# Algebraic Approach to Relativistic Landau Levels in the Symmetric Gauge

Ulrich D. Jentschura  Department of Physics and LAMOR, Missouri University of
Science and Technology, Rolla, Missouri 65409, USA

###### Abstract

We use an algebraic approach to the calculation of Landau levels for a uniform
magnetic field in the symmetric gauge with a vector potential
$\vec{A}=\tfrac{1}{2}(\vec{B}\times\vec{r})$, where $\vec{B}$ is assumed to be
constant. The magnetron quantum number constitutes the degeneracy index. An
overall complex phase of the wave function, given in terms of Laguerre
polynomials, is a consequence of the algebraic structure. The relativistic
generalization of the treatment leads to fully relativistic bispinor Landau
levels in the symmetric gauge, in a representation which writes the
relativistic states in terms of their nonrelativistic limit, and an
algebraically accessible lower bispinor component. Negative-energy states and
the massless limit are discussed. The relativistic states can be used for a
number of applications, including the calculation of higher-order quantum
electrodynamic binding corrections to the energies of quantum cyclotron
levels.

###### Contents

  1. I Introduction
  2. II Nonrelativistic Problem
  3. III Relativistic Landau Levels
  4. IV Negative–Energy States
  5. V Massless Limit
  6. VI Conclusions

##  I Introduction

This paper is about nonrelativistic and relativistic Landau levels for
spin-$1/2$ particles, pertinent to a uniform magnetic field. We aim to develop
algebraic relations, which describe the energetically degenerate Landau levels
in terms of cyclotron and magnetron quantum numbers. Furthermore, we aim to
generalize the treatment to the fully relativistic domain, where the Landau
levels become eigenstates of the magnetically coupled Dirac equation. Landau
levels, including the relativistic case, are important for many physical
processes, first and foremost perhaps, in the context of the determination of
fundamental physical constants, for quantum cyclotron processes in Penning
traps Brown and Gabrielse (1982, 1986); Jentschura (2023); Jentschura and
Moore . Among the many other application, we mention the description of the
quantum Hall effect Goerbig (2009, 2011, 2022); Ciftja (2020, 2023), 2D
quantum dots Bhuiyan and Marsiglio (2020), astrophysical phenomena occurring
the magnetars Maruyama  _et al._ (2018), and processes related to synchrotron
radiation Sokolov and Ternov (1986); Karlovets and Di Piazza .

The constant, uniform magnetic field is assumed to be directed along the $z$
axis, and used in the form
$\vec{B}_{\mathrm{T}}=B_{\mathrm{T}}\,\hat{\mathrm{e}}_{z}$, where we use the
subscript $\mathrm{T}$ to denote the possible application to a Penning trap
Brown and Gabrielse (1982, 1986). Landau levels are normally calculated Landau
(1937) in the Landau gauge $\vec{A}=-By{\hat{\rm e}}_{x}$ [see Eq. (112.1) of
Ref. Landau and Lifshitz (1958)]. Upon solving the corresponding Schrödinger
equation, one finds that in the Landau gauge, energy eigenstates are also
eigenstates of the $x$ component of the momentum component $p_{x}$. The
momentum component $p_{x}$ becomes a parameter which shifts the center of
oscillation for the effective harmonic oscillator potential, which acts onto
the motion in the $y$ direction [see Eq. (112.5) of Ref. Landau and Lifshitz
(1958)]. In the Landau gauge, the value of $p_{x}$ does not affect the energy
of the state and becomes a continuous degeneracy index.

Here, we use the so-called symmetric gauge Powell and Crasemann (1965);
Bhuiyan and Marsiglio (2020), where the vector potential is taken as
$\vec{A}=\tfrac{1}{2}(\vec{B}_{\mathrm{T}}\times\vec{r})$. In the symmetric
gauge, for the nonrelativistic case, the eigenfunctions and energies are in
principle known Powell and Crasemann (1965); Bhuiyan and Marsiglio (2020).
Here, we augment the theory by identifying the raising and lowering operators
of the cyclotron and magnetron motions as the determining dynamic variables of
the problem. This enables us to fully clarify the origin of the degeneracy of
the (nonrelativistic) Landau levels in the symmetric gauge, and greatly
simplify the evaluation of matrix elements of eigenstates.

We also find the fully relativistic generalization of the Landau levels,
corresponding to the relativistic quantum cyclotron states in the limit of
vanishing axial frequency. The relativistic states can be used as input for
higher-order quantum electrodynamic calculations of quantized-field effects in
Penning traps Jentschura and Moore , and any physical systems where
relativistic effects become important. Units with $\hbar=c=\epsilon_{0}=1$ are
used throughout this paper.

##  II Nonrelativistic Problem

We start our treatment of Landau levels in the symmetric gauge, inspired by an
electron in a Penning trap Brown and Gabrielse (1982, 1986); Wienczek  _et
al._ (2022), where an additional electric quadrupole potential leads to axial
confinement. In the limit of a vanishing axial frequency, the cyclotron and
magnetron levels of the Penning trap become equal to Landau levels. In
general, quantum cyclotron states are described Brown and Gabrielse (1982,
1986); Wienczek  _et al._ (2022) by the quantum numbers $k\ell ns$, where
$k=0,1,2,\dots$ is the axial excitation, $\ell=0,1,2,\dots$ is the magnetron
quantum number, $n=0,1,2,\dots$ is the cyclotron quantum number, and $s=\pm 1$
is the spin projection quantum number.

The kinetic momentum $\vec{\pi}_{\mathrm{T}}=\vec{p}-e\,\vec{A}$, in the
symmetric gauge, can be written as

$\displaystyle\vec{\pi}_{\mathrm{T}}=$
$\displaystyle\;\vec{\pi}_{\parallel}+\vec{p}_{\perp}\,,$ (1a)
$\displaystyle\vec{\pi}_{\parallel}=$
$\displaystyle\;\vec{p}_{\parallel}-\frac{e}{2}\left(\vec{B}_{\mathrm{T}}\times\vec{r}\right)\,,\quad\vec{p}_{\parallel}=p_{x}\,{\hat{\rm
e}}_{x}+p_{y}\,{\hat{\rm e}}_{y}\,,$ (1b) $\displaystyle\vec{B}_{\mathrm{T}}=$
$\displaystyle\;B_{\mathrm{T}}\,\hat{\mathrm{e}}_{z}\,.\qquad\vec{p}_{\perp}=p_{z}{\hat{\rm
e}}_{z}\,.$ (1c)

Here, $e$ is the electron charge. As already mentioned, we use the symmetric
gauge for the vector potential
$\vec{A}=\tfrac{1}{2}(\vec{B}_{\mathrm{T}}\times\vec{r})$, rather than
$\vec{A}^{\prime}=-By{\hat{\rm e}}_{x}$, which yields the same magnetic field.
The gauge transformation $\Lambda=-\tfrac{1}{2}\,B_{\mathrm{T}}xy$ leads from
$\vec{A}=\tfrac{1}{2}(\vec{B}_{\mathrm{T}}\times\vec{r})$ to
$\vec{A}^{\prime}$. We note that the physical interpretation of a wave
function can depend on the gauge (see p. 268 of Ref. Lamb (1952)). The choice
$\vec{A}=\tfrac{1}{2}(\vec{B}_{\mathrm{T}}\times\vec{r})$ leads to wave
functions which are confined in the $x$ and $y$ directions, while, with the
choice $\vec{A}^{\prime}$, one obtains unconfined wave functions in the $y$
direction.

