#  Discrete Dynamics and Supergeometry

Subhobrata Chatterjee♢, Andrew Waldron♠ & Cem Yeti̇şmi̇şoğlu♣ ♢,♠Department of
Mathematics
University of California
Davis, CA95616, USA<EMAIL_ADDRESS>♣Department of
Mathematics
İstanbul Technical University
İstanbul, 34469, Turkey<EMAIL_ADDRESS>
###### Abstract.

We formulate a geometric measurement theory of dynamical classical systems
possessing both continuous and discrete degrees of freedom. The approach is
covariant with respect to choices of clocks and canonically incorporates
laboratories. The latter are embedded symplectic submanifolds of an odd-
dimensional symplectic structure. When suitably defined, symplectic geometry
in odd dimensions is exactly the structure needed for covariance. A
fundamentally probabilistic viewpoint allows classical supergeometries to
describe discrete dynamics. We solve the problem of how to construct
probabilistic measures on supermanifolds given a (possibly odd dimensional)
supersymplectic structure. This relies on a superanalog of the Hodge star for
differential forms and a description of probabilities by convex cones. We also
show how stochastic processes such as Markov chains can be described by
supergeometry.

Keywords: Symplectic geometry, supergeometry, classical mechanics, measurement
theory, discrete systems, Markov processes

###### Contents

  1. 1 Introduction
  2. 2 States, Laboratories and Measurements
  3. 3 Dynamical Systems
  4. 4 Phase-spacetime Measurements
  5. 5 Super Geometry
  6. 6 Discrete Dynamical Systems
  7. 7 Example: Evolution of Discrete Probabilities

##  1\. Introduction

Many physical systems involve both continuous and discrete degrees of freedom.
Standard classical mechanics typically treats continuous systems. Our aim is
to give a unified treatment of dynamical systems with both discrete and
continuous degrees of freedom. We focus on models whose dynamics is locally
determined. The key to our approach is to view measurement as fundamentally
probabilistic. Dynamics is often used to describe time evolution, so we also
ensure that our approach is covariant with respect to choices of clocks, à la
Einstein’s theory of general relativity. This requires a a natural
generalization of symplectic geometry to odd dimensional manifolds.

Just as symplectic geometry describes continuous classical mechanical systems,
we claim that symplectic supergeometry describes dynamical systems with
continuous and discrete degrees of freedom. This neccesitates a probabilistic
approach.

In quantum physics, action principles such as

$S=\int\big{(}\lambda_{i}(z,\theta){\hskip
0.85358pt}dz^{i}+\ell_{a}(z,\theta){\hskip 0.85358pt}d\theta^{a}\big{)}\,,$

where $z$ and $\theta$ are respectively “Bose” and “Fermi” coordinates, are
employed to describe the quantum mechanics of particles with intrinsic spin
(see for example [1, 2, 3, 4, 5]). Almost fifty years ago, Berezin and Marinov
attempted to ascribe a physical interpretation to a classical theory described
by such an action, but could not formulate a good notion of positive definite
probability distributions [3] (see [6] for another early attempt). We shall
solve this problem.

Our treatment begins with the standard formulation of measurement theory in
classical mechanics, based on ideas dating back to Boltzmann, Maxwell and
Gibbs and later formalized by Koopman and von Neumann [7, 8]. In this
probabilistic approach the state of a system is described in terms of
probability densities which are dual to observables corresponding to the
quantities being measured. More general than the notion of a probability
density is a “probability cone” describing the space of states as a suitable
subset of the dual to the space of observables [9, 10, 11, 12, 13, 14, 15,
16]. Expectations of observables are given by the natural pairing between
elements of these two spaces. This picture is described in Section 2.

At first sight, the probabilistic picture seems very different to introductory
approaches to classical mechanics involving idealized particles and their
trajectories in some “configuration space” manifold $Z$. However, the manifold
$Z$ still plays an important rôle, but one is now interested in a bundle of
probability cones over $Z$. We outline this theory in Section 2. Most of the
ideas given there are not novel (see for example [17, 18, 19, 20, 21, 22]). A
key new concept however is the notion of a laboratory. This is required to
describe classical systems independently of any particular choice of clocks
[23]. Broadly speaking, a laboratory is an instantaneous snapshot of the
system that is described by a hypersurface in a configuration space that
includes directions corresponding to time evolution.

In the probabilistic picture the notion of time evolution is fundamentally
stochastic. However, we wish to achieve covariance with respect to choices of
time. Therefore, rather than working with an even dimensional symplectic
manifold as the mathematical model for phase-space, and a Hamiltonian function
whose Hamiltonian vector field generates dynamics in terms of parameterized
phase-space paths, a unified description of dynamics is captured by odd
symplectic manifolds: In odd dimensions, a symplectic manifold describes
generalized positions, momenta and time and may thus be viewed as a “phase-
spacetime”. The required data in this setting is minimal, videlicet a maximal
rank closed two form whose kernel defines unparameterized paths. Some authors
include the data of a volume form [24, 25] in the definition of an odd
symplectic manifold. Cosymplectic structures add the data of a closed one-form
which in turn defines a volume form [26]. We drop any requirement of a volume
form in our treatment of odd symplectic geometry; see Section 3. Phase-
spacetime hypersurfaces naturally serve as laboratories whenever they are
symplectic in the standard sense. Evolution of probabilistic states is
governed by the Liouville equation. In Section 4 we give the clock
independent/time covariantized analog of the Liouville equation.

In any stochastic approach to dynamics it is important to ensure that dynamics
evolves probabilities into probabilities, or in other words preserves
positivity. More generally, any evolution of states must be confined to the
probability cone. There is a simple mechanism guaranteeing this property
namely re-express states as a positive square of a “state function” with
respect to a suitable product; see again Section 4. This mechanism is in fact
employed in the Born interpretation of quantum mechanics, see [15]. This
manuever is critical to our supergeometric description of discrete-plus-
continuous dynamical systems.

Discrete dynamical systems can be described using supergeometry. Accomplished
supergeometers can safely skip the first half of the Section 5 exposition
which handles standard notions such as sheaves of exterior algebras,
superfunctions, supervectors, super differential forms and super integral
calculus [27, 28, 29]. A key notion is the correspondence between (split)
supermanifolds and vector bundles provided by the structure theorems of
Batchelor [30] and Koszul [31]. We also review symplectic supergeometry [32,
33, 34, 35] and introduce its odd dimensional counterpart. The latter is
necessary for time covariance of discrete-plus-continuous dynamics.

To construct probability cones on symplectic supergeometries both a measure
and product are required. The former is for pairing states and observables,
and the product defines positive squares and in turn a cone. The product is
constructed via a super Moyal product [36, 37] which implements a super-analog
of the standard Hodge pairing on exterior algebras equipped with a metric.
This is described at the end of Section 5, where our probabilistic treatment
of symplectic supergeometry is also given.

Having developed a probability theory on symplectic supergeometries, it is not
difficult to construct instantaneous (super) laboratories and study evolution
via a super Liouville equation. This is presented in Section 6. The paper
concludes with a simple example.

##  2\. States, Laboratories and Measurements

A measurement typically amounts to a set of numerical outputs characterizing
the state of a system. These ought include errors reflecting uncertainties
associated to measuring devices. Indeed, in practice one usually can only hope
to have a statistical description of a physical system. In that light, states
of a physical system should be viewed as probability distributions each of
which assigns a positive, normalizable, measure on some space of physically
observable events.

More generally, a set of positive measures may be treated as a convex cone
${\mathcal{C}}\,\not{\ni}\,0$ in a topological vector space ${\mathcal{V}}$,
meaning some subset of non-zero vectors that is closed under addition and
multiplication by positive scalars; we shall denote this by
${\mathcal{C}}<{\mathcal{V}}$. When ${\mathcal{V}}$ is equipped with a norm
$||{\hskip 0.85358pt}{\boldsymbol{\hbox to2.5pt{\hss$\cdot$\hss}}}{\hskip
0.85358pt}||$, this can be used to normalize states, and in that case we call
$({\mathcal{C}}<{\mathcal{V}},||{\hskip 0.85358pt}{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}}{\hskip 0.85358pt}||)$ a probability cone.

An observable is any linear functional $X:{\mathcal{V}}\to{\mathbb{R}}$ (say).
The expectation of the observable $X$ in the state $\Psi\in{\mathcal{C}}$ is
then

$\langle X\rangle_{\Psi}=\frac{X(\Psi)}{||\Psi||}\,.$

######  Example 2.1.

Let ${\mathcal{V}}$ be $\mathbb{R}^{2}$ equipped with the standard
$L^{1}$-norm and ${\mathcal{C}}$ be the (closed) first quadrant
$\mathbb{R}^{2}_{+}$ with the origin $(0,0)$ removed. A normalized state
$\Psi=(p_{\tt 0},p_{\tt 1})\in\mathcal{C}$ is thus a pair of probabilities
summing to unity. The expectation of the observable $X=(x_{\tt 0}\>x_{\tt 1})$
is

$\langle X\rangle_{\Psi}=x_{\tt 0}p_{\tt 0}+x_{\tt 1}p_{\tt 1}\,.$

In a probability context, the observable $X$ is called a random variable.
$\blacksquare$

The first quadrant in the above is an example of a convex polyhedral cone. Its
one-dimensional corners define vectors, the expectations of whose canonical
duals (the covectors $\begin{pmatrix}1\\!&\\!0\end{pmatrix}$ and
$\begin{pmatrix}0\\!&\\!1\end{pmatrix}$) define probabilities ($p_{\tt 0}$ and
$p_{\tt 1}$) in the standard sense. Later we will encounter more general
probability cones (see in particular Example 5.10).

Classical mechanics is most often formulated as a theory of trajectories in
some configuration/phase-space, the latter of which we will model by some
manifold $Z$. As intimated earlier, from a probabilistic viewpoint one should
supplant trajectories by suitable functions of $Z$ and somehow view these as
either states or observables. Hence we make the following definition for
states of a classical dynamical system.

######  Definition 2.2.

Let $C\to\mathcal{S}Z\to Z$ be a subbundle of some vector bundle
$V\to\mathcal{V}Z\to Z$ over a manifold $Z$, whose fibers are a cone $C\subset
V$ (and $0\in C$). We shall call non-zero sections of the bundle
${\mathcal{S}}Z$ pre-states. When the section space $\Gamma({\mathcal{V}}Z)$
is equipped with a norm function, normalizable pre-states are termed states.
The set of all states is then a normed space and is termed the state space.
$\blacklozenge$

In the above definition, we have included the zero vector in $C$ so that
sections of ${\mathcal{S}}Z$ with compact support are allowed. The
(normalizable) section space $\Gamma({\mathcal{S}}Z)$ plays the rôle of the
probability cone ${\mathcal{C}}$ discussed above, while
$\Gamma({\mathcal{V}}Z)$ corresponds to the topological vector space
${\mathcal{V}}$. Supposing $V$ is a finite dimensional vector space and that
the bundle of cones $\mathcal{S}Z\to Z$ is a subbundle of some vector bundle
$V\to\mathcal{V}Z\to Z$, we then have a natural dual vector bundle
$V^{*}\to\mathcal{V}^{*}\\!Z\to Z$, and may make the following definition.

######  Definition 2.3.

Sections of the dual bundle $V^{*}\to\mathcal{V}^{*}\\!Z\to Z$ are called
preobservables. If $Z$ is equipped with a measure $\mu$, when a preobservable
$\chi$ obeys

$\int_{Z}\mu{\hskip 0.85358pt}\chi(\Psi)<\infty$

for all states $\Psi$, then $\int_{Z}\mu{\hskip
0.85358pt}\chi({\boldsymbol{\hbox to2.5pt{\hss$\cdot$\hss}}})$ is termed an
observable. $\blacklozenge$

Given the data of a measure, “nice” preobservables determine observables. When
such is available, we shall often blur the distinction between observables and
preobservables.

######  Example 2.4.

Let $(M^{2n},\omega)$ be the pair of a compact, even-dimensional manifold and
a non-degenerate $2$-form $\omega$. Then we may take $Z=M^{2n}$,
$V={\mathbb{R}}$, $C={\mathbb{R}}_{\geq 0}$ and
${\mathcal{S}}Z=M\times{\mathbb{R}}_{\geq 0}$. The state space is the space of
positive but not everywhere vanishing functions
$\Gamma({\mathcal{S}}Z)\backslash 0$ with norm $||{\hskip
0.85358pt}{\boldsymbol{\hbox to2.5pt{\hss$\cdot$\hss}}}{\hskip
0.85358pt}||=\int\omega^{\wedge n}|{\hskip 0.85358pt}{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}}{\hskip 0.85358pt}|$. With respect to the measure
$\mu:=\omega^{\wedge n}$, any smooth function $\chi$ of $M^{2n}$ is a
preobservable because we may define a linear functional on states $\Psi$ by
the product action

$\chi(\Psi)=\chi\Psi\,.$

Note that this relies on the algebra of functions on $M^{2n}$. The expectation
of the observable $X=\int\mu{\hskip 0.85358pt}\chi({\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}})$ is

$\langle X\rangle_{\Psi,\mu}=\frac{\int_{M^{2n}}\omega^{\wedge
n}\chi\Psi}{\int_{M^{2n}}\omega^{\wedge n}\Psi}\,.$

Blurring the distinction between preobservables and observables, we may view
the latter as the set of smooth functions on $M^{2n}$. $\blacksquare$

Observe that if ${\mathcal{V}}Z$ is a normed bundle, meaning that there is a
smooth map

$|\,{\boldsymbol{\hbox to2.5pt{\hss$\cdot$\hss}}}\,|:\Gamma({\mathcal{V}}Z)\to
C_{+}^{\infty}Z$

taking values in positive functions such that $|\,{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}}\,|$ gives a fiberwise norm, then given a measure on
$Z$, we obtain the $L^{1}$-norm

$||{\hskip 0.85358pt}{\boldsymbol{\hbox to2.5pt{\hss$\cdot$\hss}}}{\hskip
0.85358pt}||=\int_{Z}\mu\,|\,{\boldsymbol{\hbox to2.5pt{\hss$\cdot$\hss}}}\,|$

on $\Gamma({\mathcal{V}}Z)$. This mechanism was used in the above example. In
some contexts a normed bundle is termed a Banach bundle.

As we are henceforth primarily concerned with probability cones constructed as
section spaces of vector bundles ${\mathcal{V}}Z$, there arise situations
where measurements may not access all of $Z$. It may even be that only some
submanifold $\Sigma\hookrightarrow Z$ is endowed with norms and measures.
Along $\Sigma$ we naturally get a vector bundle
${\mathcal{V}}\Sigma:={\mathcal{V}}Z|_{\Sigma}$ by restriction and in the same
way a bundle of cones ${\mathcal{S}}\Sigma$. This leads to the notion of a
laboratory:

######  Definition 2.5.

Any triple $(\Sigma,\mu,||{\hskip 0.85358pt}{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}}{\hskip 0.85358pt}||),$ where $\Sigma\hookrightarrow
Z$ is any submanifold equipped with a measure $\mu$ and a norm
$||\,{\boldsymbol{\hbox to2.5pt{\hss$\cdot$\hss}}}\,||$ on
$\Gamma({\mathcal{V}}\Sigma)$, is called a laboratory. $\blacklozenge$

We can now define laboratory observables
$X=\int_{\Sigma}\mu\chi({\boldsymbol{\hbox to2.5pt{\hss$\cdot$\hss}}})$ in
terms of preobservables $\chi\in\Gamma({\mathcal{V}}^{*}\\!Z)$ subject to
$\int_{\Sigma}\mu{\hskip 0.85358pt}\chi(\Psi)<\infty$ for all
$\Psi\in\Gamma({\mathcal{S}}Z)$ that are normalizable along $\Sigma$. Then the
expectation of $X$ as measured by a laboratory
$(\Sigma,\mu,||\,{\boldsymbol{\hbox to2.5pt{\hss$\cdot$\hss}}}\,||)$ with
respect to some state $\Psi\in\Gamma({\mathcal{S}}Z)$ is given by

$\langle X\rangle_{\Psi,(\Sigma,\mu,||\,{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}}\,||)}:=\frac{\int_{\Sigma}\mu{\hskip
0.85358pt}\chi(\Psi)}{||\Psi||}\,.$

In some cases laboratories may inherit measures and norms from the underlying
manifold $Z$ when this has sufficiently much data, as is the case in Example
2.4.

######  Example 2.6.

Continuing from Example 2.4, let $\Sigma^{2k}$ be any compact submanifold
along which the pullback $\omega^{*}$ of $\omega$ is non-degenerate (so that
$(\Sigma,\omega^{*}{}^{\wedge k})$ is itself symplectic). Then
$\big{(}\Sigma^{2k},\omega^{*\wedge k},\int_{\Sigma^{2k}}\omega^{*\wedge
k}|{\hskip 0.85358pt}{\boldsymbol{\hbox to2.5pt{\hss$\cdot$\hss}}}{\hskip
0.85358pt}|\big{)}$ is a laboratory. $\blacksquare$

The following is an example where $Z$ itself is not equipped with norms or
measures.

######  Example 2.7.

Carrying on from Example 2.1 take
$V={\mathbb{R}}^{2}\supset\mathbb{R}^{2}_{+}=C$ and consider the manifold
$Z={\mathbb{R}}\ni t$. Choosing laboratories
$(\Sigma=\\{t\\},\mu=1,||\,{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}}\,||=\sum_{\\{{\tt 0},{\tt
1}\\}}|\,{\boldsymbol{\hbox to2.5pt{\hss$\cdot$\hss}}}\,|)$, the observable
$X=\big{(}x_{\tt 0}(t)\>\>x_{\tt 1}(t)\big{)}$ has expectation

$\langle X\rangle_{\Psi,(\Sigma,\mu,||\,{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}}\,||)}=\frac{x_{\tt 0}(t)\pi_{\tt 0}(t)+x_{\tt
1}(t)\pi_{\tt 1}(t)}{\pi_{\tt 0}(t)+\pi_{\tt 1}(t)}\,,$

with respect to the state $\Psi=\big{(}\pi_{\tt 0}(t),\pi_{\tt 1}(t)\big{)}$
in these laboratories. $\blacksquare$

Viewing the coordinate $t$ as time, the above example includes dynamics. In
this sense, $(\pi_{\tt 0}(t),\pi_{\tt 1}(t))$ is a “dynamical state”.

##  3\. Dynamical Systems

A dynamical system is the data of a configuration space with evolution. For
continuous systems this is typically modeled by a smooth manifold equipped
with a parameterized path through each point. For the case of particles moving
in space, an important insight of Einstein was that parameterized paths could
be replaced by unparameterized ones in a spacetime manifold. This perspective
both allowed time to be treated covariantly and made manifest the geometry of
space and time. In classical mechanics, dynamics can be described by
parameterized paths in phase-space. But again, it is often propitious, for
both reasons of covariance and geometry, to consider unparameterized paths in
a phase-spacetime. In both cases, one is (at least locally) dealing with a
manifold foliated by paths.

Rather than working with a system of paths, it is useful to describe physics
modeled upon a manifold by local systems of equations. In very many situations
the data of a suitable two-form suffices for this.

######  Definition 3.1.

A symplectic manifold is a pair $(Z,\omega)$ where $Z$ is a smooth manifold
and $\omega$ is a maximally non-degenerate closed two-form. $\blacklozenge$

We also need a notion of normalizability for functions on a symplectic
manifold $Z$. When $Z$ has even dimension $2n$, we say that $\Psi\in
C^{\infty}Z$ is normalizable if $\int_{Z}\omega^{\wedge n}|\Psi|<\infty$. For
$Z$ of odd dimension $2n+1$, we call $\Psi\in C^{\infty}Z$ is normalizable if
$\int_{\Sigma}\omega^{*\wedge n}|\Psi|<\infty$ for all hypersurfaces
$\Sigma\hookrightarrow Z$ along which the pullback $\omega^{*}$ is non-
degenerate.

