Note on the energy-momentum tensor for general mixed tensor-spinor fields
skip Department of Physics
Beijing
100875
PRC \end{tabular}
This note provides an explicit proof of the equivalence of the Belinfante's energy-momentum tensor and the metric energy-momentum tensor for general mixed tensor-spinor fields.
math-ph0412064.tex
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\begin{document}
\title{Note on the energy-momentum tensor for general mixed tensor-spinor fields}
\author{\begin{tabular}{c}\bigskip Hongbao Zhang\footnote{Email: hbzhang@pkuaa.edu.cn}\\  \smallskip Department of Physics, Beijing Normal University, Beijing, 100875, PRC
\end{tabular}}
\maketitle
\begin{abstract}
This note provides an explicit proof of the equivalence of the
Belinfante's energy-momentum tensor and the metric energy-momentum
tensor for general mixed tensor-spinor fields.
\end{abstract}
\section{Introduction}
It is well known that a suitable definition for the
energy-momentum tensor has been under investigation for many
years. We require $T_{ab}$ constructed should not only provide
meaningful physical conserved quantities but also act as the
source of Einstein's gravitational field equation.

In a flat spacetime, the canonical energy-momentum tensor arises
from Noether's Theorem by employing the conserved currents
associated with translational symmetries\cite{Wald}. However, only
for the scalar field the canonical energy-momentum tensor turns
out to be symmetric, satisfying both of the requirements above.
Indeed, in the case of the Maxwell field, the canonical
energy-momentum tensor is not even gauge invariant\cite{Jack}. Of
course, for general higher spin fields, we can correct it through
Belinfante's symmetrization procedure\cite{Belin}, although this
is usually presented as an {\em ad hoc} prescription\cite{Weinb1}.

On the other hand, a completely different approach, based on the
powerful variational principle, leads to the metric
energy-momentum tensor, which naturally satisfies our requirements
above\cite{Wald,Weinb2}.

It is certainly interesting to ask if the Belinfante's
energy-momentum tensor is equivalent to the metric energy-momentum
tensor. Despite a general belief in their equivalence, to our
knowledge there exists no such an explicit proof in the literature
except that \cite{Sara} presents a detailed confirmation for
general tensor fields. The aim of this note is to generalize this
confirmation to the case of general mixed tensor-spinor fields.
Accordingly, the present work acquires its practical importance:
it justifies much work such as \cite{Vollick,YW} and references
therein, where the Belinfante's energy momentum tensor for fermion
fields is imposed as the source of Einstein's gravitation.

