# Magnetostatic waves in metallic rectangular waveguides filled with uniaxial
negative permeability media

Afshin Moradi1<EMAIL_ADDRESS>Mohammed M. Bait-Suwailam2,3
<EMAIL_ADDRESS>_1 Department of Engineering Physics, Kermanshah
University of Technology, Kermanshah, Iran
2Department of Electrical and Computer Engineering, Sultan Qaboos University,
Muscat, Oman
3Remote Sensing and GIS Research Center, Sultan Qaboos University, Muscat,
Oman _

###### Abstract

The propagation characteristics of magneto-quasistatic waves, more commonly,
known as magnetostatic waves in a long, metallic rectangular waveguide filled
with a metamaterial slab are comprehensively investigated. The metamaterial
slab consists of split-ring resonators as an anisotropic uniaxial medium with
transversal negative effective permeability. Some analytical relations and
numerical validations on the characteristics of these waves are presented. The
results include the dispersion relations, mode patterns (field distributions)
that can be supported by such media and their corresponding cutoff
frequencies, group velocities, power flows, and storage energies of
magnetostatic waves. The findings from the present research study can be
advantageous to advance the synthesis and development of negative permeability
materials with peculiar features in guiding structures.

###### pacs:

41.20.Gz, 41.20.Cv, 41.20.Jb

##  I Introduction

A metallic rectangular waveguide (as a basic guiding structure in microwaves,
radars and antenna technology) is a long hollow tube (compared with its cross
section) of rectangular cross section with four metallic walls. There are
quite a number of interesting applications that the proposed structure can be
useful and aid in the advancement of waveguides and antenna structures alike,
including but not limited to miniaturization of waveguides S.H110 ; W.L2705 ,
performance enhancement of antennas integrated with anisotropic negative
permeability media H.T1 ; Y.L737 , beamforming and filters using magnetostatic
waves-enabled antennas A.R11 , and feeding structure for antenna arrays.

Figure 1: (a) Side view of a planar metamaterial slab composed of SRRs as an
anisotropic uniaxial $\mu$-negative medium with two metallic boundaries. The
metallic interfaces separating the slab ($-a\leq z\leq a$). (b) A metallic
rectangular waveguide filled with a medium of SRRs with its appropriate
dimensions.

Let us note that the investigation of electromagnetic characteristics of
metallic waveguides loaded with various anisotropic media has been receiving
considerable interests for almost seven decades A.T.M305 ; H.S409 ; G.B685 ;
B.A.A4081 ; H.S.T134 ; S.L111 ; S.K.J346 ; K.S3429 . Also, the mode patterns
(field lines) in hollow (rectangular/circular) waveguides and cylindrical
dielectric waveguides were presented in C.L271  and T.X2599 , respectively.
Furthermore, there are many interesting works in (rectangular/circular)
waveguides filled with metamaterials R.M183901 ; R.M405 ; Y.X426 ; S.H110 ;
A.T2706 ; D.R2513 ; A.B3064 ; A.B3385 ; A.B677 . For instance, Marques et al.
R.M183901 ; R.M405  found a propagation below the cutoff frequency of a very
unusual waveguide loaded with so-called split-ring resonators (SRRs) that
usually used for synthesis of metamaterial with negative permeability
J.B.P2075 ; D.R.S4184 ; L.S ; M.M.S2894 , and then extensively studied by
Hrabar et al. in S.H110 ; S.H2587 ; S.H494 , where the results in S.H110
shows that backward propagation can occur when the longitudinal permeability
is positive and the transversal permeability is negative. In fact, SRRs are an
array of non-magnetic conducting rings that are arranged in a periodic
fashion, which exhibit a strong resonant response on the magnetic component of
electromagnetic field. As a result, these elements show an effective negative
permeability A.B.S .

Figure 2: Variation of $\mu_{yy}$ with respect to the frequency, when
$\omega_{0}=4$GHz and $F=0.56$. The bandwidth of operation is right above the
resonance frequency $\omega_{0}=4$GHz to $\omega_{0}/\sqrt{1-F}\approx 6$GHz.

Although the electromagnetic characteristics of metallic waveguides loaded by
SRRs metamaterials have already been studied by several researchers S.H110 ;
R.M183901 ; R.M2572 ; J.D.B1451 ; P.M36622 ; S.H2587 ; S.H494 ; M.W.F2955 ;
F.Y.M3400 , to the best of our knowledge, the authors considered only the
propagation of TEm0 modes related to the waveguide axis, i.e., the $x$-axis in
Fig. 1 (for instance, see S.H110 ; R.M183901 ; F.Y.M3400 ). Furthermore, in
the general case the axis of the waveguide does not coincide with the axis of
TM and TE decomposition, and thus, a change of coordinates should be performed
and analyzed.

In earlier studies, the subject of electrostatic waves in metallic rectangular
waveguides loaded by hyperbolic metamaterials and/or microwire metamaterials
as anisotropic $\varepsilon$-negative media was extensively investigated and
analyzed in A.M24522 ; A.M2024 . The analysis of magnetostatic waves on
circular waveguides filled with anisotropic $\mu$-negative media were carried
out in A.M143901 ; A.M178 . In this study, we investigate the propagation of
magneto-quasistatic TEy waves (i.e., magneto-quasistatic TE waves related to
the $y$-axis in Fig. 1) in a long, metallic rectangular waveguide filled with
a metamaterial slab composed of SRRs as an anisotropic uniaxial $\mu$-negative
medium. It is worth noting here that the characteristics of such magnetostatic
waves were not tackled by neither earlier studies nor in recent works
A.M143901 ; A.M178 .

Such magnetostatic waves do not exist in the case of a hollow metallic
waveguide or a metallic waveguide filled with an isotropic medium. Physically,
the magnetostatic waves (in general, quasi-static waves) owe their existence
to the anisotropic property of the system and deserves particular attention.
In fact, in the absence of anisotropic $\mu$-negative properties, these slow
magnetic waves disappear and cannot propagate.

Figure 3: Dispersion characteristics of the first ten bulk magnetostatic modes
of a planar slab of SRRs with $a=25$mm, $\omega_{0}=4$GHz and $F=0.56$ and
metallic boundaries, using Eq. (9). The modes $n=0,2,4,6,8$ (red curves) are
even (symmetric) modes and $n=1,3,5,7,9$ (blue curves) are odd (anti-
symmetric) modes. The slope of curves is positive in the frequency range
$\omega_{0}<\omega<\omega_{0}/\sqrt{1-F}$ for all mode orders. Therefore,
these modes called backward waves. For a backward wave, the directions of
group velocity (power flow) and phase velocity (phase propagation) are
mutually opposite.

Moreover, we note here that the presented study investigates thoroughly the
dispersion characteristics and modal solutions of magnetostatic waves that
exist in long waveguide structures filled with planar $\mu$-negative SRR
media, which earlier studies, for example S.H110 ; A.M24522  did not
investigate. In other words, our contribution in this research work is the
presentation of a complete analytical and numerical solutions of magnetostatic
waves in long metamateirals-based waveguide structures. In fact, the presented
graphical solutions in this study adds more physical insights into the modal
solution of magentostatic waves in engineered waveguide structures. Also, we
should stress here that the important topics of energy density and power flow
of magnetostatic waves in long waveguide structures filled with planar
$\mu$-negative SRR media are the subjects of the last part of the present
study.

Figure 4: Dispersion characteristics of the two modes ($n=1$ when $a=18.5$mm,
and $n=2$ when $a=35.5$mm) of a planar slab of SRRs with $\omega_{0}=4$GHz and
$F=0.56$ and metallic boundaries. Solid curves show the magnetostatic result
using Eq. (9). Dashed curves show the rigorous electromagnetic analyze.

##  II Spectroscopy of magnetostatic modes of a planar slab of SRRs

One type of magnetostatic waveguide is a planar slab of height $2a$ with two
metallic boundaries filled with a medium of SRRs as an anisotropic uniaxial
metamaterial with transversal negative effective permeability, as shown in
panel (a) of Fig. 1. To simplify the analysis of the structure, at first we
reduce the problem to a 2D one (its length in the $z$-direction is infinite)
so that $\partial/\partial z=0$. We now investigate the behaviors of
magnetostatic bulk modes of a planar slab of SRRs, based on the field
analysis. Let us assume that the waves are traveling in the $x$-direction, and
the structure is infinite in that direction as illustrated in panel (a) of
Fig. 1. We shall obtain the general expression for dispersion relation of the
magnetostatic waves. For the present problem, the relative permeability tensor
of the present medium is (see Eq. (6.1) in Chap. 6 of L.S )

$\underline{\mu}(\omega)=\left(\begin{array}[]{ccc}1&0&0\\\ 0&\mu_{yy}&0\\\
0&0&1\\\ \end{array}\right)\;,$ (1)

where $\mu_{yy}$ is the relative permeability in the $y$-direction and has the
form

$\mu_{yy}=1-\frac{F\omega^{2}}{\omega^{2}-\omega_{0}^{2}}\;,$ (2)

in the lossless case A.M634 . Here $\omega_{0}$ is the resonance frequency of
the SRRs and $F$ with $0<F<1$ is a measure of the strength of the interaction
between the SRRs and the magnetic field. One can see that Eq. (2) contains a
resonance frequency $\omega_{0}$ that allows $\mu_{yy}$ to be negative over a
bandwidth of nearly 2 GHz, depending on the applied field frequency $\omega$,
as shown in Fig. 2, when $\omega_{0}=4$GHz and $F=0.56$.

