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Connections and Loops Within SQC

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Quantum Cosmology - The Supersymmetric Perspective - Vol. 2

Part of the book series: Lecture Notes in Physics ((LNP,volume 804))

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Abstract

This is an area of SQC where there exists a wide range of possibilities for exploration. Some progress has been made, in which cosmological models were investigated either through a reduction of SUGRA formulated in terms of connections (or loops) [1–19] or through a (hidden) N = 2 SUSY framework based upon Bianchi models described ab initio with connections [20,21]. Further motivation will surely arise from recent developments in loop quantum cosmology [22–43], inheriting some of the main features and principles from loop quantum gravity [44–66]. In this chapter, we explain why the outlook is so promising in this area.

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Notes

  1. 1.

    The constraint \(\mathcal{J}^{AB}\) enforces SL(2,\(\mathbb{C}\)) covariance (see Appendix A) and generalizes the Gauss constraint of vacuum general relativity.

  2. 2.

    It is possible to proceed differently. We have to resort to a description in Euclidean terms, where all the Ashtekar variables become real. But non-polynomial expressions resurface, although there are claims that a satisfactory quantization is possible [3].

  3. 3.

    The (scientifically subversive) reader may nevertheless question the use of ‘reality conditions’, and rightly so. There are several works on this possibility (either in general relativity or N = 1 SUGRA). The innovative setting known as the Barbero–Sawaguchi canonical transformation uses real Ashtekar variables, but at a price. One loses the polynomiality of the Hamiltonian constraint and has to deal with a more complicated form of Dirac brackets (although these can nevertheless be made simpler by switching to new variables) [15].

  4. 4.

    Here \(\mathcal{D}_{i}\) is the covariant derivative with connection \({{\relax\ifmmode\mathsf{A}\else\textsf{A}\fi}}_{iA}{}^{B}\) and curvature

    $$\tilde{{\mathcal{R}}}_{ijA}{}^{B}\equiv 2\partial_{[i}{{\relax\ifmmode\mathsf{A}\else\textsf{A}\fi}}_{j]A}{}^{B} +2{{\relax\ifmmode\mathsf{A}\else\textsf{A}\fi}}_{[i|A}{}^{C}{{\relax\ifmmode\mathsf{A}\else\textsf{A}\fi}}_{j]C}{}^{B}\;.$$
  5. 5.

    For higher N SUGRA, see [5, 9, 10, 13, 16, 18, 19].

  6. 6.

    The action is nevertheless not complex, due to the fact that \(R_{[\lambda \mu \nu ]\rho }=0\). In addition, the imaginary part of the Lagrangian becomes a total derivative when the \({{\relax\ifmmode\mathsf{A}\else\textsf{A}\fi}}_{ABi}\) equation is satisfied.

  7. 7.

    This means that we will employ Poisson variables instead of having to resort to the Dirac brackets, since we will have no second-class constraints of the type discussed in Chap. 4 of Vol. I.

  8. 8.

    Or almost polynomial, as the h −1 term indicates in some expressions. Nonetheless, suitable redefinitions of the Lagrange multipliers can bring the full set of constraints into the desired polynomial form. In summary, we have to use

    $$\psi _{A0}^{(1)} \equiv h^{-1/2}n_{A}{}^{A^{\prime }}{\overline \psi}{}_{A^\prime 0}\;,$$
    ((6.32))
    $${\overline {\mathcal{S}}}{}^{A^{\prime }} \longrightarrow \mathcal{S}^{(1)A}\equiv \left( {{\relax\ifmmode\mathsf{E}\else\textsf{E}\fi}}^{j}{{\relax\ifmmode\mathsf{E}\else\textsf{E}\fi}}^{k}\mathcal{D}_{[j}\psi _{k]}\right) ^{A}\;,$$
    ((6.33))
    $$\mathcal{H}^{(1)AB} \equiv \mathcal{H}^{AB}+\left( 2h^{-1}{{\relax\ifmmode\mathsf{E}\else\textsf{E}\fi}}_{C}{}^{A}{}_i\pi ^{Bi}\right) \mathcal{S}^{(1)C}\;,$$
    ((6.33))

    to get a full polynomial set of constraints in N = 1 SUGRA.

  9. 9.

    We recall that, in N = 1 SUGRA, only a zero or negative cosmological constant is allowed [70]. Note the different use signs and constants with regard to [4, 8, 13, 14].

  10. 10.

    In general terms,

    $$S_\textrm{CS}\sim \frac{1}{\varUpsilon^{2}}\int \varepsilon ^{ijk}\left( {{\relax\ifmmode\mathsf{A}\else\textsf{A}\fi}}_{i}{}^{AB}\partial _{j}{{\relax\ifmmode\mathsf{A}\else\textsf{A}\fi}}_{kAB}+\frac{2}{3}{{\relax\ifmmode\mathsf{A}\else\textsf{A}\fi}}_{i}{}^{AB}{{\relax\ifmmode\mathsf{A}\else\textsf{A}\fi}}_{jB}^{\;C}{{\relax\ifmmode\mathsf{A}\else\textsf{A}\fi}}_{kCA}+2\varUpsilon\psi _{i}^{A}\mathcal{D}_{j}\psi _{kA}\right)\;.$$
  11. 11.

    The reader will recall that only even powers appear because of the SL(2,\(\mathbb{C}\)) invariance. There is no mixing of fermionic number (in this case), so the quantum constraints (6.69), (6.70), and (6.71) can be solved order by order.

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

    Paulo Vargas Moniz

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Moniz, P.V. (2010). Connections and Loops Within SQC . In: Quantum Cosmology - The Supersymmetric Perspective - Vol. 2. Lecture Notes in Physics, vol 804. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11570-7_6

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