Modern topics in relativistic spin dynamics and magnetism Andrew James
Steinmetz Doctor of Philosophy Physics DEPARTMENT OF PHYSICS 2023

October 27th, 2023 Johann Rafelski John Rutherfoord Shufang Su Sean Fleming
Stefan Meinel

subtex/acknowledgements subtex/dedication

###### Contents

  1. Publications and author contributions
  2. 1 The importance of spin
    1. 1.1 Quantum magnetic dipoles and wave equations
      1. 1.1.1 Anomalous magnetic moment
      2. 1.1.2 Dirac and Dirac-Pauli equations
    2. 1.2 Klein-Gordon-Pauli equation
      1. 1.2.1 Features of the KGP Lagrangian
      2. 1.2.2 The special case of g = 2
    3. 1.3 A few words on cosmology
  3. 2 Dynamics of charged particles with arbitrary magnetic moment
    1. 2.1 Homogeneous magnetic fields
    2. 2.2 Hydrogen-like atoms
      1. 2.2.1 Non-relativistic Coulomb problem energies
      2. 2.2.2 g-Lamb Shift between 2S and 2P orbitals
      3. 2.2.3 g-Fine structure effects within P orbitals
    3. 2.3 Particles in strong electromagnetic fields
      1. 2.3.1 Strong homogeneous magnetic fields
      2. 2.3.2 High-Z hydrogen-like atoms
    4. 2.4 Combination of mass and magnetic moment
    5. 2.5 Classical relativistic spin dynamics
      1. 2.5.1 Covariant magnetic potential and modified Lorentz force
      2. 2.5.2 Equivalency of Gilbertian and Ampèrian Stern-Gerlach forces
  4. 3 Dynamic neutrino flavor mixing through transition moments
    1. 3.1 Electromagnetic characteristics of neutrinos
    2. 3.2 Neutrino flavor mixing and electromagnetic fields
      1. 3.2.1 Effective Majorana neutrino Lagrangian
      2. 3.2.2 Chiral properties of the relativistic Pauli dipole
    3. 3.3 Electromagnetic-flavor mixing for two generations
      1. 3.3.1 Separating electromagnetic-mass mixing into two rotations
      2. 3.3.2 Effective electromagnetic-mass eigenvalues
    4. 3.4 Strong field (degenerate mass) and weak field limits
  5. 4 Matter-antimatter origin of cosmic magnetism
    1. 4.1 Short survey of magnetism in the universe
    2. 4.2 Electron-positron abundance
    3. 4.3 Theory of thermal matter-antimatter plasmas
      1. 4.3.1 Eigenstatess of magnetic moment in cosmology
      2. 4.3.2 Magnetized fermion partition function
      3. 4.3.3 Boltzmann approach to electron-positron plasma
      4. 4.3.4 Non-relativistic limit of the magnetized partition function
      5. 4.3.5 Electron-positron chemical potential
    4. 4.4 Relativistic paramagnetism of electron-positron gas
      1. 4.4.1 Evolution of electron-positron magnetization
      2. 4.4.2 Dependency on g-factor
      3. 4.4.3 Magnetization per lepton
    5. 4.5 Polarization potential and ferromagnetism
      1. 4.5.1 Hypothesis of ferromagnetic self-magnetization
      2. 4.5.2 Matter inhomogeneities in the cosmic plasma
  6. 5 Outlook and key results
  7. 6 Future research efforts
    1. 6.1 Klein-Gordon-Pauli extensions to non-Abelian fields
    2. 6.2 Electromagnetic forced neutrino CP violation
      1. 6.2.1 Jarlskog invariant
      2. 6.2.2 Toy model: CP violation amplification
      3. 6.2.3 Electric dipole moments and CP symmetry
    3. 6.3 Spin in 5D Kaluza-Klein theory
  8. References
  9. A Magnetic dipole moment in relativistic quantum mechanics
  10. B Strong fields and neutral particle magnetic moment dynamics
  11. C Relativistic dynamics of point magnetic moment
  12. D Dynamic fermion flavor mixing through transition dipole moments
  13. E A Short Survey of Matter-Antimatter Evolution in the Primordial Universe
  14. F Matter-antimatter origin of cosmic magnetism

###### List of Figures

  1. 1.1 A schematic of the universe’s evolution since the Big Bang. The region of interest studied in this dissertation is emphasized (in the highlighted box) to contain a dense nearly charge neutral matter-antimatter plasma.
  2. 2.1 Diagram organizing the KGP-Landau levels for particles with zero z-component momentum. The Landau levels $n$ and spin $s$ serve to split levels while the anomaly $a$ controls the degeneracy among levels.
  3. 2.2 The $n=0$, $s=1/2$ ground state for a KGP electron given by Eq. (2.16) with $g/2-1=\alpha/2\pi$ in a homogeneous magnetic field. We consider the particle with no $z$-direction momentum. The particle state (solid red) and antiparticle (dashed red) are presented.
  4. 2.3 The $n=0$, $s=1/2$ ground state for a DP electron given by Eq. (2.19b) with $g/2-1=\alpha/2\pi$ in a homogeneous magnetic field. We consider the particle with no $z$-direction momentum. The particle state (solid red) and antiparticle (dashed red) are presented.
  5. 2.4 The KGP $1S_{1/2}$ (lower red curves) and $2P_{1/2}$ (upper green curves) energy levels for $g$-factor values $g\\!=\\!\\{1.8,\ 1.9977,\ 2.0023,\ 2.2\\}$ are shown for large $Z$ hydrogen-like atoms. The curves for the Dirac $g\\!=\\!2$ case for (lower dashed blue) $1S_{1/2}$ and (upper dashed blue) $2P_{1/2}$ are also presented.
  6. 2.5 The KGP $1S_{1/2}$ and $2P_{1/2}$ states are plotted for hydrogen-like energies for $220>Z>0$. The spread of lines corresponds to a spread of $g$-factor values: $4>g>0$. Integer multiples of $Z=137/2$ are marked with vertical dashed lines. The separation between any two adjacent curves is $|g_{i}-g_{i+1}|=0.05$. The unique curve which dives towards and stops at the boundary $E=0$ is the Dirac $g=2$ ground state.
  7. 2.6 A close up of the KGP $1S_{1/2}$ and $2P_{1/2}$ states between $2.0625>g>1.9375$. The separation between any two adjacent curves is $|g_{i}-g_{i+1}|=0.003125$. The value $Z=137$ is marked with a vertical dashed line.
  8. 2.7 Energy level curves are plotted for total angular momentum quantum number $j=1/2,\,3/2$ with the relativistic principle quantum number $n_{r}=0$. The separation between any two adjacent curves is $|g_{i}-g_{i+1}|=0.2$. Dashed vertical lines indicate integer multiples of $Z=137/2$.
  9. 2.8 The $n=0$, $s=1/2$ ground state for a IKGP proton given by Eq. (2.68) with $g=5.58$ in a homogeneous magnetic field. The magnetic minimum is well visible for particles with larger anomalous moment such as proton. We consider the particle with no $z$-direction momentum. The particle state (solid red) and antiparticle (dashed red) are presented. The magnetic field scale is ${B}^{\prime}_{S}=(m^{2}_{p}/m^{2}_{e})B_{S}$.
  10. 4.1 Qualitative plot of the primordial magnetic field strength over cosmic time. All figures are printed in temporal sequence in the expanding universe beginning with high temperatures (and early times) on the left and lower temperatures (and later times) on the right.
  11. 4.2 Number density of electron $e^{-}$ and positron $e^{+}$ to baryon ratio $n_{e^{\pm}}/n_{B}$ as a function of photon temperature in the universe. See text for further details. In this work we measure temperature in units of energy (keV) thus we set the Boltzmann constant to $k_{B}=1$. Figure courtesy of Cheng Tao Yang.
  12. 4.3 Organizational schematic of matter-antimatter $(\sigma)$ and polarization $(s)$ states with respect to the sign of the non-relativistic magnetic dipole energy $U_{\rm Mag}$ obtainable from Eq. (4.4).
  13. 4.4 The chemical potential over temperature $\mu/T$ is plotted as a function of temperature with differing values of spin potential $\eta$ and magnetic scale $b_{0}$.
  14. 4.5 The magnetization ${\mathfrak{M}}$, with $g\\!=\\!2$, of the primordial $e^{+}e^{-}$ plasma is plotted as a function of temperature. Figure made in collaboration with Cheng Tao Yang.
  15. 4.6 The magnetization $\mathfrak{M}$ as a function of $g$-factor plotted for several temperatures with magnetic scale $b_{0}=10^{-3}$ and polarization fugacity $\xi=1$.
  16. 4.7 The magnetic moment per lepton $|\vec{m}|_{z}$ along the field axis as a function of temperature. Figure made in collaboration with Cheng Tao Yang.
  17. 4.8 The spin potential $\eta$ and chemical potential $\mu$ are plotted under the assumption of self-magnetization through a nonzero spin polarization in bulk of the plasma. Figure made in collaboration with Cheng Tao Yang.
  18. 4.9 The number density $n_{e^{\pm}}$ of polarized electrons and positrons under the self-magnetization model for differing values of $b_{0}$. Figure courtesy of Cheng Tao Yang.

###### List of Tables

  1. 1.1 The $g$-factor of various particles found in Tiesinga et al. (2021); Workman et al. (2022).
  2. 4.1 Organizational schematic of matter-antimatter $(\sigma)$ and polarization $(s)$ states with respect to the chemical $\mu$ and polarization $\eta$ potentials as seen in Eq. (4.32). Companion to Table 4.3.
  3. 6.1 Time (T), parity (P) and PT symmetries of electric $\bm{E}$, magnetic $\bm{B}$, spin $\bm{s}$ three-vectors and the inner products which describe the dipole Hamiltonian terms.

subtex/abstract

## Publications and author contributions

I, Andrew James Steinmetz, in the course of satisfying the University of
Arizona Department of Physics’s requirements for a Ph.D. doctoral
dissertation, prepared the following publications which are reprinted in full
in the appendices. These articles are not ordered chronologically, but in the
contextual order of presentation in this document. My contribution to each
work is described under each item.

  * •

Appendix A \- “Magnetic dipole moment in relativistic quantum mechanics” by
Steinmetz, Formanek, and Rafelski (2019) is a study and comparison of DP and
KGP wave equations for homogeneous magnetic fields and hydrogen-like atoms. I
performed all computation, writing, and figure making in preparation of the
first draft and approved the final draft before submission. I acknowledge the
help and consultation of Martin Formanek (MF) and Johann Rafelski (JR) in
research, writing and editing.

  * •

Appendix B \- “Strong fields and neutral particle magnetic moment dynamics” by
Formanek, Evans, Rafelski, Steinmetz, and Yang (2018) is an overview of our
research group’s efforts in studying neutral particle dynamics in
electromagnetic fields. I wrote Section 2.1 in collaboration with MF. I
consulted and helped lead author MF and co-authors Stefan Evans (SE) and Cheng
Tao Yang (CTY) in editing and revising the overall manuscript.

  * •

Appendix C \- “Relativistic dynamics of point magnetic moment” by Rafelski,
Formanek, and Steinmetz (2018) introduces a new covariant formulation of
classical spin dynamics and unifies Gilbertian and Ampèrian dipoles. I wrote
Section 3 in collaboration with JR and MF and aided in the computation in
Section 5.1. I otherwise consulted in the research, writing, and editing
process of this publication.

  * •

Appendix D \- “Dynamic fermion flavor mixing through transition dipole
moments” by Rafelski, Steinmetz, and Yang (2023c) is a study of Majorana
neutrino flavor mixing in electromagnetic fields and proposes a novel
dynamical EM-mass basis for propagating neutrinos. The article was written
originally via invitation of JR by Gerhard Buchalla, Dieter Lüst and Zhi-Zhong
Xing as a memorial chapter in a book dedicated to Harald Fritzsch. I performed
all computation and writing in preparation of the first draft and approved the
final draft before submission. I acknowledge the help and consultation of JR
and CTY in research, writing and editing.

  * •

Appendix E \- “A Short Survey of Matter-Antimatter Evolution in the Primordial
Universe” by Rafelski, Birrell, Steinmetz, and Yang (2023a) is a 50 page long
review with many novel results describing the role of antimatter in the early
universe. I supervised (in collaboration with CTY) the document creation,
combining the writing contributions of all authors (including myself, Jeremiah
Birrell (JB), CTY, and JR) into one coherent presentation. I also coordinated
with all authors in formatting and editing the technical figures in this
review by JB, CTY, and JR.

  * •

Appendix F \- “Matter-antimatter origin of cosmic magnetism” by Steinmetz,
Yang, and Rafelski (2023) proposes a model of para-magnetization driven by the
large matter-antimatter (electron-positron) content of the early universe. I
carried out all writing in preparation of the first draft and approved the
final draft before submission. Computation and figure making was done in
collaboration with CTY who contributed key results and five technical figures.
I acknowledge the help and consultation of CTY and JR in research, writing and
editing.

This is not a total catalogue of my research efforts, but lists the works that
form the foundation of Chapter 2, Chapter 3 and Chapter 4 of this
dissertation. Where noted, these chapters also contain sections of complete
yet unpublished work. Chapter 6 contains brief discussions of still-in-
progress research efforts to be completed after submission of this
dissertation.

I was also co-author on the following publications which are not used
extensively in this dissertation and are not reprinted as appendices. They are
listed in chronological order below. In these three works I consulted with MF,
CTY and JR in research and editing making content clarifying contributions to
these manuscripts:

  * •

“Classical neutral point particle in linearly polarized EM plane wave field”
by Formanek, Steinmetz, and Rafelski (2019) explores the dynamical equations
presented in Appendix C for neutral particles with magnetic moment.

  * •

“Radiation reaction friction: Resistive material medium” by Formanek,
Steinmetz, and Rafelski (2020) introduces a novel model of relativistic
covariant friction within a medium.

  * •

“Motion of classical charged particles with magnetic moment in external plane-
wave electromagnetic fields” by Formanek, Steinmetz, and Rafelski (2021b) is a
followup to the above 2019 work and Appendix C for charged particles with
magnetic moment.

  * •

“Decomposition of Fermi gas into zero and finite temperature distributions
with examples” by Yang, Formanek, Steinmetz, and Rafelski (2023) is a
mathematical methods paper detailing a novel analytic form of the finite
temperature behavior of the Fermi-Dirac distribution function. The cold
magnetized gas is analyzed as an example.

##  Chapter 1 The importance of spin

All fundamental particles known in physics have a non-zero quantized spin
angular momentum with the exception of the Higgs boson which is a scalar with
spin-0. All other confirmed elementary particles (such as electrons, quarks,
photons, etc…) have values of either spin-1/2 or spin-1. Particles with even
values of spin are known as bosons while half-integer particles with spin are
called fermions. Composite particles (such as atomic nuclei) can exhibit more
exotic spin values and fundamental particles with higher spins such as
spin-3/2 or spin-2 graviton are commonly predicted in beyond-standard-model
(BSM) physics.

In the realm of the Poincaré group of spacetime symmetry (rotations, boosts
and translations) transformations, each particle can be uniquely labeled by
two distinct Casimir invariants: mass and spin. These two operators commute
with all generators of the Poincaré group and act as labels which represent a
particle. Therefore in a relativistic context, mass and spin are of
fundamental importance on equal footing.

If a particle is electrically charged, then by virtue of its spin it will have
a magnetic dipole moment. Most neutral particles with spin, though not all,
will also have magnetic dipoles though for more complex reasons. Therefore the
magnetic behavior of particle is an important window into probing one of the
most fundamental properties in physics. As quantum mechanics is not well
described in terms of forces or accelerations (except in the context of
Ehrenfest-style equations), there is no simple operator description of torque
and spin-forces despite having played a key role in the development of quantum
mechanics. For a short historical overview of spin and its relationship to
angular momentum, see Ohanian (1986).

This introduction serves to motivate the fundamental concepts of spin,
magnetic moment and electromagnetism which have played a crucial role in the
history physics and will be explored in the subsequent research chapters.
Magnetic (and electric) dipoles, anomalous magnetic moments (AMM), and the
Dirac and Dirac-Pauli wave equations which describe spin-1/2 fermions are
covered in Section 1.1. The Klein-Gordon-Pauli equation is introduced in
Section 1.2. Lastly, Section 1.3 covers topics in $\Lambda\mathrm{CDM}$
cosmology which are particular relevance to Chapter 4. This chapter will also
serve to establish notation and mathematical conventions. SI units will be
used unless otherwise stated.

###  1.1 Quantum magnetic dipoles and wave equations

In classical theory, when charges rotate or circulate in some manner, a
magnetic field is produced characterized by the magnetic dipole moment of the
system. An Ampèrian loop of wire with a current is the quintessential example.
This concept can be transplanted into quantum theory for spinning particles
where the natural size of the magnetic moment of a particle (in this context a
charged lepton) is given by the magneton value

$\displaystyle\mu_{\ell}\equiv\frac{e\hbar}{2m_{\ell}}$ (1.1)

where the lepton (denoted by $\ell$) has charge $e$ and mass $m_{\ell}$.

A quick word on notation: Euclidean three-vectors and matrices will be denoted
by boldface font. If indices are specifically printed, they will be done so
using Latin indices such as $s_{i}$. Inner products of three-vectors will be
noted via $\bm{a}\cdot\bm{b}=a_{i}b_{i}$ using Einstein summation notation
where repeated indices are summed over. For electrons, Eq. (1.1) is referred
to as the Bohr magneton $\mu_{B}$. The non-relativistic spin operator $\bm{S}$
for a spin-1/2 particle is defined as

$\displaystyle\bm{S}=\frac{\hbar}{2}\bm{\sigma}=\frac{\hbar}{2}\left(\sigma_{1},\,\sigma_{2},\,\sigma_{3}\right)^{\mathrm{T}}\,,$
(1.2)

where $\bm{\sigma}$ is the three-vector comprised of the familiar $2\times 2$
Pauli matrices which act upon two-component spinors
$\chi=(\chi_{1},\chi_{2})^{\mathrm{T}}$. Spinor indices will be suppressed or
noted with Latin indices. The algebra defined by the commutators of the Pauli
matrices serves as a representation of $SU(2)$ group structure

$\displaystyle\\{\sigma_{i},\sigma_{j}\\}=2\delta_{ij}\,,\qquad[\sigma_{i},\sigma_{j}]=2i\varepsilon_{ijk}\sigma_{k}\,,$
(1.3)

where $\varepsilon_{ijk}$ is the totally antisymmetric Levi-Civita symbol and
$\delta_{ij}$ is the Kronecker delta.