Under the magnetostatic approximation with the magnetostatic equations
$\nabla\times\textbf{H}=0$, $\nabla\cdot\textbf{B}=0$ and
$\nabla\times\textbf{E}=-\partial\textbf{B}/\partial t$ D.D.S , the magnetic
field H can be represented by the gradient of a magnetostatic potential
$\psi$, as $\textbf{H}=-\nabla\psi$ A.M045801 . The Maxwell’s equation
$\nabla\cdot\textbf{B}=0$ with $\textbf{B}=\underline{\mu}\cdot\textbf{H}$
gives the wave equation for the magnetostatic potential of an unbounded medium
of SRRs

$\left[\dfrac{\partial^{2}}{\partial
x^{2}}+\mu_{yy}\dfrac{\partial^{2}}{\partial y^{2}}\right]\psi=0\;,$ (3)

where all the field quantities are assumed to have the harmonic time
dependence of the form $\exp(j\omega t)$. Note that $\omega$ is angular
frequency of a magnetostatic wave in the system.

In deriving the magnetostatic mode spectrum of a planar slab of SRRs, we use
the following general separated-variable solution for traveling waves in the
$+x$-direction

$\psi(x,y)=\left(A\sin\kappa y+B\cos\kappa y\right)\exp(-j\beta x)\;.$ (4)

Substituting this into Eq. (3), we get

$\beta^{2}+\mu_{yy}\kappa^{2}=0\;.$ (5)

Eq. (5) shows that a homogeneous bulk magnetostatic plane wave corresponding
to real $\beta$ and $\kappa$ is possible only when $\mu_{yy}$ is negative.
Field existing within this waveguide must be characterized by zero normal
components of magnetic field at the metallic walls. However, the boundary
conditions at the two metallic walls can be written as

$\dfrac{\partial\psi}{\partial y}\bigg{|}_{y=-a,a}=0\;.$ (6)

Now, by applying the mentioned boundary conditions we can find the
relationship between the constants $A$ and $B$, as

$A\cos\kappa a\pm B\sin\kappa a=0\;.$ (7)

To satisfy Eq. (7) the categories of even and odd modes can be considered. For
an even mode we have $A=0$ and then $\kappa a=n\pi$ (with $n=0,1,2,\cdots$) is
the dispersion relation for magnetostatic waves with symmetric potential
functions. For an odd mode we have $B=0$ and therefore $\kappa
a=\left(n+1/2\right)\pi$ is the dispersion relation for magnetostatic waves
with anti-symmetric potential functions. The general dispersion relation can
be written as

$\kappa a=n\dfrac{\pi}{2}\;,$ (8)

where this equation corresponds to an even mode for $n$ even, and to an odd
mode for $n$ odd. Now, from Eqs. (2), (5) and (8), we obtain

Figure 5: Dispersion curves of magnetostatic modes ($m,n$) of a metallic
square waveguide ($h=2a=w$) filled with a medium of SRRs, using Eq. (18) for
$\omega_{0}=4$GHz, $F=0.56$, $n=1,2,3$, and $m=0,1,2,3$. In panel (a) solid
black, blue, red and green curves show the magnetostatic modes ($0,1$),
($1,1$), ($2,1$) and ($3,1$), respectively, when $a=18$mm. In panel (b) solid
black, blue, red and green curves show the magnetostatic modes ($0,2$),
($1,2$), ($2,2$) and ($3,2$), respectively, when $a=35$mm. In panel (c) solid
black, blue, red and green curves show the magnetostatic modes ($0,3$),
($1,3$), ($2,3$) and ($3,3$), respectively, when $a=55$mm.

$\omega=\omega_{0}\left(\dfrac{1+\left(\dfrac{n\pi}{2\beta
a}\right)^{2}}{1+(1-F)\left(\dfrac{n\pi}{2\beta
a}\right)^{2}}\right)^{1/2}\;.$ (9)

If $\beta\rightarrow 0$, then $\omega\rightarrow\omega_{0}/\sqrt{1-F}$. If
$\beta\rightarrow\infty$, then $\omega\rightarrow\omega_{0}$. This means that
this system represents a backward bulk magnetostatic band filter with the
angular frequency bands $1<\omega/\omega_{0}<1/\sqrt{1-F}$, as illustrated in
Fig. 2. Furthermore, all backward modes have equal cutoff frequency
$(\omega_{c})_{n}=\omega_{0}/\sqrt{1-F}$, except $n=0$ mode. Actually, for
$n=0$ the dispersion curve degenerates into a line and the wave stops
propagating.

Using Eq. (9), the dispersion characteristics of magnetostatic modes of a
planar slab of SRRs with $a=25$mm, $\omega_{0}=4$GHz and $F=0.56$ L.S  for
various values of $n$ are depicted in Fig. 2. One can see an infinite number
of bulk magnetostatic waves with different values of $n$ which depend on the
conditions of excitation. Also, we see that the frequency of a magnetostatic
mode decreases monotonically with wavenumber throughout the allowed frequency
range $\omega_{0}<\omega<\omega_{0}/\sqrt{1-F}$. Furthermore, in Fig. 4 the
magnetostatic results shown in Fig. 2 is compared with the results using the
rigorous electromagnetic analyze for modes $n=1$ and $2$. In fact, by using
the full set of Maxwell’s equations, Eq. (5) for TE0n modes related to the
$y$-axis should be read as (see Appendix)

$\beta^{2}+\mu_{yy}\left[\kappa^{2}-\dfrac{\omega^{2}}{c^{2}}\right]=0\;,$
(10)

where $c$ is speed of light in free space. The comparison of the magnetostatic
results using Eq. (9), with Eq. (10) demonstrates that for the present example
the magnetostatic waves almost disappear for $\beta<50$ radian/m and cannot
propagate. Physically, the existence of the magnetostatic waves (and also the
electrostatic waves A.M24522 ) is tied to the resonance frequency of the
system, where the propagation constant (wavenumber $\beta$) becomes infinity,
here almost for $\beta>50$ radian/m. Near the resonance frequency, where the
phase velocity $v_{\mathrm{ph}}$ [using Eq. (10)] goes to zero, i.e.,

$v_{\mathrm{ph}}=\dfrac{\omega}{\beta}=\dfrac{c}{\beta\sqrt{\mu_{yy}}}\sqrt{\beta^{2}+\mu_{yy}\kappa^{2}}=0\;,$
(11)

which leads to Eq. (5), the magnetostatic waves can propagate in the system.
In fact, the existence of the magnetostatic waves indicates the velocity of
light in free space must be much more than the phase velocity of these waves
D.D.S ; A.M .

##  III Spectroscopy of magnetostatic modes of a rectangular waveguides filled
with a medium of SRRs

Now, let us consider a metallic rectangular waveguide of lateral dimensions
$w$ and $h$ in the $y$ and $z$ directions, respectively, as shown in panel (b)
of Fig. 1. Initially assume that the waveguide is of infinite length and is
filled with a medium of SRRs. It is our purpose to determine the various
magnetostatic modes that can exist inside this rectangular magnetostatic
waveguide. In this case, the wave equation for the magnetostatic potential
inside the waveguide can be written as

$\left[\dfrac{\partial^{2}}{\partial
x^{2}}+\mu_{yy}\dfrac{\partial^{2}}{\partial
y^{2}}+\dfrac{\partial^{2}}{\partial z^{2}}\right]\psi=0\;.$ (12)

Again, the solution to Eq. (12) can be obtained by using the separation of
variables method. In general, the solution to $\psi(x,y,z)$ for traveling
magnetostatic waves in the $+x$-direction can be written as

$\psi(x,y,z)=\left(C\sin\kappa y+D\cos\kappa y\right)\\\
\times\left(E\sin\alpha z+F\cos\alpha z\right)\exp(-j\beta x)\;.$ (13)

Note that $C$, $D$, $E$, and $F$ are constants. Substituting this equation
into Eq. (12), we get

Figure 6: Snapshot of the magnetic field patterns for the magnetostatic modes
$(m=[0,1,2,3],n=[1,2,3])$ of a metallic square waveguide ($h=w$) filled with a
medium of SRRs.