Notation and conventions adopted by Chapter 13 in \cite{Wald} are
followed in the rest of our work.
\section{The three operations' applications to derivatives}
Fix, once and for all, a 4-dimensional manifold $M$, with a spinor
structure $\{\epsilon_{AB},\bar{\epsilon}_{A'B'}\}$\footnote{Here,
we shall take the soldering form $\sigma^a{}_{AA'}$ as the
fundamental variable, with the metric given by
$g^{ab}=\sigma^a{}_{AA'}\sigma^b{}_{BB'}\epsilon^{AB}\bar{\epsilon}^{A'B'}$\cite{Ashtekar}.}.
Following \cite{Sara}, we would like to introduce three convenient
operations: acute-operation, grave-operation, and hat-operation,
each of which means a mapping, from mixed tensor-spinor fields of
rank $(i,j|k,l;m,n)$ to fields of rank
$(i+1,j+1|k+1,l+1;m+1,n+1)$. For example, given a tensor-spinor
field of rank $(1,1|1,1;1,1)$ $\psi^a{}_b{}^{EF'}{}_{GH'}$, then
the three operations are defined as
\begin{equation}
\acute{\psi}^{ac}{}_{bd}{}^{EF'II'}{}_{GH'JJ'}=\frac{1}{4}(\psi^c{}_b{}^{EF'}{}_{GH'}\delta^a{}_d\epsilon^I{}_J\bar{\epsilon}^{I'}{}_{J'}-\psi^a{}_d{}^{EF'}{}_{GH'}\delta^c{}_b\epsilon^I{}_J\bar{\epsilon}^{I'}{}_{J'}),
\end{equation}
\begin{eqnarray}
\grave{\psi}^{ac}{}_{bd}{}^{EF'II'}{}_{GH'JJ'}=&\frac{1}{8}(\psi^a{}_b{}^{IF'}{}_{GH'}\delta^c{}_d\epsilon^E{}_J\bar{\epsilon}^{I'}{}_{J'}+\psi^a{}_b{}^{EI'}{}_{GH'}\delta^c{}_d\epsilon^I{}_J\bar{\epsilon}^{F'}{}_{J'}\nonumber\\
&-\psi^a{}_b{}^{EF'}{}_{JH'}\delta^c{}_d\epsilon^I{}_G\bar{\epsilon}^{I'}{}_{J'}-\psi^a{}_b{}^{EF'}{}_{GJ'}\delta^c{}_d\epsilon^I{}_J\bar{\epsilon}^{I'}{}_{H'}),
\end{eqnarray}
and
\begin{equation}
\hat{\psi}^{ac}{}_{bd}{}^{EF'II'}{}_{GH'JJ'}=\acute{\psi}^{ac}{}_{bd}{}^{EF'II'}{}_{GH'JJ'}+\grave{\psi}^{ac}{}_{bd}{}^{EF'II'}{}_{GH'JJ'}.
\end{equation}
It is easy to generalize these operations to fields of other
ranks; and know that acute-operation on pure spinor fields
vanishes, since there is no tensor index to be replaced.
Similarly, grave-operation on pure tensor fields vanishes;
especially, the three operations on a scalar field all vanish. One
checks these operations: i)are additive, ii)satisfy the Leibnitz
rule under out products, iii)commute with tensor and spinor
contraction, and iv)commute with the complex-conjugation.

When there is no danger of confusion, we shall suppress the
unnecessary indices. For instance, we shall denote
$\psi^a{}_b{}^{EF'}{}_{GH'}$ by $\psi$, and
$\hat{\psi}^{ac}{}_{bd}{}^{EF'II'}{}_{GH'JJ'}$ by
$\hat{\psi}^c{}_d{}^{II'}{}_{JJ'}$.

To show the convenience of introducing these operations, we would
like to present some examples, which are also useful to the rest
of our work. Firstly, given a soldering form $\sigma^b{}_{BB'}$,
let $\nabla_a$ be the unique covariant derivative operator
associated with $\sigma^b{}_{BB'}$, then we have\cite{Wald}
\begin{eqnarray}
(\nabla_a\nabla_b-\nabla_b\nabla_a)\psi&=&-(R_{abc}{}^d\epsilon_I{}^J\bar{\epsilon}_{I'}{}^{J'}\acute{\psi}^c{}_d{}^{II'}{}_{JJ'}+\delta_c{}^dR_{abII'}{}^{JJ'}\grave{\psi}^c{}_d{}^{II'}{}_{JJ'})\nonumber\\
&=&-(R_{abc}{}^d\epsilon_I{}^J\bar{\epsilon}_{I'}{}^{J'}+\delta_c{}^dR_{abII'}{}^{JJ'})\hat{\psi}^c{}_d{}^{II'}{}_{JJ'}.
\end{eqnarray}
Furthermore, let $\xi^a$ be a Killing vector field with respect to
$\sigma^b{}_{BB'}$, then we have\cite{Geroch}
\begin{eqnarray}
\mathcal{D}(\xi)\psi=(\pounds_\xi-\xi^a\nabla_a)\psi&=&-(\nabla_c\xi^d\epsilon_I{}^J\bar{\epsilon}_{I'}{}^{J'}\acute{\psi}^c{}_d{}^{II'}{}_{JJ'}+\delta_c{}^d\nabla_{II'}\xi^{JJ'}\grave{\psi}^c{}_d{}^{II'}{}_{JJ'})\nonumber\\
&=&-(\nabla_c\xi^d\epsilon_C{}^D\bar{\epsilon}_{C'}{}^{D'}+\delta_c{}^d\nabla_{II'}\xi^{JJ'})\hat{\psi}^c{}_d{}^{II'}{}_{JJ'}.\label{Lie}
\end{eqnarray}
Especially, $D(\xi)\psi$ vanishes for all translation Killing
vector fields in a flat spacetime. By
$\nabla_a\nabla_b\xi^c=\xi^dR_{dab}{}^c$\cite{Wald}, we have
\begin{eqnarray}
(\pounds_\xi\nabla_a-\nabla_a\pounds_\xi)\psi&=&\xi^b(\nabla_b\nabla_a-\nabla_a\nabla_b)\psi\nonumber\\
&&+\nabla_a(\nabla_c\xi^d\epsilon_I{}^J\bar{\epsilon}_{I'}{}^{J'}+\delta_c{}^d\nabla_{II'}\xi^{JJ'})]\hat{\psi}^c{}_d{}^{II'}{}_{JJ'}\nonumber\\
&=&-\xi^b(R_{bac}{}^d\epsilon_I{}^J\bar{\epsilon}_{I'}{}^{J'}+\delta_c{}^dR_{baII'}{}^{JJ'})\hat{\psi}^c{}_d{}^{II'}{}_{JJ'}\nonumber\\
&&+\xi^b(R_{bac}{}^d\epsilon_I{}^J\bar{\epsilon}_{I'}{}^{J'}+\delta_c{}^dR_{baII'}{}^{JJ'})\hat{\psi}^c{}_d{}^{II'}{}_{JJ'}\nonumber\\
&=&0,
\end{eqnarray}
which means the Lie-derivative operator via a Killing vector field
commutes with the covariant derivative operator.