The relativistic theory of spin-1/2 fermions however necessitates a four-
component spinor $\psi=(\psi_{1},\psi_{2},\psi_{3},\psi_{4})^{\mathrm{T}}$
which as Dirac famously noted accommodates the required degrees of freedom for
particles and antiparticles of both spin up $(\uparrow)$ and spin down
$(\downarrow)$ eigenstates. The Hamiltonian density (in the Dirac
representation) for the magnetic dipole moment interaction is given by

$\displaystyle\mathcal{H}_{\mathrm{int}}=\frac{e\hbar}{2m_{\ell}}\psi^{{\dagger}}\begin{pmatrix}-\bm{\sigma}\cdot\bm{B}&i\bm{\sigma}\cdot\bm{E}/c\\\
-i\bm{\sigma}\cdot\bm{E}/c&\bm{\sigma}\cdot\bm{B}\end{pmatrix}\psi\,,$ (1.4)

where $\psi^{{\dagger}}$ is the complex conjugate transpose of the $\psi$
spinor. The electric $\bm{E}$ and magnetic $\bm{B}$ fields are defined in
terms of the scalar potential $V$ and vector potential $\bm{A}$ in the usual
way.

$\displaystyle\bm{E}=-\bm{\nabla}V-\frac{\partial\bm{A}}{\partial
t}\,,\qquad\bm{B}=\bm{\nabla}\times\bm{A}\,.$ (1.5)

In the non-relativistic limit for particle states, the lower (antiparticle)
components of $\psi$ are suppressed by $|\bm{p}|/mc$. We can approximate the
particle states in terms of two-component spinors $\chi$ to first order as

$\displaystyle\psi\approx\left(\chi,\
\frac{\bm{\sigma}\cdot\bm{\pi}}{2m_{\ell}c}\chi\right)^{\mathrm{T}}\,,\qquad\bm{\pi}=\bm{p}-e\bm{A}\,.$
(1.6)

A more rigorous method of obtaining non-relativistic Hamiltonian can be found
in Foldy and Wouthuysen (1950). The operator $\bm{\pi}$ is the kinetic
momentum operator written in terms of canonical momentum $\bm{p}$ and vector
potential $\bm{A}$. Making use of the identity

$\displaystyle\sigma_{i}\sigma_{j}=\delta_{ij}+i\varepsilon_{ijk}\sigma_{k}\,,$
(1.7)

we insert Eq. (1.6) into Eq. (1.4) yielding to order $\mathcal{O}(1/m^{3})$

$\displaystyle\mathcal{H}_{\mathrm{int}}\approx-\chi^{{\dagger}}\left(\frac{e\hbar}{2m_{\ell}}\bm{\sigma}\cdot\bm{B}+\frac{ie\hbar}{4m_{\ell}^{2}c^{2}}\Big{[}(\bm{\sigma}\cdot\bm{E}),(\bm{\sigma}\cdot\bm{\pi})\Big{]}\right)\chi$
(1.8)
$\displaystyle\mathcal{H}_{\mathrm{int}}\approx-\chi^{{\dagger}}\left(\frac{e\hbar}{2m_{\ell}}\bm{\sigma}\cdot\bm{B}+\frac{e\hbar^{2}}{4m_{\ell}^{2}c^{2}}\bm{\nabla}\cdot\bm{E}+\frac{e\hbar}{4m_{\ell}^{2}c^{2}}\bm{\sigma}\cdot\left(\bm{E}\times\bm{\pi}-\bm{\pi}\times\bm{E}\right)\right)\chi\,.$
(1.9)

Keeping only up to first order, the dipole interaction Eq. (1.4) reduces to

$\displaystyle\mathcal{H}_{\mathrm{int}}\approx-\frac{e\hbar}{2m_{\ell}}\chi^{{\dagger}}\bm{\sigma}\cdot\bm{B}\chi\,,$
(1.10)

which is the expected non-relativistic quantum dipole term. The second and
third terms in Eq. (1.9) can be interpreted as a Darwin term
$\sim\bm{\nabla}\cdot\bm{E}$ sensitive to charge density and spin orbit
coupling $\sim\bm{\sigma}\cdot(\bm{E}\times\bm{p})$. We will return to
relativistic notation and concepts in Section 1.1.2.

The magnetic moment operator $\bm{\mu}$, as suggested by Eq. (1.10) is defined
in terms of the Pauli matrices as

$\displaystyle\bm{\mu}=g\left(\frac{e\hbar}{2m_{\ell}}\right)\frac{\bm{\sigma}}{2}=g\mu_{\ell}\frac{\bm{\sigma}}{2}\,,\qquad\mu\equiv\frac{g}{2}\mu_{\ell}\,,$
(1.11)

where $\mu$ is the ‘total magneton’ value representing the full magnetic
moment. The parameter $g$ in Eq. (1.11) is the gyromagnetic ratio (or
$g$-factor) of the particle. The ‘natural’ value is $g\\!=\\!2$. While this
prediction is normally attributed to the Dirac equation, it justified from the
construction of the kinetic energy operator in the Schrödinger-Pauli equation;
see Section 1.2.2 and Sakurai (1967).

In non-relativistic quantum mechanics, the time-dependant Schrödinger-Pauli
(SP) equation (with Hamiltonian $H_{\mathrm{SP}}$) for a charged particle is
given by

$\displaystyle{H}_{\mathrm{SP}}\chi=\left(\frac{1}{2m_{\ell}}\bm{\pi}^{2}-\bm{\mu}\cdot\bm{B}+e{V}\right)\chi=i\hbar\frac{\partial}{\partial
t}\chi\,,\qquad\bm{\pi}=\bm{p}-e{\bm{A}}\,,$ (1.12)

where $\chi$ is again a two-component spinor. It is well known that Eq. (1.12)
is obtainable from the Dirac equation (see Section 1.1.2) in the non-
relativistic limit.

Before moving on, we will verify that the SP Eq. (1.12) contains within it an
expression of the Stern-Gerlach force which was used to first provide evidence
of the quantization of angular momentum (Gerlach and Stern, 1922). To
accomplish this, we will work in the Heisenberg representation where operators
obey the following equation of motion

$\displaystyle
i\hbar\frac{d\bm{O}}{dt}=[\bm{O},H]+i\hbar\frac{\partial\bm{O}}{\partial
t}\,,$ (1.13)

To obtain a ‘force’ in quantum mechanics we need to find the time derivative
of the kinematic momentum operator $\bm{\pi}$ which is given by

$\displaystyle\frac{d\bm{\pi}}{dt}=-\frac{i}{\hbar}[\bm{\pi},H_{\mathrm{SP}}]+\frac{\partial\bm{\pi}}{\partial
t}=-\frac{i}{\hbar}\left[\bm{\pi},\frac{(\bm{\sigma}\cdot\bm{\pi})^{2}}{2m}+eV\right]+\frac{\partial\bm{\pi}}{\partial
t}\,,$ (1.14) $\displaystyle\frac{\partial\bm{\pi}}{\partial
t}=-\frac{\partial e\bm{A}}{\partial
t}\,,\qquad[\pi_{i},\pi_{j}]=ie\hbar\varepsilon_{ijk}B_{k}\,,\qquad[\pi_{i},B_{j}]=-i\hbar\nabla_{i}B_{j}\,.$
(1.15)

After some derivation and making use of the identities in Eq. (1.15), we
arrive at the quantum analog of the Lorentz force for particles with spin

$\displaystyle\boxed{\frac{d\bm{\pi}}{dt}=e\bm{E}+\frac{e}{2m_{\ell}}(\bm{\pi}\times\bm{B}-\bm{B}\times\bm{\pi})+\frac{e\hbar}{2m_{\ell}}\sigma_{i}\bm{\nabla}B_{i}}\,.$
(1.16)

The last term in the expression is the Stern-Gerlach force which is sensitive
to inhomogeneous magnetic fields. We also note this equation is suggestive of
the ‘Ampérian’ dipole force which is in the direction of the gradient
$\bm{\nabla}$ rather than the ‘Gilbertian’ type of dipole force which is in
the direction of the field $\bm{B}$; see Section 2.5. Eq. (1.16) can be
connected to our classical understanding by taking the expectation value and
casting it as an Ehrenfest-style theorem (Ehrenfest, 1927).

####  1.1.1 Anomalous magnetic moment

In nature there is no particle with exactly $g\\!=\\!2$. As seen in Table 1.1,
composite particles often deviate from $g\\!=\\!2$ greatly as the $g$-factor
of a composite particle is related to its internal composition. In the case of
the neutron and proton, the internal quarks themselves are responsible in a
nontrivial fashion (Chang et al., 2015). The comparison between three listed
isotopes of hydrogen also displays how magnetic moments can ‘cancel out’ or
add together: While the deuterium nucleus value of $g$ is suppressed by the
extra neutron, the two neutrons in the tritium nucleus balance one another
returning the ratio into one manifestly similar to the proton. This reasoning
however only works as a heuristic and non-perturbative Lattice QCD
computations (Detmold et al., 2019) are needed to obtain the magnetic moments
of hadrons with great accuracy.

When $g\neq 2$ (which is true for all physical particles with magnetic moment;
composite of otherwise) the anomalous magnetic moment (AMM) can be defined via

$\displaystyle a\equiv\frac{g}{2}-1\,,\qquad
a\frac{e\hbar}{2m_{\ell}}\rightarrow\delta\mu\equiv\mu-\mu_{\ell}\,,$ (1.17)

where $a$ is the anomaly parameter. We also introduce $\delta\mu$ as the
anomalous magneton which will be helpful in our proposal to connect mass and
magnetic moment in Section 2.4 and Section 3.1.

particle | category |  $g$-factor
---|---|---
electron | elementary | -2.002 319 304 362 56(35)
muon | elementary | -2.002 331 8418(13)
tau | elementary | -2.036(34)
neutron | composite | -3.826 085 45(90)
proton | composite | 5.585 694 6893(16)
deuteron | composite | 0.857 438 2338(22)
triton | composite | 5.957 924 931(12)
Table 1.1: The $g$-factor of various particles found in Tiesinga et al.
(2021); Workman et al. (2022).

The anomalous magnetic moment of a particle can arise from a variety of
physical sources with the most famous being the one-loop vacuum polarization
contribution to the electron first computed by Schwinger (1951). In that work,
the first correction to $g$ is given by

$\displaystyle
a_{e}=\frac{\alpha}{2\pi}\,,\qquad\alpha\equiv\frac{1}{4\pi\varepsilon_{0}}\frac{e^{2}}{\hbar
c}\,,$ (1.18)

where $\alpha$ is the fine structure constant with an approximate value of
$1/137$. The measurement of the electron’s $g$-factor is among the most
precise measurements in all of physics (Tiesinga et al., 2021). Precision
measurements of the muon’s anomalous magnetic moment are rapidly improving
(Aguillard et al., 2023). This makes the study of magnetic moment, and spin,
an exciting area of physical research as new developments continue today.

####  1.1.2 Dirac and Dirac-Pauli equations

While it is always beneficial to be well-appraised of non-relativistic
mechanics, nature is intrinsically relativistic and therefore this
dissertation must be as well. The relativistic generalization of Eq. (1.12) is
the Dirac equation given by

$\displaystyle\left(\gamma_{\alpha}\left(i\hbar\partial^{\alpha}-eA^{\alpha}\right)-m_{\ell}c\right)\psi=0\,,$
(1.19)
$\displaystyle\pi^{\alpha}=i\hbar{\widetilde{\nabla}}^{\alpha}=i\hbar\partial^{\alpha}-eA^{\alpha}\,.$
(1.20)

The wave function $\psi$ in Eq. (1.19) is understood to be a four-component
spinor and $\widetilde{\nabla}^{\alpha}$ in Eq. (1.20) is the covariant
derivative. $\pi^{\alpha}$ is the four-vector version of the kinetic momentum
versus the four-momentum $p^{\alpha}=i\hbar\partial^{\alpha}$. Four-vectors
and tensors in this work will be denoted by Greek indices. Inner products of
four-vectors will be noted by $a\cdot
b=a^{\alpha}\eta_{\alpha\beta}b^{\beta}=a^{\alpha}b_{\alpha}$ again following
Einstein notation. The four-derivative $\partial^{\alpha}$ and four-potential
$A^{\alpha}$ are defined as

$\displaystyle\partial^{\alpha}=\left(\frac{1}{c}\frac{\partial}{\partial
t},\,-\bm{\nabla}\right)\,,\qquad
A^{\alpha}=\left(\frac{V}{c},\,\bm{A}\right)\,.$ (1.21)

We have written the Dirac equation here in the covariant form where
$\gamma^{\alpha}$ are the gamma matrices which obey the anticommuting Clifford
algebra

$\displaystyle\\{\gamma_{\alpha},\gamma_{\beta}\\}=\gamma_{\alpha}\gamma_{\beta}+\gamma_{\beta}\gamma_{\alpha}=2\eta_{\alpha\beta}\,,$
(1.22) $\displaystyle\eta_{\alpha\beta}=\mathrm{diag}(+1,-1,-1,-1)\,,$ (1.23)

where $\eta_{\alpha\beta}$ is the flat spacetime Minkowski metric tensor
defined with a positive time metric signature. The metric tensor is also
responsible for raising and lowering covariant and contravariant indices e.g.
$a_{\alpha}=\eta_{\alpha\beta}a^{\beta}$. As $\gamma^{\alpha}$ are also spinor
matrices, the commutator in Eq. (1.22) carries implicit spinor indices which
here computes to the $4\times 4$ identity matrix $\mathbbm{1}_{4}$ (which is
suppressed). We also introduce the ‘fifth’ gamma matrix $\gamma^{5}$ which
anticommutes with $\gamma^{\alpha}$ and the following standard conventions
following Itzykson and Zuber (1980)

$\displaystyle\bm{\alpha}=\gamma^{0}\bm{\gamma}\,,\indent\bm{\Sigma}=\gamma^{5}\bm{\alpha}\,,\indent\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}\,,\indent\gamma^{2}_{5}=1\,.$
(1.24)

As mentioned before, Eq. (1.19) predicts $g\\!=\\!2$ which is a standard
calculation in many textbooks. The most straight-forward manner to generalize
the Dirac equation allowing for an anomalous magnetic moment is to add a Pauli
term proportional to the anomalous parameter $a$. While in most texts, the
anomaly is given in terms of $g-2$ or $a$, we wish to keep our equations
generalized to fermions of any given charge $e$ and magnetic moment $\mu$.

Therefore we make use of the substitution in Eq. (1.17) and write the Dirac-
Pauli (DP) equation as

$\displaystyle\left(\gamma_{\alpha}\left(i\hbar\partial^{\alpha}-eA^{\alpha}\right)-m_{\ell}c-\delta\mu\frac{1}{2c}\sigma_{\alpha\beta}F^{\alpha\beta}\right)\psi=0\,,$
(1.25)

where the antisymmetric spin tensor $\sigma_{\alpha\beta}$ is defined in terms
of the commutator of the gamma matrices

$\displaystyle\sigma_{\alpha\beta}=\frac{i}{2}\left[\gamma_{\alpha},\gamma_{\beta}\right]=\frac{i}{2}\left(\gamma_{\alpha}\gamma_{\beta}-\gamma_{\beta}\gamma_{\alpha}\right)\,.$
(1.26)

In the above we printed only the magnetic moment term; some remarks about
electric dipole moments (EDM) and CP violation can be found in Section 6.2.3.

Exact solutions to the DP equation are relatively scarce due to the
complicating nature of the anomalous term. The most extensively studied
solutions are those with high symmetries or constant external fields (Thaller,
2013). When the anomalous part $\delta\mu$ is zero, the Dirac equation is
recovered. $F^{\alpha\beta}$ is the standard antisymmetric electromagnetic
field tensor defined by

$\displaystyle
F^{\alpha\beta}=\partial^{\alpha}A^{\beta}-\partial^{\beta}A^{\alpha}=\begin{pmatrix}0&-E_{1}/c&-E_{2}/c&-E_{3}/c\\\
E_{1}/c&0&-B_{3}&B_{2}\\\ E_{2}/c&B_{3}&0&-B_{1}\\\
E_{3}/c&-B_{2}&B_{1}&0\end{pmatrix}\,.$ (1.27)

The electromagnetic field tensor can also be defined in terms of the
commutators of the covariant derivative Eq. (1.20) as

$\displaystyle\left[\widetilde{\nabla}^{\alpha},\widetilde{\nabla}^{\beta}\right]=\frac{ie}{\hbar}F^{\alpha\beta}\,.$
(1.28)

It is also useful to define the Hodge dual of the electromagnetic field tensor

$\displaystyle
F_{\alpha\beta}^{*}=\frac{1}{2}\varepsilon_{\alpha\beta\mu\nu}F^{\mu\nu}=\begin{pmatrix}0&-B_{1}&-B_{2}&-B_{3}\\\
B_{1}&0&-E_{3}/c&E_{2}/c\\\ B_{2}&E_{3}/c&0&-E_{1}/c\\\
B_{3}&-E_{2}/c&E_{1}/c&0\end{pmatrix}\,,$ (1.29)

where we use the four-dimensional fully antisymmetric Levi-Civita pseudo-
tensor $\varepsilon_{\alpha\beta\mu\nu}$ with the $\varepsilon_{0123}=+1$
convention. The contracted portion $\sigma_{\alpha\beta}F^{\alpha\beta}$ in
the Pauli term in Eq. (1.25) can be further expressed as

$\displaystyle\frac{1}{2}\sigma_{\alpha\beta}F^{\alpha\beta}=i\bm{\alpha}\cdot\bm{E}/c-\bm{\Sigma}\cdot\bm{B}=i\gamma^{0}\bm{\gamma}\cdot\bm{E}/c-\gamma^{5}\gamma^{0}\bm{\gamma}\cdot\bm{B}\,,$
(1.30)

which captures that relativistic magnetic moments should be sensitive to
electric as well as magnetic fields as required by Lorentz transformations of
the $\bm{E}$ and $\bm{B}$ fields. We note that Eq. (1.30) is the matrix which
appears in Eq. (1.4) specifically in the Dirac representation of $\bm{\alpha}$
and $\bm{\Sigma}$. This should be unsurprising if one considers how the non-
relativistic dipole form must generalize under Lorentz boosts which mix
electric and magnetic fields.

The DP equation can be obtained from perturbative QED as an effective field
theory for leptons due to vacuum polarization; see standard texts Itzykson and
Zuber (1980); Schwartz (2014). However, if a particle’s anomalous magnetic
moment is not sourced by perturbative QFT, then the Pauli term introduced in
Eq. (1.25) must be added by hand _ad hoc_ or obtained via non-perturbative
means such as Lattice calculations (Aoyama et al., 2020). This is the case for
the hadronic contribution to anomalous magnetic moment of leptons as well as
any composite particle such as the proton or neutron whose moment is
determined by internal structure (Hewett et al., 2012; Green et al., 2015).