$\alpha^{2}+\beta^{2}+\mu_{yy}\kappa^{2}=0\;.$ (14)

Here, the boundary conditions on the problem are that normal components of
magnetic field vanish at the four metallic walls. Hence, we have

$\dfrac{\partial\psi}{\partial
y}\bigg{|}_{y=0,w}=0\;,\;\;\;\;\;\;\;\;\;\;\dfrac{\partial\psi}{\partial
z}\bigg{|}_{z=0,h}=0\;.$ (15)

To satisfy the boundary conditions for magnetostatic modes, we choose $C=0=E$,
$\kappa=n\pi/w$, and $\alpha=m\pi/h$, where each integer $m$ and $n$, except
$m=0=n$, specifies a mode. Hence, the magnetic potential is

$\psi(x,y,z)=A_{mn}\cos\left(\dfrac{n\pi y}{w}\right)\cos\left(\dfrac{m\pi
z}{h}\right)\exp(-j\beta x)\;,$ (16)

with $m=0,1,2,\cdots$; $n=0,1,2,\cdots$; $m=0=n$ excepted. In Eq. (16) the
case $m=0=n$ is excluded because for that $\psi$ is a constant and all the
components of magnetic field vanish; thus a trivial solution. The dispersion
equation for the modes can be written as

$\beta^{2}=-\left(\dfrac{m\pi}{h}\right)^{2}-\mu_{yy}\left(\dfrac{n\pi}{w}\right)^{2}\;.$
(17)

From the above dispersion relation it is easy to conclude that $n=0$ is not
acceptable and should be excluded. Since there are infinite combinations of
$m$ and $n$ ($n\neq 0$), an infinite number of magnetostatic modes can be
found. Substituting Eq. (2) into Eq. (17), we find

$\omega=\omega_{0}\left(\dfrac{1+\left(\dfrac{n\pi}{\beta
w}\right)^{2}+\left(\dfrac{m\pi}{\beta
h}\right)^{2}}{1+(1-F)\left(\dfrac{n\pi}{\beta
w}\right)^{2}+\left(\dfrac{m\pi}{\beta h}\right)^{2}}\right)^{1/2}\;.$ (18)

Also, for the cutoff frequency of a given $(m,n)$ mode, we find

$\left(\omega_{c}\right)_{mn}=\omega_{0}\left(\dfrac{\left(\dfrac{n\pi}{w}\right)^{2}+\left(\dfrac{m\pi}{h}\right)^{2}}{(1-F)\left(\dfrac{n\pi}{w}\right)^{2}+\left(\dfrac{m\pi}{h}\right)^{2}}\right)^{1/2}\;.$
(19)

For $m=0$ and $n=1,2,3,\cdots$ we have
$\left(\omega_{c}\right)_{0n}=\omega_{0}/\sqrt{1-F}$.

The dispersion curves for magnetostatic modes $(m,n)$ ($n=1,2,3$ and
$m=0,1,2,3$) of a metallic square waveguide ($h=2a=w$) filled with a medium of
SRRs with $\omega_{0}=4$GHz and $F=0.56$ are depicted Fig. 5. Note that by
using the full set of Maxwell’s equations, Eq. (17) for TEmn modes related to
the $y$-axis should be read as (see Appendix)

$\beta^{2}=-\left(\dfrac{m\pi}{h}\right)^{2}-\mu_{yy}\left[\left(\dfrac{n\pi}{w}\right)^{2}-\dfrac{\omega^{2}}{c^{2}}\right]\;.$
(20)

The mode patterns (field lines) are also of interest, where Fig. 6 shows
sketches of cross-sectional mode patterns for some of the magnetostatic modes
of a square waveguide, where results agree quite well with the results using
the full set of Maxwell’s equations for the large values of propagation
constant (not shown here). In a rectangular waveguide, the field lines are
bent from the location shown, but they retain the same general appearance. By
comparing the results in figure 4 of C.L271  with Fig. 6 in the present work,
one can conclude that the magnetic field pattern of magnetostatic mode $(m,n)$
is similar with magnetic field pattern of TE mode for a hollow waveguide.

Another part of the present analytical study is to find the power flow
associated with the magnetostatic waves that may lead to further insight. To
find the power that flows in the waveguide, the time-averaged of the power
density directed along the axis of the waveguide is needed first. Then, we can
obtain the power flow along the waveguide by integrating the power density in
the axial direction over the cross-section of the waveguide. The power density
along the $x$-direction that is delivered by a magnetostatic mode can be
written as D.D.S ; K.Y2170

$S_{x}=-\dfrac{\mu_{0}}{2}{\rm Re}\left[\psi\dfrac{\partial}{\partial
t}\dfrac{\partial}{\partial x}\psi^{\ast}\right]\;,$ (21)

in the complex-number representation, where ∗ denotes complex conjugation, and
${\rm Re}$ denotes taking the real part. Use of (16) into (21), the
$x$-directed power density for the magnetostatic modes can be written as

$S_{x}=-\dfrac{\mu_{0}\omega k}{2}|A_{mn}|^{2}\cos^{2}\left(\dfrac{n\pi
y}{w}\right)\cos^{2}\left(\dfrac{m\pi z}{h}\right)\;.$ (22)

The power transmitted along the waveguide can be found by integrating Eq. (22)
over the guide cross-section $\mathcal{A}=hw$. This gives

$P_{mn}=\int_{0}^{h}\int_{0}^{w}S_{x}dydz=-\dfrac{\mu_{0}\omega
k}{8}hw|A_{mn}|^{2}\;.$ (23)

Also, the time-averaged of energy density distribution associated with the
waves can be written as K.Y2170

$u=\dfrac{\mu_{0}}{4}\nabla\psi^{\ast}\cdot\left[\dfrac{d\left(\omega\underline{\mu}\right)}{d\omega}\cdot\nabla\psi\right]\;.$
(24)

After substitution Eq. (16) into Eq. (24), we obtain

$u=\dfrac{\mu_{0}k^{2}}{4}|A_{mn}|^{2}\cos^{2}\left(\dfrac{n\pi
y}{w}\right)\cos^{2}\left(\dfrac{m\pi z}{h}\right)\\\
+\dfrac{\mu_{0}}{4}\left(\dfrac{n\pi}{w}\right)^{2}\dfrac{d\left(\omega\mu_{yy}\right)}{d\omega}|A_{mn}|^{2}\sin^{2}\left(\dfrac{n\pi
y}{w}\right)\cos^{2}\left(\dfrac{m\pi z}{h}\right)\\\
+\dfrac{\mu_{0}}{4}\left(\dfrac{m\pi}{h}\right)^{2}|A_{mn}|^{2}\cos^{2}\left(\dfrac{n\pi
y}{w}\right)\sin^{2}\left(\dfrac{m\pi z}{h}\right)\;,$ (25)

The associated storage energy is obtained by integrating Eq. (25) over the
cross-section $\mathcal{A}$ of the waveguide, as

$U_{mn}=\dfrac{\mu_{0}}{16}wh\omega|A_{mn}|^{2}\left(\dfrac{n\pi}{w}\right)^{2}\dfrac{d\mu_{yy}}{d\omega}\;.$
(26)

Note that using the simple formula $v_{\mathrm{g}}=d\omega/dk$ and Eq. (17),
the group velocity of magnetostatic modes of the system can be simply
computed. However, in the absence of the damping effects, the group velocity
of the waves is also equal with the ratio of the power flow to the storage
energy, such as

$v_{\mathrm{g}}=\dfrac{P_{mn}}{U_{mn}}=-2k\left(\dfrac{n\pi}{w}\right)^{-2}\left(\dfrac{d\mu_{yy}}{d\omega}\right)^{-1}\;,$
(27)

that is a manifestation of self-consistency and general validity of presented
results in the magnetostatic theory.

##  IV Conclusion

In summary, we have investigated the existence of magnetostatic waves in a
long, metallic rectangular waveguide filled with a metamaterial slab composed
of SRRs as an anisotropic uniaxial $\mu$-negative medium. We have obtained
general expression for dispersion equation of the magneto-quasistatic TE waves
and then presented the mode patterns in such magnetostatic waveguides.
Physically, the present magnetostatic modes owe their existence to the
anisotropic property of the system, where in the absence of this property,
these magnetostatic modes are disappeared. We have verified the obtained
results by showing that group velocity of the waves is the same as energy
velocity (i.e., the ratio of the power flow to the storage energy). Also, we
have carried out rigorous electromagnetic analysis, which agree quite well
with the magnetostatic results for the large propagation constant. Because of
the possibility of magnetostatic waves propagation in a rectangular waveguides
filled with a metamaterial composed of SRRs, they may be used in the
development of new waveguides using guided magnetostatic waves.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

Afshin Moradi: Project administration (lead); Conceptualization (lead);
Investigation (lead); Methodology (lead); Validation (lead); Formal analysis
(lead); Software (lead); Writing - original draft (lead); Writing – review and
editing (equal). Mohammed M. Bait-Suwailam: Conceptualization (supporting);
Validation (supporting); Formal analysis (supporting); Writing – review and
editing (equal).

### DATA AVAILABILITY

The data that supports the findings of this study are available within the
article.

## Appendix: The rigorous electromagnetic analysis

In the general case, let us consider a biaxial medium whose principal axes
coincide with the axes of the Cartesian coordinate system $(x,y,z)$. The
relative permittivity and permeability tensors $\underline{\varepsilon}$ and
$\underline{\mu}$ can be represented by diagonal matrices

$\underline{\varepsilon}(\omega)=\left(\begin{array}[]{ccc}\varepsilon_{xx}&0&0\\\
0&\varepsilon_{yy}&0\\\ 0&0&\varepsilon_{zz}\\\ \end{array}\right)\;,$ (A-1)

and

$\underline{\mu}(\omega)=\left(\begin{array}[]{ccc}\mu_{xx}&0&0\\\
0&\mu_{yy}&0\\\ 0&0&\mu_{zz}\\\ \end{array}\right)\;.$ (A-2)

Now, contrary to the assumption in S.H110 ; F.Y.M3400 , where the authors
considered only the propagation of TEm0 modes related to the waveguide axis
(i.e., the $x$-axis in Fig. 1), let us suppose that the axis of decomposition
of TM and TE modes is along the $y$-axis.