Furthermore, the choice
$\vec{A}=\tfrac{1}{2}(\vec{B}_{\mathrm{T}}\times\vec{r})$ is generalizable to
the case of a nonvanishing additional electric quadrupole trap field Brown and
Gabrielse (1982, 1986). The nonrelativistic Hamiltonian is given as follows,

$\displaystyle H_{0}=$
$\displaystyle\;\frac{(\vec{\sigma}\cdot\vec{\pi}_{\mathrm{T}})^{2}}{2m}=H_{\parallel}+\frac{\omega_{c}}{2}\sigma_{z}+\frac{p_{z}^{2}}{2m}\,,$
(2) $\displaystyle\omega_{c}=$
$\displaystyle\;-\frac{e\,B_{\mathrm{T}}}{m}=\frac{|e|\,B_{\mathrm{T}}}{m}\,.$
(3)

Here, $|e|=-e$ is the modulus of the electron charge. The spin-independent
part of the Hamiltonian relevant for the $xy$ plane is

$\displaystyle H_{\parallel}=$
$\displaystyle\;\frac{\vec{\pi}_{\parallel}^{\,2}}{2m}=\frac{\vec{p}_{\parallel}^{\,2}}{2m}+\frac{\omega_{c}}{2}L_{z}+\frac{m\omega_{c}^{2}}{8}\,\rho^{2}$
$\displaystyle=$
$\displaystyle\;\omega_{c}\left(a_{c}^{\dagger}\,a_{c}+\frac{1}{2}\right)\,,$
(4a)

where the $a_{c}^{\dagger}$ and $a_{c}$ are the raising and lowering operators
of the cyclotron motion. We have set the electric quadrupole potential of the
trap equal to zero, wherefore, in the sense of Ref. Brown and Gabrielse (1982,
1986), the magnetron frequency vanishes. The cyclotron ($c$) and magnetron
(${\rm m}$) raising ($a_{c}^{\dagger}$ and ($a_{\rm m}^{\dagger}$) and
lowering operators ($a_{c}$ and $a_{\rm m}$) are given as follows,

$\displaystyle a_{c}=$
$\displaystyle\;\frac{1}{\sqrt{2}}\,\left({\tilde{a}}_{0}\,p_{x}-{\rm
i}{\tilde{a}}_{0}\,p_{y}-\frac{{\rm
i}\,x}{2\,{\tilde{a}}_{0}}-\frac{y}{2\,{\tilde{a}}_{0}}\right)\,,$ (5a)
$\displaystyle a^{\dagger}_{c}=$
$\displaystyle\;\frac{1}{\sqrt{2}}\,\left({\tilde{a}}_{0}\,p_{x}+{\rm
i}{\tilde{a}}_{0}\,p_{y}+\frac{{\rm
i}\,x}{2\,{\tilde{a}}_{0}}-\frac{y}{2\,{\tilde{a}}_{0}}\right)\,,$ (5b)
$\displaystyle a_{\rm m}=$
$\displaystyle\;\frac{1}{\sqrt{2}}\,\left({\tilde{a}}_{0}\,p_{x}+{\rm
i}{\tilde{a}}_{0}\,p_{y}-\frac{{\rm
i}\,x}{2\,{\tilde{a}}_{0}}+\frac{y}{2\,{\tilde{a}}_{0}}\right)\,,$ (5c)
$\displaystyle a_{\rm m}^{\dagger}=$
$\displaystyle\;\frac{1}{\sqrt{2}}\,\left({\tilde{a}}_{0}\,p_{x}-{\rm
i}{\tilde{a}}_{0}\,p_{y}+\frac{{\rm
i}\,x}{2\,{\tilde{a}}_{0}}+\frac{y}{2\,{\tilde{a}}_{0}}\right)\,.$ (5d)

In these formulas, the generalized (magnetic) Bohr radius is (we temporarily
restore factors of $\hbar$ and $c$)

${\tilde{a}}_{0}=\sqrt{\frac{mc^{2}}{\hbar\omega_{c}}}\,\frac{\hbar}{mc}=\frac{\hbar}{\alpha_{c}mc}=\sqrt{\frac{\hbar}{|e|B_{\mathrm{T}}}}=\ell_{B}\,,$
(6)

where we have set $\alpha_{c}=\sqrt{\hbar\omega_{c}/(mc^{2})}$ ($\alpha_{c}$
is a generalized fine-structure constant), and $\ell_{B}$ is the magnetic
length Goerbig (2009). The relations (5) can be inverted,

$\displaystyle p_{x}=$
$\displaystyle\;\frac{1}{2\sqrt{2}{\tilde{a}}_{0}}\,\left[a_{c}+a_{c}^{\dagger}+a_{\rm
m}+a_{\rm m}^{\dagger}\right]\,,$ (7a) $\displaystyle p_{y}=$
$\displaystyle\;\frac{{\rm
i}}{2\sqrt{2}{\tilde{a}}_{0}}\,\left[a_{c}-a_{c}^{\dagger}-a_{\rm m}+a_{\rm
m}^{\dagger}\right]\,,$ (7b) $\displaystyle x=$ $\displaystyle\;\frac{{\rm
i}{\tilde{a}}_{0}}{2\sqrt{2}}\,\left[a_{c}-a_{c}^{\dagger}+a_{\rm m}-a_{\rm
m}^{\dagger}\right]\,,$ (7c) $\displaystyle y=$
$\displaystyle\;-\frac{{\tilde{a}}_{0}}{2\sqrt{2}}\,\left[a_{c}+a_{c}^{\dagger}-a_{\rm
m}-a_{\rm m}^{\dagger}\right]\,.$ (7d)

The nonvanishing commutators are as follows,

$[a_{c},a^{\dagger}_{c}]=[a_{\rm m},a^{\dagger}_{\rm m}]=1\,.$ (8)

All other commutators vanish, which means that, in particular, the cyclotron
and magnetron excitation numbers, together with the $z$ component of the
momentum and the spin projection quantum number $s$, form a complete set of
commuting observables for the nonrelativistic spin-independent problem. The
eigenfunctions are given as follows,

$\displaystyle H_{\mathrm{NR}}\,\psi_{kn\ell s}(\vec{r})=$
$\displaystyle\;E_{\mathrm{NR}}\,\psi_{kn\ell s}(\vec{r})\,,$ (9a)
$\displaystyle E_{\mathrm{NR}}=$
$\displaystyle\;\omega_{c}\,\left(n+\frac{s+1}{2}\right)+\frac{k^{2}}{2m}\,.$
(9b)

Here, $k\in\mathbbm{R}$ is a continuous quantum number characterizing the
momentum component in the $z$ direction. The energy eigenvalue is independent
of the magnetron quantum number $\ell$. The momentum in the $z$ component and
the spin component of the eigenstate can be split off,

$\displaystyle\psi_{k\ell ns}(\vec{r})=$
$\displaystyle\;\psi_{n\ell}(\vec{\rho})\,\frac{{\rm e}^{{\rm
i}kz}}{\sqrt{2\pi}}\,\chi_{s}\,,$ (10a) $\displaystyle\vec{\rho}=$
$\displaystyle\;x\,\hat{\mathrm{e}}_{x}+y\,\hat{\mathrm{e}}_{y}\,,$ (10b)
$\displaystyle\chi_{+1}=$ $\displaystyle\;\left(\begin{array}[]{c}1\\\
0\end{array}\right)\,,\qquad\chi_{-1}=\left(\begin{array}[]{c}0\\\
1\end{array}\right)\,.$ (10g)

According to Refs. Brown and Gabrielse (1982, 1986); Wienczek  _et al._
(2022), one uses the quantum numbers $k\ell ns$ (in this, alphabetically
inspired, sequence) by convention. However, for the spin-independent, two-
dimensional, nonrelativistic case, we use the designation $n\ell$, in
correspondence with the hydrogen atom where the first index indicates the
quantum number which determines the energy (for a comprehensive discussion,
see Chap. 4 of Ref. Jentschura and Adkins (2022)). One may search for the
general formula for the nonrelativistic eigenfunction
$\psi_{n\ell}(\vec{\rho})$, which fulfills the equation

$H_{\parallel}\,\psi_{n\ell}(\vec{\rho})=E_{n}\,\psi_{n\ell}(\vec{\rho})\,,\qquad
E_{n}=\omega_{c}\left(n+\frac{1}{2}\right)\,,$ (11)

where $H_{\parallel}$ is given in Eq. (4). The solution reads as follows,

$\displaystyle\psi_{n\ell}(\vec{\rho})=$
$\displaystyle\;\frac{2^{-\tfrac{1}{2}(|n-\ell|+1)}}{\sqrt{\pi}\,{\tilde{a}}_{0}}\,\sqrt{\frac{\min(n,\ell)!}{\max(n,\ell)!}}\,\left(\frac{\rho}{{\tilde{a}}_{0}}\right)^{|n-\ell|}$
$\displaystyle\;\times{\rm
i}^{|n-\ell|}\,L^{|n-\ell|}_{\min(n,\ell)}\left(\frac{1}{2}\,\left(\frac{\rho}{{\tilde{a}}_{0}}\right)^{2}\right)\,$
$\displaystyle\;\times{\rm e}^{{\rm
i}(n-\ell)\varphi}\,\exp\left[-\frac{1}{4}\,\left(\frac{\rho}{{\tilde{a}}_{0}}\right)^{2}\right]\,.$
(12)

Here, $\rho=|\vec{\rho}|$ and $\varphi=\arctan(y/x)$. The complex phase ${\rm
i}^{|n-\ell|}$ is a consequence of the application of the raising operators on
the ground states [see also Eq. (13a)].

