# Fermionic Wigner functional theory

Filippus S. Roux<EMAIL_ADDRESS>orcid:0000-0001-9624-4189 Manor House 7,
104 Priory Road, Lynnwood Manor 0081, Pretoria, South Africa

###### Abstract

A Grassmann functional phase space is formulated for the definition of
fermionic Wigner functionals. The formulation follows a stepwise process,
starting with the identification of suitable fermionic operators that are
analogues to bosonic quadrature operators. The Majorana operators do not
suffice for this purpose. Instead, a set of fermionic Bogoliubov operators are
used. The eigenstates of these operators are shown to provide a complete
orthogonal basis, provided that the dual space is defined by augmenting the
Hermitian conjugation with a spin transformation. These bases serve as
quadrature bases in terms of which the Wigner functionals can be defined in a
way that is analogues to the bosonic case.

##  I Introduction

Quantum field theory Peskin and Schroeder (1995) has been very successful in
formulating theories for most of the fundamental dynamics of nature. The only
exception is a quantum description of gravity. The states used for
phenomenological calculations in quantum field theory are usually rather
simple — tensor products of pure single-particle number states for the
different particle fields that are to take part in the interactions. It is not
the purpose of quantum field theory to represent exotic quantum states.

On the other hand, formulations of non-relativistic quantum physics that are
used for quantum information theory, such as those used in quantum optics,
have been developed to a high level, allowing the description of exotic
quantum measurements applied to exotic quantum states. Such descriptions often
use phase space representations for such exotic states in terms of ladder
operators or quadrature operators Braunstein and Van Loock (2005); Weedbrook
et al. (2012); Adesso et al. (2014), leading to a quasi-Moyal formalism. The
dynamics in these applications are relatively simple compared to the
fundamental dynamics of nature as represented in terms of quantum field
theory.

Progress in fundamental physics (specifically with respect to a quantum
description of gravity) would require the merger of these two disciplines to
obtain a formalism that can successfully represent both the fundamental
dynamics and arbitrary complex quantum states. A promising approach is to
generalize the Moyal formalism Groenewold (1946); Moyal (1949); Curtright and
Zachos (2012) to obtain one in which states and operations are represented as
distributions on a functional phase space, taking over the role of the path
integral domain. The formal resemblance between functional phase space
integrals and path integrals implies that the former would allow a similar
development as found for the quantum field theory based on the latter.

The development of a functional Moyal formalism for bosonic fields has so far
been quite successfully based on a generalization of quantum optics
Mrowczynski and Mueller (1994); Roux (2018); Berra-Montiel and Molgado (2020);
Mrówczyński (2013). The relativistic formulation of a functional phase space
for the massless photon field does not present significant challenges. The
development of a functional Moyal formalism for fermionic fields is more
challenging. In addition to the early work by Cahill and Glauber Cahill and
Glauber (1999), there are a number of fermion phase space formulations based
on the Glauber-Sudarshan $P$-distribution Corney and Drummond (2006); Dalton
et al. (2016); Polyakov (2016); Dalton et al. (2017) and the Husimi
$Q$-distribution Joseph et al. (2018) in the context of condensed matter
physics. There are also fermion phase space formulations based on the Wigner
function Mrówczyński (2013). These formulations are generally done for finite
dimensional systems, leading to phase spaces composed of a finite number of
the two-dimensional phase spaces, each representing a one-dimensional system.

In this paper, we consider infinite dimensional systems where the phase space
becomes a functional phase space and the distributions are defined as
functionals, leading to a functional Moyal formalism for fermion fields. For
this purpose, we choose to work with the Wigner functionals. Such a functional
Moyal formalism for fermion fields based on Wigner functionals has been
presented before Mrówczyński (2013). However, it is based on an axiomatic
approach that takes the existence of eigenstates of field operators for
granted. Here, we show the full development and compute the eigenstates for
the pertinent operators explicitly.

A functional Moyal formulation with fermionic Wigner functionals is frustrated
by the fact that the fermionic analogue of the bosonic quadrature operators
(i.e., the Majorana operators) do not share the properties of their bosonic
equivalents: they do not commute with themselves. More importantly, their
eigenspaces behave differently from those of the bosonic quadrature operators.
Here, we consider a set of fermionic Bogoliubov operators, instead of the
Majorana operators. It is shown that they behave more closely to the bosonic
quadrature operators and therefore serve as a preparation for the development
of a fermionic Wigner functional Moyal formalism. These fermionic Bogoliubov
operators consist of linear combinations of ladder operators together with a
spin transformation. We obtain their eigenstates and compute the inner
products among these eigenstates. The results are surprising in that the dual
space of these states do not follow from the usual Hermitian adjoint. Instead,
it requires a spin transformation in addition to the Hermitian adjoint, which
we call a fermionic adjoint. It follows that the set of fermionic Bogoliubov
operators are self-adjoint with respect to this fermionic adjoint.

After identifying suitable fermionic quadrature bases, we proceed to formulate
the fermionic Wigner functional theory. It is shown that the Weyl
transformation for the Wigner functionals allows transformations between
fermionic density operators and Grassmann Wigner functionals. The Moyal star
product is also derived allowing quantum operations to be represented purely
in terms of fermionic Wigner functionals.

##  II Background

###  II.1 Bosonic operators

Since we intend to develop the fermionic phase space representation in analogy
to the phase space associated with bosonic Wigner distributions, it helps to
review bosonic Wigner functions. We start by considering only the particle-
number degrees of freedom, ignoring all other degrees of freedom. In such a
case, the pair of ladder operators — the annihilation operator $\hat{a}$ and
its Hermitian adjoint, the creation operator $\hat{a}^{{\dagger}}$ — obeys the
simple commutation relations
$[\hat{a},\hat{a}]=[\hat{a}^{{\dagger}},\hat{a}^{{\dagger}}]=0$ and
$[\hat{a},\hat{a}^{{\dagger}}]=1$. The quadrature operators are defined as the
sum and difference of the ladder operators

$\hat{q}=\frac{1}{\sqrt{2}}(\hat{a}+\hat{a}^{{\dagger}}),~{}~{}~{}\hat{p}=\frac{-i}{\sqrt{2}}(\hat{a}-\hat{a}^{{\dagger}}).$
(1)

They are Hermitian operators, obeying another set of simple commutation
relations $[\hat{q},\hat{q}]=[\hat{p},\hat{p}]=0$ and $[\hat{q},\hat{p}]=i$,
which follow unambiguously from those of the ladder operators. Their
eigenstates, which are solutions of the eigen-equations

$\hat{q}\left|{q}\right\rangle=\left|{q}\right\rangle
q,~{}~{}~{}\hat{p}\left|{p}\right\rangle=\left|{p}\right\rangle p,$ (2)

with nondegenerate real-valued eigenvalues $q$ and $p$, represent orthogonal
bases. The eigen-bases of the respective quadrature operators are mutually
unbiased — the inner product $\langle{q}|{p}\rangle=\exp(iqp)$ implies that
$|\langle{q}|{p}\rangle|^{2}=1$.

The Wigner function of a state is given in terms of the quadrature eigenstates
by

$W_{\hat{\rho}}(q,p)=\int\left\langle{q+\tfrac{1}{2}x}\right|\hat{\rho}\left|{q-\tfrac{1}{2}x}\right\rangle\exp(-ixp)\
\text{d}x,$ (3)

where $\hat{\rho}$ is the density operator of the state. Such a Wigner
function can be computed for any operator. The properties of the quadrature
operators and their eigenbases are crucial for the definition of the Wigner
function.

Bogoliubov operators are also produced as linear combinations of ladder
operators,

$\hat{b}=u\hat{a}+v\hat{a}^{{\dagger}},~{}~{}~{}\hat{b}^{{\dagger}}=v^{*}\hat{a}+u^{*}\hat{a}^{{\dagger}}.$
(4)

Unlike the quadrature operators, they are not Hermitian. The defining property
of the Bogoliubov operators is that they obey the same commutation relations
as the ladder operators from which they are constructed:
$[\hat{b},\hat{b}]=[\hat{b}^{{\dagger}},\hat{b}^{{\dagger}}]=0$ and
$[\hat{b},\hat{b}^{{\dagger}}]=1$. It thus follows that $|u|^{2}-|v|^{2}=1$.
For $u=1$ and $v=0$, the Bogoliubov operators are equal to the original ladder
operators.

Incorporating the other degrees of freedom (spin and spatiotemporal degrees of
freedom), represented by a spin index $s$ and a dependence on the wave vector
$\mathbf{k}$, the eigen-equations for the quadrature operators become

$\displaystyle\begin{split}\hat{q}_{s}(\mathbf{k})\left|{q}\right\rangle=&\left|{q}\right\rangle
q_{s}(\mathbf{k}),\\\
\hat{p}_{s}(\mathbf{k})\left|{p}\right\rangle=&\left|{p}\right\rangle
p_{s}(\mathbf{k}).\end{split}$ (5)

The eigenvalue now becomes a spectral eigenvalue function. Each such function
is associated with a unique eigenstate. Hence, the phase space becomes a
functional phase space, and the Wigner function becomes a Wigner functional.
It is represented by

$\displaystyle W[q,p]=$ $\displaystyle\int\exp\left[-i\sum_{s}\int
p_{s}(\mathbf{k})x_{s}(\mathbf{k})\frac{\text{d}^{3}k}{(2\pi)^{3}\omega}\right]$
$\displaystyle\times\left\langle{q+\tfrac{1}{2}x}\right|\hat{\rho}\left|{q-\tfrac{1}{2}x}\right\rangle\
\mathcal{D}[x],$ (6)

where $\mathcal{D}[x]$ represents a functional integration measure over the
space of all finite energy functions $x_{s}(\mathbf{k})$. Apart from the
complexities introduced by the functional nature of the resulting formalism,
all the properties of the operators and the states remain essentially the
same, thus leading to a successful formulation for the states and operations
of massless bosonic fields Roux (2018).

For the benefit of the subsequent discussion, we summarize the properties that
we associate with bosonic quadrature operators and bosonic Bogoliubov
operators. For bosonic quadrature operators they are:

  * •

bQ1: Quadrature operators are Hermitian; their eigenstates exist, are
orthogonal and the associated eigenvalues are real.

  * •

bQ2: The pair of quadrature operators do not commute; their commutation
relation produces an imaginary valued quantity.

  * •

bQ3: Quadrature operators commute with themselves.

  * •

bQ4: The eigenbases of the respective quadrature operators are mutually
unbiased.

The properties of Bogoliubov operators are:

  * •

bB1: Bogoliubov operators are not Hermitian; they are pairwise Hermitian
adjoints of each other.

  * •

bB2: A pair of Bogoliubov operators do not commute; their commutation
relations have the same form as those of the ladder operators from which they
are constructed.

  * •

bB3: Bogoliubov operators commute with themselves.

The third property of the respective sets follows trivially from the
commutation relations of the ladder operators. We state these trivial
properties explicitly because they become nontrivial in the fermionic case
when we replace “commute” by “anti-commute.”

###  II.2 Majorana operators

Now we leave the bosonic case behind and hence only consider fermionic
operators, unless explicitly stated otherwise. We use the same notation for
the fermionic case. Since anti-fermionic ladder operators anti-commute with
fermionic ladder operators, we do not consider the anti-fermionic fields here.
Without all but the particle-number degrees of freedom, the fermionic ladder
operators obey the simple anti-commutation relations

$\begin{split}\\{\hat{a},\hat{a}\\}=\\{\hat{a}^{{\dagger}},\hat{a}^{{\dagger}}\\}=0,~{}~{}~{}\\{\hat{a},\hat{a}^{{\dagger}}\\}=1.\end{split}$
(7)

where $\hat{a}$ and its Hermitian adjoint $\hat{a}^{{\dagger}}$ represent the
fermionic annihilation and creation operators, respectively. Without any other
degrees of freedom, the fermionic Hilbert space is two-dimensional, consisting
of linear combinations of the vacuum $\left|{\text{vac}}\right\rangle$ and the
single-particle state $\left|{1}\right\rangle$. There are only four basic
operations

$\displaystyle\begin{split}\hat{a}\left|{\text{vac}}\right\rangle=&0,~{}~{}~{}\hat{a}\left|{1}\right\rangle=\left|{\text{vac}}\right\rangle,\\\
\hat{a}^{{\dagger}}\left|{\text{vac}}\right\rangle=&\left|{1}\right\rangle,~{}~{}~{}\hat{a}^{{\dagger}}\left|{1}\right\rangle=0.\end{split}$
(8)

We are seeking fermionic operators produced as linear combinations of the
fermionic ladder operators, such that they behave in a way that is analogues
to the bosonic quadrature operators. The expected properties for such
fermionic quadrature operators are:

  * •

fQ1: Fermionic quadrature operators are Hermitian; their eigenstates exist,
are orthogonal and the associated eigenvalues are real.

  * •

fQ2: Pairs of fermionic quadrature operators do not anti-commute; their anti-
commutation relation produces an imaginary valued quantity.

  * •

fQ3: Each fermionic quadrature operator anti-commutes with itself.

  * •

fQ4: A pair of fermionic quadrature operators have eigenbases that are
mutually unbiased.

To this end, we consider the Hermitian operators defined in terms of the sum
and difference of the fermionic ladder operators

$\hat{m}=\frac{1}{\sqrt{2}}(\hat{a}+\hat{a}^{{\dagger}}),~{}~{}~{}\hat{n}=\frac{-i}{\sqrt{2}}(\hat{a}-\hat{a}^{{\dagger}}),$
(9)

They are the Majorana operators. However, we find that they produce the
commutation relations

$\\{\hat{m},\hat{m}\\}=\\{\hat{n},\hat{n}\\}=1,~{}~{}~{}\\{\hat{m},\hat{n}\\}=0.$
(10)

Therefore, the Majorana operators do not satisfy conditions fQ2 and fQ3.

Since they are Hermitian, we expect the Majorana operators to have eigenstates
that are orthogonal with real eigenvalues. Due to the two-dimensional Hilbert
space, these eigenstates are linear combinations of
$\left|{\text{vac}}\right\rangle$ and $\left|{1}\right\rangle$. For $\hat{m}$,
they are

$\left|{m_{\pm}}\right\rangle=\frac{1}{\sqrt{2}}(\left|{\text{vac}}\right\rangle\pm\left|{1}\right\rangle),$
(11)

with eigenvalues $\pm 1/\sqrt{2}$, respectively, and for $\hat{n}$,

$\left|{n_{\pm}}\right\rangle=\frac{1}{\sqrt{2}}(\left|{\text{vac}}\right\rangle\mp
i\left|{1}\right\rangle),$ (12)

with eigenvalues $\pm 1/\sqrt{2}$, respectively. The eigenstates of each
Majorana operator are orthogonal and the two sets are mutually unbiased.
Therefore, these operators satisfy fQ1 and fQ4. We can use either of the two
sets of eigenstates as an orthogonal basis for the Hilbert space.

When we include the other degrees of freedom, the dimension of the Hilbert
space increases. For instance, if we include spin, the Hilbert space becomes
four-dimensional, with basis elements $\left|{\text{vac}}\right\rangle$,
$\left|{1_{1}}\right\rangle$, $\left|{1_{2}}\right\rangle$, and
$\left|{1_{1}}\right\rangle\left|{1_{2}}\right\rangle$, where the subscripts
represent the two spins. The ladder operators are represented as $\hat{a}_{s}$
and $\hat{a}_{s}^{{\dagger}}$, where $s=1,2$ is the spin index. Operators
associated with different degrees of freedom anti-commute. Therefore, the
anti-commutation relations become

$\\{\hat{a}_{r},\hat{a}_{s}\\}=\\{\hat{a}^{{\dagger}}_{r},\hat{a}_{s}^{{\dagger}}\\}=0,~{}~{}~{}\\{\hat{a}_{r},\hat{a}_{s}^{{\dagger}}\\}=\delta_{r,s}.$
(13)

The Majorana operators now include spin

$\displaystyle\begin{split}\hat{m}_{s}=&\frac{1}{\sqrt{2}}(\hat{a}_{s}+\hat{a}_{s}^{{\dagger}}),\\\
\hat{n}_{s}=&\frac{-i}{\sqrt{2}}(\hat{a}_{s}-\hat{a}_{s}^{{\dagger}}).\end{split}$
(14)

They still do not satisfy conditions fQ2 and fQ3.

