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Canonical Quantization of N = 1 Supergravity

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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 where our journey of exploration will really begin. The last two chapters introduced the essential features of a quantum mechanical description of the early universe and the use of fermions in a supersymmetric setting. On the one hand, the analysis of general relativity from a Hamiltonian perspective [1–7] has brought new insights into the very origin of the universe1 (as well as many other domains of research such as spacetime singularities, gravitationally driven chaos, primordial perturbations).

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Notes

  1. 1.

    In fact, we were given a methodology with which to probe the universe from a non-perturbative and quantum mechanical standpoint. Concepts such as the wave function and the creation of the universe, associated with a transition to classical states, were then extensively explored [14–21].

  2. 2.

    Since the early 1970s, a considerable range of publications about SUSY and SUGRA can be found in the literature in the context of particle physics interactions. Fascinating results have been achieved [22, 12], notably finite values for specific interactions that were not renormalizable using just perturbative general relativity.

  3. 3.

    In N = 1 SUGRA, we will find fourteen constraints associated with the invariances under gene- ric changes of coordinates (four), local Lorentz rotations (six), and SUSY transformations (four). Moreover, hypersurface generators (of deformations) in the spacetime are obtained from the anticommutator of two (local) SUSY generators, hence widening the SUSY context, where a translation is retrieved from the anticommutation of two SUSY transformations (see Sect. 3.1).

  4. 4.

    For N = 1 SUGRA, we will have additional fermionic terms in the generators, compared with those in Chap. 2, as well as new generators. The geometry associated with N = 1 SUGRA (a new theory of gravitation, and with expected benefits within the quantum regimes) will be remarkably different from the Riemannian setting of general relativity.

  5. 5.

    In this context, the Christoffel symbol in (2.9) is actually \(\Gamma_{\mu\nu}^{(0)\,\lambda}\), as the reader has surely noticed [see also (4.3) and Sect. 3.4.3 for further explanation].

  6. 6.

    Recall that a is the ‘flat’ index of the tetrad and runs from 0 to 3 (see also Appendix A). Indices a, b, c,… are raised and lowered with \(\eta^{ab}\) and \(\eta_{ab}\), respectively, where \(\eta^{ab}\) has the signature \((-,+,+,+)\). Spacetime indices \(\mu,\nu\) are raised and lowered with \(g^{\mu\nu}\) and \(g_{\mu\nu}\), respectively. The relation between the spacetime metric and the tetrad elements can be written as \( g_{\mu\nu}=\eta_{ab}e^a{}_{\mu}e^b{}_\nu\), \(\eta^{ab}=g^{\mu\nu}e^a{}_{\mu}e^b{}_{\nu}\).

  7. 7.

    Although in some cases for the 2-spinor notation, just one equation is sufficient for working purposes, the other just being the Hermitian conjugate.

  8. 8.

    It is important to emphasize, if it is not already clear, that we formulate a CSK theory but with torsion determined and explicitly written in terms of a \(\psi^{[a]}_\mu\) gravitino matter sector.

  9. 9.

    To simplify, until further notice, we follow [34] and set \({\textbf{k}}^2 =1\).

  10. 10.

    It is worth pointing out the approach of [28, 34]. The authors use \(h^{1/2}\psi_{a}\) instead of \(\psi_{\mu}\), as well as the time gauge, retrieving \(\left[ p^{ia}(x),h^{1/2}\psi_{b}(x^{\prime})\right]_{\textrm{D}}=\left[ p^{ia}(x),p^{{jb}}(x^{\prime})\right]_{\textrm{D}}=0\). What is the time gauge? In brief, it consists in removing three of the six SO(3,1) degrees of freedom, leaving only the SO(3) spatial rotations.

  11. 11.

