Complex-Valued Electromagnetic Fields in Matter: Their Relevance to Electromagnetic Field Theories of Conscious Experience

Abstract

It has previously been shown that complex-valued electromagnetic fields can substantially increase the symmetry of Maxwell’s Equations (MEs). They are consistent with known experimental findings in classical electrodynamics and result in some interesting predictions. For example, complex MEs predict the existence of magnetic monopoles that would have escaped detection in past experimental searches for them. This paper extends the basic complex-valued MEs for use inside matter. The increased symmetry of the extended MEs is demonstrated by an electromagnetic duality transformation analogous to that of the standard MEs and a fundamentally new type of duality transform. A derived wave equation unexpectedly shows that the imaginary-valued portion of waves inside of matter propagates without attenuation or reduced speed. Demonstrating the existence of the imaginary-valued field components predicted by this theory could have substantial implications for understanding physical and biological phenomena. To illustrate this, ways in which imaginary-valued field components would contribute to existing electromagnetic field theories of consciousness are described. The ability of complex-valued fields to account for disparate phenomena (failure of past experimental searches to find magnetic monopoles; several poorly understood features of subjective time and memory) increases the probability of their existence.

1. Introduction

Past work within classical electrodynamics has generally assumed that electromagnetic fields have only real-valued components and that Maxwell’s equations (MEs) are essentially complete. However, many physicists have long suspected that these equations are not complete due to their well-known asymmetries [1,2]. These asymmetries have motivated, for example, extensive past experimental searches for the existence of magnetic monopoles and novel reformulations of MEs [3]. Of specific interest here is the recent demonstration that MEs can be made more symmetric by allowing electromagnetic fields to have complex-valued components rather than solely real-valued components as they currently have [4,5]. These field components’ newly added imaginary-valued parts are assumed to be unobservable and extend into time. When such complex-valued fields are used with MEs, the modified equations exhibit increased symmetry by implying that both electric and magnetic charges exist, that electric and magnetic charges are the same entity rather than distinct entities, that space and time are both three dimensional, and that electromagnetic fields extend not only into space but also into time. The complex-valued MEs remain consistent with existing experimental findings, explaining how magnetic charge/monopoles can exist but be undetected in past experimental searches.

This current paper aims to modify these existing complex-valued MEs to facilitate describing electromagnetic fields inside of conducting matter. The motivation for doing this is to allow one to explore how the imaginary-valued components of these fields would affect existing electromagnetic field theories of consciousness (EMFTCs). These theories argue that conscious experience is closely associated with the brain’s endogenous electromagnetic fields. Viewing the brain as a volume conductor, the derived complex-valued MEs for conducting matter can be used to, for the first time, determine how the imaginary components of the fields would impact existing EMFTCs. Investigating this issue in the following is limited solely to classical electrodynamics; no consideration of quantum field theory or general relativity, among others, is undertaken. Therefore, the work reported here should be viewed as only a first step in exploring this issue.

A recent generalization of MEs to handle complex-valued fields [4,5] was developed only within free space. This current paper extends past work in two ways. First, it inquires what the implications of adopting these free-space MEs would be for use in matter generally and in conducting matter specifically. Past work on complex-valued fields has not addressed this issue. The main contributions are establishing that (1) the complex-valued MEs in matter remain consistent with experimental data in classical electrodynamics, (2) they exhibit mathematical symmetries (duality transformations) that do not exist in standard real-valued MEs, (3) they predict the existence of imaginary-valued electromagnetic wave components that remain stationary in space but propagate through time, and (4) these latter imaginary-valued wave components do not have diminished velocity inside of matter as in the familiar real-valued components. The second extension to previous work is to show how these electrodynamics results would, surprisingly, contribute to explaining several properties of conscious experience that are poorly understood based on existing EMFTCs. The main contributions in this context include showing that the imaginary components of fields would account for (1) why the subjective experience of the present has duration, (2) why we subjectively perceive a flow of time that does not occur in modern physical theories, and (3) why the brain has such an enormous memory capacity for past conscious experiences.

What is striking about these results is that they all follow from trying to maximize the symmetry of MEs. The increased symmetry of the complex MEs in matter relative to the standard version is demonstrated by deriving electromagnetic duality equations for the former. These transformations are symmetry operations under which the complex MEs are invariant. In addition to a “within domain” transformation like those described in past work, a new type of “cross-domain” transformation is described. Thus, the increased symmetries in complex-valued MEs directly explain why past experimental searches for magnetic monopoles have been unsuccessful and for several poorly understood aspects of conscious experience. The ability of complex-valued fields to account for such different phenomena increases the probability that they exist.

The remainder of this paper begins by briefly summarizing some recent past work generalizing MEs to accommodate complex-valued fields (Section 2). The complex-valued MEs are then extended to encompass field behaviors in conducting matter while remaining consistent with both the original standard MEs and experimental findings (Section 3). The symmetry of the extended MEs is formally represented by duality transformations, symmetry operations that leave the extended equations unchanged. A wave equation is derived from the complex MEs in matter. It is found to predict the existence of waves that exhibit unusual properties, especially their propagation through time and their relationship to an observer at rest in space (Section 4). The consequences of such fields for existing electromagnetic field theories of consciousness are then given (Section 5). While speculative, it is found that the existence of complex-valued fields would enhance the ability of any EMFTC to account for some currently puzzling aspects of the subjective experience of time and of the recall of past conscious experiences. This is highly significant because it shows that complex-valued fields would not only explain the negative results of the numerous past experimental searches for magnetic monopoles but would also have substantial implications for understanding additional physical and biological phenomena. Finally, the conclusions and limitations of this work are discussed (Section 6). Mathematical derivations for results are in a Supplemental Information file.

2. Methods: Accommodating Complex-Valued Electromagnetic Fields

Classical electrodynamics is typically expressed in terms of an electric field vector E_x and a magnetic field vector B_x (bold font indicates vectors), each of which has three real-valued components. The behavior of these fields is characterized by the standard MEs (Figure 1a), where ρ is electric charge density and J is electric current density. These equations have a well-known asymmetry involving the existence of electric charge but not of magnetic charge (red 0′s in Figure 1a), suggesting to many physicists that these equations are incomplete. This has often been remedied theoretically by introducing hypothetical magnetic charge density ρ_m and magnetic current density J_m (Figure 1b, red font), but extensive experimental search for magnetic charge over many years has consistently failed to find it. The increased symmetry of these hypothetical extended MEs relative to the standard form is demonstrated by an electromagnetic duality transformation, such as that in Figure 1c (taken from [2], p. 49, with θ = π/2), under which the extended MEs are invariant (${\mathit{x}}^{\prime }=\mathit{x}$ and $t^{\prime }=t$ are implicit in Figure 1c). In other words, this duality transformation is a symmetry operation that, applied to any of the MEs in Figure 1b, leaves that equation unchanged (examples of this process are given later in this paper).

While the real-valued fields E_x(x,t) and B_x(x,t) are a function of both space and time, they do not have components that extend into time. A few past studies have challenged the restriction to solely real-valued fields by proposing that electromagnetic fields may have complex-valued components [6,7,8,9,10,11]. These previous studies differ from the current work in using formulations such as ${\mathit{E}}_{\mathit{x}}\pm ic{\mathit{B}}_{\mathit{x}}$, in using a novel type of magnetic charge ${\rho }_m$ and current J_m densities (as $\rho ={\rho }_e+i{\rho }_m$, J = J_e + i J_m), and/or in using more complicated differential operators. None of this past work has explicitly examined the idea that the imaginary components of the fields extend into time, as is done in the current paper.

The approach considered here to complex-valued fields, performed solely within classical electrodynamics, is guided by trying to maximize the symmetry of the standard MEs without violating known experimental findings [4,5]. These fields have the following forms:

$$\mathit{E}=\left[\begin{array}{c}{E}_{x1}+i{E}_{t1}\\ E_{x2}+i{E}_{t2}\\ E_{x3}+i{E}_{t3}\end{array}\right]={\mathit{E}}_{\mathit{x}}+i{\mathit{E}}_{\mathit{t}}$$

where${\mathit{E}}_x=\left[\begin{array}{c}{E}_{x1}\\ E_{x2}\\ E_{x3}\end{array}\right]$and i${\mathit{E}}_t=i\left[\begin{array}{c}{E}_{t1}\\ E_{t2}\\ E_{t3}\end{array}\right]$ are the usual real-valued electric field vectors in classical electrodynamics and a new imaginary-valued vector, respectively (E_t itself is real-valued). Analogous considerations apply to B, B_x, and iB_t. The imaginary portions of E and B are assumed to be unobservable and to extend into time rather than space. While it may seem likely that inserting such complex-valued fields into MEs would lead to inconsistencies or predictions of physical phenomena that do not occur, this was found not to be the case.

It proves useful to visualize the six real-valued components of the complex vectors E and B as existing in two 3D spaces, as illustrated in Figure 2. The real-valued portion of these vectors is pictured as existing in familiar 3D space, or r-space, while the imaginary portion is pictured as existing in a separate 3D temporal space, or t-space. The latter t-space reflects the viewpoint that 1D conventional clock time t actually has an underlying 3D basis [5]. While the notion that time is, in this sense, three-dimensional may seem to be a radical suggestion, there have been many past proposals in the physics literature since the 1970s that time is actually multidimensional. Some recent examples have arisen in investigating the unification of quantum mechanics and gravity [12], two-time physics [13], cosmological modeling [14], Dirac’s quantization condition [15], and quantum gravity [16]. The important implication for this current work is that complex-valued electromagnetic fields not only exist in time just as in standard physics but that their components also extend into time, unlike in classical electrodynamics.