To obtain the eigenstates, we can consider the different spins separately.
These eigenstates are

$\displaystyle\begin{split}\left|{m_{\pm},s}\right\rangle=&\frac{1}{\sqrt{2}}(\left|{\text{vac}}\right\rangle\pm\left|{1_{s}}\right\rangle),\\\
\left|{n_{\pm},s}\right\rangle=&\frac{1}{\sqrt{2}}(\left|{\text{vac}}\right\rangle\mp
i\left|{1_{s}}\right\rangle),\end{split}$ (15)

with eigenvalues $\pm 1/\sqrt{2}$ in both cases. A basis for the full Hilbert
space is obtained from all the tensor products of these eigenstates, leading
to complete orthogonal bases for $\hat{m}_{s}$ and $\hat{n}_{s}$,
respectively. These two bases are mutually unbiased. As a result, they satisfy
fQ1 and fQ4.

The inclusion of the spatiotemporal degrees of freedom does not resolve the
issues. If the spatiotemporal degrees of freedom are treated as a finite
number of discrete modes, their inclusion follows the same pattern as the
inclusion of spin with a larger dimension. The discrete basis is constructed
as tensor products of all the eigenstates for the individual modes. If the
number of modes is increased to infinity, the resulting bases become similar
to those associated with the Fock states. Such bases that are constructed as
tensor products of infinitely many eigenstates of discrete modes are often
cumbersome to work with.

It is therefore preferable to treat the (infinite dimensional) spatiotemporal
degrees of freedom as continuous degrees of freedom. The ladder operators are
represented by $\hat{a}(u)$ and $\hat{a}^{{\dagger}}(u)$, where $u$ is a
generic continuous variable representing the spatiotemporal degrees of
freedom. The anti-commutation relations are now given by

$\displaystyle\begin{split}\\{\hat{a}(v),\hat{a}(u)\\}=&\\{\hat{a}^{{\dagger}}(v),\hat{a}^{{\dagger}}(u)\\}=0,\\\
\\{\hat{a}(v),\hat{a}^{{\dagger}}(u)\\}=&\delta(u-v).\end{split}$ (16)

The Hilbert space is spanned by an infinite dimensional basis, consisting of
anti-symmetric tensor products of single-particle states
$\left|{u}\right\rangle$ and the vacuum $\left|{\text{vac}}\right\rangle$. The
states can have arbitrary numbers of particles, provided that no two of then
share the same argument. Operating on the single-particle states, the
operators produce

$\displaystyle\begin{split}\hat{a}(u)\left|{v}\right\rangle=&\
\left|{\text{vac}}\right\rangle\delta(u-v),\\\
\hat{a}^{{\dagger}}(u)\left|{v}\right\rangle=&\
\left\\{\begin{array}[]{ccc}\left|{u}\right\rangle\left|{v}\right\rangle=-\left|{v}\right\rangle\left|{u}\right\rangle&\text{for}&u\neq
v\\\ 0&\text{for}&u=v\end{array}\right..\end{split}$ (17)

The Majorana operators are now represented by

$\displaystyle\begin{split}\hat{m}(u)=&\frac{1}{\sqrt{2}}[\hat{a}(u)+\hat{a}^{{\dagger}}(u)],\\\
\hat{n}(u)=&\frac{-i}{\sqrt{2}}[\hat{a}(u)-\hat{a}^{{\dagger}}(u)].\end{split}$
(18)

A general eigenstate of these Majorana operators can be expressed as an
expansion of the form

$\displaystyle\left|{\psi}\right\rangle=$
$\displaystyle\left|{\text{vac}}\right\rangle\mu_{0}+\int\left|{u}\right\rangle\mu_{1}(u)du+\int\left|{u}\right\rangle\left|{v}\right\rangle\mu_{2}(u,v)dudv$
$\displaystyle+\int\left|{u}\right\rangle\left|{v}\right\rangle\left|{w}\right\rangle\mu_{3}(u,v,w)dudvdw+...,$
(19)

where $\mu_{n}(...)$ are coefficient functions that are antisymmetric with
respect to the interchange of any two variables. To determine the coefficient
functions, we substitute the expansion into the eigen-equation

$\hat{m}(t)\left|{\psi}\right\rangle=\left|{\psi}\right\rangle\lambda(t),$
(20)

where $t$ is the continuous variable. Then we separate the result into
different equations for the different numbers of particles. For the vacuum, we
get

$\mu_{1}(t)=\mu_{0}\lambda(t),$ (21)

which gives $\mu_{1}$ in terms of $\mu_{0}$. Next, we consider terms with
single-fermion states, for which we get

$2\int\left|{v}\right\rangle\mu_{2}(t,v)dv+\left|{t}\right\rangle\mu_{0}=\int\left|{u}\right\rangle\mu_{1}(u)\lambda(t)du.$
(22)

When we use the expression for $\mu_{1}$ in Eq. (21), the resulting expression
for $\mu_{2}$ becomes

$\mu_{2}(t,v)=\tfrac{1}{2}\mu_{0}\lambda(v)\lambda(t)-\tfrac{1}{2}\delta(v-t)\mu_{0}.$
(23)

However, there is a problem with the symmetry of the expression. The first
term can be made antisymmetric by assuming the eigenvalue function
$\lambda(v)$ is a Grassmann function, but the last term cannot be made anti-
symmetric and would therefore have to be dropped. Without such a term, the
eigen-equation cannot be solved. Similar issues with the symmetry appear in
all the expressions for larger numbers of particles.

It may seem that this issue can be avoided by including both discrete degrees
of freedom, such as spin, and continuous degrees of freedom. The same
procedure then produces an expression for $\mu_{2}$ where the second term has
the same symmetric form multiplied by $\delta_{r,s}$. Hence, the symmetry
problem persists. Even in such a general case, the Majorana operators do not
have eigenstates.

We conclude that the Majorana operators with continuous variables do not have
a full set of eigenstates for all the degrees of freedom. As a result, none of
the conditions fQ1, fQ2, fQ3 and fQ4, are satisfied. The Majorana operators
are therefore not suitable equivalents of the quadrature operators found in
the bosonic case.

##  III Fermionic Bogoliubov operators

If we compare the analysis for the eigenstates of the Majorana operators with
continuous variables with the equivalent analysis for the bosonic quadrature
operators in Roux (2018), we can identify the source of the issue. In Roux
(2018), the Dirac delta terms in the coefficient functions eventually led to
an operator of the form

$\hat{a}_{R}=\int\hat{a}^{2}(u)\ \text{d}u,$ (24)

where $\hat{a}(u)$ is the bosonic annihilation operator. The equivalent
fermionic operator would obviously vanish due to the anti-commuting property,
unless some degree of freedom can be used to introduce an anti-symmetry. Such
an anti-symmetry can be obtained if the linear combination of fermionic ladder
operators incorporates a relative transformation of their degrees of freedom.
However, such a relative transformation implies that the resulting linear
combination would not be Hermitian. Since the Hermiticity of the operator is
the defining property of a quadrature operator, we are led to abandon the
consideration of a fermionic equivalent of the quadrature operators and rather
consider the fermionic equivalent of the Bogoliubov operators.

The properties of fermionic Bogoliubov operators are expected to be the
following:

  * •

fB1: Fermionic Bogoliubov operators are not Hermitian; they are pairwise
Hermitian adjoints of each other.

  * •

fB2: Pairs of fermionic Bogoliubov operators do not anti-commute; their anti-
commutation relations have the same form as those of the fermionic ladder
operators from which they are constructed.

  * •

fB3: Fermionic Bogoliubov operators anti-commute with themselves.

From this point onward, we’ll use the fermion ladder operators with all
degrees of freedom (spin and three-dimensional wave vectors), as in quantum
field theory. As stated before, we’ll exclude the anti-fermion ladder
operators since they anti-commute with the fermion ladder operators. The
Lorentz convariant anti-commutation relations for the fermion ladder operators
are

$\displaystyle\begin{split}\\{\hat{a}_{s}(\mathbf{k}),\hat{a}_{r}^{{\dagger}}(\mathbf{k}^{\prime})\\}=&\\{\hat{a}_{s}(\mathbf{k}),\hat{a}_{r}^{{\dagger}}(\mathbf{k}^{\prime})\\}=0,\\\
\\{\hat{a}_{s}(\mathbf{k}),\hat{a}_{r}^{{\dagger}}(\mathbf{k}^{\prime})\\}=&(2\pi)^{3}E_{\mathbf{k}}\delta_{s,r}\delta(\mathbf{k}-\mathbf{k}^{\prime}),\end{split}$
(25)

with

$E_{\mathbf{k}}=\sqrt{m^{2}+|\mathbf{k}|^{2}}.$ (26)

Note that $\mathbf{k}$ is the wave vector (as opposed to the momentum).
Therefore, the “energy” $E_{\mathbf{k}}$ and the “mass” $m$ both have the
units of inverse distance.

There are various ways to define fermionic Bogoliubov operators. Here, we
proceed with the line of argument presented above. Therefore, we apply a spin
transformation to one of the ladder operators and consider linear combinations
of it together with the other ladder operator. So, we define the following two
operators

$\displaystyle\begin{split}\hat{g}_{s}(\mathbf{k})=&\frac{1}{\sqrt{2}}\left[\hat{a}_{s}(\mathbf{k})+\varepsilon_{s,r}\hat{a}_{r}^{{\dagger}}(\mathbf{k})\right],\\\
\hat{h}_{s}(\mathbf{k})=&\frac{i}{\sqrt{2}}\left[\hat{a}_{s}(\mathbf{k})-\varepsilon_{s,r}\hat{a}_{r}^{{\dagger}}(\mathbf{k})\right],\end{split}$
(27)

where $\varepsilon_{s,r}$ is an antisymmetric spin matrix
($\varepsilon_{s,r}=-\varepsilon_{r,s}$). Moreover,
$\varepsilon_{s,r}^{*}=-\varepsilon_{s,r}$ and
$\varepsilon_{s,r}\varepsilon_{r,t}=\delta_{s,t}$. For two-dimensional spin
degrees of freedom, $\varepsilon_{s,r}$ is defined as the Pauli $y$-matrix

$\varepsilon_{r,s}=\sigma^{y}_{r,s}=\left(\begin{array}[]{cc}0&-i\\\
i&0\end{array}\right).$ (28)

We follow the summation convention where repeated indices are summed over.

The Hermitian adjoints of these operators are

$\displaystyle\begin{split}\hat{g}_{s}^{{\dagger}}(\mathbf{k})=&\frac{1}{\sqrt{2}}\left[\hat{a}_{s}^{{\dagger}}(\mathbf{k})-\varepsilon_{s,r}\hat{a}_{r}(\mathbf{k})\right]\equiv
i\varepsilon_{s,r}\hat{h}_{r}(\mathbf{k}),\\\
\hat{h}_{s}^{{\dagger}}(\mathbf{k})=&\frac{-i}{\sqrt{2}}\left[\hat{a}_{s}^{{\dagger}}(\mathbf{k})+\varepsilon_{s,r}\hat{a}_{r}(\mathbf{k})\right]\equiv-i\varepsilon_{s,r}\hat{g}_{r}(\mathbf{k}).\end{split}$
(29)

So, the Hermitian adjoint of one is related to the other.

###  III.1 Fermionic adjoint

Here, we define a fermionic adjoint that includes a spin transformation with
the Hermitian adjoint, so that

$\displaystyle\begin{split}\hat{a}_{s}(\mathbf{k})\rightarrow&\
\varepsilon_{s,r}\hat{a}_{r}^{{\dagger}}(\mathbf{k})\triangleq\hat{a}_{s}^{{\ddagger}}(\mathbf{k}),\\\
\hat{a}_{s}^{{\dagger}}(\mathbf{k})\rightarrow&\
-\varepsilon_{s,r}\hat{a}_{r}(\mathbf{k}).\end{split}$ (30)

The symbol ${\ddagger}$ is used to represent the fermionic adjoint. The
fermionic Bogoliubov operators defined in Eq. (27) are fermionic self-adjoint:

$\displaystyle\begin{split}\hat{g}_{s}^{{\ddagger}}(\mathbf{k})=&\frac{1}{\sqrt{2}}\left\\{\varepsilon_{s,r}\hat{a}_{r}^{{\dagger}}(\mathbf{k})-\varepsilon_{s,r}\left[-\varepsilon_{r,t}\hat{a}_{t}(\mathbf{k})\right]\right\\}\\\
=&\frac{1}{\sqrt{2}}\left[\varepsilon_{s,r}\hat{a}_{r}^{{\dagger}}(\mathbf{k})+\hat{a}_{s}(\mathbf{k})\right]\equiv\hat{g}_{s}(\mathbf{k}),\\\
\hat{h}_{s}^{{\ddagger}}(\mathbf{k})=&\frac{-i}{\sqrt{2}}\left\\{\varepsilon_{s,r}\hat{a}_{r}^{{\dagger}}(\mathbf{k})+\varepsilon_{s,r}\left[-\varepsilon_{r,t}\hat{a}_{t}(\mathbf{k})\right]\right\\}\\\
=&\frac{-i}{\sqrt{2}}\left[\varepsilon_{s,r}\hat{a}_{r}^{{\dagger}}(\mathbf{k})-\hat{a}_{t}(\mathbf{k})\right]\equiv\hat{h}_{s}(\mathbf{k}).\end{split}$
(31)

###  III.2 Anti-commutation relations

The various anti-commutation relations among $\hat{g}$, $\hat{h}$ and their
Hermitian adjoints are

$\displaystyle\begin{split}\left\\{\hat{g}_{r}(\mathbf{k}),\hat{g}_{s}(\mathbf{k}^{\prime})\right\\}=&\left\\{\hat{g}_{r}^{{\dagger}}(\mathbf{k}),\hat{g}_{s}^{{\dagger}}(\mathbf{k}^{\prime})\right\\}=0,\\\
\left\\{\hat{h}_{r}(\mathbf{k}),\hat{h}_{s}(\mathbf{k}^{\prime})\right\\}=&\left\\{\hat{h}_{r}^{{\dagger}}(\mathbf{k}),\hat{h}_{s}^{{\dagger}}(\mathbf{k}^{\prime})\right\\}=0,\\\
\left\\{\hat{g}_{r}(\mathbf{k}),\hat{h}_{s}(\mathbf{k}^{\prime})\right\\}=&\left\\{\hat{g}_{r}^{{\dagger}}(\mathbf{k}),\hat{h}_{s}^{{\dagger}}(\mathbf{k}^{\prime})\right\\}\\\
=&i(2\pi)^{3}E_{\mathbf{k}}\varepsilon_{r,s}\delta(\mathbf{k}-\mathbf{k}^{\prime}),\\\
\left\\{\hat{g}_{r}(\mathbf{k}),\hat{g}_{s}^{{\dagger}}(\mathbf{k}^{\prime})\right\\}=&\left\\{\hat{h}_{r}(\mathbf{k}),\hat{h}_{s}^{{\dagger}}(\mathbf{k}^{\prime})\right\\}\\\
=&(2\pi)^{3}E_{\mathbf{k}}\delta_{r,s}\delta(\mathbf{k}-\mathbf{k}^{\prime}),\\\
\left\\{\hat{g}_{r}(\mathbf{k}),\hat{h}_{s}^{{\dagger}}(\mathbf{k}^{\prime})\right\\}=&\left\\{\hat{h}_{r}(\mathbf{k}),\hat{g}_{s}^{{\dagger}}(\mathbf{k}^{\prime})\right\\}=0.\end{split}$
(32)

It shows that these operators satisfy all requirements for fermionic
Bogoliubov operators fB1, fB2 and fB3.

