Some Curious Consequences of the Minimal Length Uncertainty Relation

por Tatsu Takeuchi (Viginia Tech (USA))

jueves 7 jun. 2012 11:30 → 12:30 Europe/Madrid

Sala Seminarios IFIC (Edf. Institutos de Investigación)

In theories of quantum gravity, it is expected that spacetime distances smaller than the Planck length would not be resolvable. In quantum mechanical language, gravitational effects are expected to deform the uncertainty relation between position and momentum from the canonical Heisenberg form to Delta (x) >= (hbar/2) (1/Delta(p) + beta Delta(p)), which would imply that Delta (x) is bounded from below by Delta (x) (min) = hbar \sqrt{beta}. Indeed, this "minimal length uncertainty relation" has been shown to hold in perturbative string theory, which is the top candidate theory of quantum gravity. Canonical non-relativistic quantum mechanics, on the other hand, is completely oblivious of the existence of this minimal length, even though we expect it to be the infrared limit of quantum gravity (or string theory, if it is indeed the theory of everything). One way to incorporate the "minimal length" into non-relativistic quantum mechanics is to deform the commutation relation between position and momentum to [x,p]= i hbar (1+beta p^2). In this talk, I will discuss the consequences of this deformation on the eigenstates and eigenvalues of the harmonic oscillator hamiltonian, and present several curious puzzles that have been uncovered.