A comparison of Eq. (II) with the literature is indicated. In Eq. (2) of Ref.
Bellandi  _et al._ (1976), one obtains a corresponding result. The spin-
independent wave function obtained in Ref. Bellandi  _et al._ (1976) is equal
to our wave function if one replaces $k$ (in the notation of Ref. Bellandi
_et al._ (1976) by $\min(n,\ell)$, and and $m$ (in the notation of Ref.
Bellandi  _et al._ (1976)) by $n-\ell$ (in our notation). Furthermore, one may
point out that the wave functions given in Eq. (2) of Ref. Bellandi  _et al._
(1976) are specialized to the case $\omega_{c}=2$. In Ref. Ciftja (2020), this
restriction is lifted. The parameter $l_{0}$ defined in Eq. (28) of Ref.
Ciftja (2020) is equal to our parameter ${\tilde{a}}_{0}$,

In Eq. (A31) of Ref. Bhuiyan and Marsiglio (2020), the authors obtain a
corresponding result. Their parameter $\ell$ (which, in the conventions of
Ref. Bhuiyan and Marsiglio (2020) can become negative) is equal to our
$n-\ell$.

Our formalism clearly identifies the cyclotron excitation, which is linked to
the quantum number $n$ that determines the energy, and the magnetron quantum
number $\ell$, which does not shift the energy but shifts the orbit further
outward with increasing $\ell$ for given $n$ (see Figs. 1 and 2). When one
increases $\ell$ by one, the wave function acquires a phase factor $\exp(-{\rm
i}\varphi)$. The cyclotron and magnetron excitations can be associated with
the action of mutually commuting raising operators, acting on the ground
state, and the quantum numbers $n$ and $\ell$ belong to mutually commuting
observables.

Indeed, the nonrelativistic eigenfunctions can be obtained from the ground
state by the operation of the raising operators,

$\displaystyle\psi_{n\ell}(\vec{\rho})=$
$\displaystyle\;\frac{1}{\sqrt{n!\,\ell!}}\,(a_{c}^{\dagger})^{n}\,(a_{\rm
m}^{\dagger})^{\ell}\,\psi_{00}(\vec{\rho})\,,$ (13a)
$\displaystyle\psi_{00}(\vec{\rho})=$
$\displaystyle\;\frac{1}{{\tilde{a}}_{0}\,\sqrt{2\pi}}\,\exp\left(-\frac{\rho^{2}}{4{\tilde{a}}_{0}^{2}}\right)\,.$
(13b)

The number operators count the number of cyclotron and magnetron excitations,

$a_{c}^{\dagger}\,a_{c}\,|n,\ell\rangle=n\,|n,\ell\rangle\,,\qquad a_{\rm
m}^{\dagger}\,a_{\rm m}\,|n,\ell\rangle=\ell\,|n,\ell\rangle\,.$ (14)

The wave functions $\psi_{n\ell}(\vec{\rho})$ are eigenfunctions of the
Hamiltonian $H_{\parallel}$, while their energy eigenvalue only depends on the
cyclotron quantum number $n$, and is independent of the magnetron quantum
number $\ell$. The magnetron quantum number $\ell=0,1,2,\dots$ acts as a
degeneracy index for the Landau level. Here, the magnetron degeneracy index is
countable and leads to a more intuitive form of the wave function, which is
normalizable to unity. Wave functions are orthonormal,

$\int{\rm
d}^{2}\rho\,\psi^{*}_{n\ell}(\vec{\rho})\,\psi_{n^{\prime}\ell^{\prime}}(\vec{\rho})=\delta_{nn^{\prime}}\,\delta_{\ell\ell^{\prime}}\,.$
(15)

A total of four example wave functions are plotted in Figs. 1 and 2.
Furthermore, one has the relation

$\int{\rm d}^{3}r\,\psi^{\dagger}_{k\ell
ns}(\vec{\rho})\,\psi_{k^{\prime}\ell^{\prime}n^{\prime}s^{\prime}}(\vec{\rho})=\delta(k-k^{\prime})\,\delta_{nn^{\prime}}\,\delta_{\ell\ell^{\prime}}\,\delta_{ss^{\prime}}\,.$
(16)

(For the spinor case, we need to use $\psi^{\dagger}$ because it is a two-
component wave function.) With increasing magnetron quantum number, the states
spread further away from the origin, as is evident from the expectation value

$\int{\rm
d}^{2}\rho\,\rho^{2}\,|\psi_{n\ell}(\vec{\rho})|^{2}=(2+2n+2\ell){\tilde{a}}_{0}^{2}\,.$
(17)

This trend is depicted in Fig. 2.

Figure 1:  We display the probability density
$|\psi_{(n=0),(\ell=0)}(\vec{\rho})|^{2}$ of the cyclotron ground state, as
given in Eq. (II). It is radially symmetric and nonvanishing at the origin.
Figure 2:  The cyclotron states with $n=2$ all have the same energy
$\tfrac{5}{2}\omega_{c}$. The probability density
$|\psi_{(n=2)\ell}(\vec{\rho})|^{2}$ vanishes at the origin. With increasing
magnetron excitation $\ell=0,6,12$, the states spread away from the origin.
The azimuthal dependence of the wave function is given by the factor
$\exp[{\rm i}(n-\ell)\,\varphi]$. The plot surfaces are shaded according to
the imaginary part of the wave function, with lighter regions indicating a
positive imaginary part, and darker regions indicating a negative imaginary
part of the wave function.

##  III Relativistic Landau Levels

Before we indulge in the calculations, let us remember that the problem of a
relativistic electron coupled to a uniform electric or magnetic field, has
been discussed in the literature before, notably, in a comprehensive treatise
given in Ref. Demeur (1953). In unnumbered equations between Eqs. (10) and
(11) of Sec. 3 of Chap. 3 of Ref. Demeur (1953), the relativistic eigenstate
of the magnetically coupled Dirac equation is given in terms of parabolic
cylinder functions (Weber functions, denoted as $D_{n}$), The vector potential
is taken in the Landau gauge. Alternative discussions of the relativistic
states are given in Sec. 4 of Chap. 26 of Ref. Sokolov and Ternov (1986) and
in Sec. 6 of Chap. 2 of Ref. Bagrov and Gitman (1990). Our goal here is to
find a representation of the relativistic eigenstates which allows us to
clearly identify the connections of the nonrelativistic limit with the fully
relativistic state.

We here aim to find the relativistic Landau levels in the symmetric gauge. Let
us recall the nonrelativistic (NR) Hamiltonian and its eigenstates,

$\displaystyle H_{\mathrm{NR}}\psi_{\mathrm{NR}}=$
$\displaystyle\;E_{\mathrm{NR}}\psi_{\mathrm{NR}}\,,\quad
H_{\mathrm{NR}}=\frac{(\vec{\sigma}\cdot\vec{\pi}_{\mathrm{T}})^{2}}{2m}\,,$
(18) $\displaystyle\qquad E_{\mathrm{NR}}=$
$\displaystyle\;\omega_{c}\left(n+\frac{s+1}{2}\right)+\frac{k^{2}}{2m}\,.$
(19)

The kinetic momentum in the trap is
$\vec{\pi}_{\mathrm{T}}=\vec{p}-e\vec{A}_{\mathrm{T}}$, while the vector
potential $\vec{A}_{\mathrm{T}}$ for the uniform magnetic field of the trap
can be written as

$\vec{\pi}_{\mathrm{T}}=\vec{p}-e\vec{A}_{\mathrm{T}}\,,\qquad\vec{A}_{\mathrm{T}}=\tfrac{1}{2}(\vec{B}_{\mathrm{T}}\times\vec{r})\,.$
(20)

The Dirac Hamiltonian for a uniform magnetic field is

$\displaystyle H_{\mathrm{D}}=$
$\displaystyle\;\vec{\alpha}\cdot\vec{\pi}_{\mathrm{T}}+\beta\,m\,,$ (21)
$\displaystyle\vec{\alpha}=$
$\displaystyle\;\left(\begin{array}[]{cc}0&\vec{\sigma}\\\
\vec{\sigma}&0\end{array}\right)\,,\quad\beta=\left(\begin{array}[]{cc}\mathbbm{1}_{2\times
2}&0\\\ 0&-\mathbbm{1}_{2\times 2}\\\ \end{array}\right)\,,$ (26)

where we use the Dirac matrix in the Dirac representation. We search for
bispinor solutions of the form