A foliation by paths yields a line subbundle of the tangent bundle. At each
point this is determined by the span of the tangent vector to the path
determined by the foliation. Thus to describe dynamics, we would like to
assign a distinguished line to each tangent space of $(Z,\omega)$.

######  Definition 3.2.

A dynamical phase-space is a pair, denoted $(Z,\omega,L)$, given by a
symplectic manifold $(Z,\omega)$ and a line subbundle $L\subset TZ$ subject to
the compatibility condition that in a neighborhood of every point in $Z$,
there is some non-vanishing $\rho\in\Gamma(L)$ such that

$\mathcal{L}_{\rho}\omega=0\,.$

When $L$ is equipped with a global ray subbundle $R\subset L$, the dynamical
phase-space $(Z,\omega,L)$ is termed directed. $\blacklozenge$

An even-dimensional symplectic manifold equipped with a suitably generic
Hamiltonian determines, at least locally, a dynamical phase-space through the
tangent lines to integral curves of the corresponding Hamiltonian vector
field.

Directed dynamical phase-spaces enjoy an everywhere forward notion of time
evolution. Moreover their corresponding line bundles $L$ are trivial, meaning
$L=M\times{\mathbb{R}}$. Often we will not be concerned with the distinction
between the directed and non-directed cases, since our main aim is to
establish local theories of measurement.

A vector $\rho$ such that ${\mathcal{L}}_{\rho}\omega=0$ for some line
subbundle $L\subset TZ$ is called symplectic, and flows generated from such
are termed symplectomorphisms. In Definition 3.1 we quite deliberately did not
specify the dimension parity of $Z$. When this is even, we relabel $Z$ by $M$
and call $(M,\omega)$ a phase-space (oftentimes this is taken as the
definition of a symplectic manifold), while the odd case is termed a phase-
spacetime. The manifold $M$ describes generalized positions and momenta, while
$Z$ in addition covariantly incorporates time.

######  Example 3.3.

Let $H(p,q,t)$ be a time dependent Hamiltonian on the phase-space
$M=({\mathbb{R}}^{2}\ni(p,q),dp\wedge dq=:\bar{\omega})$. This data can be
rephrased as a phase-spacetime $Z=(M\times{\mathbb{R}}\ni t,\omega)$ by
viewing $H$ as a function of $Z$ and setting

$\omega:=\Big{(}dp+\frac{\partial H}{\partial
q}dt\Big{)}\wedge\Big{(}dq-\frac{\partial H}{\partial p}dt\Big{)}\,.$

The line bundle $L\subset TZ$ is given by the span of the vector field

$\rho=\frac{\partial}{\partial t}+\frac{\partial H}{\partial
p}\frac{\partial}{\partial q}-\frac{\partial H}{\partial
q}\frac{\partial}{\partial p}\,.$

Notice that $\omega(\rho,{\boldsymbol{\hbox to2.5pt{\hss$\cdot$\hss}}})=0$, so
clearly ${\mathcal{L}}_{\rho}\omega=0$. Of course, the above symplectic form
$\omega$ is the pullback of the symplectic form $\Omega:=dp\wedge dq-dE\wedge
dt$ on ${\mathbb{R}}^{4}\ni(p,q,E,t)$ to the “on-shell” hypersurface
$E=H(p,q,t)$. $\blacksquare$

In fact any phase-space $(M,\omega)$ can be treated as a section of the
trivial line bundle $Z=M\times\mathbb{R}$ equipped with the symplectic form
$\pi^{*}\omega$ where $\pi:Z\to M$ denotes the canonical projection. The pair
$(Z,\pi^{*}\omega)$ is a phase-spacetime as is any contact manifold [23, 38].
Not all phase-spacetimes can be obtained so easily.

Three symplectic manifolds appear in Example 3.3: $({\mathbb{R}}^{4},\Omega)$,
$({\mathbb{R}}^{3},\omega)$ and $({\mathbb{R}}^{2},\bar{\omega})$. To describe
dynamics, the first requires the choice of some hypersurface ($E=H(p,q,t)$
say) along which evolution is described in terms of unparameterized paths. The
second is in some sense optimal, since the symplectic form $\omega$ determines
unparameterized paths describing evolution. The third describes dynamics in
terms of parameterized paths for which a choice of Hamiltonian is required.
Indeed, in odd dimensions, the data of a symplectic form is primal in the
following sense.

######  Proposition 3.4.

Every phase-spacetime $(Z,\omega)$ canonically defines a dynamical phase-
space.

###### Proof.

Let $\rho$ denote any non-zero vector field solving

(3.1) $\iota_{\rho}\omega=0\,.$

Because $\omega$ is a maximal rank two-form, it defines a linear map
$\Gamma(TZ)\to\Gamma(T^{*}Z)$ whose kernel is everywhere one-dimensional. By
our definition, the vector $\rho$ need not be everywhere non-vanishing and
therefore such a vector can be constructed on any patch in $Z$. Next, using
Cartan’s magic formula, we compute

$\mathcal{L}_{\rho}\omega=d\iota_{\rho}\omega+\iota_{\rho}d\omega=0\,.$

The term $\iota_{\rho}d\omega$ vanishes in the above because the symplectic
form is closed. Thus patch-wise, and hence everywhere,
$\operatorname{span}(\rho)$ defines a line subbundle $L$ with the desired
$\mathcal{L}_{\rho}\omega=0$ property. ∎

Our primary focus is now on phase-spacetimes. In particular we need to set up
measurement theory and dynamics for states.

##  4\. Phase-spacetime Measurements

As discussed earlier, the data of a phase-spacetime $(Z,\omega)$ encodes the
dynamics of a system. Therefore one might expect that knowledge of the state
of a system at one instant of “generalized time” should determine the state
for all times. In other words, a state on a phase-spacetime should encode the
information of the system’s state at all times.

######  Definition 4.1.

Let $(Z,\omega)$ be a phase-spacetime. Then any normalizable function $0$ $<$
$\Psi\in C^{\infty}Z$ subject to

${\mathcal{L}}_{\rho}\Psi=0\,,$

for all vector fields $\rho$ obeying $\omega(\rho,{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}}{\hskip 0.85358pt})=0$, is termed a dynamical state.
$\blacklozenge$

Next we discuss how dynamical states can be measured. The symplectic form
$\omega$ on a phase-spacetime does not define a volume form, so expectations
of observables cannot be defined as was done in Example 2.4. This is a feature
not a bug, since experiments are often viewed as taking place at some instant
in “time”.

######  Definition 4.2.

Let $(Z,\omega)$ be a phase-spacetime. Then any hypersurface $\Sigma$ such
that the pullback of $\omega$ is non-degenerate is termed an instantaneous
laboratory. $\blacklozenge$

An instantaneous laboratory gives a generalized notion of “when” one plans to
measure a dynamical state. Observables encode what quantity is being measured.
This ought be specified independently of any particular choice of laboratory.
We note that one can also consider more general laboratories given by even
dimensional submanifolds along which $\omega$ is non-degenerate. This is
useful when one plans to measure only some subset of observable quantities,
such as is the case in statistical mechanics. The ideas presented below
generalize straightforwardly to these cases.

In a phase-spacetime context we will deliberately blur the distinction between
preobservables and observables, and hence make the follow definition.

######  Definition 4.3.

Let $(Z^{2n+1},\omega)$ be a phase-spacetime. Then any function $X\in
C^{\infty}Z$ such that $\int_{\Sigma}\omega^{*\wedge n}\Psi X<\infty$ for all
instantaneous laboratories $\Sigma$ and dynamical states $\Psi$, is called an
observable. $\blacklozenge$

The data of a dynamical state and an instantaneous laboratory suffices to
compute expectations of observables at any “generalized instant in time”.

######  Definition 4.4.

Let $(Z^{2n+1},\omega)$ be a phase-spacetime, $\Psi$ be a dynamical state and
$\Sigma$ an instantaneous laboratory. Then the expectation of $X$ as measured
by the laboratory $\Sigma$ with respect to the state $\Psi$ is defined to be

$\langle X\rangle_{\Psi,\Sigma}:=\frac{\int_{\Sigma}\omega^{*\wedge
n}\,X\Psi}{\int_{\Sigma}\omega^{*\wedge n}\,\Psi}\,.$

$\blacklozenge$

######  Example 4.5 (Massless relativistic particle).

We can describe a photon moving in the spacetime

$ds^{2}=-dt^{2}+h_{ij}(x,t)dx^{i}dx^{j}$

as follows. For $(Z,\omega)$ we take $Z={\mathbb{R}}^{7}_{\star}\ni(0\neq
p_{i},x^{i},t)$ and

$\omega=\Big{(}dp_{i}-p^{k}\Gamma_{k}{}^{j}{}_{i}p_{j}{\hskip
0.85358pt}\frac{dt}{E}\Big{)}\wedge\Big{(}dx^{i}-p^{i}{\hskip
0.85358pt}\frac{dt}{E}\Big{)}\,.$

In the above indices are raised and lowered with the spatial metric $h_{ij}$,
$\Gamma_{k}{}^{j}{}_{i}$ are the corresponding Christoffel symbols, and the
photon energy

$E:=\sqrt{p_{i}h^{ij}p_{j}}\,.$

For simplicity let us make the simple spatial metric ansatz

$h_{ij}(x,t)=f(t)^{2}\,\delta_{ij}\,,$

which for monotonic increasing $f(t)$ describes an expanding universe. Let us
consider an instantaneous laboratory $\Sigma=\\{(p_{i},x^{i},0)\\}$. The
measure $\mu$ obtained by pulling back $\omega$ is given by

$\frac{1}{3!}{\hskip 0.85358pt}\mu=\bigwedge_{i=1..3}dp_{i}\wedge dx^{i}\,.$

Now suppose that the state of the system in the initial laboratory is given by
some non-negative function $\psi(p,x)$ that has unit $L^{1}$-norm with respect
to $\mu$.

Now, it is easy to check that the vector

$\rho=f(t)\left(\frac{\partial}{\partial t}+\hat{p}^{{\hskip
0.85358pt}i}\frac{\partial}{\partial x^{i}}\right)\,,$

is symplectic, where $\hat{p}$ is a unit vector with respect to the spatial
Euclidean metric which we from now on use to raise and lower indices. Because
$f$ is monotonic, we may introduce a new variable $\tau$ such that
$d\tau/dt=f^{-1}$. It is easy to solve ${\mathcal{L}}_{\rho}\Psi=0$ with this
initial data, giving

$\Psi=\psi(p,x-\hat{p}\tau)\,.$

The expectation of an observable $X(p,x,\tau)$ with respect to a constant
$\tau=T$ laboratory is then given by

$\langle
X\rangle_{\Psi,\tau=T}=\int\big{(}\prod_{i=1..3}dp_{i}dx^{i}\big{)}X(p,x,T)\psi(p,x-\hat{p}{\hskip
0.85358pt}T)\,.$

In the case where $\psi$ is a unit mass function strongly peaked around a
point $(p,x)=(P,X)$, one has

$\langle p_{i}\rangle_{\Psi,\tau=T}\approx P_{i}\>\mbox{ and }\langle
x^{i}\rangle_{\Psi,\tau=T}\approx\hat{P}^{i}T\,.$

$\blacksquare$

Observe that in the above example, positivity of the initial state $\psi(p,x)$
implies the same for the dynamical state $\Psi$ everywhere in $Z$ so that
indeed $\Psi\in\Gamma({\mathcal{S}}Z)$. This follows generally because the
dynamical state is obtained by Lie dragging the initial state along the
direction defined by the line bundle $L$.

Now suppose we are given a one-parameter family of laboratories
$\Sigma_{\tau}$ on $(Z^{2n+1},\omega)$. Because these are necessarily
transverse to the line bundle $L$, we may at least locally realize these as
the exponential map of some section of $\rho\in\Gamma(L)$, id est

$\Sigma_{\tau}=\operatorname{exp}(\tau\rho)(\Sigma)\,,$

where $\Sigma:=\Sigma_{0}$. This yields an evolution law for expectations

$\frac{d(\langle
X\rangle_{\Psi,\Sigma_{\tau}})}{d\tau}\Big{|}_{\tau=0}=\frac{\int_{\Sigma}\omega^{*\wedge
n}({\mathcal{L}_{\rho}X)\Psi}}{\int_{\Sigma}\omega^{*\wedge n}\Psi}\,.$

The above result is easily established remembering that $\rho$ preserves both
$\omega$ and $\psi$.

We next ask the silly question, why was the dynamical state $\Psi$ a positive
function? The silly answer is that $\Psi$ must be a square

(4.1) $\Psi=\Phi^{2}\,,$

since this is manifestly positive. In more general models involving states
described by sections of more complicated vector bundles, this question and
its answer are no longer so trivial, but can be addressed via generalized
“squares”.

######  Definition 4.6.

Let $(Z,\omega)$ be a phase-spacetime. The fundamental state space is the set
of smooth functions $\Phi\neq 0$ whose squares are normalizable, and that are
subject to

(4.2) ${\mathcal{L}}_{\rho}\Phi=0$

for all vector fields $\rho$ obeying $\omega(\rho,{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}}{\hskip 0.85358pt})=0$. Such a function is termed a
state function. $\blacklozenge$

######  Lemma 4.7.

The square of any state function is a dynamical state.

###### Proof.

Let $\Phi$ be a state function. We only need establish that
${\mathcal{L}}_{\rho}\Phi^{2}=0$ for any vector field $\rho$ obeying
$\omega(\rho,{\boldsymbol{\hbox to2.5pt{\hss$\cdot$\hss}}}{\hskip
0.85358pt})=0$. This is a direct consequence of the Leibniz rule for Lie
derivatives. ∎

Finally, observe that when expressed in terms of state functions, expectations
of observables take a form highly reminiscent of that for quantum mechanical
systems:

$\langle X\rangle_{\Psi,\Sigma}:=\frac{\int_{\Sigma}\omega^{*\wedge n}\,\Phi
X\Phi}{\int_{\Sigma}\omega^{*\wedge n}\,\Phi^{2}}\,.$

State functions are in this sense classical analogs of quantum mechanical wave
functions and Equation (4.2) is the analog of the Schrödinger equation.

##  5\. Super Geometry

We are primarily interested in discrete dynamical systems associated to
symplectic supermanifolds. The space of functions of $m$ Grassmann variables
is a vector space of dimension $2^{m}$. This suggests studying independent
coin flips or Bernoulli random variables. Indeed consider an $m$-bit computer.
In a perfect world the state of this computer could be described by an
$m$-element list of ones and zeros:

$\underbrace{{\tt 110010010000111111011}\cdots{\tt 001110001101}}_{m\ \rm
terms}$

However in principle, even classical measurements of discrete systems are
error-prone, so in general we only know a list of $2^{m}$ probabilities.
Moreover, observables are then random variables assigning a real number (say)
to each of $2^{m}$ configurations of ${\tt 0}$’s and ${\tt 1}$’s. This data
can encoded by a superfield: Introducing Grassmann variables $\\{\theta_{\tt
1},\ldots,\theta_{\tt m}\\}$, the single superfield

$X(\theta)=x_{{\tt 000}\cdots{\tt 00}}+x_{{\tt 100}\cdots{\tt 00}}\theta_{\tt
1}+\cdots+x_{{\tt 101100}\cdots{\tt 00}}\theta_{\tt 1}\theta_{\tt
3}\theta_{\tt 4}+\cdots+x_{{\tt 11}\cdots{\tt 11}}\theta_{\tt 1}\theta_{\tt
2}\cdots\theta_{\tt m-1}\theta_{\tt m}$

succinctly encodes elements $(x_{{\tt 000}\cdots{\tt 00}},\ldots,x_{{\tt
111}\cdots{\tt 11}})$ of the set $[0,1]^{2^{m}}$. The question of how to write
down dynamical states for dynamical systems that include discrete degrees of
freedom is the main goal of the remainder of this article. For the simplest
case of a two state discrete system, for an observable
$X(\theta)=\frac{x_{{\tt 0}}+x_{{\tt 1}}}{2}+\frac{x_{{\tt 0}}-x_{{\tt
1}}}{2}{\hskip 0.85358pt}\theta$, a state might be given as

$\Psi=p_{\tt 0}(1+\theta)+p_{\tt 1}(1-\theta)\,,$

where the probabilities $p_{\tt 0}$ and $p_{\tt 1}$ sum to unity (the
combinations of Grassmann variables appearing here are to do with the
construction of pure states). The expectation $\langle X\rangle_{\Psi}$ of $X$
with respect to the state $\Psi$ should then be

$p_{\tt 0}x_{\tt 0}+p_{\tt 1}x_{\tt
1}\stackrel{{\scriptstyle?}}{{=}}\frac{\int_{\mathcal{M}}X\Psi}{\int_{\mathcal{M}}\Psi}\,.$

The natural question is whether this expectation can be expressed as a
suitable integration over a symplectic supermanifold ${\mathcal{M}}$. We would
also like to develop the corresponding notions of instantaneous laboratories
and dynamics. Thus we now describe pertinent aspects of supermanifold
geometry; a more detailed account can be found in [27, 39]. Many
supergeometers can safely proceed directly to Section 5.5.

###  5.1. Supermanifolds

An $(n,m)$ supermanifold $\mathcal{M}$ is a pair $(M,\mathcal{A})$ where $M$
is a smooth, dimension $n$ manifold termed the body, and $\mathcal{A}$ is a
$\mathbb{Z}_{2}$-graded commutative sheaf of functions on $M$; $m$ is called
the odd-dimension of $\mathcal{M}$. By definition, there is a surjective map
$\epsilon:\mathcal{A}\to C^{\infty}M$ compatible with restrictions to open
sets. That is, for each open set $U\subseteq M$, we have a surjective
homomorphism $\epsilon_{U}:\mathcal{A}U\to C^{\infty}U$, where
$\mathcal{A}U:=\mathcal{A}|_{U}$. Moreover $\mathcal{A}$ satisfies a local
triviality condition: given an open cover $\\{U_{\alpha}\\}$ of $M$, there are
local isomorphisms

(5.1) $\mathcal{A}U_{\alpha}\cong
C^{\infty}U_{\alpha}\otimes\Lambda\mathbb{R}^{m}\,.$

In physics, the sheaf of functions $\mathcal{A}$ corresponds to the space of
superfields. Hence we will slightly abuse notations and also use
${\mathcal{A}}$ to denote the space of superfunctions
$\Gamma(M,{\mathcal{A}})$. These form a supercommutative algebra and the
output of the map $\epsilon$ is called the body of a superfield.

Note that if $M$ is any smooth manifold, then the pair $(M,C^{\infty}M)$
defines a supermanifold with a trivial $\mathbb{Z}_{2}$-grading. A less
trivial example is $(\mathbb{R},C^{\infty}\mathbb{R})$ where
$C^{\infty}\mathbb{R}=C_{+}^{\infty}\mathbb{R}\oplus C_{-}^{\infty}\mathbb{R}$
and $\pm$ denote even and odd functions, respectively. Another important
example is the exterior bundle $\Lambda M$ for any $M$.