##  IV Eigenstates

###  IV.1 Bosonized operators

In the attempt to determine the eigenstates for Majorana operators with
continuous degrees of freedom, we found that the eigenfunctions need to be
Grassmann functions to satisfy the requirement for anti-symmetry in fermionic
states consisting of more than one fermion. These Grassmann eigenfunctions
would also parameterize their associated eigenstates. Therefore, in analogy to
the bosonic case Roux (2018), we expect that these eigenstates are represented
in terms of bosonized (or Grassmann even) spectral operators, defined as

$\displaystyle\begin{split}\hat{A}=&\int
A_{s}^{*}(\mathbf{k})\hat{a}_{s}(\mathbf{k})\
\text{d}k_{E}=A^{*}\diamond\hat{a},\\\
\hat{A}^{{\dagger}}=&\int\hat{a}_{s}^{{\dagger}}(\mathbf{k})A_{s}(\mathbf{k})\
\text{d}k_{E}=\hat{a}^{{\dagger}}\diamond A,\end{split}$ (33)

where $A_{s}(\mathbf{k})$ is a Grassmann spectral function,

$\text{d}k_{E}=\frac{\text{d}^{3}k}{(2\pi)^{3}E_{\mathbf{k}}},$ (34)

and a $\diamond$-contraction notation is defined. The combinations of the
fermion operators and the Grassmann spectral functions produce operators that
follow commutation relations instead of anti-commutation relations and are
therefore referred as being “bosonized.” Indeed, we find that
$[\hat{A},\hat{B}]=[\hat{A}^{{\dagger}},\hat{B}^{{\dagger}}]=0$, while

$[\hat{A},\hat{B}^{{\dagger}}]=\int A_{s}^{*}(\mathbf{k})B_{s}(\mathbf{k})\
\text{d}k_{E}=A^{*}\diamond B.$ (35)

Moreover, we also have commutation relations between spectral operators and
fermion ladder operators

$\displaystyle\begin{split}[\hat{a}_{s}(\mathbf{k}),\hat{A}]=&\
[\hat{A}^{{\dagger}},\hat{a}_{s}^{{\dagger}}(\mathbf{k})]=0,\\\
[\hat{a}_{s}(\mathbf{k}),\hat{A}^{{\dagger}}]=&\ A_{s}(\mathbf{k}),\\\
[\hat{A},\hat{a}_{s}^{{\dagger}}(\mathbf{k})]=&\
A_{s}^{*}(\mathbf{k}).\end{split}$ (36)

In addition to the spectral operators, the definition of the eigenstates also
needs operators of the form

$\displaystyle\begin{split}\hat{R}=&\frac{1}{2}\int\hat{a}_{s}(\mathbf{k})\varepsilon_{s,r}\hat{a}_{r}(\mathbf{k})\
\text{d}k_{E}=\tfrac{1}{2}\hat{a}\ominus\hat{a},\\\
\hat{R}^{{\dagger}}=&\frac{1}{2}\int\hat{a}_{s}^{{\dagger}}(\mathbf{k})\varepsilon_{s,r}\hat{a}_{r}^{{\dagger}}(\mathbf{k})\
\text{d}k_{E}=\tfrac{1}{2}\hat{a}^{{\dagger}}\ominus\hat{a}^{{\dagger}},\end{split}$
(37)

where $\varepsilon$ is the spin transform matrix defined in Eq. (28), and we
introduce a simpler notation where $\ominus$ represents contractions that
involve $\varepsilon$. The commutation relations between these operators and
fermion ladder operators are

$\displaystyle\begin{split}\left[\hat{a}_{r}(\mathbf{k}),\hat{R}^{{\dagger}}\right]=&\
\varepsilon_{r,s}\hat{a}_{s}^{{\dagger}}(\mathbf{k}),\\\
\left[\hat{R},\hat{a}_{r}^{{\dagger}}(\mathbf{k})\right]=&\
\hat{a}_{s}(\mathbf{k})\varepsilon_{s,r}.\end{split}$ (38)

The commutations among these bosonized operators produce additional operators.
To complete the algebra, we also need to define spectral operators that
include $\varepsilon$:

$\displaystyle\begin{split}\hat{A}_{\varepsilon}=&\int
A_{s}(\mathbf{k})\varepsilon_{s,r}\hat{a}_{r}(\mathbf{k})\
\text{d}k_{E}=A\ominus\hat{a},\\\
\hat{A}_{\varepsilon}^{{\dagger}}=&\int\hat{a}_{r}^{{\dagger}}(\mathbf{k})\varepsilon_{r,s}A_{s}^{*}(\mathbf{k})\
\text{d}k_{E}=\hat{a}^{{\dagger}}\ominus A^{*},\end{split}$ (39)

and a symmetrized number operator

$\displaystyle\hat{s}=$
$\displaystyle\frac{1}{2}\int\hat{a}_{s}^{{\dagger}}(\mathbf{k})\hat{a}_{s}(\mathbf{k})-\hat{a}_{s}(\mathbf{k})\hat{a}_{s}^{{\dagger}}(\mathbf{k})\
\text{d}k_{E}$ $\displaystyle=$
$\displaystyle\hat{a}^{{\dagger}}\diamond\hat{a}-\tfrac{1}{2}\Omega,$ (40)

where the divergent constant $\Omega$ is given by

$\Omega=\int\delta(0)\ \text{d}^{3}k.$ (41)

The commutation relations among all these bosonized operators are summarized
in Appen. A.

###  IV.2 Eigen-equations and their solutions

The Bogoliubov operators defined in Eq. (27) are not normal. Therefore, their
left-eigenstates are not the Hermitian adjoints of their right-eigenstate. As
a result, we’ll distinguish between the right- and left-eigenstates with
subscript $R$ and $L$, respectively. The eigen-equations for the right-
eigenstates are

$\displaystyle\begin{split}\hat{g}_{s}(\mathbf{k})\left|{g_{R}}\right\rangle=&\left|{g_{R}}\right\rangle
g_{s}(\mathbf{k}),\\\
\hat{g}_{s}^{{\dagger}}(\mathbf{k})\left|{\bar{g}_{R}}\right\rangle=&\left|{\bar{g}_{R}}\right\rangle\bar{g}_{s}(\mathbf{k}),\\\
\hat{h}_{s}(\mathbf{k})\left|{h_{R}}\right\rangle=&\left|{h_{R}}\right\rangle
h_{s}(\mathbf{k}),\\\
\hat{h}_{s}^{{\dagger}}(\mathbf{k})\left|{\bar{h}_{R}}\right\rangle=&\left|{\bar{h}_{R}}\right\rangle\bar{h}_{s}(\mathbf{k}),\end{split}$
(42)

and for the left-eigenstates, they are

$\displaystyle\begin{split}\left\langle{g_{L}}\right|\hat{g}_{s}(\mathbf{k})=&g_{s}^{*}(\mathbf{k})\left\langle{g_{L}}\right|,\\\
\left\langle{\bar{g}_{L}}\right|\hat{g}_{s}^{{\dagger}}(\mathbf{k})=&\bar{g}_{s}^{*}(\mathbf{k})\left\langle{\bar{g}_{L}}\right|,\\\
\left\langle{h_{L}}\right|\hat{h}_{s}(\mathbf{k})=&h_{s}^{*}(\mathbf{k})\left\langle{h_{L}}\right|,\\\
\left\langle{\bar{h}_{L}}\right|\hat{h}_{s}^{{\dagger}}(\mathbf{k})=&\bar{h}_{s}^{*}(\mathbf{k})\left\langle{\bar{h}_{L}}\right|.\end{split}$
(43)

The eigenstates are defined in terms of rendering operators that produce the
eigenstates from the vacuum. These rendering operators are expressed as
exponentiated bosonized operators. For the right-eigenstates, they are
generically given by

$\left|{\psi_{R}}\right\rangle=F_{0}\exp\left(\hat{A}^{{\dagger}}\pm\hat{R}^{{\dagger}}\right)\left|{\text{vac}}\right\rangle,$
(44)

where $F_{0}$ is a prefactor to be determined. The form of the rendering
operator for each of the Bogoliubov operators is determined by substituting
the generic form of the rendering operator into the eigen-equation to
determine the correct sign and the relationship between spectral function and
the Grassmann eigenfunction in the equation. From the detail calculations,
shown in Appens. B and C, we obtain in this way the follow expressions for the
right-eigenstates in terms of their rendering operators

$\displaystyle\begin{split}\left|{g_{R}}\right\rangle=&G_{R}\exp\left(\hat{A}^{{\dagger}}-\hat{R}^{{\dagger}}\right)\left|{\text{vac}}\right\rangle,\\\
\left|{\bar{g}_{R}}\right\rangle=&\bar{G}_{R}\exp\left(\hat{A}^{{\dagger}}+\hat{R}^{{\dagger}}\right)\left|{\text{vac}}\right\rangle,\\\
\left|{h_{R}}\right\rangle=&H_{R}\exp\left(\hat{A}^{{\dagger}}+\hat{R}^{{\dagger}}\right)\left|{\text{vac}}\right\rangle,\\\
\left|{\bar{h}_{R}}\right\rangle=&\bar{H}_{R}\exp\left(\hat{A}^{{\dagger}}-\hat{R}^{{\dagger}}\right)\left|{\text{vac}}\right\rangle,\end{split}$
(45)

where $G_{R}$, $\bar{G}_{R}$, $H_{R}$, and $\bar{H}_{R}$ are prefactors that
we still need to determine. The relationships between the Grassmann parameter
functions in the respective spectral operators and the respective
eigenfunctions are

$\displaystyle\begin{split}A_{r}(\mathbf{k})=&\sqrt{2}g_{r}(\mathbf{k}),\\\
A_{r}(\mathbf{k})=&-\sqrt{2}\varepsilon_{r,s}\bar{g}_{s}(\mathbf{k})=\sqrt{2}\bar{g}_{s}(\mathbf{k})\varepsilon_{s,r},\\\
A_{r}(\mathbf{k})=&-i\sqrt{2}h_{r}(\mathbf{k}),\\\
A_{r}(\mathbf{k})=&i\sqrt{2}\varepsilon_{r,s}\bar{h}_{s}(\mathbf{k})=-i\sqrt{2}\bar{h}_{s}(\mathbf{k})\varepsilon_{s,r}.\end{split}$
(46)

The same procedure is used for the left-eigenstates of
$\hat{g}_{s}(\mathbf{k})$, $\hat{g}_{s}^{{\dagger}}(\mathbf{k})$,
$\hat{h}_{s}(\mathbf{k})$, and $\hat{h}_{s}^{{\dagger}}(\mathbf{k})$. The
results are given by

$\displaystyle\begin{split}\left\langle{g_{L}}\right|=&G_{L}\left\langle{\text{vac}}\right|\exp\left(\hat{A}+\hat{R}\right),\\\
\left\langle{\bar{g}_{L}}\right|=&\bar{G}_{L}\left\langle{\text{vac}}\right|\exp\left(\hat{A}-\hat{R}\right),\\\
\left\langle{h_{L}}\right|=&H_{L}\left\langle{\text{vac}}\right|\exp\left(\hat{A}-\hat{R}\right),\\\
\left\langle{\bar{h}_{L}}\right|=&\bar{H}_{L}\left\langle{\text{vac}}\right|\exp\left(\hat{A}+\hat{R}\right),\end{split}$
(47)

where $G_{L}$, $\bar{G}_{L}$, $H_{L}$, and $\bar{H}_{L}$ are prefactors, and
the spectral functions are related to the eigenfunctions by

$\displaystyle\begin{split}A_{r}^{*}(\mathbf{k})=&\sqrt{2}\varepsilon_{r,s}g_{s}^{*}(\mathbf{k})=-\sqrt{2}g_{s}^{*}(\mathbf{k})\varepsilon_{s,r},\\\
A_{r}^{*}(\mathbf{k})=&\sqrt{2}\bar{g}_{r}^{*}(\mathbf{k}),\\\
A_{r}^{*}(\mathbf{k})=&i\sqrt{2}\varepsilon_{r,s}h_{s}^{*}(\mathbf{k})=-i\sqrt{2}h_{s}^{*}(\mathbf{k})\varepsilon_{s,r},\\\
A_{r}^{*}(\mathbf{k})=&i\sqrt{2}\bar{h}_{r}^{*}(\mathbf{k}).\end{split}$ (48)

###  IV.3 Adjoints

Comparing Eqs. (45) and (47), we find the following relationships among the
states and their Hermitian adjoints (assuming their parameter functions are
the same):

$\displaystyle\begin{split}\left\langle{g_{L}}\right|=&\left(\left|{\bar{g}_{R}}\right\rangle\right)^{{\dagger}},~{}~{}~{}\left\langle{\bar{g}_{L}}\right|=\left(\left|{g_{R}}\right\rangle\right)^{{\dagger}},\\\
\left\langle{h_{L}}\right|=&\left(\left|{\bar{h}_{R}}\right\rangle\right)^{{\dagger}},~{}~{}~{}\left\langle{\bar{h}_{L}}\right|=\left(\left|{h_{R}}\right\rangle\right)^{{\dagger}}.\end{split}$
(49)

On the other hand, with the fermionic adjoint, we have

$\displaystyle\begin{split}\left\langle{g_{L}}\right|=&\left(\left|{g_{R}}\right\rangle\right)^{{\ddagger}},~{}~{}~{}\left\langle{\bar{g}_{L}}\right|=\left(\left|{\bar{g}_{R}}\right\rangle\right)^{{\ddagger}},\\\
\left\langle{h_{L}}\right|=&\left(\left|{h_{R}}\right\rangle\right)^{{\ddagger}},~{}~{}~{}\left\langle{\bar{h}_{L}}\right|=\left(\left|{\bar{h}_{R}}\right\rangle\right)^{{\ddagger}}.\end{split}$
(50)

The fermionic adjoint converts the left-eigenstates into the right-eigenstates
of the same operator, and visa versa.

##  V Inner products

An important aspect of the properties of the eigenstates computed above is the
various inner products between the left- and right-eigenstates. It will reveal
whether there are any sets that are mutually orthogonal or mutually unbiased.
It will also provide an opportunity to define suitable prefactors.

###  V.1 Commuting the operators

To compute all the inner products between the left- and right-eigenstates, we
consider a generic inner product, given by

$\displaystyle\langle{f_{L}}|{f_{R}}\rangle=$ $\displaystyle
F_{L}F_{R}\left\langle{\text{vac}}\right|\exp\left(\hat{A}+c_{1}\hat{R}\right)$
$\displaystyle\times\exp\left(\hat{B}^{{\dagger}}+c_{2}\hat{R}^{{\dagger}}\right)\left|{\text{vac}}\right\rangle.$
(51)

Here $F_{L}$ and $F_{R}$ are unknown prefactors, $\hat{B}^{{\dagger}}$ is a
spectral operator with parameter function $B_{s}(\mathbf{k})$, and
$c_{1,2}=\pm 1$, depending on which eigenstates are considered.

