Generalization of the Bargmann-Wigner

approach to constructing relativistic fields111Based on a talk presented by
A.P. Isaev at the International Conference on Particle Physics and Cosmology
(professor V.A. Rubakov memorial conference), October 02-07, 2023, Yerevan,
Armenia.

I.L. Buchbinder1, S.A. Fedoruk1, A.P. Isaev1,2, M.A. Podoinitsyn1

1Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow
region, Russia,

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

2Physics Faculty, Lomonosov Moscow State University, Russia

We review the method for constructing local relativistic fields corresponding
to the Bargmann-Wigner wave functions that describe the unitary irreducible
representations of the $4D$ Poincaré group. The method is based on the use of
the generalized Wigner operator connecting the wave functions of induced
representations and local relativistic fields. Applications of this operator
for constructing massive local relativistic fields as well as massless
helicity local fields and massless local infinite spin fields are considered.

PACS: 11.10.-z, 11.30.-j, 11.30.Cp, 03.65.Pm

Keywords: unitary representations, massive particles, massless particles,
relativistic fields

##  1 Introduction

The Poincaré group is the mathematical expression of full relativistic
symmetry. Therefore, the study of various aspects of the Poincaré group and
its representations is closely related to describing the properties of
fundamental objects in modern physics. It is sufficient to say that just
irreducible unitary representations of the Poincaré group underly the
classification of elementary particles in mass and spin.

The fundamentals of describing unitary irreducible representations of the $4D$
Poincaré group were laid down in the eminent paper by Wigner [1]111Earlier,
this subject, in a slightly less general form, was also studied by Majorana
[2] and Dirac [3].. The approach to describing irreducible representations of
the Poincaré group was further developed in the works by Wigner and Bargmann
[4, 5], where a general construction of such representations was proposed.

As is well known, irreducible unitary representations of the Poincaré group
are divided into two classes: massive and massless. Massless representations
in turn are subdivided into helicity representations and representations with
continuous (infinite) spin. It is generally believed that only helicity
representations are physically acceptable. However, recently continuous
(infinite) spin representations have attracted attention as well (see e.g. [6,
7, 8, 9] and the references therein).

Following Wigner, all unitary irreducible representations of the Poincaré
group can be constructed as induced ones from unitary irreducible
representations of the test momentum stability subgroup (Wigner little group).
As it is known, these representations are realized on the Bargmann-Wigner wave
functions [4], [5]. However, in coordinate space, these functions are non-
locally transformed under the action of the Lorentz group. Therefore, in their
initial form, the Bargmann-Wigner wave functions can not be used to construct
relativistic field theories which are formulated in terms of local fields. The
problem of constructing bosonic relativistic fields on the basis of the
Bargmann-Wigner wave function was discussed in our recent papers [10, 11, 12],
where the method of deriving the corresponding relativistic fields was
developed. We were used the ideas of papers [13, 14]. The central object of
such a construction is the generalized Wigner operator that transforms the
Bargmann-Wigner wave functions into local relativistic fields. The form of
this operator is different for different classes of representations: it is a
matrix acting on tensors for massive representations, an integral operator for
representations with continuous (infinite) spin, and for helicity ones it is
given in terms of distributions. In fact, this generalized Wigner operators
provide the most general relativistic equations and expressions for local
relativistic fields corresponding to unitary irreducible representations of
the Poincaré group. The paper under consideration is a brief review of our
approach.

The work is organized as follows. In Section 2, we recall the basic notion
concerning the irreducible representations of the Poincaré group. Subsection
3.1 is devoted to describing the general construction of the induced Wigner
representations. In Subsection 3.2, we discuss in detail the case of massive
representations, their realization in terms of local relativistic fields and
free equations of motion for these fields. Subsection 3.3 is devoted to the
explicit construction of Bargmann-Wigner wave functions for massless
representations with continuous (infinite) spin. In Subsection 3.4, the
generalized Wigner operator is introduced and its properties are discussed. In
Sections 4 and 5, we find an explicit form of the generalized Wigner operators
in the case of massless representations with continuous (infinite) spin and in
the case of helicity representations, respectively. Then with the help of
these operators we construct the corresponding local relativistic fields. We
also demonstrate here that in the case of helicity representations, the
approach under consideration automatically leads to a description of the
corresponding relativistic fields in terms of potentials defined up to gauge
transformations.

##  2 Little group and the fundamental Wigner operators

In this section we briefly describe the basic notation and conventions used in
the paper.

We begin with the well known one-to-one correspondence between the set of
coordinates $x_{\mu}\in\mathbb{R}^{1,3}$ and the set $\mathcal{H}$ of
Hermitian matrices $X=x_{\mu}\sigma^{\mu}\in\mathcal{H}$, where
$\sigma^{0}=I_{2}$ is the $2\times 2$ unit matrix and $\sigma^{i}$, $i=1,2,3$
are the Pauli $\sigma$-matrices. The action of the group
$\mathrm{SL}(2,\mathbb{C})$ on the set $\mathcal{H}$

$X\to X^{\prime}=AXA^{\dagger}\,,\qquad X,X^{\prime}\in\mathcal{H}\,,\quad
A\in\mathrm{SL}(2,\mathbb{C})$ (2.1)

leads to the following group homomorphism
$\mathrm{SL}(2,\mathbb{C})\to\mathrm{SO}^{\uparrow}(1,3)$:

$A\,\sigma^{\mu}A^{\dagger}=\sigma^{\nu}\;\Lambda_{\nu}^{\;\;\mu}(A)\,,\qquad
A\in\mathrm{SL}(2,\mathbb{C})\,,\quad\Lambda_{\mu}{}^{\nu}(A)\in\mathrm{SO}^{\uparrow}(1,3)\,.$
(2.2)

Let $p\,\in\,{\mathbb{R}^{1,3}}$ be the 4-momentum of the relativistic
particle and let $\overset{{}_{\mathrm{\;o}}}{p}\,\in\,{\mathbb{R}^{1,3}}$ be
some singled out test 4-momentum. The momentum coordinates $p_{\mu}$ and
$\overset{{}_{\mathrm{\;o}}}{p}_{\mu}$ are transformed one into another by the
Wigner operator $A_{(p)}$ according to the relation

$A_{(p)}(\overset{{}_{\mathrm{\;o}}}{p}\,\sigma)A_{(p)}^{\dagger}=(p\,\sigma)\quad\Rightarrow\quad\Lambda_{\mu}{}^{\nu}(A_{(p)})\,\overset{{}_{\mathrm{\;o}}}{p}_{\nu}=p_{\mu}\,,$
(2.3)

where we have used the notation $(p\sigma)\,{:=}\,p_{\mu}\sigma^{\mu}$. The
transformations (2.3) are in fact the definition of the operators
$A_{(p)}\in\mathrm{SL}(2,\mathbb{C})$ which transform some test momentum
$\overset{{}_{\mathrm{\;o}}}{p}$ into the arbitrary momentum $p$. It is well
known that the above transformations preserve the square of the 4-momentum
$p_{\mu}p^{\mu}$, i.e.
$p_{\mu}p^{\mu}=\overset{{}_{\mathrm{\;o}}}{p}_{\mu}\overset{{}_{\mathrm{\;o}}}{p}^{\mu}$.
Further we will consider the case of massive particles, where the 4-momentum
is time-like, $p_{\mu}p^{\mu}=\bm{m}^{2}\neq 0$, and the case of massless
particles, when the 4-momentum is light-like, $p_{\mu}p^{\mu}=0$.

Among the transformations of the Lorentz group $\mathrm{SL}(2,\mathbb{C})$
there are those that preserve the test momentum
$\overset{{}_{\mathrm{\;o}}}{p}$, i.e.:

$h(\overset{{}_{\mathrm{\;o}}}{p}\,\sigma)h^{\dagger}=(\overset{{}_{\mathrm{\;o}}}{p}\,\sigma)\,.$
(2.4)

Such matrices $h\in\mathrm{SL}(2,\mathbb{C})$ form the Wigner little group
$G_{0}$ of the test momentum
$\overset{{}_{\mathrm{\;o}}}{p}\,\in\,{\mathbb{R}^{1,3}}$.

As follows from (2.3) and (2.4), the Wigner operators $A_{(p)}$ are defined up
to right multiplication by the elements of $G_{0}$ and parametrize the coset
space $\mathrm{SL}(2,\mathbb{C})/G_{0}$. According to (2.2)–(2.3), the left
action of the group $\mathrm{SL}(2,\mathbb{C})$ on this coset is defined by

$A\,A_{(p)}=A_{(\Lambda p)}\,h_{A,p}\,,\qquad h_{A,p}\in G_{0}\,.$ (2.5)

This relation, rewritten as

$h_{A,p}=A_{(\Lambda p)}^{-1}\,A\,A_{(p)}\quad\Rightarrow\quad
h_{A,\Lambda^{-1}p}=A_{(p)}^{-1}\,A\,A_{(\Lambda^{-1}p)}\,,$ (2.6)

determines the element $h_{A,p}$ of the Wigner little group $G_{0}$, where the
labels $A,p$ in the notation $h_{A,p}$ indicate its dependence on
$A\in\mathrm{SL}(2,\mathbb{C})$ and the 4-momentum $p$.

Unitary irreducible representations of the Poincaré group are induced by
unitary irreducible representations of the Wigner little group $G_{0}$, the
structure of which is determined by the mass parameter of the particle being
considered, i.e., massive particle with $\bm{m}\neq 0$ or massless one with
$\bm{m}=0$.

