A new Einstein-nonlinear electrodynamics solution in $2+1$ dimensions

We introduce a class of solutions in <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> -dimensional Einstein–Power–Maxwell theory for a circularly symmetric electric field. The electromagnetic field is considered with an angular component given by <math><mrow><msub><mi>F</mi><mrow><mi mathvariant="italic">μ</mi><mi mathvariant="italic">ν</mi></mrow></msub><mo>=</mo><msub><mi>E</mi><mn>0</mn></msub><msubsup><mi mathvariant="italic">δ</mi><mrow><mi mathvariant="italic">μ</mi></mrow><mi>t</mi></msubsup><msubsup><mi mathvariant="italic">δ</mi><mrow><mi mathvariant="italic">ν</mi></mrow><mi mathvariant="italic">θ</mi></msubsup></mrow></math> for <math><mrow><msub><mi>E</mi><mn>0</mn></msub><mo>=</mo></mrow></math> constant. First, we show that the metric for zero cosmological constant and the Power–Maxwell Lagrangian of the form of <math><msqrt><mfenced close="|" open="|" separators=""><msub><mi>F</mi><mrow><mi mathvariant="italic">μ</mi><mi mathvariant="italic">ν</mi></mrow></msub><msup><mi>F</mi><mrow><mi mathvariant="italic">μ</mi><mi mathvariant="italic">ν</mi></mrow></msup></mfenced></msqrt></math> coincides with the solution given in <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> -dimensional gravity coupled with a massless, self-interacting real scalar field. With the same Lagrangian and a non-zero cosmological constant we obtain a non-asymptotically flat wormhole solution in <math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math> dimensions. The confining motions of massive charged and chargeless particles are investigated too. Secondly, another interesting solution is given for zero cosmological constant together with the conformal invariant condition. The formation of a timelike naked singularity for this particular case is investigated within the framework of the quantum mechanics. Quantum fields obeying the Klein–Gordon and Dirac equations are used to probe the singularity and test the quantum mechanical status of the singularity.