Next, we commute the exponentiated operators in Eq. (51) so that they appear
in normal order, producing a more complex expression with new operators
generated by the current ones. The expected operators include $\hat{s}$,
$\hat{A}_{\varepsilon}^{{\dagger}}$ and $\hat{B}_{\varepsilon}$. The general
expression has the form

$\displaystyle\exp(\hat{A})\exp(c_{1}\hat{R})\exp(\hat{B}^{{\dagger}})\exp(c_{2}\hat{R}^{{\dagger}})$
$\displaystyle=$
$\displaystyle\exp(h_{0})\exp(h_{1}\hat{B}^{{\dagger}})\exp(h_{2}\hat{A}_{\varepsilon}^{{\dagger}})\exp(h_{3}\hat{R}^{{\dagger}})\exp(h_{4}\hat{s})$
$\displaystyle\times\exp(h_{5}\hat{B}_{\varepsilon})\exp(h_{6}\hat{A})\exp(h_{7}\hat{R}),$
(52)

where $h_{0}$, $h_{1}$, $h_{2}$, $h_{3}$, $h_{4}$, $h_{5}$, $h_{6}$ and
$h_{7}$ are unknown constants. We introduce an auxiliary variable $t$ into the
exponents on the left-hand side and convert the unknown constants into unknown
functions of $t$ on the right-hand side. The equation then reads

$\displaystyle\exp(t\hat{A})\exp(tc_{1}\hat{R})\exp(t\hat{B}^{{\dagger}})\exp(tc_{2}\hat{R}^{{\dagger}})$
$\displaystyle=$
$\displaystyle\exp[h_{0}(t)]\exp[h_{1}(t)\hat{B}^{{\dagger}}]\exp[h_{2}(t)\hat{A}_{\varepsilon}^{{\dagger}}]$
$\displaystyle\times\exp[h_{3}(t)\hat{R}^{{\dagger}}]\exp[h_{4}(t)\hat{s}]\exp[h_{5}(t)\hat{B}_{\varepsilon}]$
$\displaystyle\times\exp[h_{6}(t)\hat{A}]\exp[h_{7}(t)\hat{R}],$ (53)

The unknown functions become zero for $t=0$, and for $t=1$, we recover the
original expression.

Substituting Eq. (53) into Eq. (51) and evaluating the contractions with the
vacuum state, we get

$\langle{f_{L}}|{f_{R}}\rangle=F_{L}F_{R}\exp[h_{0}(t)]\exp\left[\tfrac{1}{2}h_{4}(t)\Omega\right],$
(54)

where we used the fact that any exponentiated normal ordered operator
$\hat{O}$ produces
$\exp(K\hat{O})\left|{\text{vac}}\right\rangle=\left|{\text{vac}}\right\rangle$
and
$\left\langle{\text{vac}}\right|\exp(K\hat{O})=\left\langle{\text{vac}}\right|$,
regardless of the value of $K$. The expression in Eq. (40) is used to express
$\hat{s}$ in normal order. Therefore

$\left\langle{\text{vac}}\right|\exp(K\hat{s})\left|{\text{vac}}\right\rangle=\exp\left(\tfrac{1}{2}K\Omega\right).$
(55)

Now we apply a derivative with respect to $t$ on both sides of Eq. (53) and
remove as many of the exponentiated operators as possible by operating with
the respective inverse operators on the right-hand sides of both sides of the
equations. The result reads

$\displaystyle\hat{A}+c_{1}\hat{R}+\exp(t\hat{A})\exp(tc_{1}\hat{R})\hat{B}^{{\dagger}}\exp(-tc_{1}\hat{R})\exp(-t\hat{A})+\exp(t\hat{A})\exp(tc_{1}\hat{R})c_{2}\hat{R}^{{\dagger}}\exp(-tc_{1}\hat{R})\exp(-t\hat{A})$
$\displaystyle=$
$\displaystyle\partial_{t}h_{0}+\partial_{t}h_{1}\hat{B}^{{\dagger}}+\partial_{t}h_{2}\hat{A}_{\varepsilon}^{{\dagger}}+\partial_{t}h_{3}\hat{R}^{{\dagger}}+\exp(h_{1}\hat{B}^{{\dagger}})\exp(h_{2}\hat{A}_{\varepsilon}^{{\dagger}})\exp(h_{3}\hat{R}^{{\dagger}})\partial_{t}h_{4}\hat{s}\exp(-h_{3}\hat{R}^{{\dagger}})\exp(-h_{2}\hat{A}_{\varepsilon}^{{\dagger}})\exp(-h_{1}\hat{B}^{{\dagger}})$
$\displaystyle+\exp(h_{1}\hat{B}^{{\dagger}})\exp(h_{2}\hat{A}_{\varepsilon}^{{\dagger}})\exp(h_{3}\hat{R}^{{\dagger}})\exp(h_{4}\hat{s})\partial_{t}h_{5}\hat{B}_{\varepsilon}\exp(-h_{4}\hat{s})\exp(-h_{3}\hat{R}^{{\dagger}})\exp(-h_{2}\hat{A}_{\varepsilon}^{{\dagger}})\exp(-h_{1}\hat{B}^{{\dagger}})$
$\displaystyle+\exp(h_{1}\hat{B}^{{\dagger}})\exp(h_{2}\hat{A}_{\varepsilon}^{{\dagger}})\exp(h_{3}\hat{R}^{{\dagger}})\exp(h_{4}\hat{s})\partial_{t}h_{6}\hat{A}\exp(-h_{4}\hat{s})\exp(-h_{3}\hat{R}^{{\dagger}})\exp(-h_{2}\hat{A}_{\varepsilon}^{{\dagger}})\exp(-h_{1}\hat{B}^{{\dagger}})$
$\displaystyle+\exp(h_{1}\hat{B}^{{\dagger}})\exp(h_{2}\hat{A}_{\varepsilon}^{{\dagger}})\exp(h_{3}\hat{R}^{{\dagger}})\exp(h_{4}\hat{s})\partial_{t}h_{7}\hat{R}\exp(-h_{4}\hat{s})\exp(-h_{3}\hat{R}^{{\dagger}})\exp(-h_{2}\hat{A}_{\varepsilon}^{{\dagger}})\exp(-h_{1}\hat{B}^{{\dagger}}).$
(56)

Using the identities derived in Appen. B and summarized in Eq. (121), we
convert the products of operators into sums of operators. It leads to

$\displaystyle\hat{A}+c_{1}\hat{R}+\hat{B}^{{\dagger}}+tA^{*}\diamond
B+tc_{1}\hat{B}_{\varepsilon}+c_{2}\hat{R}^{{\dagger}}+tc_{2}\hat{A}_{\varepsilon}^{{\dagger}}+\tfrac{1}{2}t^{2}c_{2}A^{*}\ominus
A^{*}-tc_{1}c_{2}\hat{s}-t^{2}c_{1}c_{2}\hat{A}-t^{2}c_{1}^{2}c_{2}\hat{R}$
$\displaystyle=$
$\displaystyle\partial_{t}h_{0}+\partial_{t}h_{1}\hat{B}^{{\dagger}}+\partial_{t}h_{2}\hat{A}_{\varepsilon}^{{\dagger}}+\partial_{t}h_{3}\hat{R}^{{\dagger}}+\partial_{t}h_{4}\hat{s}-h_{1}\partial_{t}h_{4}\hat{B}^{{\dagger}}-h_{2}\partial_{t}h_{4}\hat{A}_{\varepsilon}^{{\dagger}}-2h_{3}\partial_{t}h_{4}\hat{R}^{{\dagger}}+\exp(-h_{4})\partial_{t}h_{5}\hat{B}_{\varepsilon}$
$\displaystyle-h_{1}\exp(-h_{4})\partial_{t}h_{5}B\ominus
B+h_{2}\exp(-h_{4})\partial_{t}h_{5}A^{*}\diamond
B+h_{3}\exp(-h_{4})\partial_{t}h_{5}\hat{B}^{{\dagger}}-h_{1}\exp(-h_{4})\partial_{t}h_{6}A^{*}\diamond
B$
$\displaystyle+\exp(-h_{4})\partial_{t}h_{6}\hat{A}-h_{2}\exp(-h_{4})\partial_{t}h_{6}A^{*}\ominus
A^{*}-h_{3}\exp(-h_{4})\partial_{t}h_{6}\hat{A}_{\varepsilon}^{{\dagger}}+\exp(-2h_{4})\partial_{t}h_{7}\hat{R}-h_{1}\exp(-2h_{4})\partial_{t}h_{7}\hat{B}_{\varepsilon}$
$\displaystyle+\tfrac{1}{2}h_{1}^{2}\exp(-2h_{4})\partial_{t}h_{7}B\ominus
B+h_{2}\exp(-2h_{4})\partial_{t}h_{7}\hat{A}-h_{1}h_{2}\exp(-2h_{4})\partial_{t}h_{7}A^{*}\diamond
B+h_{3}\exp(-2h_{4})\partial_{t}h_{7}\hat{s}$
$\displaystyle-\tfrac{1}{2}h_{2}^{2}\exp(-2h_{4})\partial_{t}h_{7}A^{*}\ominus
A^{*}-h_{1}h_{3}\exp(-2h_{4})\partial_{t}h_{7}\hat{B}^{{\dagger}}-h_{2}h_{3}\exp(-2h_{4})\partial_{t}h_{7}\hat{A}_{\varepsilon}^{{\dagger}}-h_{3}^{2}\exp(-2h_{4})\partial_{t}h_{7}\hat{R}^{{\dagger}}.$
(57)

This equation can be divided into separate equations for the different
operators. With some simplifications, we obtain a set of eight differential
equations, one for each of the $h$-functions. They are given by

$\displaystyle\begin{split}\partial_{t}h_{0}=&A^{*}\diamond
B(t+Th_{1})(1-c_{1}h_{2})\\\ &+B\ominus
B(tc_{1}h_{1}+\tfrac{1}{2}c_{1}Th_{1}^{2})\\\ &+A^{*}\ominus
A^{*}(\tfrac{1}{2}t^{2}c_{2}+Th_{2}-\tfrac{1}{2}c_{1}Th_{2}^{2}),\\\
\partial_{t}h_{1}=&1-tc_{1}c_{2}h_{1}-tc_{1}h_{3}-c_{1}Th_{1}h_{3},\\\
\partial_{t}h_{2}=&tc_{2}-tc_{1}c_{2}h_{2}+Th_{3}-c_{1}Th_{2}h_{3},\\\
\partial_{t}h_{3}=&c_{2}-2tc_{1}c_{2}h_{3}-c_{1}Th_{3}^{2},\\\
\partial_{t}h_{4}=&-tc_{1}c_{2}-c_{1}Th_{3},\\\
\partial_{t}h_{5}=&tc_{1}\exp(h_{4})+c_{1}Th_{1}\exp(h_{4}),\\\
\partial_{t}h_{6}=&T(1-c_{1}h_{2})\exp(h_{4}),\\\
\partial_{t}h_{7}=&c_{1}T\exp(2h_{4}),\end{split}$ (58)

where $T=1-c_{1}c_{2}t^{2}$. These differential equations can be solved in
sequence to obtain the solutions for the $h$-functions. These solutions are

$\displaystyle\begin{split}h_{0}(t)=&\frac{\left(A^{*}\diamond
B+\tfrac{1}{2}c_{1}tB\ominus B+\tfrac{1}{2}c_{2}tA^{*}\ominus
A^{*}\right)t^{2}}{1+c_{1}c_{2}t^{2}},\\\
h_{1}(t)=&\frac{t}{1+c_{1}c_{2}t^{2}},~{}~{}~{}h_{2}(t)=\frac{c_{2}t^{2}}{1+c_{1}c_{2}t^{2}},\\\
h_{3}(t)=&\frac{c_{2}t}{1+c_{1}c_{2}t^{2}},~{}~{}~{}h_{4}(t)=-\ln\left(1+c_{1}c_{2}t^{2}\right),\\\
h_{5}(t)=&\frac{c_{1}t^{2}}{1+c_{1}c_{2}t^{2}},~{}~{}~{}h_{6}(t)=\frac{t}{1+c_{1}c_{2}t^{2}},\\\
h_{7}(t)=&\frac{c_{1}t}{1+c_{1}c_{2}t^{2}}.\end{split}$ (59)

Substituted into Eq. (54), these solutions produce

$\displaystyle\langle{f_{L}}|{f_{R}}\rangle=$
$\displaystyle\frac{F_{L}F_{R}}{\left(1+c_{1}c_{2}t^{2}\right)^{\Omega/2}}\exp\left(\frac{t^{2}A^{*}\diamond
B}{1+c_{1}c_{2}t^{2}}\right.$
$\displaystyle\left.+\frac{\tfrac{1}{2}c_{1}t^{3}B\ominus
B}{1+c_{1}c_{2}t^{2}}+\frac{\tfrac{1}{2}c_{2}t^{3}A^{*}\ominus
A^{*}}{1+c_{1}c_{2}t^{2}}\right).$ (60)

It represents a general expression for the inner products among the different
eigenstates.

We use Eqs. (46) and (48) to represent $B_{s}(\mathbf{k})$ and
$A_{s}^{*}(\mathbf{k})$ in terms of the parameter functions for the respective
eigenstates. We use $g_{s}(\mathbf{k})$, $\bar{g}_{s}(\mathbf{k})$,
$h_{s}(\mathbf{k})$, and $\bar{h}_{s}(\mathbf{k})$ for the respective right-
eigenstates and $\gamma_{s}^{*}(\mathbf{k})$,
$\bar{\gamma}_{s}^{*}(\mathbf{k})$, $\theta_{s}^{*}(\mathbf{k})$, and
$\bar{\theta}_{s}^{*}(\mathbf{k})$ for the respective left-eigenstates to
distinguish them. The self-contractions $B\ominus B$ for the respective
parameter functions are

$\displaystyle\begin{split}B\ominus B=&2g\ominus g,\\\ B\ominus
B=&-2\bar{g}\ominus\bar{g},\\\ B\ominus B=&-2h\ominus h,\\\ B\ominus
B=&2\bar{h}\ominus\bar{h},\end{split}$ (61)

and those for $A^{*}\ominus A^{*}$ are

$\displaystyle\begin{split}A^{*}\ominus
A^{*}=&-2\gamma^{*}\ominus\gamma^{*},\\\ A^{*}\ominus
A^{*}=&2\bar{\gamma}^{*}\ominus\bar{\gamma}^{*},\\\ A^{*}\ominus
A^{*}=&2\theta^{*}\ominus\theta^{*},\\\ A^{*}\ominus
A^{*}=&-2\bar{\theta}^{*}\ominus\bar{\theta}^{*}.\end{split}$ (62)

###  V.2 Inner products with common signs

Considering those inner products for which $c_{1}=c_{2}$, we have two sets.
Those with $c_{1}=c_{2}=1$ are: $\langle{g_{L}}|{\bar{g}_{R}}\rangle$,
$\langle{\bar{h}_{L}}|{\bar{g}_{R}}\rangle$, $\langle{g_{L}}|{h_{R}}\rangle$,
and $\langle{\bar{h}_{L}}|{h_{R}}\rangle$, and for $c_{1}=c_{2}=-1$ they are:
$\langle{\bar{g}_{L}}|{g_{R}}\rangle$, $\langle{h_{L}}|{g_{R}}\rangle$,
$\langle{\bar{g}_{L}}|{\bar{h}_{R}}\rangle$, and
$\langle{h_{L}}|{\bar{h}_{R}}\rangle$. For $t=1$, they produce expressions of
the form

$\langle{f_{L}}|{f_{R}}\rangle=\frac{F_{L}F_{R}}{2^{\Omega/2}}\exp(\tfrac{1}{2}A^{*}\diamond
B\pm\tfrac{1}{4}B\ominus B\pm\tfrac{1}{4}A^{*}\ominus A^{*}),$ (63)

where the sign is given by $c_{1}=c_{2}$.