A supermanifold locally looks like the trivialization of a vector bundle. We
call this the local model and denote it by
$\mathbb{R}^{n|m}:=(\mathbb{R}^{n},C^{\infty}\mathbb{R}^{n}\otimes\Lambda\mathbb{R}^{m})$.
Of course $\mathbb{R}^{n|m}$ is itself a supermanifold. The sheaf of functions
on $\mathbb{R}^{n|m}$ is a section space of the vector bundle
$\Lambda\mathbb{R}^{m}\to\mathbb{R}^{n}\times\Lambda\mathbb{R}^{m}\to\mathbb{R}^{n}$,
id est

$C^{\infty}\mathbb{R}^{n}\otimes\Lambda\mathbb{R}^{m}=\Gamma(\mathbb{R}^{n}\times\Lambda\mathbb{R}^{m})\,.$

These can be encoded by “functions” $F(x,\theta)$ of $n$ bosonic/even
coordinates $\\{x^{i}\\}$ and $m$ fermionic/odd/Grassmann “coordinates”
$\\{\theta^{a}\\}$, respectively, where the latter are subject to the relation

$\theta^{a}\theta^{b}+\theta^{b}\theta^{a}=0\,,$

and

$F(x,\theta):=f(x)+f_{a}(x)\theta^{a}+f_{ab}(x)\theta^{a}\theta^{b}+\cdots+f(x)_{a_{1}\cdots
a_{m}}\theta^{a_{1}}\cdots\theta^{a_{m}}\,.$

The coefficients above are real-valued functions of ${\mathbb{R}}^{n}$.

One often wants to consider supermanifold morphisms which are defined as
follows: Let $\mathcal{M}=(M,\mathcal{A})$ and $\mathcal{N}=(N,\mathcal{B})$
be supermanifolds. A supermanifold morphism

$\Phi:\mathcal{M}\to\mathcal{N}$

is a pair of maps $(f,\varphi)$ where $f:M\to N$ is a manifold morphism and

$\varphi:\mathcal{B}\to f_{*}\mathcal{A}$

is a morphism of sheaves over $N$. The map $f_{*}$ is constructed by
considering preimages of open sets in $N$, and is often termed the sheaf
pushforward [27].

On a supermanifold, there is a canonical, globally-defined body map between
sheaves

$i:(M,C^{\infty}M)\to\mathcal{M}\,,$

given by the pair $(\text{id}_{M},\epsilon_{M})$. The body map pulls back
superfields and superdifferential forms (the latter will be defined below) on
$\mathcal{M}$ to the body manifold $M$. The body of a superfield
$F\in\mathcal{A}M$ is denoted by $F_{0}:=\epsilon_{M}(F)$ (and similarly for
forms).

We are particularly interested in the case where we can describe supergeometry
using vector bundles as this melds well with our previous discussion of
probabilistic states, see in particular Definition 2.2. For that we need a map
projecting fibers to the base. However a projection

$\pi:\mathcal{M}\to(M,C^{\infty}M)\>\mbox{ such that }\>\pi\circ
i=\text{id}_{M}$

may not, in general, exist globally [27]. When it does, $\mathcal{M}$ is
called a split supermanifold. A seminal theorem due to Batchelor guarantees
that real supermanifolds are always split [30]; this relies on partitions of
unity for the base manifold $M$.

A split supermanifold $\mathcal{M}$ is isomorphic to an exterior bundle

$\pi:\mathcal{E}M=M\ltimes\Lambda V^{*}\to M$

associated to a vector bundle

$\mathcal{V}M:=M\ltimes V\to M\,.$

Points in $\mathcal{V}M$ may be loosely thought of as “superpoints” in
$\mathcal{M}$, while the sections of $\mathcal{E}M$ correspond to superfields
on $\mathcal{M}$. It is possible to define a precise notion of superpoints
using the so-called “functor of points” [40], but this will not play an
important rôle here. Because of our interest in vector bundles, in what
follows always work with split supermanifolds.

The isomorphism between a split supermanifold $\mathcal{M}$ and a vector
bundle $(\mathcal{E}M,\pi)$ cannot be realized canonically; a given choice of
such is termed a splitting. We will label splittings by their corresponding
projection $\pi$. Certain supermanifold (auto)morphisms
$\mathcal{M}\to\mathcal{M}$ neither preserve the projection $\pi$ nor
correspond to bundle automorphisms of $\mathcal{E}M$. Moreover, the vector
bundle $\mathcal{E}M$ has an additional $\mathbb{Z}_{m+1}$-grading because it
is an associated exterior bundle of $\mathcal{V}M$. This grading is not
preserved by such morphisms. While a supermanifold $\mathcal{M}$ itself need
not have a canonical $\mathbb{Z}_{m+1}$-grading, there is a canonical
(splitting independent) descending filtration. This is described below, but
first we develop the vector bundle picture and discuss superdiffeomorphisms.

The body $M$, defined as the quotient of the total space $\mathcal{E}M$ by the
projection $\pi$, is independent of any splitting. Because we want to work
independently of the choice of a splitting, we could instead consider the
following quotient

$\mathcal{M}:=\Bigg{(}\bigsqcup_{\\{\Phi:\mathcal{M}\to\mathcal{M}\\}}(\mathcal{E}M,\pi\circ\Phi)\Bigg{)}\Bigg{/}\sim\,,$

and take this as a working definition of a supermanifold. In the above, the
quotient is by the equivalence relation
$(\mathcal{E}M,\pi_{1})\sim(\mathcal{E}M,\pi_{2})$ if and only if
$\pi_{1}=\pi_{2}\circ\Phi$ for any supermanifold automorphism $\Phi$ on
$\mathcal{M}$. In particular we focus on the case that $\mathcal{M}$ is a
smooth supermanifold, and quite generally assume that all structures are
smooth. In this case, we term such supermanifold automorphisms,
superdiffeomorphisms of $\mathcal{M}$. Note that by construction $\mathcal{M}$
is itself a vector bundle with base $M$ and projection obtained by suitably
quotienting a disjoint union of projections.

For computations, we may always choose a particular representative
$(\mathcal{V}M,\pi)$, or equivalently $(\mathcal{E}M,\pi)$; observe this does
not rely on a particular choice of local coordinates. On the other hand, on a
trivializing chart with coordinates $x^{i}$ on $M$, let us denote the fiber
coordinates of $\mathcal{V}M$ and $\mathcal{E}M$ by $y^{a}$ and $\theta^{a}$,
respectively. Then, on intersecting trivializing covers, the respective bundle
automorphisms take the form

(5.2) $\tilde{x}^{i}=\tilde{x}^{i}(x)\,,\quad\tilde{y}^{a}=g^{a}_{\
b}(x)y^{b}\,,\quad\tilde{\theta}^{a}=g^{a}_{\ b}(x)\theta^{b}\,,\quad g\in
GL(m,\mathbb{R})\,.$

Here, the fiber coordinates of $\mathcal{V}M$ are Grassmann even, whereas that
of $\mathcal{E}M$ are odd. This is an example of the parity reversal operation
$\Pi$, that acts as $\Pi y^{a}=\theta^{a}$, and changes the Grassmann even
grading of the coordinates $y^{a}$ to the odd one of $\theta^{a}$. In some
contexts, it is useful to think of these odd coordinates as covectors
$\theta^{a}=dy^{a}$.

In general, the parity reversal $\Pi$ is defined on a $\mathbb{Z}_{2}$-graded
vector space $V=V_{0}\oplus V_{1}$ by

$(\Pi V)_{0}=V_{1}\,,\qquad(\Pi V)_{1}=V_{0}\,,$

where subscripts $0,1$ denote even and odd subspaces, respectively. In
particular, if $V$ is purely even, id est $V=V_{0}$, then polynomial functions
on $\Pi V$ can be identified with elements of the exterior algebra of the dual
$V^{*}$, that is

$\mathbb{R}[\Pi V]\cong\Lambda V^{*}\,.$

Consequently, by applying the parity reversal fiberwise to the vector bundle
$\mathcal{V}M$ and considering functions on the total space, we end up with
the section space of $\mathcal{E}M$. This observation will be important when
we discuss super vectors on $\mathcal{M}$ and their relation to derivations on
$\Pi\mathcal{V}M$.

Given data such as a particular superfield on a supermanifold, we may pick a
representative $(\mathcal{E}M,\pi)$, and then consider bundle invariant
expressions (respecting (5.2)) built therefrom. In turn we can explicitly
compute the superdiffeomorphism transformation of such expressions in local
coordinates $(x,\theta)$ to verify whether they are supermanifold invariants.
For instance any superfield $F\in\Gamma(\mathcal{E}M)$ has a decomposition

$F(x,\theta)=F_{0}(x)+\hat{F}(x,\theta)\,,$

where

$\hat{F}=\sum_{a}\theta^{a}\hat{F}_{a}(\theta)=:O(\theta)\,.$

Here the notation $O(\theta^{k})$ denotes terms of polynomial degree at least
$k\in\\{0,\ldots,m\\}$ in the odd coordinates $\theta$. In these coordinates
the invariantly defined body of $F$ is given by $F_{0}(x)\in C^{\infty}M$
while the soul $\hat{F}$ depends on the choice of splitting $\pi$.
Superdiffeomorphisms themselves are locally represented by a set of
superfields

(5.3)
$\tilde{x}^{i}=\tilde{x}^{i}(x,\theta)\,,\quad\tilde{\theta}^{a}=\tilde{\theta}^{a}(x,\theta)\,.$

Note that the new coordinates $(\tilde{x},\tilde{\theta})$ may have more
general dependence on the original ones than that of the bundle gluings (5.2).
Also, by dint of invertibility, $\tilde{x}$ must be an even polynomial of
$O(\theta^{0})=:O(1)$, and $\tilde{\theta}$ is an odd polynomial of
$O(\theta)$.

It follows from the structure of superdiffeomorphisms described in (5.3), that
the algebra of superfields $\mathcal{A}$, and hence the supermanifold
$\mathcal{M}$ itself, has a decreasing filtration:

(5.4)
$\mathcal{A}=\mathcal{A}_{0}\supset\mathcal{A}_{1}\supset\cdots\supset\mathcal{A}_{m}\,,$

where $m$ is the odd dimension of $\mathcal{M}$, and in any choice of
coordinates $\mathcal{A}_{k}=O(\theta^{k})$. Observe that given a superfield
$F\in\mathcal{A}_{k}$, then $F_{k}\in\mathcal{A}_{k}/\mathcal{A}_{k+1}$ is
invariantly defined. The body $F_{0}$ of $F\in\mathcal{A}$ is a particular
example. A choice of splitting $(\mathcal{E}M,\pi)$ further induces a
$\mathbb{Z}_{m+1}$-grading associated with this filtration. Indeed, the choice
of a splitting is equivalent to such a $\mathbb{Z}_{m+1}$-grading [31].

Super vectors will be central to our study of dynamics; these are defined as
sections of the tangent sheaf ${\hskip 0.85358pt}T\mathcal{M}$. The latter is
the subsheaf of graded derivations of the sheaf of endomorphisms of
$\mathcal{A}$. Super vector fields form a rank $n+m$, free
$\mathcal{A}M$-module. They also form a Lie superalgebra under the
supercommutator (inherited from the usual composition of endomorphisms) of
super vector fields denoted by $[{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}},{\boldsymbol{\hbox to2.5pt{\hss$\cdot$\hss}}}\\}$.
We will also need superdifferential forms. Super covectors are sections of the
cotangent sheaf $T^{*}\\!\mathcal{M}$. The latter is defined as the pointwise
$\mathcal{A}M$-dual of the tangent sheaf $T\mathcal{M}$. Sections of
$T^{*}\\!\mathcal{M}$ are called superdifferential one-forms; these also form
a rank $n+m$, free $\mathcal{A}M$-module. Higher forms are defined below.

To describe super vectors and covectors/one-forms on a representative vector
bundle $(\mathcal{E}M,\pi)$, we denote the tangent sheaf by $T_{\pi}\\!{\hskip
0.85358pt}\mathcal{M}$. Note in particular that this does not equal the space
$T\mathcal{E}M$. Similarly we denote the cotangent sheaf by
$T^{*}_{\pi}\\!{\hskip 0.85358pt}\mathcal{M}$. On a trivializing patch with
coordinates $(x,\theta)$, super vectors and covectors are spanned by
$\\{\frac{\partial}{\partial x},\frac{\partial}{\partial\theta}\\}$, and their
duals $\\{dx,d\theta\\}$, respectively, with coefficients taking values in
sections of $\mathcal{E}M$. A super vector $v\in T_{\pi}\\!{\hskip
0.85358pt}\mathcal{M}$ and a super covector $\alpha\in T^{*}_{\pi}\\!{\hskip
0.85358pt}\mathcal{M}$ are then locally expressed as

$\displaystyle v$ $\displaystyle=v^{i}(x,\theta)\frac{\partial}{\partial
x^{i}}+v^{a}(x,\theta)\frac{\partial}{\partial\theta^{a}}\,,$
$\displaystyle\alpha$
$\displaystyle=\alpha_{i}(x,\theta)dx^{i}+\alpha_{a}(x,\theta)d\theta^{a}\,.$

Vectors $\frac{\partial}{\partial x}$ along the base are Grassmann even, and
vectors $\frac{\partial}{\partial\theta}$ along the fiber are Grassmann odd.
Note that on the body $f(x)\in C^{\infty}M$ of some superfield, vectors
$\frac{\partial}{\partial x}$ act as derivations in the standard way. Also
note that the vector $v^{a}\frac{\partial}{\partial\theta^{a}}$ acts as
interior multiplication on the fiber viewed as the exterior algebra $\Lambda
V^{*}$. The coordinate one-form $dx$ is assigned odd Grassmann parity while
$d\theta$ is even so that, for $\alpha$ and $v$ Grassmann even,

$\alpha(v)=\alpha_{i}v^{i}+\alpha_{a}v^{a}\in\Gamma({\mathcal{E}}M)\,.$

For even super vectors $v$, there is a well-defined notion of a body $v_{0}$.
Namely, for any function $f_{0}$ on the body,

$v_{0}f_{0}:=(vf)_{0}\,,$

where $f$ is any superfunction with body $f_{0}$.

There also exists a general notion of superflows along super vectors, see for
example [41, 42]. We only require a simple version of this: Given an even
supervector $v\in\Gamma(T\mathcal{M})$ and a chart $(U,\mathcal{A}U)$, for
short times $t\in(-\delta,\delta)\subset{\mathbb{R}}$, we can define a
supermanifold morphism between charts

$\operatorname{exp}(tv):(U,\mathcal{A}U)\to(U_{t},\mathcal{A}{U_{t}})\,,$

by requiring the following properties: $(i)$
$\operatorname{exp}(0v)=\operatorname{Id}$, (ii)
$\operatorname{exp}\big{(}(t+t^{\prime})v\big{)}=\operatorname{exp}(tv)\circ\operatorname{exp}(t^{\prime}v)$,
and $(iii)$ given any superfunction $f$ and any element $X$ of the dual
${\mathcal{A}}^{*}$ to the space of superfunctions,

$\frac{dX(f_{t})}{dt}=X(vf_{t})\,,$

where $f_{t}:=\operatorname{exp}(-tv)^{*}f$ is defined by pulling back the
second map ${\mathcal{A}}U_{t}\to{\mathcal{A}U}$ of the supermanifold
morphism. Notice that the map $U\to U_{t}$ is defined by the standard
exponential map $\operatorname{exp}(tv_{0})$. Short time existence and
uniqueness of $\operatorname{exp}(tv)$ is established in [42]. Formally,
$\operatorname{exp}(tv)$ acts on functions according to Taylor expansion in
powers of derivations. In local coordinates
$v=v^{i}(x,\theta)\frac{\partial}{\partial
x^{i}}+v^{a}(x,\theta)\frac{\partial}{\partial\theta^{a}}$, the open set
$U_{t}:=\operatorname{exp}(tv_{0}^{i}(x)\frac{\partial}{\partial x^{i}})(U)$
and $\mathcal{A}U\ni f(x^{i},\theta^{a})\mapsto
f(x^{i}+tv^{i},\theta^{a}+tv^{a})$. For suitable $v$, this (short time) map
extends to all of $\mathcal{M}$; main properties of the usual exponential map
hold for the super case too because it defines a one-parameter family of
superdiffeomorphisms [43].

On a representative vector bundle $(\mathcal{E}M,\pi)$, there are invariant
notions of vertical super vectors and horizontal super covectors. Sections of
the vertical subbundle $\text{Ver}{\hskip 0.85358pt}T_{\pi}\\!{\hskip
0.85358pt}\mathcal{M}:=\text{ker}{\hskip 0.85358pt}d\pi\subset
T_{\pi}\\!{\hskip 0.85358pt}\mathcal{M}$ of the tangent bundle are termed
vertical super vectors. The fibers of $\text{Ver}{\hskip
0.85358pt}T_{\pi}\\!{\hskip 0.85358pt}\mathcal{M}$ are isomorphic to that of
$\Pi\mathcal{V}^{*}M$. Furthermore, a super covector $\alpha$ satisfying
$\alpha(v)=0$ for any vertical super vector $v$, is termed a horizontal super
one-form. In fact we have an exact sequence

$0\rightarrow\Gamma(\mathcal{E}M\otimes\text{Ver}{\hskip
0.85358pt}T_{\pi}\\!{\hskip
0.85358pt}\mathcal{M})\rightarrow\Gamma(T_{\pi}\\!{\hskip
0.85358pt}\mathcal{M})\xrightarrow{d\pi}\Gamma(\mathcal{E}M\otimes
TM)\rightarrow 0\,.$

Super covectors decompose according to the dual exact sequence.

A superaffine connection $\nabla$ on a supermanifold is a map

$\nabla:\Gamma(T{\mathcal{M}})\to\Gamma(T^{*}{\mathcal{M}}\otimes
T{\mathcal{M}})$

subject to superanalogs of the usual properties:

$\nabla_{fU}V=f\nabla_{U}V\,,\qquad\nabla_{U}(fV)=(Uf)V+f\nabla_{U}V\,,$

for any (even) superfield $f$ and supervectors $U$ and $V$. A seminal result
of Koszul shows that a superaffine connection determines a splitting [31]. For
this let $F\in{\mathcal{A}_{k}}$ (see Equation (5.4)). Then we search for an
even super vector $X$ such that

$XF=kF+G$

for some $G\in{\mathcal{A}_{k+1}}$. Note that for ${\mathbb{R}}^{n|m}$ the
“Eulerian” vector $X=\theta^{a}\frac{\partial}{\partial\theta_{a}}$ obeys the
above stipulation. Koszul shows that upon additionally demanding

$\nabla_{X}X=X\,,$

the vector $X$ both exists and is unique. Then one can study the space of odd
super vectors $U$ such that

$\nabla_{X}U=-U\,.$

The exponential map of $\nabla$ in these directions determines vertical fibers
of the splitting associated to $\nabla$. Eigenvalues of the vector $X$ yield a
${\mathbb{Z}}_{m+1}$-grading on sections of the corresponding vector bundle.
We term the latter the Koszul bundle of $\nabla$, or a Koszul splitting.

Arbitrary rank tensor fields on $\mathcal{M}$ are defined by considering
tensor products of $T\mathcal{M}$ and $T^{*}\\!\mathcal{M}$ over
$\mathcal{A}M$. In particular superdifferential forms are defined by graded
skew tensor products of $T^{*}\\!\mathcal{M}$ whose section space is denoted
by $\Omega\mathcal{M}$. Note that while superdifferential forms can only
depend polynomially on the Grassmann odd differentials $dx$, the same is not
true for the even differentials $d\theta$. In principle, the precise
functional dependence on the $d\theta$ depends on the physical problem of
interest, see for example [44, 45]. Here we focus on formal power series.

The space of superdifferential forms is graded by form degree:

(5.5) $\Omega\mathcal{M}=\bigoplus_{k\in\mathbb{N}}\Omega^{k}\mathcal{M}\,.$

In local coordinates, the section spaces $\Omega^{k}\mathcal{M}$ are
eigenspaces of the Euler operator $dx^{i}\frac{\partial}{\partial
dx^{i}}+d\theta^{a}\frac{\partial}{\partial d\theta^{a}}$ (some authors denote
$\frac{\partial}{\partial dx^{i}}$ by $\iota_{\frac{\partial}{\partial
x^{i}}}$ and $\frac{\partial}{\partial d\theta^{i}}$ by
$\iota_{\frac{\partial}{\partial\theta^{i}}}$). There is also a
$\mathbb{Z}_{2}$-grading by the total Grassmann parity.