For the benefit of the reader of the present work, let us note that Meng et
al. F.Y.M3400  analyzed rectangular waveguides loaded with anisotropic
metamaterials to assess the controllability of transmission characteristics of
the involved electromagnetic waves. In this way, by assuming the existence of
TE waves related to the waveguide axis in the mentioned system, at first, they
considered the propagation of TEmn modes (see Eqs. (4), (6), (7), (8), and (9)
in F.Y.M3400 ) and in the following, they investigated the propagation of TEm0
modes. However, in the general case, hybrid wave propagation should be
expected for the rectangular waveguides loaded with anisotropic metamaterials,
and therefore TEmn waves are unable to propagate. Therefore, Eqs. (4), (6),
(7), (8), and (9) in F.Y.M3400  are incorrect (see A.MTAP ).

Substituting electric and magnetic fields E and H describing a wave traveling
in the $x$-direction

$\textbf{E}(x,y,x)=\left(\begin{array}[]{ccc}E_{x}(y,z)\\\ E_{y}(y,z)\\\
E_{z}(y,z)\\\ \end{array}\right)e^{j(\omega t-\beta x)}\;,$ (A-3)
$\textbf{H}(x,y,x)=\left(\begin{array}[]{ccc}H_{x}(y,z)\\\ H_{y}(y,z)\\\
H_{z}(y,z)\\\ \end{array}\right)e^{j(\omega t-\beta x)}\;,$ (A-4)

into Maxwell’s equations

$\nabla\times\textbf{E}=-j\omega\mu_{0}\underline{\mu}\textbf{H}\;,$ (A-5)
$\nabla\times\textbf{H}=j\omega\varepsilon_{0}\underline{\varepsilon}\textbf{E}\;,$
(A-6)

we obtain

$\dfrac{\partial E_{z}}{\partial y}-\dfrac{\partial E_{y}}{\partial
z}=-j\omega\mu_{0}\mu_{xx}H_{x}\;,$ (A-7) $j\beta E_{z}+\dfrac{\partial
E_{x}}{\partial z}=-j\omega\mu_{0}\mu_{yy}H_{y}\;,$ (A-8) $j\beta
E_{y}+\dfrac{\partial E_{x}}{\partial y}=j\omega\mu_{0}\mu_{zz}H_{z}\;,$ (A-9)
$\dfrac{\partial H_{z}}{\partial y}-\dfrac{\partial H_{y}}{\partial
z}=j\omega\varepsilon_{0}\varepsilon_{xx}E_{x}\;,$ (A-10) $j\beta
H_{z}+\dfrac{\partial H_{x}}{\partial
z}=j\omega\varepsilon_{0}\varepsilon_{yy}E_{y}\;,$ (A-11) $j\beta
H_{y}+\dfrac{\partial H_{x}}{\partial
y}=-j\omega\varepsilon_{0}\varepsilon_{zz}E_{z}\;.$ (A-12)

Thus, following the method of determinants, we obtain $E_{x}$, $E_{z}$,
$H_{x}$, and $H_{z}$ with respect to components $E_{y}$ and $H_{y}$, as

$E_{x}=\dfrac{j}{\kappa^{2}-\varepsilon_{xx}\mu_{zz}\dfrac{\omega^{2}}{c^{2}}}\left(\beta\dfrac{\partial
E_{y}}{\partial y}-\omega\mu_{0}\mu_{zz}\dfrac{\partial H_{y}}{\partial
z}\right)\;,$ (A-13)
$H_{x}=\dfrac{j}{\kappa^{2}-\varepsilon_{zz}\mu_{xx}\dfrac{\omega^{2}}{c^{2}}}\left(\beta\dfrac{\partial
H_{y}}{\partial y}+\omega\varepsilon_{0}\varepsilon_{zz}\dfrac{\partial
E_{y}}{\partial z}\right)\;,$ (A-14)
$E_{z}=\dfrac{1}{\kappa^{2}-\varepsilon_{zz}\mu_{xx}\dfrac{\omega^{2}}{c^{2}}}\left(\omega\beta\mu_{0}\mu_{xx}H_{y}-\dfrac{\partial^{2}E_{y}}{\partial
y\partial z}\right)\;,$ (A-15)
$H_{z}=\dfrac{-1}{\kappa^{2}-\varepsilon_{xx}\mu_{zz}\dfrac{\omega^{2}}{c^{2}}}\left(\omega\beta\varepsilon_{0}\varepsilon_{xx}E_{y}+\dfrac{\partial^{2}H_{y}}{\partial
y\partial z}\right)\;,$ (A-16)

where we have used $\partial^{2}/\partial y^{2}=-\kappa^{2}$. Substituting
Eqs. (A-13) and (A-15) into Eq. (A-8), and Eqs. (A-14) and (A-16) into Eq.
(A-11) we get

$\dfrac{\mu_{zz}}{\kappa^{2}-\varepsilon_{xx}\mu_{zz}\dfrac{\omega^{2}}{c^{2}}}\dfrac{\partial^{2}H_{y}}{\partial
z^{2}}-\left(\mu_{yy}+\dfrac{\mu_{xx}\beta^{2}}{\kappa^{2}-\varepsilon_{zz}\mu_{xx}\dfrac{\omega^{2}}{c^{2}}}\right)H_{y}=\\\
\dfrac{\omega\beta\varepsilon_{0}\left(\varepsilon_{xx}\mu_{zz}-\varepsilon_{zz}\mu_{xx}\right)}{\left(\kappa^{2}-\varepsilon_{xx}\mu_{zz}\dfrac{\omega^{2}}{c^{2}}\right)\left(\kappa^{2}-\varepsilon_{zz}\mu_{xx}\dfrac{\omega^{2}}{c^{2}}\right)}\dfrac{\partial^{2}E_{y}}{\partial
y\partial z}\;,$ (A-17)
$\dfrac{\varepsilon_{zz}}{\kappa^{2}-\varepsilon_{zz}\mu_{xx}\dfrac{\omega^{2}}{c^{2}}}\dfrac{\partial^{2}E_{y}}{\partial
z^{2}}-\left(\varepsilon_{yy}+\dfrac{\varepsilon_{xx}\beta^{2}}{\kappa^{2}-\varepsilon_{xx}\mu_{zz}\dfrac{\omega^{2}}{c^{2}}}\right)E_{y}=\\\
\dfrac{\omega\beta\mu_{0}\left(\varepsilon_{xx}\mu_{zz}-\varepsilon_{zz}\mu_{xx}\right)}{\left(\kappa^{2}-\varepsilon_{xx}\mu_{zz}\dfrac{\omega^{2}}{c^{2}}\right)\left(\kappa^{2}-\varepsilon_{zz}\mu_{xx}\dfrac{\omega^{2}}{c^{2}}\right)}\dfrac{\partial^{2}H_{y}}{\partial
y\partial z}\;.$ (A-18)

In general, Eqs. (A-17) and (A-18) may be satisfied by the same value of
$\beta$ if $E_{y}\neq 0$ and $H_{y}\neq 0$. Hence, the so-called anomalous
modes having all the six field components would be possible. However, let us
explore the possibility of TEy and TMy wave propagation. If we assume
$E_{y}=0$ (TEy modes), then Eqs. (A-17) and (A-18), reduce to

$\dfrac{\mu_{zz}}{\kappa^{2}-\varepsilon_{xx}\mu_{zz}\dfrac{\omega^{2}}{c^{2}}}\dfrac{\partial^{2}H_{y}}{\partial
z^{2}}-\left(\mu_{yy}+\dfrac{\mu_{xx}\beta^{2}}{\kappa^{2}-\varepsilon_{zz}\mu_{xx}\dfrac{\omega^{2}}{c^{2}}}\right)H_{y}=0\;,$
(A-19)
$\dfrac{\omega\beta\mu_{0}\left(\varepsilon_{xx}\mu_{zz}-\varepsilon_{zz}\mu_{xx}\right)}{\left(\kappa^{2}-\varepsilon_{xx}\mu_{zz}\dfrac{\omega^{2}}{c^{2}}\right)\left(\kappa^{2}-\varepsilon_{zz}\mu_{xx}\dfrac{\omega^{2}}{c^{2}}\right)}\dfrac{\partial^{2}H_{y}}{\partial
y\partial z}=0\;.$ (A-20)

Equations (A-19) and (A-20) cannot be simultaneously satisfied by the same
value of $\beta$ unless in two special cases. In the first case, decoupling
takes place if

$\dfrac{\partial H_{y}}{\partial y}=0\;,\;\;\;\mathrm{or}\;\;\;\dfrac{\partial
H_{y}}{\partial z}=0\;.$ (A-21)

In the second case, decoupling takes place if material parameters satisfy the
condition

$\varepsilon_{xx}\mu_{zz}=\varepsilon_{zz}\mu_{xx}\;,$ (A-22)

that is simply possible for our present system, because we have
$\varepsilon_{xx}=\varepsilon_{zz}=1$ and $\mu_{xx}=\mu_{zz}=1$. Therefore, we
obtain

$\dfrac{\partial^{2}H_{y}}{\partial
z^{2}}-\left(\beta^{2}+\mu_{yy}\left[\kappa^{2}-\dfrac{\omega^{2}}{c^{2}}\right]\right)H_{y}=0\;.$
(A-23)

If we set $\partial/\partial z=0$, we get Eq. (10), and by using
$\partial^{2}/\partial z^{2}=-\alpha^{2}$, Eq. (20) can be obtained.

