arXiv subjects

Arkadiusz Jadczyk

Publications and source records attributed to Arkadiusz Jadczyk.

At least 19 recordsLinked to original sources

arXiv2024

The explicit form of the unitary representation of the Poincar\'e group for vector-valued wave functions (massive and massless), with applications to photon's localization and position operators

We geometrically derive the explicit form of the Unitary representation of the Poincare group and use it to apply speed-of-light boosts to simple polarization basis to end up with Hawton-Baylis photon position operator with commuting components. We give explicit formulas for other photon boost eigenmodes. We investigate the underlying affine connections on the light cone in momentum space and find that while Pryce connection is metric semi-symmetric, the flat Hawton-Baylis connection is not semi-symmetric. Finally we discuss localizability of photon states localized on closed loops and show that photon states on the circle, both unnormalized improper states and finite norm wave packet smeared over washer-like regions are strictly localized with respect to Hawton-Baylis operators with commuting components and also with respect to the noncommutative Jauch-Piron-Amrein POV measure.

quant-ph↗

arXiv2022

Random walk on quantum blobs

We describe the action of the symplectic group on the homogeneous space of squeezed states (quantum blobs) and extend this action to the semigroup. We then extend the metaplectic representation to the metaplectic (or oscillator) semigroup and study the properties of such an extension using Bargmann-Fock space. The shape geometry of squeezing is analyzed and noncommuting elements from the symplectic semigroup are proposed to be used in simultaneous monitoring of noncommuting quantum variables - which should lead to fractal patterns on the manifold of squeezed states.

quant-ph↗

arXiv2021

On the bundle of Clifford algebras over the space of quadratic forms

For each quadratic form Q in Quad(V) over a given vector space over a field R we have the Clifford algebra Cl(V,Q) defined as the quotient T(V)/I(Q) of the tensor algebra T(V) over the two-sided ideal generated by expressions of the form $x x-Q(x),x in V. In the present paper we consider the whole family Cl(V,Q) in a geometric way as a Z2-graded vector bundle over the base manifold Quad(V). Bilinear forms F from Bil(V) act on this bundle providing natural bijective linear mappings lambda_F between Clifford algebras for different Cl(V,Q). Alternate (or antisymmetric) forms induce vertical automorphisms, which we propose to consider as 'gauge transformations'. We develop here the formalism of N. Bourbaki, which generalizes the well known Chevalley's isomorphism Cl(V,Q)->End(Wedge(V)->Wedge(V). In particular we realize the Clifford algebra twisting gauge trnsformations induced by antisymmetric bilinear forms as exponentials of contractions with elements of $Wedge(V^* representing these forms. Throughtout all this paper we intentionally avoid using the so far accepted term "Clifford algebra of a bilinear form" (known otherwise as "Quantum Clifford algebra"), which we consider as possibly misleading, as it does not represent any well defined mathematical object. Instead we show explicitly how any given Clifford algebra Cl(Q) can be naturally realized as acting via endomorphisms of any other Clifford algebra Cl(Q') if Q'=Q+Q_F,and Q_F(x)=F(x,x). Possible physical meaning of such transformations are also mentioned.

math-ph↗

arXiv2017

Lorentz transformation from an elementary point of view

Elementary methods are used to examine some nontrivial mathematical issues underpinning the Lorentz transformation. Its eigen-system is characterized through the exponential of a $G$-skew symmetric matrix, underlining its unconnectedness at one of its extremes (the hyper-singular case). A different yet equivalent angle is presented through Pauli coding which reveals the connection between the hyper-singular case and the shear map.

math-ph↗

arXiv2015

Asymptotic formula for quantum harmonic oscillator tunneling probabilities

Using simple methods of asymptotic analysis it is shown that for a quantum harmonic oscillator in n-th energy eigenstate the probability of tunneling into the classically forbidden region obeys an unexpected but simple asymptotic formula: the leading term is inversely proportional to the cube root of n.

quant-ph↗

arXiv2015

Comment on `The minimum-uncertainty squeezed states for atoms and photons in cavity'

In a recent work, Kryuchkov, Suslov and Vega-Guzman [20013 J. Phys. B: At. Mol. Opt. Phys. 46 104007] described a multi-parameter family of minimum-uncertainty states satisfying the time-dependent Schrodinger equation for the harmonic oscillator. We show how a different parametrization and a proper geometrical insight reduces a complicated set of equations to two simple circular motions.

quant-ph↗

arXiv2014

A Comment on 'On the Rotation Matrix in Minkowski Space-time' by Ozdemir and Erdogdu

We comment on the article by M. Ozdemir and M. Erdogdu. We indicate that the exponential map onto the Lorentz group can be obtained in two elementary ways. The first way utilizes a commutative algebra involving a conjugate of a semi-skew-symmetric matrix, and the second way is based on the classical epimorphism from SL(2,C) onto SO_0(3,1)

physics.gen-ph↗

arXiv2014

What is Time in Quantum Mechanics?

