11institutetext:  Department of Physics, College of Science, Qassim
University, P.O. Box 6644, Buraidah 51452, Saudi Arabia

Quantum Formalism Phenomenological theories Classical electromagnetism

# Electrodynamics with quaternionic mass

A. I<EMAIL_ADDRESS>
(today; today)

###### Abstract

New electrodynamics with quaternionic mass is found to yields interesting
results. The quaternionic mass involves longitudinal as well as transverse
(vector) masses. Because of these two masses, an application of a magnetic
field in a system induces electric charge and current densities in the system.
Frank Wilczek axion electrodynamics is derived from this new electrodynamics
bearing a quantum signature. A new version of Maxwell-Proca equations that are
invariant under the gauge transformations is derived from the new
electrodynamics. New electric and magnetic fields that are different from the
standard formalism are proposed to describe massive electrodynamics. Axion
electrodynamics are shown to be equivalent to Proca electrodynamics. The
energy and momentum conservation equations are not influenced by the
quaternionic mass.

###### pacs:

03.65.Ca

###### pacs:

74.20.De

###### pacs:

03.50.De

##  1 Introduction

Maxwell’s theory had laid down the foundations of electromagnetic phenomena we
now know [1]. Maxwell’s equations were firstly expressed in quaternions in
terms of 20 variables. By eliminating all scalars in the original Maxwell’s
equations, Heaviside and Gibbs reduced them to 4 equations which we now know.
Further, Proca extended Maxwell’s equations to include massive vector bosons
[2]. Quaternionic representation of a physical quantity generalizes the real
representation to generalized complex representation. These include the
particle electromagnetic field and electric current density which are the main
constituents of Maxwell equations.

The modern understanding of particle interactions is made via the exchange of
a mediator (gauge boson). The mass of the gauge boson is generated by
employing the spontaneous symmetry breaking prescription (the Higgs
mechanism). The electromagnetic gauge boson is the photon. The photon is
treated as a massless boson to save the gauge invariance of Maxwell’s theory.
The photon has a dual nature if it behaves like a particle. Its electrodynamic
equations in this instance should be governed by quantum mechanics in addition
to Maxwell equations. For such a description, we apply quantum rules for the
massive photon, firstly treating the electromagnetic field as its wavefunction
once, and secondly the electromagnetic potential as its wavefunction.

Using quaternions, one can extend Maxwell’s equation to involve an additional
scalar and a massive photon [3]. The mass of the particle is normally
represented by a real number. If we allow the photon’s mass to be represented
by a quaternion, additional interesting consequences will arise. The photon
with mass requires quantum mechanics and electrodynamics to be combined in one
framework. The new derived quantum electrodynamics yields the axion
electrodynamics proposed recently by Frank Wilczek [4, 5]. Maxwell-Proca
equations that are invariant under the gauge transformations are found to
result from this formalism. Other new sets of modified Maxwell’s equations are
derived.

We would like in this work to explore these interesting consequences by
allowing the photon mass to be a quaternion. The resulting equations are
capable of explaining some of the experimental facts now observed in the field
of condensed matter physics [6]. The application of a magnetic field is found
to induce electric charge and current densities in the system.

This paper is organized as follows: we introduce in Section 2 the concept of
quaternionic mass and work out the electrodynamics associated with it. In
Section 3, we apply the quaternionic momentum eigen-value equation (Dirac-
type) to derive the quantum mechanics of a massive photon, where the
electromagnetic field is considered to be its wavefunction. This leads to a
system of equations similar to Maxwell’s electrodynamics. A transformation is
made to yield the corresponding electrodynamics that involves Maxwell-Proca
equations and their generalization. Section 4 explores a Dirac-like quantum
mechanical model of the massive photon by associating a quaternionic
wavefunction consisting of scalar and vector components. This model is found
to generalize the axion and Proca electrodynamics. We introduce new
electrodynamics of based on the momentum eigen-value equation in Section 6
where the electromagnetic potential is taken as the photon’s wavefunction. The
derived equations are found to generalize Maxwell-Proca equations too. A
Dirac-like quantum mechanics for massive photon employing the electromagnetic
potential as its wavefunction is studied in Section 7. We end our paper with
concluding remakes in Section 8.

##  2 Electrodynamics with quaternionic mass

We have studied earlier massive electrodynamics using quaternion formalism
[3]. In this study we treat the particle mass as a real quantity. Massive
electrodynamics was also studied by Proca but his theory was not gauge
invariant [2]. However, our present formalism is gauge invariant. We would
like now to consider massive electrodynamics where the mass is quaternionic.
To this end, one writes222$\tilde{F}=\tilde{\nabla}\tilde{A}$ , where
$\tilde{A}=(\frac{i}{c}\varphi\,,\vec{A})$ and
$\tilde{\nabla}=(\frac{i}{c}\frac{\partial}{\partial t}\,,\vec{\nabla})$.

$\tilde{P}\tilde{F}=c\tilde{M}\tilde{F}\,,\,\,\,\,\tilde{P}=\left(\frac{i}{c}\,E\,,\vec{p}\right),\,\,\,\,\tilde{F}=\left(\Lambda\,,\vec{B}-\frac{i}{c}\,\vec{E}\right),\qquad\tilde{M}=(im_{\ell}\,,\vec{m}_{t})\,,$
(1)

where $\vec{E}$ and $\vec{B}$ are the electric and magnetic fields, and
$\Lambda$ expresses the Lorenz gauge condition,

$\vec{\nabla}\cdot\vec{A}+\frac{1}{c^{2}}\,\frac{\partial\varphi}{\partial
t}=-\Lambda\,.$ (2)

The product of two quaternions, $\tilde{A}=(a_{0}\,,\vec{a})$ and
$\tilde{B}=(b_{0}\,,\vec{b})$ is given by [7]

$\tilde{A}\tilde{B}=(a_{0}b_{0}-\vec{a}\cdot\vec{b}\,,a_{0}\vec{b}+\vec{a}b_{0}+\vec{a}\times\vec{b})\,.$
(A)

Expanding eq.(1) using the above quaternionic product rule, we obtain

$\vec{\nabla}\cdot\vec{E}=\frac{\partial\Lambda}{\partial
t}-\frac{c^{2}}{\hbar}\vec{m}_{t}\cdot\vec{B}\,,$ (3)
$\vec{\nabla}\cdot\vec{B}=\frac{1}{\hbar}\,\vec{m}_{t}\cdot\vec{E}+\frac{m_{\ell}c}{\hbar}\,\Lambda\,,$
(4) $\vec{\nabla}\times\vec{E}=-\frac{\partial\vec{B}}{\partial
t}-\frac{c^{2}}{\hbar}\Lambda\vec{m}_{t}-\frac{m_{\ell}c}{\hbar}\,\vec{E}-\frac{c^{2}}{\hbar}\,\vec{m}_{t}\times\vec{B}\,,$
(5)

and

$\vec{\nabla}\times\vec{B}=\frac{1}{c^{2}}\frac{\partial\vec{E}}{\partial
t}-\vec{\nabla}\Lambda+\frac{\vec{m}_{t}}{\hbar}\,\times\vec{E}-\frac{m_{\ell}c}{\hbar}\,\vec{B}\,.$
(6)

The above system of equations is recently derived for real mass in the
framework of coupling the quaternionic Dirac field with photons [8]. A
complementary electrodynamics is also derived in [3]. It is interesting to
notice that eqs.(3) - (6) are gauge invariant. The above electrodynamics
reduces to least modified electrodynamics recently derived electrodynamics
when $\vec{m}_{t}=0$ and $m_{\ell}=0$ [9]. As evident from eqs.(3) and (6),
one can define an electric charge density induced by the vector mass as
$\rho_{a}=-\frac{\vec{m}_{t}\cdot\vec{B}}{\mu_{0}\hbar}$ and an electric
current density that is perpendicular to the electric field by
$\frac{\vec{m}}{\mu_{0}}\times\vec{E}$ . The existence of these two quantities
give rise to interesting results occurring in topological insulators [6]. The
above electrodynamics is invariant under duality transformations if
$\Lambda=0$.

Notice that a massive particle has a vanishing transverse mass when it is
completely free. The particle interacts with a momentum given by
$\vec{p}=c\,\vec{m}_{t}$. Thus, when comparing this with the minimal coupling
method for a charged particle interacting with an electromagnetic field,
where, $\vec{p}\,^{\prime}=\vec{p}+q\vec{A}$, then $c\,\vec{m}_{t}=q\vec{A}$.
Therefore, the interaction of the electromagnetic field with any other field
defined by $\chi$ will be governed by eqs.(3) - (6) such that
$c\,\vec{m}=\hbar\vec{\nabla}\chi$ .

Thus, eqs.(3) - (6) show that the masses are the source of the electromagnetic
field. This helps us understand why the neutron has electromagnetic properties
although it is neutral. This is truly possible if one relates the transverse
mass to the particle’s spin. The above mass interaction of the electromagnetic
fields allows gravity to be included too. However, in Einstein’s theory of
gravitation, the influence of gravity on a theory is to use the covariant
derivatives instead of the normal derivatives via the so-called the General
Covariance Principle [10]. This replacement brings about some extra terms in
the theory that would represent the interaction of the initial fields with
gravity.

The energy conservation equation associated with eqs.(3) - (6) is given by

$\frac{\partial u}{\partial t}+\vec{\nabla}\cdot\vec{S}=0\,,\qquad
u=\frac{B^{2}+\Lambda^{2}}{2\mu_{0}}+\frac{1}{2}\,\varepsilon_{0}E^{2}\,,\qquad\vec{S}=\mu_{0}^{-1}(\vec{E}\times\vec{B}+\Lambda\vec{E})\,,$
(7)

which is independent of the masses, $\vec{m}_{t}$ and $m_{\ell}$. Equation (7)
reveals that there is no energy loss due to the system described by the
quaternionic mass. Therefore, these two masses are but some kinds of gauge
fields.

The momentum conservation equation of the above electrodynamics system can be
obtained from eqs.(3) - (6) which reads

$\frac{\partial}{\partial
t}\,\left(\vec{E}\times\vec{B}-\Lambda\,\vec{E}\right)_{i}+\partial_{j}\left(\frac{\delta_{ij}}{2}(E^{2}+c^{2}B^{2}-c^{2}\Lambda^{2})+\varepsilon_{ijk}c^{2}(\Lambda\,B_{k})-(E_{i}E_{j}+c^{2}B_{i}B_{j})\right)=0\,,$
(7a)

which can be expressed as

$-\frac{\partial g_{i}}{\partial t}+\partial_{j}\,\sigma_{ij}=0\,,$ (7b)

where

$\begin{split}\sigma_{ij}=\varepsilon_{0}(E_{i}E_{j}+c^{2}B_{i}B_{j})-\frac{\delta_{ij}}{2}\,\varepsilon_{0}(E^{2}+c^{2}B^{2}-c^{2}\Lambda^{2})-\varepsilon_{ijk}\,\mu_{0}^{-1}\,\Lambda\,B_{k})\,,\\\
g_{i}=\varepsilon_{0}\left(\vec{E}\times\vec{B}-\Lambda\,\vec{E}\right)_{i}\,.\end{split}$
(7c)

Here $g_{i}$ and $\sigma_{ij}$ are the momentum density and stress tensor of
the electromagnetic field involving the effect of the scalar $\Lambda$ . It is
now apparent that the scalar field $\Lambda$ increases the energy density of
the electromagnetic field at the expense of losing momentum density, and vice
versa. Once again the mass of the particle does not influence the momentum
conservation equation. The system appears as if it has zero mass. It is
pertinent to mention that the energy gained by $\Lambda$, eq.(7), is lost by
the momentum, eq.(7a). The stress tensor in eq.(7c) is no longer symmetric. A
method to obtain a symmetric stress is provided by Belinfante [11]. Recall
that the energy-momentum tensor obtained from the electromagnetic Lagrangian
density was not symmetric too. The above energy and momentum conservation
equations agree with our electrodynamics with vanishing masses
($\vec{m}_{t}=0$ and $m_{\ell}=0$) [9]. A manipulation with eqs.(3) - (6)
reveals that the scalar $\Lambda$ satisfies the Klein-Gordon equation,

$\frac{1}{c^{2}}\frac{\partial^{2}\Lambda}{\partial
t^{2}}-\nabla^{2}\Lambda+\left(\frac{Mc}{\hbar}\right)^{2}\Lambda=0\,,$ (8)

where

$M^{2}=m_{t}^{2}-m_{\ell}^{2}\,.$ (9)

Remarkably, one can define the mass $M$ as a rest-mass of the field.
Therefore, eqs.(3) - (6) reflect the duality of the massive photon where the
electromagnetic field of the massive photon shows matter as well as wave
aspects. The two masses should be related to the wave and particle natures of
the massive photon. Interestingly, a massless particle (field) has
$|\vec{m}_{t}|=m_{\ell}$ , and not necessarily vanishing two masses. This will
clearly show that mass has another genuine meaning.

###  2.1 Axion electrodynamics and quantum mechanics

The axion is a particle proposed to resolve the strong CP problem in quantum
chromodynamics (QCD), but lately is thought to contribute to the dark matter
in the universe. Wilczek found the electrodynamics describing the evolution of
axions by the modified Maxwell’s equations [4, 5]

$\vec{\nabla}\cdot\vec{E}=-\kappa\vec{\nabla}a\cdot\vec{B}\,,\qquad\qquad\vec{\nabla}\times\vec{E}=-\frac{\partial\vec{B}}{\partial
t}\,,$ (10)
$\vec{\nabla}\cdot\vec{B}=0\,,\qquad\qquad\vec{\nabla}\times\vec{B}=\frac{1}{c^{2}}\frac{\partial\vec{E}}{\partial
t}+\kappa\left(\frac{\partial a}{\partial
t}\,\vec{B}+\vec{\nabla}a\,\times\vec{E}\right)\,,$ (11)

where $a$ and $\kappa$ are the axion field, and the axion–photon coupling,
respectively. It is interesting to see that eqs.(3) - (6) are analogous to the
axion-electrodynamics describing the axion-photon interaction, if one sets [4,
12]

$\vec{m}_{t}=\hbar g_{a\gamma}\,\vec{\nabla}a\,,\qquad m_{\ell}c=-\hbar
g_{a\gamma}\frac{\partial a}{\partial
t}\,\,\qquad\Rightarrow\qquad\vec{m}_{t}=-m_{\ell}c\left(\frac{\vec{\nabla}a}{\dot{a}}\right)\,,\qquad\dot{a}=\frac{\partial
a}{\partial t}\,,$ (12)

Notice that [12] uses $g_{a\gamma}a=\frac{\kappa}{c}\,\theta$ instead. Hence,
eqs.(3) - (6) can be seen as the quantized axion electrodynamics. They also
generalize the axion electrodynamics to allow magnetic current and charge
densities. One can compare the axion electrodynamics studied by Tiwari
adopting a duality perspective with our present formalism [13]. His equations,
eqs.[39] - [42], are in agreement with our equations, eqs.(3) - (6), upon
substituting $g\vec{W}\rightarrow\frac{\vec{m}_{t}}{\hbar}$ and
$gW_{0}\rightarrow\frac{m_{\ell}c}{\hbar}$, $\vec{W}$ and $W_{0}$ are the
duality gauge four-vector, and $g$ is the coupling constant of this field with
the electromagnetic field.

