Q}_{0}$ is the identity.

Together, these structures must satisfy for each $f,g\in\mathcal{P}$,

  1. (i)

(von Neumann’s condition) $\lim_{\hbar\to
0}\lVert\mathcal{Q}_{\hbar}(f)\mathcal{Q}_{\hbar}(g)-\mathcal{Q}_{\hbar}(fg)\rVert_{\hbar}=0$;

  2. (ii)

(Dirac’s condition) $\lim_{\hbar\to
0}\lVert\frac{i}{\hbar}\big{[}\mathcal{Q}_{\hbar}(f),\mathcal{Q}_{\hbar}(g)\big{]}-\mathcal{Q}_{\hbar}\big{(}\\{f,g\\}\big{)}\rVert_{\hbar}=0$;

  3. (iii)

(Rieffel’s condition) the map
$\hbar\mapsto\lVert\mathcal{Q}_{\hbar}(f)\rVert_{\hbar}$ is continuous on $I$;

  4. (iv)

(Deformation condition) for each $\hbar\in I$, the map $\mathcal{Q}_{\hbar}$
is injective, its image $\mathcal{Q}_{\hbar}[\mathcal{P}]$ is closed under the
product in $\mathfrak{A}_{\hbar}$, and $\mathcal{Q}_{\hbar}[\mathcal{P}]$ is
dense in $\mathfrak{A}_{\hbar}$.

Every strict deformation quantization satisfying mild technical conditions
defines a continuous bundle of C*-algebras (See Landsman [19, §II.1.2] or
Steeger and Feintzeig [31, Appendix A]). The algebra of sections is generated
by the maps $[\hbar\mapsto\mathcal{Q}_{\hbar}(f)]$ for each $f\in\mathcal{P}$
and the maps $\phi_{\hbar}$ are given concretely as evaluation of the sections
at a particular value $\hbar\in I$.

In the opposite direction of quantization, the classical limit can be
understood as the process of restricting a continuous bundle of C*-algebras
obtained from quantization back to the commutative C*-algebra at $\hbar=0$.
Steeger and Feintzeig [31] show that the fiber algebra $\mathfrak{A}_{0}$ at
$\hbar=0$ can be reconstructed from a given continuous bundle of C*-algebras
$((\mathfrak{A}_{\hbar},\phi_{\hbar})_{\hbar\in I},\mathfrak{A})$ over
$I=(0,1]$ containing only information about the quantum theory for $\hbar>0$.
To do so, consider the closed two-sided ideal $K_{0}=\\{a\in\mathfrak{A}\ |\
\lim_{\hbar\to 0}\lVert\phi_{\hbar}(a)\rVert_{\hbar}=0\\}$ of sections
vanishing at $\hbar=0$. Steeger and Feintzeig [31, §4] show that the quotient

$\displaystyle\mathfrak{A}_{0}=\mathfrak{A}/K_{0}$ (1)

is the unique limit point C*-algebra at $\hbar\to 0$ of the bundle up to
*-isomorphism and that the quotient map
$\phi_{0}:\mathfrak{A}\to\mathfrak{A}/K_{0}$ defines the unique evaluation map
at the fiber over $\hbar=0$. This procedure allows one to reconstruct the
fiber algebra $\mathfrak{A}_{0}$ of the classical theory at $\hbar=0$ from the
bundle of quantum algebras for $\hbar>0$.

We now have enough tools to define the action of quantization and the
classical limit on objects. Quantization associates a Poisson algebra
$\mathcal{P}$ of functions on $M$ to a non-commutative C*-algebra
$\mathfrak{A}_{\hbar}$ obtained by strict deformation quantization. On the
other hand, the classical limit associates a non-commutative C*-algebra
$\mathfrak{A}_{\hbar}$ to the unique commutative algebra $\mathfrak{A}_{0}$
obtained as the $\hbar\to 0$ limit of some continuous bundle, which is an
algebra of functions on the phase space. In order to draw structural
comparisons, we further need a way to associate morphisms of classical and
quantum models with one another.

One can take the classical limit of a morphism as follows. Consider two
continuous bundles of C*-algebras
$((\mathfrak{A}_{\hbar},\phi_{\hbar})_{\hbar\in I},\mathfrak{A})$ and
$((\mathfrak{B}_{\hbar},\psi_{\hbar})_{\hbar\in I},\mathfrak{B})$ over
$I=(0,1]$ representing quantum systems for $\hbar>0$. Suppose one has a family
of morphisms
$(\alpha_{\hbar}:\mathfrak{A}_{\hbar}\to\mathfrak{B}_{\hbar})_{\hbar\in I}$ of
the fibers for $\hbar>0$ that lift to a *-homomorphism
$\alpha:\mathfrak{A}\to\mathfrak{B}$ of the algebras of sections commuting
with the evaluation maps in the sense that

$\displaystyle\alpha_{\hbar}\circ\phi_{\hbar}=\psi_{\hbar}\circ\alpha$ (2)

for each $\hbar>0$. Steeger and Feintzeig [31, §5] show that in this situation
the morphism is appropriately continuous in $\hbar$ so that there is a unique
limit morphism $\alpha_{0}:\mathfrak{A}_{0}\to\mathfrak{B}_{0}$ obtained by
factoring through the quotient $\mathfrak{A}/K_{0}$ and thus satisfying

$\displaystyle\alpha_{0}\circ\phi_{0}=\psi_{0}\circ\alpha.$ (3)

This provides a direct way to associate morphisms of a model of quantum
physics with morphisms of a model of classical physics through the classical
limit.

Now we will encode the conditions under which one can take the classical limit
of a morphism of a fiber at a given value $\hbar>0$. We will associate with
each family of quantization maps $\\{\mathcal{Q}_{\hbar}\\}_{\hbar\in I}$ a
collection of _rescaling maps_
$\big{\\{}R^{\mathcal{Q}}_{\hbar\to\hbar^{\prime}}:\mathcal{Q}_{\hbar}[\mathcal{P}]\to\mathcal{Q}_{\hbar^{\prime}}[\mathcal{P}]\big{\\}}_{\hbar,\hbar^{\prime}\in
I}$ defined by

$\displaystyle
R^{\mathcal{Q}}_{\hbar\to\hbar^{\prime}}=\mathcal{Q}_{\hbar^{\prime}}\circ(\mathcal{Q}_{\hbar})^{-1}$
(4)

for any $\hbar,\hbar^{\prime}\in I$.

######  Definition 3 (morphisms).

Suppose one has two strict deformation quantizations
$(\mathfrak{A}_{\hbar},\mathcal{Q}_{\hbar})_{\hbar\in I}$ and
$(\mathfrak{B}_{\hbar},\mathcal{Q}^{\prime}_{\hbar})_{\hbar\in I}$ over
$I\subseteq\mathbb{R}$ of $\mathcal{P}$ and $\mathcal{P}^{\prime}$,
respectively. A *-homomorphism
$\alpha_{\hbar}:\mathfrak{A}_{\hbar}\to\mathfrak{B}_{\hbar}$ between the fiber
algebras at a fixed value $\hbar\neq 0\in I$ is called

  1. (i)

_smooth_ if
$\alpha_{\hbar}\big{[}\mathcal{Q}_{\hbar}[\mathcal{P}]\big{]}\subseteq\mathcal{Q}^{\prime}_{\hbar}[\mathcal{P}^{\prime}]$;

  2. (ii)

_scaling_ if for every $\hbar^{\prime}\neq 0$, the map

$\displaystyle\alpha_{\hbar^{\prime}}=R^{\mathcal{Q}^{\prime}}_{\hbar\to\hbar^{\prime}}\circ\alpha_{\hbar}\circ
R^{\mathcal{Q}}_{\hbar^{\prime}\to\hbar}$ (5)

extends continuously to a *-homomorphism
$\mathfrak{A}_{\hbar^{\prime}}\to\mathfrak{B}_{\hbar^{\prime}}$.

