The Essence of Supersymmetric Quantum Cosmology

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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