Conditional Symmetries and the Canonical Quantization of Constrained Minisuperspace Actions: the Schwarzschild case

Abstract

A conditional symmetry is defined, in the phase-space of a quadratic in velocities constrained action, as a simultaneous conformal symmetry of the supermetric and the superpotential. It is proven that such a symmetry corresponds to a variational (Noether) symmetry.The use of these symmetries as quantum conditions on the wave-function entails a kind of selection rule. As an example, the minisuperspace model ensuing from a reduction of the Einstein - Hilbert action by considering static, spherically symmetric configurations and r as the independent dynamical variable, is canonically quantized. The conditional symmetries of this reduced action are used as supplementary conditions on the wave function. Their integrability conditions dictate, at a first stage, that only one of the three existing symmetries can be consistently imposed. At a second stage one is led to the unique Casimir invariant, which is the product of the remaining two, as the only possible second condition on $\Psi$. The uniqueness of the dynamical evolution implies the need to identify this quadratic integral of motion to the reparametrisation generator. This can be achieved by fixing a suitable parametrization of the r-lapse function, exploiting the freedom to arbitrarily rescale it. In this particular parametrization the measure is chosen to be the determinant of the supermetric. The solutions to the combined Wheeler - DeWitt and linear conditional symmetry equations are found and seen to depend on the product of the two "scale factors"

20 pages, LaTeX2e source file, no figures

References in corpus (5)

• Hamiltonian dynamics and Noether symmetries in Extended Gravity Cosmology
• Lie and Noether symmetries of geodesic equations and collineations
• Two dimensional dynamical systems which admit Lie and Noether symmetries
• The General Solution of Bianchi Type III Vacuum Cosmology
• Ultimate generalization of Noether's theorem in the realm of Hamiltonian point dynamics

Cited by in corpus (31)

• Lie - point and variational symmetries in minisuperspace Einstein's gravity
• FLRW metric $f(R)$ cosmology with a perfect fluid by generating integrals of motion
• Conditional symmetries in axisymmetric quantum cosmologies with scalar fields and the fate of the classical singularities
• Noether analysis of Scalar-Tensor Cosmology
• Minisuperspace Canonical Quantization of the Reissner-Nordstrom Black Hole via Conditional Symmetries
• A late time accelerated FRW model with scalar and vector fields via Noether symmetry
• Quantum Black Hole Wave Packet: Average Area Entropy and Temperature Dependent Width
• Crossing the phantom divide line as an effect of quantum transitions
• Canonical Quantization of the BTZ Black Hole using Noether Symmetries
• Decoupling of the re-parametrization degree of freedom and a generalized probability in quantum cosmology
• Variational Contact Symmetries of Constraint Lagrangians
• Reduced Lagrangians and analytic solutions in Einstein-æther Cosmology
• Schrödinger Symmetry in Gravitational Mini-Superspaces
• Exact Kantowski-Sachs spacetimes in Einstein-Aether Scalar field theory
• Wheeler-DeWitt equation and Lie symmetries in Bianchi scalar-field cosmology
• On the equivalence between different canonical forms of F(R) theory of gravity
• Similarity solutions for the Wheeler-DeWitt equation in $f\left(R\right)$-cosmology
• Lie symmetry approach to the time-dependent Karmarkar condition
• Dynamical symmetries of homogeneous minisuperspace models
• Duality transformation and conformal equivalent scalar-tensor theories
• Quantum cosmology of a Bianchi III LRS geometry coupled to a source free electromagnetic field
• A Study of the Approximate Singular Lagrangian -- Conditional Noether Symmetries and First Integrals
• Discrete spectrum of the quantum Reissner - Nordström geometry
• Lie symmetries for systems of evolution equations
• Correlations across horizons in quantum cosmology
• Higher dimensional black holes as constrained systems
• Symmetries of Differential equations and Applications in Relativistic Physics
• Quantum fate of timelike naked singularity with scalar hair
• Classical and quantum (2+1)-dimensional spatially homogeneous string cosmology
• Time-covariant Schrödinger equation and invariant decay probability: The $\Lambda$-Kantowski-Sachs universe
• Hyperbolic trajectories around Black Holes