    This means that the constraints we will use (\({\mathcal{H}}_{\bot},{\mathcal{H}}_{i}, {\mathcal{S}} ,{\mathcal{J}}^{{ab}}\)) contain only the canonical variables and no Lagrange multipliers. This secures a simpler Hamiltonian, and more importantly, it means that the constraints will satisfy \(\big[C_{m},C_{n}\big]_{\textrm{D}}=\rho_{mn}^{k}C_{k}\), which is essential when applying the Teitelboim procedure [34, 35, 40, 41].

  12. 12.

    If a minimally coupled action, with ≀ instead of \(D_{\nu }\), is employed, then SUSY is lost, therefore confirming what was indicated in Chap. 3 concerning the form of the N = 1 SUGRA action.

  13. 13.

    Remember, the derivative \(\nabla_{j}\) contains torsion in the form of terms quadratic in the gravitino.

  14. 14.

    As remarked in [34], this mechanism is specific to SUGRA, whereby the gauge symmetry is neither purely external or internal, and caused by the specific derivative coupling when taking SUSY into a curved space.

  15. 15.

    The \(\rho_{{mn}}^{k}\) are the structure constants of the algebra, which can be functionals of \(q\equiv\{e,\psi\}\) and the corresponding canonically conjugate momenta p.

  16. 16.

    The Hamiltonian H in (4.37) does not contain arbitrary rotations in a self-evident manner, so the presence of \({\mathcal{J}}^{{ab}}\) in the Teitelboim procedure cures this aspect with an appropriate geometrical structure.

  17. 17.

    \({\mathcal{H}}_{a}={\mathcal{H}}_{\mu }e^\mu{}_{a}\) are the generators of localized translations, being the projection of the surface deformations \({\mathcal{H}}_{\mu }\) along the tetrad directions \(e^\mu{}_{a}\).

  18. 18.

    \(\mathcal{N}\) and \({\mathcal{N}}^i\) of (A.47) are Lagrange multipliers, as are \(\psi^{\, A}_0\) and \(\overline{\psi}_0^{A^{\prime}}\).

  19. 19.

    Note the equivalent representation in terms of a symmetrized version (see Appendix A and (4.19), (4.20), (4.21), (4.22), and (4.23)):

    $$\pi^{\mathit{ij}}\equiv-\frac{1}{2}p^{(\mathit{ij})},\qquad p^{\mathit{ij}}=-e^{AA^{\prime}j}p_{AA^{\prime}}^i\,.$$
    ((4.68))
  20. 20.

    The momentum conjugate to \(\psi^{A^{\prime}}_i\) is represented by \(\tilde{\pi}_{A^{\prime}}^i\), as it is minus the Hermitian conjugate of \(\pi_{A}^i\) [26, 27].

  21. 21.

    See Appendix B. The symbol ≅ for weak equality is employed here.

  22. 22.

    Note that \(^{(3)}\,R\) is the 3D scalar curvature (A.96). Moreover, \(^{(3\textrm{s})}\nabla_k\) and \(^{(3)}\nabla_k\) correspond to the usual spatial covariant derivative without torsion and with torsion, respectively (see Appendix A).

  23. 23.

    Although (4.80) and (4.84) are apparently somewhat odd, using [28] \(\psi^A \leftarrow \varphi^{A}_i=h^{1/4}\psi^{A}_i\), the Dirac bracket would be zero, but with the price that the Hamiltonian would get more complicated, and worse, we could not obtain a wave functional \(\Psi[e,\psi]\) with independent variables [25, 27].

  24. 24.

