Related papers: Reductions in finite-dimensional quantum mechanics…
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…
Hereunder we continue the study of the representation theory of the algebra of permutation operators acting on the $n$-fold tensor product space, partially transposed on the last subsystem. We develop the concept of partially reduced…
Given a unitary operator in a finite dimensional complex Hilbert space, its unitary reduction to a subspace is defined. The application to quantum graphs is discussed. It is shown how the reduction allows to generate the scattering matrices…
Based on a recent purely geometric construction of observables for the spatial diffeomorphism constraint, we propose two distinct quantum reductions to spherical symmetry within full 3+1-dimensional loop quantum gravity. The construction of…
We extend some results of group representation theory and von Neumann algebras to the quaternionic Hilbert space case, proving the double commutant theorem (whose quaternionic proof requires a different procedure) and extend to the…
This survey article is concerned with the modeling of the kinematical structure of quantum systems in an algebraic framework which eliminates certain conceptual and computational difficulties of the conventional approaches. Relying on the…
A new functional calculus, developed recently for a fully non-perturbative treatment of quantum gravity, is used to begin a systematic construction of a quantum theory of geometry. Regulated operators corresponding to areas of 2-surfaces…
Mathematical core of quantum mechanics is the theory of unitary representations of symmetries of physical systems. We argue that quantum behavior is a natural result of extraction of "observable" information about systems containing…
The residual finite-dimensionality of a $\mathrm{C}^*$-algebra is known to be encoded in a topological property of its space of representations, stating that finite-dimensional representations should be dense therein. We extend this…
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 analysis of symmetry in quantum systems is of utmost theoretical importance, useful in a variety of applications and experimental settings, and is difficult to accomplish in general. Symmetries imply conservation laws, which partition…
We propose the construction of equations of motion based on symmetries in quantum-mechanical systems, using Heisenberg's uncertainty principle as a minimal foundation. From canonical operators, two spaces of conjugate operators are…
Symmetry algebras of quantum many-body systems with locality can be understood using commutant algebras, which are defined as algebras of operators that commute with a given set of local operators. In this work, we show that these symmetry…
Unlike standard quantum mechanics, dynamical reduction models assign no particular a priori status to `measurement processes', `apparata', and `observables', nor self-adjoint operators and positive operator valued measures enter the…
It often goes unnoticed that, even for a finite number of degrees of freedom, the canonical commutation relations have many inequivalent irreducible unitary representations; the free particle and a particle in a box provide examples that…
States of a quantum mechanical system are represented by rays in a complex Hilbert space. The space of rays has, naturally, the structure of a K\"ahler manifold. This leads to a geometrical formulation of the postulates of quantum mechanics…
A unital $C^*$-algebra is called $N$-subhomogeneous if its irreducible representations are finite dimensional with dimension at most $N$. We extend this notion to operator systems, replacing irreducible representations by boundary…
In Quantum Physics, a measurement is represented by a projection on some closed subspace of a Hilbert space. We study algebras of operators that abstract from the algebra of projections on closed subspaces of a Hilbert space. The properties…
As established by Sol\`er, Quantum Theories may be formulated in real, complex or quaternionic Hilbert spaces only. St\"uckelberg provided physical reasons for ruling out real Hilbert spaces relying on Heisenberg principle. Focusing on this…
Motivated by the sharp contrast between classical and quantum physics as probability theories, in these lecture notes I introduce the basic notions of operator algebras that are relevant for the algebraic approach to quantum physics.…