$H_{\mathrm{D}}\,\Psi=E_{\mathrm{D}}\,\Psi\,,\qquad\Psi=\left(\begin{array}[]{c}\psi_{1}\\\
\psi_{2}\end{array}\right)\equiv\Psi_{k\ell ns}\,.$ (27a) We note that $\Psi$
denotes the relativistic bispinor wave function, while $\psi$ is the
nonrelativistic counterpart. We use the following ansatz for the relativistic
eigenstates,
$\psi_{1}=\mathcal{N}\,\psi_{\mathrm{NR}}\,,\qquad\psi_{2}=\mathcal{N}\,\frac{\vec{\sigma}\cdot\vec{\pi}_{\mathrm{T}}}{E_{D}+m}\psi_{\mathrm{NR}}\,.$
(27b)

Acting with $H_{\mathrm{D}}$ on $\Psi$, one obtains

$\displaystyle H_{\mathrm{D}}\Psi=$
$\displaystyle\;\mathcal{N}\,\left(\begin{array}[]{cc}m\mathbbm{1}_{2\times
2}&\vec{\sigma}\cdot\vec{\pi}_{\mathrm{T}}\\\
\vec{\sigma}\cdot\vec{\pi}_{\mathrm{T}}&-m\mathbbm{1}_{2\times 2}\\\
\end{array}\right)\left(\begin{array}[]{c}\psi_{\mathrm{NR}}\\\
\frac{\vec{\sigma}\cdot\vec{\pi}_{\mathrm{T}}}{E_{\mathrm{D}}+m}\psi_{\mathrm{NR}}\end{array}\right)$
(32) $\displaystyle=$
$\displaystyle\;\mathcal{N}\,\left(\begin{array}[]{cc}m\left(1+\dfrac{2E_{\mathrm{NR}}}{E_{\mathrm{D}}+m}\right)\psi_{\mathrm{NR}}\\\\[4.79333pt]
\vec{\sigma}\cdot\vec{\pi}_{\mathrm{T}}\left(1-\dfrac{m}{E_{\mathrm{D}}+m}\right)\psi_{\mathrm{NR}}\end{array}\right)$
(35) $\displaystyle\mathop{=}^{\mbox{!}}$
$\displaystyle\;\mathcal{N}\,E_{\mathrm{D}}\,\left(\begin{array}[]{c}\psi_{\mathrm{NR}}\\\\[4.79333pt]
\dfrac{\vec{\sigma}\cdot\vec{\pi}_{\mathrm{T}}}{E_{\mathrm{D}}+m}\psi_{\mathrm{NR}}\end{array}\right)=E_{D}\,\Psi\,.$
(38)

From the secular equation for the upper component, one obtains

$E_{\mathrm{D}}=m\left(1+\frac{2E_{\mathrm{NR}}}{E_{\mathrm{D}}+m}\right)=m+\frac{2mE_{\mathrm{NR}}}{E_{\mathrm{D}}+m}\,.$
(39)

This can be rewritten as follows,

$E_{\mathrm{D}}-m=\frac{2mE_{\mathrm{NR}}}{E_{\mathrm{D}}+m}\,,\qquad
E_{\mathrm{D}}^{2}-m^{2}=2mE_{\mathrm{NR}}\,.$ (40)

The Dirac energy $E_{\mathrm{D}}$ is obtained as follows,

$\displaystyle E_{\mathrm{D}}=$
$\displaystyle\;\sqrt{m^{2}+2mE_{\mathrm{NR}}}=m\sqrt{1+\frac{2E_{\mathrm{NR}}}{m}}$
$\displaystyle=$
$\displaystyle\;m\sqrt{1+\frac{\omega_{c}}{m}\left(2n+s+1\right)+\frac{k^{2}}{m^{2}}}\,.$
(41)

Somewhat surprisingly, the state with quantum numbers $n=0$ and $s=-1$, and
$k=0$, has exact rest mass energy even in the fully relativistic formalism. A
check of the eigenvector property for the lower component is successful,

$\left(1-\frac{m}{E_{\mathrm{D}}+m}\right)=\frac{E_{\mathrm{D}}}{E_{\mathrm{D}}+m}=E_{\mathrm{D}}\,\frac{1}{E_{\mathrm{D}}+m}\,.$
(42)

In view of the Taylor expansion
$\sqrt{1+\epsilon}=1+\frac{\epsilon}{2}-\frac{\epsilon^{2}}{8}+\frac{\epsilon^{3}}{16}+\mathcal{O}(\epsilon^{4})$,
we can write for positive energy in the limit $k\to 0$,

$E_{\mathrm{D}}(k\to
0)=m+\frac{\omega_{c}}{2m}\left(2n+s+1\right)\\\\[0.4878pt]
-\frac{1}{8}\left(\frac{\omega_{c}}{m}\left(2n+s+1\right)\right)^{2}+\frac{1}{16}\left(\frac{\omega_{c}}{m}\left(2n+s+1\right)\right)^{3}\,.$
(43)

This is consistent with Eqs. (90), (96) and (105) of Ref. Wienczek  _et al._
(2022). We set $\omega_{c}=\alpha_{c}^{2}m$, where $\alpha_{c}$ is the
cyclotron coupling constant. The normalization of the relativistic states
according to : $\int{\rm d}^{3}r\,|\psi_{\mathrm{NR}}|^{2}=\int{\rm
d}^{3}r|\Psi|^{2}=1$ leads to the relation

$\displaystyle\int{\rm d}^{3}r\;\Psi^{\dagger}(\vec{r})\,\Psi(\vec{r})=$
$\displaystyle\;\mathcal{N}^{2}\left[1+\frac{2mE_{\mathrm{NR}}}{(E_{\mathrm{D}}+m)^{2}}\right]\mathop{=}^{\mbox{!}}1\,,$
(44) $\displaystyle\mathcal{N}=$
$\displaystyle\;\left[1+\frac{2mE_{\mathrm{NR}}}{(E_{\mathrm{D}}+m)^{2}}\right]^{-1/2}\,.$
(45)

In the above formulas for the relativistic states, the dependence on the
quantum numbers $k\ell ns$ has been suppressed for notational simplicity.
Restoring the quantum numbers, one finds that

$\int{\rm d}^{3}r\,\Psi^{\dagger}_{k\ell
ns}(\vec{\rho})\,\Psi_{k^{\prime}\ell^{\prime}n^{\prime}s^{\prime}}(\vec{\rho})=\delta(k-k^{\prime})\,\delta_{nn^{\prime}}\,\delta_{\ell\ell^{\prime}}\,\delta_{ss^{\prime}}\,.$
(46)

For completeness, it is perhaps useful to remark that, if desired, one can
easily write the entire operator $\vec{\sigma}\cdot\vec{\pi}_{\mathrm{T}}$ in
terms of cyclotron, magnetron and spin ladder operators, and express the lower
component $\psi_{2}$ of the fully relativistic state, in terms of
nonrelativistic eigenstates of raised and lowered quantum numbers.

##  IV Negative–Energy States

In Dirac theory, there is an energy gap between positive-energy states with
$E\geq m$ and negative-energy states with $E\leq-m$. It is relatively
straightforward to check that the states [cf. Eqs. (27a) and (27b)]

$\Phi=\mathcal{N}\left(\begin{array}[]{c}-\dfrac{\vec{\sigma}\cdot\vec{\pi}_{\mathrm{T}}}{E_{\mathrm{D}}+m}\,\psi_{\mathrm{NR}}\\\\[12.91663pt]
\psi_{\mathrm{NR}}\end{array}\right)$ (47)

with the same normalization factor $\mathcal{N}$ as for positive-energy states
[see Eq. (45)], has the property

$H_{\mathrm{D}}\,\Phi=-E_{\mathrm{D}}\Phi$ (48)

and thus constitutes the negative-energy (antiparticle) state of Dirac theory.
As compared to the positive-energy state, the upper and lower components of
the Dirac bispinor are interchanged, and the upper spinor receives a minus
sign.