The smoothness condition says that a morphism preserves the additional
structure of the collection of quantized smooth functions on which the Poisson
bracket is defined. Insofar as the information that certain quantities are
smooth, in addition to being merely continuous, is part of the physical
theory, structure-preserving maps between quantum models should encode this
structure. The scaling condition says that a morphism preserves the algebraic
structure regardless of the numerical value of $\hbar$, where the rescaling
maps are used to shift the morphism to different values of $\hbar^{\prime}>0$
in order to make the comparison. Insofar as the numerical value of Planck’s
constant $\hbar$ in a strict deformation quantization depends on a system of
units (See Feintzeig [9]), it is merely a conventional choice, which the
status of a map as preserving the structure of the model should not depend on.
Indeed, since it is typical in a strict deformation quantization—and it holds
true in all cases considered in this paper—that all of the algebras
$\mathfrak{A}_{\hbar}$ for $\hbar>0$ are *-isomorphic, we can understand all
of the algebras for the quantum theory in different systems of units
(different values of $\hbar>0$) as having the same structure. By understanding
morphisms of a quantum theory as *-homomorphisms satisfying the scaling
condition, we are only requiring that these structure-preserving maps respect
this structural sameness in different systems of units.555In fact, we leave it
as an open question whether there even exist morphisms between fiber
C*-algebras of a strict deformation quantization that do not satisfy the
scaling condition in cases of interest. We have not been able to find
morphisms between the fiber C*-algebras used in §3-4 that fail the scaling
condition.

If a *-homomorphism
$\alpha_{\hbar}:\mathfrak{A}_{\hbar}\to\mathfrak{B}_{\hbar}$ is scaling, then
the construction surrounding Eq. (2) provides a lift of the family
$(\alpha_{\hbar})_{\hbar\in I}$ to a morphism of the algebra of sections of
the bundle and produces a unique limit morphism $\alpha_{0}$ satisfying Eq.
(3). If $\alpha_{\hbar}$ is smooth, then it follows that $\alpha_{0}$
preserves the privileged Poisson subalgebra _and_ the Poisson bracket defined
on it [31, Prop. 5.5].

We pause here to consider a possible objection to the conditions we require on
morphisms in Def. 3. It is well-known that some quantizations on compact phase
spaces lead to situations where the quantization maps are only defined for
discrete values of $\hbar$. So one might worry that the scaling condition is
too restrictive and may not be satisfied for all values of $\hbar>0$. In the
rest of this paper, we will restrict attention to quantizations where $\hbar$
takes continuous values in $(0,1]$ for convenience. So in our examples, the
scaling condition does require that morphisms are translatable to all
continuous values of $\hbar>0$. However, our definition is general enough to
accommodate other base spaces, even ones with only discrete values of $\hbar$.
In general, one may define strict quantizations and continuous bundles of
C*-algebras for locally compact base spaces. Moreover, the construction of the
classical limit from Steeger & Feintzeig [31] also applies for more general
locally compact base spaces, as long as the base space carries the additional
structure of a metric. In situations where the base space is different than
$(0,1]$, the scaling condition only imposes a requirement for those values of
$\hbar\neq 0$ in the base space on which the quantization maps are defined.
The central idea of the scaling condition is only to require that morphisms be
translatable among all of the allowed values of $\hbar$ by the rescaling maps.
As long as morphisms can be translated to all the values of $\hbar$ in the
base space (whether discrete or continuous), those morphisms can be lifted and
extended to a limit point of the base space by the construction of Steeger and
Feintzeig. As such, the scaling condition is a plausible condition on
morphisms, and indeed we do not know of any morphisms that fail to satisfy the
scaling condition.

Now that we have tools for understanding quantization and the classical limit,
we will proceed to characterize two categories of models of classical physics
that can be quantized functorially, and whose quantization we will demonstrate
provides a categorical equivalence.

##  3 The C*-Weyl Algebra for Linear Phase Spaces

One standard method for quantizing classical theories via the C*-Weyl algebra
applies to systems whose phase space is the dual space (i.e., the collection
of continuous linear functionals) $V^{\prime}$ of a topological vector space
$V$ with a symplectic form $\sigma$ (i.e., a non-degenerate, bilinear,
antisymmetric map $V\times V\to\mathbb{R}$). In this case, the Poisson
*-algebra $\Delta(V,0)\subseteq C_{b}(V^{\prime})$ is generated by linear
combinations of the functions $W_{0}(f):V^{\prime}\to\mathbb{C}$ for each
fixed $f\in V$ defined by

$\displaystyle W_{0}(f)(F)=e^{iF(f)}$ (6)

for all $F\in V^{\prime}$. The Poisson bracket on $\Delta(V,0)$ is defined by
the linear extension of

$\displaystyle\\{W_{0}(f),W_{0}(g)\\}=\sigma(f,g)W_{0}(f+g)$ (7)

for all $f,g\in V$. This algebra $\Delta(V,0)$ is norm dense in the C*-algebra
$AP(V^{\prime})$ of continuous almost periodic functions on the phase space
$V^{\prime}$. This structure specifies the classical model.

The corresponding quantum model is obtained through the exponentiated Weyl
form of the canonical commutation relations, which define for each $\hbar>0$ a
C*-algebra $\mathcal{W}(V,\hbar\sigma)$. A dense *-subalgebra
$\Delta(V,\hbar\sigma)$ is generated freely by linearly independent elements
of the form $W_{\hbar}(f)$ for $f\in V$ with multiplication and involution
operations specified by

$\displaystyle W_{\hbar}(f)W_{\hbar}(g)$
$\displaystyle=e^{-\frac{i}{2}\hbar\sigma(f,g)}W_{\hbar}(f+g)$ (8)
$\displaystyle W_{\hbar}(f)^{*}$ $\displaystyle=W_{\hbar}(-f)$ (9)

for all $f,g\in V$. There is a unique maximal C*-norm on
$\Delta(V,\hbar\sigma)$ and the C*-Weyl algebra
$\mathcal{W}(V,\hbar\sigma)=\overline{\Delta(V,\hbar\sigma)}$ is defined as
the completion of this dense *-subalgebra with respect to the C*-norm [See 30,
25, 26, 5].

In the special case where $V=\mathbb{R}^{2n}$, one can understand
$\mathcal{W}(V,\hbar\sigma)$ through the standard Schrödinger representation
$\pi_{S}$ on $\mathcal{H}_{S}=L^{2}(\mathbb{R}^{n})$. In this case, if we let
$Q^{\hbar}_{j}$ and $P^{\hbar}_{j}$ denote the position and momentum operators

$\displaystyle(Q^{\hbar}_{j}\psi)(x)$ $\displaystyle=x_{j}\cdot\psi(x)$ (10)
$\displaystyle(P^{\hbar}_{j}\psi)(x)$
$\displaystyle=i\hbar\frac{\partial}{\partial x_{j}}\psi(x)$ (11)

for all $\psi\in L^{2}(\mathbb{R})$, then $\pi_{S}$ is the continuous linear
extension of the representation

$\displaystyle\pi_{S}(W_{\hbar}(a,b))=e^{i\sum_{j=1}^{n}a_{j}\cdot
P^{\hbar}_{j}+ib_{j}\cdot Q^{\hbar}_{j}}$ (12)

so that $\mathcal{W}(V,\hbar\sigma)$ can be understood as the C*-algebra
generated by exponentials of configuration and momentum quantities.

The quantization maps
$\mathcal{Q}_{\hbar}:\Delta(V,0)\to\mathcal{W}(V,\hbar\sigma)$ are given for
$\hbar\in[0,1]$ by the linear extension of

$\displaystyle\mathcal{Q}_{\hbar}(W_{0}(f))=W_{\hbar}(f)$ (13)

for all $f\in V$. These quantization maps define a strict deformation
quantization [6] on $M=V^{\prime}$ for the Poisson *-algebra
$\mathcal{P}=\Delta(V,0)\subseteq\mathfrak{A}_{0}=AP(V^{\prime})$ and fiber
C*-algebras $\mathfrak{A}_{\hbar}=\mathcal{W}(V,\hbar\sigma)$ for $\hbar>0$.

