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A theory of principal bundles possessing quantum structure groups and classical base manifolds is presented. Structural analysis of such quantum principal bundles is performed. A differential calculus is constructed, combining differential…
As time passes, once simple quantum states tend to become more complex. For strongly coupled k-local Hamiltonians, this growth of computational complexity has been conjectured to follow a distinctive and universal pattern. In this paper we…
We propose a general scheme for the "logic" of elementary propositions of physical systems, encompassing both classical and quantum cases, in the framework given by Non Commutative Geometry. It involves Baire*-algebras, the non-commutative…
The algebraic formulation of the quantum group gauge models in the framework of the $R$-matrix approach to the theory of quantum groups is given. We consider gauge groups taking values in the quantum groups and noncommutative gauge fields…
We review the problem of finding a general framework within which one can construct quantum theories of non-standard models for space, or space-time. The starting point is the observation that entities of this type can typically be regarded…
Hybrid classical-quantum models are computational schemes that investigate the time evolution of systems, where some degrees of freedom are treated classically, while others are described quantum-mechanically. First, we present the…
The canonical answer to the question posed is "Yes." -- tacitly assuming that quantum theory and the concept of spacetime are to be unified by `quantizing' a theory of gravitation. Yet, instead, one may ponder: Could quantum mechanics arise…
For a more general notion of Cartan connection we define characteristic classes, we investigate their relation to usual characteristic classes.
When compared to quantum mechanics, classical mechanics is often depicted in a specific metaphysical flavour: spatio-temporal realism or a Newtonian "background" is presented as an intrinsic fundamental classical presumption. However, the…
The characterization of physical systems requires a comprehensive understanding of quantum effects. One aspect is a proper quantification of the strength of such quantum phenomena. Here, a general convex ordering of quantum states will be…
We explore the possibility of replacing point set topology by higher category theory and topos theory as the foundation for quantum general relativity. We discuss the BC model and problems of its interpretation, and connect with the…
We introduce the notion of characters of comodules over coribbon Hopf algebras. The case of quantum groups of type $A_n$ is studied. We establish a characteristic equation for the quantum matrix and a q-analogue of Harish-Chandra-…
The geometry of the symplectic structures and Fubini-Study metric is discussed. Discussion in the paper addresses geometry of Quantum Mechanics in the classical phase space. Also, geometry of Quantum Mechanics in the projective Hilbert…
Interpretational problems with quantum mechanics can be phrased precisely by only talking about empirically accessible information. This prompts a mathematical reformulation of quantum mechanics in terms of classical mechanics. We survey…
Standard particle theory is based on quantized matter embedded in a classical geometry. Here, a complementary model is proposed, based on classical matter -- massive bodies, without quantum properties -- embedded in a quantum geometry. It…
We introduce quantum versions of Manin pairs and Manin triples and define quantum moment maps in this context. This provides a framework that incorporates quantum moment maps for actions of Lie algebras and quantum groups for any quantum…
Precise rules are developed in order to formalize the reasoning processes involved in standard non-relativistic quantum mechanics, with the help of analogies from classical physics. A classical or quantum description of a mechanical system…
Some very elementary ideas about quantum groups and quantum algebras are introduced and a few examples of their physical applications are mentioned.
It is shown that quantum mechanics is a plausible statistical description of an ontology described by classical electrodynamics. The reason that no contradiction arises with various no-go theorems regarding the compatibility of QM with a…
Quantum machine learning is often motivated by the idea that quantum systems can expose useful high-dimensional structure that is difficult to access with classical models. We isolate one central component of this claim: the fixed…