16,370 research outputs found
The vanishing ideal of a finite set of points with multiplicity structures
Given a finite set of arbitrarily distributed points in affine space with
arbitrary multiplicity structures, we present an algorithm to compute the
reduced Groebner basis of the vanishing ideal under the lexicographic ordering.
Our method discloses the essential geometric connection between the relative
position of the points with multiplicity structures and the quotient basis of
the vanishing ideal, so we will explicitly know the set of leading terms of
elements of I. We split the problem into several smaller ones which can be
solved by induction over variables and then use our new algorithm for
intersection of ideals to compute the result of the original problem. The new
algorithm for intersection of ideals is mainly based on the Extended Euclidean
Algorithm.Comment: 12 pages,12 figures,ASCM 201
Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory
On the basis of a thorough discussion of the Batalin-Vilkovisky formalism for
classical field theory presented in our previous publication, we construct in
this paper the Batalin-Vilkovisky complex in perturbatively renormalized
quantum field theory. The crucial technical ingredient is a proof that the
renormalized time-ordered product is equivalent to the pointwise product of
classical field theory. The renormalized Batalin-Vilkovisky algebra is then the
classical algebra but written in terms of the time-ordered product, together
with an operator which replaces the ill defined graded Laplacian of the
unrenormalized theory. We identify it with the anomaly term of the anomalous
Master Ward Identity of Brennecke and D\"utsch. Contrary to other approaches we
do not refer to the path integral formalism and do not need to use
regularizations in intermediate steps.Comment: 34 page
Canonical Quantization of Spherically Symmetric Gravity in Ashtekar's Self-Dual Representation
We show that the quantization of spherically symmetric pure gravity can be
carried out completely in the framework of Ashtekar's self-dual representation.
Consistent operator orderings can be given for the constraint functionals
yielding two kinds of solutions for the constraint equations, corresponding
classically to globally nondegenerate or degenerate metrics. The physical state
functionals can be determined by quadratures and the reduced Hamiltonian system
possesses 2 degrees of freedom, one of them corresponding to the classical
Schwarzschild mass squared and the canonically conjugate one representing a
measure for the deviation of the nonstatic field configurations from the static
Schwarzschild one. There is a natural choice for the scalar product making the
2 fundamental observables self-adjoint. Finally, a unitary transformation is
performed in order to calculate the triad-representation of the physical state
functionals and to provide for a solution of the appropriately regularized
Wheeler-DeWitt equation.Comment: 43 page
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