$\bullet$ Massive case. The test momentum $\overset{{}_{\mathrm{\;o}}}{p}$ can
be chosen as the rest particle momentum

$||\overset{{}_{\mathrm{\;o}}}{p}_{\mu}||=(\overset{{}_{\mathrm{\;o}}}{p}_{0},\overset{{}_{\mathrm{\;o}}}{p}_{1},\overset{{}_{\mathrm{\;o}}}{p}_{2},\overset{{}_{\mathrm{\;o}}}{p}_{3})=(\bm{m},0,0,0)$
(2.7)

and the Wigner little group is a unitary group: $G_{0}\simeq\mathrm{SU}(2)$.
In this case, formula (2.6) can also be written in the form

$h_{A,p}=(h_{A,p})^{-1\dagger}=A^{\dagger}_{(\Lambda
p)}A^{-1\dagger}A^{-1\dagger}_{(p)}\,.$ (2.8)

$\bullet$ Massless case. The test momentum
$\overset{{}_{\mathrm{\;o}}}{p}\,\in\,{\mathbb{R}^{1,3}}$ can be fixed as the
light-cone particle momentum

$||\overset{{}_{\mathrm{\;o}}}{p}_{\mu}||=(\overset{{}_{\mathrm{\;o}}}{p}_{0},\overset{{}_{\mathrm{\;o}}}{p}_{1},\overset{{}_{\mathrm{\;o}}}{p}_{2},\overset{{}_{\mathrm{\;o}}}{p}_{3})=(E,0,0,E)\,,\;\;\;\;\;\;\;\;{(\overset{{}_{\mathrm{\;o}}}{p}\,\sigma)=\begin{pmatrix}2E&0\\\
0&0\end{pmatrix}\,.}$ (2.9)

The Wigner little group is formed by the matrices

$h=\begin{pmatrix}1&b_{1}+{\sf i}b_{2}\\\
0&1\end{pmatrix}\,\begin{pmatrix}e^{{\sf i}\theta/2}&0\\\ 0&e^{-{\sf
i}\theta/2}\end{pmatrix}\,,$ (2.10)

where $\theta\in[0,2\pi)$ and $\vec{b}=(b_{1},b_{2})\in\mathbb{R}^{2}$, and is
isomorphic to the group of motions of two-dimensional Euclidean space
$G_{0}\simeq\mathrm{ISO}(2)$.

##  3 Generalized Wigner operators and relativistic fields

Irreducible representations of the Poincaré group as well as its covering
group are classified according to the eigenvalues of the corresponding Casimir
operators. The Poincaré algebra in four dimensions has two Casimir operators:
the squared momentum $\hat{P}^{2}$ and the squared Pauli-Lubanski vector
$\hat{W}^{2}$. For the cases under consideration, all possible unitary
irreducible representations are determined by the following eigenvalues of
these operators [1, 4, 5].

  * •

Massive representations.
On the spaces of these representations, the following conditions are
fulfilled:

$\hat{P}^{2}=\bm{m}^{2}\,,\qquad\hat{W}^{2}=-\bm{m}^{2}j(j+1)\,,$ (3.1)

where $\bm{m}>0$ is the real parameter describing the particle mass, and
$j\in\mathbb{Z}_{\geq 0}/2$ is the particle spin.

  * •

Massless representations.
In these representations, the Casimir operators are equal to:

$\hat{P}^{2}=0\,,\qquad\hat{W}^{2}=-\bm{\mu}^{2}\,,$ (3.2)

where $\bm{\mu}\in\mathbb{R}_{\geq 0}$ is the mass dimension parameter.
Depending on the value of this parameter, the massless representations belong
to one of the two classes, i.e. they are subdivided into

    * –

either the helicity representations with $\bm{\mu}=0$;

    * –

or the continuous (infinite) spin representations with $\bm{\mu}>0$.

Note also that the parameter $\bm{\mu}$, which defines the massless continuous
spin representations (3.2), corresponds to the radius of the circle in the
auxiliary two-dimensional Euclidean space that arises when considering the
small group (2.10). In addition, we emphasize that the continuous spin
representation contains an infinite number of massless states with all
possible integer or half-integer helicities222For these representations
helicities are not invariants under the action of the Lorentz group
$\mathrm{SL}(2,\mathbb{C})$.. The name “continuous spin representation” was
developed historically after its introduction in [1, 4, 5] and is associated
with the possibility of describing this discrete set as a Fourier series with
respect to the angular variable which is similar to the angle $\theta$ in
(2.10). To avoid any misunderstandings, we also use the name “infinite spin
representations” for such representations.

###  3.1 Bargmann-Wigner wave functions

The main method in constructing unitary irreducible representations $U$ of the
Poincaré covering group $\mathrm{ISL}(2,\mathbb{C})$ is based on the Wigner
scheme [1, 4, 5]. Within this scheme, all unitary irreducible representations
(UIR’s) are realized on the Bargmann-Wigner wave functions $\Phi_{M}(p)$,
which depend on the four-momentum $p_{\mu}$ on the mass shell
$p^{\mu}p_{\mu}=\bm{m}^{2}$ and have an additional (multi)index $M$ that runs
over a finite or infinite number of values. This index is related to the
unitary irreducible representation of the Wigner little group $G_{0}$.

Representation $U(A)$ of the element $A\in SL(2,\mathbb{C})$ acts on the
Bargnmann-Wigner wave function as follows:

$\Phi^{\prime}_{M}(p):=[U(A)\Phi]_{M}(p)=\mathcal{D}_{MN}(h_{A,\Lambda^{-1}p})\Phi_{N}(\Lambda^{-1}p)\,,$
(3.3)

where the matrix $\Lambda$ is related to $A$ by (2.2), $\mathcal{D}_{MN}$ is
UIR of the Wigner little group $G_{0}$ and the element $h_{A,\Lambda^{-1}p}$
is determined by relation (2.6). One can check directly that
$U(A\,B)=U(A)\,U(B)$. The representation (3.3) of the Poincaré group
$\mathrm{ISL}(2,\mathbb{C})$, constructed in this way, is induced by the
unitary representation of the Wigner little group $G_{0}$.

However, the transformation law (3.3) becomes non-local for the space-time
field (field in coordinate representation) obtained from the Bargmann-Wigner
wave function $\Phi_{M}(p)$ using the Fourier transform. To find local
relativistic fields, it is necessary to modify the Bargmann-Wigner wave
function before performing the Fourier transform. In the next subsection we
will briefly describe how it can be done for the case of massive
representations (for a detailed discussion of them, see [13, 12, 15]).

###  3.2 Massive representations

In the massive case, UIR of the Wigner little group $G_{0}=\mathrm{SU}(2)$ is
characterized by the spin quantum number $j\in\mathbb{Z}_{\geq 0}/2$. This
representation is realized on the Bargmann-Wigner wave functions
$\Phi_{a_{1}\cdots a_{2j}}(p)=\Phi_{(a_{1}\cdots a_{2j})}(p)$ which are the
$\mathrm{SU}(2)$-spinors of rank $2j$, where $a=1,2$ is the
$\mathrm{SU}(2)$-index and the parentheses $(a_{1}\dots a_{2j})$ mean total
symmetrization over all indices. In this case, the transformation law (3.3) is
written as

$\Phi^{\prime}_{a_{1}\dots a_{2j}}(p):=[U(A)\Phi]_{a_{1}\dots
a_{2j}}(p)=(h_{A,\Lambda^{-1}p})_{a_{1}}^{\;b_{1}}\cdots(h_{A,\Lambda^{-1}p})_{a_{2j}}^{\;b_{2j}}\;\Phi_{b_{1}\dots
b_{2j}}(\Lambda^{-1}p)\,.$ (3.4)

Let us move from the Bargmann-Wigner wave functions $\Phi_{a_{1}\cdots
a_{2j}}(p)$ to the new wave functions whose transformation law, in contrast to
the transformation law (3.4), uses the matrices which do not depend on the
argument $p$. These new wave functions are the multicomponent fields labeled
by the indices $q,\,r$ and defined as follows:

$\Psi^{(q,r)}{}^{{\phantom{(}}\\!\\!\dot{\beta_{1}}...\dot{\beta_{r}}}_{{\phantom{(}}\\!\\!\alpha_{1}...\alpha_{q}}(p)=(A_{(p)})_{\alpha_{1}}^{\,b_{1}}\cdots(A_{(p)})_{\alpha_{q}}^{\,b_{q}}\,(\tilde{A}^{-1\dagger}_{(p)})^{\dot{\beta}_{1}b_{q+1}}\cdots(\tilde{A}^{-1\dagger}_{(p)})^{\dot{\beta}_{r}b_{2j}}\Phi_{b_{1}\dots
b_{2j}}(p)\,,$ (3.5)

where $q=0,1,\ldots,2j$, $r=2j-q$ and the following notation333In addition to
the relativistic Pauli matrices $(\sigma^{\mu})_{\alpha\dot{\beta}}$ we also
use the matrices
$(\tilde{\sigma}^{\mu})^{\dot{\alpha}\beta}=(\tilde{\sigma}^{0},\tilde{\sigma}^{i})=(\sigma^{0},-\sigma^{i})$.
$\tilde{A}_{(p)}^{-1\dagger}:=A_{(p)}^{-1\dagger}\tilde{\sigma}_{0}$ is used.
The constructed fields (3.5) are symmetric with respect to indices of the same
type:
$\Psi^{(q,r)}{}^{{\phantom{(}}\\!\\!\dot{\beta_{1}}...\dot{\beta_{r}}}_{{\phantom{(}}\\!\\!\alpha_{1}...\alpha_{q}}(p)=\Psi^{(q,r)}{}^{(\dot{\beta_{1}}...\dot{\beta_{r}})}_{(\alpha_{1}...\alpha_{q})}(p)$.
An important property of the fields (3.5) is their transformation law

