5 research outputs found
More on coupling coefficients for the most degenerate representations of SO(n)
We present explicit closed-form expressions for the general group-theoretical
factor appearing in the alpha-topology of a high-temperature expansion of
SO(n)-symmetric lattice models. This object, which is closely related to
6j-symbols for the most degenerate representation of SO(n), is discussed in
detail.Comment: 9 pages including 1 table, uses IOP macros Update of Introduction and
Discussion, References adde
Coupling coefficients of SO(n) and integrals over triplets of Jacobi and Gegenbauer polynomials
The expressions of the coupling coefficients (3j-symbols) for the most
degenerate (symmetric) representations of the orthogonal groups SO(n) in a
canonical basis (with SO(n) restricted to SO(n-1)) and different semicanonical
or tree bases [with SO(n) restricted to SO(n'})\times SO(n''), n'+n''=n] are
considered, respectively, in context of the integrals involving triplets of the
Gegenbauer and the Jacobi polynomials. Since the directly derived
triple-hypergeometric series do not reveal the apparent triangle conditions of
the 3j-symbols, they are rearranged, using their relation with the
semistretched isofactors of the second kind for the complementary chain
Sp(4)\supset SU(2)\times SU(2) and analogy with the stretched 9j coefficients
of SU(2), into formulae with more rich limits for summation intervals and
obvious triangle conditions. The isofactors of class-one representations of the
orthogonal groups or class-two representations of the unitary groups (and, of
course, the related integrals involving triplets of the Gegenbauer and the
Jacobi polynomials) turn into the double sums in the cases of the canonical
SO(n)\supset SO(n-1) or U(n)\supset U(n-1) and semicanonical SO(n)\supset
SO(n-2)\times SO(2) chains, as well as into the_4F_3(1) series under more
specific conditions. Some ambiguities of the phase choice of the complementary
group approach are adjusted, as well as the problems with alternating sign
parameter of SO(2) representations in the SO(3)\supset SO(2) and SO(n)\supset
SO(n-2)\times SO(2) chains.Comment: 26 pages, corrections of (3.6c) and (3.12); elementary proof of
(3.2e) is adde
On the Implementation of the Canonical Quantum Simplicity Constraint
In this paper, we are going to discuss several approaches to solve the
quadratic and linear simplicity constraints in the context of the canonical
formulations of higher dimensional General Relativity and Supergravity
developed in our companion papers. Since the canonical quadratic simplicity
constraint operators have been shown to be anomalous in any dimension D>2,
non-standard methods have to be employed to avoid inconsistencies in the
quantum theory. We show that one can choose a subset of quadratic simplicity
constraint operators which are non-anomalous among themselves and allow for a
natural unitary map of the spin networks in the kernel of these simplicity
constraint operators to the SU(2)-based Ashtekar-Lewandowski Hilbert space in
D=3. The linear constraint operators on the other hand are non-anomalous by
themselves, however their solution space will be shown to differ in D=3 from
the expected Ashtekar-Lewandowski Hilbert space. We comment on possible
strategies to make a connection to the quadratic theory. Also, we comment on
the relation of our proposals to existing work in the spin foam literature and
how these works could be used in the canonical theory. We emphasise that many
ideas developed in this paper are certainly incomplete and should be considered
as suggestions for possible starting points for more satisfactory treatments in
the future.Comment: 30 pages, 2 figures. v2: Journal version. Comparison to existing
approaches added. Discussion extended. References added. Sign error in
equation (2.15) corrected. Minor clarifications and correction