When we combine the self-contractions in Eqs. (61) and (62) with the
appropriate signs given by $c_{1}$ and $c_{2}$ we find that they are all
negative. We can therefore remove them from the exponent by defining the
prefactors as the following functionals with the positive exponentiated self-
contractions times a divergent constant to remove the remaining constant
factor:

$\displaystyle\begin{split}G_{L}[\gamma]=&2^{\Omega/4}\exp\left(\tfrac{1}{2}\gamma^{*}\ominus\gamma^{*}\right),\\\
H_{L}[\theta]=&2^{\Omega/4}\exp\left(\tfrac{1}{2}\theta^{*}\ominus\theta^{*}\right),\\\
\bar{G}_{L}[\bar{\gamma}]=&2^{\Omega/4}\exp\left(\tfrac{1}{2}\bar{\gamma}^{*}\ominus\bar{\gamma}^{*}\right),\\\
\bar{H}_{L}[\bar{\theta}]=&2^{\Omega/4}\exp\left(\tfrac{1}{2}\bar{\theta}^{*}\ominus\bar{\theta}^{*}\right),\\\
G_{R}[g]=&2^{\Omega/4}\exp\left(\tfrac{1}{2}g\ominus g\right),\\\
H_{R}[h]=&2^{\Omega/4}\exp\left(\tfrac{1}{2}h\ominus h\right),\\\
\bar{G}_{R}[\bar{g}]=&2^{\Omega/4}\exp\left(\tfrac{1}{2}\bar{g}\ominus\bar{g}\right),\\\
\bar{H}_{R}[\bar{h}]=&2^{\Omega/4}\exp\left(\tfrac{1}{2}\bar{h}\ominus\bar{h}\right).\end{split}$
(64)

With these definitions, together with the contractions $A^{*}\diamond B$ in
terms of the parameter functions in Eqs. (46) and (48), the inner products
with $c_{1}=c_{2}=1$ become

$\displaystyle\begin{split}\langle{g_{L}}|{\bar{g}_{R}}\rangle=&\exp\left(\gamma^{*}\diamond\bar{g}\right),\\\
\langle{\bar{h}_{L}}|{\bar{g}_{R}}\rangle=&\exp\left(-i\bar{\theta}^{*}\ominus\bar{g}\right),\\\
\langle{g_{L}}|{h_{R}}\rangle=&\exp\left(i\gamma^{*}\ominus h\right),\\\
\langle{\bar{h}_{L}}|{h_{R}}\rangle=&\exp\left(\bar{\theta}^{*}\diamond
h\right),\end{split}$ (65)

and those with $c_{1}=c_{2}=-1$ become

$\displaystyle\begin{split}\langle{\bar{g}_{L}}|{g_{R}}\rangle=&\exp\left(\bar{\gamma}^{*}\diamond
g\right),\\\ \langle{h_{L}}|{g_{R}}\rangle=&\exp\left(-i\theta^{*}\ominus
g\right),\\\
\langle{\bar{g}_{L}}|{\bar{h}_{R}}\rangle=&\exp\left(i\bar{\gamma}^{*}\ominus\bar{h}\right),\\\
\langle{h_{L}}|{\bar{h}_{R}}\rangle=&\exp\left(\theta^{*}\diamond\bar{h}\right).\end{split}$
(66)

If all the parameter functions are real-valued Grassmann functions, then all
these inner products produce unitary functionals of these parameter functions,
because, for example

$\displaystyle\begin{split}\left(\gamma\diamond\bar{g}\right)^{{\dagger}}=&\bar{g}\diamond\gamma=-\gamma\diamond\bar{g},\\\
\left(i\gamma\ominus h\right)^{{\dagger}}=&-ih\ominus\gamma=-i\gamma\ominus
h.\end{split}$ (67)

The modulus squares of these inner products are thus equal to 1. Hence, with
real-valued Grassmann parameter functions, the respective eigenstates in these
inner products represent mutually unbiased bases.

###  V.3 Inner products with opposite signs

The inner products where the signs in the eigenstates are opposite
($c_{1}=-c_{2}$) include all the fermionic self-overlaps
$\langle{g_{L}}|{g_{R}}\rangle$, $\langle{\bar{g}_{L}}|{\bar{g}_{R}}\rangle$,
$\langle{h_{L}}|{h_{R}}\rangle$, and
$\langle{\bar{h}_{L}}|{\bar{h}_{R}}\rangle$, and the four remaining inner
products $\langle{\bar{h}_{L}}|{g_{R}}\rangle$,
$\langle{h_{L}}|{\bar{g}_{R}}\rangle$, $\langle{\bar{g}_{L}}|{h_{R}}\rangle$
and $\langle{g_{L}}|{\bar{h}_{R}}\rangle$. With opposite signs for $c_{1}$ and
$c_{2}$, the expression in Eq. (60) becomes

$\displaystyle\langle{f_{L}}|{f_{R}}\rangle=$
$\displaystyle\frac{F_{L}F_{R}}{\left(1-t^{2}\right)^{\Omega/2}}\exp\left(\frac{t^{2}A^{*}\diamond
B}{1-t^{2}}\right.$ $\displaystyle\left.+\frac{c_{1}}{2}\frac{t^{3}B\ominus
B}{1-t^{2}}+\frac{c_{2}}{2}\frac{t^{3}A^{*}\ominus A^{*}}{1-t^{2}}\right).$
(68)

We can’t set $t=1$, because the expression is singular at $t=1$. Therefore, we
set $t=1-\epsilon$ and then consider the limit $\epsilon\rightarrow 0$. The
$t$-dependent factors become

$\displaystyle\begin{split}\frac{1}{1-t^{2}}\rightarrow&\
\frac{1}{2\epsilon-\epsilon^{2}}\approx\frac{1}{2\epsilon},\\\
\frac{t^{2}}{1-t^{2}}\rightarrow&\
\frac{1}{2\epsilon}-1+\frac{1}{2(2-\epsilon)}\approx\frac{1}{2\epsilon}-\frac{3}{4},\\\
\frac{t^{3}}{1-t^{2}}\rightarrow&\
\frac{1}{2\epsilon}-1-\frac{1}{2(2-\epsilon)}+\epsilon\approx\frac{1}{2\epsilon}-\frac{5}{4}.\end{split}$
(69)

The general expression for the inner product becomes

$\displaystyle\langle{f_{L}}|{f_{R}}\rangle=$
$\displaystyle\lim_{\epsilon\rightarrow
0}\frac{F_{L}F_{R}}{(2\epsilon)^{\Omega/2}}\exp\left[\left(\frac{1}{2\epsilon}-\frac{3}{4}\right)A^{*}\diamond
B\right.$
$\displaystyle+\frac{c_{1}}{2}\left(\frac{1}{2\epsilon}-\frac{5}{4}\right)B\ominus
B$
$\displaystyle\left.+\frac{c_{2}}{2}\left(\frac{1}{2\epsilon}-\frac{5}{4}\right)A^{*}\ominus
A^{*}\right].$ (70)

We initially retain the subleading order in $\epsilon$ in the exponent,
because a part of the exponent is cancelled by the exponentiated factors in
the prefactors in Eq. (64). While the combination of the self-contractions in
Eqs. (61) and (62) with the appropriate signs given by $c_{1}$ and $c_{2}$ for
the current inner product make them all positive, the subleading order parts
of the $t$-dependent factors are negative, leading to the cancellations. After
the cancellations, we can remove the remaining subleading order terms. The
appropriate expressions for the contractions $A^{*}\diamond B$ are obtained
from those in Eqs. (46) and (48).

The resulting expressions for these eight inner products all have the form

$\langle{f_{L}}|{f_{R}}\rangle=p[f]\triangleq\lim_{\epsilon\rightarrow
0}\frac{1}{\epsilon^{\Omega/2}}\exp\left(\frac{1}{2\epsilon}f\ominus
f\right).$ (71)

For the fermionic self-overlaps, we get

$\displaystyle\begin{split}\langle{g_{L}}|{g_{R}}\rangle=&p\left[\gamma^{*}-g\right],\\\
\langle{\bar{g}_{L}}|{\bar{g}_{R}}\rangle=&p\left[\bar{\gamma}^{*}-\bar{g}\right],\\\
\langle{h_{L}}|{h_{R}}\rangle=&p\left[\theta^{*}-h\right],\\\
\langle{\bar{h}_{L}}|{\bar{h}_{R}}\rangle=&p\left[\bar{\theta}^{*}-\bar{h}\right].\end{split}$
(72)

The expressions for the remaining four inner products are

$\displaystyle\begin{split}\langle{\bar{h}_{L}}|{g_{R}}\rangle=&p\left[\bar{\theta}^{*}+i\varepsilon\cdot
g\right],\\\
\langle{\bar{g}_{L}}|{h_{R}}\rangle=&p\left[\bar{\gamma}^{*}-i\varepsilon\cdot
h\right],\\\
\langle{g_{L}}|{\bar{h}_{R}}\rangle=&p\left[\gamma^{*}-i\varepsilon\cdot\bar{h}\right],\\\
\langle{h_{L}}|{\bar{g}_{R}}\rangle=&p\left[\theta^{*}+i\varepsilon\cdot\bar{g}\right].\end{split}$
(73)

Note that, for the contraction $f\ominus f$, any spin transformation in the
first $f$ needs to be converted to $\varepsilon\cdot
g\rightarrow-g\cdot\varepsilon$.

##  VI Orthogonality

The general expression for the inner product in Eq. (71) resembles that for
the Dirac delta functional in the bosonic case Roux (2018). It is tempting to
conclude that these inner products produce Grassmann Dirac delta functionals.
However, to be a Dirac delta functional, the inner product must satisfy

$\int W[f]\langle{f}|{f^{\prime}}\rangle\mathcal{D}[f]=\Lambda W[f^{\prime}],$
(74)

where $W[f]$ is an arbitrary functional of $f$, and $\Lambda$ is an
orthogonality constant. In Appen. D, we show that it is indeed the case.
Therefore, Eq. (71) represents an orthogonality condition. Formally, we’ll
represent it as

$\displaystyle\langle{f}|{f^{\prime}}\rangle=$
$\displaystyle\lim_{\epsilon\rightarrow
0}\frac{1}{\epsilon^{\Omega/2}}\exp\left[\frac{1}{2\epsilon}(f-f^{\prime})\ominus(f-f^{\prime})\right]$
$\displaystyle\triangleq$ $\displaystyle\Lambda\delta[f-f^{\prime}],$ (75)

where $\delta[f-f^{\prime}]$ is a Grassmann Dirac delta functional.

The inner products associated with the fermionic self-overlaps are then
represented as

$\displaystyle\begin{split}\langle{g_{L}}|{g_{R}}\rangle=&\Lambda\delta[\gamma^{*}-g],\\\
\langle{\bar{g}_{L}}|{\bar{g}_{R}}\rangle=&\Lambda\delta[\bar{\gamma}^{*}-\bar{g}],\\\
\langle{h_{L}}|{h_{R}}\rangle=&\Lambda\delta[\theta^{*}-h],\\\
\langle{\bar{h}_{L}}|{\bar{h}_{R}}\rangle=&\Lambda\delta[\bar{\theta}^{*}-\bar{h}].\end{split}$
(76)

Here, the arguments of the Dirac delta functional contain the differences
between one parameter function and the complex conjugate of the other. It
suggests that these parameter functions should be real valued.

To represent the orthogonality conditions for all the different eigenstates in
terms of real-valued Grassmann parameter functions, we modify the notation so
that we can distinguish between the type of eigenstate and the parameter
function. The real-valued Grassmann parameter function, represent by $q$’s or
$p$’s, are shown as a functional dependence in the state. The expressions for
the inner products that produce the orthogonality conditions are then
represented as

$\displaystyle\begin{split}\left\langle\left.{g_{L}[q_{2}]}\right|{g_{R}[q_{1}]}\right\rangle=&\Lambda\delta[q_{1}-q_{2}],\\\
\left\langle\left.{\bar{g}_{L}[p_{4}]}\right|{\bar{g}_{R}[p_{3}]}\right\rangle=&\Lambda\delta[p_{3}-p_{4}],\\\
\left\langle\left.{h_{L}[p_{2}]}\right|{h_{R}[p_{1}]}\right\rangle=&\Lambda\delta[p_{1}-p_{2}],\\\
\left\langle\left.{\bar{h}_{L}[q_{4}]}\right|{\bar{h}_{R}[q_{3}]}\right\rangle=&\Lambda\delta[q_{3}-q_{4}],\\\
\left\langle\left.{g_{L}[q_{2}]}\right|{\bar{h}_{R}[q_{3}]}\right\rangle=&\Lambda\delta[q_{4}-i\varepsilon\cdot
q_{2}],\\\
\left\langle\left.{\bar{g}_{L}[p_{4}]}\right|{h_{R}[p_{1}]}\right\rangle=&\Lambda\delta[p_{1}-i\varepsilon\cdot
p_{3}],\\\
\left\langle\left.{\bar{h}_{L}[q_{4}]}\right|{g_{R}[q_{1}]}\right\rangle=&\Lambda\delta[q_{1}+i\varepsilon\cdot
q_{3}],\\\
\left\langle\left.{h_{L}[p_{2}]}\right|{\bar{g}_{R}[p_{3}]}\right\rangle=&\Lambda\delta[p_{4}+i\varepsilon\cdot
p_{2}].\end{split}$ (77)

They separate into two groups: those inner products involving the $p$’s and
those involving the $q$’s.

In terms of the new notation, the inner products in Eq. (65) become

$\displaystyle\begin{split}\langle{g_{L}[q_{2}]}|{\bar{g}_{R}[p_{3}]}\rangle=&\exp\left(q_{2}\diamond
p_{3}\right),\\\
\langle{\bar{g}_{L}[p_{4}]}|{g_{R}[q_{1}]}\rangle=&\exp\left(p_{4}\diamond
q_{1}\right),\\\
\langle{h_{L}[p_{2}]}|{\bar{h}_{R}[q_{3}]}\rangle=&\exp\left(p_{2}\diamond
q_{3}\right),\\\
\langle{\bar{h}_{L}[q_{4}]}|{h_{R}[p_{1}]}\rangle=&\exp\left(q_{4}\diamond
p_{1}\right),\\\
\langle{h_{L}[p_{2}]}|{g_{R}[q_{1}]}\rangle=&\exp\left(-ip_{2}\ominus
q_{1}\right),\\\
\langle{\bar{h}_{L}[q_{4}]}|{\bar{g}_{R}[p_{3}]}\rangle=&\exp\left(-iq_{4}\ominus
p_{3}\right),\\\
\langle{g_{L}[q_{2}]}|{h_{R}[p_{1}]}\rangle=&\exp\left(iq_{2}\ominus
p_{1}\right),\\\
\langle{\bar{g}_{L}[p_{4}]}|{\bar{h}_{R}[q_{3}]}\rangle=&\exp\left(ip_{4}\ominus
q_{3}\right).\end{split}$ (78)

All these inner products produce contractions between $p$’s and $q$’s. It
follows form Eq. (67) that the eigenstates parameterized in terms of the $q$’s
represent bases that are mutually unbiased with respect to the bases composed
of eigenstate parameterized in terms of the $p$’s.

We also see that

$\displaystyle\begin{split}\left|{g_{R}[i\varepsilon\cdot
q]}\right\rangle=&\left|{\bar{h}_{R}[q]}\right\rangle,~{}~{}~{}\left|{\bar{h}_{R}[-i\varepsilon\cdot
q]}\right\rangle=\left|{g_{R}[q]}\right\rangle,\\\
\left|{\bar{g}_{R}[i\varepsilon\cdot
p]}\right\rangle=&\left|{h_{R}[p]}\right\rangle,~{}~{}~{}\left|{h_{R}[-i\varepsilon\cdot
p]}\right\rangle=\left|{\bar{g}_{R}[p]}\right\rangle,\\\
\left\langle{g_{L}[-iq\cdot\varepsilon]}\right|=&\left\langle{\bar{h}_{L}[q]}\right|,~{}~{}~{}\left\langle{\bar{h}_{L}[iq\cdot\varepsilon]}\right|=\left\langle{g_{L}[q]}\right|,\\\
\left\langle{\bar{g}_{L}[-ip\cdot\varepsilon]}\right|=&\left\langle{h_{L}[p]}\right|,~{}~{}~{}\left\langle{h_{L}[ip\cdot\varepsilon]}\right|=\left\langle{\bar{g}_{L}[p]}\right|.\end{split}$
(79)

Since $i\varepsilon$ is a real valued matrix, the spin-transformed real valued
parameter functions remain real valued. Therefore, these relationships show
that we only need to consider half of the eigenstates. The other half is
directly related to them via spin transformations of the parameter functions.