Complex-valued MEs for these fields can be written as

$$\nabla ·\mathit{E}={\displaystyle \frac{1}{{ϵ}_0}}\rho$$

(1)

$$\nabla \times \mathit{E}={-c\mu }_{0}i\mathit{J}-{\displaystyle \frac{\partial \mathit{B}}{\partial t}}$$

(2)

$$\nabla ·\mathit{B}={c\mu }_{0}i\rho$$

(3)

$$\nabla \times \mathit{B}={\mu }_0\mathit{J}+{\displaystyle \frac{1}{c^2}}{\displaystyle \frac{\partial \mathit{E}}{\partial t}}$$

(4)

where SI units are used [5]. Symbols ${ϵ}_0$, ${\mu }_0$, c, $\rho$ and J indicate the usual permittivity, permeability, and speed of light in empty space, and charge and current density, respectively. These “complexified” equations, derived based on maximizing the symmetry of the standard MEs, differ from Maxwell’s original equations in several ways. First, the electric and magnetic field vectors here are functions of location in a spacetime whose points are represented by $\mathit{s}=\mathit{x}+ic\mathit{t}$, where x in r-space and t in t-space are both 3D real-valued vectors in ${\mathbb{R}}^3$, and vector t is associated with time. The real-valued variable t explicit in Equations (1)–(4) is familiar clock time, where an increment of clock time corresponds to a distance traversed in t-space, as explained further in Section 4.1. Second, complex-valued fields E and B not only exist in r-space, but their imaginary components also extend into t-space and, hence, into time, a concept that does not exist in classical electrodynamics. Third, the first imaginary-valued terms on the right-hand sides of Equations (2) and (3) replace zeros in the standard MEs, making the complex-valued equations more symmetric in a way reminiscent of Figure 1b but now using imaginary values $i\rho$ and $i\mathit{J}$ for charge and current densities rather than ρ_m and J_m. Finally, the complex differential operator is defined as $\nabla ={\nabla }_{\mathit{x}}+i{\displaystyle \frac{1}{c}}{\nabla }_t$, where ${\nabla }_x=\left[\begin{array}{c}{\displaystyle \frac{\partial }{\partial x_1}}\\ {\displaystyle \frac{\partial }{\partial x_2}}\\ {\displaystyle \frac{\partial }{\partial x_3}}\end{array}\right]$ and ${\nabla }_t=\left[\begin{array}{c}{\displaystyle \frac{\partial }{\partial t_1}}\\ {\displaystyle \frac{\partial }{\partial t_2}}\\ {\displaystyle \frac{\partial }{\partial t_3}}\end{array}\right]$. The complex divergence $\nabla ·$and curl $\nabla \times$ operations in Equations (1)–(4) are reduction vector operators that are “reduced” to a linear sum of the corresponding standard ${\mathbb{R}}^3$ operations in r-space and t-space. Specifically, if $\mathit{C}={\mathit{C}}_{\mathit{x}}+i{\mathit{C}}_{\mathit{t}}$ is a continuous differentiable vector field in ${\mathbb{C}}^3$ (such as E or B), then the reduction divergence, curl, gradient, and Laplacian for complex-valued fields are defined as follows, respectively:

$$\nabla ·\mathbf{C}={\nabla }_{\mathit{x}}·{\mathit{C}}_{\mathit{x}}+i{\displaystyle \frac{1}{c}}{\nabla }_{\mathit{t}}·{\mathbf{C}}_{\mathit{t}}$$

(5)

$$\nabla \times \mathbf{C}={\nabla }_{\mathit{x}}\times {\mathit{C}}_{\mathit{x}}+i{\displaystyle \frac{1}{c}}{\nabla }_{\mathit{t}}\times {\mathit{C}}_{\mathit{t}}$$

(6)

$$\nabla T={\nabla }_{\mathit{x}}{T}_x+i{\displaystyle \frac{1}{c}}{\nabla }_t{T}_t$$

(7)

$${\nabla }^2\mathit{C}=\nabla ·\nabla \mathbf{C}={\nabla }_{\mathit{x}}^2{\mathit{C}}_{\mathit{x}}+i{\displaystyle \frac{1}{c^2}}{\nabla }_t^2{\mathbf{C}}_{\mathit{t}}$$

(8)

where $T=T_{\mathit{x}}+i{T}_{\mathit{t}}$ is a continuously differentiable scalar field in ${\mathbb{C}}^3.$

Several consequences follow from these complex-valued MEs [5], and some of these are briefly listed here. Equations (1)–(4) are consistent with the standard real-valued MEs and thus with past experimental findings; they do not predict new r-space phenomena that have not been observed. As noted earlier, the three imaginary components of the fields indicate the existence of a 3D t-space that is hypothesized to underly the existence of familiar 1D clock time t. There is a temporal correspondence given by $dt=\left|d\mathit{t}\right|$, where dt is an increment of observable clock time, and $\left|d\mathit{t}\right|$ is the distance between two sequential events in t-space. Equations (1)–(4) imply that—unlike the fields — charge and current only exist in r-space; they do not extend into t-space. Therefore, $\rho$ and J are purely real-valued (for example, if it were the case that $\rho ={\rho }_x+i{\rho }_t$, then it would imply that ${\nabla }_x·B_x\ne 0$, which is inconsistent with experimental data and, hence, it must be that ${\rho }_t=0$), while $i\rho$ and $i\mathit{J}$ are purely imaginary-valued. The new right-side terms ${c\mu }_{0}i\rho$ and ${-c\mu }_{0}i\mathit{J}$ imply the existence of magnetic charge, as discussed in [4,5]. The implied magnetic monopoles differ from previously proposed ones (and from dyons) in that their magnetic fields are in t-space and hence unobservable, explaining why past experimental searches for magnetic monopoles have failed to find them. Every charged particle having electric fields extending into r-space (electrons, protons, etc.) also has unobserved magnetic fields extending into t-space. The increased symmetry of Equations (1)–(4) relative to the standard MEs is formally represented by an electromagnetic duality transformation, as follows:

$${\mathit{E}}^{\prime }=c\mathit{B},{\mathit{B}}^{\prime }=-{\displaystyle \frac{1}{c}}\mathit{E},{\rho }^{\prime }=i\rho ,{\mathit{J}}^{\prime }=i\mathit{J}$$

(9)

under which each of Equations (1)–(4) remains invariant (i.e., Equation (9) is a symmetry operation that, when applied to MEs (1)–(4), leaves those equations unchanged). This duality transformation is analogous to that for the extended real-valued MEs in Figure 1b,c (${\mathit{x}}^{\prime }=\mathit{x}$ and ${\mathit{t}}^{\prime }=\mathit{t}$ are again implicit in Equation (9)), but is now simpler in spite of involving complex-valued fields because it does not have to handle separate ${\rho }_m$ and J_m entities. Finally, Equations (1)–(4) predict the existence of electromagnetic waves that propagate not only through r-space but also through t-space [4,5]. As explained previously in [5], Equations (1)–(4) are falsifiable.

3. Complex-Valued Maxwell’s Equations in Matter

In adopting the complex-valued MEs above for use in matter, we take the same approach used in modifying the standard MEs for their use in matter (assumed to be homogeneous, isotropic, and linear), but instead now using complex fields. The increased symmetry of the complex-valued MEs in matter is formally demonstrated not only by a “within-domains” electromagnetic duality transformation analogous to that long known for the extended MEs (Figure 1c) and for the previous complex-valued MEs described above (Equation (9)) but also by a fundamentally new type of “between-domains” duality transformation.

3.1. Complex Maxwell’s Equations in Linear Homogeneous Matter

One can adapt the standard real-valued MEs for use in linear homogeneous isotropic matter using the rules ${ϵ}_0\to ϵ$ and ${\mu }_0\to \mu$; i.e., by replacing ${ϵ}_0$ with $ϵ$ and ${\mu }_0$ with $\mu$ ([1], p. 401; [2], pp. 44–45). Quantities $ϵ$ and $\mu$ are the electrical permittivity and magnetic permeability of the underlying medium, respectively. We do the same thing here, but now, starting from the complex MEs (1)–(4). Care must be taken in doing this because $c={\displaystyle \frac{1}{\sqrt{{ϵ}_0{\mu }_0}}}$ is transformed by these rules to be $c\to {\displaystyle \frac{1}{\sqrt{ϵ\mu }}}\to {\displaystyle \frac{\sqrt{{ϵ}_0{\mu }_0}}{\sqrt{{ϵ}_0{\mu }_0}}}{\displaystyle \frac{1}{\sqrt{ϵ\mu }}}\to {\displaystyle \frac{1}{\sqrt{{ϵ}_0{\mu }_0}}}{\displaystyle \frac{\sqrt{{ϵ}_0{\mu }_0}}{\sqrt{ϵ\mu }}}\to {\displaystyle \frac{c}{n}}$, where $n\equiv \sqrt{{\displaystyle \frac{ϵ\mu }{{ϵ}_0{\mu }_0}}}$ is the medium’s index of refraction. Thus, there is an implicit third replacement rule $c\to {\displaystyle \frac{c}{n}}$ indicating that wherever c occurs, it should be replaced by the speed of light ${\displaystyle \frac{c}{n}}$ inside matter. This includes modifying the definition of $\nabla ={\nabla }_{\mathit{x}}+i{\displaystyle \frac{1}{c}}{\nabla }_t$, which establishes the following:

$$\nabla ={\nabla }_{\mathit{x}}+i{\displaystyle \frac{n}{c}}{\nabla }_t$$

(10)

This is instead used exclusively in the following (also altering Equations (5)–(8) accordingly). When applied to the complex-valued MEs (1)–(4), this transformation gives the following:

$$\nabla ·\mathit{E}={\displaystyle \frac{1}{ϵ}}\rho$$

(11)

$$\nabla \times \mathit{E}=-{\displaystyle \frac{c}{n}}\mu i\mathit{J}-{\displaystyle \frac{\partial \mathit{B}}{\partial t}}$$

(12)

$$\nabla ·\mathit{B}={\displaystyle \frac{c}{n}}\mu i\rho$$

(13)

$$\nabla \times \mathit{B}=\mu \mathit{J}+ϵ\mu {\displaystyle \frac{\partial \mathit{E}}{\partial t}}$$

(14)

for complex Maxwell’s equations in matter, where $ϵ\mu$ on the right side of Equation (14) follows from ${\displaystyle \frac{n^2}{c^2}}=ϵ\mu$, and $\nabla$ is as in Equation (10). While fields $\mathit{D}=ϵ\mathit{E}$ and $\mathit{H}={\displaystyle \frac{1}{\mu }}\mathit{B}$ are often used in adopting the standard real-valued MEs for use in linear matter, this convention is avoided here to minimize the number of field symbols that appear. Equations (11)–(14) remain consistent with both the original standard MEs and with their complexified version (Equations (1)–(4)) when $ϵ={ϵ}_0$, $\mu ={\mu }_0$.