###  5.2. Differential Calculus.

For a representative vector bundle $(\mathcal{E}M,\pi)$, we denote the sheaf
of superdifferential forms by $\Omega_{\pi}\\!{\hskip 0.85358pt}\mathcal{M}$.
The grading determined by the choice $(\mathcal{E}M,\pi)$ allows us to
consider separately the operators $dx^{i}\frac{\partial}{\partial dx^{i}}$ and
$d\theta^{a}\frac{\partial}{\partial d\theta^{a}}$, with respective
eigenvalues $p$ and $q$. This gives a further decomposition on top of that of
Equation (5.5),

$\Omega^{k}_{\pi}\\!{\hskip 0.85358pt}\mathcal{M}=\bigoplus_{p+q=k,\ p,q\geq
0}\Omega^{p,q}_{\pi}\\!\mathcal{M}\,.$

In local coordinates $\Omega^{p,q}_{\pi}\\!\mathcal{M}$ is the space of
$(p+q)$-forms with $p$ bosonic and $q$ fermionic generators $dx$ and
$d\theta$, respectively.

The exterior derivative, interior product, and Lie derivative of
superdifferential forms can be defined along similar lines to their bosonic
counterparts. One may define these in local coordinates for a representative
vector bundle and then verify that this defines superdiffeomorphism invariant
operations. In particular the exterior derivative
$d_{T}:\Omega^{k}_{\pi}\\!{\hskip
0.85358pt}\mathcal{M}\to\Omega^{k+1}_{\pi}\\!{\hskip 0.85358pt}\mathcal{M}$ is
defined by

(5.6) $d_{T}:=dx^{i}\frac{\partial}{\partial
x^{i}}+d\theta^{a}\frac{\partial}{\partial\theta^{a}}\,.$

It is a nilpotent operator, videlicet $d_{T}^{2}=0$. In calculations we often
denote the respective summands on the right hand side above by $d$ and
$\hat{d}$. The choice of splitting thus leads to maps
$\Omega^{p,q}_{\pi}\\!\mathcal{M}\stackrel{{\scriptstyle
d}}{{\to}}\Omega^{p+1,q}_{\pi}\\!\mathcal{M}$ and
$\Omega^{p,q}_{\pi}\\!\mathcal{M}\stackrel{{\scriptstyle\hat{d}}}{{\to}}\Omega^{p,q+1}_{\pi}\\!\mathcal{M}$
for the corresponding representative $\mathcal{E}M$.

Given an even super vector field $v\in\Gamma(T_{\pi}\\!{\hskip
0.85358pt}\mathcal{M})$, the interior product maps $\Omega^{k}_{\pi}\\!{\hskip
0.85358pt}\mathcal{M}$ to $\Omega^{k-1}_{\pi}\\!{\hskip 0.85358pt}\mathcal{M}$
according to

$\iota_{v}:=v^{i}\frac{\partial}{\partial
dx^{i}}+v^{a}\frac{\partial}{\partial d\theta^{a}}\,.$

Once again this defines a pair of maps $\iota_{v^{i}\frac{\partial}{\partial
x^{i}}}$ and $\iota_{v^{a}\frac{\partial}{\partial\theta^{a}}}$ with domains
$\Omega^{p,q}_{\pi}\\!\mathcal{M}$. The Lie derivative with respect to a super
vector $v$ acting on superdifferential forms is defined, à la Cartan’s magic
formula, by

$\mathcal{L}_{v}:=d_{T}{\hskip 0.85358pt}\iota_{v}+\iota_{v}{\hskip
0.85358pt}d_{T}\,.$

Finally, for any supermanifold ${\mathcal{M}}$ , there is a canonical, top
degree polyvector

${\mathscr{X}}\in\Gamma(\Lambda^{m}T{\mathcal{M}})\,,$

that we term the Eulerian top field: Given a splitting and corresponding
representative vector bundle $\mathcal{E}M$, there is a canonical fiber-
density valued, top polyvector

(5.7) $\epsilon:=\frac{1}{m!}{\hskip 0.85358pt}\epsilon^{a_{1}\cdots
a_{m}}\frac{\partial}{\partial\theta^{a_{1}}}\cdots\frac{\partial}{\partial\theta^{a_{m}}}\in\Gamma\big{(}(\Lambda^{m}\mathcal{V}_{\pi}M)^{-1}\otimes\Lambda^{m}\Pi\mathcal{V}^{*}_{\pi}M\big{)}\,,$

where $\epsilon^{a_{1}\cdots a_{m}}$ denotes the totally anti-symmetric Levi-
Civita symbol. The above is not canonical to the supermanifold ${\mathcal{M}}$
itself. Nor is the fiberwise Euler operator

$\theta^{a}\frac{\partial}{\partial\theta^{a}}\in\Gamma(\Pi\mathcal{V}_{\pi}M\otimes\Pi\mathcal{V}_{\pi}^{*}M)\,,$

(whose eigenvalues correspond to the ${\mathbb{Z}}_{m+1}$ grading à la
Koszul). However, combining these ingredients—taking the top skew power of the
fiber Euler operator—does give an invariant object

(5.8) ${\mathscr{X}}:=\frac{1}{m!}\theta^{a_{1}}\cdots{\hskip
0.85358pt}\theta^{a_{m}}\frac{\partial}{\partial\theta^{a_{1}}}\cdots\frac{\partial}{\partial\theta^{a_{m}}}\in\Gamma(\Lambda^{m}\Pi\mathcal{V}\otimes\Lambda^{m}\Pi\mathcal{V}^{*})\subset\Gamma(\Lambda^{m}T{\mathcal{M}})\,.$

###  5.3. Integral Calculus.

Let us begin with the supermanifold $\mathbb{R}^{n|m}$. A superfield
$F\in\mathcal{A}\mathbb{R}^{n|m}$ is called super integrable if its
coefficients in a $\theta$-expansion are $\mathbb{R}^{n}$-integrable. Its
Berezin integral is given by [46]

$\int_{\mathbb{R}^{n|m}}F:=\int_{\mathbb{R}^{n}}d^{n}x\frac{\partial}{\partial\theta^{m}}\cdots\frac{\partial}{\partial\theta^{1}}F(x,\theta)\,.$

In the case $n=m$, the supermanifold $\mathbb{R}^{n|n}$ can be viewed as the
vector bundle $\Lambda\mathbb{R}^{n}\to{\mathbb{R}}^{n}$. A superfield $F$
then amounts to a (possibly inhomogeneous) differential form
$F\in\Omega\mathbb{R}^{n}$. The Berezin integral computes the integral over
$\mathbb{R}^{n}$ of its top form component.

The Berezin integral depends on an $\mathbb{R}^{n|m}$-integration measure
$\mu=d^{n}x\frac{\partial}{\partial\theta^{m}}\cdots\frac{\partial}{\partial\theta^{1}}$.
Under a superdiffeomorphism
$\varphi:(x,\theta)\mapsto(\tilde{x},\tilde{\theta})$, this transforms by a
(right) factor of the superdeterminant defined by

$\operatorname{sdet}(\varphi)=\frac{\operatorname{det}(\varphi_{00}-\varphi_{01}\varphi_{11}^{-1}\varphi_{10}^{T})}{\operatorname{det}(\varphi_{11})}=\frac{\operatorname{det}(\varphi_{00})}{\operatorname{det}(\varphi_{11}-\varphi_{10}^{T}\varphi_{00}^{-1}\varphi_{01})}\,,$

where the above matrices are $\varphi_{00}:=\frac{\partial
x}{\partial\tilde{x}}$, $\varphi_{01}:=\frac{\partial
x}{\partial\tilde{\theta}}$,
$\varphi_{10}:=\frac{\partial\theta}{\partial\tilde{x}}$, and
$\varphi_{11}:=\frac{\partial\theta}{\partial\tilde{\theta}}$.

On a general supermanifold $\mathcal{M}$, one can integrate sections of a
distinguished line bundle $\text{Ber}\mathcal{M}$ called the Berezinian of
$\mathcal{M}$ [39]. On a representative $(\mathcal{E}M,\pi)$, the Berezinian
admits the decomposition

$\text{Ber}_{\pi}\mathcal{M}=\Lambda^{n}_{\pi}M\otimes\Lambda^{m}\Pi\mathcal{V}^{*}_{\pi}M\,.$

Locally, a section of $\text{Ber}_{\pi}\mathcal{M}$ is generated by
$d^{n}x\frac{\partial}{\partial\theta^{m}}\cdots\frac{\partial}{\partial\theta^{1}}$
in concordance with the Berezin measure $\mu$ on $\mathbb{R}^{n|m}$. Thus,
under a superdiffeomorphism, a section of the Berezinian transforms by a
factor of the superdeterminant of the Jacobian. Berezin integrals can then be
performed by trivializing sections of the Berezinian over a partition of
unity; see for example [27, 39].

###  5.4. Hermitean Superfields

Given a choice of splitting, there is a notion of a hermitean superfield (see
for example [47]) defined as follows. First one considers the space of
complex-valued superfields

$F\in\Gamma({\mathcal{E}}M)\oplus
i\Gamma({\mathcal{E}}M)=:\Gamma_{\mathbb{C}}({\mathcal{E}}M)\,.$

Then we define an (antilinear) involution
$\dagger:\Gamma_{\mathbb{C}}({\mathcal{E}}M)\to\Gamma_{\mathbb{C}}({\mathcal{E}}M)$
via the map

$F^{\dagger}=\big{(}\oplus_{k=0}^{m}F_{k})^{\dagger}:=\oplus_{k=0}^{m}(-1)^{\lfloor\frac{k}{2}\rfloor}F^{*}_{k}\,,$

where the direct sum above is that of the ${\mathbb{Z}}_{k+1}$ grading and $*$
denotes complex conjugation. In local (real) Grassmann coordinates
$\theta^{a}$, the above amounts to defining

$(\theta^{a})^{\dagger}=\theta^{a}\>\mbox{ and
}\>(\theta^{a}\theta^{b})^{\dagger}:=(\theta^{b})^{\dagger}(\theta^{a})^{\dagger}=-\theta^{a}\theta^{b}\,.$

A complex-valued vector field that obeys

$F=F^{\dagger}$

is termed a hermitean superfield. The above involution induces corresponding
actions on super tensors where the grading is obtained by counting $\theta$’s,
$d\theta$’s and $\frac{\partial}{\partial\theta^{a}}$’s. For example,
$(d\theta^{a}\wedge d\theta^{b})^{\dagger}=-d\theta^{a}\wedge d\theta^{b}$.
This ensures that a hermitean super $k$-form integrated along a
$k$-dimensional manifold obeys a reality condition.

######  Example 5.1.

Consider the spinning particle model of Berezin and Marinov [3]:

$S[\theta(t)]=i\int
dt(\delta_{ab}\theta^{a}\dot{\theta^{b}}+\epsilon_{abc}v^{a}\theta^{b}\theta^{c})\,,$

where $\theta^{a}\in{\mathbb{R}}^{0|3}$ and $v^{a}\in{\mathbb{R}}^{3}$. Since
$(\theta^{a}\dot{\theta}^{b})^{\dagger}=-\theta^{a}\dot{\theta}^{b}$, the
above action is “real”. Hence the one-form
$i\big{(}\delta_{ab}\theta^{a}d{\theta^{b}}+\epsilon_{abc}v^{a}\theta^{b}\theta^{c}dt)$
is hermitean. $\blacksquare$

Note that in physics the above notion of hermiticity is called a real
superfield. Observe however that the product of two hermitean superfields is
not necessarily hermitean (for example
$(\theta^{1}\theta^{2})^{\dagger}=-\theta^{1}\theta^{2}$), so the subset of
hermitean superfields in $\Gamma_{\mathbb{C}}({\mathcal{E}}M)$ is not closed
under multiplication. One can also consider supermanifolds given by a sheaf of
complex exterior algebras $\Lambda^{m}{\mathbb{C}}$ over a base manifold $M$
and extra data of an involution $\dagger$ [47]. We prefer to work with the
above, real supermanifold-based notion of hermiticity. Hermitean superfields
will play an important rôle when defining an inner product in Section 5.6.

###  5.5. Symplectic Supermanifolds

A symplectic supermanifold $(\mathcal{M},\Omega)$ is an $(n,m)$-supermanifold
$\mathcal{M}$ equipped with a maximally non-degenerate, closed, Grassmann-
even, superdifferential two-form $\Omega$. By maximally non-degenerate we mean
that the subbundle of $T\mathcal{M}$ defined by the super-vector solution
space of

$\iota_{v}\Omega=0\,,$

has rank zero or one, as a module over the ring of superfunctions on
${\mathcal{M}}$. Equivalently, the subbundle $\operatorname{ker}\flat\subset
T\mathcal{M}$ defined by the kernel of the musical map

$\displaystyle\flat:T\mathcal{M}$ $\displaystyle\to T^{*}\mathcal{M}\,,$
$\displaystyle v\;$ $\displaystyle\mapsto\;\iota_{v}\Omega$

is Grassmann even and has rank either zero or one. For physical applications
we will need supermanifolds equipped with a hermitean symplectic form. Again
these are classified by the rank of the musical map $\flat$ (but now over
${\mathbb{C}}$). To distinguish the two cases, following Section 3, we term
supermanifolds equipped with a hermitean symplectic form with even or odd
dimensional body manifolds, super phase-spaces and super phase-spacetimes, and
typically denote them by $({\mathcal{M}},\Omega)$ and
$({\mathcal{Z}},\Omega)$, respectively. Exactly as in the bosonic case, a
hermitean supersymplectic form $\Omega$ on a super phase-space is non-
degenerate, whereas on a super phase-spacetime it admits a one dimensional
nontrivial (Grassmann even) kernel.

######  Example 5.2 (Rothstein structures).

Rothstein [48] shows that real (non-degenerate) super symplectic manifolds
correspond in a natural way to the data $({\mathcal{V}}M,\eta,\nabla,\omega)$
of a vector bundle ${\mathcal{V}}M$, a bundle metric $\eta$ with a compatible
bundle connection $\nabla$ as well as a symplectic form $\omega$ on the base.
A careful account of Rothstein structures necessitates a discussion of the
Ehresmann description of bundle connections on $T{\mathcal{M}}$. These details
are not needed in a local coordinate description. We focus on a special case
due to Sasaki [49]: Let $(M,g)$ be an even dimensional Riemannian manifold and
${\mathcal{M}}=(M,\Omega M)$. In local coordinates $x^{\mu}$ for $M$, the
fibers of ${\mathcal{M}}$ are generated by (odd) covectors
$\theta^{\mu}=dx^{\mu}$. Superfields on ${\mathcal{M}}$ are differential forms
on $M$.

Let us define an operator mapping $\Omega{\mathcal{M}}\to\Omega{\mathcal{M}}$
by

${\nabla}:=d-\Gamma^{\mu}_{\
\nu}\theta^{\nu}\frac{\partial}{\partial\theta^{\mu}}\,,$

where $\Gamma^{\mu}{}_{\nu}$ denotes the (one-form valued) Levi-Civita
connection coefficients of $g_{\mu\nu}$ and
$d:=dx^{\mu}\frac{\partial}{\partial x^{\mu}}$. Accordingly we can
covariantize the exterior derivative operator

$d_{T}:=d+d\theta^{\mu}\frac{\partial}{\partial\theta^{\mu}}={\nabla}+(\nabla\theta^{\mu})\frac{\partial}{\partial\theta^{\mu}}\,,$

where $\nabla\theta^{\mu}:=d\theta^{\mu}+\Gamma^{\mu}_{\ \nu}\theta^{\nu}$.
Now let $\omega_{0}$ be any symplectic form on $M$. Then the above data can be
packaged as a supersymplectic two-form

$\Omega=\omega_{0}+g_{\mu\nu}\nabla\theta^{\mu}\nabla\theta^{\nu}-R_{\mu\nu}\theta^{\mu}\theta^{\nu}\,,$

where indices are manipulated using $g_{\mu\nu}$ and the (two-form valued)
Riemann-tensor
$R^{\rho}{}_{\sigma}:=d\Gamma^{\rho}{}_{\sigma}+\Gamma^{\rho}{}_{\kappa}\Gamma^{\kappa}_{\sigma}$.
Non-degeneracy and closed-ness of $\Omega$ are easily verified. In particular,
locally we have that $\omega=d\lambda$ and in turn

$\Omega=d_{T}\big{(}\lambda+\theta_{\mu}\nabla\theta^{\mu}\big{)}\,.$

Note that Rothstein has a stronger result constructing local Darboux models
for super phase-spaces [48]. Also note that
$\omega_{0}+i(g_{\mu\nu}\nabla\theta^{\mu}\nabla\theta^{\nu}-R_{\mu\nu}\theta^{\mu}\theta^{\nu})$
defines a hermitean symplectic form and, in turn, a super phase-space.
$\blacksquare$

It is useful to further analyze the closure condition on the supersymplectic
form $\Omega$. On a representative vector bundle $(\mathcal{E}M,\pi)$, we have
the decomposition

$\Omega=\omega+A+\eta\,,$

where $\omega\in\Omega^{2,0}_{\pi}\\!\mathcal{M}$,
$A\in\Omega^{1,1}_{\pi}\\!\mathcal{M}$ and
$\eta\in\Omega^{0,2}_{\pi}\\!\mathcal{M}$, respectively. The body $\omega_{0}$
of $\Omega$ is a two-form on $M$. We will primarily be interested in the
generic case where $\omega_{0}$ itself is a symplectic form on $M$. Therefore
in the case that $\Omega$ is degenerate, any degeneracy is encoded in the
symplectic body manifold $(M,\omega_{0})$ as $\omega_{0}$ itself must be
maximally non-degenerate. Because $\Omega$ is an even supermatrix, it is
invertible if and only if both $\omega$ and $\eta$ are invertible.
Equivalently the symmetric bilinear form $\eta$ on the Grassmann distribution
is always invertible. Decomposing $d_{T}=d+\hat{d}$ in concord with Equation
(5.6), closedness $0=d_{T}\Omega$ amounts to a quartet of equations:

$\displaystyle 0=d\omega\in\Omega^{3,0}_{\pi}\\!\mathcal{M}\,,\qquad
0=\hat{d}\omega+dA\in\Omega^{2,1}_{\pi}\\!\mathcal{M}\,,\quad$ $\displaystyle
0=\hat{d}A+d\eta\in\Omega^{1,2}_{\pi}\\!\mathcal{M}\,,\qquad
0=\hat{d}\eta\in\Omega^{0,3}_{\pi}\\!\mathcal{M}\,.$

In turn, closedness of $\Omega$ implies the following Hodge-like decomposition
(see [50]).

######  Theorem 5.3.

Let $(\mathcal{M},\Omega)$ be a symplectic supermanifold and
$(\mathcal{E}M,\pi)$ be a representative vector bundle such that
$\Omega=\omega+A+\eta$ where $\omega\in\Omega^{2,0}_{\pi}\\!\mathcal{M}$,
$A\in\Omega^{1,1}_{\pi}\\!\mathcal{M}$ and
$\eta\in\Omega^{0,2}_{\pi}\\!\mathcal{M}$. Then

(5.10) $\Omega=\omega_{0}+\frac{1}{1+\hat{d}^{-1}d}\,\hat{d}(\beta+\gamma)\,,$

where the body $\omega_{0}\in\Omega^{2}M$ of $\omega$ is $d$-closed,
$\hat{d}^{-1}$ is any partial inverse of $\hat{d}$, and
$\beta\in\Omega^{1,0}_{\pi}\\!\mathcal{M}$ and
$\gamma\in\Omega^{0,1}_{\pi}\\!\mathcal{M}$ are arbitrary Grassmann-odd one-
forms. In the above, the operator $1/(1+\hat{d}^{-1}d)$ acting on a section of
$\Omega^{p,q}_{\pi}\\!\mathcal{M}$ stands for the formal power series
$\sum_{\ell=0}(-\hat{d}^{-1}d)^{\ell}$ where $\hat{d}^{{\hskip 0.85358pt}-1}$
is defined to be the zero operator when acting on
$\Omega^{k,0}_{\pi}{\mathcal{M}}$.

