From a 4-Rank Totally Antisymmetric Field Strength to Two Dual Electromagnetic Fields in Four Time and Four Space Dimensions

Abstract

A 4-rank “electromagnetic” gauge field strength in four-time and four-space dimensions (4+4-dimensions) is considered. We show that by necessitating such a four-rank gauge field we satisfy the Grassmann–Plücker relations which allows to choose an ansatz (for a basic basis) such that it is broken into two dual electromagnetic fields – one in a 1+3-world and the other in a 3+1-world. An interesting aspect of this mechanism is that the electromagnetic field 1+3-world turns out to be dual to the electromagnetic field in the 3+1-world.

1. Introduction

It is known that the Grassmann Plücker relations [1, 2, 3, 4, 5, 6] of totally antisymmetric forms determine the Plücker coordinates which mean that such a totally antisymmetric form is decomposable (see Ref. [3] and references therein). Physically, the Plücker embedding can be found in a Grassmannian sigma model in $\mathit{SU}\left(2\right)$ Yang Mills model [7] and in spherically symmetric instantons of the scale invariant $\mathit{SU}\left(2\right)$ gauged Grassmannian model in $d=4$ [8]. When this developments are applied to $2$-rank antisymmetric gauge field in four-dimensions, it is found that the corresponding electromagnetic field strength can be written in terms of the true degrees of freedom [9].

In this work, we make a number of remarks on Plücker coordinates associated with $4$-rank totally antisymmetric gauge fields strength $F^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}$ (differential 4-form) in $\left(4+4\right)$-dimensions. When such a gauge field satisfies the Grassmann–Plücker relations can be, of course, decomposable in terms of more elementary quantities. Surprisingly, for a particular case of ansatz for such elementary basic quantities, the field equations for $F^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}$ lead to both the electromagnetic field equation for $F^{\mathit{\mu \nu }}$ in $\left(1+3\right)$-dimensions and the electromagnetic field equation for $G^{\mathit{ij}}$ in $\left(3+1\right)$-dimensions. An interesting aspect of this mechanism is that the source of $F^{\mathit{\mu \nu }}$ is determined in part by $G^{\mathit{ij}}$ and dually the source of $G^{\mathit{ij}}$ is determined in part by $F^{\mathit{\mu \nu }}$.

2. Field equations for a 4-rank field strength

Let us start considering a totally antisymmetric gauge field strength

$$F^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}=F^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}\left(x^{\sigma },y^i\right)$$

in $\left(4+4\right)$-dimensions, which we shall assume is a function of $\left(1+3\right)$-coordinates $x^{\sigma }$ and $\left(3+1\right)$-coordinates $y^i$. Suppose that $F^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}$ satisfies the Maxwell type equations:

$${\partial }_{\widehat{\beta }}{F}^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}=0$$E1

and

$${{\partial }_{\widehat{\beta }}}^{\ast }{F}^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}=0,$$E2

where ${}^{\ast }{F}^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}$ is the dual gauge field defined as

$${}^{\ast }{F}^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}=\frac{1}{4!}{\mathrm{ϵ}}^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }\widehat{\sigma }\widehat{\rho }\widehat{\gamma }\widehat{\eta }}{F}_{\widehat{\sigma }\widehat{\rho }\widehat{\gamma }\widehat{\eta }}.$$E3

In general, we raise and lower indices with a flat Minkowski metric ${\eta }_{\widehat{\mu }\widehat{\nu }}$, which in $\left(4+4\right)$-dimensions takes the form

$${\eta }_{\widehat{\mu }\widehat{\nu }}=\mathit{diag}\left(-\mathrm{1,1,1,1},-1,-1,-1,1\right).$$E4

The $\mathrm{ϵ}$-symbol is a totally antisymmetric symbol (Levi–Civita symbol) defined as

$${\mathrm{ϵ}}^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }\widehat{\sigma }\widehat{\rho }\widehat{\gamma }\widehat{\eta }}\in \left\{-\mathrm{1,0,1}\right\}.$$E5

