A theory of quantum gravity constitutes a foremost aspiration in theoretical physics and the inclusion of supersymmetry seems to yeld significant benefits. Research in supersymmetric quantum cosmology using canonical methods started about 10 years ago [1]. Since then, many papers relating to the subject have appeared in the literature (cf. ref. [2] and references therein).

Supersymmetry relates bosons with
fermions
[3]. Its promotion to a gauge symmetry
has resulted in an elegant field theory: * supergravity*
Supersymmetry plays an important role when dealing with
divergences. Furthermore,
it
would be sensible
to consider bosons and fermions
on an equivalent
level within a early universe.

The canonical formulation of (pure) N=1 supergravity was presented in [1] In finding a physical state it is sufficient to solve the Lorentz and supersymmetry constraints: The wave function will subsequently obey the Hamiltonian constraints. In fact, supergravity constitutes a ``square root'' of gravity. Different approaches of canonical methods in supersymmetric quantum cosmology can be found in the literature: Ashtekar and loop variables [4], ADM metric representation with either s-models [5], matrix representation for gravitinos [6] and a differential operator approach for fermions [1,2]. Progress within the latter has provided interesting physical insights and the following summarizes recent results.

Minisuperspace models have proved to be a very valuable tool. FRW models are the simplest ones but Bianchi models enable us to include more gravitino modes. An important feature is that fermion number is conserved. Hence, each fermionic sector may be treated separately.

. Quantum states may be represented by *F [b, f] *,
*b* denotes bosonic fields and *f* fermionic
fields. The
two-component spinor form of
the tetrad (graviton) and the spin-3/2 gravitino field can be represented by
*b* and *f*, respectively.
The wave function may then be expanded in even powers of
*f*.

Prior to ref. [7] solutions for Bianchi class-A
were * only* present
in the empty and fermionic filled
sectors.
But this curious result was joined by yet
another disturbing one.
When a cosmological constant was added *no*
physical states were found. Regarding
the k = +1 FRW model, a bosonic
state was found, namely the Hartle-Hawking solution for a
De Sitter case. Moreover, one could not find a
wormhole (Hawking-Page) or no-boundary (Hartle-Hawking)
states
in the same spectrum of solutions.
Finding one or the other
depended on homogeneity conditions imposed on
the gravitino [8]. Furthermore, these solutions
were shown to have no counterpart in the
full theory [9]: states with * finite* number of
fermions are * impossible* there.

The cause was the use of too restrictive middle fermionic sectors. These problems were then subdued in ref. [7] and nontrivial solutions were found in all sectors. Furthermore, these physical states have direct analogues in full supergravity. The wormhole and Hartle-Hawking solutions were obtained in the empty and 4-fermion sector, respectively. In the presence of a cosmological constant solutions have the form of exponentials of the N=1 supersymmetric Chern-Simons functional [7].

The introduction of matter led to new results. A scalar supermultiplet, constituted by complex scalar fields and their spin 1/2 partners was considered in ref. [10,11] for FRW models. A vector supermultiplet, formed by gauge vector fields and fermionic partners, was added in ref. [12] A Bianchi type-IX model with scalar supermultiplet was studied in [13].

A wormhole
state was found in ref. [10] but not in ref.
[11]. The reason for this
discrepancy
was discussed in [14]. When all matter fields were included,
the only allowed physical state was *F=0* [12].
This
motivated further research present in [15] where
solely vector supermultiplets were present in a FRW model.
Overall, these results strongly suggest that the
treatments of supermatter require further analysis.

Models with a richer structure can be found from
* extended* supergravities.
N=1 supergravity is the simplest
theory
with one real massless gravitino. N=0 corresponds
to ordinary general
relativity. N=2 supergravity realises Einstein's dream
of
unifying gravity with electromagnetism. This theory contains
2 gravitinos besides the gravitational and Maxwell fields [3].
These are supergravity theories
with more gravitinos
and
have
additional symmetries coupling several physical
variables.
For Bianchi class-A models in N=2 supergravity
[16]
it was found that
the presence of Maxwell fields leads to a non-conservation
of the fermionic number. This then implies a
mixing between Lorentz invariant
sectors in the wave function.

Further challenges in supersymmetric quantum cosmology still prevail [2]:

**a)** Why there are *no* physical states in
FRW models with generic gauged supermatter [12]
but solutions exist when soley vector supermultiplets are
included [13] ;

**b)** Obtain conserved currents in supersymmetric
minisuperspaces;

** c)** The validity of the
minisuperspace approximation in supersymmetric models.

[1] P. D. D'Eath, Phys. Rev. D 29 (1984) 2199 and ref. therein.

[2] P. Moniz, *Supersymmetric Quantum Cosmology*,
Int. J. Mod. Phys. - A (review), DAMTP--R95/53 and
ref. therein.

[3] P. van Nieuwenhuizen, Phys. Rep. 68 (1981) 189.

[4] H-J. Matschull, Class. Q. Grav. 11 (1994) 2395; R. Ganbini, O. Obregon. J. Pullin, hep-th/9508036.

[5] R. Graham and J. Bene, Phys. rev. D49 (1994) 799.

[6] A. Macias, O. Obregon and J. Socorro, Int. J. Mod. Phys.-A8 (1993) 4291.

[7] R. Graham and A. Csordas, Phys. Rev. Lett. 74 (1995) 4129; Phys. Rev. D52 (1995) 5653.

[8] R. Graham and H. Luckock, Phys. Rev. D49, R4981 (1994).

[9] S. Carroll, D. Freedman, M. Ortiz and D. Page, Nuc. Phys. B423, 3405 (1994).

[10] L.J. Alty, P.D. D'Eath and H.F. Dowker, Phys. Rev. D46 4402 (1992).

[11] A.D.Y. Cheng and P.R.L.V. Moniz, Int. J. Mod. Phys. D4 (1995) 189.

[12] A.D.Y. Cheng, P.D. D'Eath and P.R.L.V. Moniz, Class. Quantum Grav. 12 (1995) 1343

[13] P. Moniz, Int. J. Mod. Phys-A vol. 11 (1996).

[14] P. Moniz, Gen. Rel. Grav. Vol.28 (1996) 97

[15] P. Moniz, Quantization of FRW model in N=1 supergravity with gauged fields, DAMTP Report R95/36

[16] A.D.Y. Cheng and P.V. Moniz, Canonical quantization of Bianchi class A models in N=2 supergravity, Mod. Phys. Lett.-A to appear

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