metric superfieldsMetricSuperfields2008-08-15 21:57:122009-02-02 05:34:05Topicmathematical programs in quantum gravitysupergravity fieldsupergravityquantum gravitysuperspacerelativistic QFT and quantum field theories% this is the default PlanetMath preamble. as your knowledge
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\def\>{\rangle}This is a topic entry on metric superfields in quantum supergravity
and the mathematical cncepts related to spinor and tensor fields.
\section {Metric superfields: spinor and tensor fields}
Because in supergravity both spinor and tensor fields are being
considered, the gravitational fields are represented in terms of
\emph{tetrads}, $e^a_\mu(x),$ rather than in terms of the general
relativistic metric $g_{\mu \nu}(x)$. The connections between
these two distinct representations are as follows:
\begin{equation}
g_{\mu\nu}(x)=\eta_{ab}~ e^a_\mu (x)e^b_\gamma(x)~,
\end{equation}
with the general coordinates being indexed by $\mu,\nu,$ etc.,
whereas local coordinates that are being defined in a locally
inertial coordinate system are labeled with superscripts a, b,
etc.; $ \eta_{ab}$ is the diagonal matrix with elements +1, +1,
+1 and -1. The tetrads are invariant to two distinct types of
symmetry transformations--the local Lorentz transformations:
\begin{equation}
e^a_\mu (x)\longmapsto \Lambda^a_b (x) e^b_\mu (x)~,
\end{equation}
(where $\Lambda^a_b$ is an arbitrary real matrix), and the general
coordinate transformations:
\begin{equation}
x^\mu \longmapsto (x')^\mu(x) ~.
\end{equation}
In a weak gravitational field the tetrad may be represented as:
\begin{equation}
e^a_\mu (x)=\delta^a_\mu(x)+ 2\kappa \Phi^a_\mu (x)~,
\end{equation}
where $\Phi^a_\mu(x)$ is small compared with $\delta^a_\mu(x)$ for
all $x$ values, and $\kappa= \surd 8\pi G$, where G is Newton's
gravitational constant. As it will be discussed next, the
supersymmetry algebra (SA) implies that the graviton has a
fermionic superpartner, the hypothetical \emph{gravitino}, with
helicities $\pm$ 3/2. Such a self-charge-conjugate massless
particle as the gravitiono with helicities $\pm$ 3/2 can only have
\emph{low-energy} interactions if it is represented by a Majorana
field $\psi _\mu(x)$ which is invariant under the gauge
transformations:
\begin{equation}
\psi _\mu(x)\longmapsto \psi _\mu(x)+\delta _\mu \psi(x) ~,
\end{equation}
with $\psi(x)$ being an arbitrary Majorana field as defined by
Grisaru and Pendleton (1977). The tetrad field $\Phi _{\mu
\nu}(x)$ and the graviton field $\psi _\mu(x)$ are then
incorporated into a term $H_\mu (x,\theta)$ defined as the
\emph{metric superfield}. The relationships between $\Phi _{\mu _
\nu}(x)$ and $\psi _\mu(x)$, on the one hand, and the components
of the metric superfield $H_\mu (x,\theta)$, on the other hand,
can be derived from the transformations of the whole metric
superfield:
\begin{equation}
H_\mu (x,\theta)\longmapsto H_\mu (x,\theta)+ \Delta _\mu
(x,\theta)~,
\end{equation}
by making the simplifying-- and physically realistic-- assumption
of a weak gravitational field (further details can be found, for
example, in Ch.31 of vol.3. of Weinberg, 1995). The interactions
of the entire superfield $H_\mu (x)$ with matter would be then
described by considering how a weak gravitational field,
$h_{\mu_\nu}$ interacts with an energy-momentum tensor $T^{\mu
\nu}$ represented as a linear combination of components of a real
vector superfield $\Theta^\mu$. Such interaction terms would,
therefore, have the form:
\begin{equation}
I_{\mathcal M}= 2\kappa \int dx^4 [H_\mu \Theta^\mu]_D ~,
\end{equation}
($\mathcal M$ denotes `matter') integrated over a four-dimensional
(Minkowski) spacetime with the metric defined by the superfield
$H_\mu (x,\theta)$. The term $\Theta^\mu$, as defined above, is
physically a \emph{supercurrent} and satisfies the conservation
conditions:
\begin{equation}
\gamma^\mu \mathbf{D} \Theta _\mu = \mathbf{D} ~,
\end{equation}
where $\mathbf{D}$ is the four-component super-derivative and $X$
denotes a real chiral scalar superfield. This leads immediately to
the calculation of the interactions of matter with a weak
gravitational field as:
\begin{equation}
I_{\mathcal M} = \kappa \int d^4 x T^{\mu \nu}(x)h_{\mu \nu}(x) ~,
\end{equation}
It is interesting to note that the gravitational actions for the
superfield that are invariant under the generalized gauge
transformations $H_\mu \longmapsto H _\mu + \Delta _\mu$ lead to
solutions of the Einstein field equations for a homogeneous,
non-zero vacuum energy density $\rho _V$ that correspond to either
a de Sitter space for $\rho _V>0$, or an anti-de Sitter space for
$\rho _V <0$. Such spaces can be represented in terms of the
hypersurface equation
\begin{equation}
x^2_5 \pm \eta _{\mu,\nu} x^\mu x^\nu = R^2 ~,
\end{equation}
in a quasi-Euclidean five-dimensional space with the metric
specified as:
\begin{equation}
ds^2 = \eta _{\mu,\nu} x^\mu x^\nu \pm dx^2_5 ~,
\end{equation}
with '+' for de Sitter space and '-' for anti-de Sitter space,
respectively.
\textbf{Note}
The presentation above follows the exposition by S. Weinberg in his book
on ``Quantum Field Theory'' (2000), vol. 3, Cambridge University Press (UK),
in terms of both concepts and mathematical notations.