Therefore we can describe the AMM as an added Lagrangian interaction term

$\displaystyle\mathcal{L}_{\mathrm{DP,AMM}}=-{\bar{\psi}}\left(\delta\mu\frac{1}{2}\sigma_{\alpha\beta}F^{\alpha\beta}\right)\psi\,,$
(1.31)

where ${\bar{\psi}}=\psi^{\dagger}\gamma^{0}$ is the Dirac adjoint. While the
focus of this dissertation is not on quantum field theory (QFT), it is
valuable to note that the Pauli Lagrangian term in Eq. (1.31) is considered
5-dimensional as the $\psi$ fields have natural units of
$[\mathrm{length}]^{-3/2}$ as determined from the Dirac Lagrangian

$\displaystyle\mathcal{L}_{\mathrm{D}}/c=\bar{\psi}\left(i\hbar\gamma_{\alpha}\widetilde{\nabla}^{\alpha}-m_{\ell}c\right)\psi\,,\qquad\mathcal{L}_{\mathrm{DP}}=\mathcal{L}_{\mathrm{D}}+\mathcal{L}_{\mathrm{DP,AMM}}\,.$
(1.32)

To demonstrate, we note that the electromagnetic field tensor has natural
units of $F^{\alpha\beta}\sim[\mathrm{length}]^{-2}$. Therefore the product
$\psi\sigma_{\alpha\beta}F^{\alpha\beta}\psi$ has natural units of
$[\mathrm{length}]^{-5}$ and the coefficient of Eq. (1.31) (given by
$\delta\mu$) has to compensate with $\delta\mu\sim[\mathrm{length}]^{1}$. This
makes the DP Lagrangian unsuitable for renormalization which is an essential
feature required for well-behaved QFTs. While this does not stop us using DP
as an effective QFT with some natural cutoff scale responsible for the
anomalous moment, it does reduce the usefulness of the equation as a general
description of quantum dipole moments.

As such, there is no reason to expect non-perturbative sources of magnetic
moment to strictly adhere to the DP form. Additionally, the DP equation has
the physically inelegant consequence of splitting the spin dynamics of
fermions into (a) natural $g\\!=\\!2$ behavior (see Section 1.2.2) encompassed
by the spinor structure of the Dirac equation and (b) the anomalous behavior
contained in the Pauli term.

###  1.2 Klein-Gordon-Pauli equation

While the DP equation is more commonly studied, there exists an alternative
wave equation which describes the magnetic behavior of fermions called the
Klein-Gordon-Pauli (KGP) equation. This first introduced by Fock (1937) and
found usefulness in the quantum electrodynamics (Feynman, 1951) and in
studying weak interactions (Feynman and Gell-Mann, 1958) due to the ease of
describing chiral states.

The KGP equation is generally considered to be the ‘square’ of the Dirac
equation as unlike the Dirac or DP equations, it is a second order equation
wave equation for the four-component spinor $\Psi$

$\displaystyle\left((i\hbar\partial^{\alpha}-eA^{\alpha})^{2}-m_{\ell}^{2}c^{2}-g\mu_{\ell}m_{\ell}\frac{1}{2}\sigma_{\alpha\beta}F^{\alpha\beta}\right)\Psi=0\,.$
(1.33)

The initial benefit of the KGP formulation is that the wave equation fully
commutes with $\gamma^{5}$ making eigen-functions explicitly good chiral
states.

Eq. (1.33) is mathematically similar to the Klein-Gordon equation which
describes charged scalar particles. In the same manner as scalar-QED, the
squared covariant derivative contains a $e^{2}A^{2}$ term which in QFT results
in the presence of a 4-vertex seagull interaction (Schwartz, 2014) at tree-
level.

It is important to emphasize that the KGP Eq. (1.33) and DP Eq. (1.25) are
distinct wave equations which do not share solutions except when $g\\!=\\!2$
whereas both reduce to the Dirac Eq. (1.19). We will clarify on the
relationship between the KGP and Dirac equations here by rewriting the Dirac
equation in Eq. (1.19) as

$\displaystyle\mathcal{D}_{\pm}=i\hbar\gamma_{\alpha}\widetilde{\nabla}^{\alpha}\pm
m_{\ell}c\,,\qquad\mathcal{D}_{-}\psi=0\,,$ (1.34)

with a ‘Dirac operator’ $\mathcal{D}_{\pm}$ defined in terms of positive and
negative mass. This operator has the following properties

$\displaystyle\mathcal{D}_{-}=-\gamma^{5}\mathcal{D}_{+}\gamma^{5}\,,\qquad[\mathcal{D}_{+},\mathcal{D}_{-}]=0\,.$
(1.35)

Ignoring the proportionality factor of $\sqrt{\hbar/m_{\ell}c}$ which
accommodate the units of $\psi$ versus $\Psi$, we can complete the square of
the Dirac equation via the substitution

$\displaystyle\psi\rightarrow\mathcal{D}_{+}\Psi\,,\qquad\mathcal{D}_{-}\psi\rightarrow\mathcal{D}_{-}\mathcal{D}_{+}\Psi\,,$
(1.36)
$\displaystyle\mathcal{D}_{+}\mathcal{D}_{-}\Psi=\left(-\hbar^{2}\widetilde{\nabla}^{2}-m_{\ell}^{2}c^{2}-e\hbar\frac{1}{2}\sigma_{\alpha\beta}F^{\alpha\beta}\right)\Psi=0\,.$
(1.37)

This procedure yields the KGP equation for $g\\!=\\!2$. This algebraic ‘square
root’ will be elaborated on further in Section 1.2.2.

For $g\\!\neq\\!2$ the relationship between the DP and KGP equation becomes
more complicated. Instead of a clean algebraic separation, the substitution
between $\psi$ and $\Psi$ requires an infinite series expansion resulting from
the non-local inverse substitution

$\displaystyle\Psi\rightarrow\frac{1}{\mathcal{D}_{+}}\psi=\frac{1}{m_{\ell}c}\left(1-\frac{\hbar}{m_{\ell}c}i\gamma_{\alpha}\widetilde{\nabla}^{\alpha}-\frac{\hbar^{2}}{m_{\ell}^{2}c^{2}}\left(\gamma_{\alpha}\widetilde{\nabla}^{\alpha}\right)^{2}+\ldots\right)\psi\,.$
(1.38)

The expansion in Eq. (1.38) is considered non-local because it requires an
infinite number of initial conditions to determine.

While this procedure ‘square roots’ the KGP equation $(\mathrm{\sqrt{KGP}})$,
the resulting AMM Pauli Lagrangian Eq. (1.31) picks up an infinite number of
derivative and field terms which makes the theory rather unpalatable.

$\displaystyle\mathcal{L}_{\sqrt{\mathrm{KGP}}}=\mathcal{L}_{\mathrm{D}}+\mathcal{L}_{\mathrm{\sqrt{KGP},AMM}}\,,$
(1.39)
$\displaystyle\mathcal{L}_{\mathrm{\sqrt{KGP},AMM}}=-{\bar{\psi}}\left(\delta\mu\frac{1}{2}\sigma_{\alpha\beta}F^{\alpha\beta}\left(1-\frac{\hbar}{m_{\ell}c}i\gamma_{\alpha}\widetilde{\nabla}^{\alpha}-\frac{\hbar^{2}}{m_{\ell}^{2}c^{2}}\left(\gamma_{\alpha}\widetilde{\nabla}^{\alpha}\right)^{2}+\ldots\right)\right)\psi\,.$
(1.40)

We note each term in Eq. (1.40) is preceded by powers of the reduced Compton
wavelength $\lambda_{\mathrm{C}}\\!\equiv\\!\hbar/m_{\ell}c$ therefore the
$\sqrt{\mathrm{KGP}}$ model still might be of interest to study assuming the
physical system that admits a reasonable cutoff.

While the first term present in Eq. (1.40) is indeed the correct
$\mathcal{L}_{\mathrm{DP,AMM}}$ term, the resulting non-local behavior
ultimately breaks the unitarity of the theory making it unsuitable as a
fundamental particle theory (Veltman, 1998). While the above is suggestive
that there exists no unitary transform between the KGP and DP wave equations,
we do not claim it as an absolute proof. If a generalized description of
$g\\!\neq\\!2$ magnetic moment exists as a good quantum field theory, then
likely non-minimal electromagnetic terms are required to maintain both
renormalization and unitarity.

####  1.2.1 Features of the KGP Lagrangian

Before continuing to specific physical problems, we consider how current
conversation functions in the KGP formulation of fermions and how it might
differ from the Dirac current
$\mathcal{J}_{\mathrm{D}}^{\mu}\propto-i\bar{\psi}\gamma^{\mu}\psi$. The KGP
equation can be obtained from a Lagrangian not dissimilar to the Klein-Gordon
Lagrangian (Delgado-Acosta et al., 2011, 2015; Espin, 2015) and has the
expression

$\displaystyle\mathcal{L}_{\mathrm{KGP}}/c^{2}=\left(i\hbar{\widetilde{\nabla}}^{\mu}\right)^{{\dagger}}\bar{\Psi}h_{\mu\nu}\left(i\hbar{\widetilde{\nabla}}^{\mu}\right)\Psi-m^{2}c^{2}\bar{\Psi}\Psi\,,\qquad
h_{\mu\nu}=\eta_{\mu\nu}-i\frac{g}{2}\sigma_{\mu\nu}\,.$ (1.41)

The matrix $h_{\mu\nu}$ acts an ‘effective’ metric which has been modified to
account for the presence of an AMM. We note that the field $\Psi$ must have
units $[\mathrm{length}]^{-1}$ such that the Lagrangian density itself has
natural units of $[\mathrm{length}]^{-4}$.

In comparison to the DP AMM Lagrangian Eq. (1.31), the KGP magnetic moment
Lagrangian obtained from Eq. (1.41) is

$\displaystyle\mathcal{L}_{\mathrm{KGP,MM}}/c^{2}=-{\bar{\Psi}}\left(\frac{g}{2}\mu_{\ell}m_{\ell}\sigma_{\alpha\beta}F^{\alpha\beta}\right)\Psi\,.$
(1.42)

While they are mathematically similar both being ‘Pauli terms’, there are some
important differences. Here the combination of fields $\Psi$ and
$F^{\alpha\beta}$ have natural units $[\mathrm{length}]^{-4}$ and the coupling
coefficient $\mu m_{\ell}\\!\propto\\!g$ is manifestly dimensionless. This
means the KGP Lagrangian is at first inspection renormalizable which is an
improvement over the DP Lagrangian (Rafelski et al., 2023b). Literature
however suggests that the KGP Lagrangian requires additional fermion self-
interactions $\mathcal{L}_{\mathrm{int}}\sim\mathcal{O}(\bar{\Psi}\Psi)^{2}$
to be fully renormalizable (unless $g\\!=\\!0,\pm 2$) which are not forbidden
at tree-level (Angeles-Martinez and Napsuciale, 2012; Vaquera-Araujo et al.,
2013).

The conserved current obtained from Eq. (1.41) can be expressed as

$\displaystyle\mathcal{J}^{\mu}=-\frac{1}{c^{2}}\frac{\partial\mathcal{L}}{\partial
eA_{\mu}}\equiv\mathcal{J}^{\mu}_{\mathrm{Conv}}+\mathcal{J}^{\mu}_{\mathrm{Mag}}\,,$
(1.43)
$\displaystyle\mathcal{J}^{\mu}=\bar{\Psi}\left(i\hbar{\widetilde{\nabla}}^{\mu}\right)\Psi+\left(i\hbar{\widetilde{\nabla}}^{\mu}\right)^{{\dagger}}\bar{\Psi}\Psi+i\frac{g}{2}\bar{\Psi}\sigma^{\mu\nu}\left(i\hbar{\widetilde{\nabla}}_{\nu}\right)\Psi+i\frac{g}{2}\left(i\hbar{\widetilde{\nabla}}_{\nu}\right)^{{\dagger}}\bar{\Psi}\sigma^{\nu\mu}\Psi\,.$
(1.44)

The conserved current Eq. (1.43) can be interpreted as the sum of a convection
current $\mathcal{J}_{\mathrm{Conv}}$ and magnetization current
$\mathcal{J}_{\mathrm{Mag}}$.

$\displaystyle\mathcal{J}^{\mu}_{\mathrm{Conv}}$
$\displaystyle=\bar{\Psi}\left(i\hbar{\widetilde{\nabla}}^{\mu}\right)\Psi+\left(i\hbar{\widetilde{\nabla}}^{\mu}\right)^{{\dagger}}\bar{\Psi}\Psi\,,$
(1.45) $\displaystyle\mathcal{J}^{\mu}_{\mathrm{Mag}}$
$\displaystyle=\frac{g}{2}\hbar{\partial}_{\nu}\left(\bar{\Psi}\sigma^{\nu\mu}\Psi\right)\;.$
(1.46)

This is nearly identical to the Gordon Decomposition of the Dirac current
$\mathcal{J}_{\mathrm{D}}^{\mu}$, with the exception that the magnetization
current is proportional to $g$-factor.

The covariant derivative happens to simplify as
$\widetilde{\nabla}\rightarrow\partial$ in Eq. (1.46) such that the current is
a divergence of the spin density $\bar{\Psi}\sigma_{\mu\nu}\Psi$. Because of
the antisymmetry of $\sigma_{\mu\nu}$ the spin tensor, the magnetization
current is conserved independently of the the charge current. That both are
independently conserved indicates conservation in both charge $(e)$ and
magnetic moment $(\mu)$.

####  1.2.2 The special case of g = 2

There is a strong predilection in nature towards $g\\!=\\!2$ which can be
explained by the requirements of kinetic operator in quantum mechanics. Rather
than taking the non-relativistic limit of the Dirac equation, $g\\!=\\!2$ can
also be derived as a consequence of replacing the definition of the inner
product for vectors which accounts for spinor structure via an argument
attributed to R. P. Feynman; see footnote in Chap. 3.2 of Sakurai (1967).

The Schrödinger equation can be converted into the SP Eq. (1.12) via the
following replacement

$\displaystyle\bm{\pi}^{2}\rightarrow(\bm{\sigma}\cdot\bm{\pi})^{2}=\pi_{i}\sigma_{i}\sigma_{j}\pi_{j}\,.$
(1.47)

Because the $2\times 2$ Pauli matrices $\sigma_{i}$ all anticommute, we can
write down the relation

$\displaystyle(\bm{\sigma}\cdot\bm{a})(\bm{\sigma}\cdot\bm{b})=\bm{a}\cdot\bm{b}+i\bm{\sigma}\cdot(\bm{a}\times\bm{b})\,.$
(1.48)

The non-relativistic kinetic energy (KE) Hamiltonian from Eq. (1.12) then
reads as

$\displaystyle{H}_{\mathrm{SP,KE}}=\frac{1}{2m}\left(\bm{\sigma}\cdot\bm{\pi}\right)^{2}=\frac{1}{2m}\bm{\pi}^{2}+i\bm{\sigma}\cdot(\bm{\pi}\times\bm{\pi})=\frac{1}{2m}\bm{\pi}^{2}-\frac{e\hbar}{2m}\bm{\sigma}\cdot\bm{B}\,.$
(1.49)

As the kinetic momentum operator $\bm{\pi}$ does not self-commute, its cross
product is non-zero resulting in a magnetic moment term with magneton size
$\mu_{\ell}=e\hbar/2m_{\ell}$ and $g\\!=\\!2$. Therefore, we see there is
conceptual value in replacing the inner dot product with a more intricate
algebraic structure; in this case:
$\delta_{ij}\rightarrow\sigma_{i}\sigma_{j}$.

The natural gyromagnetic ratio then appears to arise from the $SU(2)$ Lie
algebra representation that the Pauli matrices describe and electromagnetic
minimal coupling. The natural scale of the magnetic moment can be interpreted
as originating from group symmetry requirements on charged particles.

An almost identical argument that $g$-factor arises from spin-structure and
electromagnetic coupling can be made for the relativistic case as well. First
we consider the quantum analog to the energy-momentum relation

$\displaystyle\eta_{\alpha\beta}p^{\alpha}p^{\beta}\Phi=m^{2}c^{2}\Phi\,.$
(1.50)

Eq. (1.50) as written evaluates to the Klein-Gordon equation on scalar field
$\Phi$ where the four-momentum is written in the position basis
$p^{\alpha}\rightarrow i\hbar\partial^{\alpha}$. Much like the non-
relativistic example, we can introduce spin by replacing the momentum inner
product with one sensitive to a Clifford algebra (Weinberg, 2005). Rather than
the Pauli matrices, the relativistic replacement utilizes the gamma matrices
yielding

$\displaystyle\eta_{\alpha\beta}\rightarrow\gamma_{\alpha}\gamma_{\beta}\,,\qquad\gamma_{\alpha}\gamma_{\beta}p^{\alpha}p^{\beta}\Psi$
$\displaystyle=m^{2}c^{2}\Psi\,.$ (1.51)

Here $\Psi$ is understood to be a four-component spinor unlike in Eq. (1.50).
The corresponding $4\times 4$ matrix contraction identity analog to Eq. (1.48)
is then

$\displaystyle\gamma_{\alpha}\gamma_{\beta}a^{\alpha}b^{\beta}=\eta_{\alpha\beta}a^{\alpha}b^{\beta}-i\sigma_{\alpha\beta}a^{\alpha}b^{\beta}\,.$
(1.52)

In both the relativistic and non-relativistic cases, the distinction between
spin-1/2 and spinless particles is only made apparent in the kinematics in the
presence of electromagnetic fields. For minimal coupling
$\pi^{\alpha}=p^{\alpha}-eA^{\alpha}$ we take advantage of the fact that any
tensor product of vectors can be decomposed as a sum of commuting (symmetric)
and anticommuting (antisymmetric) parts

$\displaystyle\pi^{\alpha}\pi^{\beta}=\frac{1}{2}\left\\{\pi^{\alpha},\pi^{\beta}\right\\}+\frac{1}{2}\left[\pi^{\alpha},\pi^{\beta}\right]\,,\qquad\gamma^{\alpha}\gamma^{\beta}=\frac{1}{2}\left\\{\gamma^{\alpha},\gamma^{\beta}\right\\}+\frac{1}{2}\left[\gamma^{\alpha},\gamma^{\beta}\right]\,.$
(1.53)

From the above and Eq. (1.51) and Eq. (1.28) we obtain

$\displaystyle\gamma_{\alpha}\gamma_{\beta}\pi^{\alpha}\pi^{\beta}\Psi=\left(\eta_{\alpha\beta}\pi^{\alpha}\pi^{\beta}-\frac{e\hbar}{2}\sigma_{\alpha\beta}F^{\alpha\beta}\right)\Psi=m^{2}c^{2}\Psi\,.$
(1.54)

Eq. (1.54) is the square of the Dirac equation with precisely $g\\!=\\!2$ but
in a different sense than the argument established in Eq. (1.37). Rather than
squaring the Dirac equation, from this perspective, we are enlarging the
structure of the energy-momentum relation. The spin-1 Proca equations and
spin-3/2 Rarita-Schwinger equations can also be justified via this line of
reasoning with different replacements for the field and inner-product
definition.

How $g\\!\neq\\!2$ AMM ‘breaks’ the inner product substitution is seen more
explicitly when we write the effective metric tensor $h_{\mu\nu}$ from Eq.
(1.41) as

$\displaystyle
h_{\mu\nu}=\frac{1}{2}\\{\gamma_{\mu},\gamma_{\nu}\\}+\frac{1}{2}(1+a)[\gamma_{\mu},\gamma_{\nu}]=\gamma_{\mu}\gamma_{\nu}+\frac{a}{2}[\gamma_{\mu},\gamma_{\nu}]\,.$
(1.55)

The anomalous part in Eq. (1.55) is inconveniently unable to be packaged as
the elementary tensor product of two four-vectors like in Eq. (1.51). The same
issue occurs with any anomalous EDM. We suggest that more elaborate algebraic
structures might accommodate such terms more naturally though we leave that to
future work.