In fact, the $\mathrm{ϵ}$-symbol has values $+1$ or $-1$ depending on even or odd permutations of ${\mathrm{ϵ}}^{\mathrm{12...8}}$, respectively, otherwise the $\mathrm{ϵ}$-symbol is zero. It is verified that the relation

$${\mathrm{ϵ}}^{{\widehat{\mu }}_1\dots {\widehat{\mu }}_8}{\mathrm{ϵ}}_{{\widehat{\nu }}_1\dots {\widehat{\nu }}_8}=-{\delta }_{{\widehat{\nu }}_1\dots {\widehat{\nu }}_8}^{{\widehat{\mu }}_1\dots {\widehat{\mu }}_8},$$E6

where ${\delta }_{{\widehat{\nu }}_1\dots {\widehat{\nu }}_8}^{{\widehat{\mu }}_1\dots {\widehat{\mu }}_8}$ is a generalized Kronecker delta [10].

Let us assume that the Grassmann–Plücker relations (see Ref. [3] and references therein) hold for $F^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}$, namely

$$F^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }[\widehat{\beta }}{F}^{\widehat{\sigma }\widehat{\rho }\widehat{\gamma }\widehat{\eta }]}=0.$$E7

Here, the bracket $\left[\widehat{\beta }\widehat{\sigma }\widehat{\rho }\widehat{\gamma }\widehat{\eta }\dots \right]$ means totally antisymmetric. This implies that $F^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}$ is decomposable. In other words, this means that $F^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}$ can be written as

$$F^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}=\frac{1}{4!}{\epsilon }^{\widehat{A}\widehat{B}\widehat{C}\widehat{D}}{v}_{\widehat{A}}^{\widehat{\mu }}{v}_{\widehat{B}}^{\widehat{\nu }}{v}_{\widehat{C}}^{\widehat{\alpha }}{v}_{\widehat{D}}^{\widehat{\beta }}.$$E8

Here, the elementary quantities $v_{\widehat{A}}^{\widehat{\mu }}=v_{\widehat{A}}^{\widehat{\mu }}\left(x^{\sigma },y^i\right)$ can be considered as basic basis elements and ${\epsilon }^{\widehat{A}\widehat{B}\widehat{C}\widehat{D}}$ is an ‘internal’ four-dimensional $\mathrm{ϵ}$-symbol.

Moreover, from (2) one learns that

$$F_{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}={\partial }_{\widehat{\mu }}{A}_{\widehat{\nu }\widehat{\alpha }\widehat{\beta }]}.$$E9

So one sees that $A_{\widehat{\nu }\widehat{\alpha }\widehat{\beta }}$ is a totally antisymmetric gauge field which under the transformation

$$A_{\widehat{\nu }\widehat{\alpha }\widehat{\beta }}\to A_{\widehat{\nu }\widehat{\alpha }\widehat{\beta }}+{\partial }_{\widehat{\nu }}{\mathrm{\Omega }}_{\widehat{\alpha }\widehat{\beta }]},$$E10

the $4$-rank tensor $F_{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}$ becomes invariant.

3. Kaluza-Klein type ansatz

In general, one finds that $F^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}$ can be written as

$$\begin{array}{l}{F}^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}=F^{\widehat{\mu }\widehat{\nu }}{G}^{\widehat{\alpha }\widehat{\beta }}-F^{\widehat{\mu }\widehat{\alpha }}{G}^{\widehat{\nu }\widehat{\beta }}+F^{\widehat{\mu }\widehat{\beta }}{G}^{\widehat{\nu }\widehat{\alpha }}\\ +G^{\widehat{\mu }\widehat{\nu }}{F}^{\widehat{\alpha }\widehat{\beta }}-G^{\widehat{\mu }\widehat{\alpha }}{F}^{\widehat{\nu }\widehat{\beta }}+G^{\widehat{\mu }\widehat{\beta }}{F}^{\widehat{\nu }\widehat{\alpha }},\end{array}$$E11

where

$$F^{\widehat{\mu }\widehat{\nu }}=\frac{1}{4!}{\mathrm{ϵ}}^{\mathit{ab}}{v}_a^{\widehat{\mu }}{v}_b^{\widehat{\nu }}$$E12

and

$$G^{\widehat{\mu }\widehat{\nu }}=\frac{1}{4!}{\mathrm{ϵ}}^{\mathit{AB}}{v}_A^{\widehat{\mu }}{v}_B^{\widehat{\nu }}.$$E13