The proof of the above theorem relies on the following standard (see for
example [51]) lemma and its corollary.

######  Lemma 5.4.

Any $\hat{d}$-closed form is $\hat{d}$-exact.

###### Proof.

It suffices to establish the claim for a $\hat{d}$-closed form
$f\in\Omega^{r,p}_{\pi}\\!\mathcal{M}$. Thanks to the
$\mathbb{Z}_{m+1}$-grading on $\mathcal{E}M$, we may focus on the coefficient
of $f_{p,q}$ with $q$ Grassmann generators. Because $\hat{d}$ is a fiber-wise
operator, we can employ local coordinates for which

$f_{p,q}=f(x,dx)_{a_{1}\cdots a_{q}b_{1}\cdots
b_{p}}\theta^{a_{1}}\cdots\theta^{a_{q}}d\theta^{b_{1}}\cdots
d\theta^{b_{p}}\,.$

Here the coefficients are totally anti-symmetric in the $a$ indices and
totally symmetric in the $b$ indices. In Young tableaux notation, we can
decompose $f$ as

$\ytableausetup{aligntableaux=center,smalltableaux,boxsize=1em}\ytableaushort{{\scriptscriptstyle
a_{1}},{\scriptscriptstyle
a_{2}},{\none[\scriptscriptstyle\vdots]},{\scriptscriptstyle
a_{q}}}\>\otimes\>\ytableaushort{{\scriptscriptstyle b_{1}}{\scriptscriptstyle
b_{2}}{\none[\scriptscriptstyle\cdots]}{\scriptscriptstyle
b_{p}}}=\ytableaushort{{\scriptscriptstyle b_{1}}{\scriptscriptstyle
b_{2}}{\none[\scriptscriptstyle\cdots]}{\scriptscriptstyle
b_{p}},{\scriptscriptstyle
a_{1}},{\none[\scriptscriptstyle\vdots]},{\scriptscriptstyle
a_{q}}}\>\oplus\>\ytableaushort{{\scriptscriptstyle a_{1}}{\scriptscriptstyle
b_{1}}{\none[\scriptscriptstyle\cdots]}{\scriptscriptstyle
b_{p}},{\scriptscriptstyle
a_{2}},{\none[\scriptscriptstyle\vdots]},{\scriptscriptstyle a_{q}}}\ \ .$

Closedness of $f$ says that right tableau in the last equality vanishes.

Now consider $f_{p-1,q+1}\in\Omega^{r,p-1}_{\pi}\\!\mathcal{M}$ with $q+1$
Grassmann generators. Then the exact form
$\hat{d}f_{p-1,q+1}\in\Omega^{r,p}_{\pi}\\!\mathcal{M}$ with $q$ Grassmann
generators has the tensor structure

$\ytableausetup{aligntableaux=center,smalltableaux,boxsize=1.2em}\ytableaushort{{\scriptscriptstyle
a_{1}}{\scriptscriptstyle
b_{1}}{\none[\scriptscriptstyle\cdots]}{\scriptscriptstyle
b_{p\\!-\\!1}},{\scriptscriptstyle
a_{2}},{\none[\scriptscriptstyle\vdots]},{\scriptscriptstyle\scriptscriptstyle
a_{q\\!+\\!1}}}\ \ ,$

which matches that of any $\hat{d}$-closed $f$. ∎

######  Corollary 5.5.

Let $f\in\Omega^{r,p}_{\pi}\\!\mathcal{M}$ and
$g\in\Omega^{r,p+1}_{\pi}\\!\mathcal{M}$ be $\hat{d}$-closed. Then any
solution to equation

$\hat{d}f=g$

has the form $f=\hat{d}^{-1}g+\hat{d}h$ where $\hat{d}^{-1}$ is a partial
inverse of $\hat{d}$, and $h\in\Omega^{r,p-1}_{\pi}\\!\mathcal{M}$ is an
arbitrary form.

###### Proof.

Existence of partial inverses follows because $\hat{d}$ is a fiber-wise
operator, so we are dealing with a finite dimensional, non-zero linear map.
Moreover $g$ is in the image of $\hat{d}$ by Lemma 5.4. Thus $\hat{d}^{-1}g$
solves the displayed equation. Moreover the difference of two solutions is
necessarily $\hat{d}$-exact. The result now follows by again employing Lemma
5.4.

∎

###### Proof of Theorem 5.3.

Writing $\omega=\omega_{0}+\hat{\omega}$, the first equation in
$(\ref{quartet1})$ implies the body $\omega_{0}$ of $\omega$ is $d$-closed.
Also $\hat{d}\omega_{0}$ necessarily vanishes so the second equation says

$\hat{d}{\hskip 0.85358pt}\hat{\omega}+dA=0\,.$

Hence $dA$ is $\hat{d}$-closed, so using Corollary 5.5 we have

$\hat{\omega}=-\hat{d}^{-1}dA\,.$

(There is no possibility to add a $\hat{d}$-exact piece because
$\hat{\omega}\in\Omega^{2,0}_{\pi}\\!\mathcal{M}$.) Now noting that $d\eta$ is
$\hat{d}$-closed, we can solve the third equation using Corollary 5.5,

$A=-\hat{d}^{-1}d\eta+\hat{d}\beta\,,$

for some Grassmann-odd $\beta\in\Omega^{1,0}_{\pi}\\!\mathcal{M}$. Finally
using Lemma 5.4, the fourth equation can be solved as

$\eta=\hat{d}\gamma\,,$

where $\gamma\in\Omega^{0,1}_{\pi}\\!\mathcal{M}$ is an arbitrary Grassmann-
odd form. Orchestrating the above gives

$\Omega=\omega_{0}+(1-\hat{d}^{{\hskip 0.85358pt}-1}d){\hskip
0.85358pt}\hat{d}\beta+\big{(}1-\hat{d}^{{\hskip
0.85358pt}-1}d+(\hat{d}^{{\hskip 0.85358pt}-1}d)^{2}\big{)}{\hskip
0.85358pt}\hat{d}\gamma\,.$

The operator $\hat{d}^{{\hskip 0.85358pt}-1}d$ is, respectively, two and
three-step nilpotent when acting on $\hat{d}\beta$ and $\hat{d}\gamma$. The
result displayed in the theorem is a succinct rewriting of the above display
based on this latter fact. ∎

###  5.6. Supersymplectic Inner Products

It is no longer obvious how to employ the simple mechanism in Equation (4.1)
where states were expressed as—manifestly positive—squares of functions,
because the algebra of superfunctions is Grassmann. One might require
positivity of the body of superfields, but this imposes no conditions on
higher filtered components. To resolve this difficulty, we observe the
following: Odd generators on a supermanifold can be viewed as generalized one-
forms. Moreover, on a Riemannian manifold, the Hodge pairing yields an inner
product on differential forms. Therefore we aim to employ the supersymplectic
form to mimic the Hodge construction. (Note that a supergravity motivated
Hodge construction on supermanifolds is also studied in [50].) One further
important observation is that there is a direct correspondence between the
Grassmann analog of the Moyal star product and the Hodge construction.
Ironically therefore, we will employ star products (whose origins are quantum
mechanical) to describe classical measurement.

In fact, even for a pair of compactly supported functions $f$ and $g$ on
${\mathbb{R}}^{2n}\ni\xi^{i}$, with its standard symplectic form $\omega_{\rm
std}=\frac{1}{2}{\hskip 0.85358pt}\omega_{ij}{\hskip 0.85358pt}d\xi^{i}\wedge
d\xi^{j}$ ($\omega_{ij}$ is a constant matrix and $\omega^{ij}$ is its
inverse), and accompanying Moyal star product

$f\star g=f\operatorname{exp}\left(\frac{1}{2}{\hskip
0.85358pt}\overset{\leftarrow}{\partial}_{\xi^{i}}{\hskip
0.85358pt}{\omega^{ij}{\hskip
0.85358pt}\overset{\rightarrow}{\partial}_{\xi^{j}}}\right)g\,,$

one has that [36]

$\int_{{\mathbb{R}}^{2n}}(\omega_{\rm std})^{\wedge
n}fg=\int_{{\mathbb{R}}^{2n}}(\omega_{\rm std})^{\wedge n}f\star g\,.$

The right hand side of the above can also be viewed as a quantum mechanical
trace of the operators $\hat{f}$ and $\hat{g}$ obtained by the Moyal
quantization of the corresponding classical functions (see [52]). In summary,
our aim is to generalize the right hand side of the above display to
symplectic supermanifolds.

####  5.6.1. Supersymplectic volume form

Note that on any super phase-space $(\mathcal{M},\Omega)$, the symplectic form
$\Omega$ is non-degenerate and defines a natural section
$\operatorname{Ber}(\Omega)$ of the Berezinian $\text{Ber}\mathcal{M}$. On a
representative $(\mathcal{E}M,\pi)$, $\operatorname{Ber}(\Omega)$ is locally
trivialized as

(5.11) $dx^{1}\cdots
dx^{2n}\frac{\partial}{\partial\theta^{m}}\cdots\frac{\partial}{\partial\theta^{1}}\circ\text{sdet}^{1/2}(\Omega)\in\Gamma(\text{Ber}_{\pi}\mathcal{M})\,,$

where

$\operatorname{sdet}(\Omega)=\frac{\operatorname{det}(\omega-A\eta^{-1}A^{T})}{\operatorname{det}(\eta)}=\frac{\operatorname{det}(\omega)}{\operatorname{det}(\eta-A^{T}\omega^{-1}A)}\,,$

and $\Omega=\omega+A+\eta$ with $\omega\in\Omega^{2,0}_{\pi}\\!\mathcal{M}$,
$A\in\Omega^{1,1}_{\pi}\\!\mathcal{M}$ and
$\eta\in\Omega^{0,2}_{\pi}\\!\mathcal{M}$. Note that the integral
$\int_{\mathcal{M}}\operatorname{Ber}(\Omega)$ vanishes for the standard
supersymplectic form on any patch in $\mathbb{R}^{n|m}$, so clearly must be
modified to obtain a notion of supersymplectic volumes. For that we introduce
a canonical superfield $\Theta\in\mathcal{A}_{m}$ on a symplectic
supermanifold $(\mathcal{M},\Omega)$ that plays the rôle of a vertical volume
form.

Our construction relies on a set of contraction operations between arbitrary
tensor powers of bilinear forms and suitable sets of vectors on $\mathcal{M}$.
Let $B\in\Gamma({\otimes^{2}}T^{*}\mathcal{M})$ be any covariant bilinear form
on $\mathcal{M}$ and $k$ a positive integer. Then define
$B^{k}\in\Gamma({\otimes^{2k}}T^{*}\mathcal{M})$ by

$B^{k}(X_{1},\ldots,X_{k};Y_{1},\ldots,Y_{k}):=\frac{1}{k!}{\hskip
0.85358pt}B(X_{1},Y_{1}){\hskip 0.85358pt}\cdots B(X_{k},Y_{k})\,,$

where
$(X_{1},\cdots,X_{k}),(Y_{1},\cdots,Y_{k})\in\Gamma(T\mathcal{M})\times\cdots\times\Gamma(T\mathcal{M})$
(note that $B^{k}$ differs from $B^{\otimes k}$ only by a permutation of its
arguments). The tensors $B^{k}$ descend to any representative vector bundle
$\mathcal{E}M$ and act on density-valued tensors in the obvious way. Observe
that, applied to the top polyvector (5.7), we then obtain a fiber density on
$\mathcal{E}M$ given by

$\Omega^{m}(\epsilon,\epsilon)=\frac{1}{m!}{\hskip
0.85358pt}\epsilon^{a_{1}\cdots a_{m}}\Omega_{a_{1}b_{1}}\cdots{\hskip
0.85358pt}{\hskip 0.85358pt}\Omega_{a_{m}b_{m}}\epsilon^{b_{1}\cdots
b_{m}}=\det\eta\,.$

When fed the Eulerian top vector (5.8) and top polyvector, $\Omega^{m}$
produces

$\Omega^{m}(\epsilon,{\mathscr{X}})=\det\eta\ \theta^{1}\cdots{\hskip
0.85358pt}\theta^{m}\,.$

Hence we have

(5.12)
$\Theta(\Omega):=\Omega^{m}(\epsilon,{\mathscr{X}})/\Omega^{m}(\epsilon,\epsilon)^{1/2}=\sqrt{\det\eta}\>{\hskip
0.85358pt}\theta^{1}\cdots{\hskip 0.85358pt}\theta^{m}\,.$

It can be explicitly checked using Equation (5.3) that the above is a
superfield (defined independently of the choice of splitting), indeed

$\Theta\in{\mathcal{A}}_{m}\,.$

We dub the canonically defined superfield $\Theta$ the vertical volume field.
We can now define the symplectic supervolume of $({\mathcal{M}},\Omega)$ by

${\rm
Vol}_{\Omega}({\mathcal{M}}):=\int_{\mathcal{M}}\operatorname{Ber}(\Omega){\hskip
0.85358pt}\Theta(\Omega)\,,$

when this integral exists. For a patch ${\mathcal{U}}$ in
$({\mathbb{R}}^{n|m},\Omega_{\rm std})$ over a compact set
$U\subset{\mathbb{R}}^{n}$, the above returns the standard Euclidean volume of
$U$. Note that using (5.11) and (5.12), for any superfield $F$, in a choice of
splitting $\Omega=\omega+A+\eta$, one has

(5.13)
$\int_{\mathcal{M}}\text{Ber}(\Omega)\Theta(\Omega)F=\int_{M}\omega_{0}^{\wedge\frac{\text{dim}M}{2}}F_{0}\,.$

We also require the following invariance property of the above measure with
respect to supersymplectomorphisms.

######  Lemma 5.6.

Let $(\mathcal{M},\Omega)$ be a super phase-space. Suppose
$X\in\Gamma(T\mathcal{M})$ obeys $\mathcal{L}_{X}\Omega=0$. Then, for any
compactly supported $F\in\mathcal{A}$

$\int_{\mathcal{M}}\operatorname{Ber}(\Omega){\hskip
0.85358pt}\Theta(\Omega)\,{\mathcal{L}}_{X}F=0\,.$

###### Proof.

Making a choice of splitting, we decompose $\Omega=\omega+A+\eta$ in the usual
way. Similarly, $X=x+\chi$ where $x$ is a derivation along the base. The body
of $\mathcal{L}_{X}\Omega$ must vanish separately and is given by

$\mathcal{L}_{x_{0}}\omega_{0}\,,$

where $\omega_{0}\in\Omega^{2}M$ and $x_{0}\in\Gamma(TM)$. But using (5.13),
we deduce that

$\int_{\mathcal{M}}\operatorname{Ber}(\Omega){\hskip
0.85358pt}\Theta(\Omega)\,{\mathcal{L}}_{X}F=\int_{M}\omega_{0}^{\wedge\frac{\dim
M}{2}}{\hskip 0.85358pt}\,{\mathcal{L}}_{x_{0}}F_{0}\,,$

where $F_{0}$ is the body of $F$. The result now follows from the compact
support of $F_{0}$. ∎

Finally note that the above construction, as well as the corresponding analog
of Lemma 5.6, applies to the case where $\Omega$ is a hermitean
supersymplectic form.

####  5.6.2. Super stars

The supersymplectic generalization of the Moyal star product can be employed
to quantize the algebra of superfunctions[53, 54, 55, 56, 57, 58]. Indeed
given the data of a suitable superaffine connection, it is possible to define
a super Moyal-$\filledstar$ product [59, 60, 61]. However what we require is a
fiberwise inner product in order to extract classical probabilities from
superfunctions. A simple example is illustrative.

######  Example 5.7.

Consider the super manifold ${\mathbb{R}}^{0|2}$ equipped with a hermitean
supersymplectic form

$\Omega=i\big{(}d\theta^{1}\wedge d\theta^{1}+d\theta^{2}\wedge d\theta^{2})$

and a hermitean superfield

$\Lambda\mathbb{R}^{2}\ni
f_{0}+f_{1}\theta^{1}+f_{2}\theta^{2}-if_{12}\theta^{1}\theta^{2}=F\,,$

where $f_{0},f_{1},f_{2},f_{12}\in{\mathbb{R}}$. We now define the
map/representation

$\sigma(F)=\begin{pmatrix}f_{0}+f_{12}&\bar{f}\\\
f&f_{0}-f_{12}\end{pmatrix}=\sigma(F)^{\dagger}\,,$

where $f:=f_{1}+if_{2}$ and the above dagger is the standard involution on
complex $2\times 2$ matrices. Given $F,G\in{\mathcal{A}}$ we can then define a
star product by demanding

$\sigma(F\filledstar G):=\sigma(F)\,\sigma(G)\,,$

in which case we have that

$\filledstar:=\mathbb{1}\,+\stackrel{{\scriptstyle\leftarrow}}{{\frac{\partial}{\partial\theta^{1}}}}{\hskip
0.85358pt}\stackrel{{\scriptstyle\rightarrow}}{{\frac{\partial}{\partial\theta^{1}}}}\,+\,\stackrel{{\scriptstyle\leftarrow}}{{\frac{\partial}{\partial\theta^{2}}}}{\hskip
0.85358pt}\stackrel{{\scriptstyle\rightarrow}}{{\frac{\partial}{\partial\theta^{2}}}}\,+\,\stackrel{{\scriptstyle\leftarrow}}{{\frac{\partial}{\partial\theta^{2}}}}\stackrel{{\scriptstyle\leftarrow}}{{\frac{\partial}{\partial\theta^{1}}}}{\hskip
0.85358pt}\stackrel{{\scriptstyle\rightarrow}}{{\frac{\partial}{\partial\theta^{1}}}}\stackrel{{\scriptstyle\rightarrow}}{{\frac{\partial}{\partial\theta^{2}}}}{\hskip
0.85358pt}={\hskip
0.85358pt}\operatorname{exp}\Big{(}\frac{1}{i}\stackrel{{\scriptstyle\leftarrow}}{{\partial}}_{A}\Omega^{AB}\stackrel{{\scriptstyle\rightarrow}}{{\partial}}_{B}\\!\Big{)}\,.$

Note that $\theta^{1}\filledstar\theta^{1}=1=\theta^{2}\filledstar\theta^{2}$,
$\theta^{1}\filledstar\theta^{2}=\theta^{1}\theta^{2}=-\theta^{2}\filledstar\theta^{1}$.
Indeed, this star product encodes the multiplication rule for matrices, and
thus is necessarily associative. The (super) Moyal product for general
hermitean superfields $F,G$ yields

$F\filledstar
G=f_{0}g_{0}+f_{1}g_{1}+f_{2}g_{2}+f_{12}g_{12}+O(\theta^{1}\\!,\theta^{2})\,.$

In turn

$\displaystyle\int_{\mathcal{M}}\text{Ber}(\Omega){\hskip
0.85358pt}\Theta(\Omega){\hskip 0.85358pt}F\filledstar G$
$\displaystyle=\partial_{\theta^{2}}\partial_{\theta^{1}}\big{(}\theta^{1}\theta^{2}\big{(}f_{0}g_{0}+f_{1}g_{1}+f_{2}g_{2}+f_{12}g_{12}+O(\theta^{1}\\!,\theta^{2})\big{)}$
$\displaystyle=f_{0}g_{0}+f_{1}g_{1}+f_{2}g_{2}+f_{12}g_{12}=:(F,G)_{\Omega}\,.\hskip
56.9055pt$

The above is a bona fide inner product. Hence, both obviously and importantly
for constructing states,

$\int_{\mathcal{M}}\text{Ber}(\Omega){\hskip 0.85358pt}\Theta(\Omega){\hskip
0.85358pt}F\filledstar F=f_{0}^{2}+|f|^{2}+f_{12}^{2}\geq 0\,.\hskip
56.9055pt$

$\blacksquare$

The above example demonstrates that, on a purely fermionic vector space
equipped with a non-degenerate, hermitean, symplectic form $\Omega$, the Moyal
product of hermitean superfields integrated against the supersymplectic volume
density $\operatorname{Ber}(\Omega)\Theta(\Omega)$ yields an inner product.
This is precisely the Hodge pairing (coming from the symmetric bilinear form
$-i{\hskip 0.85358pt}\Omega|_{\Lambda\mathbb{R}^{m}}$) of $F$ and $G$ viewed
as generalized differential forms. This construction can be applied mutatis
mutandis to any purely fermionic vector space ${\mathbb{R}}^{0|m}$; see
Section 5.7. Of course instantaneous super laboratories (see Section 6.2) are
more general supermanifolds than purely fermi vector spaces. To generalize the
above construction we need the following technical lemma.

