Abstract
The d’Alembertian □ϕ = 0 has the solution ϕ = f(v)/r, where f is a function of a null coordinate v, and this allows creation of a divergent singularity out of nothing. In scalar-Einstein theory a similar situation arises both for the scalar field and also for curvature invariants such as the Ricci scalar. Here what happens in canonical quantum gravity is investigated. Two minispace Hamiltonian systems are set up: extrapolation and approximation of these indicates that the quantum mechanical wave function can be finite at the origin.
Highlights
Physicists have wondered what happens at the origin of the reciprocal potential 1/r, which is ubiquitous and, for example, occurs in electromagnetism and gravitation
The scalar field is usually quantized directly so it is hard to compare with the exact solution (1.1)
The content of a scalar field is so configured that it cancels out the energy of gravitons, giving no overall energy which would agree with the classical case
Summary
Physicists have wondered what happens at the origin of the reciprocal potential 1/r, which is ubiquitous and, for example, occurs in electromagnetism and gravitation. The physical motivation is that the system is described by two variables, the scalar field and the killing potential, each of which in turn is relaxed. The solution has two characteristic scalars: the scalar field and the homothetic Killing potential, and expressing the solution in terms of these leads to one variable problems. This can be pictured as what happens when there is one quantum degree of freedom introduced into the system corresponding to fluctuations in the homothetic Killing vector away from its classical properties; classical fluctuation have been discussed by Frolov [11]. Conventions used are signature 2 þ þ þ, indices and arguments of functions left out when the ellipsis is clear, V to describe a scalar field potential and U to describe the Wheeler– DeWitt potential, f for the scalar field in a scalar-Einstein solution, j for a source scalar field, field equations Gmn 1⁄4 Gmn À 8pkTmn m, n, . . . are space – time coordinates, A, B, . . . are field variables
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