Similarly, if we assume that (TMy modes), then Eqs. (A-17) and (A-18), reduce
to

$\dfrac{\omega\beta\varepsilon_{0}\left(\varepsilon_{xx}\mu_{zz}-\varepsilon_{zz}\mu_{xx}\right)}{\left(\kappa^{2}-\varepsilon_{xx}\mu_{zz}\dfrac{\omega^{2}}{c^{2}}\right)\left(\kappa^{2}-\varepsilon_{zz}\mu_{xx}\dfrac{\omega^{2}}{c^{2}}\right)}\dfrac{\partial^{2}E_{y}}{\partial
y\partial z}=0\;,$ (A-24)
$\dfrac{\varepsilon_{zz}}{\kappa^{2}-\varepsilon_{zz}\mu_{xx}\dfrac{\omega^{2}}{c^{2}}}\dfrac{\partial^{2}E_{y}}{\partial
z^{2}}-\left(\varepsilon_{yy}+\dfrac{\varepsilon_{xx}\beta^{2}}{\kappa^{2}-\varepsilon_{xx}\mu_{zz}\dfrac{\omega^{2}}{c^{2}}}\right)E_{y}=0\;.$
(A-25)

Again, Eqs. (A-24) and (A-25) cannot be simultaneously satisfied unless
material parameters satisfy the condition shown by Eq. (A-22), or if

$\dfrac{\partial E_{y}}{\partial y}=0\;,\;\;\;\mathrm{or}\;\;\;\dfrac{\partial
E_{y}}{\partial z}=0\;.$ (A-26)

For our present system, we have
$\varepsilon_{xx}=\varepsilon_{yy}=\varepsilon_{zz}=1$ and
$\mu_{xx}=\mu_{zz}=1$, then we obtain

$\dfrac{\partial^{2}E_{y}}{\partial
z^{2}}-\left(\kappa^{2}+\beta^{2}-\dfrac{\omega^{2}}{c^{2}}\right)E_{y}=0\;.$
(A-27)

This equation describes the propagation of TMy waves in hollow waveguides and
is well-known in many electromagnetic textbooks. In this case, the system
cannot support quasi-static waves.

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# Beables, Primitive Ontology and Beyond: How Theories Meet the World

Andrea Oldofredi Contact Information: Université de Lausanne, Section de
Philosophie, 1015 Lausanne, Switzerland. E-mail<EMAIL_ADDRESS>
###### Abstract

Bohm and Bell’s approaches to the foundations of quantum mechanics share
notable features with the contemporary Primitive Ontology perspective and
Esfeld and Deckert minimalist ontology. For instance, all these programs
consider ontological clarity a necessary condition to be met by every
theoretical framework, promote scientific realism also in the quantum domain
and strengthen the explanatory power of quantum theory. However, these
approaches remarkably diverge from one another, since they employ different
metaphysical principles leading to conflicting _Weltanschaaungen_. The
principal aim of this essay is to spell out the relations as well as the main
differences existing among such programs, which unfortunately remain often
unnoticed in literature. Indeed, it is not uncommon to see Bell’s views
conflated with the PO programme, and the latter with Esfeld and Deckert’s
proposal. It will be our task to clear up this confusion.

_Keywords_ : Quantum Mechanics; David Bohm; John Bell; Local Beables;
Primitive Ontology

> _To appear in the volume “Quantum Mechanics and Fundamentality: Naturalizing
> Quantum Theory between Scientific Realism and Ontological Indeterminacy”,
> edited by Valia Allori, Springer Nature._

###### Contents

  1. 1 Introduction
  2. 2 David Bohm, Pluralism and Infinitism
  3. 3 Local and Non-local Beables
  4. 4 The Primitive Ontology Approach
  5. 5 Beyond the PO: Minimalism and Fundamental Ontology
  6. 6 Conclusion

##  1 Introduction

Quantum Mechanics (QM) is one of the most efficient descriptions of the
inherent structure of matter and its behavior. As is known, however, this
theory represents elementary objects in a way that drastically changed our
conception of reality, speaking about intrinsically indeterminate systems,
non-local interactions, wave-particle dualities, _etc._. Indeed, it is still a
pivotal philosophical challenge to understand what scientific image of the
world it provides—i.e. to comprehend its ontology, and how the macroscopic
realm can be explained from it. Referring to this, it is worth noting that
quantum theory is affected by several conceptual and technical conundra;
hence, we cannot consider it the final word about ontological matters,
notwithstanding its empirical success. In particular, scholars agree that QM
contains physically ill-defined notions within its axioms, and inconsistencies
among its dynamical laws—resulting in the (in)famous quantum measurement
problem.

Against this background, the Primitive Ontology (PO) approach has been
advanced to overcome these issues. Today, in fact, we have several PO theories
that resolve the quantum puzzles and that can be considered serious
alternatives to the standard formulation of QM, as for instance Bohmian
Mechanics (BM), dynamical reduction models as the Ghirardi-Rimini-Weber (GRW)
theories, _etc._. Although the expression “primitive ontology” has been
introduced in Dürr et al., (1992), the roots of this programme can be found in
Bohm and Bell’s foundational works. Following their steps, the modern
formulation of the PO perspective requires that well-defined physical theories
explain the macroscopic regime through a more fundamental, microscopic
picture, postulating a clear primitive ontology—i.e. specifying a set of
theoretical entities representing real objects moving in physical space—and a
consistent dynamics. Moreover, elaborating on the notion of primitive
ontology, Esfeld and Deckert, (2017) extended and modified this concept
proposing an atomistic ontology that is taken to be valid at every
energy/length scale between the classical regime and the quantum field
theoretical level.

Bohm and Bell’s approaches share notable features with the modern PO
perspectives; for instance, they consider ontological clarity a necessary
condition to be met by every theoretical framework, promote scientific realism
also in the quantum domain and strengthen the explanatory power of quantum
theory.111I assume that the reader has some familiarity with this literature.
However, these positions remarkably diverge from one another, since they
employ different metaphysical principles leading to conflicting
_Weltanschaaungen_. The principal aim of this essay is to spell out the
relations and the main differences existing among such programs, which
unfortunately remain often unnoticed in literature. Indeed, it is not uncommon
to see Bell’s methodological approach conflated with the PO programme, and the
latter with Esfeld and Deckert’s views. It will be our task to clear up this
confusion.

The paper is organized as follows: section 2 analyzes Bohm’s reflections on
the nature of theoretical entities and physical laws, taking into account also
his metaphysical infinitism. Section 3 focuses on Bell’s theory of local
beables. In section 4 the modern formulation of the PO programme is presented
underlying the main metaphysical and methodological differences w.r.t. Bohm
and Bell’s approaches. Section 5 introduces Esfeld and Deckert’s minimalist
ontology and illustrates how this perspective diverges under important
respects with all the previous proposals. Finally, section 6 concludes the
paper.

##  2 David Bohm, Pluralism and Infinitism

Ontological clarity is an essential feature of physical theories according to
Bohm. On his view the problems of the standard interpretation of QM come from
its obscure metaphysical content. Bohm was deeply unsatisfied by the lack of
explanation of measurement results, since quantum theory excludes a precise
characterization of individual systems as well as an accurate description of
their dynamical evolution “without proving that such a renunciation is
necessary” (Bohm, (1952):168). Furthermore, he argued that the empirical
robustness of QM and its contingent mathematical structure are not sufficient
to exclude a priori other ontologically clearer formulations.

To tame these issues, Bohm joined the foundational debate proposing an
alternative interpretation of QM where quantum particles have definite
positions and velocities, and are dynamically guided by a real $\psi$-field,
challenging the metaphysical indeterminacy of the standard formulation. Hence,
postulating a dual ontology of particles and fields, he showed that an
individual description of quantum systems following continuous trajectories in
space was not only mathematically possible, but also physically consistent.
Moreover, this theory is able to explain measurement outcomes without invoking
mysterious collapses of $\psi$, strengthening the explanatory power of quantum
theory. In addition, Bohm extended his approach to electromagnetism employing
fields—different from $\psi$—as fundamental entities, showing that he was open
to implement various ontologies in different theories (cf. Appendix A of Bohm,
(1952)). The common trait between such theories is their metaphysical clarity,
since Bohm always specified which variables represent matter and how they
dynamically behave.

In addition, he claimed that QM—as any other theory—has a limited validity and
that “at distances of the order of 10-13cm or smaller and for times of the
order of this distance divided by the velocity of light or smaller, present
theories become so inadequate that it is generally believed that they are
probably not applicable” (Bohm, (1952), footnote 6). At these regimes he
expected that new ontologies and new theories will be discovered. More
generally, Bohm thought that physical theories have limited domains of
application at precise length/energy scales, and remarked that what is
measured at a particular level—thereby what exists at that scale—depends on
the theory at hand:

> our epistemology is determined to a large extent by the existing theory. It
> is therefore not wise to specify the possible forms of future theories in
> terms of purely epistemological limitations deduced from existing theories
> (Bohm, (1952):188).

