Searcharxiv⌕ Search

arXiv subjects

Maciej Przanowski

Publications and source records attributed to Maciej Przanowski.

At least 19 recordsLinked to original sources
arXiv2025

Curvature and conformal curvature dynamics formalisms and their applications in linearized gravity

Tensorial, spinorial and helicity formalisms of the curvature and conformal curvature dynamics are developed. Equations of linearized gravity within that formalisms are given. Gravitational radiation in linearized gravity in terms of curvature dynamics is investigated. Equivalence of the Białynicki-Birula formula for the gravitational energy in linearized gravity and the Landau-Lifschitz formula is proved. Analogous result is found for the momentum in linearized gravity.

gr-qc↗
arXiv2023

Linear graviton as a quantum particle

Wave function of a single linear graviton and its interpretation are proposed. The evolution equation for this function is given. A Hermitian operator with mutually commuting components canonically conjugated to the momentum operator of the linear graviton is found.

gr-qc↗
arXiv2023

Construction of a photon position operator with commuting components from natural axioms

A general form of the photon position operator with commuting components fulfilling some natural axioms is obtained. This operator commutes with the photon helicity operator, is Hermitian with respect to the Bialynicki-Birula scalar product and defined up to a unitary transformation preserving the transversality condition. It is shown that, using the procedure analogous to the one introduced by T. T. Wu and C. N. Yang for the case of the Dirac magnetic monopole, the photon position operator can be defined by a flat connection in some trivial vector bundle over $\mathbb{R}^3 \setminus \{(0,0,0)\}$. This observation enables us to reformulate quantum mechanics of a~single photon on $(\mathbb{R}^{3} \setminus \{(0,0,0)\}) \times \mathbb{C}^2$.

quant-ph↗
arXiv2021

The geometrical interpretation of the photon position operator

It is shown that the photon position operator $\hat{\vec{X}}$ with commuting components can be written in the momentum representation as $\hat{\vec{X}}=i \hat{\vec{D}}$, where $\hat{\vec{D}}$ is a flat connection in the tangent bundle $T(\mathbb{R}^3 \setminus \{ (0,0,k_3) \in \mathbb{R}^3 : k_3 \geq 0\})$ over $\mathbb{R}^3 \setminus \{ (0,0,k_3) \in \mathbb{R}^3 : k_3 \geq 0\}$ equipped with the Cartesian structure. Moreover, $\hat{\vec{D}}$ is such that the tangent $2$-planes orthogonal to the momentum are parallelly propagated with respect to $\hat{\vec{D}}$ and, also, $\hat{\vec{D}}$ is an anti-Hermitian operator with respect to the scalar product $\langle \mathbfΨ | \hat{H}^{-2s} |\mathbfΦ \rangle$. The eigenfunctions $\mathbfΨ_{\vec{X}} (\vec{x})$ of the position operator $\hat{\vec{X}}$ are found.

quant-ph↗
arXiv2020

The Weyl -- Wigner -- Moyal Formalism on a Discrete Phase Space. II. The Photon Wigner Function

Classical model of light in helicity formalism is presented. Then quantum point of view at photons -- construction and interpretation of photon wave function is proposed. Quantum mechanics of photon is investigated. The Białynicki -- Birula scalar product $\langle {\bf Ψ}_{1}|{\bf Ψ}_{2}\rangle_{BB}$ and the generalized Hermitian conjugation $\widehatγ^{+\hspace{-0.45em}+}$ of linear operator $\widehatγ$ are discussed. Quantum description of light on a phase space is developed. A photon Wigner function is built.

quant-ph↗
arXiv2019

The Weyl-Wigner-Moyal formalism on a discrete phase space. I. A Wigner function for a nonrelativistic particle with spin

The Weyl-Wigner-Moyal formalism for quantum particle with discrete internal degrees of freedom is developed. A one to one correspondence between operators in the Hilbert space $L^{2}(\mathbb{R}^{3})\otimes{\mathcal{H}}^{(s+1)}$ and functions on the phase space $\mathbb{R}^{3}\times\mathbb{R}^{3}\times \{0,...,s\} \times\{0,...,s\}$ is found. The expressions for the Stratonovich-Weyl quantizer, star product and Wigner functions of such systems for arbitrary values of spin are obtained in detail. As examples the Landau levels and the corresponding Wigner functions for a spin $\frac{1}{2}$ nonrelativistic particle as well as the magnetic resonance for a spin $\frac{1}{2}$ nonrelativistic uncharged particle are analysed.

quant-ph↗
arXiv2018

On twisting type $[\textrm{N}] \otimes [\textrm{N}]$ Ricci flat complex spacetimes with two homothetic symmetries