$[U(A)\Psi^{(q,r)}]{}^{{\phantom{(}}\\!\\!\dot{\beta_{1}}...\dot{\beta_{r}}}_{{\phantom{(}}\\!\\!\alpha_{1}...\alpha_{q}}(p)=A^{\;\;\gamma_{1}}_{\alpha_{1}}\cdots
A^{\;\;\gamma_{q}}_{\alpha_{q}}\;\;\bigl{(}A^{\dagger-1}\bigr{)}^{\dot{\beta_{1}}}_{\;\;\dot{\kappa}_{1}}\cdots\bigl{(}A^{\dagger-1}\bigr{)}^{\dot{\beta_{r}}}_{\;\;\dot{\kappa}_{r}}\;\;\Psi^{(q,r)}{}^{{\phantom{(}}\\!\\!\dot{\kappa}_{1}...\dot{\kappa}_{r}}_{{\phantom{(}}\\!\\!\gamma_{1}...\gamma_{q}}(\Lambda^{-1}\cdot
p)\;,$ (3.6)

which is a direct consequence of the transformation law (3.4) and relations
(2.6), (2.8). Here $A\in\mathrm{SL}(2,\mathbb{C})$. Since the transformation
matrix in (3.6) does not depend on the 4-momentum $p$, the constructed fields
$\Psi^{(q,r)}{}^{{\phantom{(}}\\!\\!\dot{\beta_{1}}...\dot{\beta_{r}}}_{{\phantom{(}}\\!\\!\alpha_{1}...\alpha_{q}}(p)$
are the standard local spin-tensor fields in the momentum representation. The
operators $A_{(p)}^{\otimes q}(\tilde{A}_{(p)}^{-1\dagger})^{\otimes r}$,
$r+q=2j$, which transform the Bargmann-Wigner wave functions
$\Phi_{a_{1}\cdots a_{2j}}(p)$ into local relativistic fields
$\Psi^{(q,r)}{}^{{\phantom{(}}\\!\\!\dot{\beta_{1}}...\dot{\beta_{r}}}_{{\phantom{(}}\\!\\!\alpha_{1}...\alpha_{q}}(p)$,
are called the spin $j$ Wigner operators.

The fields
$\Psi^{(q,r)}{}^{{\phantom{(}}\\!\\!\dot{\beta_{1}}...\dot{\beta_{r}}}_{{\phantom{(}}\\!\\!\alpha_{1}...\alpha_{q}}(p)$
at fixed spin $j=(q+r)/2$ are not independent of mass-shell: they
automatically satisfy the Dirac-Pauli-Fierz equations

$\begin{array}[]{l}(p\,\tilde{\sigma})^{\dot{\gamma}_{1}\alpha_{1}}\Psi^{(q,r)}{}^{{\phantom{(}}\\!\\!\dot{\beta_{1}}...\dot{\beta_{r}}}_{{\phantom{(}}\\!\\!\alpha_{1}...\alpha_{q}}(p)=\bm{m}\,\Psi^{(q-1,r+1)}{}^{{\phantom{(}}\\!\\!\dot{\gamma}_{1}\dot{\beta_{1}}...\dot{\beta_{r}}}_{{\phantom{(}}\\!\\!\alpha_{2}...\alpha_{q}}(p)\;,\;\;\;\;(r=0,\dots,2j-1)\;,\\\\[5.69046pt]
(p\,\sigma)_{\gamma_{1}\dot{\beta}_{1}}\Psi^{(q,r)}{}^{{\phantom{(}}\\!\\!\dot{\beta_{1}}...\dot{\beta_{r}}}_{{\phantom{(}}\\!\\!\alpha_{1}...\alpha_{q}}(p)=\bm{m}\,\Psi^{(q+1,r-1)}{}^{{\phantom{(}}\\!\\!\dot{\beta_{2}}...\dot{\beta_{r}}}_{{\phantom{(}}\\!\\!\gamma_{1}\alpha_{1}...\alpha_{q}}(p)\;,\;\;\;\;(r=1,\dots,2j)\;.\end{array}$
(3.7)

which follow from (2.3) and (2.7). In particular, for $j=\frac{1}{2}$, eqs.
(3.7) are two Weyl projections of the spin-$\frac{1}{2}$ Dirac equation. These
and other aspects of such a description of massive particle states are given
in detail in [12, 15]. Other methods of deducing relativistic field equations
from the first principles were also considered in [16].

###  3.3 Massless representations

UIR’s of the group $\mathrm{ISO}(2)$, being the Wigner little group $G_{0}$ in
the massless case, are well known (see e.g. [17, 18]). They are characterized
by a real nonnegative parameter $\bm{\rho}$ and realized on the functions
$\Phi(\varphi)$ defined on the circle of the radius $\bm{\rho}$ on the plane
parameterized by the vector coordinates
$\vec{t}_{\varphi}=((t_{\varphi})_{1},(t_{\varphi})_{2})=(\bm{\rho}\cos\varphi,\bm{\rho}\sin\varphi)$.
Induced UIR’s of the Poincaré group $\mathrm{ISL}(2,\mathbb{C})$ are realized
on the wave functions $\Phi(p,\varphi)$, which also depend on the 4-momentum
$p_{\mu}$. In the massless case, the formula (3.3) of
$\mathrm{SL}(2,\mathbb{C})$ takes the following form:

$\begin{array}[]{rcl}\Phi^{\prime}(p,\varphi)&:=&\displaystyle[U(A)\Phi](p,\varphi)=\int\limits_{0}^{2\pi}d\varphi^{\prime}\
\mathcal{D}_{\varphi\varphi^{\prime}}(\theta_{A,\Lambda^{-1}p},\vec{b}_{A,\Lambda^{-1}p})\,\Phi(\Lambda^{-1}p,\varphi^{\prime})\\\\[6.0pt]
&\equiv&\displaystyle
e^{-i\,\vec{b}_{A,\Lambda^{-1}p}\cdot\vec{t}_{\varphi}}\,\Phi(\Lambda^{-1}p,\varphi-\theta_{A,\Lambda^{-1}p})\,,\end{array}$
(3.8)

where the operator
$\mathcal{D}_{\varphi\varphi^{\prime}}(\theta_{A,\Lambda^{-1}p},\vec{b}_{A,\Lambda^{-1}p})$
is the representative of the little group element
$h_{A,\Lambda^{-1}p}\in\mathrm{ISO}(2)$ in unitary representation (see e.g.
[17]). The dependence of the element $h\in\mathrm{ISO}(2)$ (parametrized by
the pair $(\theta,\vec{b})$ as in (2.10)) on $A$, $p$ was discussed in [10].
The matrix element $\mathcal{D}_{\varphi\varphi^{\prime}}(\theta,\vec{b})$ has
the following explicit form:

$\mathcal{D}_{\varphi\varphi^{\prime}}(\theta,\vec{b})=e^{-i\,\vec{b}_{\mathstrut}\cdot\vec{t}_{\varphi}}\delta(\varphi-\varphi^{\,\prime}-\theta)\,,$
(3.9)

where $\delta(\varphi)$ is the periodic Dirac $\delta$-function on a circle
and
$\vec{b}_{\mathstrut}\cdot\vec{t}_{\varphi}=\bm{\rho}\,\bigl{(}b_{1}\cos\varphi+b_{2}\sin\varphi\bigr{)}$.

There is another equivalent formulation of the induced massless Poincaré group
representations (see e.g. [6], [10]), which is obtained after expanding the
Bargmann-Wigner wave function $\Phi(p,\varphi)$ into a Fourier series:

$\Phi(p,\varphi)=\sum_{n=-\infty}^{\infty}\Phi_{n}(p)e^{in\varphi}\,.$ (3.10)

In the space of functions $\Phi_{n}(p)$, the transformation law (3.8) has the
form:

$[U(A)\Phi]_{n}(p)=\sum_{m=-\infty}^{\infty}\mathcal{D}_{nm}(\theta_{A,\Lambda^{-1}p},\vec{b}_{A,\Lambda^{-1}p})\,\Phi_{m}(\Lambda^{-1}p)\,,$
(3.11)

where the matrix elements $\mathcal{D}_{nm}(\theta,\vec{b})$ of the little
group $\mathrm{ISO}(2)$ are

$\mathcal{D}_{nm}(\theta,\vec{b})=(-ie^{i\beta})^{m-n}\,e^{-im\theta}J_{(m-n)}(b{\bm{\rho}})\;.$
(3.12)

Here the real numbers $\beta$ and $b$ are the polar coordinates of the
2-vector $\vec{b}=(b_{1},b_{2})=b\,(\cos\beta,\sin\beta)$ and $J_{(n)}(x)$ are
the Bessel functions of integer order $n$.

The induced unitary Wigner representations (3.8) and (3.11) of the Poincaré
group $\mathrm{ISL}(2,\mathbb{C})$ are irreducible for $\bm{\rho}\neq 0$ and
infinite-dimensional. The corresponding representations ${\cal D}$ of
$\mathrm{ISO}(2)$ written for the discrete bases (3.10)-(3.12) are decomposed
into a direct sum of one-dimensional irreducible representations of the
subgroup $\mathrm{SO}(2)\subset\mathrm{ISO}(2)$. The action of the group
$\mathrm{ISO}(2)$ in the representation ${\cal D}$ in the limit $\bm{\rho}\to
0$ is reduced to the action of its subgroup $\mathrm{SO}(2)$. Thus, in the
limit $\bm{\rho}\to 0$, the representation ${\cal D}$ of $\mathrm{ISO}(2)$
becomes reducible and is decomposed into a sum of infinite number of one
dimensional UIR’s. In the limit $\bm{\rho}\to 0$ expression (3.11) is used in
the description of the $\mathrm{ISL}(2,\mathbb{C})$ helicity representations
with finite numbers of relativistic spin states (see Section 5 below).