Next, given another soldering form $\tilde{\sigma}^b{}_{BB'}$ and
the associated covariant derivative operator $\tilde{\nabla}_a$,
we have\cite{HHZ}
\begin{equation}
(\tilde{\nabla}_a-\nabla_a)\psi=C^d{}_{ac}\epsilon_I{}^J\bar{\epsilon}_{I'}{}^{J'}\acute{\psi}^c{}_d{}^{II'}{}_{JJ'}+\delta_c{}^d\Gamma_{aII'}{}^{JJ'}\grave{\psi}^c{}_d{}^{II'}{}_{JJ'},\label{Variation1}
\end{equation}
where
\begin{equation}
C^d{}_{ac}=\frac{1}{2}\tilde{g}^{db}(\nabla_a\tilde{g}_{cb}+\nabla_c\tilde{g}_{ab}-\nabla_b\tilde{g}_{ac}),\label{Variation2}
\end{equation}
and
\begin{equation}
\Gamma_{aII'}{}^{JJ'}=\frac{1}{2}[(\nabla_a\tilde{\sigma}^b{}_{IA'}+C^b{}_{ac}\tilde{\sigma}^c{}_{IA'})\tilde{\sigma}_b{}^{JA'}\bar{\epsilon}_{I'}{}^{J'}+(\nabla_a\tilde{\sigma}^b{}_{AI'}+C^b{}_{ac}\tilde{\sigma}^c{}_{AI'})\tilde{\sigma}_b{}^{AJ'}\epsilon_I{}^J].\label{Variation3}
\end{equation}
\section{The equivalence of the two energy-momentum tensors}
Start with the lagrangian $\mathcal{L}$ for any mixed
tensor-spinor field $\psi$, which is a local function of the
spinor structure $\{\epsilon_{AB},\bar{\epsilon}_{A'B'}\}$, the
soldering form $\sigma^a{}_{AA'}$, the tensor-spinor field $\psi$
and its first derivative, i.e.,
\begin{equation}
\mathcal{L}_p=\mathcal{L}[\epsilon_{AB}(p),\bar{\epsilon}_{A'B'}(p),\sigma^a{}_{AA'}(p),\psi(p),\nabla_a\psi(p)].
\end{equation}

