## Abstract

Symmetries in physical systems are defined in terms of conserved Noether Currents of the associated Lagrangian. In electrodynamic systems, global symmetry is defined through conservation of charges, which is reflected in gauge symmetry; however, loss of charges from a radiating system can be interpreted as localized loss of the Noether current which implies that electrodynamic symmetry has been locally broken. Thus, we propose that global symmetries and localized symmetry breaking are interwoven into the framework of Maxwell's equations which appear as globally conserved and locally non-conserved charges in an electrodynamic system and define the geometric topology of the electromagnetic field. We apply the ideas in the context of explaining radiation from dielectric materials with low physical dimensions. We also briefly look at the nature of reversibility in electromagnetic wave generation which was initially proposed by Planck, but opposed by Einstein and in recent years by Zoh.

This article is part of the theme issue ‘Celebrating 125 years of Oliver Heaviside's ‘Electromagnetic Theory’’.

## 1. Introduction

Spontaneous symmetry breaking where the ground-state breaks the system's symmetry plays the main role in generation of cooper pairs in superconductivity [1–3], Goldstone Bosons [4,5], acquisition of mass through Higgs mechanism in the Standard Model [6–9] and generation of phonons in crystals [10]. In a related phenomenon called explicit symmetry breaking, the defining equations of motion represented by the Lagrangian break the symmetry [10]. Initial work in the field of symmetry in physical systems was done by Curie who proposed that in any physical phenomenon, the element of symmetry of its cause is also present in the resulting effect [11]. He further argued that the asymmetry associated with an effect is also present in the causes. Symmetries and the associated conserved physical quantities defined by the Noether Theorem play a central role in physics [12]. For example, temporal symmetry of the Lagrangian of a system is associated with energy conservation [13]. Similarly, symmetry of the electromagnetic field is expressed through gauge invariance [14].

In the current article, we propose that transformations of force fields from one form to the other are linked with loss of invariance of at least one physical quantity resulting in a localized symmetry breaking of the associated field in the localized region of space and time which generates a Noether current which is globally conserved. In that context, we argue that loss of invariance in the temporal domain of a force field causes loss of invariance in the spatial domain of a corresponding physical quantity and vice-versa. For example, potential energy of a body is transformed into change in momentum when the symmetry of the potential energy surface is locally broken by some external agent. These observations converge towards our novel discovery of the fact that in Maxwell's equations on radiation, the globally conserved Noether current is generated as a consequence of localized symmetry breaking of an electromagnetic field. At a global level, it plays a critical role in defining the global symmetries expressed through gauge invariance and generation of electromagnetic radiation.

According to Maxwell, radiation is generated due to acceleration of electrons [15,16]. This aspect was further highlighted and put on a simpler mathematical framework by Heaviside [17,18]. An important contribution of Heaviside was reformulation of the initial Maxwell's equations in quaternion form into a vector form comprising of four equations in terms of electric and magnetic fields and charges (static and dynamic) [19].

There are some physical instances where electron acceleration does not result in radiation, in apparent violation of the Maxwell's theoretical prediction. For example, an infinitely long wire placed at an infinitesimal distance, from a conductive ground plane, excited by a time varying voltage source, is a classical example where electrons are accelerated but the overall radiation field is cancelled due to opposing currents induced in the ground plane. A closely spaced transmission line, excited by a time varying voltage source, is another such example, where there is no measurable level of radiation despite electron acceleration as the charges flowing in the wires are in opposite directions which cancels the radiation field.

An important consequence of Maxwell's work is that charges at uniform velocities do not emit electromagnetic radiation. However, there are physical conditions, where electromagnetic radiation is emitted from charges moving at constant velocities. For example, a charged particle moving through a dielectric material at a uniform velocity leads to polarization and radiation. Similarly, in Cherenkov radiation, electrons moving at superluminal velocities in a medium generate electromagnetic radiation [20–22].

Maxwell and Heaviside's work on radiation was put on a stronger physical foundation by J.J. Thomson, whose perspective on radiation was related to the topological changes in the nature of electric lines of field [23]. He argued that acceleration changes the geometric pattern of field lines. The prior theoretical work on radiation were validated through the experimental work by Hertz [24,25], who proved the existence of electromagnetic waves. The other pioneers in the field of early work on wireless communication were Bose [26], Tesla [27] and Marconi [28], who worked on the empirical aspects of generation, transmission and reception of electromagnetic waves. One of the key findings by Marconi was the fact that radiation efficiency significantly increases when a metallic pole, being used as a transmitting antenna, is held vertically over a horizontal plane [29]. The current work aims to unify the diverse aspects of radiation explored along theoretical and empirical dimensions through a common thread based on broken symmetries of force fields.