Our next step is to consider some particular cases. In principle one may assume a kind of Kaluza–Klein ansatz for $v_{\widehat{A}}^{\widehat{\mu }}$, namely

$$v_{\widehat{A}}^{\widehat{\mu }}=\left(\begin{array}{ll}{v}_a^{\mu }& v_A^{\mu }\\ 0& v_A^i\end{array}\right).$$E14

However, in this case, one looses the symmetry between the $\left(1+3\right)$-world and the $\left(3+1\right)$-world. So one shall assume that

$$v_{\widehat{A}}^{\widehat{\mu }}=\left(\begin{array}{ll}{v}_a^{\mu }& 0\\ 0& v_A^i\end{array}\right).$$E15

In this case, one gets

$$F^{\mathit{\mu i}}=0$$E16

and

$$F^{\mathit{ij}}=0.$$E17

And one also obtains

$$G^{\mathit{\mu i}}=0$$E18

and

$$G^{\mathit{\mu \nu }}=0.$$E19

Hence, using Eq. (11) one learns that the only nonvanishing component of $F^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}$ is

$$F^{\mathit{\text{μνij}}}=F^{\mathit{\mu \nu }}\left(x^{\sigma },y^k\right)G^{\mathit{ij}}\left(x^{\lambda },y^k\right),$$E20

where the “electromagnetic” fields $F_{\mathit{\mu \nu }}$ and $G_{\mathit{ij}}$ corresponds to the $\left(1+3\right)$-world and $\left(3+1\right)$-world, respectively.

4. Inhomogeneous Maxwell field equations

From (1) one knows that

$${\partial }_{\beta }{F}^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\beta }+{\partial }_j{F}^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }j}=0.$$E21

Thus, the relevant equations that can be obtained from Eq. (21) are

$${\partial }_{\nu }{F}^{\mathit{\text{μνij}}}=0$$E22

and

$${\partial }_j{F}^{\mathit{\text{μνij}}}=0.$$E23

Hence, using Eq. (20) one learns that

$$\left({\partial }_{\nu }{F}^{\mathit{\mu \nu }}\right)G^{\mathit{ij}}+\left({\partial }_{\nu }{G}^{\mathit{ij}}\right)F^{\mathit{\mu \nu }}=0$$E24

and

$$\left({\partial }_j{G}^{\mathit{ij}}\right)F^{\mathit{\mu \nu }}+\left({\partial }_j{F}^{\mathit{\mu \nu }}\right)G^{\mathit{ij}}=0.$$E25

These equations can also be written as

$${\partial }_{\nu }{F}^{\mathit{\mu \nu }}=J^{\mu }$$E26

and

$${\partial }_j{G}^{\mathit{ij}}=J^i.$$E27

Here,

$$J^{\mu }=-F^{\mathit{\mu \nu }}{\partial }_{\nu }ln\psi$$E28

and

$$J^i=-G^{\mathit{ij}}{\partial }_{j}ln\xi ,$$E29

with

$$\psi ={\left(\frac{1}{2}{G}^{\mathit{ij}}{G}_{\mathit{ij}}\right)}^{1/2}$$E30

and

$$\xi ={\left(\frac{1}{2}{F}^{\mathit{\mu \nu }}{F}_{\mathit{\mu \nu }}\right)}^{1/2}.$$E31

Another interesting way to write (26) and (27) is

$${\partial }_{\nu }\left(\psi F^{\mathit{\mu \nu }}\right)=0$$E32

and

$${\partial }_j\left(\xi G^{\mathit{ij}}\right)=0,$$E33

respectively.

It is evident that (26) (and (27)) or (32) (and (33)) are inhomogeneous Maxwell field type equations. Consequently, $F^{\mathit{\mu \nu }}$ can be identified with the electromagnetic field strength in the $\left(1+3\right)$-world and $G^{\mathit{ij}}$ with the dual-mirror electromagnetic field strength in the $\left(3+1\right)$-world. However, (26) or (32) establishes something else that the source of the electromagnetic field $\psi$ arises in part from the $\left(3+1\right)$-world via$G^{\mathit{ij}},$ and the eqs. (27) and (33) indicate that the source of the mirror electromagnetic field emerges from the $\left(1+3\right)$-world via$F^{\mathit{\mu \nu }}$. This process shows a duality between the $\left(1+3\right)$-world and the $\left(3+1\right)$-world. An important point is that both electromagnetic fields $F^{\mathit{\mu \nu }}$ and $G^{\mathit{ij}}$ are part according to (18) of the $4$-rank gauge field strength $F_{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}$ which “lives” in a $\left(4+4\right)$-world.

