A coordinateless form of the Dirac equation
N.G.Marchuk
We present a so called Dirac-type tensor equation (DTTE). This equation is written in coordinateless form with the aid of differential operators $d$ and $\delta$. A wave function of DTTE belongs to a minimal left ideal of the algebra of exterior forms with respect to the Clifford product. We show that a coordinate form of DTTE is identical to the Dirac equation in a fixed coordinate system.
math-ph0307042.tex
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\begin{document}
\setcounter{page}{1}
\baselineskip=16pt

\author{N.G.Marchuk}

\title{A coordinateless form of the Dirac equation}

\maketitle


Steklov Mathematical Institute, Gubkina st.8, Moscow 119991, Russia

nmarchuk@mi.ras.ru,   www.orc.ru/\~{}nmarchuk



\begin{abstract}
We present a so called Dirac-type tensor equation (DTTE). This equation
is written in coordinateless form with the aid of differential operators
$d$ and $\delta$. A wave function of DTTE  belongs to a minimal left
ideal of the algebra of exterior forms with respect to the Clifford
product. We show that a coordinate form of DTTE is identical to the
Dirac equation in a fixed coordinate system.
\end{abstract}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $\R^{1,3}$  be the Minkowski space with coordinates $x^\mu$ ($\mu=0,1,2,3$)
and $\eta$ be the Minkowski matrix
$\eta=\|\eta_{\mu\nu}\|=\|\eta^{\mu\nu}\|=\diag(1,-1,-1,-1)$.
The Dirac equation for an electron has the form
\begin{equation}
\gamma^\mu(\partial_\mu\psi+ia_\mu\psi)+im\psi=0,
\label{Dirac:eq}
\end{equation}
where $\gamma^\mu$ are complex valued
matrices such that
$$
\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=2\eta^{\mu\nu}{\bf 1},
\quad \psi=\psi(x)=(\psi^1,\ldots,\psi^4)^{\rm T}
$$
and ${\bf 1}$ is the identity matrix of fourth order. Summation convention
over repeating indices is assumed and
$\partial_\mu=\partial/\partial
x^\mu$.


In 1962 E.K\"ahler
\cite{Kahler} suggests the following equation for the electron
(the Dirac-K\"ahler equation)
$$
(d-\delta)\Phi+iA\Phi+im\Phi=0,\quad \Phi\in\Lambda,
\quad A\in\Lambda_1^\R.
$$
The equation is written in invariant (coordinateless) form.
Several types of algebra-geometric objects are used in this equation.

\noindent {\bf Exterior forms.} Suppose that in the Minkowski space
$\R^{1,3}$ with coordinates
$x^\mu$ we have basis coordinates vectors
$e_\mu$ and basis covectors
$e^\mu=\eta^{\mu\nu}e_\nu$.
Let $\Lambda_k$ be the set of exterior $k$-forms
$\frac{1}{k!}\phi_{\mu_1\ldots\mu_k}e^{\mu_1}\ww e^{\mu_k}
\in\Lambda_k$, where $\phi_{\mu_1\ldots\mu_k}=\phi_{[\mu_1\ldots\mu_k]}$
are complex valued antisymmetric covariant rang-$k$ tensors.
We take $\Lambda=\Lambda_0\oplus\ldots\oplus\Lambda_4$.
Complex dimensions of the spaces
$\Lambda_k$ and
$\Lambda$ are equal to $1,4,6,4,1$ and 16 respectively.

\noindent{\bf Hodge star operation}
$\star\,:\,\Lambda_k\to\Lambda_{4-k}$ identifies  $\Lambda_k$ with
$\Lambda_{4-k}$
$$
\star\ss{k}{\Phi}=
\frac{1}{k!(4-k)!}\varepsilon_{\mu_1\ldots\mu_4}\phi^{\mu_1\ldots\mu_k}
e^{\mu_{k+1}}\ww e^{\mu_4},\quad
\star\star\ss{k}{\Phi}=(-1)^{k+1}\ss{k}{\Phi},
$$
where $\varepsilon_{\mu_1\ldots\mu_4}$ is the completely antisymmetric tensor
and $\varepsilon_{0123}=1$.