## 2. Symmetry in electrostatics

The Lagrangian of an electrodynamic system is represented by electric and magnetic fields [30], hence, the geometric patterns of lines of electromagnetic fields can also be used to study the associated symmetries in the localized region of the given electrodynamic system. For example, in electrostatics, for a static point charge, symmetry is observed in electric field lines which are directed away from the centre of the charge and possess rotational symmetry (figure 1*a*) [31,32]. The Lagrangian density of a static point charge *Q* within a sphere of radius *R* is expressed by
2.1where *ϵ* is the permittivity of the medium, and *E* is the electric field. Here, d*L*_{d}/d*t* and d*L*_{d}/d*θ* are equal to zero where, *t* is time and *θ* is angular displacement from the centre of the charge, which indicates symmetry of the Lagrangain. The symmetry of a dynamic system is associated with a conserved quantity, which is charge in this case. In electrostatics, the net value of charge does not change with time. The symmetry is also expressed through Gauss's Law
2.2where ** D** is electric flux density and

*ρ*is the total charge density enclosed by the given Gaussian surface. It could be argued that for a system of more than one charge, the geometric symmetry of the electric lines of fields is broken as shown in figure 1

*b*, however, the symmetry defined by equation (2.2) is maintained around a Gaussian surface

*S*, enclosing the charges, though

**and**

*D**ρ*vary over the surface such that , where

*Q*is the total charge enclosed by the surface. In addition to this, the temporal variation of the Lagrangian is zero, which also indicates symmetry.

Symmetries in electromagnetism are integral parts of the current scientific literature, however, an interpretation of the physics of the system in terms of geometric symmetry of the electric lines of field and conservation of the associated charges at local and global level can establish the condition of generation of radiation or its absence from an electrodynamic system. This offers a novel perspective which supplements the theoretical prediction of Maxwell, according to which charges at rest and at uniform velocities do not radiate and charges under acceleration result in radiation. For example, in the context of charging of a parallel plate capacitor, with infinitely large plates with an infinitesimal separation, the conservation of charges defined by equation (2.2) remains valid at a local as well as a global level besides the symmetry of the electric field, which confirms the absence of radiation. A charge moving at a uniform velocity in a dielectric material cannot be expressed in terms of the symmetry of the electric field as defined by equation (2.2); besides this, its Lagrangian density does not remain invariant along temporal and spatial dimensions, so, we can associate a radiation field with it. Thus, our symmetry based interpretation offers a novel dimension to our current understanding of electromagnetic phenomenon while complementing the existing understanding based on the foundations of Maxwell, Heaviside and Thomson.

## 3. Localized symmetry breaking in electromagnetism

The relationship between symmetries and conservation was discovered by Emmy Noether [12], which is widely referred to as the Noether theorem. It defines a relationship between global symmetries and globally conserved currents in a physical system. However, the Noether theorem is largely silent about localized symmetry breaking and the role of non-conserved currents, within the localized region of space and time, which play an important role in defining global symmetries of a physical system. Its connection to Maxwell's equations and role in radiation can lead to a deeper and refined understanding of electrodynamic systems.

A mass kept at the highest point of a gravitational potential surface as shown in figure 2*a* has a finite value of potential energy. At the highest point, the potential energy surface has symmetry, i.e. the Lagrangian has spatial symmetry, , along a spatial dimension *r* [30]. The Noether current in this context is momentum, which is generated only when . In the context of the ball shown in figure 2*b*, there is a net force due to non-zero value of the potential gradient. In other words, the localized symmetry breaking of the Lagrangian generates a change in momentum of force expressed through the Euler Lagrange equation
3.1where is rate of change of position of the particle along the dimension *r*. The change in momentum with respect to time gives a sense of symmetry breaking of momentum ** p** along temporal dimension expressed in terms of non-zero value of d

**/d**

*p**t*. The discussion converges towards the point that localized symmetry breaking of one physical quantity along spatial dimension breaks the symmetry of a related physical quantity along the temporal dimension. We extend the concept to electromagnetic force fields, where it leads to the condition of generation of radiation fields under broken symmetries.