######  Lemma 5.8.

Let $(\mathcal{M},\Omega)$ be a super phasespace, $U$ and $V$ be supervectors,
and $(\mathcal{E}M,\pi)$ be a representative vector bundle where
$\Omega=\omega+A+\eta$ with $\omega\in\Omega^{2,0}_{\pi}\\!\mathcal{M}$,
$A\in\Omega^{1,1}_{\pi}\\!\mathcal{M}$ and
$\eta\in\Omega^{0,2}_{\pi}\\!\mathcal{M}$. Then the pairing

$\Theta(\Omega){\hskip 0.85358pt}\eta(U,V)\in\mathcal{A}$

defines a superfunction on ${\mathcal{M}}$.

###### Proof.

We need to establish independence of the choice of splitting
$(\mathcal{E}M,\pi)$. On a trivializing chart $(X^{A})=(x^{i},\theta^{a})$ for
$\mathcal{E}M$, on which $U=u^{i}\frac{\partial}{\partial
x^{i}}+\mu^{a}\frac{\partial}{\partial\theta^{a}}$ and
$V=v^{i}\frac{\partial}{\partial
x^{i}}+\nu^{a}\frac{\partial}{\partial\theta^{a}}$, the pairing reads

$\sqrt{\det\eta}{\hskip 0.85358pt}{\hskip
0.85358pt}\theta^{1}\cdots\theta^{m}\eta_{ab}\mu^{a}\nu^{b}\,.$

The transformation of vectors and the bilinear form under an arbitrary
superdiffeomorphism $X^{A}\mapsto\tilde{X}^{A}(X)$ is controlled by the super
Jacobian

$J^{A}{}_{B}=\frac{\partial
X^{A}}{\partial\tilde{X}^{B}}\,,\qquad(J^{-1})^{A}{}_{B}=\frac{\partial\tilde{X}^{A}}{\partial
X^{B}}\,,$

videlicet,

$\nu^{a}\mapsto\tilde{\nu}^{a}=J^{a}{}_{B}V^{B}\,,\qquad\eta_{ab}\mapsto\tilde{\eta}_{ab}=(J^{-1})^{C}{}_{a}\Omega_{CD}(J^{-1})^{D}{}_{b}\,.$

The invariance of $\Theta(\Omega){\hskip 0.85358pt}\eta(U,V)$ follows from
that of $\Theta(\Omega)$, noting that $J^{i}{}_{a}=O(\theta^{1})$, and
$J^{a}{}_{b}=O(\theta^{0})$. In particular

$\Theta(\Omega)J^{a}{}_{D}(J^{-1})^{C}{}_{a}=\Theta(\Omega)\delta_{d}{}^{c}\,,$

where $D=(j,d)$ and $C=(i,c)$ (and any terms with $D=j$ or $C=i$ vanish). ∎

This lemma brings us one step closer to a vertical Moyal product on super
phase-spaces. Firstly note that the above result extends directly to hermitean
supersymplectic forms. To proceed further we introduce—by way of a technical
assumption—the additional data of a flat, torsion-free superaffine connection
$\nabla$ preserving the supersymplectic form, id est

$\nabla\Omega=0=[\nabla,\nabla\\}\,.$

In principle it might be that such a connection only exists locally, but this
suffices for a local theory of measurement. Moreover, recall from the
discussion in Section 5.1 that, a lá Koszul, this also gives the data of a
splitting, and in turn a canonical involution $\dagger$ as defined in Section
5.4. Indeed, given $\nabla$, it is canonical to choose the splitting given by
the corresponding Koszul bundle
${\mathcal{E}}^{\\!\nabla}\\!M:=(\mathcal{E}M,\pi_{\nabla})$. In that case we
will often label the splitting choice by $\pi_{\nabla}$ or simply $\nabla$.

Now consider $\nabla^{k}F,\nabla^{k}G\in\Gamma(\otimes^{k}T^{*}\mathcal{M})$
where $F,G\in\mathcal{A}$, and $k$ is a positive integer. Using the Koszul
splitting $\mathcal{E}M$, we may decompose these as

(5.14) $\nabla^{k}F=f_{k}+\varphi_{k}\,,\qquad\nabla^{k}G=g_{k}+\gamma_{k}\,,$

where
$f_{k},g_{k}\in\Omega^{k}_{\pi}\\!\mathcal{M}\setminus\Omega^{0,k}_{\pi}\\!\mathcal{M}$
and $\varphi_{k},\gamma_{k}\in\Omega^{0,k}_{\pi}\\!\mathcal{M}$. Then the
proof of Lemma 5.8 can be employed to show that

(5.15) $\Theta(\Omega){\hskip
0.85358pt}\Omega^{-k}(\varphi_{k},\gamma_{k}):=\Theta(\Omega)\left(F\reflectbox{$\vec{\reflectbox{$\nabla$}}$}{}^{k}{\hskip
0.85358pt}\eta^{-k}{\hskip 0.85358pt}\vec{\nabla}^{k}G\right)\in\mathcal{A}$

is also a well-defined superfield. Here we used the notation

(5.16) $F\reflectbox{$\vec{\reflectbox{$\nabla$}}$}{}^{k}{\hskip
0.85358pt}\eta^{-k}{\hskip
0.85358pt}\vec{\nabla}^{k}G=\left(\nabla_{a_{1}}\cdots\nabla_{a_{k}}F\right)\eta^{a_{1}b_{1}}\cdots\eta^{a_{k}b_{k}}\left(\nabla_{b_{1}}\cdots\nabla_{b_{k}}G\right)\,.$

The above establishes the following inner product result.

######  Lemma 5.9.

Let $(M,\Omega,\nabla)$ be an $(n,m)$-dimensional super phase-space equipped
with a flat, torsion-free superaffine connection preserving $\Omega$. Let
$(\mathcal{E}M,\pi_{\nabla})$ be the Koszul bundle with $\Omega=\omega+A+\eta$
where $\omega\in\Omega^{2,0}_{\nabla}\\!\mathcal{M}$,
$A\in\Omega^{1,1}_{\nabla}\\!\mathcal{M}$ and
$\eta\in\Omega^{0,2}_{\nabla}\\!\mathcal{M}$. Then for any
$F,G\in\mathcal{A}$,

$(F,G)_{\Omega,\nabla}:=\sum_{k=0}^{m}c_{k}\int_{\mathcal{M}}\text{\rm
Ber}(\Omega)\Theta(\Omega)\left(F\reflectbox{$\vec{\reflectbox{$\nabla$}}$}{}^{k}{\hskip
0.85358pt}\eta^{-k}{\hskip 0.85358pt}\vec{\nabla}^{k}G\right)$

is an inner product when $c_{k}\in\mathbb{R}_{>0}$ and $\eta$ is a bundle
metric.

Lemma 5.8 also implies that the action of the inverse $\eta^{-1}$, seen as a
bidifferential operator acting on superfields $F,G\in\mathcal{A}$ defined by

$\Theta(\Omega){\hskip 0.85358pt}F{\hskip 0.85358pt}\eta^{-1}{\hskip
0.85358pt}G\in\mathcal{A}\,,$

is also invariant. Moreover, given the data $({\mathcal{M}},\Omega,\nabla)$,
we can define a vertical Moyal star product
$\filledstar_{\scriptscriptstyle\nabla}$ canonical to this data as well by

(5.17)
$F\filledstar_{\scriptscriptstyle\nabla}G:=\left(\operatorname{exp}\eta^{-1}\right)(F,G)\,,$

where the action of $\eta^{-k}$ in the (finite) expansion of
$\operatorname{exp}\eta^{-1}$ is defined by decomposing $F,G$ using (5.14) and
then employing (5.15). This amounts to particular—possibly negative—choices of
the coefficients $c_{k}$ appearing in Lemma 5.9. While the above display is
not an invariantly defined superfield, we do have that

$\Theta(\Omega)F\filledstar_{\scriptscriptstyle\nabla}G\in\mathcal{A}\,.$

Because $\nabla$ is flat and torsion-free, the usual associativity property of
the star product holds

$(F\filledstar_{\scriptscriptstyle\nabla}G)\filledstar_{\scriptscriptstyle\nabla}H=F\filledstar_{\scriptscriptstyle\nabla}(G\filledstar_{\scriptscriptstyle\nabla}H)\,.$

If in addition to associativity, one desires a star product subject to a
positivity condition and an inner product, then the above construction must be
extended to hermitean supersymplectic forms.

###  5.7. Supersymplectic Probabilities

Motivated by the silly Equation (4.1), we wish to define probability states as
“star-squares”. More precisely, on a super phase-space
$({\mathcal{M}},\Omega,\nabla)$ equipped with a flat, torsion-free superaffine
connection preserving $\Omega$, we can construct a probability cone

(5.18)
${\mathcal{C}}=\Big{\\{}\Psi\in\Gamma({\mathcal{E}}^{\\!\nabla}\\!M)\,\Big{|}\,\Psi=\Phi\star_{\scriptscriptstyle\nabla}\Phi\mbox{
and }\Phi\in{\mathcal{A}}\Big{\\}}\,.$

To explain this assertion, we continue our running Example 5.7.

######  Example 5.10.

Let the hermitean superfield $F$ be given as in Example 5.7. Then

(5.19) $\frac{F\filledstar
F}{||F||^{2}}=\mu_{0}+\mu_{1}\theta^{1}+\mu_{2}\theta^{2}-i\mu_{12}\theta^{1}\theta^{2}\,,$

where

$||F||^{2}:=f_{0}^{2}+f_{1}^{2}+f_{2}^{2}+f_{12}^{2}\,\,\mbox{ and
}\mu_{0}=1\,,\>\mu_{1}=\frac{2f_{0}f_{1}}{||F||^{2}}\,,\>\mu_{2}=\frac{2f_{0}f_{2}}{||F||^{2}}\,,\>\mu_{12}=\frac{2f_{0}f_{12}}{||F||^{2}}\,.$

Employing standard spherical coordinates $(r,\theta,\varphi,\chi)$ on
${\mathbb{R}}^{4}\ni(f_{0},f_{1},f_{2},f_{12})$ yields

$\mu_{1}=\sin(2\theta)\cos\varphi\,,\quad\mu_{2}=\sin(2\theta)\sin\varphi\cos\chi\,,\quad\mu_{3}=\sin(2\theta)\sin\varphi\sin\chi\,,$

and hence $\mu_{1}^{2}+\mu^{2}_{2}+\mu_{12}^{2}=\sin^{2}(2\theta)$ with
$\theta\in[0,\pi]$. It follows that the vector
$\vec{\mu}=(\mu_{1},\mu_{2},\mu_{12})$ lies in the unit ball in
${\mathbb{R}}^{3}$ and labels rays
$[\mu_{0}:\mu_{1}:\mu_{2}:\mu_{12}]=[1:\vec{\mu}]\in{\mathbb{R}}^{4}/{\mathbb{R}}_{>0}$.
This describes the probability cone.

Now consider an observable

$X=x_{0}+x_{1}\theta^{1}+x_{2}\theta^{2}-ix_{12}\theta^{1}\theta^{2}=X^{\dagger}\,,$

given by any hermitean superfield. Then (in this example we suppress the
Koszul connection since implicitly we are using $\nabla=d$ throughout) the
corresponding expectation with respect to the state $\Psi=F\filledstar F$ is

(5.20) $\langle
X\rangle_{\Psi,\Omega}=\frac{\int_{\mathcal{M}}\text{Ber}(\Omega){\hskip
0.85358pt}\Theta(\Omega){\hskip 0.85358pt}F\filledstar X\filledstar
F}{\int_{\mathcal{M}}\text{Ber}(\Omega){\hskip 0.85358pt}\Theta(\Omega){\hskip
0.85358pt}F\filledstar F}=x_{0}+\vec{\mu}{\hskip 0.85358pt}\hbox
to2.5pt{\hss$\cdot$\hss}{\hskip 0.85358pt}\vec{x}\in{\mathbb{R}}\,.$

Here $\vec{x}=(x_{1},x_{2},x_{12})$ and $\vec{\mu}$ is as above. Finally, one
might ask how to express this result in terms of probabilities taking values
in $[0,1]$ and summing to unity (after all, the components of $\vec{\mu}$ are
elements of $[-1,1]$). For this, special observables/random variables,
typically called indicator functions, can be employed as follows.

First, from Equation (5.19) we see that any normalized state takes the form

$\Psi=1+\nu_{1}\theta^{1}+\nu_{2}\theta^{2}-i\nu_{12}\theta^{1}\theta^{2}\,,$

where $\vec{\nu}$ is in the unit ball. This defines a convex cone in
${\mathbb{R}}^{4}$ over a three-ball. Such a cone is is not finitely
generated. Nonetheless we may define a four-state discrete system whose states
are subset of the set of all probabilities for four independent events. There
are of course many ways to do this, for example, we can embed the three-ball
in a regular tetrahedron which in turn defines a larger convex cone with four
generators. For that consider the following four hermitean superfields

$\displaystyle\psi_{1}$ $\displaystyle=1+\sqrt{8}{\hskip
0.85358pt}\theta^{1}+i\theta^{1}\theta^{2}\,,\qquad\qquad\quad\>\psi_{2}=1-\sqrt{2}\theta^{1}+\sqrt{6}\theta^{2}+i\theta^{1}\theta^{2}\,,$
$\displaystyle\psi_{3}$
$\displaystyle=1-\sqrt{2}\theta^{1}-\sqrt{6}\theta^{2}+i\theta^{1}\theta^{2}\,,\qquad\psi_{4}=1-3i\theta_{1}\theta_{2}\,.$

Then it can be checked that any normalized state $\Psi$ can be expressed as

$\Psi=p_{1}\psi_{1}+p_{2}\psi_{2}+p_{3}\psi_{3}+p_{4}\psi_{4}\,,$

so long as

(5.21) $p_{1}+p_{2}+p_{3}+p_{4}=1\>\mbox{ and
}\>6({p_{1}}-{p_{2}})^{2}+2(p_{1}+p_{2}-2p_{4})^{2}+(4p_{3}-1)^{2}\leq 1\,.$

The two relations above constrain each of the four probabilities $p_{i}$
($i=1,\ldots,4$) to be separately in the interval $[0,\frac{1}{2}]$. (Note
that it is not difficult to write down an explicit map between the four
probabilities $p_{i}$, corresponding to this particular choice of random
events, and the coefficients $(f_{0},f_{1},f_{2},f_{12})$ of the underlying
state function $F$.)

To call $p_{i}$ probabilities we ought identify the corresponding events. The
four observables $X_{i}:=\frac{\psi_{i}+2}{12}$ obey
$(\psi_{i},X_{j})=\delta_{ij}$ with respect to the inner product introduced
below in Equation (5.23). More importantly they resolve unity, meaning

$X_{1}+X_{2}+X_{3}+X_{4}=1\,,$

and have expectations

$\langle X_{i}\rangle_{\Psi}=p_{i}\,.$

In this sense they are indicator functions. $\blacksquare$

Next we construct the Moyal product for the super manifold
${\mathbb{R}}^{0|m}$ which will later appear as fibers for more general super
phase-spaces. The construction of this product follows the previous example
but now relies on a Clifford algebra (see [37]). The latter is generated by
$\\{\gamma^{a}|a=1,\ldots,m\\}$ where

$\\{\gamma^{a},\gamma^{b}\\}=2\delta^{ab}\operatorname{Id}\,.$

For our purposes it suffices to view the generators $\gamma^{a}$ as a set of
$2^{\lfloor\frac{m}{2}\rfloor}\times 2^{\lfloor\frac{m}{2}\rfloor}$,
hermitean, trace-free, matrices and $\operatorname{Id}$ is represented by the
identity matrix (such a representation certainly exists, see for example
[62]). Now we need to define an invertible map $\sigma$ from superfields to
the Clifford algebra. Consider the hermitean superfield

$F=f_{0}+f_{a}\theta^{a}+\frac{1}{2!i}{\hskip
0.85358pt}f_{ab}\theta^{a}\theta^{b}+\frac{1}{3!i}{\hskip
0.85358pt}f_{abc}\theta^{a}\theta^{b}\theta^{c}+\frac{1}{4!}{\hskip
0.85358pt}f_{abcd}\theta^{a}\theta^{b}\theta^{c}\theta^{d}+\frac{1}{5!}{\hskip
0.85358pt}f_{abcde}\theta^{a}\theta^{b}\theta^{c}\theta^{d}\theta^{e}+\cdots\,.$

Then we set

(5.22)
$\sigma(1)=\operatorname{Id}\,,\quad\sigma(\theta^{a})=\gamma^{a}\,,\quad\sigma(\theta^{a}\theta^{b})\stackrel{{\scriptstyle
a\neq
b}}{{=}}\gamma^{a}\gamma^{b}\,,\quad\sigma(\theta^{a}\theta^{b}\theta^{c})\stackrel{{\scriptstyle
a\neq b\neq c}}{{=}}\gamma^{a}\gamma^{b}\gamma^{c}\,,\>\ldots\,,$

and define $\sigma(F)$ by linearity. In turn we obtain a star product on
superfields $F$ and $G$ by setting

$\sigma(F\filledstar G):=\sigma(F)\sigma(G)\,,$

where the right hand side is matrix multiplication. Note that

$\theta^{a}\filledstar\theta^{b}+\theta^{b}\filledstar\theta^{a}=2\delta^{ab}\,.$

Moreover, if $F$ is a hermitean superfield, one has that

$\sigma(F)^{\dagger}=\sigma(F)\,,$

where $\dagger$ here is complex conjugation composed with matrix
transposition. Hence the matrix trace, $\operatorname{tr}\sigma(F)^{2}$, is a
non-negative real number. The following result shows that this star product
can be used to construct a probability cone on the space of hermitean
superfields on ${\mathbb{R}}^{0|m}$.

######  Proposition 5.11.

Let ${C}$ be the space of superfields $\Psi$ on ${\mathbb{R}}^{0|m}$ such that

$\Psi=F\star F\,,$

where $F$ is any hermitean superfield. With respect to the norm defined by

$||F||^{2}:=\frac{1}{2^{\lfloor\frac{m}{2}\rfloor}}\operatorname{tr}\sigma(F)^{2}\,$

the set ${C}$ is a probability cone.