\noindent {\bf The differential} $d:\Lambda_k\to\Lambda_{k+1}$
$$
d\Phi=e^\mu\wedge\partial_\mu\Phi, \quad d^2=0.
$$

\noindent {\bf Codifferential} $\delta:\Lambda_k\to\Lambda_{k-1}$
$$
\delta=\star d\star,\quad \delta^2=0.
$$

\noindent {\bf The Clifford product} $UV\in\Lambda$ of exterior forms
$U,V\in\Lambda$ is defined by the rules
\begin{description}
\item[(i)] $\Lambda$ is an associative algebra with identity element $1$
with respect to the Clifford product.
\item[(ii)] $e^\mu e^\nu=e^\mu\wedge e^\nu+\eta^{\mu\nu}$.
\item[(iii)] $e^{\mu_1}\ldots e^{\mu_k}=e^{\mu_1}\ww e^{\mu_k}\quad
\hbox{for}\quad \mu_1<\cdots<\mu_k$.
\end{description}

It follows from this rules that
\begin{description}
\item[1)]
$
e^\mu e^\nu+e^\nu e^\mu=2\eta^{\mu\nu}.
$
\item[2)] $
A^2=a_\mu a^\mu\in\Lambda_0  \quad\hbox{for}\quad A=a_\mu e^\mu\in\Lambda_1,
$
\item[3)] $
A\Phi=A\wedge\Phi-\star(A\wedge\star\Phi)
\quad\hbox{for}\quad A\in\Lambda_1,\Phi\in\Lambda.
$
\end{description}

The identity 1) is the main identity in the Clifford algebra
$\cl(1,3)$.

The operations of differential, codifferential, and Clifford product
take tensors (exterior forms) to tensors. Hence the Dirac-K\"ahler
equation is a tensor equation.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let us take a set of four linear independent orthonormal covectors
$h_\mu{}^a$. This set is called a {\em tetrad}.
$$
h_\mu{}^a h^{\mu b}=\eta^{ab},\quad \partial_\mu h_\nu{}^a=0.
$$
Also we take tetrad 1-forms
$h^a=h_\mu{}^a e^\mu$.
We use Greed indices as tensorial and Latin indices as nontensorial
(tetrad) indices. We raise and lower  Greek and Latin indices with the
aid of the Minkowski matrix
$h_{\nu a}=\eta_{ab}h_\nu{}^b$, $h^{\nu a}=\eta^{\mu\nu}h_\mu{}^a$.

Consider the operation of Clifford conjugation
$*\,:\,\Lambda_k\to\Lambda_k$. For
$U\in\Lambda_k$
$$
U^*=(-1)^{\frac{k(k-1)}{2}}\bar{U},
$$
where $\bar{U}$ is the exterior form with complex conjugated components.

Denote $E=h^0$. Now we can define the operation of Hermitian conjugation
of exterior forms
$\dagger\,:\,\Lambda_k\to\Lambda_k$
$$
U^\dagger=E U^* E \quad\hbox{for}\quad U\in\Lambda.
$$
Evidently,
$$
(UV)^\dagger=V^\dagger U^\dagger,\quad U^{\dagger\dagger}=U,
\quad i^\dagger=-i.
$$
An exterior form
$U\in\Lambda$ is called {\em Hermitian} if
$U^\dagger=U$, and anti-Hermitian if $U^\dagger=-U$.

An element $t$ of an algebra is called {\em an idempotent} (or
projector) if
$t^2=t$. Any idempotent generates the left ideal. A primitive idempotent
generates the minimal left ideal.

Let
$t\in\Lambda$ be an idempotent of
$\Lambda$ with respect to Clifford product such that
$$
t^2=t,\quad \partial_\mu t=0, \quad t^\dagger=t
$$
and
$$
\I(t)=\{Ut\,:\,U\in\Lambda\}
$$
be the minimal left ideal of $\Lambda$.

Let us introduce {\em a Dirac-type tensor equation} (DTTE) in
coordinateless form
\begin{equation}
(d-\delta)\Psi+iA\Psi+im\Psi=0,\quad \Psi\in\I(t),\quad
A\in\Lambda_1^\R.
\label{Dtt0}
\end{equation}
The exterior form $\Psi$ in DTTE belongs to the minimal left ideal
$\I(t)\subset\Lambda$. And this is the only difference between the
Dirac-K\"ahler equation and DTTE.
In the sequel we connect DTTE with the Dirac equation
(\ref{Dirac:eq}).

