#  Deconjugation of canonical variables and the Koopman-von Neumann theory

Mustafa Amin<EMAIL_ADDRESS>Mark A. Walton<EMAIL_ADDRESS>
###### Abstract

The Koopman-von Neumann (KvN) theory is one where the dynamical momentum is
not canonically conjugate to position, i.e., position and momentum are
deconjugated. From this point of view, we show that the KvN theory arises from
quantum mechanics, extracting classical equations of motion from quantum ones.
However, preserving the canonical structure of the theory requires introducing
“auxiliary” canonical conjugates to position and momentum. We show that using
the KvN formulation to study the interaction between quantum and classical
systems forces the auxiliary variables to take on a physical role. While
giving rise to classical behaviour, the KvN theory might be more than
classical.

## Introduction

This paper grew out of an attempt to understand the underlying structure of
the Koopman-von Neumann (KvN) formulation of classical mechanics. In that
formulation, physical quantities are represented as operators acting on
Hilbert space, much like the operator formulation of quantum mechanics.
However, a tension arises between the classical nature of the theory and the
canonical structure of operator dynamics. On one hand, the physical operators
must commute in order to give classical vanishing uncertainty relations. On
the other hand, the natural dynamical bracket for operators is the commutator,
which would identically vanish for the classical physical operators.
Consistency then requires the introduction of auxiliary variables that are
canonically conjugate to the physical position and momentum, but are typically
considered unphysical.

Unlike canonical theories, the KvN formulation then contains an object that is
conjugate to the physical position, but is not itself the physical momentum.
Similarly, there exists an object that is conjugate to the physical momentum,
but is not the physical position. This inspires the separation of the concepts
of conjugate momentum and dynamical momentum, which are usually identified as
one and the same.

Here we examine the process of deconjugating the dynamical momentum from its
associated position in quantum mechanics. The analysis presented here shows
that the KvN theory of a classical system can be “extracted” from its quantum
analogue in a manner akin to, but not the same as, taking the classical limit.
In this respect, we find hints that the auxiliary variables of the theory
might not be easily dismissed as unphysical. The KvN formulation seems to
describe more than just classical mechanics.

The possibility that the auxiliary variables are physical is reinforced when
we let a KvN system interact with a quantum one. If the KvN formulation is
used to describe quantum-classical interactions, we show that the extra
variables show up in the equations of motion of the manifestly physical ones.
The mixing of physical and auxiliary variables is clearly demonstrated in the
Heisenberg picture.

This paper is organized as follows. We set up some useful notation in Sec. 1
and a summary of the canonical formulation in Sec. 2. We then introduce and
discuss the process of deconjugation in Sec. 3 along with the resulting
equations of motion. The KvN formulation and its relationship to the classical
limit are discussed in Sec. 4 through the lens of deconjugation. Finally, we
show in Sec. 5 the role of the auxiliary variables in the physical equations
of motion of a classical-quantum system.

##  Notation

Square brackets $[u,v]$ are used to denote the commutator of two variables
$uv-vu$. Double square brackets denote the commutator divided by $i\hbar$

$\displaystyle[\\![u,v]\\!]=\frac{1}{i\hbar}[u,v]~{}.$ (1.1)

This notation is useful for comparison with classical mechanics. Note that,
for variables $u$ and $v$ that are not explicit functions of $\hbar$, it can
be shown that

$\displaystyle[u,v]=O(\hbar)$ $\displaystyle\implies\lim_{\hbar\to
0}[u,v]=0~{},$ (1.2) $\displaystyle[\\![u,v]\\!]=O(\hbar^{0})$
$\displaystyle\implies\lim_{\hbar\to 0}[\\![u,v]\\!]\neq 0~{}.$ (1.3)

We will use an underline to signify an expression where the fundamental
variables $q$ and $p$ commute

$\displaystyle qp\neq pq~{},\quad\underline{qp}=\underline{pq}~{}.$ (1.4)

This implies

$\displaystyle\underline{f(q,p)}=f(q,p)\big{|}_{qp=pq}~{}.$ (1.5)

Note that a vanishing expression implies the vanishing of its underline, but
not the other way around

$\displaystyle f(q,p)=0$ $\displaystyle\implies\underline{f(q,p)}=0~{},$ (1.6)
$\displaystyle\underline{f(q,p)}=0$ $\displaystyle\centernot\implies
f(q,p)=0~{}.$ (1.7)