The complex MEs in matter have a within-domain duality transformation T_wd of the following:

$${\mathit{x}}^{\prime }=\mathit{x},{\mathit{t}}^{\prime }=\mathit{t},{\mathit{E}}^{\prime }={\displaystyle \frac{c}{n}}\mathit{B},{\mathit{B}}^{\prime }=-{\displaystyle \frac{n}{c}}\mathit{E},{\rho }^{\prime }=i\rho ,{\mathit{J}}^{\prime }=i\mathit{J}$$

(15)

Under this, each of the MEs (11)–(14) remains invariant. The relations ${\mathit{x}}^{\prime }=\mathit{x}$ and ${\mathit{t}}^{\prime }=\mathit{t}$ included in Equation (15) are usually implicit in specifying within-domain duality transformations (e.g., Figure 1c and Equation (9)), but here they are made explicit for later contrast with the between-domains transformations. This transformation, or symmetry operation, is interpreted in the same way as described for Equation (9) above. In other words, take the left side of one of the complex-valued MEs (11)–(14) in its primed form, apply the duality transformation equations (15) to it, use the original unprimed MEs on that, and then convert back to the primed form with Equations (15): the final equation in primed form is the same as the original unprimed equation. For example, applying this transformation to Equation (14) gives the following:

$$\begin{array}{cc}{\nabla }^{\prime }\times {\mathit{B}}^{\prime }& =-{\displaystyle \frac{n}{c}}\nabla \times \mathit{E}=-{\displaystyle \frac{n}{c}}\left(-{\displaystyle \frac{c}{n}}\mu i\mathit{J}-{\displaystyle \frac{\partial \mathit{B}}{\partial t}}\right)\\ & =\mu i\left(-i{\mathit{J}}^{\prime }\right)+{\displaystyle \frac{n}{c}}{\displaystyle \frac{\partial \left(\frac{n}{c}{\mathit{E}}^{\prime }\right)}{\partial t}}=\mu {\mathit{J}}^{\prime }+{\left({\displaystyle \frac{n}{c}}\right)}^2{\displaystyle \frac{\partial {\mathit{E}}^{\prime }}{\partial t}}\\ & =\mu {\mathit{J}}^{\prime }+ϵ\mu {\displaystyle \frac{\partial {\mathit{E}}^{\prime }}{\partial t}}\end{array}$$

This leaves Equation (14) invariant. Similar results apply to the remaining equations, as shown in Supplemental Information S1. Transformation T_wd is analogous to and reduces to Equation (9) when n = 1 and is termed “within-domain” because it involves transforming between field vectors that are both within the r-space domain (${\mathit{E}}_x^{\prime }={\displaystyle \frac{c}{n}}{\mathit{B}}_x$ and ${\mathit{B}}_x^{\prime }=-{\displaystyle \frac{n}{c}}{\mathit{E}}_x$) and between those that are both within the t-space domain (${\mathit{E}}_t^{\prime }={\displaystyle \frac{c}{n}}{\mathit{B}}_t$ and ${\mathit{B}}_t^{\prime }=-{\displaystyle \frac{n}{c}}{\mathit{E}}_t$).

Since the MEs (11)–(14) in matter incorporate complex fields, they actually represent two sets of equations, arrived at by equating their real parts and their imaginary parts individually. For example, Equation (13) gives the following:

$$\nabla ·\mathit{B}=\left({\nabla }_{\mathit{x}}+i{\displaystyle \frac{n}{c}}{\nabla }_t\right)·\left({\mathit{B}}_{\mathit{x}}+i{\mathit{B}}_{\mathit{t}}\right)={\nabla }_{\mathit{x}}·{\mathit{B}}_{\mathit{x}}+i{\displaystyle \frac{n}{c}}{\nabla }_{\mathit{t}}·{\mathit{B}}_{\mathit{t}}={\displaystyle \frac{c}{n}}\mu i\rho$$

Additionally, setting their real and imaginary parts to be equal gives the two following equations:

$${\mathbf{\nabla }}_{\mathit{x}}·{\mathit{B}}_{\mathit{x}}=0$$

(16)

$${\nabla }_{\mathit{t}}·{\mathit{B}}_{\mathit{t}}={\displaystyle \frac{1}{ϵ}}\rho$$

(17)

where each of Equations (16) and (17) is in ${\mathbb{R}}^3$, with the first involving r-space and the second involving t-space. Equation (17) follows from ${\displaystyle \frac{n^2}{c^2}}=ϵ\mu$ as described above. Using this same procedure with all four complex-valued MEs in matter gives the following:

$${\nabla }_x·{\mathit{E}}_{\mathit{x}}={\displaystyle \frac{1}{ϵ}}\rho$$

(18)

$${\nabla }_x\times {\mathit{E}}_{\mathit{x}}=-{\displaystyle \frac{\partial {\mathit{B}}_x}{\partial t}}$$

(19)

$${\nabla }_x·{\mathit{B}}_{\mathit{x}}=0$$

(20)

$${\nabla }_x\times {\mathit{B}}_{\mathit{x}}=\mu \mathit{J}+ϵ\mu {\displaystyle \frac{\mathbf{\partial }{\mathit{E}}_{\mathit{x}}}{\mathbf{\partial }\mathit{t}}}$$

(21)

These characterize field behavior in r-space; they are the same as those commonly used for MEs in linear matter ([1], pp. 341–342; [2], pp. 584–585). They are thus consistent with existing experimental results in observable r-space. This same procedure also results in four equations as follows:

$${\nabla }_t·{\mathit{E}}_{\mathit{t}}=0$$

(22)

$${\nabla }_t\times {\mathit{E}}_{\mathit{t}}=-{\displaystyle \frac{1}{ϵ}}\mathit{J}-{\displaystyle \frac{c}{n}}{\displaystyle \frac{\partial {\mathit{B}}_{\mathit{t}}}{\partial t}}$$

(23)

$${\nabla }_t·{\mathit{B}}_{\mathit{t}}={\displaystyle \frac{1}{ϵ}}\rho$$

(24)

$${\nabla }_t\times {\mathit{B}}_{\mathit{t}}={\displaystyle \frac{n}{c}}{\displaystyle \frac{\partial {\mathit{E}}_{\mathit{t}}}{\partial t}}$$

(25)

These characterize the unobservable fields E_t and B_t in t-space inside of matter.

Comparing Equations (18)–(21) and (22)–(25), it is evident that the separate r-space and t-space equations are essentially symmetric with respect to one another if one reverses the roles of the electric and magnetic fields in going between r-space and t-space. Thus, unlike with the extended real-valued MEs (Figure 1b,c), with complex MEs in matter, there is an additional novel type of between-domains duality transformation T_bd,

$${\mathit{x}}^{\prime }={\displaystyle \frac{c}{n}}\mathit{t},$$

(26)

$${\mathit{t}}^{\prime }={\displaystyle \frac{n}{c}}\mathit{x},$$

(27)

$${\mathit{E}}^{\prime }=-i{\displaystyle \frac{c}{n}}\mathit{B},$$

(28)

$${\mathit{B}}^{\prime }=i{\displaystyle \frac{n}{c}}\mathit{E},$$

(29)

$${\rho }^{\prime }=\rho ,$$

(30)

$${\mathit{J}}^{\prime }=\mathit{J},$$

(31)

This, like T_wd, leaves the full complex-valued MEs (11)–(14) in matter invariant, as shown in the first part of Supplemental Information S2. This is to be interpreted as a symmetry operation in the same way as with the previous duality transformations described above. Transformation T_bd is termed “between-domains” because it transforms vectors between the domains of r-space and t-space (${\mathit{E}}_x^{\prime }={\displaystyle \frac{c}{n}}{\mathit{B}}_t$ and ${\mathit{B}}_x^{\prime }=-{\displaystyle \frac{n}{c}}{\mathit{E}}_t$; and ${\mathit{E}}_t^{\prime }=-{\displaystyle \frac{c}{n}}{\mathit{B}}_x$ and ${\mathit{B}}_t^{\prime }={\displaystyle \frac{n}{c}}{\mathit{E}}_x$) rather than between vectors in the same domains as in T_wd. Equations (26) and (27) implicitly imply three additional rules as follows:

$${\nabla }_x^{\prime }={\displaystyle \frac{n}{c}}{\nabla }_t,$$

(32)

$${\nabla }_t^{\prime }={\displaystyle \frac{c}{n}}{\nabla }_x,$$

(33)

$${\nabla }^{\prime }={\displaystyle \frac{n}{c}}{\nabla }_t+i{\nabla }_x=i{\nabla }^{*}$$

(34)

These are part of T_bd, with * indicating the complex conjugate. Equation (32) follows from (26) because ${\displaystyle \frac{\partial }{\partial x_j}}={\displaystyle \frac{\partial }{\partial \left(\frac{c}{n}{t}_j\right)}}={\displaystyle \frac{n}{c}}{\displaystyle \frac{\partial }{\partial t_j}}$ for j in {1, 2, 3}. Equation (33) follows similarly from (27), and Equation (34) follows from Equations (10), (32) and (33).

A significant implication of the between-domains duality transformation is that it gives the following cross-domain mapping:

$$\mathit{x}⟹{\displaystyle \frac{c}{n}}\mathit{t},$$

(35)

$${\mathit{E}}_x⟹{\displaystyle \frac{c}{n}}{\mathit{B}}_t,$$

(36)

$${\mathit{B}}_x⟹-{\displaystyle \frac{n}{c}}{\mathit{E}}_t$$

(37)

This lets one convert equations in r-space into their dual equations in t-space. Equations (35)–(37) come from the real parts of Equations (26), (28) and (29), respectively. Equation (35) implies that ${\nabla }_x⟹{\displaystyle \frac{n}{c}}{\nabla }_t$ for the same reason given above for Equation (32). For example, this cross-domain mapping converts the set of Equations (18)–(21) in r-space into the corresponding set of Equations (22)–(25) in t-space and vice versa using the inverse mapping. As a specific example, when applied to Equation (18), this cross-domain mapping gives the following:

$${\nabla }_x·{\mathit{E}}_{\mathit{x}}={\displaystyle \frac{1}{ϵ}}\rho ⟹{\displaystyle \frac{n}{c}}{\nabla }_t·\left({\displaystyle \frac{c}{n}}{\mathit{B}}_t\right)={\displaystyle \frac{1}{ϵ}}\rho ⟹{\nabla }_t·{\mathit{B}}_t={\displaystyle \frac{1}{ϵ}}\rho$$

The resulting equation is Equation (24). Similar application of the cross-domain mapping to the rest of these equations give Equations (22)–(25), as shown in the second part of Supplemental Information S2.

