superfields
In general, superfields are physically understood as quantized gravity fields that admit a highly reducible representation of a supersymmetry
algebra. The problem of specifying a supergravity
theory can be then defined as a search for those representations
that allow the construction of consistent local actions, perhaps considered as either quantum group, or quantum groupoid, actions. Extending quantum symmetries
to include quantized gravity fields-specified as `superfields'- is called supersymmetry in quantum gravity theories. A first approach to supersymmetry relied on a curved `superspace' (Wess and Bagger,1983 [2]) and is analogous to supersymmetric gauge theories (see, for example, Sections 27.1 to 27.3 of Weinberg, 1995).
Because in supergravity both spinor
and tensor fields are being
considered, The Gravitational Fields
are represented in terms of
tetrads,
rather than in terms of Einstein's
general
relativistic metric
. The connections between
these two distinct representations are as follows:
 |
(0.1) |
with the general coordinates being indexed by
etc.,
whereas local coordinates that are being defined in a locally
inertial coordinate system
are labeled with superscripts a, b,
etc.;
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:
 |
(0.2) |
(where
is an arbitrary real matrix), and the general
coordinate transformations:
 |
(0.3) |
In a weak gravitational field the tetrad may be represented as:
 |
(0.4) |
where
is small compared with
for
all
values, and
, 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 `gravitino', with
helicities
3/2. Such a self-charge-conjugate massless
particle as the `gravitiono' with helicities
3/2 can only have
low-energy interactions if it is represented by a Majorana
field
which is invariant under the gauge
transformations:
 |
(0.5) |
with
being an arbitrary Majorana field as defined by
Grisaru and Pendleton (1977). The tetrad field
and the graviton field
are then
incorporated into a term
defined as the
metric superfield. The relationships between
and
, on the one hand, and the components
of the metric superfield
, on the other hand,
can be derived from the transformations of the whole metric
superfield:
 |
(0.6) |
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
with matter would be then
described by considering how a weak gravitational field,
interacts with an energy-momentum tensor
represented as a linear combination of components of a real
vector
superfield
. Such interaction terms would,
therefore, have the form:
![$\displaystyle I_{\mathcal M}= 2\kappa \int dx^4 [H_\mu \Theta^\mu]_D ~,$](img30.png) |
(0.7) |
(
denotes `matter') integrated over a four-dimensional
(Minkowski) spacetime
with the metric defined by the superfield
. The term
, as defined above, is
physically a supercurrent and satisfies the conservation
conditions:
 |
(0.8) |
where
is the four-component super-derivative and
denotes a real chiral scalar
superfield. This leads immediately to
the calculation of the interactions of matter with a weak
gravitational field as:
 |
(0.9) |
It is interesting to note that the gravitational actions for the
superfield that are invariant under the generalized gauge
transformations
lead to
solutions of the Einstein field equations for a homogeneous,
non-zero vacuum energy density
that correspond to either
a de Sitter space for
, or an anti-de Sitter space for
. Such spaces can be represented in terms of the
hypersurface equation
 |
(0.10) |
in a quasi-Euclidean five-dimensional space with the metric
specified as:
 |
(0.11) |
with '
' for de Sitter space and '
' for anti-de Sitter space,
respectively.
- 1
-
S. Weinberg.: The Quantum Theory of Fields. Cambridge, New York and Madrid:
Cambridge University Press, Vols. 1 to 3, (1995-2000).
- 2
-
J. Wess and J. Bagger: Supersymmetry and Supergravity,
Princeton University Press, (1983).
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