Recall that M.~Riesz
in 1947 was the first who consider Dirac spinors as
elements of a minimal left ideal of the Clifford algebra (a partial case
of pure spinors was considered by E.~Cartan in 1938). M.~Riesz deals
with the spinor representation of the Dirac equation.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Now we want to consider the coordinate form of DTTE instead of the
coordinateless form. For this we take the idempotent
$$
t=\frac{1}{4}(1+h^0)(1+ih^1 h^2)\in\Lambda
$$
such that $\quad t^2=t$, $\partial_\mu t=0$, $t^\dagger=t$.
It is easy to check that the following 1-forms:
\begin{equation}
t_1=t,\quad t_2=-h^1 h^3 t,\quad t_3=h^0 h^3 t,\quad t_4=h^0 h^1 t
\label{tttt}
\end{equation}
form a basis of the minimal left ideal
$\I(t)$. With the aid of this basis we define the map
$\gamma:\Lambda\to \M(4,\C)$, where $\M(4,\C)$ is the algebra
of complex valued matrices of fourth order. Namely, for
$U\in\Lambda$
$$
Ut_\n=\gamma(U)^\k_\n t_\k,
$$
where $\gamma(U)^\k_\n$ are the elements of the matrix $\gamma(U)$
(an upper index enumerates rows and a lower index enumerates columns)
Note that we take the basis
(\ref{tttt}) such that
$$
\gamma(U^\dagger)=(\gamma(U))^\dagger,
$$
where $U^\dagger$ is the Hermitian conjugated form $U^\dagger=EU^*E$,
and
$(\gamma(U))^\dagger$ is the Hermitian conjugated matrix
(transposed matrix with complex conjugated elements).

Also we use the map that takes elements of the left ideal
$\Omega\in\I(t)$ to ket vector
$|\Omega\ra\in \C^4$.
If $\Omega=\omega^\k t_\k\in\I(t)$, then
$$
|\Omega\ra=(\omega^1,\ldots, \omega^4)^{\rm T}.
$$
For $U,V\in\Lambda,\Omega\in\I(t)$ we have
$$
|U\Omega\ra=\gamma(U)|\Omega\ra, \quad
\gamma(UV)=\gamma(U)\gamma(V),
$$
i.e, $\gamma$ is a matrix representation of elements of
$\Lambda$.
It can be shown that
$$
d-\delta=e^\mu\partial_\mu.
$$
Let us rewrite DTTE in the form
\begin{equation}
\Omega\equiv e^\mu(\partial_\mu\Psi+ia_\mu\Psi)+im\Psi=0
\label{'""'}
\end{equation}
and get the ket vector
\begin{equation}
|\Omega\ra=\gamma^\mu(\partial_\mu\psi+ia_\mu\psi)+im\psi=0,
\label{Omega1}
\end{equation}
where $\psi=|\Psi\ra$,
$\gamma^\mu=\gamma(e^\mu)$.
From the identity
$e^\mu e^\nu+e^\nu e^\mu=2\eta^{\mu\nu}$ we have
$\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=2\eta^{\mu\nu}{\bf 1}$.

Hence we get the theorem.

\theorem. The coordinate form (\ref{Omega1}) of DTTE coincides with
the Dirac equation
(\ref{Dirac:eq}).\par


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}

\bibitem{25.} Marchuk N.G., A concept of Dirac-type tensor equations,
        Nuovo Cimento, 117B, 12, (2002), pp.1357-1388.\par


\bibitem{Dirac} Dirac P.A.M., Proc. Roy. Soc. Lond. A117 (1928) 610.

\bibitem{Ivanenko} Ivanenko D., Landau L., Z. Phys., 48 (1928)340.\par
\bibitem{Kahler} K\"ahler E., Randiconti di Mat. (Roma) ser. 5, 21, (1962) 425.\par

\end{thebibliography}


\end{document}


math-ph0307042.extracted_bib
\begin{thebibliography}{99}

\bibitem{25.} Marchuk N.G., A concept of Dirac-type tensor equations,
        Nuovo Cimento, 117B, 12, (2002), pp.1357-1388.\par


\bibitem{Dirac} Dirac P.A.M., Proc. Roy. Soc. Lond. A117 (1928) 610.

\bibitem{Ivanenko} Ivanenko D., Landau L., Z. Phys., 48 (1928)340.\par
\bibitem{Kahler} K\"ahler E., Randiconti di Mat. (Roma) ser. 5, 21, (1962) 425.\par

\end{thebibliography}