An important contribution in the case of electric lines of force of a momentarily accelerated electron was done by J.J. Thomson, who argued that the electric lines of field of an accelerated charge undergo a distortion as illustrated in figure 3*a* [33]. Although Thomson's picture was highly intuitive and led to a better understanding of the nature of geometric lines of force associated with charges under radiation, for a long time there was no effort to develop a mathematical model corresponding to distortions in the electric lines of field from the framework of broken symmetries. Interpretation of the distortions in electric lines of force associated with an accelerated charge as a consequence of breaking of the translational symmetry of the electric field in the radial direction within a localized region of space and time due to rotation of the electric field, which results in generation of electromagnetic waves as shown in figure 3*b*, was published recently [34,35]. Thus, although a periodic acceleration of the dipole configuration of charged particles results in electromagnetic radiation which is associated with generation of rotating electric field in space is well known, its correlation to loss of temporal and spatial invariance of the associated Lagrangian has remained unnoticed, for long. In fact, it is an instance of explicit symmetry breaking, where the defining equations of motion represented by the Lagrangian break the symmetry [36].

The fact that translational symmetry of the electric field is explicitly broken, resulting in the generation of time varying magnetic field and radiation, is expressed by Maxwell's equation
3.2where ** E** is electric field intensity and

**is magnetic flux density.**

*B*The above arguments also apply in the case of radiation from a dipole antenna connected to a parallel two wire transmission line excited by a time varying voltage source (figure 3*c*). If the parallel section of the transmission line is assumed to be infinitely long with an infinitesimally small separation, the electric field has translational symmetry along the transmission line which is broken along the flared end where rotational electric field is generated, resulting in radiation. Here, it is assumed that the currents not aligned to the axis of the transmission line at the supply end are shielded. This, in essence, assumes the complete absence of common mode current components in the parallel section of the wire. These aspects have also been highlighted by Ianconescu & Vulfin, who show that in the case of an infinite transmission line, there is no far field radiation, but there is a non-radiative field which decays as inverse square of distance from the axis of the wire as in the case of an electrostatic field [37], rather than inverse of distance as associated with radiation) [35]. However, a two wire transmission line radiates when one takes into account the currents which do not lie in the axis of the transmission line. For example, current flowing at the supply end and low end of the transmission line leads to a loop antenna like effect [37].

The concept of explicit symmetry breaking has also been applied in experiments on generation of tunable broadband radiation from a superconducting ring where a pulse of laser light breaks the symmetry of cooper pairs, resulting in electromagnetic radiation [38]. Under explicit symmetry breaking, the momentum current associated with the test charge loses its conserved value within a localized region of space and time as the corresponding Noether current is the driver of electromagnetic radiation.

An important goal of the current article is to underscore the fact that localized symmetry breaking is expressed within the framework of Maxwell's equation on radiation, and it denotes a close relationship between dependence of the force fields along temporal and spatial dimensions. The rotation of the electric field expressed by curl of the vector ** E** indicates that the symmetry of the electric field along a set of dimensions in the physical space has been broken. Thus, the equation essentially implies that broken symmetry of magnetic flux lines in the temporal dimension results in symmetry breaking along a set of spatial dimensions. In a curved space around a mass, the rotation of the electric lines of force would show an additional curvature. For a uniform plane wave propagating along the

*z*-axis, with the electric and magnetic field lines aligned along

*x*- and

*y*-axes respectively, equation (3.2) takes the form [31] 3.3Here, the finite value of rate of change of magnetic flux density can be interpreted as broken symmetry of the magnetic flux lines along the temporal dimension. Here d

*E*

_{x}/d

*B*

_{y}is also a measure of the degree of curvature of space around a mass. If the space is flat, the fields are orthogonal to each other in the context of a plane polarized electromagnetic wave. However, when the space is curved, the orthogonality is lost. In metamaterials, by artificially engineering a given material, the fields are changed in order to change the existing symmetry [39].

Equation (3.2) can be transformed into a line integral using Stoke's theorem resulting in the expression , where *ϕ* is the flux density. The finite value of d*ϕ*/d*t* as the magnetic flux lines vary can be interpreted as localized symmetry breaking of the magnetic flux lines along the temporal dimension. Similarly, the electric field intensity ** E** due to a charge

*q*in free space can also be interpreted as localized symmetry breaking of the electric potential

*V*along the spatial dimension.