Finally, in Sec. 3, we will use the symbols $\tilde{q}$, $\tilde{p}$ and
$\tilde{H}$ that correspond to the usual $-\lambda_{p}$, $\lambda_{q}$ and the
Liouvillian $L_{H}$ in the literature on KvN theory. These variables will be
discussed in context. Any variable $u$ should be understood as a function
$u(q,p)$ of $q$ and $p$ only, while a tilde over a symbol $\tilde{u}$ implies
that it is a function $\tilde{u}(q,\tilde{p},\tilde{q},p)$ .

##  Summary of the Hamiltonian formulation

In a Hamiltonian theory, one usually starts with pairs of variables $(q,p)$, a
Hamiltonian $H(q,p)$ and a dynamical bracket $[\\![\cdot\,,\cdot]\\!]$ with
which a set of fundamental relations are defined

$\displaystyle[\\![q,p]\\!]=1~{},\quad[\\![q,q]\\!]=[\\![p,p]\\!]=0~{}.$ (2.1)

The dynamical bracket $[\\![\cdot\,,\cdot]\\!]$ is a binary operation between
the dynamical variables $(u,v,w,\cdots)$ that obeys the following conditions:

$\displaystyle[\\![v,u]\\!]$ $\displaystyle=-[\\![u,v]\\!]~{},$ (2.2)
$\displaystyle[\\![u+v,w]\\!]$ $\displaystyle=[\\![u,w]\\!]+[\\![v,w]\\!]~{},$
(2.3) $\displaystyle[\\![[\\![u,v]\\!],w]\\!]$
$\displaystyle=[\\![[\\![u,w]\\!],v]\\!]+[\\![u,[\\![v,w]\\!]]\\!]~{},$ (2.4)
$\displaystyle[\\![uv,w]\\!]$
$\displaystyle=[\\![u,w]\\!]v+u[\\![v,w]\\!]~{}.$ (2.5)

Those are the antisymmetry, linearity, Jacobi and Leibniz properties. In
addition, the bracket vanishes between dynamical variables and scalars
$[\\![u,c]\\!]=0$ . Using the fundamental relations and the Leibniz property,
it is easy to show that the bracket acts like a derivative (e.g.,
$[\\![q^{n},p]\\!]=nq^{n-1}$), so we can define

$\displaystyle\partial_{q}u$ $\displaystyle:=[\\![u,p]\\!]~{},$ (2.6)
$\displaystyle\partial_{p}u$ $\displaystyle:=[\\![q,u]\\!]~{}$ (2.7)

even for noncommutative variables.

The Hamiltonian $H$ defines the time evolution of the system. In the
Heisenberg picture, the system evolves through the dynamical variables

$\displaystyle\frac{d}{dt}u=[\\![u,H]\\!]+\frac{\partial}{\partial t}u~{},$
(2.8)

while in the Schrödinger picture, the dynamical variables are static and the
system evolves through its state $\rho$ according to the von Neumann equation

$\displaystyle[\\![\rho,H]\\!]+\frac{\partial}{\partial t}\rho=0~{}.$ (2.9)

The Heisenberg and Schrödinger pictures are related through the equality of
the time dependence of expectations values in a given state $\rho$

$\displaystyle\langle u(t)\rangle_{\rho}=\langle u\rangle_{\rho(t)}~{}.$
(2.10)

The left hand side of the above equation is calculated in the Heisenberg
picture and the right hand side is in the Schrödinger picture. Using either
pictures, the Ehrenfest equation can be derived as

$\displaystyle\frac{d}{dt}\langle
u\rangle=\left\langle[\\![u,H]\\!]\right\rangle+\left\langle\frac{\partial}{\partial
t}u\right\rangle~{}.$ (2.11)

This equation for the evolution of expectation values is a consequence of the
Hamiltonian formulation and the definition of expectation values.

It is clear that a central role is played by the algebraic properties of the
dynamical bracket along with the fundamental relations. In fact, none of the
above formal structure is specific to either classical or quantum dynamics.
The same formal equations hold for classical mechanics replacing the re-scaled
commutator $[\\![\cdot\,,\cdot]\\!]$ with the Poisson bracket
$\\{\cdot\,,\cdot\\}$ and the state $\rho$ would be the Liouville phase-space
distribution. Of course, in a classical theory, all variables commute.
Further, the formal structure above is also valid for phase-space quantum
mechanics with the appropriate identification of variables and their product.