Furthermore, compelling arguments can be made that all elementary particles of
any spin must have a natural gyromagnetic factor of $g\\!=\\!2$; though we
mention a competing idea is Belinfante’s conjecture of $g\\!=\\!1/s$. To
paraphrase the arguments by Ferrara, Porrati, and Telegdi (1992), $g\\!=\\!2$
is likely the natural scale for particles of any spin because:

  1. 1.

The W boson, as the only known higher spin charged elementary particle, has at
tree level $g\\!=\\!2$ via a Proca-like equation.

  2. 2.

The relativistic TBMT torque equation is the same for any classical spin value
and is most simple when the anomalous moment is zero.

  3. 3.

For arbitrary spin, $g\\!=\\!2$ facilitates finite Compton scattering cross
sections without additional physical requirements.

  4. 4.

For charged interacting particles with arbitrary spin, open bosonic and super-
symmetric string theory predicts $g\\!=\\!2$.

We would like to add the additional argument that rotating charged black holes
described by the Kerr-Newman metric also have a magnetic dipole moment with
fixed $g\\!=\\!2$ character (Carter, 1968). This illustrates that in some
sense the spin of a black hole is ‘particle-like’ and dissimilar to the
orbital Ampèrian motion of matter which has an orbital $g$-factor of
$g_{\mathrm{L}}\\!=\\!1$; see Section 2.1.

While the above provide a nice justification for why particles should tend to
this specific $g$-factor, the reality is no particle has exactly $g\\!=\\!2$
with all of them displaying some form of anomaly. The charged leptons come the
closest to the natural value, but famously have vacuum polarization
contributions (Schwinger, 1951) from QED, non-perturbative hadronic
contributions (Jegerlehner, 2017), and potentially BSM interactions (Knecht,
2004) contributing to their anomalous magnetic dipole moment.

While the perturbative approach has proven to be exceedingly successful for
the charged leptons, it is not appropriate for particles whose moments are
dramatically different from $g\\!=\\!2$ or if the origin of the anomaly comes
from internal structure such as the hadrons whose moments are determined by
non-perturbative QCD (Pacetti et al., 2015) and not weakly coupled $\alpha\sim
1/137\ll 1$ EM vacuum structure.

###  1.3 A few words on cosmology

This section introduces some necessary concepts which will be useful in
describing the magnetization of the electron-positron primordial plasma in
Chapter 4. We operate under the $\Lambda$ Cold Dark Matter
$(\Lambda\mathrm{CDM})$ model of cosmology where the contemporary universe is
approximately 69% dark energy, 26% dark matter, 5% baryons, and $<1$% photons
and neutrinos in energy density (Davis and Lineweaver, 2004; Aghanim et al.,
2020). The standard picture of the universe’s evolution is outlined in Figure
1.1.

The Friedmann-Lemaître-Robertson-Walker (FLRW) line element and metric
(Weinberg, 1972) in spherical coordinates is

$\displaystyle
ds^{2}=dt^{2}-a^{2}(t)\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}d\theta^{2}+r^{2}\sin\theta^{2}d\phi^{2}\right]\,,$
(1.56) $\displaystyle g_{\alpha\beta}=\begin{pmatrix}1&0&0&0\\\
0&-\frac{a^{2}(t)}{1-kr^{2}}&0&0\\\ 0&0&-a^{2}(t)r^{2}&0\\\
0&0&0&-a^{2}(t)r^{2}\sin\theta^{2}\end{pmatrix}\,.$ (1.57)

The Gaussian curvature $k$ informs the spatial shape of the universe with the
following possibilities: infinite flat Euclidean $(k=0)$, finite spherical but
unbounded $(k=+1)$, or infinite hyperbolic saddle-shaped $(k=-1)$. Observation
indicates our universe is flat or nearly so.

Figure 1.1: A schematic of the universe’s evolution since the Big Bang. The
region of interest studied in this dissertation is emphasized (in the
highlighted box) to contain a dense nearly charge neutral matter-antimatter
plasma.

The scale factor $a(t)$ denotes the change of proper distances $L(t)$ over
time as

$\displaystyle L(t)=L_{0}\frac{a_{0}}{a(t)}\rightarrow L(z)=L_{0}(1+z)\,,$
(1.58)

where $z$ is the redshift and $L_{0}$ the comoving length. In an expanding (or
contracting) universe which is both homogeneous and isotropic. This implies
volumes change with $V(t)=V_{0}/a^{3}(t)$ where $V_{0}=L_{0}^{3}$ is the
comoving Cartesian volume. The evolutionary expansion of the universe is then
traditionally defined in terms of the Hubble parameter $H(t)$ following the
conventions in Weinberg (1972)

$\displaystyle H(t)^{2}\equiv\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi
G_{N}}{3}\rho_{\mathrm{total}},\qquad\rho_{\mathrm{total}}(t)=\rho_{\Lambda}+\rho_{\mathrm{DM}}(t)+\rho_{\mathrm{Baryons}}(t)+\ldots$
(1.59) $\displaystyle\frac{\ddot{a}}{a}=-qH^{2},\qquad
q\equiv-\frac{a\ddot{a}}{\dot{a}^{2}},\qquad\dot{H}=-H^{2}(1+q).$ (1.60)

where $G_{N}$ is the Newtonian constant of gravitation. Eq. (1.59) and Eq.
(1.60) are also known as the Friedmann equations. The total density
$\rho_{\mathrm{total}}$ is the sum of all contributions from any form of
matter, radiation or field. This includes but is not limited to: dark energy
$(\Lambda)$, dark matter (DM), baryons (B), leptons $(\ell,\nu)$ and photons
$(\gamma)$. Depending on the age of the universe, the relative importance of
each group changes as each dilutes different under expansion with dark energy
infamously remaining constant in density and accelerating the universe today.

The parameter $q$ is the cosmic deceleration parameter, which for historical
reasons is positive for $q>0$. This sign convention was chosen before the
discovery of dark energy under the tacit assumption that the universe would be
decelerating. The value of $q$ depends on the energy content of the universe:
The early universe was radiation dominated $(q=1)$, subsequently matter
dominated $(q=1/2)$, and lastly the contemporary universe is undergoing a
transition from matter to dark energy dominated approaching the asymptotic
value of $q=-1$; see Rafelski and Birrell (2014).

We can consider the expansion to be an adiabatic process (Abdalla et al.,
2022) which results in a smooth shifting of the relevant dynamical quantities.
As the universe undergoes isotropic expansion, the temperature decreases as

$\displaystyle T(t)=T_{0}\frac{a_{0}}{a(t)}\rightarrow T(z)=T_{0}(1+z)\,,$
(1.61)

where $z$ is the redshift. The entropy within a comoving volume is kept
constant until gravitational collapse effects become relevant. The comoving
temperature $T_{0}$ is given by the the present CMB temperature
$T_{0}=2.7{\rm\ K}\simeq 2.3\times 10^{-4}\text{ eV}$ (Aghanim et al., 2020),
with contemporary scale factor $a_{0}=1$.

As the universe expands, redshift reduces the momenta of particles lowering
their contribution to the energy content of the universe. This cosmic redshift
is written as

$\displaystyle p_{i}(t)=p_{i,0}\frac{a_{0}}{a(t)}\,.$ (1.62)

Momentum (and the energy of massless particles $E=pc$) scales with the same
factor as temperature. The energy of massive free particles in the universe
however scales differently based on their momentum (and thus temperature).

When hot and relativistic, particle energy decreases inversely with scale
factor like radiation. As the particles transition to non-relativistic (NR)
energies, they decrease with the inverse square of the scale factor

$\displaystyle E(t)=E_{0}\frac{a_{0}}{a(t)}\xrightarrow{\mathrm{NR}}\
E_{0}\frac{a_{0}^{2}}{a(t)^{2}}\,.$ (1.63)

This occurs because of the functional dependence of energy on momentum in the
relativistic $E\sim p$ versus non-relativistic $E\sim p^{2}$ cases.

##  Chapter 2 Dynamics of charged particles with arbitrary magnetic moment

In Section 1.1, we addressed two different models of introducing anomalous
magnetic moment in QM:

  1. (a)

the Dirac-Pauli (DP) first order equation which is the Dirac equation where
$g$-factor is precisely fixed to the $g\\!=\\!2$, with the addition of an
incremental Pauli term; and

  2. (b)

the Klein-Gordon-Pauli (KGP) second order equation which “squares” the Dirac
equation and thereafter allows the magnetic moment $\bm{\mu}$ to vary
independently of charge and mass, unlike Dirac theory.

These two approaches coincide when the anomaly $a$ vanishes. However, all
particles that have magnetic moments differ from the Dirac value $g\\!=\\!2$,
either due to their composite nature or due to the quantum vacuum fluctuation
effect.

We find that even a small magnetic anomaly has a large effect in the limit of
strong fields generated by massive magnetar stars (Kaspi and Beloborodov,
2017). Therefore it is not clear that the tacit assumption of $g\\!=\\!2$ in
the case of strong fields (Rafelski et al., 1978; Greiner et al., 2012a;
Rafelski et al., 2017) is prudent (Evans and Rafelski, 2018). This argument is
especially applicable to tightly bound composite particles such as protons and
neutrons where the large anomalous magnetic moment can be taken as an external
prescribed parameter unrelated to the elementary quantum vacuum fluctuations.
It is then of particular interest to study the dynamical behavior of these
particles in fields of magnetar strength. This interest carries over to the
environment of strong fields created in focus of ultra-intense laser pulses
and the associated particle production processes (Dunne, 2014; Hegelich et
al., 2014). We consider also precision spectroscopic experiments and recognize
consequences even in the weak coupling limit.

This chapter reviews our work done in exploring relativistic dynamics with
arbitrary magnetic dipoles in both a quantum mechanical and classical context.
Section 2.1 and Section 2.2 and covers analytic solutions for the Klein-
Gordon-Pauli (KGP) equation in the presence of homogeneous magnetic fields and
the Coulomb problem for hydrogen-like atoms. Comparisons with the Dirac-Pauli
(DP) and Dirac solutions are made and novel consequences for strong fields are
discussed in Section 2.3. Section 2.4 explores extensions to KGP which combine
mass and magnetic moment into a dynamical mass which is sensitive to
electromagnetic fields. This work is primarily based on Steinmetz et al.
(2019).

Relativistic classical spin dynamics is discussed in Section 2.5 and is based
on our work in Rafelski et al. (2018). We propose in Section 2.5.1 a covariant
form of magnetic dipole potential which modifies the Lorentz force, extends
the Thomas-Bargmann-Michel-Telegdi (TMBT) equation, and reproduces the Stern-
Gerlach force in the non-relativistic limit. Section 2.5.2 demonstrates that
this magnetic potential also serves to unify both the Ampèrian and Gilbertian
pictures of dipole moments.

###  2.1 Homogeneous magnetic fields

The case of the homogeneous magnetic field, sometimes referred to as the
Landau problem, provides a stepping stone in which to examine the consequences
of quantum spin dynamics in a concrete analytical fashion. We present here an
abbreviated analysis and the full treatment of this solution in terms of
Ladder operators can be found in Steinmetz et al. (2019) while alternative
approaches are shown in texts such as Itzykson and Zuber (1980). We assume a
constant magnetic field in the $z$-direction

$\displaystyle\bm{B}=(0,0,B)\,.$ (2.1)

For our choice of gauge, there are two common options: (a) the Landau
$\bm{A}_{\mathrm{L}}$ gauge and (b) the symmetric $\bm{A}_{\mathrm{S}}$ gauge

$\displaystyle\bm{A}_{\mathrm{L}}=B(0,x,0)\,,\indent\bm{A}_{\mathrm{S}}=\frac{B}{2}(-y,x,0)\,.$
(2.2)

As the system has a manifest rotational symmetry perpendicular to the
direction of the homogeneous field, we will choose the symmetric gauge
$\bm{A}_{\mathrm{S}}$ which preserves this symmetry explicitly.

Before we examine relativistic wave equations, it will be helpful to first
consider the non-relativistic Schrödinger-Pauli case as the KGP-Landau problem
can be written as equivalent to the Schrödinger-Pauli Hamiltonian. We consider
energy eigenstates of and electron with $m_{e}$ described by Eq. (1.12) under
Eq. (2.1) as

$\displaystyle\chi\rightarrow\chi_{\mathrm{E}}\exp\left(-\frac{iEt}{\hbar}\right)\,,\qquad\left(\frac{1}{2m_{e}}\bm{\pi}^{2}-\bm{\mu}\cdot\bm{B}\right)\chi_{\mathrm{E}}=E\chi_{\mathrm{E}}\,,$
(2.3)

where $\mu$ is the magnitude of the magnetic moment as defined in Eq. (1.11).
Eq. (2.3) can be further rewritten using angular momentum $\bm{L}$ and the
symmetric gauge Eq. (2.2) as

$\displaystyle\left(\frac{1}{2m_{e}}\bm{p}^{2}+\frac{e^{2}B^{2}}{8m_{e}}(x^{2}+y^{2})-\frac{eB}{2m_{e}}L_{3}-\mu
B\sigma_{3}\right)\chi_{\mathrm{E}}=E\chi_{\mathrm{E}}\,,$ (2.4)
$\displaystyle\bm{L}=\bm{r}\times\bm{p}\,,\qquad
L_{i}=\varepsilon_{ijk}x_{j}p_{k}\,.$ (2.5)

The above can be broken into a set of three mutually commuting Hamiltonian
operators:

  * (a)

Free particle Hamiltonian (Free)

  * (b)

Quantum harmonic oscillator (HO)

  * (c)

Zeeman interaction (ZI)

given by

$\displaystyle\mathrm{(a)}\quad$ $\displaystyle
H_{\mathrm{Free}}=\frac{p_{3}^{2}}{2m}\,,$ (2.6)
$\displaystyle\mathrm{(b)}\quad$ $\displaystyle
H_{\mathrm{HO}}\,=\frac{1}{2m_{e}}\left(p_{1}^{2}+p_{2}^{2}\right)+\frac{1}{2}m_{e}\omega^{2}(x^{2}+y^{2})\,,$
(2.7) $\displaystyle\mathrm{(c)}\quad$ $\displaystyle H_{\mathrm{ZI}}\
\,=-\mu_{B}B\frac{L_{3}}{\hbar}-\mu B\sigma_{3}\,,$ (2.8)

which forms the total Hamiltonian

$\displaystyle
H_{\mathrm{total}}=H_{\mathrm{Free}}+H_{\mathrm{HO}}+H_{\mathrm{Mag.}}\,.$
(2.9)

The cyclotron frequency appears in Eq. (2.7) as
$2\omega=\omega_{\mathrm{C}}=eB/m_{e}$. We note that the Zeeman Eq. (2.8) is
usually expressed as

$\displaystyle
H_{\mathrm{ZI}}=-\frac{e}{2m}\left(g_{\mathrm{L}}\bm{L}+g\bm{S}\right)\cdot\bm{B}\,,\qquad
g_{\mathrm{L}}=1\,,$ (2.10)

with $\bm{S}$ defined in Eq. (1.2). We see explicitly that the orbital
gyromagnetic ratio $g_{\mathrm{L}}$ which is a coefficient to the angular
momentum operator $\bm{L}$ is unity unlike for spin. We refer back to our
comment about black hole rotation in Section 1.2.2 as spin-like rather than
orbital-like in terms of its magnetic behavior. As all the above Hamiltonian
operators are mutually commuting, the energy eigenvalue of the total
Hamiltonian is the sum of the individual energy eigenvalues. Our remaining
goal will be to convert the KGP eigenvalue equation into the above three non-
relativistic Hamiltonian operators.

We now return to the KGP equation and write expand Eq. (1.33) for the Landau
problem with energy eigenstates $\Psi_{\mathrm{E}}$ yielding

$\displaystyle\left(\frac{E^{2}}{c^{2}}-m_{e}^{2}c^{2}-\bm{p}^{2}-\frac{1}{4}e^{2}B^{2}\left(x^{2}+y^{2}\right)+eBL_{3}+2\mu
m_{e}B\sigma_{3}\right)\Psi_{\mathrm{E}}=0\;.$ (2.11)

We introduce the substitutions

$\displaystyle E\to m^{\prime}c^{2}\,,\quad\frac{E^{2}-m_{e}^{2}c^{4}}{2E}\to
E^{\prime}\;,$ (2.12)

and recast KGP Eq. (2.11) into a Schrödinger-style Hamiltonian equation

$\displaystyle\left(\frac{1}{2m^{\prime}}\bm{p}^{2}+\frac{e^{2}B^{2}}{8m^{\prime}}(x^{2}+y^{2})-\frac{eB}{2m^{\prime}}L_{3}-\mu\left(\frac{m_{e}}{m^{\prime}}\right)B\sigma_{3}\right)\Psi_{\mathrm{E}}=E^{\prime}\Psi_{\mathrm{E}}\,,$
(2.13)

which matches the non-relativistic Hamiltonian presented in Eq. (2.9).

The energy eigenvalues of Eq. (2.9) are given by

$\displaystyle E^{\prime}_{n,s}(p_{3},B)$
$\displaystyle=\frac{p_{3}^{2}}{2m^{\prime}}+\frac{e\hbar
B}{m^{\prime}}\left(n+\frac{1}{2}\right)-\mu
B\left(\frac{m_{e}}{m^{\prime}}\right)s\,,$ (2.14)

where $n\in 1,2,3\ldots$ is the Landau orbital quantum number and $s\in\pm 1$
is the spin quantum number. The physical relativistic energies can be obtained
by undoing the substitutions in Eq. (2.12) yielding from Eq. (2.14)

$\displaystyle E^{2}_{n,s}(p_{3},B)=m_{e}^{2}c^{4}+p_{3}^{2}c^{2}+2e\hbar
c^{2}B\left(n+\frac{1}{2}\right)-2\mu Bm_{e}c^{2}s\,,$ (2.15) $\displaystyle
E_{n,s}=\pm\sqrt{m_{e}^{2}c^{4}+p_{3}^{2}c^{2}+2e\hbar
c^{2}B\left(n+\frac{1}{2}\right)-2\mu Bm_{e}c^{2}s}\;.$ (2.16)

This expression for the relativistic Landau levels is the same as found by
Weisskopf (1936) for the Dirac equation setting $g\\!=\\!2$ in Eq. (2.16). The
Landau orbital part and spin portions can be combined when the magnetic moment
is expressed in terms of $e/m$, but the form in Eq. (2.16) keeps it
generalized for the case of neutral particles.

Restricting ourselves to the positive energy spectrum, the non-relativistic
reduction of Eq. (2.16) can be carried out in powers of $1/m$ in the large
mass limit yielding

$\displaystyle E_{n,s}|_{\mathrm{NR}}$
$\displaystyle=m_{e}c^{2}+\frac{p_{3}^{2}}{2m_{e}}+\mu_{B}B\left(2n+1-\frac{g}{2}s\right)-\frac{p_{3}^{4}}{8m_{e}^{3}c^{2}}$
(2.17)
$\displaystyle\\!-\\!\frac{p_{3}^{2}}{2m_{e}}\frac{\mu_{B}B}{m_{e}c^{2}}\left(2n+1-\frac{g}{2}s\right)\\!-\\!\frac{\mu_{B}^{2}B^{2}}{2m_{e}c^{2}}\left(2n+1-\frac{g}{2}s\right)^{2}\\!+\\!\mathcal{O}(1/m_{e}^{5})\;,$

which contains the expected terms such as the non-relativistic kinetic energy
in the z-direction, the first relativistic correction to kinetic energy, the
Landau energies, and cross terms that behave like modifications to the mass of
the particle.