$\mathcal{HH}$ spaces of type $[\textrm{N}] \otimes [\textrm{N}]$ with twisting congruence of null geodesics defined by the 4-fold undotted and dotted Penrose spinors are investigated. It is assumed that these spaces admit two homothetic symmetries. The general form of the homothetic vector fields are found. New coordinates are introduced which enable us to reduce the $\mathcal{HH}$ system of PDEs to one ODE on one holomorphic function. In a special case this is a second-order ODE and its general solution is explicitly given. In the generic case one gets rather involved fifth-order ODE.

gr-qc↗
arXiv2016

From the discrete Weyl -- Wigner formalism for symmetric ordering to a number -- phase Wigner function

The general Weyl -- Wigner formalism in finite dimensional phase spaces is investigated. Then this formalism is specified to the case of symmetric ordering of operators in an odd -- dimensional Hilbert space. A respective Wigner function on the discrete phase space is found and the limit, when the dimension of Hilbert space tends to infinity, is considered. It is shown that this limit gives the number -- phase Wigner function in quantum optics. Analogous results for the `almost' symmetric ordering in an even -- dimensional Hilbert space are obtained. Relations between the discrete Wigner functions introduced in our paper and some other discrete Wigner functions appearing in literature are studied.

quant-ph↗
arXiv2015

On some properties of number-phase Wigner function

It is shown that the number-phase Wigner function defines uniquely the respective density operator. Relations between the Glauber-Sudarshan distribution $\mathcal{P}(α)$ and the number-phase Wigner function is found. This result is then generalised to the case of the Cahil-Glauber distributions $\mathcal{W}^{(s)}(α)$, $-1\leq s \leq 1$.

math-ph↗
arXiv2015

Uncertainty relations in quantum optics. Is the photon intelligent?

The Robertson -- Schrödinger, Heisenberg -- Robertson and Trifonov uncertainty relations for arbitrary two functions $f_{1}$ and $f_{2}$ depending on the quantum phase and the number of photons respectively, are given. Intelligent states and states which minimize locally the product of uncertainties $(Δf_{1})^{2}\cdot (Δf_{2})^{2}$ or the sum $(Δf_{1})^{2}+(Δf_{2})^{2}$ are investigated for the cases $f_{1}=ϕ,\exp{(iϕ)}, \exp{(-iϕ)}, \cosϕ, \sinϕ$ and $f_{2}=n$.

quant-ph↗
arXiv2013

From the Weyl quantization of a particle on the circle to number-phase Wigner functions

A generalized Weyl quantization formalism for a particle on the circle investigated in \cite{1} is developed. A Wigner function for the state $\hat{\varrho}$ and the kernel $\mathcal{K}$ for a particle on the circle is defined and its properties are analyzed. Then it is shown how this Wigner function can be easily modified to give the number-phase Wigner function in quantum optics. Some examples of such number-phase Wigner function are considered.

math-ph↗
arXiv2013

Killing Symmetries in $\mathcal{H}$-Spaces with $Λ$

All Killing symmetries in complex $\mathcal{H}$-spaces with $Λ$ in terms of the Plebański - Robinson - Finley coordinate system are found. All $\mathcal{H}$-metrics with $Λ$ admitting a null Killing vector are explicitly given. It is shown that the problem of non-null Killing vector reduces to looking for solution of the Boyer - Finley - Plebański (Toda field) equation

gr-qc↗
arXiv2013

Generalized Weyl quantization on the cylinder and quantum phase

Generalized Weyl quantization formalism for the cylindrical phase space $S^1 \times \mathbb{R}^1$ is developed. It is shown that the quantum observables relevant to the phase of linear harmonic oscillator or electromagnetic field can be represented within this formalism by the self-adjoint operators on the Hilbert space $L^2(S^1)$.

math-ph↗
arXiv2011

Notes on thermodynamics in special relativity

Foundations of thermodynamics in special theory of relativity are considered. We argue that from the phenomenological point of view the correct relativistic transformations of heat and absolute temperature are given by the formulae proposed by H. Ott, H. Arzeliès and C. M\oller. It is shown that the same transformation rules can be also found from the relativistic Gibbs distribution for ideal gas. This distribution has been recently verified by the computer simulations. Phenomenological and statistical thermometers in relativistic thermodynamics are analysed.

cond-mat.stat-mech↗
arXiv2006

Null Kähler structures, Symmetries and Integrability

We review the integrable systems which arise as symmetry reductions of Plebanski's heavenly equations, and their generalisations. We also show that all four-dimensional null Kahler-Einstein (or type N hyper-heavenly) metrics with symmetry can be found from solutions to a variable coefficient generalisation of the dispersionless Kadomtsev-Petviashvili equation.

gr-qc↗