One can define a category of classical models with linear phase spaces, as
follows. This category will form the domain of a quantization functor.

######  Definition 4.

We denote the following category by $\mathbf{LinClass}$:

  * •

_Objects_ are pairs $\big{(}AP(V^{\prime}),\Delta(V,0)\big{)}$, where
$AP(V^{\prime})$ is the C*-algebra of almost periodic functions on the dual to
a topological vector space $V$, and $\Delta(V,0)$ is the dense Poisson
*-subalgebra with Poisson bracket defined by a symplectic form $\sigma$.

  * •

_Arrows_ are *-homomorphisms $\alpha_{0}:AP(V^{\prime})\to AP(U^{\prime})$ for
symplectic topological vector spaces $(V,\sigma)$ and $(U,\sigma^{\prime})$
that are _smooth_ in the sense that

$\displaystyle\alpha_{0}\big{[}\Delta(V,0)\big{]}\subseteq\Delta(U,0)$ (14)

and _Poisson_ in the sense that

$\displaystyle\alpha_{0}\big{(}\\{A,B\\}_{V}\big{)}=\big{\\{}\alpha_{0}(A),\alpha_{0}(B)\big{\\}}_{U}$
(15)

for all $A,B\in\Delta(V,0)$.

Note that this category is general enough to include infinite-dimensional
phase spaces representing linear classical field theories. The morphisms in
this category preserve the structure of classical models at $\hbar=0$ as
symplectic phase spaces.

Similarly, one can define a category of quantum models corresponding to these
linear phase spaces.

######  Definition 5.

We denote the following category by $\mathbf{LinQuant}$:

  * •

_Objects_ are strict quantizations
$\big{(}\mathcal{W}(V,\hbar\sigma),\Delta(V,\hbar\sigma),\mathcal{Q}_{\hbar}\big{)}_{\hbar\in(0,1]}$
of $\mathcal{P}=\Delta(V,0)$, as defined in Eq. (13).

  * •

_Arrows_ are smooth, scaling *-homomorphisms
$\alpha_{1}:\mathfrak{A}_{1}\to\mathfrak{B}_{1}$, where
$\mathfrak{A}_{1}=\mathcal{W}(V,\sigma)$ and
$\mathfrak{B}_{1}=\mathcal{W}(U,\sigma^{\prime})$ are the C*-Weyl algebras at
$\hbar=1$ for symplectic topological vector spaces $(V,\sigma)$ and
$(U,\sigma^{\prime})$, respectively.

The morphisms in this category thus preserve the structure of the fully
quantized models as non-commutative C*-algebras of operators at $\hbar=1$.

The following proposition characterizing classical morphisms in
$\mathbf{LinClass}$ is essential for the definition of the quantization
functor.

######  Proposition 1.

If $\alpha_{0}:AP(V^{\prime})\to AP(U^{\prime})$ is a morphism in
$\mathbf{LinClass}$ for symplectic topological vector spaces $V$ and $U$, then
there is a character $\chi:V\to\mathbb{C}$ on the additive group $V$ and an
additive, symplectic, origin-preserving transformation $T:V\to U$ such that

$\displaystyle\alpha_{0}(W_{0}(f))=\chi(f)W_{0}(Tf)$ (16)

for each $f\in V$.

###### Proof.

First, note that $AP(V^{\prime})\cong C(\hat{V})$, where $\hat{V}$ is the
compact space of all characters on $V$ with the topology of pointwise
convergence, and similarly for $AP(U^{\prime})\cong C(\hat{U})$. The
isomorphism here is defined as follows. For each $f\in V$, consider the
function $\eta(f)\in C(\hat{V})$ defined by

$\displaystyle\eta(f)(\chi)=\chi(f)$ (17)

for all $\chi\in\hat{V}$. Then the isomorphism $AP(V^{\prime})\cong
C(\hat{V})$ is given by the continuous linear extension of

$\displaystyle W_{0}(f)\in AP(V^{\prime})\mapsto\eta(f)\in C(\hat{V})$ (18)

for all $f\in V$ [5, Thm. 4-3, p. 2903]. In what follows, we will use the
symbol $W_{0}(f)$ for both the element of $AP(V^{\prime})$ and for the element
$\eta(f)\in C(\hat{V})$.

By Gelfand duality [24, §C.2-3], there is a unique map
$\hat{T}:\hat{U}\to\hat{V}$, continuous in the topologies of pointwise
convergence on $\hat{U}$ and $\hat{V}$, such that

$\displaystyle(\hat{T}\chi)(f)=W_{0}(f)(\hat{T}\chi)=\alpha_{0}(W_{0}(f))(\chi)$
(19)

for every $f\in V$ and $\chi\in\hat{U}$. From this, define
$\hat{T}^{\prime}:\hat{U}\to\hat{V}$ by

$\displaystyle\hat{T}^{\prime}\chi=(\hat{T}\chi)\cdot(\hat{T}e)^{-1}$ (20)

for all $\chi\in\hat{U}$, where $e\in\hat{U}$ is the identity element of the
character group, i.e., the function $e(f)=1$ for all $f\in U$.

We further know that $V\cong\hat{\hat{V}}$ (and similarly for $U$), where
$\hat{\hat{V}}$ is the group of continuous characters on $\hat{V}$, when the
latter is given the topology of pointwise convergence. The existence of an
isomorphism here follows from Pontryagin duality [29, p. 27] because $V$ is
locally compact when considered with the discrete topology (in which $\hat{V}$
is the collection of all continuous characters on $V$), even though $V$ may
_not_ be locally compact in its original vector space topology.

The identification $V\cong\hat{\hat{V}}$ allows us to define $T:V\to U$ by

$\displaystyle(Tf)(\chi)=(\hat{T}^{\prime}\chi)(f)=\frac{(\hat{T}\chi)(f)}{(\hat{T}e)(f)}$
(21)

for all $f\in V$ and $\chi\in\hat{U}$.

It follows by direct calculation that $T$ is additive, symplectic and origin-
preserving, i.e.,

$\displaystyle T(f+g)$ $\displaystyle=Tf+Tg$ (22) $\displaystyle\sigma(f,g)$
$\displaystyle=\sigma^{\prime}(Tf,Tg)$ (23) $\displaystyle T(0)$
$\displaystyle=0$ (24)

for all $f,g\in V$. Moreover, it follows that for $f\in V$,

$\displaystyle(\hat{T}e)(f)\cdot
W_{0}(Tf)(\chi)=(\hat{T}\chi)(f)=\alpha_{0}(W_{0}(f))(\chi)$ (25)

for every $\chi\in\hat{U}$. Hence, we have shown

$\displaystyle\alpha_{0}(W_{0}(f))=(\hat{T}e)(f)W_{0}(Tf)$ (26)

for all $f\in V$. ∎

This proposition vindicates the earlier remark that morphisms in
$\mathbf{LinClass}$ preserve the structure of classical symplectic phase
spaces. It also provides a way to lift a morphism to any value $\hbar>0$, as
follows.

######  Corollary 1.

Suppose $\alpha_{0}:AP(V^{\prime})\to AP(U^{\prime})$ is a morphism in
$\mathbf{LinClass}$ for symplectic topological vector spaces $(V,\sigma)$ and
$(U,\sigma^{\prime})$. Suppose $\alpha_{0}$ is associated with the continuous
character $\chi:V\to\mathbb{C}$ and additive, symplectic, origin-preserving
transformation $T:V\to U$ as in Eq. (16). For any $\hbar>0$, define the map
$\alpha_{\hbar}:\mathcal{W}(V,\hbar\sigma)\to\mathcal{W}(U,\hbar\sigma^{\prime})$
as the continuous linear extension of

$\displaystyle\alpha_{\hbar}(W_{\hbar}(f))=\chi(f)W_{\hbar}(Tf)$ (27)

for all $f\in V$. Then $\alpha_{\hbar}$ is a *-homomorphism; in particular
$\alpha_{1}$ is smooth and scaling, and thus is a morphism in
$\mathbf{LinQuant}$.666This is a slight generalization of the results in [6,
§D], which treats the case where $T$ is bijective and linear. One can easily
check that these conditions are not necessary for $\alpha_{\hbar}$ as given in
Eq. (27) to define a *-homomorphism.