##  Deconjugation

Using the fundamental relations stated in the previous section, we can show
that the value of the dynamical variable $u$ after a translation $q\to
q+\epsilon$ is given by

$\displaystyle u(q+\epsilon,p)=e^{\epsilon[\\![\cdot\,,p]\\!]}u(q,p)~{}.$
(3.1)

That is, because $p$ is the canonical conjugate of $q$ (i.e.,
$[\\![q,p]\\!]=1$), it is the generator of translation along $q$. The role of
the conjugate is independent of the dynamics (equations of motion) and the
Hamiltonian.

On the other hand, the Heisenberg equation of motion for $q$ relates $\dot{q}$
and $p$

$\displaystyle\dot{q}=[\\![q,H]\\!]=v(q,p)~{}.$ (3.2)

The dot indicates a total time derivative. For example, the simple Hamiltonian
$H=p^{2}/2m+V(q)$ gives $\dot{q}=p/m$ . It is the appearance of $p$ in the
Hamiltonian that lands it on the right hand side of (3.2) . The role played by
the variable $p$ in this equation is that of dynamical momentum. We call it
“dynamical” because it arises from the equations of motion which depend on the
form of the Hamiltonian.

Conjugate momentum and dynamical momentum are different concepts and need not
be identical. To demonstrate the difference, consider a system of two degrees
of freedom with the usual fundamental relations (with $i,j\in{1,2}$)

$\displaystyle[\\![q_{i}\,p_{j}]\\!]=\delta_{ij}~{},\quad[\\![q_{i},q_{j}]\\!]=[\\![p_{i},p_{j}]\\!]=0~{}.$
(3.3)

Let the Hamiltonian be given by

$\displaystyle H=\frac{p_{1}p_{2}}{m}+k\,q_{1}q_{2}~{},$ (3.4)

then the equations of motion are

$\displaystyle\dot{q}_{1}$
$\displaystyle=p_{2}/m~{},\quad\dot{q}_{2}=p_{1}/m~{},$ (3.5)
$\displaystyle\dot{p}_{1}$
$\displaystyle=-k\,q_{2}~{},\quad\dot{p}_{2}=-k\,q_{1}~{}.$ (3.6)

In this example, we see that $p_{1}$ is the conjugate momentum of $q_{1}$
while its dynamical momentum is $p_{2}$ . The Heisenberg equations of motion
were used here because they provide a straightforward illustration of the
different concepts of conjugate and dynamical momentum. That distinction,
however, is not specific to the Heisenberg picture. Observe that the dynamical
momentum can be defined as the one resulting from the bracket
$[\\![q,H]\\!]=v(q,p)$ without imposing the time dependence of $q$ or $p$.

If we wish to have a theory where the conjugate momentum is not the same as
the dynamical momentum, then we need to deconjugate the momentum $p$ from its
dynamically associated $q$ . As demonstrated above, this requires twice as
many variables as those of the standard theory. For example, in the KvN
theory, the need to separate the two types of momenta arises from the desire
to formulate classical mechanics in the language of operators. There, the
natural dynamical bracket is the commutator, but maintaining the classical
nature of the theory requires observables to commute. Here we will see that
this problem can be formulated in more general terms.

The canonical Hamiltonian theory was defined in Sec. 2. Let us now define a
deconjugation of that theory. We shall denote the canonical conjugate of $q$
as $\tilde{p}$ and the canonical conjugate of $p$ as $\tilde{q}$ . To avoid
confusion, we will use the “tilde-bracket” $[\\![\cdot\,,\cdot]\\!]_{\\!\sim}$
for the new deconjugated relations and keep the original bracket
$[\\![\cdot\,,\cdot]\\!]$ for the standard relations. The deconjugated
fundamental relations are

$\displaystyle[\\![q,\tilde{p}]\\!]_{\\!\sim}=[\\![\tilde{q},p]\\!]_{\\!\sim}=1~{},\quad[\\![q,p]\\!]_{\\!\sim}=[\\![\tilde{q},\tilde{p}]\\!]_{\\!\sim}=0~{},$
(3.7)

and all other brackets vanish. The number of variables in the theory has
doubled. But this doubling is simply a result of deconjugation or the
separation of conjugate ($\tilde{p}$) and dynamical ($p$) momenta, not an
addition of truly independent degrees of freedom.

