Related papers: Hilbert Transformation and $r\mathrm{Spin}(n)+\mat…

In this paper we study the Hilbert transformations over $L^2(\mathbb{R})$ and $L^2(\mathbb{T})$ from the viewpoint of symmetry. For a linear operator over $L^2(\mathbb{R})$ commutative with the ax+b group we show that the operator is of the…

One of the main goals of these notes is to explain how rotations in reals^n are induced by the action of a certain group, Spin(n), on reals^n, in a way that generalizes the action of the unit complex numbers, U(1), on reals^2, and the…

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…

In quantum mechanics, symmetry groups can be realized by projective, as well as by ordinary unitary, representations. For the permutation symmetry relevant to quantum statistics of N indistinguishable particles, the simplest properly…

It is shown that since the geometric spinors are elements of Clifford algebras, they must have the same transformation properties as any other Clifford number. In general, a Clifford number $\Phi$ transforms into a new Clifford number…

We discuss some basic properties of Lie group representations in rigged Hilbert spaces. In particular, we show that a differentiable representation in a rigged Hilbert space may be obtained as the projective limit of a family of continuous…

The line geometric model of 3-D projective geometry has the nice property that the Lie algebra sl(4) of 3-D projective transformations is isomorphic to the bivector algebra of CL(3,3), and line geometry is closely related to the classical…

In the Hilbert space of n qubits, we introduce the symplectic space (n odd) and the orthogonal space (n even) via the spin-flip operator. Under this mathematical structure we discuss some properties of n qubits, including homomorphically…

We discuss a Clifford algebra framework for discrete symmetry groups (such as reflection, Coxeter, conformal and modular groups), leading to a surprising number of new results. Clifford algebras allow for a particularly simple description…

Exact diagonalization and other numerical studies of quantum spin systems are notoriously limited by the exponential growth of the Hilbert space dimension with system size. A common and well-known practice to reduce this increasing…

In this paper we combine methods from projective geometry, Klein's model, and Clifford algebra. We develop a Clifford algebra whose Pin group is a double cover of the group of regular projective transformations. The Clifford algebra we use…

There are six different mathematical formulations of the symmetry group in quantum mechanics, among them the set of pure states $\mathbf{P}$ -- i.e., the set of one-dimensional projections on a complex Hilbert space $H$ -- and the…

Let $\mathbb S$ be a Clifford module for the complexified Clifford algebra $\mathbb{C}\ell(\mathbb R^n)$, $\mathbb S'$ its dual, $\rho$ and $\rho'$ be the corresponding representations of the spin group ${\rm Spin}(n)$. The group $G= {\rm…

Symmetries impose structure on the Hilbert space of a quantum mechanical model. The mathematical units of this structure are the irreducible representations of symmetry groups and I consider how they function as conceptual units of…

In this work we explore the structure of Clifford algebras and the representations of the algebraic spinors in quantum information theory. Initially we present an general formulation through elements of left minimal ideals in tensor…

For Connes' spectral triples, the group of automorphisms lifted to the Hilbert space is defined and used to fluctuate the metric. A few commutative examples are presented including Chamseddine and Connes' spectral unification of gravity and…

Spin space groups, formed by operations where the rotation of the spins is independent of the accompanying operation acting on the crystal structure, are appropriate groups to describe the symmetry of magnetic structures with null…

We introduce the symplectic group $\mathrm{Sp}_2(G, \sigma)$ associated to a Lie subgroup $G$ of a (possibly noncommutative) associative algebra $A$ equipped with an anti-involution $\sigma$. Our construction recovers several classical Lie…

A signature epsilon=(p,q) dependent transposition anti-involution T of real Clifford algebras Cl_{p,q} for non-degenerate quadratic forms was introduced in [arXiv.1005.3554v1]. In [arXiv.1005.3558v1] we showed that, depending on the value…

Let $(\tau,V_\tau)$ be a spinor representation of $\mathrm{Spin}(n)$ and let $(\sigma,V_\sigma)$ be a spinor representation of $\mathrm{Spin}(n-1)$ that occurs in the restriction $\tau_{\mid \mathrm{Spin}(n-1)}$. We consider the real…