arXiv subjects

Jakub Káninský

Publications and source records attributed to Jakub Káninský.

5 recordsLinked to original sources

arXiv2021

Quantum Mechanical Observables under a Symplectic Transformation of Coordinates

We consider a general symplectic transformation (also known as linear canonical transformation) of quantum-mechanical observables in a quantized version of a finite-dimensional system with configuration space isomorphic to $ \mathbb{R}^{q} $. Using the formalism of rigged Hilbert spaces, we define eigenstates for all the observables. Then we work out the explicit form of the corresponding transformation of these eigenstates. A few examples are included at the end of the paper.

quant-ph↗

arXiv2020

Discrete Black Holes

The classical spacetime is usually described by a differentiable manifold with infinitely many degrees of freedom. Occasionally though, it is useful to consider an approximation whose number of degrees of freedom is finite. There are several discrete models of spacetime like that, some of which have been used to build a (simplified) representation of a black hole. We will shortly revisit these discrete black hole models. Then we limit ourselves to one particular case and show how it can be inhabited by quantum matter fields. It is suggested that the field dynamics should be described by the framework of discrete canonical evolution, and we point out some of the most significant implications of this approach.

gr-qc↗

arXiv2020

Models of Discrete Linear Evolution for Quantum Systems

Discrete canonical evolution is a key tool for understanding the dynamics in discrete models of spacetime, in particular those represented by a triangular Regge lattice. We consider a finite-dimensional system whose evolution is realized by a series of discrete-time evolution steps governed by Hamiltonian equations of motion that are linear in the canonical coordinates. The evolution is allowed to be irregular, which produces constraints as well as non-uniqueness of solutions. We provide two independent and fundamentally different approaches to canonical quantization of this system, including detailed description of the evolution maps, measurement and path integrals. It is argued that some irregular discrete systems may be most naturally described by a non-unitary quantum evolution. The formalism is then applied to a simple yet physically relevant model of massless scalar field on a two-dimensional spacetime lattice.

gr-qc↗

arXiv2020

Discrete Linear Canonical Evolution

This work builds on an existing model of discrete canonical evolution and applies it to the general case of a linear dynamical system, i.e., a finite-dimensional system with configuration space isomorphic to $ \mathbb{R}^{q} $ and linear equations of motion. The system is assumed to evolve in discrete time steps. The most distinctive feature of the model is that the equations of motion can be irregular. After an analysis of the arising constraints and the symplectic form, we introduce adjusted coordinates on the phase space which uncover its internal structure and result in a trivial form of the Hamiltonian evolution map. For illustration, the formalism is applied to the example of massless scalar field on a two-dimensional spacetime lattice.

math-ph↗

arXiv2017

Probabilistic Spacetimes

Probabilistic Spacetime is a simple generalization of the classical model of spacetime in General Relativity, such that it allows to consider multiple metric field realizations endowed with probabilities. The motivation for such a generalization is a possible application in the context of some quantum gravity approaches, particularly those using the path integral. It is argued that this model might be used to describe simplified geometry, resulting e.g. from discretization, while keeping the continuous manifold; or it may be used as an effective description of a probabilistic geometry resulting from a full-fledged quantum gravity computation.

gr-qc↗