##  V Massless Limit

In the limit $m\to 0$, the gap between positive-energy and negative-energy
states vanishes. We recall that

$\omega_{c}m=|e|B_{\mathrm{T}}=\ell_{B}^{-2}\,.$ (49)

The Dirac energy (III) attains the massless limit

$\displaystyle E_{\mathrm{D}}\mathop{=}^{m\to
0}\sqrt{\ell_{B}^{-2}\left(2n+s+1\right)+k^{2}}\,.$ (50)

The square-root dependence on the quantum numbers confirms the corresponding
dependence, obtained on the basis of the Weyl equation, given in Eq. (2.23) of
Ref. Goerbig (2009). The appropriate zero-mass limit of the normalization
factor is

$\mathcal{N}\mathop{=}^{m\to 0}\frac{1}{\sqrt{2}}\,,$ (51)

and the positive-energy solution is

$\Psi\mathop{=}^{m\to
0}\left(\begin{array}[]{c}\dfrac{1}{\sqrt{2}}\,\psi_{\mathrm{NR}}\\\\[14.2083pt]
\dfrac{1}{\sqrt{2}}\,\dfrac{\vec{\sigma}\cdot\vec{\pi}_{\mathrm{T}}}{E_{D}}\psi_{\mathrm{NR}}\end{array}\right)\,.$
(52)

The negative-energy solution is

$\Phi\mathop{=}^{m\to
0}\left(\begin{array}[]{c}-\dfrac{1}{\sqrt{2}}\,\dfrac{\vec{\sigma}\cdot\vec{\pi}_{\mathrm{T}}}{E_{D}}\psi_{\mathrm{NR}}\\\\[14.2083pt]
\dfrac{1}{\sqrt{2}}\,\psi_{\mathrm{NR}}\end{array}\right)\,.$ (53)

##  VI Conclusions

We have considered the calculation of both nonrelativistic as well as fully
relativistic Landau levels in the symmetric gauge where
$\vec{A}=\tfrac{1}{2}(\vec{B}_{\mathrm{T}}\times\vec{r})$, based on an
algebraic approach. The spin-independent part of the nonrelativistic states
finds a natural form in term of cyclotron quantum number $n$, and magnetron
quantum number $\ell$. We find a universal representation of the spinless
nonrelativistic quantum state wave function with quantum numbers $n$
(cyclotron) and $\ell$ (magnetron) in Eq. (II). After including the spin and
the axial motion, we find the nonrelativistic spinor state with quantum
numbers $n$ and $\ell$, and $k$ (axial) and $s$ (spin), as given in Eq. (10).
For the nonrelativistic wave function $\psi=\psi_{k\ell ns}$, the spin-
independent form is given in Eq. (II), and the nonrelativistic spinor form is
given in Eq. (10). The orthonormality relations are given in Eqs. (15) (for
the spin-independent part) and in Eq. (16) (for the spinor wave function).
While the magnetron quanta are raised from the ground state by the magnetron
raising operator $a_{\rm m}^{\dagger}$ [see Eq. (14)], the magnetron quantum
number $\ell$ does not enter the formula for the energy of the Landau level
[see Eqs. (11) and (9)]. Inverting the relations given in Eq. (5), one obtains
a representation of the $x$, $y$, $p_{x}$ and $p_{y}$ operators in terms of
the raising and lowering operators of the cyclotron and magnetron motions [see
Eq. (7)].

In the context of the quantum Hall effect, Landau levels have been considered
in both the Landau as well as the symmetric gauges Goerbig (2009). They are
also relevant to quantum cyclotron states in Penning traps Brown and Gabrielse
(1982, 1986). Indeed, in the limit of vanishing axial confinement, the quantum
cyclotron states approach the Landau levels in the symmetric gauge and our
formulas are essential elements in the discussion of higher-order quantum
electrodynamic corrections to quantum cyclotron energy levels Jentschura and
Moore .

Based on an essential generalization of the nonrelativistic problem (see Sec.
II and Ref. Goerbig (2009)), we calculate the fully relativistic Landau levels
in the symmetric gauge, in Sec. III. These relativistic states constitute
solutions of the magnetically coupled relativistic Dirac equation. The same
quantum numbers $k$, $\ell$, $n$ and $s$, that we found for the
nonrelativistic problem, characterize the relativistic state. The relativistic
wave function is given in Eqs. (27a) and (27b), with the normalization given
in Eq. (44). In the nonrelativistic limit, the relativistic wave functions
reduce to the nonrelativistic states discussed in Sec. II. The fully
relativistic form is needed as an essential ingredient in the calculation of
quantum electrodynamic corrections to quantum cyclotron energy levels
Jentschura and Moore . We mention relativistic quantum dynamics in
synchrotrons as a further potential area of application. The massless limit of
the Dirac bispinor solutions is discussed in Sec. V

## Acknowledgments

The author acknowledge insightful conversations with Istvá Nándori, and
Professor Gerald Gabrielse. Support from the National Science Foundation
(grant PHY–2110294) is gratefully acknowledged. Furthermore, support from the
Templeton Foundation (Fundamental Physics Black Grant, Subaward 60049570 of
Grant ID #61039), is also gratefully acknowledged.

## References

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# Federated Learning on Patient Data for Privacy-Protecting Polycystic Ovary
Syndrome Treatment

Lucía Morris
Stanford University
Stanford, USA 94305
<EMAIL_ADDRESS>
Tori Qiu⋆
Stanford University
Stanford, USA 94305
<EMAIL_ADDRESS>
Nikhil Raghuraman⋆
Stanford University
Stanford, USA 94305
<EMAIL_ADDRESS>
All authors contributed equally.

###### Abstract

The field of women’s endocrinology has trailed behind data-driven medical
solutions, largely due to concerns over the privacy of patient data. Valuable
datapoints about hormone levels or menstrual cycling could expose patients who
suffer from comorbidities or terminate a pregnancy, violating their privacy.
We explore the application of Federated Learning (FL) to predict the optimal
drug for patients with polycystic ovary syndrome (PCOS). PCOS is a serious
hormonal disorder impacting millions of women worldwide, yet it’s poorly
understood and its research is stunted by a lack of patient data. We
demonstrate that a variety of FL approaches succeed on a synthetic PCOS
patient dataset. Our proposed FL models are a tool to access massive
quantities of diverse data and identify the most effective treatment option
while providing PCOS patients with privacy guarantees. Our code is open-
sourced at https://github.com/toriqiu/fl-pcos.

##  1 Introduction

###  1.1 Privacy Concerns with PCOS Patient Data

Biologic hyperandrogenism is a key criterion for polycystic ovary syndrome
(PCOS) diagnosis and is determined by blood work. These blood panels measure
various hormones associated with PCOS and provide a snapshot of the patient’s
overall reproductive health. Blood panel results are sensitive, as they can be
used to extract information about pregnancies, abortions, and miscarriages.
For example, using the hormone levels revealed by blood panel data, it’s
possible to access a patient’s pregnancy history by detecting menstrual
irregularities and using luteinizing hormone (LH) and follicle-stimulating
hormone (FSH) to diagnose pregnancy [1], [2]. These hormone levels are
especially distinct during early pregnancy [3]. A patient file containing
reports of mood disorders or comorbidities, if extracted, could also impact
the patient’s job prospects and insurance rates [4]. Considering the intimate
information PCOS data can reveal about a patient’s medical history, privacy is
essential when handling it. In situations where medical providers are no
longer the sole repositories of patient data, it is not difficult for a third-
party server to obtain medical records [5].

###  1.2 Federated Learning Meets the Challenges of PCOS Research

The field of gynecology research largely relies on primary data collection
methods and randomized controlled trials (RCTs) for studying diagnosis and
treatment. Both these modes of inquiry suffer from a lack of large-scale data.
RCTs conducted in OB/GYN studies typically enroll an average study population
of 200 female patients [6], which is miniscule compared to the millions of
data points available in aggregated databases, even after filtering for a
study’s inclusion criteria. However, RCTs are particularly informative because
randomizing exposure to treatment eliminates selection bias and rules out
confounding variables, establishing causality of a treatment. Despite the
advantages of randomization, RCT findings are not generalizable. Their small
sample sizes preclude analysis of correlations between hormonal levels and
treatment, and they lack guarantees on non-IID data. HIPAA Privacy Rule
prohibits U.S. healthcare providers and insurance companies from using or
disclosing protected health information without explicit patient consent [7].
Voluntary disclosures are unlikely as widespread anxieties of data security
and privacy persist within the patient population [8], and even anonymized
data can be re-identifiable via triangulation with other data sets [9], [10].
Not only does the need for patient consent severely limit data availability,
it is only practical for cases in which consent granted by the patient is
unconditional, since recalling data from those who obtained it is practically
unenforceable. These factors complicate the study of PCOS treatment. Ideally,
study of PCOS treatment can be conducted at scale with minimal data retention
to protect the privacy of subjects.

We propose federated learning (FL) as a method for improving PCOS treatment
outcomes. In particular, we develop models to learn optimal oral contraceptive
pill (OCP) treatments based on blood panel data and historical success of
treatment for similar hormone profiles. We argue that FL is a promising
solution to improve the PCOS treatment process given the privacy concerns
around patient blood panel data and the lack of comprehensive datasets
relevant to PCOS [11]. Isolated data from smaller studies can introduce sample
bias in which demographics (e.g. ethnicity, age, etc.) skew the predictions,
affecting the accuracy of treatment prediction for certain sites. FL allows
access to thousands of valuable data points across a wide geographic area
without the need for a centralized database or sharing of data between
institutions. Using FL to determine personalized medical treatment bypasses
the high risks of direct intervention on patients and the expenses of
conducting clinical trials [12]. With FL, the advantages of large datasets can
be harnessed for PCOS research without the security threats historically
associated with learning at this scale.