3.2. Complex Maxwell Equations in Conducting Matter

We now modify the complex MEs (11)–(14) in matter to apply to ohmic conducting materials specifically. In r-space, any free charge inside of a good conductor quickly dissipates, leaving $\rho =0$ inside the conductor. This reduces the two divergence equalities in Equations (11)–(14) to $\nabla ·\mathit{E}=0$ and $\nabla ·\mathit{B}=0$ inside the conductor. Further, the current density in r-space is taken to be linearly proportional to the electric field:

$$\mathit{J}={\sigma }_x{\mathit{E}}_x$$

(38)

This is according to Ohm’s law, where ${\sigma }_x$ is the familiar conductivity of the underlying medium in r-space. Thus, ${\sigma }_x{\mathit{E}}_x$ can be substituted for $\mathit{J}$ in $\nabla \times \mathit{B}=\mu \mathit{J}+ϵ\mu {\displaystyle \frac{\partial \mathit{E}}{\partial t}}$ as is done in standard classical electrodynamics because $\mu \mathit{J}$ applies just to r-space. However, handling $i\mathit{J}$ in $\nabla \times \mathit{E}=-{\displaystyle \frac{c}{n}}\mu i\mathit{J}-{\displaystyle \frac{\partial \mathit{B}}{\partial t}}$ is problematic because $i\mathit{J}$ applies solely to t-space. Directly substituting $i{{\sigma }_x\mathit{E}}_x$ for $i\mathit{J}$ in this latter case is inconsistent with the cross-domain mapping (35)–(37) Applying Equations (35)–(37) to Ohm’s law (Equation (38)) and letting ${\sigma }_x⟹{\sigma }_t$ (allowing for the possibility that conductivity ${\sigma }_t$ in t-space may differ from ${\sigma }_x$ in r-space) gives the following:

$$i\mathit{J}={\displaystyle \frac{c}{n}}{\sigma }_t{i\mathit{B}}_t$$

(39)

This is an analog of Ohm’s law for t-space. These modifications to Equations (11)–(14) give the following set of complex Maxwell equations in conducting matter:

$$\nabla ·\mathit{E}=0$$

(40)

$$\nabla \times \mathit{E}=-{\displaystyle \frac{{\sigma }_t}{ϵ}}{i\mathit{B}}_t-{\displaystyle \frac{\partial \mathit{B}}{\partial t}}$$

(41)

$$\nabla ·\mathit{B}=0$$

(42)

$$\nabla \times \mathit{B}=\mu {\sigma }_x{\mathit{E}}_x+ϵ\mu {\displaystyle \frac{\partial \mathit{E}}{\partial t}}$$

(43)

The symmetry of these equations is again formally indicated by specifying a within-domain duality transformation $T_{wd}^c$

$${\mathit{x}}^{\prime }=\mathit{x},{\mathit{t}}^{\prime }=\mathit{t},{\mathit{E}}^{\prime }={\displaystyle \frac{c}{n}}\mathit{B},{\mathit{B}}^{\prime }=-{\displaystyle \frac{n}{c}}\mathit{E},{\rho }^{\prime }=i\rho ,{\mathit{J}}^{\prime }=i\mathit{J}$$

(44)

for conducting matter under which Equations (40)–(43) remain invariant, as shown in the first part of Supplemental Information S3. Equation (44) is again a symmetry operation, and is the same as Equation (15) except that (44) is now supplemented by Ohm’s law and its t-space analog (Equations (38) and (39)) in applying the transformation.

Equating the real parts and imaginary parts in each of the MEs (40)–(43) for conductors again gives four equations:

$${\nabla }_x·{\mathit{E}}_{\mathit{x}}=0$$

(45)

$${\nabla }_x\times {\mathit{E}}_{\mathit{x}}=-{\displaystyle \frac{\partial {\mathit{B}}_{\mathit{x}}}{\partial t}}$$

(46)

$${\nabla }_x·{\mathit{B}}_{\mathit{x}}=0$$

(47)

$${\nabla }_x\times {\mathit{B}}_{\mathit{x}}=\mu {\sigma }_x{\mathit{E}}_x+ϵ\mu {\displaystyle \frac{\mathbf{\partial }{\mathit{E}}_{\mathit{x}}}{\mathbf{\partial }\mathit{t}}}$$

(48)

These characterize the field behavior in r-space inside conductors. These four equations are exactly the same as those in widely used textbooks for standard real-valued MEs in conductors ([1], p. 413; [2], p. 607). Therefore, Equations (40)–(43) remain consistent with existing experimental results inside of conductors in observable r-space. Equating the imaginary portions of Equations (40)–(43) also results in four equations:

$${\nabla }_t·{\mathit{E}}_{\mathit{t}}=0$$

(49)

$${\nabla }_t\times {\mathit{E}}_{\mathit{t}}=-{\displaystyle \frac{c}{n}}{\displaystyle \frac{{\sigma }_t}{ϵ}}{\mathit{B}}_t-{\displaystyle \frac{c}{n}}{\displaystyle \frac{\partial {\mathit{B}}_{\mathit{t}}}{\partial t}}$$

(50)

$${\nabla }_t·{\mathit{B}}_{\mathit{t}}=0$$

(51)

$${\nabla }_t\times {\mathit{B}}_{\mathit{t}}={\displaystyle \frac{n}{c}}{\displaystyle \frac{\partial {\mathit{E}}_{\mathit{t}}}{\partial t}}$$

(52)

These characterize the unobservable fields E_t and B_t in t-space inside of the conductors.

As in the previous subsection, there is an additional between-domains duality transformation for conductors $T_{bd}^c$ given by the following:

$${\mathit{x}}^{\prime }={\displaystyle \frac{c}{n}}\mathit{t},{\mathit{t}}^{\prime }={\displaystyle \frac{n}{c}}\mathit{x},{\mathit{E}}^{\prime }=-i{\displaystyle \frac{c}{n}}\mathit{B},{\mathit{B}}^{\prime }=i{\displaystyle \frac{n}{c}}\mathit{E},{\rho }^{\prime }=\rho ,{\mathit{J}}^{\prime }=\mathit{J}$$

(53)

This, like $T_{wd}^c$, leaves the MEs (40)–(43) in conducting matter invariant, as shown in the second part of Supplemental Information S3. This transformation is unchanged from that of matter in general, as given in Equations (26)–(31), and also implies that${\nabla }^{\prime }=i{\nabla }^{*}$.

A significant implication of the between-domains duality transformation$T_{bd}^c$ is that it gives a cross-domain mapping as follows:

$$\mathit{x}⟹{\displaystyle \frac{c}{n}}\mathit{t}{,\mathit{E}}_x⟹{\displaystyle \frac{c}{n}}{\mathit{B}}_t,{\mathit{B}}_x⟹-{\displaystyle \frac{n}{c}}{\mathit{E}}_t,{\sigma }_x⟹{\sigma }_t$$

(54)

This is consistent with Equations (35)–(37) but adds the rule ${\sigma }_x⟹{\sigma }_t$. This lets one convert equations in r-space into their dual equations in t-space. Like before, Equation (54) implies that ${\nabla }_x⟹{\displaystyle \frac{n}{c}}{\nabla }_t$. As an example, this cross-domain mapping converts the set of Equations (45)–(48) in r-space into the corresponding set of Equations (49)–(52) in t-space, as shown in the final part of Supplemental Information S3.

4. Complex-Valued Wave Equation Inside of Conducting Matter

In this section, complex-valued wave equations for the inside of conducting matter are derived from Equations (40)–(43) while retaining consistency with standard wave equations for conductors in r-space, and a plane wave solution is examined. Before deriving the new wave equations, it is useful for what follows to first characterize the underlying spacetime involved.

4.1. The Underlying Spacetime

Electromagnetic fields having three imaginary components suggest the existence of an underlying complex-valued spacetime ${\mathbb{C}}^3$ [4,5]. Points s in this spacetime are represented by $\mathit{s}=\mathit{x}+ic\mathit{t}$, where $\mathit{x}=\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]$ in r-space and $\mathit{t}=\left[\begin{array}{c}{t}_1\\ t_2\\ t_3\end{array}\right]$ in t-space are 3D real-valued vectors in ${\mathbb{R}}^3$, and vector t is associated with time, as discussed in Section 2. Each unobservable time component t_j is measured in seconds, and then, like x_j, the quantity c t_j is measured in meters. We take the 3D vector t in t-space to underly our familiar measurable notion of 1D clock time t. Just like we associate a distance with the extent to which an object moves along a 1D trajectory through a 3D r-space, we can associate the measured passage of clock time with the extent to which an entity moves along a 1D trajectory in a 3D t-space. Let $d\mathit{s}=d\mathit{x}+icd\mathit{t}$ be the spacetime displacement between two infinitesimally separated events, and let the following define the distances occurring in r-space and t-space between those two events:

$$\left|d\mathit{x}\right|={\left({d{x}_1}^2+{d{x}_2}^2+{d{x}_3}^2\right)}^{{\displaystyle \frac{1}{2}}}$$

(55)

$$\left|d\mathit{t}\right|={\left({d{t}_1}^2+{d{t}_2}^2+{d{t}_3}^2\right)}^{{\displaystyle \frac{1}{2}}}$$

(56)

We adopt a temporal correspondence $dt=\left|d\mathit{t}\right|$ between the measured clock time $dt$ separating the two events and the distance $\left|d\mathit{t}\right|$ separating the two events in t-space. This asserts that an increment of clock time $dt$ corresponds to the “distance” $\left|d\mathit{t}\right|$ along a trajectory through t-space, just as $dx$ relates to the distance $\left|d\mathit{x}\right|$ along a trajectory through r-space. It implies that dt is always positive, as observed. Finally, we define the complex velocity $\mathit{v}$ as follows:

$$\mathit{v}={\displaystyle \frac{d\mathit{s}}{dt}}={\displaystyle \frac{d\mathit{x}}{dt}}+ic{\displaystyle \frac{d\mathit{t}}{dt}}={\mathit{v}}_{\mathit{x}}+i{\mathit{v}}_{\mathit{t}}$$

(57)

Here ${\mathit{v}}_{\mathit{x}}$ is the conventional velocity in r-space, and ${\mathit{v}}_{\mathit{t}}=c{\displaystyle \frac{\mathit{d}\mathit{t}}{\mathit{d}t}}$ in t-space is a temporal velocity, which is also measured in m/s.