The KGP-Landau levels above the ground state lose their (accidental)
degeneracy for $g\neq 2$. This is shown schematically in Figure 2.1. The
anomaly also causes the ground state to be pushed downward, such that
$E^{2}<m^{2}$; if the anomaly and the magnetic field are large enough, states
above the ground state are also pushed below the rest mass energy of the
particle.

Figure 2.1: Diagram organizing the KGP-Landau levels for particles with zero
z-component momentum. The Landau levels $n$ and spin $s$ serve to split levels
while the anomaly $a$ controls the degeneracy among levels.

However, we recognize a periodicity considering the energy as a function of
$g$. We recall that in Eq. (2.16) $n=0,1,2\ldots$. As $g$ varies, each time
$gs/2$ crosses an integer value, for a different value of $n$ the energy
eigenvalue $E$ repeat as a function of changing $g$. All possible values of
energy $E$ are reached (at fixed $m$ and $p^{2}_{3}$) for $-2\leq g\leq 2$.
Moreover, while for almost all $g\neq 2$ the degeneracy is completely broken,
this periodicity implies that energy degeneracy is restored for values Evans
(2022); Evans and Rafelski (2022)

$\displaystyle
g_{k}/2=1+k\,,\qquad\lambda_{\mathrm{L}}^{\prime}=\lambda_{\mathrm{L}}-ks\;,\qquad\lambda_{\mathrm{L}}=n+\frac{1}{2}-s\;,$
(2.18)

where $k=0,\pm 1,\pm 2,\ldots$ The Landau levels Eq. (2.15) contain an
infinite number of degenerate levels bounded from below. Certain states change
the sign of the magnetic energy and their total energies become unphysical in
the limit that $g_{k}B$ becomes large; for even $k$ there are $k/2$ such
states and for odd $k$ there are $(k+1)/2$.

It is useful to compare the KGP solution to the the Landau levels for the DP
equation which we obtained by Tsai and Yildiz (1971). The DP-Landua energy
eigenstates are given by

$\displaystyle E^{2}_{n,s}(p_{3},B)|_{\mathrm{DP}}=$
$\displaystyle\left(\\!\\!\sqrt{\displaystyle m_{e}^{2}c^{4}\\!+\\!2e\hbar
c^{2}B\left(n+\frac{1}{2}-s\right)}-\frac{eB\hbar}{2m_{e}}(g-2)s\\!\right)^{2}\\!\\!\\!+p_{3}^{\\!\\!2}c^{2},$
(2.19a) $\displaystyle E_{n,s}|_{\mathrm{DP}}=\pm$
$\displaystyle\sqrt{\\!\left(\\!\\!\sqrt{\displaystyle m_{e}^{2}c^{4}+2e\hbar
c^{2}B\left(n+\frac{1}{2}-s\right)}\\!-\\!\frac{eB\hbar}{2m_{e}}(g-2)s\\!\right)^{\\!\\!2}\\!\\!\\!+p_{3}^{2}c^{2}}\;,$
(2.19b)

which in our opinion fails Dirac’s principle of mathematical beauty when
compared to the KGP result Eq. (2.16). Both Eqs. (2.16) and (2.19b) have the
correct non-relativistic reduction at the lowest order though, the latter
obscures the physical interpretation.

The most egregious issue with the DP-Landau levels is that, in a perturbative
expansion, it includes cross terms between the $g\\!=\\!2$ magnetic moment and
anomalous terms in $a=(g-2)/2$; thus the result does not depend on the
particle magnetic moment alone; there is a functional dependence on the
magnetic anomaly $a$. The presence of these cross terms implies that above
first order the results cannot be given in terms of the full magnetic moment
alone. In comparison, for the KGP-Landau levels Eq. (2.19b), the entire effect
of magnetic moment is contained in a single term.

###  2.2 Hydrogen-like atoms

The Coulomb problem, or sometimes referred to as the Kepler problem, provides
us an important application of any quantum theory to explore. As the hydrogen-
like atoms are the the most well understood atomic system in physics, any non-
minimal behavior especially for high-$Z$ systems can lead to consequences for
the resulting spectral lines. We take the Coulomb potential to be

$eV_{\mathrm{C}}\equiv eA^{0}=\frac{Z\alpha\hbar c}{r}\;,\qquad\bm{A}=0\;.$
(2.20)

The KGP-Coulomb problem with arbitrary magnetic moment can be solved
analytically and we will briefly sketch out the solution and its consequences.

For energy states $\Psi=e^{-iEt/\hbar}\Psi_{\mathrm{E}}$ the KGP equation
yields the following differential equation

$\displaystyle\left(\frac{E^{2}-m^{2}c^{4}}{\hbar^{2}c^{2}}+\frac{Z^{2}\alpha^{2}}{r^{2}}+\frac{2E}{\hbar
c}\frac{Z\alpha}{r}+\frac{1}{r}\frac{\partial\bm{L}^{2}}{\partial
r^{2}}r-\frac{\bm{L}^{2}/\hbar^{2}}{r^{2}}-\frac{g}{2}Z\alpha\frac{i\bm{\alpha}\cdot\bm{\hat{r}}}{r^{2}}\right)\Psi_{\mathrm{E}}=0\;.$
(2.21)

We recast the squared angular momentum operator $\bm{L}^{2}$ with the Dirac
spin-alignment operator

$\displaystyle\mathcal{K}=\gamma^{0}\left(1+\bm{\Sigma}\cdot\frac{\bm{L}}{\hbar}\right)\;,\indent\bm{L}^{2}/\hbar^{2}=\mathcal{K}\left(\mathcal{K}-\gamma^{0}\right)\;.$
(2.22)

The operator $\mathcal{K}$ commutes with $\bm{\alpha}\cdot\bm{\hat{r}}$ and
its eigenvalues are given as either positive or negative integers
$\kappa=\pm(j+1/2)$ where $j$ is the total angular momentum quantum number. By
grouping all terms proportional to $1/r^{2}$, we see the effective angular
momentum eigenvalues take on non-integer values which in the limit of
classical mechanics corresponds to orbits which do not close. The non-integer
eigenvalues depends explicitly on $g$-factor.

The difficulty of this equation is that the effective angular momentum
operator is non-diagonal in spinor space due to the presence of
$\bm{\alpha}\cdot\bm{\hat{r}}$ which mixes upper and lower components. The
effective radial potential within Eq. (2.21) is then

$\displaystyle\boxed{V_{\mathrm{eff}}=-\frac{2E}{\hbar
c}\frac{Z\alpha}{r}-\frac{Z^{2}\alpha^{2}}{r^{2}}+\frac{g}{2}Z\alpha\frac{i\bm{\alpha}\cdot\bm{\hat{r}}}{r^{2}}}\,.$
(2.23)

We emphasize that the distinguishing characteristic which separates the KGP
solutions (and Dirac for $g\\!=\\!2$) from the Klein-Gordon solutions is the
last term in the effective potential Eq. (2.23). This is the dipole-charge
interaction term which exists only because of the relativistic expression for
the magnetic dipole and is entirely absent non-relativistically.

Following the procedure of Martin and Glauber (1958), we introduce the
operator

$\displaystyle\mathfrak{L}=-\gamma^{0}\mathcal{K}-\frac{g}{2}Z\alpha(i\bm{\alpha}\cdot\bm{\hat{r}})\;,$
(2.24)

but with the novel modification that $g$-factor directly appears in the second
term. This operator is also sometimes referred to as the Temple operator,
therefore we will refer to it as the $g$-Temple operator. This operator then
commutes with the spin-alignment operator $\mathcal{K}$ and has eigenvalues

$\displaystyle\Lambda=\pm\sqrt{\kappa^{2}-\displaystyle\frac{\displaystyle
g^{2}}{4}Z^{2}\alpha^{2}}\;,$ (2.25)

where the absolute values are denoted as $\lambda=|\Lambda|$. The angular
momentum contributions to Eq. (2.21) can then be replaced by

$\displaystyle\mathcal{K}(\mathcal{K}-\gamma^{0})-Z^{2}\alpha^{2}-\frac{g}{2}Z\alpha(i\bm{\alpha}\cdot\bm{\hat{r}})=\mathfrak{L}(\mathfrak{L}+1)+\left(\frac{g^{2}}{4}-1\right)Z^{2}\alpha^{2}.$
(2.26)

If the $g$-factor is taken to be $g\\!=\\!2$, then the differential Eq. (2.21)
reverts to the one discussed in Martin and Glauber (1958). The coefficient
$g^{2}/4-1$ will be commonly seen to precede new more complicated terms, which
conveniently vanish for $|g|=2$ demonstrating that as function of $g$ there is
a “cusp” (Rafelski et al., 2023b) for $|g|=2$. This will become especially
evident when we discuss strongly bound systems which behave very differently
for $|g|<2$ versus $|g|>2$.

We omit further derivation which can be found in Steinmetz et al. (2019). We
find the resulting energy levels of the KGP-Coulomb equation to be

$\displaystyle
E_{\pm\lambda}^{n_{r},j}=mc^{2}\left[1+\displaystyle\frac{Z^{2}\alpha^{2}}{\left(n_{r}+1/2+\nu\right)^{2}}\right]^{-1/2}\,,$
(2.27) $\displaystyle\nu=\sqrt{(\lambda\pm
1/2)^{2}+\left(\frac{g^{2}}{4}-1\right)Z^{2}\alpha^{2}}\,,\qquad\lambda=\sqrt{\displaystyle(j+1/2)^{2}-\frac{\displaystyle
g^{2}}{4}Z^{2}\alpha^{2}}\;.$ (2.28)

where $n_{r}$ is the node quantum number which takes on the values
$n_{r}=0,1,2,\ldots$. Eq. (2.27) is the same “Sommerfeld-style” expression for
energy that we can obtain from the Dirac or KG equations. The difference
between them arises from the expression of the relativistic angular momentum
which depends on $g$-factor for the KGP equation. The KGP eigenvalues Eq.
(2.27) for arbitrary spin were obtained by Niederle and Nikitin (2006) using a
tensor approach.

In the limit that $g\rightarrow 2$ for the Dirac case the expressions for
$\lambda$ and $\nu$ reduce to

$\displaystyle\lim_{g\rightarrow
2}\lambda=\sqrt{\displaystyle(j+1/2)^{2}-Z^{2}\alpha^{2}},$ (2.29a)
$\displaystyle\lim_{g\rightarrow 2}\nu_{\pm\lambda}=\lambda\pm 1/2\;.$ (2.29b)

This procedure requires taking the root of perfect squares; therefore, the
sign information is lost in Eq. (2.29a). As long as $Z^{2}\alpha^{2}<3/4$ we
can drop the absolute value notation as $\nu$ is always positive. The energy
is then given by

$\displaystyle
E_{\pm\lambda}^{n_{r},j}=mc^{2}\left[1+\displaystyle\frac{Z^{2}\alpha^{2}}{\left(n_{r}\begin{smallmatrix}+1\\\
+0\end{smallmatrix}+\sqrt{\displaystyle(j+1/2)^{2}-Z^{2}\alpha^{2}}\right)^{2}}\right]^{-1/2}\;.$
(2.30)

The $\begin{smallmatrix}+1\\\ +0\end{smallmatrix}$ notation is read as the
upper value corresponding to the $+\lambda$ states and the lower value
corresponding to the $-\lambda$ states.

The ground state energy (with: $n_{r}=0,\ \Lambda<0,\ j=1/2$) is therefore

$\displaystyle E^{0,1/2}_{-\lambda(j=1/2)}=mc^{2}\sqrt{1-Z^{2}\alpha^{2}}\;,$
(2.31)

as expected for the Dirac-Coulomb ground state. Eq. (2.30) reproduces the
Dirac-Coulomb energies and also contains a degeneracy between states of
opposite $\lambda$ sign, same $j$ quantum number and node quantum numbers
offset by one

$\displaystyle E^{n_{r}+1,j}_{-\lambda}=E^{n_{r},j}_{+\lambda}\;,$ (2.32)

which corresponds to the degeneracy between $2S_{1/2}$ and $2P_{1/2}$ states.
There is no degeneracy for the $E^{0,j}_{-\lambda}$ states.

In the limit that $g\rightarrow 0$, which is the KG case, the expressions are
given by

$\displaystyle\lim_{g\rightarrow 0}\lambda=j+1/2,$ (2.33a)
$\displaystyle\lim_{g\rightarrow
0}\nu_{\pm\lambda}=\sqrt{\left(j\begin{smallmatrix}+1\\\
+0\end{smallmatrix}\right)^{2}-Z^{2}\alpha^{2}}\;,$ (2.33b)

which reproduces the correct expressions for the energy levels for the Klein-
Gordon case

$\displaystyle
E_{\pm\lambda}^{n_{r},j}=mc^{2}\left[1+\displaystyle\frac{Z^{2}\alpha^{2}}{\left(n_{r}+1/2+\displaystyle\sqrt{\left(j\begin{smallmatrix}+1\\\
+0\end{smallmatrix}\right)^{2}-Z^{2}\alpha^{2}}\right)^{2}}\right]^{-1/2}\;,$
(2.34)

except that in this limit we are still considering the total angular moment
quantum number $j$ rather than orbital momentum quantum number $\ell$. It is
interesting to note that the KG-Coulomb problem’s energy formula contains
$\ell+1/2$, which matches identically to our half-integer $j$ values;
therefore, this artifact of spin, untethered and invisible by the lack of
magnetic moment, does not alter the energies of the states. The degeneracy in
energy levels are given by

$\displaystyle E^{n_{r},j+1}_{-\lambda}=E^{n_{r},j}_{+\lambda}\;,$ (2.35)

with levels of opposite $\lambda$ sign, same node quantum number and shifted
$j$ values by one. In a similar fashion to the Dirac case, here we have no
degeneracy for $E^{n_{r},1/2}_{-\lambda}$ states.

####  2.2.1 Non-relativistic Coulomb problem energies

The first regime of interest to understand the effect of variable $g$ in the
KGP-Coulomb problem is the non-relativistic limit characterized by the weak
binding of low-Z atoms. We now will convert from $n_{r}$, $j$ and $\pm\lambda$
to the familiar quantum numbers of $n$, $j$ and $\ell$ allowing for easy
comparison with the hydrogen spectrum in standard notation. We start by
expanding Eq. (2.27) in powers of $Z\alpha$ to compare to the known hydrogen
spectrum.

To order $\mathcal{O}(Z^{4}\alpha^{4})$ the energy levels are given by

$\displaystyle\frac{E^{n_{r},j}_{\pm\lambda}}{mc^{2}}=1$
$\displaystyle-\frac{1}{2}\frac{Z^{2}\alpha^{2}}{(n_{r}+1/2+(\nu_{\pm\lambda})|_{Z=0})^{2}}+\frac{(\nu_{\pm\lambda})|_{Z=0}^{\prime}Z^{3}\alpha^{3}}{(n_{r}+1/2+(\nu_{\pm\lambda})|_{Z=0})^{3}}$
(2.36) $\displaystyle+\frac{1}{2}\frac{(3/4-3(\nu_{\pm\lambda})|_{Z=0}^{\prime
2})Z^{4}\alpha^{4}}{(n_{r}+1/2+(\nu_{\pm\lambda})|_{Z=0})^{4}}+\frac{1}{2}\frac{(\nu_{\pm\lambda})|_{Z=0}^{\prime\prime}Z^{4}\alpha^{4}}{(n_{r}+1/2+(\nu_{\pm\lambda})|_{Z=0})^{3}}+\mathcal{O}(Z^{6}\alpha^{6})\;,$

where primed $\nu_{\pm\lambda}$ indicate derivatives with respect to
$Z\alpha$. These derivatives evaluate to

$\displaystyle(\nu_{\pm\lambda})|_{Z=0}=j+1/2\pm 1/2,$ (2.37)
$\displaystyle(\nu_{\pm\lambda})|_{Z=0}^{\prime}=0,$
$\displaystyle(\nu_{\pm\lambda})|_{Z=0}^{\prime\prime}=\frac{(g^{2}/4-1)}{j+1/2\pm
1/2}-\frac{g^{2}/4}{j+1/2}\;.$

Eq. (2.36) then simplifies to

$\displaystyle\frac{E^{n_{r},j}_{\pm\lambda}}{mc^{2}}=1$
$\displaystyle-\frac{1}{2}\frac{Z^{2}\alpha^{2}}{\left(n_{r}+j\begin{smallmatrix}+3/2\\\
+1/2\end{smallmatrix}\right)^{2}}+\frac{3}{8}\frac{Z^{4}\alpha^{4}}{\left(n_{r}+j\begin{smallmatrix}+3/2\\\
+1/2\end{smallmatrix}\right)^{4}}$ (2.38)
$\displaystyle+\frac{1}{2}\left(\frac{(g^{2}/4-1)}{j\begin{smallmatrix}+1\\\
+0\end{smallmatrix}}-\frac{g^{2}/4}{j+1/2}\right)\frac{Z^{4}\alpha^{4}}{\left(n_{r}+j\begin{smallmatrix}+3/2\\\
+1/2\end{smallmatrix}\right)^{3}}+\mathcal{O}(Z^{6}\alpha^{6})\;.$

In the non relativistic limit, the node quantum number corresponds to the
principle quantum number via $n_{r}=n^{\prime}-j-1/2$ with
$n^{\prime}=1,2,3\ldots$ Using Eq. (2.38) and Eq. (2.24) we see that in the
non relativistic limit $+\lambda$ corresponds to $\kappa>0$ or anti-aligned
spin-angular momentum with $j=\ell-1/2$ and $\ell\geq 1$. Conversely
$-\lambda$ corresponds to $\kappa<0$ or aligned spin-angular momentum with
$j=\ell+1/2$.

