van Veenendaal, Michel
The systematic use of alternative normalization constants for 3j symbols can lead to a more natural expression of quantities, such as vector products and spherical tensor operators. The redefined coupling constants directly equate tensor products to the inner and outer products without any additional square roots. The approach is extended to…
Bitencourt, Ana Carla P; Littlejohn, Robert G; Anderson, Roger; Aquilanti, Vincenzo
The Wigner $3j$ symbols of the quantum angular momentum theory are related to the vector coupling or Clebsch-Gordan coefficients and to the Hahn and dual Hahn polynomials of the discrete orthogonal hyperspherical family, of use in discretization approximations. We point out the important role of the Regge symmetries for defining the screen where images of the coefficients are projected, and for discussing their asymptotic properties and semiclassical behavior. Recursion relationships are formulated as eigenvalue equations, and exploited both for computational purposes and for physical interpretations.
Alesci, Emanuele; Mäkinen, Ilkka
We introduce a new technique for dealing with the matrix elements of the Hamiltonian operator in loop quantum gravity, based on the use of intertwiners projected on coherent states of angular momentum. We give explicit expressions for the projections of intertwiners on the spin coherent states in terms of complex numbers describing the unit vectors which label the coherent states. Operators such as the Hamiltonian can then be reformulated as differential operators acting on polynomials of these complex numbers. This makes it possible to describe the action of the Hamiltonian geometrically, in terms of the unit vectors originating from the angular momentum coherent states, and opens up a way towards investigating the semiclassical limit of the dynamics via asymptotic approximation methods.
Chernega, V. N.; Manko, O. V.; Manko, V. I.; Seilov, Z.
The Clebsch-Gordan coefficients of the group SU(2) are shown to satisfy new inequalities. They are obtained using the properties of Shannon and Tsallis entropies. The inequalities associated with the Wigner 3-j symbols are obtained using the relation of Clebsch-Gordan coefficients with probability distributions interpreted either as distributions for composite systems or distributions for noncomposite systems. The new inequalities were found for Hahn polynomials and hypergeometric functions
Mironov, A; Morozov, And
Operator product expansion (OPE) of two operators in two-dimensional conformal field theory includes a sum over Virasoro descendants of other operator with universal coefficients, dictated exclusively by properties of the Virasoro algebra and independent of choice of the particular conformal model. In the free field model, these coefficients arise only with a special "conservation" relation imposed on the three dimensions of the operators involved in OPE. We demonstrate that the coefficients for the three unconstrained dimensions arise in the free field formalism when additional Dotsenko-Fateev integrals are inserted between the positions of the two original operators in the product. If such coefficients are combined to form an $n$-point conformal block on Riemann sphere, one reproduces the earlier conjectured $\\beta$-ensemble representation of conformal blocks, thus proving this (matrix model) version of the celebrated AGT relation. The statement can also be regarded as a relation between the $3j$-symbols of...
Cremmer, E; Roussel, J F; Gervais, Jean-Loup
The F and B matrices associated with Virasoro null vectors are derived in closed form by making use of the operator-approach suggested by the Liouville theory, where the quantum-group symmetry is explicit. It is found that the entries of the fusing and braiding matrices are not simply equal to quantum-group symbols, but involve additional coupling constants whose derivation is one aim of the present work. Our explicit formulae are new, to our knowledge, in spite of the numerous studies of this problem. The relationship between the quantum-group-invariant (of IRF type) and quantum-group-covariant (of vertex type) chiral operator-algebras is fully clarified, and connected with the transition to the shadow world for quantum-group symbols. The corresponding 3-j-symbol dressing is shown to reduce to the simpler transformation of Babelon and one of the author (J.-L. G.) in a suitable infinite limit defined by analytic continuation. The above two types of operators are found to coincide when applied to states with L...
Reaction stereodynamics can be studied in quantum mechanics using alternative representations of the S matrix. In this paper we employ the equations for the orthogonal transformations (expressed in terms of Wigner 3j symbols) that convert the S matrix from the body fixed (vertical bar jΩ>) representation into the stereodirected one (vertical bar νΩ>). This representation is characterized by the introduction of the steric quantum number ν, which in the vector model of quantum mechanics is put into correspondence with given precession cones of attack of the incoming atom on the diatomic molecule for the reactants' channels, and of cones of escape for the departing atom away from the diatomic molecule for the products' channels. The angles of aperture of such cones are determined from the uncertainty principle. As the ν quantum number increases (semiclassical limit), the grid of discrete values of the precession cones more finely scans the angle between the Jacobi vectors. Using a time-independent hyperspherical coordinate method we have generated the full S matrix including all open reactive and inelastic channels for two potential energy surfaces corresponding to the F + H2 → HF + H reaction and they have been used to calculate, via vertical bar jΩ>→ vertical bar νΩ> matrix transformations, the attack and exit cumulative reaction probabilities. During the calculations, we have distinguished between ortho-H2 and para-H2. Clear stereodynamical effects have being identified, in particular, regarding the reaction entrance channel, that F-atom attacks are preferred at the transition state (bent) geometry, while for the exit channel the H-atom departs in a collinear geometry by the H-end side of HF
Concepts of symmetry in physics have had a long history, particularly if they are of a geometric or crystallographic origin, yet in classical physics they had a somewhat esoteric position. This situation changed radically when in the XX Century we passed from classical to quantum mechanics. In the former a state for a system of particles was given by a number of points in phase space and the transformation groups related with symmetries mainly gave the invariance of concepts such as energy or angular momentum. In the latter the state is characterized by a vector in Hilbert space in which the transformations had a representation. Eugene Wigner was the right man (for his mathematical ability and physical intuition) at the right place and time (Germany, in the twenties) to take full advantage of this new situation. His first interest was atomic spectroscopy (then a very active field) and the fact that its basic states were related with irreducible representation of the orthogonal group in three dimensions O(3). The German version of his book on ``Group theory and Application" published in 1931 established, as he quotes ``that almost all rules of spectroscopy follow from the symmetry of the problem". His later extension to the direct product of two or more representations led to his development of the 3-j symbol, that he explicitly derived, and his interest in the properties of 6-j, 9-j, etc. His awareness of the time inversion as an antiunitary operator, and the analysis of its combination with the unitary representations of other symmetries, proved fundamental for deriving the features of time reversed reactions from their direct behavior. His interest in space reflection and the concept of parity led to important selection rules, and was of relevance even in weak interactions where parity is not a good symmetry. His later interest in nuclear physics, solid state, elementary particles etc., was almost never without a component of the role of symmetry in these