Hence, quantization defines a functor as follows.

######  Definition 6.

The functor $Q_{W}:\mathbf{LinClass}\to\mathbf{LinQuant}$ is defined by the
map on objects

$\displaystyle\Big{(}AP(V^{\prime}),\Delta(V,0)\Big{)}$
$\displaystyle\mapsto\Big{(}\mathcal{W}(V,\hbar\sigma),\Delta(V,\hbar\sigma),\mathcal{Q}_{\hbar}\Big{)}_{\hbar\in(0,1]}$
(28)

where each classical model is associated with its strict deformation
quantization via Eq. (13), and the map on arrows

$\displaystyle\Big{[}\alpha_{0}:AP(V^{\prime})\to AP(U^{\prime})\Big{]}$
$\displaystyle\mapsto\Big{[}\alpha_{1}:\mathcal{W}(V,\sigma)\to\mathcal{W}(U,\sigma^{\prime})\Big{]}$
(29)

where each morphism is associated with its quantized counterpart via Prop. 1
and Cor. 1.

In the other direction, the classical limit also defines a functor as follows.
Recall that each strict quantization
$\big{(}\mathcal{W}(V,\hbar\sigma),\Delta(V,\hbar\sigma),\mathcal{Q}_{\hbar}\big{)}_{\hbar\in(0,1]}$
defines a continuous bundle of C*-algebras with a C*-algebra of continuous
sections that we will denote by $\mathcal{W}\big{(}V,(0,1]\sigma\big{)}$. This
algebra contains a subalgebra $\Delta\big{(}V,(0,1]\sigma\big{)}$ generated by
the sections of the form

$\displaystyle[\hbar\mapsto W_{\hbar}(f)]$ (30)

for fixed $f\in V$ and all $\hbar\in(0,1]$. Moreover,
$\mathcal{W}\big{(}V,(0,1]\sigma\big{)}$ contains a closed two-sided ideal
$K_{0}(V)=\\{a\in\mathcal{W}(V,(0,1]\sigma)\ |\ \lim_{\hbar\to
0}\lVert\phi_{\hbar}(a)\rVert_{\hbar}=0\\}$.

######  Definition 7.

The functor $L_{W}:\mathbf{LinQuant}\to\mathbf{LinClass}$ is defined on
objects by

$\displaystyle\Big{(}\mathcal{W}(V,\hbar\sigma),\Delta(V,\hbar\sigma),\mathcal{Q}_{\hbar}\Big{)}_{\hbar\in(0,1]}\mapsto\Big{(}\mathcal{W}\big{(}V,(0,1]\sigma\big{)}/K_{0}(V),\
\Delta\big{(}V,(0,1]\sigma\big{)}/K_{0}(V)\Big{)}$ (31)

where each quantum model is associated with its classical limit
$\mathcal{W}\big{(}V,(0,1]\sigma\big{)}/K_{0}(V)\cong AP(V^{\prime})$ via the
construction surrounding Eq. (1), and the map on arrows

$\displaystyle\Big{[}\alpha_{1}:\mathcal{W}(V,\sigma)\to\mathcal{W}(U,\sigma^{\prime})\Big{]}\mapsto[\alpha_{0}:AP(V^{\prime})\to
AP(U^{\prime})\Big{]}$ (32)

where each morphism is associated with its classical limit via the
construction surrounding Eqs. (2)-(3).

With these functors now explicitly defined, we have the following result.

######  Theorem 1.

The functors

$\displaystyle Q_{W}:\mathbf{LinClass}\leftrightarrows\mathbf{LinQuant}:L_{W}$
(33)

provide a categorical equivalence.

###### Proof.

We shall establish the equivalent standard for categorical equivalence
provided in [1, p. 172-3] by defining two natural transformations
$\eta:1_{\mathbf{LinClass}}\to L_{W}\circ Q_{W}$ and
$\phi:1_{\mathbf{LinQuant}}\to Q_{W}\circ L_{W}$.

To define $\eta$, we recall that it follows from [31] that for any object
$AP(V^{\prime})$ in $\mathbf{LinClass}$, there is an isomorphism

$\displaystyle\eta_{V}:AP(V^{\prime})\to\mathcal{W}(V,(0,1]\sigma)/K_{0}(V)=L_{W}\circ
Q_{W}(AP(V^{\prime}))$ (34)

generated by the linear, continuous extension of

$\displaystyle\eta_{V}(W_{0}(f))=[\hbar\mapsto\mathcal{Q}_{\hbar}(W_{0}(f))]+K_{0}(V)$
(35)

for any $f\in V$. One can easily check that for any arrow
$\alpha_{0}:AP(V^{\prime})\to AP(U^{\prime})$ in $\mathbf{LinClass}$, the
following diagram commutes:

${AP(V^{\prime})}$${L_{W}\circ
Q_{W}(AP(V^{\prime}))}$${AP(U^{\prime})}$${L_{W}\circ
Q_{W}(AP(U^{\prime}))}$$\scriptstyle{\eta_{V}}$$\scriptstyle{\alpha_{0}}$$\scriptstyle{L_{W}\circ
Q_{W}(\alpha_{0})}$$\scriptstyle{\eta_{U}}$

This establishes that $\eta_{V}$ is a natural isomorphism.

To define $\phi$, we recall that
$Q_{W}(AP(V^{\prime}))=\mathcal{W}(V,\sigma)$, so we can use the isomorphism

$\displaystyle\phi_{V}=Q_{W}(\eta_{V}):\mathcal{W}(V,\sigma)\to Q_{W}\circ
L_{W}(\mathcal{W}(V,\sigma)).$ (36)

One can easily check that for any arrow
$\alpha_{1}:\mathcal{W}(V,\sigma)\to\mathcal{W}(U,\sigma^{\prime})$ in
$\mathbf{LinQuant}$, the following diagram commutes:

${\mathcal{W}(V,\sigma))}$${Q_{W}\circ
L_{W}(\mathcal{W}(V,\sigma))}$${\mathcal{W}(U,\sigma^{\prime})}$${Q_{W}\circ
L_{W}(\mathcal{W}(U,\sigma^{\prime}))}$$\scriptstyle{\phi_{V}}$$\scriptstyle{\alpha_{1}}$$\scriptstyle{Q_{W}\circ
L_{W}(\alpha_{1})}$$\scriptstyle{\phi_{U}}$

This establishes that $\phi_{V}$ is a natural isomorphism. ∎

This shows that the functors $Q_{W}$ and $L_{W}$ provide a one-to-one
correspondence between the structure-preserving maps of each model in
$\mathbf{LinClass}$ and $\mathbf{LinQuant}$.777See [3] for closely related
results on automorphisms of the polynomial Weyl algebras. Hence, this shows a
sense in which, relative to the structure encoded in these choices of
categories, classical and quantum models have shared structure, when compared
with these choices of functors.

We close this section by emphasizing that the morphisms considered in the
categories $\mathbf{LinClass}$ and $\mathbf{LinQuant}$ are significantly
constrained by the restriction that they be additive maps between vector
spaces, as established in Prop. 1. It would be of great interest to extend the
categorical equivalence result to a wider class of morphisms.

##  4 Rieffel’s Quantization for Actions of $\mathbb{R}^{d}$

While quantization via the Weyl algebra is a prominent example, it has limited
applications as well as technical issues.888For more discussion of issues with
the Weyl algebra, see [12, 13, 8, 10, 11]. In this section we will consider a
quantization prescription for a different algebra. To do so, we will restrict
attention to finite-dimensional phase spaces, so we will lose the generality
of the Weyl algebra for representing field theories. But we will allow
ourselves to consider phase spaces that are more generally manifolds and not
necessarily linear spaces.