With all this input we arrive at

$\displaystyle\frac{E^{n,j}_{\begin{smallmatrix}\kappa>0\\\
\kappa<0\end{smallmatrix}}}{mc^{2}}=1$
$\displaystyle-\frac{1}{2}\frac{Z^{2}\alpha^{2}}{\left(n^{\prime}\begin{smallmatrix}+1\\\
+0\end{smallmatrix}\right)^{2}}+\frac{3}{8}\frac{Z^{4}\alpha^{4}}{\left(n^{\prime}\begin{smallmatrix}+1\\\
+0\end{smallmatrix}\right)^{4}}$ (2.39)
$\displaystyle+\frac{1}{2}\frac{(g^{2}/4-1)}{j\begin{smallmatrix}+1\\\
+0\end{smallmatrix}}\frac{Z^{4}\alpha^{4}}{\left(n^{\prime}\begin{smallmatrix}+1\\\
+0\end{smallmatrix}\right)^{3}}-\frac{1}{2}\frac{g^{2}/4}{j+1/2}\frac{Z^{4}\alpha^{4}}{\left(n^{\prime}\begin{smallmatrix}+1\\\
+0\end{smallmatrix}\right)^{3}}+\mathcal{O}(Z^{6}\alpha^{6})\;.$

Lastly we recast, for the $\kappa>0$ states, the principle quantum number as
$n^{\prime}+1\rightarrow n$ with $n\geq 2$ and we simply relabel
$n^{\prime}\rightarrow n$ for $\kappa<0$ states. This allows Eq. (2.39) to be
completely written in terms of $n$, $j$, and $\ell$ as

$\displaystyle\frac{E^{n,j}_{\ell}}{mc^{2}}=1$
$\displaystyle-\frac{1}{2}\frac{Z^{2}\alpha^{2}}{n^{2}}+\frac{3}{8}\frac{Z^{4}\alpha^{4}}{n^{4}}$
(2.40)
$\displaystyle+\frac{1}{2}\frac{(g^{2}/4-1)}{\ell+1/2}\frac{Z^{4}\alpha^{4}}{n^{3}}-\frac{1}{2}\frac{g^{2}/4}{j+1/2}\frac{Z^{4}\alpha^{4}}{n^{3}}+\mathcal{O}(Z^{6}\alpha^{6})\;,$

where it is understood that $n-\ell\geq 1$, this condition allows us to write
what was previously described in Eq. (2.39) as two distinct spectra now as a
single energy spectra. In the limit $g\rightarrow 2$ or $g\rightarrow 0$ the
correct expansion to order $Z^{4}\alpha^{4}$ of the Dirac or KG energies are
obtained. In the following we explore some consequences of our principal non-
relativistic result, Eq. (2.40).

####  2.2.2 g-Lamb Shift between 2S and 2P orbitals

The breaking of degeneracy in Eq. (2.40) between states of differing $\ell$
orbital quantum number, but the same total angular momentum $j$ and principle
quantum number $n$ is responsible for the Lamb shift due to anomalous magnetic
moment. The only term in Eq. (2.40) (up to order $Z^{4}\alpha^{4}$) that
breaks the degeneracy between the $E^{n,j}_{\ell=j+1/2}$ and
$E^{n,j}_{\ell=j-1/2}$ states for $n\geq 2$ is the fourth term. This is
unsurprising as it depends exclusively on quantum number $\ell$ and $n$. The
lowest order Lamb shift due to anomalous magnetic moment is then

$\displaystyle\frac{\Delta E_{\mathrm{gLamb}}^{n,j}}{mc^{2}}$
$\displaystyle=E^{n,j}_{\ell=j-1/2}-E^{n,j}_{\ell=j+1/2}=\left(g^{2}/8-1/2\right)\left(\frac{1}{j}-\frac{1}{j+1}\right)\frac{Z^{4}\alpha^{4}}{n^{3}}\;.$
(2.41)

For the $2S_{1/2}$ and $2P_{1/2}$ states Eq. (2.41) reduces to

$\displaystyle\frac{\Delta E_{\mathrm{gLamb}}^{2S_{1/2}-2P_{1/2}}}{mc^{2}}$
$\displaystyle=\left(g^{2}/8-1/2\right)\frac{Z^{4}\alpha^{4}}{6}=\left(a+a^{2}/2\right)\frac{Z^{4}\alpha^{4}}{6}\;.$
(2.42)

Our result in Eq. (2.41) and Eq. (2.42) is sensitive to
$g^{2}/8-1/2=a+a^{2}/2$. Traditionally the Lamb shift due to an anomalous
lepton magnetic moment is obtained perturbatively (Itzykson and Zuber, 1980)
by considering the DP equation which is sensitive to $g/2-1=a$ the shift takes
on the expression at lowest order

$\displaystyle\frac{\Delta E_{\mathrm{gLamb,DP}}^{2S_{1/2}-2P_{1/2}}}{mc^{2}}$
$\displaystyle=\left(\frac{g-2}{2}\right)\frac{Z^{4}\alpha^{4}}{6}=a\frac{Z^{4}\alpha^{4}}{6}\;.$
(2.43)

It is of experimental interest to resolve this discrepancy between the first
order DP equation and the second order fermion formulation KGP. We recall the
present day values

$\displaystyle a_{e}$ $\displaystyle=1159.65218091(26)\times
10^{-6}\simeq\frac{\alpha}{2\pi}\;,$ (2.44a) $\displaystyle a_{\mu}-a_{e}$
$\displaystyle=6.2687(6)\times 10^{-6}\;.$ (2.44b)

The largest contribution to the anomalous moment for charged leptons is, as
indicated the lowest order QED Schwinger result $a=\alpha/2\pi$. For the KGP
approach, the anomalous $g$-factor mixes contributions of different powers of
fine structure $\alpha$. Precision values for the fundamental constants are
taken from Tiesinga et al. (2021). For the $2S_{1/2}-2P_{1/2}$ states, the
shift is

$\displaystyle\frac{\Delta
E_{\mathrm{gLamb}}^{2S_{1/2}-2P_{1/2}}}{mc^{2}}=\frac{Z^{4}\alpha^{5}}{12\pi}+\frac{Z^{4}\alpha^{6}}{48\pi^{2}}\;.$
(2.45)

The scale of the discrepancy between KGP and DP for the hydrogen atom is then

$\displaystyle\Delta$ $\displaystyle
E_{\mathrm{gLamb,KGP}}^{2S_{1/2}-2P_{1/2}}\\!-\\!\Delta
E_{\mathrm{gLamb,DP}}^{2S_{1/2}-2P_{1/2}}\\!=\\!\frac{\alpha^{6}mc^{2}}{48\pi^{2}}$
(2.46) $\displaystyle\;\;=\\!1.62881214\times
10^{-10}\;\mathrm{eV}=39.3845030\;\mathrm{kHz}\;,$

without taking into account the standard corrections such as reduced mass,
recoil, radiative, or finite nuclear size; for more information on those
corrections please refer to Jentschura and Pachucki (1996); Eides et al.
(2001); Tiesinga et al. (2021). It is to be understood that the corrections
presented here are illustrative of the effect magnetic moment has on the
spectroscopic levels, but that further work is required to compare these to
experiment: for example we look here on behavior of point particles only.

While the discrepancy is small for the hydrogen system, it is $\approx 40$ kHz
and will be visible in this or next generation’s spectroscopic experiments.
The discrepancy is also non-negligible for hydrogen-like exotics such as
proton-antiproton because the proton $g$-factor is much larger

$\displaystyle g_{p}=5.585694702(17)\;,\qquad a_{p}=1.792847351(9)\;.$ (2.47)

The discrepancy for the proton-antiproton system is

$\displaystyle\Delta E_{\mathrm{gLamb,KGP}}^{2S_{1/2}-2P_{1/2}}-\Delta
E_{\mathrm{gLamb,DP}}^{2S_{1/2}-2P_{1/2}}$
$\displaystyle=0.71268151\;\mathrm{eV}\;.$ (2.48)

####  2.2.3 g-Fine structure effects within P orbitals

The fifth term in Eq. (2.40), which depends on $j$ and $n$, will shift the
levels due to an anomalous moment, but does not contribute to the Lamb shift.
Rather this expression, which contains the spin-orbit $\vec{L}\cdot\vec{S}$
coupling, is responsible for the fine structure splittings. From Eq. (2.40)
the fine structure splitting is given by

$\displaystyle\frac{\Delta E_{\mathrm{gFS}}^{n,\ell}}{mc^{2}}$
$\displaystyle=E^{n,j=\ell+1/2}_{\ell}-E^{n,j=\ell-1/2}_{\ell}=\left(g^{2}/8\right)\left(\frac{1}{\ell}-\frac{1}{\ell+1}\right)\frac{Z^{4}\alpha^{4}}{n^{3}}\;.$
(2.49)

The splitting between the $2P_{3/2}$ and $2P_{1/2}$ states is therefore

$\displaystyle\frac{\Delta E_{\mathrm{gFS}}^{2P_{3/2}-2P_{1/2}}}{mc^{2}}$
$\displaystyle=\left(g^{2}/8\right)\frac{Z^{4}\alpha^{4}}{16}=\left(1/2+a+a^{2}/2\right)\frac{Z^{4}\alpha^{4}}{16}\;.$
(2.50)

In comparison the fine structure dependence on $g$-factor in the DP equation
is given as

$\displaystyle\frac{\Delta E_{\mathrm{gFS,DP}}^{2P_{3/2}-2P_{1/2}}}{mc^{2}}$
$\displaystyle=\left(\frac{g-1}{2}\right)\frac{Z^{4}\alpha^{4}}{16}=\left(1/2+a\right)\frac{Z^{4}\alpha^{4}}{16}\;.$
(2.51)

Just as in the case of the Lamb shift, we find that the KGP and DP equations
disagree for fine structure splitting. For the hydrogen atom this discrepancy
is

$\displaystyle\Delta$ $\displaystyle
E_{\mathrm{gFS,KGP}}^{2P_{3/2}-2P_{1/2}}\\!-\\!\Delta
E_{\mathrm{gFS,DP}}^{2P_{3/2}-2P_{1/2}}\\!=\\!\frac{\alpha^{6}mc^{2}}{128\pi^{2}}$
(2.52) $\displaystyle\;\;=\\!6.10804553\times
10^{-11}\;\mathrm{eV}=14.7691885\;\mathrm{kHz}\;,$

and for proton-antiproton, the fine structure splitting discrepancy is

$\displaystyle\Delta E_{\mathrm{gFS,KGP}}^{2P_{3/2}-2P_{1/2}}-\Delta
E_{\mathrm{gFS,DP}}^{2P_{3/2}-2P_{1/2}}$
$\displaystyle=0.26725557\;\mathrm{eV}\;.$ (2.53)

For fine structure of the muonic-hydrogen system, the KGP-DP discrepancy is

$\displaystyle\Delta E_{\mathrm{gFS,KGP}}^{2S_{1/2}-2P_{1/2}}-\Delta
E_{\mathrm{gFS,DP}}^{2S_{1/2}-2P_{1/2}}$ $\displaystyle=\\!1.272774\times
10^{-8}\;\mathrm{eV}\;$ (2.54)

We can make a general observation that non minimal magnetic coupling, such as
we have studied in the DP and KGP cases, enlarge energy level splittings. The
above shows that these discrepancies will remain when calculating within more
realistic finite nuclear size context.

###  2.3 Particles in strong electromagnetic fields

Care must be taken when interpreting the results presented in strong
electromagnetic fields; see Gonoskov et al. (2022); Fedotov et al. (2023);
Sainte-Marie (2023). Strong EM fields are produced in heavy-ion collisions
forming quark-gluon-plasma (Grayson et al., 2022). For physical electrons the
AMM interaction is the result of vacuum fluctuations whose strength also
depends on the strength of the field. For example in the large magnetic field
limit a QED computation shows that the ground state is instead of Eq. (2.16)
given by Jancovici (1969)

$\displaystyle E_{0}\approx mc^{2}+\frac{\alpha}{4\pi}mc^{2}\
\mathrm{ln}^{2}\left(\frac{2e\hbar B}{m^{2}c^{3}}\right)\,,$ (2.55)

which even for enormous magnetic fields does not deviate significantly from
the rest mass-energy of the electron. Further the AMM radiative corrections
approach zero for higher Landau levels (Ferrer et al., 2015; Hackebill, 2022).
Therefore the AMM in the case of electrons does not have a significant effect
in highly magnetized environments such as those found in astrophysics
(magnetars).

The situation is different for composite particles such as the proton, neutron
and light nuclei whose anomalous magnetic moments are dominated by their
internal structure and not by vacuum fluctuations. In this situation we expect
that the AMM interaction in high magnetic fields remains significant.
Therefore, asking whether the DP or KGP equations better describes the
dynamics of composite hadrons and atomic nuclei in presence of magnetar
strength fields is a relevant question despite the standard choice in
literature being the DP equation (Broderick et al., 2000). The same question
can be asked for certain exotic hydrogen-like atoms where the particles have
anomalous moments which can be characterized as an external parameter.

####  2.3.1 Strong homogeneous magnetic fields

The magnetic moment anomaly can flip the sign of the magnetic energy for the
least excited states causing the gap between particle and antiparticle states
to decrease with magnetic field strength. Setting $p_{z}=0$ in Eq. (2.15), we
show in Figure 2.2 that the energy of the lowest KGP Landau eigenstate
$n=0,s=1/2$ reaches zero where the gap between particle and antiparticle
states vanishes for the field

$\displaystyle B_{\mathrm{crit}}^{e}=\frac{{B}_{S}^{e}}{a_{e}}\simeq
861{B}_{S}^{e}=3.8006\times 10^{16}\;\mathrm{G}\;,$ (2.56a) $\displaystyle
B_{\mathrm{crit}}^{p}=\frac{{B}_{S}^{p}}{a_{p}}\simeq\frac{1}{1.79}{B}_{S}^{p}=8.3138\times
10^{19}\;\mathrm{G}\;,$ (2.56b)

where ${B}_{S}$ is the so-called Schwinger critical field (Schwinger, 1951).

$\displaystyle{B}_{S}^{e}\equiv\frac{{m_{e}^{2}}c^{2}}{e\hbar}=\frac{m_{e}c^{2}}{2\mu_{B}}=4.4141\times
10^{13}\;\mathrm{G}\;,$ (2.57a)
$\displaystyle{B}_{S}^{p}\equiv\frac{{m_{p}^{2}}c^{2}}{e\hbar}=\frac{m_{p}c^{2}}{2\mu_{N}}=1.4882\times
10^{20}\;\mathrm{G}\;.$ (2.57b)

The numerical results are evaluated for the anomalous moment of the electron
and proton, given by Eq. (2.44a) and Eq. (2.47). At the critical field
strength $B_{\mathrm{crit}}$ the Hamiltonian loses self-adjointness and the
KGP loses its predictive properties. The Schwinger critical field Eq. (2.57a)
denotes the boundary when electrodynamics is expected to behave in an
intrinsically nonlinear fashion, and the equivalent electric field
configurations become unstable (Labun and Rafelski, 2009). However, it is
possible that the vacuum is stabilized by strong magnetic fields (Evans and
Rafelski, 2018).

Figure 2.2: The $n=0$, $s=1/2$ ground state for a KGP electron given by Eq.
(2.16) with $g/2-1=\alpha/2\pi$ in a homogeneous magnetic field. We consider
the particle with no $z$-direction momentum. The particle state (solid red)
and antiparticle (dashed red) are presented.

The critical magnetic fields as shown in Eq. (2.56a) appear in discussion of
magnetars (Kaspi and Beloborodov, 2017). The magnetar field is expected to be
more than 100-fold that of the Schwinger critical magnetic field which is on
the same order of magnitude as $B_{\textrm{crit}}$ for an electron. While the
critical field for a proton exceeds that of a magnetar, the dynamics of
protons (and neutrons) in such fields is nevertheless significantly modified.
A correct description of magnetic moment therefore has relevant consequences
to astrophysics.

Figure 2.3 shows analogous reduction in particle/antiparticle energy gap for
the DP equation. In this case the vanishing point happens at a larger magnetic
field strength. This time the solutions continue past this point, but require
allowing the states to cross into the opposite continua which we consider
nonphysical. We are not satisfied with either model’s behavior though the KGP
description is preferable. However, it is undesirable that both KGP and DP
solutions loose physical meaning and vacuum stability in strong magnetic
fields.

Figure 2.3: The $n=0$, $s=1/2$ ground state for a DP electron given by Eq.
(2.19b) with $g/2-1=\alpha/2\pi$ in a homogeneous magnetic field. We consider
the particle with no $z$-direction momentum. The particle state (solid red)
and antiparticle (dashed red) are presented.

####  2.3.2 High-Z hydrogen-like atoms

For the case of $g\\!=\\!2$ hydrogen-like systems with large $Z$ nuclei, there
is extensive background related to the long study of the solutions of the
Dirac equation (Rafelski et al., 1978; Greiner et al., 2012a; Rafelski et al.,
2017). For $g\neq 2$ and $1/r$ singular potential we refer back to the exact
expression for the KGP energy levels in Eq. (2.27). In the situation of
critical electric fields, states lose self-adjointness for large $Z$ in both
the Dirac $g\\!=\\!2$ case (Gesztesy et al., 1985) and for KGP $g\\!\neq\\!2$.
Thaller (2013) notes that the DP-Coulomb solutions retain self-adjointness via
‘diving’ states. For KGP $|g|<2$, states vanish similar to Dirac energy levels
for the $1/r$ singular potential, but if $|g|\\!>\\!2$ there is merging of
particle to particle states (and antiparticle to antiparticle) for states of
the same total angular momentum quantum number $j$, but opposite spin
orientations.

This merging state behavior can be seen in Figure 2.4, which shows the meeting
of the $1S_{1/2}$ and $2P_{1/2}$ states when $|g|\\!>\\!2$. For $|g|\\!<\\!2$
there is no state merging, but the solution is discontinuous in the sense that
even for $1S_{1/2}$ we see a maximum allowed value of $Z$ at a finite energy.
This is reminiscent of the Dirac $g\\!=\\!2$ behavior we are familiar with for
the $2P_{1/2}$ state (see upper dashed blue line in Figure 2.4) and many other
$g\\!=\\!2$ eigenstates. We know from study of numerical solutions of the
Dirac equation that the regularization of the Coulomb potential by a finite
nuclear size removes this singular behavior. It remains to be seen how this
exactly works in the context of the KGP equation allowing for the magnetic
anomaly.

Figure 2.4: The KGP $1S_{1/2}$ (lower red curves) and $2P_{1/2}$ (upper green
curves) energy levels for $g$-factor values $g\\!=\\!\\{1.8,\ 1.9977,\
2.0023,\ 2.2\\}$ are shown for large $Z$ hydrogen-like atoms. The curves for
the Dirac $g\\!=\\!2$ case for (lower dashed blue) $1S_{1/2}$ and (upper
dashed blue) $2P_{1/2}$ are also presented.

Thaller (2013) presented numerically computed DP equation energy levels for
large $Z$ hydrogen like atoms. These numerical solutions involve crossings in
energy levels between states with the same total angular quantum number $j$,
but differing spin orientations such as $1S_{1/2}$ and $2P_{1/2}$; these
states also have the behavior of diving into the antiparticle lower continuum
even for $1/r$-potential. These features are not present for the KGP-Coulomb
solution. However, there is a similarity between the numerical solutions of
the DP equation and our analytical KGP solutions, because for $|g|>2$ the
merging states as described above correspond to the crossing states in the DP
solution.

The DP equation also allows for the so-called ‘super-positronium’ states as
described by Barut and Kraus (1975, 1976). Such states represent resonances
due to the magnetic interaction that reside incredibly close to the center of
the atom i.e $\sqrt{\langle r^{2}\rangle}\approx a\alpha\hbar/mc$, but this
feature is absent from the KGP formation of the Coulomb problem as all KGP-
Coulomb wave functions which can be normalized can be successfully matched to
their Dirac ($g\\!=\\!2$) companions.