The standard Lorentz transformation (boost) is readily generalized to the ${\mathbb{C}}^3$ spacetime described above, and the generalized spacetime interval $d{s}^2=c^{2}d{t}^2-d{x}^2$ is invariant under this transformation [5]. In standard four-vector special relativity, there is a universal speed constraint that, in an inertial reference frame, relates the speed with which an object is moving through time to its speed in space ([17], pp. 86–87). The invariant interval generalized to a ${\mathbb{C}}^3$ spacetime implies that the following analogous universal speed constraint exists for the ${\mathbb{C}}^3$ spacetime used here [5].

$$v_x^2+v_{\tau }^2=c^2$$

(58)

Informally, Equation (58) implies that an entity’s total speed in spacetime, i.e., the rate at which one is moving through both space and time (left-hand side), is always a constant. The temporal velocity ${\mathit{v}}_{\mathit{\tau }}=c{\displaystyle \frac{\mathit{d}\mathit{\tau }}{\mathit{d}t}}$ through t-space is based on the ratio of a differential displacement $d\mathit{\tau }$ in the t-space of the object’s proper inertial reference frame to dt, a real-valued clock time increment in the observer’s reference frame measured during this displacement $d\mathit{\tau }$ [5]. Speed $v_{\tau }$ can thus be viewed informally as how fast the object is aging from the viewpoint of an observer at rest in the r-space of an inertial reference frame where the object is moving with speed $v_x$ in the observer’s frame. The universal speed constraint implies that there is an upper limit of c on the speed $v_{\tau }$ that any object can have in t-space (just like in r-space), that any object “at rest” in an inertial frame’s r-space must have an associated temporal speed $v_{\tau }$ = c (it is moving through t-space at the speed of light), and that photons traveling through r-space with speed $v_x$= c have an associated speed $v_{\tau }$ = 0 (consistent with standard special relativity in which photons are not aging).

4.2. The Complex Wave Equation in Conductors

We now derive wave equations for E and B inside of conductors. The derivation, given in Supplemental Information S4, is performed in the usual fashion by taking the curl of Equations (41) and (43), except now using the complex curl $\nabla \times$ where $\nabla ={\nabla }_{\mathit{x}}+i{\displaystyle \frac{n}{c}}{\nabla }_t$, as defined by Equation (10). The resulting wave equations are the following:

$${\nabla }^2\mathit{E}=\left(\mu {\sigma }_x{\displaystyle \frac{\mathbf{\partial }{\mathit{E}}_{\mathit{x}}}{\mathbf{\partial }\mathit{t}}}+ϵ\mu {\displaystyle \frac{{\partial }^2{\mathit{E}}_x}{{\partial t}^2}}\right)+i\left(\mu {\sigma }_t{\displaystyle \frac{\mathbf{\partial }{\mathit{E}}_{\mathit{t}}}{\mathbf{\partial }\mathit{t}}}+ϵ\mu {\displaystyle \frac{{\partial }^2{\mathit{E}}_t}{{\partial t}^2}}\right)$$

(59)

$${\nabla }^2\mathit{B}=\left(\mu {\sigma }_x{\displaystyle \frac{\mathbf{\partial }{\mathit{B}}_{\mathit{x}}}{\mathbf{\partial }\mathit{t}}}+ϵ\mu {\displaystyle \frac{{\partial }^2{\mathit{B}}_x}{{\partial t}^2}}\right)+i\left(\mu {\sigma }_t{\displaystyle \frac{\mathbf{\partial }{\mathit{B}}_{\mathit{t}}}{\mathbf{\partial }\mathit{t}}}+ϵ\mu {\displaystyle \frac{{\partial }^2{\mathit{B}}_t}{{\partial t}^2}}\right)$$

(60)

These wave equations are more complicated than those typically seen in matter with real-valued fields as they allow the possibility that conductivity in t-space may differ from that in r-space, i.e., possibly ${\sigma }_t\ne {\sigma }_x$. If one finds that ${\sigma }_t={\sigma }_x=\sigma$, then these equations each reduce to a simpler familiar form of the wave equation, e.g., ${\nabla }^2\mathit{E}=\mu \sigma {\displaystyle \frac{\partial \mathit{E}}{\partial t}}+ϵ\mu {\displaystyle \frac{{\partial }^2\mathit{E}}{{\partial t}^2}}$.

Equating the real parts and the imaginary parts of each of these equations again gives separate equations for r-space and t-space. For example, Equation (59) implies the following in r-space and t-space, respectively:

$${\nabla }_x^2{\mathit{E}}_{\mathit{x}}=\mu {\sigma }_x{\displaystyle \frac{\partial {\mathit{E}}_{\mathit{x}}}{\partial t}}+ϵ\mu {\displaystyle \frac{{\partial }^2{\mathit{E}}_x}{{\partial t}^2}}$$

(61)

$${\nabla }_t^2{\mathit{E}}_{\mathit{t}}={\displaystyle \frac{{\sigma }_t}{ϵ}}{\displaystyle \frac{\partial {\mathit{E}}_{\mathit{t}}}{\partial t}}+{\displaystyle \frac{{\partial }^2{\mathit{E}}_t}{{\partial t}^2}}$$

(62)

The r-space Equation (61) is the same as that found in contemporary electrodynamics textbooks ([1], p. 413; [2], p. 607), and it reverts to the standard wave equation for an empty vacuum if one takes ${\sigma }_x=0$.

Equations (61) and (62) indicate that wave propagation unexpectedly differs in r-space and t-space. In r-space, the factor $ϵ\mu$ immediately preceding ${\displaystyle \frac{{\partial }^2{\mathit{E}}_x}{{\partial t}^2}}$ on the right side of Equation (61) corresponds to the reciprocal of the squared speed ${\left|{\displaystyle \frac{d\mathit{x}}{dt}}\right|}^2$ with which the wave propagates through r-space. Thus, ${\left|{\displaystyle \frac{d\mathit{x}}{dt}}\right|}^2={\displaystyle \frac{1}{ϵ\mu }}={\displaystyle \frac{c^2}{n^2}}$. Since velocity ${\mathit{v}}_{\mathit{x}}={\displaystyle \frac{\mathit{d}\mathit{x}}{\mathit{d}\mathit{t}}}$, the speed of wave propagation in r-space is $v_x=\left|{\mathit{v}}_{\mathit{x}}\right|=\left|{\displaystyle \frac{d\mathit{x}}{dt}}\right|=c/n$ m/s, which is consistent with experimental observations. Analogously, Equation (62) indicates that the imaginary portions of electromagnetic waves in t-space also propagate through time, a concept that does not exist in standard electrodynamics. The factor 1 s/s immediately preceding ${\displaystyle \frac{{\partial }^2{\mathit{E}}_t}{{\partial t}^2}}$ on the right side of Equation (62) is the reciprocal of the squared rate ${\left|{\displaystyle \frac{d\mathit{t}}{dt}}\right|}^2$ with which the wave is propagating through t-space with respect to clock time t; hence, $\left|{\displaystyle \frac{d\mathit{t}}{dt}}\right|=1$ s/s. Since the temporal velocity ${\mathit{v}}_{\mathit{t}}=c{\displaystyle \frac{\mathit{d}\mathit{t}}{\mathit{d}\mathit{t}}}$ scales this quantity by c, the speed of wave propagation in t-space is $v_t=\left|{\mathit{v}}_{\mathit{t}}\right|=c\left|{\displaystyle \frac{d\mathit{t}}{dt}}\right|=c$ m/s. Thus, surprisingly, electromagnetic waves propagate through the t-space with a speed of c m/s even inside the conducting materials.

Figure 3 illustrates this wave propagation from the perspective of an imagined observer o at rest at the origin of r-space in an inertial reference frame where the observer emits a pulse of electromagnetic radiation (e.g., a light pulse inside of salt water) at the observer’s location at t = t_o. In r-space (Figure 3a, in red), a spherical wave propagates outward with speed $v_x=c/n$ m/s, as is well known, while observer o remains at the origin. The wavefront’s radius is given by $r={\displaystyle \frac{c}{n}}t$. In t-space (Figure 3b), there are two differences. First, while a spherical wave again propagates outward, it now has speed $v_t=c$ m/s; hence, the wavefront’s radius is given by $r=ct$. This suggests that t-space is best viewed as having the properties of a vacuum, even inside a material object. This is consistent with the earlier inference (Section 2) that, unlike electromagnetic fields, while electric charge exists in the t-space, it does not extend into the t-space (it only exists at a point in the t-space). Second, by the universal speed constraint described above, the observer at rest in r-space is also moving at speed $v_{\tau }=c$ m/s through t-space in the (arbitrary) direction of the dotted arrow in Figure 3b. Thus, the imagined observer is moving through the t-space along with the wavefront. This is fundamentally different from standard electrodynamics, where there is no concept that waves propagate through time (t-space). It is challenging to visualize the hyper-ellipsoid wavefront in the six real-valued dimensions of this spacetime. Figure 3c shows a cross-section of this wavefront in the plane formed by the x_j and t_j axes (j = 1, 2, or 3) where the observer is at rest in r-space and axis t_j is chosen to be the direction that the observer is moving in t-space. Points (x_j, t_j) on this curve are characterized by ${\displaystyle \frac{1}{{\left(c/n\right)}^2}}{x}_j^2+{\displaystyle \frac{1}{c^2}}{\left(c{t}_j\right)}^2=t^2$, an ellipse with eccentricity $1-{\displaystyle \frac{1}{n^2}}$, where n is the material’s index of refraction. Observer o, at rest in r-space with the t_j axis chosen in the direction that o is moving in t-space, is seen to be moving along with the wavefront in t-space. The horizontal dashed line through the observer in Figure 3c is tangent to the wavefront. If the observer emits a sequence of three light pulses at the times indicated by red ×’s in Figure 3d, the wavefronts in the observer’s reference frame would be as pictured. From the observer’s perspective, the unobservable local t-space fields are oscillating and accumulating locally as a single combined wavefront.

4.3. Plane Wave Solution to the Wave Equation and Its Implications

Consideration of a simple plane wave solution to Equation (59) provides further insight into the nature of t-space and wave propagation inside of it. Suppose that a single isolated source at rest at the origin of the r-space inside of a conducting medium emits a monochromatic electromagnetic pulse that generates a hyper-ellipsoid wave propagating through the medium’s r-space and t-space. Sufficiently distant from this source, a small portion of the wavefront can be approximated locally by a monochromatic sinusoidal plane wave given by the following:

$$\mathit{E}={\mathit{E}}_o{e}^{i\varphi }$$

(63)

where ${\mathit{E}}_o$ is a constant 3D real-valued vector. With the standard MEs in r-space, complex exponential forms like this are frequently used as a computational convenience to represent plane wave solutions, but the imaginary part of the solution is ultimately discarded. Here, we are not discarding the imaginary portion because we are working with complex-valued fields.

