116,962 research outputs found
General Relativity without paradigm of space-time covariance, and resolution of the problem of time
The framework of a theory of gravity from the quantum to the classical regime
is presented. The paradigm shift from full spacetime covariance to spatial
diffeomorphism invariance, together with clean decomposition of the canonical
structure, yield transparent physical dynamics and a resolution of the problem
of time. The deep divide between quantum mechanics and conventional canonical
formulations of quantum gravity is overcome with a Schr\"{o}dinger equation for
quantum geometrodynamics that describes evolution in intrinsic time. Unitary
time development with gauge-invariant temporal ordering is also viable. All
Kuchar observables become physical; and classical spacetime, with direct
correlation between its proper times and intrinsic time intervals, emerges from
constructive interference. The framework not only yields a physical Hamiltonian
for Einstein's theory, but also prompts natural extensions and improvements
towards a well behaved quantum theory of gravity. It is a consistent canonical
scheme to discuss Horava-Lifshitz theories with intrinsic time evolution, and
of the many possible alternatives that respect 3-covariance (rather than the
more restrictive 4-covariance of Einstein's theory), Horava's ``detailed
balance" form of the Hamiltonian constraint is essentially pinned down by this
framework.Comment: 11 page
From Quantum Mechanics to Quantum Field Theory: The Hopf route
We show that the combinatorial numbers known as {\em Bell numbers} are
generic in quantum physics. This is because they arise in the procedure known
as {\em Normal ordering} of bosons, a procedure which is involved in the
evaluation of quantum functions such as the canonical partition function of
quantum statistical physics, {\it inter alia}. In fact, we shall show that an
evaluation of the non-interacting partition function for a single boson system
is identical to integrating the {\em exponential generating function} of the
Bell numbers, which is a device for encapsulating a combinatorial sequence in a
single function. We then introduce a remarkable equality, the Dobinski
relation, and use it to indicate why renormalisation is necessary in even the
simplest of perturbation expansions for a partition function. Finally we
introduce a global algebraic description of this simple model, giving a Hopf
algebra, which provides a starting point for extensions to more complex
physical systems
Canonical ordering for graphs on the cylinder, with applications to periodic straight-line drawings on the flat cylinder and torus
We extend the notion of canonical ordering (initially developed for planar
triangulations and 3-connected planar maps) to cylindric (essentially simple)
triangulations and more generally to cylindric (essentially internally)
-connected maps. This allows us to extend the incremental straight-line
drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case)
and of Kant (in the -connected case) to this setting. Precisely, for any
cylindric essentially internally -connected map with vertices, we
can obtain in linear time a periodic (in ) straight-line drawing of that
is crossing-free and internally (weakly) convex, on a regular grid
, with and ,
where is the face-distance between the two boundaries. This also yields an
efficient periodic drawing algorithm for graphs on the torus. Precisely, for
any essentially -connected map on the torus (i.e., -connected in the
periodic representation) with vertices, we can compute in linear time a
periodic straight-line drawing of that is crossing-free and (weakly)
convex, on a periodic regular grid
, with and
, where is the face-width of . Since ,
the grid area is .Comment: 37 page
Dill: An Algorithm and a Symbolic Software Package for Doing Classical Supersymmetry Calculations
An algorithm is presented that formalizes different steps in a classical
Supersymmetric (SUSY) calculation. Based on the algorithm Dill, a symbolic
software package, that can perform the calculations, is developed in the
Mathematica programming language. While the algorithm is quite general, the
package is created for the 4-D, N=1 model. Nevertheless, with little
modification, the package could be used for other SUSY models. The package has
been tested and some of the results are presented.Comment: 42 pages, LaTe
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