The time varying electric flux density under the rotation of magnetic field intensity is expressed by
3.4where ** H** is magnetic field intensity and

**is electric flux density. Equation (3.4) takes the form of 3.5for the uniform plane wave, and it indicates its symmetry breaking of electric flux density along the dimension of time which, in turn, drives symmetry breaking of magnetic field intensity along the spatial dimension. The temporal and spatial dependence of force fields during their transformation and generation of locally non-conserved, but globally conserved currents, under symmetry breaking, along with their role in radiation offers a new degree of freedom in understanding electrodynamic systems.**

*D*## 4. Charge in an electromagnetic field

According to the symmetry principle of electromagnetism, an electron in a simple harmonic motion generates a plane polarized electromagnetic wave; similarly, a plane polarized electromagnetic wave generates a simple harmonic motion in an electron [32]. Here, we argue that this aspect of symmetry in electromagnetism partially breaks down as the rotational component of electric field of electromagnetic wave excites the angular momentum of the electron which generates circularly polarized radiation under acceleration. As force ** F** on a charged particle with a charge

*q*in an electric field

**is given by**

*E**q*

**, we can multiply both sides of equation (3.2) by**

*E**q*to get the following expression: 4.1The left-hand side of the equation (4.1) can be transformed into a line integral using Stoke's theorem , where

*U*is the electric potential energy. Its derivative along the dimension

*r*leads to,

**= ∂**

*F**U*/∂

*r*or ∂

*V*/∂

*r*, where

*V*is the electric potential. The force exerted by the electric field

**on another charge**

*E**q*generates a momentum

**which can be expressed as**

*p***=**

*F**q*

**= d**

*E***/d**

*P**t*. The expression defined by equation (4.1) expresses the fact that a charged particle in an electromagnetic field shows rotational motion and the radiation generated by it is not linearly but circularly polarized.

The equations discussed earlier can also be expressed using the electromagnetic field tensor [32], *F*_{μν} = ∂*A*_{ν}/∂*x*_{μ} − ∂*A*_{μ}/∂*x*_{ν} where ** A** is magnetic vector potential,

*x*is a dimension in space,

*μ*and

*ν*are subscripts denoting spatial components. The symmetry breaking of the electromagnetic field tensor can be expressed as ∂

*F*_{μν}/∂

*x*=

*J*_{μ}, where

*J*_{μ}is the four current vector. The finite value of spatial variation of ∂

*F*_{μν}indicates its symmetry breaking along the spatial dimension which results in corresponding breaking in symmetry along the temporal domain as expressed by the finite value of four current vector

*J*_{μ}.

The explicit symmetry breaking of the electric and magnetic field during electromagnetic wave propagation is also reflected in the non-conserved values of its Lagrangian within localized regions of space and time. The Euler–Lagrange equation for the electromagnetic Lagrangian density can be stated as follows:
4.2Here, the Lagrangian has subscripts, *α* and *β*, denoting components in space and , which indicates that symmetry breaking in the spatial domain results in generation of Noether current in the localized region of space and time [32]. As the electromagnetic wave moves into a certain localized region of space and time, the Lagrangian of the electromagnetic field loses its temporal symmetry within that region. It acquires a positive value as the wave moves in and slowly starts getting reduced and eventually reduces to zero as the wave moves out of the region.

Symmetry is never absolute and is represented in terms of the invariance of a system with a well-defined transformation group called the symmetry group [40]. The phenomenon of Symmetry breaking expresses relationships of transformation between a group and its sub-groups. It indicates that the initial symmetry is broken to one of its sub-groups [40]. In the context of static charges, the translational or rotational symmetry of the electric lines of field can also be interpreted as a broken symmetry of a higher order. For example, a stack of ferroelectric material having randomly oriented dipole moments has rotational symmetry as the net dipole moment is invariant with respect to rotation. When it is inserted within the plates of a charged parallel plate capacitor where the electric field lines have translational symmetry, the net polarization induced in it is *P*_{e} = *χ*_{e}*ϵ*** E** where

*χ*

_{e}is electric susceptibility and

*ϵ*is permittivity [41]. This can also be interpreted as explicit breaking of rotational symmetry of the polarization field resulting in generation of translational symmetry. Similarly, induced magnetism in ferromagnetic material under a strong magnetic field can be considered as explicit breaking of rotational symmetry and induction of translational symmetry [42].