    See Appendix A and Sect. 3.5. We recall that the indices \( {\mathsf{I}}\) and \({\mathsf{J}}{^\ast}\equiv \overline {\mathsf{J}} \) are KNhler indices, and that the KNhler metric herein is \( {\mathsf{G}}_{{\mathsf{I J}}^\ast} = {\mathsf{G}}_{{\mathsf{I \overline J}}}={\mathsf{K}}_{I J^{\ast}} \) on the space of \(\phi^{\mathsf{I}}\), \(\phi^{{\mathsf{J}}^{\ast}}\) (the KNhler manifold), where \( {\mathsf{K}}_{{\mathsf{I J}}^{\ast}} \) is a shorthand for \(\partial^2 {\mathsf{K}}/\partial \phi^{\mathsf{I}} \partial \phi^{{\mathsf{J}}^{\ast}}\), with \( {\mathsf{K}}\) the KNhler potential (see Note 3.8). Each index (a) corresponds to an independent (holomorphic) Killing vector field \(X^{(a)}\) of the KNhler geometry. The corresponding Killing equation implies that there exist real scalar functions \( D^{(a)} ( \phi^I, \phi^{I^{\ast}})\) known as Killing potentials.

  25. 25.

    For later use, we recall the relevant definitions (see also Sects. 3.3.1 and 3.5):

    • \({\mathsf{\Gamma}}^{\mathsf{I}}_{{\mathsf{JK}}}\) is a Christoffel symbol derived from the KNhler metric.

    • \({\mathsf{P}}\) is henceforth a complex scalar-field-dependent analytic potential energy term, which will relate to the superpotential.

    • \(\check{\textrm{D}}_\textrm{{I}}\equiv \frac{\partial}{\partial \phi^{{\mathsf{I}}}} + \frac{\partial {\mathsf{K}} }{ \partial \phi^{{\mathsf{I}}}}.\).

    • \(\tilde{{\textrm{D}}}_\mu \equiv \partial_\mu + \omega_\mu + \frac{1}{4} \left(\frac{\partial {\mathsf{K}}}{\partial \phi^{\mathsf{J}}} \hat {\textrm{D}}_\mu\phi^{\mathsf{J}} - \frac{\partial {\mathsf{K}}}{\partial\overline\phi{}^{{\mathsf{J}}^{\ast}}} \hat {\textrm{D}}_\mu\overline\phi{}^{{\mathsf{J}}^{\ast}}\right) + \frac{1}{2} A^{(a)}_\mu \textrm{Im} F^{(a)}.\).

    • \(\hat {\textrm{D}}_\mu \phi^{\mathsf{I}} \equiv \partial_\mu \phi^{\mathsf{I}} - v^{(a)}_\mu X^{{\mathsf{I}}(a)},\qquad F^{(a)} \equiv X^{I(a)}\frac{\partial {\mathsf{K}}}{\partial \phi^{\mathsf{I}}} + \textrm{i} D^{(a)}.\).

    • \(C^{\mu\nu}{}_{A}{}{B} \equiv \frac{1}{4} \left(e^\mu_{AA^{\prime}}e^{\nu BA^{\prime}} - e_{AA^{\prime}}^\nu e^{\mu BA^{\prime}}\right).\).

    • \(\hat{f}_{\mu\nu}^{(a)} \equiv f^{(a)}_{\mu\nu} - \textrm{i}\left\{\psi^A_{[\mu}e_{\nu]AA^{\prime}} \overline\lambda{}^{(a)A^{\prime}} -\overline\psi{}_{A^{\prime}[\mu} e^{AA}_{\nu]}{^{\prime}}{\lambda^{(a)}_A}\right\}.\).

  26. 26.

    Here \({\mathsf{K}}^{1/2}_{{\mathsf{I J}}^{\ast}}\) denotes a ‘square-root’ of the KNhler metric, obeying \( {\mathsf{K}}^{1/2}_{{\mathsf{I J}}^\ast} \delta^{{\mathsf{K J}}^\ast} {\mathsf{K}}^{1/2}_{{\mathsf{K L}}^\ast} = {\mathsf{K_{I L}}}\). This may be found by diagonalizing \( {\mathsf{K_{I J^\ast}}}\) via a unitary transformation, assuming that the eigenvalues are all positive. One needs to assume that there is an ‘identity metric’ \(\delta^{{\mathsf{K J}}^\ast}\) defined over the whole KNhler manifold.