###### Proof.

The star-squared property is clearly preserved by scalar multiplication by a
positive real number, so to show that ${C}$ is a probability cone we need to
establish the convexity property

$C+{C}\subseteq{C}\,.$

This means the sum of a pair $\filledstar$-squares must itself be a
$\filledstar$-square. Since $\filledstar$ is defined by the matrix product of
hermitean matrices, it suffices to show that the sum of squares of two
hermitan matrices is itself the square of another hermitean matrix. The result
then follows by standard linear algebra. ∎

We can also construct an inner product between hermitean superfields

$(F,G):=\frac{1}{D}\operatorname{tr}\big{(}\sigma(F)\sigma(G)\big{)}=\frac{1}{D}\operatorname{tr}\sigma(F\filledstar
G)=f_{0}g_{0}+\sum_{k=1}^{m}\big{(}\prod_{i=1}^{k}\sum_{a_{i}=1}^{m}\big{)}\frac{1}{k!}{\hskip
0.85358pt}f_{a_{1}\cdots a_{k}}g_{a_{1}\cdots a_{k}}\,,$

where $D:=2^{\lfloor\frac{m}{2}\rfloor}$. This normalization ensures that the
unit superfield $F=1$ has unit norm. The above construction relies on a
hermitean symplectic form (or fiberwise metric)

$\Omega=i\delta_{ab}d\theta^{a}\wedge d\theta^{b}\,.$

The star product can be expressed as
$\filledstar=\operatorname{exp}\Big{(}\frac{1}{i}\stackrel{{\scriptstyle\leftarrow}}{{\partial}}_{A}\Omega^{AB}\stackrel{{\scriptstyle\rightarrow}}{{\partial}}_{B}\\!\Big{)}$.
Moreover it is easy to check (this follows almost directly from the
computation given in Example 5.7) that the above inner product can be obtained
from the superintegral

(5.23) $(F,G)_{\Omega}=\int_{{\mathbb{R}}^{0|m}}\text{Ber}(\Omega){\hskip
0.85358pt}\Theta(\Omega){\hskip 0.85358pt}F\filledstar G\,.$

It is interesting to develop a bound analogous to that given for the vector
$\vec{\mu}$ in Example 5.19. A good reference for general theory of convex
cones applied to probabilities is [63].

######  Proposition 5.12.

Let $F$ be a hermitean superfield on ${\mathbb{R}}^{0|m}$. Then

${\mathscr{U}}:=\frac{F\filledstar F}{||F||^{2}}-1\,,$

where $||F||^{2}:=(F,F)_{\Omega}$, obeys the bound

$||{\mathscr{U}}||^{2}\leq(2^{\scriptscriptstyle\frac{1}{2}\lfloor\frac{m}{2}\rfloor}+1)^{2}-\frac{1}{5}\,.$

###### Proof.

Note that ${\mathscr{U}}$ is the soul of $\check{\Psi}:=\frac{F\filledstar
F}{||F||^{2}}$ and plays the rôle of $\vec{\mu}$ in our earlier example.
Decomposing the normalized state function

$\frac{F}{||F||}=\check{f}_{0}+{\mathscr{F}}$

into its body and soul, we have that
$\check{f}_{0}^{2}+||{\mathscr{F}}||^{2}=1$ because
$(1,\theta^{a_{1}}\cdots\theta^{a_{k}})_{\Omega}=0$ (the trace of any product
of distinct gamma matrices vanishes). Thus we may write

$\check{f}_{0}=\cos\theta\,\mbox{ and
}{\mathscr{F}}=\sin\theta\check{\mathscr{F}}$

for some angle $\theta$, where $||\check{\mathscr{F}}||=1$. In turn

$\check{\Psi}=1-\sin^{2}\theta+\sin(2\theta)\check{\mathscr{F}}+\sin^{2}\theta\check{\mathscr{F}}^{\filledstar
2}\,.$

Hence

${\mathscr{U}}\filledstar{\mathscr{U}}=\sin^{2}(2\theta)\check{\mathscr{F}}^{\filledstar
2}-2\sin(2\theta)\sin^{2}\theta(\check{\mathscr{F}}-\check{\mathscr{F}}^{\filledstar
3})+\sin^{4}\theta{\hskip 0.85358pt}(1-\check{\mathscr{F}}^{\filledstar
2})^{\filledstar 2}\,,$

so

$\displaystyle||{\mathscr{U}}||^{2}$
$\displaystyle=\sin^{4}\theta+(\sin^{2}(2\theta)-2\sin^{4}\theta)||\check{\mathscr{F}}||^{2}+2\sin(2\theta)\sin^{2}\theta\,\big{(}\check{\mathscr{F}}^{\filledstar
3}\big{)}_{0}+\sin^{4}\theta{\hskip
0.85358pt}||\check{\mathscr{F}}^{\filledstar 2}||^{2}$
$\displaystyle=\sin^{2}(2\theta)+\sin^{4}\theta{\hskip 0.85358pt}({\hskip
0.85358pt}||\check{\mathscr{F}}^{\filledstar
2}||^{2}-1)+2\sin(2\theta)\sin^{2}\theta\,\big{(}\check{\mathscr{F}}^{\filledstar
3}\big{)}_{0}\,.$

Since $\check{\mathscr{F}}$ is represented by a hermitean matrix,

$||\check{\mathscr{F}}^{\filledstar
2}||^{2}=\frac{1}{D}\operatorname{tr}\big{(}\sigma(\check{\mathscr{F}})^{4}\big{)}\leq\frac{1}{D}\Big{(}\operatorname{tr}\big{(}\sigma(\check{\mathscr{F}})^{2}\big{)}\Big{)}^{2}=D\,$

and

$\Big{|}\big{(}\check{\mathscr{F}}^{\filledstar
3}\big{)}_{0}\Big{|}=\Big{|}\frac{1}{D}\operatorname{tr}\big{(}\sigma(\check{\mathscr{F}})^{3}\big{)}\Big{|}\leq\sqrt{D}\,.$

Hence

$||{\mathscr{U}}||^{2}\leq\big{|}D\sin^{4}(\theta)+\sin^{2}(2\theta)-\sin^{4}\theta\big{|}+2\sqrt{D}\leq(\sqrt{D}+1)^{2}-\frac{1}{5}\,.$

∎

We have just shown how to produce a probability cone along fibers but still
need to show that Equation (5.18) actually defines a probability cone, so we
now return to the data of a super phase-space $({\mathcal{M}},\Omega,\nabla)$
equipped with a flat, torsion-free superaffine connection preserving $\Omega$.
The Koszul bundle ${\mathcal{E}}^{\nabla\\!}M=\Lambda{\mathcal{V}}M$ of
$\nabla$ is equipped with a fiberwise inner product coming from the bundle
metric $\eta$, for which we can consider the corresponding bundle of frames.
This is an $O(m)$ principal bundle over the base $M$. In turn there is (at
least locally) an associated Clifford bundle ${\mathcal{C}}M$ over $M$ with
Clifford map $\gamma$

$\Gamma({\mathcal{V}}M)\ni v\longmapsto\gamma(v)\in\Gamma({\mathcal{C}}M)\,,$

where

$\gamma(v)\gamma(u)+\gamma(u)\gamma(v)=2\eta(v,u)\operatorname{Id}\,,$

for any pair of sections $v$ and $u$. Here $\operatorname{Id}$ is the unit
section of ${\mathcal{C}}M$. The Clifford map extends to act on tensor powers
of vertical vectors according to

$\gamma(v_{1}\otimes\cdots\otimes
v_{k})=\gamma(v_{1})\cdots\gamma(v_{k})\in\Gamma({\mathcal{C}}M)\,.$

In turn this defines $\gamma(p)\in\Gamma({\mathcal{C}M})$ for any section
$p\in\Gamma(\Lambda^{k}{\mathcal{V}}M)$. Indeed any section of
${\mathcal{C}}M$ can uniquely decomposed as a sum
$\sum_{k=0}^{m}\gamma(p_{k})$ where
$p_{k}\in\Gamma(\Lambda^{k}{\mathcal{V}}M)$. Note that
$\gamma(f):=f\operatorname{Id}$ when $f\in C^{\infty}M=\Gamma(\Lambda^{0}M)$.

As discussed earlier, because $\nabla$ is flat and torsion-free, given a
superfield $F$ we may employ the inverse of
$\eta\in\Omega^{0,2}_{\nabla}\\!\mathcal{M}$ to make a section
$\eta^{-k}(\nabla^{k}F)\>\in\Gamma(\Lambda^{k}{\mathcal{V}}M)$. (Note that
this only uses the vertical part of $\nabla^{k}F$.) In turn, fed to $\gamma$,
we get a section

$\gamma(\nabla^{k}F):=\gamma\big{(}\eta^{-k}(\nabla^{k}F)\big{)}\in\Gamma({\mathcal{C}}M)\,.$

We may now define an invertible map from superfields $F$ to Clifford bundle
sections

$\sigma(F)=\gamma\big{(}(\operatorname{exp}\eta^{-1}\nabla
F)_{0}\big{)}=\operatorname{Id}F_{0}+\gamma(\nabla
F)+\frac{1}{2!}\gamma(\nabla^{2}F)+\cdots+\frac{1}{m!}\gamma(\nabla^{m}F)\,.$

Thus Clifford multiplication can be used to define a vertical star product via

$\sigma(F\filledstar_{\nabla}G)=\sigma(F)\sigma(G)\,.$

It is not difficult to check (by applying our earlier discussion of
${\mathbb{R}}^{0|m}$ along fibers) that the Clifford definition of the star
product agrees with the one given earlier, id est,

$F\filledstar_{\nabla}G=\left(\operatorname{exp}\eta^{-1}\right)(F,G)\,.$

The Berezinian integral then gives us a manifestly positive inner product
between hermitean superfields $F$ and $G$ on ${\mathcal{M}}$,

$(F,G)_{\Omega,\nabla}:=\int_{\mathcal{M}}\text{Ber}(\Omega)\Theta(\Omega)F\filledstar_{\nabla}G=\int_{M}\omega_{0}^{\wedge\frac{\text{dim}M}{2}}\Big{(}F_{0}G_{0}+\sum_{k=1}^{m}\big{(}\eta^{-k}(\nabla^{k}F,\nabla^{k}G)\big{)}_{0}\Big{)}\,,$

where $\big{(}\eta^{-k}(\nabla^{k}F,\nabla^{k}G)\big{)}_{0}$ is the function
on $M$ defined by the body of Equation (5.16).

The probability cone ${\mathcal{C}}$ of Equation (5.18) is now defined. That
this is indeed a probability cone in fact follows directly from the fiberwise
examples given above. In the language of Definition 2.2, the vector bundle
${\mathcal{V}}Z$ is ${\mathcal{E}}^{\scriptscriptstyle\nabla}M$. The vector
space $V$ is the space $\Lambda{\mathbb{R}}^{m}$ while $C$ is the space of
elements of the form $\sigma^{-1}\big{(}\sigma(F)^{2}\big{)}$ where
$F\in\Lambda{\mathbb{R}}^{m}$. Pre-states are superfunctions, id est sections
$\Psi$ of ${\mathcal{E}}^{\scriptscriptstyle\nabla}M$ expressible as a
$\filledstar_{\scriptscriptstyle\nabla}$-square
$\Phi\filledstar_{\scriptscriptstyle\nabla}\Phi$. When
$\int_{\mathcal{M}}\operatorname{Ber}(\Omega)\Theta(\Omega)\Phi\filledstar_{\scriptscriptstyle\nabla}\Phi<\infty$,
the pre-state $\Psi$ is a bona fide state.

To summarize, on a super phase-space $(M,\Omega)$ equipped with a flat,
torsion-free connection $\nabla$, the expectation of an observable given by a
hermitean superfield $X$ with respect to a state $\Psi\in{\mathcal{C}}$
defined by the state function $\Phi$ is given by

$\langle
X\rangle_{\Psi}:=\frac{\int_{\mathcal{M}}\operatorname{Ber}(\Omega)\Theta(\Omega)\,\Phi\filledstar_{\nabla}\\!X\filledstar_{\nabla}\\!\Phi}{\int_{\mathcal{M}}\operatorname{Ber}(\Omega)\Theta(\Omega)\,\Phi\filledstar_{\nabla}\Phi}\,.$

We are now ready to describe dynamics.

##  6\. Discrete Dynamical Systems

###  6.1. Super Dynamics

Analogous to the bosonic case, dynamics on a symplectic supermanifold shall be
generated by Grassmann even super vectors. On a symplectic supermanifold
$(\mathcal{M},\Omega)$, a symplectic super vector is any super vector $P$ that
satisfies

(6.1) $\mathcal{L}_{P}\Omega=0\,.$

In analogy with the bosonic case, we make the following definition.

######  Definition 6.1.

A dynamical super phase-space is a triple $(\mathcal{M},\Omega,L)$ where
$(\mathcal{M},\Omega)$ is a symplectic supermanifold and $L\subset
T\mathcal{M}$ is a Grassmann-even line subbundle whose sections are symplectic
super vectors. $\blacklozenge$

Given an even superfield $H$ on a super phase-space, one can define the
corresponding Hamiltonian super vector field $X_{H}$ by

$\flat X_{H}=d_{T}H\,.$

Clearly $X_{H}$ is a symplectic super vector. Once again (see Example 3.3),
this data can be succinctly repackaged in terms of a super phase-spacetime:

######  Example 6.2.

Let $(\mathcal{M},\Omega)$ be a super phase-space such that
$\Omega=d_{T}\Lambda$ and let $H$ be an even superfield. Calling

$\mathcal{Z}=\mathcal{M}\times\stackunder{\mathbb{R}}{\stackrel{{\scriptstyle\rotatebox{90.0}{$\scriptscriptstyle\in$}}}{{t}}}\stackrel{{\scriptstyle\pi}}{{\to}}\mathcal{M}\,,$

we consider the super 1-form

$A=\pi^{*}\Lambda+Hdt\in\Omega^{1}\mathcal{Z}\,.$

The pair $(\mathcal{Z},d_{T}A)$ defines a super phase-spacetime. Moreover the
supervector

$i^{t}_{*}\Omega^{-1}(d_{T}H,{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}})+\partial_{t},$

where $i^{t}$ is the inclusion $\mathcal{M}\to\mathcal{M}\times\\{t\\}$,
defines a line bundle whose sections are symplectic supervectors and thus in
turn defines a dynamical super phase-space. $\blacksquare$

Quite generally, the supersymplectic form of a super phase-spacetime
$(\mathcal{Z},\Omega)$ encodes a line bundle $L=\operatorname{ker}\Omega$,
sections of which determine generators of dynamics on $\mathcal{Z}$. Note that
the proof of Proposition 3.4 applies directly here as well. When the line
bundle $L$ is orientable, there is a notion of future directed super dynamics

The body $(Z,\omega_{0})$ of $(\mathcal{Z},\Omega)$ is itself a symplectic
manifold. When $\mathcal{Z}$ is a super phase-spacetime it follows, for any
$\rho$ subject to $\iota_{\rho}\omega_{0}=0$, that
$(Z,\omega_{0},\operatorname{span}\rho)$ is a dynamical phase-space. The
following proposition shows how generators of dynamics $\rho_{0}$ on
$(Z,\omega_{0})$ lift to super generators on $\mathcal{Z}$.

######  Proposition 6.3.

Let $(\mathcal{Z},\Omega,L)$ be a super phase-spacetime, and
$(\mathcal{E}Z,\pi)$ be a representative vector bundle where
$\Omega=\omega+A+\eta$ with $\omega\in\Omega^{2,0}_{\pi}\\!\mathcal{Z}$,
$A\in\Omega^{1,1}_{\pi}\\!\mathcal{Z}$ and
$\eta\in\Omega^{0,2}_{\pi}\\!\mathcal{Z}$. Suppose that

$\omega_{0}(\rho_{0},{\boldsymbol{\hbox to2.5pt{\hss$\cdot$\hss}}})=0$

for some $0\neq\rho_{0}\in\Gamma(TZ)$. Assuming $\eta$ is invertible, the
general solution of Equation (6.1) is then given by:

$P=\rho+\eta^{-1}A(\rho,{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}})\,,\quad\rho=\Big{(}\frac{\operatorname{Id}}{\operatorname{Id}+\omega_{0}^{-1}\circ\hat{\omega}}\Big{)}(\rho_{0},{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}}{\hskip 0.85358pt})\,.$

In the above $\hat{\omega}:=\omega-\omega_{0}-A\circ\eta^{-1}\circ A$ and
$\omega_{0}^{-1}$ is any choice of partial inverse of $\omega_{0}$.

###### Proof.

Decomposing

$P=\rho+\rho_{V}\,$

into its body $\rho$ and soul $\rho_{V}$, Equation (6.1) produces

$\omega(\rho,{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}})+A({\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}},\rho_{V})=0=A(\rho,{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}})+\eta(\rho_{V})\,.$

Solving the second of these equations for $\rho_{V}$ and back-substituting in
the first yields

(6.2) $\bar{\omega}(\rho)=0\,,$

where

$\bar{\omega}:=\omega-A\circ\eta^{-1}\circ A\,.$

Upon expanding in the ${\mathbb{Z}}_{m+1}$-grading determined by the splitting
$\pi$, Equation (6.2) becomes a block lower triangular matrix system of
equations whose diagonal is $\omega_{0}$, thus let us write

$\hat{\omega}=\bar{\omega}-\omega_{0}\,,$

where $\hat{\omega}$ is nilpotent. Moreover, because $\bar{\omega}$ is skew,
$\bar{\omega}(\rho_{0},{\boldsymbol{\hbox to2.5pt{\hss$\cdot$\hss}}})$
annihilates $\rho_{0}$, so the covector
$\big{(}\omega_{0}^{-1}\circ\hat{\omega}\big{)}(\rho_{0},{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}})$ is well-defined given a partial inverse (note
that the image of $\omega_{0}$ is the kernel of $\rho_{0}$ viewed as a linear
map $\Gamma(T^{*}Z)\to C^{\infty}Z$). Now suppose $F$ is any even superfield.
Then an easy inspection of this matrix system of equations shows that the
general solution to Equation (6.2) is

$\rho=\Big{(}\frac{\operatorname{Id}}{\operatorname{Id}+\omega_{0}^{-1}\circ\hat{\omega}}\Big{)}({F}\rho_{0},{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}}{\hskip 0.85358pt})\,,$

where $\omega_{0}^{-1}$ is any choice of partial inverse mapping
$\ker\rho_{0}\subset\Gamma(T^{*}Z)$ to a choice of subspace of
$\Gamma(TZ)\not\ni\rho_{0}$.

∎

We have now assembled the super geometric technology required to handle
dynamical systems involving both discrete and continuous degrees of freedom.

###  6.2. Super Laboratories, States and Measurements

Motivated by our earlier discussion of computer bits and superfields, we now
focus on the data of a super phase-spacetime $({\mathcal{Z}},\Omega)$. Our
first task is to develop a notion of an instantaneous laboratory along the
lines discussed in Section 2. First, if ${\mathcal{Z}}=(Z,{\mathcal{A}})$ is a
supermanifold and $M\hookrightarrow Z$ is a hypersurface in $Z$, we call the
induced supermanifold $(M,{\mathcal{A}}|_{M})$ a superhypersurface. Note that
an exponential map $\operatorname{exp}(tv)$ on a supermanifold ${\mathcal{Z}}$
restricts to superhypersurfaces in the natural way. In particular, this
induces the standard exponential map on the body, giving
$M_{t}=\operatorname{exp}(tv_{0})M$. We may now make the following definition.

