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Hans de Vries

Science Advisor

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90 years have gone by since P.A.M. Dirac published his equation in 1928. Some of its most basic consequences however are only discovered just now. (At least I have never encountered this before). We present the

This representation is arguably the most fundamental one because it is constructed using the generators that act on the Dirac spinors (in the chiral form).

$$\begin{array}{lrcl}

\mbox{mass dimension 1:}~~~~ & \mathbf{A} &=& \gamma^\mu A^\mu \\

\mbox{mass dimension 2:}~~~~ & \mathbf{F} &=& \vec{K}\cdot\vec{E}-\vec{J}\cdot\vec{B} \\

\mbox{mass dimension 3:}~~~~ & \mathbf{J}\, &=& \gamma^\mu\,j^\mu \\

\end{array}$$

Using the standard chiral gamma matrices and the boost and rotation generators.

$$\gamma^o ~=~

\left( \begin{array}{rr}

~0&I~ \\ ~I&0~

\end{array} \right)

~~~~~~~~

\gamma^i\, ~=~

\left(\, \begin{array}{cc}

\!0&\!\sigma_i\! \\ \!\!\!\!-\sigma_i&\!\!0\!

\end{array} \right)

~~~~~~~~

\gamma^5 ~=~

\left(\! \begin{array}{rr}

\!-I&0~ \\ ~0&I~

\end{array}\! \right)

$$

$$

~J^i ~=~

\left(\, \begin{array}{rr}

\!\!\!\!-i\sigma_i\!\!&0 \\ ~0&\!\!\!\!-i\sigma_i\!\!

\end{array}~ \right)

~~~~~~~~

K^i ~=~

\left(\, \begin{array}{cc}

\!\!\!-\sigma_i\!\!&0 \\ ~0&\sigma_i\!

\end{array}~\right)$$

The Lorentz force operator field ##\mathbf{F}## that acts on the bi-spinor field is defined combining the boost generator ##\vec{K}## with the electric field ##\vec{E}## and the rotation generator ##\vec{J}## with the magnetic field ##\vec{B}##, with a (historical) minus sign.

We can now write the laws of the Electromagnetic field, going from one mass dimension to another, by applying the differential operator matrix ##/\!\!\!{\partial} =\gamma^\mu\partial_\mu = \sqrt{\Box}## on the operator field matrices defined above.

$$/\!\!\!{\partial}\mathbf{A}^{\!\dagger} = \mathbf{F}~~~~~~ ~~~/\!\!\!{\partial}\mathbf{F} = \mathbf{J}^\dagger$$

The complex conjugate transpose ##\mathbf{A}^{\!\dagger}=\gamma^\mu A_\mu ## is the covariant form of ##\mathbf{A}##.

In the first step we have applied the conservation law ##\partial_\mu A^\mu\!=\!0## on the diagonal and in the second step we find

The equations above do not only provide a complete description of the electromagnetic field. They also symmetrize the transitions between mass dimensions. One can say that the expressions takes the square root of,

$$\Box\,\mathbf{A} ~=~

\mathbf{J}$$

and halfway we find the Lorentz Force operator field. Going in the opposite direction from higher to lower mass dimensions we see that it splits the propagator ##\Box^{-1}## into two more convenient ##(\gamma^\mu\partial_\mu)^{-1}## propagators.

We can write for the Lagrangian of the Electromagnetic field in vacuum.

$$\mathcal{L} ~=~ \tfrac12\mathbf{F}\mathbf{F}$$

Where ##\mathcal{L}## is a matrix operator field invariant under Lorentz transform.

To find the equations of motions: The four Maxwell equations. We write:

$$\mathcal{L} ~=~ \tfrac12/\!\!\!{\partial}\mathbf{A}^{\!\dagger}/\!\!\!{\partial}\mathbf{A}^{\!\dagger}~~~~

\underrightarrow{\mbox{ Euler Lagrange }}~~~~

/\!\!\!{\partial}/\!\!\!{\partial}\mathbf{A}^{\!\dagger} ~=~ /\!\!\!{\partial}\mathbf{F} ~=~0$$

Thus the equations of motion, the four Maxwell equations, are given by ##/\!\!\!{\partial}\mathbf{F}=0##.

If we work out the Lorentz invariant Lagrangian operator field we get.

$$\mathcal{L} ~~=~~ \tfrac12\mathbf{F}\mathbf{F}

~~=~~ \tfrac12\left(E^2-B^2\right)\!I

~+~ \left(\vec{E}\cdot\vec{B}\right)i\gamma^5

$$

The Lorentz scalar ##\tfrac12(E^2-B^2)## of the electromagnetic field is associated with the diagonal matrix ##I## while the pseudo scalar ##\vec{E}\cdot\vec{B}## of the electromagnetic field is associated with the pseudo scalar generator ##i\gamma^5##. If we include sources to the Lagrangian we can write.

$$\mathcal{L} ~=~ \tfrac12(

\mathbf{F}\mathbf{F}

+ (\mathbf{J}\mathbf{A}

+ \mathbf{A}\mathbf{J})^\dagger)

~=~ \left(\tfrac12(E^2-B^2) + j_\mu A_\mu\right)\! I

+ \left(\vec{E}\cdot\vec{B}\right)i\gamma^5

$$

The Lorentz scalars on the diagonal ##I## lead to the inhomogeneous Maxwell equations while the pseudo-scalar associated with ##i\gamma^5## provides the homogeneous Maxwell equations.

$$

\begin{array}{lll}

\tfrac12(E^2-B^2) + j_\mu A_\mu & \rightarrow & \mbox{inhomogenious Maxwell equations}\\