Figure 2.5: The KGP $1S_{1/2}$ and $2P_{1/2}$ states are plotted for hydrogen-
like energies for $220>Z>0$. The spread of lines corresponds to a spread of
$g$-factor values: $4>g>0$. Integer multiples of $Z=137/2$ are marked with
vertical dashed lines. The separation between any two adjacent curves is
$|g_{i}-g_{i+1}|=0.05$. The unique curve which dives towards and stops at the
boundary $E=0$ is the Dirac $g=2$ ground state. Figure 2.6: A close up of the
KGP $1S_{1/2}$ and $2P_{1/2}$ states between $2.0625>g>1.9375$. The separation
between any two adjacent curves is $|g_{i}-g_{i+1}|=0.003125$. The value
$Z=137$ is marked with a vertical dashed line.

Because analytical solutions of the DP-Coulomb problem are not available,
unlike our results for KGP, it is hard to pinpoint precisely the origin of the
diving and crossing state behavior. However, we can hypothesize that the
problems arise due to the pathological structure of DP equation where the
magnetic anomaly rather than full magnetic moment appears. On the other hand
KGP framework for large $Z$ shows interesting and well-behaved analytical
behavior.

Figure 2.7: Energy level curves are plotted for total angular momentum quantum
number $j=1/2,\,3/2$ with the relativistic principle quantum number $n_{r}=0$.
The separation between any two adjacent curves is $|g_{i}-g_{i+1}|=0.2$.
Dashed vertical lines indicate integer multiples of $Z=137/2$.

We further can explore the $g$-dependency on the $1S_{1/2}$ and $2P_{1/2}$
states by plotting a spectrum of $g$-factor values as is done in Figure 2.5.
Purple regions are where $g\rightarrow 0$ and the energies resembles the
familiar Klein-Gordon case. As $g\to 0$, small changes in $g$-factor only lead
to modest changes in the energies for large $Z$ systems. The black curves
represent where $g\rightarrow 4$. A unique feature of $g>2$ fermions is that
after a certain point, certain states become _less_ bound with increasing $Z$.
These rising curves represent spin anti-aligned $-\lambda$ levels which become
nonphysical (e.g the slope becomes vertical) and merge with their spin aligned
$+\lambda$ counterparts precisely where those states also become nonphysical.

The cyan curves in Figure 2.5 are where $g\approx 2$ and the states resemble
the Dirac case. At exactly $g\\!=\\!2$ there is a unique behavior where the
$1S_{1/2}$ state dashes downward and terminates at $Z\\!=\\!1/\alpha$. This
path is unique and does not occur for any $g\neq 2$. Additionally, the
$g\\!=\\!2$ state does not smoothly connect with the $g\approx 2$ solutions
for large $Z$ hydrogen-like atoms. This is more visible in Figure 2.6 (note
the same purple-cyan-black color scheme is used but for different $g$-factor
values) which plots a variety of $g$-factor values near $g\approx 2$. More
specifically, small changes in $g$-factor lead to large deviations in the
energies which is represented by the lack of dense lines near $g\\!=\\!2$
indicating the “cusp-like” nature of the Dirac case for very small anomalies.

The pattern of merging states with the same angular momentum first discussed
in Figure 2.4 repeats itself for higher values of $j$ total angular momentum.
We show this explicitly in Figure 2.7 where mergers between $j=1/2$ states
occur in the left lobe while mergers between $j=3/2$ states occur in the right
lobe for $g>2$.

###  2.4 Combination of mass and magnetic moment

While we have thus far focused on the DP and KGP models for magnetic dipole
moments, the non-uniqueness of spin dynamics allows us to invent further non-
linear EM models which all in the non-relativistic limit yield the non-
relativistic QM magnetic dipole Hamiltonian. One such extension to quantum
spin dynamics is to note the close relationship mass and magnetic moment share
in the KGP formalism which we describe below.

Neutral particle electromagnetic interactions are of interest (Bruce and Diaz-
Valdes, 2020; Bruce, 2021b, a) allowing for a variety of model-building. The
‘Landau’ energies for neutral particles (_i.e._ neutron with mass
$m_{\mathrm{N}}$) in Eq. (2.16) simplifies to

$\displaystyle E_{n,s}|_{e=0}\rightarrow
E_{s}(\bm{p},B)=\sqrt{m_{\mathrm{N}}^{2}c^{4}+\bm{p}^{2}c^{2}-2\mu
Bm_{\mathrm{N}}c^{2}s}\;,$ (2.58)

which is just the free particle motion with a magnetic dipole energy. We note
the correspondence between the quantized Landau orbitals and continuous
transverse momentum $\bm{p}^{2}=p_{3}^{2}+p_{\mathrm{T}}^{2}$. We can define
an ‘effective’ polarization mass given by

$\displaystyle\tilde{m}^{2}_{s}(B)=m^{2}c^{4}-2\mu Bmc^{2}s\,.$ (2.59)

This effective polarization mass (which also is easily defined for charged
particles) will find use when we consider cosmic thermodynamics and plasmas in
Chapter 4.

Inspired by Eq. (2.59), we can write a unified dipole-mass as

$\displaystyle\boxed{\widetilde{m}(\bm{E},\bm{B})=m+\mu\frac{\sigma_{\alpha\beta}F^{\alpha\beta}}{2c^{2}}}\,,$
(2.60)

which satisfies the wave equation

$\displaystyle\left((i\hbar\partial_{\mu}-eA_{\mu})^{2}-\widetilde{m}^{2}(\bm{E},\bm{B})c^{2}\right)\Psi=0\,,$
(2.61)
$\displaystyle\left(\left(i\hbar\partial_{\mu}-eA_{\mu}\right)^{2}-\left(mc+\mu\frac{\sigma^{\alpha\beta}F_{\alpha\beta}}{2c}\right)^{2}\right)\Psi=0\;.$
(2.62)

This modified KGP formulation then requires spin sensitive and explicit
electromagnetic component to the charged lepton mass. In terms of QFT, Eq.
(2.62) results in higher order vertex diagrams coupling fermions to photons.
This ansatz is inspired by classical theory where charged particles should be
understood to derive at least some of their mass from their electromagnetic
fields. Dynamical mass is the driving motivation behind our work in Chapter 3
(Rafelski et al., 2023c) in the context of dynamic neutrino flavor mixing. It
is also a useful mathematical tool in Chapter 4 (Steinmetz et al., 2023) in
the context of cosmic magnetism.

We note that Eq. (2.60) suggests that there may by some perturbative
connection between particle mass and magnetic moment. This relationship would
only manifest in strong fields where non-minimal coupled electromagnetism may
be large. To this end we propose the following as a possible ansatz

$\displaystyle\widetilde{m}(\bm{E},\bm{B})=m\,\exp\left(\frac{\mu}{m}\frac{\sigma_{\alpha\beta}F^{\alpha\beta}}{2c^{2}}\right)\,,$
(2.63)

which for weak fields $F\rightarrow 0$ reduces to the prior form Eq. (2.60).
We will not explore Eq. (2.63) further in this work.

The approach in Eq. (2.60) is superficially similar to the model proposed by
Frenkel (1926) in classical mechanics by giving the particle a spin dependent
mass of the form $m\sim\Sigma_{\mu\nu}F^{\mu\nu}$ where $\Sigma_{\mu\nu}$ is
the covariant generalization of the classical magnetic and electric dipole; a
more detailed exploration can be found in Formanek (2020). Eq. (2.62) is
however distinct in that the mass is allowed off-diagonal components in spinor
space (a subspace which doesn’t exist classically). The dipole-mass Eq. (2.60)
is off-diagonal in spinor space in the Dirac representation and no longer
commutes likes a scalar.

Eq. (2.60) also differs from the regular KGP equation by the presence of an
additional quadratic interaction which we can evaluate using Eq. (1.30) in the
Dirac representation as

$\displaystyle\delta
V=-\frac{\mu^{2}}{4}\left(\gamma_{\alpha}\gamma_{\beta}F^{\alpha\beta}\right)^{2}=\mu^{2}\begin{pmatrix}i\bm{\sigma}\cdot\bm{E}/c&-\bm{\sigma}\cdot\bm{B}\\\
-\bm{\sigma}\cdot\bm{B}&i\bm{\sigma}\cdot\bm{E}/c\end{pmatrix}\begin{pmatrix}i\bm{\sigma}\cdot\bm{E}/c&-\bm{\sigma}\cdot\bm{B}\\\
-\bm{\sigma}\cdot\bm{B}&i\bm{\sigma}\cdot\bm{E}/c\end{pmatrix}\,,$ (2.64)
$\displaystyle\delta
V=\mu^{2}\begin{pmatrix}\bm{B}^{2}-\bm{E}/c^{2}&-i\bm{E}\cdot\bm{B}\\\
-i\bm{E}\cdot\bm{B}&\bm{B}^{2}-\bm{E}/c^{2}\end{pmatrix}=2\mu^{2}\begin{pmatrix}\mathcal{S}&-i\mathcal{P}\\\
-i\mathcal{P}&\mathcal{S}\end{pmatrix}\,.$ (2.65)

We define the invariants of the electromagnetic field tensor $F^{\alpha\beta}$
in Eq. (2.65) letting us write the above more compactly as

$\displaystyle\mathcal{S}=\frac{1}{2}(\bm{B}^{2}-\bm{E}^{2}/c^{2})\,,\qquad\mathcal{P}=\bm{E}\cdot\bm{B}/c\,,\qquad\delta
V=2\mu^{2}\left(\mathcal{S}-i\gamma^{5}\mathcal{P}\right)\,.$ (2.66)

We note that $\sigma_{\alpha\beta}F^{\alpha\beta}/2$ can also be written in
terms of its four eigenvalues

$\displaystyle\lambda_{1}=+\mathcal{S}+i\mathcal{P}\,,\qquad\lambda_{2}=+\mathcal{S}-i\mathcal{P}\,,\qquad\lambda_{3}=-\mathcal{S}+i\mathcal{P}\,,\qquad\lambda_{4}=-\mathcal{S}-i\mathcal{P}\,.$
(2.67)

This represents only one possible non-linear extension to electromagnetism in
relativistic quantum mechanics of which there are a family of extensions
(Foldy, 1952).

For the homogeneous magnetic field Eq. (2.62) can be solved in much the same
way as the KGP equation in Section 2.1. One obtains energy eigenvalues by
noting the simple shift that occurs in the mass of $m^{2}\rightarrow
m^{2}+\mu^{2}{B}^{2}$ quadratic in the magnetic field. Quadratic (spin
independent) contributions are often referred to as scalar polarization
(Holstein and Scherer, 2014). The resulting energy levels are

$\displaystyle
E_{n,s}(\bm{B})=\sqrt{m_{e}^{2}c^{4}+\mu^{2}{B}^{2}+p_{3}^{2}c^{2}+2e\hbar
c^{2}B\left(n+\frac{1}{2}\right)-2\mu Bm_{e}c^{2}s}$ (2.68)

Interestingly in ultra-high magnetic fields $(B\gg{B}_{S})$, Eq. (2.68)
approximates

$\displaystyle E\approx\mu B\;.$ (2.69)

This is not dissimilar to the non-relativistic case where the magnetic energy
is simply proportional to the magnetic field.

Figure 2.8: The $n=0$, $s=1/2$ ground state for a IKGP proton given by Eq.
(2.68) with $g=5.58$ in a homogeneous magnetic field. The magnetic minimum is
well visible for particles with larger anomalous moment such as proton. We
consider the particle with no $z$-direction momentum. The particle state
(solid red) and antiparticle (dashed red) are presented. The magnetic field
scale is ${B}^{\prime}_{S}=(m^{2}_{p}/m^{2}_{e})B_{S}$.

The most striking feature is that the ground state remains physical for all
values of magnetic field when an anomalous moment is included. The self-
adjointness of the system is not lost for some critical magnetic field
strength. It can be then thought that the magnetic field provides a
stabilizing influence on the system. Rather, there exists a “magnetic minimum”
located for $n=0,s=1/2$ at

$\displaystyle B_{\mathrm{min}}$
$\displaystyle=\frac{4mc^{2}}{g^{2}\mu_{B}}a\;,$ (2.70)

which for the electron is

$\displaystyle B_{\mathrm{min}}^{e}$
$\displaystyle=\frac{8a_{e}}{g_{e}^{2}}B_{S}=1.02126\times
10^{11}\;\mathrm{G}\;.$ (2.71)

The minimum for a proton is in comparison

$\displaystyle B_{\mathrm{min}}^{p}$
$\displaystyle=\frac{8a_{p}}{g_{p}^{2}}\frac{m_{p}^{2}}{m_{e}^{2}}B_{S}=6.841\times
10^{19}\;\mathrm{G}\;,$ (2.72)

which can be seen in Figure 2.8. Here it is understood that for the
calculation of the proton’s magnetic minimum, the nuclear mass and magneton
was used rather than the electron Bohr magneton. For large enough g-factor,
excited states also contain a minimum, but for any nonzero anomalous moment
the ground state always does.

###  2.5 Classical relativistic spin dynamics

We turn away from quantum mechanical approaches to inspect the classical
analogue of spin dynamics. Considering the Poincaré group of space-time
symmetry transformations (Weinberg, 2005; Greiner and Müller, 2012), it was
established by Wigner that elementary particles are group representations and
can be characterized by eigenvalues of the Poincaré group’s two Casimir
operators:

$\displaystyle C_{1}\equiv p_{\mu}p^{\mu}=p^{2}=m^{2}c^{2}\,,$ (2.73)
$\displaystyle C_{2}\equiv w_{\mu}w^{\mu}=w^{2}\,,\qquad
w_{\alpha}=\frac{1}{2}\varepsilon_{\alpha\beta\mu\nu}M^{\beta\mu}p^{\nu}\,,$
(2.74)

where $w^{\alpha}$ is the Pauli-Lubanski pseudo-vector. Here $M^{\alpha\beta}$
is the relativistic tensor expression for the angular momentum defined via

$\displaystyle
M^{\alpha\beta}=x^{\alpha}p^{\beta}-x^{\beta}p^{\alpha}+S^{\alpha\beta}$
(2.75)

where $S^{\alpha\beta}$ is the spin angular momentum tensor. The spin tensor
$S^{\alpha\beta}$ can be understood via the classical (Cl.) four-spin in the
rest frame (RF) as

$\displaystyle s^{\alpha}|_{\mathrm{RF}}=(0,\bm{s}_{\mathrm{Cl.}})\,,\qquad
s_{\alpha}s^{\alpha}=-\bm{s}_{\mathrm{Cl.}}^{2}\,,\qquad
s_{\alpha}=\frac{1}{2mc}\varepsilon_{\alpha\beta\mu\nu}S^{\beta\mu}p^{\nu}\,,$
(2.76)

where $\bm{s}_{\mathrm{Cl.}}$ is the classical Euclidean three-spin (not to be
confused with the quantum operator $\bm{s}$). We also note that to make the
units correct, the Pauli-Lubanski pseudo-vector and the four-spin are
proportional by a factor of $\sqrt{C_{1}}=mc$. Quantum mechanically (Ohlsson,
2012) $S^{\alpha\beta}$ appears as $\sim\sigma^{\alpha\beta}$ which we’ve
already identified as the spin tensor in Dirac theory defined via the
$\gamma^{\alpha}$ matrices.

####  2.5.1 Covariant magnetic potential and modified Lorentz force

We are interested in elementary particles with electric charge $e$, and
elementary magnetic dipole charge $d=\mu/|\bm{s}_{\mathrm{Cl.}}|$. Therefore
the covariant dynamics must be extended beyond the Lorentz force to
incorporate the Stern–Gerlach (SG) force. To achieve a generalization we
introduce (Rafelski et al., 2018) the covariant magnetic potential

$\displaystyle B_{\alpha}(x,s)\equiv F_{\alpha\beta}^{*}s^{\beta}$ (2.77)

As $s_{\alpha}$ is a pseudo-vector; the product in Eq. (2.77) results in a
polar 4-vector $B_{\alpha}$. We note that the magnetic dipole potential
$B_{\alpha}$ by construction in terms of the antisymmetric field pseudo-vector
$F_{\alpha\beta}^{*}$ satisfies

$\displaystyle\partial_{\alpha}B^{\alpha}=0\,,\qquad s\cdot B=0\rightarrow
B\cdot\frac{ds}{d\tau}+s\cdot\frac{dB}{d\tau}\,,$ (2.78)

where $\tau$ is the proper time.

The zeroth component of the covariant potential in the rest frame Eq. (2.77)
reproduces the classical magnetic dipole energy given by

$\displaystyle
U_{\mathrm{Mag.}}=dB^{0}=dF^{0\beta}s_{\beta}=-\bm{\mu}_{\mathrm{Cl.}}\cdot\bm{B}\,,\qquad\bm{\mu}_{\mathrm{Cl.}}=\mu\frac{\bm{s}_{\mathrm{Cl.}}}{|\bm{s}_{\mathrm{Cl.}}|}\,.$
(2.79)

We can then define a covariant magnetic field tensor from the potential Eq.
(2.77) which generalizes the covariant Lorentz force $F_{\mathrm{L}}^{\alpha}$
as

$\displaystyle
G^{\alpha\beta}=\partial^{\alpha}B^{\beta}-\partial^{\beta}B^{\alpha}=\partial^{\alpha}F^{*\beta\gamma}s_{\gamma}-\partial^{\beta}F^{*\alpha\gamma}s_{\gamma}\,,$
(2.80) $\displaystyle
F_{\mathrm{L}}^{\alpha}\equiv\frac{dp^{\alpha}}{d\tau}=eF^{\alpha\beta}u_{\beta}+dG^{\alpha\beta}u_{\beta}\,,$
(2.81)

where $u_{\alpha}$ is the four-velocity and as previously stated, $e$ and $d$
are the electric and dipole charges. While the first term in Eq. (2.81) is the
standard Lorentz force, the second term is a covariant formulation of the SG
force.

Because the spin precession is sensitive to the force on a particle, the
presence of a SG force will induce precession terms which are second order in
spin. The torque on the magnetic moment of the particle can be determined via
the properties of the four-spin. Namely $s^{\alpha}$ is orthogonal (Schwinger,
1974) to the four-velocity yielding

$\displaystyle u\cdot\frac{ds}{d\tau}+\frac{du}{d\tau}\cdot s$ (2.82)

The spin torque equations can be obtained (Thomas, 1926; Bargmann et al.,
1959) by inserting the Lorentz force (in our case the modified Lorentz force)
that corresponds to Eq. (2.81) yielding

$\displaystyle\frac{\mathrm{d}s^{\mu}}{\mathrm{d}\tau}$
$\displaystyle=(1+\tilde{a})\frac{e}{m}F^{\mu\nu}s_{\nu}-\tilde{a}\frac{e}{m}u^{\mu}\left(u_{\alpha}F^{\alpha\beta}s_{\beta}\right)/c^{2}$
$\displaystyle+(1+\tilde{b})\frac{d}{m}G^{\mu\nu}s_{\nu}-\tilde{b}\frac{d}{m}u^{\mu}\left(u_{\alpha}G^{\alpha\beta}s_{\beta}\right)/c^{2}\,.$
(2.83)

The constants $\tilde{a}$ and $\tilde{b}$ are arbitrary allowing for extra
terms not forbidden by special relativity. With $d=0$, Eq. (2.83) are known as
the Thomas-Bargmann-Michel-Telegdi (TMBT) equations. In the standard
derivation of relativistic spin precession, in the TBMT equation, the
$\tilde{a}$ constant is associated with the anomalous magnetic moment.