The new fundamental relations (3.7) imply that now
$[\\![\cdot\,,\tilde{p}]\\!]_{\\!\sim}$ plays the role of
$[\\![\cdot\,,p]\\!]$ in the original formulation and similarly for
$\tilde{q}$ when acting on $u(q,p)$ :

$\displaystyle\begin{split}\left(\partial_{q}u\right)_{\\!\sim}&=[\\![u,\tilde{p}]\\!]_{\\!\sim}=\underline{[\\![u,p]\\!]}=\underline{\partial_{q}u}~{},\\\
\left(\partial_{p}u\right)_{\\!\sim}&=[\\![\tilde{q},u]\\!]_{\\!\sim}=\underline{[\\![q,u]\\!]}=\underline{\partial_{p}u}~{}.\end{split}$
(3.8)

We use a tilde subscript to indicate that a quantity is calculated in the
tilde-theory. The fact that $q$ and $p$ commute in the tilde-formulation
implies some loss of structure. We can assume that the tilde-variables
($\tilde{q}$,$\tilde{p}$) encode the missing noncommutative details. If this
is true, then those extra variables might not be unphysical as is usually
assumed, though the original theory cannot be uniquely recovered from its
deconjugated theory.

The original Hamiltonian $H(q,p)$ cannot give rise to the equations of motion,
because its tilde-bracket with $q$ or $p$ vanishes. We must construct a new
tilde-Hamiltonian $\tilde{H}(q,\tilde{p},\tilde{q},p)$ to provide such
equations. If we choose the tilde-Hamiltonian such that $(\dot{q})_{\\!\sim}$
is a function of $(q,\tilde{p},\tilde{q},p)$, then it would not be clear which
variable plays the role of the dynamical momentum of $q$ . Thus, the tilde-
variables must disappear from the equation of motion for $q$, and similarly
for $p$ . Then the tilde-Hamiltonian must be linear in $\tilde{q}$ and
$\tilde{p}$ . Further, our goal is to merely deconjugate an existing canonical
theory, not to introduce another canonical one with twice the degrees of
freedom. Then we ask that the tilde equations of motion for $q$ and $p$ be
identical to their original counterparts, save for the commutativity:

$\displaystyle\begin{split}(\dot{q})_{\\!\sim}&=[\\![q,\tilde{H}]\\!]_{\\!\sim}=\underline{[\\![q,H]\\!]}=\underline{\partial_{p}H}=\underline{\dot{q}}~{},\\\
(\dot{p})_{\\!\sim}&=[\\![p,\tilde{H}]\\!]_{\\!\sim}=\underline{[\\![p,H]\\!]}=-\underline{\partial_{q}H}=\underline{\dot{p}}~{}.\end{split}$
(3.9)

For these equations to hold, we write the tilde-Hamiltonian as

$\displaystyle\tilde{H}=\tilde{p}\underline{\partial_{p}H}+\underline{\partial_{q}H}\tilde{q}+\underline{\alpha_{H}(q,p)}~{},$
(3.10)

where $\alpha_{H}(q,p)$ is arbitrary. The placement of $\tilde{q}$ and
$\tilde{p}$ in this equation is not unique. For example, the first term could
have been $\underline{\partial_{p}H}\tilde{p}$. The choice of ordering and of
$\alpha_{H}$, however, do not affect the equation of motion for any variable
$u(q,p)$ as shown below, but will affect those of tilde-variables.