  27. 27.

    In this case, the tetrad component n a of the normal vector \( n^\mu\) is restricted by \( n^a = \delta^a_0 \Leftrightarrow e^0{}_{i} = 0\). Thus, the original Lorentz rotation freedom becomes replaced by that of spatial rotations. In the time gauge, the geometry is described by the triad \(e^a{}_{i}\) (\(a = 1, 2, 3\)), and the conjugate momentum is \( p_a{^i}\). Notice that

    $$\pi^{ij} \equiv - \frac{1}{2} p^{(ij)} =\frac{1}{2} e^{AA^{\prime}(i}p^{j)}_{AA^\prime}=-\frac{1}{2}e^{a(i}p_{a}^{j)}\,,$$

    where the last equality follows from the time gauge conditions.

  28. 28.

    \(\pi^{n (a)} \) is the momentum conjugate to \( A^{(a)}_n\).

  29. 29.

    In particular, notice again the form of the extended spatial derivatives:

    $$^{(3 \textrm{s})} \tilde{D}_j \psi^A{}_{k} =\partial_j \psi^A{}_{k} + ^{(3 s)} \omega^A{}_{B j} \psi^B{}_{k} + \frac{1}{4} \big({\mathsf{K}}_{\mathsf{K}} \hat {{\textrm{ D}}}_j \phi^{\mathsf{K}} -{\mathsf{K}}_{{\mathsf{K}}^{\ast}}\hat {{\textrm{ D}}}_j \phi^{{\mathsf{K}}^{\ast}} \big) \psi^A{}_{k} + \frac{1}{2} v^{(a)}_j \textrm{Im}\,F^{(a)} \psi^A{}_{k}\,,$$

    where \(\hat {D}_i \phi^{\mathsf{K}} = \partial_i\phi^{\mathsf{K}}-v^{(a)}_i X^{{\mathsf{K}} (a)}\), and \( X^{{\mathsf{K}} (a)}\) is the a th Killing (KNhler) vector field [13].

References

  1. R. Arnowitt, S. Deser, and C.W. Misner: The dynamics of general relativity, gr-qc/0405109 (1962)

    Google Scholar 

  2. B.S. DeWitt: Quantum theory of gravity. I. The canonical theory. Phys. Rev. 160, 1113–1148 (1967)

    Article  ADS  Google Scholar 

  3. C.M. Dewitt and J.A. Wheeler: Battelle Rencontres, 1967 Lectures in Mathematics and Physics (Seattle), W.A. Benjamin, Inc., New York (1968)

    Google Scholar 

  4. C. Kiefer. Quantum Gravity, 2nd edn., International Series of Monographs on Physics 136, Clarendon Press, Oxford (2007)

    Google Scholar 

  5. C.W. Misner: Minisuperspace. In J.R. Klauder (Ed.), Magic Without Magic, Freeman, San Francisco (1972) pp. 441–473

    Google Scholar 

  6. M.P. Ryan and L.C. Shepley: Homogeneous Relativistic Cosmologies. Princeton Series in Physics, Princeton University Press, Princeton, NJ (1975)

    Google Scholar 

  7. J.A. Wheeler: Superspace. In Gilbert, R.D. and Newton, R. (Eds.), Analytic Methods in Mathematical Physics, Gordon and Breach, New York (1970) pp. 335–378

    Google Scholar 

  8. C.V. Johnson: D-Branes. Cambridge University Press, Cambridge, MS (2003)

    MATH  Google Scholar 

  9. M. Kaku: Introduction to Superstrings and M-Theory.Springer, New York (1999)

    Book  MATH  Google Scholar 

  10. M. Kaku: Strings, Conformal Fields, and M-Theory. Springer, New York (2000)

    Book  MATH  Google Scholar 

  11. B. Zwiebach: A First Course in String Theory. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  12. P. Van Nieuwenhuizen: Supergravity. Phys. Rep. 68, 189–398 (1981)