One can consider the interaction of the quantum particle (electron) with the
axion field by substituting eq.(12) in the quaternionic momentum eigen-value
equation with a quaternionic mass defined by
$\tilde{P}\tilde{\Psi}^{*}=c\tilde{M}\tilde{\Psi}^{*}$ [14]. This substitution
yields

$\frac{1}{c^{2}}\,\frac{\partial\psi_{0}}{\partial
t}+\vec{\nabla}\cdot\vec{\psi}=0\,,$ (13) $\frac{\partial\vec{\psi}}{\partial
t}+\vec{\nabla}\psi_{0}+c^{2}g_{ap}\vec{\nabla}a\times\vec{\psi}=0\,\,,$ (14)
$\vec{\nabla}\times\vec{\psi}-g_{ap}\,\left(\frac{\partial a}{\partial
t}\,\vec{\psi}+(\vec{\nabla}a)\,\psi_{0}\right)=0\,,$ (15)

and

$\left(\frac{\partial a}{\partial
t}\right)\,\psi_{0}+c^{2}(\vec{\nabla}a)\cdot\vec{\psi}=0\,,$ (16)

where $g_{ap}$ describes the axion-particle (electron) coupling. Equation (12)
shows that the axion field imparted momentum and energy to the particle
(electron) given, respectively by,

$c\,\vec{m}_{t}=c\hbar g_{ap}\vec{\nabla}a\,,\qquad\qquad\qquad
m_{\ell}c^{2}=-\hbar cg_{ap}\frac{\partial a}{\partial t}\,,$ (17)

where $\vec{m}_{t}$ is a constant vector. Remarkably, the interaction of the
axion field with a quantum particle is made by the minimal coupling
prescription by setting $q\vec{A}\rightarrow c\vec{m}_{t}$ and
$q\varphi\rightarrow m_{\ell}c^{2}$ .

##  3 Quaternionic Maxwell equation

Let us now assume the massive electromagnetic field, $\tilde{F}$ satisfies the
Dirac’s momentum eigen-value equation. We further let the massive photon to
have a quaternionic mass. To this end one writes

$\tilde{P}\,\tilde{\gamma}\,\tilde{F}=c\tilde{M}\tilde{F}\,,\qquad\tilde{\gamma}=(i\beta\,,\vec{\gamma})\,.$
(18)

Expanding eq.(18) and equating the real and imaginary parts in the two sides
of the resulting equations, yield

$\vec{\nabla}\cdot(-c\vec{\gamma}\Lambda+\beta\vec{E}+c\vec{\gamma}\times\vec{B})=-\frac{\partial}{\partial
t}(\beta\Lambda-\frac{\vec{\gamma}}{c}\cdot\vec{E})-\frac{m_{\ell}c^{2}}{\hbar}\,\Lambda+\frac{c}{\hbar}\vec{m}_{t}\cdot\vec{E}\,,$
(19)
$\vec{\nabla}\cdot\,(c\beta\vec{B}-\vec{\gamma}\times\vec{E})=\frac{\partial}{\partial
t}(\vec{\gamma}\cdot\vec{B})-\frac{c^{2}}{\hbar}\,\vec{m}_{t}\cdot\vec{B}\,,$
(20)
$-\vec{\nabla}\times(-c\vec{\gamma}\Lambda+\beta\vec{E}+c\vec{\gamma}\times\vec{B})+\vec{\nabla}(c\vec{\gamma}\cdot\vec{B})-\frac{1}{c}\,\frac{\partial}{\partial
t}(c\beta\vec{B}-\vec{\gamma}\times\vec{E})=-\frac{c}{\hbar}\,\vec{m}_{t}\times\vec{E}+\frac{m_{\ell}c^{2}}{\hbar}\,\vec{B}\,,$
(21)

and

$-\vec{\nabla}\times(c\beta\vec{B}-\vec{\gamma}\times\vec{E})+\vec{\nabla}(c\beta\Lambda-\vec{\gamma}\cdot\vec{E})+\frac{1}{c}\,\frac{\partial}{\partial
t}(-c\vec{\gamma}\,\Lambda+\beta\vec{E}+c\vec{\gamma}\times\vec{B})=\frac{c^{2}}{\hbar}\,\vec{m}_{t}\Lambda-\frac{m_{\ell}c}{\hbar}\,\vec{E}-\frac{c^{2}}{\hbar}\,\vec{m}_{t}\times\vec{B}\,.$
(22)

The above equations represent the electric field of the photon having wave and
particle natures. To this description one defines

$\vec{E}_{M}=\vec{E}+c\beta\vec{\gamma}\times\vec{B}-c\beta\vec{\gamma}\Lambda\,,\qquad\qquad\vec{B}_{M}=\vec{B}-\frac{\beta\vec{\gamma}}{c}\times\vec{E}\,,\qquad\qquad\Lambda_{M}=\Lambda-\frac{\beta\vec{\gamma}}{c}\cdot\vec{E}\,,$
(23)

as the effective electric and magnetic fields, and the scalar field, so that
eqs.(19) - (22) become

$\vec{\nabla}\cdot\vec{E}_{M}=-\frac{\partial\Lambda_{M}}{\partial
t}-\mu\,c\,\Lambda_{M}\,,$ (24)
$\vec{\nabla}\cdot\,\vec{B}_{M}=-\frac{1}{c}\frac{\partial}{\partial
t}(\beta\vec{\gamma}\cdot\vec{B}_{M})-\beta\mu\,\vec{\gamma}\cdot\vec{B}_{M}\,,$
(25) $\vec{\nabla}\times\vec{E}_{M}=-\frac{\partial\vec{B}_{M}}{\partial
t}+\vec{\nabla}(c\beta\vec{\gamma}\cdot\vec{B}_{M})-\beta\,c\vec{B}_{M}\,,$
(26)

and

$\vec{\nabla}\times\vec{B}_{M}=\frac{1}{c^{2}}\,\frac{\partial\vec{E}_{M}}{\partial
t}+\vec{\nabla}\Lambda_{M}+\frac{\beta\mu}{c}\,\vec{E}_{M}\,,$ (27)

where

$\vec{\gamma}\,m_{\ell}=\beta\,\vec{m}_{t}\,.$ (28)

Recall that in Dirac’s theory, one has $\vec{v}=c\,\beta\,\vec{\gamma}$ and
the eigen-value of $\beta$ are $\pm\,1$, so that eqs.(23) becomes

$\begin{split}\vec{E}_{M}=\vec{E}+\vec{v}\times\vec{B}-\vec{v}\,\Lambda\,,\qquad\qquad\vec{B}_{M}=\vec{B}-\frac{\vec{v}}{c^{2}}\times\vec{E}\,,\\\
\Lambda_{M}=\Lambda-\frac{\vec{v}}{c^{2}}\cdot\vec{E}\,,\qquad\qquad\vec{v}\cdot\vec{B}=0\,.\end{split}$
(29)

Interestingly, eq.(28) now reads

$m_{\ell}\vec{v}=c\,\vec{m}_{t}\,,$ (28a)

that reveals the momentum of the particle is equal to that of a wave. This
indicates that $\vec{m}_{t}$ represent a wave’s mass whereas $m_{\ell}$ a
particle’s mass. This is a mathematical realization of the wave-particle
duality principle suggested by de Broglie. The relation in eq.(28a) was
already found when a quantum mechanical aspect of a particle is considered
[15].

If we now apply eq.(29) in eqs.(24) - (27), one finds

$\vec{\nabla}\cdot\vec{E}_{M}=-\frac{\partial\Lambda_{M}}{\partial
t}\mp\mu\,c\,\Lambda_{M}\,,$ (30) $\vec{\nabla}\cdot\,\vec{B}_{M}=0\,,$ (31)
$\vec{\nabla}\times\vec{E}_{M}=-\frac{\partial\vec{B}_{M}}{\partial
t}\mp\mu\,c\,\vec{B}_{M}\,,$ (32)

and

$\vec{\nabla}\times\vec{B}_{M}=\frac{1}{c^{2}}\,\frac{\partial\vec{E}_{M}}{\partial
t}+\vec{\nabla}\Lambda_{M}\pm\frac{\mu}{c}\,\vec{E}_{M}\,,\qquad\mu=\frac{m_{\ell}c}{\hbar}\,.$
(33)

The above equations are but the massive field Maxwell’s equation in a moving
frame. They are the quantized Maxwell’s equations that were recently derived
[3]. The electromagnetic field in the above equations has four components
describing the particle and antiparticle states.

##  4 Maxwellian quantum mechanics with quaternionic mass

Let us now consider a Dirac-like quaternionic momentum eigen-value equation,
and consider the mass to be quaternionic. A real mass equation is recently
considered and shown to lead to Maxwell-like equations of quantum mechanics
[3]. The Dirac-like quaternionic momentum eigen-value equation with
quaternionic mass is expressed as

$\tilde{P}\,\tilde{\gamma}\,\tilde{\Psi}=c\tilde{M}\,\tilde{\Psi}\,,\qquad\qquad\tilde{\gamma}=(i\beta\,,\vec{\gamma}),\qquad\qquad\tilde{\Psi}=(\frac{i}{c}\,\psi_{0}\,,\vec{\psi})\,,$
(34)

where $\vec{\gamma}$ and $\beta$ are the Dirac matrices [16]. The quaternionic
wavefunction $\tilde{\Psi}$ represents the state of the particle. It has four
components that is analogous to Dirac’s spinors. It is more akin to the scalar
and vector representation of the electromagnetic field.

Expanding eq.(34) and equating the real and imaginary parts in the two sides
of the resulting equations to each other, yield

$\vec{\nabla}\cdot\vec{E}_{d}=-\frac{\partial\Lambda_{d}}{\partial
t}-\mu\,\psi_{0}-\frac{c^{2}}{\hbar}\,\vec{m}_{t}\cdot\vec{\psi}\,,$ (35)
$\vec{\nabla}\cdot\vec{B}_{d}=0\,,$ (36)
$\vec{\nabla}\times\vec{E}_{d}=-\frac{\partial\vec{B}_{d}}{\partial
t}-\frac{c^{2}}{\hbar}\,\vec{m}_{t}\times\vec{\psi}\,,$ (37)
$\vec{\nabla}\times\vec{B}_{d}=\frac{1}{c^{2}}\frac{\partial\vec{E}_{d}}{\partial
t}+\vec{\nabla}\Lambda_{d}-\mu\,\vec{\psi}-\frac{\vec{m}_{t}}{\hbar}\,\psi_{0}\,,$
(38)

where

$\vec{E}_{d}=-(c\beta\vec{\psi}+\vec{\gamma}\,\psi_{0})\,,\qquad\vec{B}_{d}=\vec{\gamma}\times\vec{\psi}\,,\qquad\Lambda_{d}=\frac{\beta}{c}\,\psi_{0}+\vec{\gamma}\cdot\vec{\psi}\,.$
(39)

Equation (35) - (39) show a deep relationship between Dirac and Maxwell
fields. Equation (39) shows how the matter and electromagnetic fields are
related. Notice that $\vec{E}_{d}\cdot\vec{B}_{d}=0$. In electrodynamics, the
electric and magnetic fields are defined by the derivatives of the scalar and
vector potentials. Here the matter _electroinertial_ ($\vec{E}_{d}$) and
_magnetoinertial_ ($\vec{B}_{d}$) fields are constructed directly from the
particle wavefunctions. It is interesting to see that for $\vec{m}_{t}=$,
$m_{\ell}=0$ and $\Lambda=0$, we obtain Maxwell - Proca - like equations with
$\vec{\psi}\rightarrow\vec{A}$ and $\psi_{0}\rightarrow\varphi$ .

The right-hand sides of eqs.(35) - (39) express the interaction of the quantum
particle fields with the external field via $\vec{m}$. They can also be
compared with eqs.(3) - (6) of the Maxwell fields. Interestingly, eq.(35) -
(38) were recently derived but with different definitions of the electric and
magnetic fields [17].

In quantum electrodynamics, the electric current and charged density in
Maxwell’s equations are defined by the Dirac’s spinor (wavefunction) of the
electron. It is thus clear that particle mass can give rise to electromagnetic
interactions. It is interesting to see $\vec{E}_{d}$ and $\vec{B}_{d}$ are
invariant under the transformation

$\vec{\psi}\,^{\prime}=\vec{\psi}\pm\frac{\gamma}{c}\,\psi_{0}\,,\qquad\qquad\psi_{0}^{\prime}=\psi_{0}\pm\beta\,\psi_{0}\,.$
(40)

Equations (35) - (38) are invariant under the above transformation if
$\Lambda=0$, and

$m_{\ell}\,\vec{\gamma}=-\vec{m}_{t}\,\beta\,.$ (41)

Since $\vec{\gamma}=\beta\vec{\alpha}$, then eq.(41) yields

$m_{\ell}\,\vec{\alpha}=\vec{m}_{t}I\,$ (41a)

where in Dirac’s theory, $\vec{v}=c\vec{\alpha}$ [16] and $I$ is an identity
matrix. Therefore, the longitudinal and transverse masses can enter Dirac
equation via the above relation. Now eq.(41a) can be expressed as

$m_{\ell}\,\vec{v}=\vec{m}_{t}c\,.$ (41b)

Interestingly, eq.(41b) expresses the equality of matter-wave momentum and
particle momentum as demonstrated by de Broglie’s hypothesis. In this case,
$\vec{m}_{t}$ represents the wave vector mass and $m_{\ell}$ is the particle
mass.

