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SQC N=1 SUGRA (Fermionic Differential Operator Representation)

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Quantum Cosmology - The Supersymmetric Perspective - Vol. 1

Part of the book series: Lecture Notes in Physics ((LNP,volume 803))

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Abstract

This is a very important chapter. It may seem rather long and perhaps over-detailed, but the reader should have no doubt that it is all quite necessary. All journeys of exploration, including those involved with research, have to start somewhere, and many steps may be needed before an interesting vantage point is reached. SQC is such a case and this chapter is part of the route toward that goal. A significant amount of the results and research in SQC have been produced within the methodology we will describe here (see [1], but also [2–6]). It follows directly from the reduction of 4D canonical quantum N = 1 SUGRA (indirectly related to a higher dimensional superstring theory) to a cosmological minisuperspace with time-dependent (1D) variables and momenta represented by differential operators of the canonically conjugate variables [7, 8].

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Notes

  1. 1.

    Consult Sects. 4.1 and 4.2 of Vol. II for the canonical quantization of the full theory. There we identify the characteristics of physical states [12, 4, 13] (Sect. 4.1), discussing the route to a semiclassical perspective [14] and how one might investigate (cosmologically) testable directions (Sect. 4.2).

  2. 2.

    It is also important to stress that auxiliary fields are also required to balance the number of fermionic and bosonic degrees of freedom. However, these auxiliary fields can be neglected in the end (see [15, 2, 21, 16, 17] for more details).

  3. 3.

    This means that we truncate the general decomposition (see Sect. 5.2) \(\psi^A{}_{BB^{\prime}} = e_{B\,B^{\prime}}{}^{i}\psi^A{}_{i}\) in (5.191) at the spin-1/2 mode level, i.e., with \(\beta^A = 3 n^{AA^{\prime}} \overline\psi{}_{A^{\prime}}/4 \sim\overline\psi{^A}\).

  4. 4.

    The reader should note that \({\mathsf{P}}\) will henceforth play the role of \(\mathsf{W}\) as indicated in Chap. 3 of Vol. II. The corresponding literature in SQC has mainly used \({\mathsf{P}}\) instead (see also [36]).

  5. 5.

    Please notice that we are not introducing a Fourier-like expansion in the variables \(a,\phi,\overline{\phi}\). This would lead to a midisuperspace scenario [39, 40], a setting situated somewhere between a SQC minisuperspace and the full theory of supergravity. Instead, we restrict ourselves to a strongly truncated supersymmetric minisuperspace and employ the expansion as a simplifying assumption in our equations.

  6. 6.

    However, in the case of larger gauge group, some of the gauge symmetries will survive. These will give rise, in the one-dimensional model, to local internal symmetries with a reduced gauge group. Therefore, a gauge constraint can be expected to play an important role in that case. It would be particularly interesting to study such a model (see Chap. 8).

  7. 7.

    The introduction of fermions in ordinary quantum cosmological models with gauge fields led to additional non-trivial ansätze for the fermionic fields [63]. These involve restrictions from group theory, rather than just imposing time dependence.

  8. 8.

    In the present case, the first term is absent due to our choice of gauge group.

  9. 9.

    Notice that the above expressions correspond to a gauge group SU(2) and hence a compact Kähler manifold, which implies that the analytical potential \({\mathsf{P}(\phi^{I})}\) is zero.

  10. 10.

    Other approaches and frameworks will be described in the next chapter and in Vol. II.

  11. 11.

    See Sect. 4.1.3 concerning the Teitelboim procedure.

  12. 12.

    Moreover, it should be noticed that \(^{(3\textrm{s})}D_j\), the spatial covariant derivative on the gravitino, is explicitly present in the supersymmetry constraints (see Appendix A).

  13. 13.

    In the representation \(\overline \Psi(e^{AA^{\prime}}{}_{i}, \overline \psi{}^{A^{\prime}}{}_{i})\), these have of the form \(\overline \psi{}^6\), \(\overline \psi{}^4\), \(\overline \psi{}^2\), and \( \overline \psi{}^0\), the superscript denoting the fermionic order.

  14. 14.

    Meaning that only time dependence is required for the effective degrees of freedom (see the discussion in Sect. 5.1.5). It should be noted that this simple ansatz is not invariant under homogeneous supersymmetry transformations. To obtain an ansatz that is invariant under supersymmetry, one must use a non-diagonal triad \(e^a{}_{i} = b^a{}_{b} E^b{}_{i}\), where b ab is symmetric (\(a, b = 1, 2, 3\) here), combined with supersymmetry, homogeneous spatial coordinate, and local Lorentz transformations.

  15. 15.

    A term \((\beta^A \gamma_{ABC})^2 = \beta^A \gamma_{ABC}\beta^D \gamma_D{}^{BC}\) can be rewritten, using the anticommutation of the β and γ terms, as \(\beta^E\,\beta_E \varepsilon^{AD} \gamma_{ABC} \gamma_D{}^{BC} \sim (\beta_E \beta^E)(\gamma_{ABC} \gamma^{ABC})\). Similarly, any quartic in γ ABC can be rewritten as a multiple of \((\gamma_{ABC} \gamma^{ABC})^2\). Since there are only four independent components of \(\gamma_{ABC} = \gamma_{(ABC)}\), only one independent quartic can be made from γ ABC , and it is sufficient to check that \((\gamma_{ABC} \gamma^{ABC})^2\) is non-zero. Now \(\gamma_{ABC} \gamma^{ABC} = 2\gamma_{000} \gamma_{111} - 6 \gamma_{100} \gamma_{011}\). Hence \((\gamma_{ABC} \gamma^{ABC})^2\) includes a non-zero quartic term \(\gamma_{000} \gamma_{100} \gamma_{110} \gamma_{111}\) [2].

  16. 16.

    Only the class A models allow the spatial sections to be compactified by factoring if necessary by a discrete subgroup of isometries [74, 76].

  17. 17.

    Regarding the \(k = +1\) FRW model, a bosonic state was found, namely the Hartle–Hawking solution for an anti-de Sitter case (see Sect. 5.1.2).

  18. 18.

    For the case of an FRW model without supermatter, and due to the restriction of the gravitino field to its spin-1/2 mode component, the former ansatz for the wave function remains valid.

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  1. Depto. Fisica, Universidade da Beira Interior, Rua Marquês d’Avila e, Bolama, 6200-307, Covilhã, Portugal

    Paulo Vargas Moniz

  2. CENTRA-IST, Lisboa, Portugal

    Paulo Vargas Moniz

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Moniz, P.V. (2010). SQC N=1 SUGRA (Fermionic Differential Operator Representation) . In: Quantum Cosmology - The Supersymmetric Perspective - Vol. 1. Lecture Notes in Physics, vol 803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11575-2_5

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