The general symmetry breaking of a force field in either space or time which results in symmetry breaking of a corresponding quantity in the other dimension is unified in the general theory of relativity where both the symmetries are simultaneously broken. The presence of mass explicitly breaks the symmetry of space–time fabric (figure 4*a*). This is expressed in the basic equation [43,44]
4.3where *G*_{μν} indicates the curvature of space–time surface which is an indication of the total potential energy gradient of the system, *T*_{μν} indicates the energy momentum tensor and can be assumed to quantify the total kinetic energy components. Although space and time occur implicitly in both the sides of the equation, space has a dominant role in the potential energy component and time has the key role in the kinetic energy component. When there is no mass present, *R*_{μν} = 0 represents zero curvature of space indicating translational symmetry and a zero value of energy-momentum tensor. Figure 4*b* further illustrates the explicit symmetry breaking of the space–time curve in the form of elevations and depressions around a given mass.

## 5. Broken symmetry in electrodynamic systems, gauge symmetry and radiation

Symmetries in electromagnetism are defined in terms of gauge invariance [14]. Here, we attempt to link the temporal and spatial symmetry breaking of electric and magnetic field through the magnetic vector potential and establish its correlation to gauge invariance. We argue that such interactions involve localized symmetry breaking of the magnetic vector potential which is essentially linked to gauge invariance and global symmetries. The magnetic vector potential ** A** is expressed by [15]
5.1where

*μ*is permeability,

*ω*is frequency and

*j*indicates the imaginary component in the complex plane. It can be interpreted as comprising two symmetry breaking terms–the broken symmetry of the vector magnetic potential along spatial and temporal dimension. In equation (5.1), the first term denotes near-field effects and the second term denotes the radiation term.

An important goal of this discussion is to establish that the localized symmetry breaking of the electromagnetic field is closely associated with gauge symmetry at a global level. The magnetic flux density ** B** is related to magnetic vector potential by the relation [31,32]
5.2The expression of equation (5.2) can be interpreted from a novel perspective by considering that the translational symmetry breaking of magnetic vector potential along spatial dimensions results in its rotation, leading to the generation of magnetic flux density. The rotational symmetry breaking of the magnetic vector potential is associated with its invariance under the transformation

**=**

*A***+**

*A***∇**

*ψ*, where

*ψ*is a scalar field. This leads to the invariance of

**, expressed as,**

*B***=**

*B***∇**× (

**+**

*A***∇**

*ψ*) = ∇ ×

**. In physical terms, as the magnetic flux density is generated as a consequence of translational symmetry breaking of the magnetic vector potential, it is not influenced by any additional changes in its symmetry along a specific spatial dimension.**

*A*In electrodynamics, an increase in the magnetic vector potential ** A**, by

**∇**

*ψ*is also associated with a corresponding decrease in the corresponding electric potential

*V*, expressed as

*V*− ∂

*ψ*/∂

*t*, which is the basic condition of gauge invariance [31,32]. For a uniform plane wave propagating in free space, the electric field is expressed by

**= − ∂**

*E***/∂**

*A***. It establishes the relationship between temporal symmetry breaking of the magnetic vector potential and generation of an electric field, which remains invariant to any temporal changes in the value of electric potential**

*t**V*, expressed by

*V*− ∂

*ψ*/∂

*t*. Thus, the localized symmetry breaking of the magnetic vector potential,

**, along the spatial and temporal dimensions is an essential condition of the invariance of**

*A***and**

*E***which implies the invariance of the associated Lagrangian defining an electromagnetic wave and generation of global symmetries.**

*B*The ideas outlined above provide new insights in understanding the phenomenon of radiation from a diverse set of systems including piezoelectric and dielectric materials, which are insulators and do not have free charge carriers. Our current understanding of dielectric antennas is governed by the empirical and theoretical work by Long, McAllister and Shen who used a cylindrical structure of length *L* and radius *R* mounted on a ground plane whose radiation pattern was used to validate the theoretical assumption that the antenna has a boundary comprising a perfect magnetic conductor [45]. The electromagnetic field patterns were expressed using Bessel functions while arguing that the dielectric material behaved like a magnetic dipole antenna [45]. The magnetic wall model fails in offering a theoretical framework towards calculation of input impedance of dielectric antennas which play a key role in defining the radiation efficiency [46]. Besides this, it assumes the presence of an infinite ground plane which does not satisfy the operation of such devices at an empirical level [47]. Under empirical conditions, dielectric antennas have been found to have radiation patterns similar to electric dipole, magnetic dipole and quadrupole antennas, depending on shapes and mounting patterns and feeding geometries [48].