The corresponding Maxwell’s equations analogous to eqs.(35) - (38) are
obtained by the substitution, $\vec{E}_{d}\rightarrow\vec{E}$,
$\vec{B}_{d}\rightarrow\vec{B}$, $\Lambda_{d}\rightarrow\Lambda$,
$\psi_{0}\rightarrow\frac{m_{\ell}c}{\hbar}\,\varphi$, and
$\vec{\psi}\rightarrow\frac{m_{\ell}c}{\hbar}\,\vec{A}$. This substitution
yields

$\vec{\nabla}\cdot\vec{E}=-\frac{\partial\Lambda}{\partial
t}-\mu^{2}\varphi-\mu\,c^{2}\frac{\vec{m}_{t}}{\hbar}\,\cdot\vec{A}\,,$ (42)
$\vec{\nabla}\cdot\vec{B}=0\,,$ (43)
$\vec{\nabla}\times\vec{E}=-\frac{\partial\vec{B}}{\partial
t}-\mu\,c^{2}\frac{\vec{m}_{t}}{\hbar}\times\vec{A}\,,$ (44)
$\vec{\nabla}\times\vec{B}=\frac{1}{c^{2}}\frac{\partial\vec{E}}{\partial
t}+\vec{\nabla}\Lambda-\mu^{2}\vec{A}-\mu\frac{\,\vec{m}_{t}}{\hbar}\,\varphi\,.$
(45)

Equations (42) - (45) reduce to the Maxwell-Proca equations if we set,
$\Lambda=0$ and $\vec{m}_{t}=0$ [2]. It is shown by Tajmar that Proca
electrodynamics is appropriate to account for the Meissner effect in
superconductivity [18]. Equations (42) - (45) satisfy the following equation

$\vec{\nabla}\times\vec{J}_{m}+\frac{\partial}{\partial
t}\,(\mu_{0}\vec{J}_{e})=0\,,\qquad\vec{J}_{e}=\frac{\vec{m}_{t}}{\mu_{0}\hbar}\times\vec{E}+\frac{\mu}{\mu_{0}}\,\vec{B}\,,\qquad\vec{J}_{m}=\mu\,\vec{E}-\frac{c^{2}}{\hbar}\Lambda\vec{m}_{t}-\frac{c^{2}}{\hbar}\,\vec{m}_{t}\times\vec{B}\,,$
(B)

where $\vec{J}_{e}$ and $\vec{J}_{m}$ are the electric and magnetic currents
associated with the above system. It is apparent from eqs.(42) - (45) that
electric charge, electric and magnetic currents are induced. These are given
by

$\rho^{e}_{A}=-\frac{\mu}{\mu_{0}\hbar}\,\vec{m}_{t}\cdot\vec{A}\,,\qquad\vec{J}^{m}_{A}=\mu\,c^{2}\frac{\vec{m}_{t}}{\hbar}\times\vec{A}\,,\qquad\vec{J}^{e}_{A}=-\mu\frac{\,\vec{m}_{t}}{\mu_{0}\hbar}\,\varphi\,.$
(45a)

The above electrodynamics, eqs.(42) - (45), is invariant under the gauge
transformations

$\vec{A}\,^{\prime}=\vec{A}-\vec{\nabla}f\,,\qquad\qquad\varphi\,^{\prime}=\varphi+\frac{\partial
f}{\partial t}\,,$ (a)

provided that

$\vec{m}_{t}\times\vec{\nabla}f=0\,,\qquad
m_{\ell}c\vec{\nabla}f-\vec{m}_{t}\,\frac{\partial f}{\partial t}=0\,,\qquad
c\vec{m}_{t}\cdot\vec{\nabla}f-m_{\ell}\frac{\partial f}{\partial t}=0\,.$ (b)

This is satisfied if the scalar function, $f$, satisfies the wave equation

$\frac{1}{c^{2}}\frac{\partial^{2}f}{\partial t^{2}}-\nabla^{2}f=0\,.$ (c)

Mathematical consistency of eq.(b) requires that
$|\vec{m}_{t}|^{2}=m_{\ell}^{2}$ . Inasmuch as the transverse mass
$\vec{m}_{t}$ expresses the interaction of the particle with the external
field, the Proca - Maxwell equations with quaternionic mass should read, with
$\Lambda=0$ ,

$\vec{\nabla}\cdot\vec{E}=-\mu^{2}\varphi-\mu\,\frac{c^{2}\vec{m}_{t}}{\hbar}\,\cdot\vec{A}\,,$
(46) $\vec{\nabla}\cdot\vec{B}=0\,,$ (47)
$\vec{\nabla}\times\vec{E}=-\frac{\partial\vec{B}}{\partial
t}-\mu\,\frac{c^{2}\vec{m}_{t}}{\hbar}\,\times\vec{A}\,,$ (48)
$\vec{\nabla}\times\vec{B}=\frac{1}{c^{2}}\frac{\partial\vec{E}}{\partial
t}-\mu^{2}\vec{A}-\mu\frac{\,\vec{m}_{t}}{\hbar}\,\varphi\,.$ (49)

Here the vector boson field is described by its longitudinal and transverse
masses. Furthermore, the mass scale in Proca - Maxwell theory is
$\mu=\frac{m_{\ell}c}{\hbar}$ [2]. These suggested equations are new to the
best of my knowledge. Equation (46) - (49) reduce to the standard Maxwell’s
equation in vacuum when $m_{\ell}=0$ and $\vec{m}_{t}=0$ . However, if we let
$c\vec{m}_{t}\rightarrow q\vec{A}$ and $m_{\ell}c^{2}\rightarrow q\varphi$, we
will obtain a nonlinear electrodynamics. These are

$\vec{\nabla}\cdot\vec{E}=-\frac{q^{2}\varphi^{3}}{\hbar^{2}c^{2}}-\frac{q^{2}}{\hbar^{2}}\,\varphi
A^{2}\,,$ (50) $\vec{\nabla}\cdot\vec{B}=0\,,$ (51)
$\vec{\nabla}\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}\,,$ (52)
$\vec{\nabla}\times\vec{B}=\frac{1}{c^{2}}\frac{\partial\vec{E}}{\partial
t}-\frac{2q^{2}\varphi^{2}}{\hbar^{2}c^{2}}\,\vec{A}\,.$ (53)

These will represent the coupling of the electromagnetic field with massive
photons. Equations (50) and (53) suggest non-linear electric gauge charge and
current densities given by

$\rho_{g}=-\frac{\varepsilon_{0}q^{2}\varphi^{3}}{\hbar^{2}c^{2}}-\frac{\varepsilon_{0}q^{2}}{\hbar^{2}}\,\varphi
A^{2}\,,\qquad\qquad\vec{J}_{g}=-\frac{2\varepsilon_{0}q^{2}}{\hbar^{2}}\,\varphi^{2}\vec{A}\,.$
(i)

Applying the Lorenz gauge condition in eqs.(50) and (53) one finds

$\frac{1}{c^{2}}\frac{\partial^{2}\varphi}{\partial
t^{2}}-\nabla^{2}\varphi+ac^{2}\varphi\,A^{2}+a\varphi^{3}=0\,,\,\,\frac{1}{c^{2}}\frac{\partial^{2}\vec{A}}{\partial
t^{2}}-\nabla^{2}\vec{A}+2a\varphi^{2}\vec{A}=0\,,\qquad
a=\frac{q^{2}}{\hbar^{2}c^{2}}\,.$ (ii)

Recall that the theory of superconductivity was explained by employing the
non-linear Schrodinger’s equation with linear London’s equations of
superconductivity. It would be interesting to explore the application of the
above non-linear massive electrodynamics with the non-linear Schrodinger’s
equation. We remark that the above system of equations will very soon find its
application in the field of condensed matter and the allied sciences.

##  5 Massive photon with quaternionic mass

It is recently shown that the quaternionic momentum eigen - value equation
with the quaternion photon’s field, $\tilde{A}$, led to similar equations as
that due to Maxwell quaternionic equation [19]. Let us study this theory with
quaternionic mass by writing

$\tilde{P}^{*}\tilde{A}=c\tilde{M}\tilde{A}\,,\qquad\qquad\tilde{A}=\left(\frac{i}{c}\,\varphi\,,\vec{A}\right),$
(54)

where $\tilde{M}$ is the photon quaternionic mass. We assume here the photon
to be massive. It has longitudinal and transverse masses.

Expanding eq.(54) and equate the real and imaginary parts in the two sides of
the resulting equations to each other yield

$\vec{\nabla}\cdot\vec{A}+\frac{1}{c^{2}}\,\frac{\partial\varphi}{\partial
t}=0\,,\qquad\qquad c\vec{m}_{t}\cdot\vec{A}+m_{\ell}\varphi=0\,,$ (55)

and

$\vec{E}=\frac{c^{2}\vec{m}_{t}}{\hbar}\,\times\vec{A}\,,\qquad\qquad\vec{B}=\frac{\vec{m}_{t}}{\hbar}\,\varphi+\frac{m_{\ell}c}{\hbar}\,\vec{A}\,.$
(56)

It is known that the electric and magnetic fields of a massless photon are
zero. But eq.(56) reveals that the photon has neither electric field nor
magnetic fields not because it has no charge, but because it has no mass.

Manipulating eqs.(55) and (56), one finds the following equations

$\vec{\nabla}\cdot\vec{E}=-\frac{c^{2}\vec{m}_{t}}{\hbar}\cdot\vec{B}\,,\qquad\qquad\vec{\nabla}\cdot\vec{B}=0\,,$
(57) $\vec{\nabla}\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}\,,$ (58)

and

$\vec{\nabla}\times\vec{B}=\frac{1}{c^{2}}\,\frac{\partial\vec{E}}{\partial
t}+\frac{\vec{m}_{t}}{\hbar}\times\vec{E}+\frac{m_{\ell}c}{\hbar}\,\vec{B}\,,$
(59)

Equations (57) - (59) can be compared with eqs.(3) - (6) with $\Lambda=0$.
They are similar to eqs.(10) and (11) of Wilczek axion electrodynamics [4, 5].
It is thus very interesting that the transverse mass, $\vec{m}_{t}$ couples to
the electric and magnetic fields giving rise to electric charge density
besides an electric current density. The electric current density,
$\vec{J}_{e}=\frac{\vec{m}_{t}}{\mu_{0}\hbar}\times\vec{E}+\frac{m_{\ell}c}{\mu_{0}\hbar}\,\vec{B}$,
is non-dissipative.

The above electrodynamics is found to be obtained by considering an axion
field interacting with photons. Axion electrodynamics is found to be a
promising description for topological insulators [6]. Recall that the CP
violation in QCD was explained as due to the presence of the axion field.
Owing to, eqs.(57) - (59), such a violation is a consequence of the photon
wave-particle nature, particularly, when the transverse mass (dynamical mass)
of the photon does not vanish. Note that eqs.(57) - (59) are not invariant
under CP because of the fact that $m_{\ell}\neq 0$. Hence, one may attribute
the non-invariance of CP transformation to the photon being massive. Eqs.(57)
- (59) violate the Time - Reversal symmetry too.

Interestingly, eqs.(57) - (59) are invariant under the gauge transformations.
The electric and magnetic fields in eqs.(57) - (59) satisfy the Klein-Gordon
equation

$\frac{1}{c^{2}}\frac{\partial^{2}\vec{E}}{\partial
t^{2}}-\nabla^{2}\vec{E}+\left(\frac{Mc}{\hbar}\right)^{2}\vec{E}=0\,,\qquad\qquad\frac{1}{c^{2}}\frac{\partial^{2}\vec{B}}{\partial
t^{2}}-\nabla^{2}\vec{B}+\left(\frac{Mc}{\hbar}\right)^{2}\vec{B}=0\,.$ (56a)

Equations (57) - (59), with the aid of eqs.(55) and (56), yield

$\vec{\nabla}\cdot\vec{E}=-\left(\frac{Mc}{\hbar}\right)^{2}\varphi\,,\qquad\qquad\vec{\nabla}\cdot\vec{B}=0\,,$
(56b) $\vec{\nabla}\times\vec{E}=-\frac{\partial\vec{B}}{\partial
t}\,,\qquad\qquad\vec{\nabla}\times\vec{B}=\frac{1}{c^{2}}\,\frac{\partial\vec{E}}{\partial
t}-\left(\frac{Mc}{\hbar}\right)^{2}\vec{A}\,.$ (56c)

Interestingly, these equations are but the Maxwell-Proca equations of a
massive boson whose mass is $M$. These new Maxwell-Proca equations are but
invariant under the gauge transformations. Remarkably, the axion
electrodynamics is equivalent to the new Maxwell-Proca electrodynamics
provided one sets $\vec{m}\propto\vec{\nabla}a$ and $m_{\ell}\propto\partial
a/\partial t\,.$

We remark here that in the standard model, the masses of the gauge fields are
set to zero because the presence of the gauge’s mass spoils the gauge
symmetry. However, with the new formulation, such an impasse can be lifted and
a theory with massive fields can be proposed.

The electric and magnetic fields in eq.(56) are invariant under gauge
transformation, eq.(a), provided

$\vec{\nabla}(m_{\ell}cf)-\frac{\partial}{\partial
t}(\vec{m}_{t}f)=0\,,\qquad\vec{\nabla}\cdot(\vec{m}_{t}f)-\frac{1}{c^{2}}\frac{\partial}{\partial
t}(m_{\ell}cf)=0\,,\qquad\vec{\nabla}\times(\vec{m}_{t}f)=0\,.$ (C)

The above system of equations is satisfied if the function $f$ satisfies the
wave equation

$\frac{1}{c^{2}}\frac{\partial^{2}f}{\partial t^{2}}-\nabla^{2}f=0\,.$ (D)

Intriguingly, eq.(C) suggests that the gauge transformations can be viewed as
a translation of the scalar and vector potentials to read

$\vec{A}\,^{\prime}=\vec{A}-\frac{c}{\hbar}\,\vec{m}_{t}f\,,\qquad\qquad\varphi\,^{\prime}=\varphi+\frac{m_{\ell}c^{2}}{\hbar}f\,.$
(E)

Remarkably, there are no electric and magnetic fields associated with the
additional scalar and vector potentials in eq.(E).