For a flat spacetime $(M,\eta_{ab})$, let $\xi^a$ be a Killing
vector field, then we have
\begin{equation}
\nabla_a(\xi^a\mathcal{L})=\xi^a\nabla_a\mathcal{L}=\pounds_\xi\mathcal{L}=\frac{\partial
\mathcal{L}}{\partial \psi}\pounds_\xi\psi+\frac{\partial
\mathcal{L}}{\partial \nabla_a\psi}\pounds_\xi\nabla_a\psi,
\end{equation}
where we have used\cite{Geroch}
\begin{equation}
\pounds_\xi\sigma^a{}_{AA'}=0,
\end{equation}
and
\begin{equation}
\pounds_\xi\epsilon_{AB}=0.
\end{equation}
Employing $\pounds_\xi\nabla_a\psi=\nabla_a\pounds_\xi\psi$, and
the equation of motion for $\psi$, i.e.,
\begin{equation}
\frac{\partial \mathcal{L}}{\partial \psi}=\nabla_a\frac{\partial
\mathcal{L}}{\partial \nabla_a\psi},
\end{equation}
we obtain
\begin{equation}
\nabla_a(\frac{\partial \mathcal{L}}{\partial
\nabla_a\psi}\pounds_\xi\psi-\xi^a\mathcal{L})=0.\label{Noether}
\end{equation}
Noting that $D(\xi)\psi$ vanishes for all translation Killing
vector fields, Eq.(\ref{Noether}) implies the canonical
energy-momentum tensor
\begin{equation}
T_\mathcal{C}^{ab}=\frac{\partial \mathcal{L}}{\partial
\nabla_a\psi}\nabla^b\psi-\eta^{ab}\mathcal{L}\label{Can}
\end{equation}
is conserved, i.e., $\nabla_aT_\mathcal{C}^{ab}=0$. Next, let
$\{x^\mu\}$ be the Lorentzian coordinate system, and
$\xi^{\mu\nu}$ be rotation or boost Killing vector fields, i.e.,
\begin{equation}
\xi^{\mu\nu}_b=x^\mu (dx^\nu)_b-x^\nu (dx^\mu)_b,\label{Belin1}
\end{equation}
then Eq.(\ref{Noether}) can be written as
\begin{equation}
\nabla_a[\frac{\partial \mathcal{L}}{\partial
\nabla_a\psi}\mathcal{D}(\xi^{\mu\nu})\psi+T_\mathcal{C}^{ab}\xi^{\mu\nu}_b]=0.\label{Belin2}
\end{equation}
Using $\nabla_aT_\mathcal{C}^{ab}=0$, Eq.(\ref{Belin1}) and
Eq.(\ref{Belin2}) imply
\begin{equation}
T_\mathcal{C}^{[\mu\nu]}=-\frac{1}{2}\nabla_a[\frac{\partial
\mathcal{L}}{\partial
\nabla_a\psi}\mathcal{D}(\xi^{\mu\nu})\psi].\label{Symmetry1}
\end{equation}
Define the tensors
\begin{equation}
N^{abc}=\frac{\partial\mathcal{L}}{\partial\nabla_a\psi}{\cal
D}(\xi^{\mu\nu})\psi(\frac{\partial}{\partial
x^{\mu}})^b(\frac{\partial}{\partial x^{\nu}})^c,\label{Symmetry3}
\end{equation}
which is antisymmetric for the last two indices; and
\begin{equation}
F^{abc}=\frac{1}{2}(N^{abc}-N^{cab}+N^{bca}),\label{Symmetry2}
\end{equation}
which satisfies $F^{cab}=F^{[ca]b}$ and $F^{cab}=F^{c[ab]}$. Then
the conserved and symmetric Belinfante's energy-momentum tensor
can be constructed as
\begin{equation}
T_\mathcal{B}^{ab}=T_\mathcal{C}^{ab}+\nabla_cF^{cab}=T_\mathcal{C}^{(ab)}+\nabla_cN^{(ab)c}.\label{Bel}
\end{equation}
For convenience, we shall write $N^{abc}$ explicitly as
\begin{equation}
N^{abc}=-2\frac{\partial {\cal
 L}}{\partial\nabla_a\psi}(\delta_e{}^{[b}\eta^{c]d}\epsilon_I{}^J\bar{\epsilon}_{I'}{}^{J'}+\delta_e{}^d\sigma^{[b}{}_{II'}\sigma^{c]JJ'})\hat{\psi}^e{}_d{}^{II'}{}_{JJ'},\label{N}
\end{equation}
where Eq.(\ref{Lie}), Eq.(\ref{Belin1}), and Eq.(\ref{Symmetry3})
have been employed.

