Related papers: Duality of quantum geometries
We undertake a study of the notion of a quantum graph over arbitrary finite-dimensional $C^*$-algebras $B$ equipped with arbitrary faithful states. Quantum graphs are realised principally as either certain operators on $L^2(B)$, the quantum…
The motion of a quantum particle constrained to a two-dimensional non-compact Riemannian manifold with non-trivial metric can be described by a flat-space Schroedinger-type equation at the cost of introducing local mass and metric and…
We derive a generalized wave-particle duality relation for arbitrary multipath quantum interference phenomena. Beyond the conventional notion of the wave nature of a quantum system, i.e., the interference fringe visibility, we introduce a…
A quantum principal bundle is constructed for every Coxeter group acting on a finite-dimensional Euclidean space $E$, and then a connection is also defined on this bundle. The covariant derivatives associated to this connection are the…
Quantum theory of the free Maxwell field in Minkowski space is constructed using a representation in which the self dual connection is diagonal. Quantum states are now holomorphic functionals of self dual connections and a decomposition of…
This paper is an introduction to diagrammatic methods for representing quantum processes and quantum computing. We review basic notions for quantum information and quantum computing. We discuss topological diagrams and some issues about…
Using tangent bundle geometry we construct an equivalent reformulation of classical field theory on flat spacetimes which simultaneously encodes the perspectives of multiple observers. Its generalization to curved spacetimes realizes a new…
The notion of quantum matrix pairs is defined. These are pairs of matrices with non-commuting entries, which have the same pattern of internal relations, q-commute with each other under matrix multiplication, and are such that products of…
Motivated by the expectation that relativistic symmetries might acquire quantum features in Quantum Gravity, we take the first steps towards a theory of ''Doubly'' Quantum Mechanics, a modification of Quantum Mechanics in which the…
Following the definition of quantum differential operators given by Lunts and Rosenberg in (Sel. math., New ser. 3 (1997) 335--359), we show that the ring of quantum differential operators on the affine line is the ring generated by x and…
It is demonstrated that in gate-based quantum computing architectures quantum walk is a natural mathematical description of quantum gates. It originates from field-matter interaction driving the system, but is not attached to specific qubit…
We reconsider differential geometry from the point of view of the quantum theory of non-relativistic spinning particles, which provides examples of supersymmetric quantum mechanics. This enables us to encode geometrical structure in…
We describe an approach to the quantization of (2+1)--dimensional gravity with topology R x T^2 and negative cosmological constant, which uses two quantum holonomy matrices satisfying a q--commutation relation. Solutions of diagonal and…
Double field theory was developed by theoretical physicists as a way to encompass $T$-duality. In this paper, we express the basic notions of the theory in differential-geometric invariant terms, in the framework of para-Kaehler manifolds.…
It is natural to consider a quantum system in the continuum limit of space-time configuration. Incorporating also, Einstein's special relativity, leads to the quantum theory of fields. Non-relativistic quantum mechanics and classical…
Different approaches are compared to formulation of quantum mechanics of a particle on the curved spaces. At first, the canonical, quasi-classical and path integration formalisms are considered for quantization of geodesic motion on the…
The geometry of the classical phase space C of a finite number of degrees of freedom determines the possible duality symmetries of the corresponding quantum mechanics. Under duality we understand the relativity of the notion of a quantum…
Geometrically, quantum mechanics is defined by a complex line bundle $L_\hbar$ over the classical particle phase space $T^*{R}^3\cong{R}^6$ with coordinates $x^a$ and momenta $p_a$, $a,...=1,2,3$. This quantum bundle $L_\hbar$ is endowed…
A quantum set is defined to be simply a set of nonzero finite-dimensional Hilbert spaces. Together with binary relations, essentially the quantum relations of Weaver, quantum sets form a dagger compact category. Functions between quantum…
Quantum mechanics is formulated as a geometric theory on a Hilbert manifold. Images of charts on the manifold are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations in this…