To construct a local relativistic field corresponding to the Bargmann-Wigner
wave function $\Phi(p,\varphi)$, we use a generalization of the method applied
in Subsection 3.2 in the case of massive representations.

###  3.4 Generalized Wigner operators in the infinite spin case

In the massive case, relation (3.5) between the local relativistic fields and
the Bargmann-Wigner wave functions is served by the Wigner operators
$A_{(p)}^{\otimes q}(\tilde{A}_{(p)}^{-1\dagger})^{\otimes r}$. In the general
case of massless representations including infinite-dimensional
representations of continuous spin, it is required to use an infinite-
dimensional generalization of this correspondence.

The main role in this correspondence is played by the Wigner operator
$A_{(p)}$ whose matrices parameterize the coset-space
$\mathrm{SL}(2,\mathbb{C})/G_{0}$. After restoring the indices, the left
action (2.5) of the element $A\in\mathrm{SL}(2,\mathbb{C})$ on
$\mathrm{SL}(2,\mathbb{C})/G_{0}$ is written as

$A_{\alpha}{}^{\beta}\,(A_{(p)})_{\beta}{}^{c}=(A_{(\Lambda
p)})_{\alpha}{}^{b}\,(h_{A,p})_{b}{}^{c}\,,$ (3.13)

where the matrix $(A_{(p)})_{\alpha}{}^{a}$ of the Wigner operator has the
$\mathrm{SL}(2,\mathbb{C})$-index $\alpha$ and $G_{0}$-index $a$ (in
particular, for the massive case, we have $G_{0}=\mathrm{SU}(2)$). Thus, the
operators $A_{(p)}^{\otimes q}(\tilde{A}_{(p)}^{-1\dagger})^{\otimes r}$ play
the role of a bridge that converts the little subgroup indices $a,b,...$ of
$G_{0}$-type into the relativistic indices $\alpha,\beta,...$ of
$\mathrm{SL}(2,\mathbb{C})$-type.

An analogue of such a transformation for the Bargmann-Wigner wave function
$\Phi(p,\varphi)$ introduced in (3.8) for the massless case is given by the
following formula:

$\Psi(p,y)\ =\
\int\limits_{0}^{2\pi}d\varphi\,\mathcal{A}(p,y,\varphi)\,\Phi(p,\varphi)\,,$
(3.14)

where additional variables $y$ denote a certain set of variables that play the
role of the relativistic $\mathrm{SL}(2,\mathbb{C})$-type indices while
$\varphi$ is of the $\mathrm{ISO}(2)$-type ($G_{0}$-type) variable. The
transform (3.14) can be thought as an infinite-dimensional version of the
transition (3.5): it converts the index $\varphi$ of the Wigner little group
$\mathrm{ISO}(2)$ into some (as yet undefined) relativistic index $y$. We call
the integral operator with the kernel $\mathcal{A}(p,y,\varphi)$ in the right-
hand side of (3.14) the generalized Wigner operator.

In the next section, the $\mathrm{SL}(2,\mathbb{C})$-type variables $y$ are
taken to be the components of the Lorentz vector
$\eta^{\mu}\in\mathbb{R}^{1,3}$, or the components of a pair of Weyl spinors
$u^{\alpha}$, $\bar{u}^{\dot{\alpha}}$. The requirement of the locality of the
$\mathrm{SL}(2,\mathbb{C})$ transformations of the relativistic fields
$\Psi(p,\eta)$, or $\Psi(p,u,\bar{u})$, leads to the definition of the
explicit form of the generalized Wigner operators in (3.14) for these two
cases.

##  4 Relativistic fields with infinite spin

###  4.1 Additional vector variables

Let us consider variables $y$ in (3.14) as additional vector coordinates
$\eta_{\mu}\in\mathbb{R}^{1,3}$ of the fields. The field $\Psi(p,\eta)$ must
have the local transformation law under the action of
$A\in\mathrm{SL}(2,\mathbb{C})$ (independent of $p$), as it was in (3.6). In
the case when, a relativistic field is characterized by its dependence on the
auxiliary vector variables $\eta_{\mu}$ instead of external indices, this
transformation law takes the form

$\Psi^{\prime}(p,\eta)\ :=\ [U(A)\Psi](p,\eta)\ =\
\Psi(\Lambda^{-1}p,\Lambda^{-1}\eta)\,,$ (4.1)

where the matrices $A$ and $\Lambda$ are related by (2.2) and momentum $p$ is
on the mass shell $p^{2}=0$. Taking into account (3.8) and (3.14), the
condition (4.1) leads to the equation for the kernel
$\mathcal{A}(p,\eta,\varphi)$:

$\mathcal{A}(\Lambda^{-1}p,\Lambda^{-1}\eta,\varphi)\ =\ \int
d\varphi^{\prime}\mathcal{A}(p,\eta,\varphi^{\prime})\,\mathcal{D}_{\varphi^{\prime}\varphi}(\theta_{A,\Lambda^{-1}p},\vec{b}_{A,\Lambda^{-1}p})\,.$
(4.2)

This equation on the generalized Wigner operator is an infinite-dimensional
counterpart of the matrix coset transformation (3.13).

As shown in [19, 20, 10], the infinitesimal form of a relation (4.2) is
represented as three differential equations for the kernel
$\mathcal{A}(p,\eta,\varphi)$, which can be solved exactly. There are two
types of solutions of these equations, which can be called non-singular and
singular [19, 20, 10].

  * •

Non-singular solution.
The operator $\mathcal{A}(p,\eta,\varphi)$ that satisfies (4.2) has the
following explicit form:

$\mathcal{A}(p,\eta,\varphi)=e^{\,i{\bm{\mu}}\,\eta\cdot\varepsilon_{(1)}(\varphi)/(\eta\cdot
p)}\,f(\eta\\!\cdot\\!\eta,\eta\\!\cdot\\!p)\,,$ (4.3)

where $f(\eta\\!\cdot\\!\eta,\eta\\!\cdot\\!p)$ is an arbitrary function of
two variables $\eta\\!\cdot\\!\eta:=\eta^{\mu}\eta_{\mu}$ and
$\eta\\!\cdot\\!p:=\eta^{\mu}p_{\mu}$. The dimensional constant $\bm{\mu}$ is
related to the dimensionless constant $\bm{\rho}$ by the relation

$\bm{\mu}=E\bm{\rho}\,,$ (4.4)

where $E$ is defined by nonzero components of the massless test momentum
$\overset{{}_{\mathrm{\;o}}}{p}$ in (2.9). In (4.3), the vector
$\varepsilon^{\mu}_{(1)}(\varphi)=\varepsilon^{\mu}_{(1)}\cos\varphi-\varepsilon^{\mu}_{(2)}\sin\varphi$
is constructed from the polarization vectors $\varepsilon_{(1)}$ and
$\varepsilon_{(2)}$ which are determined by the conditions
$\varepsilon_{(1)}\\!\cdot\\!\varepsilon_{(1)}=\varepsilon_{(2)}\\!\cdot\\!\varepsilon_{(2)}=-1$,
$p\\!\cdot\\!\varepsilon_{(1)}=p\\!\cdot\\!\varepsilon_{(2)}=\varepsilon_{(1)}\\!\cdot\\!\varepsilon_{(2)}=0$.

  * •

Singular solution.
In this case, the solution of (4.2) can be written as follows:

$\mathcal{A}(p,\eta,\varphi)\ =\
\delta(\eta\\!\cdot\\!p)\,\delta(\eta\\!\cdot\\!{\varepsilon}_{(2)}(\varphi))\,e^{i{\bm{\mu}}\,\eta\cdot{\varepsilon}/(\eta\cdot{\varepsilon}_{(1)}(\varphi))}\,g(\eta\\!\cdot\\!{\varepsilon}_{(1)}(\varphi))\,,$
(4.5)

where $g(\eta\\!\cdot\\!{\varepsilon}_{(1)}(\varphi))$ is an arbitrary
function and
$\varepsilon^{\mu}_{(2)}(\varphi)=\varepsilon^{\mu}_{(1)}\sin\varphi+\varepsilon^{\mu}_{(2)}\cos\varphi$.
In (4.5), the vector ${\varepsilon}^{\mu}$ obeys the conditions
$\varepsilon\\!\cdot\\!p=1$,
$\varepsilon\\!\cdot\\!\varepsilon=\varepsilon\\!\cdot\\!\varepsilon_{(1)}=\varepsilon\\!\cdot\\!\varepsilon_{(1)}=0$
and forms tetrads with the vectors $p$, $\varepsilon_{(1)}$,
$\varepsilon_{(2)}$.

Two expressions (4.3) and (4.5) for the generalized Wigner operators produce
two types of relativistic fields $\Psi(p,\eta)$, according to the transform
(3.14), where one can use the expansion (3.10) for the Bargmann-Wigner wave
function. By construction, the fields $\Psi(p,\eta)$ describe physical states
with the eigenvalues (3.2) of the Casimir operators. Indeed, the first
eigenvalue in (3.2) is evident. The second eigenvalue was checked in [10],
where we used the explicit formula for the square of the Pauli-Lubanski
operator $\hat{W}^{2}$ acting in the space of the fields $\Psi(p,\eta)$. All
these statements are fulfilled for both non-singular and singular solutions.