~~~(\,\vec{E}~\,\cdot~\vec{B}~) & \rightarrow & \mbox{homogenious Maxwell equations}

\end{array}

$$

The Lorentz transform operator used for chiral bi-spinor fields is.

$$

\mbox{$\Lambda$}

~=~

\exp\left(~\tfrac12 \vec{\phi}\cdot\vec{J} ~+~\tfrac12 \vec{\varpi}\cdot\vec{K}~\right)

$$

This operator is also used to transform all the electromagnetic operator fields in the same way, including the axial current field ##\mathbf{J}_{^{\!A}}\!=i\gamma^\mu\gamma^5\,j^\mu_{^{\!A}}## associated with the two homogenious Maxwell equations.

$$

\begin{array}{lcccll}

\mbox{Bi-spinor fermion field:} &&

\psi^{'} & = & ~~\mbox{$\Lambda\,\mathbf{\psi}$} & \\ \\

\mbox{Potential operator field:} &&

\mathbf{A}^{\!'} & = & \mbox{$\Lambda^{^{\!-\dagger}} \,\mathbf{A} \,~\Lambda^{^{\!\dagger}}$} & \\

\mbox{Force operator field:} &&

\mathbf{F}^{ '} & = & \mbox{$\Lambda^{^{\!-\dagger}} \,\mathbf{F} \,~\Lambda^{^{\!\dagger}}$} & \\

\mbox{Current operator field:} &&

\mathbf{J}^{ '} & = & \mbox{$\Lambda^{^{\!-\dagger}} \,\mathbf{J} ~~\Lambda^{^{\!\dagger}}$} & \\

\mbox{Axial operator field:} &&

\,\mathbf{J}^{ '}_{^{\!A}} & = & \mbox{$\Lambda^{^{\!-\dagger}} \,\mathbf{J}_{^{\!A}}\,\Lambda^{^{\!\dagger}}$} & \\

\mbox{Matrix Lagrangian:} &&

\mathcal{L}^{ '} & = & \mbox{$\Lambda^{^{\!-\dagger}} \,\mathcal{L}\,~\Lambda^{^{\!\dagger}}$} &

=~\mathcal{L}\\

\end{array}

$$

Where the ##\dagger## and -##\dagger## denote the conjugate transpose of ##\Lambda## and the conjugate transpose of ##\Lambda^{\!-1}##. The generators used to construct the operator field determine how the field behaves under a Lorentz transform.

There is a PDF file here, and a Mathematica file here.