    Article  MathSciNet  ADS  Google Scholar 

  13. J. Wess and J. Bagger: Supersymmetry and Supergravity. Princeton University Press, Princeton, NJ (1992)

    Google Scholar 

  14. L.Z. Fang and R. Ruffini (Eds.): Quantum cosmology. Advanced Series in Astrophysics and Cosmology 3, World Scientific, Singapore (1987)

    Google Scholar 

  15. J.J. Halliwell: Introductory lectures on quantum cosmology. To appear in Proc. of Jerusalem Winter School on Quantum Cosmology and Baby Universes, Jerusalem, Israel, 27 December 1989–4 January 1990

    Google Scholar 

  16. J.B. Hartle: Quantum cosmology. In New Haven 1985, Proceedings, High Energy Physics, Vol. 2 (1985) pp. 471–566

    ADS  Google Scholar 

  17. C. Kiefer: Quantum cosmology. Expectations and results. Ann. Phys. 15, 316–325 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. C. Kiefer and T.P. Singh: Quantum gravitational corrections to the functional Schrödinger equation. Phys. Rev. D 44, 1067–1076 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  19. D.N. Page: Lectures on quantum cosmology. In Proceedings of Banff Summer Institute on Gravitation, Banff, Canada, 12–15 August 1990, Mann, R.B. et al. (Eds.), World Scientific, Singapore (1991)

    Google Scholar 

  20. T. Vachaspati and A. Vilenkin: On the uniqueness of the tunneling wave function of the universe. Phys. Rev. D 37, 898 (1988)

    Article  MathSciNet  ADS  Google Scholar 

  21. D.L. Wiltshire: An introduction to quantum cosmology. In Cosmology: The Physics of the Universe, Proceedings of the 8th Physics Summer School, A.N.U., Jan–Feb, 1995, By B. Robson, N. Visvanathan, and W.S. Woolcock (Eds.), World Scientific, Singapore (1996) pp. 473–531. gr-qc/0101003 (1995)

    Google Scholar 

  22. P. Binetruy: Supersymmetry, Oxford University Press, Oxford (2006)

    MATH  Google Scholar 

  23. R. Casalbuoni: On the quantization of systems with anticommuting variables. Nuovo Cim. A 33, 115 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  24. L. Castellani, P. Van Nieuwenhuizen, and M. Pilati: First order tetrad gravity in Dirac’s Hamiltonian formalism. Phys. Rev. D 26, 352 (1982)

    Article  MathSciNet  ADS  Google Scholar 

  25. P.D. DEath: Quantum supergravity via canonical quantization. In Oxford 1984, Proceedings, Quantum Concepts in Space and Time (1984) pp. 341–350

    Google Scholar 

  26. P.D. DEath: Supersymmetric Quantum Cosmology. Cambridge University Press, Cambridge, (1996) p. 252

    Chapter  Google Scholar 

  27. P.D. D’Eath: The canonical quantization of supergravity. Phys. Rev. D 29, 2199 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  28. S. Deser, J.H. Kay, and K.S. Stelle: Hamiltonian formulation of supergravity. Phys. Rev. D 16, 2448 (1977)

    Article  ADS  Google Scholar 

  29. E.S. Fradkin and T.E. Fradkina: Quantization of relativistic systems with boson and fermion first and second class constraints. Phys. Lett. B 72, 343 (1978)

    Article  ADS  Google Scholar 

  30. E.S. Fradkin and M.A. Vasiliev: Hamiltonian formalism, quantization and S matrix for supergravity. Phys. Lett. B 72, 70 (1977)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. A. Macias, O. Obregon, and M.P. Ryan: Quantum cosmology. The supersymmetric square root. Class. Quant. Grav. 4, 1477–1486 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  32. J.E. Nelson and C. Teitelboim: Hamiltonian for the Einstein–Dirac field. Phys. Lett. B 69, 81–84 (1977)

