相关论文: Quantising on a category

There are two ways to turn a categorical model for pure quantum theory into one for mixed quantum theory, both resulting in a category of completely positive maps. One has quantum systems as objects, whereas the other also allows classical…

In ordinary quantum field theory, one can define the algebra of observables in a given region in spacetime, but in the presence of gravity, it is expected that this notion ceases to be well-defined. A substitute that appears to make sense…

We introduce a constructive procedure that maps all spatial correlations of a broad class of states into temporal correlations between general quantum measurements. This allows us to present temporal phenomena analogous to genuinely…

Ideas from deformation quantization applied to algebras with one generator lead to methods to treat a nonlinear flat connection. It provides us elements of algebras to be parallel sections. The moduli space of the parallel sections is…

The goal of this paper is to summarise the first steps in developing a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum…

We show that it is possible to represent various descriptions of Quantum Mechanics in geometrical terms. In particular we start with the space of observables and use the momentum map associated with the unitary group to provide an unified…

In gauge theories, physical histories are represented by space-time connections modulo gauge transformations. The space of histories is thus intrinsically non-linear. The standard framework of constructive quantum field theory has to be…

We review the definition of geometric quantization, which begins with defining a mathematical framework for the algebra of observables that holds equally well for classical and quantum mechanics. We then discuss prequantization, and go into…

In this report we present a new modelling framework for concepts based on quantum theory, and demonstrate how the conceptual representations can be learned automatically from data. A contribution of the work is a thorough category-theoretic…

A general result relating skew monoidal structures and monads is proved. This is applied to quantum categories and bialgebroids. Ordinary categories are monads in the bicategory whose morphisms are spans between sets. Quantum categories…

The framework of generalized probabilistic theories is a powerful tool for studying the foundations of quantum physics. It provides the basis for a variety of recent findings that significantly improve our understanding of the rich physical…

These lecture notes cover 13 sessions and are presented as an e-print, intended to evolve over time. Quantum invariants do more than distinguish topological objects; they build bridges between topology, algebra, number theory and quantum…

In this paper, we investigate the connection between Classical and Quantum Mechanics by dividing Quantum Theory in two parts: - General Quantum Axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoint…

All gauge theories need ``something fixed'' even as ``something changes.'' Underlying the implementation of these ideas all major physical theories make indispensable use of an elaborately designed spacetime model as the ``something…

We generalize the small object argument in order to allow for its application to proper classes of maps (as opposed to sets of maps in Quillen's small object argument). The necessity of such a generalization arose with appearance of several…

Quantum theory is a mathematical formalism to compute probabilities for outcomes happenning in physical experiments. These outcomes constitute events happening in space-time. One of these events represents the fact that a system located in…

Applying (enriched) categorical structures we define the notion of ordered sheaf on a quantaloid Q, which we call `Q-order'. This requires a theory of semicategories enriched in the quantaloid Q, that admit a suitable Cauchy completion.…

For a rigid tensor abelian category $T$ over a field $k$ we introduce a notion of a normal quotient $q:T\to Q$. In case $T$ is a Tannaka category, our notion is equivalent to Milne's notion of a normal quotient. More precisely, if $T$ is…

A useful general concept of bialgebroid seems to be resolving itself in recent publications; we give a treatment in terms of modules and enriched categories. We define the term "quantum category". The definition of antipode for a…

The most standard description of symmetries of a mathematical structure produces a group. However, when the definition of this structure is motivated by physics, or information theory, etc., the respective symmetry objects might become more…