The method of quantization due to Rieffel [27, 28] applies to classical
systems whose phase space is a manifold $M$ with a diffeomorphic action
$\beta$ of the Lie group $\mathbb{R}^{d}$. In what follows, we will assume the
group $\mathbb{R}^{d}$ acts freely on $M$. Furthermore, we assume the Lie
group carries a symplectic form $\sigma$ on $\mathbb{R}^{d}$, which
corresponds to an antisymmetric matrix $J_{jk}$ on the vector space
$\mathbb{R}^{d}$, understood as the Lie algebra of the Lie group
$\mathbb{R}^{d}$. In this case the Poisson *-algebra
$C_{c}^{\infty}(M)\subseteq C_{b}(M)$ is the collection of smooth, compactly
supported functions on the phase space. This algebra $C_{c}^{\infty}(M)$ is
norm dense in the C*-algebra $C_{0}(M)$ of continuous functions vanishing at
infinity on the phase space. The action of $\mathbb{R}^{d}$ on $M$ defines an
automorphic action $\tau$ of $\mathbb{R}^{d}$ on $C_{0}(M)$ by

$\displaystyle\tau_{x}(f)=f\circ\beta_{x}.$ (37)

The subalgebra $C_{c}^{\infty}(M)$ carries a corresponding infinitesimal
action of the Lie algebra $\mathbb{R}^{d}$ by smooth vector fields $\xi_{X}$
for $X\in\mathbb{R}^{d}$ given by

$\displaystyle\xi_{X}(f)=\frac{\partial}{\partial t}_{|t=0}\tau_{tX}(f)$ (38)

for all $f\in C_{c}^{\infty}(M)$. The Poisson bracket on $C_{c}^{\infty}(M)$
is then defined from the infinitesimal action of the Lie algebra and the
symplectic form $\sigma$ for all $f,g\in C_{c}^{\infty}(M)$ by

$\displaystyle\\{f,g\\}=\sum_{j,k}J_{jk}\xi_{X_{j}}(f)\xi_{X_{k}}(g),$ (39)

where the vectors $\\{X_{k}\\}_{k=1}^{d}$ form a basis for the Lie algebra
$\mathbb{R}^{d}$. This definition does not depend on the chosen basis, and one
can check that it satisfies the conditions of a Poisson bracket on $M$. This
structure specifies the classical model.999The methods developed by Rieffel
[27] for quantization apply much more broadly, even to deforming products on
non-commutative C*-algebras carrying actions of $\mathbb{R}^{d}$. The methods
have been further generalized by Landsman [17, 18, 19, 20] to cases where the
construction is employed locally, including Riemannian manifolds, principal
bundles, and Lie groupoids. Also, Bieliavsky and Gayral [4] have provided a
generalization of the quantization prescription for a much wider class of
group actions.

The corresponding quantum model is obtained by deforming the product on
$C_{c}^{\infty}(M)$. Define $\mathcal{P}_{\hbar}(M)$ to be the vector space
with involution $C_{c}^{\infty}(M)$ with the new multiplication operation
$\star_{\hbar}$, sometimes called the Moyal product, defined by

$\displaystyle
f_{\hbar}\star_{\hbar}g_{\hbar}=\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}\tau_{x}(f)\tau_{y}(g)e^{i\hbar\sigma(x,y)}$
(40)

where we use the notation $f_{\hbar},g_{\hbar}\in\mathcal{P}_{\hbar}(M)$ only
to distinguish these from the identical elements $f,g\in C_{c}^{\infty}(M)$.
Rieffel [28] shows that this expression can be made well-defined in terms of
oscillatory integrals, and that one can define a C*-norm on
$\mathcal{P}_{\hbar}(M)$ so that the completion

$\displaystyle\mathfrak{A}_{\hbar}(M)=\overline{\mathcal{P}_{\hbar}(M)}$

with respect to this norm is a C*-algebra. We note that each C*-algebra
$\mathfrak{A}_{\hbar}(M)$ also carries a strongly continuous group action of
$\mathbb{R}^{d}$, which we denote by $\tau^{\hbar}$, picked out as the unique
continuous extension of the group action $\tau$ on $\mathcal{P}_{\hbar}$ [See
28, Thm. 5.11, p. 44]. Likewise, it follows from [28, Thm. 7.1] that there is
an infinitesimal action of the Lie algebra, which we denote by $\xi^{\hbar}$
on the subalgebra $\mathcal{P}_{\hbar}(M)$.

For example, in the special case where $M=\mathbb{R}^{2n}$ with the group
action $\tau$ for $d=2n$ by translations, we have
$\mathfrak{A}_{\hbar}(\mathbb{R}^{2n})\cong\mathcal{K}(L^{2}(\mathbb{R}^{n}))$.
We can also understand this algebra through the standard Schrödinger
representation of $\mathfrak{A}_{\hbar}(\mathbb{R}^{2n})$ on
$L^{2}(\mathbb{R}^{n})$, which we now denote $\tilde{\pi}_{S}$, given by the
continuous extension of

$\displaystyle\tilde{\pi}_{S}(f_{\hbar})=\int_{\mathbb{R}^{2n}}\frac{d^{n}ad^{n}b}{(2\pi)^{n}}(\mathcal{F}f)(a,b)\pi_{S}(W_{\hbar}(a,b))$
(41)

for $f_{\hbar}\in\mathcal{P}_{\hbar}(\mathbb{R}^{2n})$. Here, $\mathcal{F}f$
denotes the Fourier transform of the function $f\in
C_{c}^{\infty}(\mathbb{R}^{2n})$ and $\pi_{S}(W_{\hbar}(a,b))$ is the
Schrödinger representation of the element $W_{\hbar}(a,b)$ in the Weyl algebra
$\mathcal{W}(\mathbb{R}^{2n},\sigma)$ as given by Eq. (12).

The quantization maps
$\mathcal{Q}_{\hbar}:C_{c}^{\infty}(M)\to\mathfrak{A}_{\hbar}(M)$ are given
for $\hbar\in[0,1]$ by

$\displaystyle\mathcal{Q}_{\hbar}(f)=f_{\hbar}$ (42)

for all $f\in C_{c}^{\infty}(M)$. These quantization maps define a strict
deformation quantization on $M$ with $\mathcal{P}=C_{c}^{\infty}(M)$ and fiber
algebras $\mathfrak{A}_{\hbar}(M)$ for $\hbar>0$.

To define our categories of classical and quantum models suitable for Rieffel
quantization, we will need to specify when a morphism of a C*-algebra (either
$C_{0}(M)$ or $\mathfrak{A}_{\hbar}(M)$) is compatible with a group action.
Suppose we have a *-homomorphism $\alpha:\mathfrak{A}\to\mathfrak{B}$ between
two C*-algebras $\mathfrak{A}$ and $\mathfrak{B}$ carrying group actions by
$\mathbb{R}^{d}$ and $\mathbb{R}^{d^{\prime}}$, respectively. We now denote
the infinitesimal action of the Lie algebra by $\xi$ (corresponding to the
action $\xi$ or $\xi^{\hbar}$, as above.) We will call $\alpha$ _compatible
with the group actions_ if for each $X\in\mathbb{R}^{d^{\prime}}$, there is a
$Y\in\mathbb{R}^{d}$ such that $\xi_{X}\circ\alpha=\alpha\circ\xi_{Y}$ on the
domain of $\xi_{X}$ and $\xi_{Y}$ (i.e., on $C_{c}^{\infty}(M)$ or
$\mathcal{P}_{\hbar}$).

Now we define a category of classical models suitable for Rieffel
quantization.

######  Definition 8.

We denote the following category by $\mathbf{RClass}$:

  * •

_Objects_ are triples $(C_{0}(M),C^{\infty}_{c}(M),\tau)$, where $C_{0}(M)$ is
the C*-algebra of continuous functions vanishing at infinity on a manifold
$M$, and $C_{c}^{\infty}(M)$ is a dense *-subalgebra. Further $\tau$ is a
strongly continuous, free action of $\mathbb{R}^{d}$ on $C_{0}(M)$ arising
from a family of diffeomorphisms by Eq. (37), which defines the Poisson
bracket on $C_{c}^{\infty}(M)$ by Eq. (39).

