Related papers: On the Dolbeault--Dirac Operator of Quantized Symm…
In order to extend the spectral action principle to non-compact spaces, we propose a framework for spectral triples where the algebra may be non-unital but the resolvent of the Dirac operator remains compact. We show that an example is…
We describe quantum and classical Hamiltonian dynamics in a common Hilbert space framework, that allows the treatment of mixed quantum-classical systems. The analysis of some examples illustrates the possibility of entanglement between…
A Dirac operator D on the standard Podles sphere is defined and investigated. It yields a spectral triple such that |D|^{-z} is of trace class for Re z>0. Commutators with the Dirac operator give the distinguished 2-dimensional covariant…
Compatibility of symmetric quantization of the Dirac equation in a curved space with general covariance gives a special representation of the spin connections in which their dot product with the Dirac gamma matrices becomes equal to the…
We explore the relation between noncommutative geometry, in the spectral triple formulation, and quantum mechanics. To this aim, we consider a dynamical theory of a noncommutative geometry defined by a spectral triple, and study its…
The purpose of this article is to apply the concept of the spectral triple, the starting point for the analysis of noncommutative spaces in the sense of A.~Connes, to the case where the algebra $\cA$ contains both bosonic and fermionic…
It is shown that the calculation of Dirac operator for the spherical coordinate system with spherical Dirac matrices and using the spin connection formalism is in the contradiction with the definition of standard Dirac operator in the…
We consider optical systems where propagation of light can be described by a Dirac-like equation with $PT$-symmetric Hamiltonian. In order to construct exactly solvable configurations, we extend the confluent Crum-Darboux transformation for…
In this paper we show that for a Koszul Calabi-Yau algebra, there is a shifted bi-symplectic structure in the sense of Crawley-Boevey-Etingof-Ginzburg, on the cobar construction of its co-unitalized Koszul dual coalgebra, and hence its DG…
Motivated by Wigner's theorem, a canonical construction is described that produces an Atiyah-Singer Dirac operator with both unitary and anti-unitary symmetries. This Dirac operator includes the Dirac operator for KR-theory as a special…
We investigate Dirac's bra-ket formalism based on a rigged Hilbert space for a non-Hermite quantum system with a positive-definite metric. First, the rigged Hilbert space, characterized by positive-definite metric, is established. With the…
We study Heisenberg's matrix mechanics within an algebraic pre-Hilbert framework of arbitrary finite dimension. The commutator of the position and momentum matrices naturally generates a third Hermitian operator whose unbounded character…
S^4 is not a complex manifold, but it is sufficient to remove one point to make it complex. Using supersymmetry methods, we show that the Dolbeault complex (involving the holomorphic exterior derivative and its Hermitian conjugate) can be…
We use quantum harmonic analysis for densely defined operators to provide a simplified proof of the Berger-Coburn theorem for boundedness of Toeplitz operators. In addition, we revisit compactness and Schatten-class membership of densely…
This paper is a follow-up to a recent article about the essential spectrum of Toeplitz operators acting on the Bergman space over the unit ball. As mentioned in the said article, some of the arguments can be carried over to the case of…
This work is a contribution to the area of Strict Quantization (in the sense of Rieffel) in the presence of curvature and non-Abelian group actions. More precisely, we use geometry to obtain explicit oscillatory integral formulae for…
By breaking the spin symmetry in the relativistic domain, a powerful tool in physical sciences was lost. In this work, we examine an alternative of spin symmetry for systems described by the many-electron Dirac-Coulomb Hamiltonian. We show…
The Dirac operator for a manifold Q, and its chirality operator when Q is even dimensional, have a central role in noncommutative geometry. We systematically develop the theory of this operator when Q=G/H, where G and H are compact…
The spectra of the second quantization and the symmetric second quantization of a strict Hilbert space contraction are computed explicitly and shown to coincide. As an application, we compute the spectrum of the nonsymmetric…
In the high-energy physics literature one finds statements such as ``matrix algebras converge to the sphere''. Earlier I provided a general precise setting for understanding such statements, in which the matrix algebras are viewed as…