In his view this is a direct objection against the empiricist basis of QM,
which follows the positivistic principle of not accepting entities that cannot
be currently observed. According to Bohm, this is a poor working hypothesis,
since the history of physics showed the fruitfulness to assume the existence
of certain items before their empirical discovery, and the atomic theory is a
pivotal example. On the other hand, he rejected another perspective, later
called “mechanistic philosophy”, for which reality can be fully explained
starting from a fixed set of entities, and a restricted set of laws—something
close to what philosophers call foundationalism.222In section 5 we will see
that Esfeld and Deckert, (2017) proposed a view similar to the mechanistic
philosophy described by Bohm. It should be noted, however, that according to
these authors physical laws can change in time in order to accomodate the
discovery of new phenomena. He warns us not to expect such a knowledge
“because there are almost certainly more elements in existence than we
possibly can be aware of at any particular stage of scientific development.
Any specified element, however, can in principle ultimately be discovered, but
never all of them” (Bohm, (1952):189). This is an hint of the metaphysical
infinitism endorsed in _Causality and Chance in Modern Physics_ to which we
now turn.

In Bohm, (1957) we find detailed objections against mechanistic philosophy.
Firstly, history of physics disconfirms the basic tenets of this view, since
the revolutions that occurred from Newton to this day radically changed the
entities and the laws of our theories. Moreover, future frameworks will be as
revolutionary as QM was w.r.t. classical physics. Secondly, the assumptions
concerning the final character of any particular ontology are neither
necessary, nor provable, because future theories may demonstrate its limited
validity.333Referring to this Bohm writes: “Newton’s laws of motion, regarded
as absolute and final for over two hundred years, were eventually found to
have a limited domain of validity, these limits having finally been expressed
with the aid of the quantum theory and the theory of relativity” (Bohm,
(1957):90). Finally, foundationalism contravenes the scientific method, since
the latter imposes that every object and law must be continuously subjected to
verification. This process of testing may end up in contradiction with new
discoveries or new domains of science. Looking at how physics evolved, says
Bohm, such contradictions not only systematically appeared, but also led to a
deeper comprehension of the world.

Contrary to such view, he proposes a form of infinitism. In essence, Bohm
stated that physical sciences and experimental data push us to a conception of
nature composed by an infinity of different entities, which do not depend
ontologically on a fixed set of absolutely fundamental objects (Bohm, (1957):
91). According to this view of science, physical theories do not always lead
us closer to a fundamental ground, but instead show the infinite complexity of
nature. Furthermore, he believes that empirical data cannot a priori provide
any justification to metaphysical restrictions concerning a particular set of
entities to be chosen as absolutely ontologically independent. On the
contrary, conforming to Bohm’s infinitism, scientific practice always
discloses new entities, laws and phenomena which contribute to our continuous,
never-ending process of understanding the structure of reality.

However, although Bohm denied the existence of a fundamental level, he firmly
believed that every theory must be ontologically unambiguous in its domain of
application. Therefore, theoretical frameworks must provide a clear ontology
to be applied at the relevant energy/length scale, meaning that the entities
forming the basic ontology of a given theory can be considered _relatively_
fundamental. Referring to this, he stated that

> [a]ny given set of qualities and properties of matter and categories of laws
> that are expressed in terms of these qualities and properties is in general
> applicable only within limited contexts, over limited ranges of conditions
> and to limited degrees of approximation, these limits being subject to
> better and better determination with the aid of further scientific research
> (Bohm, (1957):91).

Thus, Bohm claimed that a well-defined physical theory should provide a clear
ontological picture for the domain in which it is a reliable description of
physical phenomena. Nonetheless, its ontology can be substantially modified
with the progress of scientific research. This certainly exemplifies Bohm’s
scientific pluralism and his heterodox metaphysical views w.r.t. to the
dominant paradigm towards the interpretation of QM (cf. van Strien, (2019)).

As we can see, the seeds of the PO programme—i.e. the requirements of
ontological clarity and explanatory robustness—have been sowed. They will
germinate in Bell’s theory of local beables which we are going to discuss.

##  3 Local and Non-local Beables

It is well-known that Bell was disappointed by the lack of ontological clarity
in QM, which in his opinion was the source of its inconsistency and
redundancy. Inconsistency because the dynamical laws of the theory contradict
each other444These laws are the Schrödinger equation and the collapse
postulate.; redundancy because some central notions contained in its axioms,
e.g.  _observation_ and _observable_ , should be derived from more fundamental
concepts, the beables of the theory:

> [t]he concept of ‘observable’ lends itself to very precise _mathematics_
> when identified with ‘self-adjoint operator’. But physically, it is a rather
> woolly concept. It is not easy to identify precisely which physical
> processes are to be given the status of ‘observations’ and which are to be
> relegated to the limbo between one observation and another. So it could be
> hoped that some increase in precision might be possible by concentration on
> the _be_ ables, which can be described in ‘classical terms’, because they
> are there (Bell, (1987):52).

According to Bell, a well-defined physical theory $T$ must postulate a clear
ontology, or in his jargon, a set of local beables. These are the theoretical
entities of $T$ referring to real objects ascribed to bounded regions of
space-time, i.e. they correspond to the elements of reality that the theory
postulates to exist independently on any observation. It is worth noting that
the beables cannot be derived from other more fundamental notions of $T$;
hence, they establish what is fundamental _in the context of T_.

Moreover, together with dynamical laws555Remarkably, according to Bell in the
context of a given theory $T$ there is a distinction between physical and non-
physical entities: the former denote the beables of the theory, the latter are
the mathematical structures needed to formulate dynamical laws for the
variables representing matter in space; cf. Bell, (1987), Chapter 7., the
beables explain the physical phenomena lying within the domain of application
of the theory under consideration, connecting its formal structure to the
macroscopic world accessible to us:

> [t]he beables must include the settings of switches and knobs on
> experimental equipment, the currents in coils, and the reading of
> instruments. ‘Observables’ must be _made_ , somehow, out of beables. The
> theory of local beables should contain, and give precise physical meaning
> to, the algebra of local observables (_ibid_.).

But how do we select a particular set of beables for a theory? In answering
this question Bell’s pluralist attitude becomes manifest, since he claimed
that the only necessary requirements for an ontology is to state explicitly
which variables represent matter, and to provide unambiguous explanation of
macroscopic observations.666To this regard Bell writes that “what is essential
is to be able to define the positions of things, including the positions of
pointers or (the modern equivalent) of ink of computer output” (Bell,
(1987):175). Cf. also Bell, (1987), Chapter 5.

These liberal requirements do not impose any strict limitation to the
selection of a certain ontology. Such a freedom of choice is justified because
(i) there are always several adequate options to account for the macroscopic
data that a certain theory must explain, (ii) physical theories have a
provisional character. Therefore, according to Bell, there is no one-to-one
relation between the beables of a theory and the physical world, meaning that
we cannot definitively establish whether a certain ontology is the correct,
ultimate description of reality. To this regard, Bell used “the term ‘beable’
rather than some more committed term like ‘being’ or ‘beer’ to recall the
essentially tentative nature of any physical theory. Such a theory is at best
a _candidate_ for the description of nature. Terms like ‘being’, ‘beer’,
‘existent’, etc., would seem to me lacking in humility. In fact ‘beable’ is
short for ‘maybe-able’ ” (Bell, (1987):174).

Indeed, there are various examples of beables implemented in different
classical and quantum theories, as for instance electromagnetic fields, point
particles, matter density fields, strings _etc._.777Maudlin, (2016) underlines
that a similar choice is available also for the specification of the space-
time structure of a given theory. Bell himself, although he was a supporter of
the pilot-wave theory (Bell, (1987), Chapter 17), proposed a rival picture for
non-relativistic QM, namely a GRW theory with a flash ontology (_ibid._ ,
Chapter 22). Moreover, he extended Bohm’s approach to Quantum Field Theory
(QFT) implementing an ontology of fermion number density (_ibid._ , Chapter
19). Such beables have no classical analogues, implying an ontological
discontinuity between the classical and the quantum regime; however, such a
discontinuity was tolerated by him. Similarly, Bell’s approach allows that a
single theory can postulate the existence of different kinds of objects, i.e.
it can provide a multi-category ontology, as for instance in classical
electromagnetism, where we find both point particles and fields. Thus, the
theory of local beables does not require that reality must be reduced to a
unique category of physical entities. These facts certainly denote his liberal
views about the ontology of physical theories, in close analogy with Bohm’s
pluralist perspective. Finally, it is interesting to note the Bell admitted
the existence of _non-local_ beables.888Cf. Bell, (1987):53. This fact shows
that according to him, the general category of what exists is divided into two
subsets: local and non-local beables. The former must be defined in
3-dimensional space and associated with bounded portion of spacetime; while
the latter can be either defined in high-dimensional spaces or in the usual 3D
space. For instance, he considered the state vector defined in configuration
space as a proper beable in his essay _Beables for quantum field theory_.
Another important example is given by the multi-field interpretation of the
$\psi$ field in Bohm’s theory. According to this approach, given a
$N$-particle system one projects the $\psi$ values defined on configuration
space into multi-field values in 3-dimensional space, i.e. one associates a
particular field value not with individual points (as one would do with usual
fields), but with $N$-tuples of points. In this manner, the multi-filed
determines the motion of a configuration of particles in physical space. This
new object is a beable because it is considered a real physical item living in
3-dimensional space, and it is non-local since its value is specified for a
configuration of $N$ points and not for a single point.999For details on this
particular perspective cf. Hubert and Romano, (2018). Clearly, such type of
beables have been admitted also by Bohm himself since he explicitly considered
the wave function a real physical field, as said in the previous section.
Another recent example of non-local beables can be found in Smolin, (2015).