Henceforth, we use a reduced set of eigenstates, thanks to the relationships
in Eq. (79). The four eigenstates that we select are those represented by
$g$’s. They are related via Hermitian and fermionic adjoints, as shown in Eqs.
(49) and (50).

##  VII Completeness

Based on the form of the orthogonality conditions — the first two expressions
in Eq. (77) — the expected forms of the associated completeness conditions are

$\displaystyle\int\left|{g_{R}[q]}\right\rangle\left\langle{g_{L}[q]}\right|\
\mathcal{D}_{\Lambda}[q]=$ $\displaystyle\mathds{1},$ (80)
$\displaystyle\int\left|{\bar{g}_{R}[p]}\right\rangle\left\langle{\bar{g}_{L}[p]}\right|\
\mathcal{D}_{\Lambda}[p]=$ $\displaystyle\mathds{1}.$ (81)

We absorb a factor of $\Lambda^{-1}$ into the Grassmann functional integration
measure $\mathcal{D}_{\Lambda}[q]=\Lambda^{-1}\mathcal{D}[q]$. The identity
operator for the Hilbert space is denoted by $\mathds{1}$.

To test whether Eqs. (80) and (81) represent valid resolutions of the
identity, we overlap them by two arbitrary eigenstates and see whether the
result gives the inner product between these eigenstates. For Eq. (80), the
four inner products among the eigenstates give

$\displaystyle\left\langle{g_{L}[q_{2}]}\right|\mathds{1}\left|{g_{R}[q_{1}]}\right\rangle=$
$\displaystyle\int\left\langle\left.{g_{L}[q_{2}]}\right|{g_{R}[q]}\right\rangle\left\langle\left.{g_{L}[q]}\right|{g_{R}[q_{1}]}\right\rangle\
\mathcal{D}_{\Lambda}[q]=\Lambda^{2}\int\delta[q-q_{2}]\delta[q_{1}-q]\
\mathcal{D}_{\Lambda}[q]=\Lambda\delta[q_{1}-q_{2}],$ (82)
$\displaystyle\left\langle{g_{L}[q_{2}]}\right|\mathds{1}\left|{\bar{g}_{R}[p_{1}]}\right\rangle=$
$\displaystyle\int\left\langle\left.{g_{L}[q_{2}]}\right|{g_{R}[q]}\right\rangle\left\langle\left.{g_{L}[q]}\right|{\bar{g}_{R}[p_{1}]}\right\rangle\
\mathcal{D}_{\Lambda}[q]=\Lambda\int\delta[q-q_{2}]\exp\left(q\diamond
p_{1}\right)\ \mathcal{D}_{\Lambda}[q]=\exp\left(q_{2}\diamond p_{1}\right),$
(83)
$\displaystyle\left\langle{\bar{g}_{L}[p_{2}]}\right|\mathds{1}\left|{g_{R}[q_{1}]}\right\rangle=$
$\displaystyle\int\left\langle\left.{\bar{g}_{L}[p_{2}]}\right|{g_{R}[q]}\right\rangle\left\langle\left.{g_{L}[q]}\right|{g_{R}[q_{1}]}\right\rangle\
\mathcal{D}_{\Lambda}[q]=\Lambda\int\exp\left(p_{2}\diamond
q\right)\delta[q_{1}-q]\ \mathcal{D}_{\Lambda}[q]=\exp\left(p_{2}\diamond
q_{1}\right),$ (84)
$\displaystyle\left\langle{\bar{g}_{L}[p_{2}]}\right|\mathds{1}\left|{\bar{g}_{R}[p_{1}]}\right\rangle=$
$\displaystyle\int\left\langle\left.{\bar{g}_{L}[p_{2}]}\right|{g_{R}[q]}\right\rangle\left\langle\left.{g_{L}[q]}\right|{\bar{g}_{R}[p_{1}]}\right\rangle\
\mathcal{D}_{\Lambda}[q]=\int\exp\left(p_{2}\diamond
q\right)\exp\left(q\diamond p_{1}\right)\
\mathcal{D}_{\Lambda}[q]=\Lambda\delta[p_{1}-p_{2}].$ (85)

The Grassmann Dirac delta functionals alleviate the evaluation of the
integrations in Eqs. (82)-(84). They produce the expected results, thanks to
the orthogonality conditions. We conclude that Eq. (80) is valid and that the
eigenstates in terms of which the identity is resolved represent a complete
basis. The expression in Eq. (85) then follows as a definition for the
Grassmann Dirac delta functional in terms of a Grassmann functional integral.
The inner products obtained from Eq. (81) follow the same pattern and
therefore indicates that those eigenstates also represent a complete basis.

##  VIII Grassmann Fourier analysis

###  VIII.1 Grassmann Dirac delta functionals

The result in Eq. (85) can be used as a definition of the Grassmann Dirac
delta functional. A similar expression is obtained from Eq. (81). The two
definitions produced by Eqs. (80) and (81) can be represented as

$\displaystyle\begin{split}\int\exp(q\diamond p)\
\mathcal{D}_{\Lambda}[p]=&\Lambda\delta[q],\\\ \int\exp(q\diamond p)\
\mathcal{D}_{\Lambda}[q]=&\Lambda\delta[p].\end{split}$ (86)

If we redefine the field variables in the first expression by interchanging
$p\leftrightarrow q$, and change the order in the exponent, we get

$\int\exp(-q\diamond p)\ \mathcal{D}_{\Lambda}[q]=\Lambda\delta[p].$ (87)

It shows that the Grassmann Dirac delta functional is symmetric, just like its
bosonic counterpart.

###  VIII.2 Grassmann functional Fourier transform

We can now use the above to represent functionals of the Grassmann functional
phase space in terms of Grassmann functional Fourier integrals. The Fourier
transform of a functional $W[q]$ is given by

$\tilde{W}[p]=\int W[q]\exp(p\diamond q)\ \mathcal{D}_{\Lambda}[q].$ (88)

The inverse Fourier transform then gives

$\displaystyle\int\tilde{W}[p]\exp(q^{\prime}\diamond p)\
\mathcal{D}_{\Lambda}[p]$ $\displaystyle=$ $\displaystyle\int
W[q]\exp(p\diamond q)\exp(q^{\prime}\diamond p)\ \mathcal{D}_{\Lambda}[p,q]$
$\displaystyle=$ $\displaystyle\int W[q]\Lambda\delta[q-q^{\prime}]\
\mathcal{D}_{\Lambda}[q]$ $\displaystyle=$ $\displaystyle\int
W[q]\delta[q-q^{\prime}]\ \mathcal{D}[q]=W[q^{\prime}].$ (89)

Note that the $\Lambda$’s cancel as required.

In summary, the Grassmann functional Fourier transform and its inverse are
given by

$\displaystyle\begin{split}\tilde{W}[p]=&\int W[q]\exp(p\diamond q)\
\mathcal{D}_{\Lambda}[q],\\\ W[q]=&\int\tilde{W}[p]\exp(q\diamond p)\
\mathcal{D}_{\Lambda}[p].\end{split}$ (90)

The two expressions are exactly symmetric — one is obtained from the other by
interchanging $p\leftrightarrow q$.

##  IX Fermionic Wigner functionals

Everything is now in place to provide a representation of fermionic states in
terms of Wigner functionals defined on a Grassmann functional phase space.
First, we simplify the notation.

The quadrature bases in terms of which the Wigner functionals are defined, are
those represented by $g$. These eigenstates are now simply denoted by their
parameter functions. Incorporating the prefactors defined in Eq. (64), we have
the following explicit expressions for these eigenstates

$\displaystyle\begin{split}\left|{q}\right\rangle\triangleq&\left|{g_{R}[q]}\right\rangle\\\
=&\mathcal{N}\exp\left(\tfrac{1}{2}q\ominus
q+\sqrt{2}\hat{a}^{{\dagger}}\diamond
q-\hat{R}^{{\dagger}}\right)\left|{\text{vac}}\right\rangle,\\\
\left\langle{q}\right|\triangleq&\left\langle{g_{L}[q]}\right|\\\
=&\mathcal{N}\left\langle{\text{vac}}\right|\exp\left(\tfrac{1}{2}q\ominus
q-\sqrt{2}q\ominus\hat{a}+\hat{R}\right),\\\
\left|{p}\right\rangle\triangleq&\left|{\bar{g}_{R}[p]}\right\rangle\\\
=&\mathcal{N}\exp\left(\tfrac{1}{2}p\ominus
p-\sqrt{2}\hat{a}^{{\dagger}}\ominus
p+\hat{R}^{{\dagger}}\right)\left|{\text{vac}}\right\rangle,\\\
\left\langle{p}\right|\triangleq&\left\langle{\bar{g}_{L}[p]}\right|\\\
=&\mathcal{N}\left\langle{\text{vac}}\right|\exp\left(\tfrac{1}{2}p\ominus
p+\sqrt{2}p\diamond\hat{a}-\hat{R}\right),\end{split}$ (91)

where $\mathcal{N}=2^{\Omega/4}$.

The inner products among these bases elements are

$\displaystyle\begin{split}\left\langle\left.{q}\right|{q^{\prime}}\right\rangle=&\Lambda\delta[q-q^{\prime}],\\\
\left\langle\left.{p}\right|{p^{\prime}}\right\rangle=&\Lambda\delta[p-p^{\prime}],\\\
\left\langle\left.{q}\right|{p}\right\rangle=&\exp\left(q\diamond p\right),\\\
\left\langle\left.{p}\right|{q}\right\rangle=&\exp\left(p\diamond
q\right).\end{split}$ (92)

where $\Lambda$ is a divergent orthogonality constant. The first two represent
orthogonality conditions and the last two indicate that the bases are mutually
unbiased: we have $|\langle{q}|{p}\rangle|^{2}=1$ because $p\diamond
q=-q\diamond p$. These bases resolve the identity as follows

$\displaystyle\begin{split}\mathds{1}=&\int\left|{q}\right\rangle\left\langle{q}\right|\
\mathcal{D}_{\Lambda}[q],\\\
\mathds{1}=&\int\left|{p}\right\rangle\left\langle{p}\right|\
\mathcal{D}_{\Lambda}[p],\end{split}$ (93)

which shows that they are complete bases. The dual states are given by the
fermionic adjoints

$\left\langle{q}\right|=\left(\left|{q}\right\rangle\right)^{{\ddagger}},~{}~{}~{}\left\langle{p}\right|=\left(\left|{p}\right\rangle\right)^{{\ddagger}}.$
(94)

For $q=p$, two mutually unbiased bases are related via the Hermitian adjoints:

$\left\langle{p}\right|=\left(\left|{q}\right\rangle\right)^{{\dagger}},~{}~{}~{}\left\langle{q}\right|=\left(\left|{p}\right\rangle\right)^{{\dagger}},$
(95)

which does not have an equivalent in the bosonic case.

The Grassmann functional Fourier transform and its inverse are given by Eq.
(90). The Grassmann Dirac delta functional is obtained from

$\displaystyle\begin{split}\int\exp(\pm q\diamond p)\
\mathcal{D}_{\Lambda}[q]=&\Lambda\delta[p],\\\ \int\exp(\pm q\diamond p)\
\mathcal{D}_{\Lambda}[p]=&\Lambda\delta[q].\end{split}$ (96)

###  IX.1 Functional operator identities

The fermionic quadrature operators are now identified as the fermionic
Bogoliubov operators

$\displaystyle\begin{split}\hat{q}_{s}(\mathbf{k})\triangleq&\frac{1}{\sqrt{2}}\left[\varepsilon_{s,r}\hat{a}_{r}^{{\dagger}}(\mathbf{k})+\hat{a}_{s}(\mathbf{k})\right]=\hat{g}_{s}(\mathbf{k}),\\\
\hat{p}_{s}(\mathbf{k})\triangleq&\frac{1}{\sqrt{2}}\left[\hat{a}_{s}^{{\dagger}}(\mathbf{k})-\varepsilon_{s,r}\hat{a}_{r}(\mathbf{k})\right]=\hat{g}_{s}^{{\dagger}}(\mathbf{k}).\end{split}$
(97)

They are related by Hermitian conjugation as indicated by the $g$-definitions.
These operators can be represented in terms of their eigenstates as

$\displaystyle\begin{split}\hat{q}_{s}(\mathbf{k})=&\int\left|{q}\right\rangle
q_{s}(\mathbf{k})\left\langle{q}\right|\ \mathcal{D}_{\Lambda}[q],\\\
\hat{p}_{s}(\mathbf{k})=&\int\left|{p}\right\rangle
p_{s}(\mathbf{k})\left\langle{p}\right|\ \mathcal{D}_{\Lambda}[p],\\\
\end{split}$ (98)

By operating with the identity operators resolved in the opposite bases, the
elements of the different bases can be represented in terms of each other

$\displaystyle\begin{split}\left|{q}\right\rangle=\mathds{1}\left|{q}\right\rangle=&\int\left|{p}\right\rangle\exp\left(p\diamond
q\right)\ \mathcal{D}_{\Lambda}[p],\\\
\left|{p}\right\rangle=\mathds{1}\left|{p}\right\rangle=&\int\left|{q}\right\rangle\exp\left(q\diamond
p\right)\ \mathcal{D}_{\Lambda}[q].\end{split}$ (99)

###  IX.2 Definition of the Wigner functional

In analogy with the bosonic definition, together with the knowledge of the
fermionic quadrature bases, we can now define the fermionic Wigner functional
as

$\displaystyle W[q,p]\triangleq$
$\displaystyle\int\left\langle{q+\tfrac{1}{2}x}\right|\hat{\rho}\left|{q-\tfrac{1}{2}x}\right\rangle$
$\displaystyle\times\exp\left(p\diamond x\right)\ \mathcal{D}_{\Lambda}[x],$
(100)

where $\hat{\rho}$ (nominally a density operator) can represent any operator
defined on the Hilbert space of all fermionic states incorporating all the
spin, spatiotemporal and particle-number degrees of freedom. The density
operator $\hat{\rho}$ can be represented as a density ‘matrix’, which we refer
to as a density functional

$\rho[q_{1},q_{2}]\triangleq\left\langle{q_{1}}\right|\hat{\rho}\left|{q_{2}}\right\rangle.$
(101)

In the case of pure states, the density functional becomes a product of wave
functionals

$\rho[q_{1},q_{2}]=\psi[q_{1}]\psi^{*}[q_{2}],$ (102)

where $\psi[q]=\langle{q}|{\psi}\rangle$.

###  IX.3 Quadrature and ladder operators

The simplest cases of Wigner functionals are those for the quadrature
operators. Using the eigen-equation for the quadrature operator
$\hat{q}_{s}(\mathbf{k})$, we obtain

$\displaystyle W_{\hat{q}}[q,p]=$
$\displaystyle\int\left\langle{q+\tfrac{1}{2}x}\right|\hat{q}_{s}(\mathbf{k})\left|{q-\tfrac{1}{2}x}\right\rangle\exp\left(p\diamond
x\right)\ \mathcal{D}_{\Lambda}[x]$ $\displaystyle=$
$\displaystyle\int\left\langle\left.{q+\tfrac{1}{2}x}\right|{q-\tfrac{1}{2}x}\right\rangle\left[q_{s}(\mathbf{k})-\tfrac{1}{2}x_{s}(\mathbf{k})\right]$
$\displaystyle\times\exp\left(p\diamond x\right)\ \mathcal{D}_{\Lambda}[x]$
$\displaystyle=$
$\displaystyle\int\Lambda\delta[x][q_{s}(\mathbf{k})-\tfrac{1}{2}x_{s}(\mathbf{k})]\exp\left(p\diamond
x\right)\ \mathcal{D}_{\Lambda}[x]$ $\displaystyle=$ $\displaystyle
q_{s}(\mathbf{k}).$ (103)

For the Wigner functional of $\hat{p}_{s}(\mathbf{k})$, we represent it as in
Eq. (98). Then

$\displaystyle W_{\hat{p}}[q,p]=$
$\displaystyle\int\left\langle{q+\tfrac{1}{2}x}\right|\hat{p}_{s}(\mathbf{k})\left|{q-\tfrac{1}{2}x}\right\rangle\exp\left(p\diamond
x\right)\ \mathcal{D}_{\Lambda}[x]$ $\displaystyle=$
$\displaystyle\int\left\langle\left.{q+\tfrac{1}{2}x}\right|{p^{\prime}}\right\rangle
p_{s}^{\prime}(\mathbf{k})\left\langle{p^{\prime}}\left|{q-\tfrac{1}{2}x}\right.\right\rangle$
$\displaystyle\times\exp\left(p\diamond x\right)\
\mathcal{D}_{\Lambda}[p^{\prime},x]$ $\displaystyle=$ $\displaystyle\int
p_{s}^{\prime}(\mathbf{k})\exp\left[x\diamond(p^{\prime}-p)\right]\
\mathcal{D}_{\Lambda}[p^{\prime},x]$ $\displaystyle=$ $\displaystyle\int
p_{s}^{\prime}(\mathbf{k})\Lambda\delta[p^{\prime}-p]\
\mathcal{D}_{\Lambda}[p^{\prime}]=p_{s}(\mathbf{k}).$ (104)

where we used Eqs. (92) and (96). Note that the order of $x$ and $p$ in the
definition of the Wigner functional needs to be as given, otherwise we will
get the negative of what we have here.