##  6 New electromagnetic fields

We now employ the Dirac’s momentum eigen-value equation for a massive photon
described by the quaternionic wavefunction, $\tilde{A}$. Let us further treat
the mass as a quaternion quantity. This then reads

$\tilde{P}^{*}\tilde{\gamma}\,\tilde{A}^{*}=c\tilde{M}\tilde{A}^{*}\,.$ (60)

Expand eq.(60) and equate the real and imaginary parts in the two sides of the
resulting equations to each other, yield

$\vec{\nabla}\cdot(-c\beta\vec{A}+\vec{\gamma}\,\varphi)+\frac{\partial}{\partial
t}\left(\beta\frac{\varphi}{c}-\vec{\gamma}\cdot\vec{A}\right)=-\frac{m_{\ell}c}{\hbar}\,\varphi+\frac{c^{2}}{\hbar}\vec{m}_{t}\cdot\vec{A}\,,$
(61) $\vec{\nabla}\cdot\,(\vec{\gamma}\times\vec{A})=0\,,$ (62)
$-\frac{1}{c}\,\vec{\nabla}\times\left(-c\beta\vec{A}+\vec{\gamma}\,\varphi\right)+\frac{1}{c}\frac{\partial}{\partial
t}(\vec{\gamma}\times\vec{A})=-\frac{c}{\hbar}\,\vec{m}\times\vec{A}\,,$ (63)

and

$-\vec{\nabla}\times(\vec{\gamma}\times\vec{A})-\frac{1}{c^{2}}\frac{\partial}{\partial
t}\left(-c\beta\vec{A}+\vec{\gamma}\varphi\right)-\vec{\nabla}\left(\beta\frac{\varphi}{c}-\vec{\gamma}\cdot\vec{A}\right)=-\frac{m_{\ell}c}{\hbar}\,\vec{A}+\frac{\vec{m}_{t}}{\hbar}\varphi\,.$
(64)

If we define the electric and magnetic fields as

$\vec{E}=\mu(-c\beta\vec{A}+\vec{\gamma}\,\varphi)\,,\qquad\qquad\vec{B}=-\mu\vec{\gamma}\times\vec{A}\,,\qquad\qquad\Lambda=\mu\left(\beta\frac{\varphi}{c}-\vec{\gamma}\cdot\vec{A}\right)\,,$
(65)

then eqs.(61) - (64), read

$\vec{\nabla}\cdot\vec{E}=-\frac{\partial\Lambda}{\partial
t}-\mu^{2}\varphi+\frac{c^{2}\mu}{\hbar}\,\,\vec{m}_{t}\cdot\vec{A}\,,$ (66)
$\vec{\nabla}\cdot\,\vec{B}=0\,,$ (67)
$\vec{\nabla}\times\vec{E}=-\frac{\partial\vec{B}}{\partial
t}+\frac{c^{2}\mu}{\hbar}\,\,\vec{m}_{t}\times\vec{A}\,,$ (68)

and

$\vec{\nabla}\times\vec{B}=\frac{1}{c^{2}}\frac{\partial\vec{E}}{\partial
t}+\vec{\nabla}\Lambda-\mu^{2}\vec{A}+\mu\,\frac{\vec{m}_{t}}{\hbar}\,\varphi\,.$
(69)

The electric and magnetic fields in eq.(65) are linear functions of $\vec{A}$
and $\varphi$ are new to electrodynamics. They, therefore, represent a new
option for electrodynamics. They can be compared with eqs.(42) - (45). Since
we use the Dirac matrices in these definitions, we suggest calling these
fields the Dirac-Maxwell electromagnetic fields. Interestingly, eqs.(66) -
(69) reveal the interaction of the vector mass with the electromagnetic
potentials. They represent a generalization of the Maxwell-Proca equations for
vector mass and $\Lambda\neq 0$ . Recall that only a charged particle has the
opportunity to interact with the electromagnetic field, however, a massive
photon can interact with its vector mass somehow. The presence of such an
interaction could help explain how the neutron, inside a nucleus, interacts
with the neighboring protons.

The electric, magnetic fields and scalar function ($\Lambda$) in eq.(65) are
invariant under the gauge transformations provided

$c\beta\vec{\nabla}f+\vec{\gamma}\frac{\partial f}{\partial
t}=0\,,\qquad\vec{\gamma}\times\vec{\nabla}f=0\,,\qquad\beta\frac{\partial
f}{\partial t}+c\vec{\gamma}\cdot\vec{\nabla}f=0\,.$ (70)

The above equations are satisfied if $f$ satisfies the wave equation

$\nabla^{2}f-\frac{1}{c^{2}}\frac{\partial^{2}f}{\partial t^{2}}=0\,.$ (71)

Therefore, eqs.(66) - (69) are gauge invariant too with the conditions

$cm_{\ell}\vec{\nabla}f+\vec{m}_{t}\frac{\partial f}{\partial t}=0\,,\qquad
m_{\ell}\frac{\partial f}{\partial
t}+c\vec{m}_{t}\cdot\vec{\nabla}f=0\,,\qquad\vec{m}_{t}\times\vec{\nabla}f=0\,.$
(72)

Equations (70) and (72) imply that

$m_{\ell}\vec{\gamma}=\vec{m}_{t}\beta\,,$ (73)

which due to Dirac velocity prescription, $\vec{v}=\beta\vec{\gamma}$, yields

$m_{\ell}\vec{v}=\vec{m}_{t}c\,.$ (74)

It agrees once again with eq.(41b).

##  7 Concluding remarks

Different scenarios in which the photon mass is quaternionic are investigated.
The massive electrodynamics with quaternionic mass yields quantized and
generalized axion electrodynamics. The use of the quaternionic mass in massive
electrodynamics is found to be equivalent to introducing interaction with the
electromagnetic field. Interestingly, the energy conservation equation of the
massive electrodynamics is not influenced by the presence of the two masses. A
system of equations describing the interaction of the axion field with the
quantum particle is provided.

The interaction of the quantum particle with the electromagnetic field can be
obtained from that of a free particle with quaternionic mass provided one
defines $c\,\vec{m}_{t}\equiv q\vec{A}$ and $m_{\ell}c^{2}\equiv q\varphi$. A
new equation supporting the de Broglie hypothesis is derived where a new
concept of vector mass for a wave is introduced. Therefore, the mass besides
being the source of the matter-wave could also be a source of the
electromagnetic field. Interestingly, the quaternionic Dirac equation with
quaternionic mass yields the Proca-Maxwell equations. New definitions for the
electric and magnetic fields are proposed. The presence of a quaternionic mass
in Maxwell’s theory is shown to give the same effects as that attributed to
axions. It is shown that the new massive electrodynamics yields Maxwell-Proca
equations that are also shown to be equivalent to axion electrodynamics.

## References

  * [1] Griffiths, D. J., _Introduction to Electrodynamics_ , Cambridge University Press, (2017).
  * [2] Proca, A., _Sur la Theorie ondulatoire des electrons positifs et negatifs_ , J. Phys. Radium 7, 347 (1936).
  * [3] Arbab, A. I., _Quantized Maxwell’s equations_ , Optik 136, 64 (2017).
  * [4] Wilczek, F., _Two applications of axion electrodynamics_ , Phys. Rev. Lett. 58, 1799 (1987).
  * [5] Sikivie, P., _Experimental Tests of the “Invisible” Axion_ , Phys. Rev. Lett. 51, 1415 (1983).
  * [6] Xiao-Liang Qi, X., Zhang, S., _Topological insulators and superconductors_ , Rev. Mod. Phys. 83, 1057 (2011).
  * [7] Hamilton, W. R., _Elements of Quaternions,_ Vol. I et II (First edition 1866; second edition edited and expanded by C.J. Joly 1899-1901; reprinted by Chelsea Publishing, New York, 1969) 1185 pp.
  * [8] Arbab, A. I., _Coupling of a biquaternionic Dirac field to a bosonic field_ , Theor. Math. Phys. 203, 631 (2020).
  * [9] Arbab, A. I., _Do we need to modify Maxwell’s equations?_ , https://arxiv.org/abs/1801.01534v2.
  * [10] Weinberg, S., _Gravitation and cosmology_ , Wiley, (1972).
  * [11] Belinfante, J., _On the current and the density of the electric charge, the energy, the linear momentum and the angular momentum of arbitrary fields_ , Physica 7, 449 (1940).
  * [12] Ouellet, J. and Bogorad, Z., _Solutions to axion electrodynamics in various geometries_ , Phys. Rev. D 99, 055010 (2019).
  * [13] Tiwari, S. C., _Axion electrodynamics in the duality perspective_ , Modern Physics Letters A, 30, 1550204 (2015).
  * [14] Arbab, A. I., _Quantum mechanics with quaternionic mass_ , (unpublished) (2020).
  * [15] Arbab, A. I., Widatallah, H. M., Khalafalla, M. A. H., _Toward a Complete Wave-Particle Duality: Do Matter Waves Have Inertia?_ , Advances in Natural Science, 5, 32 (2012).
  * [16] Bjorken, J. D., Drell, S. D., _Relativistic Quantum Fields_ , McGraw-Hill College (1965).
  * [17] Arbab, A. I., _Maxwellian quantum mechanics_ , Optik 136, 382 (2017).
  * [18] Tajmar, M., _Electrodynamics in Superconductors Explained by Proca Equations_ , Phys. Lett. A 372, 3289 (2008).
  * [19] Arbab, A. I., _Electric and Magnetic Fields Due to Massive Photons and Their Consequences_ , Progress In Electromagnetics Research M, Vol. 34, 153 (2014).


# What is the modular Fourier transform?

Jackson Walters  Arlington, VA<EMAIL_ADDRESS>https://jacksonwalters.com

###### Abstract.

The discrete Fourier transform can be viewed as an isomorphism of the group
algebra of a cyclic group to its decomposition in terms of irreducible
representations when the characteristic does not divide the order of the
group. We describe the DFT when $p|N$ by factoring $x^{N}-1$ over finite
fields. For the symmetric group, the Fourier transform is implemented using
representations, Specht modules $S^{\lambda}$ for partitions $\lambda$, and
Maschke’s theorem. In the modular case, we construct the Peirce decomposition
using central primitive orthogonal idempotents $e_{i}$ in SageMath, yielding a
change-of-basis matrix which generalizes the DFT.

##  1\. Introduction

The Fourier transform is an old and powerful tool used in variety of
applications, from signal processing to fundamental physics. It is central in
mathematics. At its core, it is a change of basis that allows us to view a
signal changing in time in terms of static frequencies. This article aims to
build intuition for an extension of the usual Fourier transform which takes
place over the field of real or complex numbers to a more general field of
potentially positive characteristic. Further, we implement the so called
“modular Fourier transform” in SageMath for finite abelian groups and for the
symmetric group $S_{m}$ for $m\leq 6$. This requires computing the dimensions
of the simple modules $D_{p}^{\lambda}$ of the symmetric group. We do so by
computing the rank of the Gram matrix over $\mathbb{F}_{p}$. The modular
Fourier transform for a field $K$ with $\text{char}(K)>0$ uses the isomorphism
from the group algebra $K\left[G\right]$ to its block decomposition
$\bigoplus_{i}B_{i}$, where $B_{i}=K\left[G\right]e_{i}$ and $1=\sum_{i}e_{i}$
is a decomposition of the identity into primitive orthogonal idempotents. This
generalizes the usual Fourier transform which uses Mashke’s theorem to
decompose $K\left[G\right]=\bigoplus_{i}End(V_{i})$ for irreducible
representations $V_{i}$ of $G$ so long as $\text{char}(K)\nmid|G|$.

## Acknowledgements

I would like to extend my sincere thanks to Dave Benson, Ezra Brown, Yves de
Cornulier, Jill Pipher, Sridhar Ramesh, Steve Rosenberg, Travis Scrimshaw,
Jared Weinstein, and Mark Wildon for helpful comments and conversation.

##  2\. The Discrete Fourier Transform

The article will primarily concerned with the Fourier transform for finite
groups, the simplest example being abelian groups which gives rise to the
discrete Fourier transform (DFT). This is a map from
$\mathbb{C}^{N}\rightarrow\mathbb{C}^{N}$ which can be represented by a matrix
of order four. The DFT has wide applicability in engineering, and is amenable
to implementation on a computer.

The DFT generalizes readily to the realm of quantum computing. We view a
collection of $n$ qubits $c_{0}|0\rangle+c_{1}|1\rangle\in\mathbb{C}^{2}$
which are concatenated in space to correspond to the Hilbert space
$\mathcal{H}\cong\left(\mathbb{C}|0\rangle+\mathbb{C}|1\rangle\right)^{n}\cong\mathbb{C}^{2^{n}}$.
Then the Fourier transform is a unitary matrix
$U:\mathcal{H}\rightarrow\mathcal{H}$, where we label the eigenstates using
binary notation, e.g. for $n=3$,
$|0\rangle\otimes|1\rangle\otimes|1\rangle=|011\rangle$ can be written as
$|3\rangle$ in the familiar decimal, and we interpret the label modulo
$N=2^{n}$. We can then define maps such as
$|x\rangle\mapsto|a^{x}(mod\,N)\rangle$ which is useful in applications like
Shor’s algorithm, where we use the inverse Fourier transform to extract the
periodicity of that modular exponentiation viewed as a function
$\mathbb{Z}/N\mathbb{Z}\rightarrow\mathbb{Z}/N\mathbb{Z}$.

This technique is referred to as “Fourier sampling”. It is important that the
base field, $\mathbb{C}$, is of characteristic zero. This means that we can
always invert the order of the cyclic group, $N$, to define the inverse
Fourier transform. However, we have thus far suppressed the representation
theory and harmonic analysis. Note that $\mathbb{Z}/N\mathbb{Z}$ is an abelian
group, so its representations are one dimensional. So we’re looking at maps
$\chi:\mathbb{Z}/N\mathbb{Z}\rightarrow\mathbb{C}$. This means that
$\chi(N)=\chi(1)^{N}=1$, which means $\chi(1)$ is an $N^{th}$ root of unity.
Since these are one dimensional, they are irreducible. Thus, there are $N$
rep’ns given by $\chi_{k}(n)=\exp(\frac{-2\pi ikn}{N})$.

Note that $\chi_{k}$ are functions on the group $\mathbb{Z}/N\mathbb{Z}$.
Thus, for each group element $n\in\\{0,\ldots,N-1\\}$ we have a complex number
$\chi_{k}(n)\in\mathbb{C}$. Recall that the space of class functions on a
group form a vector space. Since the group is abelian, each conjugacy class
has one element, so $\chi_{k}$ may be viewed as a vector in
$V^{*}\cong\mathbb{C}^{N}$. Separately, we have the group alegbra
$V=\mathbb{C}\left[\mathbb{Z}/N\mathbb{Z}\right]\cong\mathbb{C}^{N}$ as a
vector space. We can multiply these “vectors” by viewing group elements as a
basis and extending the multiplication as we do with polynomials. Label the
irreducible representations $V_{k}\cong\mathbb{C}$. Each element
$n\in\mathbb{Z}/N\mathbb{Z}$ may be viewed as an endomorphism $n:V\rightarrow
V$ by multiplication. That is, if $s=\sum_{j}c_{j}j\in V$ then $n\cdot
s=\sum_{k}c_{j}(n\cdot j)$ where the product $n\cdot j=n+j\mod N$ is in
$\mathbb{Z}/N\mathbb{Z}$. Therefore we have a linear action of
$\mathbb{Z}/N\mathbb{Z}$ on $V$, which is invertible since $-n$ is the
multiplicative inverse of $n$. Thus, each $n\in GL(V)$, so this is a
representation of the group. $V$ can be decomposed into irreducible
representations $V\bigoplus_{k}V_{k}$ and the action of $n$ diagonalized as
$n_{k}\in End(V_{k})$. We thus have the decomposition
$V=\mathbb{C}\left[\mathbb{Z}/N\mathbb{Z}\right]\cong\bigoplus_{k}End(V_{k})\cong\bigoplus_{k}\mathbb{C}\chi_{k}=V^{*}$.