###  4.2 Additional spinor variables

Let us now consider the case when the components of the commuting Weyl spinor
$u_{\alpha}$, $\bar{u}_{\dot{\alpha}}=(u_{\alpha})^{\dagger}$,
$\alpha,\dot{\alpha}=1,2$ are taken as additional variables $y$ in (3.14). The
local relativistic field $\Psi(p,u,\bar{u})$ constructed by using the
generalized Wigner operator in the procedure described above was found in [10]
and has the following form:

$\Psi(\pi,\bar{\pi},u,\bar{u})=\int\limits_{0}^{2\pi}d\varphi\,\exp\left\\{-i\,{\bm{\mu}}\,\Bigl{(}\frac{u^{\alpha}\lambda_{\alpha}}{u^{\beta}\pi_{\beta}}\,e^{-i\varphi}+\frac{\bar{u}^{\dot{\alpha}}\bar{\lambda}_{\dot{\alpha}}}{\bar{u}^{\dot{\beta}}\bar{\pi}_{\dot{\beta}}}\,e^{i\varphi}\Bigr{)}\right\\}\mathrm{f}(u^{\gamma}\pi_{\gamma}\,e^{\frac{i}{2}\varphi},\bar{u}^{\dot{\gamma}}\bar{\pi}_{\dot{\gamma}}\,e^{-\frac{i}{2}\varphi})\,\Phi(p,\varphi)\,,$
(4.6)

where $\mathrm{f}$ is an arbitrary function of two conjugated scalar
variables. In (4.6), the commuting Weyl spinor $\pi_{\alpha}$ is actually a
twistor which defines massless 4-momentum through the Cartan-Penrose relation

$(p\,\sigma)_{\alpha\dot{\beta}}=\bigl{(}A_{(p)}\,(\overset{{}_{\mathrm{\;o}}}{p}\,\sigma)\,A_{(p)}^{\dagger}\bigr{)}_{\alpha\dot{\beta}}=\pi_{\alpha}\bar{\pi}_{\dot{\beta}}\,.$
(4.7)

Another commuting Weyl spinor $\lambda_{\alpha}$ is determined by the
normalization condition $\pi^{\alpha}\lambda_{\alpha}=1$, that is, a pair of
spinors $\pi_{\alpha}$, $\lambda_{\alpha}$ forming the basis in the
2-component spinor space. Thus, for the choice (2.9), one can take the
parametrization:
$A_{(p)}=\begin{pmatrix}\pi_{1}/\sqrt{2E}\,,&\lambda_{1}\sqrt{2E}\\\
\pi_{2}/\sqrt{2E}\,,&\lambda_{2}\sqrt{2E}\\\ \end{pmatrix}$.

The square of the Pauli-Lubanski vector $\hat{W}^{2}$ on the space of the
fields $\Psi(\pi,\bar{\pi},u,\bar{u})$ has the following representation (see
e.g. [7]):

$\hat{W}^{2}=(u^{\alpha}\pi_{\alpha}\bar{\pi}_{\dot{\alpha}}\bar{u}^{\dot{\alpha}})\Bigl{(}\frac{\partial}{\partial\bar{u}^{\dot{\beta}}}\,\bar{\pi}^{\dot{\beta}}\pi^{\beta}\frac{\partial}{\partial
u^{\beta}}\Bigr{)}\,.$ (4.8)

By making use of the explicit form (4.6) for $\Psi(\pi,\bar{\pi},u,\bar{u})$,
it can be shown that

$\hat{W}^{2}\,\Psi(\pi,\bar{\pi},u,\bar{u})=-{\bm{\mu}}^{2}\,\Psi(\pi,\bar{\pi},u,\bar{u})\,.$
(4.9)

Therefore, the UIR’s of the group $\mathrm{ISL}(2,\mathbb{C})$ with infinite
spin are realized on the field $\Psi(\pi,\bar{\pi},u,\bar{u})$. For more
details about the equations of motion for the field
$\Psi(\pi,\bar{\pi},u,\bar{u})$ see [21, 22, 10].

##  5 Relativistic gauge fields

Another important example of massless UIR’s of the Poincaré covering group is
massless helicity representations. Such representations are physically
motivated and wildly used in higher spin field theory (see, e.g., the review
[23]).

Helicity representations are induced from UIR’s of the subgroup
$\mathrm{SO}(2)\subset\mathrm{ISO}(2)$, or from UIR’s of $\mathrm{ISO}(2)$
with trivial realizations of the 2-dimensional translations. All UIR’s of the
$\mathrm{SO}(2)$ group are one-dimensional and characterized by an integer or
half-integer number corresponding to particle helicity. In what follows we
will be interested only in representations with integer helicities and
demonstrate that the approach of the generalized Wigner operators [11]
reproduces well-known expressions for massless fields of higher helicities.
One of the advantages of this method is obtaining a group theoretical
description of massless helicity representations in terms of field potentials
defined up to gauge transformations.

Helicity representations are induced by the UIR’s of the Wigner little group
$ISO(2)$ at $\bm{\rho}=0$. In this limit, the transformation low (3.11) takes
the form

$[U(A)\Phi]_{n}(p)=e^{-in\theta_{\\!\\!A,\Lambda^{-1}p}}\,\Phi_{n}(\Lambda^{-1}p)\,,\;\;\;\;\;\forall\;n\in\mathbb{Z}\;,$
(5.1)

where the property $J_{m-n}(0)=\delta_{m,n}$ for the Bessel functions was
used. Thus, at $\bm{\rho}=0$ the unitary representations of the Poincaré group
are decomposed into a direct sum of UIR’s induced from the one-dimensional
UIR’s of $SO(2)$ which are described by the functions $\Phi_{n}(p)$. Here the
number $n$ corresponds to particle helicity.

Relativistic local fields are constructed from the Wigner wave functions
$\Phi_{n}(p)$ by using the transform (3.14) written in the discrete basis
(3.10). It turns out that to construct a massless relativistic field
$\Psi_{n}(p)$ with spin $n>0$, two Bargmann-Wigner wave functions with
opposite helicities $n$ and $-n$ should be used. In fact, this corresponds to
the construction of a massless irreducible representation of the extended
Poincaré group, which includes discrete transformations $P$ and $T$ (see e.g.
[24, 25]). Thus, in accordance with (3.14), the local field of the helicity
$n$ particle has the form444The case $n=0$ is considered separately.

$\Psi_{n}(p,\eta)\ =\ \mathcal{A}(p,\eta,n)\Phi_{n}(p)\ +\
\mathcal{A}(p,\eta,-n)\Phi_{-n}(p)\,,\quad n\in\mathbb{Z}_{>0}\,.$ (5.2)

The explicit form of the generalized Wigner operator $\mathcal{A}(p,\eta,n)$
in the discrete basis for the case $\bm{\rho}=0$ was found in [11] and has the
form

$\mathcal{A}(p,\eta,n)=\left\\{\begin{array}[]{lll}\displaystyle\delta(\eta\\!\cdot\\!p)\,(\varepsilon_{(+)}\\!\cdot\\!\eta)^{n}\,\;\;\;\;\;{\mbox{with}}\;\;n>0\,,\\\\[10.0pt]
\displaystyle\delta(\eta\\!\cdot\\!p)\,(\varepsilon_{(-)}\\!\cdot\\!\eta)^{-n}\,\;\;\;{\mbox{with}}\;\;n<0\,,\end{array}\right.$
(5.3)

where $\varepsilon_{(\pm)}=\varepsilon_{(2)}\pm i\varepsilon_{(1)}$. Then, the
substitution of (5.3) into (5.2) gives the expression for the massless field
of the helicity $n$:

$\Psi_{n}(p,\eta)\ =\ \delta(\eta\\!\cdot\\!p)\,F_{n}(p,\eta)\,,$ (5.4)

where

$F_{n}(p,\eta)\ =\ F^{{}_{(+)}}_{n}(p,\eta)\ +\
F^{{}_{(-)}}_{n}(p,\eta)\,,\qquad
F^{{}_{(\pm)}}_{n}(p,\eta)=(\varepsilon_{(\pm)}\cdot\eta)^{n}\,\Phi_{\pm
n}(p)\,.$ (5.5)

The explicit form of the functions (5.5) reproduces automatically the
equations of motion of the field $F_{n}(p,\eta)$:

$p^{2}\,F_{n}(p,\eta)=0\,,\quad\left(p\cdot\\!\frac{\partial}{\partial\eta}\right)\,F_{n}(p,\eta)=0\,,\quad\left(\frac{\partial}{\partial\eta}\cdot\\!\frac{\partial}{\partial\eta}\right)\,F_{n}(p,\eta)=0\,,$
(5.6)
$\left(\eta\cdot\\!\frac{\partial}{\partial\eta}\right)\,F_{n}(p,\eta)=n\,F_{n}(p,\eta)\,.$
(5.7)

where the properties of polarization vectors were used. The last equation
determines the degree of homogeneity for the field $F_{n}(p,\eta)$ in the
variables $\eta$. In addition, the presence in the definition of
$\Psi_{n}(p,\eta)$ of the field $F_{n}(p,\eta)$ together with the
$\delta$-function $\delta(\eta\cdot p)$ leads to the following equivalence
relation:

$F_{n}(p,\eta)\ \ \sim\ \
F_{n}(p,\eta)+(p\cdot\eta)\,\epsilon_{n-1}(p,\eta)\,,$ (5.8)

where $\epsilon_{n-1}(p,\eta)$ are homogeneous functions in $\eta$ of degree
$(n-1)$, such that they satisfy equations (5.6). Relation (5.8) is essentially
a gauge transformation with the parameters $\epsilon_{n-1}(p,\eta)$. Thus, the
construction (5.4)-(5.5) reproduces the description of gauge fields with all
integer spins $n>0$ in terms of distributions (see [6, 26]). A method similar
to that described in this section but using additional spinor variables was
discussed in [27].

##  6 Conclusion

Let us briefly formulate the results.

In this paper, we have presented the main aspects of constructing a field
realization of the $4D$ Poincaré group UIR’s. We considered all possible
physically interesting cases of UIR’s including massive and massless
representations. The basic fundamentals of our approach were the Wigner method
for constructing induced representations. At the same time, the key point was
to find a relation between the Bargmann-Wigner wave functions, on which
induced representations are implemented, and relativistic fields which are
locally transformed under the Poincaré group.