    Article  ADS  Google Scholar 

  33. J.E. Nelson and C. Teitelboim: Hamiltonian formulation of the theory of interacting gravitational and electron fields. Ann. Phys. 116, 86 (1978)

    Article  ADS  Google Scholar 

  34. M. Pilati: The canonical formulation of supergravity. Nucl. Phys. B 132, 138 (1978)

    Article  MathSciNet  ADS  Google Scholar 

  35. R. Tabensky and C. Teitelboim: The square root of general relativity. Phys. Lett. B 69, 453 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  36. A.J. Hanson, T. Regge, and C. Teitelboim: Constrained Hamiltonian systems. RX-748 (1976)

    Google Scholar 

  37. M. Henneaux and C. Teitelboim: Quantization of Gauge Systems. Princeton University Press, Princeton, NJ (1992)

    MATH  Google Scholar 

  38. M. Pilati: Another geometrical representation of supergravity. Nuovo Cim. A 57, 361 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  39. T. Regge and C. Teitelboim: Role of surface integrals in the Hamiltonian formulation of general relativity. Ann. Phys. 88, 286 (1974)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  40. C. Teitelboim: Supergravity and square roots of constraints. Phys. Rev. Lett. 38, 1106–1110 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  41. C. Teitelboim: Surface integrals as symmetry generators in supergravity theory. Phys. Lett. B 69, 240–244 (1977)

    Article  MathSciNet  ADS  Google Scholar 

  42. C. Teitelboim: Spin, supersymmetry, and square roots of constraints. To appear in Proceedings of Current Trends in Theory of Fields, Tallahasse, FL, 6–7 April 1978

    Google Scholar 

  43. C. Teitelboim: In A. Held (Ed.): General Relativity and Gravitation, Vol. 1 (1981) pp. 195–225

    Google Scholar 

  44. C. Teitelboim: How commutators of constraints reflect the spacetime structure. Ann. Phys. 79, 542–557 (1973)

    Article  ADS  Google Scholar 

  45. F.W. Hehl, P. von der Heyde, G.D. Kerlick, and J.M. Nester: General relativity with spin and torsion: Foundations and prospects. Rev. Mod. Phys. 48(3), 393–416 (1976)

    Article  ADS  Google Scholar 

  46. J.M. Charap and J.E. Nelson: Canonical general relativity: The primary constraint algebra. J. Phys. A 16, 3355–3360 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  47. J.M. Charap and J.E. Nelson: The canonical Hamiltonian for vierbein general relativity. Class. Quant. Grav. 3, 1061 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  48. S. Deser and C.J. Isham: Canonical vierbein form of general relativity. Phys. Rev. D 14,2505 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  49. E.S. Fradkin and G.A. Vilkovisky: Quantization of relativistic systems with constraints. Equivalence of canonical and covariant formalisms in quantum theory of gravitational field. CERN-TH-2332 (1977)

    Google Scholar 

  50. P.A.M. Dirac: Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  51. P.A.M. Dirac: The theory of gravitation in Hamiltonian form. Proc. R. Soc. Lond. A 246, 333–343 (1958)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  52. S. Weinberg: The Quantum Theory of Fields. 2. Modern Applications. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  53. A. Lerda, J.E. Nelson, and T. Regge: Covariant canonical formalism for supergravity. Phys. Lett. B 161, 294 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  54. C. Rovelli: Quantum Gravity. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  55. P.D. D’Eath and J.J. Halliwell: Fermions in quantum cosmology. Phys. Rev. D 35, 1100 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  56. H.P. Nilles: Supersymmetry, supergravity and particle physics. Phys. Rep. 110, 1 (1984)

    Article  ADS  Google Scholar 

  57. R. Penrose and W. Rindler: Spinors and Space–Time. 1. Two-Spinor Calculus and Relativistic Fields. Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge (1984)