It is worth noting that although we resort to Killing vector
fields to construct Belinfante's energy momentum tensor; according
to Eq.(\ref{Can}), Eq.(\ref{Bel}) and Eq.(\ref{N}),
$T_\mathcal{B}^{ab}$ itself does not depend on these Killing
vector fields. $T_\mathcal{B}^{ab}$ depends on the spinor
structure, the soldering form, the tensor-spinor field and its
first derivative instead, thus $T_\mathcal{B}^{ab}$ constructed in
the flat spacetime can naturally extend to general curved
spacetime: only replace everywhere the soldering form and the
covariant derivative operator associated with $\eta_{ab}$ by those
associated with $g_{ab}$.

On the other hand, performing the variation of the action
\begin{equation}
S=\int\mathcal{L}\sqrt{-g}\mathbf{d^4x},
\end{equation}
we have
\begin{eqnarray}
\delta S=&&\int[\frac{\partial
\mathcal{L}}{\partial\sigma^a{}_{AA'}}\delta\sigma^a{}_{AA'}+\frac{\partial
\mathcal{L}}{\partial\nabla_a\psi}\delta\nabla_a)\psi-\frac{1}{2}\mathcal{L}g_{ab}\delta
g^{ab}]\sqrt{-g}\mathbf{d^4x}\nonumber\\
=&&\int\frac{1}{2}[(\frac{\partial
\mathcal{L}}{\partial\sigma^a{}_{AA'}}\sigma_{bAA'}-\mathcal{L}g_{ab})\delta
g^{ab}+2\frac{\partial
\mathcal{L}}{\partial\nabla_a\psi}\delta\nabla_a)\psi]\sqrt{-g}\mathbf{d^4x}.
\end{eqnarray}
Here, without loss of generalization, with respect to the
variation of  the metric, the variation of the soldering form has
been gauge-fixed as
\begin{equation}
\delta\sigma^a{}_{AA'}=\frac{1}{2}\sigma_{bAA'}\delta
g^{ab}.\label{Gauge}
\end{equation}
Noting that
\begin{equation}
\acute{\mathcal{L}}^c{}_d{}^{II'}{}_{JJ'}=0=\frac{1}{4}\frac{\partial
\mathcal{L}}{\partial\sigma^a{}_{AA'}}\delta^a{}_d\sigma^c{}_{AA'}\epsilon^I{}_J\bar{\epsilon}^{I'}{}_{J'}+\frac{\partial
\mathcal{L}}{\partial
\psi}\acute{\psi}^c{}_d{}^{II'}{}_{JJ'}+\frac{\partial
\mathcal{L}}{\partial
\nabla_a\psi}[\nabla_a\acute{\psi}^c{}_d{}^{II'}{}_{JJ'}-\frac{1}{4}\delta^c{}_a\nabla_d\psi)\epsilon^I{}_J\bar{\epsilon}^{I'}{}_{J'}],
\end{equation}
we have
\begin{equation}
\frac{\partial
\mathcal{L}}{\partial\sigma^a{}_{AA'}}\sigma_{bAA'}=\frac{\partial\mathcal{L}}{\partial
\nabla^b\psi}\nabla_a\psi-\nabla_c(\frac{\partial\mathcal{L}}{\partial
\nabla_c\psi}\acute{\psi}_{ba}{}^{II'}{}_{JJ'})\epsilon_I{}^J\bar{\epsilon}_{I'}{}^{J'}.
\end{equation}
Therefore, the variation of the action can be written as
\begin{equation}
\delta S=\int\frac{1}{2}[T_{\mathcal{C}(ab)}\delta
g^{ab}-\nabla_c(\frac{\partial\mathcal{L}}{\partial