**Covariant QED representation of the Electromagnetic field**.**1 - Definition of the fundamental operator fields**This representation is arguably the most fundamental one because it is constructed using the generators that act on the Dirac spinors (in the chiral form).

$$\begin{array}{lrcl}

\mbox{mass dimension 1:}~~~~ & \mathbf{A} &=& \gamma^\mu A^\mu \\

\mbox{mass dimension 2:}~~~~ & \mathbf{F} &=& \vec{K}\cdot\vec{E}-\vec{J}\cdot\vec{B} \\

\mbox{mass dimension 3:}~~~~ & \mathbf{J}\, &=& \gamma^\mu\,j^\mu \\

\end{array}$$

Using the standard chiral gamma matrices and the boost and rotation generators.

$$\gamma^o ~=~

\left( \begin{array}{rr}

~0&I~ \\ ~I&0~

\end{array} \right)

~~~~~~~~

\gamma^i\, ~=~

\left(\, \begin{array}{cc}

\!0&\!\sigma_i\! \\ \!\!\!\!-\sigma_i&\!\!0\!

\end{array} \right)

~~~~~~~~

\gamma^5 ~=~

\left(\! \begin{array}{rr}

\!-I&0~ \\ ~0&I~

\end{array}\! \right)

$$

$$

~J^i ~=~

\left(\, \begin{array}{rr}

\!\!\!\!-i\sigma_i\!\!&0 \\ ~0&\!\!\!\!-i\sigma_i\!\!

\end{array}~ \right)

~~~~~~~~

K^i ~=~

\left(\, \begin{array}{cc}

\!\!\!-\sigma_i\!\!&0 \\ ~0&\sigma_i\!

\end{array}~\right)$$

The Lorentz force operator field ##\mathbf{F}## that acts on the bi-spinor field is defined combining the boost generator ##\vec{K}## with the electric field ##\vec{E}## and the rotation generator ##\vec{J}## with the magnetic field ##\vec{B}##, with a (historical) minus sign.

**2 - Covariant representation of the E.M. field laws**We can now write the laws of the Electromagnetic field, going from one mass dimension to another, by applying the differential operator matrix ##/\!\!\!{\partial} =\gamma^\mu\partial_\mu = \sqrt{\Box}## on the operator field matrices defined above.

$$/\!\!\!{\partial}\mathbf{A}^{\!\dagger} = \mathbf{F}~~~~~~ ~~~/\!\!\!{\partial}\mathbf{F} = \mathbf{J}^\dagger$$

The complex conjugate transpose ##\mathbf{A}^{\!\dagger}=\gamma^\mu A_\mu ## is the covariant form of ##\mathbf{A}##.

In the first step we have applied the conservation law ##\partial_\mu A^\mu\!=\!0## on the diagonal and in the second step we find

**all four**of Maxwell's laws, the inhomogeneous ##\partial_\mu F^{\mu\nu}\!=\!j^\nu## as well as the homogeneous ##~\partial_\mu\! *\!\!F^{\mu\nu}\!=j^\nu_{^A}=\!0##.The equations above do not only provide a complete description of the electromagnetic field. They also symmetrize the transitions between mass dimensions. One can say that the expressions takes the square root of,

$$\Box\,\mathbf{A} ~=~

\mathbf{J}$$

and halfway we find the Lorentz Force operator field. Going in the opposite direction from higher to lower mass dimensions we see that it splits the propagator ##\Box^{-1}## into two more convenient ##(\gamma^\mu\partial_\mu)^{-1}## propagators.

We can write for the Lagrangian of the Electromagnetic field in vacuum.

$$\mathcal{L} ~=~ \tfrac12\mathbf{F}\mathbf{F}$$

Where ##\mathcal{L}## is a matrix operator field invariant under Lorentz transform.

To find the equations of motions: The four Maxwell equations. We write:

$$\mathcal{L} ~=~ \tfrac12/\!\!\!{\partial}\mathbf{A}^{\!\dagger}/\!\!\!{\partial}\mathbf{A}^{\!\dagger}~~~~

\underrightarrow{\mbox{ Euler Lagrange }}~~~~

/\!\!\!{\partial}/\!\!\!{\partial}\mathbf{A}^{\!\dagger} ~=~ /\!\!\!{\partial}\mathbf{F} ~=~0$$

Thus the equations of motion, the four Maxwell equations, are given by ##/\!\!\!{\partial}\mathbf{F}=0##.

If we work out the Lorentz invariant Lagrangian operator field we get.

$$\mathcal{L} ~~=~~ \tfrac12\mathbf{F}\mathbf{F}

~~=~~ \tfrac12\left(E^2-B^2\right)\!I

~+~ \left(\vec{E}\cdot\vec{B}\right)i\gamma^5

$$

The Lorentz scalar ##\tfrac12(E^2-B^2)## of the electromagnetic field is associated with the diagonal matrix ##I## while the pseudo scalar ##\vec{E}\cdot\vec{B}## of the electromagnetic field is associated with the pseudo scalar generator ##i\gamma^5##. If we include sources to the Lagrangian we can write.

$$\mathcal{L} ~=~ \tfrac12(

\mathbf{F}\mathbf{F}

+ (\mathbf{J}\mathbf{A}

+ \mathbf{A}\mathbf{J})^\dagger)

~=~ \left(\tfrac12(E^2-B^2) + j_\mu A_\mu\right)\! I

+ \left(\vec{E}\cdot\vec{B}\right)i\gamma^5

$$

The Lorentz scalars on the diagonal ##I## lead to the inhomogeneous Maxwell equations while the pseudo-scalar associated with ##i\gamma^5## provides the homogeneous Maxwell equations.

$$

\begin{array}{lll}

\tfrac12(E^2-B^2) + j_\mu A_\mu & \rightarrow & \mbox{inhomogenious Maxwell equations}\\

