LOG#092. Group theory(XII).

In the next group theory threads we are going to study the relationship between Special Relativity, electromagnetic fields and the complex group $SO(3,\mathbb{C})$ .

There is a close interdependence of the following three concepts:

$\mbox{Lorentz symmetry}\leftrightarrow \mbox{EM fields}\leftrightarrow \mbox{Minkovski spacetime}$

The classical electromagnetic fields $E=(E_x,E_y,E_z)$ and $B=(B_x,B_y,B_z)$ can be in fact combined into a complex six dimensional (6D) vector, sometimes called SIXTOR or Riemann-Silberstein vector:

$\boxed{\Psi=\dfrac{1}{\sqrt{2}}\left(E+iB\right)}$

and where the numerical prefactor is conventional ( you can give up for almost every practical purposes).

$E=(E_x,E_y,E_z)=\begin{pmatrix}E_x\\ E_y\\E_z\end{pmatrix}$

$B=(B_x,B_y,B_z)=\begin{pmatrix}B_x\\ B_y\\B_z\end{pmatrix}$

Moreover, we have

$X=x^\mu e_\mu=(ct,x,y,z)=(x^0,x^1,x^2,x^2)=\begin{pmatrix}x^0\\ x^1\\ x^2\\ x^3\end{pmatrix}$

where $x^0=ct$ and $\eta_{\mu\nu}=\mbox{diag}(-1,1,1,1)$ so

$x_\mu =\eta_{\mu\nu}x^\nu=(x^0,x^1,x^2,x^3)^T\eta=(x^0,\mathbf{x})^T\eta=(-x^0,x^1,x^2,x^3)$

and where we have used natural units $c=1$ for simplicity.

The Maxwell-Faraday equation reads:

$\partial_tB=-\nabla\times E=-\mbox{curl}E$

The Ampère circuital law in vacuum reads:

$\partial_tE=+\nabla\times B$

These two equations can be combined into a single equation using the Riemann-Silberstein vector or sixtor $\Psi$ :

$\dfrac{\partial\Psi}{\partial t}=-i\nabla \times \Psi=\dot{\Psi}$

Check:

A) $\partial_t (E+iB)=-i\nabla\times (E+iB)=-i\nabla\times E+\nabla\times B$

B) Comparing both sides in A), we easily get $\partial_t E=+\nabla\times B$ and $\partial_t B=-\nabla\times E$

We can take the divergence of the time derivative of the sixtor:

$\nabla\cdot \dot{\Psi}=\partial_t (\nabla\cdot\Psi)=-i\nabla\cdot (\nabla\times\Psi)=0$

Therefore, $\nabla\cdot E=0$ and $\nabla\cdot B=0$ hold in the absence of electric and magnetic charges on any section of a Minkovski spacetime, and everywhere! The presence of electric charges and the absence of magnetic charges, the so-called magnetic monopoles, breaks down the gauge symmetry of

$\partial_t \Psi=-i\nabla\times \Psi$

$\Psi\longrightarrow e^{i\alpha}\Psi$ $\forall \alpha\in \mathbb{R}$

Introducing 3 matrices $S_1,S_2,S_3$ with the aid of the 3D Levi-Civita tensor $\varepsilon_{abc}=\varepsilon^{abc}$ , the completely antisymmetric tensor with 3 indices such that $\varepsilon_{abc}=+1$ and $\varepsilon_{abc}=-\varepsilon_{bac}=-\varepsilon_{acb}$ we can write these matrices as follows:

$(S_b)_ac=-i\varepsilon_{bac}$ so

$S_a=\begin{pmatrix}0 & 0 & 0\\ 0 & 0 & i\\ 0 & -i & 0\end{pmatrix}$ $S_b=\begin{pmatrix}0 & 0 & -i\\ 0 & 0 & 0\\ i & 0 & 0\end{pmatrix}$ $S_c=\begin{pmatrix}0 & i & 0\\ -i & 0 & 0\\ 0 & 0 & 0\end{pmatrix}$

If $\partial_k\equiv \dfrac{\partial}{\partial x^{k}}$ for $k=1,2,3$ , then

$\partial_t \Psi=S_k\partial_k\Psi$

We can define matrices $\Gamma^\mu$ so $\Gamma^0=I_3=\mbox{diag}(1,1,1)$

and then $\Gamma_k=S_k=-\Gamma^k$ for $k=1,2,3$ . Experts in Clifford/geometric algebras will note that these matrices are in fact “Dirac matrices” $\Gamma=\gamma$ up to a conventional sign.

In fact, you can admire the remarkable similarity between the sixtor equation AND the Dirac equation as follows:

$\partial_t\Psi=S_k\partial_k\Psi\leftrightarrow \boxed{i\Gamma^\mu \partial_\mu\Psi=0}$

In summary: the sixtor equation is a Dirac-like equation (but of course the electromagnetic field is not a fermion!).

The equation for $\Psi^*$ , since $\Psi^*=\dfrac{1}{\sqrt{2}}(E-iB)$ , will be the feynmanity

$i\overline{\Gamma}^\mu \partial_\mu\partial\Psi^*=0$

where $\overline{\Gamma}^\mu=(I_3,S_1,S_2,S_3)$

Let us define the formal adjoint field $\Psi^+=\Psi^{*T}$ and the 4 components of a “density-like” quantity

$j^\mu=T^{0\mu}=\Psi^+\Gamma^\mu\Psi$

Then, we can recover the classical result that says that the energy density and the Poynting vector of the electromagnetic field is

$T^{00}=\dfrac{E}{V}=\Omega=\dfrac{1}{2}(E^2+B^2)$

$J^k=T^{0k}=(E\times B)_k$

These equations provide an important difference between the Dirac equation for a massive spin $1/2$ true (anti)particle and the electromagnetic massless spin $1$ photon, because you can observe that in the former case you HAVE:

$i\gamma^\mu\partial_\mu\Psi-m\Psi=0$

and you HAVE

$i\Gamma^\mu\partial_\mu \Psi=0$

in the latter (the electromagnetic field has not mass term!). In fact, you also have that for a Dirac field the current is defined to be:

$J^\mu_{Dirac}=\Psi^+\gamma^0\gamma^\mu\Psi=\overline{\Psi}\gamma^\mu\Psi$

and it transforms like a VECTOR field under Lorentz transformations, while the previous current $T^{0\mu}=T^{\mu 0}$ are the components of some stress-energy-momentum $T^{\mu\nu}$ !!!! They are NOT the same thing!

In fact, $(\Psi,\Psi^*)$ transform under the $(1,0)$ and $(0,1)=(1,0)^*$ (complex conjugated) representation of the proper Lorentz group.

Remark: Belinfante coined the term “undor” when dealing with fields transforming according to some specific representations of the Lorentz group.

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The Spectrum Of Riemannium

Mobilis in mobili: Physmatics Chronicles!

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