In sum, from our discussion we can say that Bohm and Bell’s101010It is worth
stressing that Bell did not explicitly endorsed a metaphysical infinitism as
Bohm did. Nonetheless, the similarities between their views are evident.
reflections concerning the ontology of physical theories led to the following
ideas:

  * •

the beables of a theory $T$ are fundamental in the context of the framework
under consideration, i.e. they may be non-fundamental in another, deeper
theory;

  * •

the beables must explain all the physical phenomena lying within the domain of
application of $T$;

  * •

several ontologies can be proposed to explain the same set of physical
phenomena;

  * •

it is possible to implement different beables within the same theoretical
framework postulating a multi-category ontology;

  * •

beables can be non-local and defined either in 3D space or in higher-
dimensional spaces;

  * •

finally, it is possible to implement different beables at different
energy/length scales (scientific pluralism).

Since the PO programme evolved from the work of these physicists, let us then
see what has been kept and what has been left behind.

##  4 The Primitive Ontology Approach

The PO approach is a normative perspective about the construction of physical
theories since it provides a set of requirements that any theoretical
framework should met to be considered empirically and metaphysically
consistent. Indeed, according to the proponents of this view, well-defined
theories share a common architecture deriving from these
constraints.111111Here I follow the modern presentation of this perspective
contained in Allori, (2013, 2015). NB: other proponents of the PO approach as
for instance R. Tumulka, S. Goldstein or N. Zanghì may have different opinions
concerning this programme and its main goals. For details about the common
structure of PO theories see Allori et al., (2008).

Following Bell121212For the sake of conceptual accuracy, it should be stressed
that the PO approach not only is rooted in Bohm and Bell’s works, but it is
also inspired by Einstein, de Broglie and Schrödinger’s reflections on and
critiques to the ontology of QM. These physicists, indeed, explicitly rejected
the instrumental philosophy of standard quantum theory and tried to provide an
unambiguous metaphysical picture for this theoretical framework. Given the
space limit, here I cannot elaborate of their influence on the PO approach.
However, proponents of this perspective in several occasions—as for instance
Nino Zanghì in many conversations and Valia Allori (private
communication)—emphasize their influence., also in this programme the
mathematical structure of a theory $T$ is divided into two subcategories.
Firstly, there are “primitive” variables provided with a physical meaning;
they represent matter and refer to real objects precisely localized and moving
in 3-dimensional space (or in space-time), which is generally considered a
real substance as well. These are the PO of the theory, the fundamental
entities postulated by $T$ which constitute the building blocks of macroscopic
reality. Remarkably, the PO defines the observable quantities of $T$—i.e. the
properties of physical systems—and its symmetries, since it establishes which
entities remain invariant under symmetry transformations.131313For technical
details see Allori et al., (2008). More on this later. On the other hand, $T$
contains mathematical structures that are responsible for the dynamical
evolution of the primitive ontology, without representing material objects—the
non-primitive (or nomological) variables (cf. footnote 5). An example is given
by the wave function in QM: in every PO theory $\psi$ is not considered a
physical substance, but rather an essential mathematical tool useful to
formulate empirically adequate laws for the dynamical history of the primitive
ontology, similarly to the parameters of mass and charge.141414Arguments
against wave function realism can be found in Allori, (2013, 2015, 2021).

Furthermore, the PO has explanatory power since the macroscopic reality is
metaphysically dependent on the primitive ontology of particular theories and
its dynamical laws. This dependence relation can be intuitively characterized
as mereological composition: the entities of a PO interact and form nuclei and
atoms, which in turn aggregate into molecules and more complex biological
organisms, arriving eventually to the macroscopic realm.151515The exact rules
of mereological composition as well as the dependence relation based on them
have not yet been investigated in detail in the context of PO. This will be
the subject for future research. For instance, according to BM, GRWm or GRWf,
the macroscopic regime is literally composed by particles, matter density
fields or flashes respectively, meaning that such PO theories explain the
existence of macroscopic objects in terms of the motion and interaction in
space of their different mereological primitives. Thus, the properties and the
behavior of macroscopic objects depend on—i.e. are reduced to—the properties
and interactions of their fundamental constituents. Clear examples of these
explanations are macroscopic measurement outcomes. For instance, in BM
measurements of spin are explained via—and reduced to—particles’ positions and
their dynamical evolution as shown in Bell, (1987) Chapter 17 and Dürr et al.,
2004b . This example is generalizable (i) to every other PO theory and (ii) to
every observable, since these frameworks provide rigorous descriptions of the
physical processes taking place in measurements situations, and then also
detailed explanations of the obtained results.161616For technical details cf.
Dürr et al., 2004b  and Goldstein et al., (2012). All these are remarkable
virtues, because observation constitutes the only connection between theory
and experience.

To conclude this brief presentation, it is worth stressing that the PO
approach puts very few constraints on the selection of the primitive
variables: the latter must be microscopic, since they must explain the
macroscopic regime, and located in 3-dimensional space.171717The reader may
refer to Allori, (2015) Sections 4 and 5 for arguments in favor of the
requirement of “microscopicality” and 3-dimensionality. Referring to this,
indeed, Allori states that

> there is no rule to determine the primitive ontology of a theory. Rather, it
> is just a matter of understanding how the theory was introduced, it has
> developed, and how its explanatory scheme works (Allori, (2013):65).

The definition of the primitive variables of a given theoretical framework
depends on the metaphysical assumptions used to construct it, and may vary
from a theory to another, as witnessed by the several PO theories available.
However, the primitive ontology is never chosen _a posteriori_ , i.e. it is
not read off from the mathematics of a theory. Rather, the PO itself provides
an interpretation of the formalism: in the process of theory construction, a
scientist selects a particular PO and will use the appropriate mathematical
structures to implement her choice.

Let us now turn to the differences between the PO programme w.r.t. the
previous approaches. To this regard, Allori, (2013, 2015, 2021) stress two
related crucial points181818To my knowledge Allori is the first that pointed
clearly out the differences between the PO and local beables. In literature,
however, such differences are often neglected and these approaches are
frequently conflated. See for instance Esfeld, (2014) or Tumulka, (2016).:
firstly, not every local beable is necessarily a PO. For instance,
electromagnetic fields in classical electrodynamics can be interpreted as
_nomological_ variables instead of primitive ones given their role in the
theory: not only there is asymmetry between particles and fields—i.e. the
particles can generate fields, but not vice versa—but also the latter act as
mediators of particles’ interactions. These facts provide indications that
fields are not fundamental entities. Consequently, E and H are not part of the
PO, i.e. they do not represent matter in space, but describe how particles
move. Thus, they are part of the explanatory machinery of the theory. Hence,
some local beables can be nomological, while PO must be exclusively material.

Secondly, if every local beable of a given theory $T$ is considered part of
the PO, then $T$ may lose symmetry properties. A simple example is provided by
the symmetry of time reversal in electromagnetism: if E and H would be real
physical entities, one would expect that, under time-reversal, they would
still represent possible ways in which fields may be. However, the
transformations $\textbf{E}(t)\rightarrow\textbf{E}(-t)$ and
$\textbf{H}(t)\rightarrow\textbf{H}(-t)$ are not solutions of the Maxwell’s
equations. On the contrary, particles’ trajectories are invariant under time
reversal. Thus, if E and H are considered part of the PO, then classical
electromagnetism would lose the symmetry of time-reversal. Hence, Allori
concludes the PO preserves the symmetries of the theory, while sometimes local
beables do not.191919This is another elegant argument against the reality of
fields. More details are given in Allori, (2021) Sections 4.2 and 4.3.

In addition, one can also find other important differences between these
perspectives. Firstly, a given PO theory $T$ defines a set of entities which
are considered fundamental _tout court_ , since they are considered the
building blocks of nature. According to the PO perspective, indeed, our
macroscopic realm is completely ontologically reduced to the primitive
variables of $T$ and its dynamical laws, which are considered the fundamental
elements of reality.202020To this regard, Allori writes that in the PO
approach “macroscopic objects are thought to be fundamentally composed of the
microscopic entities the PO specifies. As such, the PO approach is
(ontologically) reductionist, at least to the extent that it allows to make
sense of claims like the PO being “the building blocks of everything else”,
and of the idea that macroscopic regularities are obtained entirely from the
microscopic trajectories of the PO” (Allori, (2018):71-72). On the contrary,
Bohm stated that a particular ontology should be valid and reliable only
within the domain of application of a given theory. Similarly, Bell stressed
that we may have two different sets of local beables at different
energy/length scales, implying that a certain theory $T$ postulate an ontology
which is _theory dependent_ and only relatively fundamental.