In allowing for spin precession sourced by a Stern-Gerlach dipole force, an
additional constant $\tilde{b}$ must be introduced. The terms in Eq. (2.80)
involving the $G$ tensor are spin precession directly originating from dipole
forces. In homogeneous electromagnetic fields, Eq. (2.80) reduces to the
standard TBMT equation. These dynamical torque equations have found use in
describing neutral and charged systems classically (Formanek et al., 2019,
2021b) and inspired further efforts to improve covariant dynamics (Formanek,
2020). The incorporation of electric dipole moments (EDM) (Asenjo and Koch,
2023) into the above formalism would be a novel extension of our work here to
further generalize the TBMT equations.

####  2.5.2 Equivalency of Gilbertian and Ampèrian Stern-Gerlach forces

We can identify the covariant formulation described in Eq. (2.81) with the SG
force in two different ways which define different interpretations of the
magnetic dipole. Explicitly magnetic forces in the non-relativistic limit
manifest in two variants:

$\displaystyle\bm{F}_{\mathrm{G}}=(\bm{\mu}\cdot\bm{\nabla})\bm{B}\ $
$\displaystyle:\ \mathrm{Gilbert\ dipole}\,,$ (2.84)
$\displaystyle\bm{F}_{\mathrm{A}}=\bm{\nabla}(\bm{\mu}\cdot\bm{B})\ $
$\displaystyle:\ \mathrm{Amp\grave{e}re\ dipole}\,.$ (2.85)

The Gilbert model describes the dipole moment of two magnetic monopole
charges; and the Ampère dipole is generated by a loop of electric current.
They differ in the directionality of the Euclidean three-force: The Gilbertian
dipole force is in the direction of magnetic field while the Ampèrian force is
in the direction of gradient. The two forces are related via

$\displaystyle\bm{\nabla}(\bm{\mu}\cdot\bm{B})=(\bm{\mu}\cdot\bm{\nabla})\bm{B}+\bm{\mu}\times(\bm{\nabla}\times\bm{B})\,,$
(2.86)

with the assumption that the dipole is spatially independent. The second term
in the above equation can be rewritten using Ampère’s law

$\displaystyle\bm{\mu}\times(\bm{\nabla}\times\bm{B})=\bm{\mu}\times\Big{(}\mu_{0}\bm{J}+\frac{1}{c^{2}}\frac{\partial\bm{E}}{\partial
t}\Big{)}$ (2.87)

where $\mu_{0}$ is understood to be the vacuum permeability. The important
takeaway is that the Ampèrian (gradient direction) and Gilbertian (field
direction) forces are related via a term ultimately sensitive to the external
current $\bm{J}$ of the system.

Before we show how this manifests in the covariant formulation, we recall the
orthogonality of four-velocity and four-spin given in Eq. (2.82). This
property allows us to write the covariant Lorentz force in Eq. (2.81) in two
equivalent forms which we will call the Ampèrian $F_{\mathrm{L,A}}^{\alpha}$
and Gilbertian $F_{\mathrm{L,G}}^{\alpha}$ covariant expressions for reasons
that will become obvious soon

$\displaystyle F_{\mathrm{L,A}}^{\alpha}$
$\displaystyle=eF^{\alpha\beta}u_{\beta}+ds_{\gamma}u_{\beta}(\partial^{\alpha}F^{*\beta\gamma}-\partial^{\beta}F^{*\alpha\gamma})\,,$
(2.88) $\displaystyle F_{\mathrm{L,G}}^{\alpha}$
$\displaystyle=eF^{\alpha\beta}u_{\beta}-d(s_{\gamma}\partial^{\gamma})F^{*\alpha\beta}u_{\beta}+ds_{\gamma}T^{\gamma\alpha\beta}u_{\beta}\,.$
(2.89)

The tensor $T^{\alpha\mu\nu}$ is the completely antisymmetric dual tensor to
electric four-current

$\displaystyle
T^{\alpha\mu\nu}=\partial^{\alpha}F^{*\mu\nu}+\partial^{\mu}F^{*\nu\alpha}+\partial^{\nu}F^{*\alpha\mu}\,.$
(2.90)

We will now show that the expression $F_{\mathrm{L,A}}^{\alpha}$ depends on an
Ampèrian dipole and the proper time derivative of the fields. Recognizing that
$d/d\tau=u\cdot\partial$, Eq. (2.88) can be rewritten as

$\displaystyle
F_{\mathrm{L,A}}^{\alpha}=eF^{\alpha\beta}u_{\beta}+ds_{\gamma}\left(\partial^{\alpha}F^{*\beta\gamma}u_{\beta}-\frac{d}{d\tau}F^{*\alpha\gamma}\right)\,.$
(2.91)

We evaluate the four-force in the instantaneous rest frame (RF) of the
particle such that $u^{\alpha}=(c,0)$, $d/d\tau=\partial/\partial t$ and
$s^{\alpha}=(0,\bm{s}_{\mathrm{Cl.}})$. Noting that
$\bm{\mu}=dc\bm{s}_{\mathrm{Cl.}}$ this yields

$\displaystyle F_{\mathrm{L,A}}^{0}|_{\mathrm{RF}}$
$\displaystyle=-\frac{1}{c}\bm{\mu}\cdot\left(\frac{\partial\bm{B}}{\partial
t}-\frac{\partial\bm{B}}{\partial t}\right)=0\,,$ (2.92)
$\displaystyle\bm{F}_{\mathrm{L,A}}|_{\mathrm{RF}}$
$\displaystyle=e\bm{E}+\bm{\nabla}(\bm{\mu}\cdot\bm{B})-\frac{1}{c^{2}}\bm{\mu}\times\frac{\partial\bm{E}}{\partial
t}\,.$ (2.93)

As we see in Eq. (2.93), the second term corresponds to the Ampèrian SG force
written in Eq. (2.85).

We can accomplish the same procedure for $F_{\mathrm{L,G}}^{\alpha}$ which
explicitly depends on a covariant Gilbertian dipole and spin coupling to
current. The four-force yields

$\displaystyle F_{\mathrm{L,G}}^{0}|_{\mathrm{RF}}$ $\displaystyle=0$ (2.94)
$\displaystyle\bm{F}_{\mathrm{L,G}}|_{\mathrm{RF}}$
$\displaystyle=e\bm{E}+(\bm{\mu}\cdot\bm{\nabla})\bm{B}+\left(\bm{\nabla}(\bm{\mu}\cdot\bm{B})-(\bm{\mu}\cdot\bm{\nabla})\bm{B}-\frac{1}{c^{2}}\bm{\mu}\times\frac{\partial\bm{E}}{\partial
t}\right)$ (2.95)

If we cancel out the second and forth terms in Eq. (2.95), we re-obtain the
Ampèrian three-force written in Eq. (2.93).

However, making use of the expressions in Eq. (2.86) and Ampère’s circuital
law Eq. (2.87), we can rewrite the above terms yielding

$\displaystyle\bm{F}_{\mathrm{L,G}}|_{\mathrm{RF}}$
$\displaystyle=e\bm{E}+(\bm{\mu}\cdot\bm{\nabla})\bm{B}+\bm{\mu}\times\mu_{0}\bm{J}$
(2.96)

which is the Gilbertian expression for the SG force as given in Eq. (2.84).
Our covariant formulation then requires that

$\displaystyle\boxed{\bm{F}_{\mathrm{L,A}}|_{\mathrm{RF}}\equiv\bm{F}_{\mathrm{L,G}}|_{\mathrm{RF}}}$
(2.97)

Thus we find an equivalence between the Gilbertian and Ampèrian dipoles from
the covariant introduction of intrinsic magnetic dipole in our covariant
dynamics.

##  Chapter 3 Dynamic neutrino flavor mixing through transition moments

We proposed in Rafelski et al. (2023c) that neutrino flavors are remixed when
exposed to strong EM fields travelling as a superposition distinct from the
vacuum propagation of free neutrinos. Neutrino mixing is an important topic
for studying BSM physics as flavor mixing only occurs in the presence of
massive neutrinos allowing for the misalignment between the flavor basis which
participates in left-chiral $SU(2)_{L}$ weak interactions and the mass basis
which are the propagating neutrino states.

We discuss the neutrino anomalous magnetic moment (AMM) in Section 3.1. We
narrow our analysis for Majorana neutrinos which are allowed only transition
magnetic moments which couple different flavors electromagnetically, but do
not violate CPT symmetry. Transition moments however notably break lepton
number conservation. We discuss the standard flavor mixing program in Section
3.2 and the effective Lagrangian density in Section 3.2.1 containing both
Majorana mass and transition moments. In Section 3.2.2 we discuss the
chirality of the relativistic Pauli dipole.

The two-flavor neutrino model is evaluated explicitly in Section 3.3 and the
remixed electromagnetic-mass eigenstates are obtained in Section 3.3.1. We
obtain in Section 3.3.2 an EM-mass basis, distinct from flavor and free-
particle mass basis, which mixes flavors as a function of EM fields. The case
of two nearly degenerate neutrinos is studied in detail in Section 3.4.
Moreover we show solutions relating to full EM field tensor which result in
covariant expressions allowing for both magnetic and electrical fields.

###  3.1 Electromagnetic characteristics of neutrinos

We study the connection between Majorana neutrino transition magnetic dipole
moments (Fujikawa and Shrock, 1980; Shrock, 1980, 1982) and neutrino flavor
oscillation. Neutrino electromagnetic (EM) properties have been considered
before (Schechter and Valle, 1981; Giunti and Studenikin, 2015; Popov and
Studenikin, 2019; Dvornikov, 2019; Chukhnova and Lobanov, 2020) including the
effect of oscillation in magnetic fields (Lim and Marciano, 1988; Akhmedov,
1988; Pal, 1992; Elizalde et al., 2004; Akhmedov and Martínez-Miravé, 2022).
The influence of transition magnetic moments on solar neutrinos is expected
(Martínez-Miravé, 2023), but difficult to measure due to the lack of knowledge
of solar magnetism near the core.

The case of transition moments has the mathematical characteristics of an off-
diagonal mass which is distinct from normal direct dipole moment behavior. EM
field effects are also distinct from weak interaction remixing within matter,
i.e. the Mikheyev-Smirnov-Wolfenstein effect (Wolfenstein, 1978; Mikheyev and
Smirnov, 1985; Mikheev and Smirnov, 1986; Smirnov, 2003).

The size of the neutrino magnetic dipole moment can be constrained as follows:
The lower bound is found by higher order standard model interactions with the
minimal extension of neutrino mass $m_{\nu}$ included (Fujikawa and Shrock,
1980; Shrock, 1980, 1982). The upper bound is derived from reactor, solar and
astrophysical experimental observations (Giunti et al., 2016; Canas et al.,
2016; Studenikin, 2016; Aristizabal Sierra et al., 2022). The bounds are
expressed in terms of the electron Bohr magneton $\mu_{B}$ as

$\displaystyle\frac{e\hbar G_{F}m_{\nu}c^{2}}{8\pi^{2}\sqrt{2}}\sim
10^{-20}\mu_{B}<\mu_{\nu}^{\mathrm{eff}}<10^{-10}\mu_{B}\,,\qquad\mu_{B}=\frac{e\hbar}{2m_{e}}$
(3.1)

where $G_{F}$ is the Fermi constant and $\mu_{\nu}^{\mathrm{eff}}$ is the
effective and characteristic size of the neutrino magnetic moment. In Eq.
(3.1), the lower bound was estimated using a characteristic mass of
$m_{\nu}\sim 0.1~{}\mathrm{eV}$. From cosmological studies, the sum of
neutrino masses is estimated (Aghanim et al., 2020) to be $\sum_{i}m_{i}<0.12$
eV; the effective electron (anti)neutrino mass is bounded (Aker et al., 2022)
by $m_{e}^{\nu}<0.8$ eV.

###  3.2 Neutrino flavor mixing and electromagnetic fields

Oscillation of neutrino flavors observed in experiment is in general
interpreted as being due to a difference in neutrino mass and flavor
eigenstates. This misalignment between the two representations is described as
rotation of the neutrino flavor $N$-vector where $N=3$ is the observed number
of generations. The unitary mixing matrix $V_{\ell k}$ allows for the change
of basis between mass $(k)$ and flavor $(\ell)$ eigenstates via the transform

$\displaystyle\nu_{\ell}=V_{\ell
k}\nu_{k}\,\rightarrow\begin{pmatrix}\nu_{e}\\\ \nu_{\mu}\\\
\nu_{\tau}\end{pmatrix}=\begin{pmatrix}V_{e1}&V_{e2}&V_{e3}\\\ V_{\mu
1}&V_{\mu 2}&V_{\mu 3}\\\ V_{\tau 1}&V_{\tau 2}&V_{\tau
3}\end{pmatrix}\begin{pmatrix}\nu_{1}\\\ \nu_{2}\\\ \nu_{3}\end{pmatrix}\,,$
(3.2)

where $\nu_{\ell}$ is the neutrino four-spinor written in the flavor basis
while in the mass basis we use $\nu_{k}$ with $k\in 1,2,3$.

The parameterization of the components of the mixing matrix depends on the
Dirac or Majorana-nature of the neutrinos. First we recall the Dirac neutrino
mixing matrix $U_{\ell k}$ in the standard parameterization (Xing, 2014;
Schwartz, 2014)

$\displaystyle U_{\ell
k}=\begin{pmatrix}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta}\\\
-s_{12}c_{23}-c_{12}s_{13}s_{23}e^{i\delta}&c_{12}c_{23}-s_{12}s_{13}s_{23}e^{i\delta}&c_{13}s_{23}\\\
s_{12}s_{23}-c_{12}s_{13}c_{23}e^{i\delta}&-c_{12}s_{23}-s_{12}s_{13}c_{23}e^{i\delta}&c_{13}c_{23}\end{pmatrix}\,,$
(3.3)

where $c_{ij}=\mathrm{cos}(\theta_{ij})$ and
$s_{ij}=\mathrm{sin}(\theta_{ij})$. In this convention, the three mixing
angles $(\theta_{12},\theta_{13},\theta_{23})$ are understood to be the Euler
angles for generalized rotations and $\delta$ is the CP-violating complex
phase.

For the Majorana case we must allow a greater number of complex phases:
Majorana neutrinos allow up to two additional complex phases $\rho$ and
$\sigma$ which along with $\delta$ participate in CP-violation. A
parameterization is achieved by introducing an additional phase matrix
$P_{kk^{\prime}}$

$\displaystyle V_{\ell k}=U_{\ell k^{\prime}}P_{k^{\prime}k}\,,$ (3.4)
$\displaystyle P_{kk^{\prime}}=\mathrm{diag}(e^{i\rho},e^{i\sigma},1)\,.$
(3.5)

The mixing matrix $V_{\ell k}$ defined in Eq. (3.4) can then be used to
transform the symmetric mass matrix $M_{\ell\ell^{\prime}}$ from the flavor
basis into the diagonal mass basis

$\displaystyle
M_{\ell\ell^{\prime}}=\begin{pmatrix}m_{ee}^{\nu}&m_{e\mu}^{\nu}&m_{e\tau}^{\nu}\\\
m_{e\mu}^{\nu}&m_{\mu\mu}^{\nu}&m_{\mu\tau}^{\nu}\\\
m_{e\tau}^{\nu}&m_{\mu\tau}^{\nu}&m_{\tau\tau}^{\nu}\end{pmatrix}\,,\qquad
M_{\ell\ell^{\prime}}^{\mathrm{T}}=M_{\ell\ell^{\prime}}\,,$ (3.6)
$\displaystyle V_{\ell
k}^{\mathrm{T}}M_{\ell\ell^{\prime}}V_{\ell^{\prime}k^{\prime}}=M_{kk^{\prime}}=m_{k}\delta_{kk^{\prime}}=\mathrm{diag}(m_{1},m_{2},m_{3})\,.$
(3.7)

We note the Majorana mass matrix is symmetric due to the anticommuting nature
of the neutrino fields $\bar{\nu}\nu=-\nu^{\mathrm{T}}\bar{\nu}^{\mathrm{T}}$
and is in general complex (Adhikary et al., 2013; Giunti and Kim, 2007) though
it will be taken to be fully real in this work. There are many interesting
models for mass matrices which were pioneered by Fritzsch and Xing (1996,
1998, 2000); Xing (2001) in the leptonic sector. The masses $m_{k}$ are taken
to be real and positive labelling the free propagating states of the three
neutrinos.

####  3.2.1 Effective Majorana neutrino Lagrangian

Given the mass matrix defined in Eq. (3.6), the Majorana mass term in the
Lagrangian can be written in the flavor basis as

$\displaystyle-\mathcal{L}_{\mathrm{mass}}^{\mathrm{Maj.}}=\frac{1}{2}\bar{\nu}_{\ell}M_{\ell\ell^{\prime}}\nu_{\ell^{\prime}}=-\frac{1}{2}\nu_{L,\ell}^{\mathrm{T}}C^{{\dagger}}M_{\ell\ell^{\prime}}\nu_{L,\ell^{\prime}}+\mathrm{h.c.}\,,$
(3.8)

where the Majorana fields are written as
$\nu=\nu_{L}+C(\bar{\nu}_{L})^{\mathrm{T}}$. The field $\nu_{L}$ refers to
left-handed Weyl four-component spinors. Charged conjugated fields are written
as $\nu^{c}=C(\bar{\nu})^{\mathrm{T}}$. The charge conjugation operator $C$ is
defined in the usual way in Itzykson and Zuber (1980); p.692.

Given these conventions, we can extend our consideration to include the
electromagnetic interaction of neutrinos which is possible if neutrinos are
equipped with a magnetic moment matrix $\mu_{\ell\ell^{\prime}}$. We allow for
a fixed _external_ electromagnetic field tensor
$F^{\alpha\beta}_{\mathrm{ext}}(x^{\mu})$ which imparts a force on the
neutrino fields. We emphasize that $F^{\alpha\beta}_{\mathrm{ext}}$ is not
dynamical in our formulation and consists of real functions over four-position
and not field operators.

We generalize the AMM Pauli Lagrangian in Eq. (1.31) to account for the
Majorana fields in the flavor basis as

$\displaystyle-\mathcal{L}_{\mathrm{AMM}}^{\mathrm{Maj.}}=\frac{1}{2}\bar{\nu}_{\ell}\left(\mu_{\ell\ell^{\prime}}\frac{1}{2}\sigma_{\alpha\beta}F^{\alpha\beta}_{\mathrm{ext}}\right)\nu_{\ell^{\prime}}=-\frac{1}{2}\nu_{L,\ell}^{\mathrm{T}}C^{{\dagger}}\left(\mu_{\ell\ell^{\prime}}\frac{1}{2}\sigma_{\alpha\beta}F^{\alpha\beta}_{\mathrm{ext}}\right)\nu_{L,\ell^{\prime}}+\mathrm{h.c.}$
(3.9)