Time of arrival in quantum mechanics is discussed in two versions: the classical axiomatic "time of arrival operator" introduced by J. Kijowski and the EEQT method. It is suggested that for free particles the two methods may lead to the same result. On the other hand the EEQT method can be easily geometrized within the framework of Galilei-Newton general relativistic quantum mechanics developed by M. Modugno and collaborators, and it can be applied to non-free evolutions. The way of geometrization of irreversible quantum dynamics based on dissipative Liouville equation is suggested.

quant-ph↗

arXiv2011

START in a five-dimensional conformal domain

In this paper we give a brief review of the pseudo-Riemannian geometry of the five-dimensional homogeneous space for the conformal group O(4,2). Its topology is described and its relation to the conformally compactified Minkowski space is described. Its metric is calculated using a generalized half-space representation. Compactification via Lie-sphere geometry is outlined. Possible applications to Jaime Keller's START theory may follow by using its predecessor - the 5-optics of Yu. B. Rumer. The point of view of Rumer is given extensively in the last section of the paper. Keywords. Kaluza,Klein, Rumer, conformal symmetry, hyperbolic space, START, fifth dimension, action coordinate, 5-optics

math-ph↗

arXiv2011

Geometry and Shape of Minkowski's Space Conformal Infinity

We review and further analyze Penrose's 'light cone at infinity' - the conformal closure of Minkowski space. Examples of a potential confusion in the existing literature about it's geometry and shape are pointed out. It is argued that it is better to think about conformal infinity as of a needle horn supercyclide (or a limit horn torus) made of a family of circles, all intersecting at one and only one point, rather than that of a 'cone'. A parametrization using circular null geodesics is given. Compactified Minkowski space is represented in three ways: as a group manifold of the unitary group U(2) a projective quadric in six-dimensional real space of signature (4,2) and as the Grassmannian of maximal totally isotropic subspaces in complex four--dimensional twistor space. Explicit relations between these representations are given, using a concrete representation of antilinear action of the conformal Clifford algebra Cl(4,2) on twistors. Concepts of space-time geometry are explicitly linked to those of Lie sphere geometry. In particular conformal infinity is faithfully represented by planes in 3D real space plus the infinity point. Closed null geodesics trapped at infinity are represented by parallel plane fronts (plus infinity point). A version of the projective quadric in six-dimensional space where the quotient is taken by positive reals is shown to lead to a symmetric Dupin's type `needle horn cyclide' shape of conformal infinity.

math-ph↗

arXiv2011

Gravitation on a Homogeneous Domain

Among all plastic deformations of the gravitational Lorentz vacuum \cite{wr1} a particular role is being played by conformal deformations. These are conveniently described by using the homogeneous space for the conformal group SU(2,2)/S(U(2)x U(2)) and its Shilov boundary - the compactified Minkowski space \tilde{M} [1]. In this paper we review the geometrical structure involved in such a description. In particular we demonstrate that coherent states on the homogeneous Kae}hler domain give rise to Einstein-like plastic conformal deformations when extended to \tilde{M} [2].

math-ph↗

arXiv2011

Compactified Minkowski Space: Myths and Facts

Minkowski space, conformal group, compactification, conformal infinity, conformal inversion, light cone at infinity, SU(2,2), SO(4,2), Hodge star operator, Clifford algebra, spinors, twistors, antilinear operators, exterior algebra, bivectors, isotropic subspaces, null geodesics, Lie spheres, Dupin cyclides, gravitation} \abstract{Maxwell's equations are invariant not only under the Lorentz group but also under the conformal group. Irving E. Segal has shown that while the Galilei group is a deformation of the Poincaré group, and the Poincaré group is a deformation of the conformal group, the conformal group ends the road, it is {\em rigid\}. There are thus compelling mathematical and physical reasons for promoting the conformal group to the role of the fundamental symmetry of space--time, more important than the Poincaré group that formed the group-theoretical basis of special and general theories of relativity. While the action of the conformal group on Minkowski space is singular, it naturally extends to a nonsingular action on the compactified Minkowski space, often referred to in the literature as "Minkowski space plus light-cone at infinity". Unfortunately in some textbooks the true structure of the compactified Minkowski space is sometimes misrepresented, including false proofs and statements that are simply wrong. In this paper we present in, a simple way, two different constructions of the compactified Minkowski space, both stemming from the original idea of Roger Penrose, but putting stress on the mathematically subtle points and relating the constructions to the Clifford algebra tools. In particular the little-known antilinear Hodge star operator is introduced in order to connect real and complex structures of the algebra. A possible relation to Waldyr Rodrigues' idea of gravity as a plastic deformation of Minkowski's vacuum is also indicated.

math-ph↗

arXiv2010

Some comments on projective quadrics subordinate to pseudo--Hermitian spaces

We study in some detail the structure of the projective quadric Q' obtained by taking the quotient of the isotropic cone in a standard pseudo-Hermitian space H_{p,q} with respect to the positive real numbers R^+ and, further, by taking the quotient Q'/U(1). The case of signature (1,1) serves as an illustration. Q'/U(1) is studied as a compactification of RxH_{p-1,q-1}.