The original Hamiltonian can, in general, be any function of $q$ and $p$.
Therefore, Eqn. (3.10) defining the tilde-Hamiltonian can be generalized to
obtain the “tilde-image” of a general original variable $u(q,p)$

$\displaystyle\tilde{u}(q,\tilde{p},\tilde{q},p)=\tilde{p}\underline{\partial_{p}u}+\underline{\partial_{q}u}\tilde{q}+\underline{\alpha_{u}(q,p)}~{}.$
(3.11)

The tilde-image is defined up to an arbitrary function $\alpha_{u}(q,p)$ with
the condition $\alpha_{q}=\alpha_{p}=0$ . Taking the tilde-bracket of an
original variable $u(q,p)$ with the tilde-image of another variable
$\tilde{v}(q,\tilde{p},\tilde{q},p)$, and using (3.8), we get

$\displaystyle[\\![u,\tilde{v}]\\!]_{\\!\sim}=[\\![\tilde{u},v]\\!]_{\\!\sim}=\underline{\partial_{q}u\partial_{p}v}-\underline{\partial_{q}v\partial_{p}u}~{}.$
(3.12)

The tilde-bracket of $u(q,p)$ and $\tilde{v}(q,\tilde{p},\tilde{q},p)$ is just
the Poisson bracket of $u(q,p)$ and $v(q,p)$ with commuting $q$ and $p$ .
Equations (3.11) and (3.12) define the correspondence between the original
canonical theory and its deconjugation.

The Poisson bracket appears, and tilde-variables are absent from it. This is
by design. We chose the tilde-image $\tilde{u}$ of a general variable $u$ in
the original theory to be linear in $\tilde{q}$ and $\tilde{p}$ precisely for
that to happen. Creating tilde-images of the canonical theory is, of course,
not the only way to have functions of $\tilde{q}$ and $\tilde{p}$ , but we
will not be concerned with those.

While the equations of motion for the original variables are independent of
$\tilde{q}$ and $\tilde{p}$, the reverse is not generally true. For example,
the simple Hamiltonian

$\displaystyle H=\frac{p^{2}}{2m}+V(q)$ (3.13)

gives rise to the tilde-Hamiltonian

$\displaystyle\tilde{H}=\frac{\tilde{p}p}{m}+V^{\prime}(q)\tilde{q}+\underline{\alpha_{H}(q,p)}~{},$
(3.14)

and the equations of motion are

$\displaystyle\dot{q}$ $\displaystyle=p/m~{},\quad\dot{p}=-V^{\prime}(q)~{},$
(3.15) $\displaystyle\dot{\tilde{q}}$
$\displaystyle=\tilde{p}/m+\underline{\partial_{p}\alpha_{H}}~{},\quad\dot{\tilde{p}}=-V^{\prime\prime}(q)\tilde{q}-\underline{\partial_{q}\alpha_{H}}~{}.$
(3.16)

This one-sided relationship between the original and the tilde-variables is
interesting. One (visible) set of variables acts on, but is not acted-upon by,
the other (hidden) set. As we will see in Sec. 5, this will not remain the
case when mixing canonical (quantum) and deconjugated (classical) systems.
Further, since we are restricted to using tilde-Hamiltonians that are linear
in $\tilde{q}$ and $\tilde{p}$, we don’t have any direct interaction between
tilde-variables.

For completeness, we can find the tilde-Lagrangian given by a Legendre
transform of the tilde-Hamiltonian

$\displaystyle\tilde{L}=\tilde{p}\dot{q}+p\dot{\tilde{q}}-\tilde{H}~{}.$
(3.17)

For the simple tilde-Hamiltonian (3.14), we get

$\displaystyle\tilde{L}=m\dot{q}\dot{\tilde{q}}-V^{\prime}(q)\tilde{q}-\alpha_{H}(q,p)~{}.$
(3.18)

Finally, a note on conserved quantities after deconjugation. From the
equations of motion for $q$ and $p$ (3.9), we see immediately that if $p$ (or
$q$) is conserved in the original canonical theory, then it is also conserved
(under $\tilde{H}$) in the deconjugated one. Further, it is easy to show that
$\tilde{p}$, too, will be conserved for a certain choice of $\alpha_{H}$ in
$\tilde{H}$ . To generalize the relationship between conserved quantities in
the canonical and the deconjugated theories, we make use of McCoy’s formula
[1]

$\displaystyle[\\![u,v]\\!]=\sum_{k=1}^{\infty}\frac{(-i\hbar)^{k-1}}{k!}\left(\partial_{q}^{k}u\partial_{p}^{k}v-\partial_{q}^{k}v\partial_{p}^{k}u\right)~{}.$
(3.19)

If $[\\![u,H]\\!]=0$, then underlining (setting $qp=pq$) the terms in the
expansion, we get

