Related papers: Spin, Statistics, and Reflections, I. Rotation Inv…
There is a unique finite group that lies inside the 2-dimensional unitary group but not in the special unitary group, and maps by the symmetric square to an irreducible subgroup of the 3-dimensional real special orthogonal group. In an…
A massless scalar field in two spacetime dimensions splits into two independent sectors of left and right-moving modes on the light cone. At the quantum level, these two sectors carry a representation of the group of affine transformations…
The models of spin systems defined on Euclidean space provide powerful machinery for studying a broad range of condensed matter phenomena. While the non-relativistic effective description is sufficient for most of the applications, it is…
In quantum gravity the unitary evolution does not follow from the Wheeler-DeWitt dynamics equation as it follows from the Schr\"odinger equation in non-relativistic quantum mechanics. Therefore we can define a spin-foam model based on…
We show that a generating function for open $r$-spin enumerative invariants produces a universal unfolding of the polynomial $x^r$. Further, the coordinates parametrizing this universal unfolding are flat coordinates on the Frobenius…
We consider a family of quantum loop models in 2+1 spacetime dimensions with marginally long-ranged and statistical interactions mediated by a U$(1)$ gauge field, both purely in 2+1 dimensions and on a surface in a 3+1 dimensional bulk…
This paper is devoted to the representations of the groups $SO (2,1)$ and $ISO (2,1)$. Those groups have an important role in cosmology, elementary particle theory and mathematical physics. Irreducible unitary representations of the…
For classical groups we show the isomorphism of the Knapp-Stein $R$-group, which describes the structure of parabolically induced representations, and the Arthur $R$-group of the parameter associated to the inducing representation by the…
Rotation representations are foundational in fields such as computer graphics, robotics, and machine learning, where precise and efficient modeling of 3D orientations is critical. This paper comprehensively investigates diverse…
Coherent states are introduced and their properties are discussed for all simple quantum compact groups. The multiplicative form of the canonical element for the quantum double is used to introduce the holomorphic coordinates on a general…
Starting from the defining transformations of complex matrices for the SO(4) group, we construct the fundamental representation and the tensor and spinor representations of the group SO(4). Given the commutation relations for the…
We construct a class of spin foam models describing matter coupled to gravity, such that the gravitational sector is described by the unitary irreducible representations of the appropriate symmetry group, while the matter sector is…
Spinor structure and internal symmetries are considered within one theoretical framework based on the generalized spin and abstract Hilbert space. Complex momentum is understood as a generating kernel of the underlying spinor structure. It…
This paper is a continuation and elaboration of our work quant-ph/0206057 (Nucl. Phys. B, 1968, 7, 79) where some approach to the variable-mass problem were proposed. Here we have found a concret realization of irreducible representations…
The Bohr-Sommerfeld quantization rule and the Gamow formula for the width of quasistationary level are generalized by taking into account the relativistic effects, spin and Lorentz structure of interaction potentials. The relativistic…
A gauge theory of the Lorentz group with a mass-dimension one gauge field coupling to matter of any spin is developed. As a completely new feature the "Vierbein" assuring local gauge invariance enters not as an independent dynamical field,…
There are exactly three finite subgroups of SU(2) that act irreducibly in the spin 1 representation, namely the binary tetrahedral, binary octahedral and binary icosahedral groups. In previous papers I have shown how the binary tetrahedral…
We investigate the structure of SO(3)-invariant quantum systems which are composed of two particles with spins j_1 and j_2. The states of the composite spin system are represented by means of two complete sets of rotationally invariant…
We investigate classical spin systems in $d\geq 1$ dimensions whose transfer operator commutes with the action of a nonamenable unitary representation of a symmetry group, here ${\rm SO}(1,N)$; these systems may alternatively be interpreted…
Covariant or invariant functions under a compact linear group can be expressed in terms of functions defined in the orbit space of the group. The semialgebraic relations defining the orbit spaces of all finite coregular real linear groups…