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This is a review of the aspirations and disappointments of the canonical quantization of geometry. I compare the two chief ways of looking at canonical gravity, geometrodynamics and connection dynamics. I capture as much of the classical…
We treat continuous histories within the histories approach to generalised quantum mechanics. The essential tool is the `history group': the analogue, within the generalised history scheme, of the canonical group of single-time quantum…
We study in detail the algebraic structures underlying quantum circuits generated by CNOT gates. Our results allow us to propose polynomial-time heuristics to reduce the number of gates used in a given CNOT circuit and we also give…
We propose a new formalism of quantum subsystems which allows to unify the existing and new methods of reduced description of quantum systems. The main mathematical ingredients are completely positive maps and correlation functions. In this…
We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local…
We formulate a theory of generalized Fock spaces which underlies the different forms of quantum statistics such as ``infinite'', Bose-Einstein and Fermi-Dirac statistics. Single-indexed systems as well as multi-indexed systems that cannot…
The unitary Clifford algebras are described here for the first time, and arise from the intersection of the orthogonal and common symplectic (Weyl) Clifford algebras of the complexification of the canonical phase space. The convergence of…
Generalized coherent states for shape invariant potentials are constructed using an algebraic approach based on supersymmetric quantum mechanics. We show this generalized formalism is able to: a) supply the essential requirements necessary…
We study deterministic and quantum dynamics from a constructive "finite" point of view, since the introduction of a continuum, or other actual infinities in physics poses serious conceptual and technical difficulties, without any need for…
Harnessing the potential computational advantage of quantum computers for machine learning tasks relies on the uploading of classical data onto quantum computers through what are commonly referred to as quantum encodings. The choice of such…
We provide a general construction of quantum generalized master equations with memory kernel leading to well defined, that is completely positive and trace preserving, time evolutions. The approach builds on an operator generalization of…
In this paper the q-deformed $W$ algebra $\WW_q$ is constructed, whose nontrivial quantum group structure is presented.
The phase diagram of a simple area-preserving map, which was motivated by the quantum dynamics of cold atoms, is explored analytically and numerically. Periodic orbits of a given winding ratio are found to exist within wedge-shaped regions…
We investigate the coupled dynamics of charge and energy in interacting lattice models with dipole conservation. We formulate a generic hydrodynamic theory for this combination of fractonic constraints and numerically verify its…
Generalized Fourier transformation between the position and the momentum representation of a quantum state is constructed in a coordinate independent way. The only ingredient of this construction is the symplectic (canonical) geometry of…
A list of generic $U_q(sl_2)$-module algebra structures on the Laurent polynomial algebra over the quantum plane with uncountably many isomorphism classes is produced. Also, a complete list of such structures is presented in which the…
A new, configuration-space picture of a formalism of group quantization, the GAQ formalism, is presented in the context of a previous, algebraic generalization. This presentation serves to make a comprehensive discussion in which other…
In their seminal paper (Phys. Rev.121, 920 (1961)) Sudarshan, Mathews and Rau investigated properties of the dynamical $A$ and $B$ maps acting on $n$ dimensional quantum systems. Nature of the dynamical maps in open quantum system…
It is shown, that the only reason for possibility of using microcanonical ensemble is that there are probabilistic processes in microworld, that are not described by quantum mechanic. On a simple example it is demonstrated, that canonical…
Periodically driven flows are fundamental models of chaotic behavior and the study of their transport properties is an active area of research. A well-known analytic construction is the augmentation of phase space with an additional time…