The Wigner functionals of the ladder operators follow directly from these two
results since they are linear combinations of the quadrature operators, as
given by

$\displaystyle\begin{split}\hat{a}_{s}(\mathbf{k})=&\frac{1}{\sqrt{2}}\left[\hat{q}_{s}(\mathbf{k})-\varepsilon_{s,r}\hat{p}_{r}(\mathbf{k})\right],\\\
\hat{a}_{s}^{{\dagger}}(\mathbf{k})=&\frac{1}{\sqrt{2}}\left[\varepsilon_{s,r}\hat{q}_{r}(\mathbf{k})+\hat{p}_{s}(\mathbf{k})\right].\end{split}$
(105)

The Wigner functionals of linear combinations of operators are the linear
combinations of the Wigner functionals of those operators. Hence,

$\displaystyle\begin{split}W_{\hat{a}}[q,p]=&\frac{1}{\sqrt{2}}\left[q_{s}(\mathbf{k})-\varepsilon_{s,r}p_{r}(\mathbf{k})\right],\\\
W_{\hat{a}^{{\dagger}}}[q,p]=&\frac{1}{\sqrt{2}}\left[\varepsilon_{s,r}q_{r}(\mathbf{k})+p_{s}(\mathbf{k})\right].\end{split}$
(106)

###  IX.4 Weyl transformations

The Weyl transformation represents the inverse process whereby the fermion
density functional is reproduced from the Grassmann Wigner functional

$\displaystyle\rho[q_{1},q_{2}]=$ $\displaystyle\int
W\left[\tfrac{1}{2}q_{1}+\tfrac{1}{2}q_{2},p\right]$
$\displaystyle\times\exp[(q_{1}-q_{2})\diamond p]\ \mathcal{D}_{\Lambda}[p].$
(107)

The Fourier transform of Eq. (107) with respect to $x=q_{1}-q_{2}$ reproduces
the expression for the Grassmann Wigner functional in Eq. (100). The Weyl
transformations can also be used to retrieve the density operator

$\displaystyle\hat{\rho}=$ $\displaystyle\int\left|{q_{1}}\right\rangle
W\left[\tfrac{1}{2}q_{1}+\tfrac{1}{2}q_{2},p\right]$
$\displaystyle\times\exp[p\diamond(q_{1}-q_{2})]\left\langle{q_{2}}\right|\
\mathcal{D}_{\Lambda}[p,q_{1},q_{2}]$ $\displaystyle=$
$\displaystyle\int\left|{q+\tfrac{1}{2}x}\right\rangle W[q,p]$
$\displaystyle\times\exp(p\diamond x)\left\langle{q-\tfrac{1}{2}x}\right|\
\mathcal{D}_{\Lambda}[p,q,x].$ (108)

It follows that the trace of an operator is given by

$\text{tr}\\{\hat{A}\\}=\int W_{\hat{A}}[q,p]\ \mathcal{D}_{\Lambda}[q,p].$
(109)

The normalization of a density operator thus implies

$\text{tr}\\{\hat{\rho}\\}=\int W_{\hat{\rho}}[q,p]\
\mathcal{D}_{\Lambda}[q,p]=1.$ (110)

###  IX.5 Characteristic functional

The characteristic functional is the Grassmann functional Fourier transform of
the Wigner functional with respect to both $q$ and $p$. It reads

$\chi[\xi,\zeta]=\int W[q,p]\exp(\xi\diamond q-p\diamond\zeta)\
\mathcal{D}_{\Lambda}[q,p].$ (111)

The Wigner functional is recovered from the characteristic functional via the
inverse Grassmann functional Fourier transform

$W[q,p]=\int\chi[\xi,\zeta]\exp(p\diamond\zeta-\xi\diamond q)\
\mathcal{D}_{\Lambda}[\zeta,\xi].$ (112)

###  IX.6 Grassmann star-products

The way Grassmann Wigner functionals are combined to represent the products of
operators leads to the definition of the Grassmann star-product or Grassmann
Moyal product. In Appen. E, we derive integral expressions for the star-
products of two and three Wigner functionals, respectively. For two Wigner
functionals, it reads

$\displaystyle W_{\hat{A}\hat{B}}=$ $\displaystyle\frac{1}{2^{2\Omega}}\int
W_{\hat{A}}\left[q_{1},p_{1}\right]W_{\hat{B}}\left[q_{2},p_{2}\right]$
$\displaystyle\times\exp\left[2(q-q_{2})\diamond p_{1}+2(q_{1}-q)\diamond
p_{2}\right.$ $\displaystyle\left.+2(q_{2}-q_{1})\diamond p\right]\
\mathcal{D}_{\Lambda}[q_{1},p_{1},q_{2},p_{2}].$ (113)

The star-product of three Wigner functionals is

$\displaystyle W_{\hat{A}\hat{B}\hat{C}}=$ $\displaystyle\int
W_{\hat{A}}\left[\frac{q+q_{a}+q_{b}}{2},\frac{p+p_{a}+p_{b}}{2}\right]W_{\hat{B}}\left[q_{a},p_{a}\right]$
$\displaystyle\times
W_{\hat{C}}\left[\frac{q+q_{a}-q_{b}}{2},\frac{p+p_{a}-p_{b}}{2}\right]$
$\displaystyle\times\exp\left[(q-q_{a})\diamond
p_{b}-q_{b}\diamond(p-p_{a})\right]$
$\displaystyle\times\mathcal{D}_{\Lambda}[q_{a},p_{a},q_{b},p_{b}].$ (114)

##  X Conclusion

A Wigner functional formalism is developed for fermions. It is defined on a
Grassmann functional phase space. Much of the development involves the
identification of suitable quadrature bases. We showed that the eigenstates of
the Majorana operators are not suitable for this purpose. Instead, we found
that a set of fermionic Bogoliubov operators behaves in a way that is
analogues to the bosonic quadrature operators, provided that the Hermitian
adjoint is replaced by a so-called fermionic adjoint, which incorporates a
spin transformation.

The eigenstates of the fermionic Bogoliubov operators are obtained with the
aid of a closed algebra of bosonized operators, constructed from the fermion
ladder operators and Grassmann spectral functions. The resulting eigenstates
are respectively associated with Grassmann eigenvalue functions carrying spin
and spatiotemporal degrees of freedom. These eigenvalue functions eventually
serve as the field variables of the Grassmann functional phase space. The
justification for identifying the eigenstates of these fermionic Bogoliubov
operators as suitable quadrature bases comes from a thorough calculation of
all the inner products among these eigenstates and by showing that these
eigenstates are complete orthogonal bases.

The resulting fermionic quadrature bases are similar to their bosonic
counterparts, but with some differences. The most important difference is the
fact that the dual space is not given by the Hermitian adjoint, but by the
fermionic adjoint. The eigenstates do not satisfy orthogonality relationships
with their Hermitian adjoints. However, the left and right eigenstates of the
operators, which are related via the fermionic adjoint, do satisfy
orthogonality relationships provided that the parameter functions are real
valued. The Hermitian adjoint (without the spin transformation) converts the
kets of one of the quadrature bases into the bras of the other quadrature
basis. It thus relates the two mutually unbiased bases.

The inner products that lead to orthogonality relationships also produce a
divergent orthogonality constant, which we denote by $\Lambda$. The exact
nature of $\Lambda$ is not yet revealed in this analysis. In the functional
integrals, it becomes a normalization constant $\Lambda^{-1}$, which we
incorporate into the functional integration measure, where it needs to cancel
any $\Lambda$ that would accompany a Dirac delta functional produced by an
inner product. In calculations of physical quantities, all $\Lambda$’s must
cancel.

The completeness relations for the fermionic quadrature bases, together with
their orthogonality, leads to a Fourier relationship between the two mutually
unbiased bases. It allows for the definition of a functional Fourier
transformation that transforms functionals of field variables associated with
one quadrature basis into those associated with the other quadrature basis.

The Grassmann Wigner functional is then defined with the aid of the quadrature
basis and the functional Fourier transform. It can represent any operator
defined on the Hilbert space of the fermions. A Grassmann Weyl transformation
is defined, which can convert a Grassmann Wigner functional back into an
operator. The Grassmann characteristic functional is given by the functional
Fourier transform of a Grassmann Wigner functional with respect to both field
variables. We also derive functional integral expressions for the star product
that combine Wigner functionals of operators to obtain the Wigner functional
for the product of these operators.

##  Appendix A Commutation relations

The complete algebra of bosonized operators obeys the following commutation
relations. Those among spectral operators are

$\displaystyle\begin{split}[\hat{A},\hat{B}^{{\dagger}}]=&A^{*}\diamond
B,~{}~{}~{}[\hat{A}_{\varepsilon},\hat{B}^{{\dagger}}]=A\ominus B,\\\
[\hat{A},\hat{B}_{\varepsilon}^{{\dagger}}]=&A^{*}\ominus
B^{*},~{}~{}~{}[\hat{A}_{\varepsilon},\hat{B}_{\varepsilon}^{{\dagger}}]=-B^{*}\diamond
A.\end{split}$ (115)

Those including $\hat{R}$ and $\hat{R}^{{\dagger}}$, are

$\displaystyle\begin{split}[\hat{A},\hat{R}^{{\dagger}}]=&\hat{A}_{\varepsilon}^{{\dagger}},~{}~{}~{}[\hat{R},\hat{A}_{\varepsilon}^{{\dagger}}]=-\hat{A},\\\
[\hat{R},\hat{A}^{{\dagger}}]=&\hat{A}_{\varepsilon},~{}~{}~{}[\hat{A}_{\varepsilon},\hat{R}^{{\dagger}}]=-\hat{A}^{{\dagger}},\\\
[\hat{R},\hat{R}^{{\dagger}}]=&-\hat{s},\end{split}$ (116)

and those including $\hat{s}$, are

$\displaystyle\begin{split}[\hat{A},\hat{s}]=\hat{A},~{}~{}~{}[\hat{A}_{\varepsilon},\hat{s}]=&\hat{A}_{\varepsilon},~{}~{}~{}[\hat{R},\hat{s}]=2\hat{R},\\\
[\hat{s},\hat{A}^{{\dagger}}]=\hat{A}^{{\dagger}},~{}~{}~{}[\hat{s},\hat{A}_{\varepsilon}^{{\dagger}}]=&\hat{A}_{\varepsilon}^{{\dagger}},~{}~{}~{}[\hat{s},\hat{R}^{{\dagger}}]=2\hat{R}^{{\dagger}}.\end{split}$
(117)

##  Appendix B Products with exponentiated operators

To compute the products with exponentiated operators, we use the identity

$\displaystyle\exp(\hat{X})\hat{Y}\exp(-\hat{X})=$
$\displaystyle\hat{Y}+[\hat{X},\hat{Y}]+\frac{1}{2!}[\hat{X},[\hat{X},\hat{Y}]]$
$\displaystyle+\frac{1}{3!}[\hat{X},[\hat{X},[\hat{X},\hat{Y}]]]+...\ .$ (118)

The products with exponentiated operators involving only spectral operators
are

$\displaystyle\begin{split}\exp(c\hat{A})\hat{B}^{{\dagger}}\exp(-c\hat{A})=&\hat{B}^{{\dagger}}+cA^{*}\diamond
B,\\\
\exp(c\hat{A}_{\varepsilon})\hat{B}^{{\dagger}}\exp(-c\hat{A}_{\varepsilon})=&\hat{B}^{{\dagger}}+cA\ominus
B,\\\
\exp(c\hat{A})\hat{B}_{\varepsilon}^{{\dagger}}\exp(-c\hat{A})=&\hat{B}_{\varepsilon}^{{\dagger}}+cA^{*}\ominus
B^{*},\\\
\exp(c\hat{A}_{\varepsilon})\hat{B}_{\varepsilon}^{{\dagger}}\exp(-c\hat{A}_{\varepsilon})=&\hat{B}_{\varepsilon}^{{\dagger}}-cB^{*}\diamond
A,\end{split}$ (119)

where $c$ is an arbitrary complex constant. Including $\hat{R}$ and
$\hat{R}^{{\dagger}}$, we have

$\displaystyle\begin{split}\exp(c\hat{R})\hat{A}^{{\dagger}}\exp(-c\hat{R})=&\hat{A}^{{\dagger}}+c\hat{A}_{\varepsilon},\\\
\exp(c\hat{R})\hat{A}_{\varepsilon}^{{\dagger}}\exp(-c\hat{R})=&\hat{A}_{\varepsilon}^{{\dagger}}-c\hat{A},\\\
\exp(c\hat{A})\hat{R}^{{\dagger}}\exp(-c\hat{A})=&\hat{R}^{{\dagger}}+c\hat{A}_{\varepsilon}^{{\dagger}}\\\
&+\tfrac{1}{2}c^{2}A^{*}\ominus A^{*},\\\
\exp(c\hat{A}_{\varepsilon})\hat{R}^{{\dagger}}\exp(-c\hat{A}_{\varepsilon})=&\hat{R}^{{\dagger}}-c\hat{A}^{{\dagger}}\\\
&-\tfrac{1}{2}c^{2}A\ominus A,\\\
\exp(c\hat{R})\hat{R}^{{\dagger}}\exp(-c\hat{R})=&\hat{R}^{{\dagger}}-c\hat{s}-c^{2}\hat{R},\end{split}$
(120)

and then with $\hat{s}$, we have

$\displaystyle\begin{split}\exp(c\hat{A})\hat{s}\exp(-c\hat{A})=&\hat{s}+c\hat{A},\\\
\exp(c\hat{A}_{\varepsilon})\hat{s}\exp(-c\hat{A}_{\varepsilon})=&\hat{s}+c\hat{A}_{\varepsilon},\\\
\exp(c\hat{R})\hat{s}\exp(-c\hat{R})=&\hat{s}+2c\hat{R},\\\
\exp(c\hat{s})\hat{A}^{{\dagger}}\exp(-c\hat{s})=&\exp(c)\hat{A}^{{\dagger}},\\\
\exp(c\hat{s})\hat{A}_{\varepsilon}^{{\dagger}}\exp(-c\hat{s})=&\exp(c)\hat{A}_{\varepsilon}^{{\dagger}},\\\
\exp(c\hat{s})\hat{R}^{{\dagger}}\exp(-c\hat{s})=&\exp(2c)\hat{R}^{{\dagger}}.\end{split}$
(121)