###  1.3 Contributions of Our Approach

Current ML research on PCOS focuses on its diagnosis [13], [14],[15], with
much of this research employing the same publicly available dataset of 541
points from Kerala, India [16], [17]. Our key contributions are as follows:

  * •

We devise the first application of ML to PCOS drug recommendation, focusing on
treatment rather than diagnosis. We learn correlations between seven hormonal
metrics associated with PCOS and their compatibility with five common OCPs.
Using these correlations, our models predict the most effective intervention.

  * •

We generate synthetic data to reflect the client pool of a deployed FL model,
significantly expanding the dataset size from existing studies. The larger
dataset helps avoid overfitting to a small group and captures the complete
demographics of patients who are impacted by predictive care.

  * •

We demonstrate a variety of FL approaches achieve excellent performance on the
task of drug recommendation. Our models are evaluated on IID data, non-IID
data, clients with identical dataset sizes, and clients with differing dataset
sizes.

##  2 Background and Related Work

###  2.1 PCOS Lacks a Clear Treatment Process

PCOS is the most common hormonal disorder in women of reproductive age,
impacting an estimated 10% of women worldwide. PCOS is named for the painful,
fluid-filled cysts that form on the ovaries, which can damage organ function
and in some cases require surgical removal. Other common symptoms of PCOS
include irregular menstruation, infertility, hirsutism, acne, mood disorders,
weight gain, and an increased risk for Type II diabetes and heart disease
[18].

The precise cause of PCOS is unknown and no cure exists. Using OCPs to
regulate hormone levels is the primary method of symptom management among
patients [19]. OCPs suppress secretion of luteinizing hormone (LH) and
increase sex hormone-binding globulin levels to decrease androgen and
testosterone levels, correlating with fewer PCOS symptoms [20]. The American
Society of Reproductive Medicine recommends OCPs as the primary treatment
option but provides no guidelines as to which OCPs are effective for various
presentations of the disease [21]. There are over 200 OCPs on the market, and
the birth control industry has an estimated value of over $13 billion [22].
OCPs containing progestins with lower androgenic activity are most effective
in treating PCOS symptoms [19]; examples include Apri, Cyclen, Tri-cyclen,
Yaz, and Diane-35.

Typically, a PCOS patient will use a trial and error approach to test various
OCPs for a 3-month period each before she hopefully finds a solution [23]. The
U.S. healthcare system spends approximately $4 billion diagnosing and treating
PCOS annually, and patients can spend over a year finding treatment [18].
While gynecologists consult with the patient to provide the best OCP
recommendation, the highly variable presentation of PCOS, particularly across
ethnic groups, in conjunction with how poorly understood the condition is, can
make selecting an effective OCP seem arbitrary. Prescribing the most effective
OCP option for PCOS patients is a frustrating and expensive process in need of
a more streamlined solution.

###  2.2 Federated Learning Overview

FL is a subfield of privacy-preserving ML which enables multiple clients to
train a model collaboratively without exchanging datasets with one another.
Training consists of a number of communication rounds. In each communication
round, a central server randomly selects a subset of the clients and
broadcasts the global model to each of them. Each selected client trains on
its local dataset and sends its newly learned model parameters back to the
server. Then, the server selects global parameters by aggregating the local
parameters in a manner that approximately minimizes overall loss.

We consider four FL approaches: FedAvg [24], FedAvgM [25], FedProx [26], and
FedAdam [27]. We designate the first approach as a baseline and select the
latter three approaches to handle heterogeneous client datasets. Technical
details of these approaches can be found in the Appendix.

###  2.3 Federated Learning Applied to Medicine

FL models optimize dataset diversity, prediction accuracy, and data privacy,
making FL the ideal method to harness medical data for improving patient
outcomes. The highly distributed nature of medical data limits the application
of non-federated ML algorithms, which require storage of data on a central
server for training. In practice, this means non-federated models have smaller
datasets to train on and lack sufficient training signal. FL models offer
improved accuracy by aggregating training results from local datasets into a
global model. Importantly, the local data storage units — i.e., client devices
— in FL models only exchange summary statistics such as each local model’s
parameters and dataset size. Privacy is facilitated because local devices can
train on their data and transfer training results back to the central server
without directly uploading the underlying data. In contrast, non-federated ML
approaches house the model and data on a single server, meaning they require
access to a massive, central dataset in order to train. These datasets are
typically provided by telemedicine companies, such as GoodRx, which are
unregulated by HIPAA security rules, and health information companies, which
collect proprietary medical data to achieve clinical data interoperability
[28], [29]. By sending the model to clients instead of sending the data to the
server, FL reduces risk of corruption from a central data controller.

##  3 Experiments

As a proof of concept, we develop FL models which learn OCP treatments based
on synthetic patient data. Evaluation of this model is designed to serve as a
stepping stone for future FL models learned on real patient data. Using
synthetic data is necessary as currently no real patient datasets exist for
the task of predicting PCOS treatment.

###  3.1 Generation of Synthetic Data

####  3.1.1 Patient Profiles

For each dataset, we randomly generate patient profiles for $12$ clients.
These clients represent gynecology practices or hospitals. Each patient
profile contains blood test results of seven key metrics associated with PCOS:
luteinizing hormone (LH) to follicle-stimulating hormone (FSH) ratio, total
testosterone level, dehydroepiandrosterone (DHEA-S) level, prolactin level,
androstenedione level, estradiol level, and anti-müllerian hormone (AMH) level
[30]. The profile also includes which OCP was effective in treating the
patient. For each metric, we identified the range associated with PCOS
diagnosis (see Table 6, column 2) and then split the total range into various
sub-ranges to be correlated with one of five OCP options (see Table 7). These
sub-ranges were determined arbitrarily as existing literature lacks data on
metric distribution. Not every OCP correlates with a specific metric; for
example, Apri works best for patients with a LH-FSH ratio within $2-2.5$, but
lacks correlation with the patient’s prolactin or estradiol levels.

####  3.1.2 Generation of Random IID and Non-IID Data

We consider datasets where distribution across each client is IID and datasets
where distribution is non-IID. Investigating effects of non-IID data is
necessary due to demographic differences across gynecology practices and the
varied distribution of PCOS phenotypes [31]. To generate IID data, we assume
that clients prescribe each OCP with equal probability. To create synthetic
patient profiles, we first select an OCP for each patient independently and
uniformly at random. We then sample hormone metrics uniformly at random for
each profile using the range associated with the patient’s designated OCP (see
Table 7). To generate non–IID data, for each client, we first sample a
multinomial probability distribution using a Dirichlet distribution with
$\alpha=\mathbf{1}$ as the conjugate prior, as in [25]. For each patient, we
sample an OCP from this multinomial probability distribution and, as in the
IID case, sample hormone metrics uniformly at random using the range
associated with the selected OCP. Using a different probability distribution
over OCPs for each client allows us to model differences between practices and
hospitals.

We inject random noise into our data to mimic confounding variables that
impact the optimal OCP treatment. With probability $0.1$, after sampling all
hormone levels for a patient, we change their assigned OCP to one sampled
uniformly at random from all $5$ available OCP options. This makes our dataset
more difficult, capping a model’s maximum accuracy at $0.92$. A model achieves
this maximum accuracy if it correctly predicts treatment for all unchanged
patients and randomly guesses for the $10\%$ of changed patients (with $0.2$
accuracy).

####  3.1.3 Variation in Dataset Size

Clientele and patient numbers vary between gynecology practices. We generate
training datasets containing 12,500 patient profiles per client and training
datasets of varying sizes per client, where the number of patients is sampled
randomly from $Uniform(200,20000)$. Additionally, we create validation and
test datasets for each client that are 25% the size of the test dataset.