It is shown that such a connection is provided by the generalized Wigner
operator, which explicitly relates the Bargmann-Wigner wave functions and
local relativistic fields. In the massive case, this operator is described by
the finite-dimensional matrix and the local fields obtained are automatically
satisfied by the well-known Dirac-Pauli-Fierz equations. In the massless case
including both helicity and infinite spin representations, the generalized
Wigner operator is either an infinite-dimensional matrix or the kernel of the
integral transform (3.14). Infinite spin fields are defined in the space with
additional commuting coordinates which can be vectorial or spinorial. In the
case of vectorial additional coordinates, we reproduce the infinite spin
fields studied in [19, 20]. In the case of spinor additional variables, we
obtain twistor infinite spin fields [21, 22]. In the case of massless helicity
representations, the generalized Wigner operator contains the
$\delta$-function that allows us to introduce local fields in form of
potentials which are defined up to gauge transformations.

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# Race Bias Analysis of Bona Fide Errors
in face anti-spoofing

Latifah Abduh & Ioannis Ivrissimtzis
Department of Computer Science, Durham University, UK
e-mail:{latifah.a.abduh<EMAIL_ADDRESS>
###### Abstract

The study of bias in Machine Learning is receiving a lot of attention in
recent years, however, few only papers deal explicitly with the problem of
race bias in face anti-spoofing. In this paper, we present a systematic study
of race bias in face anti-spoofing with three key characteristics: the focus
is on analysing potential bias in the bona fide errors, where significant
ethical and legal issues lie; the analysis is not restricted to the final
binary outcomes of the classifier, but also covers the classifier’s scalar
responses and its latent space; the threshold determining the operating point
of the classifier is considered a variable. We demonstrate the proposed bias
analysis process on a VQ-VAE based face anti-spoofing algorithm, trained on
the Replay Attack and the Spoof in the Wild (SiW) databases, and analysed for
bias on the SiW and Racial Faces in the Wild (RFW), databases. The results
demonstrate that race bias is not necessarily the result of different mean
response values among the various populations. Instead, it can be better
understood as the combined effect of several possible characteristics of the
response distributions: different means; different variances; bimodal
behaviour; existence of outliers.

_K_ eywords Face presentation attacks $\cdot$ face anti-spoofing $\cdot$
fairness $\cdot$ race bias.

##  1 Introduction

Face recognition is the method of choice behind some of the most widely
deployed biometric authentication systems, currently supporting a range of
applications, from passport control at airports, to mobile phone or laptop
login. A key weaknesses of the technology, preventing it from being employed
in security sensitive applications in uncontrolled environments, as for
example ATM machines for money withdrawal, is its vulnerability to
presentation attacks, where imposters attempt to gain wrongful access by
presenting in front of the system’s camera a photo, or a video, or by wearing
a mask resembling a registered person. As a solution to this problem,
algorithms for presentation attack detection (PAD) are developed, that is,
binary classifiers trained to distinguish between the bona fide samples coming
from live subjects, and those coming from imposters.

The large variety in the types of possible presentation attacks, and the large
variation in the environmental conditions under which they might take place,
make PAD a particularly challenging problem. However, the current state-of-
the-art, utilising the power of deep learning, comprises classifiers with
excellent accuracy rates, and a satisfactory generalisation power to at least
a limited number of previously unseen attacks. Cross-database generalisation
is still problematic, however, it is debatable if this is a real obstacle to
the deployment of PAD algorithms in practical applications, since such
algorithms as usually embedded in specific face recognition systems, with
given camera specifications and configurations.

Here, we deal with the problem of race bias in face anti-spoofing algorithms.
It is a topic that has attracted considerably less research interest than
accuracy and generalisation power, despite the fact that it raises ethical,
legal, and regulatory considerations, which, by their own, can prevent
adoption in specific applications. Addressing this gap, the aim of this paper
is to provide a framework for studying the question: Does the classifier work
equally well on people from all races?.

The proposed race bias analysis process has three key characteristics. First,
the focus is on the bona fide error, that is, on genuine people wrongly
classified as imposters. Bias in this type of error has significant ethical,
legal and regulatory ramifications, and as it has recently point out “creates
customer annoyance and inconvenience, and this is also where bias can occur in
PAD systems”, [1]. Secondly, we analyse various stages of the classification
process. Not just the final binary outcome, but also the scalar responses of
the network prior to thresholding, and before that the representation of the
face image in the network’s latent space. Thirdly, we treat the value of the
threshold that determines the classifier’s operating point on the ROC curve as
a user-defined variable. We do not assume it is fixed by the vendor of the
biometric verification system through a black-box process.

In the rest of the paper, we demonstrate the application of the proposed bias
analysis approach on a face anti-spoofing algorithm based on the recently
proposed Vector Quantized Variational Autoencoder (VQ-VAE) architecture, [2].
The network is trained and validated on the Replay Attack and the SiW
databases, and tested for racial bias on bona fide samples from the SiW and
the RFW databases. Hypotheses are tested using the chi-squared test on the
binary outcomes, the Mann–Whitney U test on the scalar responses, and
Hartigan’s Dip for testing bimodality in the response distributions. To test
for bias in the latent space of the VQ-VAE network, we train an SVM with
encoding vectors from two races, and measure its performance as a binary
classifier.

The contributions of the paper are summarised as follows:

  * •

A demonstration that race bias can be attributed to several characteristics of
the response distributions: different means; different variances; bimodality;
outliers.

  * •

A demonstration that non-specialised databases, such as RFW, can be used to
analyse face anti-spoofing algorithms.

  * •

A VQ-VAE based network for face anti-spoofing.

The rest of the paper is organised as follows. In Section 2, we review the
relevant literature. In Section 3, we describe the VQ-VAE face anti-spoofing
algorithm and the databases we used. In Section 4, we present the bias
analysis on the SiW database, and in Section 5 the bias analysis on the RFW
database. We briefly conclude in Section 6.

##  2 Background

We briefly review the area of face anti-spoofing, and then studies of bias in
machine learning, and PAD in particular.

###  2.1 Face anti-spoofing

The earlier machine learning approaches to PAD were based on handcrafted
features [3, 4], with Histogram of Oriented Gradient (HOG) [5], and Local
Binary Patterns (LBP) [3], among the most popular.

More recent approaches were based on CNNs, [6, 7], or combinations of various
deep network types [8], leading to the current state of the art being based on
various forms of deep learning [9, 10, 11, 12, 13, 8, 14, 15], such as Central
Difference Convolutional Networks (CDCN) [11, 10], or transformers [16].
Following some earlier approaches [17, 18], the current state of the art
algorithms also utilise depth information [19, 13, 15, 20], which can be
estimated by a independently trained neural network, while the use of GANs to
estimate Near Infrared (NIR) information was proposed [14].

The experiment presented in this paper is based on a one class trained
autoencoder. Anomaly detection is a popular approach [21, 22, 23, 24, 25],
offering good generalisation to unseen attacks. In [26], images from face
recognition datasets were added to the two-class training set of an
autoencoder, and improved cross-database generalisation was reported. A
similar behaviour was reported in [27] when images from the in-the-wild were
added to the training set.

###  2.2 Databases

In this paper, the first training set is from Replay-Attack [3], a database
consisting of 50 subjects of three types of ethnicities, 76% Caucasian, 22%
Asian, and 2% African. Our second training set is from SiW [18], a database
consisting of 165 subjects, of four types of ethnicities, 35% of Asian and 35%
Caucasian and 23% Indian, and 7% African American. The bias analysis is
performed on SiW with the subject annotated for ethnicity type by us, and the
already annotated RFW database [28].

Regarding other databases, NUAA [29] was one of the first face anti-spoofing
large databases. It is rarely used these days as its low quality of imagery
poses an unfair challenge in the cross-database validation of algorithms. MSU
MFSD [30] consists of 55 subjects, captured by four different devices, while
OULU [31] has again 55 subjects captured by six different mobile devices. WMCA
[32] contains 72 subjects and information is captured in RGB, depth, infrared,
and thermal. CASIA-SURF [33] consists of 1000 subjects captured in RGB, depth,
and infrared.

The first face anti-spoofing database to include explicit ethnic labels was
CASIA-SURF CeFA [34], which has 1,607 in three ethnicities, captured in three
modalities. In this paper, for bias analysis we use the RFW [28], which
includes four types of ethnicities, Caucasian, Asian, Indian, and African. RFW
does not specialise in face anti-spoofing, and it is more widely used in the
bias analysis literature.

###  2.3 Bias in machine learning

In [35], several high profile cases of machine learning bias are documented;
Google search results appeared to be biased towards women in 2015; Hewlett-
Packard’s software for web cameras struggled to recognize dark skin tones; and
Nikon’s camera software was inaccurately identifying Asian people as blinking.

Thus, given also the ethical, legal, and regulatory issues associated with the
problem of bias within human populations, there is a considerable amount of
research on the subject, especially in face recognition (FR). A recent
comprehensive survey can be found in [36], where the significant sources of
bias [37, 38] are categorised and discussed, and the negative effect of bias
on downstream learning tasks is pointed out. We also note that while the
current deep learning based FR algorithms are under intense scrutiny for
potential bias [39], this is due to their wider deployment in real life
applications, rather than any evidence that they are more biased than
traditional approaches.

In one of the earliest studies of bias in FR, predating deep learning, [40]
reported differences in the performance on humans of Caucasian and East Asian
decent between Western and East Asia developed algorithms. In [41], several
deep learning based FR algorithms are analysed and a small amount of bias is
detected in all of them. Then, the authors show how this bias can be exploited
to enhance the power of malicious morphing attacks to FR based security
systems.

