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Closed-form solutions of the Wheeler–DeWitt equation in a scalar-vector field cosmological model by Lie symmetries

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Abstract

We apply as selection rule to determine the unknown functions of a cosmological model the existence of Lie point symmetries for the Wheeler–DeWitt equation of quantum gravity. Our cosmological setting consists of a flat Friedmann–Robertson–Walker metric having the scale factor a(t), a scalar field with potential function \(V(\phi )\) minimally coupled to gravity and a vector field of its kinetic energy is coupled with the scalar field by a coupling function \(f(\phi )\). Then, the Lie symmetries of this dynamical system are investigated by utilizing the behavior of the corresponding minisuperspace under the infinitesimal generator of the desired symmetries. It is shown that by applying the Lie symmetry condition the form of the coupling function and also the scalar field potential function may be explicitly determined so that we are able to solve the Wheeler–DeWitt equation. Finally, we show how we can use the Lie symmetries in order to construct conservation laws and exact solutions for the field equations.

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Notes

  1. Recently in the literature a vector field which satisfies the condition (1) has been termed as “Noether Gauge Symmetry”, see [5–10]. This is incorrect terminology, since condition (1) is that which has been introduced by E. Noether in her original work. The function, g, of (1) is a boundary term (not a gauge function) introduced to allow for the infinitessimal transformations which in the value of the Action Integral produced by the infinitesimal change in the boundary of the domain caused by the infinitesimal transformation of the variables in the Action Integral.

  2. In the following we consider \(\omega \ne 0\) and \(f_{0}=1.\)

  3. There is also the Lie symmetry \(X_{B}=B\left( a,\phi ,\zeta \right) \partial _{\Psi }\), where \(B\left( a,\phi ,\zeta \right) \) is a solution of (32). However since \(X_{B}\) is a trivial symmetry we will omit it.

  4. The symbolic package Sym for Mathematica have been used to test the resutls [47].

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Acknowledgments

AP acknowledges Prof. PGL Leach, Sivie Govinder, as also DUT for the hospitality provided and the UKNZ of South Africa for financial support while part of this work carried out during his visits in South Africa. The research of AP was supported by FONDECYT postdoctoral grant no. 3160121.

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Authors and Affiliations

  1. Instituto de Ciencias Físicas y Matemáticas, Universidad Austral de Chile, Valdivia, Chile

    Andronikos Paliathanasis

  2. Department of Physics, Tehran Central Branch, Islamic Azad University, Tehran, Iran

    Babak Vakili

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  1. Andronikos Paliathanasis
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  2. Babak Vakili
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Correspondence to Andronikos Paliathanasis.

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Paliathanasis, A., Vakili, B. Closed-form solutions of the Wheeler–DeWitt equation in a scalar-vector field cosmological model by Lie symmetries. Gen Relativ Gravit 48, 13 (2016). https://doi.org/10.1007/s10714-015-2010-5

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