  * •

_Arrows_ are *-homomorphisms $\alpha_{0}:C_{0}(M)\to C_{0}(N)$ for manifolds
$M$ and $N$ that are compatible with the group actions, that are _smooth_ in
the sense that

$\displaystyle\alpha_{0}\big{[}C_{c}^{\infty}(M)\big{]}\subseteq
C_{c}^{\infty}(N)$ (43)

and that are _Poisson_ in the sense that

$\displaystyle\alpha_{0}\big{(}\\{A,B\\}_{M}\big{)}=\big{\\{}\alpha_{0}(A),\alpha_{0}(B)\big{\\}}_{N}$
(44)

for all $A,B\in C_{c}^{\infty}(M)$.

Note that this category is general enough to include non-linear phase spaces.
The morphisms in this category preserve the Poisson structure of classical
models at $\hbar=0$ as phase spaces.

Similarly, we define a category of quantum models corresponding to Rieffel’s
quantization prescription

######  Definition 9.

We denote the following category by $\mathbf{RQuant}$:

  * •

_Objects_ are strict quantizations
$\big{(}\mathfrak{A}_{\hbar}(M),\mathcal{P}_{\hbar}(M),\mathcal{Q}_{\hbar},\tau^{\hbar}\big{)}_{\hbar\in(0,1]}$
of $\mathcal{P}=C_{c}^{\infty}(M)$, as given by the discussion around Eq. (40)
and (42).

  * •

_Arrows_ are smooth, scaling *-homomorphisms
$\alpha_{1}:\mathfrak{A}_{1}(M)\to\mathfrak{A}_{1}(N)$ that are compatible
with the group actions, where $\mathfrak{A}_{1}(M)$ and $\mathfrak{A}_{1}(N)$
are the C*-algebras at $\hbar=1$ obtained as the Rieffel quantizations of
$C_{c}^{\infty}(M)$ and $C_{c}^{\infty}(N)$, respectively.

The morphisms in this category preserve the structure of the fully quantized
models as non-commutative C*-algebras of operators at $\hbar=1$.

The following proposition characterizing classical morphisms in
$\mathbf{RClass}$ is essential for the definition of the quantization functor.

######  Proposition 2.

If $\alpha_{0}:C_{0}(M)\to C_{0}(N)$ is a morphism in $\mathbf{RClass}$ for
manifolds $M$ and $N$ with actions of $\mathbb{R}^{d}$ and
$\mathbb{R}^{d^{\prime}}$, respectively, then there is a smooth map
$\varphi:N\to M$ such that

$\displaystyle\alpha_{0}(f)=f\circ\varphi$ (45)

for each $f\in C_{0}(M)$, and a symplectic map
$S:\mathbb{R}^{d^{\prime}}\to\mathbb{R}^{d}$ such that

$\displaystyle\alpha_{0}\circ\tau_{SX}=\tau_{X}\circ\alpha_{0}$ (46)

for all $X\in\mathbb{R}^{d^{\prime}}$.

###### Proof.

It follows from Gelfand duality [24, §C.2-3] that there is a continuous map
$\varphi:N\to M$ such that

$\displaystyle\alpha_{0}(f)=f\circ\varphi$ (47)

for all $f\in C_{0}(M)$.

To show that $\varphi$ is smooth, we claim that whenever $f\in C^{\infty}(M)$,
it follows that $f\circ\varphi\in C^{\infty}(N)$. To show this, we suppose
$f\in C^{\infty}(M)$ and aim to show that $f\circ\varphi$ is smooth in an open
neighborhood of each $p\in N$. For any such $p\in N$, we know there is some
open neighborhood $O$ of $\varphi(p)\in M$ and some function $\tilde{f}\in
C_{c}^{\infty}(M)$ such that $\tilde{f}_{|O}=f_{|O}$. Therefore, since
$\alpha_{0}$ is smooth, we have that
$\tilde{f}\circ\varphi=\alpha_{0}(\tilde{f})\in C_{c}^{\infty}(N)$. Moreover,
it follows that $\varphi^{-1}[O]$ is an open neighborhood of $p\in N$ and
$(\tilde{f}\circ\varphi)_{|\varphi^{-1}[O]}=(f\circ\varphi)_{|\varphi^{-1}[O]}$.
Hence, $f\circ\varphi$ is smooth in an open neighborhood of $p$, and since
$p\in N$ was arbitrary, it follows that $f\circ\varphi$ is smooth on $N$.
Finally, it follows from the fact that $f\in C^{\infty}(M)$ was arbitrary that
$\varphi$ is smooth.101010This is a small extension of the well-known fact
known as “Milnor’s exercise” [16, Cor. 35.10]. We have simply extended the
correspondence between algebra homomorphisms and smooth maps from algebras of
the type $C^{\infty}(M)$ to algebras of the type $C_{c}^{\infty}(M)$ for a
manifold $M$.

To show that $\varphi$ corresponds to a symplectic map
$S:\mathbb{R}^{d^{\prime}}\to\mathbb{R}^{d}$, we draw on the fact that
$\alpha_{0}$ is Poisson, meaning that

$\displaystyle\alpha_{0}\big{(}\\{f,g\\}_{M}\big{)}=\big{\\{}\alpha_{0}(f),\alpha_{0}(g)\big{\\}}_{N}$
(48)

for all $f,g\in C_{c}^{\infty}(M)$, which implies

$\displaystyle\\{f,g\\}_{M}\circ\varphi=\\{f\circ\varphi,g\circ\varphi\\}_{N}.$
(49)

Given $X\in\mathbb{R}^{d^{\prime}}$, since $\alpha_{0}$ is compatible with the
group actions, there is an element $SX\in\mathbb{R}^{d}$ satisfying

$\displaystyle\xi_{X}\circ\alpha_{0}=\alpha_{0}\circ\xi_{SX}.$ (50)

Moreover the value $SX\in\mathbb{R}^{d}$ in Eq. (50) is unique since the group
actions are free. So this can be rewritten as a local condition that at each
$p\in N$, where we denote the differential of $\varphi$ at $p\in N$ by
$d\varphi_{p}:T_{p}N\to T_{\varphi(p)}M$. We have that for each $f\in
C_{c}^{\infty}(M)$,

$\displaystyle d\varphi_{p}\big{(}(\xi_{X})_{|p}\big{)}(f)$
$\displaystyle=(\xi_{X})_{|p}(f\circ\varphi)=(\xi_{X})_{|p}(\alpha_{0}(f))=\big{(}(\xi_{X}\circ\alpha_{0})(f)\big{)}(p)$
(51)
$\displaystyle=\big{(}(\alpha_{0}\circ\xi_{SX})(f)\big{)}(p)=\big{(}(\xi_{SX})(f)\big{)}\big{(}\varphi(p)\big{)}=(\xi_{SX})_{|\varphi(p)}(f).$
(52)

In other words, we have that $\xi_{X}$ and $\xi_{SX}$ are $\varphi$-related.
This implies that we have defined a linear map
$S:\mathbb{R}^{d^{\prime}}\to\mathbb{R}^{d}$ for all
$X\in\mathbb{R}^{d^{\prime}}$. The fact that $S$ is linear follows from the
linearity of the differential $d\varphi_{p}$ at each point $p\in N$ and the
linearity of the map $X\mapsto\xi_{X}$. It now follows from the definition of
the Poisson bracket that $S$ is symplectic. Moreover, it follows from the
compatibility with the group actions that

$\displaystyle\alpha_{0}\circ\tau_{SX}=\tau_{X}\circ\alpha_{0},$ (53)

as desired. ∎

This proposition vindicates the earlier remark that morphisms in
$\mathbf{RClass}$ preserve the structure of classical phase spaces. It also
provides a way to lift a morphism to any value $\hbar>0$, as follows. Since
the map $\varphi$ corresponding to $\alpha_{0}$ is a smooth map, it follows
from the compatibility of $\alpha_{0}$ with the group action that the
pushforward $\varphi_{*}$ lifts to a symplectic transformation
$S:\mathbb{R}^{d^{\prime}}\to\mathbb{R}^{d}$. It then follows from Prop. 10.4
of [28, p. 70] that each morphism $\alpha_{0}$ can be quantized.