~~~(\,\vec{E}~\,\cdot~\vec{B}~) & \rightarrow & \mbox{homogenious Maxwell equations}

\end{array}

$$

**3 - Lorentz transform of the operator fields**The Lorentz transform operator used for chiral bi-spinor fields is.

$$

\mbox{$\Lambda$}

~=~

\exp\left(~\tfrac12 \vec{\phi}\cdot\vec{J} ~+~\tfrac12 \vec{\varpi}\cdot\vec{K}~\right)

$$

This operator is also used to transform all the electromagnetic operator fields in the same way, including the axial current field ##\mathbf{J}_{^{\!A}}\!=i\gamma^\mu\gamma^5\,j^\mu_{^{\!A}}## associated with the two homogenious Maxwell equations.

$$

\begin{array}{lcccll}

\mbox{Bi-spinor fermion field:} &&

\psi^{'} & = & ~~\mbox{$\Lambda\,\mathbf{\psi}$} & \\ \\

\mbox{Potential operator field:} &&

\mathbf{A}^{\!'} & = & \mbox{$\Lambda^{^{\!-\dagger}} \,\mathbf{A} \,~\Lambda^{^{\!\dagger}}$} & \\

\mbox{Force operator field:} &&

\mathbf{F}^{ '} & = & \mbox{$\Lambda^{^{\!-\dagger}} \,\mathbf{F} \,~\Lambda^{^{\!\dagger}}$} & \\

\mbox{Current operator field:} &&

\mathbf{J}^{ '} & = & \mbox{$\Lambda^{^{\!-\dagger}} \,\mathbf{J} ~~\Lambda^{^{\!\dagger}}$} & \\

\mbox{Axial operator field:} &&

\,\mathbf{J}^{ '}_{^{\!A}} & = & \mbox{$\Lambda^{^{\!-\dagger}} \,\mathbf{J}_{^{\!A}}\,\Lambda^{^{\!\dagger}}$} & \\

\mbox{Matrix Lagrangian:} &&

\mathcal{L}^{ '} & = & \mbox{$\Lambda^{^{\!-\dagger}} \,\mathcal{L}\,~\Lambda^{^{\!\dagger}}$} &

=~\mathcal{L}\\

\end{array}

$$

Where the ##\dagger## and -##\dagger## denote the conjugate transpose of ##\Lambda## and the conjugate transpose of ##\Lambda^{\!-1}##. The generators used to construct the operator field determine how the field behaves under a Lorentz transform.

There is a PDF file here, and a Mathematica file here.

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