Supersymmetric Minisuperspace with Nonvanishing Fermion Number

Abstract

The Lagrangian of W = 1 supergravity is dimensionally reduced to one (timelike) dimension assuming spatial homogeneity of any Bianchi type within class A of the classification of Ellis and McCallum. The algebra of the supersymmetry generators, the Lorentz generators, the diffeomorphism generators, and the Hamiltonian generator is determined and found to close. In contrast to earlier work, infinitely many physical states with nonvanishing even fermion number are found to exist in these models, indicating that minisuperspace models in supergravity may be just as useful as in pure gravity. PACS numbers: 04.60.Kz, 04.65.+e Minisuperspace models have long served as an enormously useful testing ground for new ideas in quantum gravity, ranging from explorations of its mathematical structure to investigation in the many open problems of quantum cosmology (see, e.g. , Refs. [1 -3]). More recently, also the quantum theory of supersymmetric minisuperspace models has attracted the interest of many authors (see, e.g., Refs. [4 -16]) for similar reasons. Our own interest was first aroused by the chaotic classical nature of the Bianchi type IX models and the discovery [7] that supersymmetric versions have simple explicit analytical solutions in the empty and filled fermion sectors which can be interpreted as wormhole states [9] and, in other cases, as Hartle-Hawking no-boundary states

Hartle-Hawking state in supersymmetric minisuperspace

“…Due to the different ordering chosen, the algebra (6)-(9) differs slightly from a corresponding result given in [11], but both forms are, of course, fully consistent.…”

“…Subsequently it was shown [10] that the particular SO(3) symmetry of Bianchi type IX permits an alternative homogeneity ansatz for the Rarita-Schwinger field, and that its application replaces the permitted 'worm-hole' state in the empty or filled fermion sector by a 'no-boundary' state in the same sector. In a recent paper [11] we reexamined the supersymmetric minisuperspace models of Bianchi type in class A [12] without matter fields and showed that, contrary to previous expectations, they posses infinitely many physical states. Hence, the question of the existence and form of a 'no-boundary' state in such models must be reconsidered.…”

“…Hence, the question of the existence and form of a 'no-boundary' state in such models must be reconsidered. In the present paper we (i) apply the theory of [11] to the supersymmetric Bianchi type IX model without matter, and with the conventional homogeneity condition for the Rarita-Schwinger field, and (ii) construct the Hartle-Hawking 'no-boundary'-state for that model explicitely. The dependence of that state on the 3-metric turns out to be the same as in [10] (see also [13]), where the alternative homogeneity condition was applied.…”

“…Here we shall deviate from ref. [11] and adopt the choice of the ordering as displayed explicitely in eq. (1), while in [11] we ordered the kernel Cα β pr to the right of p + pa before quantizing.…”

“…[11] and adopt the choice of the ordering as displayed explicitely in eq. (1), while in [11] we ordered the kernel Cα β pr to the right of p + pa before quantizing. While, at least so far, no reason of principle is visible to prefer one choice of ordering over the other (or over any mixed ordering in between), the ordering chosen here will actually simplify in an essential way the form of eq.…”