######  Corollary 2.

If $\alpha_{0}:C_{0}(M)\to C_{0}(N)$ is a morphism in $\mathbf{RClass}$ for
manifolds $M$ and $N$, then the restriction of $\alpha_{0}$ to
$\mathcal{P}_{\hbar}(M)=C_{c}^{\infty}(M)$ continuously extends to a
*-homomorphism
$\alpha_{\hbar}:\mathfrak{A}_{\hbar}(M)\to\mathfrak{A}_{\hbar}(N)$ for each
$\hbar>0$. Moreover, the map $\alpha_{1}$ so defined is smooth and scaling,
and thus is a morphism in $\mathbf{RQuant}$.111111The fact that $\alpha_{1}$,
so defined, is compatible with the group action follows from [28, Thm. 7.1].

Hence, quantization defines a functor as follows.

######  Definition 10.

The functor $Q_{R}:\mathbf{RClass}\to\mathbf{RQuant}$ is defined on objects by

$\displaystyle\Big{(}C_{0}(M),C_{c}^{\infty}(M),\tau\Big{)}\mapsto\Big{(}\mathfrak{A}_{\hbar}(M),\mathcal{P}_{\hbar}(M),\mathcal{Q}_{\hbar},\tau^{\hbar}\Big{)}_{\hbar\in(0,1]}$
(54)

where each classical model is associated with its strict deformation
quantization via Eq. (42), and the map on arrows

$\displaystyle\Big{[}\alpha_{0}:C_{0}(M)\to
C_{0}(N)\Big{]}\mapsto\Big{[}\alpha_{1}:\mathfrak{A}_{1}(M)\to\mathfrak{A}_{1}(N)\Big{]}$
(55)

where each morphism is associated with its quantized counterpart via Prop. 2
and Cor. 2.

In the other direction, the classical limit also defines a functor as follows.
Recall that each strict quantization
$\big{(}\mathfrak{A}_{\hbar}(M),\mathcal{P}_{\hbar}(M),\mathcal{Q}_{\hbar}\big{)}_{\hbar\in(0,1]}$
defines a continuous bundle of C*-algebras with a C*-algebra of continuous
sections that we will denote by $\mathfrak{A}_{(0,1]}(M)$. The algebra
contains a subalgebra $\mathcal{P}_{(0,1]}(M)$ generated by the sections of
the form

$\displaystyle[\hbar\mapsto f_{\hbar}]$ (56)

for fixed $f\in C_{c}^{\infty}(M)$ and all $\hbar\in(0,1]$. Moreover,
$\mathfrak{A}_{(0,1]}(M)$ contains a closed two-sided ideal
$K_{0}(M)=\\{a\in\mathfrak{A}_{(0,1]}(M)\ |\lim_{\hbar\to
0}\lVert\phi_{\hbar}(a)\rVert_{\hbar}=0\\}$.

######  Definition 11.

The functor $L_{R}:\mathbf{RQuant}\to\mathbf{RClass}$ is defined on objects by

$\displaystyle\Big{(}\mathfrak{A}_{\hbar}(M),\mathcal{P}_{\hbar}(M),\mathcal{Q}_{\hbar},\tau^{\hbar}\Big{)}_{\hbar\in(0,1]}\mapsto\Big{(}\mathfrak{A}_{(0,1]}(M)/K_{0}(M),\mathcal{P}_{(0,1]}(M)/K_{0}(M),\tau^{0}\Big{)}$
(57)

where each quantum model is associated with its classical limit
$\mathfrak{A}_{(0,1]}(M)/K_{0}(M)\cong C_{0}(M)$ via the construction
surrounding Eq. (1) and where $\tau^{0}$ is the classical limit of the scaling
morphism $\tau^{1}$, which is specified in the same way the functor maps all
arrows. The functor acts on arrows by

$\displaystyle\Big{[}\alpha_{1}:\mathfrak{A}_{1}(M)\to\mathfrak{A}_{1}(N)\Big{]}\mapsto\Big{[}\alpha_{0}:C_{0}(M)\to
C_{0}(N)\Big{]}$ (58)

where each morphism is associated with its classical limit via the
construction surrounding Eqs. (2)-(3). Steeger and Feintzeig [31, Prop. 5.5]
establish that $\alpha_{0}$ so defined is Poisson, and hence is a morphism in
$\mathbf{RClass}$.121212The fact that $\alpha_{0}$, so defined, is compatible
with the group action follows from [28, Thm. 7.1].

With these functors now explicitly defined, we have the following result.

######  Theorem 2.

The functors

$\displaystyle Q_{R}:\mathbf{RClass}\leftrightarrows\mathbf{RQuant}:L_{R}$
(59)

provide a categorical equivalence.

###### Proof.

The proof proceeds exactly as for Thm. 1. We shall establish the equivalent
standard for categorical equivalence provided in [1, p. 172-3] by defining two
natural transformations $\eta:1_{\mathbf{RClass}}\to L_{R}\circ Q_{R}$ and
$\phi:1_{\mathbf{RQuant}}\to Q_{R}\circ L_{R}$.

To define $\eta$, we recall that it follows from [31] that for any object
$C_{0}(M)$ in $\mathbf{RQuant}$, there is an isomorphism

$\displaystyle\eta_{M}:C_{0}(M)\to\mathfrak{A}_{(0,1]}(M)/K_{0}(M)=L_{R}\circ
Q_{R}(C_{0}(M))$ (60)

generated by the linear, continuous extension of

$\displaystyle\eta_{V}(f)=[\hbar\mapsto\mathcal{Q}_{\hbar}(f)]+K_{0}(M)$ (61)

for any $f\in C_{c}^{\infty}(M)$. One can easily check that for any arrow
$\alpha_{0}:C_{0}(M)\to C_{0}(N)$ in $\mathbf{RClass}$, the following diagram
commutes:

${C_{0}(M)}$${L_{R}\circ Q_{R}(C_{0}(N))}$${C_{0}(N)}$${L_{R}\circ
Q_{R}(C_{0}(N))}$$\scriptstyle{\eta_{M}}$$\scriptstyle{\alpha_{0}}$$\scriptstyle{L_{W}\circ
Q_{W}(\alpha_{0})}$$\scriptstyle{\eta_{N}}$

This establishes that $\eta_{V}$ is a natural isomorphism.

To define $\phi$, we recall that $Q_{R}(C_{0}(M))=\mathfrak{A}_{1}(M)$, so we
can use the isomorphism

$\displaystyle\phi_{M}=Q_{R}(\eta_{M}):\mathfrak{A}_{1}(M)\to Q_{R}\circ
L_{R}(\mathfrak{A}_{1}(M)).$ (62)

One can easily check that for any arrow
$\alpha_{1}:\mathfrak{A}_{1}(M)\to\mathfrak{A}_{1}(N)$ in $\mathbf{RQuant}$,
the following diagram commutes:

${\mathfrak{A}_{1}(M)}$${Q_{R}\circ
L_{R}(\mathfrak{A}_{1}(M))}$${\mathfrak{A}_{1}(N)}$${Q_{R}\circ
L_{R}(\mathfrak{A}_{1}(N))}$$\scriptstyle{\phi_{M}}$$\scriptstyle{\alpha_{1}}$$\scriptstyle{Q_{R}\circ
L_{R}(\alpha_{1})}$$\scriptstyle{\phi_{N}}$

This establishes that $\phi_{V}$ is a natural isomorphism. ∎

Just as in the previous section, this shows that the functors $Q_{R}$ and
$L_{R}$ provide a one-to-one correpsondence between the structure-preserving
maps of each model in $\mathbf{RClass}$ and $\mathbf{RQuant}$. Hence, this
shows a sense in which, relative to the structure encoded in these choices of
categories, classical and quantum models share structure, when compared with
these choices of functors.