Earlier, it was pointed out by Sinha & Amaratunga that the assumptions by Long *et al.* violate the foundational aspects of radiating systems as a time varying excitation fed to a dielectric material generates oscillations of bound charges which is essentially oscillation of electrical dipoles which are inversely proportional to the cube of distance [35,38]. The electric field ** E**, associated with a Hertzian dipole in spherical coordinates along the dimensions

*r*,

*θ*and

*ϕ*, is expressed by [49] 5.3and 5.4Here,

*I*is the current in the antenna,

*L*is its length,

*η*is the intrinsic impedance of space,

*k*is the wavenumber associated with the wave and . The other component of electric field is

*E*

_{ϕ}= 0. Oscillating dipoles generate an electric field which corresponds to the 1/

*r*

^{3}term in the antenna field equation [49]. Considering this, the power generated by a dielectric antenna should hold a vanishingly small value in the far field region, which defies empirical observations.

A dielectric antenna has a capacitive reactance, which acts as a reservoir of electrons and is mounted on a finite ground plane, whose inductive reactance provides a framework for motion of charges between the capacitive and inductive elements. Hence, at a fundamental level, dielectric antennas work like simple capacitor–inductor antennas where electrons swing between the two elements resulting in radiation [34,35]. Thus, the physical operation of dielectric antennas can be understood by considering the role of non-conserved Noether currents in the system generated as a consequence of electrodynamic asymmetry which facilitates transformation of current into electromagnetic wave. The magnetic vector potential which determines the radiation pattern can be found by solving the inhomogeneous Helmholtz equation, **∇**^{2}** A** +

*ω*

^{2}

*μϵ*

**=**

*A***, where the current density**

*J***is the non-conserved Noether current, driving radiation.**

*J*The vector magnetic potential ** A**, at a point

*r*and time

*t*in a medium of permeability

*μ*, is expressed as [32] 5.5where

**(**

*J**r*′,

*t*′) is current at time

*t*′ and a position

*r*′ flowing along a length

*dr*′. The time

*t*′ is called retarded time defined by

*t*′ =

*t*− |

*r*−

*r*′|/

*c*, where

*c*is the speed of light in the given medium. For a wire of length

*L*oriented along the

*z*-direction, carrying a current

*I*, the magnetic vector potential can be expressed as,

**(**

*A**x*,

*y*,

*z*) =

*a*_{z}

*μIL*e

^{−jkr}/(4

*πr*) [32]. An important point which we wish to make in the current work, which has not been addressed in the literature on electromagnetism until now, is the fact that the factor

*IL*plays a key role in determining the radiation field, which has a much higher value in a metallic antenna as the electrons flow along the length

*L*determined by the antenna's spatial dimension. In a dielectric antenna, in the absence of a ground plane, under electric excitation, the capacitive current can be made high, which is equivalent to the current in a metallic antenna in terms of flow of charge per unit time, however, in such an antenna, due to the fact that the charges are bound and the net displacement around the mean position,

*L*, has atomic dimensions, which is miniscule in comparison to a metallic antenna of a similar size, the charges can move freely along the spatial dimensions of the antenna. The result is that the overall value of

*IL*is much lower in a dielectric antenna resulting in low radiation efficiency. This point further strengthens the arguments on the role of ground plane in dielectric antennas which facilitate radiation by providing a longer path of current flow.