##  4 Results

Figure 1: Accuracies observed across FL algorithms. From left to right, these plots depict accuracy on IID data with the same dataset size, non-IID data with the same dataset size, and non-IID data with different dataset sizes. Accuracies for each communication round are obtained by taking the weighted average of accuracies across all clients. Table 1: Accuracies recorded after the last communication round Dataset/Algorithm | FedAvg | FedAvgM | FedProx | FedAdam
---|---|---|---|---
IID data + same dataset size | 0.9211 | 0.9200 | 0.873 | 0.9201
Non-IID data + same dataset size | 0.9181 | 0.9186 | 0.8606 | 0.8676
Non-IID data + dif dataset size | 0.9214 | 0.9209 | 0.9105 | 0.9220

###  4.1 Discussion of Results

We train a model on four different FL algorithms: FedAvg, FedAvgM, FedProx,
and FedAdam, which are described in Tables 2, 3, 4, and 5, respectively.
First, all four approaches achieve the maximum performance of 92% accuracy on
most datasets (see Table 1), meaning the model correctly solves the drug
prediction task and randomly guesses on the 10% of noisy patients. Performance
is also comparable between IID and non-IID data, despite FL performance
traditionally degrading on non-IID data. This shows promise for training on
real-world gynecology data, which is likely to consist of non-IID datasets of
different sizes. In all four algorithms we test, non-IID data distributions
with identical dataset sizes consistently perform slightly worse than the
other two dataset types.

Second, only a limited amount of training is needed to achieve the maximum
performance of 92% accuracy (see Figure 1). While some algorithms (FedProx)
are slower than others (FedAvg) to converge, all tend to converge by the
eighth round of training. This demonstrates that the task can be solved with
limited computational resources. With approximately 92% accuracy, these basic
models predict an OCP that is $4.6$ times more likely to be effective for the
patient than randomly selecting a treatment (see Random Assignment in Figure
1). This provides a sufficient proof of concept that FL can significantly
improve PCOS patient outcomes, reducing the need for patients to try out
multiple drugs and incur the associated costs.

##  5 Conclusions

###  5.1 Limitations

Our results should be interpreted cautiously since we learn on synthetic data.
This data may not match that of real gynecology practices, particularly with
respect to the number of clients, size of client datasets, and feature
distributions. In comparison to our $12$ clients and eight communication
rounds (with six clients randomly selected per round), typical cross-device FL
systems select around $50-5,000$ clients per round and take $500-10,000$
rounds to converge [32].

The models must begin naively as currently no substantial research exists
linking these hormone metrics with OCP performance. When deployed, the model
can reveal features which consistently demonstrate low correlations with OCP
effectiveness, and this non-correlation can be validated by analysis of
medical professionals. The patient profile can be enhanced to include
additional factors such as age, BMI, and the presence or absence of PCOS-
associated symptoms such as mood disorders, acne, hirsutism, and irregular
menstrual cycles. Including patient-reported features into the model improves
predictions to better reflect patient disposition and key symptom concerns.

Finally, what the model considers to be the best OCP for the patient — the
last option they tried at least six months ago — may be an incorrect
assumption for some patient profiles. High healthcare costs can prevent the
patient from scheduling a follow-up appointment when their initial OCP is
ineffective, or they may forego the optimal OCP because of insurance and
pricing constraints.

###  5.2 Further Directions

Challenges that may hinder model performance in realistic settings include
partially labeled data and adversarial attacks. Patient data supplied by
multiple sites may contain missing or incomplete observations, and deciding
how to discard or note absence of an observation during training is a relevant
area to explore next [33], [34], [35]. While our selected algorithms address
data heterogeneity, they assume the clients and server are trustworthy actors.
Algorithms which address privacy, such as FedPerm [36], can handle a server
which attempts to extract clients’ data by inspecting model updates sent by
participants. A client may also reconstruct another client’s private data via
gradient inversion attacks [37] or poison the global model by sending
manipulated parameter updates. FL is highly compatible with additional privacy
features such as homomorphic encryption and secure aggregation, which prevent
the server or clients from inferring information about data based on model
updates [38], [39], [40]. The security of PCOS patient data in our model can
be improved by incorporating these techniques; for example, Salvia is an
extension that can securely aggregate updates received from clients and
tolerate various percentages of corrupt users [41] [42].

Our work reports only aggregated model performance across all local clients.
In a deployment setting, it is important to consider how performance varies
among demographic groups. Approaches such as q-FedAvg [43], which we did not
consider, handle situations in which the model underperforms on certain
clients. Assuming certain hormonal metrics vary by demographic, models must
ensure equitable outcomes for each client’s patient population. Our
aggregation function obtains a global accuracy by weighting each client’s
accuracy by its dataset size, meaning each patient profile is equally
influential to training and the overall classification error is prioritized.
Incorporating an algorithm to improve subpopulation accuracy, such as
MultiAccuracyBoost [44], offers our model a fairness guarantee [45].

There is also the question of system heterogeneity, which our experiments do
not address. Resource-constrained devices may lag due to connectivity issues
or variable speeds, and hardware constraints limit the amount of data that can
be stored in memory. Given FedAvg’s policy of dropping client devices that
fail to complete the required number of epochs within a certain timeframe,
important data is excluded during training and fair resource distribution
across devices is compromised. Further testing should be conducted to ensure
the model accurately accommodates devices with varying computational
capacities; for example, testing can employ a complete implementation of
FedProx, which incorporates partial updates made by “straggler” devices [24].

Women’s reproductive health is an area of medicine with massive amounts of
untapped data and serious privacy requirements. Even if our model, when
deployed on real PCOS patient data, does not detect pertinent correlations
between hormone metrics and the effectiveness of OCP treatments, we hope to
illuminate FL as an approach to address the many other frontiers in women’s
health research.

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##  Appendix A Appendix

###  A.1 Federated Learning Algorithms

In our work, we explore four FL approaches to robustly evaluate the
applications of FL to PCOS patient data. All FL approaches that we consider
follow the procedure described in the “FL Overview” section and summarized in
Algorithm 1 below. Some notation is adapted from [27].

Algorithm 1 Federated Learning Approaches

$T$, the number of communication rounds, $N$, the total number of training
examples across all clients, and $K$, the number of local SGD epochs.

Initialize $W$, the global model weights

for $t$ = 0, …, $T-1$ do

Sample a set $C$ of clients

for client $i\in C$ do

Broadcast $W$ to client $i$

Let $\mathcal{L}_{i}$ be the loss function

Let $n_{i}$ be the number of local training examples

$w_{i,opt}=\texttt{optimize}(\mathcal{L}_{i},K)$ $\triangleright$ Run SGD
locally for $K$ epochs

$\Delta_{i}=w_{i,opt}-W$ $\triangleright$ Delta from global model

$p_{i}=n_{i}/N$

end for

$\Delta=\sum_{i=1}^{|C|}p_{i}\Delta_{i}$ $\triangleright$ Average the deltas

Update the global model $W$ using $\Delta$

end for

Table 2: Steps required for one communication round of FedAvg. Notation defined in Algorithm 1. Loss function on $i$th client | $\mathcal{L}_{i}=f_{i}(w_{i})$
---|---
Optimal weights on $i$th client | $w_{i,opt}=\texttt{optimize}(\mathcal{L}_{i},K)$
Delta from the global model on $i$th client | $\Delta_{i}=w_{i,opt}-W$
Delta averaged across all clients | $\Delta=\sum_{i=1}^{|C|}p_{i}\Delta_{i}$
Global model update | $W\leftarrow W+\Delta$

####  A.1.1 FedAvg (see Table 2)

FedAvg (Federated Averaging) is the simplest FL approach, in which a single
global model is broadcast from a server to a randomly selected subset of
clients in each communication round [24]. Each client performs a few epochs of
stochastic gradient descent (SGD) using its own local data on its own copy of
the global model. Finally, the updated global models are broadcast back to the
server, which updates the global model as an average of all client models,
weighted by the number of training examples on each client.

We use FedAvg as our baseline approach for minimizing loss across clients.
While FedAvg demonstrates strong performance on the average distribution of
all clients, the technique does not explicitly address cases of non-IID data
between clients. This approach raises concern because patient data
distributions likely differ across clients, particularly due to patient
demographics. We therefore experiment with three additional approaches
designed to address data heterogeneity. Note that in our setting,
$f_{i}(w_{i})$ refers to a cross entropy loss.

Table 3: Steps required for one communication round of FedAvgM. Notation defined in Algorithm 1. Steps that differ from those of FedAvg are bolded. Loss function on $i$th client | $\mathcal{L}_{i}=f_{i}(w_{i})$
---|---
Optimal weights on $i$th client | $w_{i,opt}=\texttt{optimize}(\mathcal{L}_{i},K)$
Delta from the global model on $i$th client | $\Delta_{i}=w_{i,opt}-W$
Delta averaged across all clients | $\Delta=\sum_{i=1}^{|C|}p_{i}\Delta_{i}$
Global model update |  $v\leftarrow\beta v+\Delta$ $W\leftarrow W+v$ where $\beta$ is a hyperparameter

####  A.1.2 FedAvgM (see Table 3)

Inspired by the success of applying momentum updates to SGD [46], FedAvgM
(Federated Averaging with Server Momentum) updates the global model by
accumulating the gradients broadcast from the clients with a momentum term
(Equation 3) [25]. FedAvgM is applicable to our setting as the momentum term
potentially makes global updates less susceptible to noisy variations in local
model updates caused by non-IID data.