5. Dual Maxwell field equations

Moreover, from Eq. (20) one can also show that the only nonvanishing components of the dual gauge field strength ${}^{\ast }{F}^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}$ are ${}^{\ast }{F}^{\mathit{\text{μνij}}}$ which can be written as

$${}^{\ast }{F}^{\mathit{\text{μνij}}}{=}^{\ast }{F}^{\mathit{\mu \nu }}{\left(x^{\sigma },y^k\right)}^{\ast }{G}^{\mathit{ij}}\left(x^{\lambda },y^k\right),$$E34

where

$${}^{\ast }{F}^{\mathit{\mu \nu }}=\frac{1}{2!}{\mathrm{ϵ}}^{\mathit{\text{μναβ}}}{F}_{\mathit{\alpha \beta }}$$E35

and

$${}^{\ast }{G}^{\mathit{ij}}=\frac{1}{2!}{\mathrm{ϵ}}^{\mathit{\text{ijkl}}}{G}_{\mathit{kl}}.$$E36

From the field eq. (2), one obtains

$${{\partial }_{\nu }}^{\ast }{F}^{\mathit{\mu \nu }}=-{\partial }_{\nu }{\left(ln{\psi }^{\ast }\right)}^{\ast }{F}^{\mathit{\mu \nu }}$$E37

and

$${{\partial }_j}^{\ast }{G}^{\mathit{ij}}=-{\partial }_j{\left(ln{\xi }^{\ast }\right)}^{\ast }{G}^{\mathit{ij}}.$$E38

Here,

$${\psi }^{\ast }={\left({\frac{1}{2}}^{\ast }{G}^{\mathit{ij}\ast }{G}_{\mathit{ij}}\right)}^{1/2}$$E39

and

$${\xi }^{\ast }={\left({\frac{1}{2}}^{\ast }{F}^{\mathit{\mu \nu }\ast }{F}_{\mathit{\mu \nu }}\right)}^{1/2}.$$E40

One finds that (37) and (38) can also be written as

$${\partial }_{\nu }\left({}^{\ast }{\psi }^{\ast }{F}^{\mathit{\mu \nu }}\right)=0$$E41

and

$${\partial }_j\left({}^{\ast }{\xi }^{\ast }{G}^{\mathit{ij}}\right)=0,$$E42

respectively.

6. Final remarks

Let us make some final remarks. Usually, for describing different phenomena in our universe, one time and three-space dimensions ($\left(1+3\right)$-dimensions) are the chosen number of real dimensions. But the question emerges, why $\left(1+3\right)$-dimensions? why not $\left(3+1\right)$-dimensions? or why not $\left(4+4\right)$-dimensions? Unfortunately (or fortunately) until now these questions are an open theoretical problem; as far as one knows, nobody knows the answer. It turns out that looking for a possible solution one stumbling with the discovery that there is a triality relation between the signatures $\left(1+9\right),$$\left(5+5\right)$ and $\left(9+1\right)$ [11, 12]. This means that by triality the $\left(5+5\right)$-dimensional world can always be related to the other basic signatures $\left(1+9\right)$ and $\left(9+1\right)$. It turns out that $\left(5+5\right)$-world is a common signature to both type $\mathit{IIA}$ strings and type $\mathit{IIB}$ strings. From this perspective, one may say that the $\left(4+4\right)$-world can be considered as the transverse world of the $\left(5+5\right)$-world (see Refs [11, 12] and references therein). Moreover, it turns out that in $\left(4+4\right)$-dimensions, there are a number of remarkable mathematical and physical results that are worth mentioning. Mathematically, it has been suggested that the mathematical structures of oriented matroid theory [13] (see Refs. [14, 15, 16, 17, 18, 19, 20, 21, 22] and references therein) and surreal number theory [23, 24] (see also Refs [25, 26] and references therein) may provide interesting routes for a connection with the $\left(4+4\right)$-world. Physically, the Dirac equation in $\left(4+4\right)$-dimensions is consistent with Majorana–Weyl spinors which give exactly the same number of components as the complex spinor of $\frac{1}{2}$-spin particles such as the electron or the quarks (see Refs. [12, 27]). Second, the most general Kruskal–Szekeres transformation of a black-hole coordinates in $\left(1+3\right)$-dimensions leads to eight-regions (instead of the usual four-regions), which can be better described in $\left(4+4\right)$-dimensions [28]. Third, it also has been shown [29] that duality