So how does the Fourier transform look in this interpretation? Well, consider
the element $x=\left(x_{1},\ldots,x_{N}\right)\in\mathbb{C}^{N}$. We map it to
$\hat{x}_{k}=\sum_{n=0}^{N-1}x_{n}\exp\left(\frac{-2\pi ikn}{N}\right)$.

Let’s rewrite this formula a bit.
$\hat{x}_{k}=\sum_{n=0}^{N-1}x_{n}\exp\left(\frac{-2\pi
k}{N}\right)^{n}=\sum_{n=0}^{N-1}x_{n}\chi_{k}(n)$. We see that our “Fourier
coefficients” are just sums of the components of $x$ against the values of the
irreducible characters. We obtain $N$ coefficients, and this is exactly the
discrete Fourier transform for the group $\mathbb{Z}/N\mathbb{Z}$. This
formula has the nice property of being invertible, with inverse given by
$x_{n}=\frac{1}{N}\sum_{k=0}^{N-1}\hat{x}_{k}\chi_{k}(-n)$, whose proof uses
the orthogonality of characters.

This is a bit of a special case, in which the number of representations equals
the order of the group, i.e. every representation is degree 1. This only
occurs for abelian groups. For nonabelian groups, we need to use matrices.

##  3\. Fourier Transform for Non-abelian Groups

Let $G$ be a finite group and let $\hat{G}$ denote the set of irreducible
representations of $G$, also a finite set. Over a field of characteristic
zero, we have Maschke’s theorem:

######  Theorem 1.

Let $G$ be a finite group and $K$ a field whose characteristic does not divide
the order of $G$. Then $K\left[G\right]$, the group algebra of $G$, is
semisimple.

This theorem tells us that as a $K\left[G\right]$-module (multiply
polynomials) we may decompose $K\left[G\right]$ into a direct sum of simple
$K\left[G\right]$-modules. That is, $K\left[G\right]=\bigoplus_{i}M_{i}$. It
turns out these simple $M_{i}$ are just the endomorphism algebras of
irreducible representations, i.e. $K\left[G\right]=\bigoplus_{i}End(V_{i})$.
Thus, an element of the group algebra may be viewed as a “block diagonal”
matrix acting on itself, with each block corresponding to an irreducible
representation.

By analogy, we may form the “matrix Fourier coefficients” as sums of
coefficients times irreducible representations. Let $f:G\rightarrow K$ be a
function on the group, or equivalently an element of the group algebra $f\in
K\left[G\right]$. Define for $\rho\in\hat{G}$

$\hat{f}(\rho)=\sum_{a\in G}f(a)\rho(a)$

where $f(a)\in K$ and each $\rho(a)$ is a $d_{\rho}\times d_{\rho}$ matrix.
This formula is invertible yielding:

$f(a)=\frac{1}{|G|}\sum_{g\in\hat{G}}d_{\rho}\text{Tr}(\rho(a^{-1})\hat{f}(\rho))$

We notice a few things. It is clear we require $\text{char}(K)\nmid|G|$, both
in order to comply with the hypothesis of Mashke’s Theorem, and to avoid
division by zero. We need to know the dimension numbers of irreducible
representations, $d_{\rho}$. We are dealing with matrices, so we are taking
the matrix trace now in the sum.

##  4\. The Symmetric Group over $\mathbb{C}$

For the symmetric group over $\mathbb{C}$, Maschke’s theorem applies. The
irreducible representations are labeled by Young tableaux $\lambda$
corresponding to partitions of $m$. For each $\lambda$, we obtain a simple
modules $V^{\lambda}$ with basis given by

$E_{T}=\sum_{\sigma\in Q_{T}}\epsilon(\sigma)\\{\sigma(T)\\}\in V^{\lambda}$

where $Q_{T}$ is the subgroup of permutations, preserving (as sets) all
columns of $T$ and $\epsilon(\sigma)$ is the sign of the permutation $\sigma$.
The Specht module of the partition $\lambda$ is the module generated by the
elements $E_{T}$ as $T$ runs through all tableaux of shape $\lambda$. The
dimension of the Specht module $V^{\lambda}$ is the number of standard Young
tableaux of shape $\lambda$. It is given by the hook length formula. Over
fields of characteristic 0 the Specht modules are irreducible, and form a
complete set of irreducible representations of the symmetric group.

We have everything we need to compute the non-abelian Fourier transform, which
is implemented in [FTSG] for the symmetric group (though it generalizes to any
finite group given a labeling for the representations).

⬇

#define a permutation group of size n!

n=4

G=SymmetricGroup(n)

#define the Fourier transform at the representation spc

#which is the Specht module corresponding to partition

def f_hat(f,partition):

spc = SymmetricGroupRepresentation(partition, ’specht’)

return sum(f(sigma)*spc.representation_matrix(Permutation(sigma)) for sigma in
G)

def fourier_coeffs(f):

return {partition:f_hat(f,partition) for partition in Partitions(n)}

#define the inverse Fourier transform

#given a collection of matrices for each representation rho_i

def fourier_inv(sigma,f_hat):

value=0

for partition in Partitions(n):

spc = SymmetricGroupRepresentation(partition, ’specht’)

repn_mat = spc.representation_matrix(Permutation(sigma).inverse())

d_part = repn_mat.ncols() #should extract directly from parition of rep’n

fourier_coeff = f_hat[partition] #fourier coefficient

value += d_part*(repn_mat*fourier_coeff).trace()

return (1/G.order())*value

##  5\. When Maschke’s Theorem Fails, $p\mid|G|$

There is a well-known isomorphism of categories between
$K\left[G\right]$-modules are representations of the group $G$ over the field
$K$. The “regular representation” where $k\left[G\right]$ acts on itself by
multiplication decomposes into a sum of irreducible representations. When the
characteristic of the field $p$ does not divide the order of the group $|G|$
we can use Maschke’s theorem as stated above. The key element of the proof is
the construction of an “average” or invariant inner product, $\langle
x,y\rangle_{\text{inv}}=\frac{1}{|G|}\sum_{g\in G}\langle g\cdot x,g\cdot
y\rangle$. Note that typically we require $K$ be algebraically closed, since
otherwise a more complicated statement holds: $K\left[G\right]$ is a product
of matrix algebras over division rings over $K$. In the case of finite fields,
any division ring over $F_{p}$ must be a field by Wedderburn’s little theorem,
and the only finite fields of characteristic $p$ are the extensions $F_{q}$
where $q=p^{\nu}$.

When $p\mid|G|$, the situation is more complicated and is the subject of
modular representation theory. In our case, we are primarily concerned with
cyclic groups and the symmetric group $S_{m}$. In general, we no longer have
Maschke’s theorem giving us a decomposition of the group algebra into simple
modules. Instead, we have a decomposition into “blocks”:

$K\left[G\right]=\bigoplus_{i}K\left[G\right]e_{i}$

where $\sum_{i}e_{i}=1$ is a decomposition of the identity into central
primitive orthogonal idempotents. Far from being abstract placeholders, in
certain cases these idempotents are readily computable.

##  6\. The Discrete Fourier Transform over a Ring

We implement this procedure in SageMath code. [SAGE] [DFTCG]
.

The discrete Fourier transform as developed above works over the complex
numbers. However, the basic formula only uses a homomorphism from a cyclic
group to a field, $\chi_{k}(n)=\exp(\frac{2\pi n}{N})$. We just need an
element $\alpha$ such that $\alpha^{N}=1$, and $\sum_{j=0}^{N-1}\alpha^{jk}=0$
for $0\leq k<n$. Then we may form a map
$\left(v_{0},\ldots,v_{N-1}\right)\mapsto\left(f_{0},\ldots,f_{N-1}\right)$
where

$f_{k}=\sum_{j=0}^{N-1}v_{j}\alpha^{jk}$

If $R$ is an integral domain (which includes fields), then we only require
$\alpha$ be a primitive $N^{th}$ root of unity, i.e. $\alpha^{k}\neq 1$ for
$1\leq k<N$.

When $R=\mathbb{F}_{p}$, note that
$\mathbb{F}_{p}^{*}\cong\mathbb{Z}/(N-1)\mathbb{Z}$, that is the
multiplicative group of a finite field is cyclic. By Fermat’s little theorem,
$\alpha^{p-1}=1$. Thus, we require $N|p-1$. This implies that $p>N+1$, which
means that the characteristic we choose must be bigger than the size of the
list we are transforming. This is quite a constraint!

We also require that $p\nmid N$, to ensure that $N$ is invertible in
$\mathbb{F}_{p}$. Otherwise we cannot define the inverse Fourier transform.

⬇

n=16; p=17

#finite field of size p

K = GF(p)

assert K(n) != K(0) #ensure n is invertible

assert n.divides(p-1) #ensure a primitive n-th root of unity exists

#find an n-th root in finite field of size p

def primitive_root(K,n):

for a in K:

if a**n == 1 and all(a**k != 1 for k in range(1,n)):

return a

#list to be transformed

v = [K(i) for i in range(n)]

alpha = primitive_root(K,n)

#DFT over a ring with primitive root alpha

def fourier_transform(v,alpha):

return [sum(v[j]*alpha**(j*k) for j in range(len(v))) for k in range(len(v))]

f=fourier_transform(v,alpha); f

[1, 8, 2, 15, 7, 4, 6, 5, 9, 13, 12, 14, 11, 3, 16, 10]

#define the inverse Fourier transform

def inverse_fourier_transform(f,alpha):

return [K(1/len(f))*sum(f[k]*alpha**(-j*k) \

for k in range(len(f))) for j in range(len(f))]

inverse_fourier_transform(f,alpha)

[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]

##  7\. Cyclic Groups over $\mathbb{F}_{p}$

We now consider the group algebra
$\mathbb{F}_{p}\left[\mathbb{Z}/N\mathbb{Z}\right]$.

The question of how to classify all finite-dimensional representations of
$\mathbb{Z}/N\mathbb{Z}$ over an arbitrary field $F$ can be studied using the
structure theorem for finitely-generated modules over a principal ideal
domain, in this case $F\left[x\right]$. The structure theorem asserts that any
finitely-generated module is uniquely a finite direct sum of modules of the
form $F\left[x\right]/p(x)^{r}$ where $p\in F\left[x\right]$ is irreducible
and $r$ is a non-negative integer.

If $T$ is an operator acting on $F^{k}$ for some $n$, then $F^{k}$ becomes a
finitely-generated module over $F\left[x\right]$ with $x$ acting by $T$. $T$
gives a representation of the cyclic group $\mathbb{Z}/N\mathbb{Z}$ if and
only if $T^{N}=1$, in which case the summands $F\left[x\right]/q(x)^{r}$ in
the decomposition of $F^{k}$ must have the property that $q(x)^{r}|x^{N}-1$.

###  7.1. Case $p\nmid N$

If $F$ has characteristic 0 or has characteristic $p$ and $p\nmid N$, then
$x^{N}-1$ is separable over $F$, hence $r\leq 1$ and $F^{k}$ is a direct sum
of irreducible representations, all of which are of the form
$F\left[T\right]/q(T)$ where $q$ is an irreducible factor of $x^{N}-1$ over
$F$.

In module language, that is $F\left[\mathbb{Z}/N\mathbb{Z}\right]\cong
F\left[x\right]/(x^{N}-1)\cong\bigoplus_{k}F\left[x\right]/q_{k}(x)$, where
$q_{k}(x)$ are irreducible polynomials over $F$.

What we are really doing is factoring $x^{N}-1$ in $F_{p}\left[x\right]$. We
can first factor this into a product of cyclotomic polynomials $\Phi_{d}(x)$
in $\mathbb{Z}\left[x\right]$. That is, $x^{N}-1=\prod_{d|N}\Phi_{d}(x)$.
Further, we wish to factor $\Phi_{d}(x)$ in $F_{p}$. This is equivalent to
factoring the the prime ideal $(p)$ in
$\mathbb{Z}\left[\zeta\right]=\mathbb{Z}\left[x\right]/\Phi_{d}(x)$. It turns
out $\Phi_{d}=P_{1}\ldots P_{g}$ where $P_{i}$ each have residue degree $f$,
and $fg=\varphi(d)$ where $\varphi$ is Euler’s totient function, and $f$ is
the order of $p\mod d$.

Note that solutions to $x^{N}-1=0$ in $F_{p}$ are just roots of unity in a
finite field. Since $x^{p-1}-1=0$ in any field of order $p$, we require that
$N|p-1$ to have solutions. Further, when there are solutions there are
$gcd(N,p-1)$ of them.

####  7.1.1. Examples

$N=6,p=7$

In this case, $N|p-1$, so we expect to have a primitive $6^{th}$ root of unity
in $F_{7}$. Once we have such a root $\alpha$, all $k^{th}$ powers of $\alpha$
are roots of $x^{N}-1$ as well, so we factor

$x^{6}-1=(x+1)(x+2)(x+3)(x+4)(x+5)(x+6)$

$N=10,p=7$

$x^{10}-1=\prod_{d|10}\Phi_{d}(x)=\prod_{1,2,5,10}\Phi_{d}(x)=(x-1)(x+1)(x^{4}+x^{3}+x^{2}+x+1)(x^{4}-x^{3}+x^{2}-x+1)$

How do $\Phi_{5}(x)$ and $\Phi_{10}(x)$ factor in $F_{p}\left[x\right]$? Well,
$\varphi(5)=5-1=4$. $7\equiv 2\mod 5$, so $7$ has order 4 $\mod 5$. Thus, when
$d=5$, $f=4$, so $g=1$, thus there is only one polynomial in the
factorization, the cyclotomic polynomial. When $d=10$, $\varphi(10)=4$.
$7^{2}=49\equiv-1\mod 10$, so $7^{4}\equiv 1\mod 10$, thus 7 has order 4 $\mod
10$, so $f=4$, and $g=1$. Therefore our factorization is complete.