$\displaystyle\sum_{k=1}^{\infty}\frac{(-i\hbar)^{k-1}}{k!}\left(\underline{\partial_{q}^{k}u\partial_{p}^{k}H}-\underline{\partial_{q}^{k}H\partial_{p}^{k}u}\right)=0.$
(3.20)

If $u$ and $H$ do not contain $\hbar$ explicitly, the underlined terms do not
contain $\hbar$’s at all since $q$ and $p$ are set to commute. Then each term
in the expansion (3.20) must vanish individually. Noticing that the first term
is nothing but the tilde-bracket of $u$ and $\tilde{H}$, we get the result

$\displaystyle[\\![u,H]\\!]=0\implies[\\![u,\tilde{H}]\\!]_{\\!\sim}=0~{}.$
(3.21)

The reverse is generally not true

$\displaystyle[\\![u,\tilde{H}]\\!]_{\\!\sim}=0\centernot\implies[\\![u,H]\\!]=0~{}.$
(3.22)

Thus, conservation in the canonical theory implies conservation in its
deconjugation, but not vice versa111In fact, any generator $G$ of
transformation in the canonical theory will have its role played by its tilde-
image $\tilde{G}$ in the deconjugated theory. This emphasises that tilde-
images are not truly “new” or separate variables.. This is expected if the
canonical theory is quantum and the deconjugated one represents classical
mechanics.

The general arguments of this section are valid for systems of any number of
degrees of freedom as long as all degrees of freedom are deconjugated. In Sec.
5 we shall see that deconjugating only a subset of the degrees of freedom is,
in general, problematic.

##  KvN is deconjugated QM

The Koopman-von Neumann operator formulation of classical mechanics [2, 3, 4]
uses the same tilde-commutation relations (3.7) . Indeed, the generator of
time evolution (3.10) obtained here appears in the KvN literature. The
deconjugation view presented here allows us to see KvN not only as a
formulation of classical mechanics, but as one extracted directly from quantum
mechanics. Looking at McCoy’s formula (3.19), we see that the tilde-bracket of
any variable $u(q,p)$ and the tilde-image of any other variable $v(q,p)$ is
the same as the underlined $\hbar^{0}$ term in the expansion for the quantum
bracket $[\\![u,v]\\!]$ . In other words, the KvN bracket can be obtained by
setting the commutator of $q$ and $p$ to zero (underlining) then taking
$\hbar\to 0$. This is reminiscent of phase-space quantum mechanics viewed as a
deformation of classical mechanics.

Let us examine this further. The KvN formulation describes classical mechanics
in terms of operators, often called the “language of quantum mechanics”. It is
sometimes compared to the phase-space formulation of quantum mechanics which
uses the “language of classical mechanics”, functions on phase space. This
analogy, however, might be superficial. While phase-space QM takes simple CM
and deforms it until it becomes quantum, an operator description of a
commutative theory (classical mechanics) requires additional complications:
the inclusion of tilde-variables.

Deconjugation (KvN) finds classical equations of motion and uncertainty
relations out of quantum ones, but at the cost of introducing new quantum
equations and uncertainties:

$\displaystyle\begin{split}\Delta q\Delta\tilde{p}&\geq\hbar/2~{},\\\
\Delta\tilde{q}\Delta p&\geq\hbar/2~{}.\end{split}$ (4.1)

A rich quantum structure still exists. The tilde-variables are not easily
dismissed as unphysical, they generate transformations and can be conserved
quantities as mentioned in the previous section. In the next section, we show
that the tilde-variables do not remain “hidden” when a KvN system interacts
with a quantum one.

##  Partial deconjugation and Q-C systems

We have seen that the process of deconjugation alters the canonical relations
yet reproduces the original equations of motion modulo the ordering of $q$ and
$p$. This is true when the process is applied to all degrees of freedom.
However, the equations of motion will be altered when deconjugating only a
subset of the degrees of freedom. As we discuss in this section, this might
affect the use of the KvN theory in the study of quantum-classical
interactions. For a treatment of quantum-classical systems using KvN theory,
see [5] and references therein.