$${\sigma }^2\leftrightarrow \frac{1}{{\sigma }^2},$$E43

of a Gaussian distribution in terms of the standard deviation $\sigma$ of $4$-space coordinates associated with the de Sitter space (anti-de Sitter) and the vacuum zero-point energy yields to a Gaussian of $4$-time coordinates of the same vacuum scenario. Moreover, loop quantum gravity in $\left(4+4\right)$-dimensions [30, 31] admits a self-duality curvature structure analogue to the traditional $\left(1+3\right)$-dimensions.

In the above sense, the contribution of this work adds to the fact that “electromagnetic” field in a $\left(4+4\right)$-world described by a $4$-rank totally antisymmetric field strength $F^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}$ can be broken into two electromagnetic field strengths; the field strength $F^{\mathit{\mu \nu }}$ associated with the $\left(1+3\right)$-world and the field strength $G^{\mathit{ij}}$ associated with the $\left(3+1\right)$-world. An interesting aspect of this result is that there is a hidden duality symmetry feature of $F^{\mathit{\mu \nu }}$ and $G^{\mathit{ij}}$ in the sense that $G^{\mathit{ij}}$ contribute to the source of $F^{\mathit{\mu \nu }}$ and vice versa.

Finally, it is worth mentioning that $4$-rank totally antisymmetric field strength $F^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}$ in $\left(1+10\right)$-dimensions are a key mathematical notion in $3$-brane theory which, it is known, is an important part in the so-called $M$-theory (see Ref. [15] and references therein). In fact, in Ref. [9] it is shown how totally antisymmetric fields can be related to $p$-brane. For the case of the field strength $F^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}$_, one uses (6) and writes

$$F^{\widehat{\nu }\widehat{\alpha }\widehat{\beta }\widehat{A}}=\frac{1}{4!}{\epsilon }^{\widehat{B}\widehat{C}\widehat{D}\widehat{A}}{v}_{\widehat{B}}^{\widehat{\nu }}{v}_{\widehat{C}}^{\widehat{\alpha }}{v}_{\widehat{D}}^{\widehat{\beta }}$$E44

and assume that $F^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}=F^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}\left(x^{\sigma },y^i,{\xi }^{\widehat{A}}\right)$. Moreover, instead of (1) one considers the field equation:

$${\partial }_{\widehat{A}}{F}^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{A}}=0,$$E45

where ${\partial }_{\widehat{A}}=\frac{\partial }{{\xi }^{\widehat{A}}}$. This expression implies that due to (44) one can write

$$v_{\widehat{B}}^{\widehat{\mu }}={\partial }_{\widehat{B}}{X}^{\widehat{\mu }}.$$E46

Thus, substituting this result into (8) leads to

$$F^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}=\frac{1}{4!}{\epsilon }^{\widehat{A}\widehat{B}\widehat{C}\widehat{D}}{\partial }_{\widehat{A}}{X}^{\widehat{\mu }}{\partial }_{\widehat{B}}{X}^{\widehat{\nu }}{\partial }_{\widehat{C}}{X}^{\widehat{\alpha }}{\partial }_{\widehat{D}}{X}^{\widehat{\beta }}.$$E47

In turn, this expression can be used to write the Schild type action for the $3$-brane in target $\left(4+4\right)$-dimensions, namely

$$S=\int {\left(F^{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}{F}_{\widehat{\mu }\widehat{\nu }\widehat{\alpha }\widehat{\beta }}\right)}^{1/2}{d}^4\xi .$$E48

So it may be interesting for further work to continue relating our present approach with $p$-brane theory and $M$-theory.

Acknowledgments

I would like to thank the Mathematical, Computational & Modeling Sciences Center of the Arizona State University where part of this work was developed.

Classification

Pacs numbers: 04.20.Jb, 04.50.-h, 04.60.-m, 11.15.-q.