Note that we may use the Chinese remainder theorem to obtain an isomorphism to
the decomposition into summands: [DFTCG]

$F_{p}\left[x\right]/(x^{N}-1)\cong\prod_{d|N,i}F_{p}\left[x\right]/(P_{i}(x))$

The map is just given by $f\mapsto f\mod P_{i}(x)$ for each of the $g$
$P_{i}$’s occuring in the factorization of the cyclotomic polynomials. The
inverse map is given by multiplying the factors in the variable $x$, and
taking the residue $\mod x^{N}-1$.

We may want to find an extension $F_{q}$ of $F_{p}$ where $q=p^{\nu}$ which
splits $x^{N}-1$. This is always possible, since we just need to find a
primitive $N^{th}$ root, which occurs when $N|q-1$. In this case, we have

$x^{N}-1=(x-\alpha^{0})(x-\alpha^{1})\ldots(x-\alpha^{N-1})$

where $\alpha^{jN}=1$ for all $0\leq j\leq N-1$. In this case, the Fourier
transform becomes the usual map

$f_{k}:x\mapsto\sum_{j}x_{j}\alpha^{jk}$

⬇

#h is an element of F_p[C_N]

#that is, h = h0+h1*x+h2x^2+...+h_{N-1}x^N-1

#we allow h as a list of N numbers modulo p

#h = [h0,h1,h2,...,h_{N-1}]

def discrete_fourier_transform(h,p,splitting_field=False):

#length of list is size of N

N = len(h)

#define the polynomial ring F_p[x]

R = PolynomialRing(GF(p),’x’)

#name the generator x an element of R

x = R.0

#define the polynomial x^N-1

f = x**N-1; assert f in R

if splitting_field:

K.<a> = f.splitting_field()

#define the polynomial ring over extended base field

R = PolynomialRing(K,’x’)

#name the generator x an element of R

x = R.0

#define the polynomial x^N-1

f = x**N-1; assert f in R

#define the quotient ring F_p[x]/(x^N-1)

S = R.quotient(x^N - 1, ’z’)

#transform the list of coefficients of h into a polynomial in R=F_p[x]

h = sum(h[i]*x**i for i in range(N)); assert h in S

#factor f in F_p[x], save as list of factors and multiplicities

f_factors = list(f.factor())

#implement the Chinese remainder theorem mapping

#S=F_p[x]/(x^N-1) \--> \prod_i R/(factor_i^mult_i)

h_transform=[list(R.quotient(f_factors[i][0]**f_factors[i][1])(h)) \

for i in range(len(f_factors))]

return h_transform

def inv_discrete_fourier_transform(hhat,p,splitting_field=False):

N = sum(len(l) for l in hhat)

#define the polynomial ring F_p[x]

R = PolynomialRing(GF(p),’x’)

#name the generator x an element of R

x = R.0

#define the polynomial x^N-1

f = x**N-1; assert f in R

if splitting_field:

K.<a> = f.splitting_field()

#define the polynomial ring over extended base field

R = PolynomialRing(K,’x’)

#name the generator x an element of R

x = R.0

#define the polynomial x^N-1

f = x**N-1; assert f in R

S = R.quotient(x^N - 1, ’x’)

f_factors = list(f.factor())

#perform inverse of Chinese remainder theorem

#for each modulus N_i = N/n_i, where n_i is the modulus of each factor

#Bezout’s theorem applies, so we get M_i*N_i + m_i*n_i = 1

#a solution x = \sum_{i=1}^k a_i*M_i*N_i, where a_i are the remainders

n = [f_factors[i][0]**f_factors[i][1] for i in range(len(f_factors))]

#get coefficients M_i, m_i from N_i, n_i

M = [xgcd(f/n[i],n[i])[1] for i in range(len(n))]

#get remainders as polynomials in R

a = [sum(hhat[i][j]*x**j for j in range(len(hhat[i]))) for i in
range(len(hhat))]

inv_transform = sum(a[i]*M[i]*(f/n[i]) for i in range(len(a)))

return list(S(inv_transform))

###  7.2. Case $p\mid N$

Writing $N=p^{s}l$ where $p\nmid l$ we have

$x^{N}-1=(x^{l}-1)^{p^{s}}$

so it follows that $r\leq p^{s}$ (but now it’s possible that $r>1$). If $r>1$,
then the corresponding representation $F\left[T\right]/q(T)^{r}$ is
indecomposable and not irreducible, where $q$ is an irreducible factor of
$x^{l}-1$ over $F$. The irreducible representations occur precisely when
$r=1$.

If $V$ is an irreducible representation of $\mathbb{Z}/N\mathbb{Z}$, and
$T:V\rightarrow V$ is the action of a generator, then

$T^{p^{s}l}-1=(T^{l}-1)^{p^{s}}=0$

Thus $T^{l}-1$ is an intertwining operator which is not invertible, so by
Schur’s lemma it is equal to zero.

In this case, we may still factor $x^{l}-1$ as before, we just obtain
multiplicities on all factors, so the representations are now indecomposable
rather than irreducible. Thus, we have the isomorphism

$F_{p}\left[x\right]/(x^{N}-1)\cong F_{p}\left[x\right]/(P_{i}(x)^{p^{s}})$

##  8\. The Symmetric Group over $\mathbb{F}_{p}$

We first need to compute the compute the dimension numbers for the simple
modules $D^{\mu}$, which can be obtained by computing the rank of the Gram
matrix over $\mathbb{F}_{p}$ for the polytabloid basis [MRTCC] [DMRSG].

We need to understand the structure of the blocks in terms of simple modules.
The blocks have Jordan-Hölder composition series with simple modules $D^{\mu}$
for $p$-regular partitions $\lambda$ occurring as simple factors. The number
of occurrences of $D^{\mu}$ in the series for $S^{\lambda}$ is usually denoted
$d^{\lambda\mu}$, and the matrix $D$ is called the decomposition matrix. These
are computed beginning on pg. 98 in [GJ78] [GJ88]. The top composition factor
is $D^{\lambda}=S^{\lambda}/(S^{\lambda}\cap(S^{\lambda})^{\perp})$.

The symmetric group algebra can be decomposed into blocks
$F_{p}\left[S_{N}\right]=\bigoplus_{i}F_{p}\left[S_{N}\right]e_{i}$ where
$1=\sum_{i}e_{i}$ is a decomposition of the identity into primitive, central,
orthogonal idempotents. This is known as the Peirce decomposition.

The idempotents are computed by Murphy [GM83] as sums involving the Jucys-
Murphy elements, and equivalence classes of partitions based on their
$p$-core. The $p$-core of a partition is obtained by removing as many rim
$p$-hooks from the edge of a partition as possible. The number removed is the
weight. $p$-cores $\gamma$ parameterize the blocks of the symmetric group.
[MONC]

Once we have computed the idempotents, we can write
$F_{p}\left[S_{N}\right]e_{i}=\text{span}_{F_{p}}\\{\sigma e_{i}|\sigma\in
S_{N}\\}$ to find a basis for the block. Putting these bases together, we
obtain a second basis for the symmetric group algebra different from the basis
$\\{\sigma|\sigma\in S_{N}\\}$. The change of basis matrix is the modular
Fourier transform. [DFTSG]

⬇

#implements modular Fourier transform

#project v onto each block U_i = F_p[S_n]*e_i

#using \pi_i: v |--> v*e_i as a projection

#this is just a change of basis

def modular_fourier_transform(p,n):

#instantiate group algebra

SGA_GFp_n = SymmetricGroupAlgebra(GF(p),n)

#compute the primitive central orthogonal idempotents

idempotents = SGA_GFp_n.central_orthogonal_idempotents()

#create a spanning set for the block corresponding to an idempotent

spanning_set = lambda idem: [SGA_GFp_n(sigma)*idem for sigma in
SGA_GFp_n.group()]

#compute the blocks as submodules of the given spanning set

blocks = [SGA_GFp_n.submodule(spanning_set(idem)) for idem in idempotents]

#compute the list of basis vectors lifed to the SGA from each block

block_decomposition_basis = flatten([[u.lift() \

for u in block.basis()] for block in blocks])

#the elements of the symmetric group are ordered

#giving the map from the standard basis

sym_group_list = list(SGA_GFp_n.group())

change_of_basis_matrix = []

for b in block_decomposition_basis:

coord_vector = [0]*len(sym_group_list)

for pair in list(b):

coord_vector[sym_group_list.index(pair[0])] = pair[1]

change_of_basis_matrix.append(coord_vector)

return matrix(GF(p),change_of_basis_matrix).transpose()

###  8.1. Examples

####  8.1.1. $p=2$, $n=3$

As expected, we obtain a matrix of order 4 despite the fact that $2\mid 6$.

⬇

modular_fourier_transform(2,3)

[1 0 0 0 1 0]

[0 1 0 0 0 1]

[0 0 1 0 0 1]

[0 0 0 1 1 0]

[1 0 0 1 1 0]

[0 1 1 0 0 1]

modular_fourier_transform(2,3)^2

[0 0 0 1 0 0]

[0 0 1 0 0 0]

[0 1 0 0 0 0]

[1 0 0 0 0 0]

[0 0 0 0 1 0]

[0 0 0 0 0 1]

modular_fourier_transform(2,3)^4

[1 0 0 0 0 0]

[0 1 0 0 0 0]

[0 0 1 0 0 0]

[0 0 0 1 0 0]

[0 0 0 0 1 0]

[0 0 0 0 0 1]

####  8.1.2. $p=3$, $n=4$

In this case, we obtain a matrix of order 488488.

⬇

print(modular_fourier_transform(3,4).str())

[1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0]

[0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0]

[0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0]

[0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0]

[0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0]

[1 2 2 1 1 0 0 0 0 2 2 2 2 2 0 0 0 0 0 0 0 0 0 1]

[0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0]

[2 1 0 0 0 1 0 0 0 2 2 0 0 0 2 0 0 0 1 0 0 0 0 0]

[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0]

[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1]

[0 0 2 0 1 0 1 0 0 0 0 2 0 2 0 2 0 0 0 0 1 0 0 0]

[1 2 2 0 1 0 0 1 0 1 1 1 0 1 0 0 2 0 0 0 0 1 0 0]

[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0]

[0 0 2 1 0 0 0 0 1 0 0 2 2 0 0 0 0 2 0 0 1 0 0 0]

[1 0 2 0 0 2 1 0 1 2 0 2 0 0 2 2 0 2 0 0 0 0 0 1]

[1 0 2 0 0 2 0 1 1 1 0 1 0 0 1 0 2 1 0 0 0 0 1 0]

[1 2 1 1 1 2 1 0 1 1 1 2 1 1 1 1 0 1 1 0 0 0 0 0]

[1 2 2 0 1 2 0 1 1 2 2 2 0 2 2 0 1 2 0 1 0 0 0 0]

[1 1 2 2 2 1 2 2 2 2 1 2 1 1 1 1 2 1 0 0 0 0 0 1]

[2 0 1 0 2 1 2 2 2 2 0 2 0 2 2 2 1 2 0 0 0 0 1 0]

[2 1 1 2 2 0 0 2 2 2 2 2 2 2 0 0 1 2 0 0 0 1 0 0]

[0 0 1 2 2 0 2 0 2 0 0 1 1 1 0 1 0 1 0 0 1 0 0 0]

[2 0 1 0 2 0 0 2 2 1 0 1 0 1 0 0 2 1 0 1 0 0 0 0]

[2 0 2 2 2 0 2 0 2 2 0 1 2 2 0 2 0 2 1 0 0 0 0 0]

#A fast way for computing the order of M is thus

#to compute the characteristic polynomial P_L of M

#factor it over F_p and check then if prime-divisors of p^k-1

#(for k the degree of an involved irreducible polynomial)

#divide the order

def order_finite_field(M):

L = M.nrows()

p = len(M.base_ring())

P_L = M.charpoly(); P_L

char_poly_factored = P_L.factor(); char_poly_factored

degree_list = [item[0].degree() for item in char_poly_factored]

U = p^(L-1)*prod(p^k-1 for k in degree_list)

for div in divisors(U):

if M^div == matrix.identity(L):

return div

order_finite_field(modular_fourier_transform(3,4))

488488

print((modular_fourier_transform(3,4)^488488).str())

[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

[0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

[0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0]

[0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0]

[0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0]

[0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0]

[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0]

[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0]

[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0]

[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0]

[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0]

[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0]

[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0]

[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0]

[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0]

[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0]

[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1]

###  8.2. $p=7$, $n=4$

Here $p\nmid n!$, and we obtain a matrix of (almost) entirely 0’s and $\pm
1$’s.

⬇

#note: this matrix has (almost) only has 1’s and -1’s

print(modular_fourier_transform(7,4).str())

[1 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1]

[1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 6]

[1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 6]

[1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1]

[1 0 0 0 0 1 0 0 0 0 6 0 0 6 0 0 0 0 1 0 0 0 0 1]

[1 1 6 6 1 1 0 0 0 0 0 6 6 0 6 6 6 6 6 0 0 0 0 6]

[1 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 6]

[1 6 1 0 0 0 1 0 0 0 1 0 0 0 6 6 0 0 0 6 0 0 0 1]

[1 0 0 0 0 0 0 1 0 0 6 0 0 6 0 0 0 0 0 0 1 0 0 1]

[1 0 0 0 0 0 0 0 1 0 0 6 6 0 0 0 0 0 0 0 0 1 0 6]

[1 0 0 6 0 1 0 1 0 0 0 0 1 0 0 0 6 0 6 0 6 0 0 6]

[1 1 6 6 0 1 0 0 1 0 0 0 0 1 1 1 1 0 1 0 0 6 0 1]

[1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 1]

[1 0 0 6 1 0 0 0 0 1 0 0 1 0 0 0 6 6 0 0 0 0 6 6]

[1 1 0 6 0 0 6 1 0 1 0 6 6 0 6 0 6 0 0 6 6 0 6 6]

[1 1 0 6 0 0 6 0 1 1 6 0 0 6 1 0 1 0 0 1 0 6 1 1]

[1 1 6 5 1 1 6 1 0 1 1 0 0 0 1 1 2 1 1 1 1 0 1 1]

[1 1 6 6 0 1 6 0 1 1 0 1 0 0 6 6 6 0 6 6 0 1 6 6]

[1 5 1 2 6 6 1 6 6 6 0 6 6 0 2 1 2 1 1 1 1 6 1 6]

[1 6 0 1 0 6 1 6 6 6 6 0 0 6 6 0 6 0 6 6 6 1 6 1]

[1 6 1 1 6 6 0 0 6 6 0 0 0 1 6 6 6 6 6 0 0 1 6 1]

[1 0 0 1 6 6 0 6 0 6 0 0 1 0 0 0 1 1 1 0 1 0 1 6]

[1 6 0 1 0 6 0 0 6 6 0 1 0 0 1 0 1 0 1 0 0 6 1 6]

[1 6 0 2 6 6 0 6 0 6 1 0 0 0 6 0 5 6 6 0 6 0 6 1]

##  9\. Applications

When $p>n$, we are no longer in the modular case but can still compute these
matrices. When $p$ is very large, one can use the matrix to encode a vector by
multiplication, and decode by multiplying by the inverse. This would appear to
be a nascent symmetric key cryptosystem with key $p$. We make no claims to its
security or efficiency.