An inertial coordinate system is selected such that the observer is at rest in r-space, including along coordinate x_j, but is moving in the direction of coordinate t_j in t-space for arbitrary j in {1, 2, 3} as in Figure 3c, such that ${\displaystyle \frac{\partial t_j}{\partial t}}=1$. Consider a plane wave tangent to the portion of the hyper-ellipsoid wave that is propagating along with the observer (horizontal dotted line through the observer in Figure 3c). Accordingly, we take the following to be the wave phase (ignoring a phase constant for simplicity)

$$\varphi =k{x}_j-\omega t_j$$

(64)

as is often done in standard classical electrodynamics in r-space within conducting matter. Here, k is a complex number $k=k_x+i{\delta }_k$, where $k_x$ is the familiar spatial wave number, and${\delta }_k$ is a spatial attenuation coefficient. Analogously, in t-space, we take $\omega$ to be a complex number $\omega ={\omega }_x+i{\delta }_{\omega }$, where ${\omega }_x$ is the wave’s familiar angular frequency, and ${\delta }_{\omega }$ is a temporal attenuation coefficient. Quantities${\delta }_k$ and${\delta }_{\omega }$ are called the attenuation coefficients because when Equation (63) is written as follows, they indicate how quickly the wave attenuates as it propagates in r-space and t-space:

$$\mathit{E}={\mathit{E}}_o{e}^{-{\delta }_k{x}_j+{\delta }_{\omega }{t}_j}{e}^{i\left(k_x{x}_j-{\omega }_x{t}_j\right)}$$

(65)

Substituting plane wave Equation (63) into the wave Equation (59), the derivation in Supplemental Information S5 shows that it is a solution of the wave equation if the following four constraints can be satisfied:

$$\mathrm{C}1:k_x^2-{\delta }_k^2=ϵ\mu {\omega }_x^2-ϵ\mu {\delta }_{\omega }^2-\mu {\sigma }_x{\delta }_{\omega }$$

(66)

$$\mathrm{C}2:k_x{\delta }_k={\displaystyle \frac{1}{2}}\mu {\sigma }_x{\omega }_x+ϵ\mu {\omega }_x{\delta }_{\omega }$$

(67)

$$\mathrm{C}3:\mu {\sigma }_t{\omega }_x=0$$

(68)

$$\mathrm{C}4:{\mu \sigma }_t{\delta }_{\omega }=0$$

(69)

The two constraints C1 and C2 pertaining to r-space are informative when they are compared to the generally accepted corresponding constraints derived from plane wave solutions for the standard real-valued MEs, which are the following:

$$k_x^2-{\delta }_k^2=ϵ\mu {\omega }_x^2$$

(70)

$$k_x{\delta }_k={\displaystyle \frac{1}{2}}\mu {\sigma }_x{\omega }_x$$

(71)

These are given for r-space in classical electrodynamics ([1], Eq. 9.124; [2], Eq. 17.106). These well-established constraints give the correct results for the dependency of $k_x$ and ${\delta }_k$ on ${\omega }_x$ in r-space. Comparing Equations (66), (67) to Equations (70), (71), respectively, the latter are seen to be embedded in the former. It follows that both $ϵ\mu {\delta }_{\omega }^2+\mu {\sigma }_x{\delta }_{\omega }=0$ and $ϵ\mu {\omega }_x{\delta }_{\omega }=0$ must hold. The only way that these latter two equations can both hold is if the following is satisfied since all the other quantities in the equations are non-zero:

$${\delta }_{\omega }=0$$

(72)

This gives an additional difference between wave portions in r-space and t-space: the portions of electromagnetic waves in t-space do not rapidly attenuate inside a conductor like the portions in r-space do. Further, it follows from constraint C3 that the conductivity of t-space must be the following as $\mu$ and ${\omega }_x$ are both non-zero:

$${\sigma }_t=0$$

(73)

Constraint C4 is automatically satisfied by this value of ${\sigma }_t$.

The observations that ρ and J do not have imaginary components (Section 2), $v_{\tau }=c$, ${\sigma }_t=0$, and ${\delta }_{\omega }=0$, are consistent with the conclusion that wave propagation in t-space effectively acts like it is occurring in a vacuum, even inside of conducting matter. Thus, the t-space components of radiated wave packets would not undergo rapid attenuation or dispersion like the r-space components. This does not mean that the t-space portion of waves does not weaken with distance from their source inside of matter. For example, with a point source, the resulting spherical t-space wavefront (Figure 3b) field strength would weaken in proportion to ${\displaystyle \frac{1}{r}}$, where r is the distance in t-space from the source.

5. Implications for Electromagnetic Field Theories of Consciousness

In making MEs more symmetric, complex-valued fields provide a novel explanation for how magnetic monopoles can exist but have not been found in past experimental searches. It is thus natural to inquire whether complex-valued fields in matter might provide novel explanations for additional outstanding scientific puzzles. Here, we consider the poorly understood and enormously important unsolved problem of determining the physical basis of consciousness. It has previously been suggested that to understand consciousness scientifically, we will need to develop new physical concepts [18], that these new concepts should focus on extending the standard model of electromagnetic fields [19], and that complex-valued fields may play an important role in this [20]. Any novel insights into conscious experience based on the properties of complex-valued electromagnetic fields derived in the previous sections would be remarkable, given that these properties were not derived from neuroscientific or psychological knowledge about consciousness.

The rest of this section considers how the existence of fields that include imaginary-valued components would impact the explanatory power of existing EMFTCs. While conscious experience will be attributed in the following to all of the complex fields hypothesized to exist in the brain and not to just the r-space fields as past EMFTCs have done, the possible implications of the t-space portion of electromagnetic waves are emphasized as there is nothing like this in standard classical electrodynamics. This is an especially significant extension to existing EMFTCs because the imaginary-valued field components are taken to extend into time rather than space, and time is arguably the key to understanding conscious experience [21,22].

5.1. Extending Past Electromagnetic Field Theories of Consciousness

Microscopically, the human brain and its endogenous electromagnetic fields that arise from neural and synaptic activity exhibit a remarkable complexity [23]. The brain’s endogenous fields are generated by its extensive ionic content as well as intracellular, extracellular, and transmembrane ionic current flows at synapses and during action potentials. In contrast, macroscopically (which is all that is considered here), the brain’s electromagnetic properties can be approximated as if the brain is a multi-compartment volume conductor [23,24,25]. This macroscopic level is widely used in modeling the brain’s biophysical fields resulting from the synchronized activity and synaptic potentials of numerous neurons, such as the fields detected by electroencephalography (EEG). Approximating the brain’s electromagnetic fields in this linearized way generally leads to reasonable agreement with macroscopic experimental findings [24].

There are numerous correlations between macroscopic electromagnetic phenomena occurring in the brain and a person’s state of consciousness. For example, unconscious states (deep non-REM sleep, coma, general anesthesia) are associated with very low EEG frequencies of 0.5–4 Hz (delta band), while consciousness is largely associated with higher frequencies in the 8–100 Hz range (alpha, beta, and gamma bands) [24]. Additional arguments for the relevance of electromagnetic fields to conscious experience include their preeminence in the brain compared to other fundamental physical forces (gravity, weak/strong nuclear) and their ability to explain the unity of conscious experience [26,27,28,29]. While historically, the brain’s electromagnetic fields were viewed as an epiphenomenon of the underlying neural activity and computations, this view is no longer tenable due to experimental demonstrations that these fields exert causal influences on the activity and synchronization of neurons [23,30].

Given these correlations, not surprisingly, there are a number of electromagnetic field theories of consciousness (EMFTCs). Such theories attribute consciousness to the brain’s endogenous (r-space) fields rather than to the brain’s underlying material structures and activity. In this sense, they provide a “physical dualism” alternative to the non-physical “ghost in the machine” of Cartesian dualism. While some of these theories incorporate concepts from quantum electrodynamics, the vast majority are based solely on classical real-valued electrodynamics, e.g., [30,31,32,33,34,35,36]. A recent lucid review of past work on EMFTCs is available [27]. These past EMFTCs collectively provide a compelling argument that the brain’s endogenous electromagnetic fields are in some way responsible for phenomenal consciousness. Previous EMFTCs have assumed that the standard real-valued MEs are complete, and none of them have considered the possible implications of complex-valued fields with imaginary-valued components that extend into time.

Figure 4 illustrates the tripartite nature of the entities considered here in extending EMFTCs to accommodate complex-valued fields: the material brain, the near zone fields, and the radiated far zone fields. The underlying material brain is taken to be a biological “machine”, just as other body organs (kidneys, lungs, heart, etc.) are viewed as being non-mysterious machines by contemporary biologists. The neuronal activation dynamics and synaptic changes of the brain’s underlying neural circuitry are viewed as the computational machinery (an automaton) responsible for many aspects of cognitive processes, short-term working memory, and learning, as is widely accepted (Figure 4a). These material mechanisms also generate and tap into the brain’s r-space endogenous electromagnetic fields to which consciousness is attributed by EMFTCs. In extending EMFTCs to complex-valued fields, the brain’s interactions with the fields continue to be the basis of ongoing conscious experience as in past EMFTCs, but now also include the imaginary-valued field components in t-space (Figure 4b). Following standard practice in physics and engineering, we can separate the brain’s fields into those localized in the near zone (near field, reactive field) where they are generated and those that are in the far zone (far field, radiation field) from where they are radiated. The material brain is assumed to interact with all of these electromagnetic fields to produce conscious experience. Interactions with the radiated far zone fields in t-space are taken to contribute to the memory of past conscious experiences (Figure 4c). The existence of imaginary-valued field components will be seen below to greatly increase the explanatory power of EMFTCs.