\nabla_c\psi}\acute{\psi}_{ba}{}^{II'}{}_{JJ'})\epsilon_I{}^J\bar{\epsilon}_{I'}{}^{J'}\delta
g^{ab}+2\frac{\partial
\mathcal{L}}{\partial\nabla_a\psi}\delta\nabla_a)\psi]\sqrt{-g}\mathbf{d^4x}
\end{equation}
According to Eq.(\ref{Variation1}), Eq.(\ref{Variation2}),
Eq.(\ref{Variation3}), and Eq.(\ref{Gauge}), we have
\begin{eqnarray}
\delta\nabla_a)\psi&=&\delta
C^d{}_{ac})\epsilon_I{}^J\bar{\epsilon}_{I'}{}^{J'}\acute{\psi}^c{}_d{}^{II'}{}_{JJ'}+\delta_c{}^d\delta\Gamma_{aII'}{}^{JJ'}\grave{\psi}^c{}_d{}^{II'}{}_{JJ'}\nonumber\\
&=&\frac{1}{2}g_{ec}(g_{af}\nabla^d\delta
g^{ef}-\nabla_a\delta g^{de}-g_{af}\nabla^e\delta g^{df})\epsilon_I{}^J\bar{\epsilon}_{I'}{}^{J'}\acute{\psi}^c{}_d{}^{II'}{}_{JJ'}\nonumber\\
&&+\delta_c{}^dg_{ab}\nabla^e\delta
g^{bf})\sigma_{[fII'}\sigma_{e]}{}^{JJ'}\grave{\psi}^c{}_d{}^{II'}{}_{JJ'}.
\end{eqnarray}
Thus, making partial integrations and employing the Stokes
theorem, we have
\begin{eqnarray}
 \delta S=&&\int\frac{1}{2}[T_{\mathcal{C}(ab)}\delta
g^{ab}-\nabla_c(\frac{\partial\mathcal{L}}{\partial
\nabla_c\psi}\acute{\psi}_{ba}{}^{II'}{}_{JJ'})\epsilon_I{}^J\bar{\epsilon}_{I'}{}^{J'}\delta
g^{ab}+\nabla_c(\frac{\partial\mathcal{L}}{\partial
\nabla_c\psi}\acute{\psi}_{ba}{}^{II'}{}_{JJ'})\epsilon_I{}^J\bar{\epsilon}_{I'}{}^{J'}\delta
g^{ab}\nonumber\\
&&-2\nabla^c(\frac{\partial\mathcal{L}}{\partial
\nabla^b\psi}\acute{\psi}_{[ac]}{}^{II'}{}_{JJ'})\epsilon_I{}^J\bar{\epsilon}_{I'}{}^{J'}\delta
g^{ab}-2\nabla^c(\frac{\partial\mathcal{L}}{\partial
\nabla^a\psi}\sigma_{[bII'}\sigma_{c]}{}^{JJ'}\grave{\psi}^c{}_d{}^{II'}{}_{JJ'})\delta_c{}^d\delta g^{ab}]\sqrt{-g}\mathbf{d^4x}\nonumber\\
=&&\int\frac{1}{2}(T_{\mathcal{C}(ab)}+\nabla^cN_{(ab)c})\delta
g^{ab}\sqrt{-g}\mathbf{d^4x}.
\end{eqnarray}
Obviously, if we define the metric energy-momentum tensor
$T_{\mathcal{M}ab}$ as $\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta
g^{ab}}$, we have
\begin{equation}
T_{\mathcal{B}ab}=T_{\mathcal{M}ab},
\end{equation}
which completes our proof.
\section*{Acknowledgement}
It is my pleasure to acknowledge Prof. R. Geroch for instructive
suggestions and helpful discussions throughout the whole work. I
would also like to give my thanks to Dr. B. Zhou for his
encouragement and discussions. In addition, this work was supported
in part by NSFC(grant 10205002) and NSFC(grant 10373003).
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\end{document}

math-ph0412064.extracted_bib
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