Boundary conditions for quantum mechanics on cones and fields around cosmic strings
Oct 1, 199042 pages
Published in:
- Commun.Math.Phys. 139 (1991) 103-140
DOI:
Report number:
- DAMTP-R-90-21
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Abstract: (Springer)
We study the options for boundary conditions at the conical singularity for quantum mechanics on a two-dimensional cone with deficit angle ≦ 2π and for classical and quantum scalar fields propagating with a translationally invariant dynamics in the 1+3 dimensional spacetime around an idealized straight infinitely long, infinitesimally thin cosmic string. The key to our analysis is the observation that minus-the-Laplacian on a cone possesses a one-parameter family of selfadjoint extensions. These may be labeled by a parameterR with the dimensions of length—taking values in [0, ∞). ForR=0, the extension is positive. WhenR≠0 there is a bound state. Each of our problems has a range of possible dynamical evolutions corresponding to a range of allowedR-values. They correspond to either finite, forR=0, or logarithmically divergent, forR≠0, boundary conditions at zero radius. Non-zeroR-values are a satisfactory replacement for the (mathematically ill-defined) notion of δ-function potentials at the cone's apex.- cosmic string
- quantum mechanics
- dimension: 2
- boundary condition
- field theory: scalar
- quantum gravity
- dimension: 3
- Klein-Gordon equation
- superconductivity: string
- cosmic string: superconductivity
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