As noted above, extensive experimental observations associate conscious experience with brain electromagnetic field oscillations roughly in the 8–100 Hz range. This EEG frequency range is found not only in humans but also in all tested mammals, from mice to large aquatic animals, independent of large variations in brain size [37,38]. This suggests that, for currently unknown reasons, these frequencies are especially relevant to the existence of conscious experience. The wavelengths λ_x in r-space associated with radiated waves at such ultra/extremely low frequencies are huge relative to the brain’s size in r-space, roughly on the order of ${\lambda }_x\approx {10}^6$ m. Thus, the radiated r-space portion of waves is of little interest in considering the brain’s conscious experiences, and these radiated r-space waves are not incorporated into past EMFTCs nor into the complex-valued theory considered here. However, in t-space, the situation is quite different because the brain’s size in t-space is relatively enormous: 1 s corresponds to c meters (see first paragraph of Section 4.1). Therefore, a wavelength of 10⁶ m corresponds to a “temporal distance” of 10 msec. Thus, the wavelengths associated with temporal portions of radiated waves in the brain are on the same time scale at which state changes of neurocomputational and cognitive processes are occurring [39,40,41]. This suggests that the t-space portion of waves would be able to transfer information at a scale relevant to consciously experienced cognition, as indicated in Figure 4c. Unlike the r-space portion of electromagnetic waves, the t-space portion of radiated fields is thus unexpectedly relevant to conscious experience, and it turns out to play a central role in considering complex-valued EMFTCs in the following (hence “t-space” but not “r-space” in Figure 4c).

It is assumed in the following that the endogenous fields are not only generated by the brain but that they can also causally affect the material brain’s activation dynamics. While this is experimentally verified bidirectionally for r-space field components [23,30], it is currently only a theoretical hypothesis for the t-space field components of complex-valued MEs, as indicated by the dotted arrows in Figure 5a. An unexpected prediction of the complex-valued MEs (Equations (40)–(43)) is that the t-space portion of the brain’s radiated fields move through time as an accumulating wavefront at the same speed as the material brain (as suggested by Figure 3c,d where the observer o is now the brain). What this implies is that, at each instance of time, the brain’s newly radiated waves merge with those from the past, accumulating as a single “wave packet” in time rather than space, as illustrated in Figure 5b. We will refer to this local accumulating t-space portion of endogenously generated electromagnetic waves as the temporal wavefront in the following. The temporal wavefront remains localized in the brain. It persists over long time durations because of the lack of attenuation and dispersion seen with t-space portions of electromagnetic waves even inside of matter (see Section 4.3). It is relevant to conscious experience because it moves at the same speed c through time as the resting brain (one second per second) and not through r-space in the brain’s reference frame. Thus, to the extent that complex-valued electromagnetic fields correspond to conscious experience, such a persistent temporal wavefront is a remarkable physical correlate of consciousness.

We next consider how the inclusion of imaginary-valued field components would contribute to explaining some poorly understood properties of conscious experience. In the following, these puzzling properties are divided into those involving the subjective perception of time (Section 5.2) and those involving the recall of past conscious experiences (Section 5.3). To the author’s knowledge, no past EMFTC has explicitly addressed either of these two issues.

5.2. Properties of the Subjective Experience of Time

Cognitive psychologists distinguish an additional concept of time different from the objective time t (chronometric time) used in physics. This subjective time t_s refers to the consciously experienced passage of time, which has been extensively studied in psychology and involves multiple cortical and subcortical structures [21,42]. The passage of subjective time is linearly associated with the passage of objective clock time [43]. The subjective awareness of time is closely associated with consciousness—we experience subjective time passing when we are awake but not when we are completely unconscious. The neurobiological basis and physical basis of experienced subjective time are currently poorly understood in the context of existing EMFTCs. Here, we consider two puzzling properties of subjective time perception that are not fully explained by existing EMFTCs.

Property #1, The Duration of Now: One puzzling aspect of subjective experience is that, according to cognitive psychologists, the present moment, sometimes referred to as the conscious experience of Now (or the specious present), is not an instantaneous point that separates the past from the future. The subjective Now actually extends over an interval of objective time, having a duration estimated as lasting 100 msec up to a couple of seconds [42,44,45]. This non-instantaneous duration of consciousness is reasonably argued to be necessary for perceiving motion and for coherently processing temporal sequences in general [21]. However, this substantial duration of Now is a challenge to current EMFTCs because the real-valued fields involved, while they exist in time (being functions of time as well as space), do not extend into time. They are instantaneous. In contrast, the complex fields considered here automatically provide at least a partial explanation for how the current moment of consciousness can span an interval of time: their imaginary components actually extend into time. Thus, associating the imaginary-valued field components with conscious experience naturally predicts the existence of a window of duration for this experience.

Property #2, The Flow of Time: A second puzzling aspect of a person’s subjective experience of time is that it involves a perceived continuous flow of time: conscious experience involves the sense that the present moment Now is steadily moving through time. In contrast to this, the dominant view in physics today is that, while there is an arrow of time, the “flow of time” does not actually exist in physical spacetime. Instead, we exist in a static 4D “block universe” in which the past, present, and future all exist, and the sense that a person has of a flow of time is something that is generated by the conscious mind [46,47,48]. What could be the physical basis of such a conscious experience of time flow that is not explained by existing physical laws? Existing EMFTCs do not provide an explanation for this. However, EMFTCs could account for this phenomenon if their fields were modified to have complex-valued components. The temporal wavefront, radiated by the brain’s oscillating neural activity and advancing through time inside the brain at the same rate as our conscious experience (Figure 3c), could account for this subjective experience of a flow of time. Such waves in t-space are literally flowing through time, and any EMFTC that attributes conscious experience to endogenous brain fields having imaginary-valued components could thus explain the subjective experience of the flow of time. Further, this temporal wavefront would also be important for self-awareness and a sense of personal identity, consistent with previous arguments that the perception of the self is intimately linked to time perception [21,49].

5.3. Properties of Recalling Previous Conscious Experiences

Cognitive psychologists view the recall of past consciously experienced events as primarily involving episodic memory [50]. Episodic memory, which can retain information over years or a lifetime, is critically dependent upon the brain’s hippocampal regions, which provide an indexing function for memory traces believed to be stored mainly in cortical regions [51,52]. Episodic memory stores not only what the past conscious episode involved but also the location in spacetime of when and where each episode occurred, possibly based on hippocampal time and place cells that form a map of spacetime and are important for retrieving remembered events [53,54,55,56]. Recalling past conscious experiences activates the same cortical regions that the original consciously experienced events activated [57] and is, in part, reconstructive [58].

The critical point here is that past conscious experiences are currently assumed by psychologists and neuroscientists to be stored solely in the brain’s current structure at the time of recall, primarily encoded as synaptic strengths. After all, where else could they be stored? Electromagnetic fields in r-space in existing EMFTCs could not help with this storage. The r-space near zone fields rapidly change over time; hence, they cannot carry information over the long term. The radiated r-space far zone fields cannot contribute to long-term memory storage either because they quickly leave the brain, their wavelength in r-space is enormous, and they rapidly attenuate and undergo dispersion. In contrast, EMFTCs that incorporate complex-valued fields provide another previously unrecognized possibility. The t-space temporal wavefront (Figure 5b), continuously modulated by ongoing activity in the brain, provides a potential mechanism for transmitting information into the future because it is unattenuated and is not dispersive, as was shown in Section 4.3. Thus, it remains in the brain for a long time. Such a mechanism would contribute to understanding several features of recalling past conscious experiences that are not fully explained by existing EMFTCs, as follows.

Property #3, Mental Time Travel: It is well documented that during recall of past conscious experiences, one subjectively re-experiences a temporal flow of the original consciously experienced events plus a feeling of traveling back in time to the original events, something that psychologists call mental time travel [58,59]. The temporal wavefront radiated by the brain’s neuroelectric activity during conscious experiences may play a central role in explaining this phenomenon. These t-space waves, continuously generated by numerous accelerated charges throughout the brain (sudden changes in trans-membrane ion flows during synaptic activity, action potentials, etc.), would thus not only serve as the basis of the subjective flow of time as explained above but would also carry very detailed specific information about the brain’s fields at the time that the waves are generated. As explained above, both the brain in its resting r-space inertial reference frame (by Equation (58)) and the accumulated, modulated intra-brain t-space portion of electromagnetic waves that the brain generates are moving together in the same direction through t-space (and thus time) at a speed of c m/s (Figure 3c, where observer o is the brain). This is only possible because, unlike the r-space portion of a wave inside of matter, the t-space portion in the brain travels at speed c, is unattenuated, and does not undergo dispersion, even inside of the brain (see Section 4.1).

This suggests the radical possibility that, instead of past conscious experiences being stored exclusively as memory traces in the brain’s structure (synapses, etc.) as is currently thought, they are also encoded in the spatiotemporal patterns of the temporal wavefronts that are propagating forward through time within the brain. For example, in this context, the recall of past conscious visual experiences could be viewed as a type of perceptual process whereby the neocortex (colloquially, the “mind’s eye”) selectively examines and replays the accumulated t-space wave activity in the brain. As explained in Section 5.1 (Figure 5b), past internally radiated t-space waves would all be arriving at the brain at the current moment in time and thus can be consciously re-experienced. In effect, this would involve literally “seeing” the consciously experienced past events again, analogous to how we perceive events occurring in r-space based on electromagnetic waves impinging on the retina. The information conveyed by the modulated temporal wavefront would be detected by their persistent influences on spatially distributed neocortical activity. Recall of past conscious experiences would thus be based on reactivating the same cortical regions (e.g., V1–V4) in a fashion similar to their activation during the initial conscious experiences, consistent with past experimental findings [57,60]. The re-experiencing of the electromagnetic fields that occurred during past events would account for the psychologist’s well-documented characterization of episodic memory as mental time travel because that is, in effect, what is happening. In a sense, it is as if the mind actually extends back through time.

The predicted lack of attenuation and dispersion of the t-space wave portions even inside of matter (Section 4.3) makes the t-space portion of waves an ideal mechanism for long-term memory storage. However, in spite of this, slow weakening of patterned temporal wavefronts would still occur, assuming spherical waves and interference effects between multiple sources. This is consistent with experimental findings in psychology that the probability of forgetting consciously experienced events increases with their temporal distance in the past [61]. Further, experimental studies by psychologists have shown that recalling past episodic memories increases their future retention and recallability [62,63], something that has been assumed to be solely due to synaptic and other structural changes in the brain. Such findings are, however, also consistent with the theoretical idea presented here that the accumulating temporal wavefront plays a role in episodic memory. When a past experience is recalled, it enters conscious experience again as the temporal wavefront reactivates these past experiences in the same regions of the brain where they originally occurred. It would accordingly be linearly added to its previous representation in the temporal wavefront via superposition, thereby strengthening its long-term retention in memory.