##  Appendix C Eigenstates

Combining the commutation relations in Eqs. (36) and Eq. (38), we have

$[\hat{a}_{r},\hat{A}^{{\dagger}}\pm\hat{R}^{{\dagger}}]=A_{r}\pm\varepsilon_{t,r}\hat{a}_{r}^{{\dagger}},~{}~{}~{}[\hat{A}\pm\hat{R},\hat{a}_{r}^{{\dagger}}]=A_{r}^{*}\pm\hat{a}_{s}\varepsilon_{s,t}.$
(122)

These commutation relations, together with Eq. (118), allow us to obtain

$\displaystyle\begin{split}\hat{a}_{s}\exp(\hat{A}^{{\dagger}}\pm\hat{R}^{{\dagger}})=&\exp(\hat{A}^{{\dagger}}\pm\hat{R}^{{\dagger}})\left(\hat{a}_{s}+A_{s}\pm\varepsilon_{s,r}\hat{a}_{r}^{{\dagger}}\right),\\\
\exp(\hat{A}\pm\hat{R})\hat{a}_{r}^{{\dagger}}=&\left(\hat{a}_{r}^{{\dagger}}+A_{r}^{*}\pm\hat{a}_{s}\varepsilon_{s,r}\right)\exp(\hat{A}\pm\hat{R}).\end{split}$
(123)

Considering the generic operators

$\hat{u}_{r}=a_{1}\hat{a}_{r}+a_{2}\varepsilon_{r,s}\hat{a}_{s}^{{\dagger}},~{}~{}~{}\hat{v}_{r}=b_{1}\varepsilon_{r,s}\hat{a}_{s}+b_{2}\hat{a}_{r}^{{\dagger}},$
(124)

where $a_{1}$, $a_{2}$, $b_{1}$, and $b_{2}$ are arbitrary constants, we
obtain for the eigen-equations

$\displaystyle\begin{split}\hat{u}_{r}\left|{u}\right\rangle=&F_{0}[a_{1}\hat{a}_{r}\exp(\hat{A}^{{\dagger}}+c_{0}\hat{R}^{{\dagger}})+a_{2}\varepsilon_{r,s}\hat{a}_{s}^{{\dagger}}\exp(\hat{A}^{{\dagger}}+c_{0}\hat{R}^{{\dagger}})]\left|{\text{vac}}\right\rangle\\\
=&F_{0}\exp(\hat{A}^{{\dagger}}+c_{0}\hat{R}^{{\dagger}})(a_{1}\hat{a}_{r}+a_{1}A_{r}+a_{1}c_{0}\varepsilon_{r,s}\hat{a}_{s}^{{\dagger}}+a_{2}\varepsilon_{r,s}\hat{a}_{s}^{{\dagger}})\left|{\text{vac}}\right\rangle\\\
=&F_{0}\exp(\hat{a}^{{\dagger}}+c_{0}\hat{R}^{{\dagger}})\left|{\text{vac}}\right\rangle
a_{1}A_{r}=\left|{u}\right\rangle
u_{r}~{}~{}~{}\text{for}~{}~{}~{}a_{1}c_{0}=-a_{2},\\\
\hat{v}_{r}\left|{v}\right\rangle=&F_{0}[b_{1}\varepsilon_{r,s}\hat{a}_{s}\exp(\hat{A}^{{\dagger}}+c_{0}\hat{R}^{{\dagger}})+b_{2}\hat{a}_{r}^{{\dagger}}\exp(\hat{A}^{{\dagger}}+c_{0}\hat{R}^{{\dagger}})]\left|{\text{vac}}\right\rangle\\\
=&F_{0}\exp(\hat{A}^{{\dagger}}+c_{0}\hat{R}^{{\dagger}})(b_{1}\varepsilon_{r,s}\hat{a}_{s}+b_{1}\varepsilon_{r,s}A_{s}+b_{1}c_{0}\varepsilon_{r,s}\varepsilon_{s,t}\hat{a}_{t}^{{\dagger}}+b_{2}\hat{a}_{r}^{{\dagger}})\left|{\text{vac}}\right\rangle\\\
=&F_{0}\exp(\hat{a}^{{\dagger}}+c_{0}\hat{R}^{{\dagger}})\left|{\text{vac}}\right\rangle
b_{1}\varepsilon_{r,s}A_{s}=\left|{u}\right\rangle
u_{r}~{}~{}~{}\text{for}~{}~{}~{}b_{1}c_{0}=-b_{2},\end{split}$ (125)

where $c_{0}=\pm 1$. For the left-eigenstates, we have

$\displaystyle\begin{split}\left\langle{u}\right|\hat{u}_{r}=&F_{0}^{*}\left\langle{\text{vac}}\right|[\exp(\hat{A}+c_{0}\hat{R})a_{1}\hat{b}_{r}+\exp(\hat{A}+c_{0}\hat{R})a_{2}\varepsilon_{r,s}\hat{b}_{s}^{{\dagger}}]\\\
=&F_{0}^{*}\left\langle{\text{vac}}\right|(a_{1}\hat{b}_{r}+a_{2}\varepsilon_{r,s}\hat{b}_{s}^{{\dagger}}+a_{2}\varepsilon_{r,s}A_{s}^{*}-c_{0}a_{2}\varepsilon_{r,s}\varepsilon_{s,t}\hat{b}_{t})\exp(\hat{A}+c_{0}\hat{R})\\\
=&a_{2}\varepsilon_{r,s}A_{s}^{*}F_{0}^{*}\left\langle{\text{vac}}\right|\exp(\hat{A}+c_{0}\hat{R})=u_{r}^{*}\left\langle{u}\right|~{}~{}~{}\text{for}~{}~{}~{}c_{0}a_{2}=a_{1},\\\
\left\langle{v}\right|\hat{v}_{r}=&F_{0}^{*}\left\langle{\text{vac}}\right|[\exp(\hat{A}+c_{0}\hat{R})b_{1}\varepsilon_{r,s}\hat{b}_{s}+\exp(\hat{A}+c_{0}\hat{R})b_{2}\hat{b}_{r}^{{\dagger}}]\\\
=&F_{0}^{*}\left\langle{\text{vac}}\right|[b_{1}\varepsilon_{r,s}\hat{b}_{s}+b_{2}\hat{b}_{r}^{{\dagger}}+b_{2}A_{r}^{*}-c_{0}b_{2}\varepsilon_{r,s}\hat{b}_{s}]\exp(\hat{A}+c_{0}\hat{R})\\\
=&b_{2}A_{r}^{*}F_{0}^{*}\left\langle{\text{vac}}\right|\exp(\hat{A}+c_{0}\hat{R})=v_{r}^{*}\left\langle{v}\right|~{}~{}~{}\text{for}~{}~{}~{}c_{0}b_{2}=b_{1}.\end{split}$
(126)

##  Appendix D Grassmann Dirac delta functionals

Assuming a generic form for the inner product given by

$\langle{f}|{f^{\prime}}\rangle=\lim_{\epsilon\rightarrow
0}\frac{1}{\epsilon^{\Omega/2}}\exp\left[\frac{1}{2\epsilon}(f-f^{\prime})\ominus(f-f^{\prime})\right],$
(127)

we have

$\displaystyle\int W[f]\langle{f}|{f^{\prime}}\rangle\mathcal{D}[f]=$
$\displaystyle\lim_{\epsilon\rightarrow 0}\frac{1}{\epsilon^{\Omega/2}}\int
W[f]\exp\left[\frac{1}{2\epsilon}(f-f^{\prime})\ominus(f-f^{\prime})\right]\
\mathcal{D}[f]$ $\displaystyle=$ $\displaystyle\lim_{\epsilon\rightarrow
0}\frac{1}{\epsilon^{\Omega/2}}\int
W[f_{0}+f^{\prime}]\exp\left(\frac{1}{2\epsilon}f_{0}\ominus f_{0}\right)\
\mathcal{D}[f_{0}]$ $\displaystyle=$ $\displaystyle\lim_{\epsilon\rightarrow
0}\frac{1}{2^{\Omega/2}\epsilon^{\Omega}}\int
W[\sqrt{2\epsilon}f_{1}+f^{\prime}]\exp\left(f_{1}\ominus f_{1}\right)\
\mathcal{D}[f_{1}]$ $\displaystyle=$ $\displaystyle
W[f^{\prime}]\lim_{\epsilon\rightarrow
0}\frac{1}{2^{\Omega/2}\epsilon^{\Omega}}\int\exp\left(f_{1}\ominus
f_{1}\right)\mathcal{D}[f_{1}]=\Lambda W[f^{\prime}],$ (128)

where

$\Lambda=\lim_{\epsilon\rightarrow
0}\frac{1}{2^{\Omega/2}\epsilon^{\Omega}}\int\exp\left(f\ominus
f\right)\mathcal{D}[f].$ (129)

Here, we first shifted $f^{\prime}$ into $f$, making it emerge in the argument
of $W[f]$. Then we absorbed the $\epsilon$-factor into a new field variable
$f_{0}$, using the change of Grassmann integration variable given by

$\int W[\theta]\mathcal{D}[\theta]=\int
W[c\theta^{\prime}]\frac{1}{c}\mathcal{D}[\theta^{\prime}].$ (130)

Hence, the constant emerges from the Grassmann measure, but as the inverse of
what it would have been for non-Grassmann measures. As a result, it does not
cancel the $\epsilon$-factor in front. In the limit, the argument of the
functional becomes independent of the integration field. Therefore, we can
remove it from the functional integral and the limit. The rest of the
expression is the limit and the Grassmann functional integral without
$W[f^{\prime}]$, all of which we combine into a new constant $\Lambda$.

To evaluate the Grassmann functional integral for the constant in Eq. (129),
we consider a two-dimensional spin matrix. Then we can write

$f\ominus f=if_{2}\diamond f_{1}-if_{1}\diamond f_{2}=-i2f_{1}\diamond f_{2}.$
(131)

Assuming that the cardinality of the Grassmann numbers is $\Omega$ (counting
all the degrees of freedom including spin), we have

$\displaystyle\int\exp\left(-i2f_{1}\diamond f_{2}\right)\mathcal{D}[f]=$
$\displaystyle(-i2)^{\Omega/2}$ $\displaystyle=$ $\displaystyle
2^{\Omega/2}\exp(-i\tfrac{1}{4}\pi\Omega).$ (132)

We discard the global phase factor. So, the constant is

$\Lambda=\lim_{\epsilon\rightarrow 0}\frac{1}{\epsilon^{\Omega}}.$ (133)

In the limit, it is a divergent constant (cardinality number) independent of
the Grassmann parameter functions.

##  Appendix E Grassmann star-products

The way Grassmann Wigner functionals are combined to represent the products of
operators leads to the definition of the Grassmann star-product or Grassmann
Moyal product. The Wigner functional for the product of two operators, each of
which is expressed as a Weyl transformation of a Wigner function, gives

$\displaystyle W_{\hat{A}\hat{B}}[q,p]=$
$\displaystyle\int\left\langle\left.{q+\tfrac{1}{2}x}\right|{q_{a}}\right\rangle
W_{\hat{A}}\left[\frac{q_{a}+q_{b}}{2},p_{1}\right]\exp[(q_{a}-q_{b})\diamond
p_{1}]\left\langle\left.{q_{b}}\right|{q_{c}}\right\rangle
W_{\hat{B}}\left[\frac{q_{c}+q_{d}}{2},p_{2}\right]\exp[(q_{c}-q_{d})\diamond
p_{2}]$
$\displaystyle\times\left\langle{q_{d}}\left|{q-\tfrac{1}{2}x}\right.\right\rangle\exp(p\diamond
x)\ \mathcal{D}_{\Lambda}[p_{1},p_{2},q_{a},q_{b},q_{c},q_{d},x]$
$\displaystyle=$
$\displaystyle\Lambda^{3}\int\delta\left[q+\tfrac{1}{2}x-q_{a}\right]\delta\left[q_{b}-q_{c}\right]\delta\left[q_{d}-q+\tfrac{1}{2}x\right]W_{\hat{A}}\left[\frac{q_{a}+q_{b}}{2},p_{1}\right]W_{\hat{B}}\left[\frac{q_{c}+q_{d}}{2},p_{2}\right]$
$\displaystyle\times\exp[(q_{a}-q_{b})\diamond p_{1}+(q_{c}-q_{d})\diamond
p_{2}-x\diamond p]\
\mathcal{D}_{\Lambda}[p_{1},p_{2},q_{a},q_{b},q_{c},q_{d},x]$ $\displaystyle=$
$\displaystyle\int
W_{\hat{A}}\left[\tfrac{1}{2}q+\tfrac{1}{2}q_{b}+\tfrac{1}{4}x,p_{1}\right]W_{\hat{B}}\left[\tfrac{1}{2}q+\tfrac{1}{2}q_{b}-\tfrac{1}{4}x,p_{2}\right]$
$\displaystyle\times\exp\left[\left(q+\tfrac{1}{2}x-q_{b}\right)\diamond
p_{1}+\left(q_{b}-q+\tfrac{1}{2}x\right)\diamond p_{2}-x\diamond p\right]\
\mathcal{D}_{\Lambda}[p_{1},p_{2},q_{b},x]$ $\displaystyle=$ $\displaystyle
2^{-2\Omega}\int
W_{\hat{A}}\left[q_{1},p_{1}\right]W_{\hat{B}}\left[q_{2},p_{2}\right]$
$\displaystyle\times\exp\left[2(q-q_{2})\diamond p_{1}+2(q_{1}-q)\diamond
p_{2}-2(q_{1}-q_{2})\diamond p\right]\
\mathcal{D}_{\Lambda}[q_{1},p_{1},q_{2},p_{2}].$ (134)

For the product of three operators, it then follows that

$\displaystyle W_{\hat{A}\hat{B}\hat{C}}[q,p]=$ $\displaystyle
2^{-4\Omega}\int\exp\left[2(q-q_{2})\diamond p_{1}+2(q_{1}-q)\diamond
p_{2}+2(q_{2}-q_{1})\diamond p\right]$
$\displaystyle\times\exp\left[2(q_{2}-q_{4})\diamond
p_{3}+2(q_{3}-q_{2})\diamond p_{4}+2(q_{4}-q_{3})\diamond p_{2}\right]$
$\displaystyle\times
W_{\hat{A}}\left[q_{1},p_{1}\right]W_{\hat{B}}\left[q_{3},p_{3}\right]W_{\hat{C}}\left[q_{4},p_{4}\right]\
\mathcal{D}_{\Lambda}[q_{1},p_{1},q_{2},p_{2},q_{3},p_{3},q_{4},p_{4}]$
$\displaystyle=$ $\displaystyle 2^{-2\Omega}\int\exp\left[2q\diamond
p_{1}-2q_{1}\diamond p+2q_{3}\diamond(p-p_{1})-2(q-q_{1})\diamond
p_{3}\right]$ $\displaystyle\times
W_{\hat{A}}\left[q_{1},p_{1}\right]W_{\hat{B}}\left[q_{3},p_{3}\right]W_{\hat{C}}\left[q-q_{1}+q_{3},p-p_{1}+p_{3}\right]\
\mathcal{D}_{\Lambda}[q_{1},p_{1},q_{3},p_{3}]$ $\displaystyle=$
$\displaystyle\int\exp\left[(q-q_{a})\diamond
p_{b}+q_{b}\diamond(p_{a}-p)\right]W_{\hat{A}}\left[\tfrac{1}{2}q+\tfrac{1}{2}q_{a}+\tfrac{1}{2}q_{b},\tfrac{1}{2}p+\tfrac{1}{2}p_{a}+\tfrac{1}{2}p_{b}\right]$
$\displaystyle\times
W_{\hat{B}}\left[q_{a},p_{a}\right]W_{\hat{C}}\left[\tfrac{1}{2}q+\tfrac{1}{2}q_{a}-\tfrac{1}{2}q_{b},\tfrac{1}{2}p+\tfrac{1}{2}p_{a}-\tfrac{1}{2}p_{b}\right]\
\mathcal{D}_{\Lambda}[q_{a},p_{a},q_{b},p_{b}].$ (135)

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