Consider a system of two degrees of freedom $q_{1}$ and $q_{2}$ with canonical
momenta $p_{1}$ and $p_{2}$. The equations of motion in the standard
formulation with a Hamiltonian $H(q_{i},p_{i})$ are

$\displaystyle\dot{q}_{i}=[\\![q_{i},H]\\!]~{},\quad\dot{p}_{i}=[\\![p_{i},H]\\!]~{},$
(5.1)

where $i=1,2$ . Now suppose that we choose to deconjugate only $q_{1}$ and
$p_{1}$, while keeping the standard canonical relation between $q_{2}$ and
$p_{2}$ intact. The new canonical relations are

$\displaystyle\begin{split}[\\![q_{1},\tilde{p}_{1}]\\!]_{\\!\sim}=[\\![\tilde{q}_{1},p_{1}]\\!]_{\\!\sim}=[\\![q_{2},p_{2}]\\!]_{\\!\sim}&=1~{},\\\
[\\![q_{1},p_{1}]\\!]_{\\!\sim}&=0~{}.\end{split}$ (5.2)

All other brackets vanish. Following the procedure in Sec. 3, we set the
tilde-Hamiltonian to

$\displaystyle\tilde{H}=\tilde{p}_{1}\underline{\partial_{p_{1}}H}+\underline{\partial_{q_{1}}H}\tilde{q}_{1}+\underline{H}+\underline{\alpha_{H}(q_{1},p_{1})}~{}.$
(5.3)

Here the underlines mark expressions in which $q_{1}$ and $p_{1}$ are set to
commute. This tilde-Hamiltonian gives the correct equations of motion for
$q_{1}$ and $p_{1}$:

$\displaystyle(\dot{q}_{1})_{\\!\sim}=\underline{\partial_{p_{1}}H}~{},\quad(\dot{p}_{1})_{\\!\sim}=-\underline{\partial_{q_{1}}H}~{}.$
(5.4)

However, for $q_{2}$ and $p_{2}$ (the still-conjugated part of the system) we
get

$\displaystyle\begin{split}(\dot{q}_{2})_{\\!\sim}&=[\\![q_{2},\tilde{H}]\\!]_{\\!\sim}\\\
&=[\\![q_{2},\underline{H}]\\!]+\tilde{p}_{1}[\\![q_{2},\underline{\partial_{p_{1}}H}]\\!]+[\\![q_{2},\underline{\partial_{q_{1}}H}]\\!]\,\tilde{q}_{1}~{},\\\
\end{split}$ (5.5)

and

$\displaystyle\begin{split}(\dot{p}_{2})_{\\!\sim}&=[\\![p_{2},\tilde{H}]\\!]_{\\!\sim}\\\
&=[\\![p_{2},\underline{H}]\\!]+\tilde{p}_{1}[\\![p_{2},\underline{\partial_{p_{1}}H}]\\!]+[\\![p_{2},\underline{\partial_{q_{1}}H}]\\!]\,\tilde{q}_{1}~{}.\end{split}$
(5.6)

We get extra terms, proportional to $\tilde{q}_{1}$ and $\tilde{p}_{1}$, in
the equations of motion for $q_{2}$ and $p_{2}$. Preserving the original
equations of motion for the deconjugated part of the system spoils those of
the non-deconjugated part.

Before discussing the implications of this, we can first investigate the
possibility of restoring the correct equations of motion dynamically; that is,
through the equations of motion for $\tilde{q}_{1}$ and $\tilde{p}_{1}$.
Consider the simple Hamiltonian

$\displaystyle
H=\frac{p_{1}^{2}}{2m_{1}}+\frac{p_{2}^{2}}{2m_{2}}+V(q_{1},q_{2})~{}.$ (5.7)

The tilde-Hamiltonian then is

$\displaystyle\tilde{H}=\frac{\tilde{p}_{1}p_{1}}{m_{1}}+\frac{p_{2}^{2}}{2m_{2}}+\partial_{q_{1}}V+V+\alpha_{H}(q_{1},p_{1})~{}.$
(5.8)

The equations of motion for $q_{2}$ and $p_{2}$ become

$\displaystyle(\dot{q}_{2})_{\\!\sim}=\frac{p_{2}}{m_{2}}~{},\quad(\dot{p}_{2})_{\\!\sim}=-\partial_{q_{2}}V-\partial_{q_{2}}\partial_{q_{1}}V\,\tilde{q}_{1}~{}.$
(5.9)