Secondly, the PO approach does not admit different ontologies at diverse
scales.212121If BM would be the correct description of reality at a
fundamental level, we would have ontological continuity between the classical
and the quantum scale. Referring to this, not only Allori is inclined to
postulate particles as the fundamental PO, but also she argues that the
ontological discontinuity between classical mechanics and GRW theories is an
argument against the latter (cf. Allori, (2018)). Clearly, I cannot speak for
other proponents of this programme. Indeed, if either GRWm or GRWf were the
correct description of the world, the non-fundamental particle-like character
of classical mechanics would emerge and explained in terms of the fundamental
GRW PO. However, since we do not have direct access to the fine-grained nature
of the primitive variables, we can consider the flashes/matter fields as if
they were particles for explanatory purposes, recovering the appearance of
classical corpuscles:

> [a]t the level of microphysics we may have flashes or a continuous field,
> but at some mesoscopic level they produce trajectories as if they are
> produced by particles. So, even if the microscopic PO is not one of
> particles, there is a mesoscopic scale in which they behave as _if_ they are
> in the sense that from that level up to the macroscopic level the
> explanation is the same as if they were particles (Allori, (2018): 73).

In this case, then, we would know that at the fundamental level the PO is not
corpuscular, even though we would keep talking of particles for explanatory
aims. Clearly, Bohm and Bell pluralist attitude seems to be lost in favor of a
foundationalist perspective. Consequently, infinitism is abandoned as well.

Thirdly, contrary to the multiple-category ontologies endorsed by Bohm and
Bell, PO theories explain our manifest image of the world postulating
exclusively a one-category ontology, i.e. the macroscopic regime is reduced to
a single type of physical objects and their evolution. However, this reduction
may be _non-eliminative_ : for instance, as state above, in the case a GWR-
type theory would postulate the correct PO, particles would be stil considered
an effective description of reality from a mesoscopic to a macroscopic scale.
Thus, for all explanatory purposes a classical particle ontology can be
maintained.

Finally, as we have seen in the previous sections, Bohm and Bell conceived the
existence of non-local beables, whereas in the PO perspective these cannot be
considered material entities. Thus, they cannot be part of the primitive
ontology of a theory, and must be included within its nomological variables.
In sum, if for Bell and Bohm what physically exists can be defined in terms of
local and/or non-local beables, the PO can only be a subset of the local
beables. Thus, the PO programme imposes stricter criteria to determine the
fundamental elements of a theory w.r.t. the preceding approaches.

In this section we saw that the PO approach diverges from the views of its
pioneers, since it employs different metaphysical assumptions and
methodological principles that lead to diverse pictures of reality. Thus, we
should be aware of these dissimilarities to not conflate these approaches. To
conclude the essay, let us now consider the most recent development of the PO
programme.

##  5 Beyond the PO: Minimalism and Fundamental Ontology

Starting from the modern formulation of the PO programme, Esfeld and Deckert
radicalized the notion of primitive ontology eliminating the theoretical
dependence of the PO: “to our mind, it is inappropriate to speak of the
ontology of this or that physical theory. Ontology is about what there is”
(Esfeld and Deckert, (2017):12-13). Hence, they move from the notion of
“primitive ontology of a theory $T$” to ontology _tout court_ , developing
their metaphysical proposal independently of any particular theoretical
framework.

To achieve this result, the authors introduce a background independent
atomistic ontology which seeks to recover the predictions of classical
mechanics, QM and QFT via the definition of (i) different dynamical laws to be
applied at the relevant energy/length scales, and (ii) appropriate typicality
measures.222222For technical and metaphysical details of this proposal see
Esfeld and Deckert, (2017), Chapter 2. More specifically, they formulate a
relationalist ontology of propertyless, finite and permanent _matter points_
uniquely individuated by their mutual distance relations—the only property of
these points is position. The change of these relations, which is taken as a
primitive fact, constitutes the dynamical aspect of the proposal. Referring to
this, the authors suggest a general dynamics of matter points in terms of the
first derivative of a given configuration $\Delta$ with respect to time $t$:

$\displaystyle
v_{t}(\Delta_{t})=\frac{d}{dt}\Delta_{t},\forall{t}\in\mathbb{R}.$

Clearly, to describe actual modifications of distance relations at various
length scales, one has to provide the correct laws of motion. Here parameters
as masses, charges, fields, potentials _etc._ enter into the scene: they are
useful mathematical tools used to implement the evolution of matter points,
consequently they are not part of the PO but have an essential explanatory
function. An explicit example is provided by the velocity field appearing in
the guidance equation of BM (cf. Esfeld and Deckert, (2017), equation 3.6), as
well as those of classical mechanics and QFT (cf. equations 3.3 and 4.4
respectively).

Indeed, following this metaphysical project, it is always the dynamics that
changes, while the ontology of matter points remains unaltered in every theory
change from the classical regime to QFT—and possibly beyond it in more
fundamental theories as e.g. quantum gravity. Such an ontology, thus, is
primitive in a new sense: it is the fundamental ontology of our physical
world. These matter points are (i) ontologically independent from any other
entity, and (ii) constitute the mereological basis of every object in the
universe at every scale. Hence, contrary to Bohm and Bell approaches, Esfeld
and Deckert’s project excludes the possibility to have different ontologies at
different energy/length scales. Here the PO is chosen _a priori_ in virtue of
the guiding metaphysical principle of ontological parsimony. This ontology, in
fact, is the simplest possibility to explain our macroscopic reality providing
the minimal ingredients to recover spacetime and matter in motion (once the
correct dynamical equations are given).232323Esfeld and Deckert argue that
parsimony has been a guiding principle in both philosophy and physics in their
historical evolution. Indeed, given a class of theories explaining the very
same set of physical phenomena, we prefer the one(s) able to explain them with
the lowest number of entities and laws.

This proposal captures the idea of fundamentality expressed in terms of _well-
foundedness_ (cf. Tahko, (2018)): a dependence chain is said to be well-
founded if and only if it terminates with entities which do not ontologically
depend on other items. In the case of Esfeld and Deckert’s project, matter
points close the ontological dependence chain, and constitute the ultimate
ingredients of reality. Well-foundedness, in turn, is closely related to
metaphysical foundationalism, which can be defined in full generality by
saying that every non-fundamental entity is dependent on some fundamental
item—in this case matter points—that fully account for its being and reality.
Hence, Esfeld and Deckert’s project expresses the idea that reality has an
ultimate foundation of matter points to which everything else is ontologically
reduced—the notion of a relative fundamentality as conceived by Bohm and
Bell’s is no longer maintained.242424It is obvious that this project rejects
Bohm’s infinitism and his objections against a foundational metaphysics.

Let me conclude this section underlying other important differences w.r.t. the
PO programme. Firstly, according to Esfeld and Deckert, space-time and its
geometry emerge from the configuration of matter points, contrary to the
substantivalist perspective of the PO approach, where the primitive ontology
of a theory is interpreted as a decoration of space.252525This expression is
used explicitly in Allori et al., (2008). Secondly, while the PO perspective
allows the postulation of several primitive ontologies, these authors accept
the existence only of particles since in their opinion (i) almost all the
evidence coming from the most advanced experimental research is given in
corpuscular terms, and (ii) atomism can be retained not only at the classical
level, but also in QM and QFT in virtue of the existence of BM and Bohmian
QFTs. Thus, they exclude every non-particle primitive ontology. Interestingly,
if the PO is a subset of the local beables, then such an atomistic ontology is
a subset of the possible POs; hence, Esfeld and Deckert’s project imposes
stricter criteria to determine what is real w.r.t. the PO approach. Thirdly,
these authors postulate that the number of matter points remain constant in
time. This is an important difference w.r.t. the PO programme where theories
with a variable number of entities have been given. For instance, the number
of particles may not be constant in time, as we can see in particular Bohmian
QFTs. This fact leads to significant differences when QFT and its
phenomenology is taken into account.262626In Dürr et al., 2004a  it is
postulated an ontology with a variable particle number. For a discussion of
the various Bohmian QFTs with a particle ontology cf. Oldofredi, (2018).

##  6 Conclusion

Starting from Bohm’s infinitism and Bell’s theory of local beables, we arrived
to the PO programme and its most recent development, i.e. Esfeld and Deckert’s
minimalist ontology. Although the initial steps of this journey are in open
contradiction w.r.t. the latest works proposed within the PO community, there
is still some confusion in literature, where it is not uncommon to conflate
Bell’s methodological approach with the PO programme, and the latter with
Esfeld and Deckert’s views. Thus, in this essay I underlined the main
differences among these projects.

In particular, we saw that Bell’s methodological approach shows important
similarities with Bohm’s ideas, sharing the belief about the provisional
character of physical theories and a pluralist attitude towards their
ontology. Then the PO programme has been presented emphasizing some important
differences w.r.t. Bell’s and Bohm’s views. Elaborating on Allori’s works, I
have pointed out new divergences between these two schools of thought.
Finally, I considered Esfeld and Deckert’s minimalist ontology, showing (i)
the radical difference of this project w.r.t. Bohm and Bell’s views, and (ii)
its subtle dissimilarities with the tenets of the PO programme. In conclusion,
I hope to have clarified at least some of the differences existing among such
views, and to have shown to the reader that these approaches employ divergent
metaphysical and methodological principles leading to contrasting pictures of
reality. Acknowledgements: I warmly thank the Editor of this volume, Valia
Allori, for her kind invitation to contribute to this book. I would like to
thank Olga Sarno for helpful comments on previous versions of this manuscript
and the reviewers for their positive and constructive feedback. This work is
financially supported by the Swiss National Science Foundation (Grant No.
105212-175971).

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