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Jerzy Kijowski

Publications and source records attributed to Jerzy Kijowski.

At least 19 recordsLinked to original sources
arXiv2023

Arrival time in quantum mechanics (demonstrated in geometrical order)

A geometric construction of the arrival time in conventional quantum mechanics is presented. It is based on a careful mathematical analysis of different quantization procedures for classical observables as functions of positions and momenta. A class of observables is selected which possess a unique (if any) quantized version. A simple criterion for existence of such a quantized version is formulated. These mathematical results are then applied to the classical "arrival time" observable.

quant-ph↗
arXiv2020

Gauge-invariant quadratic approximation of quasi-local mass and its relation with Hamiltonian for gravitational field

Gauge invariant, Hamiltonian formulation of field dynamics within a compact region $Σ$ with boundary $\partial Σ$ is given for the gravitational field linearized over a Kottler metric. The boundary conditions which make the system autonomous are discussed and the corresponding Hamiltonian functional $\mathcal{H}_\text{Invariant}$ is calculated. It is shown that, under specific boundary conditions, the quasi-local Geroch--Hawking mass $\mathcal{H}_\text{Hawking}$ reduces to $\mathcal{H}_\text{Invariant}$ in the weak field approximation. This observation is a quasi-local version of the classical Brill--Deser result.

gr-qc↗
arXiv2020

Localizing Energy in Fierz-Lanczos theory

We calculate energy carried by the massless spin-2 field using Fierz-Lanczos representation of the theory. For this purpose Hamiltonian formulation of the field dynamic is thoroughly analyzed. Final expression for the energy is very much analogous to the Maxwell energy in electrodynamics (spin-1 field) and displays the locality property. Known as a ``super-energy'' in gravity theory, this quantity differs considerably from the well understood gravitational field energy (represented in linear gravity by the quadratic term in Taylor expansion of the A.D.M. mass) which cannot be localized.

gr-qc↗
arXiv2016

On the existence of rigid spheres in four-dimensional spacetime manifolds

This paper deals with the generalization of usual round spheres in the flat Minkowski spacetime to the case of a generic four-dimensional spacetime manifold $M$. We consider geometric properties of sphere-like submanifolds in $M$ and introduce conditions on external curvature and torsion, which lead to a definition of a {\em rigid sphere}. The main result is a local existence theorem concernig such spheres. For this purpose we apply the surjective implicit function theorem. The proof is based on a detailed analysis of the linearized problem and leads to an eight-parameter family of solutions in case when the metric tensor $g$ of $M$ is from a certain neighbourhood of the flat Minkowski metric. This contribution continues the study of rigid spheres in (Class. Quantum Grav. \textbf{30} (2013), 175010, doi:10.1088/0264-9381/30/17/175010, 18 pp.).

gr-qc↗
arXiv2016

Universality of the Einstein theory of gravitation

We show that generalizations of general relativity theory, which consist in replacing the Hilbert Lagrangian $L_{Hilbert} = \frac 1{16π} \sqrt{|g|} R$ by a generic scalar density $L=L(g_{μν}, R^λ_{μνκ})$ depending upon the metric $g_{μν}$ and the curvature tensor $R^λ_{μνκ}$, are equivalent to the conventional Einstein theory for a (possibly) different metric tensor $\tilde{g}_{μν}$ and (possibly) a different set of matter fields. The simple proof of this theorem relies on a new approach to variational problems containing metric and connection.

gr-qc↗
arXiv2015

Wigner function of a qubit

We show that real polarization method can be effectively used to geometrically quantize physical systems with compact phase space, like the spin. Our method enables us to construct a wave function of a qubit in both position and momentum representations and also its Wigner function. These results can be used in quantum informatics.

math-ph↗
arXiv2014

Symplectic structures related with higher order variational problems

In this paper we derive the symplectic framework for field theories defined by higher-order Lagrangians. The construction is based on the symplectic reduction of suitable spaces of iterated jets. The possibility of reducing a higher-order system of PDEs to a constrained first-order one, the symplectic structures naturally arising in the dynamics of a first-order Lagrangian theory, and the importance of the Poincaré-Cartan form for variational problems, are all well-established facts. However, their adequate combination corresponding to higher-order theories is missing in the literature. Here we obtain a consistent and truly finite-dimensional canonical formalism, as well as a higher-order version of the Poincaré-Cartan form. In our exposition, the rigorous global proofs of the main results are always accompanied by their local coordinate descriptions, indispensable to work out practical examples.