We see that if the “unphysical” $\tilde{q}_{1}$ is dynamically zero for all
times, the original equations of motion for $p_{2}$ are recovered. For the
simple Hamiltonian above, this is possible since the equation of motion for
$\tilde{q}_{1}$ combined with that of $\tilde{p}_{1}$ gives

$\displaystyle\ddot{\tilde{q}}_{1}=\frac{1}{m_{1}}\left(-\partial_{q_{1}}^{2}V\,\tilde{q}_{1}+\dot{p}_{1}-\partial_{q_{1}}\alpha_{H}\right)+\frac{d}{dt}\partial_{p_{1}}\alpha_{H}~{}.$
(5.10)

We can set the arbitrary function to be $\alpha_{H}=-p_{1}^{2}/2m_{1}$, which
gives

$\displaystyle\ddot{\tilde{q}}_{1}=-\frac{1}{m_{1}}\partial_{q_{2}}^{2}V\,\tilde{q}_{1}~{}.$
(5.11)

Now, if we assert that $\tilde{q}_{1}$ is an unphysical variable, then we can
choose its initial conditions so that $\tilde{q}_{1}(t)=0$. This choice
restores the original equation of motion for $p_{2}$.

Treating $\tilde{q}_{1}$ and $\tilde{p}_{1}$ this way may be problematic,
however. As mentioned before, even though they are not the physical position
and momentum, they are still the generators of translation in the momentum and
configuration spaces, respectively. Further, should we think of those
variables as unphysical if the choice of their initial conditions affects the
physical degrees of freedom? Even if we choose to ignore these conceptual
problems, the trick used above is not possible for more general Hamiltonians
where the potential is a function $V(q_{1},p_{1},q_{2},p_{2})$ of both
positions and momenta.

It is possible that the role of the auxiliary variables in a mixed system
becomes obscured in the Schrödinger picture, but analyzing the problem in the
Heisenberg picture makes it clear. The issue needs to be confronted.

The auxiliary variables surface when KvN and quantum systems are mixed. But
this happens more generally, whenever deconjugated and canonical systems are
combined. For example, part of a classical system can be deconjugated such
that we have the fundamental relations (5.2), with quantum brackets replaced
by Poisson brackets. Partial deconjugation makes auxiliary variables appear in
the physical equations of motion even for a classical-classical system.

The difficulties discussed in this section should be considered when studying
quantum-classical dynamics using the KvN formulation. The classical part of
the system is a deconjugated part while the quantum remains canonical. Even
though the “original” equations of motion for the system are unknown, we see
from this section that the KvN path to quantum-classical equations of motion
might not be straightforward.

## Conclusion

The no-go theorems for quantum-classical mixing (see [6] and references
therein) reveal a clash between the canonical structures of classical and
quantum mechanics. Although this incompatibility can be circumvented for
certain classes of systems [7, 8], it is recognized that the mixing is not
generally possible.

We do not consider here the option of mixing canonical quantum mechanics with
a theory that is not canonical. Selecting the kind of non-canonical system is
difficult, as is the crucial task of showing it has a strong relation to
classical mechanics.

The impediment to canonical hybrid quantum-classical systems is the failure of
the quantum-classical bracket to satisfy the Leibniz and Jacobi properties
necessary for consistent canonical dynamics. At first glance, the Koopman-von
Neumann approach seems to bypass the inconsistency issue of the dynamical
bracket. It uses the commutator, which automatically satisfies the required
conditions.

However, we have shown here that the KvN theory is deconjugated canonical
quantum theory. Classical equations of motion do appear, but at the cost of
introducing auxiliary variables. KvN theory is more than just an operator
formulation of classical mechanics. It is not physically clear that the
auxiliary variables can be ignored.

Even if it may be possible to argue away the auxiliary variables as unphysical
for a single system, they are indispensable in a hybrid quantum-KvN system
(see also [4]).

We see that the mixing of classical and quantum systems remains somewhat
problematic even in an operator formulation of both systems. Using the
operator commutator for both quantum and classical systems raises new issues.

## Acknowledgements

This research was supported in part by a Discovery Grant (M.A.W.) from the
Natural Sciences and Engineering Research Council of Canada (funding Reference
No. RGPIN-2022-04225).

## References

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