Property #4, One-Step Learning: Remarkably, people exhibit the ability to continually store conscious experiences immediately in memory following only a single occurrence of each experience, something that is often called one-step learning in the neural computation literature. It is very difficult to believe that one-step storage of detailed, temporally extended experiences occurs solely due to synaptic changes in real time like this (something that is assumed by existing EMFTCs which incorporate only r-space fields), given the extensive evidence from artificial neurocomputational systems over many decades. For example, the powerful deep learning artificial neural networks currently used in AI require numerous repeated exposures to data in order to store information via synaptic changes, while computational memory models based on one-step learning methods are quite limited in capacity. The brain’s ability to record the massive amounts of information contained in conscious experiences that occur over a lifetime solely via one-step synaptic changes is thus hardly credible. The retention of past conscious experiences in modulated portions of t-space temporal wavefronts would help explain people’s ability to store these experiences immediately in memory following the single occurrence of event sequences, i.e., one-step learning. Such a storage mechanism would be automatic and immediate because it is built into the physical generation of the radiated t-space fields. Gradual loss of episodic memories over time and the loss of details concerning the oldest conscious experiences would, of course, occur, but these are consistent with the weakening of spherical electromagnetic waves over time as well as with expected interference effects between different t-space wave patterns.

Property #5, Enormous Memory Capacity: It is widely recognized that episodic memory has an enormous storage capacity, something that is currently attributed mainly to storing long-term memories as changes to hippocampal and cortical synaptic strengths. As an example of this capacity, people are able to retain detailed information about thousands of images, each seen briefly just once over several hours of time during psychological testing [64,65]. The ability of the biological brain to store this massive amount of information about conscious experiences that occur over a lifetime solely via one-step synaptic changes strains credulity. For example, working memory, which in contrast is convincingly attributed to fast synaptic changes and persistent neuronal activity in the neocortex [57,66,67], is only able to transiently retain a few recent consciously experienced events at any one time [66,68,69,70,71]. Further, the brain simultaneously supports many other memory systems (semantic, procedural, etc.), which are also presumed to occur via synaptic modifications. Existing EMFTCs do not address this issue. However, the perceptual view outlined above of episodic memory based upon a modulated endogenous temporal wavefront would help to explain the enormous memory capacity of human memory for past conscious experiences occurring over a lifetime. Instead of having to store all of this information in synaptic connections (even with generative aspects of episodic memory), much of it would be stored in the spatiotemporal patterns of the radiated far zone t-space electromagnetic fields.

Property #6, Memory Effects of Callosal Sectioning: The hypothesized perceptual mechanisms underlying episodic memory described above would also better explain neuropsychologists’ findings in split-brain subjects than existing EMFTCs. The individual cerebral hemispheres of people who have had their corpus callosum sectioned are well-known to have different episodic memories [72]. This loss of the unity of remembered conscious experiences is traditionally attributed to the structural disconnection of the hemispheres from one another. However, this phenomenon is problematic for past EMFTCs that attribute conscious experience to solely r-space electromagnetic fields. The spatially distributed r-space fields in the two hemispheres following callosal sectioning would still intermingle across the new interhemispheric cerebrospinal-filled gap. Thus, they would not obviously be expected to produce this loss of unity in recalling past conscious experiences. However, with complex-valued fields as proposed here, associating the memory of past conscious experiences with the t-space temporal wavefront in the brain would not face this difficulty. The imaginary-valued portion of each hemisphere’s temporal wavefront would be expected to be largely restricted spatially to the hemisphere from which they were generated (they are not propagating in r-space), consistent with the observed different recall of past conscious experiences by each of the two isolated hemispheres.

6. Discussion

The work on complex-valued electromagnetic fields considered here originally began with a relatively simple question: What would the consequences be of maximizing the symmetry of Maxwell’s equations (MEs) by assuming that electromagnetic fields have complex-valued components? One might anticipate that using complex-valued fields with MEs (rather than just the usual real-valued components) would necessarily lead to predictions that are inconsistent with existing experimental data. It turns out that this is not necessarily the case, and what results from extending MEs in this way is intriguing. For example, complex-valued fields allow one to make MEs much more symmetric by introducing magnetic charge while remaining consistent with the numerous past experimental searches that have been unable to find magnetic monopoles [4,5]. This is because any electrically charged particle (electrons, protons, etc.) is also magnetically charged, but the magnetic fields involved extend into time (t-space) rather than into space (r-space). As discussed previously [4,5], such magnetic monopoles are thus qualitatively simpler than many of those proposed in the past. They do not require the existence of a new type of particle, and their magnetic fields in t-space would not have been detected in the past experimental searches looking for particles with magnetic fields in r-space. While imaginary-valued field components are only a theoretical concept at present, their existence is, in principle, falsifiable ([5], Section 5.1), but there is currently no known experimental evidence for this. Further, their imaginary components are intuitively plausible in the context of current theory that considers space and time to be a unified spacetime. Thus, a priori, one would expect electromagnetic fields to extend into time as well as space.

The current paper extends previous work on complex-valued electromagnetic fields by deriving a form of MEs suitable for application inside of conducting matter. This was performed using the same methods as are typically used in adopting standard real-valued MEs for application in linear conducting matter (replacing ${ϵ}_0$ with $ϵ$ and ${\mu }_0$ with $\mu$, using Ohm’s law, deriving a wave equation using $\nabla \times \nabla \times$, etc.) except now using complex-valued fields. The resulting complex-valued MEs for inside matter have several significant implications. They remain consistent with the standard real-valued MEs in matter but are more symmetric, as shown by electromagnetic duality transformations (symmetry operations). These equations and the wave equations derived from them also imply three interesting properties, as follows.

First, the derived complex-valued MEs exhibit not only a within-domains electromagnetic duality transformation $T_{wd}^c$ but also a new type of between-domains duality transformation $T_{bd}^c$. The first of these, $T_{wd}^c$, swaps field vectors ${\mathit{E}}_x$ and B_x that are both within the domain of r-space and field vectors ${\mathit{E}}_t$ and B_t, which are both within the domain of t-space. The second transformation, $T_{bd}^c$, swaps field vectors ${\mathit{E}}_x$ and B_t that are in the different domains of r-space and t-space, and it similarly swaps field vectors ${\mathit{E}}_t$ and B_x that are in different domains. The latter $T_{bd}^c$ is novel in that it has no analog in standard real-valued electrodynamics extended to have magnetic monopoles (Figure 1b,c). Its significance is that it defines a cross-domain mapping that allows one to translate r-space equations into their dual form in t-space.

Second, a wave equation derived from the complex-valued MEs in matter predicts that electromagnetic waves will propagate in both r-space (space) and t-space (time), the latter having no analog in classical electrodynamics. The r-space portion of the wave has the same properties in matter as do waves associated with standard real-valued MEs: reduced propagation speed of c/n, rapid attenuation in conducting matter, etc. In contrast, the complex-valued MEs predict that inside matter, the t-space portion of the wave will propagate just as it would in a vacuum, retaining a speed of c and not exhibiting rapid attenuation or dispersion.

Third, as an example of how electromagnetic fields incorporating imaginary-valued components might have concrete, real-world consequences, it was shown that the existence of complex-valued fields in matter would substantially increase the explanatory power of any electromagnetic field theory that attributes conscious experience to the endogenous fields in the brain. When the brain is viewed as a volume conductor, complex-valued fields can contribute to explaining several poorly understood aspects of consciousness that have been experimentally established by cognitive scientists and neuroscientists. These aspects include the temporal duration of consciousness, the subjective sense of time flow that is not accounted for in current physics theory, the notion of recalling past experiences as mental time travel, the one-step learning process of conscious events, the vast storage capacity of episodic memory, and the compartmentalization of memory after callosal sectioning. This explanatory power is remarkable given that the complex-valued MEs were not developed based on addressing aspects of consciousness but instead on trying to increase their symmetry. While none of these observations directly confirms the existence of complex-valued fields, collectively they support the plausibility of such fields. The existence of complex-valued fields would have substantial implications for issues such as whether some non-human animals are conscious and how one might go about creating an artificial consciousness [73].

The work described in this paper has some limitations. The idea of complex-valued electromagnetic fields is currently a purely theoretical concept. It is based solely on the philosophy that symmetry in natural laws is an important foundational principle of physics. While the idea of complex-valued fields is, in principle, testable (see [5]), at the present time, there is neither experimental evidence confirming the existence of imaginary-valued field components nor refuting their existence. Further, the results described here were derived solely within the scope of classical electrodynamics and thus are only an initial step in analyzing complex-valued electromagnetic fields in matter. Classical theory can, at best, provide an approximation to reality, and thus, the extension of this work to quantum field theory as it relates to the properties of matter is extremely important. Such extensions, which would be quite difficult, could address important issues, including their implications for quantum tunneling effects, entanglement, and quantum cosmology. Finally, like most past work on the physical and biological nature of conscious experience, the analysis of the implications of complex-valued fields for electromagnetic field theories of consciousness (Section 5) is speculative. It is only intended to show that the results presented here have the potential to lead to a deeper understanding of human perception of time and the nature of human memory.

7. Conclusions

This research is asking what the implications would be of maximizing the symmetry of Maxwell’s equations (MEs) by assuming that electromagnetic fields have complex-valued components. Previous related work [4,5] has shown that, in free space, such a generalization of MEs would explain how magnetic monopoles can exist yet not be found in past experimental searches for them, and that electromagnetic waves would not only propagate through real-valued space but also through an associated t-space defined by the imaginary-valued temporal components of the fields. In the current paper, the complex-valued MEs modified for inside of matter lead to several conclusions. First, the complex-valued MEs for free space can be used for inside of matter while remaining consistent with the standard real-valued MEs in matter and, hence, with existing experimental results in classical electrodynamics. Second, the resulting complex-valued MEs in matter are more symmetric than the corresponding standard MEs, as demonstrated via the derivation of symmetry operations (electromagnetic duality transformations T_wd, T_bd, etc.) under which the complex-valued equations are invariant. Third, and completely unexpected, inside of matter, the portions of electromagnetic waves propagating through t-space defined by the imaginary components of the fields advance at the speed of light c and are not slowed like the real-valued portions of waves. Fourth, the existence of imaginary components would greatly expand the explanatory power of electromagnetic field theories of consciousness by accounting for several properties of the subjective experience of time and the recall of past conscious experiences that have been experimentally determined by cognitive neuroscientists. Finally, these conclusions are limited in that they are based on purely theoretical ideas about the importance of symmetry in the foundations of physics, and they are derived solely within the scope of classical electrodynamics. Thus, future work is needed to experimentally falsify/verify the theoretical ideas considered here and to extend the analysis to quantum field theory.