math-ph↗

arXiv2010

On Conformal Infinity and Compactifications of the Minkowski Space

Using the standard Cayley transform and elementary tools it is reiterated that the conformal compactification of the Minkowski space involves not only the "cone at infinity" but also the 2-sphere that is at the base of this cone. We represent this 2-sphere by two additionally marked points on the Penrose diagram for the compactified Minkowski space. Lacks and omissions in the existing literature are described, Penrose diagrams are derived for both, simple compactification and its double covering space, which is discussed in some detail using both the U(2) approach and the exterior and Clifford algebra methods. Using the Hodge * operator twistors (i.e. vectors of the pseudo-Hermitian space H_{2,2}) are realized as spinors (i.e., vectors of a faithful irreducible representation of the even Clifford algebra) for the conformal group SO(4,2)/Z_2. Killing vector fields corresponding to the left action of U(2) on itself are explicitly calculated. Isotropic cones and corresponding projective quadrics in H_{p,q} are also discussed. Applications to flat conformal structures, including the normal Cartan connection and conformal development has been discussed in some detail.

physics.gen-ph↗

arXiv2008

The Theory of Kairons

In relativistic quantum mechanics wave functions of particles satisfy field equations that have initial data on a space--like hypersurface. We propose a dual field theory of ``wavicles'' that have their initial data on a time--like worldline. Propagation of such fields is superluminal, even though the Hilbert space of the solutions carries a unitary representation of the Poincare group of mass zero. We call the objects described by these field equations ``Kairons''. The paper builds the field equations in a general relativistic framework, allowing for a torsion. Kairon fields are section of a vector bundle over space-time. The bundle has infinite--dimensional fibres.

math-ph↗

arXiv2007

Comments on ``Spin Connection Resonance in Gravitational General Relativity''

We comment on a recent article of M.W.Evans, Acta Physica Polonica B38 (2007) 2211. We point out that the equations underlying Evans' theory are highly problematic. Moreover, we demonstrate that the so-called ``spin connection resonance'', predicted by Evans, cannot be derived from the equation he used. We provide an exact solution of Evans' corresponding equation and show that is has definitely no resonance solutions.

physics.class-ph↗

arXiv2007

Quantum fractals on n-spheres. Clifford Algebra approach

Using the Clifford algebra formalism we extend the quantum jumps algorithm of the Event Enhanced Quantum Theory (EEQT) to convex state figures other than those stemming from convex hulls of complex projective spaces that form the basis for the standard quantum theory. We study quantum jumps on n-dimensional spheres, jumps that are induced by symmetric configurations of non-commuting state monitoring detectors. The detectors cause quantum jumps via geometrically induced conformal maps (Mobius transformations) and realize iterated function systems (IFS) with fractal attractors located on n-dimensional spheres. We also extend the formalism to mixed states, represented by "density matrices". As a numerical illustration we study quantum fractals on the circle, two--sphere (octahedron), and on three-dimensional sphere (hypercube-tesseract, 24 cell, 600 cell,and 120 cell). The invariant measure on the attractor is approximated by the powers of the Markov operator. In the appendices we calculate the Radon-Nikodym derivative of the SO(n+1) invariant measure on S^n under SO(1,n+1) transformations and discuss the Hamilton's "icossian calculus" as well as its application to quaternionic realization of the binary icosahedral group that is at the basis of the 600 cell and its dual, the 120 cell. As a by-product of this work we obtain several Clifford algebraic results, such as a characterization of positive elements in a Clifford algebra Cl(n+1) as generalized Lorentz boosts, and their action as Moebius transformation on n-sphere, and a decomposition of any element of Spin^+(1,n+1) into a boost and a rotation, including the explicit formula for the pullback of the O(n+1) invariant Riemannian metric with respect to the associated Mobius transformation.

quant-ph↗

arXiv2006

Quantum Fractals. Geometric modeling of quantum jumps with conformal maps

Positive matrices in SL(2,C) have a double physical interpretation; they can be either considered as "fuzzy projections" of a spin 1/2 quantum system, or as Lorentz boosts. In the present paper, concentrating on this second interpretation, we follow the clues given by Pertti Lounesto and, using the classical Clifford algebraic methods, interpret them as conformal maps of the "heavenly sphere" S^2. The fuzziness parameter of the first interpretation becomes the "boost velocity" in the second one. We discuss simple iterative function systems of such maps, and show that they lead to self--similar fractal patterns on S^2. The final section of this paper is devoted to an informal discussion of the relations between these concepts and the problems in the foundations of quantum theory, where the interplay between different kinds of algebras and maps may enable us to describe not only the continuous evolution of wave functions, but also quantum jumps and "events" that accompany these jumps. Paper dedicated to the memory of Pertti Lounesto.

quant-ph↗