We close this section by emphasizing that the morphisms considered in the
categories $\mathbf{RClass}$ and $\mathbf{RQuant}$ are significantly
constrained by the restriction that they satisfy the condition of Eq. (46),
which was established in Prop. 2. We can think of this condition as the
requirement that morphisms are smooth maps that are “almost equivariant” for
the group actions. It would be of great interest to extend the categorical
equivalence result to a wider class of morphisms.

##  5 Conclusion

We have established that in two cases—the quantization of linear spaces using
the C*-Weyl algebra and the quantization of phase spaces with actions of
$\mathbb{R}^{d}$ through Rieffel’s prescription—quantization and the classical
limit each define functors that together form a categorical equivalence. This
shows a precise sense in which for each of these cases, the algebraic
structure of observables in classical physics corresponds with the algebraic
structure of observables in quantum physics.

The models and quantization procedures we have used are among the simplest
mathematically, so the present results serve primarily as a proof of concept.
There are open questions concerning whether the results can be extended either
to a broader class of objects or to a broader class of arrows. Can the results
concerning Rieffel’s quantization be extended to quantization by actions of
non-Abelian groups via the prescription of Bieliavsky and Gayral [4]? Can the
results concerning quantization of *-homomorphisms be extended to the
quantization of more general Hilbert bimodules from symplectic dual pairs via
the prescription of Landsman [22, 23]? We leave these questions for future
research. We hope that the current paper establishes merely the possibility
for positive results and provides reason to be interested in quantization as a
categorical equivalence.

## Acknowledgements

Funding was provided by the National Science Foundation (Grant No. 571
2043089).

## Conflict of Interest Statement

On behalf of all authors, the corresponding author states that there is no
conflict of interest.

## Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

## References

  * [1] S. Awodey.  Category Theory.  Oxford University Press, New York, 2nd edition, 2010.
  * [2] J. Baez, T. Bartels, and J. Dolan.  Property, structure, and stuff.  Quantum Gravity Seminar, University of California, Riverside. http://math.ucr.edu/home/baez/qg-spring2004/, 2004.
  * [3] A. Belov-Kanel, A. Elishev, and J.-T. Yu.  On Automorphisms of the Tame Polynomial Automorphism Group in Positive Characteristic.  https://arxiv.org/abs/2103.12784, March 2021.
  * [4] P. Bieliavsky and V. Gayral.  Deformation Quantization for Actions of Kählerian Lie Groups.  Memoirs of the American Mathematical Society. American Mathematical Society, Providence, RI, 2015.
  * [5] E. Binz, R. Honegger, and A. Rieckers.  Construction and uniqueness of the C*-Weyl algebra over a general pre-symplectic space.  Journal of Mathematical Physics, 45(7):2885–2907, 2004.
  * [6] E. Binz, R. Honegger, and A. Rieckers.  Field-theoretic Weyl Quantization as a Strict and Continuous Deformation Quantization.  Annales de l’Institut Henri Poincaré, 5:327–346, 2004.
  * [7] J. Dixmier.  C*-Algebras.  North Holland, New York, 1977.
  * [8] B. Feintzeig.  The classical limit of a state on the Weyl algebra.  Journal of Mathematical Physics, 59:112102, 2018.
  * [9] B. Feintzeig.  The classical limit as an approximation.  Philosophy of Science, 87(4):612–539, 2020.
  * [10] B. Feintzeig, J. Manchak, S. Rosenstock, and J. Weatherall.  Why be regular? Part I.  Studies in the History and Philosophy of Modern Physics, 65:122–132, 2019.
  * [11] B. Feintzeig and J. Weatherall.  Why be regular? Part II.  Studies in the History and Philosophy of Modern Physics, 65:133–144, 2019.
  * [12] H. Grundling.  A group algebra for inductive limit groups. continuity problems of the canonical commutation relations.  Acta Applicandae Mathematicae, 46:107–145, 1997.
  * [13] H. Grundling and K.-H. Neeb.  Full regularity for a C*-algebra of the Canonical Commutation Relations.  Reviews in Mathematical Physics, 21:587–613, 2009.
  * [14] R. Honegger, A. Rieckers, and L. Schlafer.  Field-Theoretic Weyl Deformation Quantization of Enlarged Poisson Algebras.  Symmetry, Integrability and Geometry: Methods and Applications, 4:047–084, 2008.
  * [15] E. Kirchberg and S. Wasserman.  Operations on continuous bundles of C*-algebras.  Mathematische Annalen, 303:677–697, 1995.
  * [16] Ivan Kolář, Jan Slovák, and Peter W. Michor.  Natural Operations in Differential Geometry.  Springer Berlin Heidelberg, 1993.
  * [17] N. P. Landsman.  Deformations of Algebras of Observables and the Classical Limit of Quantum Mechanics.  Reviews in Mathematical Physics, 5(4), 1993.
  * [18] N. P. Landsman.  Strict deformation quantization of a particle in external gravitational and Yang-Mills fields.  Journal of Geometry and Physics, 12:93–132, 1993.
  * [19] N. P. Landsman.  Mathematical Topics Between Classical and Quantum Mechanics.  Springer, New York, 1998.
  * [20] N. P. Landsman.  Lie Groupoid C*-algebras and Weyl Quantization.  Communications in Mathematical Physics, 206(2):367–381, 1999.
  * [21] N. P. Landsman.  Bicategories of operator algebras and Poisson manifolds.  In Mathematical physics in mathematics and physics. Quantum and operator algebraic aspects. Proceedings of a conference, Siena, Italy, June 20–24, 2000. Dedicated to Sergio Doplicher and John E. Roberts on the occasion of their 60th birthday, pages 271–286. AMS, American Mathematical Society, Providence, RI, 2001.
  * [22] N. P. Landsman.  Operator Algebras and Poisson Manifolds Associated to Groupoids.  Communications in Mathematical Physics, 222:97–116, 2001.
  * [23] N. P. Landsman.  Quantization as a Functor.  In T. Voronov, editor, Quantization, Poisson Brackets and beyond, pages 9–24. Contemp. Math., 315, AMS, Providence, 2003.
  * [24] N. P. Landsman.  Foundations of Quantum Theory: From Classical Concepts to Operator Algebras.  Springer, 2017.
  * [25] J. Manuceau, M. Sirugue, D. Testard, and A. Verbeure.  The Smallest C*-algebra for the Canonical Commutation Relations.  Communications in Mathematical Physics, 32:231–243, 1974.
  * [26] D. Petz.  An Invitation to the Algebra of Canonical Commutation Relations.  Leuven University Press, Leuven, 1990.
  * [27] M. Rieffel.  Deformation Quantization of Heisenberg manifolds.  Communications in Mathematical Physics, 122:531–562, 1989.
  * [28] M. Rieffel.  Deformation quantization for actions of $\mathbb{R}^{d}$.  Memoirs of the American Mathematical Society. American Mathematical Society, Providence, RI, 1993.
  * [29] W. Rudin.  Fourier Analysis on Groups.  Wiley & Sons, Inc., 1962.
  * [30] J. Slawny.  On factor representations and the C*-algebra of canonical commutation relations.  Communications in Mathematical Physics, 24(2):151–170, jun 1972\.
  * [31] J. Steeger and B. Feintzeig.  Extensions of bundles of C*-algebras.  Reviews in Mathematical Physics, 33(8):2150025, 2021.


where $M_{\rm c,max}$ is the Mach number evaluated at $T=T_{\rm c,max}$. The
‘unbound radius’ we are after is the transition radius where surfaces of
$Be=0$ and $T=T_{\rm c,max}$ no longer coincide in Fig. 9. It lies interior to
the sonic surface, so a lower limit is found by setting $M_{\rm c,max}=1$ in
the above expression. This defines a characteristic radius

$R_{\rm u}\equiv 2\,r_{0}\,{\rm
HEP}_{0}\frac{\gamma-1}{\gamma+1}\frac{T_{0}}{T_{\rm c,max}}.$ (B2)

Eliminating $r_{0}\,{\rm HEP}_{0}$ in favor of $R_{\rm IC}$ using Eq. (4)
gives the expression quoted in Eq. (11).