In quantum computing, the technique of Fourier sampling is used prominently in
Shor’s algorithm for the abelian group $\mathbb{Z}/N\mathbb{Z}$. In graph
isomorphism, the hidden subgroup is the automorphism group of a graph, which
is a subgroup of the symmetric group. Strong Fourier sampling fails for the
symmetric group over $\mathbb{C}$, but to our knowledge has not been explored
over finite fields. [SFS]

## References

  * [SAGE] SageMath, the Sage Mathematics Software System (Version 10.1), The Sage Developers, 2023,https://www.sagemath.org
  * [FTSG] https://github.com/jacksonwalters/fourier-transform-symmetric-group
  * [DFTCG] https://github.com/jacksonwalters/dft-finite-field/blob/main/cyclic_group_dft.ipynb
  * [MRTCC] https://web.math.princeton.edu/~charchan/ModularRepresentationsSymmetricGroupSeminar.pdf
  * [DMRSG] https://github.com/jacksonwalters/dim-modular-repn-symmetric-group
  * [GJ78] Gordon James, _The Representation Theory of the Symmetric Group_. 1978
  * [GJ88] Gordon James, _The Decomposition Matrices of $GL_{n}(q)$ for n $\leq$ 10_. 1988
  * [GM83] G. E. Murphy, _The idempotents of the symmetric group and Nakayama’s conjecture_. 1983
  * [DFTSG] https://github.com/jacksonwalters/dft-finite-field/blob/main/symmetric_group_dft.ipynb
  * [IDEM] https://github.com/jacksonwalters/dft-finite-field/blob/main/central_primitive_orthogonal_idempotents.ipynb
  * [SFS] Moore, Russell, Schulman. _The Symmetric Group Defies Strong Fourier Sampling_. 2008
  * [MONC] Wildon, _Notes on Murphy Operators and Nakayama’s Conjecture_

##  Appendix A Computing Central Orthogonal Idempotents

We provide a direct implementation of [GM83] to compute central primitive
orthogonal idempotents [IDEM], but it is significantly slower than the built
in method to compute idempotents.

⬇

timeit.timeit("central_orthogonal_idempotents(3,5)", number=1,
globals=globals())

14.356356750002305

timeit.timeit("SymmetricGroupAlgebra(GF(3),5) \

.central_orthogonal_idempotents()", number=1, globals=globals())

0.02521254200109979


# Characterizing the Narrowband Ambiguity Function of Multi-Tone Sinusoidal
Frequency Modulated Waveforms

David A. Hague  Sensors and Sonar Systems Department
Naval Undersea Warfare Center
<EMAIL_ADDRESS>
Abstract: This paper characterizes the mainlobe and sidelobe structure of the
Multi-Tone Sinusoidal Frequency Modulated (MTSFM) transmit waveform’s
narrowband Ambiguity Function (AF) for active sonar applications. The MTSFM
waveform’s modulation function is represented as a Fourier series. The Fourier
coefficients form a discrete set of parameters that are modified to synthesize
waveforms with novel characteristics. The contour of the AF’s mainlobe is well
approximated as a coupled ellipse known as the Ellipse Of Ambiguity (EOA). The
EOA parameters determine the mainlobe width in range and Doppler as well as
the degree of coupling between them. This coupling factor, known as the Range-
Doppler Coupling Factor (RDCF), determines whether a waveform is Doppler
sensitive or Doppler tolerant. This paper derives exact closed form
expressions for the EOA parameters of the MTSFM’s AF. The MTSFM’s design
coefficients allow for fine control of its AF mainlobe width in range and
Doppler as well as its RDCF. This fine control facilitates designing waveforms
that can smoothly trade-off between possessing Doppler sensitive and Doppler
tolerant characteristics. Additionally, this paper introduces a method to
control the sidelobe structure of the MTSFM’s Auto Correlation Function (ACF)
while maintaining the waveform’s AF mainlobe shape. This is achieved using a
numerical optimization technique that minimizes the ratio of $\ell_{2}$-norms
of the ACF mainlobe and sidelobe regions subject to constraints on the EOA
parameters. Simulations demonstrate the effectiveness of this optimization
technique.

Keywords: Multi-Tone Sinusoidal Frequency Modulation, Waveform Diversity,
Active Sonar.

##  1 Introduction

Waveform diversity, the ability to optimize waveforms to maximize system
performance, possibly in a dynamically adaptive manner, has been an active
topic of research in the radar community for over two decades [1]. This field
of research has been enabled by the development of several parameterized
modulation techniques such as Phase-Coding (PC) and Frequency Shift-Keying
(FSK) [2] which facilitate the synthesis of waveforms with novel
characteristics. Waveform diversity has increasingly become a topic of
interest to the active sonar community with diverse sets of waveforms being
employed for Multi-Beam Echo Sounding (MBES) [3] and Multiple-Input Multiple
Output (MIMO) sonar applications [4]. These efforts highlight the need for
continued development of parameterized waveform models to further enable
waveform diversity for active sonar systems.

Recently, the author developed the Multi-Tone Sinusoidal Frequency Modulated
(MTSFM) waveform as a novel parameterized FM waveform model [5]. The MTSFM
waveform’s phase and frequency modulation functions are composed of a finite
Fourier series. The Fourier coefficients representing the waveform’s
instantaneous phase and frequency functions are utilized as a discrete set of
adjustable parameters. To date, most efforts have focused on the design of
Doppler sensitive MTSFM waveforms that possess a “Thumbtack-Like” Ambiguity
Function (AF). However, the parameterization of the MTSFM model suggests that
a variety Doppler tolerant designs should also be feasible.

This paper describes a method to jointly control the MTSFM waveform’s AF
mainlobe shape and the sidelobe structure of its Auto Correlation Function
(ACF). The AF mainlobe is characterized using the Ellipse of Ambiguity (EOA)
model. The MTSFM’s design coefficients allow for fine control of its AF
mainlobe width. This facilitates synthesizing waveforms that can smoothly
trade-off between possessing Doppler sensitive and Doppler tolerant
characteristics. Additionally, this paper demonstrates a numerical
optimization technique that minimizes an $\ell_{2}$ norm on the mainlobe and
sidelobe regions of the waveform’s ACF subject to constraints on the MTSFM’s
EOA parameters. This allows for controlling the sidelobe structure of the
MTSFM’s ACF while maintaining the waveform’s AF mainlobe shape. The rest of
this paper is organized as follows: Section 2 introduces the MTSFM waveform
model and the metrics used to describe the AF mainlobe and ACF sidelobe
structure; Section 3 demonstrates how the MTSFM’s parameters can be utilized
to shape its AF mainlobe and reduce ACF sidelobes via several illustrative
design examples; finally, Section 4 concludes the paper.

##  2 MTSFM Waveform and the Ambiguity Function

In general, a basebanded FM waveform is expressed in continuous time as

$s\left(t\right)=\frac{\operatorname{rect}\left(t/T\right)}{\sqrt{T}}e^{j\varphi\left(t\right)}$
(1)

where $T$ is the waveform’s duration and $\varphi\left(t\right)$ is its phase
modulation function. The $1/\sqrt{T}$ term normalizes the signal energy to
unity. The waveform’s frequency modulation function $m\left(t\right)$ maps its
instantaneous frequency as a function of time and is expressed as
$m\left(t\right)=\left(1/2\pi\right)\partial\varphi(t)/\partial t$. The MTSFM
waveform’s frequency and phase modulation functions are expressed as a finite
Fourier series

$\displaystyle m\left(t\right)$
$\displaystyle=\frac{a_{0}}{2}+\sum_{\ell=1}^{L}a_{\ell}\cos\left(\dfrac{2\pi\ell
t}{T}\right)+b_{\ell}\sin\left(\dfrac{2\pi\ell t}{T}\right),$ (2)
$\displaystyle\varphi\left(t\right)$ $\displaystyle=\pi
a_{0}t+\sum_{\ell=1}^{L}\alpha_{\ell}\sin\left(\dfrac{2\pi\ell
t}{T}\right)-\beta_{\ell}\cos\left(\dfrac{2\pi\ell t}{T}\right)$ (3)

where $a_{\ell}$ and $b_{\ell}$ are the Fourier coefficients and
$\alpha_{\ell}=a_{\ell}T/\ell$ and $\beta_{\ell}=b_{\ell}T/\ell$ are the
waveform’s modulation indices. The Fourier coefficients of the MTSFM
waveform’s modulation function, and correspondingly, the modulation indices of
the waveform’s instantaneous phase, form a discrete set of adjustable
parameters that can be modified to synthesize waveforms with a variety of
unique properties [5, 6]. The narrowband Ambiguity Function (AF) measures the
response of the waveform’s MF to its Doppler shifted versions and is defined
as [7]

$\chi\left(\tau,\nu\right)=\int_{-\infty}^{\infty}s\left(t\right)s^{*}\left(t+\tau\right)e^{j2\pi\nu
t}dt$ (4)

where $\nu$ is the Doppler frequency shift. Lastly, the ACF is the zero
Doppler cut of the AF $R\left(\tau\right)=\chi\left(\tau,0\right)$. There are
a series of design metrics that characterize the mainlobe and sidelobe
structure of a waveform’s AF and ACF shape. These metrics are described below.

###  2.1 Mainlobe Structure

The AF mainlobe structure determines a waveform’s ability to estimate the
range and Doppler of a target and to resolve multiple targets in range and
Doppler. The AF mainlobe can be approximated by a second order Taylor series
expansion [7]. The EOA is the contour of the mainlobe approximation at some
height $1-\xi$ which is always a coupled ellipse and is expressed as

$1-|\chi\left(\tau,\nu\right)|^{2}=\xi=\beta_{rms}^{2}\tau^{2}+2\rho\tau\nu+\tau_{rms}^{2}\nu^{2}$
(5)

where $\beta_{rms}^{2}$ is the waveform’s RMS bandwidth and determines time-
delay (range) sensitivity, $\tau_{rms}^{2}$ is the RMS pulse length which
determines Doppler sensitivity, and $\rho$ is the RDCF for the AF mainlobe.
The RMS bandwidth is expressed as [7]

$\beta_{rms}^{2}=\left(2\pi\right)^{2}\int_{\infty}^{\infty}\left(f-f_{0}\right)^{2}|S\left(f\right)|^{2}df=\dfrac{1}{T}\int_{-T/2}^{T/2}\left[\dot{\varphi}\left(t\right)\right]^{2}dt-\left|\dfrac{1}{T}\int_{-T/2}^{T/2}j\dot{\varphi}\left(t\right)dt\right|^{2}$
(6)

where $f_{0}$ is the waveform’s spectral centroid $\langle f\rangle$,
$S\left(f\right)$ is the waveform’s Fourier transform, and
$\dot{\varphi}\left(t\right)$ is the first time derivative of the waveform’s
instantaneous phase. The RMS pulse length term is expressed as [7]

$\tau_{rms}^{2}=4\pi^{2}\int_{\Omega_{t}}\left(t-t_{0}\right)^{2}|s\left(t\right)|^{2}dt$
(7)

where $t_{0}$ is the first time moment $\langle t\rangle$ of the waveform’s
complex envelope $|s\left(t\right)|^{2}$ and is zero for waveforms such as (1)
that are even-symmetric in time. The RDCF $\rho$ is expressed as [7]

$\displaystyle\rho$
$\displaystyle=-2\pi\Im\Biggl{\\{}\int_{\Omega_{t}}ts\left(t\right)\dot{s}^{*}\left(t\right)dt\Biggr{\\}}=2\pi\int_{-T/2}^{T/2}t\dot{\varphi}\left(t\right)dt$
(8)

where $\Im\\{\\}$ denotes the imaginary component of the integral. The RMS
bandwidth and RDCF EOA parameters are solely dependent upon the frequency
modulation function $\dot{\varphi}\left(t\right)$. If the waveform’s
modulation function is known, $\beta_{rms}^{2}$ and $\rho$ can be calculated
in exact closed form which along with $\tau_{rms}^{2}$ provides full
characterization of the waveform’s AF mainlobe structure [7]. Furthermore, if
the waveform’s modulation function is composed of a discrete set of
parameters, those parameters can be optimized to design waveforms with
specific EOA parameters thus uniquely shaping the waveform’s AF mainlobe
structure [6].

###  2.2 Sidelobe Structure

This paper focuses specifically on the sidelobe structure of the ACF. Two of
the most common metrics are the Peak-to-Sidelobe Level Ratio (PSLR) and the
Integrated Sidelobe Level (ISL) [1]. The PSLR is expressed as

$\text{PSLR}=\dfrac{\underset{\Delta\tau\leq|\tau|\leq
T}{\text{max}}\bigl{\\{}\left|R\left(\tau\right)\right|^{2}\bigr{\\}}}{\underset{0\leq|\tau|\leq\Delta\tau}{\text{max}}\bigl{\\{}\left|R\left(\tau\right)\right|^{2}\bigr{\\}}}=\underset{\Delta\tau\leq|\tau|\leq
T}{\text{max}}\bigl{\\{}\left|R\left(\tau\right)\right|^{2}\bigr{\\}}$ (9)

where $\Delta\tau$ is the null of the ACF mainlobe thus establishing the ACF’s
null-to-null mainlobe width as $2\Delta\tau$. Note that the rightmost
expression in (9) results from the assumption that the waveform is unit energy
and thus the maximum value of $|R\left(\tau\right)|^{2}$ is unity which occurs
at $\tau=0$. The ISL is the ratio of the $\ell_{2}$-norms of the mainlobe and
sidelobe regions of the ACF expressed as

$\text{ISL}~{}=\dfrac{\int_{\Delta\tau}^{T}\left|R\left(\tau\right)\right|^{2}d\tau}{\int_{0}^{\Delta\tau}\left|R\left(\tau\right)\right|^{2}d\tau}.$
(10)

Note that the integration is performed only over positive time-delays since
the ACF is even-symmetric in $\tau$. A lower ISL corresponds to an ACF with
lower overall sidelobe levels but does not necessarily translate to a lower
PSLR.