Our analysis of radiating systems in terms of non-conserved Noether current and broken symmetries can help in analysing empirical results on radiating structures with low degrees of freedom. For example, in a recent work published on the problem of antenna miniaturization in wireless communication by Nan *et al.* [50], they reported that bulk acoustic waves in a magneto electric (ME) nano plate resonator (NPR) structure induce oscillation of the magnetization of ferromagnetic thin film leading to electromagnetic radiation which can realize antenna size miniaturization by 1–2 orders of magnitude. According to the Chu–Harrington (CH) limit which determines the antenna size for a given bandwidth, the antenna size reduction reduces the bandwidth and its efficiency [51,52]. Hence, a technology on antenna miniaturization must demonstrate efficiency measurements over a given bandwidth under standardized test conditions. Surpassing the CH limit over a given bandwidth is the only indicator of antenna miniaturization and Nan *et al*. in the given article [50] mention explicitly that their antennas do not surpass the CH limit, so the question of antenna miniaturization as claimed by the authors in the first part of the paper is refuted by themselves in the latter part. An analysis in terms of the Noether current associated with the system can shed novel insights on radiation from such a structure.

The theoretical foundation of the operation of ME antennas operating on electroacoustic effect addressed by Nan *et al.* [50] remains unclear as they point out that far-field radiation is a consequence of oscillating magnetic dipole moments. In a magnetic loop antenna of radius *a* carrying a current *I*, the magnetic flux density is defined as [49]
5.6and
5.7The other component of magnetic field, in the *ϕ* direction, is *H*_{ϕ} = 0. In equations (5.6) and (5.7), the 1/*r*^{3} term is the magnetic flux density generated by the oscillating magnetic dipoles which falls off inversely as cube root of distance and can induce current in test antennas during mid-field measurements giving the appearance of a lossy antenna. It can account for low radiation efficiency of 0.28% measured by the ME antenna reported by Nan *et al.* The radiation efficiency matches the value of radiation efficiency of 2%, in terms of orders of magnitude in piezoelectric films under symmetric excitation explored by Sinha & Amaratunga which effectively feeds excitation into the oscillating dipoles [35], creating a system where the corresponding Noether current is fed back to the ground plane, instead of being radiated out in space.

In the context of piezoelectric antennas, asymmetric excitation where the piezoelectric structure is used as a capacitive element on an inductive ground plane, the antenna efficiency increases as an electrodynamic asymmetry is created which provides a framework for loss of the corresponding Noether current, but this leads to miniaturization by a factor of 3–5 [34,35] and not 100 as Nan *et al*. claim. Thus, the results on antenna miniaturization as claimed by Nan *et al.* [50] need further validation as the antenna efficiency is in accordance with Harrington Chu limit. However, there is some near-field coupling between the RF field and ME NPR system which induces time varying oscillation of the magnetic moment, leading to generation of magnetic field in the near-field region which induces a finite current in the test antenna leading to a lossy antenna like effect.

Thus, an analysis of radiating structures by considering the flow of Noether current within the system can help in a more refined understanding of the mechanism of radiation from systems of low spatial dimensions. It becomes more important in electrodynamic systems which are much below the operational wavelength and parasitic elements start behaving as effective radiators.

## 6. The nature of irreversibility

In this section, we establish a relationship between the role of Noether current and broken symmetries in physical systems in defining a mathematical framework towards understanding irreversibility. One of the mysteries of physics is that the laws of physics have time reversal symmetry, however, irreversibility is intertwined with the fabric of real physical systems. Since the days of Boltzmann, scientist have looked at irreversibility from a statistical perspective, which considers irreversibility as an evolution of the number of microstates leading to an increase in entropy as the initial states become less probable. Thus, a closed system in a highly ordered state evolves towards a state of disorder as its entropy increases. Brian Greene in his book, The fabric of the cosmos: space, time and the structure of reality, writes, ‘No one has ever discovered any fundamental law which might be called the Law of the Spilled Milk or the Law of the Splattered Egg’ [53]. Lebowitz writes on Scholaropedia, ‘It is only secondary laws, which describe the behavior of macroscopic objects containing many, many atoms, such as the second law of thermodynamics, which explicitly contain this time asymmetry. The obvious question then is; how does one go from a time symmetric description of the dynamics of atoms to a time asymmetric description of the evolution of macroscopic systems made up of atoms’ [54]. According to the second law of thermodynamics, entropy of an isolated system tends to increase leading to an increase in its phase space volume [55], but Liouville's theorem states that the phase space volume is a conserved physical quantity [56]. This leads to an unresolved puzzle on the reconciliation of the entropy increase with the evolution of microstates and the onset of irreversibility in thermodynamics with time reversal symmetry and reversibility integrated with Newtonian mechanics which is referred to as the Loschmidt paradox [57]. The explanation of the Loschmidt paradox is statistical rather than physical and is usually defined in terms of the coarse graining of the phase space according to which the overall volume of phase space is conserved, but it evolves under Markov criterion into coarse-grained cells of phase space [58]. Currently, a mathematical framework which could define irreversibility in a macroscopic system does not exist. Here, we will see that a geometric analysis of trajectories of the Noether current in a system can lead to new insights in defining macroscopic irreversibility.