math.DG↗
arXiv2013

Rigid spheres in Riemannian spaces

Choice of an appropriate (3+1)-foliation of spacetime or a (2+1)-foliation of the Cauchy space, leads often to a substantial simplification of various mathematical problems in General Relativity Theory. We propose a new method to construct such foliations. For this purpose we define a special family of topological two-spheres, which we call "rigid spheres". We prove that there is a four-parameter family of rigid spheres in a generic Riemannian three-manifold (in case of the flat Euclidean three-space these four parameters are: 3 coordinates of the center and the radius of the sphere). The rigid spheres can be used as building blocks for various ("spherical", "bispherical" etc.) foliations of the Cauchy space. This way a supertranslation ambiguity may be avoided. Generalization to the full 4D case is discussed. Our results generalize both the Huang foliations (cf. \cite{LHH}) and the foliations used by us (cf. \cite{JKL}) in the analysis of the two-body problem.

gr-qc↗
arXiv2012

Fractional Fourier Transform and Geometric Quantization

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 the phase-space: no linear structure is necessary. It is shown that the "fractional Fourier transform" provides a simple example of this construction. As an application of this techniques we show that for any linear Hamiltonian system, its quantum dynamics can be obtained exactly as the lift of the corresponding classical dynamics by means of the above transformation. Moreover, it can be deduced from the free quantum evolution. This way new, unknown symmetries of the Schrödinger equation can be constructed. It is also argued that the above construction defines in a natural way a connection in the bundle of quantum states, with the base space describing all their possible representations. The non-flatness of this connection would be responsible for the non-existence of a quantum representation of the complete algebra of classical observables.

quant-ph↗
arXiv2011

Thermo--hydrodynamics As a Field Theory

The field theoretical description of thermo-hydrodynamics is given. It is based on the duality between the physical space--time and the "material space-time" which we construct here. The material space appearing in a natural way in the canonical formulation of the hydrodynamics is completed with a material time playing role of the field potential for temperature. Both Lagrangian and Hamiltonian formulations, the canonical structure, Poisson bracket, Nöther theorem and conservation laws are discussed.

math-ph↗
arXiv2010

Dynamics of a self gravitating light-like matter shell with spherical symmetry

A novel Hamiltonian description of the dynamics of a spherically symmetric, light-like, self-gravitating shell is presented. It is obtained via the systematic reduction of the phase space with respect to the Gauss-Codazzi constraints, model and rare procedure in the Canonical Gravity. The Hamiltonian of the system (numerically equal to the value of the ADM mass) is explicitly calculated in terms of the gauge-invariant "true degrees of freedom", i.e. as a function on the reduced phase space. A geometric interpretation of the momentum canonically conjugate to the shell's radius is given. Models of matter compatible with the shell dynamics are found. A transformation between the different time parameterizations of the shell is calculated. The presented model may become a new toy model of quantum gravity.

gr-qc↗
arXiv2009

Hamiltonian description of radiation phenomena: Trautman-Bondi energy and corner conditions

Cauchy initial value problem on a hyperboloid is proved to define a Hamiltonian system, provided the radiation data at null infinity are also taken into account, as a part of Cauchy data. The "Trautman-Bondi mass", supplemented by the "already radiated energy" assigned to radiation data, plays role of the Hamiltonian function. This approach leads to correct description of the corner conditions.

gr-qc↗
arXiv2007

A geometric analysis of the Maxwell field in a vicinity of a multipole particle and new special functions

A method of solving Maxwell equations in a vicinity of a multipole particle (moving along an arbitrary trajectory) is proposed. The method is based on a geometric construction of a trajectory-adapted coordinate system, which simplifies considerably the equations. The solution is given in terms of a series, where a new family of special functions arises in a natural way. Singular behaviour of the field near to the particle may be analyzed this way up to an arbitrary order. Application to the self-interaction problems in classical electrodynamics is discussed.

physics.class-ph↗
arXiv2007

Energy-minimizing two black holes initial data

An attempt to construct the ``ground state'' vacuum initial data for the gravitational field surrounding two black holes is presented. The ground state is defined as the gravitational initial data minimizing the ADM mass within the class of data for which the masses of the holes and their distance are fixed. To parameterize different geometric arrangements of the two holes (and, therefore, their distance) we use an appropriately chosen scale factor. A method for analyzing the variations of the ADM mass and the masses (areas) of the horizons in terms of gravitational degrees of freedom is proposed. The Misner initial data are analyzed in this context: it is shown that they do not minimize the ADM mass.

gr-qc↗
arXiv2007

A simple renormalization group approximation of the groundstate properties of interacting bosonic systems

We present a new, simple renormalization group method of investigating groundstate properties of interacting bosonic systems. Our method reduces the number of particles in a system, which makes numerical calculations possible for large systems. It is conceptually simple and easy to implement, and allows to investigate the properties unavailable through mean field approximations, such as one- and two-particle reduced density matrices of the groundstate. As an example, we model a weakly interacting 1D Bose gas in a harmonic trap. Compared to the mean-field Gross-Pitaevskii approximation, our method provides a more accurate description of the groundstate one-particle density matrix. We have also obtained the Hall-Post lower bounds for the groundstate energy of the gas. All results have been obtained by the straightforward numerical diagonalization of the Hamiltonian matrix.

cond-mat.stat-mech↗