Table 4: Steps required for one communication round of FedProx. Notation defined in Algorithm 1. Steps that differ from those of FedAvg are bolded. Loss function on $i$th client |  $\mathcal{L}_{i}=f_{i}(w_{i})+\frac{\mu}{2}||W-w_{i}||^{2}$ where $\mu$ is a hyperparameter
---|---
Optimal weights on $i$th client | $w_{i,opt}=\texttt{optimize}(\mathcal{L}_{i},K)$
Delta from the global model on $i$th client | $\Delta_{i}=w_{i,opt}-W$
Delta averaged across all clients | $\Delta=\sum_{i=1}^{|C|}p_{i}\Delta_{i}$
Global model update | $W\leftarrow W+\Delta$
Table 5: Steps required for one communication round of FedAdam. Notation defined in Algorithm 1. Steps that differ from those of FedAvg are bolded. Loss function on $i$th client | $\mathcal{L}_{i}=f_{i}(w_{i})$
---|---
Optimal weights on $i$th client | $w_{i,opt}=\texttt{optimize}(\mathcal{L}_{i},K)$
Delta from the global model on $i$th client | $\Delta_{i}=w_{i,opt}-W$
Delta averaged across all clients | $\Delta=\sum_{i=1}^{|C|}p_{i}\Delta_{i}$
Global model update |  $m\leftarrow\beta_{1}m+(1-\beta_{1})\Delta$ $v\leftarrow\beta_{2}v+(1-\beta_{2})\Delta^{2}$ $W\leftarrow W+\eta\frac{m}{\sqrt{v}+\tau}$ where $\beta_{1}$, $\beta_{2}$, $\eta$, and $\tau$ are hyperparameters

####  A.1.3 FedProx (see Table 4)

FedProx improves on FedAvg’s tendency to favor certain devices’ performance,
introducing a proximal term in the loss that penalizes large changes to
weights in the global model [26]. This prevents any single client from
deviating from the global model too much, making global updates less
susceptible to noisy variations between non-IID clients. The original FedProx
algorithm also includes a mechanism for incorporating partial updates from
client device failures due to system heterogeneity, but for our purposes, we
only consider the proximal term which handles data heterogeneity.

####  A.1.4 FedAdam (see Table 5)

FedAdam belongs to a family of FL algorithms, termed FedOpt, which improve on
FedAvg [27]. FedAdam uses global updates inspired by the Adam optimizer [47]
as opposed to SGD updates, and it applies to our use case because adaptive
optimization methods respond well to mild heterogeneity between clients.

###  A.2 Synthetic Data Correlations

Tables 6 and 7 contain the correlation ranges that we used to generate our
synthetic data.

Table 6: Chart of synthetic correlation for each blood panel. Each blood panel metric from the patient profile is mapped to the typical range of that metric in a PCOS patient, followed by the synthetic correlation of the metric with its most effective OCP option. Patient Profile Metric | Range of PCOS Diagnosis | Implanted Correlations
---|---|---
LH-FSH Ratio | 2-3.5 [48] |  Apri: 2-2.5 Yaz: 2.6-3 Cyclen: 3.1-3.5
Total Testosterone | 86-150 ng/dl [49] |  Cyclen: 86-100.9 ng/dl Tri-cyclen: 101-110.9 ng/dl Diane-35: 111-120.9 ng/dl Apri: 121-130.9 ng/dl Yaz: 131-150 ng/dl
DHEA-S | 200-430 ug/dl [50] |  Apri: 200-300.9 ug/dl Cyclen: 301-350.9 ug/dl Tri-cyclen: 351-400.9 ug/dl Diane-35: 401-430 ug/dl
Prolactin | 25-40 ng/ml [51] |  Diane-35: 25-30.9 ng/ml Cyclen: 31-35.9 ng/ml Yaz: 36-40 ng/ml
Androstenedione | 0.4-2.7 ng/ml [49] |  Tri-cyclen: 0.4-0.7 ng/ml Yaz: 0.8-1.0 ng/ml Apri: 1.1-1.5 ng/ml Cyclen: 1.6-2.0 ng/ml Diane-35: 2.1-2.7 ng/ml
Estradiol | 60-120 pg/ml [52] |  Yaz: 60-80.9 pg/ml Cyclen: 81-100.9 pg/ml Tri-cyclen: 101-120 pg/ml
Anti-Mullerian Hormone (AMH) | 5-10 mcg/L [53] |  Yaz: 5-6.5 mcg/L Diane-35: 6.6-8 mcg/L Apri: 8.1-10 mcg/L
Table 7: Chart of synthetic correlation for each OCP option. Each OCP option is mapped to the range of blood panel metrics used for synthetic data generation. [NC] denotes no synthetic correlation exists in this dataset between the OCP and hormonal metric. OCP | Model Correlation Ranges
---|---
Apri |  LH-FSH ratio: 2-2.5 Total testosterone: 121-130.9 ng/dl DHEA-S: 200-300.9 ug/dl Prolactin: [NC] 25-40 ng/ml Androstenedione: 1.1-1.5 ng/ml Estradiol: [NC] 60-120 pg/ml Anti-Mullerian (AMH): 8.1-10 mcg/L
Cyclen |  LH-FSH ratio: 3.1-3.5 Total testosterone: 86-100.9 ng/dl DHEA-S: 301-350.9 ug/dl Prolactin: 31-35.9 ng/ml Androstenedione: 1.6-2.0 ng/ml Estradiol: 81-100.9 pg/ml Anti-Mullerian (AMH): [NC] 5-10 mcg/L
Tri-cyclen |  LH-FSH ratio: [NC] 2-3.5 Total testosterone: 101-110.9 ng/dl DHEA-S: 351-400.9 ug/dl Prolactin: [NC] 25-40 ng/ml Androstenedione: 0.4-0.7 ng/ml Estradiol: 101-120 pg/ml Anti-Mullerian (AMH): [NC] 5-10 mcg/L
Yaz |  LH-FSH ratio: 2.6-3 Total testosterone: 131-150 ng/dl DHEA-S: [NC] 200-430 ug/dl Prolactin: 36-40 ng/ml Androstenedione: 0.8-1.0 ng/ml Estradiol: 60-80.9 pg/ml Anti-Mullerian (AMH): 5-6.5 mcg/L
Diane-35 |  LH-FSH ratio: [NC] 2-3.5 Total testosterone: 111-120.9 ng/dl DHEA-S: 401-430 ug/dl Prolactin: 25-30.9 ng/ml Androstenedione: 2.1-2.7 ng/ml Estradiol: [NC] 60-120 pg/ml Anti-Mullerian (AMH): 6.6-8 mcg/L

###  A.3 Experimental Details

We use the Flower framework [41], a scalable FL framework that handles
heterogeneous data sources and offers several built-in FL algorithms. We
implement a simple fully-connected neural network as our backbone
architecture. The architecture consists of two hidden layers, each of which
contains between $10$ and $20$ neurons. We apply a ReLU nonlinearity between
each layer, and after the final layer, we apply a softmax function to
normalize the predicted class scores. We train the model for eight
communication rounds in total. In the first communication round, two randomly-
selected clients participate, and in later communication rounds, half of the
clients are randomly selected to participate. Participating clients train for
three epochs on their local datasets per round. Each client’s copy of the
neural network trains using a cross entropy loss and SGD with momentum with
the following hyperparameters: learning rate of $0.0008$, momentum of $0.87$,
and batch size of $32$.

The FedProx algorithm adds a proximal term to the loss when training each
client. We set the coefficient on this proximal term to $0.3$. Additionally,
we set $\beta=0.8$ in FedAvgM. In FedAdam, we set $\beta_{1}=0.9$,
$\beta_{2}=0.99$, $\eta=0.1$, and $\tau=1e^{-9}$.

We report results of our approaches on multiple datasets to evaluate the
effects of both IID and non-IID data distributions and differing dataset
sizes. These datasets include:

  1. 1.

IID data among all $12$ clients where each client has $12,500$ (train)
patients and a quarter as many test patients.

  2. 2.

Non-IID data where each client has $12,500$ (train) patients and a quarter as
many test patients.

  3. 3.

Non-IID data where each client has between $200$ and $20,000$ (train) patients
and a quarter as many test patients.

We normalized all hormone levels to a range between $-0.5$ and $0.5$ to
facilitate stable training.