The first-order relationship between space and time in transformation of force fields is a characteristic of a reversible process. For example, the mechanical potential change d*U*_{m} along a given spatial dimension, *r*, leads to a gain of momentum d*p* over a time d*t*, which is expressed as, d*U*_{m}/d*r* = d*p*/d*t*. It leads to a momentum gain, as a particle is dropped from a given height. In a thermodynamic system, at equilibrium, the net momentum change is zero, i.e. over a given closed volume *V* . We cannot define a conserved current over a given volume which also means that spatial trajectories of particles in the phase space diagram are not uniquely defined.

At microscopic dimensions, we can define a surface, *S*, through which a definite momentum traverses at a certain instant of time, so it can be considered to be reversible during those specific instants of time. However, if the system is in equilibrium, the sum of momentum components vanish over a period of time d*t* and we cannot define a definite conserved current. Thus, in a process like diffusion or heat transfer, we cannot define reversibility. For example, diffusion is given as [55]
6.1where *C* is the concentration of particles and *K*_{D} is the diffusion constant. Here the first term ∇^{2}*C* leads to the second term, *K*_{D}∂*C*/∂*t*, but the second term does not lead to the first term. A microscopic model of each of the particles follows the Euler–Lagrange equation and is reversible. However, irreversibility is only a macroscopic model of a set of multiple microscopic processes which are fundamentally reversible. Similarly, heat transfer at a given temperature *T*, is expressed by [55]
6.2where *K*_{T} is thermal conductivity. Such systems do not have global conserved currents, neither do they have locally non-conserved physical quantities associated with transformation of force fields. Hence, the relationship between changes in space and time corresponding to the changes in a set of physical quantities, which can be considered to be an independent marker of reversibility of a process, do not exist in such processes.

The points mentioned above are of interest with regard to a historical debate on the nature of reversibility of generation of a photon. While Planck argued that emission of a photon is a reversible process [59], Einstein pointed out that it is an irreversible process, as the reversible process of a spontaneous absorption of a photon is not observed [60]. The debate was further expanded by Zeh, who supported Einstein's viewpoint [61]. However, an analysis in terms of the relationship between transformation of force fields and their spatial and temporal dependance implies that generation of an electromagnetic wave is a reversible process as the space and time are correlated by interdependent fields and the Noether current is lost locally but is conserved globally. In an irreversible process, the Noether current appears to be lost as a consequence of the statistical interactions of the given macroscopic system. For example, a hot furnace generates photons and slowly cools down. The process of heat transfer can be modelled according to equation (6.2) and it can appear to be an irreversible process as the globally conserved current loses the character of a vector field. In other words, the statistical nature of the system annihilates the vector fields at a macroscopic level, where the corresponding Noether current is annihilated at a macroscopic level.

## 7. Conclusion

We have shown that explicit symmetry breaking of a force field along the spatial domain is associated with explicit symmetry breaking along the spatial dimension within a localized region of space and time and vice versa. This is particularly reflected in electromagnetism where broken symmetries of electric and magnetic field are associated with temporal and spatial changes of corresponding force fields. In electrodynamic systems, radiation is generated as a consequence of loss of the corresponding Noether current, in the localized region of space and time. However, at a global level, the symmetries are maintained through conservation of charged and gauge invariance. The logic has been used to define the fundamental nature of a reversible process while arguing that in an irreversible process the globally conserved Noether current appears to be lost due to statistical addition of the corresponding vector field.

## Data accessibility

This article has no additional data.

## Author's contributions

All authors have read and approved the manuscript.

## Competing interests

We declare we have no competing interests.

## Funding

DS acknowledges support from SUTD-MIT Fellowship.

## Footnotes

One contribution of 13 to a theme issue ‘Celebrating 125 years of Oliver Heaviside's ‘Electromagnetic Theory’’.

- Accepted September 6, 2018.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.