3 research outputs found
Moduli-space structure of knots with intersections
It is well known that knots are countable in ordinary knot theory. Recently,
knots {\it with intersections} have raised a certain interest, and have been
found to have physical applications. We point out that such knots --equivalence
classes of loops in under diffeomorphisms-- are not countable; rather,
they exhibit a moduli-space structure. We characterize these spaces of moduli
and study their dimension. We derive a lower bound (which we conjecture being
actually attained) on the dimension of the (non-degenerate components) of the
moduli spaces, as a function of the valence of the intersection.Comment: 15 pages, latex-revtex, no figure
Time-of-arrival in quantum mechanics
We study the problem of computing the probability for the time-of-arrival of
a quantum particle at a given spatial position. We consider a solution to this
problem based on the spectral decomposition of the particle's (Heisenberg)
state into the eigenstates of a suitable operator, which we denote as the
``time-of-arrival'' operator. We discuss the general properties of this
operator. We construct the operator explicitly in the simple case of a free
nonrelativistic particle, and compare the probabilities it yields with the ones
estimated indirectly in terms of the flux of the Schr\"odinger current. We
derive a well defined uncertainty relation between time-of-arrival and energy;
this result shows that the well known arguments against the existence of such a
relation can be circumvented. Finally, we define a ``time-representation'' of
the quantum mechanics of a free particle, in which the time-of-arrival is
diagonal. Our results suggest that, contrary to what is commonly assumed,
quantum mechanics exhibits a hidden equivalence between independent (time) and
dependent (position) variables, analogous to the one revealed by the
parametrized formalism in classical mechanics.Comment: Latex/Revtex, 20 pages. 2 figs included using epsf. Submitted to
Phys. Rev.