##  3 Shaping the MTSFM’s AF Mainlobe and Sidelobe Structure

The MTSFM’s EOA parameters are expressed in terms of the Fourier coefficients
in its frequency modulation function as [5, 6]

$\displaystyle\beta_{rms}^{2}$
$\displaystyle=2\pi^{2}\sum_{\ell}^{L}a_{\ell}^{2}+b_{\ell}^{2},$ (11)
$\displaystyle\tau_{rms}^{2}$ $\displaystyle=\dfrac{\pi^{2}T^{2}}{3},$ (12)
$\displaystyle\rho$
$\displaystyle=-2\pi^{2}T\sum_{\ell=1}^{L}b_{\ell}\dfrac{\cos\left(\pi\ell\right)}{\pi\ell}.$
(13)

It’s important to note several observations regarding the results shown in
(11)-(13). First, as with all rectangularly tapered waveforms, the Doppler
sensitivity is proportional to the square of the waveform duration $T$ times a
constant ($\pi^{2}/3$). Second, both (11) and (13) are controlled by the
MTSFM’s coefficients. Lastly, the RDCF (13) is a function only of the odd
harmonic coefficients in (2). This is consistent with the results in [7]
describing how waveforms with even symmetric modulation functions possess an
AF mainlobe with RDCF of zero.

###  3.1 The MTSFM’s AF Mainlobe Structure - A Simple Design Example

Consider the following design examples that synthesize MTSFM waveforms with
two sine Fourier coefficients $b_{1}$ and $b_{2}$ with an RMS bandwidth
equality constraint. Using (11), the second coefficient $b_{2}$ can be
expressed in terms of the first coefficient $b_{1}$ as

$b_{2}=\pm\sqrt{\left(\dfrac{\beta_{rms}^{2}}{2\pi^{2}}-b_{1}^{2}\right)}.$
(14)

The RDCF $\rho$ is expressed as

$\displaystyle\rho$ $\displaystyle=\pi T\left[2b_{1}-b_{2}\right]=\pi
T\left[2b_{1}\mp\sqrt{\left(\dfrac{\beta_{rms}^{2}}{2\pi^{2}}-b_{1}^{2}\right)}\right].$
(15)

Figure 1 displays $b_{2}$ and $\rho$ as a function of $b_{1}$. Note that
$\rho$ is normalized to $\tilde{\rho}=\rho/\left(\beta_{rms}\tau_{rms}\right)$
so as to mimic a correlation coefficient. This normalized RDCF $\tilde{\rho}$
takes on the values $-1\leq\tilde{\rho}\leq 1$ where $\pm 1$ corresponds to an
AF mainlobe that is perfectly negatively and positively coupled respectively.
The values of $b_{1}$ and $b_{2}$ that satisfy the RMS bandwidth equality
constraint form a circle and the corresponding RDCF values $\tilde{\rho}$ form
a coupled ellipse. With only two MTSFM coefficients, $\tilde{\rho}$ takes on
all the values between $\approx\pm 0.8717$. As shown in the Appendix, the $L$
MTSFM coefficients that maximize $\tilde{\rho}$ are

$b_{\ell}=\dfrac{-\sqrt{2}\beta_{rms}}{2\pi\sqrt{\sum_{\ell^{\prime}}\left[\dfrac{\cos\left(\pi\ell^{\prime}\right)}{\pi\ell^{\prime}}\right]^{2}}}\dfrac{\cos\left(\pi\ell\right)}{\pi\ell}.$
(16)

The resulting $\tilde{\rho}_{\text{max}}$ using these optimal MTSFM
coefficients for increasing $L$ are shown in panel (c) of Figure 1. The MTSFM
is capable of possessing essentially any realizable RDCF $-1<\tilde{\rho}<1$.
Waveforms with variable RDCF values could have practical applications in
active sonar systems. Doppler sensitive waveforms with long duration require a
large number of Doppler shifted matched filters to process Doppler shifted
target echoes. This can substantially increase the complexity of the sonar
receiver [8]. Utilizing a waveform with variable Doppler sensitivity could
smoothly trade off between Doppler sensitivity and receiver complexity.

Figure 1: MTSFM coefficients that satisfy the RMS bandwidth equality
constraint (a), the corresponding RDCF values $\tilde{\rho}$ of these MTSFM
coefficients (b), and the optimal $\tilde{\rho}_{\text{max}}$ for a MTSFM with
$L$ harmonics in its modulation function (c).

###  3.2 Controlling the MTSFM’s ACF Sidelobe Structure

The ACF sidelobes of the MTSFM can be reduced while maintaining its AF
mainlobe structure. This is done via optimizing the ISL metric subject to
constraints on the RMS bandwidth and RDCF of an initial MTSFM waveform.
Formally, the optimization problem is stated as

$\displaystyle\underset{b_{\ell}}{\text{min}}\text{~{}ISL}\left(\\{b_{\ell}\\}\right)\text{s.t.}$
$\displaystyle\left(1-\delta\right)\beta^{(0)}_{rms}\leq\beta_{rms}\leq\left(1+\delta\right)\beta^{(0)}_{rms},$
(17)
$\displaystyle\left(1-\epsilon\right)\rho^{(0)}\leq\rho\leq\left(1+\epsilon\right)\rho^{(0)}$

where $\beta^{(0)}_{rms}$ and $\rho^{(0)}$ are the initial seed waveform’s RMS
bandwidth and RDCF respectively and $\delta=0.1$ and $\epsilon=0.05$ are
unitless bound parameters. The following simulation utilizes four waveforms
whose initial Fourier coefficients are listed in Table 1. The waveform
coefficients and pulse-lengths were scaled such that the resulting waveforms’
RMS bandwidths were equal to that of an LFM with a time-bandwidth product of
$T\Delta f=200$ where $\Delta f$ is the waveform’s swept bandwidth. Note that
since waveform III’s RDCF was $0.0$, the RDCF constraint was not used during
optimization. These seed MTSFM waveforms were zero-padded with an additional
126 Fourier coefficients to further increase the degrees of freedom in the
optimization problem. This has been shown to produce waveforms with lower ACF
sidelobes [5]. The optimization problem in (17) was solved using MATLAB’s
optimization toolbox and the resulting waveforms’ ISL and PSLR values for
these simulations are shown in Table 2. Also shown in Table 2 are the ratios
of the optimized waveforms’ $\beta_{rms}^{2}$ and $\rho$ to that of the
initial waveforms denoted as $\beta_{rms}/\beta_{rms}^{(0)}$ and
$\rho/\rho^{(0)}$ respectively. All four optimized waveforms exhibit clearly
lower ISL and PSLR values. Additionally, the AF mainlobe EOA parameters of all
four waveforms stay relatively close to the values of the initial seed
waveforms.

##  4 Conclusion

The closed form expressions for the MTSFM waveform’s EOA parameters allows for
fine control of its resulting AF mainlobe shape. The Fourier coefficients of
the MTSFM specifically allow for finely adjusting the RDCF. This in turn
facilitates designing MTSFM waveforms with adjustable Doppler tolerance
introducing a design tradeoff between receiver complexity and Doppler
resolution. Additionally, the sidelobe levels of the resulting waveform’s ACF
can be reduced by minimizing the ISL. The RMS bandwidth and RDCF constraints
ensure that the resulting optimized waveform will largely retain its AF
mainlobe shape. Future efforts will focus on deriving the MTSFM’s broadband AF
EOA parameters and extend the optimization routines to shape the broadband AF
sidelobes over regions in the range-Doppler plane.

Waveform | $\sqrt{2}\pi b_{1}/\beta_{rms}$ | $\sqrt{2}\pi b_{2}/\beta_{rms}$ | $\tilde{\rho}$
---|---|---|---
I | 0.8944 | -0.4473 | 0.8717
II | 0.1292 | -0.9916 | 0.4873
III | -0.4472 | -0.8944 | 0.0000
IV | -0.8944 | 0.4473 | -0.8717
Table 1: List of MTSFM waveforms with their normalized design coefficients $b_{1}$ & $b_{2}$ and their normalized RDCF $\tilde{\rho}$ taken directly from Figure 1. Waveform | Init./Opt. ISL (dB) | Init./Opt. PSLR (dB) | $\beta_{rms}/\beta_{rms}^{(0)}$ | $\rho/\rho^{(0)}$
---|---|---|---|---
I | -0.01/-15.04 | -8.89/-28.44 | 1.099 | 1.011
II | 3.91/-10.86 | -6.19/-21.67 | 1.099 | 0.953
III | 0.10/-11.74 | -11.22/-26.73 | 1.099 | ——-
IV | 2.54/-14.66 | -8.0/-30.44 | 1.099 | 1.018
Table 2: List of the performance characteristics of the initial and optimized
MTSFM waveforms. The optimized MTSFM waveforms possess substanitally lower
PSLR and ISL values than the initial seed waveforms while largely retaining
their AF mainlobe shape.

##  5 Acknowledgements

This research was supported by the Office of Naval Research (ONR) award
#N0001423WX00345.

## References

  * [1] S. D. Blunt and E. L. Mokole, “Overview of radar waveform diversity,” _IEEE Aerospace and Electronic Systems Magazine_ , vol. 31, no. 11, pp. 2–42, 2016.
  * [2] E. M. N. Levanon, _Radar Signals_. Wiley-Interscience, 2004.
  * [3] A. Blachet, A. Austeng, J. Aparicio, A. J. Hunter, and R. E. Hansen, “Multibeam echosounder with orthogonal waveforms: Feasibility and potential benefits,” _IEEE Journal of Oceanic Engineering_ , vol. 46, no. 3, pp. 963–978, 2021.
  * [4] R. E. Hansen, “Mimo synthetic aperture sonar: Benefits and limitations,” _Electronics Letters_ , vol. 59, no. 2, p. e12723, 2023. [Online]. Available: https://ietresearch.onlinelibrary.wiley.com/doi/abs/10.1049/ell2.12723
  * [5] D. A. Hague, “Adaptive transmit waveform design using multitone sinusoidal frequency modulation,” _IEEE Transactions on Aerospace and Electronic Systems_ , vol. 57, no. 2, pp. 1274–1287, 2021.
  * [6] D. A. Hague, “Target resolution properties of the multi-tone sinusoidal frequency modulated waveform,” in _2018 IEEE Statistical Signal Processing Workshop (SSP)_ , 2018, pp. 752–756.
  * [7] D. W. Ricker, _Echo Signal Processing_. Kluwer, 2003.
  * [8] Y. Doisy, L. Deruaz, S. P. Beerens, and R. Been, “Target doppler estimation using wideband frequency modulated signals,” _IEEE Transactions on Signal Processing_ , vol. 48, no. 5, pp. 1213–1224, 2000.

##  6 Derivation of MTSFM Coefficients that Optimize $\rho$ with an RMS
Bandwidth Equality Constraint

The waveform design problem is to find the set of MTSFM coefficients
$b_{\ell}$ that maximizes the RDCF $\rho$ subject to an equality constraint on
the waveform’s RMS bandwidth $\beta_{rms}^{2}$. Noting that maximizing $\rho$
is equivalent to minimizing $-\rho$, the optimization problem is formally
stated as

$\underset{b_{\ell}}{\text{min}}\Biggl{\\{}2\pi^{2}T\sum_{\ell}b_{\ell}\dfrac{\cos\left(\pi\ell\right)}{\pi\ell}\Biggr{\\}}~{}\text{s.t.}~{}2\pi^{2}\sum_{\ell}b_{\ell}^{2}=\beta_{rms}^{2}.$
(18)

Since (18) imposes an equality constraint, it can be solved using the method
of Lagrange multipliers. The Lagrangian function is expressed as

$L\left(b_{\ell},\lambda\right)=2\pi^{2}T\sum_{\ell}b_{\ell}\dfrac{\cos\left(\pi\ell\right)}{\pi\ell}+\lambda\left(2\pi^{2}\sum_{\ell}b_{\ell}^{2}-\beta_{rms}^{2}\right).$
(19)

Taking the gradient of (19) with respect to $b_{\ell}$ and setting equal to
$0$ results in the following expressions for each coefficient $b_{\ell}$ in
terms of $\lambda$

$b_{\ell}=\dfrac{-T}{2\lambda}\dfrac{\cos\left(\pi\ell\right)}{\pi\ell}$ (20)

Inserting (20) back into the RMS bandwidth constraint results in the
expression

$\beta_{rms}^{2}=2\pi^{2}\sum_{\ell}\left[\dfrac{-T}{2\lambda}\dfrac{\cos\left(\pi\ell\right)}{\pi\ell}\right]^{2}=\dfrac{2\pi^{2}T^{2}}{4\lambda^{2}}\sum_{\ell}\left[\dfrac{\cos\left(\pi\ell\right)}{\pi\ell}\right]^{2}.$
(21)

and solving in terms of $\lambda$ yields

$\lambda=\dfrac{\pi
T}{\sqrt{2}\beta_{rms}}\sqrt{\sum_{\ell}\left[\dfrac{\cos\left(\pi\ell\right)}{\pi\ell}\right]^{2}}.$
(22)

From here on, the
$\sqrt{\sum_{\ell}\left[\cos\left(\pi\ell\right)/\pi\ell\right]^{2}}$ will be
represented in terms of the variable $\ell^{\prime}$ rather than $\ell$ to
avoid confusion with the indices of the MTSFM coefficients $b_{\ell}$.
Inserting the expression for $\lambda$ (22) back into the expression for
$b_{\ell}$ (20) results in the MTSFM design coefficients $b_{\ell}$ that
maximize the RDCF $\rho$.

$b_{\ell}=\dfrac{-\sqrt{2}\beta_{rms}}{2\pi\sqrt{\sum_{\ell^{\prime}}\left[\dfrac{\cos\left(\pi\ell^{\prime}\right)}{\pi\ell^{\prime}}\right]^{2}}}\dfrac{\cos\left(\pi\ell\right)}{\pi\ell}.$
(23)

Reassuringly, for the case where $L\rightarrow\infty$, the MTSFM coefficients
$b_{\ell}$ become those of an up-sweeping LFM waveform. When
$L\rightarrow\infty$, the square-root term in (23) simplifies to
$\pi/\sqrt{6}$. Setting $\beta_{rms}=\pi\Delta f/\sqrt{3}$, the RMS bandwidth
of the LFM waveform, results in the expression

$b_{\ell}=\dfrac{\sqrt{12}\pi\Delta
f}{\sqrt{3}2\pi}\dfrac{\cos\left(\pi\ell\right)}{\pi\ell}=\Delta
f\dfrac{\cos\left(\pi\ell\right)}{\pi\ell}$ (24)

which are the Fourier coefficients to an up-sweeping LFM waveform’s modulation
function.