######  Definition 6.4.

An instantaneous super laboratory $({\mathcal{M}},\Omega^{*})$ is any
superhypersurface of a super phase-spacetime $({\mathcal{Z}},\Omega)$ along
which the pullback $\Omega^{*}$ is non-degenerate. $\blacklozenge$

Next we choose a suitable set of observables. For that we employ the algebra
formed by the space of (hermitean) superfunctions ${\mathcal{A}}$ on a super
phase-spacetime ${\mathcal{Z}}=(Z,{\mathcal{A}})$. Moreover, given a choice of
splitting ${\mathcal{E}}Z$, these belong to the space of sections of the
vector bundle ${\mathcal{E}}Z$. Given a measure, this aligns with the
definition given in Section 2. States are then identified with sections of a
convex cone inside the dual bundle $\mathcal{E}Z^{*}$. This gives a notion of
positivity. The space of (normalizable) superfunctions in the kernel of
$\mathcal{L}_{P}$, for any $P$ that obeys $\Omega(P,{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}})=0$, is the fundamental state space, which we will
denote by $\mathcal{A}_{\Omega}$.

######  Lemma 6.5.

Let $(\mathcal{Z},\Omega,\nabla)$ be a super phase-spacetime equipped with a
flat, torsion-free superaffine connection preserving $\Omega$. Moreover
suppose that $P\in\ker\Omega$. Then there exists a canonical
$\filledstar$-product on $\mathcal{A}_{\Omega}$.

###### Proof.

First note that $\Omega(P,{\boldsymbol{\hbox to2.5pt{\hss$\cdot$\hss}}})=0$
implies

$\Omega(\nabla P,\hbox to2.5pt{\hss$\cdot$\hss})=0\,.$

Thus $\nabla P$ must be proportional to $P$, or in other words $\nabla$
preserves the kernel of $\Omega$.

Now let $F,G$ be state functions. Hence $d_{T}F$ and $d_{T}G$ are both in the
kernel of $\iota_{P}$. Then observe that

$\Omega(X,{\boldsymbol{\hbox to2.5pt{\hss$\cdot$\hss}}})=d_{T}F$

has a unique solution modulo addition of any vector field in the kernel of
$\Omega$. Hence $\Omega^{-1}(d_{T}F,d_{T}G)$ is also well-defined.

Now consider the (covariant) tensor

$X:=\nabla\cdots\nabla d_{T}F\,.$

We wish to establish $X({\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}},\ldots,P,\ldots,{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}})=0$ for any $P\in\ker\Omega$. Since $\nabla$ is
torsion-free and flat, it suffices to show $X({\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}},\ldots,{\boldsymbol{\hbox
to2.5pt{\hss$\cdot$\hss}}},P)=0$. This follows from a simple inductive
argument. The base case for $X=\nabla d_{T}F$ looks like

$\iota_{P}\nabla d_{T}F=[\iota_{P},\nabla]d_{T}F\propto\iota_{P}d_{T}F=0.$

The proportionality statement follows from the assumption that $\nabla P$ is
in the span of $P$. The induction step follows similarly.

A canonical $\filledstar$-product is given by

$F\filledstar G:=\left(\operatorname{exp}\Omega^{-1}\right)(F,G)$

where $\Omega^{-k}(F,G)$ is defined in the obvious way. That the above defines
a star product again follows from usual derivation of the standard super Moyal
star product because $\nabla$ is torsion-free and flat, and thus super-
commuting (see for example [64, 65, 52]). ∎

In fact for our classical measurement theory, we only need a vertical star
operation. This is in fact simpler to construct than the super Moyal star
given above because, as discussed earlier, in a choice of splitting, $\Omega$
is fiberwise invertible.

######  Definition 6.6.

Let $(\mathcal{Z},\Omega,\nabla)$ be a super phase-spacetime equipped with a
flat, torsion-free superaffine connection preserving $\Omega$ and $F,G$ be
superfunctions on $\mathcal{Z}$. Then given a superhypersurface $\Sigma$, we
define the inner product of $F$ and $G$ by

$(F,G)_{\Omega,\Sigma}:=\int_{\Sigma}\text{Ber}(\Omega)\Theta(\Omega)F\filledstar_{\scriptscriptstyle\nabla}G\,,$

where
$F\filledstar_{\scriptscriptstyle\nabla}G:=\left(\operatorname{exp}\eta^{-1}\right)(F,G)$
and $\eta^{-k}(F,G)$ is as defined in Equation (5.16). $\blacklozenge$

The above definition is a super phase-spacetime analog of Definition 5.9; its
well-definedness (with respect to superdiffeomorphisms) follows from the
discussion preceding that definition. Next we study how the above inner
product behaves with respect to superdynamics. First we need a technical
lemma.

######  Lemma 6.7.

Let $(\mathcal{Z},\Omega,\nabla)$ be a super phase-spacetime equipped with a
flat, torsion-free superaffine connection preserving $\ker\Omega$. Then
locally there exists a vector $P\in\ker\Omega$ such that

$\nabla P=0\,.$

###### Proof.

Since $\nabla$ preserves $\ker\Omega$, there exists a (everywhere non-
vanishing on any contractible patch) vector $P^{\prime}\in\ker\Omega$ such
that ,

$\nabla P^{\prime}=\Upsilon\otimes P^{\prime},$

for some super one-form $\Upsilon$. Using that $\nabla$ is torsion-free and
flat, we observe

$0=(\nabla\wedge\Upsilon)\otimes P^{\prime}\,,$

where we used $\Upsilon\wedge\Upsilon=0$. Hence $\Upsilon$ is $\nabla$-closed
and in turn $d_{T}\Upsilon=0$ since $\nabla$ is torsion-free. So locally

$\Upsilon=d_{T}F\,,$

for some even superfunction $F$. Now consider $P:=e^{-F}P^{\prime}$. It
follows that

$\nabla P=(-d_{T}F+\Upsilon)\otimes P^{\prime}=0\,.$

∎

The inner product between superfunctions satisfies a conservation law:

######  Lemma 6.8.

Let $(\mathcal{Z},\Omega,\nabla)$ be a super phase-spacetime equipped with a
flat, torsion-free superaffine connection preserving $\Omega$. Moreover let
$\mathcal{M}_{t}$ be a smooth 1-parameter family of superhypersurfaces such
that $\mathcal{M}_{t}:=\operatorname{exp}(tP)(\mathcal{M})$, where
$P\in\ker\Omega$ and $\nabla P=0$. Then for any pair of superfields $F,G$
their inner product obeys

$\frac{d(F,G)_{\Omega,\mathcal{M}_{t}}}{dt}\Big{|}_{t=0}=(\mathcal{L}_{P}F,G)_{\Omega,\mathcal{M}}+(F,\mathcal{L}_{P}G)_{\Omega,\mathcal{M}}\,.$

###### Proof.

Firstly, recall that given a vector field $v$ and a $2n$-form $\mu$, then

$\frac{d}{dt}\left(\int_{M_{t}}\mu\right)\Big{|}_{t=0}=\int_{M}\mathcal{L}_{v}\mu\,,$

where $M_{t}:=e^{tv}(M)\hookrightarrow Z^{2n+1}$ is a 1-parameter family of
hypersurfaces generated by $v$.

Let us now begin with the right hand side of the claimed conservation law, for
which we consider the decomposition of $\Omega=\omega+A+\eta$ corresponding to
the Koszul bundle of $\nabla$. Using (5.13) we have

$(\mathcal{L}_{P}F,G)_{\Omega,\mathcal{M}}+(F,\mathcal{L}_{P}G)_{\Omega,\mathcal{M}}=\int_{M}\omega_{0}^{\wedge
n}\left[\mathcal{L}_{P}F\filledstar_{\scriptscriptstyle\nabla}G+F\filledstar_{\scriptscriptstyle\nabla}\mathcal{L}_{P}G\right]_{0}\,.$

We next expand $\filledstar_{\scriptscriptstyle\nabla}$ order by order; a
general term looks like (see (5.17) and (5.16))

(6.3)
$\eta^{-k}(\mathcal{L}_{P}\nabla^{k}F,\nabla^{k}G)+\eta^{-k}(\nabla^{k}F,\mathcal{L}_{P}\nabla^{k}G)\,,$

where we have used that $P$ is parallel with respect to the torsion-free and
flat connection $\nabla$. Furthermore, $\eta$ and hence $\eta^{-1}$ is
preserved up to $\mathcal{O}(\theta^{0})$ under flows of $P$. To see this note
that, $\mathcal{L}_{P}\Omega=0\Rightarrow\omega_{0}^{\wedge
n}\Theta\mathcal{L}_{P}\Omega=0\Rightarrow(\mathcal{L}_{P}\eta)|_{\theta=0}=0$.
By the Leibniz rule, it now follows that the body of the general term (6.3) is

$\mathcal{L}_{\rho_{0}}\left[\eta_{0}^{-k}\left((\nabla^{k}F)_{0},(\nabla^{k}G)_{0})\right)\right]\,,$

where the vector field $\rho_{0}$ is the body of $P$.

We now consider the left hand side of the claimed conservation law. Applying
Equation (5.13) followed by the first display of this proof gives

$\frac{d(F,G)_{\Omega,\mathcal{M}_{t}}}{dt}\Big{|}_{t=0}=\int_{M}\mathcal{L}_{\rho_{0}}\left(\omega^{\wedge
n}F\filledstar_{\scriptscriptstyle\nabla}G\right)_{0}\,.$

The proof is completed by recalling how the exponential map acts on
superhypersurfaces and their base manifolds, and then using
${\mathcal{L}}_{\rho_{0}}\omega_{0}=0$ in conjunction with (5.17) and (5.16).

∎

The above result gives an evolution rule for expectations of observables with
respect to changes of laboratory.

######  Corollary 6.9.

Let $(\mathcal{Z},\Omega,\nabla)$ be a super phase-spacetime equipped with a
flat, torsion-free superaffine connection preserving $\Omega$. Moreover let
$\mathcal{M}_{t}$ be a smooth 1-parameter family of superhypersurfaces such
that $\mathcal{M}_{t}:=\operatorname{exp}(tP)(\mathcal{M})$, where
$P\in\ker\Omega$ and $\nabla P=0$. Then the evolution of the expectation of an
observable $X\in\mathcal{A}Z$ with respect to a state function
$\Phi\in\mathcal{A}Z$ is given by

$\frac{d}{dt}\langle
X\rangle_{\Phi,\mathcal{M}_{t}}=\langle\mathcal{L}_{P}X\rangle_{\Phi,\mathcal{M}_{t}}\,.$

###### Proof.

To see this, note that the state function $\Phi$ obeys the Liouville equation
implying that total probabilities are preserved. Hence the only contribution
to the expectation can come from the change in the observable itself. ∎

##  7\. Example: Evolution of Discrete Probabilities

We now demonstrate our key ideas with a two-bit system evolving in time. Our
basic data is the supermanifold
$\mathcal{Z}=\mathbb{R}^{1|2}\ni(t,\theta^{a})$ with $a=1,2$. For the flat,
torsion-free affine connection we take

$\nabla=d_{T}$

in the coordinate system $(t,\theta^{a})$. Our dynamics shall be described by
the action functional of [3]

$S=\int\tfrac{d\tau}{i}\big{(}\eta_{ab}(t)\theta^{a}\dot{\theta^{b}}-\frac{1}{2}H(t)\epsilon_{ab}\theta^{a}\theta^{b}\dot{t}\big{)}\,.$

Assuming $\eta$ is invertible, this is the most general super phase-spacetime
dynamics on ${\mathcal{Z}}$. In the above the dot denotes differentiation with
respect to the worldline parameter $\tau$ and $\epsilon_{ab}$ is the two-
dimensional Levi-Civita symbol. Better said, this action is an integral of a
hermitean super one-form

$\xi=i\eta_{ab}(t)\theta^{a}d\theta^{b}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}+\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\frac{i}{2}H(t)\epsilon_{ab}\theta^{a}\theta^{b}dt\,.$

Here we used that, for a one-form $d\tau$ (given by $d_{T}$ of a bosonic
function), one has $d\tau{\hskip 0.85358pt}\dot{\theta}=d\theta$ (this choice
of sign ensures that the variation of the above action agrees with the
dynamics of the super symplectic form below). This gives the symplectic form

$\Omega=d_{T}\xi=i\eta_{ab}(t)d\theta^{a}d\theta^{b}-i\big{(}H(t)\epsilon_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}ba\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}+\eta^{\prime}_{ab}(t)\big{)}\theta^{b}d\theta^{a}dt\,.$

The data $(\mathbb{R}^{1|2},\Omega)$ defines a super phase-spacetime. Dynamics
is determined by the supervector (conferatur Proposition (6.3))

$P=\frac{\partial}{\partial
t}+\theta^{a}M^{b}{}_{a}\frac{\partial}{\partial\theta^{b}}\,,$

where

$M^{b}{}_{a}:=-\frac{1}{2}\eta^{bc}(t)\big{(}H(t)\epsilon_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}ac\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}+\eta^{\prime}_{ca}(t)\big{)}\,,$

and $\eta^{ab}$ is the inverse of $\eta_{ab}$. Also $\prime$ denotes
$t$-differentiation.

Now we consider a state function $\Phi$ given by a hermitean superfield

$\Phi=\phi+\phi_{a}\theta^{a}-\tfrac{i}{2}\varphi\epsilon_{ab}\theta^{a}\theta^{b}\,,$

(where $\epsilon_{12}=1$) subject to

$P\Phi=0\,.$

Here the four functions of $t$ given by $(\phi,\phi_{a},\varphi)$ are all real
and must obey

(7.1)
$\phi^{\prime}=0\,,\quad\phi_{a}^{\prime}+M^{b}{}_{a}\phi_{b}=0\,,\quad\varphi^{\prime}+M^{a}{}_{a}\varphi=0\,.$

Note that $M^{a}{}_{a}=-\frac{1}{2}\eta^{ab}\eta^{\prime}_{ab}$.

We now consider the super laboratory $\Sigma_{T}$ given by superhypersurface
$t=T$. The pullback of $\Omega$ is then

(7.2) $\Omega^{*}=i\eta_{ab}(T)d\theta^{a}d\theta^{b}\,.$

Before computing the star product, we solve the Equations (7.1). For that we
impose boundary conditions for $\Phi(T)$ along $\Sigma_{T}$ given by
$\Phi(T)=(\phi^{\rm i},\phi_{a}^{\rm i},\varphi^{\rm i})$. Then

$\phi(t)=\phi^{\rm i}\,,\quad\phi_{a}(t)=U(t,T)^{b}{}_{a}\,\phi_{b}^{\rm
i}\,,\quad\varphi(t)=u(t,T)\varphi(T)\,,$

where
$U(t,T)^{b}{}_{a}=\Big{[}\operatorname{P}\operatorname{exp}\big{(}-\int^{t}_{T}M\big{)}\Big{]}^{b}{}_{a}$,
$u(t,T)=\operatorname{exp}\Big{(}{-\int^{t}_{T}M^{a}{}_{a}}\Big{)}=\operatorname{exp}\Big{(}{\frac{1}{2}\int^{t}_{T}\eta^{ab}\eta^{\prime}_{ab}}\Big{)}$,
and $\operatorname{P}$ denotes path ordering.

To compute the vertical star product of state functions we first need the
Clifford map $\gamma$. For that we introduce orthonormal frames
$e_{a}{}^{\bar{a}}$ for the vertical distribution on $T^{*}{\mathcal{M}}$
determined by the Koszul connection, subject to

$\eta_{ab}(t)d\theta^{a}d\theta^{b}=\delta_{\bar{a}\bar{b}}e_{a}{}^{\bar{a}}(t)e_{b}{}^{\bar{b}}(t)d\theta^{a}d\theta^{b}\,.$

We also define $\theta^{\bar{a}}(t):=e_{a}{}^{\bar{a}}(t)\theta^{a}$ so that

$\Phi=\phi+\phi_{\bar{a}}\theta^{\bar{a}}-\tfrac{i}{2}\bar{\varphi}\,\epsilon_{\bar{a}\bar{b}}\theta^{\bar{a}}\theta^{\bar{b}}\,,$

where
$\phi_{\bar{a}}:=e^{a}{}_{\bar{a}}\phi_{a}\,,\quad\bar{\varphi}:=(\det{}^{-1}e){\hskip
0.85358pt}\varphi$, and $\epsilon_{\bar{1}\bar{2}}=1$. We may now employ the
map $\sigma$ of Example 5.7 along each fiber. This gives an invertible map
from hermitean superfunctions to hermitean matrices taking values in functions
of $t$,

$\sigma(\Phi)=\begin{pmatrix}\phi+\bar{\varphi}&\phi_{\bar{1}}-i\phi_{\bar{2}}\\\
\phi_{\bar{1}}+i\phi_{\bar{2}}&\phi-\bar{\varphi}\end{pmatrix}\,,$

such that the star product is computed via matrix multiplication:

$\Phi\filledstar_{\scriptscriptstyle\nabla}\Phi=\sigma^{-1}\big{(}\sigma(\Phi)^{2}\big{)}=\phi^{2}+\phi_{\bar{1}}^{2}+\phi_{\bar{2}}^{2}+\bar{\varphi}^{2}+2\phi(\phi_{\bar{1}}\theta^{\bar{1}}+\phi_{\bar{2}}\theta^{\bar{2}})-i\phi\bar{\varphi}\epsilon_{\bar{a}\bar{b}}\theta^{\bar{a}}\theta^{\bar{b}}\,.$

Hence, once again, a hermitean superfield $\Psi$ is a vertical star square of
a non-zero state function precisely when its body is positive. As discussed
earlier (see Example 5.10) this condition is both a necessary and sufficient
one.

###  7.1. Measurements

We can now construct expectations of observables. For that let
$X=x+x_{a}\theta^{a}-\frac{i}{2}\chi\epsilon_{ab}\theta^{a}\theta^{b}$ be a
hermitean superfield, which we may also expand in the adapted Grassmann
coordinates $\theta^{\bar{a}}(t)$,

$X=x+x_{\bar{a}}\theta^{\bar{a}}-\tfrac{i}{2}\bar{\chi}\,\epsilon_{\bar{a}\bar{b}}\theta^{\bar{a}}\theta^{\bar{b}}\,.$

The pullback in Equation (7.2) in terms of the adapted $\theta^{\bar{a}}$ is

$\Omega^{*}=i\delta_{\bar{a}\bar{b}}d\theta^{\bar{a}}d\theta^{\bar{b}}\,.$

Hence the expectation of the observable $X$ in the $t=T$ superlaboratory with
respect to a state $\Psi=\Phi\filledstar_{\scriptscriptstyle\nabla}\Phi$ is
given by (see Equation (5.20))

(7.3) $\langle
X\rangle_{\Psi,T}=x(T)+\frac{2\phi(\phi_{\bar{1}}x_{\bar{1}}+\phi_{\bar{2}}x_{\bar{2}}+\bar{\varphi}\bar{\chi})}{\phi^{2}+\phi_{\bar{1}}^{2}+\phi_{\bar{2}}^{2}+\bar{\varphi}^{2}}\Big{|}_{t=T}=x(T)+\frac{2\phi(\phi_{a}\eta^{ab}x_{b}+\det^{-1}\\!\eta\,{\hskip
0.85358pt}\varphi\chi)}{\phi^{2}+\phi_{a}\eta^{ab}\phi_{b}+\det^{-1}\\!\eta\,{\hskip
0.85358pt}\varphi^{2}}\Big{|}_{t=T}\,.$

To simplify the remainder of our discussion, let us assume $\eta$ and $H$ are
$t$-independent. It follows that $u(T,t)=1$ and

(7.4)
$U(T,t)=\cos\big{(}\tfrac{(t-T)H}{2\surd\det\eta}\big{)}\operatorname{Id}+\sqrt{\det\eta}\,\sin\big{(}\tfrac{(t-T)H}{2\surd\det\eta}\big{)}{\hskip
0.85358pt}\eta^{-1}\epsilon\,.$