In [42], the authors compute cluster validation measures on the clusters of
the various demographics inside the whole population, aiming at measuring the
algorithm’s potential for bias. Their result is negative, and they argue for
the need of more sophisticated clustering approaches. We note that in our
paper, an investigation in the latent space of the potential for bias, by
measuring the discriminative power of SVMs over the various ethnicities,
returned a similarly negative result. In [43], the aim is the detection of
bias by analysing the activation ratios at the various layers of the network.
Similarly to our work, their target application is the detection of race bias
on a binary classification problem, gender classification in their case. Their
result is positive in that they report a correlation between the measured
activation ratios and bias in the final outcomes of the classifier. However,
it is not clear if their method can be used to measure and assess the
statistical significance of the expected bias.

In Cavazos et al. [44], similarly to our approach, most of the analysis
assumes a one-sided error cost, in their case the false acceptance rate, and
the decision thresholds are treated as user defined variables. However, the
analytical tools they used, mostly visual inspection of ROC curves, do not
allow for a deep study of the distributions of the similarity scores, while,
here, we give a more detailed analysis of the distribution of the responses,
which is the equivalent of the similarity scores. In Pereira and Marcel [45],
a fairness metric is proposed, which can be optimised over the decision
thresholds, but again, there is no in-depth statistical analysis of the
scores, as we do here for the responses, and thus they offer a more limited
insight.

####  2.3.1 Bias in Presentation Attack Detection

The literature on bias in presentation attacks is more sparse. Race bias was
the key theme in the competition of face anti-spoofing algorithm on the CASIA-
SURF CeFA database [46]. Bias was assessed by the performance of the algorithm
under a cross-ethnicity validation scenario. Standard performance metrics,
such as APCER, BPCER and ACER we reported. In [47], the standard CNN models
Resnet 50 and VGG16, were compared for gender bias against the debiasing-VAE
proposed in [48], and several performance metrics were reported. In a recent
white paper by the ID R&D company, which develops face anti-spoofing software,
the results of a large scale bias assessment experiment conducted by Bixelab,
a NIST accredited independent laboratory [1]. Similarly to our approach, they
focus on the bona fide errors, and their aim is the BPCER error metric to be
below a prespecified threshold across all demographics.

Regarding other biometric identification modalities, [49] studied gender bias
in iris PAD algorithms. They reported three error metrics, APCER, BPCER, and
HTER, finding that female users would be less protected against iris PAD
attacks.

##  3 Experimental setup

In this section, we briefly describe and validate the two classifiers that
will be analysed for bias in the subsequent sections. They have the same VQ-
VAE based architecture [2], and their only difference is in the training; in
the first, the network is trained on Replay Attack, and in the second on SiW.
We chose the VQ-VAE architecture because of some recently reported impressive
results, on various computer vision problems, even though it has not been
employed yet in the field of face anti-spoofing.

###  3.1 The VQ-VAE network

The VQ-VAE architecture comprises three parts. A learned codebook, which is
used to discretise the continuous latent vectors to a set of discrete latent
variables; each continuous vector is replaced with its nearest vector in the
codebook. An encoder that maps the input to a sequence $z$ of discrete codes.
A decoder that transforms the sequence $z$ of discrete vectors back to an
image. The final binary classification decision is taken by comparing the MSE
between the input image and the reconstructed one against a threshold
determining the operating point of the algorithm.

Our encoder consists of two convolutional layers with kernel size 4, stride
step 2, padding 1, and followed by a ReLU, one convolutional layer with kernel
size 3, stride step 1, padding 1 and followed by two residual blocks, which
are implemented as ReLU, $3\times 3$ conv, ReLU, $1\times 1$ conv for each
block. The decoder is a symmetrical structure of the encoder that uses
transposed convolutions. The encoder may output a 16x16 grid of vectors, and
each one is quantized, using a codebook of size 512, before being fed to the
decoder. The weight factor $\beta$ was set to 0.25.

The ADAM optimiser was used with a learning rate of 1e-3, and the model was
trained for 100 epochs with a batch size of 16. The total loss consisted of
three components: reconstruction loss, codebook loss, and commitment loss. The
code was written entirely in Python, on the Pytorch platform. The experiments
were run on an Intel Core i7 CPU, with 64 GB of RAM and an Nvidia GTX 1650.

Figure 1: The architecture of the proposed VQ-VAE.

Figure 2: Top: The responses of the RA trained network on each race of the SiW
testset. From left to right: African, Asian, Caucasian, Indian. Bottom: The
responses of the SiW trained network.

###  3.2 Data preparation

For faces detection we used the Multi-Task Cascade Convolutional Neural
Network (MTCNN) [50]. The detected faces were rotationally aligned, making the
line between the two eye centres horizontal, and cropped at $64\times 64$
resolution. Table 1 summarises the sizes of the datasets. Notice that the
training sets consisted of bona fide data, only.

Table 1: Sizes of training and test datasets. The test sets are equally split between clients and imposters. The SiW test set, consists of 400 samples from each race. | Training | Test
---|---|---
RA | 7465 | 400
SiW | 124,000 | 1600

###  3.3 Network validation

We validated the classifiers on RA and SiW, first under an intra-database
protocol, and then with a cross-database protocol. When testing on SiW, we
also report error rates for each race separately.

  * Intra-database protocol: on an independent validation set we compute the threshold corresponding to the Equal Error Rate (EER). We use this threshold to compute the Half-Total Error Rate (HTER) on the test set. On SiW, we use a single threshold for all races.

  * Cross-database protocol: here, thresholds estimated on one database do not work on the other. So, we just report the EERs computed directly on the test sets.

Table 2 shows the error computed under the testing protocol. The intra-
database error is significantly smaller in RA than in SiW (.055 vs .169),
indicating an easier classification task, as RA has less diversity in poses
and expressions. However, as expected, the error on SiW is smaller when we
train on SiW rather than RA (.169 vs .208).

Table 2: Error rates computed under the testing protocol. | RA test | SiW test
---|---|---
RAtr | .055 [thr = .015 (HTER)] | .208 [thr = .031 (EER)]
SiWtr | .180 [thr = .065 (EER)] | .169 [thr = .054 (HTER)]

Table 3 shows HTERs for each race of the SiW testset. The thresholds are taken
from the corresponding tests of Table 2. We note that the SiW trained network
is performing better on the SiW testset, not only in total, but also on each
race separately. For each classifier and each race, Figure 2 shows the
histograms of the responses and the corresponding thresholds.

Table 3: HTERs computed for each race separately on the SiW testset. | Af | As | Ca | In
---|---|---|---|---
HTER RAtr (thr = .031) | .135 | .247 | .260 | .192
HTER SiWtr (thr = .054) | .115 | .202 | .210 | .150

##  4 Bias analysis on SiW

Here, we use the SiW testset of Section 3. The binary outcomes of the
classifiers are analysed with the chi-squared test, the scalar responses with
the Mann–Whitney U test [51], and finally, the encoding vectors are analysed
by using them to train and test an SVM on the task of race classification. The
bias analysis process is summarised in Fig. 3.

Figure 3: The bias analysis process. The binary outcome analysis is shown in
purple, the scalar responses analysis in blue, and the latent space analysis
in orange.

###  4.1 Statistical analysis of the binary outcomes

First, we analyse the binary outcomes corresponding to the thresholds of
Section 3, that is, 0.031 for the RA-trained classifier and 0.054 for the SiW
trained classifier. For each pair of races, we formed the $2\times 2$
contingency tables, and applied the chi-squared test, computing p-values for
the hypothesis that the probability of misclassification of a bona fide sample
from the race with the most misclassifications is higher.

The results are summarised in Table 4. In several cases, the p-values are low,
meaning that for any reasonable threshold of statistical significance the bias
hypothesis is accepted. In other cases, p-values above 0.05 mean that bias has
not been detected.

Table 4: p-values of the chi-squared tests for the thresholds in Section 3. | Af-As | Af-Ca | Af-In | As-Ca | As-In | Ca-In
---|---|---|---|---|---|---
RAtr | .0000 | .0001 | .1465 | .2532 | .0000 | .0000
SiWtr | .1158 | .0104 | .0147 | .0000 | .0000 | 1.0

Next, we treat the thresholds, that is, the operating points on the ROC curve,
as a variable. In Figs. 4, 5, we plot the p-value as a function of the
response, for the two classifiers and the six pairs of races. We note that,
over the range of thresholds, there are several disconnected intervals
corresponding to high bias, which means that threshold optimisation for low
bias should not assume unique solutions, as it is often implicit in the
literature.

Figure 4: For each pair of races, graphs of the p-value as a function of the
threshold. The classifier was trained on RA.

Figure 5: For each pair of races, graphs of the p-value as a function of the
threshold. The classifier was trained on RA.

###  4.2 Statistical analysis of the scalar responses

For an insight in the behaviour of the graphs of the p-values we analyse the
scalar responses of the algorithm. Figs. 6, 7 show a comparison of the
histograms of the responses for each pair of races. We note the complex
behaviour of the density functions, which induce a complex bias behaviour. In
particular, bias at certain thresholds can be caused by differences in the
means of the responses; differences in their variances; response bimodality;
or outliers.

Figure 6: For each pair of races in SiW, the histogram of the responses on
bona-fide images. The classifier was trained on RA.

Figure 7: For each pair of races in SiW, the histogram of the responses of
bona-fide images. The classifier was trained on SiW.

Regarding the means of the responses, since the Shapiro-Wilk test rejected the
hypothesis of normal distributions, we opted for the Mann–Whitney U test.
Table 5 shows the computed p-values on each pair of races, for the hypothesis
that the values of randomly selected responses from the two populations are
different.