    Google Scholar 

  58. R. Penrose and W. Rindler: Spinors and Space–Time. 2. Spinor and Twistor Methods in Space–Time Geometry. Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge (1986)

    Google Scholar 

  59. C. Kiefer, T. Luck, and P. Moniz: The semiclassical approximation to supersymmetric quantum gravity. Phys. Rev. D 72, 045006 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  60. P.V. Moniz: Supersymmetric quantum cosmology. Shaken not stirred. Int. J. Mod. Phys. A 11, 4321–4382 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  61. P.A.M. Dirac: Generalized Hamiltonian dynamics. Proc. R. Soc. Lond. A 246, 326–332 (1958)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  62. R. Casalbuoni: Relativity and supersymmetries. Phys. Lett. B 62, 49 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  63. R. Casalbuoni: The classical mechanics for Bose–Fermi systems. Nuovo Cim. A 33, 389 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  64. P.D. D’Eath, S.W. Hawking, and O. Obregon: Supersymmetric Bianchi models and the square root of the Wheeler–DeWitt equation. Phys. Lett. B 300, 44–48 (1993)

    Article  MathSciNet  ADS  Google Scholar 

  65. M. Wulf: Non-closure of constraint algebra in N = 1 supergravity. Int. J. Mod. Phys. D 6,107–118 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  66. S.M. Carroll, D.Z. Freedman, M.E. Ortiz, and D.N. Page: Bosonic physical states in N = 1 supergravity? gr-qc/9410005 (1994)

    Google Scholar 

  67. S.M. Carroll, D.Z. Freedman, M.E. Ortiz, and D.N. Page: Physical states in canonically quantized supergravity. Nucl. Phys. B 423, 661–687 (1994)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  68. P. D’Eath: What local supersymmetry can do for quantum cosmology. Prepared for Workshop on Conference on the Future of Theoretical Physics and Cosmology in Honor of Steven Hawking’s 60th Birthday, Cambridge, England, 7–10 January 2002

    Google Scholar 

  69. P.D. D’Eath: Physical states in N = 1 supergravity. Phys. Lett. B 321, 368–371 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  70. P.D. D’Eath: Canonical formulation and finiteness of N = 1 supergravity with supermatter. Prepared for 7th Marcel Grossmann Meeting on General Relativity (MG 7), Stanford, CA, 24–30 July 1994

    Google Scholar 

  71. G. Esposito and A.Y. Kamenshchik: One-loop divergences in simple supergravity. Boundary effects. Phys. Rev. D 54, 3869–3881 (1996)

    Article  MathSciNet  ADS  Google Scholar 

  72. M.T. Grisaru, P. Van Nieuwenhuizen, and J.A.M. Vermaseren. One loop renormalizability of pure supergravity and of Maxwell–Einstein theory in extended supergravity. Phys. Rev. Lett. 37, 1662 (1976)

    Article  MathSciNet  ADS  Google Scholar 

  73. A.D.Y. Cheng, P.D. D’Eath, and P.R.L.V. Moniz. Canonical quantization of N = 1 supergravity with supermatter: The general case and a FRW model. Grav. Cosmol. 1, 1 (1995)

    ADS  MATH  Google Scholar 

  74. L. Brink, S. Deser, B. Zumino, P. Di Vecchia, and P.S. Howe: Local supersymmetry for spinning particles. Phys. Lett. B 64, 435 (1976)

    Article  ADS  Google Scholar 

  75. L. Brink, M. Henneaux, and C. Teitelboim: Covariant Hamiltonian formulation of the superparticle. Nucl. Phys. B 293, 505–540 (1987)

    Article  MathSciNet  ADS  Google Scholar 

  76. C.A.P. Galvao and C. Teitelboim: Classical supersymmetric particles. J. Math. Phys. 21, 1863 (1980)

    Article  MathSciNet  ADS  Google Scholar 

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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). Canonical Quantization of N = 1 Supergravity. 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_4

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