Table 5: p-values of the Mann–Whitney U test on each pair of races. | Af-As | Af-Ca | Af-In | As-Ca | As-In | Ca-In
---|---|---|---|---|---|---
RAtr | .0000 | .0015 | .0085 | .0052 | .0000 | .0000
SiWtr | .0001 | .0078 | .0000 | .1560 | .0000 | .0000

Regarding bimodality, we used Hartigan’s Dip Test [52], with 50 bins, to test
each race for bimodality. We note that for the 200 samples we have from each
race, a statistical significance of 95% corresponds to a critical value of
0.037. Table 6 shows the computed dip values, together with the means and
standard deviations for each population.

We note that the biggest difference in mean response is between Asian and
Indian from the SiW trained classifier, while the corresponding standard
deviations are similar. Thus, the bias we observe in the corresponding diagram
in Fig 5 is due to the classifier’s higher mean response on Asians compared to
Indians.

On the other hand, the smallest difference in mean response is between Asians
and Caucasians, again under the SiW trained classifier. Thus, the bias we
observe in the corresponding diagram in Fig 5, which for very small threshold
values is statistically significant, is due to different variances. We note
here, that while at such low threshold values a standalone classifier wouldn’t
be very useful, it is still an interesting case when constructing ensembles of
weak classifiers.

Finally, we notice that all Hartigan’s Dip Test values are below the
significance threshold, and thus, all populations should be considered
unimodal. In particular, that means that some very high responses on African
people, especially from the RA trained classifier, should be treated as
outliers. We note that, nevertheless, against all the other three races, these
outliers create a second, or third region of high bias, and in those regions
samples from the African population are treated less favourably.

Table 6: SiW testset: means, standard deviations, and Hartigan’s dip values for the responses of the RA and SiW trained classifiers. | Af | As | Ca | In
---|---|---|---|---
RAtr $\mu$ | .0257 | .0291 | .0274 | .0223
SiWtr $\mu$ | .0418 | .0446 | .0438 | .0355
RAtr $s.d.$ | .0126 | .0084 | .0104 | .0079
SiWtr $s.d.$ | .0142 | .0109 | .0122 | .0096
RAtr dip | .0221 | .0360 | .0335 | .0349
SiWtr dip | .0299 | .0233 | .0366 | .0324

###  4.3 Discrete latent space

The proposed VQ-VAE encodes each image with a 32768 number long sequence of
integers in [0..1023], corresponding to indices of the vectors in the
codebook. Using these latent space representations of the images as inputs,
for each pair of races, we train an SVM with an RBF kernel of Gaussian type.
Table 7 shows AUC values for these SVM classifiers.

Table 7: AUC values for an SVM trained on the VQ-VAE’s latent space encodings of the images. SiW testset. | Af-As | Af-Ca | Af-In | As-Ca | As-In | Ca-In
---|---|---|---|---|---|---
RAtr | .88 | .87 | .74 | .71 | .78 | .83
SiWtr | .74 | .85 | .76 | .72 | .76 | .83

We note that in some cases the SVMs perform quite well, meaning that the VQ-
VAE encodings of the face images contain enough information to discriminate
between races, despite the fact that VQ-VAEs themselves were not trained with
race labels. That means that there is a potential for biased behaviour,
however, on the other hand, the statistical significance of the result and its
practical implications are not clear, and a deeper investigation is required.

##  5 Bias analysis on RFW

In this section, we apply the same bias analysis on a testset from the RFW
database, consisting of 200 images from each race. We note that this time the
race labels come as part of the database, rather than being annotated by us.
Another difference from SiW is that the RFW database is not a specialised face
anti-spoofing database. Thus, as we do not have imposter images, we do not
have empirically established thresholds, as for example the ones corresponding
to EER values. For the part of the analysis requiring specific thresholds, we
use thresholds corresponding to some predetermined TPR values.

####  5.0.1 Statistical analysis of the binary outcomes

Table 8 shows the thresholds corresponding to TPR values of 1%, 2%, 5%, 10%,
20%.

Table 8: Threshold values corresponding to some predetermined TPR values. | 1% | 2% | 5% | 10% | 20%
---|---|---|---|---|---
RAtr | .0460 | .0440 | .0389 | .0336 | .0279
SiWtr | .1206 | .1079 | .0971 | .0838 | .0712

Table 9 shows the results of the chi-squared tests on the binary outcomes. We
note that bias can be detected for some of the thresholds, for some pairs of
races. In Figs. 8, 9, for each classifier, and for each pair of races, we show
the histograms of responses on bona fide images, and the p-values of the chi-
squared test as a function of the threshold. We observe behaviours similar to
those from the tests on the SiW database.

Table 9: p-values of the chi-squared tests for the thresholds shown in Table 8. | Af-As | Af-Ca | Af-In | As-Ca | As-In | Ca-In
---|---|---|---|---|---|---
RAtr 1% | 1.0 | .0718 | .2171 | .1316 | .3680 | 1.0
RAtr 2% | 1.0 | .1271 | .2839 | .0741 | .1774 | 1.0
RAtr 5% | .8494 | .0299 | .1082 | .0116 | .0483 | .7487
RAtr 10% | .0356 | .0069 | .3938 | .0000 | .0021 | .0857
RAtr 20% | .1333 | .0025 | 1.0 | .0000 | .1047 | .0037
SiWtr 1% | 1.0 | 1.0 | 1.0 | 1.0 | .3680 | .6153
SiWtr 2% | .2839 | .4456 | 1.0 | 1.0 | .5001 | .7209
SiWtr 5% | .2924 | .0249 | .4143 | .31053 | 1.0 | .2152
SiWtr 10% | .1158 | .0302 | .3567 | .6501 | .6246 | .2725
SiWtr 20% | .0236 | .0062 | .01241 | .7194 | 0.9039 | .9054

Figure 8: For each pair of races, graphs of the p-value as a function of the
threshold. The classifier was trained on RA.

Figure 9: For each pair of races, graphs of the p-value as a function of the
threshold. The classifier was trained on SiW.

###  5.1 Statistical analysis of the scalar responses

Table 10 shows the p-values of the Mann–Whitney U test for each pair of races,
and Table 11 shows the means, standard deviations and dip values for each
population.

We note that Hartigan’s Dip Test detects a bimodality in the responses of the
SiW trained algorithm on Indian people, having a dip value of 0.055, above the
significance threshold of 0.037. This can also be verified by visual
inspection of the corresponding histograms of responses, which, for each pair
of races, are shown in Figs. 10, 11. We also note that this bimodality can be
detected in the behaviour of corresponding graphs of the p-values of the chi-
squared test. Indeed, in the three graphs in Fig. 9 corresponding to Indian
people, we can detect two distinct regions of higher bias, even though the
second one does not reach the level of statistical significance.

Table 10: p-values of the Mann–Whitney U test on each pair of races. | Af-As | Af-Ca | Af-In | As-Ca | As-In | Ca-In
---|---|---|---|---|---|---
RAtr | .0067 | .0253 | .3216 | .0000 | .0137 | .0044
SiWtr | .0004 | .0058 | .0062 | .2509 | .2743 | .4805
Table 11: RFW testset: means, standard deviations, and Hartigan’s dip values for the responses of the RA and SiW trained classifiers. | Af | As | Ca | In
---|---|---|---|---
RAtr $\mu$ | .0217 | .0239 | .0194 | .0214
SiWtr $\mu$ | .0509 | .0579 | .0569 | .0579
RAtr $s.d.$ | .0094 | .0101 | .0068 | .0082
SiWtr $s.d.$ | .0175 | .0220 | .0223 | .0220
RAtr dip | .0175 | .0178 | .0299 | .0216
SIWtr dip | .0114 | .0225 | .0149 | .0550

Figure 10: For each pair of races in RFW, the histogram of the responses of
bona-fide images. The classifier was trained on RA.

Figure 11: For each pair of races in RFW, the histogram of the responses of
bona-fide images. The classifier was trained on SiW.

###  5.2 Discrete latent space

Table 12 shows the AUC values of SVM classifiers trained and tested, on each
pair of races, with the Quantised Vectors. We note an even better performance
than in the case of the SiW testset, implying an even higher potential for
bias.

Table 12: AUC values for an SVM trained on the VQ-VAE’s latent space encodings of the images. RFW testset. | Af-As | Af-Ca | Af-In | As-Ca | As-In | Ca-In
---|---|---|---|---|---|---
RAtr | .95 | .97 | .94 | .89 | .89 | .86
SiWtr | .96 | .98 | .94 | .89 | .86 | .85

##  6 Conclusion

We conducted a systematic, largely empirical, study of race bias in face anti-
spoofing, based on a series of statistical and algorithmic tests. Our premises
were that we are interested in the bona fide error; that binary outcomes,
scalar responses, and latent space should all be analysed for bias; and that
the threshold determining the classifier’s operating point should not be
assumed fixed by a black-box procedure.

Our main finding is that the behaviour of race bias depends on several
characteristics of the distributions of the corresponding classifier’s
responses: differences in the means; differences in the variance; bimodal
responses; outliers. The main implication of this finding is that the
behaviour of bias can be quite complex and, for example, methods optimising
for low race bias should not assume unique solutions to the problem. More
generally, in our context, bias should be treated as something more complex
than the difference in the means of two populations, a misconception that
might be reinforced by the fact that in statistics, colloquially, the term
bias is often used to describe the component of the error attributed to the
difference in the means.

Our recommendations for future work include the theoretical study of bias,
especially under the reasonable assumption of log-normality in the
distributions of the responses. We also propose conducting an empirical but
more systematic study of the relation between the behaviour of bias detecting
algorithmic tests on the latent space, and statistically significant bias in
the binary outcomes of the classifier.

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